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Sea AI Lab 42

--- abstract: 'Bacteria living on surfaces use different types of motility mechanisms to move on the surface in search of food or to form micro-colonies. Twitching is one such form of motility employed by bacteria such as [*N. gonorrhoeae*]{}, in which the polymeric extensions known as type IV pili mediate its movement. Pili extending from cell body adheres to the surface and pulls the bacteria by retraction. The bacterial movement is decided by the two-dimensional *tug-of-war* between the pili attached to the surface. Natural surfaces in which these *micro-crawlers* dwell are generally spatially inhomogeneous and have varying surface properties. Their motility is known to be affected by the topography of the surfaces. Therefore, it is possible to control bacterial movement by designing structured surfaces which can be potentially utilized for controlling biofilm architecture. In this paper, we numerically investigate the twitching motility in a two-dimensional corrugated channel. The bacterial movement is simulated by two different models: (a) a detailed *tug-of-war* model which extensively describe the twitching motility of bacteria assisted by pili and (b) a coarse-grained *run-and-tumble* model which depicts the motion of wide-ranging self-propelled particles. The simulation of bacterial motion through asymmetric corrugated channels using the above models show rectification. The bacterial transport depends on the geometric parameters of the channel and inherent system parameters such as persistence length and self-propelled velocity. In particular, the variation of the particle current with the geometric parameters of the micro-channels show that one can optimize the particle current for specific values of these parameters.' author: - Konark Bisht - Rahul Marathe bibliography: - 'ref9719.bib' title: 'Rectification of twitching bacteria through narrow channels: A numerical simulations study' --- Introduction {#sec:1} ============ Many bacterial species dwell on surfaces. Each species employ a particular or a combination of different motility mechanisms such as swimming, darting, gliding and twitching to survey the surfaces for food and colonies [@RM2003]. Surface-dwelling bacteria such as *Neisseria gonorrhoeae* cannot actively swim in a liquid medium; instead, they possess a form of surface motility known as “*twitching motility*” [@JH1983; @JM2002]. It is mediated by the polymeric extensions emerging from the cell body known as *pili* and is characterized by intermittent and jerky movement [@JS2001; @CG2012]. Pili have various functions, but a class of these pili known as type-IV pili (T4P) assist in motility over surface [@JH1983; @JM2002; @BM2015]. T4P undergoes cycles of polymerization and depolymerization. The polymerization results in the generation of new pilus and subsequent elongation, out of which some get attached to the surface, whereas, depolymerization leads to retraction and eventual disappearance of a pilus. The retracting pilus, attached to the surface, exerts a force on the cell body and pulls the bacterium along its direction. A single pilus can generate a force exceeding $100$ pN [@AM2000; @BM2002; @NB2008]. The vectorial balance of forces exerted by various T4P in different directions as in a *tug-of-war* eventually decides the direction of motion of the bacterium [@MM2008; @CH2010; @RM2014; @VZ2014]. Unlike many other bacterial species, the [*N. gonorrhoeae* ]{}cannot sense the chemical gradient as the chemotactic gene is absent in them [@JT2015]. Due to the absence of any biases, these “*micro-crawlers*" essentially executes a persistent random walk on flat surfaces. In its natural habitat, however, the bacteria encounter surfaces having spatial inhomogeneities and different surface properties. The topography of the surface influence the motility of the bacteria as recent experimental studies indicate that [*N. gonorrhoeae* ]{}can sense the topography of the surfaces and the microscopic structures can guide their movement [@CH2009; @CM2012]. It has led to the designing of structured surfaces where the physio-chemical properties can control the bacterial movement and potential biofilm architecture [@CH2009; @BM2013]. In general, controlling microbial locomotion have potential applications in diverse areas such as diagnostics [@PD2012], therapeutic protein synthesis [@BR2010], photosynthetic biofuel production [@JG2008; @SS2009; @EE2011] and microfluidic devices [@MK2007; @MK2008]. One of the way to control the bacterial movement and have directed motion is by exploiting the so-called *active ratchets* [@CR2002; @LA2011; @YY2011; @CR2013; @JWu2015; @YF2015; @BH2015; @BA2017; @CR2017]. They are realized by the self-propelled particles (SPPs) which can be biological (e.g. bacteria) or non-biological (e.g. Janus particles) in nature [@SH2008; @GM2009; @PG2013; @JK2018]. SPPs are mobile agents which convert energy from an environment into persistent motion and are referred to as the active matter. In these ratchets, the SPPs moving through an asymmetric medium show rectification in their motion and there is a net transport of particles in a particular direction [@PG2013; @XA2014; @JW2014; @JW2015; @CR2017; @SP2019]. This asymmetrical or directed response can be accounted to the spatial and temporal asymmetry in the system [@PR2002; @LA2011; @SD2014]. The temporal asymmetry is inherent in the motion of SPPs as the presence of self-propelled velocity with orientational fluctuations drives the system out-of-equilibrium. On the other hand, the spatial asymmetry is imposed by the inhomogeneities in the medium. The boundaries of the channel limit the phase space volume of the particle resulting in the emergence of an effective entropic potential which reflects the variations in the cross-section of the channel. The modulation of the cross-section with a periodic boundary function having broken reflection symmetry induces a symmetry breaking which biases the particle motion [@PR2002; @PM2013]. Rectification has been observed in the motion of biological micro-swimmers such as *Escherichia coli*, in a chamber with an array of funnel barriers [@PG2007; @PG2008; @GL2010; @YC2015], or through ratcheting micro-channels [@SH2008; @JR2016]. The directional rotation of gears with asymmetric teeth in a suspension of bacteria [@AS2010; @RL2010] or in the presence of self-propelled robots [@HL2013] is another such example of rectification process in active matter systems due to spatial asymmetry. It holds significance in the context of using bacterial suspension to power mechanical micro-machines [@YH2006; @AG2008; @LA2009; @SK2016; @AS2018; @AS2019]. However, similar studies on the transport of surface motile twitching bacteria in the asymmetric medium are lacking. Unlike the micro-swimmers, the hydrodynamic interactions are absent for these micro-crawlers, and only steric surface interactions could play a role in the rectification process. In this paper, we numerically investigate the twitching motility of bacteria in narrow two-dimensional (2D) channels having periodic boundaries with broken reflection symmetry. The stochastic *tug-of-war* model (TWM) is implemented to mimic the twitching motility of [*N. gonorrhoeae* ]{}[@RM2014]. To characterize the motion, mean squared displacement (MSD) is computed, which identify distinct diffusion regimes for different time scales. The bacterium confined in the corrugated channel experience rectification in motion as evident by the non-zero value of mean displacement along the axis of the channel. The particle current which gives a measure of net transport of bacteria is dependent on the relative value of the persistence length to the size of the compartment of the channel. It is observed that the particle current can be optimized by careful selection of geometric parameters of the channel. We also compare our results with a coarse-grained *run-and-tumble* model (RTM), which is a generic model used to describe the motion of SPPs [@JH2007; @HB2014]. The rest of the paper is organized as follows. In Sec. \[sec:2\], the two models of twitching motility are introduced. In Sec. \[sec:3\], we first discuss the geometry of the corrugated channel used in the simulation. Next, in the section, relevant quantities such as MSD, mean displacement along the $x$ axis and spatial probability density, velocity profiles are evaluated to investigate the rectified motion of a bacterium. In the latter part, the dependence of particle current on geometric parameters of the channel for the two models is explored. Finally, in Sec. \[sec:4\], we conclude by providing a summary and discussion of our results. Models for twitching motility {#sec:2} ============================= Twitching motility using stochastic *tug-of-war* model {#sec:2a} ------------------------------------------------------ We simulate the twitching motility of [*N. gonorrhoeae* ]{}bacteria using the 2D stochastic *tug-of-war* model (TWM) described in Ref. [@RM2014]. The cell body of the bacterium is modeled as a point particle with straight rodlike pili emerging radially from the cell body in random directions. A pilus is considered to be in one of the three states: elongating, retracting or attached to the surface states. The pilus stochastically switches between these states with the rates that are estimated from the experiments. The retraction velocity and the unbinding rate of a pilus depend on the force experienced by the pilus. When the number of pili in opposite directions are different, the pilus on the side with lesser number of pili experience greater force than on the opposite side. Hence, it is more likely to unbind from the surface due to the force-dependent nature of the unbinding rate. The unbinding of pilus in the weaker side further increases the imbalance causing a sharp escalation of unbinding of pili from the weaker side resulting in a rapid motion in the direction of pili who win the *tug-of-war*. The further rebinding or unbinding of pili, dissolution of pili due to full retraction or creation of new pili, alter the force balance resulting in the change in the direction of motion. The one-dimensional (1D) tug-of-war models and two-state models have been applied to study the dynamics of molecular motors [@MM2008; @TG2011; @PMalgaretti2017]. However, there are quantitative differences in the dynamics in a 1D model than the 2D TWM used in this work. In the 1D models of molecular motors, the cargo gets pulled in two opposite directions only. This would correspond to pili extending in two opposite directions in case of twitching bacteria. However, in case of [*N. gonorrhoeae* ]{}where pili extend in all directions $[0, 2\pi)$. Therefore, it requires a 2D tug-of-war mechanism to simulate the twitching motility in [*N. gonorrhoeae*]{}. Our model also incorporates directional memory by allowing the pilus bundling and the re-elongation of fully retracted pili with some finite probability [@RM2014]. These two additional properties in the 2D model are essential to mimic the experimentally observed increase in the persistence time of [*N. gonorrhoeae* ]{}with the increasing pilus number [@CH2010; @RM2014]. The persistence time $t_p$ is the average time for which the overall direction of motion remains same. The persistence time $t_p$ is proportional to the persistence length $\ell_p$ as $t_p=\ell_p/v$, where $v$ is average speed of the bacterium. On the other hand, the average number of pili increases with the rate of pilus creation $R_{cr}$. Therefore, it can be stated that the persistence length (or time) increases with the rise in the pilus creation rate as also seen in experiments [@CH2010]. In a wild type [*N. gonorrhoeae* ]{}bacterium, a persistent length of $1-2$ $\mu$m corresponds to average seven pili [@CH2010; @RM2014]. The trajectories of different persistence length are obtained by varying the $R_{cr}$ in our simulation using TWM. Since our interest is in the study of the twitching motility in the confined narrow channels, the nature of motility near the boundaries plays a significant role in the transport properties. The boundary condition in TWM is implemented in the following manner. When the bacterium is near the boundary, pili cannot attach to surface beyond the boundary. So the pili can attach only in a direction which is towards the interior of the channel. It restricts the bacterium to take a step beyond the boundaries and is thus confined in the channel. Twitching motility as *run-and-tumble* motion {#sec:2b} --------------------------------------------- In this section, we discuss the motivation towards the coarse-grained model of the twitching motility. The motility of [*N. gonorrhoeae* ]{}have been recorded and analyzed in various experimental studies [@CH2010; @CM2012; @BM2013]. In these experiments, the bacteria crawling on glass plates were observed under a microscope, and movies tracking their positions were recorded for a duration of a few minutes. A distribution of step lengths $\ell$ was generated by analyzing the recorded tracks [@CH2010; @KB2017]. The step length distribution $P(\ell)$ was found to follow an exponential function given by $$P(\ell) = \frac{1}{\ell_p}\exp(-\ell/\ell_p). \label{eq:1}$$ The persistence length $\ell_p$ is the average distance a bacterium travels before taking a turn. A typical value of the average speed $v$ of [*N. gonorrhoeae* ]{}was reported to be 1.5–$2~\mu$ms$^{-1}$ [@CH2009]. A coarse-grained model for bacterial motility can be constructed using experimentally observed features of [*N. gonorrhoeae* ]{}walks [@KB2017]. The bacterium is modeled as a point particle executing a 2D random walk with step lengths drawn from the exponential distribution of Eq. (\[eq:1\]). The particle selects a new direction at each turn from a uniform distribution between $[0, 2\pi)$. A fixed value of average speed $v=1.5~\mu$ms$^{-1}$ is taken with the time duration to complete a step of $\ell$ length is given by $\ell/v$. The motility is characterized by straight trajectories with sudden random changes in the direction. The resulting motion is referred to as *run-and-tumble* and is quite ubiquitous in SPPs. In our model, the time duration of the tumbling event is zero, and change in the direction happens instantaneously at each turn. Using the above model of motility, we study the transport of bacteria in a corrugated channel. The boundary condition is taken in which the bacterium is reflected towards the interior in a random direction. So, when the bacterium hits the boundary wall, a new proposed direction is selected randomly from a uniform distribution \[0, $2\pi$). If it is not towards the interior, another direction is chosen until the proposed direction is towards the interior of the channel. The bacterium instantaneously turns to the new direction and is hence prevented from taking steps beyond the boundary walls. Simulation methods and results {#sec:3} ============================== We simulate the twitching motility using the two models described in section \[sec:2\] The bacterial motion is confined in a 2D corrugated channel which extends infinitely along $x$ axis and is bounded by a periodic function along $y$ axis (Fig. \[fig:1\]). Boundaries of the asymmetric channel are modeled by a boundary function having broken reflection symmetry which is widely used in literature [@RB1994; @PJ1996; @RB1997; @PR2002; @PMJCP2013; @PH2009]. The boundary function $w(x)$ is given by, $$w(x) = {\mathcal{A}}-{\mathcal{B}}\left[\sin\left(2\pi \frac{x}{L}\right) + \frac{\Delta}{4}\sin \left(4\pi \frac{x}{L}\right)\right]. \label{eq:2}$$ Here, $L$ is the spatial periodicity along $x$ axis of the channel. The extent of asymmetry in the shape of the channel is determined by the asymmetric parameter $\Delta$. For a symmetric periodic channel, $\Delta = 0$. The channel is biased towards the positive $x$ direction for $\Delta < 0$ and the negative $x$ direction for $\Delta > 0$. In this study, the simulations are primarily done for a channel biased towards the negative $x$ direction. The parameter ${\mathcal{A}}$ determines the half-width of the bottleneck whereas ${\mathcal{B}}$ controls the slope of the channel boundaries [@AB2009]. The parameters ${\mathcal{A}}$ and ${\mathcal{B}}$ are related to the minimum width $w_{min}$ and the maximum width $w_{max}$ of the channel as $$w_{min} = 2({\mathcal{A}}-{\mathcal{B}}\delta)\mbox{ and }w_{max} = 2({\mathcal{A}}+{\mathcal{B}}\delta). \label{eq:wmin}$$ where $\delta$ is obtained by finding the point of extremum of the boundary function $w(x)$ in Eq. (\[eq:2\]) and is given by, $$\delta(\Delta) = \left[\left({\frac{\Delta^2-1+\sqrt{1+2\Delta^2}}{2\Delta^2}}\right)^{1/2} +\frac{1}{4}\left(\frac{\Delta^4-2(\Delta^2-\sqrt{1+2\Delta^2}+1)}{\Delta^2}\right)^{1/2}\right]. \label{eq:delta}$$ From Eq. (\[eq:delta\]), we can note that the $\delta$ is an monotonic increasing function of $\Delta$ for $\Delta>0$. For a typical value of $\Delta = 1$, $\delta = 1.1$. Simulations are performed for the parameter values $\Delta = 1$, ${\mathcal{A}}= 1~\mu$m, ${\mathcal{B}}= 0.7~\mu$m and $L=5~\mu$m unless explicitly mentioned otherwise. For these values of the parameters, $w_{max} \simeq 3.5~\mu$m and $w_{min} \simeq 0.46~\mu$m. The ensemble averages of all physical quantities is denoted by $\langle..\rangle$ are derived by averaging over $10^4$ trajectories unless mentioned otherwise. In our simulations, the bacterium always starts from a fixed initial position $(x_0, y_0) = (0,0)$ at time $t=0$. The prototypical tracks traced by a bacterium having $R_{cr} = 20$ s$^{-1}$ and $\ell_p =1$ $\mu$m, simulated by TWM and RTM, respectively are shown in Fig. [\[fig:1\]]{}. The tracks from TWM are detailed and capture the jerky movements of the bacterium. On the other hand, tracks from RTM lacks the detailed structure and gives a coarse-grained representation of the motility. In both the cases, the movement is directed at short times and random at longer time scales. It is demonstrated by the plot of the mean squared displacement (MSD) $\langle\Delta{{\mathbf{r}}(t)}^2\rangle-\langle\Delta{\mathbf{r}}(t)\rangle^2$ as a function of time $t$ in Figs. \[fig:r2\](a) and \[fig:r2\](b), where $\Delta{\mathbf{r}}(t) = {\mathbf{r}}(t)-{\mathbf{r}}(0)$. The dynamics is characterized by MSD exponent $\beta$ defined by MSD $\sim t^\beta$. Fig. \[fig:r2\](a) show the plot of MSD calculated from the trajectories of TWM for $R_{cr}=5$, 10, 15, 20 s$^{-1}$. In the inset, the variation of $\beta$ with $t$ is shown which is evaluated from the relation $\beta(t) = \log_{10}[\langle\Delta{\mathbf{r}}^2(10t)\rangle/\langle\Delta{\mathbf{r}}^2(t)\rangle]$. The $\beta \simeq 2$ for short time scales indicates ballistic motion. The exponent gradually reduces to $\beta \simeq 1$ indicating diffusion for longer time scales. In TWM, the motion achieves the steady state after some period of time. In this initial transient state of the motion, $\beta>2$ for very small $t$. Fig. \[fig:r2\](b) depict the plot of MSD computed from RTM trajectories for $\ell_p= 0.1$, 1, $10~\mu$m. The motion changes from super-diffusive at very short time scales ($t\lesssim 0.1$ s) to sub-diffusive at intermediate time scales ($1\lesssim t\lesssim 100$ s). The motion is diffusive for longer time scales ($t\gtrsim 100$ s). We do not see much difference in the dynamics of the twitching bacteria in the channel geometry than in an unconfined surface. However, when we plot of $\langle\Delta{\mathbf{r}}(t)^2\rangle$ with $t$, we see anomalous motion for long times. The variation of $\langle\Delta{\mathbf{r}}(t)^2\rangle$ has been employed to search for signatures of rectification in the dynamics of SPPs in Refs. [@AC2015; @SP2019; @AW2019]. In Figs. \[fig:r2\](c) and \[fig:r2\](d), we plot the $\langle\Delta{\mathbf{r}}^2\rangle$ vs $t$ for TWM and RTM, respectively. The insets show the variation of exponent $\beta^\prime$ with $t$ which is defined by the relation $\langle\Delta{\mathbf{r}}^2\rangle\sim t^{\beta^\prime}$. We observe that the exponent $\beta^\prime >1$ at larger time scale in this case for bacterium having $\ell_p$ comparable to the channel dimensions which is distinct from the behavior of MSD exponent $\beta\simeq1$ for large $t$ for all values of $\ell_p$. This divergence is due to the non-zero finite value of $\langle{\mathbf{r}}\rangle$ at long times due to rectification which is accounted in the MSD calculations but not in the case of $\langle\Delta{\mathbf{r}}^2\rangle$. In order to ascertain the rectification in bacterial motion, we compute the mean displacement along $x$ axis $\langle\Delta x(t)\rangle$ where $\Delta x(t) = x(t)-x(0)$. The plot of $\langle\Delta x(t)\rangle$ vs $t$ obtained for two models TWM \[in Fig. \[fig:mx\](a)\] and RTM \[in Fig. \[fig:mx\](b)\] show finite displacement $\left(\vert\langle\Delta x(t)\rangle\vert\neq 0\right)$ for large time scales and it increase with the persistence length in both models. The plots are steep for larger values of persistence length which indicates that the particle current may have a dependence on the persistence length. It is explored in detail in the later part of this section. ![Top row shows the color map of scaled spatial probability density $\rho({\mathbf{r}})$ for (a) $R_{cr} = 10$ s$^{-1}$ in TWM and (b) $\ell_p =10~\mu$m in RTM. Bottom row depicts the corresponding profiles of average drift velocity $\overline{{\mathbf{v}}}({\mathbf{r}})$ for (c) TWM and (d) RTM where color maps shows the magnitude of drift velocity profile $\overline{{\mathbf{v}}}({\mathbf{r}})$ and the arrows depicts its direction.[]{data-label="fig:rhoV"}](rhoV.pdf) The directed transport in the asymmetric channels is further investigated by calculating the spatial probability density $\rho({\mathbf{r}})$ and time-averaged drift velocity $\overline{{\mathbf{v}}}({\mathbf{r}})$ profiles from the bacterial trajectories obtained from simulations for the time duration $t=10^6$ s. Figs. \[fig:rhoV\](a) and \[fig:rhoV\](b) depict the colour maps of $\rho({\mathbf{r}})$ for (a) TWM (for $R_{cr} = 20$ s$^{-1}$) and (b) RTM (for $\ell_p=10~\mu$m). The $\rho({\mathbf{r}})$ values are scaled by its largest values in the compartment. The $\rho({\mathbf{r}})$ profile show that the bacterium tend to spend more time near the boundaries for larger $R_{cr}$. This was also observed experimentally when [*N. gonorrhoeae* ]{}bacteria were allowed to interact with 3D obstacles [@CM2012]. It should also be noted that the twitching bacteria tend to accumulate at places where the channel cross-section is widest. The same observation was also reported for different kinds of SPPs in the theoretical study using Fick-Jacobs approximation in Ref. [@PM2017]. Since bacterium having large persistence length ($R_{cr}= 20$ s$^{-1}$ or $\ell_p$ = 10) stays near the boundaries, it experiences the asymmetric shape of the channel more often which causes an overall directed motion along the channel. On other hand, for small persistence lengths ($\ell_p=0.1~\mu$m) bacteria encounters the boundary rarely and move randomly in the interior of the channel. The average drift velocity ${\overline{\mathbf{v}}_d}$ fields are plotted in Figs. \[fig:rhoV\](c) and \[fig:rhoV\](d). The color map shows the scaled magnitude of time-averaged drift velocity profile ${\overline{\mathbf{v}}_d}({\mathbf{r}})$ in the compartment. The magnitude of ${\overline{\mathbf{v}}_d}({\mathbf{r}})$ is computed as $\vert{\overline{\mathbf{v}}_d}({\mathbf{r}})\vert = \left({\overline{\mathbf{v}}_d}({\mathbf{r}})_x^2+{\overline{\mathbf{v}}_d}({\mathbf{r}})_y^2\right)^{1/2}$ which is then scaled with its largest value in the compartment. The arrows in Figs. \[fig:rhoV\](c) and \[fig:rhoV\](d) depicts the normalized average drift velocity profiles $\hat{{\overline{\mathbf{v}}_d}}({\mathbf{r}}) ={\overline{\mathbf{v}}_d}({\mathbf{r}})/\vert{\overline{\mathbf{v}}_d}({\mathbf{r}})\vert$. In Fig. \[fig:rhoV\](c), the velocity profiles for TWM are shown which indicates the bacteria tend to diffuse away from constriction. On the other hand, in Fig. \[fig:rhoV\](d), the bacteria undergoing run-and-tumble move towards the direction of biasing of the channel for $\ell_p>L$. An essential quantity that quantifies the particle transport across the channel is the particle current. Since the geometry of our system is a quasi-1D along the $x$ axis, the particle current $J_x$ can be considered to be due to motion along the $x$ axis alone and is defined by [@PR2002; @GS2009; @BA2009], $$J_x = -\lim_{t\to\infty}\frac{\langle\Delta x(t)\rangle}{t}. \label{eq:Jx}$$ In our work, we have defined our channel to be biased toward negative $x$ direction in most cases. Therefore, we have taken a negative sign in our definition of $J_x$ so that the $J_x\gtrsim 0$ for a channel biased toward $-x$ direction which helps in visualization. The current $J_x <0$ for a channel biased toward $+x$ direction. In all our simulations, $J_x$ is calculated at time $t = 10^3$ s. This time is sufficient time to reach a steady-state such, and the current is stabilized. Due to the effect of orientational fluctuations intrinsic to self-propulsion mechanism, the bacteria changes its direction randomly, which results in particle current to be only a small fraction of the self-propelled speed $v$. The magnitude of current may depend on a number of intrinsic factors, such as persistence length or self-propelled speed, and on the geometric parameters of the channel. Next, we study the dependence of the particle current on the persistence length. In TWM, the persistence length is directly proportional to the pilus creation rate $R_{cr}$ whereas in RTM, $\ell_p$ determines the persistence of the walks (see Sec. \[sec:2\]). In Fig. \[fig:JxVsRcrlp\](a), we show the plot of the variation of $J_x$ with $R_{cr}$ and found $J_x$ to increase with the rise in $R_{cr}$. It is understandable since as $R_{cr}$ increase, the average number of pili increase, which in turn leads to an increase in persistence length. The bacterium encounters the boundary walls more for larger persistence length resulting in enhanced $J_x$. The current tends to approach a maximum value for a larger value of $R_{cr}$. Fig. \[fig:JxVsRcrlp\](b) show the plot of $J_x$ vs $\ell_p$ where we have taken a large variation of $\ell_p$. The $J_x\simeq 0$ for small values of $\ell_p$ ($\ell_p\lesssim 0.5~\mu$m) as bacterium rarely experience the asymmetric boundaries. The $J_x$ increases rapidly in the intermediate region where the $\ell_p$ is comparable to the dimension of the compartment [$0.5\lesssim \ell_p\lesssim 10~\mu$m]{}. The current then becomes independent of $\ell_p$ and saturates for larger values of $\ell_p$ ($\ell_p\gtrsim 100~\mu$m). For such a high value of persistence length ($\ell_p>>L$), the motion is increasingly restricted by the compartment walls, which makes the further increase in the persistence length ineffective. Therefore, as there is no effective increase in persistence, there is no further increase in the current. It indicates that the particle current has a strong dependence on the relative value of persistence length to that of the geometric parameters of the channel. We have also explored the dependence of particle current $J_x$ on the self-propulsion velocity $v$ in RTM. The particle current $J_x\sim v$ for large values of $\ell_p$ for sufficiently large $v$. In various applications, it is desirable to maximize the particle current. This could be achieved by taking the appropriate values of geometric parameters of the channel. We study the variation of the particle current $J_x$ with the geometric parameters such as the maximum width $w_{max}$, the minimum width $w_{min}$ and the asymmetric parameter $\Delta$ of the channel. In the following results, the variation of $J_x$ with the parameter of interest is studied keeping all other parameters fixed. The simulations are performed for $L=5~\mu$m, $\Delta = 1$, $w_{min} \simeq 0.46~\mu$m and $w_{max} \simeq 3.5~\mu$m. The dependence of the $J_x$ on the geometric parameters for TWM is shown for $R_{cr} = 10$, 15, 20 s$^{-1}$ in Fig. \[fig:5\]. The variation in $J_x$ with $w_{max}$ is shown in Fig. \[fig:5\](a). The channel is relatively flat for low values of $w_{max}$ ($w_{max}\simeq w_{min}$). As $w_{max}$ increase the channel become more corrugated resulting in increase in $J_x$. However, for large $w_{max}$, the bacterium spends most of the time in one compartment resulting in fewer number of trajectories passing from one compartment to another resulting in decrease in $J_x$. In Fig. \[fig:5\](b), we show the plot of $J_x$ vs $w_{min}$ of the channel with fixed values of all other parameters. As $w_{min}$ increase, the boundaries become flatter resulting in reduction in $J_x$. The variation with current with $\Delta$ (for ${\mathcal{A}}= 1$ $\mu$m, ${\mathcal{B}}= 0.7$ $\mu$m) in Fig. \[fig:5\](c) indicates that the current increases with the asymmetry in the shape of the channel with $J_x = 0$ for a symmetric periodic channel ($\Delta = 0$). Next, we report the results for the dependence of $J_x$ on different parameters using the RTM for $\ell_p=0.1$, 1, $10~\mu$m. Fig. \[fig:JxVsSPCG\](a) show the variation of $J_x$ with $w_{max}$. As discussed in the previous paragraph for the case of TWM, there exists a value of $w_{max}$ for which $J_x$ is maximum. In Fig. \[fig:JxVsSPCG\](b), the variation in current as a function of minimum width $w_{min}$ is plotted. The current is less ($J_x\simeq 0$) for small values of $w_{min}$ for a fixed value of $w_{max}$ ($\simeq 3.5~\mu$m) as only a small number of bacterial trajectories can pass from one compartment to another. The current then starts rising with an increase in $w_{min}$. But as $w_{min}$ increases and approaches the value of $w_{max}$, the channel becomes straighter which reduces the effect of the channel shape on the bacterial dynamics. Hence, the current attains a maximum for a value of $w_{min} = w^*_{min}$ ($w^*_{min}\simeq 0.75~\mu$m for $\ell_p=10~\mu$m) and starts decreasing with further increase in $w_{min}$ beyond this point. The current also seems to have a second maximum for $\ell_p = 10~\mu$m. This merit further examination and analysis in future for understanding it. The current $J_x$ as a function of asymmetric parameter $\Delta$ (with ${\mathcal{A}}= 1$ $\mu$m, ${\mathcal{B}}= 0.7$ $\mu$m) is shown in Fig. \[fig:JxVsSPCG\](c). The $J_x\simeq 0$ for $\Delta = 0$ which corresponds to a periodic channel with reflection symmetry. As $\Delta$ increase, the shape of the channel becomes more and more asymmetric, leading to increasing $J_x$. Here, we recall from Eq. (\[eq:delta\]) that $\delta(\Delta)$ is an monotonically increasing function for $\Delta > 0$ with $\delta(\Delta)=\delta(-\Delta)$. Therefore, from Eq. \[eq:wmin\], we note that as $\vert\Delta\vert$ increase, $w_{max}$ increases whereas $w_{min}$ decreases resulting in the reduction of current for large $\vert\Delta\vert$. The dependence of the particle current on the channel parameters are different in the TWM and RTM. This deviation could be due to the difference in the motility near the channel boundaries for the two models. The way SPPs couple with the bounding walls strongly affects its overall dynamics [@PM2017]. Also, the difference in the underlying motility mechanisms of different SPPs is known to affect its accumulation of the SPPs at the boundaries [@PM2017]. Besides, we could simulate only for a small range of persistence length in TWM due to parameter constraints. The effects of extreme values of persistence length are not observed in TWM as compared to RTM. Conclusion {#sec:4} ========== In this work, we study the twitching motility in a 2D asymmetric corrugated channel using the stochastic TWM and the RTM. The TWM quantitatively describes the motility of [*N. gonorrhoeae* ]{}resulting from the 2D *tug-of-war* between T4P. On the other hand, RTM is a coarse-grained model constructed by analyzing the experimental trajectories of the bacteria. It is a ubiquitous model to describe the motion of SPPs. The bacterial motion is simulated in the corrugated channel having boundaries with broken reflection symmetry. Due to the confinement of bacteria in the narrow channels, the motility shows anomalous diffusion at different time scales. A non-zero finite value of the mean displacement along the $x$ axis at long time scales signifies that the bacterial motion is rectified and there is a net transport of bacteria in one direction. The probability density and velocity profiles reveal the complex dynamics of the twitching bacteria in the compartment. The plots of spatial probability density show that for a bacterium having persistence length comparable to dimensions of the compartment, it dwells more often near the boundaries than the bulk. The average drift velocity is observed to be higher near the spatial constrictions. The particle current which quantifies this rectification has a dependence on the relative value of the persistence length to the dimensions of the compartment. The bacteria having the persistence length comparable to the compartment size undergo multiple collision with the boundaries of the channel resulting in a finite particle current along the channel. We study the variation of particle current for different geometric parameters. Our simulations reveal that one can optimize the particle current for a given value of persistence length by a suitable selection of the size and the shape of the compartments. We observe deviations in the features of the motility for the two models. The simulation of the motility using TWM is done for a narrow range of persistence length, which is dictated by the experimental observations. The large values of persistence length taken in the RWM cannot be replicated for TWM due to parameter constraints on TWM. We could simulate the bacterial motility using TWM only for an intermediate value of persistence length. In the overlapping range, the general features of the motility from TWM can be nearly mapped to those from RTM. However, there are few deviations which we presume could be due to differences in the boundary conditions. In both models, we see enhanced persistence as compared with the motility in the absence of the asymmetric channel. These ratchet effects can also be studied experimentally by creating corrugated channels by micro-printing on a substrate and observing the twitching motility under a microscope [@CH2009; @GM2009]. We hope that such experiments could further investigate the observations made in our numerical study. We also note that the motility of twitching bacteria in the presence of obstacles may further enhance the persistence, work in this direction is currently underway. K.B. acknowledges CSIR (IN) for financial support under Grant No. 09/086(1208)/2015-EMR-I. The authors thank IIT Delhi HPC facility for computational resources.

--- author: - 'Alexander L. Frenkel, David Halpern and Adam J. Schweiger' bibliography: - 'couette.bib' title: 'Surfactant and gravity dependent instability of two-layer channel flows: Linear theory covering all wave lengths ' --- Department of Mathematics, University of Alabama, Tuscaloosa AL 35487, USA Abstract {#abstract .unnumbered} ======== A linear stability analysis of a two-layer plane Couette flow of two immiscible fluid layers with different densities, viscosities and thicknesses, bounded by two infinite parallel plates moving at a constant relative velocity to each other, with an insoluble surfactant monolayer along the interface and in the presence of gravity is carried out. The normal modes approach is applied to the equations governing flow disturbances in the two layers. These equations, together with boundary conditions at the plates and the interface, yield a linear eigenvalue problem. When inertia is neglected the velocity amplitudes are the linear combinations of certain hyperbolic functions, and a quadratic dispersion equation for the increment, that is the complex growth rate, is obtained where coefficients depend on the aspect ratio, the viscosity ratio, the basic velocity shear, the Marangoni number ${\text{Ma}}$ that measures the effects of surfactant, and the Bond number ${\text{Bo}}$ that measures the influence of gravity. An extensive investigation is carried out that examines the stabilizing or destabilizing influences of these parameters. Since the dispersion equation is quadratic in the growth rate, there are two continuous branches of the normal modes: a robust branch that exists even with no surfactant, and a surfactant branch that, to the contrary, vanishes when ${\text{Ma}}\downarrow0$. Due to the availability of explicit forms for the growth rates, in many instances the numerical results are corroborated with analytical asymptotics. For the less unstable branch, a mid-wave interval of unstable wavenumbers (@Halpern2003) sometimes co-exists with a long-wave one. We study the instability landscape, determined by the threshold curve of the long-wave instability and the critical curve of the mid-wave instability in the ($\textrm{Ma, Bo}$)-plane. The changes of the extremal points of the critical curves with the variation of the other parameters, such as the viscosity ratio, and the extrema bifurcation points are investigated. Introduction {#sec:Intro} ============ Surfactants are surface active compounds that reduce the surface tension between two fluids, or between a fluid and a solid. @Frenkel2002 (hereafter referred to as FH) and @Halpern2003 (from now on referred to as HF) uncovered that certain stable surfactant-free Stokes flows become unstable if an interfacial surfactant is introduced. For this, the interfacial shear of velocity must be nonzero; in particular, this instability disappears if the basic flow is stopped. In contrast to the well-known instability of two viscous fluids (@Yih1967) which needs inertia effects for its existence, this instability may exist in the absence of fluid inertia. With regard to multi-fluid horizontal channel flows, this instability has been further studied in a number of papers, such as @Blyth2004b, @Pozrikidis2004a, @Blyth2004a, @Frenkel2005, @Wei2005c, @Frenkel2006, @Halpern2008, @Bassom2010, @JIE2010, @kalogirou2016, @picardo2016, and @Frenkel2017. In the latter paper, we have added gravity to the long-wave considerations of FH. Since in the absence of surfactants gravity can be either stabilizing or destabilizing depending on the flow parameters, the interaction of the Rayleigh-Taylor instability with the surfactant instability leads to interesting phenomena. In the present work, we expand the linear stability analysis of @Frenkel2017, which was confined to long waves, by including disturbances of arbitrary wavenumbers. The current paper can also be regarded as an extension of HF, who considered arbitrary wavenumbers, by incorporating the effects of gravity. As was indicated in @Frenkel2017, one can expect a rich landscape of stability properties, especially since there are two active normal modes of infinitesimal disturbances corresponding to the presence of two interfacial functions: the interface displacement function and the interfacial surfactant concentration (FH, HF). Since the growth rates of the normal modes satisfy a (complex) quadratic equation, and thus are relatively simple, in many instances numerical results may enjoy analytic (asymptotic) corroboration. The stability properties of two-layer Couette flows with both the interfacial surfactant and gravity effects for arbitrary wavenumbers were the subject of the dissertation @schweiger2013gravity. These studies are further developed and expanded in the present paper. In section \[sec:GovEqnsStabForm\], the stability problem is formulated. In section \[sec:FiniteCaseDerivationDispRelation\], the dispersion equation is obtained. The long-wave stability properties are considered in section \[sec:FiniteCaseLongwaveApprox\], while in section \[sec:FiniteCaseMidWave\] we consider normal modes of arbitrary wavelengths and study the so-called mid-wave instability (uncovered in HF but significantly modified by gravity effects). In section \[subsec:MA-Bo plane stability\], we consider the instability landscape in the (Marangoni number, Bond number)-plane that is determined by the threshold curve of the long-wave instability and the critical curve of the mid-wave instability, and study how it changes with the other parameters. Finally, section \[sec:Conclusions\] contains discussion and concluding remarks. Some of the more technical information appears in Appendices. Stability problem formulation {#sec:GovEqnsStabForm} ============================= The general framework and governing equations of the problem were given before (see @schweiger2013gravity, @Frenkel2016, @Frenkel2017) and are as follows. Two immiscible Newtonian fluid layers with different densities, viscosities and thicknesses are bounded by two infinite horizontal plates, a distance $d=d_{1}+d_{2}$ apart, with the top plate moving at a constant relative velocity, $U^{*}$,as shown in figure \[fig:FigDefinitionSketch\]. The $z^{\ast}$-axis is the spanwise, vertical, coordinate perpendicular to the moving plates, with the upper plate located at $z^{\ast}=d_{2}$ and the lower plate located at $z^{\ast}=-d_{1}$, and with $z^{\ast}=0$ determining the location of the unperturbed liquid-liquid interface. (The symbol $^{\ast}$ indicates a dimensional quantity.) The direction of the horizontal $x^{\ast}$-axis is parallel to the plates. At the interface, the surface tension, $\sigma^{\ast}$, depends on the concentration of the insoluble surfactant monolayer, $\Gamma^{\ast}$. The frame of reference is fixed at the liquid-liquid interface so that the velocity of the lower plate is $-U_{1}^{*}$, and that of the upper plate is $U_{2}^{*}$, where $U_{1}^{*}+U_{2}^{*}=U^{*}$, the velocity of the top plate relative to the bottom plate. In the base state, the horizontal velocity profiles are linear in $z^{\ast}$, the interface is flat, and the surfactant concentration is uniform. Once disturbed, the surfactant concentration is no longer uniform and the deflection of the interface is represented by the function $\eta^{\ast}(x^{\ast},t^{\ast})$ where $t^{\ast}$ represents the time. The infinitesimal disturbances may grow under the action of the Marangoni and/or gravity forces (@Frenkel2017). ![Sketch of a disturbed two-layer Couette flow of two horizontal liquid layers with different thicknesses, viscosities, and mass densities. The insoluble surfactant monolayer is located at the interface and is indicated by the dots. The (spanwise) uniform gravity field with a constant acceleration $g$ is not shown. \[fig:FigDefinitionSketch\] ](fig1rt){width="85.00000%"} The governing equations for this problem are given, for example, in @Frenkel2016, in both dimensional and dimensionless forms. (Also, the dimensionless form of these equations can be found in @Frenkel2017.) We use the following notations (with $j=1$ for the bottom liquid layer and $j=2$ for the top liquid layer): $\rho_{j}$ is the density; $\boldsymbol{v}_{j}^{\ast}=(u_{j}^{\ast},w_{j}^{\ast})$ is the fluid velocity vector with horizontal component $u_{j}^{\ast}$ and vertical component $w_{j}^{\ast}$; $p_{j}^{\ast}$ is the pressure; $\mu_{j}$ is the viscosity; and $g$ is the gravity acceleration. We assume the dependence of surface tension $\sigma^{\ast}$ on the surfactant concentration $\Gamma^{\ast}$ to be given by the Langmuir isotherm relation (e.g., @Edwards1991). For the small disturbances, $$\sigma^{\ast}=\sigma_{0}-E(\Gamma^{\ast}-\Gamma_{0}),\label{eq:nineSigmaStar}$$ where $\sigma_{0}$ is the base surface tension corresponding to the base surfactant concentration $\Gamma_{0}$ and the known constant $E\coloneqq\left.-(\partial\sigma^{\ast}/\partial\Gamma^{\ast})\right|_{\Gamma^{*}=\Gamma_{0}}$ is the elasticity parameter. We use the following dimensionless variables: $$(x,z,\eta)=\frac{(x^{\ast},z^{\ast},\eta^{\ast})}{d_{1}}\text{, }t=\frac{t^{\ast}}{d_{1}\mu_{1}/\sigma_{0}}\text{, }\boldsymbol{v}_{j}=(u_{j},w_{j})=\frac{(u_{j}^{\ast},w_{j}^{\ast})}{\sigma_{0}/\mu_{1}}\text{,}$$ $$\text{ }p_{j}=\frac{p_{j}^{\ast}}{\sigma_{0}/d_{1}}\,\text{, }\Gamma=\frac{\Gamma^{\ast}}{\Gamma_{0}}\text{, }\sigma=\frac{\sigma^{\ast}}{\sigma_{0}}\text{.}\label{eq:scales}$$ As in @Frenkel2016 [@Frenkel2017], the dimensionless velocity field of the basic Couette flow, with a flat interface, $\eta=0$, uniform surface tension, $\bar{\sigma}=1$, and corresponding surfactant concentration, $\bar{\Gamma}=1$ (where the over-bar indicates a base quantity), is $$\bar{u}_{1}(z)=sz,\;\bar{w}_{1}=0\text{, }\text{and }\bar{p}_{1}=-\text{Bo}_{1}z\text{ \ \ for \ }-1\leq z\leq0\text{,}\label{eq:u1w1p1BSprofiles}$$ $$\bar{u}_{2}(z)=\frac{s}{m}z,\;\bar{w}_{2}=0\text{, and }\bar{p}_{2}=-\text{Bo}_{2}z\text{ \ \ for \ }0\leq z\leq n,\label{eq:u2w2p2BSprofiles}$$ where $\text{Bo}_{j}:=\rho_{j}gd_{1}^{2}/\sigma_{0}$ is the Bond number of the layer $j$, $m=\mu_{2}/\mu_{1}$ is the ratio of the viscosities, and $n=d_{2}/d_{1}$ is the ratio of the thicknesses. The constant $s$ represents the base interfacial shear rate of the bottom layer, $s=D\bar{u}_{1}(0)$, where $D=d/dz$, and is used to characterize the flow instead of the relative velocity of the plates. It is straightforward to establish that $U=\mu_{1}U^{*}/\sigma_{0}=s(1+n/m)$. The disturbed state with small deviations (indicated by the tilde, $^{\sim}$) from the base flow is given by $$\eta=\tilde{\eta}\text{,}\ u_{j}=\bar{u}_{j}+{\tilde{u}}_{j}\text{, }w_{j}=\tilde{w}_{j}\text{, }p_{j}=\bar{p}_{j}+\tilde{p}_{j}\text{, }\Gamma=\bar{\Gamma}+\tilde{\Gamma}\text{.}\label{eq:transformations}$$ The normal modes are disturbances of the form $$(\tilde{\eta}\text{, }\tilde{u}_{j}\text{, }\tilde{w}_{j}\text{, }\tilde{p}_{j}\text{, }\tilde{\Gamma})=[h\text{, }\hat{u}_{j}(z)\text{, }\hat{w}_{j}(z)\text{, }\hat{f}_{j}(z)\text{, }G]e^{i\alpha x+\gamma t}\text{,}\label{eq:ModesNormal}$$ where $\hat{u}_{j}(z)$, $\hat{w}_{j}(z)$, and $\hat{f}_{j}(z)$ are the complex amplitudes that depend on the depth, $\alpha$ is the wavenumber of the disturbance, $G$ is the constant amplitude of $\tilde{\Gamma}$ ($G=\hat{\Gamma}$), $h$ is the constant amplitude of $\tilde{\eta}$ ($h=\hat{\eta}$), and (complex) $\gamma$ is the increment, $\gamma=\gamma_{R}+i\gamma_{I}$. The stability of the flow depends on the sign of the growth rate $\gamma_{R}$: if $\gamma_{R}>0$ for some normal modes then the system is unstable; and if $\gamma_{R}<0$ for all normal modes then the system is stable. The linearized governing equations for the disturbances translate into the following system for the normal mode amplitudes (See @Frenkel2016 [@Frenkel2017] for the omitted details). The continuity equation becomes $$\hat{u}_{j}=\frac{i}{\alpha}D\hat{w}_{j}.\label{eq:ConEqnDisturb}$$ Eliminating the pressure disturbances from the horizontal and vertical components of the momentum equations with neglected inertia yields the well-known Orr-Sommerfeld equations, here for the vertical velocity disturbances, $$m_{j}(D^{2}-\alpha^{2})^{2}\hat{w}_{j}=0\text{, }\label{eq:OrrSommerfeldStokesEqn}$$ where $m_{j}\coloneqq\mu_{j}/\mu_{1}$ (so that $m_{1}=1$ and $m_{2}=m$). The disturbances of the velocities are subject to the boundary conditions at the plates and at the interface. At the plates, the boundary conditions are $$D\hat{w}_{1}(-1)=0\text{, }\hat{w}_{1}(-1)=0\text{, }D\hat{w}_{2}(n)=0\text{, }\hat{w}_{2}(n)=0\text{.}\label{eq:PlateDisturbBCs}$$ The kinematic boundary condition and surfactant transport equation yield, respectively, $$\gamma h-\hat{w}_{1}=0\text{ (}z=0\text{),}\label{eq:KineDisturbBC}$$ $$\gamma G-D\hat{w}_{1}+si\alpha h=0\text{ (}z=0\text{).}\label{eq:SurfDisturbEqn}$$ (Note that equation (\[eq:SurfDisturbEqn\]) is the normal form of equation (2.9) in @Frenkel2017 which was derived in HF, and was mentioned there to be consistent with the more general equation of @Wong1996. The last term in (\[eq:SurfDisturbEqn\]) comes from the Taylor expansion of the base state fluid velocities at $z=\eta(x,t)$.) Continuity of velocity at the interface yields $$\hat{w}_{1}-\hat{w}_{2}=0\text{ (}z=0\text{)}\label{eq:continVatIntfDisturb}$$ and $$\text{ }D\hat{w}_{2}-D\hat{w}_{1}-i\alpha sh\left(\frac{1-m}{m}\right)=0\text{ (}z=0\text{).}\label{eq:continVatIntfDisturb2}$$ To obtain the linearized homogeneous normal stress condition, the pressure amplitude, $\hat{f}_{j}$, is first written in terms of $\hat{w}_{j}$. From the horizontal momentum equation it is given by $$\alpha^{2}\hat{f}_{j}=m_{j}(D^{2}-\alpha^{2})D\hat{w}_{j}\text{.}\label{eq:xMomEqnDisturbPressure}$$ The interfacial tangential stress condition is $$mD^{2}\hat{w}_{2}-D^{2}\hat{w}_{1}+\alpha^{2}(m\hat{w}_{2}-\hat{w}_{1})-\alpha^{2}G\text{Ma}=0\text{ (}z=0\text{),}\label{eq:TanDisturbBC}$$ where $$\text{Ma}:=E\Gamma_{0}/\sigma_{0}$$ is the Marangoni number, and the normal stress condition is $$mD^{3}\hat{w}_{2}-3m\alpha^{2}D\hat{w}_{2}-D^{3}\hat{w}_{1}+\text{Bo}\alpha^{2}h+3\alpha^{2}D\hat{w}_{1}+\alpha^{4}h=0\text{ (}z=0\text{),}\label{eq:NormDisturbStokesBC}$$ where ${\text{Bo}}$ is the effective Bond number $$\text{Bo}=\text{Bo}_{1}-\text{Bo}_{2}=\frac{(\rho_{1}-\rho_{2})gd_{1}^{2}}{\sigma_{0}}.\label{eq:effectiveBoNumber}$$ Note that $\text{Bo}$ can be negative, unlike the parameters $n$, $m$, $s$ and $\text{Ma}$. Equations (\[eq:OrrSommerfeldStokesEqn\])-(\[eq:continVatIntfDisturb2\]), (\[eq:TanDisturbBC\]) and (\[eq:NormDisturbStokesBC\]) form the eigenvalue boundary value problem for the disturbances, which determines the growth rate as a function of the wavenumber $\alpha$ and the parameters $s$, $m$, $n$, ${\text{Ma}}$, and ${\text{Bo}}$. The eigenvalue, the increment $\gamma$, satisfies a quadratic equation which is obtained in the next section. Dispersion relation; special points of dispersion curves {#sec:FiniteCaseDerivationDispRelation} ======================================================== For finite aspect ratio, $n$, the general solutions of (\[eq:OrrSommerfeldStokesEqn\]) are given by $$\hat{w}_{j}(z)=a_{j}\cosh(\alpha z)+b_{j}\sinh(\alpha z)+c_{j}z\cosh(\alpha z)+d_{j}z\sinh(\alpha z)\text{,}\label{eq:whatGSfinite}$$ where the coefficients $a_{j}$, $b_{j}$, $c_{j}$, and $d_{j}$ are determined by the boundary conditions up to a common normalization factor. Equation (\[eq:continVatIntfDisturb\]) yields $a_{2}=a_{1}$, which is used to eliminate $a_{2}$ from the equations. Applying the plate velocity conditions, equation (\[eq:PlateDisturbBCs\]), the coefficients $c_{1}$ and $d_{1}$ are expressed in terms of $a_{1}$ and $b_{1}$, and the coefficients $c_{2}$ and $d_{2}$ are expressed in terms of $a_{1}$ and $b_{2}$: $$\begin{aligned} \hat{w}_{1}(z) & =a_{1}\cosh(\alpha z)+b_{1}\sinh(\alpha z)+\frac{1}{\alpha}\left[-s_{\alpha}^{2}b_{1}+\left(s_{\alpha}c_{\alpha}+\alpha\right)a_{1}\right]z\cosh(\alpha z)\nonumber \\ & +\frac{1}{\alpha}\left[-\left(s_{\alpha}c_{\alpha}-\alpha\right)b_{1}+c_{\alpha}^{2}a_{1}\right]z\sinh(\alpha z)\label{eq:w1hat}\end{aligned}$$ and $$\begin{aligned} \hat{w}_{2}(z) & =a_{1}\cosh(\alpha z)+b_{2}\sinh(\alpha z)-\frac{1}{\alpha n^{2}}\left[s_{\alpha n}^{2}b_{2}+\left(s_{\alpha n}c_{\alpha n}+\alpha n\right)a_{1}\right]z\cosh(\alpha z)\nonumber \\ & +\frac{1}{\alpha n^{2}}\left[\left(s_{\alpha n}c_{\alpha n}-\alpha n\right)b_{2}+c_{\alpha n}^{2}a_{1}\right]z\sinh(\alpha z),\label{eq:w2hat}\end{aligned}$$ where $$c_{\alpha}=\cosh(\alpha)\text{, }s_{\alpha}=\sinh(\alpha)\,\text{, }c_{\alpha n}=\cosh(\alpha n)\text{, }s_{\alpha n}=\sinh(\alpha n)\,\text{.}\label{eq:coshsinh}$$ We substitute these velocity expressions into the interfacial conditions (\[eq:continVatIntfDisturb2\]), (\[eq:TanDisturbBC\]), and (\[eq:NormDisturbStokesBC\]) to obtain a linear nonhomogeneous system for $a_{1}$, $b_{1}$, and $b_{2}$. Solving this system yields $a_{1}$, $b_{1}$, and $b_{2}$ in terms of $h$ and $G$. Hence, we have the velocities $\hat{w}_{j}(z)$ in terms of $h$ and $G$. Then the kinematic boundary condition (\[eq:KineDisturbBC\]) and surfactant transport equation (\[eq:SurfDisturbEqn\]) yield a linear homogeneous system for $h$ and $G$, written in matrix form as $$\begin{bmatrix}(\gamma+A_{11}) & A_{12}\\ A_{21} & (\gamma+A_{22}) \end{bmatrix}\begin{bmatrix}h\\ G \end{bmatrix}=\begin{bmatrix}0\\ 0 \end{bmatrix}\text{,}\label{eq:dispEqnSystem}$$ where $A_{11}$, $A_{12}$, $A_{21}$, and $A_{22}$ are known functions of the wavenumber $\alpha$ and the system parameters (see Appendix \[sec:Coefficients-of-equations\]). The condition for the existence of nontrivial solutions is $\det(A)=(\gamma+A_{11})(\gamma+A_{22})-A_{12}A_{21}=0$; this yields a quadratic equation for the mode increment $\gamma$. We write this ’dispersion equation’ in the form $$F_{2}\gamma^{2}+F_{1}\gamma+F_{0}=0,\label{eq:DispersionEquation}$$ and its two solutions in the forms $$\gamma=\frac{1}{2F_{2}}\left(-F_{1}+\left[F_{1}^{2}-4F_{2}F_{0}\right]^{1/2}\right)\label{eq:QuadEqnGamma}$$ or $$\gamma=-\frac{F_{1}}{2F_{2}}+\left[\left(\frac{F_{1}}{2F_{2}}\right)^{2}-\frac{F_{0}}{F_{2}}\right]^{1/2},\label{eq:QuadEqnGammaa}$$ where $F_{2}$, $F_{1}$, and $F_{0}$ are as follows: $$\begin{aligned} \operatorname{Re}(F_{2}) & =\frac{1}{\alpha^{4}}\left\{ \left(c_{\alpha n}^{2}+\alpha^{2}n^{2}\right)\left(s_{\alpha}^{2}-\alpha^{2}\right)m^{2}+2\left(s_{\alpha}c_{\alpha}s_{\alpha n}c_{\alpha n}-\alpha^{2}n+\alpha^{4}n^{2}\right)m\right.\nonumber \\ & +\left.\left(s_{\alpha n}^{2}-\alpha^{2}n^{2}\right)\left(c_{\alpha}^{2}+\alpha^{2}\right)\right\} \text{,}\label{eq:F2Re}\\ \operatorname{Im}(F_{2}) & =0\text{,}\label{F2Im}\\ \operatorname{Re}(F_{1}) & =\frac{1}{2\alpha^{3}}\left\{ m\text{Ma}(s_{\alpha n}c_{\alpha n}+\alpha n)\left(s_{\alpha}^{2}-\alpha^{2}\right)+\text{Ma}(s_{\alpha n}^{2}-\alpha^{2}n^{2})\left(s_{\alpha}c_{\alpha}+\alpha\right)\right.\nonumber \\ & +\frac{1}{\alpha^{2}}m(s_{\alpha n}c_{\alpha n}-\alpha n)\left(s_{\alpha}^{2}-\alpha^{2}\right)\left(\text{Bo}+\alpha^{2}\right)\nonumber \\ & +\left.\frac{1}{\alpha^{2}}(s_{\alpha n}^{2}-\alpha^{2}n^{2})\left(s_{\alpha}c_{\alpha}-\alpha\right)\left(\text{Bo}+\alpha^{2}\right)\right\} \text{,}\label{eq:F1Re}\\ \operatorname{Im}(F_{1}) & =\frac{s}{\alpha^{2}}(1-m)(s_{\alpha n}c_{\alpha n}-\alpha n+n^{2}s_{\alpha}c_{\alpha}-\alpha n^{2})\text{,}\label{eq:F1Im}\\ \operatorname{Re}(F_{0}) & =\frac{\text{Ma}}{4\alpha^{4}}(s_{\alpha n}^{2}-\alpha^{2}n^{2})(s_{\alpha}^{2}-\alpha^{2})\left(\text{Bo}+\alpha^{2}\right)\text{,}\label{eq:F0Re}\\ \operatorname{Im}(F_{0}) & =-\frac{\text{Ma}}{2\alpha}s(s_{\alpha n}^{2}-s_{\alpha}^{2}n^{2})\text{.}\label{eq:F0Im}\end{aligned}$$ Because the coefficients of the quadratic equation (\[eq:DispersionEquation\]) are complex numbers, it is clear that in general the imaginary parts of the solutions $\gamma_{1}$ and $\gamma_{2}$ are non zero which signifies an oscillatory instability. One can see that the growth rate $\gamma_{R}$ (as well as the increment $\gamma$ ) has the function symmetry property $$\gamma_{R}(-n\alpha;\;ns,\;m^{-1},\;n^{-1},\;\text{Ma},\;n^{2}\text{Bo})=nm\gamma_{R}(\alpha;\;s,\;m,\;n,\;\text{Ma},\;\text{Bo}).\label{eq:gammasymmetrycondition}$$ In view of this symmetry, it is sufficient to consider stability for $n\ge1$. (See Frenkel and Halpern (2016) for comprehensive details.) We also note the following facts. All the coefficients of the quadratic equation (\[eq:DispersionEquation\]) are continuous at each point $(\alpha;s,m,n,\text{Ma},\text{Bo})$ for the physical values of $\alpha$ and the parameters. All parenthetical expressions in equations (\[eq:F2Re\]) through (\[eq:F0Im\]) containing hyperbolic functions are positive. Therefore, $F_{2}>0$, and $\textrm{Re}(F_{1})$ and $\textrm{Re}(F_{0})$ are positive for $\text{Bo}\ge0$. For $\textrm{Bo}<0$, the functions $\textrm{Re}(F_{1})$ and $\textrm{Re}(F_{0})$ are positive provided $\alpha^{2}>-{\text{Bo}}$. Also, $\textrm{Im}(F_{1})>0(<0)$ for $m<1(>1)$. Furthermore, $\operatorname{Im}(F_{0})=0$ for $n=1$, and negative for $n>1$. The zero gravity limit studied in FH and HF is recovered when ${\text{Bo}}=0$. We want to investigate the dependence of the growth rates $\gamma_{R}=\operatorname{Re}(\gamma)$ on the wavenumber $\alpha$ and the parameters $n$, $m$, $s$, ${\text{Ma}}$ and ${\text{Bo}}$ in the ranges $0<\alpha<\infty$, $1\leq n<\infty$, $0<m<\infty$, $0\le s<\infty$, $0\le\text{Ma}<\infty$ and $-\infty<\text{Bo}<\infty$. It is an elementary fact of complex analysis that there are two analytic, and therefore continuous, branches of the complex square root function in every simply connected domain not containing the origin (see e.g. @Bak2010 pages 114-115). Then, as the discriminant $$\zeta=F_{1}^{2}-4F_{0}F_{2}\label{eq:DiscZ}$$ is clearly a smooth function of $\alpha$ and the parameters, there are two continuous branches of the increment $\gamma$ (\[eq:QuadEqnGamma\]) as functions of $\alpha$ and the parameters, and correspondingly two continuous branches of the growth rate $\gamma_{R}$. If ${\text{Ma}}\downarrow0$ then $\gamma_{1}\gamma_{2}=F_{0}/F_{2}\downarrow0$ and $\gamma_{1}+\gamma_{2}=-F_{2}/F_{1}\not\rightarrow0$ and so either $\gamma_{1}\downarrow0$ or $\gamma_{2}\downarrow0$. We call the increment branch that is non-zero at ${\text{Ma}}=0$ the “robust branch,” and the other one, that vanishes as ${\text{Ma}}\downarrow0$, is named the “surfactant branch”. Correspondingly, these are the continuous robust and surfactant branches of the growth rate. In certain cases, such as the one considered in section 4.3.1 with $m=1$, it can be shown that the discriminant $\zeta$ never takes the zero value and the range of the function $\zeta$$(\alpha;s,m,n,{\text{Ma}},{\text{Bo}})$ is a simply connected domain in the complex $\zeta$-plane. Then, there are two branches of the growth rate which are continuous functions of $(\alpha;s,m,n,{\text{Ma}},{\text{Bo}})$. However, as will be seen below, the discriminant (\[eq:DiscZ\]) may become zero for some parameter values. This happens when $\textrm{Re}(\zeta)=0$ and $\textrm{Im(}\zeta)=0$. These two equations define a manifold of co-dimension two in the $(\alpha;s,m,n,{\text{Ma}},{\text{Bo}})$ space that is analogous to a branch point in the complex plane; and if we draw the line of increasing $\alpha$ from each point of this manifold, that is a ray parallel to the $\alpha$-axis, with all the parameter values fixed, we obtain the “branch cut” hypersurface. The growth rates are not defined on this branch cut, and there is a jump in the growth rate when crossing from one side of the branch cut to the other. Still, each of the two growth-rate branches is defined and continuous almost everywhere in the $\alpha$-parameter space (with the branch cut hypersurface excluded from it), and the growth-rate branches defined this way are smooth in $\alpha$. The surfactant branch of the growth rate is again defined as the one which vanishes as $\text{Ma}\downarrow0$. These considerations are given in more detail in appendix \[sec:On-the-Continuous-Branches\]. It will be seen below, as for example in figure 9, that the discriminant equal to zero corresponds to the reconnection point of the two growth rate branches, when the crossing dispersion curves of the two branches become non-crossing at a certain value of a changing parameter. There is a jump discontinuity of the growth rate in the changing parameter at its reconnection-point value, for all $\alpha$ exceeding the reconnection-point value of $\alpha$. Except for such reconnection situations, all the dispersion curves are smooth at all $\alpha$. Typical dispersion curves of stable and unstable cases look like those in figure \[fig:FigTypicalDispCurve\]. The unstable branch starts at $\alpha=0$ and $\gamma_{R}=0$, grows with $\alpha$, attains a maximum value $\gamma_{R\max}$ at some $\alpha=\alpha_{\max}$, then decreases and crosses the $\alpha$-axis so that $\gamma_{R}=0$ at some non-zero wavenumber, $\alpha_{0}$, called the marginal wavenumber. The other, stable, branch also starts at $\alpha=0$ and $\gamma_{R}=0$ but then decreases with $\alpha$. The values of $\alpha_{0}$, $\gamma_{R\max}$, and $\alpha_{\max}$ depend on the parameters $n$, $m$, $s$, ${\text{Ma}}$, and ${\text{Bo}}$. Each solution ($\gamma$;$h,G$) of the system (\[eq:dispEqnSystem\]) determines the normal-mode amplitudes (and thus the complete structure of the normal mode), since $h$ and $G$ determine the coefficients $a_{1}$, $b_{1}$, and $b_{2}$, and thus the vertical velocities $\hat{w}_{j}$ via equations (\[eq:w1hat\]) and (\[eq:w2hat\]), then the horizontal velocities $\hat{u}_{j}$ via equations (\[eq:ConEqnDisturb\]) and the pressures $\hat{f}_{j}$ via equations (\[eq:xMomEqnDisturbPressure\]). ![Typical dispersion curves of the two normal modes: (1) the unstable mode, which has a maximum growth rate $\gamma_{R}=\gamma_{R\max}$ at a wavenumber $\alpha=\alpha_{\max}$ and then decays, eventually becoming stable for $\alpha>\alpha_{0}$, and (2) the stable mode, which has negative growth rates for all wavenumbers.[]{data-label="fig:FigTypicalDispCurve"}](fig2rt){width="60.00000%"} It is pointed out in FH (i.e., for the case ${\text{Bo}}=0$) that at least one of the modes for each given $\alpha$ is stable. This result holds for ${\text{Bo}}\geq0$ as well, which is seen as follows. (However, we will see that for ${\text{Bo}}<0$ both modes are unstable sometimes.) Let the two solutions of (\[eq:QuadEqnGamma\]) be $\gamma_{1}=\gamma_{1R}+i\gamma_{1I}$ and $\gamma_{2}=\gamma_{2R}+i\gamma_{2I}$. Then the real parts of the solutions satisfy $\gamma_{1R}+\gamma_{2R}=-\operatorname{Re}(F_{1})/F_{2}<0$. The latter inequality holds because, as was discussed before, $\textrm{Re}(F_{1})>0$ when ${\text{Bo}}\ge0$. So, if one of the quantities $\gamma_{jR}$ is positive (corresponding to an unstable mode), then the other must be negative, thus giving a stable mode. In order to compute the maximum growth rate, $\gamma_{R\max}$, the wavenumber corresponding to the maximum growth rate, $\alpha_{\max}$, and the marginal wavenumber, $\alpha_{0}$, it is convenient to split the dispersion equation (\[eq:DispersionEquation\]) into its real and imaginary parts, $$F_{2}\gamma_{R}^{2}-F_{2}\gamma_{I}^{2}+\operatorname{Re}(F_{1})\gamma_{R}-\operatorname{Im}(F_{1})\gamma_{I}+\operatorname{Re}(F_{0})=0\text{, }\label{eq:ReDispEqn}$$ $$2F_{2}\gamma_{R}\gamma_{I}+\operatorname{Re}(F_{1})\gamma_{I}+\operatorname{Im}(F_{1})\gamma_{R}+\operatorname{Im}(F_{0})=0\text{.}\label{eq:ImDispEqn}$$ The imaginary part of the growth rate $\gamma_{I}$ is expressed in terms of $\gamma_{R}$ using equation (\[eq:ImDispEqn\]) (assuming ${\mbox{Re}}(F_{1})\ne0$) and then substituted it into (\[eq:ReDispEqn\]) to obtain the following quartic equation for $\gamma_{R}$, $$\begin{gathered} 4\,F_{2}{}^{3}\gamma_{R}^{4}+8F_{2}^{2}\operatorname{Re}(F_{1})\gamma_{R}^{3}+F_{2}\left[4\,F_{2}\operatorname{Re}(F_{0})+\operatorname{Im}(F_{1})^{2}+5\,\operatorname{Re}(F_{1})^{2}\right]\gamma_{R}^{2}\nonumber \\ +\operatorname{Re}(F_{1})\left[\operatorname{Re}(F_{1})^{2}+4F_{2}\operatorname{Re}(F_{0})+\operatorname{Im}(F_{1})^{2}\right]\gamma_{R}-F_{2}\operatorname{Im}(F_{0})^{2}\nonumber \\ +\operatorname{Re}(F_{1})^{2}\operatorname{Re}(F_{0\,})+\operatorname{Re}(F_{1})\operatorname{Im}(F_{1})\operatorname{Im}(F_{0})=0\text{.}\label{eq:MAXeqn}\end{gathered}$$ Since $\gamma_{R}=0$ at the marginal wavenumber, $\alpha_{0}$, equation (\[eq:MAXeqn\]) becomes $$-F_{2}\operatorname{Im}(F_{0})^{2}+\operatorname{Re}(F_{1})\operatorname{Im}(F_{1})\operatorname{Im}(F_{0})+\operatorname{Re}(F_{1})^{2}\operatorname{Re}(F_{0})=0\text{, }\label{eq:MaEquation}$$ the marginal wavenumber equation. This equation (\[eq:MaEquation\]) is a polynomial in ${\text{Ma}}$ and ${\text{Bo}}$ $$k_{20}\text{Ma}^{2}+k_{11}\text{Ma}B+k_{31}\text{Ma}^{3}B+k_{22}\text{Ma}^{2}B^{2}+k_{13}\text{Ma}B^{3}=0\text{ }\label{eq:MaEquation2}$$ where $B:=\text{Bo}+\alpha^{2}$ and the coefficients $k_{ij}$ are given in appendix \[sec:Coefficients-of-equations\]. For $\text{Ma}=0$, it transpires that these marginal wavenumber equations are not valid. However, then the coefficient $F_{0}$ of the quadratic equation (\[eq:DispersionEquation\]) vanishes, and there remains just one mode corresponding to the Rayleigh-Taylor instability whose increment $\gamma=-F_{1}/F_{2}$. For the marginal wavenumber, it follows that ${\mbox{Re}}(F_{1})=0$, which implies that $\alpha_{0}=(-\text{Bo})^{1/2}$. This corresponds to capillary forces balancing the destabilizing gravitational forces provided $\text{Bo}<0$. The wavenumber $\alpha_{\max}$ corresponding to the maximum growth rate $\gamma_{R\max}$ is obtained by simultaneously solving (\[eq:MAXeqn\]) and the equation obtained by differentiating (\[eq:MAXeqn\]) with respect to $\alpha$, taking into account that $d\gamma_{R}/d\alpha=0$ at the maximum. The latter equation is written as $$\gamma_{R}^{4}\frac{d}{d\alpha}C_{4}(\alpha)+\gamma_{R}^{3}\frac{d}{d\alpha}C_{3}(\alpha)+\gamma_{R}^{2}\frac{d}{d\alpha}C_{2}(\alpha)+\gamma_{R}\frac{d}{d\alpha}C_{1}(\alpha)+\frac{d}{d\alpha}C_{0}(\alpha)=0,\text{ }\label{eq:alphaeq:MAXeqn}$$ where $C_{j}$ denotes the coefficient of the $\gamma_{R}^{j}$ term that appears in equation (\[eq:MAXeqn\]). (For example, $C_{4}=4F_{2}^{3}$.) Long-wave approximation {#sec:FiniteCaseLongwaveApprox} ======================= As was mentioned earlier, from the long-wave approximation by FH (${\text{Bo}}=0$), three sectors in the $(n,m)$-plane were identified that characterize the stability of the flow for $n\ge1$. Based on the long-wave results of FH17, the same three sectors are found to be relevant in the presence of gravity effects: the $Q$ sector ($m>n^{2}$), the $R$ sector ($1<m<n^{2}$), and the $S$ sector ($0<m<1$). Figure \[fig:FigRegions\] shows the three sectors and their borders. Stability properties of the robust and surfactant branches can change significantly from sector to sector, and can be special on borders as well. ![Partition of the $(n,m)$-plane of the system, $n\ge1$ and $m>0$, into three sectors ($Q$, $R$, and $S$) and their borders corresponding to differences in stability properties of the flow. \[fig:FigRegions\]](fig3rt){width="60.00000%"} General asymptotics for the three sectors {#sec:FiniteCaseLongwaveApproxGeneralGammaExpressions} ----------------------------------------- ### Increments and growth rates While it is straightforward to use equation (\[eq:QuadEqnGamma\]) to evaluate and graph growth rates, the limit of long waves yields some simpler asymptotic expressions. The general growth rate (and the increment) expressions in the three sectors are given in this subsection, but additional results in each sector will be discussed in later sections. First, the coefficients $F_{2}$, $F_{1}$, and $F_{0}$ (\[eq:F2Re\])-(\[eq:F0Im\]) in the dispersion equation (\[eq:DispersionEquation\]) are expanded in a Taylor series about $\alpha=0$. The leading order terms are given in Appendix \[sec:Longwave-formulas-for-F\]. Unless $s=0$ and $\text{Bo}\ne0$, we have $\left\vert F_{1}^{2}\right\vert \gg\left\vert F_{2}F_{0}\right\vert $, provided $\alpha\ll s$, since if $s\ne0$, then $\left\vert F_{1}^{2}\right\vert \approx\operatorname{Re}\left(F_{1}^{2}\right)\sim\alpha^{2}$ and $\left\vert F_{2}F_{0}\right\vert \approx\operatorname{Im}(F_{2}F_{0})\sim\alpha^{3}$; and if $s=0$ and $\text{Bo}=0$ then $|F_{1}^{2}|\sim\alpha^{4}$ and $|F_{2}F_{0}|\sim\alpha^{6}$ (see Appendix \[sec:Longwave-formulas-for-F\]). Therefore, keeping the four leading members in the series for the second term of equation (\[eq:QuadEqnGamma\]), the two increments are $$\gamma\approx\frac{1}{2F_{2}}\left(-F_{1}\pm F_{1}\left[1+\frac{1}{2}\left(-\frac{4F_{2}F_{0}}{F_{1}^{2}}\right)-\frac{1}{8}\left(-\frac{4F_{2}F_{0}}{F_{1}^{2}}\right)^{2}+\frac{1}{16}\left(-\frac{4F_{2}F_{0}}{F_{1}^{2}}\right)^{3}\right]\right)\text{,}\label{eq:gamGeneralEqnRSQsectors}$$ or, keeping the terms necessary to obtain the growth rate $\gamma_{R}$ to the leading order, $$\gamma\approx-\frac{F_{1}}{F_{2}}+\frac{F_{0}}{F_{1}}+\frac{F_{0}^{2}F_{2}}{F_{1}^{3}}\label{eq:gamCeqn}$$ and $$\gamma\approx-\frac{F_{0}}{F_{1}}-\frac{F_{2}F_{0}^{2}}{F_{1}^{3}}-2\frac{F_{0}^{3}F_{2}^{2}}{F_{1}^{5}}\text{.}\label{eq:gamSeqn}$$ For $s\ne0$ the growth rates for the robust (\[eq:gamCeqn\]) and surfactant (\[eq:gamSeqn\]) branches are found to be, respectively, $$\gamma_{R}\approx\left({\frac{\,\varphi\left(m-{n}^{2}\right)}{4\left(1-m\right)\psi}}\text{Ma}-\frac{n^{3}(n+m)}{3\psi}\text{Bo}\right)\alpha^{2}\text{}\label{eq:gamCsmallAlphaApprox}$$ and $$\gamma_{R}\approx\frac{(n-1)\text{Ma}}{4(1-m)}\alpha^{2}+k_{S}\alpha^{4},\label{eq:gamSsmallAlphaApprox}$$ where $$\varphi={n}^{3}+3\,{n}^{2}+3\,mn+m\text{}\label{eq:phi}$$ and $$\psi={n}^{4}+4\,m{n}^{3}+6\,m{n}^{2}+4\,mn+{m}^{2}\text{.}\label{eq:psi}$$ We include the term with $k_{S}$ in equation (\[eq:gamSsmallAlphaApprox\]) because the coefficient of the $\alpha^{2}$ term vanishes when $n=1$. The expression for $k_{S}$ is given in appendix B, see equation (\[eq:ks\]). For the case $s=0$ and $\text{Bo}=0$, the growth rates for the robust (\[eq:gamCeqn\]) and surfactant (\[eq:gamSeqn\]) branches are found to be $$\gamma_{R}\approx-\frac{n^{3}}{12(m+n^{3})}\alpha^{4}$$ and $$\gamma_{R}\approx-\frac{n(m+n^{3})\text{Ma}}{\psi}\alpha^{2}$$ which is in agreement with FH. Finally, for the case $s=0$ and $\text{Bo}\ne0$, we find that $|F_{1}^{2}|\sim\alpha^{4}\sim|F_{2}F_{0}|$. So, the expansion (\[eq:gamGeneralEqnRSQsectors\]) is no longer valid. However, both modes are stable if $\text{Bo}>0$, but there is instability if $\text{Bo}<0$. Indeed, if $\text{Bo}<0$ then $F_{0}\approx\frac{1}{36}n^{4}\alpha^{4}\text{Ma}\text{Bo}<0$ (see equation (\[eq:F0reApprox\])). Therefore, the discriminant $F_{1}^{2}-4F_{0}F_{2}>F_{1}^{2}$. Then equation (\[eq:QuadEqnGamma\]) yields one of the two growth rates to be positive, so we have instability. On the other hand, if $\text{Bo}>0$, then ${\mbox{Re}}(F_{1})>0$ but the discriminant can be either positive or negative. If it is negative, then the square roots in equation (\[eq:QuadEqnGamma\]) are purely imaginary and therefore both values of $\gamma_{R}$ are negative. If the discriminant is positive, then $|\sqrt{F_{1}^{2}-4F_{0}F_{2}}|<F_{1}$, so that both values of $\gamma$ given by equation (\[eq:QuadEqnGamma\]) are negative again. These leading-order results were obtained in a different way and discussed in more detail in @Frenkel2016 and FH17. ### Marginal wavenumbers and their small $s$ asymptotics\[subsec:Marginal-wavenumbers\] When the marginal wavenumber determined by equation (\[eq:MaEquation\]) happens to be small (typically, due to the smallness of some of the three parameters s, Bo, and Ma), it is approximated by substituting the long-wave expressions for the coefficients (\[eq:F2reApprox\])-(\[eq:F0imApprox\]) into (\[eq:MaEquation\]) provided $\text{Ma}\ne0$. If $s\ne0$ is fixed, then by keeping only the two leading terms in $\alpha^{2}$, we arrive at $$\begin{gathered} \zeta_{0}+\zeta_{2}\alpha^{2}=0\label{eq:MAappEqn}\end{gathered}$$ where $\zeta_{0}$ and $\zeta_{2}$ are polynomials in ${\text{Ma}}$ and ${\text{Bo}}$ given by equations (\[eq:zeta0\]) and (\[eq:zeta2\]). Therefore, at leading order, $$\alpha_{0}=\sqrt{-(\zeta_{0}/\zeta_{2})}.$$ Clearly, for this result to be consistent, $\zeta_{0}/\zeta_{2}$ must be negative and small, which is the case for appropriate parameter values, such as, for example, those used in figures 4, 5, and 6. It is interesting to investigate the transition from instability to stability of the case $s=0$ by considering the limit $s\downarrow0$. In this we should distinguish two cases: $\text{Bo}=0$ and $\text{Bo}\ne0$. For $\text{Bo}\ne0$, the marginal wavenumber is given by $$\widetilde{\zeta_{0}}s^{2}+\zeta_{20}\alpha^{2}=0\label{eq:alpha0smalls1}$$ instead of equation (\[eq:MAappEqn\]), where, by definition the coefficients $\widetilde{\zeta_{0}}=\zeta_{0}/s^{2}$ and $\zeta_{20}=\zeta_{2}(s=0)$ (see equations (\[eq:zeta0\]) and (\[eq:zeta2\])). These coefficients are independent of $s$ and $\alpha$, and so, asymptotically $\alpha_{0}$ is proportional to $s$, with the coefficient of proportionality $\sqrt{-\widetilde{\zeta_{0}}/\zeta_{20}}$. However, for $\text{Bo}=0$, the coefficient of the $\alpha^{2}$ term in equation (\[eq:alpha0smalls1\]) vanishes, and, instead the leading order equation for the marginal wavenumber is found to be $$\widetilde{\zeta_{0}}s^{2}+\zeta_{40}\alpha^{4}=0,$$ where $\zeta_{40}=\frac{1}{324}n^{2}(m+n^{3})^{2}\text{Ma}^{2}$. Then the marginal wavenumber is asymptotically $\alpha_{0}=(-\widetilde{\zeta_{0}}/\zeta_{40})^{1/4}s^{1/2}$. Panel (c) of figure \[fig:fig4abcd\] shows these asymptotes along with the marginal wavenumbers obtained by solving equation (\[eq:MaEquation2\]) for $\text{Bo=0}$ and some positive values of $\text{Bo}$ in the $R$ sector. Panel (d) shows, for a fixed wavenumber, $\alpha=0.01$, how the instability at the larger $s$ corresponding to the (positive) growth rate (\[eq:gamCsmallAlphaApprox\]), changes to stability with the growth rate corresponding, in the leading order, to the case of $s=0$ and nonzero $\textrm{Bo}$. The growth rate that crosses the zero value at the $s$ for which $\alpha=0.01$ is the marginal wavenumber. In the analogous figure for the $Q$ sector, figure \[fig:figgrminaminvss\], the marginal wavenumber is the left endpoint of the interval of the unstable wavenumbers, which is bounded away from the zero of the wavenumber axis. There is a band of stable wavenumbers between this marginal wavenumber and the zero, and inside it there is a minimum of the growth rate, $\gamma_{Rmin}$, at the corresponding wavenumber $\alpha_{min}$; their dependencies on $s$ are plotted in panels (a) and (b), respectively. Correspondingly, panel (d) shows stability at the larger $s$, and instability at the smaller $s$, since here, in the $Q$ sector, it is the band of *stable* wavenumbers that shrinks toward zero as $s\downarrow0$. We call such cases, in which there is an interval of unstable wavenumbers bounded away from zero, the mid-wave instability, to distinguish them from the long-wave instability, in which the interval of unstable wavenumbers is bordered by zero. We study the mid-wave instability in detail below (see sections 5 and 6). By considering the formula for $\zeta_{2}$ (\[eq:zeta2\]) for sufficiently small $\textrm{Ma}$ and $\textrm{Bo}$, we see that all the terms are negligible as compared to the last one (the capillary term), and equation (\[eq:alpha0smalls1\]), after being multiplied by an appropriate factor, is interpreted as the instability term (\[eq:gamCsmallAlphaApprox\]) being balanced by the capillary effect (corresponding to the term $\alpha^{2}$ in $B=\textrm{Bo}+\alpha^{2}$, and arising from the second term of equation (\[eq:alpha0smalls1\]).) The resulting, asymptotically $s$-independent, value of the marginal wavenumber, as one can see at the larger $s$ in figure \[fig:figavssmalls\], is still small, consistent with the long-wave approximation. ![Marginal wavenumber $\alpha_{0}$ vs the shear parameter $s$, along with its asymptotics, at larger $s$, due to the capillary effects, and at smaller $s$, due to the combined gravity-surfactant effects, for $n=2$, $m=2$, $\text{Ma}=0.05$, and $\text{Bo}=-0.05$.\[fig:figavssmalls\]](fig6rt){width="65.00000%"} However, for the same fixed small values of $\textrm{Ma}$ and $\textrm{Bo}$, at sufficiently small $s$, the last, capillary, term in $\zeta_{20}$ is negligible, and the stabilization near the marginal wavenumber is due to non-capillary effects of the combined action of surfactants and gravity. It is clear that the three corresponding terms in $\zeta_{20}$ are not zero only if both the Marangoni and Bond numbers are non-zero. These (non-additively) combined surfactant-gravity effects are beyond the lubrication approximation, and can be captured only by the post-lubrication correction theory considered in @Frenkel2016. Figure \[fig:figavssmalls\] shows the numerical solution of the marginal-wavenumber equation (\[eq:MaEquation\]) without using the long-wave asymptotics, along with the larger-$s$ (capillary) and small-$s$ (gravity- and surfactant-determined, non-lubrication) approximations of the wavenumber given by the long-wave asymptotic equation (\[eq:alpha0smalls1\]). Excellent agreement is evident. ### Maximum growth rates As indicated earlier, a way to find $\gamma_{R\text{max }}$ and $\alpha_{\text{max}}$ is to solve equations (\[eq:MAXeqn\]) and (\[eq:alphaeq:MAXeqn\]). For $s\downarrow0$, numerical computations suggest that $\alpha_{\text{max}}\propto s$ if $\text{Bo}\ne0$ (just like $\alpha_{0}$) and $\alpha_{\text{max}}\propto s^{2/3}$ if $\text{Bo}=0$, and that $\gamma_{R\text{max}}\propto s^{2}$ for both $\text{Bo}\ne0$ and $\text{Bo}=0$, as one can see in figure \[fig:fig4abcd\]. We find the coefficients of these asymptotic dependencies as follows: For the case $\text{Bo}=0$, we write $\gamma_{R\text{max }}$ and $s^{2}$ as functions of $\alpha_{\text{max}}$ to the two leading orders, $$s^{2}\approx\phi_{1}\alpha^{3}+\phi_{2}\alpha^{4},\quad\gamma_{R\text{max}}\approx\psi_{1}\alpha_{\text{}}^{3}+\psi_{2}\alpha^{4},\label{eq:gammarsmalls}$$ with indeterminate coefficients $\phi_{1}$, $\phi_{2}$, $\psi_{1}$ and $\psi_{2}$. We have to use two leading orders because the leading order system for $\phi_{1}$ and $\psi_{1}$ turns out to be degenerate, and only gives one relation between $\phi_{1}$ and $\psi_{1}$. The other relation between $\phi_{1}$ and $\psi_{1}$ is found as the solvability condition for the next order non-homogeneous system for $\phi_{2}$ and $\psi_{2}$. The leading order of equation (\[eq:MAXeqn\]) consists of terms that are proportional to $\alpha^{9}$. Therefore, the terms which are nonlinear in $\gamma_{R}$ are discarded. This yields $$(d\alpha^{6})\psi_{1}\alpha^{3}+(f\alpha^{6})\phi_{1}\alpha^{3}=0,\label{eq:grsmallsbo0eq}$$ where $d=<Re(F_{1})>^{3}$ and $f=<Re(F_{1})><{\mbox{Im}}(F_{1})><{\mbox{Im}}(F_{0})>-F_{2}<{\mbox{Im}}(F_{0})>^{2}$. Here the bracketed quantities are the coefficients of powers of $s$ and $\alpha$ in the leading order terms of the corresponding “unbracketed” coefficients (A24)-(A28): $<Re(F_{1})>=\frac{1}{3}n(m+n^{3})\text{Ma}$, $<{\mbox{Im}}(F_{1})>=\frac{2}{3}n^{2}(n+1)(1-m)$, $<Re(F_{0})>=\frac{1}{36}n^{4}\text{Ma}$, and $<{\mbox{Im}}(F_{0})>=\frac{1}{6}n^{2}(1-n^{2})\text{Ma}$. When obtaining equation (\[eq:alphaeq:MAXeqn\]) by differentiating with respect to $\alpha$ at constant $\gamma_{R}$ and $s^{2}$, only the powers of $\alpha$ inside the parentheses of equation (\[eq:gammarsmalls\]) are differentiated, and this yields $$(6d\alpha^{5})\psi_{1}\alpha^{3}+(6f\alpha^{5})\phi_{1}\alpha^{3}=0.$$ So, the matrix of the coefficients of the linear homogeneous system for $\phi_{1}$ and $\psi_{1}$ $$M=\left[\begin{array}{cc} f & d\\ 6f & 6d \end{array}\right]$$ is singular, and the leading-order system yields the single relation $$\phi_{1}=-\frac{d}{f}\psi_{1}.\label{eq:a1smalls}$$ Therefore, we need to consider the next order of equation (\[eq:MAXeqn\]), proportional to $\alpha^{10}$. We obtain $$\begin{aligned} (d\alpha^{6})\psi_{2}\alpha^{4}+(f\alpha^{6})\phi_{2}\alpha^{4} & = & -5(F_{2}<{\mbox{Re}}(F_{1})>^{2}\alpha^{4})\psi_{1}^{2}\alpha^{6}\label{eq:a2b2eq1}\\ & & -(<{\mbox{Re}}(F_{1})><{\mbox{Im}}(F_{1})>^{2}\alpha^{4})\psi_{1}\alpha^{3}\phi_{1}\alpha^{3}\nonumber \\ & & -(<{\mbox{Re}}(F_{1})>^{2}<{\mbox{Re}}(F_{0})>)\alpha^{10}.\nonumber \end{aligned}$$ Differentiating the quantities inside the parentheses with respect to $\alpha$, the second equation for $\phi_{2}$ and $\psi_{2}$ is $$\begin{aligned} (6d\alpha^{5})\psi_{2}\alpha^{4}+(6f\alpha^{5})\phi_{2}\alpha^{4} & = & -5(4F_{2}<{\mbox{Re}}(F_{1})>^{2}\alpha^{3})\psi_{1}^{2}\alpha^{6}\label{eq:a2b2eq2}\\ & & -(4<{\mbox{Re}}(F_{1})><{\mbox{Im}}(F_{1})>^{2}\alpha^{3})\psi_{1}\alpha^{3}\phi_{1}\alpha^{3}\nonumber \\ & & -10(<{\mbox{Re}}(F_{1})>^{2}<{\mbox{Re}}(F_{0})>)\alpha^{9}.\nonumber \end{aligned}$$ Equations (\[eq:a2b2eq1\]) and (\[eq:a2b2eq2\]) form a nonhomogeneous linear system for $[\phi_{2},\;\psi_{2}]$ with the same matrix $M$. The condition for the solution $[\phi_{2},\;\psi_{2}]$ to exist requires that the right hand of the second equation is six times that of the first equation, which yields after eliminating $\phi_{1}$ by equation (\[eq:a1smalls\]) the following equation for $\psi_{1}$ $$\left(5F_{2}f-<{\mbox{Re}}(F_{1})>^{2}<{\mbox{Im}}(F_{1})>^{2}\right)\psi_{1}^{2}=2f<{\mbox{Re}}(F_{0})>.$$ This determines $\psi_{1}$, and then from equation (\[eq:a1smalls\]), $\phi_{1}$, namely, $$\phi_{1}=\left[\frac{8n^{2}(m+n^{3})^{6}}{3(n-1)(n+1)^{4}(n^{2}-m)\phi(16(m-1)^{2}(m+n^{3})^{2}+5(n-1)(n^{2}-m)\phi\psi)}\right]^{1/2}\text{Ma}^{3/2}$$ and $$\psi_{1}=\left[\frac{n^{4}(n-1)(n^{2}-m)\phi}{6(16n^{2}(m-1)^{2}(m+n^{3})^{2}+5(n-1)(n^{2}-m)\phi\psi)}\right]^{1/2}\text{Ma}^{1/2}.$$ Returning to the independent variable $s$, the asymptotics $$\gamma_{R\text{max}}=\frac{\psi_{1}}{\phi_{1}}s^{2},\;\alpha_{\text{max}}=\phi_{1}s^{2/3}$$ are shown in figure \[fig:fig4abcd\] along with the full dependencies for a representative set of the parameter values. For the case $\text{Bo}\ne0$, it is sufficient to consider only the leading order of equations (\[eq:MAXeqn\]) and (\[eq:alphaeq:MAXeqn\]) (proportional correspondingly to $\alpha^{8}$ and $\alpha^{7}$) to determine the coefficients $c_{1}$ and $d_{1}$ in the asymptotics $s^{2}=c_{1}\alpha^{2}$ and $\gamma_{R\text{max}}=d_{1}\alpha^{2}$. Since there are contributions from the terms of equations (\[eq:MAXeqn\]) and (\[eq:alphaeq:MAXeqn\]) with all powers of $\gamma_{R\text{max}}$, the resulting system of two quartic equations for $c_{1}$ and $d_{1}$ can only be solved numerically. The small-$s$ asymptotics, $$\gamma_{R\text{max}}=\frac{d_{1}}{c_{1}}s^{2},\;\alpha_{\text{max}}=c_{1}^{-1/2}s$$ are shown in figure \[fig:fig4abcd\] along with the full numerics. We see that the cases $\text{Bo}=0$ and $\text{Bo}\ne0$ have different powers of $s$ in the asymptotics for $\alpha_{\text{0}}$, and the same is true for $\alpha_{\text{max}}$. Figure \[fig:fig4abcd\](c) shows that as $\text{Bo}\downarrow0$, the interval of small $s$ for which $\alpha_{\text{0}}\propto s$ shrinks, and there is a crossover to the $s^{1/2}$ behavior characteristic of $\text{Bo}=0$ for an interval of larger (but still small) wavenumbers. Similarly, for $\alpha_{\text{max}}$ there is a crossover from $\alpha_{\text{max}}\propto s$ at the smallest $s$ to the $s^{2/3}$ asymptotic characteristic of $\text{Bo}=0$ for an interval of larger wavenumbers. These considerations clarify the transition from the instability at $s\ne0$ to stability at $s=0$, and the relation between the different powers in the $\alpha_{0}$ and $\alpha_{\text{max }}$ asymptotics of the $\text{Bo}\ne0$ and $\text{Bo}=0$ cases. Instability thresholds in the different sectors and nearby asymptotic behavior {#sec:FiniteCaseLongwaveApproxCregions} ------------------------------------------------------------------------------ In both the $R$ sector $(1<m<n^{2})$ and the $Q$ sector, $(m>n^{2})$, the surfactant branch (\[eq:gamSsmallAlphaApprox\]) is stable for all $\text{Bo }$ and the robust branch (\[eq:gamCsmallAlphaApprox\]) is unstable if ${\text{Bo}}<{\text{Bo}}_{cL}$, where, in view of equation (\[eq:gamCsmallAlphaApprox\]), the threshold value is $$\text{Bo}_{cL}=\frac{3\varphi(m-n^{2})}{4n^{3}(1-m)(n+m)}\text{Ma.}\label{eq:BoCritical}$$ In the $R$ sector, the Marangoni effect is destabilizing, so $\textrm{B\ensuremath{o_{cL}}}>0$; gravity renders the flow stable for $\text{Bo}>\text{Bo}_{cL}$, whereas for $\text{Bo }<\text{Bo}_{cL}$, the flow is unstable. In the $Q$ sector (and in the $S$ sector as well), the Marangoni effect is stabilizing, $\text{Bo}_{cL}<0$, and the gravity effect renders the robust branch unstable when the (negative, destabilizing) $\text{Bo}<\text{Bo}_{cL}$. From equation (\[eq:BoCritical\]) the ratio $\text{Bo}_{cL}/\text{Ma}$ is a function of $m$ and $n$ only, and its graph is a surface in the $(n,m,\text{Bo}_{cL}/\text{Ma})$-space. This surface is plotted in figure 3 of @Frenkel2016, and is discussed in detail there. The window of unstable wavenumbers, $0<\alpha<\alpha_{0}$, shrinks to zero as ${\text{Bo}}\uparrow{\text{Bo}}_{cL}$, so that the marginal wavenumber $\alpha_{0}\downarrow0$ for both the $R$ and $Q$ sectors. To obtain the asymptotic approximation for $\alpha_{0}$, we write the Bond number as $$\text{Bo}=\text{Bo}_{cL}-\ \Delta\text{}\label{eq:BoDeltaFormula}$$ with $\Delta\downarrow0$. Equation (\[eq:F2Re\]) is substituted into (\[eq:MAappEqn\]) and when retaining the leading order terms in $\Delta$ and $\alpha^{2}$ we find that $\zeta_{0}$ is proportional to $\Delta$ and $\zeta_{2}$ is a cubic polynomial in $\text{Bo}_{cL}$ (and is independent of $\Delta$, to the leading order). The solution is $$\begin{gathered} \alpha_{0}\approx\left[1+\beta_{1}\text{Bo}_{cL}+\beta_{3}\left.\text{Bo}_{cL}^{3}\right]^{-1/2}\Delta^{1/2}\text{}\right.\label{eq:MAapproxCmode}\end{gathered}$$ where the coefficients $\beta_{1}$ and $\beta_{3}$ are given by equations (\[eq:beta1\]) and (\[eq:beta3\]) in appendix \[sec:Coefficients-of-equations\]. Note here that ${\text{Ma}}$ has been written in terms of ${\text{Bo}}_{cL}$ using equation (\[eq:BoCritical\]). If ${\text{Bo}}_{cL}$ $\ll1$ (i.e., ${\text{Ma}}\ll1$) equation (\[eq:MAapproxCmode\]) simplifies to $$\alpha_{0}\approx\Delta^{1/2}\text{.}\label{eq:MAapproxCmodeDeltaBo}$$ We also find in the way described above the long-wave asymptotic dependences $$\alpha_{\text{max}}\propto\Delta^{1/2}\text{ and \ensuremath{\gamma_{R\text{max}}\propto\Delta^{2}}.}$$ For example, the relative error of the asymptotic expression (\[eq:MAapproxCmodeDeltaBo\]) for $n=m=2$, $s=1$, ${\text{Bo}}=10^{-6}$, and ${\text{Ma}}=10^{-6}$ to ${\text{Ma}}=10$ is less than $10\%$ for $\Delta<0.2$. This is illustrated in figure \[fig:figRTmaANDapp\], where $n=m=2$, $s=1$ and $\text{Ma}=1$. The asymptotics for $\gamma_{R\text{max}}$, $\alpha_{\text{max}}$ and $\alpha_{0}$ near $\text{Bo}=\text{Bo}_{c}$ are practically indistinguishable from the full numerical solutions. ![$\gamma_{R\text{max}},\alpha_{\text{max}}$ and $\alpha_{0}$ vs $\Delta$ for the same $n$, $m$ and $\text{Ma}$ as in figure \[fig:fig4abcd\], and $s=1$. The solid lines represent the full solutions, equation (\[eq:MaEquation2\]), and the dashed lines represent the asymptotics given by (\[eq:MAapproxCmode\]).\[fig:figRTmaANDapp\]](fig7rt){width="100.00000%"} In the $S$ sector ($1<n<\infty$ and $0<m<1$), the robust branch (\[eq:gamCsmallAlphaApprox\]) is stable when $\text{Bo}>\text{Bo}_{cL}$, the latter given by (\[eq:BoCritical\]), and unstable otherwise. However, equation (\[eq:gamSsmallAlphaApprox\]) for the surfactant branch does not contain the Bond number, and indicates instability. Thus the surfactant mode is unstable for any $\text{Bo}$ provided $\alpha$ is sufficiently small. However, it is easy to see that the window of unstable wavenumbers shrinks to zero as $\text{Bo}\uparrow\infty$. Indeed in this limit, equation (\[eq:MAappEqn\]) reduces to $$s^{2}(n-1)(n+1)^{2}(m-1)+36n^{3}(n+m)\alpha^{2}\text{Bo}^{2}=0\text{.}\label{eq:MAappEqnSmodeBigBo}$$ Hence the asymptotic formula for the marginal wavenumber is $$\alpha_{0}\approx\left[\frac{36s^{2}(n+1)^{2}(1-m)(n-1)}{n^{3}(n+m)}\right]^{1/2}\text{Bo}^{-1}\text{.}\label{eq:MAapproxSmode}$$ For the $Q$ sector, the instability threshold (\[eq:BoCritical\]) can be written in a different way: to state that (while the surfactant branch (\[eq:gamSsmallAlphaApprox\]) is stable for all $\text{Bo}$ and ${\text{Ma}}$), the robust branch is stable if ${\text{Ma}}$ exceeds a critical Marangoni number, ${\text{Ma}}_{cL}$ given by $$\text{Ma}_{cL}=\frac{4n^{3}(1-m)(n+m)}{3\varphi\left(m-n^{2}\right)}\text{Bo,}\label{eq:RTcritcalMa}$$ which is the reciprocal of (\[eq:BoCritical\]). When ${\text{Ma}}\uparrow{\text{Ma}}_{cL}$, the marginal wavenumber is expressed in terms of $\Delta_{M}={\text{Ma}}_{cL}-{\text{Ma}}$. From equation (\[eq:MAappEqn\]), we obtain in the same way that we derived (\[eq:MAapproxCmode\]) for the marginal wavenumber in the $R$ sector that $$\alpha_{0}\approx\left[M_{0}+M_{1}\text{Ma}_{cL}+M_{3}\text{Ma}_{cL}^{3}\right]^{-1/2}\Delta_{M}^{1/2}\text{.}\label{eq:RTmaApp}$$ where the coefficients $M_{0}$, $M_{1}$ and $M_{3}$ are given by (\[eq:M0\])-(\[eq:M3\]) in appendix B. Instabilities on the $\left(n,m\right)$-sector borders {#sec:Instabilities on borders} ------------------------------------------------------ The borders $m=1$, $m=n^{2}$, and $n=1$ are considered separately because of singularities that can occur in the expressions for the growth rates and the marginal wavenumber derived in the previous sections for the $R$, $S$, and $Q$ sectors. ### The $m=1$ border Consider first the case $m=1$ and $n\neq1$. In the long-wave limit, $F_{1}^{2}\ll\left\vert F_{2}F_{0}\right\vert $ since $F_{1}^{2}\sim\alpha^{4}$ and $|F_{2}F_{0}|\sim{\text{Ma}}\alpha^{3}$ (the truncated Taylor series for such quantities are shown in Appendix C of @schweiger2013gravity). Therefore, the roots to the dispersion equation (\[eq:QuadEqnGamma\]), are approximated by $$\gamma\approx\frac{1}{2F_{2}}\left(-F_{1}+(4F_{2}F_{0})^{1/2}\left[1+\frac{1}{2}\left(-\frac{F_{1}^{2}}{4F_{2}F_{0}}\right)\right]\right)\text{.}\label{eq:gammaIncrAPPm}$$ Hence, the growth rates of the two branches are $$\gamma_{R}=\frac{-\operatorname{Re}(F_{1})+\operatorname{Re}(\sqrt{\zeta})}{2F_{2}}\label{eq:gamWithZm}$$ where $\zeta$ is the discriminant of (\[eq:QuadEqnGamma\]). To leading order in $\alpha$, equation (\[eq:gamWithZm\]) reduces to $$\gamma_{R}\approx\frac{\operatorname{Re}(\sqrt{\zeta})}{2F_{2}}=\pm\frac{n\left[\left\vert n-1\right\vert (n+1)s\text{Ma}\right]^{1/2}}{2(n+1)^{2}}\alpha^{3/2}.\label{eq:GammaApproxSmallalpham}$$ This result does not depend on the Bond number and is the same as in FH and HF. It turns out that the next order correction, omitted in the leading order expression, depends on both the Bond number and the Marangoni number, and is proportional to $\alpha^{2}$. Note also that (\[eq:GammaApproxSmallalpham\]) is valid as $\alpha\downarrow0$ with the Marangoni number fixed but it is not valid as ${\text{Ma}}\downarrow0$ with the wavenumber fixed. We will show below that for $m=1$, the discriminant $\zeta$ in the expression for $\gamma_{R}$ is never zero, and thus there are two branches of $\gamma_{R}$ that are continuous at all parameter values and all $\alpha$, which we called the surfactant branch and the robust branch. It is unclear from equation (\[eq:GammaApproxSmallalpham\]) whether the positive growth rate corresponds to the surfactant branch or the robust branch. Recall that, as ${\text{Ma}}\downarrow0$, with $\alpha$ remaining finite, the identity of each branch is clear since, by definition, the branch that vanishes in this limit is the surfactant branch. Starting from there, each branch can be traced to the asymptotic region of small $\alpha$ and finite ${\text{Ma}}$ where equation (\[eq:GammaApproxSmallalpham\]) is valid and thus the branches will be identified there. The fact that there are two continuous branches of $\gamma(\alpha,\text{Ma)}$ (with the other parameters fixed and not shown explicitly) given by (\[eq:QuadEqnGamma\]) is seen as follows. As was discussed previously, in section \[sec:FiniteCaseDerivationDispRelation\] (see also Appendix A), in any simply connected domain not containing 0 of the complex $\zeta-$plane, there exist two distinct analytic branches of the square root function, $f(\zeta)=\zeta^{1/2}$. The $\sqrt{\zeta}$ in the expression for $\gamma_{R}$, is a composite function of $(\alpha,\text{Ma)}$ through $\zeta(\alpha,\text{Ma}$). The discriminant $\zeta$ is a single-valued continuous function of $(\alpha,\text{Ma)}$. It is easy to see that it maps the first quadrant of the $(\alpha,\text{Ma)}$-plane inside the upper half-plane $U$ of the $\zeta$-plane, which is a simply connected domain not containing 0. Indeed, when $m=1$ ($n\neq1$ and $s\neq0$), then from equation (\[eq:F1Im\]), $\textrm{Im}(F_{1})=0$, and hence $$\operatorname{Im}(\zeta)=-4F_{2}\operatorname{Im}(F_{0}).\label{eq:IMzPOSmeq1-n-neq1}$$ In view of $n>1$, we have $s_{\alpha n}>s_{\alpha}n$, and hence, from equation (\[eq:F0Im\]), $-\textrm{Im}(F_{0})>0$. Therefore, equation (\[eq:IMzPOSmeq1-n-neq1\]) yields $\operatorname{Im}(\zeta)>0$. Since the upper half-plane $U$ of the $\zeta$-plane is a simply connected domain not including 0, the square root function $\xi=f(\zeta)=\zeta^{1/2}$ in $U$ of $\zeta$ has two analytic branches. One of them maps $U$ onto the first quadrant of the $\xi-$plane, so that $\text{Re}(\sqrt{\zeta})>0$ for this branch, and thus $\text{Re}(\sqrt{(\zeta})$ is a *positive* continuous function of $(\alpha,\text{Ma})$. The other analytic branch of $\xi=\zeta^{1/2}$ has its range entirely in the third quadrant of the $\xi-$plane, so that $\text{Re}(\sqrt{\zeta})<0$ and thus $\text{Re}(\sqrt{\zeta})$ is a *negative* continuous function of $(\alpha,\text{Ma})$ Thus, there is the one branch of $\text{Re}(\sqrt{\zeta})$ that is continuous and positive at all $(\alpha,\text{Ma )}$ and the other branch of $\text{Re}(\sqrt{\zeta})$ that is continuous and negative at all $(\alpha,\text{Ma)}$. (We note that for even for arbitrary $m\ne0$, it readily follows that $\text{Im}(\zeta)>0$, provided that $\text{Ma}\downarrow0$ and $\text{Bo}>0$, since then, according to equations (\[eq:F1Re\])-(\[eq:F0Im\]), $F_{0}=0$, $\text{Re}(F_{1})>0$ and $\text{Im}(F_{1})>0$.) In the limit of $\text{Ma}\downarrow0$, the surfactant branch vanishes, $\gamma_{R}=0$, which from equation (\[eq:gamWithZm\]) means $\operatorname{Re}(\sqrt{\zeta})=\operatorname{Re}(F_{1})$. Therefore, $sgn(\operatorname{Re}(\sqrt{\zeta}))=sgn(\operatorname{Re}(F_{1}))$, where $sgn$ is the sign function. It is sufficient to consider here only small wavenumbers, from an interval $[0,\alpha_{s}]$, by choosing an arbitrary $\alpha_{s}$ such that $\alpha_{s}\ll1$ and $\alpha_{s}<|{\text{Bo}}|$. Then equation (\[eq:F1reApprox\]) (with $\text{Ma}=0$) yields $sgn(\operatorname{Re}(F_{1}))=sgn({\text{Bo}})$, so that $sgn(\operatorname{Re}(\sqrt{\zeta}))=sgn({\text{Bo}})$. As was already established, each branch of ${\mbox{Re}}(\sqrt{\zeta})$ has the same sign for all $(\alpha$,Ma$)$. Therefore, for the surfactant branch, the relation $sgn(\operatorname{Re}(\sqrt{\zeta}))=sgn({\text{Bo}})$ holds in the limit of $\alpha$ $\downarrow0$ as well. From equation (\[eq:GammaApproxSmallalpham\]), $sgn(\gamma_{R})=sgn(\operatorname{Re}(\sqrt{\zeta})$, and then for the surfactant branch, $sgn(\gamma_{R})=sgn({\text{Bo}})$. Thus, the surfactant branch is unstable for ${\text{Bo}}>0$, $\gamma_{R}\propto+\alpha^{3/2}$ and stable for ${\text{Bo}}<0$, $\gamma_{R}\propto-\alpha^{3/2}$. Consequently, the robust branch is stable (unstable) for ${\text{Bo}}>0$ (${\text{Bo}}<0$). This answers the question of identifying the stable and unstable modes as belonging to the appropriate branches. In certain limits it is possible to find a long-wave approximation to $\gamma_{R}$ that captures the growth rate behavior close to the marginal wavenumber $\alpha_{0}$. Assuming ${\text{Bo}}\gg{\text{Ma}}$, $\alpha^{2}\ll\textrm{Bo}$, and ${\text{Ma}}/{\text{Bo}}^{2}\ll\alpha\ll1$, equation (\[eq:QuadEqnGamma\]) can be simplified to yield, for the unstable surfactant branch, $$\gamma_{R}\approx\frac{27}{4}\frac{(n-1)^{2}(n+1)^{3}s^{2}\text{Ma}^{2}}{n^{5}\text{Bo}^{3}}-\frac{1}{4}\frac{n\text{Ma}}{(n+1)}\alpha^{2}\label{eq:GammaApproxLargealphameq1}$$ which is valid for $\alpha\approx\alpha_{0}$. (Note that this equation is not valid in the limit as $\alpha\downarrow0$; in the latter limit, the leading order behavior is still given by (\[eq:GammaApproxSmallalpham\])). In figure \[fig:Figm1gam\_smallandlarge\_a\_app\] the growth rate of the surfactant branch is plotted using (\[eq:QuadEqnGamma\]) along with the asymptotic expression (\[eq:GammaApproxLargealphameq1\]). One can see the dashed line approximations approaches the full dispersion curve as $\alpha\uparrow\alpha_{0}$. The long-wave $\gamma_{R}$ approximation (\[eq:GammaApproxSmallalpham\]) is not plotted in figure \[fig:Figm1gam\_smallandlarge\_a\_app\] but for the same parameter values the error is less than $1\%$ when $\alpha$ $<1.4\times10^{-9}$. ![The exact dispersion curve (\[eq:QuadEqnGamma\]) and the asymptotic expression of the growth rate around the marginal wavenumber (\[eq:GammaApproxLargealphameq1\]) of $\gamma_{R}$ for $m=1$, $n=2$, $s=1$, ${\text{Ma}}=1$, and ${\text{Bo}}=1000$. \[fig:Figm1gam\_smallandlarge\_a\_app\]](fig8rt){width="55.00000%"} An asymptotic expression for $\alpha_{0}$ is obtained by solving for $\alpha$ equation (\[eq:GammaApproxLargealphameq1\]) with $\gamma_{R}=0$: $$\alpha_{0}\approx\frac{3s\left\vert n-1\right\vert (n+1)^{2}[3\text{Ma}]^{1/2}}{n^{3}\text{Bo}^{3/2}}.\label{eq:alpha0asy}$$ The above expression is also obtained from the long-wave marginal wavenumber equation (\[eq:MAappEqn\]). This expression also suggests that gravity is not completely stabilizing since $\alpha_{0}>0$ at any positive finite value of ${\text{Bo}}$. We had the similar result that gravity, no matter how strong, cannot completely stabilize the Marangoni instability for the $S$ sector. ### The case $n=1$ Next, we consider the border $n=1$ with $m\neq1$. Just like the $m=1$ and $n\neq1$ case, the imaginary part of the discriminant $\zeta$, $\operatorname{Im}(\zeta)=2\operatorname{Re}(F_{1})\operatorname{Im}(F_{1})$, is positive (or negative) for $m<1$ $($or $m>1)$, see (\[eq:F2Re\])-(\[eq:F0Im\]). The growth rate for the robust mode is, from equation (\[eq:gamCsmallAlphaApprox\]), $$\gamma_{R}\approx-{\frac{\left(1+m\right)}{{m}^{2}+14\,m+1}}\left\{ \text{{Ma}}+\frac{1}{3}\text{Bo}\right\} {\alpha}^{2}\text{,}\label{eq:gamRn1border}$$ but, since the coefficient of the $\alpha^{2}$ term in equation (\[eq:gamSsmallAlphaApprox\]) becomes zero, we have for the surfactant branch, using equation (\[eq:ks\]) with $n=1$, $$\gamma_{R}\approx-{\frac{1}{96}}\,{\frac{\left(1+m\right)}{{s}^{2}\left(m-1\right)^{2}}}\left\{ \,{\frac{1}{2}\,}\text{{Ma}}+{\frac{1}{3}}\,\text{{Bo}}\right\} \text{{Bo}}{\,}\text{{Ma}}{\alpha}^{4}\text{.}\label{eq:gamSn1border}$$ For this case the robust and surfactant branches are long-wave stable for ${\text{Bo}}>0$. For ${\text{Bo}}<0$ both branches are unstable if the magnitude of ${\text{Bo}}$ is sufficiently large. This occurs when the leading term coefficients in (\[eq:gamRn1border\]) and (\[eq:gamSn1border\]) are positive, i.e. when ${\text{Bo}}<-3\text{Ma}$ for (\[eq:gamRn1border\]), and $-3\text{Ma}/2<\text{Bo}<0$ for (\[eq:gamSn1border\]). ### The $m=n^{2}$ border For the $m=n^{2}\neq1$ border, using the general equation (\[eq:gamGeneralEqnRSQsectors\]) to obtain the growth rates to the leading orders, we find $$\gamma_{R}\approx-\left\{ \frac{n\text{Bo}}{12(n+1)}\right\} \alpha^{2}+\left\{ \frac{n\left(2\text{Ma}+n\text{Bo}-5\right)}{60(n+1)}\right\} \alpha^{4}\text{, }\label{eq:Midwave_gamC_smalla}$$ and $$\gamma_{R}\approx-\left\{ \frac{\text{Ma}}{4(n+1)}\right\} \alpha^{2}.\text{}\label{eq:Midwave_gamS_smalla}$$ We have kept two leading orders in equation (\[eq:Midwave\_gamC\_smalla\]) because the $\alpha^{2}$ term vanishes for $\text{Bo}=0$. Equation (\[eq:Midwave\_gamS\_smalla\]) shows that the surfactant branch is always stable, and this is consistent with HF in the limit ${\text{Bo}}\rightarrow0$. Also, in this limit the robust branch, equation (\[eq:Midwave\_gamC\_smalla\]), reproduces the corresponding HF result, their equation (4.13). Also, for $\text{Bo}=0$, equation (\[eq:Midwave\_gamC\_smalla\]) recovers the long-wave dispersion relation found in FH. Finally, for the $m=1$ and $n=1$ case, the solutions to the dispersion equation (\[eq:QuadEqnGamma\]) for arbitrary wavenumber are of the form $$\gamma_{R}=\frac{-a\text{Ma}-b(\text{Bo}+\alpha^{2})\pm\lbrack a\text{Ma}-b(\text{Bo}+\alpha^{2})]}{2F_{2}\alpha^{4}}\text{,}\label{eq:gamPointCaseFull}$$ where $$a=\alpha^{2}(s_{\alpha}^{2}-\alpha^{2})(c_{\alpha}s_{\alpha}+\alpha)\text{ and }b=(s_{\alpha}^{2}-\alpha^{2})(c_{\alpha}s_{\alpha}-\alpha)\,\text{.}$$ After substituting $F_{2}$, $a$, and $b$ into (\[eq:gamPointCaseFull\]), the growth rate for the robust branch is $$\gamma_{R}=-\frac{(s_{\alpha}^{2}-\alpha^{2})(\text{Bo}+\alpha^{2})}{4\alpha(c_{\alpha}s_{\alpha}+\alpha)}\approx-\frac{1}{24}\left(\text{Bo}+\alpha^{2}\right)\alpha^{2}\text{ for }\alpha\ll1\text{,}$$ and the growth rate for the surfactant branch is $$\gamma_{R}=-\frac{\alpha(s_{\alpha}^{2}-\alpha^{2})\text{Ma}}{4(c_{\alpha}s_{\alpha}-\alpha)}\approx-\frac{1}{8}\text{Ma}\alpha^{2}\text{ for }\alpha\ll1\text{.}$$ Note that the surfactant branch is always stable but the robust branch is unstable if $\alpha^{2}<-{\text{Bo}}$. Obviously, this only occurs if ${\text{Bo}}<0$. Arbitrary wavenumbers; mid-wave instability {#sec:FiniteCaseMidWave} =========================================== In this section, results are given for arbitrary wavenumber, and comparisons are made across all parameter sectors. First, the influence of gravity on the maximum growth rate $\gamma_{R\max}$, the corresponding wavenumber $\alpha_{\max}$ and the marginal wavenumber $\alpha_{0}$ in the $R$, $S$, and $Q$ sectors are considered for fixed values of the Marangoni number. Then similar results are given to show the influence of surfactant for fixed values of the Bond number. Asymptotic results are also discussed. ![Dependence of the maximum growth rate $\gamma_{R\max}$, the corresponding wavenumber $\alpha_{\max}$, and the marginal wavenumber $\alpha_{0}$ on $\textrm{Bp}$ in the $R$, $S$ and $Q$ sectors. Here $s=1$, ${\text{Ma}}=0.1$ and the values of the $(n,m)$ pairs for the $R$ (a,d,g), $S$ (b,e,h), and $Q$ (c,f,i) sectors are $(2,2)$, $(2,0.5)$, and $(2,5)$, respectively.\[fig:Fig3maxmargRSQ\] ](fig9rt){width="95.00000%"} Effects of gravity ------------------ We first examine the influence of ${\text{Bo}}$ on the maximum growth rate $\gamma_{R\max}$, its corresponding wavenumber $\alpha_{\max}$, and the marginal wavenumber $\alpha_{0}$. Figure \[fig:Fig3maxmargRSQ\] shows plots of $\gamma_{\max}$, $\alpha_{\max}$ and $\alpha_{0}$ for a representative $(n,m)$ pair from each of the three sectors where panels (a, d, g), (b, e, h) and (c, f, i) represent the $R$, $S$ and $Q$ sectors, respectively. In the $R$ sector, panels (a, d, g) show that the system is unstable provided ${\text{Bo}}$ does not exceed a finite positive value ${\text{Bo}}_{c}$ and that $\gamma_{R\max}$, $\alpha_{\max}$, and $\alpha_{0}$ all decrease to zero as ${\text{Bo}}\downarrow{\text{Bo}}_{c}$. These findings were also observed in the long-wave limit (see section \[sec:FiniteCaseLongwaveApproxCregions\]). This instability is of the long-wave type even when the marginal wavenumber $\alpha_{0}$ is not small. However, for $m$ sufficiently close to $n^{2}$ but still in the $R$ sector, there appears a “mid-wave” instability (see figure 16 below), which is discussed below, in sections \[subsec:Surfactant-effects-in-Q-Sector\] and \[subsec:MA-Bo plane stability\]. Panels (b, e, h) show the surfactant branch is always unstable in the $S$ sector. The discontinuity in the graph of $\alpha_{\max}$ in panel (e) is discussed below with figure \[fig:Fig\_typical\_MaxCrossingDCurve\]. In the $Q$ sector, surfactants are completely stabilizing provided ${\text{Bo}}>\text{Bo}_{c}$, as shown in panels (c), (f) and (i). Note that ${\text{Bo}}_{c}<0$ agrees with the long-wave analysis (see equation (\[eq:RTcritcalMa\])). ![Dispersion curves given by (\[eq:QuadEqnGamma\]) in the $S$ sector ($n=2$, $m=0.5$) for selected values of ${\text{Bo}}$ showing occurrence of two local maxima and a jump in the global maximum.  Here $s=1$ and ${\text{Ma}}=0.1$. \[fig:Fig\_typical\_MaxCrossingDCurve\]](fig10rt){width="95.00000%"} The discontinuity that can occur in the $S$ sector is displayed in figure \[fig:Fig\_typical\_MaxCrossingDCurve\]. Panel (a) shows that for negligible ${\text{Bo}}$, one branch is long-wave unstable and the other one is stable. As the magnitude of ${\text{Bo}}$ increases the previously stable branch becomes unstable (${\text{Bo}}=-1$) and at some point the branches cross (${\text{Bo}}=-1.5,-2.3$). Panel (e) shows that as $\left\vert \text{Bo}\right\vert $ continues to increase the crossing eventually disappears at which point the upper branch has two local extrema. At some value of $\text{Bo}$, the global maximum shifts from the right local extremum (as for ${\text{Bo}}=-2.45$) to the left local extremum (as for ${\text{Bo}}=-2.67$). Finally, as ${\text{Bo}}\downarrow-\infty$, both branches are unstable in the long-wave manner, and feature a single maximum. Effects of surfactants in the $R$ and $S$ sectors\[subsec:Effects-of-surfactants in R and S\] --------------------------------------------------------------------------------------------- Here, we investigate, for a fixed value of ${\text{Bo}}$ in the $R$ and $S$ sectors, the Marangoni number ${\text{Ma}}$ dependences of the maximum growth rate $\gamma_{\max}$, the corresponding wavenumber $\alpha_{\max}$, and the marginal wavenumber $\alpha_{0}$. The $Q$ sector turns out to have somewhat different properties, which are discussed later (see figure \[fig:FIG\_RTgamNalphas2by4\]). However, it is immediately clear that in the $Q$ sector both branches are stable for ${\text{Bo}}>0$ and fixed ${\text{Ma}}$ (see panels (c), (f) and (i)) in figure \[fig:Fig3maxmargRSQ\]. Panels (a) and (b) of figure \[fig:FIG\_Bo1\_vs\_Ma\] show that $\gamma_{R\max}$ attains a maximum at some $\text{Ma}=O(1)$ in both the $R$ and $S$ sectors, and that $\gamma_{R\max}\downarrow0$ as $\text{Ma}\uparrow\infty$. Both $\alpha_{\max}$ and $\alpha_{0}$ also decrease to zero as $\text{Ma}\uparrow\infty$. However, in the $R$ sector there is a threshold value of ${\text{Ma}}$, ${\text{Ma}}_{cL}$, below which the flow is stable; while in the $S$ sector the flow is unstable for all ${\text{Ma}}>0$. Recall from the long-wave results that ${\text{Ma}}_{cL}({\text{Bo}})$ is the inverse of ${\text{Bo}}_{cL}({\text{Ma}})$ (see equations (\[eq:BoCritical\]) and (\[eq:RTcritcalMa\])). In the $S$ sector, $\alpha_{\max}$ and $\alpha_{0}$ approach some non-zero constant values and $\gamma_{R\max}\downarrow0$, showing no threshold value of ${\text{Ma}}$ for complete stabilization of the flow. The small and large ${\text{Ma}}$ asymptotics of $\alpha_{0}$ are discussed next. Panels (e) and (f) suggest that $\alpha_{0}\downarrow0$ as ${\text{Ma}}\uparrow\infty$. By substituting equations (\[eq:k20app\]) - (\[eq:k13app\]) into the marginal-wavenumber equation (\[eq:MaEquation2\]), and keeping only the dominant ${\text{Ma}}$ terms, the following expression is obtained: $$\frac{n^{2}}{324}(n^{3}+m)^{2}\text{BoMa}\alpha^{2}+\frac{s^{2}}{108}\varphi(n-1)(n+1)^{2}(m-n^{2})=0,\label{eq:MAeqnMaInfin}$$ from which $$\alpha_{0}\approx\frac{s(n+1)\sqrt{3\varphi(n-1)(n^{2}-m)}}{n(n^{3}+m)}\text{Bo}^{-1/2}\text{Ma}^{-1/2}.\label{eq:alpha0app0}$$ This is consistent with the numerically-found behavior for $\alpha_{0}$ at large ${\text{Ma}}$. As ${\text{Ma}}\downarrow0$, it is clear from panel (f) of figure \[fig:FIG\_Bo1\_vs\_Ma\] that in the $S$ sector, $\alpha_{0}$ approaches some finite non-zero value. Therefore, by keeping only the (dominant) linear ${\text{Ma}}$ terms, equation (\[eq:MaEquation2\]) reduces to $$k_{11}+k_{13}B^{2}=0,\label{eq:reducedMaEqn2Ma0}$$ where $k_{11}$ and $k_{13}$ depend on $\alpha$, as given by equations (\[eq:k11\]) and (\[eq:k13\]). However, this equation must be solved numerically for $\alpha_{0}$ since it is not necessarily small. Some other asymptotics for $\alpha_{0}$ approaching zero in the $R$ sector were discussed above in subsection \[subsec:Marginal-wavenumbers\]. ![(a,b) $\gamma_{R}{}_{\max}$, (c,d) $\alpha_{\max}$ and (e,f) $\alpha_{0}$ vs ${\text{Ma}}$ for ${\text{Bo}}=1.0$ in the $R$ sector (a,c,e) and $S$ sector (b,d,f). Here $s=1$ and the values of the $(n,m)$ pairs in the $R$ and $S$ sectors are $(2,2)$ and $(2,0.5)$, respectively.\[fig:FIG\_Bo1\_vs\_Ma\] ](fig11rt){width="95.00000%"} Panels (a), (b), (c) and (d) of figure \[fig:FIG\_Bo1\_vs\_Ma\] suggest that $\gamma_{R\max}$ and $\alpha_{\max}\downarrow0$ as ${\text{Ma}}\uparrow\infty$. In the long-wave limit and for $\text{Ma}\gg1$, the linear and constant terms of equation (\[eq:MAXeqn\]), whose coefficients are proportional to ${\text{Ma}}^{2}$ and ${\text{Ma}}^{3}$, are dominant, giving rise to the following simplified equation for $\gamma_{R}$: $$\frac{1}{27}n^{3}(m+n^{3})(n^{2}-m)\alpha^{6}\text{Ma}\gamma_{R}-\frac{1}{108}(n-1)n^{4}(n^{2}-m)s^{2}\varphi\alpha^{6}\text{Ma}^{2}\approx0.\label{eq:gamRlargeMa}$$ The latter gives $$\gamma_{R\max}\approx\frac{ns^{2}(n-1)(n+1)^{2}(m-n^{2})\varphi}{4(n^{3}+m)^{3}}\text{Ma}^{-1}\text{.}\label{eq:gamRmaxMaInfin}$$ Because $\alpha^{6}$ appears in the simplified equation above, it is convenient when solving for $\alpha_{\max}$ to subtract $\alpha$ times equation (\[eq:alphaeq:MAXeqn\]) from six times equation (\[eq:MAXeqn\]) and obtain $$\frac{8}{27}(m-1)^{2}n^{5}(n+1)^{2}(n^{3}+m)s^{2}\alpha^{4}\text{Ma}\gamma_{R}-\frac{1}{162}n^{6}(n^{3}+m)^{2}\alpha^{8}\text{BoMa}^{3}\approx0.\label{eq:sub6times}$$ Solving for $\alpha$ yields $$\alpha^{4}\sim48\frac{(m-1)^{2}(n+1)^{2}}{n(n^{3}+m)}s^{2}\text{Ma}^{-3}\text{Bo}^{-1}\gamma_{R}\text{.}\label{eq:equation2stars}$$ Equation (\[eq:gamRmaxMaInfin\]) is substituted into (\[eq:equation2stars\]), from which the following asymptotic expression for $\alpha_{\max}$ is obtained: $$\alpha_{\max}\approx\frac{\lbrack12\varphi(1-n)(m-n^{2})]^{1/4}(m-1)^{1/2}s}{(n^{3}+m)}\text{Ma}^{-3/4}\text{Bo}^{-1/4}\text{.}\label{eq:alphaMAXapp}$$ ![(a,b) $\gamma_{R}{}_{\max}$, (c,d) $\alpha_{\max}$ and (e,f) $\alpha_{0}$ as functions of ${\text{Ma}}$ for ${\text{Bo}}=-1.0$ in the $R$ sector (a,c,e) and $S$ sector (b,d,f), for the same $s$ and $(n,m)$ points as in figure \[fig:FIG\_Bo1\_vs\_Ma\]. \[fig:FIG\_Bon1\_vs\_Ma\]](fig12rt){width="85.00000%"} Panels (b) and (d) show that $\gamma_{R\max}\downarrow0$ and $\alpha_{\max}$ approaches some non-zero constant as ${\text{Ma}}\downarrow0$. Therefore, equation (\[eq:MAXeqn\]) is approximately linear for $\gamma_{R}\ll1$, $c_{10}\gamma_{R}\text{Ma}+c_{01}\approx0$ so that $$\gamma_{R}\approx-\frac{c_{01}}{c_{10}}\text{Ma}^{-1}\label{eq:gamRMa0Ssectors}$$ where the $c_{ij}$ are independent of ${\text{Ma}}$. An equation for $\alpha_{\max}$ is obtained by differentiating (\[eq:gamRMa0Ssectors\]) with respect to $\alpha$ and solving $d\gamma_{R}/d\alpha=0$ numerically for $\alpha$, which is then substituted into (\[eq:gamRMa0Ssectors\]) to obtain $\gamma_{R\max}$. In contrast to the case shown in figure \[fig:FIG\_Bo1\_vs\_Ma\] for ${\text{Bo}}>0$, the flow is unstable for all ${\text{Ma}}$ when ${\text{Bo}}<0$ in either the $R$ or $S$ sectors. Moreover, figures \[fig:FIG\_Bon1\_vs\_Ma\] (a) and (b) also show that $\gamma_{R\max}$ has a global maximum at ${\text{Ma}}=O(1)$. However, in the $S$ sector $\gamma_{R\max}$ decreases with increasing ${\text{Ma}}$ for sufficiently small ${\text{Ma}}$, up to ${\text{Ma}}={\text{Ma}}_{0}$. At ${\text{Ma}}={\text{Ma}}_{0}$ there is a jump in $\alpha_{\max}$. This behavior is due to the fact that the dispersion curve has two maxima, and at this particular value of ${\text{Ma}}$ there is a jump in the location of the global maximum, similar to that shown in figure \[fig:Fig\_typical\_MaxCrossingDCurve\]. Figure \[fig:FIG\_Bon1\_vs\_Ma\] also shows that $\gamma_{R\max}$, $\alpha_{\max}$ and $\alpha_{0}$ all approach some finite positive constant in the limits ${\text{Ma}}\uparrow\infty$ and ${\text{Ma}}\downarrow0$ for both sectors. Let us discuss the asymptotics of $\alpha_{0}$ with respect to the Marangoni number for the case of $\text{Bo}<0$. Panels (e) and (f) of figure \[fig:FIG\_Bon1\_vs\_Ma\] indicate that $\alpha_{0}$ asymptotes to non-zero constants as both ${\text{Ma}}\uparrow\infty$ and as ${\text{Ma}}\downarrow0$. The relevant values of $\alpha_{0}$ can be obtained as follows. For ${\text{Ma}}\uparrow\infty$, the dominant term in equation (\[eq:MaEquation2\]) is the ${\text{Ma}}^{3}$ term, and since $k_{13}\neq0$ this implies that $\text{Bo}+\alpha^{2}\approx0$, or $$\alpha_{0}\approx\left\vert \text{Bo}\right\vert ^{1/2}.\label{eq:alpha0app}$$ For ${\text{Bo}}=-1$, $a_{0}\approx1$ which is consistent with the numerical results shown in figures \[fig:FIG\_Bon1\_vs\_Ma\] (e) and (f). In the limit Ma $\downarrow0$, equation (\[eq:MaEquation2\]) reduces to $$\left(k_{11}+k_{13}B^{2}\right)\text{Ma}B\approx0\text{.}\label{eq:eq:k11eq:k13eqn}$$ In the $R$ sector, the solution $\alpha_{0}\approx\left\vert \text{Bo}\right\vert ^{1/2}$ is again obtained because $k_{13}$ is always positive and $k_{11}$ is the product of $(m-1)$ and a positive function, and thus $k_{11}>0$ in the $R$ sector. However, in the $S$ sector $k_{11}<0$, and $\alpha$ is a solution of $k_{11}+k_{13}$B$^{2}=0$ which is solved numerically for $\alpha$. The solution is approximately $\alpha_{0}\approx1.56$ , and agrees with figure \[fig:FIG\_Bon1\_vs\_Ma\] (f). Next, the asymptotics of $\gamma_{R\max}$ and $\alpha_{\max}$ in the limit ${\text{Ma}}\uparrow\infty$, and then in the limit ${\text{Ma}}\downarrow0$, (panels (a, b, c, d) of figure \[fig:FIG\_Bon1\_vs\_Ma\]) are discussed. In this case, the terms proportional to ${\text{Ma}}^{3}$ in equation (\[eq:MAXeqn\]) yield $$c_{03}+c_{13}\gamma_{R}\approx0,\label{eq:maxapp}$$ where the coefficients $c_{ij}$ correspond to the $\gamma_{R}^{i}{\text{Ma}}^{j}$ terms in equation (\[eq:MAXeqn\]). Therefore, $$\gamma_{R}\approx-\frac{c_{03}}{c_{13}}\approx-\frac{1}{2}\frac{\left(s_{\alpha}^{2}-\alpha^{2}\right)\left(s_{\alpha n}^{2}-\alpha^{2}n^{2}\right)\left(\text{Bo}+\alpha^{2}\right)}{\alpha\left(s_{\alpha}^{2}-\alpha^{2}\right)\left(s_{\alpha n}c_{\alpha n}+\alpha n\right)m+\alpha\left(s_{\alpha n}^{2}-\alpha^{2}n^{2}\right)\left(s_{\alpha}c_{\alpha}+\alpha\right)}\text{.}\label{eq:c03c13gameqn}$$ Again, one must solve $d\gamma_{R}/d\alpha=0$ numerically for $\alpha_{\max}$ which in turn is substituted into equation (\[eq:c03c13gameqn\]) to obtain $\gamma_{R\max}$. Figure \[fig:fig13abc\] shows the results of varying the shear parameter $s$. For any fixed $s$, the growth rate has a global maximum over the $(\alpha,\textrm{Ma})$-plane, denoted $\max\gamma_{R}$. We denote $\alpha(\max\gamma_{R})$ and $\textrm{Ma}(\max\gamma_{R})$ the values of the wavenumber and Marangoni number, respectively, at which the growth rate attains its maximum, $\max\gamma_{R}$. These quantities are plotted versus $s$ in figure \[fig:fig13abc\], for selected sampling points in the $R$ and $S$ sectors. ![The influence of $s$ on (a) the maximum of $\gamma_{R\text{max}}$ over all $\alpha$ and $\text{Ma}$ in the $R$ sector (at $n=2$ and $m=2$) and the $S$ sector (at $n=2,\;m=0.5$) for two different values of $\text{Bo}$ as indicated in the legend. Panels (b) and (c) show the corresponding $\alpha$ and $\text{Ma}$. \[fig:fig13abc\]](fig13rt){width="95.00000%"} We see that while in panels (a) and (c) the dependencies are linear, and also practically independent of the Bond number, this does not hold for the $\alpha(\max\gamma_{R})$ shown in panel (b); in particular, in all four cases shown there, it stays almost constant (of magnitude order 1) at large $s$ but falls off precipitously to zero as $s\downarrow0$. In this subsection we only had to deal with the long-wave instability because the values of $\text{Ma}$ considered are either sufficiently large or sufficiently small, or the viscosity ratio was not sufficiently close to the $R-Q$ boundary $m=n^{2}$. It turns out that for the intermediate values of $\text{Ma}$ and the appropriate values of $m$, even in the $R$ sector, a different type of instability, called the “mid-wave” instability (HF), may happen. Its definition is recalled in the next subsection where the $Q$ sector is considered, since this instability is more prevalent there. Some results on the mid-wave instability in the $R$ sector are found in section \[subsec:MA-Bo plane stability\] together with similar results for the $Q$ sector. In the $S$ sector, the mid-wave instability sometimes coexists with the long-wave instability of the robust mode. However, as far as we have observed, it is always weaker than the long-wave instability of the surfactant branch there. This is also discussed in section 6. Surfactant effects in the $Q$ sector \[subsec:Surfactant-effects-in-Q-Sector\] ------------------------------------------------------------------------------ It was shown in HF (for $\textrm{Bo}=0$) that for ${\text{Ma}}>5/2$ and $m>n^{2}$ ($Q$ sector), there is a mid-wave instability such that $\gamma_{R}>0$ for a finite $\alpha$-interval bounded away from $\alpha=0$. (Note that the mid-wave instability was called type I in @Cross1993 while the long-wave instability was called type II). In order to investigate such an instability allowing for nonzero Bond numbers, we introduce a critical Marangoni number, ${\text{Ma}}_{cM}$ that corresponds to the onset (or the turnoff) of the mid-wave instability, and let $\alpha_{cM}$ be the corresponding wavenumber. Thus, the quantities ${\text{Ma}}_{cM}$ and $\alpha_{cM}$ satisfy the equations $\gamma_{R}=0$ and $\partial\gamma_{R}/\partial\alpha=0$. In view of the quartic equation (\[eq:MAXeqn\]), ${\text{Ma}}_{cM}$ and $\alpha_{cM}$ (for a given ${\text{Bo}}$) can be found by numerically solving simultaneously equation (\[eq:MaEquation2\]), which we write in the notation used in equation (\[eq:alphaeq:MAXeqn\]), $$C_{0}(\text{Ma},\alpha,\text{Bo})=0,\label{eq:C_0}$$ along with $$\frac{\partial}{\partial\alpha}C_{0}(\text{Ma},\alpha,\text{Bo})=0.\label{eq:C_0' is zero}$$ To illustrate the change of stability with ${\text{Ma}}$, in the top panels of figure \[fig:Fig\_typical\_midwaveDCurve\], the growth rate in the $Q$ sector (for $n=2$ and $m=5$, at $s=1$) is plotted for three selected values of the Marangoni number and $\text{Bo}=-0.45$. ![Curves for the four different functions of the wavenumber in the $Q$ sector ($n=2$, $m=5$) for $s=1$ and ${\text{Bo}}=-0.45$. The stable mode corresponds to the panels of the left-hand column, and the less stable mode to the panels of the right-hand column. For the three values of ${\text{Ma}}$ given in the legend, panels (a) and (b) show the growth rates, (c) and (d) show the wave velocities, (e) and (f) show the interface-surfactant phase shifts, and (g) and (h) show the interface/surfactant amplitude ratio. The transition from the long-wave instability to stability to the mid-wave instability as ${\text{Ma}}$ increases is evident in panel (b).\[fig:Fig\_typical\_midwaveDCurve\]](fig14rt){width="95.00000%"} The numerical results show that the instability is long-wave provided $\text{Ma}<\text{Ma}_{cL}$ ($\approx2.28$ for the figure parameters). This is then followed by a region of stability when ${\text{Ma}}\in\left[\text{Ma}_{cL},\text{Ma}_{cM}\right]$, where ${\text{Ma}}_{cM}$ $\approx15.6$. For ${\text{Ma}}_{cL}\,<{\text{Ma}}<{\text{Ma}}_{m}$, $\gamma_{R}$ decreases monotonically with $\alpha$ (so that there is no $\gamma_{R\text{max}}$; such dispersion curves are not shown in the top right panel), but starting from the ${\text{Ma}}_{m}$ ($\approx3.70$), the local maximum $\gamma_{R\max}$ appears on the dispersion curves. So, the growth rate $\gamma_{R}$ has a local maximum $\gamma_{R\max}$ at some $\alpha_{max}>0~$ provided ${\text{Ma}}\geq{\text{Ma}}_{m}$; and once ${\text{Ma}}$ exceeds ${\text{Ma}}_{cM}$, $\gamma_{R\text{max}}$ becomes positive, i.e., the mid-wave instability switches on. Note that when ${\text{Ma}}>{\text{Ma}}_{cM}$ for at least some interval of $\text{Ma}$ corresponding to the mid-wave instability, there are two positive marginal wavenumbers, one on the left at $\alpha=$ $\alpha_{0L}$ and another one on the right at $\alpha=$ $\alpha_{0R}$ so that the interval of unstable wavenumbers is $\alpha_{0L}<\alpha<\alpha_{0R}$. (Cases with both finite and infinite $\text{Ma}$ intervals of mid-wave instability can be seen below in figure \[fig:Fig\_noses\](a) and are discussed in the last paragraph of section 6.3.) Although the stability properties of the normal modes are fully given by the dispersion curves (see panels (a) and (b) of figure \[fig:Fig\_typical\_midwaveDCurve\]), the normal modes have additional remarkable properties, such as the phase speed, the phase difference between the co-traveling waves of the interface and the surfactant, and the amplitude ratio of the interface to the surfactant disturbances. As an example, these quantities are plotted in figure \[fig:Fig\_typical\_midwaveDCurve\] as functions of the wavenumber $\alpha$. There, one notices a special value of the wavenumber, $\alpha_{s}$, close to 0.7, at which the phase shift of the decaying branch has a jump discontinuity. The wave speed at $\alpha_{s}$ is zero for any ${\text{Ma}}$, so all three curves intersect at the same point $(\alpha_{s},0)$; similarly, the amplitude ratio is zero, independent of $\text{Ma}$. For the other branch, in the right panels (which, as panel (b) shows, goes, as $\text{Ma}$ increases, from long-wave unstable, to stable and then to mid-wave unstable), all three growth rates are equal at the same $\alpha_{s}$, and the wave speeds are equal as well, but the amplitude ratios are non-zero and different. To explain these observations, note that the zero amplitude ratio implies that if $h=0$ and $G\ne0$, then from the first equation of (\[eq:dispEqnSystem\]) $A_{12}=0$. Its solution, with the explicit expression of $A_{12}$ from (\[eq:A12\]), yields $\alpha_{s}$ in terms of $n$ and $m$ (but independent of ${\text{Ma}}$). The second equation of (\[eq:dispEqnSystem\]) with $h=0$ yields $\gamma=-A_{22}$, which by (\[eq:A22\]), is real, negative, and proportional to $\text{Ma}$. This agrees with the left upper panel of figure \[fig:Fig\_typical\_midwaveDCurve\]. The wave speed is zero because $\textrm{Im}(\gamma)=0$. The other mode corresponds to the right panels of this figure, and must have $h\ne0$. Since $A_{12}=0$ for $\alpha=\alpha_{s}$, we must have $\gamma+A_{11}=0$. This implies $\text{Im}(\gamma)>0$, i.e. a negative wave speed value, independent of $\text{Ma}$, corresponding to the triple intersection in panel (d) of figure \[fig:Fig\_typical\_midwaveDCurve\]. The growth rate, $\gamma_{R}=-\text{Re(}A_{11})$, is seen to be negative and independent of $\text{Ma}$, which explains the triple intersection in panel (b). However, since $h\ne0$ for this branch, the amplitude ratio is found to be $$\left|\frac{h}{G}\right|=\left|\frac{A_{11}-A_{22}}{A_{21}}\right|.$$ Only $A_{22}$ depends on $\text{Ma}$, and $|h/G|$ changes with ${\text{Ma}}$, so the three curves in figure \[fig:Fig\_typical\_midwaveDCurve\] go through different points at $\alpha=\alpha_{s}$. Having noticed the existence of the normal modes in which the surfactant is disturbed, $G\neq0$, but the interface is undisturbed, $h=0$, the question arises if there exist some “opposite” modes, in which only the interface, but not the surfactant is disturbed, so that $G=0$, but $h\neq0$. We answer this question in Appendix \[sec:Are-there-normal\]. It turns out that such modes are possible, but only when $s=0$. In figure \[fig:FIG\_RTgamNalphas2by4\], $\gamma_{R\text{max}}$, $\alpha_{\text{max}}$ and $\alpha_{0}$ are plotted versus the Marangoni number for $n=2$, $m=5$, $s=1$ and for four selected values of $\text{Bo}$. If $\text{Bo}$ is sufficiently negative, as in panels (a) and (c), then $\gamma_{R}>0$ for all $\text{Ma}$. ![Plots of $\gamma_{R\max}$ (left-hand panels) and corresponding $\alpha_{\max}$, $\alpha_{0R}$, and $\alpha_{0L}$ (right-hand panels) vs ${\text{Ma}}$ in the $Q$ sector (here at $n=2$, $m=5$) for $s=1$ and the four indicated values of ${\text{Bo}}$. (For labeled points, see the text.)\[fig:FIG\_RTgamNalphas2by4\]](fig15rt){width="95.00000%"} For $\text{Ma}<\text{Ma}_{LM}$, the instability is long-wave, in other words, there is no $\alpha_{0L}$, since its definition implies that $\alpha_{0L}$ must be non-zero. However, a mid-wave instability ensues when $\text{Ma}>\text{Ma}_{LM}$, and there appears $\alpha_{0L}>0$ (as in panels (b) and (d)). Initially, $\alpha_{0L}$ increases rapidly, while $\alpha_{0R}$ decreases by a small amount, leading to the shrinkage of the interval of unstable wavenumbers. After reaching a maximum, $\alpha_{0L}$ decreases towards zero with increasing $\text{Ma}$ but never attains the zero value so that the instability does not return to the long-wave type, and the interval of unstable wavenumbers slowly expands. When $\text{Bo}=-0.51$ (see panels (e) and (f)), the stability picture up to $\text{Ma}=\text{Ma}_{cM_{1}}$ is very similar to that displayed in panels (b) and (d). The instability is long-wave provided $\text{Ma}<\text{Ma}_{LM}$. Starting at $\text{Ma}=\text{Ma}_{LM}$, corresponding to the lower left dot in panel (f), the long-wave instability disappears, and the mid-wave instability mentioned previously emerges. However, as $\text{Ma}$ continues to increase, the interval of unstable wavenumbers quickly shrinks to a single, non-zero, $\alpha$ point, indicated by the dot at $\text{Ma}=\text{Ma}_{cM_{1}}$. The flow then becomes stable, with $\gamma_{R}<0$ for a range of Marangoni numbers, $\text{Ma}_{cM_{1}}<\text{Ma}<\text{Ma}_{cM_{2}}$. Therefore, in this range, $\alpha_{0L}$ and $\alpha_{0R}$ are non-existent, but $\alpha_{\text{max}}$ is defined because $\gamma_{R}$ has a local maximum at a nonzero $\alpha$. The mid-wave instability reappears at $\text{Ma}_{cM_{2}}$, (see the right-most dot in panel (f)) starting from $\gamma_{R}=0$, which corresponds to the right-hand intersection point in panel (e). As $\text{Ma}$ increases beyond $\text{Ma}_{cM_{2}}$, the interval of unstable wavenumbers expands in both directions. In the final set of panels, (g) and (h), with $\text{Bo}=-0.1$, the flow is stable, and $\gamma_{R\text{max}}$, $\alpha_{\text{max}}$, and $\alpha_{0}$ do not exist, in the interval $\text{Ma}_{cL}\le\text{Ma}\le\text{Ma}_{m}$. This is because $\gamma_{R}$ has no local maximum at any $\alpha>0$. Note that, as with the previous set of panels, the flow is long-wave unstable for $\text{Ma}<\text{Ma}_{cL}$ (i.e., to the left of the left-most dot of panel (h)) and mid-wave unstable for $\text{Ma}>\text{Ma}_{cM}$ (to the right of the right-most dot). Thus, we have observed here, for the first time, the existence of another route to the mid-wave instability: the continuous transition from long-wave instability (see the marked point $(\text{Ma}_{LM},0)$ in panel (f) of figure \[fig:FIG\_RTgamNalphas2by4\]). Only the other route, the onset of mid-wave instability from stability, was present for the case of zero gravity (see HF). In the former scenario, the mid-wave instability has a non-zero growth rate and a final support interval from the very beginning. A detailed investigation of the boundaries between the domains of the mid-wave instability, long-wave instability and stability in the $(\text{Ma},\text{Bo})$-plane appears below in section 6. Figure \[fig:fig16abc\] shows the dependencies of $\text{max}\;\gamma_{R}$, $\alpha(\text{max}\gamma_{R})$ and $\text{Ma}(\text{max}\gamma_{R})$ on the shear parameter $s$ in the $Q$ sector similar to those shown in figure \[fig:fig13abc\] for the other two sectors. We observe that the existence of the global maximum in $\text{Ma}$ of the growth rate maxima with respect to the wavenumber is less common in the $Q$ sector, especially for $\text{Bo}>0$. At smaller values of $s$, the global maximum becomes a local one like the one in figure \[fig:FIG\_RTgamNalphas2by4\](a). This is indicated in figure \[fig:fig16abc\] as the change from the solid to the dashed curve at the negative $\text{Bo}$ and from the dashed to the dotted one at the positive $\text{Bo}$. At still smaller $s$, to the left of the end dot on each curve, there are neither global nor local maxima. ![The influence of $s$ on (a) the maximum of $\gamma_{R}$ over all $\alpha$ and $\text{Ma}$ in the $Q$ sector (here at $n=2,\;m=5$) for two different values of $\text{Bo}$, one positive and the other one negative. Panels (b) and (c) show the values of $\alpha$ and $\text{Ma}$ at which this maximum occurs. The global maxima of $\gamma_{Rmax}$ with respect to $\text{Ma}$, present at larger $s$, become local maxima between the pairs of dots on each curve. At smaller $s$, to the left of the end dot on each curve, there are neither global nor local maxima.\[fig:fig16abc\]](fig16rt){width="95.00000%"} $\text{(Ma},\text{Bo)}$-plane stability diagrams\[subsec:MA-Bo plane stability\] ================================================================================ Regions of the long-wave and mid-wave instabilities --------------------------------------------------- Here we present a detailed account of the mid-wave instability changes as the viscosity ratio is increased, starting from a value in the $R$ sector, $1<m<n^{2}$, then crossing the $m=n^{2}$ border and further growing in the $Q$ sector, $m>n^{2}$. In the $R$ sector, the robust branch is long-wave unstable provided ${\text{Bo}}<{\text{Bo}}_{cL}$ where ${\text{Bo}}_{cL}({\text{Ma}})$, as given by (\[eq:BoCritical\]), is positive. If $m<n^{2}$ and sufficiently far from the $m=n^{2}$ border, there exists just one stability boundary, given by ${\text{Bo}}={\text{Bo}}_{cL}$; it is a straight line (starting at the origin) that separates the long-wave unstable and stable regions, as shown in figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\](a). ![Stability diagrams in the $({\text{Ma}},{\text{Bo}})$-plane showing the influence of the viscosity ratio $m$ as $m\uparrow n^{2}$: (a) $m=10.25$; (b) $m=15$; (c) $m=15.45$; and (d) $m=15.75$. The solid and dashed curves represent long-wave and mid-wave instability boundaries respectively; S, L, and M denote the stable, long-wave unstable, and mid-wave unstable regions. Here $s=1$ and $n=4$. \[fig:figMidwaveR1m\_near\_n2\_n4s1\]](fig17rt){width="95.00000%"} As $m$ increases and gets sufficiently close to $m=n^{2}$, the onset of a mid-wave instability is observed for certain intervals of ${\text{Ma}}$ and ${\text{Bo}}$. In panels (b) and (c), a mid-wave instability occurs provided ${\text{Bo}}_{cL}<{\text{Bo}}\,<{\text{Bo}}_{cM}$, for a finite interval of the Marangoni numbers, ${\text{Ma}}_{LM1}<{\text{Ma}}<{\text{Ma}}_{LM2}$, as the ${\text{Bo}}_{cL}$ and ${\text{Bo}}_{cM}$ curves “intersect” each other at ${\text{Ma}}={\text{Ma}}_{LM1}$ and ${\text{Ma}}=\ {\text{Ma}}_{LM2}$. The “quasi-intersection” points, marked in the figure as filled squares, are the boundary points for the critical curve but are not the critical points themselves: the critical wavenumber decreases to zero as ${\text{Ma}}\rightarrow{\text{Ma}}_{LMj}$, but the zero value is prohibited for a critical wavenumber. When $m$ is approaching ever closer to $n^{2}$, at some $m$ the critical curve of the mid-wave instability acquires a maximum and a minimum, such as the ones in panel (d). Clearly, for each fixed ${\text{Ma}}$ of the ${\text{Ma}}$-interval ${\text{Ma}}_{LM1}<{\text{Ma}}<{\text{Ma}}_{LM2}$, there are three distinct ${\text{Bo}}$-intervals: a semi-infinite interval of stability ${\text{Bo}}>{\text{Bo}}_{cM}$; a finite interval of mid-wave instability ${\text{Bo}}_{cL}<{\text{Bo}}<{\text{Bo}}_{cM}$; and a semi-finite interval of long-wave instability ${\text{Bo}}<{\text{Bo}}_{cL}$. In figure \[fig:figMidwaveAlphasR1m\_near\_n2\_n4s1\](a), the wavenumber $\alpha_{cM}$ corresponding to ${\text{Bo}}_{cM}$ is plotted versus ${\text{Ma}}$ for the values of $m$ corresponding to panels (b) and (c) of figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\], and also for $m=15.96$, which is closer to the $m=n^{2}$ boundary value, $m=16$, than $m=15.75$ of figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\](d). With this, \[fig:figMidwaveAlphasR1m\_near\_n2\_n4s1\](a) suggests the hypothesis that in approaching the sector boundary, the larger quasi-intersection value of $\textrm{Ma}$ tends to infinity. The latter is in accordance with the stability diagram for the sector boundary value $m=16$ (see figure \[fig:Fig\_noses\_16\] below). ![(a) The critical wavenumber $\alpha_{cM}$ versus Marangoni number ${\text{Ma}}$ for the same parameter value choices as in figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\]; in particular, $n=4$ and $s=1$. (b) The marginal wavenumber $\alpha_{0}$ versus Bond number $\textrm{Bo}$ for $s=1$, $\textrm{Ma = 25}$, $n=4$, and $m=15.45$. There is mid-wave instability in the region bounded by the two semicircles on the horizontal axis, long-wave instability to the left of this region, and stability to the right of this region. \[fig:figMidwaveAlphasR1m\_near\_n2\_n4s1\] ](fig18rt){width="95.00000%"} For all these cases, $\alpha_{cM}$ attains a maximum at an ${\text{Ma}}$ such that ${\text{Ma}}_{LM1}<{\text{Ma}}<{\text{Ma}}_{LM2}$. Figure \[fig:figMidwaveAlphasR1m\_near\_n2\_n4s1\](b) shows, for the parameters of figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\](c) and $\textrm{Ma}=25$, that, as the Bond number grows, when it reaches the value ${\text{Bo}}_{cL}$, the long-wave instability changes into the mid-wave one by the left endpoint of the interval of unstable $\alpha$ departing from the zero $\alpha$ point. The unstable $\alpha$ interval continues to shrink from both ends, and finally becomes a single non-zero $\alpha$ point at $\textrm{Bo=Bo}_{cM}$, the right-most point on the curve. The maximum growth rate (not shown) decreases to zero at this point, and there is stability for larger $\textrm{Bo}$, in agreement with figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\](c). ![(a) Stability diagram in the ($\text{Ma}$$,\text{Bo}$)-plane similar to the ones shown in figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\], for a case where $m=n^{2}$ (here $m=16$) and (b) the corresponding critical wavenumber, $\alpha_{cM}$. The end points have $\textrm{Ma}=5/2$. Here $s=1$. \[fig:Fig\_noses\_16\]](fig19rt){width="95.00000%"} On the $m=n^{2}$ border (e.g., for $m=16$, $n=4$), the robust branch is long-wave unstable in the half-plane ${\text{Bo}}<0$ (with the boundary line ${\text{Bo}}_{cL}=0$), as shown in figure \[fig:Fig\_noses\_16\]. Along the ${\text{Ma}}$-axis (${\text{Bo}}=0$), the stability results of HF that show the existence of a mid-wave instability for ${\text{Ma}}>5/2$ are recovered: ${\text{Ma}}_{LM1}$ = 5/2 and ${\text{Ma}}_{LM2}$ $=\infty$. Notably, ${\text{Bo}}_{cL}({\text{Ma}})\downarrow0$ as ${\text{Ma}}\rightarrow\infty$. We also note that there is just a single extremum, a maximum, on the critical curve. ![(a) Stability diagram showing the regions of mid-wave and long-wave instability and stability defined by the curves ${\text{Ma}}_{cL}$ and ${\text{Ma}}_{cM}$ as $m$ increases in the $Q$ sector, and (b) the wavenumber corresponding to ${\text{Ma}}_{cM}$ for the indicated values of $m$. Here $n=4$ and $s=1$. \[fig:Fig\_noses\]](fig20rt){width="95.00000%"} In the $Q$ sector, ${\text{Bo}}_{cL}({\text{Ma}})<0$, as given by (\[eq:BoCritical\]). We see the threshold curves $\textrm{Bo}=\text{Bo}_{cL}(\textrm{Ma)}$ in figure \[fig:Fig\_noses\], for each value of $m$ represented there; all the threshold curves have the (${\text{Bo}}$, ${\text{Ma}}$)-origin as their left-hand end (with linear scales on both axes, all the threshold lines would start from the origin and have a negative slope). The long-wave instability occurs below each threshold curve; the region of long-wave instability is labeled with an “L” in the figure. At some point on each L-threshold curve, the critical curve of the mid-wave instability begins, going unbounded rightward, in the direction of increasing ${\text{Ma}}$; as ${\text{Ma}}\uparrow\infty$, each critical curve is asymptotic to $\textrm{Bo}=0$ (thus, in a difference with the $R$ sector, but similar to the boundary between the $R$ and $Q$ sectors, the threshold line of the long-wave instability intersects the critical curve of the mid-wave instability at a single point); however, in contrast with the boundary between the $R$ and $Q$ sectors, the critical curve approaches the axis $\textrm{Bo}=0$ from below. Also, at the threshold-critical quasi-intersection, the ${\text{Bo}}_{cL}({\text{Ma}})$ increases as ${\text{Ma}}\downarrow{\text{Ma}}_{_{LM1}}$. Since there is still a local maximum on the critical curve, just as there is one in the $R$ sector and on the inter-sector boundary, it follows that there must be at least two local minima as well. The mid-wave instability occurs below such a critical curve $\text{Bo}=\text{Bo}_{cM}$ (and above, or to the right of, the right-hand part (${\text{Ma}}>{\text{Ma}}_{LM1}$) of the corresponding threshold curve $\textrm{Bo}=\text{Bo}_{cL}(\textrm{Ma)}$). This region is labeled with an “M”. Above the critical curve, as well as above the left-hand part (${\text{Ma}}<{\text{Ma}}_{LM1}$) of the corresponding threshold curve, the flow is stable. The critical curve is given by a single-valued function $\textrm{Bo}=\textrm{Bo}_{cM}\textrm{(Ma)}$, that is seen in figure \[fig:Fig\_noses\] to have two local minima and a maximum in between them, provided the viscosity ratio $m$ is below a certain value $m_{N}$. These two minima appear to occur at the same value of ${\text{Bo}}$, and as $m$ increases all three extrema move downward, but the single maximum moves faster than the two minima. Eventually, at $m=m_{N}$, the three extrema merge into a single minimum, such as the one on the $m=36$ critical curve. In the $S$ sector, as was mentioned at the end of section 5.2, the mid-wave instability occurs for the robust mode, although it is overshadowed by the long-wave instability of the surfactant mode. It is illustrated in figure \[fig:S midwave\_mp1n10s1\] for the parameter values indicated there. ![(a) Stability diagram of the less ustable mode for $s=1$, $n=10$, and $m=0.1$. The long-wave instability is present below the solid line and absent above it, while the mid-wave instability is present between the dashed curve, and either the solid line line or the dotted curve. The top and bottom insets zoom in on the regions near the two lower pairwise intersections, marked by the square and the circle, respectively.[]{data-label="fig:S midwave_mp1n10s1"}](fig21rt){width="80.00000%"} The zoom-in, the upper inset, shows that, in contrast with the other sectors, the critical curve does not end at its intersection with the threshold curve of the linear instability, but continues below the intersection, until it meets another critical curve. On the latter curve, each point corresponds to a dispersion curve having zero growth rate at a local minimum (as will be illustrated in the next figure). The bottom inset of figure \[fig:S midwave\_mp1n10s1\] is a zoom-in near the quasi-intersection point of the lower critical curve and the threshold line, marked by a small circle, located at $\text{Ma}$ slightly above 0.32 and $\text{Bo}$ slightly above -3.45. The quasi-intersection point of the upper critical curve and the threshold line, marked by a small square, is located at $\text{Ma}$ slightly above 0.046 and $\text{Bo}$ slightly below -0.5. Figure \[fig:S dispersion curves-1\] illustrates the change of the dispersion curves of the robust mode for the same values of $n$, $m$ and $s$ as in figure \[fig:S midwave\_mp1n10s1\], and $\text{Ma}$ fixed at 0.363 for a decreasing sequence of $\text{Bo}$ values corresponding to moving in the upper inset of figure \[fig:S midwave\_mp1n10s1\] from the domain of stability (\[fig:S dispersion curves-1\](a)) to long-wave instability (\[fig:S dispersion curves-1\](b)) to the domain of coexisting long-wave and mid-wave instabilities (\[fig:S dispersion curves-1\](c)) to the lower critical curve (corresponding to the zero minimum in figure \[fig:S dispersion curves-1\](d)) and finally to the domain of long-wave instability (see panels (e) and (f) of figure \[fig:S dispersion curves-1\]). The mid-wave instability starts at a certain $\text{Bo}$ between those of panels (b) and(c) as the maximum, which is negative in panel (b), grows through the zero to positive values as in panel (c) near $\alpha=0.3$. In this process both intervals of (co-existing) long-wave instability and mid-wave instability expand, until they coalesce which corresponds to the snapshot shown in panel (d). Also, in going from panel (c) to panel (d), the local minimum increases from negative to zero value, and becomes positive, as in panel (e). Finally, this minimum disappears, and the dispersion has a single maximum, see panel (f). ![Dispersion curves for the robust mode in the $S$- sector. Here $n=10$, $m=0.1$, $s=1$, and $\text{Ma}=0.363$. The values of $\text{Bo}$ are as indicated in each panel.[]{data-label="fig:S dispersion curves-1"}](fig22rt){width="95.00000%"} Figure \[fig:S midwave Bo-alpha\] shows the salient features of the dispersion curves, such as the maximum growth rate, $\gamma_{R\text{max}}$, the corresponding wavenumber, $\alpha_{\text{max}}$, and the marginal wavenumbers, $\alpha_{0}$, $\alpha_{0L}$ and $\alpha_{0R}$, as continuous functions of the Bond number for three different values of the Marangoni number. In particular, figure \[fig:S dispersion curves-1\] corresponds to panels (e) and (f) of figure \[fig:S midwave Bo-alpha\]. For smaller values of the Marangoni number, such as $\textrm{Ma}=0.355$ in panels (c) and (d), which are to the left of the intersection of the (maximum) critical curve and the threshold curve, the mid-wave instability emerges before the long-wave instability as the value of $\text{Bo}$ becomes more negative (see the upper inset of figure \[fig:S midwave\_mp1n10s1\]). For a small range of $\text{Bo}$, both long-wave and mid-wave instabilities can coexist (indicated by the label “LM” in the upper inset of figure \[fig:S midwave\_mp1n10s1\]). This is then followed by a completely long-wave unstable regime. For still smaller Ma, such as $\textrm{Ma}=0.3$, in panels (a) and (b), we observe the emergence of the mid-wave instability, which, subsequently, turns into a long-wave instability, similar to figure \[fig:figMidwaveAlphasR1m\_near\_n2\_n4s1\](b). ![Plots of $\gamma_{R\max}$, corresponding $\alpha_{\max}$, and $\alpha_{0}$, vs $\text{Bo}$ in the $S$ sector for the indicated values of $\textrm{Ma}$. Here $n=10$, $m=0.1$ and $s=1$.Note that when there are two local maxima on the dispersion curves, $\gamma_{Rmax}$ shown here corresponds to the right-hand maximum even if it is smaller than the left one.[]{data-label="fig:S midwave Bo-alpha"}](fig23rt){width="95.00000%"} Figure \[fig: S critical alpha v Ma\] is the plot of the critical wavenumber corresponding to the two critical curves in the preceding figure. It shows, similar to the analogous figures for the other two sectors, that the critical wavenumber, $\alpha_{c}$, approaches zero at the quasi-intersection points. It also reveals that the rate of change of the critical wavenumber approaches infinity at the common point of the two critical curves. Using small wavenumber expansions as described in section \[subsec:Asymptotics-of-the critical curves near boundaries\], we obtained the cubic equation given by (\[eq:cubic for Bo\]) below, and solved it numerically to verify that at the left quasi-intersection point $\textrm{Ma=0.0458}$, and at the other one $\textrm{Ma=0.321}$. ![Critical $\alpha$ versus $\text{Ma}$ corresponding to the previous figure.[]{data-label="fig: S critical alpha v Ma"}](fig24rt){width="60.00000%"} Asymptotics of the critical curves near their boundaries\[subsec:Asymptotics-of-the critical curves near boundaries\] ---------------------------------------------------------------------------------------------------------------------- ### General considerations\[subsec:General-considerations 6-2-1\] It should be possible to establish the asymptotic behavior of the critical curves near their boundaries, in particular, the sense of the curve inclination at a finite boundary point, a priori, using only minimal numerical information. This, as already was indicated above, leads to certain conclusions about the number and sense of possible extrema, that in their turn facilitate the complete determination of the curve extrema. Near any finite quasi-intersection point, for both $R$ and $Q$ sectors, we look for the critical point coordinates in the form of generic power expansions $$\text{Ma}=\text{Ma}_{0}+\alpha^{2}\text{Ma}_{2}+\alpha^{4}\text{Ma}_{4}+...\label{eq:crit Ma expans}$$ and $$\text{Bo}=\text{Bo}_{0}+\alpha^{2}\text{Bo}_{2}+\alpha^{4}\text{Bo}_{4}+...,\label{eq:crit Bo expans}$$ where, to simplify notations, ${\text{Ma}}_{0}$ stands for ${\text{Ma}}_{LMj}$ (with $j=1,2$), etc. We substitute these expansions into the critical curve equations (\[eq:C\_0\]) and (\[eq:C\_0’ is zero\]) and require the collected coefficients of each power to vanish. Since the point (${\text{Ma}}_{0}$, ${\text{Bo}}_{0}$) lies on the threshold curve of the long-wave instability, we have ${\text{Bo}}_{0}=\kappa{\text{Ma}}_{0}$, where $\kappa$ is the coefficient of ${\text{Ma}}$ in equation (\[eq:BoCritical\]). Because of this relation, the leading orders $\alpha^{6}$ in (\[eq:C\_0\]) and $\alpha^{5}$ in (\[eq:C\_0’ is zero\]) are satisfied identically. The next order system, given by the orders $\alpha^{8}$ in (\[eq:C\_0\]) and $\alpha^{7}$ in (\[eq:C\_0’ is zero\]), is $$\begin{aligned} k_{206}\text{Ma}_{2}+k_{116}\text{Bo}_{2} & = & r_{1},\\ 6k_{206}\text{Ma}_{2}+6k_{116}\text{Bo}_{2} & = & 8r_{1},\end{aligned}$$ where the coefficients $k_{pqr}$ are functions of $(m,n,s)$ given in Appendix B, and $r_{1}$ is a cubic polynomial in $\text{Ma}_{0}$ whose coefficients are known combinations of $k_{pqr}$, and which lacks the quadratic term (cf. the discussion around equation (\[eq:cubic discrim-eq0\])). The consistency of this system requires that $r_{1}=0$, which is a cubic equation for $\text{Ma}_{0}$. Clearly $\text{Bo}_{2}=-(k_{206}/k_{116})\text{Ma}_{2}$, which simplifies to $$\text{Bo}_{2}=\kappa\text{Ma}_{2}.\label{eq:bo2 is kappa times ma2}$$ The cubic equation for $\text{Ma}_{0}$ can be examined using the well known Cardano’s formula and the underlying theory for the case with real coefficients. The inclination of a critical curve at any quasi-intersection point is $$d\text{Bo}/d\text{Ma}=(d\text{Bo}/d\alpha)/(d\text{Ma}/d\alpha).$$ Using (\[eq:crit Ma expans\]) and (\[eq:crit Bo expans\]) we get $d\text{Bo}/d\text{Ma}=\text{Bo}_{2}/\text{Ma}_{2}=\kappa$, where we have used (\[eq:bo2 is kappa times ma2\]). Thus, at the boundary point, the critical curve is tangent to the threshold curve through that quasi-intersection point. ### The $R$ sector finite critical curves and the threshold for their existence \[subsec:R sector critical curves\] We find that in the $R$ sector the cubic equation for ${\text{Ma}}_{0}$ has two distinct positive roots, corresponding to the two quasi-intersection points, for $m$ greater than some threshold value $m_{d}$, and one non-physical negative root. For $m=m_{d}$, the two positive roots merge into a single double root, which means that the interval of mid-wave instability shrinks to a single point, so that there is no mid-wave instability for $m<m_{d}$. If the cubic equation is written in the form $\text{Ma}_{0}^{3}+p\text{Ma}_{0}+q=0$, the condition for the double root is that a certain discriminant is zero, or $27q^{2}+4p^{3}=0$, whose solution for given $n$ and $s$ is $m_{d}$, the threshold value above which the mid-wave instability exists. For example, when $n=4$ and $s=1$, as in \[fig:figMidwaveR1m\_near\_n2\_n4s1\], $m_{d}=10.2783$. This value of $m$ is between those for the panels (a) and (b), as it should be. Thus, one can predict also the location of the boundaries of the critical curves in the $R$ sector. A somewhat different way for this, leading to a cubic equation for ${\text{Bo}}$, is as follows. A more explicit form of the system (\[eq:C\_0\])-(\[eq:C\_0’ is zero\]) is $$C_{0}(\textrm{Ma},\;\alpha,\;\textrm{Bo})=\textrm{Ma}(A_{1}+A_{2}\text{Ma}+A_{3}\text{Ma}^{2})=0,\label{eq:q_1is0}$$ $$\frac{\partial C_{0}}{\partial\alpha}=\textrm{Ma}(A_{1}'\text{}+A_{2}'\text{Ma}+A_{3}'\text{Ma}^{2})=0\label{eq:q_2is0}$$ where $$A_{1}=k_{11}B+k_{13}B^{3},\;A_{2}=k_{20}+k_{22}B^{2},\;A_{3}=k_{31}B,\label{eq:A_jdefinition}$$ and the prime stands for the $\alpha-$derivative. Since $\text{Ma}>0$, we divide equations (\[eq:q\_1is0\]) and (\[eq:q\_2is0\]) by $\text{Ma}$, and then the system consists of two quadratic equations, from which we obtain two different linear equations for $\text{Ma}$, one by eliminating the quadratic term, and the other by eliminating the zero-power term. The solvability condition, obtained by equating the two expressions for $\text{Ma}$, is $$(A_{1}A_{3}^{'}-A_{1}^{'}A_{3})^{2}-(A_{1}A_{2}^{'}-A_{1}^{'}A_{2})(A_{2}A_{3}^{'}-A_{2}^{'}A_{3})=0.\label{eq:Ma crit solvability cond}$$ Since $\alpha\downarrow0$ near a boundary point, we use the small-$\alpha$ expansions to find, to the leading order, the standard-form cubic equation $$\textrm{Bo}^{3}g_{3}+\textrm{Bo}g_{1}+g_{0}=0,\label{eq:cubic for Bo}$$ where the coefficients are defined as $g_{0}=k_{116}k_{206}^{2}$, $g_{1}=k_{206}(k_{118}k_{206}-k_{116}k_{208})$, and $g_{3}=k_{116}^{2}k_{318}-k_{116}k_{206}k_{228}+k_{206}^{2}k_{138}$. (One can see from the expressions for $k_{pqr}$ that here $g_{0}>0$ and $g_{3}>0$.) This cubic equation can be written in the standard form $\text{Bo}^{3}+p_{1}\text{Bo}+q_{1}=0$, with $p_{1}=g_{1}/g_{3}$ and $q_{1}=g_{0}/g_{3.}$ The viscosity value $m_{d}(n,s)$ satisfies the double-root condition $$27q_{1}^{2}+4p_{1}^{3}=0,\label{eq:cubic discrim-eq0}$$ which is essentially the same equation as the one found above using a different approach, where no explicit expressions were shown for $p$ and $q$ (in fact, it is clear from relation (\[eq:bo2 is kappa times ma2\]) that $p_{1}=\kappa^{2}p$ and $q_{1}=\kappa^{3}q$). Consider the asymptotics as $s\uparrow\infty$. Note that $p_{1}\propto s^{2}$ and $q_{1}\propto s^{2}$. Hence, (\[eq:cubic discrim-eq0\]) simplifies to $p_{1}=0$, which implies $g_{1}=0$ (provided $g_{3}\neq0$), and then, since $k_{206}>0$, it follows that $k_{206}k_{118}=k_{208}k_{116}$. Expanding, this equation involves $m$ and $n$ only: $$(1/2)\varphi(m-n^{2})(m(n(8n-3)+7)+n(n(7n-3)+8))$$ $$=(n+1)(n+m)(m^{2}(n(3n+8)+3)-4mn^{2}(n^{2}-1)-n^{4}(n(3n+8)+3)),\label{eq:s large}$$ where $\varphi$ is given by (\[eq:phi\]). For $n\uparrow\infty$, we look for solutions in the form $m\sim\chi n^{2}$ with $0<\chi<1$. The leading order is proportional to $n^{8}$, yielding $9\chi^{2}-4\chi-1=0$. The only acceptable solution is $\chi=(2+\sqrt{13})/9\approx0.623$. Note that even for $s=1$ and $n=4$, our (mentioned above) result $m_{d}=10.2783$ implies $m_{d}/n^{2}\approx0.643$ (cf. the asymptotic value $0.623$). If $s\uparrow\infty$ but $n\downarrow1$, it turns out that no appropriate solutions exist for $m_{d}$. Then the curve $m=m_{d}(n)$ should intersect the sector boundary $m=1$ at some finite $n=n_{0}>1$. Substituting $m=1$ into (\[eq:s large\]), we obtain the following equation for $n_{0}$: $(n-1)^{4}-16n^{2}=0$, which has a single acceptable solution, $n_{0}=3+\sqrt{8}$. ![Numerical solutions of equation (\[eq:cubic discrim-eq0\]) for the representative values of $s$ given in the legend.[]{data-label="fig:fig20rt"}](fig25rt){width="70.00000%"} Consider now the asymptotic case $s\downarrow0$. Here, equation (\[eq:cubic discrim-eq0\]) simplifies to the leading order equation $q_{1}=0$, and thus its numerator is also zero. But this contradicts the fact, mentioned above, that it is strictly positive. Therefore, there is no mid-wave instability for sufficiently small base shear. Fixing the value of $s$, we solve numerically equation (\[eq:cubic discrim-eq0\]) for the solution curve $m=m_{d}(n)$. In figure \[fig:fig20rt\], we show these solution curves for several representative values of $s$, ranging from small, to medium, to large. For large and small values of $s$, numerical solutions can be verified with analytical asymptotics. It is difficult to get numerical solutions for very large $s$. In particular, we obtain the point ($n=n_{0}$, $m=1$) which is approached when $s\uparrow\infty$ by the $m_{d}$ curves of the $R$ and $S$ sectors (the upper and lower branches in figure \[fig:fig20rt\]. As was established in the last paragraph of section \[subsec:General-considerations 6-2-1\], at any boundary point of a critical curve, the latter is tangent to the threshold curve through that quasi-intersection point of the two curves. Hence, since in the $R$ sector the threshold curves have positive slopes (see figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\]), the same holds for the critical curves near their boundary points. This means that the critical function $\text{Bo}_{cM}(\text{Ma})$ is increasing near its boundary points. Therefore, if there is a maximum, then there must be a minimum between this maximum and the right-end quasi-intersection point. It transpires that as $m$ rises through a certain threshold value $m_{t}$, such a maximum and a minimum appear at some ($\textrm{Ma, Bo}$). The latter is an inflection point on the $m=m_{t}$ critical curve, where the tangent is horizontal. We call it an “extrema bifurcation point” (EBP; see figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\] (c)). The EBPs, in both $R$ and $Q$ sectors, are discussed in detail below, in section \[subsec:Extrema-bifurcation-points\]. ### The $Q$ sector semi-infinite critical curves and their asymptotic behavior Turning next to the $Q$ sector, the cubic equation for $\text{Ma}_{0}$ has a single positive root and two non-physical complex conjugate roots. The physical root corresponds to the single “quasi-intersection” points in figure \[fig:Fig\_noses\](a). Since the threshold curve has $\text{Bo}=\text{Bo}_{cL}(\text{Ma})<0$ and for the critical curve $\text{Bo}\to0$ as $\text{Ma}\to\infty$, it is clear that the critical curve of the mid-wave instability lies above this threshold curve of the long-wave instability. This conclusion agrees with figure \[fig:Fig\_noses\]. For the $Q$ sector, the fact of the shared direction with the threshold curve at the boundary point of the critical curve, $d\text{Bo}/d\text{Ma}=\text{Bo}_{2}/\text{Ma}_{2}=\kappa$, (see the last paragraph of section \[subsec:General-considerations 6-2-1\]), implies that the function $\text{Bo}_{cM}(\text{Ma})$ is decreasing near the (single) quasi-intersection point. For $\text{Ma}\uparrow\infty$, postulating, from numerical results, that $\text{Bo}\rightarrow0$ and also $\alpha\rightarrow0$, we look for asymptotics $\text{Ma}=c_{1}\alpha^{-\zeta}$ (with $c_{1}\neq0$) and $\text{Bo}=d_{1}\alpha^{\xi}$ (with $d_{1}\neq0$), where $\zeta$ and $\xi$ are positive, and substitute this into the system of equations (\[eq:C\_0\]) and (\[eq:C\_0’ is zero\]). In more detail, these equations are (\[eq:MaEquation2\]), which for convenience is divided by $\text{Ma}^{2}$, and its partial derivative with respect to $\alpha$. Considering the first of these equations, it is clear that the second term is much smaller than the first one and the fourth and fifth terms are negligible in comparison with the third one. Thus, at leading order, the third term must balance the first one: $$k_{20}+k{}_{31}\text{Ma}B=0.\label{eq:c_1-1}$$ Since the $k_{20}\propto\alpha^{6}$, and $k_{31}\propto\alpha^{8}$, it follows that the product $\text{Ma}B\propto\alpha^{-2}$. Since $B=\text{Bo}+\alpha^{2}$ (which, clearly, entails that $\partial B/\partial\alpha=2\alpha$), one can see that necessarily $\xi=2$. This can be proved by showing that the assumption of $\xi<2$ or $\xi>2$ leads to a contradiction in the system consisting of (\[eq:c\_1-1\]) and $$k'_{20}+k'_{31}\text{Ma}B+k_{31}\text{Ma}(2\alpha)=0\label{eq:c_1}$$ has the power $9-\zeta$ which is greater than 5 (since $-\zeta+\xi=-2$, so $\zeta=2+\xi<4$). Therefore, the last term of (\[eq:c\_1\]) is negligible compared to the other terms, which are clearly of power $-5$. The first equation of the system yields $k_{206}+k_{318}c_{1}d_{1}=0$, and the second equation becomes $6k_{206}+8k_{318}c_{1}d_{1}=0$, which is clearly contradictory for $c_{1}d_{1}\neq0$. If we assume that $\xi>2$ then $B=\alpha^{2}$ to leading order. The first equation of the system yields $k_{206}+k_{318}c_{1}=0$ and the second equation $6k_{206}+8k_{318}c_{1}+2k_{318}c_{1}=0$. This system again has only the unacceptable solution $c_{1}=0$. Thus, we are left with $\xi=2$, and therefore $\zeta=4$. The system for $c_{1}$ and $d_{1}$ is now $$k_{206}+k_{318}(1+d_{1})c_{1}=0$$ and $$3k_{206}+k_{318}(5+4d_{1})c_{1}=0.$$ Eliminating $k_{206}$ from the last two equations yields $d_{1}=-2$. Then $$c_{1}=\frac{k_{206}}{k_{318}}=\frac{3(n-1)(m-n^{2})(n+1)^{2}}{n^{2}(n^{3}+m)^{2}}>0.$$ Therefore, $\text{Bo}=c_{1}^{1/2}d_{1}\text{Ma}^{-1/2}<0$. This is in excellent quantitative agreement with the numerical results documented in figure \[fig:Fig\_noses\](a). Local extrema of the critical curves ------------------------------------ As figure \[fig:Fig\_noses\] shows, in the $Q$ sector there is a local maximum on the critical curve for $m$ sufficiently close to $n^{2}$, just as there is one at $m=n^{2}$, the boundary between the $R$ and $Q$ sectors (see figure \[fig:Fig\_noses\_16\]). Taking into account that $\text{Bo}_{cM}(\text{Ma})$ is increasing at large $\text{Ma}$ (as it is negative and goes up to zero in the limit of infinitely increasing $\text{Ma}$), we conclude that there must be at least two local minima on the critical curve, which is also in agreement with the numerical results shown in figure \[fig:Fig\_noses\](a). For sufficiently large $m$, however, the critical curve is seen numerically to have just a single minimum. At any extremum, be it in the $R$ or the $Q$ sectors, we have $$\frac{d\text{Bo}}{d\text{Ma}}=0.\label{eq:dBo/dMa is 0}$$ Also, since substituting the solutions $\textrm{Bo(Ma)}$ and $\alpha(\textrm{Ma})$ of the system of equations (\[eq:C\_0\]) and (\[eq:C\_0’ is zero\]) for the critical curve into the left-hand side of equation (\[eq:C\_0\]) makes it true for all $\textrm{Ma}$, the total $\textrm{Ma}$-derivative of the left-hand side must be zero, i.e. $$\frac{\partial C_{0}}{\partial\text{Ma}}+\frac{\partial C_{0}}{\partial\alpha}\frac{d\alpha}{d\text{Ma}}+\frac{\partial C_{0}}{\partial\text{Bo}}\frac{d\text{Bo}}{d\text{Ma}}=0.$$ For the extremum, in view of equations (\[eq:C\_0’ is zero\]) and (\[eq:c\_1-1\]), this leads to the third equation in addition to (\[eq:C\_0\]) and (\[eq:C\_0’ is zero\]): $$\frac{\partial C_{0}}{\partial\textrm{Ma}}=0.\label{eq:dC_0/dMa is 0}$$ Thus the system of the three quadratic equations for the extrema points is $$\textrm{\ensuremath{A_{1}}+\ensuremath{A_{2}\text{Ma}}+\ensuremath{A_{3}\text{Ma}^{2}}}=0,\label{eq:q_11s0}$$ $$A_{1}'+A_{2}'\text{Ma}+A_{3}'\text{Ma}^{2}=0,\label{eq:q_21s0}$$ $$A_{1}+2A_{2}\text{Ma}+3A_{3}\text{Ma}^{2}=0.\label{eq:q_3is0}$$ Subtracting (\[eq:q\_11s0\]) from (\[eq:q\_3is0\]), we get the linear equation $$A_{2}+2A_{3}\text{Ma}=0,\label{eq:linearforMa}$$ which can be solved for $\textrm{Ma}$ in terms of the other variables, provided that $A_{3}\neq0$, i.e., since $k_{31}>0$, that $B\neq0$. On the other hand, another linear equation for $\textrm{Ma}$ is obtained by eliminating the quadratic terms by linearly combining the quadratic equations (\[eq:q\_11s0\]) and (\[eq:q\_3is0\]), $$2A_{1}+A_{2}\text{Ma}=0.\label{eq:linearforMa-1}$$ This can also be solved for $\textrm{Ma}$ in terms of the other variables (provided that $A_{2}\neq0$; also, it is easy to see that one has to assume that $B\neq0$ in order to have a nonzero $\textrm{Ma}$). The solvability condition of the over-determined system of the two linear equations for $\textrm{Ma}$, (\[eq:linearforMa\]) and (\[eq:linearforMa-1\]), is $$\begin{aligned} {\cal D} & = & 0,\label{eq:discriminant in terms of Ajs-1}\end{aligned}$$ where we have introduced the notation $$\begin{aligned} {\cal D} & = & A_{2}^{2}-4A_{1}A_{3},\label{eq:discriminant in terms of Ajs}\end{aligned}$$ which is independent of $\textrm{Ma}$. One has to distinguish the cases $B\ne0$ and $B=0$. For $B\ne0$, the solution of equation (\[eq:linearforMa\]) is $$\text{Ma}=-\frac{A_{2}}{2A_{3}}.\label{eq:Ma in terms of Ajs}$$ Substituting this into the quadratic equation (\[eq:q\_2is0\]), we have a system of two transcendental equations for $B$ and $\alpha$, which can be written in the following form: $$\begin{aligned} {\cal D} & = & 0,\label{eq:discriminant of q_1 is 0}\\ {\cal D}' & = & 0.\label{eq:discriminant prime is 0}\end{aligned}$$ In the $R$-sector, two solutions, a maximum and a minimum, are found by solving numerically the system of equations, (\[eq:discriminant of q\_1 is 0\]) and (\[eq:discriminant prime is 0\]), and then finding $\textrm{Ma}$ from (\[eq:Ma in terms of Ajs\]). In the $Q$ sector, this gives a single solution, which is a maximum for $m<m_{N}$, and a minimum for $m>m_{N}$. Also, there are, in a certain interval of viscosity ratios, solutions with $B=0$. In this case, the solvability condition (\[eq:discriminant in terms of Ajs-1\]) implies $$k_{20}=0,\label{eq:k_20 is 0}$$ which yields the wavenumber. Then the Bond number is determined uniquely as $$\text{Bo}=-\alpha^{2}.\label{eq:Bo is -alpha^2}$$ The quadratic equation (\[eq:q\_21s0\]), with the now known $\alpha$ and $\textrm{Bo}$, gives two distinct solutions for the Marangoni number if the discriminant $\zeta$ is positive, where $$\zeta=A_{2}^{'2}-4A_{1}^{'}A_{3}^{'}.\label{eq:zeta is q_2 discriminant}$$ For the case at hand we have the simplified relations $A_{2}^{'}=k_{20}^{'}$, $A_{1}^{'}=2\alpha k_{11}$, and $A_{3}^{'}=2\alpha k_{31}$. Thus, the solutions are $$\text{Ma}=\frac{-k_{20}'\pm\sqrt{k_{20}'^{2}-16\alpha^{2}k_{11}k_{31}}}{4k_{31}\alpha},\label{eq:Ma is solution of q_2}$$ corresponding to the two minima on the critical curves in the $Q$ sector (see figure \[fig:Fig\_noses\]). Figure \[fig:fig\_\_rtmovingnose\] shows the trajectories of the extrema in the $(\text{Ma},\text{Bo)}$-plane for $n=4$ as the viscosity ratio $m$ increases, starting from $m=15.45$, in the $R$ sector, reaching the $Q$ sector at $m=16$, and continuing to increase in the $Q$ sector. Consistent with the stability diagrams shown in figures \[fig:figMidwaveR1m\_near\_n2\_n4s1\] and \[fig:Fig\_noses\_16\], there are two extrema, a maximum and a minimum, for $m<16$, which is in the $R$ sector, provided $m>m_{t}=15.45$. In the $Q$ sector, there are three extrema as long as $m<m_{N}$. At $m=m_{N}$, the three extrema, one of them a maximum and two of them minima, collapse together into a single minimum, which then persists through the $Q$ sector. (Recall that we term this point the extrema bifurcation point (EBP)).  and (d), \[fig:Fig\_noses\_16\](a), and \[fig:Fig\_noses\](a)) as $m$ changes starting in the $R$ sector and increasing through the $R$ sector and after that, for $m>16$, the $Q$ sector. In the $R$ sector, for $15.45<m<16$, there is one maximum, the dashed curve, and one minimum, the solid curve, for $16<m<34.31$, there is one maximum between two minima, and finally, for $m>34.31$ there is one minimum. The arrows indicate the increase of $m$ and the dots correspond to the displayed values of $m$ next to them. The minimum moves to infinite $\textrm{Ma}$ as $m\to16$, from either side. \[fig:fig\_\_rtmovingnose\]](fig26rt){width="70.00000%"} In connection with the non-monotonic character of critical curves that have multiple local extrema, we note the following. In panels (e) and (f) of figure \[fig:FIG\_RTgamNalphas2by4\] (where $n=2$), we see that as $\textrm{Ma}$ increases, the long-wave instability gives way to stability at $\text{Ma}=\text{Ma}_{cL}$, which persists up to $\text{Ma}=\text{Ma}_{cM}$, at which point the mid-wave instability sets in, further persisting for all larger $\textrm{Ma}$. For short, we symbolically describe this sequence of Ma-intervals with different stability types as L-S-M, (where L indicates the long-wave instability, S denotes stability, and M stands for the mid-wave instability). The same stability interval sequence is obtained from figure \[fig:Fig\_noses\] (where $n=4$) if, e.g., we fix $m=25$ and $\textrm{Bo}=-0.1$, and go rightwards parallel to the ${\text{Ma}}$-axis. However, different sequences occur for other sets of parameters. For example, at $m=20$ and $\textrm{Bo}=-0.05$, we observe the sequence L-S-M-S-M; at $m=36$ and $\textrm{Bo}=-0.3$, the sequence is L-M-S-M; and at $m=25$ and $\textrm{Bo}=-0.2$, we have the longer sequence L-M-S-M-S-M. It appears that for any $\textrm{Bo}<0$, any sequence starts with L and ends with M. In contrast, for positive ${\text{Bo}}$, e.g., at $\textrm{Bo}=0.05$ and $m=18$, we have a S-M-S sequence of ${\text{Ma}}$-intervals. Extrema bifurcation points\[subsec:Extrema-bifurcation-points\] --------------------------------------------------------------- ### The EBP in the Q sector We turn now to the problem of equations determining the extrema bifurcation points. In this section, we consider those in the $Q$-sector, while those in the $R$-sector are examined in the following section. Clearly, the bifurcation point of the two minima and one maximum at $m=m_{N}$ in the $Q$ sector, which has $B=0$ (inherited, by continuity, from the $B=0$ property of the two minima existing at the smaller $m$), must satisfy, in addition to equations (\[eq:C\_0\]), (\[eq:C\_0’ is zero\]) and (\[eq:dC\_0/dMa is 0\]), the condition that the $\text{Ma}$ values of the two minima coalesce to a double root. It is clear from equation (\[eq:Ma is solution of q\_2\]) that this means $$k_{20}'^{2}-16\alpha^{2}k_{11}k_{31}=0.\label{eq:D1 is 0}$$ As we already noted above, the latter corresponds to the discriminant (\[eq:zeta is q\_2 discriminant\]) being zero, so that the two $\text{Ma}$ solutions of (\[eq:q\_21s0\]) for the two minima merge into just a single one. For the extrema bifurcation points in the $Q$ sector, it is convenient to use new variables $n_{1}=n-1$ and $m_{1}=(m-n^{2})/(n-1)$so that the $Q$ sector corresponds to the entire first quadrant. For any given ($n_{1},m_{1}$), as we already mentioned above, we can find the other properties of the EBP as follows: First, $\alpha$ is determined by solving equation (\[eq:k\_20 is 0\]) (which can be simplified, yielding that the quantity within the curly-bracket of $k_{20}$ in (\[eq:k20\]), denoted by ${\cal C}$, must vanish). This dependence $\alpha(n_{1},m_{1})$ is shown as the contour plot in figure \[fig:Level-curves-of alpha in Q\]. ![Level curves of $\alpha(n_{1},\;m_{1})$ for the extrema bifurcation points in the $Q$ sector. The numbers next to the curves are the corresponding values of $\alpha$. \[fig:Level-curves-of alpha in Q\]](fig27rt){width="70.00000%"} We observe this unique solution for the extended region of realistic $(n_{1},m_{1})$. For small $n_{1}$ and $\alpha$, we find that asymptotically $m_{1}=(4/15)\alpha{}^{2}$, independent of $n_{1}$, so that the level curves of $\alpha$ intersect the vertical axis at different heights. With corrections, the equation of the level curves at $\alpha\ll1$ and $n_{1}\ll1$ is $m_{1}=4/15\alpha^{2}(1+5n_{1}/2+2n{}_{1}^{2})$, where we have suppressed the terms which have powers of $\alpha^{2}$ higher than one or powers of $n_{1}$ higher than two. Keeping the two correction terms in the formula is necessary for predicting the flip of the sign of the level curve curvature as one switches between the linear and log scales of the $n_{1}$-axis. Having found $\alpha$ from equation (\[eq:k\_20 is 0\]), the Bond number $\text{Bo}(n_{1},m_{1})$ is given by equation (\[eq:Bo is -alpha\^2\]) with $\alpha=\alpha(n_{1},m_{1})$. Next, equation (\[eq:D1 is 0\]), after being divided through by $s^{2}$, is a linear equation in $s^{2}$, and gives $s(n_{1},m_{1})$, provided the derivative of $k_{20}$ with respect to $\alpha$ is negative. In view of $k_{20}=0$, it is enough to require that ${\cal C}'<0$. There is a strong evidence that the latter is indeed the case. At fixed $n_{1}$ and $m_{1}$, ${\cal C}>0$ and growing at sufficiently small $\alpha$; asymptotically, we find that ${\cal C}=\frac{1}{9}\varphi n^{2}\left(n_{1}+2\right)n_{1}m_{1}\alpha^{8}$. This factor attains a (positive) maximum and then monotonically decreases; asymptotically, at $\alpha\uparrow\infty$, we have ${\cal C}=-\frac{1}{32}\left(m+1\right)\exp(4\alpha n+2\alpha)$. Since ${\cal C}$ is decreasing at its zero, it is clear that ${\cal C}'<0$ at that $\alpha$, and this is confirmed by numerics. In addition, for a few fixed values of $n_{1}$, we computed $\alpha(n_{1},m_{1})$ and the corresponding ${\cal C}'$ up to large values of $m_{1}$, e.g. $m_{1}=10^{6}$, and this always showed ${\cal C}'<0$. We find analytically that the large-$m_{1}$ log-log asymptotic slope is $\text{d}\ln(-{\cal C}')/\text{d}\ln m_{1}=4(n_{1}+1)/n_{1}$ (with the logarithmic asymptotics $\text{d}\alpha/\text{d\ensuremath{\ln}}m_{1}=(1/2)n_{1}$). These asymptotic results are in excellent agreement with the numerical computations (which are not shown). Finally, $\text{Ma}(n_{1},m_{1})$ of the EBP is given by (\[eq:Ma is solution of q\_2\]) with discriminant equal to zero, so that $$\text{Ma}=\frac{-k_{20}'}{4k_{31}\alpha}.\label{eq:Ma is solution of q_2-1}$$ For example, there is a solution that corresponds to the EBP in figure \[fig:fig\_\_rtmovingnose\], with $n=4$, $s=1$, $m=m_{N}=34.31$, ${\text{Ma}}=13.97$, ${\text{Bo}}=-0.375$, and $\alpha=0.61$. These values are also consistent with figure \[fig:Fig\_noses\]. Note that in figure \[fig:Level-curves-of alpha in Q\], $\alpha(n_{1},m_{1})$ has no external parameters. The same is true of the other EBP dependencies: $\text{Bo}(n_{1},m_{1})$,; $s(n_{1},m_{1})$; and $\text{Ma}(n_{1},n_{1})$. ![Level curves of $\alpha(n_{1},\;m_{1})$ for the extrema bifurcation points in the $R$ sector. (For other curves, see the text.)\[fig:Level-curves-of alpha(n1,m1)\]](fig28rt) ### The EBP in the $R$ sector It was mentioned above, at the end of section \[subsec:R sector critical curves\], that one maximum and one minimum appear at the EBP on the critical curve in the $R$ sector corresponding to a threshold value $m_{t}$ of $m$. It is clear that for this EBP $$\frac{d^{2}\text{Bo}}{d\text{Ma}^{2}}=0$$ along the critical curve. (This bifurcation point of extrema corresponds to the inflection point with the horizontal tangent line in figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\](c).) Here $\text{Bo}(\text{Ma})$ is one of the two functions defined implicitly by the system (\[eq:C\_0\]) and (\[eq:C\_0’ is zero\]), where the other implicit function is $\alpha(\textrm{Ma})$. By the well-known formula for the derivative of an implicit function we have $$\frac{d\text{Bo}}{d\text{Ma}}=\frac{\partial(C_{0},C'_{0})}{\partial(\alpha,\text{Ma})}\left[\frac{\partial(C_{0},C'_{0})}{\partial(\text{Bo},\alpha)}\right]^{-1}.$$ We differentiate this expression with respect to $\text{Ma}$, taking into account (\[eq:C\_0\]), (\[eq:C\_0’ is zero\]) and (\[eq:dC\_0/dMa is 0\]). As a result, we obtain a fourth equation of the system for the bifurcation point of the extrema: $$\left(\frac{\partial^{2}C_{0}}{\partial\alpha\partial\textrm{Ma}}\right)^{2}-\frac{\partial^{2}C_{0}}{\partial\alpha^{2}}\frac{\partial^{2}C_{0}}{\partial\textrm{Ma}^{2}}=0,$$ which is, more explicitly, given by $$2\textrm{Ma}(A_{2}+3A_{3}\textrm{Ma})(A_{1}''+A_{2}''\text{Ma}+A_{3}''\text{Ma}^{2})-(A_{1}'+2A_{2}'\text{Ma}+3A_{3}'\text{Ma}^{2})^{2}=0.\label{eq:q_2is0-1}$$ We note that $\text{Bo}>0$ in the $R$ sector and therefore $B\ne0$. The four equations, (\[eq:discriminant of q\_1 is 0\]), (\[eq:discriminant prime is 0\]), (\[eq:Ma in terms of Ajs\]), and (\[eq:q\_2is0-1\]), are solved numerically. As an example, for $n=4$ and $s=1$, we find $\alpha=0.21$, $\text{Bo}=0.35$, $\text{Ma}=33.81$ and $m=m_{t}=15.45$. These numbers are consistent with figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\](c) and figure \[fig:fig\_\_rtmovingnose\]. Similarly to the $Q$ sector procedure used above, for the four EBP equations in the $R$ sector, an algebraic reduction is possible where a single equation is used to solve for one variable, and then the three other parameters of the EBP are found (with given values of $n$ and $m$). For this, we note the algebraic identity $k_{22}{}^{2}-4k_{13}k_{31}=0$. Hence, equation (\[eq:discriminant of q\_1 is 0\]) can be written, explicitly showing the $s$ and $B$ dependencies, as $$k_{20s}^{2}s^{2}+2B^{2}k_{sB}=0\label{eq:sB1}$$ and equation (\[eq:discriminant prime is 0\]) is written as $$k_{20s}k_{20s}'s^{2}+B^{2}k_{sB}'+4B\alpha k_{sB}=0.\label{eq:sB2}$$ Here we have defined the quantity $k_{sB}$ as $k_{sB}=k_{20s}k_{22}-2k_{11s}k_{31}$, where $k_{20s}$ and $k_{11s}$ are defined by $k_{20s}=k_{20}/s^{2}$ and $k_{11s}=k_{11}/s^{2}$. The last two displayed equations are linear equations for $s^{2}$, so $s^{2}$ is obtained explicitly in terms of the quantities $\alpha$, $m$, $n$, and $B$. Moreover, the solvability condition of the over-determined system of the two linear equations for $s^{2}$ yields (after dividing through by $Bk_{20s}$) a linear equation for $B$, $$B(k_{20s}k_{sB}'-2k_{20s}'k_{sB})+4k_{20s}\alpha k_{sB}=0,\label{eq:linear for B}$$ whose coefficients depend on $\alpha$, $m$, and $n$. Solving it (provided that the coefficient of $B$ is nonzero) yields $B$ in terms of $\alpha$, $m$, and $n$; using this expression in equation (\[eq:sB1\]), we obtain $s^{2}$ in terms of $\alpha$, $m$, and $n$, and then, from equation (\[eq:Ma in terms of Ajs\]), an expression for $\text{Ma }$ in terms of $\alpha$, $m$, and $n$. Substitute these expressions into (\[eq:q\_2is0-1\]) to obtain an equation containing $\alpha$, $m$, and $n$, which can be numerically solved for $\alpha$ giving it as a function of $n_{1}$ and $m_{1}$. Then, for the given values of $n_{1}$ and $m_{1}$, we find sequentially $B$, $s^{2}$, and $\textrm{Ma}$, in that order, using the linear-equations solutions for them described above. Thus, for given $n_{1}$ and $m_{1}$, we determine all the parameters, $\alpha$, $B$, $s$, and $\textrm{Ma}$, of the corresponding EBP in the $R$ sector. The level curves of $\alpha$ are seen in figure \[fig:Level-curves-of alpha(n1,m1)\] as the monotonically rising curves. Through the upper point of each curve passes the level curve, with the same value of $\alpha$, of the $\alpha$ function which makes identically vanish the coefficient $B_{d}$ of $B$ in equation (\[eq:linear for B\]). It is clear that the envelope of the family of level curves for $B_{d}=0$ is the locus of the upper ends of the level curves for the EBPs. (The envelope curve shown in the figure was obtained by solving the system $B_{d}=0$ and $\partial B_{d}/\partial\alpha=0$.) When approaching the envelope curve, the values of $B$ grow to infinity. The EBP level curves can be formally continued above the envelope curve, but lead to unphysical negative values of $B$ and $\text{Ma}$. (Note that the $R$ sector is completely mapped into the region of the $(n_{1},-m_{1})$-plane bounded above by the line $-m_{1}=n_{1}+2$, corresponding to $m=1$; however, this line is outside the range of figure \[fig:Level-curves-of alpha(n1,m1)\].) Summary and discussion {#sec:Conclusions} ====================== In this paper, we have considered the linear stability of two immiscible viscous fluid layers flowing in the channel between two parallel plates that may move steadily with respect to each other driving a Couette flow. The combined effects of gravity and an insoluble surfactant monolayer at the fluid interface were examined for certain flows such that the effect of inertia on their stability properties is negligible. The bulk velocity components satisfy linear homogeneous equations with constant coefficients. Therefore, their general solution, in the standard normal-mode analysis, is available with a few undetermined constants. The latter are determined, by the plate and interfacial-balance boundary conditions, in terms of the interface deflection and surfactant disturbance amplitudes. This yields a system of two algebraic linear homogeneous equations for the latter two amplitudes. Nontrivial solutions of this algebraic eigenvalue problem exist only if the increment $\gamma$, the complex “growth rate,” satisfies a quadratic equation whose coefficients are known functions of the wavenumber $\alpha$, the Marangoni number $\text{Ma}$, the Bond number $\text{Bo}$, the viscosity ratio $m$, the aspect ratio $n$, and the interfacial shear parameter $s$. The two solutions of this dispersion equation were shown to yield two continuous increment branches, defined almost everywhere in the wavenumber-parameters space (with a “branch cut” hypersurface excluded from it), and their real parts, the two continuous growth rate branches, were analyzed to infer conclusions concerning the stability of the flow. Similar to FH and subsequent papers, we call one of the branches the “robust” branch, as it is present even when ${\text{Ma}}=0$, and we call the other one, that vanishes as ${\text{Ma}}\downarrow0$, the “surfactant” branch. Thus, we have explicit formulas allowing us to readily compute the growth rates of instability for any given input values of the wavenumber and the five parameters of the problem. In the long-wave analysis of FH, three open sectors in the part of the ($n$, $m$)-plane given by $n\geq1$ and $m\geq0$, categorizing the stability of the system without gravity (${\text{Bo}}=0$), were identified: the $Q$ sector, ($m>n^{2}$), where both modes are stable; the $R$ sector, ($n^{2}>m>1$), where only the robust branch is unstable; and the $S$ sector, ( $0<m<1$), where only the surfactant mode is unstable. The same long-wave sectors were found to be relevant for non-zero ${\text{Bo}}$ in the lubrication theory of FH17. In the present paper, by using the long-wave asymptotics for the coefficients of the quadratic dispersion equation, we corroborate the lubrication approximation results of FH17 for the instability thresholds. In the $S$ sector, the surfactant mode remains unstable for all ${\text{Bo}}$, that is for arbitrarily strong stabilizing gravity; while in the $R$ sector the growth rate of the robust branch is unstable provided $\textrm{Bo}$ is below some positive threshold value $\textrm{Bo}_{c}$. In the $Q$ sector, both branches remain stable for Bo $\geq0$, but the robust branch is long-wave unstable for the smaller values of ${\text{Ma}}$ (while mid-wave unstable for larger values of ${\text{Ma}}$, so that there are longer waves that are stable, as discussed below), when $\textrm{Bo}$ is below some negative $\textrm{Bo}_{c}$. We have obtained the long-wave marginal wavenumbers and extremum growth rates which depend on the two main orders of the growth rate expression and were not considered in FH17. In particular, the small-$s$ behavior of the marginal wavenumber was obtained from the asymptotic form of our general equation for the marginal wavenumber. We have established that in the $R$ sector there are parametric situations in which the stabilizing effects, responsible for the emergence of the marginal wavenumber, are due, instead of the capillary forces, as is usual for larger $s$, to the nontrivial combined action of gravitational and surfactant forces. We also obtained the asymptotic small-$s$ behavior of the (long-wave) growth rate maximum and its corresponding wavenumber, which yielded different power laws for the cases of zero and non-zero Bond numbers. The asymptotic behavior in nearing the instability thresholds in the different sectors was established as well. The long-wave instabilities at the different borders between the three ($n,m$)-sectors were analyzed, such as the $S-R$ one, $m=1$. For the latter case, it was not clear from the small-wavenumber expression for the growth rates, equation (\[eq:GammaApproxSmallalpham\]), whether the unstable mode belonged to the surfactant branch, or, alternatively, to the robust one. We used complex analysis to show that there are indeed two separate branches of the growth rate function continuous for all wavenumbers and all the values of the parameters, one of the branches everywhere positive and the other one everywhere negative. The surfactant branch is easily identified near the wavenumber axis in the wavenumber-Marangoni number space, as the one of the two branches which vanishes in this limit of Marangoni number approaching zero, and it turns out to be positive or negative for positive or negative Bond number, respectively. The same is then true in the alternative limit, the wavenumber approaching zero at a finite Marangoni number, (corresponding to equation (\[eq:GammaApproxSmallalpham\])), since the branches keep their signs everywhere, and in particular the surfactant branch of the growth rate has the same sign near the $\textrm{Ma}$-axis as its sign near the $\alpha$-axis. In this way, we established that the unstable mode, corresponding to the positive sign in equation (\[eq:GammaApproxSmallalpham\]), belongs to the surfactant (robust) branch for positive (negative) Bond numbers (and the stable mode belongs to the other branch, in each case). For cases of arbitrary, (not necessarily small) wavenumbers, we still have explicit formulas for the stability quantities of interest, albeit more complicated and therefore, in general, studied numerically. It was found that in the $S$ sector and in the $R$ sector sufficiently far from the $Q$ sector, as well as in the $Q$ sector for sufficiently small Marangoni numbers, the dominant-mode instability has a long-wave character, in the sense that the left endpoint of the interval of unstable wavenumbers is zero. Otherwise, in particular in the $Q$ sector, for sufficiently large Marangoni numbers, the ’mid-wave’ instability may occur, in which the interval of unstable wavenumbers is bounded away from zero. These two situations were considered in turn. An interesting phenomenon, the dispersion-curve reconnection, was observed in the $S$ sector. Both branches are unstable for sufficiently negative values of Bond number, and, as $\textrm{Bo}$ decreases further, the robust-mode dispersion curve starts to cross the other dispersion curve at a single intersection point. Later in this process, at some sufficiently large value of $\left\vert \text{Bo}\right\vert $, the four parts of the two curves emanating from the intersection point recombine and detach, forming two new, non-intersecting, continuous curves, with the upper curve having two local maxima, of unequal heights. Then, as the Bond number decreases further, a jump in the global maximum may occur, as the shorter local maximum grows and eventually overcomes the other local maximum (figure \[fig:Fig\_typical\_MaxCrossingDCurve\]). The long-wave instability was studied with respect to gravity effects, as indicated by the dependencies of the characteristic dispersion quantities on the Bond number, figure \[fig:Fig3maxmargRSQ\], and, in the $S$ and $R$ sectors, with respect to the surfactant effects, as expressed in the dependencies on the Marangoni number, figure \[fig:FIG\_Bo1\_vs\_Ma\]. For the small and large values of these parameters, the relevant wavenumbers may be small, allowing for simpler asymptotics. Even when the limits of the characteristic dispersion quantities are not small, we sometimes get simplified equations which are easier to solve numerically, or, occasionally, even approximate analytic expressions, such as equation (\[eq:alpha0app\]). In the $R$ and $S$ sectors, at a fixed Bond number, the long-wave growth rate has a maximum at certain finite values of the wavenumber and the Marangoni number. We have observed, numerically, that both the maximum growth rate and its Marangoni number, grow linearly with the shear parameter $s$, starting from zero, while the corresponding wavenumber, which starts from zero as well, grows very fast at first, but then remains almost constant at larger $s$ (figure \[fig:fig13abc\]). Similar dependencies take place in the $Q$ sector as well (figure \[fig:fig16abc\]). The mid-wave instability turns out to emerge in two distinct ways (as a control parameter increases): it starts either from a stability stage, which we call the true onset of the mid-wave instability, or, alternatively, from a long-wave instability stage. The latter occurs when the left endpoint of the interval of the unstable wavenumbers, which is zero for the long-wave instability, starts moving away from zero (as shown in figure \[fig:figMidwaveAlphasR1m\_near\_n2\_n4s1\](b)), the maximum growth rate remaining positive all along. In the alternative scenario of the onset of the mid-wave instability, the maximum growth rate is equal to zero at a certain positive wavenumber, for which, therefore, the marginal wavenumber equation holds. But in view of the maximum, the partial derivative of the growth rate equals zero as well. Thus, we have a system of two equations, whose solution gives the critical values of the Marangoni number and the wavenumber asa function of the Bond number, for arbitrarily fixed values of the remaining three parameters. We follow, as the viscosity ratio is increased in the $R$ sector toward its border with the $Q$ sector, the emergence of the critical curve, and its consequent change, in the Marangoni number-Bond number plane (figure \[fig:figMidwaveR1m\_near\_n2\_n4s1\]). The critical curve has its two endpoints on the threshold curve of the long-wave instability. The latter is rightward-increasing in the $R$ sector, horizontal at the boundary with the $Q$ sector (figure \[fig:Fig\_noses\_16\]), and a decreasing curve in the $Q$ sector (figure \[fig:Fig\_noses\]). The right-side endpoint of the critical curve moves away to infinity as we cross into the $Q$ sector. The critical wavenumber is small near a critical curve endpoint, and so one can look for the critical solutions in the form of asymptotic power series. This gives rise to a cubic equation for the endpoint locations. Analysis of this equation leads to conclusions which are in agreement with the numerical observations, such that the critical curve in the $R$ sector exists only above a certain value of the viscosity ratio and has two endpoints, while there is just one single endpoint in the $Q$ sector. In all cases, the critical curve at its end point is tangent to the long-wave threshold curve. We also obtain and solve equations for the extrema of the critical curve, obtaining conclusions that agree with the numerical results. In the $R$ sector, there is a certain value of the viscosity ratio below which the critical curve has no extrema, but above which it has exactly two extrema: a maximum and a minimum. The latter disappears into the right-side infinity at the boundary with the $Q$ sector, and so we have just one extremum at this boundary, a maximum. Moving into the $Q$ sector as the viscosity ratio increases, there are at first one maximum in between two minima on the critical curve. These extrema coalesce into a single minimum at a certain value of the viscosity ratio $m$, and this minimum persists for the larger values of $m$. As we go from an arbitrary critical point to a critical extremum, one more constraint is added, which decreases the number of free parameters by one. The ’extrema bifurcation points’, at which the number of extrema changes, correspond to another reduction of the number of free parameters. Thus, for given $n$ and $m$, they determine all the other values: the wavenumber, Marangoni number, Bond number, and the shear parameter of the corresponding extrema bifurcation point (figure \[fig:Level-curves-of alpha(n1,m1)\]). Thus, figures \[fig:Fig\_typical\_midwaveDCurve\], \[fig:FIG\_RTgamNalphas2by4\], \[fig:Fig\_noses\], \[fig:fig\_\_rtmovingnose\] and \[fig:Level-curves-of alpha(n1,m1)\] represent different levels of information about the stability properties. Namely, going from one of the figures to the next, in the given order, the description gets more refined. On the other hand, the amount of data in the description decreases, in a certain sense. Figure \[fig:Fig\_typical\_midwaveDCurve\] gives the growth rates at every wavenumber, but all the parameters are fixed at certain values. So, out of the seven quantities, $\alpha$, $\gamma_{R}$,${\text{Ma}}$,${\text{Bo}}$, $m$, $n$ and $s$, six are independent variables, and just one quantity is a dependent variable. Thus, these data make up a six-dimensional hypersurface in the seven-dimensional space. Figure \[fig:FIG\_RTgamNalphas2by4\] corresponds to some five independent variables determining the values of the other two quantities, thus resulting in a five-dimensional manifold of data. Figure \[fig:Fig\_noses\] corresponds to a four-dimensional manifold, figure \[fig:fig\_\_rtmovingnose\] implies a three-dimensional manifold of data, and figure \[fig:Level-curves-of alpha(n1,m1)\] corresponds to a two-dimensional manifold parameterized by the independent variables $m$ and $n$, whose values determine $\alpha$, $\gamma_{R}$, ${\text{Ma}}$, ${\text{Bo}}$, $s$, (where $\gamma_{R}=0$ since our consideration here is confined to the critical conditions of mid-wave instability.) The envelope curve in figure \[fig:Level-curves-of alpha(n1,m1)\] corresponds to a one-dimensional curve in the seven-dimensional space of the relevant quantities. Finally, for the inflection point of the envelope curve in figure \[fig:Level-curves-of alpha(n1,m1)\], there are no independent variables, and all seven quantities are uniquely determined. There is the mid-wave instability of the robust branch in the $S$ sector too, albeit the long-wave instability of the surfactant branch is the stronger of the two there. In the $(\textrm{Ma,Bo})$-plane, in the vicinity of the threshold line of the long-wave instability, in addition to the more usual critical mid-wave curve which consists of the points that correspond to dispersion curves with zero maximum growth rate, there is, below the latter, another critical mid-wave curve, consisting of the points corresponding to dispersion curves with zero minimum growth rate (see figure \[fig:S midwave\_mp1n10s1\]). Correspondingly, as the Bond number decreases (to bigger-magnitude negative values), it is possible that at some point after the onset of the mid-wave instability, the long-wave instability starts, whose wavenumber interval is initially small and does not intersect the mid-wave interval of unstable wavenumbers. The coexistence of the mid-wave and the long-wave instabilities lasts until their intervals coalesce, corresponding to the critical curve of zero minimum growth rates in the $(\textrm{Ma,Bo})$-plane. After this coalescence, there is just one long-wave continuous interval of the unstable wavenumbers, with the dispersion curve having two positive local maxima of the growth rate at first, but just one single maximum eventually, at the most negative Bond number values. For another range of Marangoni number, an alternative scenario is possible, which differs from the one described above solely in that the long-wave instability starts first and the mid-wave one at the smaller (more negative) values of the Bond number. The consequent coalescence into purely long-wave instability is the same in both scenarios (figures \[fig:S midwave Bo-alpha\] and \[fig:S dispersion curves-1\]). The Continuous Branches of the Growth Rate Function\[sec:On-the-Continuous-Branches\] ===================================================================================== Recall that the two distinct analytic branches of the function $\sqrt{\zeta}$ exist in any simply connected domain in the complex plane that does not contain the origin ($\zeta=0$). As was mentioned in the text, it may happen for the discriminant $\zeta$ of the dispersion relation that $\zeta=0$ for some values of $\alpha$ and the parameters. This implies the two real equations, $\textrm{Re}(\zeta)=0$ and $\textrm{Im}(\zeta)=0$. The imaginary part of $\zeta$ (\[eq:DiscZ\]) is $$\begin{aligned} \operatorname{Im}(\zeta) & =\frac{s}{\alpha^{5}}\left(k_{Ma}\text{Ma}+\frac{k_{b}}{\alpha^{2}}\left(\text{Bo}+\alpha^{2}\right)\right)\label{eq:infinIMz}\end{aligned}$$ with the coefficients here $$\begin{aligned} k_{Ma} & = & (m-1)(\alpha n+\alpha n^{2}-n^{2}s_{\alpha}c_{\alpha}-s_{\alpha n}c_{\alpha n})\\ & & \times\left(m(s_{\alpha}^{2}-\alpha^{2})(\alpha n+s_{\alpha n}c_{\alpha n})+(\alpha+s_{\alpha}c_{\alpha})(s_{\alpha n}^{2}-\alpha^{2}n^{2})\right)\\ & & +2(s_{\alpha n}-ns_{\alpha})(ns_{\alpha}+s_{\alpha n})(-\alpha^{4}(-1+m)^{2}n^{2}+(mc_{\alpha n}s_{\alpha}+c_{\alpha}s_{\alpha n})^{2}\\ & & +\alpha^{2}(-c_{\alpha}n^{2}m^{2}+n(-c_{\alpha}^{2}n+m(-2+mns_{\alpha}^{2}))+s_{\alpha n}^{2}))\end{aligned}$$ and $$\begin{aligned} k_{b} & = & (m-1)(n(\alpha+\alpha n-c_{\alpha n}s_{\alpha})-c_{\alpha n}s_{\alpha n})(\alpha^{3}n(m+n)+s_{\alpha}s_{\alpha n}(mc_{\alpha n}s_{\alpha}+c_{\alpha}s_{\alpha n})\\ & & -\alpha^{2}(c_{\alpha n}^{2}s_{\alpha}+mc_{\alpha n}s_{\alpha n})-\alpha(mns_{\alpha}^{2}+s_{\alpha n}^{2}))\end{aligned}$$ As we mentioned before, the two equations $\text{Re}(\zeta)=0$ and $\text{Im}(\zeta)=0$ define a manifold of codimension two in the $\alpha$-parameter space. This manifold is analogous to a multivalued-function branch point in the complex plane. We consider the trace of this “branch manifold” in the three-dimensional space of $(\alpha,\;\text{Ma},\;\text{Bo})$, with the rest of the parameters fixed, as follows. Solving $\operatorname{Im}(\zeta)=0$ for Marangoni number yields $$\text{Ma}=-\frac{k_{b}}{\alpha^{2}k_{Ma}}(\text{Bo}+\alpha^{2})\text{.}\label{eq:MaWhenIM-eq-0}$$ Note that not all values of $($Bo$,\text{Ma})$ are appropriate here because ${\text{Ma}}$ must be positive. Similarly to the above expression for $\text{Im}(\zeta)$, we obtain $$Re(\zeta)=\frac{1}{\alpha^{10}}\left(K_{20}\text{Ma}^{2}+K_{02}\left(\text{Bo}+\alpha^{2}\right)^{2}+K_{11}\text{Ma}(\text{Bo}+\alpha^{2})+K_{00}\right),$$ where $$\begin{aligned} K_{20} & = & \frac{1}{4}\alpha^{4}(\alpha n(\alpha^{2}(m+n)+\alpha c_{\alpha n}s_{\alpha}-sms_{\alpha}^{2})-mc_{\alpha n}(s_{\alpha}^{2}-\alpha^{2})s_{\alpha n}-(\alpha+c_{\alpha}s_{\alpha})s_{\alpha n}^{2})^{2},\\ K_{02} & = & \frac{1}{4}(\alpha^{3}n(m+n)+s_{\alpha}s_{\alpha n}(mc_{\alpha n}s_{\alpha}+c_{\alpha}s_{\alpha n})-\alpha^{2}(c_{\alpha n}^{2}s_{\alpha}+mc_{\alpha n}s_{\alpha n})-\alpha(mns_{\alpha}^{2}+s_{\alpha n}^{2}))^{2},\\ K_{11} & = & \frac{1}{2}\alpha^{2}(m(s_{\alpha}^{2}-\alpha^{2})(s_{\alpha n}^{2}-\alpha^{2}n^{2})\left((\alpha+c_{\alpha}s_{\alpha})(c_{\alpha n}s_{\alpha n}-\alpha n)+(c_{\alpha}s_{\alpha}-\alpha)(\alpha n+c_{\alpha n}s_{\alpha n})\right)\\ & & +(c_{\alpha}^{2}s_{\alpha}^{2}-\alpha^{2})(s_{\alpha n}^{2}-\alpha^{2}n^{2})^{2}\\ & & +m^{2}(s_{\alpha}^{2}-\alpha^{2})^{2}(c_{\alpha n}^{2}s_{\alpha n}^{2}-\alpha^{2}n^{2})+2(s_{\alpha}^{2}-\alpha^{2})(s_{\alpha n}^{2}-\alpha^{2}n^{2})(\alpha^{4}(m-1)^{2}n^{2}\\ & & -(mc_{\alpha n}s_{\alpha}+nc_{\alpha}s_{\alpha})^{2}+\alpha^{2}(m^{2}c_{\alpha n}^{2}+n(nc_{\alpha}^{2}+m(2-mns_{\alpha}^{2}))-s_{\alpha n}^{2}))),\\ K_{00} & = & -s^{2}\alpha^{6}(m-1)^{2}(-\alpha n(1+n)+c_{\alpha n}^{2}s_{\alpha}+c_{\alpha n}s_{\alpha n})^{2}.\end{aligned}$$ To solve the system $\operatorname{Re}(\zeta)=0$ and $\operatorname{Im}(\zeta)=0$ for ${\text{Ma}}$ and ${\text{Bo}}$ as functions of $\alpha$ (with $s$, $m$, and $n$ fixed), equation (\[eq:MaWhenIM-eq-0\]) is substituted into $\operatorname{Re}(\zeta)$ which yields $$\operatorname{Re}(\zeta)=AB^{2}+C=0,\label{eq:RezABC}$$ where $B={\text{Bo}}+\alpha^{2}$, and $A$ and $C$ do not depend on ${\text{Ma}}$: $$A=\frac{1}{\alpha^{10}}\left(\frac{k_{b}^{2}}{\alpha^{4}k_{Ma}^{2}}K_{20}+K_{02}-\frac{k_{b}}{\alpha^{2}k_{Ma}}K_{11}\right),\;C=\frac{K_{00}}{\alpha^{10}}.$$ Therefore, ${\text{Bo}}={\text{Bo}}(\alpha)$, where $$\text{Bo}(\alpha)=-\alpha^{2}\pm\sqrt{-\frac{C}{A}}\text{.}\label{eq:BoACal}$$ Substituting (\[eq:BoACal\]) for ${\text{Bo}}$ into equation (\[eq:MaWhenIM-eq-0\]) yields ${\text{Ma}}$ such that $\zeta=0$ for a given $\alpha$. Only the unique value ${\text{Bo}}={\text{Bo}}(\alpha)$ that yields ${\text{Ma}}={\text{Ma}}(\alpha)$ $>0$ is admitted here. In figures \[fig:zetaeq0-BomandMamvsalpha\](a) and (b) curves ${\text{Bo}}={\text{Bo}}(\alpha)$ and ${\text{Ma}}={\text{Ma}}(\alpha)$ are plotted for various values of $m$. One can see that ${\text{Ma}}\uparrow\infty$ in the limit $\alpha\downarrow0$ for all $m$. In this limit, ${\text{Bo}}\uparrow\infty$ for $m>1$, but ${\text{Bo}}\uparrow-\infty$ for $m<1$. At $\alpha\uparrow\infty$, for all $m$, $\text{Bo}\sim-\alpha^{2}$ and ${\text{Ma}}\downarrow0$. There are no points where the discriminant is zero for $m=1$, as was shown in the main text for all parameter values (formally, in the figure, we get ${\text{Ma}}(\alpha)=0$ and ${\text{Bo}}(\alpha)=-\alpha^{2}$). This indicates that the branch manifold consists of at least two pieces, and perhaps more than two, some with $m>1$ and others with $m<1$. The same fact is reflected in the infinite discontinuities of the curves in the figure at finite values of $\alpha$, which take place provided $m>n^{2}$. Also, if we consider the ($\alpha$, ${\text{Ma}}$)-plane, with all the other parameters fixed, including $\text{Bo}$, corresponding to a horizontal line in figure \[fig:zetaeq0-BomandMamvsalpha\](a), there will be at most two branch points in the ($\alpha,{\text{Ma}}$)-plane since any horizontal line there intersects any curve at no more than two points. Therefore, in some sufficiently narrow infinite strip whose left boundary is the (vertical) ${\text{Ma}}$-axis, the discriminant is non-zero at all its points, and so there are two continuous branches, in agreement with the long-wave results in the main text. These results also show no intersections of the two dispersion curves (when the wavenumbers are small enough), which means that $\operatorname{Re}\sqrt{\zeta}$ is non-zero in a sufficiently narrow strip bordering the ${\text{Ma}}$-axis. The equation $\operatorname{Re}\sqrt{\zeta}=0$ implies that $\zeta$ is real (and negative). We have solved this equation for ${\text{Ma}}$ as a function of $\alpha$ at fixed values of ${\text{Bo}}$ (and the other parameters), and every resulting curve in the ($\alpha,{\text{Ma}}$)-plane indeed lies entirely outside some strip bordering the ${\text{Ma}}$-axis. Regarding the entire ($\alpha,{\text{Ma}}$)-plane, if we remove from it the branch points together with the infinite rays emanating from each branch point to the right and going parallel to the $\alpha$-axis, then in the remaining domain the discriminant is nowhere zero, and thus there are two continuous branches of the growth rate in this domain, smooth in $\alpha$ at each point that they are defined. Next, we note that the horizontal line ${\text{Bo}}=0$ in panel (a) of figure \[fig:zetaeq0-BomandMamvsalpha\] intersects every curve whose $m>1$. So, even in the absence of gravity, there may be intersections of the two dispersion curves. As ${\text{Ma}}$ is varied, these intersections disappear at some ${\text{Ma}}$, with the reconnection of the curve parts lying to the right of the “marginal intersection” point and consequent separation of the two “renovated” dispersion curves. This happens in the ranges of wavenumbers when both branches are stable, which was not noted in HF. Figure \[fig:The-zero-discriminant3d\] shows, as an example, the curve in the three-dimensional space which corresponds to the two dash-dotted, $m=2$, curves of figure \[fig:zetaeq0-BomandMamvsalpha\] . The coordinate box there is shown with its front, top, and right faces removed for a better view. The curve of zero discriminant starts at the back top right vertex and steadily goes downward and to the left simultaneously twisting first toward the viewer and then backward, until it ends at the back bottom left vertex. ![The curves (a) $\text{Bo}=\text{Bo}_{m}(\alpha)$ and (b) $\text{Ma}=\text{Ma}_{m}(\alpha)$ such that the discriminant $\zeta=0$ are plotted for the values of viscosity ratio $m$ indicated in the legend. \[fig:zetaeq0-BomandMamvsalpha\]](fig29rt){width="95.00000%"} ![The zero discriminant curve corresponding to the $m=2$ projection curves shown in panels (a) and (b) of figure \[fig:zetaeq0-BomandMamvsalpha\]. \[fig:The-zero-discriminant3d\]](fig30rt) Next, we demonstrate that there is always a strip $\mathcal{D}_{s}=\{0<\alpha<\alpha_{s},{\text{Ma}}>0\}$ where $\zeta\neq0$. Indeed, it appears in figure \[fig:zetaeq0-BomandMamvsalpha\](a) that any horizontal line ${\text{Bo}}={\text{Bo}}_{f}$ intersects any of the graphs of ${\text{Bo}}=\ {\text{Bo}}_{m}(\alpha)$ at no more than three points. If there are no intersections then the value of $\alpha_{s}$ is chosen completely arbitrarily. Otherwise, $\alpha_{s}$ must be smaller than the smallest $\alpha$ of the intersection points. For the purpose of this paper, the existence of $\mathcal{D}_{s}$ (and thus of the two branches of the growth rate) is sufficient with any small but finite $\alpha_{s}$. The existence of $\alpha_{s}$ is shown analytically for small values of $\alpha$. Coefficients of equations \[sec:Coefficients-of-equations\] =========================================================== The coefficients $A_{11}$, $A_{12}$, $A_{21}$, and $A_{22}$ of equation (\[eq:dispEqnSystem\]) are: $$\begin{aligned} \text{Re}(A_{11}) & =\left(m(s_{\alpha}^{2}-\alpha^{2})(s_{\alpha n}c_{\alpha n}-\alpha n)+(s_{\alpha n}^{2}-\alpha^{2}n^{2})(s_{\alpha}c_{\alpha}-\alpha)\right)\frac{1}{2\alpha^{5}F_{2}}B,\label{eq:ReA11}\\ \text{Im}(A_{11}) & =-\frac{(m-1)s}{\alpha^{2}F_{2}}\left(n^{2}(s_{\alpha}c_{\alpha}-\alpha)+s_{\alpha n}c_{\alpha n}-\alpha n\right),\\ A_{12} & =\left(s_{\alpha n}^{2}-\alpha^{2}n^{2}-mn^{2}(s_{\alpha}^{2}-\alpha^{2})\right)\frac{\text{Ma}}{2\alpha^{2}F_{2}},\label{eq:A12}\\ \text{Re}(A_{21}) & =\left(s_{\alpha n}^{2}-\alpha^{2}n^{2}-mn^{2}(s_{\alpha}^{2}-\alpha^{2})\right)\frac{1}{2\alpha^{2}F_{2}}B,\label{eq:ReA21}\\ \text{Im}(A_{21}) & =\left(m\left((s_{\alpha}c_{\alpha}-\alpha)(s_{\alpha n}c_{\alpha n}+\alpha n)+c{}_{\alpha n}^{2}(s_{\alpha}^{2}-\alpha^{2})+\alpha^{2}n^{2}s_{\alpha}^{2}\right)\right.\nonumber \\ & \left.+(s_{\alpha}c_{\alpha}+\alpha)(s_{\alpha n}c_{\alpha n}+\alpha n)+c_{\alpha}^{2}(s_{\alpha n}^{2}-\alpha^{2}n^{2})+\alpha^{2}s_{\alpha n}^{2}\right)\frac{s}{\alpha^{3}F_{2}},\label{eq:ImA21}\\ A_{22} & =\left(m((s_{\alpha}^{2}-\alpha^{2})(s_{\alpha n}c_{\alpha n}+\alpha n))+(s_{\alpha n}^{2}-\alpha^{2}n^{2})(s_{\alpha}c_{\alpha}+\alpha)\right)\frac{\text{Ma}}{2\alpha^{3}F_{2}}.\label{eq:A22}\end{aligned}$$ The coefficients $k_{20}$, $k_{11}$, $k_{31}$, $k_{22}$ and $k_{13}$ that appear in equation (\[eq:MaEquation2\]) are $$\begin{aligned} k_{20} & =\frac{s^{2}}{4\alpha^{6}}\left(s_{\alpha n}^{2}-n^{2}s_{\alpha}^{2}\right)\left\{ (m-1)\left(s_{\alpha n}c_{\alpha n}-\alpha n+{n}^{2}\left(s_{\alpha}c_{\alpha}-\alpha\right)\right)\right.\nonumber \\ & \times\left[m\left({\,s_{\alpha}^{2}-{\alpha}^{2}}\right)({s_{\alpha n}c_{\alpha n}+\alpha\,n})+(s_{\alpha n}^{2}-{\alpha}^{2}n^{2})\left({\alpha+s_{\alpha}c_{\alpha}}\right)\right]\nonumber \\ & -\left(s_{\alpha n}^{2}-{s}_{\alpha}^{2}n^{2}\right)\left[m^{2}\left({\,s_{\alpha}^{2}-{\alpha}^{2}}\right)\left({c_{\alpha n}^{2}}+{\alpha}^{2}{n}^{2}\right)\right.\nonumber \\ & \left.\left.+2m(n^{2}\alpha^{4}-n\alpha^{2}+{s_{\alpha}c_{\alpha}s_{\alpha n}c_{\alpha n})}+\left({c_{\alpha}^{2}}+{\alpha}^{2}\right)\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\right]\right\} ,\label{eq:k20}\end{aligned}$$ $$\begin{aligned} k_{11} & =\frac{s^{2}}{4\alpha^{8}}(m-1)\left(s_{\alpha n}^{2}-{s}_{\alpha}^{2}n^{2}\right)\left({s_{\alpha n}c_{\alpha n}-\alpha\,n+{n}^{2}}\left({s_{\alpha}c_{\alpha}-\alpha}\right)\right)\nonumber \\ & \times\left[m({s_{\alpha n}c_{\alpha n}-\alpha\,n})\left({\,s_{\alpha}^{2}-{\alpha}^{2}}\right)+({s_{\alpha}c_{\alpha}-\alpha})\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\right],\label{eq:k11}\end{aligned}$$ $$\begin{aligned} k_{31} & =\frac{1}{16\alpha^{10}}\left({\,s_{\alpha}^{2}-{\alpha}^{2}}\right)\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\left[({s_{\alpha}c_{\alpha}+\alpha)}\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\right.\nonumber \\ & \left.+m({s_{\alpha n}c_{\alpha n}+\alpha\,n})({s_{\alpha}^{2}-{\alpha}^{2}})\right]^{2},\label{eq:k31}\end{aligned}$$ $$\begin{aligned} k_{22} & =\frac{1}{8\alpha^{12}}\left({\,s_{\alpha}^{2}-{\alpha}^{2}}\right)\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\left[m({s_{\alpha n}c_{\alpha n}-\alpha\,n})({s_{\alpha}^{2}-{\alpha}^{2}})\right.\nonumber \\ & \left.+({s_{\alpha}c_{\alpha}-\alpha)}\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\right]\left[m({s_{\alpha n}c_{\alpha n}+\alpha\,n})({s_{\alpha}^{2}-{\alpha}^{2}})\right.\nonumber \\ & \left.+({s_{\alpha}c_{\alpha}+\alpha)}\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\right],\label{eq:k22}\end{aligned}$$ and $$\begin{aligned} k_{13} & =\frac{1}{16\alpha^{14}}\left({\,s_{\alpha}^{2}-{\alpha}^{2}}\right)\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\left[({s_{\alpha}c_{\alpha}-\alpha)}\left(s_{\alpha n}^{2}-{\alpha}^{2}n^{2}\right)\right.\nonumber \\ & \left.+m({s_{\alpha n}c_{\alpha n}-\alpha\,n})({s_{\alpha}^{2}-{\alpha}^{2}})\right]^{2}.\label{eq:k13}\end{aligned}$$ The corresponding long-wave approximations are $$k_{20}\approx k_{206}\alpha^{6}+k_{208}\alpha^{8},\label{eq:k20app}$$ where $$k_{206}=\frac{n^{4}s^{2}}{108}\varphi(n-1)(n+1)^{2}(m-n^{2})$$ and $$\begin{aligned} k_{208} & = & \frac{s^{2}}{810}(n-1)n^{4}(n+1)^{3}\left(m^{2}(n(3n+8)+3)-4m\left(n^{2}-1\right)n^{2}\right.\\ & & -\left.(n(3n+8)+3)n^{4}\right),\end{aligned}$$ $$k_{11}\approx k_{116}\alpha^{6}+k_{118}\alpha^{8},\label{eq:k11app}$$ where $$k_{116}=\frac{n^{7}s^{2}}{81}(n-1)(n+1)^{2}(m-1)(n+m)$$ and $$k_{118}=\frac{s^{2}}{1215}(m-1)(n-1)n^{7}(n+1)^{2}(m(n(8n-3)+7)+n(n(7n-3)+8)),$$ $$k_{31}\approx k_{318}\alpha^{8},\label{eq:k31app}$$ where $$k_{318}=\frac{n^{6}}{324}(n^{3}+m)^{2},$$ $$k_{22}\approx k_{228}\alpha^{8},\label{eq:k22app}$$ where $$k_{228}=\frac{n^{8}}{486}(n+m)(n^{3}+m),$$ and $$k_{13}\approx k_{138}\alpha^{8},\label{eq:k13app}$$ where $$k_{138}=\frac{n^{10}}{2916}(n+m)^{2}.$$ The coefficient of the $\alpha^{4}$ term that appears in equation (\[eq:gamSsmallAlphaApprox\]) is $$\begin{aligned} k_{S}= & \frac{\text{Ma}\left(n^{3}-4n^{2}+4n-1\right)}{60(m-1)}\nonumber \\ & +\frac{\text{Ma}^{3}}{128(m-1)^{5}n^{4}(n+1)s^{2}}(n-1)\left(m^{4}(3n+1)+2m^{3}\left(-3n^{3}-2n^{2}+4n+1\right)n\right.\nonumber \\ & +\left.4m^{2}\left(n^{3}-2n^{2}-2n+1\right)n^{3}+2m\left(n^{3}+4n^{2}-2n-3\right)n^{5}+(n+3)n^{8}\right)\nonumber \\ & +\frac{\text{Bo}\text{Ma}^{2}}{192(m-1)^{4}n(n+1)^{2}s^{2}}\left(m^{3}\left(3n^{2}-4n-3\right)\right.+m^{2}\left(2n^{3}+13n^{2}-6n-5\right)n\nonumber \\ & +\left.m\left(-5n^{3}-6n^{2}+13n+2\right)n^{3}+\left(-3n^{2}-4n+3\right)n^{5}\right)\nonumber \\ & +\text{Bo}^{2}\text{Ma}\frac{n^{2}\left(-m^{2}+m(n-1)n+n^{3}\right)}{144(m-1)^{3}(n+1)^{2}s^{2}}\label{eq:ks}\end{aligned}$$ The coefficients of the constant, quadratic and quartic terms of the marginal wavenumber equation (\[eq:MAappEqn\]) are $$\zeta_{0}=\frac{1}{108}s^{2}(n-1)(n+1)^{2}(m-n^{2})\varphi\text{Ma}+\frac{1}{81}n^{3}s^{2}(n-1)(n+1)^{2}(m-1)(n+m)\text{Bo},\label{eq:zeta0}$$ $$\begin{aligned} \zeta_{2}= & \begin{gathered}\frac{\text{Ma}}{810}(-1+n)(1+n)^{3}\left(-4mn^{2}\left(-1+n^{2}\right)+m^{2}(3+n(8+3n))-n^{4}(3+n(8+3n))\right)s^{2}\end{gathered} \nonumber \\ + & \text{Bo}\frac{(-1+m)(-1+n)n^{3}(1+n)^{2}(n(8+n(-3+7n))+m(7+n(-3+8n)))\ s^{2}}{1215}\nonumber \\ + & \textrm{Bo}\frac{n^{2}}{2916}\left(3(m+n^{3})\text{Ma}+n^{2}(m+n)\text{Bo}\right)^{2}+\frac{1}{81}s^{2}n^{3}(n-1)(n+1)^{2}(m-1)(n+m)\nonumber \\ \label{eq:zeta2}\end{aligned}$$ and $$\zeta_{4}=\frac{1}{324}n^{2}(m+n^{3})^{2}\text{Ma}^{2},\label{eq:zeta4}$$ where only the leading order term in $s$ has been retained in $\zeta_{4}$, so that $\zeta_{4}=\zeta_{40}$ of section \[subsec:Marginal-wavenumbers\]. The linear and cubic coefficients in $\text{Bo}_{c}$ of expression (\[eq:MAapproxCmode\]) are given by $$\begin{aligned} \beta_{1}=\frac{1}{15}\left(\frac{\left(m^{2}-1\right)m}{m+n}-m^{2}+\frac{2(m-1)m}{m-n^{2}}-\frac{6(m-1)\left(3mn+m+4n^{2}\right)}{3mn+m+(n+3)n^{2}}\right.\nonumber \\ \left.+(m-7)n+4m+n^{2}-2\right)\label{eq:beta1}\end{aligned}$$ and $$\beta_{3}=\frac{1}{36}\frac{n^{3}(n+m)\left\vert n-1\right\vert \psi^{2}}{\left[\varphi{s}(m-n^{2})(n+1)\right]^{2}\left\vert m-1\right\vert }\text{.}\label{eq:beta3}$$ The constant, linear and cubic coefficients in $\text{Ma}_{cL}$, $M_{0}$, $M_{1}$ and $M_{3}$, of the expression (\[eq:RTmaApp\]) are $$M_{0}=\frac{4n^{3}(m-1)(m+n)}{3\phi(m-n^{2})},\label{eq:M0}$$ $$\begin{aligned} M_{1}=\frac{1}{15}\left(\frac{m(1-m^{2})}{m+n}+\frac{2(m-1)m}{n^{2}-m}+\frac{6(m-1)\left(3mn+m+4n^{2}\right)}{\phi}\right.\nonumber \\ \left.-mn+(m-4)m-n^{2}+7n+2\right)\label{eq:M1}\end{aligned}$$ and $$M_{3}=-\frac{(n-1)\psi^{2}}{64(m-1)^{3}n^{3}(n+1)^{2}s^{2}(m+n)}.\label{eq:M3}$$ Long-wave formulas for $F_{0}$, $F_{1}$ and $F_{2}$ \[sec:Longwave-formulas-for-F\] =================================================================================== The small wavenumber approximations for the case of finite thickness, $n$, and small Marangoni number, ${\text{Ma}}$ are given here. The long-wave approximations of (\[eq:F2Re\])-(\[eq:F0Im\]) are first written as polynomials in ${\text{Ma}}$ and ${\text{Bo}}$, then the coefficients are expanded, so that keeping only the leading term in $\alpha$ , equations (\[eq:F2Re\])-(\[eq:F0Im\]) are approximately $$\begin{aligned} F_{2} & =\operatorname{Re}(F_{2})\approx\frac{1}{3}\,\psi\text{,}\label{eq:F2reApprox}\\ \operatorname{Re}(F_{1}) & \approx\frac{1}{9}n^{3}(m+n){\alpha}^{4}+\frac{1}{3}n(m+n^{3}){\alpha}^{2}\text{Ma}+\frac{1}{9}\,\,\,{n}^{3}(m+n){\alpha}^{2}\text{Bo,}\label{eq:F1reApprox}\\ \operatorname{Im}(F_{1}) & \approx\frac{2}{3}n^{2}s(n+1)(1-m)\alpha\text{,}\label{eq:F1imApprox}\\ \operatorname{Re}(F_{0}) & \approx\frac{1}{36}\,{n}^{4}{\alpha}^{6}\text{Ma}+\frac{1}{36}{n}^{4}{\alpha}^{4}\text{MaBo}\,\text{,}\label{eq:F0reApprox}\\ \operatorname{Im}(F_{0}) & \approx\frac{1}{6}n^{2}s(1-n^{2}){\alpha}^{3}\text{Ma,}\label{eq:F0imApprox}\end{aligned}$$ where $\psi$ is given by equation (\[eq:psi\]). For $m=1$, we find $$\begin{aligned} F_{2} & =\operatorname{Re}(F_{2})\approx\frac{1}{3}\,(n+1)^{4},\\ \operatorname{Re}(F_{1}) & \approx\frac{1}{9}n^{3}(n+1){\alpha}^{4}+\frac{1}{3}n(n^{3}+1){\alpha}^{2}\text{Ma}+\frac{1}{9}\,\,\,{n}^{3}(n+1){\alpha}^{2}\text{Bo},\\ \operatorname{Im}(F_{1}) & =0\text{,}\\ \operatorname{Re}(F_{0}) & \approx\frac{1}{36}\,{n}^{4}{\alpha}^{6}\text{Ma}+\frac{1}{36}{n}^{4}{\alpha}^{4}\text{MaBo}\,\text{,}\end{aligned}$$ and $$\operatorname{Im}(F_{0})\approx\frac{1}{6}n^{2}s(1-n^{2}){\alpha}^{3}\text{Ma.}$$ Normal modes with undisturbed surfactant \[sec:Are-there-normal\] ================================================================= Assuming that the surfactant is undisturbed, $G=0$, which implies that $h\ne0$, it follows from the second equation of (\[eq:dispEqnSystem\]) that $A_{21}=0$. This implies in particular that $\textrm{Im}(A_{21})=0$. However, in expression (\[eq:ImA21\]), each term is positive, since each of the expressions $s_{\alpha}c_{\alpha}-\alpha$, $s_{\alpha}^{2}-\alpha^{2}$, and $s_{\alpha n}^{2}-\alpha^{2}n^{2}$ is positive. This contradiction shows that there are no normal modes with $G=0$ if $s\ne0$. If, however, $s=0$, but $B$ is nonzero, then $\textrm{Im}(A_{21})=0$ identically. However, $\textrm{Re}(A_{21})=0$ yields, from equation (\[eq:ReA21\]), that $$m=\frac{s_{\alpha n}^{2}-\alpha^{2}n^{2}}{n^{2}(s_{\alpha}^{2}-\alpha^{2})}.\label{eq:m_AppD}$$ This equation gives a two-dimensional manifold of normal modes (parameterized with variables $n$ and $\alpha$). Thus, the normal modes with $G$$=0$ (and $h\neq0$) do exist, but only when $s=0$. Note that the first equation of the system (\[eq:dispEqnSystem\]) implies that $\gamma=-A_{11}$, and we find, making use of (\[eq:m\_AppD\]), the growth rate for this mode is $$\gamma_{R}=-\textrm{Re}(A_{11})=\left(\frac{s_{\alpha n}c_{\alpha n}-\alpha n}{n^{2}(s_{\alpha}^{2}-\alpha^{2})}+s_{\alpha}c_{\alpha}-\alpha\right)\left(s_{\alpha n}^{2}-\alpha^{2}n^{2}\right)\frac{1}{2\alpha^{5}F_{2}}B.$$ Thus, we have one nonzero branch of modes, which are the usual Rayleigh-Taylor modes for the stagnant base configuration. Also, for any negative $\textrm{Bo}$, if $B=0$, that is $\alpha^{2}=-\textrm{Bo}$, then $A_{21}=0$ without any restrictions on $m$ and $n$. We can see that $A_{11}=0$ in this case as well, so that $\gamma_{R}=0$, which indicates the marginal stability mode for the Rayleigh-Taylor instability of the stagnant base configuration.

--- abstract: 'A method to efficiently compute, in a automatic way, helicity amplitudes for arbitrary scattering processes at leading order in the Standard Model is presented. The scattering amplitude is evaluated recursively through a set of Dyson-Schwinger equations. The computational cost of this algorithm grows asymptotically as $3^n$, where $n$ is the number of particles involved in the process, compared to $n!$ in the traditional Feynman graphs approach. Unitary gauge is used and mass effects are available as well. Additionally, the color and helicity structures are appropriately transformed so the usual summation is replaced by Monte Carlo techniques. Some results related to the production of vector bosons and the Higgs boson in association with jets are also presented.' address: - | Institute of Nuclear Physics, NCSR Demokritos\ 15-310 Athens, Greece - | Institute of Nuclear Physics, NCSR Demokritos\ 15-310 Athens, Greece\ Institute of Nuclear Physics, PAS\ Radzikowskiego 152, 31-3420 Cracow, Poland author: - 'Costas G. Papadopoulos' - Małgorzata Worek title: | **MULTI-PARTICLE PROCESSES IN THE STANDARD\ \ MODEL WITHOUT FEYNMAN DIAGRAMS [^1]  [^2]** --- Introduction ============ Multi-particle and multi-jet final states are of great importance at the TeVatron and at the future LHC or $e^{+}e^{-}$ Linear Collider. They serve both as signals and as important backgrounds to many new and already discovered physics channels. As an example the production and decay of top quarks, Higgs boson(s) or SUSY particles can be mentioned. A typical background is the production of weak vector bosons in association with jets. Among others the proper evaluation of the eight jet QCD background will be needed. To describe the production process of a number of particles the corresponding amplitudes have to be constructed. This usually results in a very large number of terms, such that their automated construction and evaluation becomes the only solution. Apart from handling the number of amplitudes, which grows factorially with the number of particles, the integration over the multidimensional phase space of the final state particles represents a formidable task. In the past years various solutions to deal with these problems, implemented as different codes, have been presented. Either they are based on traditional methods of constructing Feynman diagrams or alternative methods with recursive equations are implemented [@Berends:1987cv; @Berends:1987me; @Mangano:1987xk; @Mangano:1987kp; @Berends:1988yn; @Mangano:1988kk; @Berends:1989hf; @Berends:1990ax; @Mangano:1990by; @Caravaglios:1995cd; @Draggiotis:1998gr; @Caravaglios:1998yr; @Draggiotis:2002hm]. The new formalism based on the Dyson-Schwinger equations recursively defines one-particle off-shell Green function. It does not involve any calculation of individual diagrams but various off-shell subamplitudes are regroupped in such a way that as little of the computation as possible is repeated. On the contrary, in the traditional approach, the same parts of different Feynman diagrams are recalculated all over again, see Fig.\[diagram\], increasing the number of steps that should be done in order to get the full amplitude. The recursive approach significantly decreases the factorial growth of the number of terms to be calculated with the number of particles down to $4^{n}$ or $3^{n}$ [^3]. Some examples of automatic parton level generators for any processes in the Standard Model are [@Boos:1994xb; @Pukhov:1999gg; @Boos:2004kh], [MadGraph/MadEvent]{}[@Stelzer:1994ta; @Maltoni:2002qb], [AMEGIC++]{} [@Krauss:2001iv] and the [HELAC/PHEGAS]{} package [@Kanaki:2000ey; @Kanaki:2000ms]. Codes designed for specific processes are [@Kaneko:1991ym; @Ishikawa:1993qr] as well as [Alpgen]{} [@Mangano:2002ea]. Very recently also on shell recursive equations have been proposed [@Britto:2004ap; @Britto:2005fq]. However, event generators based on this new method are not publicly available yet. In this article the algorithm based on Dyson-Schwinger recursive equations is briefly reviewed. It has been implemented as a new version of the multipurpose Monte Carlo generator [HELAC]{} in order to efficiently obtain cross sections for arbitrary multi-particle and multi-jet processes in the Standard Model. Dyson-Schwinger Recursive Equations =================================== Dyson-Schwinger equations give recursively the $n-$point Green’s functions in terms of the $1-$, $2-$,$\ldots$, $(n-1)-$point functions. These equations hold all the information for the fields and their interactions for any number of external legs and to all orders in perturbation theory. The recursive content of the Dyson-Schwinger equations for QCD has already been introduced in Ref.[@Draggiotis:2002hm] and reviewed recently in Ref.[@Papadopoulos:2005vg]. To include the electroweak sector, new vertices for leptons, the vector gauge bosons as well as for the scalar Higgs boson must be included. Additionally, the recursive equation for (anti)quarks should be rewritten to express their interaction with the electroweak gauge bosons. In order to better illustrate this idea let us present as an example the recursive equations for the Higgs boson interaction with massive particles only. Let $p_{1},p_{2},\ldots,p_{n}$ represent the external momenta involved in the scattering processs taken to be incoming. For a vector field we define a four vector $b^{\textnormal{\tiny V}}_{\mu}(P)$, which describes any sub-amplitudes from which a vector boson $V$ with momentum $P$ can be constructed. The momentum $P$ is given as a sum of external particles momenta. Accordingly we define a four-dimensional spinor $\psi^{\textnormal{\tiny F}}(P)$, which describes any sub-amplitude from which a fermion with momentum P can be constructed and by $\bar{\psi}^{\textnormal{\tiny F}}(P)$ a four-dimensional antispinor. Additionaly we have to introduces a scalar $H(P)$ for a Higgs boson. The content of Dyson-Schwinger equations in this case can be understood diagrammatically as in Fig.\[Higgs\]. The subamplitude with an off-shell Higgs boson momentum $P$ has contributions from three-bosons and four-bosons vertices plus the fermion antifermion vertex. The black blobs denote subamplitudes with the same structure. The following general recursive equations can be written down for a Higgs boson with the momentum $P$: $$\begin{aligned} H(P)&&=\sum_{i=1}^{n}\delta(P-p_{i})H(p_{i})\\ &&+\sum_{P=p_{1}+p_{2}} ig_{\textnormal{\tiny HVV}}~\Pi_{\textnormal{\tiny H}} ~b^{\textnormal{\tiny V}}_{\mu}(p_{1})b^{{\textnormal{\tiny V}}\mu} (p_{2})\epsilon(p_{1},p_{2})\\ &&+\sum_{P=p_{1}+p_{2}+p_{3}}ig_{\textnormal{\tiny HHVV}} ~\Pi_{\textnormal{\tiny H}}~H(p_{1})b^{\textnormal{\tiny V}}_{\mu}(p_{2}) b^{{\textnormal{\tiny V}}\mu}(p_{3}) \epsilon(p_{1},p_{2},p_{3})\\ &&+\sum_{P=p_{1}+p_{2}} ig_{\textnormal{\tiny HFF}}~ \Pi_{\textnormal{\tiny H}}~\bar{\psi}^{\textnormal{\tiny F}}(p_{1}) \psi^{\textnormal{\tiny F}}(p_{2})\epsilon(p_{1},p_{2})\\ &&+\sum_{P=p_{1}+p_{2}} ig_{\textnormal{\tiny HHH}} ~\Pi_{\textnormal{\tiny H}}~ H(p_{1})H(p_{2})\epsilon(p_{1},p_{2})\\ &&+\sum_{P=p_{1}+p_{2}+p_{3}}ig_{\textnormal{\tiny HHHH}} ~\Pi_{\textnormal{\tiny H}}~H(p_{1})H(p_{2})H(p_{3})\epsilon(p_{1},p_{2},p_{3})\end{aligned}$$ where the Higgs boson propagator is given by $$\Pi_{\textnormal{\tiny H}}=\frac{i}{P^{2}-m^{2}_{\textnormal{\tiny H}} -i\Gamma_{\textnormal{\tiny H}} m_{\textnormal{\tiny H}}}$$ and $\epsilon(p_{1},p_{2},p_{3})=\pm1$ is a sign function, which takes into account the sign change when two identical fermions are interchanged. The scattering amplitude can be calculated by any of the following relations, $${\cal A}(p_1,\ldots,p_n) = \left\{ \begin{array}{ll} \hat{b}^{\textnormal{\tiny V}}_\mu(P_i) b^{{\textnormal{\tiny V}}\mu}(p_i) & - \mbox{ vector bosons} \\ \\ \hat{H}(P_{i}) H(p_{i})& - \mbox{ Higgs boson} \\ \\ \hat{{\bar{\psi}}^{\textnormal{\tiny F}}}(P_i) {\psi}^{\textnormal{\tiny F}}(p_i) & - \mbox{ incoming fermion } \\\\ \psi^{\textnormal{\tiny F}}(p_i) \hat{\psi}^{\textnormal{\tiny F}}(P_i) & - \mbox{ outgoing fermion } \\ \end{array} \right.$$ where $$P_i=\sum_{j\not= i}p_j,$$ so that $P_i+p_i=0$. The functions with hat are given by the previous expressions except for the propagator term which is removed by the amputation procedure. This is because the outgoing momentum $P_{i}$ must be on shell. The initial conditions are given by $$\begin{aligned} b^{{\textnormal{\tiny V}}\mu}(p_i)&=&\varepsilon^\mu_\lambda(p_i) , ~~~\lambda=\pm 1,0 {\nonumber}\\ H(p_{i})&=&1\\ \psi^{\textnormal{\tiny F}}(p_i)&=&\left\{ \begin{array}{ll} u_\lambda(p_i) &\mbox{if $E_i\geq0$} \\ v_\lambda(-p_i) &\mbox{if $E_i\leq0$} \\ \end{array} \right. {\nonumber}\\ {{\bar{\psi}}}^{\textnormal{\tiny F}}(p_i)&=&\left\{ \begin{array}{ll} \bar{u}_\lambda(p_i) &\mbox{if $E_i\geq0$} \\ \bar{v}_\lambda(-p_i) &\mbox{if $E_i\leq0$} \\ \end{array} \right. \label{ampl}\end{aligned}$$ where the explicit form of $\varepsilon^\mu_\lambda,u_\lambda, v_\lambda,\bar{u}_\lambda,\bar{v}_\lambda$ are given in the Ref.[@Kanaki:2000ey]. In order to actually solve these recursive equations it is convenient to use a binary representation of the momenta involved [@Caravaglios:1995cd]. For a process with $n$ external particles, to the momentum $P^\mu$ defined as $$P^\mu=\sum_{i=1}^{n}p_i^\mu$$ a binary vector $\vec{m}=(m_1,\ldots,m_n)$ can be assigned, where its components take the values $0$ or $1$, in such a way that $$P^\mu=\sum_{i=1}^n m_i\;p_i^\mu\;.$$ Moreover this binary vector can be uniquely represented by the integer $$m=\sum_{i=1}^n 2^{i-1}m_i$$ where $$1\le m \le 2^{n-1}. $$ Therefore all subamplitudes can be labeled accordingly, $$\psi^{\textnormal{\tiny F}}(P)\rightarrow\psi^{\textnormal{\tiny F}}(m),$$ $$\bar{\psi}^{\textnormal{\tiny F}}(P) \rightarrow\bar{\psi}^{\textnormal{\tiny F}}(m),$$ $$b^{\textnormal{\tiny V}}_\mu(P)\to b^{\textnormal{\tiny V}}_\mu(m),$$ $$H(P)\to H(m).$$ A very convenient ordering of integers in binary representation relies on the notion of level $l$, defined simply as $$l=\sum_{i=1}^n m_i\;.$$ As it is easily seen all external momenta are of level $1$, whereas the total amplitude corresponds to the unique level $n$ integer $2^{n}-1$. This ordering dictates the natural path of the computation; starting with level-$1$ sub-amplitudes, we compute the level-$2$ ones using the Dyson-Schwinger equations and so on up to the level $n$ which is the full amplitude. Contrary to original [HELAC]{} [@Kanaki:2000ey; @Kanaki:2000ms], the computational part consists of only one step, where couplings allowed by the Lagrangian defined by fusion rules are only explored. Subsequently, the helicity configurations are set up. There are two possibilities, either exact summation over all helicity configurations is performed or Monte Carlo summation is applied. For example for a massive gauge boson the second option is achieved by introducing the polarization vector $$\varepsilon^{\mu}_{\phi}(p)=e^{i\phi}\varepsilon^{\mu}_{+}(p)+ e^{-i\phi}\varepsilon^{\mu}_{-}(p)+\varepsilon^{\mu}_{0}(p),$$ where $\phi \in (0,2\pi)$ is a random number. By integrating over $\phi$ we can obtain the sum over helicities $$\frac{1}{2\pi}\int_{0}^{2\pi}d\phi ~\varepsilon^{\mu}_{\phi}(p) (\varepsilon^{\nu}_{\phi}(p))^{*}=\sum_{\lambda=\pm} \varepsilon_{\lambda}^{\mu}(p)(\varepsilon_{\lambda}^{\nu}(p))^{*}.$$ The same idea can be applied to the helicity of (anti)fermions. Finally, the color factor is evaluated iteratively. Once again, we have two options. Either we proceed by computing all $3^{n_{q}}\times3^{n_{\bar{q}}}$ color configurations, where the gluon is treated as a quark-antiquark pair and $n_{q},n_{\bar{q}}$ is the number of quarks and antiquarks respectively, or Monte Carlo summation is applied. Only a fraction of all possible $3^{n_{q}}\times3^{n_{\bar{q}}}$ color configurations gives rise to a non zero amplitude. In the Monte Carlo approach for each event we randomly select a non vanishing color assignment for the external particles and evaluate the amplitude. An overall multiplicative coefficient must be introduced to provide the correct normalization. The weight of the event is simply proportional to the $|{\cal M}|^{2}$ multiplied by the number of non zero color configurations. Assuming that, on average, all color configurations contribute the same amount to the cross section this approach is numerically more efficient than summing each event over all colors, see Ref. [@new] for further details. For the spinor wave functions as well as for the Dirac matrices, we have chosen the 4-dimensional chiral representation which results in particularly simple expressions. All vertices in the unitary gauge have been included. Both the fixed width scheme (FWS) and the complex mass scheme (CMS) for unstable particles are implemented. The computational cost of [HELAC]{} grows like $\sim 3^n$, which essentially counts the steps used to solve the recursive equations. Obviously for large $n$ there is a tremendous saving of computational time, compared to the $n!$ growth of the Feynman graph approach. Numerical Results ================= As an example the algorithm has been used to compute total cross sections for the production of weak vector bosons and the Higgs boson in association with jets. The following Standard Model input parameters have been used [@Eidelman:2004wy]: $$\begin{aligned} &&m_{W}=80.425 ~\textnormal{GeV}, ~~~~~~~~~~~~~~~~~~\Gamma_{W}=2.124 ~\textnormal{GeV}\\ &&m_{Z}=91.188 ~\textnormal{GeV}, ~~~~~~~~~~~~~~~~~~~\Gamma_{Z}= 2.495 ~\textnormal{GeV}\\ &&G_{\mu}= 1.6637\times10^{-5} ~\textnormal{GeV}^{-5}\\ &&\sin^{2}\theta_{W}=1-m^{2}_{W}/m_{Z}^{2}.\end{aligned}$$ The electromagnetic coupling is derived from the Fermi constant $G_{\mu}$ according to $$\alpha_{\textnormal{\footnotesize em}}=\frac{\sqrt{2}G_{\mu}m^{2}_{W} \sin^{2}\theta_{W}}{\pi}.$$ All results are obtained with a fixed strong coupling constant $\alpha_{s}$ calculated at the $m_{Z}$ scale $$\alpha_{s}(m^{2}_{Z})=0.1187.$$ ----------------------------------- ------------------------------------------------------------------------------------ --------------------------------------------------------------------------------  $\textnormal{Process}$       $\sigma_{\textnormal {\tiny EXACT}}$ $\pm$ $\varepsilon$ $\textnormal{(nb)}$       $\sigma_{\textnormal{\tiny MC}}$ $\pm$ $\varepsilon$ $\textnormal{(nb)}$ $gg \rightarrow t\bar{t}H$ (0.2723 $\pm$ 0.0016)$\times 10^{-3}$ (0.2713 $\pm$ 0.0013)$\times 10^{-3}$ $u\bar{u} \rightarrow t\bar{t}H$ (0.2758 $\pm$ 0.0017)$\times 10^{-4}$ (0.2739 $\pm$ 0.0011)$\times 10^{-4}$ $d\bar{d} \rightarrow t\bar{t}H $ (0.1816 $\pm$ 0.0011)$\times 10^{-4}$ (0.1811 $\pm$ 0.0007)$\times 10^{-4}$ $c\bar{c} \rightarrow t\bar{t}H $ (0.8118 $\pm$ 0.0057)$\times 10^{-6}$ (0.8094 $\pm$ 0.0032)$\times 10^{-6}$ $s\bar{s} \rightarrow t\bar{t}H $ (0.2203 $\pm$ 0.0014)$\times 10^{-5}$ (0.2191 $\pm$ 0.0008)$\times 10^{-5}$ $b\bar{b} \rightarrow t\bar{t}H $ (0.2260 $\pm$ 0.0016)$\times 10^{-6}$ (0.2262 $\pm$ 0.0009)$\times 10^{-6}$ ----------------------------------- ------------------------------------------------------------------------------------ -------------------------------------------------------------------------------- : *Results for the total cross section for the associated production of Higgs boson (130 GeV) with a $t\bar{t}$ pair, $\sigma_{\textnormal {\tiny EXACT}}$ corresponds to summation over all possible color configurations, while $\sigma_{\textnormal{\tiny MC}}$ corresponds to Monte Carlo summation.* []{data-label="tab1"} ----------------------------------- ------------------------------------------------------------------------------------ --------------------------------------------------------------------------------  $\textnormal{Process}$       $\sigma_{\textnormal {\tiny EXACT}}$ $\pm$ $\varepsilon$ $\textnormal{(nb)}$       $\sigma_{\textnormal{\tiny MC}}$ $\pm$ $\varepsilon$ $\textnormal{(nb)}$ $u\bar{u} \rightarrow u\bar{u}H$ (0.1406 $\pm$ 0.0029)$\times 10^{-5}$ (0.1361 $\pm$ 0.0020)$\times 10^{-5}$ $u\bar{u} \rightarrow d\bar{d}H$ (0.6699 $\pm$ 0.0088)$\times 10^{-5}$ (0.6596 $\pm$ 0.0081)$\times 10^{-5}$ $ud \rightarrow udH $ (0.2280 $\pm$ 0.0043)$\times 10^{-4}$ (0.2222 $\pm$ 0.0021)$\times 10^{-4}$ $\bar{u}d \rightarrow \bar{u}dH $ (0.1241 $\pm$ 0.0027)$\times 10^{-5}$ (0.1258 $\pm$ 0.0025)$\times 10^{-5}$ $dd \rightarrow ddH $ (0.3404 $\pm$ 0.0046)$\times 10^{-5}$ (0.3477 $\pm$ 0.0032)$\times 10^{-5}$ $uu \rightarrow uuH $ (0.5132 $\pm$ 0.0081)$\times 10^{-5}$ (0.5178 $\pm$ 0.0060)$\times 10^{-5}$ ----------------------------------- ------------------------------------------------------------------------------------ -------------------------------------------------------------------------------- : *Results for the total cross section for the production of a heavy Higgs boson (200 GeV) via the vector boson fusion, $\sigma_{\textnormal {\tiny EXACT}}$ corresponds to summation over all possible color configurations, while $\sigma_{\textnormal{\tiny MC}}$ corresponds to Monte Carlo summation.* []{data-label="tab2"} The mass of an intermediate and a heavy Higgs boson and associated Standard Model tree level widths are assumed to be: $$\begin{aligned} &&m_{H}=130 ~\textnormal{GeV}, ~~~~~~~~~~~~~~~~~~~\Gamma_{H}= 0.005 ~\textnormal{GeV}\\ &&m_{H}=200 ~\textnormal{GeV}, ~~~~~~~~~~~~~~~~~~~\Gamma_{H}= 1 ~\textnormal{GeV}\\\end{aligned}$$ For the massive fermions the following masses have been applied: $$\begin{aligned} &&m_{u}=4 ~\textnormal{MeV}, ~~~~~~~~~~~~~~~~~~~m_{d}=8 ~\textnormal{MeV}\\ &&m_{s}=130 ~\textnormal{MeV}, ~~~~~~~~~~~~~~~~~m_{c}=1.35 ~\textnormal{GeV}\\ &&m_{b}=4.4 ~\textnormal{GeV},\\ &&m_{t}= 174.3 ~\textnormal{GeV}, ~~~~~~~~~~~~~~~\Gamma_{t}=1.56 ~\textnormal{GeV}\\\end{aligned}$$ The mixing of the quark generations is neglected. The CMS energy was chosen $\sqrt{s}=14$ $\textnormal{TeV}$. The following cuts were used to stay away from soft and collinear divergencies in the part of the phase space intergated over: $$p_{T_{i}} > 60 ~\textnormal{GeV}, ~~~~~~ |y_{i}|<2.5, ~~~~~~ \Delta R > 1.0$$ where $$p_{T_{i}}=\sqrt{p_{x_{i}}^{2}+p_{y_{i}}^{2}}, ~~~~~~~~ y_{i}=\frac{1}{2}\ln \left( \frac{ E_{i}+p_{z_{i}} }{ E_{i}-p_{z_{i}} }\right)$$ are the transverse momentum and rapidity of the $i$-jet respectively. Additionaly $\Delta R$ is a radius of the cone of the jet defined as $$\Delta R=\sqrt{(\Phi_{i}-\Phi_{j})^{2}+(y_{i}-y_{j})^2}$$ where $\Delta \Phi_{ij}=\Phi_{i}-\Phi_{j}$ $$\Delta \Phi_{ij}=\arccos \left( \frac{p_{x_{i}}p_{x_{j}}+p_{y_{i}}p_{y_{j}}} {p_{T_{i}} p_{T_{j}} } \right).$$ There are several parameterizations for the parton structure functions, we used [CTEQ6 PDF]{}’s parametrization [@Pumplin:2002vw; @Stump:2003yu]. For the phase space generation we used two different generators. The first one is [PHEGAS]{} [@Papadopoulos:2000tt], which automatically constructs mappings of all possible peaking structures of a given scattering process. Self-adaptive procedures like multi-channel optimization [@Kleiss:1994qy] is additionally applied exhibiting high efficiency. The Second one is a flat phase-space generator [RAMBO]{} [@Kleiss:1985gy]. In the Table \[tab1\]. results for the total cross section for the associated production of the Higgs boson of $m_{H}=130$ GeV mass, with a $t\bar{t}$ pair are presented. The $t\bar{t}H$ production channel is one of the most promising reactions to study both the top quark and the Higgs boson at the LHC, in the second case especially for the $b\bar{b}$ decay channel of the Higgs boson [@Beneke:2000hk; @Djouadi:2005gi]. As we can see from the Table \[tab1\]. the $gg\rightarrow t\bar{t}H$ process dominates due to the enhanced gluon structure function. The final state of this channel consists of $W$ bosons and four $b-$jet, two from the decay of the top quarks and two from the decay of the Higgs boson. The main background process is $gg\rightarrow W^{+}W^{-}b\bar{b}b\bar{b}$ with the contributions from all intermediate states. In Fig.\[inv1\] we present the invariant mass distribution for the $b\bar{b}$ system. Transverse momentum and rapidity distributions of $b$-jet for the background process are shown in Fig. \[pt-eta-b\]. A more powerful channel for higher Higgs boson masses is the vector boson fusion $qq\rightarrow V^{*}V^{*}\rightarrow qqH$ with $H\rightarrow W^{+}W^{-}$ decay. In the Table \[tab2\]. results for the total cross section for some production processes of a heavy Higgs boson of $m_{H}=200$ GeV mass via vector boson fusion are presented. The main background processes consist of $qq \rightarrow W^{+}W^{-}qq$ channels. As an example the distribution of the invariant mass of the $W^{+}W^{-}$ system from the $u\bar{u} \rightarrow W^{+}W^{-}d \bar{d}$ process is presented in Fig.\[inv1\]. Transverse momentum and rapidity distributions of $d$-jet and $W$ are also shown in Fig.\[pt-eta-d\] and Fig.\[pt-eta-w\]. Summary {#summary .unnumbered} ======= An efficient tool for automatic computation of helicity amplitudes and cross sections for multi-particle final states in the Standard Model has been presented. Matrix elements and cross sections are calculated iteratively by Dyson-Schwinger equations. We are free from the task of computing all Feynman diagrams for a given process, which can become impossible for the large number of particles involved. The computationally expensive procedures of summing over color and helicity configurations have been replaced by Monte Carlo summation. At this stage, the code is able to compute scattering matrix elements and cross sections for hard processes. In the next step we plan to make calculations of fully hadronic final states in $p\bar{p}$ and $pp$ collisions possible. In particular, we wish to include the the emission of secondary partons and translate the emerging partons into primordial hadrons by interfacing this package with codes like [PYTHIA]{}[@Sjostrand:2000wi] or [HERWIG]{}[@Corcella:2000bw]. This kind of multipurpose Monte Carlo generator will be of great interest in the study of the TeVatron, LHC or $e^{+}e^{-}$ Linear Collider data. Acknowledgments {#acknowledgments .unnumbered} =============== Work supported by the Polish State Committee for Scientific Research Grants number 1 P03B 009 27 for years 2004-2005 (M.W.). In addition, M.W. acknowledges the Maria Curie Fellowship granted by the the European Community in the framework of the Human Potential Programme under contract HPMD-CT-2001-00105 ([*“Multi-particle production and higher order correction”*]{}). 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[^2]: [IFJPAN-V-2005-12]{} [^3]: To reduce the computational complexity down to an asymptotic $3^n$, each four-boson vertex must be replaced with a three-boson vertex by introducing an auxiliary field represented by the antisymmetric tensor $H^{\mu\nu}$, see [@Draggiotis:2002hm] for details.

=-1cm

--- abstract: | The sequence of phase transitions and the symmetry of in particular the low temperature incommensurate and spin-Peierls phases of the quasi one-dimensional inorganic spin-Peierls system TiOX (TiOBr and TiOCl) have been studied using inelastic light scattering experiments. The anomalous first-order character of the transition to the spin-Peierls phase is found to be a consequence of the different symmetries of the incommensurate and spin-Peierls (P$2_{1}/m$) phases. The pressure dependence of the lowest transition temperature strongly suggests that magnetic interchain interactions play an important role in the formation of the spin-Peierls and the incommensurate phases. Finally, a comparison of Raman data on VOCl to the TiOX spectra shows that the high energy scattering observed previously has a phononic origin. author: - Daniele Fausti - 'Tom T. A. Lummen' - Cosmina Angelescu - Roberto Macovez - Javier Luzon - Ria Broer - Petra Rudolf - 'Paul H.M. van Loosdrecht' - Natalia Tristan - Bernd Büchner - Sander van Smaalen - Angela Möller - Gerd Meyer - Timo Taetz bibliography: - 'bibliografia1.bib' title: Symmetry disquisition on the TiOX phase diagram --- Introduction ============ The properties of low-dimensional spin systems are one of the key topics of contemporary condensed matter physics. Above all, the transition metal oxides with highly anisotropic interactions and low-dimensional structural elements provide a fascinating playground to study novel phenomena, arising from their low-dimensional nature and from the interplay between lattice, orbital, spin and charge degrees of freedom. In particular, low-dimensional quantum spin (S=1/2) systems have been widely discussed in recent years. Among them, layered systems based on a $3d^{9}$ electronic configuration were extensively studied in view of the possible relevance of quantum magnetism to high temperature superconductivity[@Imad98; @Dag99]. Though they received less attention, also spin=1/2 systems based on early transition metal oxides with electronic configuration $3d^{1}$, such as titanium oxyhalides (TiOX, with X=Br or Cl), exhibit a variety of interesting properties[@kataev2003; @imai2003]. The attention originally devoted to the layered quasi two-dimensional $3d^{1}$antiferromagnets arose from considering them as the electron analog to the high-$T_{c}$ cuprates[@Maule88]. Only recently TiOX emerged in a totally new light, namely as a one-dimensional antiferromagnet and as the second example of an inorganic spin-Peierls compound (the first being CuGeO$_{3}$)[@seidel2003; @caimi2004]. The TiO bilayers constituting the TiOX lattice are candidates for various exotic electronic configurations, such as orbital ordered[@kataev2003], spin-Peierls[@seidel2003] and resonating-valence-bond states[@Beynon1993]. In the case of the TiOX family the degeneracy of the $d$ orbitals is completely removed by the crystal field splitting, so that the only $d-$electron present, mainly localized on the Ti site, occupies a nondegenerate energy orbital[@kataev2003]. As a consequence of the shape of the occupied orbital (which has lobes oriented in the $b-$ and $c-$directions, where $c$ is perpendicular to the layers), the exchange interaction between the spins on different Ti ions arises mainly from direct exchange within the TiO bilayers, along the $b$ crystallographic direction[@kataev2003]. This, in spite of the two-dimensional structural character, gives the magnetic system of the TiOX family its peculiar quasi one-dimensional properties[@seidel2003]. Magnetic susceptibility[@seidel2003] and ESR [@kataev2003] measurements at high temperature are in reasonably good agreement with an antiferromagnetic, one-dimensional spin-1/2 Heisenberg chain model. At low temperature ($T_{c1}$) TiOX shows a first-order phase transition to a dimerised nonmagnetic state, discussed in terms of a spin Peierls state [@seidel2003; @caimi2004-1; @shaz]. Between this low temperature spin Peierls phase (SP) and the one-dimensional antiferromagnet in the high temperature phase (HT), various experimental evidence [@hemberger; @ruck2005; @imai2003; @Lemmens2003] showed the existence of an intermediate phase, whose nature and origin is still debated. The temperature region of the intermediate phase is different for the two compounds considered in this work, for TiOBr $T_{c1}=28$ K and $T_{c2}=48$ K while for TiOCl $T_{c1}=67$ K and $T_{c2}=91$ K. To summarize the properties so far reported, the intermediate phase ($T_{c1}<T_{c2}$) exhibits a gapped magnetic excitation spectrum[@imai2003], anomalous broadening of the phonon modes in Raman and IR spectra[@Lemmens2003; @caimi2004-1], and features of a periodicity incommensurate with the lattice[@palatinus2005; @smaalen2005; @Schon2006; @krim2006]. Moreover, the presence of a pressure induced metal to insulator transition has been recently suggested for TiOCl[@kun2006]. Due to this complex phase behavior, both TiOCl and TiOBr have been extensively discussed in recent literature, and various questions still remain open: there is no agreement on the crystal symmetry of the spin Peierls phase, the nature and symmetry of the incommensurate phase is not clear and the anomalous first-order character of the transition to the spin Peierls state is not explained. Optical methods like Raman spectroscopy are powerful experimental tools for revealing the characteristic energy scales associated with the development of broken symmetry ground states, driven by magnetic and structural phase transitions. Indeed, information on the nature of the magnetic ground state, lattice distortion, and interplay of magnetic and lattice degrees of freedom can be obtained by studying in detail the magnetic excitations and the phonon spectrum as a function of temperature. The present paper reports on a vibrational Raman study of TiOCl and TiOBr, a study of the symmetry properties of the three phases and gives coherent view of the anomalous first order character of the transition to the spin Peierls phase. Through pressure-dependence measurements of the magnetic susceptibility, the role of magnon-phonon coupling in determining the complex phase diagram of TiOX is discussed. Finally, via a comparison with the isostructural compound VOCl, the previously reported[@Lemmens2003; @lemmens2005] high energy scattering is revisited, ruling out a possible interpretation in terms of magnon excitations. Experiment ========== Single crystals of TiOCl, TiOBr, and VOCl have been grown by a chemical vapor transport technique. The crystallinity was checked by X-ray diffraction[@ruck2005]. Typical crystal dimensions are a few mm$^{2}$ in the $ab-$plane and 10-100 $\mu$m along the $c-$axis, the stacking direction[@smaalen2005]. The sample was mounted in an optical flow cryostat, with a temperature stabilization better than 0.1 K in the range from 2.6 K to 300 K. The Raman measurements were performed using a triple grating micro-Raman spectrometer (Jobin Yvon, T64000), equipped with a liquid nitrogen cooled CCD detector (resolution 2 [cm$^{-1}$]{} for the considered frequency interval). The experiments were performed with a 532 nm Nd:YVO$_{4}$ laser. The power density on the sample was kept below 500 W/cm$^{2}$ to avoid sample degradation and to minimize heating effects. The polarization was controlled on both the incoming and outgoing beam, giving access to all the polarizations schemes allowed by the back-scattering configuration. Due to the macroscopic morphology of the samples (thin sheets with natural surfaces parallel to the $ab-$planes) the polarization analysis was performed mainly with the incoming beam parallel to the $c-$axis ($c$(aa)$\bar{c}$, $c$(ab)$\bar{c}$ and $c$(bb)$\bar{c}$, in Porto notation). Some measurements were performed with the incoming light polarized along the $c-$axis, where the $k-$vector of the light was parallel to the $ab-$plane and the polarization of the outgoing light was not controlled. These measurements will be labeled as $x$($c\star$)$\bar{x}$. The magnetization measurements were performed in a Quantum Design Magnetic Property Measurement System. The pressure cell used is specifically designed for measurement of the DC-magnetization in order to minimize the cell’s magnetic response. The cell was calibrated using the lead superconducting transition as a reference, and the cell’s signal (measured at atmospheric pressure) was subtracted from the data. Results and Discussion ====================== The discussion will start with a comparison of Raman experiments on TiOCl and TiOBr in the high temperature phase, showing the consistency with the reported structure. Afterwards, through the analysis of Raman spectra the crystal symmetry in the low temperature phases will be discussed, and in the final part a comparison with the isostructural VOCl will be helpful to shed some light on the origin of the anomalous high energy scattering reported for TiOCl and TiOBr[@Lemmens2003; @lemmens2005]. High Temperature Phase ---------------------- The crystal structure of TiOX in the high temperature (HT) phase consists of buckled Ti-O bilayers separated by layers of X ions. The HT structure is orthorhombic with space group P$mmn$. The full representation[@Rou1981] of the vibrational modes in this space group is: $$\Gamma_{tot}=3A_{g}+2B_{1u}+3B_{2g}+2B_{2u}+3B_{3g}+2B_{3u}.$$ Among these, the modes with symmetry $B_{1u}$, $B_{2u}$, and $B_{3u}$ are infrared active in the polarizations along the $c$, $b$, and $a$ crystallographic axes[@caimi2004-1], respectively. The modes with symmetry $A_{g}$, $B_{2g}$, and $B_{3g}$ are expected to be Raman active: The $A_{g}$ modes in the polarization ($aa$), ($bb$), and ($cc$); the $B_{2g}$ modes in ($ac$) and the $B_{3g}$ ones in ($bc$). ![(Color online) Polarized Raman spectra ($A_{g}$) of TiOCl and TiOBr in the high temperature phase, showing the three $A_{g}$ modes. Left panel: ($bb$) polarization; right panel: ($aa$) polarization.[]{data-label="fig1plus"}](f1.eps){width="70mm"} Fig.\[fig1plus\] shows the room temperature Raman measurements in different polarizations for TiOCl and TiOBr, and ![(Color online) Polarization analysis of the Raman spectra in the three phases of TiOBr, taken at 3 (a), 30 (b) and 100K (c). The spectra of TiOCl show the same main features and closely resemble those of TiOBr. Table \[Tab2\] reports the frequencies of the TiOCl modes. The inset shows the TiOBr spectrum in the $x$($c\ast$)$\bar{x}$ polarization (see text).[]{data-label="fig1"}](f2.eps){width="90mm"} Fig.\[fig1\] displays the characteristic Raman spectra for the three different phases of TiOBr, the spectra are taken at 100 (a), 30 (b) and 3K (c). At room temperature three Raman active modes are clearly observed in both compounds for the $c$($aa$)$\bar{c}$ and $c$($bb$)$\bar{c}$ polarizations (Fig.\[fig1plus\]), while none are observed in the $c$($ab$)$\bar{c}$ polarization. These results are in good agreement with the group theoretical analysis. The additional weakly active modes observed at 219 [cm$^{-1}$]{} for TiOCl and at 217 [cm$^{-1}$]{} for TiOBr are ascribed to a leak from a different polarization. This is confirmed by the measurements with the optical axis parallel to the $ab$-planes ($x$($c\star$)$\bar{x}$) on TiOBr, where an intense mode is observed at the same frequency (as shown in the inset of Fig.\[fig1\](a)). In addition to these expected modes, TiOCl displays a broad peak in the $c$($bb$)$\bar{c}$ polarization, centered at around 160 [cm$^{-1}$]{} at 300K; a similar feature is observed in TiOBr as a broad background in the low frequency region at 100K. As discussed for TiOCl[@Lemmens2003], these modes are thought to be due to pre-transitional fluctuations. Upon decreasing the temperature, this “peaked” background first softens, resulting in a broad mode at $T_{c2}$ (see Fig.\[fig1\](b)), and then locks at $T_{c1}$ into an intense sharp mode at 94.5 [cm$^{-1}$]{} for TiOBr (Fig.\[fig1\](c)) and at 131.5 [cm$^{-1}$]{} for TiOCl. ----------------------------------------------- ------- ------- ------- ------- ------- ------- (a) TiOBr TiOCl VOCl Exp. Cal. Exp. Cal. Exp. Cal. $A_{g}~(\sigma_{aa},\sigma_{bb},\sigma_{cc})$ 142.7 141 203 209.1 201 208.8 329.8 328.2 364.8 331.2 384.9 321.5 389.9 403.8 430.9 405.2 408.9 405.2 $B_{2g}(\sigma_{ac})$ 105.5 157.1 156.7 328.5 330.5 320.5 478.2 478.2 478.2 $B_{3u}(IR,a)$ 77 75.7 104 94.4 93.7 417 428.5 438 428.5 425.2 $B_{3g}(\sigma_{bc})$ *60* 86.4 129.4 129.4 *216* 336.8 *219* 336.8 327.2 *598* 586.3 586.3 585.6 $B_{2u}(IR,b)$ 131 129.1 176 160.8 159.5 275 271.8 294 272.1 269.8 $B_{1u}(IR,c)$ 155.7 194.1 192.4 304.8 301.1 303.5 ----------------------------------------------- ------- ------- ------- ------- ------- ------- : (a)Vibrational modes for the high temperature phase in TiOCl, TiOBr and VOCl. The calculated values are obtained with a spring model. The mode reported in $italics$ in Table \[Tab1\] are measured in the $x$($c\star$)$\bar{x}$ polarization they could therefore have either $B_{2g}$ or $B_{3g}$ symmetry (see experimental details). \[Tab1\] ----- ------ -------------- --------------------- ------- ------- ------- (b) Mode $\nu$(TiOBr) $\nu_{Cl}/\nu_{Br}$ Ti O Br 1 142.7 1.42 0.107 0.068 1 2 329.8 1.11 1 0.003 0.107 3 389.9 1.11 0.04 1 0.071 ----- ------ -------------- --------------------- ------- ------- ------- : The ratio between the frequency of the $A_{g}$ Raman active modes measured in TiOBr and TiOCl is related to the atomic displacements of the different modes as calculated for TiOBr (all the eigenvectors are fully $c-$polarized, the values are normalized to the largest displacement). \[Tab1b\] The frequency of all the vibrational modes observed for TiOCl and TiOBr in their high temperature phase are summarized in Table \[Tab1\]. Here, the infrared active modes are taken from the literature[@caimi2004-1; @caimi2004] and for the Raman modes the temperatures chosen for the two compounds are 300K for TiOCl and 100K for TiOBr. The observed Raman frequencies agree well with previous reports[@Lemmens2003]. The calculated values reported in Table \[Tab1\] are obtained with a spring-model calculation based on phenomenological longitudinal and transversal spring constants (see Appendix). The spring constants used were optimized using the TiOBr experimental frequencies (except for the ones of the $B_{3g}$ modes due to their uncertain symmetry) and kept constant for the other compounds. The frequencies for the other two compounds are obtained by merely changing the appropriate atomic masses and are in good agreement with the experimental values. The relative atomic displacements for each mode of $A_{g}$ symmetry are shown in Table \[Tab1b\]. The scaling ratio for the lowest frequency mode (mode 1) between the two compounds is in good agreement with the calculation of the atomic displacements. The low frequency mode is mostly related to Br/Cl movement and, indeed, the ratio $\nu_{TiOCl}/\nu_{TiOBr}=1.42$ is similar to the mass ratio $\sqrt{M_{Br}}/\sqrt{M_{Cl}}$. The other modes (2 and 3) involve mainly Ti or O displacements, and their frequencies scale with a lower ratio, as can be expected. Low Temperature Phases ---------------------- Although the symmetry of the low temperature phases has been studied by X-ray crystallography, there is no agreement concerning the symmetry of the SP phase; different works proposed two different space groups, P$2_{1}/m$[@Schon2006; @palatinus2005; @smaalen2005] and P$mm2$[@sasaki2006]. The possible symmetry changes that a dimerisation of Ti ions in the $b-$direction can cause are considered in order to track down the space group of the TiOX crystals in the low temperature phases. Assuming that the low temperature phases belong to a subgroup of the high temperature orthorhombic space group P$mmn$, there are different candidate space groups for the low temperature phases. Note that the assumption is certainly correct for the intermediate phase, because the transition at $T_{c2}$ is of second-order implying a symmetry reduction, while it is not necessarily correct for the low temperature phase, being the transition at $T_{c1}$ is of first-order. ![(Color online) Comparison of the possible low temperature symmetries. The low temperature structures reported are discussed, considering a dimerisation of the unit cell due to Ti-Ti coupling and assuming a reduction of the crystal symmetry. The red rectangle denotes the unit cell of the orthorhombic HT structure. Structure (a) is monoclinic with its unique axis parallel to the orthorhombic $c-$axis (space group P$2/c$), (b) shows the suggested monoclinic structure for the SP phase (P$2_{1}/m$), and (c) depicts the alternative orthorhombic symmetry proposed for the low T phase P$mm2$.[]{data-label="fig2"}](f3.eps){width="80mm"} ----- -------------------------------------------------------------------------- (a) Space group P$2/c$ Unique axis $\perp$ to TiO plane, $C_{2h}^{4}$ 4TiOBr per unit cell $\Gamma = 7A_{g} + 6A_{u} + 9B_{g} + 11B_{u}$ $7A_{g}$ Raman active $\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy}$ $11B_{g}$ Raman active $\sigma_{xz},\sigma_{yz}$ $6A_{u}$ and $9B_{u}$ IR active (b) Space group P$2_{1}/m$ Unique axis in the TiO plane, $C_{2h}^{2}$ 4 TiOBr per unit cell $\Gamma = 12A_{g} + 5A_{u} + 6B_{g} + 10B_{u}$ $12A_{g}$ Raman active $\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy}$ $6B_{g}$ Raman active $\sigma_{xz},\sigma_{yz}$ $5A_{u}$ and $10B_{u}$ IR active (c) Space group P$mm2$ 4 TiOBr per unit cell $\Gamma = 11A_{1} + A_{2} + 4B_{1} + 5B_{2}$ $11A_{1}$ Raman active $\sigma_{xx},\sigma_{yy},\sigma_{zz}$ $A_{2}$ Raman active $\sigma_{xy}$ $4B_{1}$ and $5B_{2}$ Raman active in $\sigma_{xz}$ and $\sigma_{yz}$ ----- -------------------------------------------------------------------------- : Comparison between the possible low temperature space group.[]{data-label="Tab3"} Fig.\[fig2\] shows a sketch of the three possible low temperature symmetries considered, and Table \[Tab3\] reports a summary of the characteristic of the unit cell together with the number of phonons expected to be active for the different space groups. Depending on the relative position of the neighboring dimerised Ti pairs, the symmetry elements lost in the dimerisation are different and the possible space groups in the SP phase are P$2/c$ (Table \[Tab3\](a)), P$2_{1}/m$ (b) or P$mm2$ (c). The first two are monoclinic groups with their unique axis perpendicular to the TiO plane (along the $c-$axis of the orthorhombic phase), and lying in the TiO plane ($\parallel$ to the $a-$axis of the orthorhombic phase), respectively. The third candidate (Fig.\[fig2\](c)) has orthorhombic symmetry. The group theory analysis based on the two space groups suggested for the SP phase (P$2_{1}/m$[@palatinus2005] and P$mm2$[@sasaki2006]) shows that the number of modes expected to be Raman active is different in the two cases (Table \[Tab3\](b) and (c)). In particular, the 12 fully symmetric vibrational modes ($A_{g}$), in the P$2_{1}/m$ space group, are expected to be active in the $\sigma_{xx},\sigma_{yy},\sigma_{zz}$ and $\sigma_{xy}$polarizations, and $6 B_{g}$ modes are expected to be active in the cross polarizations ($\sigma_{xz}$ and $\sigma_{yz}$). Note that in this notation, $z$ refers to the unique axis of the monoclinic cell, so $\sigma_{yz}$ corresponds to $c(ab)c$ for the HT orthorhombic phase. For P$mm2$ the 11 $A_{1}$ vibrational modes are expected to be active in the $\sigma_{xx},\sigma_{yy},\sigma_{zz}$polarizations, and only one mode of symmetry $A_{2}$ is expected to be active in the cross polarization ($\sigma_{xy}$ or $c(ab)c$). [llcccccc]{}\ (a) &TiOBr&$A_{g} (\sigma_{xx},\sigma_{yy})$ & 94.5 & 102.7 & 142.4 & 167 & 219\ & & & 276.5 & 330 & 351 & 392 & 411$^{\ast}$\ & & $A_{g} (\sigma_{xy})$ & 175,6 & 506.5\ &TiOCl & $A_{g} (\sigma_{xx},\sigma_{yy})$ & 131.5 & 145.8 & 203.5 & 211.5 & 296.5\ & & & 305.3 & 322.6 & 365.1 & 387.5 & 431$^{\ast}$\ & &$A_{g} (\sigma_{xy})$ & 178.5 & 524.3\ [llcccccc]{}\ (b) &TiOBr (30K)& $A_{g} (\sigma_{xx},\sigma_{yy})$ & 94.5 & 142 & 221.5 & 277 & 328.5\ & & & 344.5 & 390.4\ & TiOCl (75K)& $A_{g} (\sigma_{xx},\sigma_{yy})$ & 132.8 & 206.2 & 302 & 317.2 & 364.8\ && & 380 & 420.6\ The experiments, reported in Table \[Tab2\] for both compounds and in Fig.\[fig1\] for TiOBr only, show that 10 modes are active in the $c(aa)c$ and $c(bb)c$ in the SP phase (Fig.\[fig1\](c)), and, more importantly, two modes are active in the cross polarization $c(ab)c$. This is not compatible with the expectation for P$mm2$. Hence the comparison between the experiments and the group theoretical analysis clearly shows that of the two low temperature structures reported in X-ray crystallography[@smaalen2005; @sasaki2006], only the P$2_{1}/m$is compatible with the present results. ![(Color online) The temperature dependence of the Raman spectrum of TiOBr is depicted (an offset is added for clarity). The 3 modes present at all temperatures are denoted by the label $R_{T}$. The modes characteristic of the low temperature phase (disappearing at $T_{c1}=28$ K) are labelled $L_{T}$, and the anomalous modes observed in both the low temperature and the intermediate phase are labelled $I_{T}$. The right panel (b) shows the behavior of the frequency of $I_{T}$ modes, plotted renormalized to their frequency at 45 K. It is clear that the low-frequency modes shift to higher energy while the high-frequency modes shift to lower frequency.[]{data-label="fig3"}](f4.eps){width="90mm"} As discussed in the introduction, the presence of three phases in different temperature intervals for TiOX is now well established even though the nature of the intermediate phase is still largely debated[@ruck2005; @caimi2004; @smaalen2005]. The temperature dependence of the Raman active modes for TiOBr between 3 and 50 K, is depicted in Fig.\[fig3\]. In the spin-Peierls phase, as discussed above, the reduction of the crystal symmetry[@Schon2006] increases the number of Raman active modes. Increasing the temperature above $T_{c1}$ a different behavior for the various low temperature phonons is observed. As shown in Fig.\[fig3\], some of the modes disappear suddenly at $T_{c1}$(labeled $L_{T}$), some stay invariant up to the HT phase ($R_{T}$) and some others undergo a sudden broadening at $T_{c1}$ and slowly disappear upon approaching $T_{c2}$ ($I_{T}$). The polarization analysis of the Raman modes in the temperature region $T_{c1}<T<T_{c2}$ shows that the number of active modes in the intermediate phase is different from that in both the HT and the SP phases. The fact that at $T=T_{c1}$ some of the modes disappear suddenly while some others do not disappear, strongly suggests that the crystal symmetry in the intermediate phase is different from both other phases, and indeed confirms the first-order nature of the transition at $T_{c1}$. In the X-ray structure determination [@smaalen2005], the intermediate incommensurate phase is discussed in two ways. Firstly, starting from the HT orthorhombic (P$mmn$) and the SP monoclinic space group (P$2_{1}/m$ - unique axis in the TiO planes, $\parallel$ to $a$), the modulation vector required to explain the observed incommensurate peaks is two-dimensional for both space groups. Secondly, starting from another monoclinic space group, with unique axis perpendicular to the TiO bilayers (P$2/c$), the modulation vector required is one-dimensional. The latter average symmetry is considered (in the commensurate variety) in Fig.\[fig2\](a) and Table \[Tab3\](a). In the IP, seven modes are observed in the $\sigma_{xx},\sigma_{yy}$ and $\sigma_{zz}$ geometry on both compounds (see Table \[Tab2\](b)), and none in the $\sigma_{xy}$geometry. This appears to be compatible with all the space groups considered, and also with the monoclinic group with unique axis perpendicular to the TiO planes (Table \[Tab3\](a)). Even though from the evidence it is not possible to rule out any of the other symmetries discussed, the conjecture that in the intermediate incommensurate phase the average crystal symmetry is already reduced, supports the description of the intermediate phase as a monoclinic group with a one-dimensional modulation[@smaalen2005], and moreover it explains the anomalous first-order character of the spin-Peierls transition at $T_{c1}$. ![(Color online) The average crystal symmetry of the intermediate phase is proposed to be monoclinic with the unique axis parallel to the $c-$axis of the orthorhombic phase. Hence the low temperature space group is not a subgroup of the intermediate phase, and the transition to the spin-Peierls phase is consequently of first order.[]{data-label="fig4"}](f5.eps){width="80mm"} The diagram shown in Fig.\[fig4\] aims to visualize that the space group in the spin-Peierls state (P$2_{1}/m$) is a subgroup of the high temperature P$mmn$ group, but not a subgroup of any of the possible intermediate phase space groups suggested (possible P$2/c$). This requires the phase transition at $T_{c1}$ to be of first order, instead of having the conventional spin-Peierls second-order character. Let us return to Fig.\[fig3\](b) to discuss another intriguing vibrational feature of the intermediate phase. Among the modes characterizing the intermediate phase ($I_{T}$), the ones at low frequency shift to higher energy approaching $T_{c2}$, while the ones at high frequency move to lower energy, seemingly converging to a central frequency ($\simeq$300 [cm$^{-1}$]{} for both TiOCl and TiOBr). This seems to indicate an interaction of the phonons with some excitation around 300 [cm$^{-1}$]{}. Most likely this is in fact arising from a strong, thermally activated coupling of the lattice with the magnetic excitations, and is consistent with the pseudo-spin gap observed in NMR experiments[@imai2003; @bak2007] of $\approx$430 K ($\simeq$300 [cm$^{-1}$]{}). Magnetic Interactions --------------------- As discussed in the introduction, due to the shape of the singly occupied 3$d$ orbital, the main magnetic exchange interaction between the spins on the Ti ions is along the crystallographic $b-$direction. ![(Color online) (a) Magnetization as a function of temperature measured with fields 1 T and 5 T (the magnetization measured at 1 T is multiplied by a factor of 5 to evidence the linearity). The inset shows the main magnetic interactions (see text). (b) Pressure dependence of $T_{c1}$. The transition temperature for transition to the spin-Peierls phase increases with increasing pressure. The inset shows the magnetization versus the temperature after subtracting the background signal coming from the pressure cell.[]{data-label="fig5"}](f6.eps){width="90mm"} This, however, is not the only effective magnetic interaction. In fact, one also expects a superexchange interaction between nearest and next-nearest neighbor chains ($J_{2}$ and $J_{3}$ in the insert of Fig.\[fig5\](a))[@Roberto]. The situation of TiOX is made more interesting by the frustrated geometry of the interchain interaction, where the magnetic coupling $J_{2}$ between adjacent chains is frustrated and the exchange energies can not be simultaneously minimized. Table V reports the exchange interaction values for the three possible magnetic interactions calculated for TiOBr. These magnetic interactions were computed with a DFT Broken symmetry approach[@Nood79] using an atom cluster including the two interacting atoms and all the surrounding ligand atoms, in addition the first shell of Ti$^{3+}$ ions was replaced by Al$^{3+}$ ions and also included in the cluster. The calculations were performed with the Gaussian03 package[@Frish04] using the hybrid exchange-correlation functional B3LYP[@Beck93] and the 6-3111G\* basisset. ------------------ TiOBr $J_{1}=-250$ K $J_{2}=-46.99$ K $J_{3}=11.96$ K ------------------ : Calculated Exchange interactions in TiOBr[]{data-label="Tab5"} Although the computed value for the magnetic interaction along the $b-$axis is half of the value obtained from the magnetic susceptibility fitted with a Bonner-Fisher curve accounting for a one-dimensional Heisenberg chain, it is possible to extract some conclusions from the ab-initio computations. The most interesting outcome of the results is that in addition to the magnetic interaction along the $b-$axis, there is a relevant interchain interaction ($J_{1}/J_{2}= 5.3$) in TiOBr. Firstly, this explains the substantial deviation of the Bonner-Fisher fit from the magnetic susceptibility even at temperature higher than $T_{c2}$. Secondly, the presence of an interchain interaction, together with the inherent frustrated geometry of the bilayer structure, was already proposed in literature[@ruck2005] in order to explain the intermediate phase and its structural incommensurability. The two competing exchange interactions $J_{1}$ and $J_{2}$ have different origins: the first arises from direct exchange between Ti ions, while the second is mostly due to the superexchange interaction through the oxygen ions[@Roberto]. Thus, the two exchange constants are expected to depend differently on the structural changes induced by hydrostatic pressure, $J_{1}$ should increase with hydrostatic pressure (increases strongly with decreasing the distance between the Ti ions), while $J_{2}$ is presumably weakly affected due only to small changes in the Ti–O–Ti angle (the compressibility estimated from the lattice dynamics simulation is similar along the $a$ and $b$crystallographic directions). The stability of the fully dimerized state is reduced by the presence of an interchain coupling, so that $T_{c1}$ is expected to be correlated to $J_{1}/J_{2}$. Pressure dependent magnetic experiments have been performed to monitor the change of $T_{c1}$ upon increasing hydrostatic pressure. The main results, shown in Fig.\[fig5\], indeed is consistent with this expectation: $T_{c1}$ increases linearly with pressure; unfortunately it is not possible to address the behavior of $T_{c2}$ from the present measurements. Electronic Excitations and Comparison with VOCl ----------------------------------------------- The nature of the complex phase diagram of TiOX was originally tentatively ascribed to the interplay of spin, lattice and orbital degrees of freedom[@caimi2004]. Only recently, infrared spectroscopy supported by cluster calculations excluded a ground state degeneracy of the Ti $d$ orbitals for TiOCl, hence suggesting that orbital fluctuations can not play an important role in the formation of the anomalous incommensurate phase[@ruck2005long; @Zach2006]. Since the agreement between the previous cluster calculations and the experimental results is not quantitative, the energy of the lowest $3d$ excited level is not accurately known, not allowing to discard the possibility of an almost degenerate ground state. For this reason a more formal cluster calculation has been performed using an embedded cluster approach. In this approach a TiO2Cl4 cluster was treated explicitly with a CASSCF/CASPT2 quantum chemistry calculation. This cluster was surrounded by eight Ti$^{3+}$ TIP potentials in order to account for the electrostatic interaction of the cluster atoms with the shell of the first neighboring atoms. Finally, the cluster is embedded in a distribution of punctual charges fitting the Madelung’s potential produced by the rest of the crystal inside the cluster region. The calculations were performed using the MOLCAS quantum chemistry package[@Karl2003] with a triple quality basis set; for the Ti atom polarization functions were also included. TiOCl TiOBr --------------- ----------- ----------- $xy$ 0.29-0.29 0.29-0.30 $xz$ 0.66-0.68 0.65-0.67 $yz$ 1.59-1.68 1.48-1.43 $x^{2}-r^{2}$ 2.30-2.37 2.21-2.29 : Crystal field splitting of 3$d^{1}$ Ti$^{3+}$ in TiOCl and TiOBr (eV).[]{data-label="Tab4"} The calculations reported in Table \[Tab4\], confirmed the previously reported result[@ruck2005long] for both TiOCl and TiOBr. The first excited state $d_{xy}$ is at 0.29-0.3 eV ($>3000$ K) for both compounds, therefore the orbital degrees of freedom are completely quenched at temperatures close to the phase transition. A comparison with the isostructural compound VOCl has been carried out to confirm that the phase transitions of the TiOX compounds are intimately related to the unpaired S=1/2 spin of the Ti ions. The V$^{3+}$ ions have a 3$d^{2}$ electronic configuration. Each ion carries two unpaired electrons in the external d shell, and has a total spin of 1. The crystal field environment of V$^{3+}$ ions in VOCl is similar to that of Ti$^{3+}$ in TiOX, suggesting that the splitting of the degenerate d orbital could be comparable. The electrons occupy the two lowest $t_{2g}$ orbitals, of $d_{y^2-z^2}$ (responsible for the main exchange interaction in TiOX) and $d_{xy}$ symmetry respectively. Where the lobes of the latter point roughly towards the Ti$^{3+}$ ions of the nearest chain (Table \[Tab4\]). It is therefore reasonable to expect that the occupation of the $d_{xy}$ orbital in VOCl leads to a substantial direct exchange interaction between ions in different chains in VOCl and thus favors a two-dimensional antiferromagnetic order. Indeed, the magnetic susceptibility is isotropic at high temperatures and well described by a quadratic two-dimensional Heisenberg model, and at $T_{N}=80$ K VOCl undergoes a phase transition to a two-dimensional antiferromagnet[@wied84]. ![(Color online) Raman scattering features of VOCl. (a) High energy scattering of TiOCl/Br and VOCl, and (b) temperature dependence of the vibrational scattering features of VOCl. No symmetry changes are observed at $T_{N}=80$ K.[]{data-label="fig6"}](f7.eps){width="60mm"} The space group of VOCl at room temperature is the same as that of TiOX in the high temperature phase (P$mmn$), and, as discussed in the previous section, three $A_{g}$ modes are expected to be Raman active. As shown in Fig.\[fig6\](b), three phonons are observed throughout the full temperature range ($3-300$ K), and no changes are observed at $T_{N}$. The modes observed are consistent with the prediction of lattice dynamics calculations (Table \[Tab1\]). In the energy region from 600 to 1500 [cm$^{-1}$]{}, both TiOBr and TiOCl show a similar highly structured broad scattering continuum, as already reported in literature[@Lemmens2003; @lemmens2005]. The fact that the energy range of the anomalous feature is consistent with the magnetic exchange constant in TiOCl (J=660 K) suggested at first an interpretation in terms of two-magnon Raman scattering [@Lemmens2003]. Later it was shown that the exchange constant estimated for TiOBr is considerably smaller (J=406 K) with respect to that of TiOCl while the high energy scattering stays roughly at the same frequency. Even though the authors of ref.[@lemmens2005] still assigned the scattering continuum to magnon processes, it seems clear taht the considerably smaller exchange interaction in the Br compound (J=406 K) falsifies this interpretation and that magnon scattering is not at the origin of the high energy scattering of the two compounds. Furthermore, the cluster calculation (Table \[Tab4\]) clearly shows that no excited crystal field state is present in the energy interval considered, ruling out a possible orbital origin for the continuum. These observations are further strengthened by the observation of a similar continuum scattering in VOCl (see fig. \[fig6\](a)) which has a different magnetic and electronic nature. Therefore, the high energy scattering has most likely a vibrational origin. The lattice dynamics calculations, confirmed by the experiments, show that a “high” energy mode ($\simeq$600 [cm$^{-1}$]{}) of symmetry $B_{3g}$ (Table \[Tab1\]) is expected to be Raman active in the $\sigma_{yz}$polarization. Looking back at Fig.\[fig1\], the inset shows the measurements performed with the optical axis parallel to the TiOX plane, where the expected mode is observed at 598 [cm$^{-1}$]{}. The two phonon process related to this last intense mode is in the energy range of the anomalous scattering feature and has symmetry $A_{g}$($B_{3g} \otimes B_{3g}$). The nature of the anomalies observed is therefore tentatively ascribed to a multiple-phonon process. Further detailed investigations of lattice dynamics are needed to clarify this issue. Conclusion ========== The symmetry of the different phases has been discussed on the basis of inelastic light scattering experiments. The high temperature Raman experiments are in good agreement with the prediction of the group theoretical analysis (apart from one broad mode which is ascribed to pre-transitional fluctuations). Comparing group theoretical analysis with the polarized Raman spectra clarifies the symmetry of the spin-Peierls phase and shows that the average symmetry of the incommensurate phase is different from both the high temperature and the SP phases. The conjecture that the intermediate phase is compatible with a different monoclinic symmetry (unique axis perpendicular to the TiO planes) could explain the anomalous first-order character of the transition to the spin-Peierls phase. Moreover, an anomalous behavior of the phonons characterizing the intermediate phase is interpreted as evidencing an important spin-lattice coupling. The susceptibility measurements of TiOBr show that $T_{c1}$ increases with pressure, which is ascribed to the different pressure dependence of intrachain and interchain interactions. Finally, we compared the TiOX compounds with the “isostructural” VOCl. The presence of the same anomalous high energy scattering feature in all the compounds suggests that this feature has a vibrational origin rather than a magnetic or electronic one. [*Acknowledgements*]{}  The authors are grateful to Maxim Mostovoy, Michiel van der Vegte, Paul de Boeij, Daniel Khomskii, Iberio Moreira and Markus Grüninger for valuable and insightful discussions. This work was partially supported by the Stichting voor Fundamenteel Onderzoek der Materie \[FOM, financially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)\], and by the German Science Foundation (DFG). Appendix: Details of the spring model calculation ================================================= The spring model calculation reported in the paper, was carried out using the software for lattice-dynamical calculation UNISOFT[@esc92] (release 3.05). In the calculations the Born-von Karman model was used; here the force constants are treated as model parameters and they are not interpreted in terms of a special interatomic potential. Only short range interactions between nearest neighbor ions are taken into account. Considering the forces to be central forces, the number of parameters is reduced to two for each atomic interaction: the longitudinal and transversal forces respectively defined as $L=\frac{d^2V(\bar{r}_{i,j})}{dr^2}$ and $T=\frac{1}{r}\frac{dV(\bar{r}_{i,j})}{dr}$. A custom made program was interfaced with UNISOFT to optimize the elastic constants. Our program proceeded scanning the $n$ dimensional space ($n$ = number of parameters) with a discrete grid, to minimize the squared difference between the calculated phonon frequencies and the measured experimental frequencies for TiOBr, taken from both Raman and infrared spectroscopy. The phonon frequencies of TiOCl and VOCl were obtained using the elastic constants optimized for TiOBr and substituting the appropriate ionic masses. The optimized force constants between different atoms are reported in $N/m$ in the following Table. -------- ------------- -------------------------- ------------------------- Number Ions Longitudinal (L) ($N/m$) Transversal (T) ($N/m$) 1 Ti(a)-Ti(b) 18.5 32.7 2 Ti(a)-O(a) 18.5 11.1 3 Ti(a)-O(b) 53.1 9.5 4 Ti(a)-X(a) 29.0 4.4 5 O(a)-O(b) 20.6 7.3 6 X(a)-O(a) 18.5 3.5 7 X(a)-X(b) 11.7 0.7 -------- ------------- -------------------------- ------------------------- : Elastic constants used in the spring model calculation. The label numbers refer to Fig. \[add1\], while the letters refer to the different inequivalent positions of the ions in the crystal. \[Tadd1\]

--- abstract: 'Quantum mechanics is not the unique no-signaling theory which is endowed with stronger-than-classical correlations, and there exists a broad class of no-signaling theories allowing even stronger-than-quantum correlations. The principle of information causality has been suggested to distinguish quantum theory from these nonphysical theories, together with an elegant information-theoretic proof of the quantum bound of two-particle correlations. In this work, we extend this to genuine $N$-particle correlations that cannot be reduced to mixtures of states in which a smaller number of particles are entangled. We first express Svetlichny’s inequality in terms of multipartite no-signaling boxes, then prove that the strongest genuine multipartite correlations lead to the maximal violation of information causality. The maximal genuine multipartite correlations under the constraint of information causality is found to be equal to the quantum mechanical bound. This result consolidates information causality as a physical principle defining the possible correlations allowed by nature, and provides intriguing insights into the limits of genuine multipartite correlations in quantum theory.' author: - Yang Xiang - Wei Ren title: Bound on genuine multipartite correlations from the principle of information causality --- [*Introduction*]{}  The violation of Bell inequalities [@bell; @chsh] proves that the quantum mechanics cannot be regarded as a local realistic theory. Tsirelson [@tsir] proved an upper bound on the violation of the CHSH inequality [@chsh], which means that the amount of non-locality allowed by quantum mechanics is limited. One may think that Tsirelson bound is a consequence of relativity, but Popescu and Rohrlich [@pr] showed that there exists a broad class of no-signaling theories which allow even stronger-than-quantum correlations. An example of the no-signaling theories is the Popescu and Rohrlich boxes (PR-boxes) [@pr]. This broad class of no-signaling theories possessing extremely powerful correlations are usually called post-quantum theories and modeled as no-signaling boxes (NS-boxes) [@barrett1]. These post-quantum theories have much in common with quantum mechanics, such as no-cloning [@noclon], no-broadcasting [@nobroad], monogamy of correlations [@noclon], information-disturbance trade-offs [@trade], and the security for key distribution [@key], so there is a need to find some principles at the very root of quantum theory and distinguish it from these post-quantum theories. In recent years, an intensive study has been made on this issue. In Ref. [@dam], van Dam showed that the availability of PR-boxes makes communication complexity trivial. However, communication complexity is not trivial in quantum physics and it is strongly believed that devices producing such correlations making communication complexity trivial are very unlikely to exist. Later, Brassard *et al.* proved that some post-quantum theories would lead to an implausible simplification of distributed computational tasks [@brassard1; @nonlocal; @comput; @brun1]. More recently, Barnum *et al.* [@barnum; @acin] showed that the combination of local quantum measurement assumption and relativity results in quantum correlations, and in Ref. [@acin] the authors provided a unified framework for all no-signaling theories. From an information theoretic point of view, Paw[ł]{}owski *et al.* [@pawlowski1] suggested a bold physical principle: information causality (IC), stating that communication of $n$ classical bits causes information gain of at most $n$ bits. When $n=0$ IC is just the no-signaling principle. In a bipartite scenario where each party has two inputs and two outputs, Paw[ł]{}owski *et al.* showed that IC is respected both in classical and quantum physics, but all correlations stronger than the strongest quantum correlations (Tsirelson bound) violate it, and they derived Tsirelson bound from IC. It must be noted that there are some stronger-than-quantum correlations which are not known to violate IC [@allcock; @cav]. The present work is to extend the research of understanding the quantum mechanical bound on nonlocal correlations to genuine multipartite correlations. The structure of multipartite correlations is much richer than that of bipartite correlations [@barrett1]. For example, in [@pironio] the authors dealt with a tripartite scenario where each party has two inputs and two outputs, they found that there exist $53856$ extremal no-signaling tripartite correlations which belong to $46$ inequivalent classes, and there are more than three classes which feature genuine tripartite nonlocality. So there exist many inequivalent types of genuine multipartite correlations, and in the present paper we deal only with Svetlichny genuine multipartite correlations which is relevant to Svetlichny’s inequality (SI) [@svet; @seev]. We first express SI in terms of multipartite no-signaling boxes, and then prove that the strongest Svetlichny genuine multipartite correlation leads to the maximal violation of IC. Under the constraint of IC, the maximal Svetlichny genuine multipartite correlation just equals to the quantum mechanical bound. [*Tripartite Svetlichny’s inequality*]{}  We first introduce SI of three-particle [@svet], which can distinguish between genuine three-particle correlations and two-particle correlations. A violation of SI implies the presence of genuine three-particle correlations. Consider three observers, Alice, Bob, and Carol, who share three entangled qubits. Each of the three observers can choose to measure one of two dichotomous observables. We denote $x\in\{0,1\}$ and $A\in\{-1,1\}$ as Alice’s measurement choice and outcome respectively, and similarly $y$ and $B$ ($z$ and $C$) for Bob’s (Carol’s). Thus SI can be expressed as [@svet] $$\begin{aligned} S&\equiv&|E(ABC|x=0,y=0,z=0)\nonumber\\ &&+E(ABC|x=0,y=0,z=1)\nonumber\\ &&+E(ABC|x=0,y=1,z=0)\nonumber\\ &&+E(ABC|x=1,y=0,z=0)\nonumber\\ &&-E(ABC|x=0,y=1,z=1)\nonumber\\ &&-E(ABC|x=1,y=0,z=1)\nonumber\\ &&-E(ABC|x=1,y=1,z=0)\nonumber\\ &&-E(ABC|x=1,y=1,z=1)|\leq 4,\end{aligned}$$ where $E(ABC|x,y,z)$’s represent the expectation value of the product of the measurement outcomes of the observables $x$, $y$, and $z$, and we call $S$ as Svetlichny operator. It was shown by Svetlichny [@svet] that quantum predictions violate his inequality, and the maximum violation ($S=4\sqrt{2}$) allowed in quantum mechanics can be achieved with GHZ states [@mit]. If we define $a=\frac{1-A}{2}$, $b=\frac{1-B}{2}$, and $c=\frac{1-C}{2}$, each of $E(ABC|x,y,z)$’s can be expressed in terms of probabilities, for example, $$\begin{aligned} &&E(ABC|x=0,y=0,z=0)\nonumber\\ &=&2P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x=0,y=0,z=0)-1,\nonumber \\ &&E(ABC|x=0,y=1,z=1)\nonumber\\ &=&1-2P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x=0,y=1,z=1),\nonumber \\\end{aligned}$$ where $P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x=0,y=0,z=0)$ is the probability that $a\oplus b\oplus c=xy\oplus yz\oplus xz$ under the condition $x=0,y=0,z=0$ , and $\oplus$ denotes the addition modulo $2$. So we can also write the SI as $$\begin{aligned} \frac{1}{8}\sum_{x,y,z}P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)\leq \frac{3}{4}. \label{si}\end{aligned}$$ From the above inequality we find that there is a convenient way of thinking about genuine three-particle correlations by three black boxes shared by Alice, Bob, and Carol. The correlations between inputs $x$, $y$, $z$ and outcomes $a$, $b$, $c$ are described by probability $P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)$, and we call these boxes Svetlichny boxes [@barrett1]. The maximal algebraic value $S=8$ is reached if and only if $P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)=1$ for any $x$, $y$, and $z$. It is obvious that Svetlichny boxes belong to tripartite NS-boxes, since Svetlichny boxes still satisfy the principle of no-signaling due to uniformly random local outcomes. [*Svetlichny boxes lead to violation of IC*]{}  Before elucidating that Svetlichny boxes can maximally violate IC, we first give a brief overview of IC. Suppose there are two persons, Alice and Bob, Alice has $N$ random and independent bits $(a_{1},a_{2},...,a_{N})$, and Bob receives a random variable $l\in\{1,2,...,N\}$. Alice can send $n$ classic bits to Bob, and Bob’s task is to guess the value of the $l$-th bit in Alice’s list with the help of the $n$ bits. The amount of the information about Alice’s list gained by Bob is measured by $$\begin{aligned} I\equiv \sum_{k=1}^{N}I(a_{k}:g|l=k)\geq N-\sum_{k=1}^{N}h(p_{k}), \label{i}\end{aligned}$$ where $I(a_{k}:g|l=k)$ is Shannon mutual information between $a_{k}$ and $g$ ($g$ is Bob’s guess), and $p_{k}$ is the probability that $a_{k}=g$, both computed in the case of that Bob has received $l=k$. In Eq. (\[i\]), the inequality can be proved by Fano inequality [@cover]. IC states that physically allowed theories must have $$\begin{aligned} I\leq n.\end{aligned}$$ Now we consider that there exist Svetlichny boxes shared by Alice, Bob, and Carol (see Fig.(\[fig1\])). Alice and Bob sit next to each other, at a long distance from Carol, and Alice(Bob) can send one bit to Carol. Carol’s mission is to guess the value of $x$ (Alice’s input) when she receives $z=0$ and guess the value of $y$ (Bob’s input) when she receives $z=1$. The message which sent by Alice (Bob) to Carol is $m=a\oplus b\oplus xy\oplus x$. Upon receiving the message $m$, Carol can compute her guess $g=c\oplus m=a\oplus b\oplus c\oplus xy\oplus x$. The probabilities of correct guess of $x$ and $y$ are $$\begin{aligned} p_{x}&=&\frac{1}{4}[P(a\oplus b\oplus c=0|0,0,0)+P(a\oplus b\oplus c=0|0,1,0)\nonumber\\ &&+P(a\oplus b\oplus c=0|1,0,0)+P(a\oplus b\oplus c=1|1,1,0)]\nonumber \\ \label{px}\end{aligned}$$ $$\begin{aligned} p_{y}&=&\frac{1}{4}[P(a\oplus b\oplus c=0|0,0,1)+P(a\oplus b\oplus c=1|0,1,1)\nonumber\\ &&+P(a\oplus b\oplus c=1|1,0,1)+P(a\oplus b\oplus c=1|1,1,1)]\nonumber \\ \label{py}\end{aligned}$$ The Svetlichny boxes of $P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)=1$ predict $p_{x}=p_{y}=1$, from Eq. (\[i\]) we have $I=2$ for $n=1$, so the Svetlichny boxes can maximally violate IC. [*The bound on genuine three-particle correlations*]{}  Now we proceed to show that stronger-than-quantum genuine three-particle correlations lead to the violation of IC. Since it is known that the maximal violation is obtained by the GHZ state [@seev] and in this case all probabilities $P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)$ are the same, it is a natural choice to consider the isotropic boxes and indeed this choice successfully leads to the quantum bound. The isotropic Svetlichny boxes can be written as $$\begin{aligned} P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)=\frac{1+E}{2}, \label{tnsb}\end{aligned}$$ where $0\leq E\leq1$. The Svetlichny boxes of Eq. (\[tnsb\]) has strongest genuine tripartite correlations when $E=1$, and it correspond to uncorrelated random bits when $E=0$. SI of Eq. (\[si\]) is violated as soon as $E>\frac{1}{2}$, and the quantum bound $S=4\sqrt{2}$ corresponds to $E=\frac{\sqrt{2}}{2}$. In Fig(\[fig2\]), we illustrate how to transform Svetlichny boxes to bipartite NS-boxes. If the initial Svetlichny boxes are described by probability $P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)=\frac{1+E}{2}$, the transformed bipartite NS-boxes can be described by probability $P(A\oplus c=(x\oplus y)z|x,y,z)=\frac{1+E}{2}$. So any bipartite NS-boxes of $P(a\oplus b=xy|x,y)=\frac{1+E}{2}$ can be simulated by Svetlichny boxes of $P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)=\frac{1+E}{2}$. In Ref. [@pawlowski1], the authors proved that the bipartite NS-boxes of $P(a\oplus b=xy|x,y)=\frac{1+E}{2}$ would lead to the violation of IC as soon as $E>\frac{\sqrt{2}}{2}$, thus we can conclude that the Svetlichny boxes of $P(a\oplus b\oplus c=xy\oplus yz\oplus xz|x,y,z)=\frac{1+E}{2}$ lead to the violation of IC as soon as $E>\frac{\sqrt{2}}{2}$. So we have proven that the maximal genuine three-particle correlation under the constraint of IC just corresponds to the quantum bound of violation of SI. [*The bound on genuine multipartite correlations*]{}  In Ref. [@seev], the SI of three-particle has been generalized to the case of $N$ particles. Here, by using the derivation method of Eq. (\[si\]) we express these $N$-particle SI in terms of probability, then genuine $N$-particle correlations can be modeled as $N$-particle no-signaling boxes (NNS-boxes). Suppose there are $N$ players who shared $N$ particles, each one of them performs dichotomous measurements on each of the $N$ particles. The measurement settings are represented by $x_{1}$, $x_{2}$,...$x_{N}$ respectively, with possible values $0,1$. The measurement results are represented by $a_{1}$, $a_{2}$,...$a_{N}$ respectively, and also with possible values $0,1$. Then the $N$-particle SI can be written as (proof in the Appendix) $$\begin{aligned} \frac{1}{2^{N}}\sum_{\{x_{i}\}}P\Big(\sum_{i}^{N}a_{i}=\sum_{i<j\leq N}x_{i}x_{j}|x_{1},x_{2},...,x_{N}\Big)\leq \frac{3}{4}, \label{nsi}\end{aligned}$$ where $\{x_{i}\}$ stands for an $N$-tuple $x_{1},...,x_{N}$, $\sum_{i}^{N}$ and $\sum_{i<j\leq N}$ both denote summation modula $2$, and $P$ is the probability that $\sum_{i}^{N}a_{i}=\sum_{i<j\leq N}x_{i}x_{j}$ with given $x_{1},x_{2},...,x_{N}$. The isotropic NNS-boxes can be written as a simple form: $$\begin{aligned} P\Big(\sum_{i}^{N}a_{i}=\sum_{i<j\leq N}x_{i}x_{j}|x_{1},x_{2},...,x_{N}\Big)=\frac{1+E}{2}, \label{nnsb}\end{aligned}$$ where $0\leq E\leq1$. SI of Eq. (\[nnsb\]) is violated as soon as $E>\frac{1}{2}$. The quantum bound of genuine $N$-particle correlations corresponds to $E=\frac{\sqrt{2}}{2}$, and it can be achieved with $N$-particle GHZ states [@seev]. In Fig.(\[fig3\]), we illustrate the transformation of NNS-boxes to bipartite NS-boxes. If the initial NNS-boxes is described by probability $P\Big(\sum_{i}^{N}a_{i}=\sum_{i<j\leq N}x_{i}x_{j}|x_{1},x_{2},...,x_{N}\Big)=\frac{1+E}{2}$, the transformed bipartite NS-boxes can be described by probability $P(A\oplus a_{N}=(x_{1}\oplus x_{2}\oplus ...\oplus x_{N-1})x_{N})=\frac{1+E}{2}$. So any bipartite NS-boxes of $P(a\oplus b=xy|x,y)=\frac{1+E}{2}$ can be simulated by NNS-boxes of $P\Big(\sum_{i}^{N}a_{i}=\sum_{i<j\leq N}x_{i}x_{j}|x_{1},x_{2},...,x_{N}\Big)=\frac{1+E}{2}$. This implies that the NNS-boxes of $P\Big(\sum_{i}^{N}a_{i}=\sum_{i<j\leq N}x_{i}x_{j}|x_{1},x_{2},...,x_{N}\Big)=\frac{1+E}{2}$ would lead to the violation of IC as soon as $E>\frac{\sqrt{2}}{2}$. So we have proven that the maximal genuine $N$-particle correlations under the constraint of IC just corresponds to the quantum bound of violation of SI of $N$-particle. [*Discussion*]{}   In this work we give an information-theoretical proof about the quantum bound of violations of SI, i.e. the maximal violations of SI just equal to the quantum bound due to the constraint of IC. We first employ a genuine multipartite correlation resource to simulate a bipartite correlation, and then make use of the previously known bipartite results [@pawlowski1]. We note that, while there exist many different protocols to simulate a bipartite correlation by using a genuine $N$-partite correlation, all the simulations will result in the same conclusion: if there exists a stronger-than-quantum genuine $N$-partite correlation then we can use it to simulate a bipartite correlation which can breach IC. With regard to different simulation protocols, for example, we can combine left $k$ boxes to form a new box and the remaining $N-k$ boxes are combined to form the other new box. If the initial $N$-partite no-signaling boxes is described by probability $P\Big(\sum_{i}^{N}a_{i}=\sum_{i<j\leq N}x_{i}x_{j}|x_{1},x_{2},...,x_{N}\Big)=\frac{1+E}{2}$, then the transformed bipartite no-signaling boxes is described by probability $P(A\oplus B=(x_{1}\oplus x_{2}\oplus ...\oplus x_{k})(x_{k+1}\oplus ...\oplus x_{N}))=\frac{1+E}{2}$, where $A=\sum_{i=1}^{k}{a_{i}}\oplus\sum_{1\leq i<j\leq k}{x_{i}x_{j}}$ and $B=\sum_{i=k+1}^{N}{a_{i}}\oplus\sum_{k+1\leq i<j\leq N}{x_{i}x_{j}}$. This simulation is different from the previous simulation but would lead to the same conclusion. The genuine multipartite correlations are essentially more powerful correlation resources than bipartite correlations. One Svetlichny box can simulate a PR-box, but we must use three PR-boxes to simulate a Svetlichny box [@barrett1]. So the bound of genuine multipartite correlations what the IC tells us is a genuine new and exciting result, which bears some fundamental differences from the known bipartite results. [*Appendix*]{}   Proof of inequality (9) Suppose there are $N$ players who shared $N$ particles, each one of them performs dichotomous measurements on each of the $N$ particles. The measurement settings are represented by $x_{1}$, $x_{2}$,...$x_{N}$, respectively, with possible values $0,1$. The measurement results are represented by $A_{1}$, $A_{2}$,...$A_{N}$, respectively, and with possible values $-1,1$. Then the original $N$-particle SI [@seev] can be expressed as $$\begin{aligned} S_{N}&\equiv& |\sum_{\{x_{i}\}} {v(x_{1},x_{2},...,x_{N})E(A_{1}A_{2}\cdot\cdot\cdot A_{N}|x_{1},x_{2},...,x_{N})}|\nonumber\\ &\leq& 2^{N-1}, \label{sn1}\end{aligned}$$ where $\{x_{i}\}$ stands for an $N$-tuple $x_{1},...,x_{N}$, $E(A_{1}A_{2}\cdot\cdot\cdot A_{N}|x_{1},x_{2},...,x_{N})$ represents the expectation value of the product of the measurement outcomes of the observables $x_{1},x_{2},...,x_{N}$, and $v(x_{1},x_{2},...,x_{N})$ is a sign function given by $$\begin{aligned} v(x_{1},x_{2},...,x_{N})=(-1)^{[\frac{k(k-1)}{2}]},\end{aligned}$$ where $k$ is the number of times index $1$ appears in $(x_{1},x_{2},...,x_{N})$. We can easily find that $$\begin{aligned} v(x_{1},x_{2},...,x_{N})=(-1)^{\sum_{i<j\leq N}x_{i}x_{j}},\end{aligned}$$ where $\sum_{i<j\leq N}$ denotes summation modula $2$. If we define $a_{i}=\frac{1-A_{i}}{2}$, then $$\begin{aligned} &&E(A_{1}A_{2}\cdot\cdot\cdot A_{N}|x_{1},x_{2},...,x_{N})=(-1)^{\sum_{i<j\leq N}x_{i}x_{j}}\nonumber\\ &\cdot&\Big[2P\Big(\sum_{i}^{N}a_{i}=\sum_{i<j\leq N}x_{i}x_{j}|x_{1},x_{2},...,x_{N}\Big)-1\Big], \label{sn2}\end{aligned}$$ where $\sum_{i}^{N}$ denotes summation modula $2$. From Eq. (\[sn1\]) and Eq. (\[sn2\]) we finally obtain inequality (9) in the main text. [*Acknowledgments*]{}  This work is supported by National Foundation of Natural Science in China under Grant Nos. 10947142 and 11005031. [100]{} J. S. Bell, Physics (Long Island City, N.Y.) [**1**]{}, 195(1964); J. S. Bell, *Speakable and Unspeakable in Quantum Mechanics* (Cambridge University Press, Cambridge, England, 1988). J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. [**23**]{}, 880(1969); [**24**]{}, 549(E)(1970). B. S. Cirel’son, Lett. Math. Phys. [**4**]{}, 93(1980). S. Popescu and D. Rohrlich, Found. Phys. 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--- abstract: 'In light of the recent LHC Higgs search data, we investigate the pair production of a SM-like Higgs boson around 125 GeV in the MSSM and NMSSM. We first scan the parameter space of each model by considering various experimental constraints, and then calculate the Higgs pair production rate in the allowed parameter space. We find that in most cases the dominant contribution to the Higgs pair production comes from the gluon fusion process and the production rate can be greatly enhanced, maximally 10 times larger than the SM prediction (even for a TeV-scale stop the production rate can still be enhanced by a factor of 1.3). We also calculate the $\chi^2$ value with the current Higgs data and find that in the most favored parameter region the production rate is enhanced by a factor of 1.45 in the MSSM, while in the NMSSM the production rate can be enhanced or suppressed ($\sigma_{SUSY}/\sigma_{SM}$ varies from 0.7 to 2.4).' author: - 'Junjie Cao$^{1,2}$, Zhaoxia Heng$^1$, Liangliang Shang$^1$, Peihua Wan$^1$, Jin Min Yang$^3$' title: Pair Production of a 125 GeV Higgs Boson in MSSM and NMSSM at the LHC --- Introduction ============ Based on the combined data collected at the center-of-mass energies of 7 TeV and 8 TeV, the experimental programme to probe the mechanism of electroweak symmetry breaking at the LHC has recently witnessed the discovery of a new particle around 125 GeV [@1207ATLAS-CMS]. The properties of this particle, according to the updated analyses of the ATLAS and CMS collaborations at the end of 2012 [@1212ATLAS-CMS], roughly agree with the the Standard Model (SM) prediction and thus it should play a role in both the symmetry breaking and the mass generation. However, the issue of whether this particle is the SM Higgs boson is still open, and indeed there are some motivations, such as the gauge hierarchy problem and the excess in the di-photon channel over the SM prediction [@1207ATLAS-CMS; @1212ATLAS-CMS], to consider new physics interpretation of this boson. Studies in this direction have been performed intensively in low energy supersymmetry (SUSY) and it was found that some SUSY models can naturally provide a 125 GeV Higgs boson [@Feb-Cao; @Carena-Higgsmass; @1213-125GeV-Higgs], and fit the data better than the SM [@July-Cao] (similar studies have also been performed in some non-SUSY models like the little Higgs models and two-Higgs-doublet or Higgs-triplet models [@125-other]). After the discovery of the Higgs boson, the next important task for the LHC is to test the property of this Higgs boson by measuring all the possible production and decay channels with high luminosity. Among the production channels, the Higgs pair production is a rare process at the LHC. Since it can play an important role for testing the Higgs self-couplings  [@SM-NLO-35fb; @1213-DHiggs] (the determination of the Higgs self-couplings is of great importance since it is indispensable to reconstruct the Higgs potential), it will be measured at the LHC with high luminosity. In the SM the Higgs pair production at the LHC proceeds by the parton process $gg \to h h$ through the heavy quark induced box diagrams and also through the production of an off-shell Higgs which subsequently splits into two on-shell Higgs bosons [@SM-LO-20fb; @DHiggsInSM]. The production rate is rather low for $\sqrt{s} = 14 {\rm TeV}$, about 20 fb at leading order [@SM-LO-20fb] and reaching roughly 35 fb after including the next-to-leading order QCD correction [@SM-NLO-35fb]. The capability of the LHC to detect this production process was investigated in [@bbgaga; @bbWW; @bbtautau; @DHiggs-Detect]. These analyses showed that for a 125 GeV Higgs boson the most efficient channel is $g g \to h h \to b \bar{b} \gamma \gamma$ with 6 signal events over 14 background events expected for 600 fb$^{-1}$ integrated luminosity after considering some elaborate cuts [@bbgaga] (the detection through other channels like $h h \to b \bar{b} W^+ W^-$ and $h h \to b \bar{b} \tau^+ \tau^-$ has also been studied recently [@bbWW; @bbtautau]). In principle, the capability can be further improved if the recently developed jet substructure technique [@JetSubstructure] is applied for the Higgs tagging. The Higgs pair production at the LHC may also be a sensitive probe for new physics. In supersymmetric models such as the Minimal Supersymmetric Standard Model (MSSM) [@MSSM], the pair production of the SM-like Higgs boson receives additional contributions from the loops of the third generation squarks and also from the parton process $b \bar{b} \to H_i \to h h$ with $H_i$ denoting a CP-even non-standard Higgs boson [@RunningMass; @DHiggsInMSSM]. It was found that in some cases (e.g., a light stop with a large trilinear soft breaking parameter $A_t$ and/or a large $\tan \beta$ together with moderately light $H_i$), these new contributions may be far dominant over the SM contribution, and as a result, the rate of the pair production may be enhanced by several orders [@RunningMass; @DHiggsInMSSM]. Note that since the experimental constraints (direct or indirect) on the SUSY parameter space have been becoming more and more stringent, the previous MSSM results should be updated by considering the latest constraints. This is one aim of this work. To be specific, we will consider the following new constraints: - The currently measured Higgs boson mass $m_h = 125$ GeV [@1212ATLAS-CMS]. In SUSY this mass is sensitive to radiative correction and thus the third generation squark sector has been tightly limited. - The LHC search for the third generation squarks [@1213ThirdSquark-LHC]. So far although the relevant bounds are rather weak and usually hypothesis-dependent, it becomes more and more clear that a stop lighter than about 200 GeV is strongly disfavored. - The observation of $B_s \to \mu^+ \mu^-$ by the LHCb [@1213Bsmumu]. In the MSSM it is well known that the branching ratio of $B_s \to \mu^+ \mu^-$ is proportional to $\tan^6\beta/m_H^4$ for a large $\tan \beta$ and a moderately light $H$ [@Bobeth]. Since the experimental value of $B_s \to \mu^+ \mu^-$ coincides well with the SM prediction, $\tan \beta$ as a function of $m_{H}$ has been upper bounded. - The LHC search for a non-standard Higgs boson $H$ through the process $p p \to H \to \tau^+ \tau^-$ [@Htautau]. Such the search relies on the enhanced $H \bar{b} b$ coupling and the nought signal seen by the LHC experiments implies that a broad region in the $\tan \beta-m_H$ plane has been ruled out. - The global fit of the SUSY predictions on various Higgs signals to the Higgs data reported by the ATLAS and CMS collaborations [@GlobalFit], the dark matter relic density [@WMAP] as well as the XENON2012 dark matter search results [@XENON2012] can also limit SUSY parameters in a complex way. ¡¡ Another motivation of this work comes from the fact that the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [@NMSSM] is found to be more favored by the Higgs data and the fine-tuning argument [@July-Cao]. So far the studies on the Higgs pair production in the NMSSM are still absent. So it is necessary to extend the study to the NMSSM. This paper is organized as follows. In Sec. II we briefly introduce the features of the Higgs sector in the MSSM and NMSSM. Then in Sec. III we present our results for the Higgs pair production in both models. Some intuitive understandings on the results are also presented. Finally, we summarize our conclusions in Sec. IV. Higgs sector in MSSM and NMSSM ============================== As the most economical realization of SUSY in particle physics, the MSSM [@MSSM] has been intensively studied. However, since this model suffers from some problems such as the unnaturalness of $\mu$ parameter, it is well motivated to go beyond this minimal framework. Among the extensions of the MSSM, the NMSSM as the simplest extension by singlet field [@NMSSM] has been paid much attention. The differences between the two models come from their superpotentials and soft-breaking terms, which are given by $$\begin{aligned} W_{\rm MSSM}&=& Y_u\hat{Q}\cdot\hat{H_u}\hat{U}-Y_d \hat{Q}\cdot\hat{H_d}\hat{D} -Y_e \hat{L}\cdot\hat{H_d} \hat{E} + \mu \hat{H_u}\cdot \hat{H_d}, \label{MSSM-pot}\\ W_{\rm NMSSM}&=&Y_u\hat{Q}\cdot\hat{H_u}\hat{U}-Y_d \hat{Q}\cdot\hat{H_d}\hat{D} -Y_e \hat{L}\cdot\hat{H_d} \hat{E} + \lambda\hat{H_u} \cdot \hat{H_d} \hat{S} + \frac{1}{3}\kappa \hat{S^3},\\ V_{\rm soft}^{\rm MSSM}&=&\tilde m_u^2|H_u|^2 + \tilde m_d^2|H_d|^2 + (B\mu H_u\cdot H_d + h.c.),\\ V_{\rm soft}^{\rm NMSSM}&=&\tilde m_u^2|H_u|^2 + \tilde m_d^2|H_d|^2 + \tilde m_S^2|S|^2 +(A_\lambda \lambda SH_u\cdot H_d +\frac{A_\kappa}{3}\kappa S^3 + h.c.).\end{aligned}$$ Here $\hat{H}_i$ ($i=u,d$) and $\hat{S}$ denote gauge doublet and singlet Higgs superfields respectively, $\hat{Q}$, $\hat{U}$, $\hat{D}$, $\hat{L}$ and $\hat{E}$ represent matter superfields with $Y_i$ ($i=u,d,e$) being their Yukawa coupling coefficients, $\tilde{m}_i$ ($i=u,d,S$), $B$, $A_\lambda$, and $A_\kappa$ are all soft-breaking parameters and the dimensionless parameters $\lambda$ and $\kappa$ reflect coupling strengthes of Higgs self interactions. Note the $\mu$-term in the MSSM is replaced by Higgs self interactions in the NMSSM, so when the singlet field $\hat{S}$ develops a vacuum expectation value $s$, an effective $\mu$ is generated by $\mu_{eff} = \lambda s$. Like the general treatment of the multiple-Higgs theory, one can write the Higgs fields in the NMSSM as $$\begin{aligned} H_u = \left ( \begin{array}{c} H_u^+ \\ v_u +\frac{ \phi_u + i \varphi_u}{\sqrt{2}} \end{array} \right),~~ H_d & =& \left ( \begin{array}{c} v_d + \frac{\phi_d + i \varphi_d}{\sqrt{2}}\\ H_d^- \end{array} \right),~~ S = s + \frac{1}{\sqrt{2}} \left(\sigma + i \xi \right),\end{aligned}$$ and diagonalize their mass matrices to get Higgs mass eigenstates: $$\begin{aligned} \left( \begin{array}{c} H_1 \\ H_2 \\ H_3 \end{array} \right) = U_H \left( \begin{array}{c} \phi_u \\ \phi_d\\ \sigma\end{array} \right),~ \left(\begin{array}{c} A_1\\ A_2\\ G^0 \end{array} \right) = U_A \left(\begin{array}{c} \varphi_u \\ \varphi_d \\ \xi \end{array} \right),~ \left(\begin{array}{c} H^+ \\G^+ \end{array} \right) =U_C \left(\begin{array}{c}H_u^+\\ H_d^+ \end{array} \right). \label{rotation}\end{aligned}$$ Here $H_1$, $H_2$, $H_3$ with convention $m_{H_1}<m_{H_2}<m_{H_3}$ and $A_1$, $A_2$ with convention $m_{A_1} < m_{A_2}$ denote the physical CP-even and CP-odd Higgs bosons respectively, $G^0$ and $G^+$ are Goldstone bosons eaten by $Z$ and $W$ bosons respectively, and $H^+$ is the physical charged Higgs boson. The Higgs sector in the MSSM can be treated in a similar way except that it predicts only two physical CP-even states and one physical CP-odd state, and consequently, the rotation matrices $U_H$ and $U_A$ are reduced to $2 \times 2$ matrices. ![Feynman diagrams for the pair production of the SM-like Higgs boson via gluon fusion in the MSSM and NMSSM with $H_I$ denoting a CP-even Higgs ($I=1,2$ for the MSSM and $I=1,2,3$ for the NMSSM) and $\tilde q_{i,j}$ ($i,j=1,2$) for a squark. The diagrams with initial gluons or final Higgs bosons interchanged are not shown here. For the quarks and squarks we only consider the third generation due to their large Yukawa couplings.[]{data-label="fig-diagram-total"}](fig1.ps){width="12cm"} One distinct feature of the MSSM is that $H_1$ usually acts as the SM-like Higgs boson (denoted by $h$ hereafter) and its mass is upper bounded by $m_Z$ at tree level. Obviously, to coincide with the LHC discovery of a 125 GeV boson, large radiative correction to $m_h$ is needed, which in turn usually requires the trilinear soft breaking parameter $A_t$ to be large. For example, in the case of large $m_A$ and moderate $\tan\beta$, $m_h$ is given by [@Carena-Higgsmass] $$\label{mh} m^2_{h} \simeq M^2_Z\cos^2 2\beta + \frac{3m^4_t}{4\pi^2v^2} \left[\ln\frac{m^2_{\tilde t}}{m^2_t} + \frac{X^2_t}{m^2_{\tilde t}} \left( 1 - \frac{X^2_t}{12m^2_{\tilde t}}\right)\right],$$ where the first term is the tree-level mass and the last two terms are the dominant corrections from the top-stop sector, $m_{\tilde t} = \sqrt{m_{\tilde{t}_1}m_{\tilde{t}_2}}$ ($m_{\tilde{t}_i}$ denotes stop mass with convention $m_{\tilde{t}_1} < m_{\tilde{t}_2}$) represents the average stop mass scale and $X_t \equiv A_t - \mu \cot\beta$. One can easily check that for a 500 GeV and 1 TeV stop, $|A_t|$ should be respectively about 1.8 TeV and 3.5 TeV to give $m_h \simeq 125~{\rm GeV}$. In the NMSSM, $m_h$ exhibits at least two new features [@Feb-Cao]. One is that it gets additional contribution at tree level so that $m_{h,tree}^2 = (m_Z^2 - \lambda^2 v^2 ) \cos^2 2 \beta + \lambda^2 v^2$, and for $\lambda \sim 0.7 $ and $\tan \beta \sim 1$, $m_h$ can reach 125 GeV even without the radiative correction. The other feature is that the mixing between the doublet and singlet Higgs fields can significantly alter the mass. To be more explicit, if the state $H_1$ is $h$, the mixing is to pull down the mass, while if $H_2$ acts as $h$, the mixing will push up the mass. Another remarkable character of the NMSSM is that in the limit of very small $\lambda$ and $\kappa$ (but keep $\mu$ fixed), the singlet field decouples from the theory so that the phenomenology of the NMSSM reduces to the MSSM. So in order to get a Higgs sector significantly different from the MSSM, one should consider a large $\lambda$. Throughout this work, we require $0.50 \leq \lambda \leq 0.7$ in our discussion of the NMSSM and we consider two scenarios: - NMSSM1 scenario: $H_1$ acts as the SM-like Higgs boson. For this scenario, the additional tree-level contribution to $m_h$ is canceled by the mixing effect, and if the mixing effect is dominant, the parameters in the stop sector will be tightly limited in order to give $m_h \simeq 125 {\rm ~GeV}$. - NMSSM2 scenario: $H_2$ acts as the SM-like Higgs boson. In this scenario, both the additional tree-level contribution and the mixing effect can push up the mass. So for appropriate values of $\lambda$ and $\tan \beta$, $m_h$ can easily reach 125 GeV even without the radiative correction. ![Feynman diagrams for the parton process $b\bar b\to hh$ in the MSSM and NMSSM.[]{data-label="fig-diagram-bb"}](fig2.ps){width="15cm"} Calculations and numerical results ================================== In SUSY the pair production of the SM-like Higgs boson proceeds through the gluon fusion shown in Fig.\[fig-diagram-total\] and the $b\bar b$ annihilation shown in Fig.\[fig-diagram-bb\]. These diagrams indicate that the genuine SUSY contribution to the amplitude is of the same perturbation order as the SM contribution. So the SUSY prediction on the production rate may significantly deviate from the SM result. To ensure the correctness of our calculation, we checked that we can reproduce the SM results presented in [@SM-LO-20fb] and the MSSM results in [@RunningMass]. Since the analytic expressions are quite lengthy, we do not present here their explicit forms. In our numerical calculation we take $m_t=173$ GeV, $m_b=4.2$ GeV, $m_Z=91.0$ GeV, $m_W=80.0$ GeV and $\alpha=1/128$ [@PDG], and use CT10 [@CT10] to generate the parton distribution functions with the renormalization scale $\mu_R$ and the factorization scale $\mu_F$ chosen to be $2 m_h$. The collision energy of the LHC is fixed to be 14 ${\rm TeV}$. Then we find that for $m_h=125$ GeV, the production rate in the SM is 18.7 fb for $gg\to hh$ and 0.02 fb for $b\bar{b}\to hh$ (the rates change very little when $m_h$ varies from 123 GeV to 127 GeV). For each SUSY model we use the package NMSSMTools-3.2.0 [@NMSSMTools] to scan over the parameter space and then select the samples which give a SM-like Higgs boson in the range of $125 \pm 2 ~{\rm GeV} $ and also satisfy various experimental constraints, including those listed in Section I. The strategy of our scan is same as in [@July-Cao] except for three updates. First, since the rare decay $B_s \to \mu^+ \mu^-$ has been recently observed with $Br(B_s \to \mu^+ \mu^-) = 3.2^{+1.5}_{-1.2} \times 10^{-9}$ [@1213Bsmumu], we use a double-sided limit $0.8 \times 10^{-9} \leq Br(B_s \to \mu^+ \mu^-) \leq 6.2 \times 10^{-9}$. Second, for the LHC search of the non-standard Higgs boson, we use the latest experimental data [@Htautau]. The third one is that we require stops heavier than 200 GeV [@1213ThirdSquark-LHC]. After the scan, we calculate the Higgs pair production rate in the allowed parameter space. We will demonstrate the ratio $\sigma_{SUSY}/\sigma_{SM}$ for each surviving sample. Of course, such a ratio is less sensitive to higher order QCD corrections. ![The scatter plots of the surviving samples, showing $\sigma_{SUSY}/\sigma_{SM}$ versus the SM-like Higgs boson mass. The plus ’+’ (blue) denote the results with only the gluon fusion contribution, while the circles ’$\circ$’ (pink) are for the total results. []{data-label="fig-ratio-mh"}](fig3.ps){width="15cm"} In Fig. \[fig-ratio-mh\] we show the normalized production rate as a function of the Higgs boson mass for the surviving samples in the MSSM and NMSSM (for the NMSSM we show the results for the NMSSM1 and NMSSM2 scenarios defined in Sec.II). This figure shows two common features for the three scenarios. One is that the production rate can deviate significantly from the SM prediction: in most cases the deviation exceeds $30\%$ and in some specail cases the production rate can be enhanced by one order. The other feature is that for most cases the dominant contribution to the pair production comes from the gluon fusion, which is reflected by the approximate overlap of ’$\circ$’ (pink) with ’+’ (blue). Fig. \[fig-ratio-mh\] also exhibits some difference between different scenarios. For example, in the MSSM the $b\bar{b}$ annihilation contribution can be dominant for some surviving samples, which, however, never occurs in the NMSSM. Another difference is that the NMSSM1 tends to predict a larger production rate than other scenarios. Now we explain some features of the results in Fig. \[fig-ratio-mh\]. First, we investigate the cases of the MSSM where the $b\bar{b}$ annihilation plays the dominant role in the production. We find that they are characterized by a moderately large $\tan \beta$ ($\tan \beta \sim 10$ so that the $Hb\bar{b}$ coupling is enhanced), a moderately light $H$ ($ 300{\rm ~GeV} \lesssim m_H \lesssim 400{\rm ~GeV} $) and a relatively large $Hhh$ coupling. While for the NMSSM scenarios, since we are considering large $\lambda$ case, only a relatively small $\tan \beta$ is allowed so that the $H_i b\bar{b}$ coupling is never enhanced sufficiently [@Feb-Cao]. We also scrutinize the characters of the gluon fusion contribution in the MSSM. As the first step, we compare the sbottom loop contribution with the stop loop. We find that for the surviving samples the former is usually much smaller than the latter. Next we divide the amplitude of Fig. \[fig-diagram-total\] into five parts with $M_1,M_2,M_3,M_4$ and $M_5$ denoting the contributions from diagrams (1)+(2), (3)+(4), (5), (6)+(7) and (8)+(9)+(10), respectively. For each of the amplitude, it is UV finite so we can learn its relative size directly. We find that the magnitudes of $M_2$ and $M_3$ are much larger than the others. This can be understood as follows: among the diagrams in Fig. \[fig-diagram-total\], only (3), (4) and (5) involve the chiral flipping of the internal stop, so in the limit $ m_{\tilde{t}_2}, m_{\tilde{t}_1} \gg 2 m_h$ the main parts of $M_2$ and $M_3$ can be written as $$\begin{aligned} M \sim \alpha_s^2 Y_t^2 ( c_1 \sin^2 2\theta_t \frac{A_t^2}{m_{\tilde{t}_1}^2} + c_2 \frac{A_t^2}{m_{\tilde{t}_2}^2}) \label{simpleform}\end{aligned}$$ where $Y_t$ is the top quark Yukawa coupling, $\theta_t$ and $A_t$ are respectively the chiral mixing angle and the trilinear soft breaking parameter in the stop sector, and $c_{1} $ and $c_2$ are ${\cal{O}}(1)$ coefficients with opposite signs. Since a large $A_t$ is strongly favored to predict $m_h \sim 125 {\rm ~GeV}$ in the MSSM [@Feb-Cao] and the other contributions are usually proportional to $m_t^2/m_{\tilde{t}_i}^2$ or $m_h^2/m_{\tilde{t}_i}^2$, one can easily conclude that $M_2$ and $M_3$ should be most important among the five amplitudes. In fact, we checked that without the strong cancelation between $M_2$ and $M_3$, the production rate can easily exceed 100 fb for most surviving samples. ![Same as Fig \[fig-ratio-mh\], but showing $A_t/m_{\tilde{t}_1}$ versus $m_{\tilde{t}_1}$. The samples are classified according to the value of $R=\sigma_{SUSY}(gg\to hh)/\sigma_{SM}(gg\to hh)$ with $\sigma$ denoting the hadronic cross section via $gg\to hh$.[]{data-label="fig-ratio-atmst1mst1"}](fig4.ps){width="15cm"} As a proof for the validity of Eq.(\[simpleform\]), in Fig. \[fig-ratio-atmst1mst1\] we show $A_t/m_{\tilde{t}_1}$ versus $m_{\tilde{t}_1}$, where the samples are classified according to the value of $R=\sigma_{SUSY}(gg\to hh)/\sigma_{SM}(gg\to hh)$. The left panel indicates that in the MSSM the region characterized by a light $m_{\tilde{t}_1}$ and a large $|A_t/m_{\tilde{t}_1}|$ usually predicts a large $R$. This can be understood as follows. In the MSSM with a light $\tilde{t}_1$, the other stop ($\tilde{t}_2$) must be sufficiently heavy in order to predict $m_h \sim 125 {\rm ~GeV}$ [@Feb-Cao]. Then, after expressing $\sin^2 2 \theta_t$ in terms of $A_t$ and stop masses, one can find that the first term in Eq.(\[simpleform\]) scales like $ (A_t/m_{\tilde{t}_1})^4 (m_t^2 m_{\tilde{t}_1}^2/m_{\tilde{t}_2}^4)$, and therefore its value grows rapidly with the increase of $|A_t/m_{\tilde{t}_1}|$ and is unlikely to be canceled out by the second term in Eq.(\[simpleform\]). In fact, the upper left region of the panel reflects such a behavior. This panel also indicates that even for $\tilde{t}_1$ and $\tilde{t}_2$ at TeV scale, the production rate in the MSSM may still deviate from its SM prediction by more than $30\%$. This is obvious since $|A_t|$ in Eq.(\[simpleform\]) is usually larger than stop masses [@Feb-Cao]. Finally, we note that for $m_{\tilde{t}_1} > 1 ~{\rm TeV}$, there exist some cases where the deviation is small even for $A_t/m_{\tilde{t}_1} \sim 3$. We checked that these cases actually correspond to a small mass splitting between $\tilde{t}_1$ and $\tilde{t}_2$. In such a situation, the first term in Eq.(\[simpleform\]) is proportional to $A_t^2/m_{\tilde{t}_1}^2$ (since $\theta_t \simeq \pi/4$), and its contribution to the rate is severely canceled by the second term. ![Same as Fig. \[fig-ratio-mh\], but showing $\sigma_{SUSY}/\sigma_{SM}$ versus $\chi^2$. Here only the samples satisfying 125 GeV $\le m_h \leq 126$ GeV are plotted.[]{data-label="fig-ratio-chisq"}](fig5.ps){width="15cm"} Eq.(\[simpleform\]) may also be used to explain the results of the NMSSM1 scenario. In this scenario we checked that the mixing effect on $m_h$ often exceeds the additional tree level contribution (as discussed in Sec. II), and consequently the soft breaking parameters in the stop sector are more tightly limited than the other two scenarios. For example, given the same values of $m_{\tilde{t}_1}$ and $m_{\tilde{t}_2}$ for the three scenarios, the NMSSM1 scenario usually prefers a larger $|A_t|$. Consequently, this scenario tends to predict the largest production rate according to Eq.(\[simpleform\]). As for the $R$ value in the NMSSM2 scenario, the situation is quite complex because a large $\lambda$ alone can push the value of $m_h$ up to about 125 GeV and thus the soft breaking parameters in the stop sector are not so constrained by the Higgs mass [@Feb-Cao]. But, anyway, this scenario still has the features that $R$ is maximized for a large $A_t$ and a light $\tilde{t}_1$ and that $R$ can deviate sizably from unity for TeV-scale stops. Finally, we focus on the samples which predict a SM-like Higgs boson in the best fitted mass region, $125 {\rm ~GeV} \le m_h \leq 126 {\rm ~GeV}$ [@GlobalFit]. For these samples, we calculate the $\chi^2$ value with the LHC Higgs data (for details, see [@July-Cao; @GlobalFit]) and show its correlation with the normalized rate $\sigma_{SUSY}/\sigma_{SM}$ in Fig. \[fig-ratio-chisq\]. This figure indicates that in the MSSM and NMSSM2 scenarios, there exist a lot of samples with $\chi^2$ much smaller than its SM value ($\chi^2_{SM} =16.5$), which implies that the MSSM and NMSSM2 scenarios may be favored by the current data [@July-Cao]. In contrast, the NMSSM1 scenario can only slightly improve the fit. From this figure we also see that in the favored parameter space with a small $\chi^2$ the production rate can sizably deviate from the SM prediction ( in the parameter space with a large $\chi^2$ the production rate can be several times larger than the SM value). For example, in the low $\chi^2$ region of the MSSM, the normalized rate is approximately 1.45, while in the NMSSM2 scenario the rate varies from 0.7 to 2.4. Summary and Conclusions {#Sum} ======================= Recently, the CMS and ATLAS collaborations announced the discovery of a new resonance whose property is in rough agreement with the SM Higgs boson. But the nature of this new state, especially its role in electroweak symmetry breaking, needs to be scrutinized. So the most urgent task for the LHC is to test the property of this Higgs-like boson by measuring all the possible production and decay channels with high luminosity. Among the production channels, the Higgs pair production is a rare process at the LHC. Since it can play an important role for testing the Higgs self-couplings, it will be measured at the LHC with high luminosity. In this work we studied the pair production of the SM-like Higgs boson in the popular SUSY models: the MSSM and NMSSM. To make our study realistic, we first scanned the parameter space of each model by considering various experimental constraints. Then we examined the Higgs pair production in the allowed parameter space. We found that for most cases in both models, the dominant contribution to the pair production comes from the gluon fusion process with its rate maximized at a moderately light $\tilde{t}_1$ and a large trilinear soft breaking parameter $A_t$. The production rate can be sizably enhanced relative to the SM prediction: $\sigma_{SUSY}/\sigma_{SM}$ can reach 10, and even for a TeV-scale stop it can also exceed 1.3. For each model we also calculated its $\chi^2$ with current Higgs data and found that in the most favored parameter region the value of $\sigma_{SUSY}/\sigma_{SM}$ is approximately 1.45 in the MSSM, while in the NMSSM it varies from 0.7 to 2.4. Acknowledgement {#acknowledgement .unnumbered} =============== We thank Jingya Zhu for helpful discussions. 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--- abstract: | We consider the renormalized Nelson model at a fixed total momentum $P$: ${H_{\mathrm{ren}}}(P)$; The Hamiltonian ${H_{\mathrm{ren}}}(P)$ is defined through an infinite energy renormalization. We prove that $e^{-\beta {H_{\mathrm{ren}}}(P)}$ is positivity improving for all $P\in {\mathbb{R}}^3$ and $\beta >0$ in the Fock representation. [**Mathematics Subject Classification (2010).**]{} Primary: 47A63, 47D08, 81T16; Secondary: 47N50, 81T10 [**Keywords.** ]{} Nelson model; Energy renormalization; Operator inequalities; Positivity improving semigroups. author: - | Tadahiro Miyao\ [*Department of Mathematics,*]{} [*Hokkaido University,*]{}\ [*Sapporo 060-0810, Japan*]{}\ [miyao@math.sci.hokudai.ac.jp]{} --- \[section\] \[define\][Theorem]{} \[define\][Proposition]{} \[define\][Lemma]{} \[define\][Remark]{} \[define\][Corollary]{} Introduction ============ In a celebrated paper [@Nelson], Nelson studies the Hamiltonian, which describes the interaction of $N$ particles with a massive Bose field. He constructs a model without the ultraviolet cutoff through an infinite energy renormalization. We expect that his observation provides a hint to understand renormalization procedures in more complicated models; His model is nowadays called the Nelson model, and has been actively studied. For example, Fr[ö]{}hlich studies the Nelson model at a fixed total momentum [@JFroehlich1; @JFroehlich2]; asymptotic completeness is addressed in [@Ammari; @DM]; existence of a ground state is proved in [@A.; @Arai; @HHS; @Sasaki]; functional integral representations are constructed in [@GHL; @LoMS; @MM], and so on [@AH; @GW; @Gross; @LS; @Pizzo; @Spohn]. The cutoff Nelson Hamiltonian reads $$\begin{aligned} H_{\Lambda}=-\frac{1}{2}\Delta-g\int_{{\mathbb{R}}^3} dk\frac{\chi_{\Lambda}(k)}{\sqrt{\omega(k)}} (e^{ik\cdot x} a(k)+e^{-ik\cdot x}a(k)^*) +{H_{\mathrm{f}}}\end{aligned}$$ acting in $$\begin{aligned} L^2({\mathbb{R}}^3) \otimes {\mathfrak{F}}, \end{aligned}$$ where ${\mathfrak{F}}$ is the bosonic Fock space over $L^2({\mathbb{R}}^3)$. Recall that $$\begin{aligned} {\mathfrak{F}}=\Sumoplus L_{\mathrm{sym}}^2({\mathbb{R}}^{3n}), \end{aligned}$$ where $L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})=\big\{ {\varphi}\in L^2({\mathbb{R}}^{3n})\, |\, {\varphi}(k_1,\dots, k_n)={\varphi}(k_{\sigma(1)}, \dots, k_{\sigma(n)})\ \mbox{a.e. }\forall \sigma \in \mathfrak{S}_n \big\} $ and $L^2_{\mathrm{sym}}({\mathbb{R}}^{0})={\mathbb{C}}$ (where $\mathfrak{S}_n$ is the permutation group on a set $\{1, 2, \dots , n\}$). The single particle Schr[ö]{}dinger operator $-\frac{1}{2}\Delta$ is the Hamiltonian of the free particle, where $\Delta$ is the $3$-dimensional Laplacian. The annihilation- and creation operators of the field, $a(k)$ and $a(k)^*$, satisfy the standard commutation relations: $$\begin{aligned} [a(k), a(k')^*]=\delta(k-k'),\ \ [a(k), a(k')]=0,\ \ k, k'\in {\mathbb{R}}^3. \end{aligned}$$ The field energy ${H_{\mathrm{f}}}$ is given by $$\begin{aligned} {H_{\mathrm{f}}}=\int_{{\mathbb{R}}^3} dk \omega(k)a(k)^*a(k). \end{aligned}$$ The dispersion relation $\omega(k)$ is given by $$\begin{aligned} \omega(k)=\sqrt{k^2+m^2},\ \ m>0. \end{aligned}$$ The ultraviolet cutoff fuction $\chi_{\Lambda}\ (\Lambda>0)$ is defined by $$\begin{aligned} \chi_{\Lambda}(k) =\begin{cases} 1, & |k| \le \Lambda\\ 0, & |k|> \Lambda. \end{cases} \end{aligned}$$ The prefactor $g$ is a coupling strength between the particle and the field. Without loss of generality, we may assume that $$\begin{aligned} g>0. \end{aligned}$$ The interaction is infinitesimally small relative to the free Hamiltonian. Hence, by the Kato-Rellich theorem, $H_{\Lambda}$ is self-adjoint on the domain ${\mathrm{dom}}(-\Delta) \cap {\mathrm{dom}}({H_{\mathrm{f}}})$ and bounded from below. The generator of translations is the total momentum operator $$\begin{aligned} {P_{\mathrm{tot}}}=-i \nabla+{P_{\mathrm{f}}}\end{aligned}$$ with $\displaystyle {P_{\mathrm{f}}}=\int_{{\mathbb{R}}^3} dk k a(k)^*a(k). $ The total momentum is conserved, namely, $ e^{ia\cdot P_{\mathrm{tot}}}H_{\Lambda}=H_{\Lambda} e^{ia\cdot P_{\mathrm{tot}}} $ for all $a\in {\mathbb{R}}^3$. Therefore, $H_{\Lambda}$ admits the direct integral decomposition $$\begin{aligned} UH_{\Lambda}U^*&=\int^{\oplus}_{{\mathbb{R}}^3} H_{\Lambda}(P) dP,\\ H_{\Lambda}(P)&=\frac{1}{2}(P-{P_{\mathrm{f}}})^2-g\int_{{\mathbb{R}}^3} dk \frac{\chi_{\Lambda}(k)}{\sqrt{\omega(k)}} (a(k)+a(k)^*)+{H_{\mathrm{f}}},\end{aligned}$$ where $U$ is some unitary operator on $L^2({\mathbb{R}}^3) \otimes {\mathfrak{F}}$. $H_{\Lambda}(P)$ acts in ${\mathfrak{F}}$. By the Kato-Rellich theorem again, $H_{\Lambda}(P)$ is self-adjoint on ${\mathrm{dom}}({P_{\mathrm{f}}}^2)\cap {\mathrm{dom}}({H_{\mathrm{f}}})$ and bounded from below for all $P\in {\mathbb{R}}^3$. $ H_{\Lambda}(P)$ is called the cutoff Nelson Hamiltonian at a fixed total momentum $P$. Let $$\begin{aligned} E_{\Lambda}=-g^2\int_{{\mathbb{R}}^3} dk \frac{\chi_{\Lambda}(k)}{\omega(k)\{\omega(k)+k^2/2\}}. \label{DefE_k}\end{aligned}$$ Notice that $E_{\Lambda} \to -\infty$ as $\Lambda\to \infty$. We define $$\begin{aligned} {H_{\mathrm{ren}, \Lambda}}=H_{\Lambda}-E_{\Lambda},\ \ \ \ {H_{\mathrm{ren}, \Lambda}}(P)=H_{\Lambda}(P)-E_{\Lambda}. \end{aligned}$$ Nelson’s result is stated as follows. - There exists a self-adjoint operator ${H_{\mathrm{ren}}}$ bounded from below such that ${H_{\mathrm{ren}, \Lambda}}$ converges to ${H_{\mathrm{ren}}}$ in strong resolvent sense as $\Lambda\to \infty$. - For all $P\in {\mathbb{R}}^3$, there exists a self-adjoint operator ${H_{\mathrm{ren}}}(P)$ bounded from below such that ${H_{\mathrm{ren}, \Lambda}}(P)$ converges to ${H_{\mathrm{ren}}}(P)$ in strong resolvent sense as $\Lambda\to \infty$. In this study, we are interested in the renormalized Nelson Hamiltonian at a fixed total momentum: ${H_{\mathrm{ren}}}(P)$. Following Fr[ö]{}hlich [@JFroehlich1; @JFroehlich2], we introduce a convex cone ${\mathfrak{F}}_+$ by $$\begin{aligned} {\mathfrak{F}}_+=\Sumoplus L_{\mathrm{sym}}^2({\mathbb{R}}^{3n})_+, \label{FockCone}\end{aligned}$$ where $ L_{\mathrm{sym}}^2({\mathbb{R}}^{3n})_+= \big\{ {\varphi}\in L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})\, |\, {\varphi}(k_1,\dots, k_n)\ge 0\ \mbox{a.e.} \big\}$ with $ L_{\mathrm{sym}}^2({\mathbb{R}}^{0})_+={\mathbb{R}}_+=\{r\in {\mathbb{R}}\, |\, r\ge 0\} $. To state our results, the following terminologies are needed. - A vector ${\varphi}\in {\mathfrak{F}}$ is called [*positive*]{} if ${\varphi}\in {\mathfrak{F}}_+$; - A vector ${\varphi}=\Sumoplus {\varphi}_n\in {\mathfrak{F}}$ is called [*strictly positive*]{} if ${\varphi}_n(k_1, \dots, k_n)>0$ a.e. for all $n\in {\mathbb{N}}_0=\{0\}\cup {\mathbb{N}}$; - We say that a bounded linear operator $A$ is [*positivity preserving*]{} if $A$ maps ${\mathfrak{F}}_+$ into ${\mathfrak{F}}_+:\ A{\mathfrak{F}}_+\subseteq {\mathfrak{F}}_+$; - A bounded linear operator $A$ is called [*positivity improving*]{} if $A{\varphi}$ is strictly positive whenever ${\varphi}$ is positive and ${\varphi}\neq 0$. $\diamondsuit$ Our main theorem is the following. \[Main1\] $e^{-\beta {H_{\mathrm{ren}}}(P)}$ is positivity improving for all $P\in {\mathbb{R}}^3$ and $\beta >0$. The following corollary immediately follows from Theorems \[Main1\] and \[PFF\]. Suppose that $E(P)=\inf \mathrm{spec}({H_{\mathrm{ren}}}(P))$ is an eigenvalue. Then $E(P)$ is a simple eigenvalue with a strictly positive eigenvector. - By applying methods in [@JFroehlich1; @LMS], we can prove that $E(P)$ is actually an eigenvalue, provided that $|P|<1$. - Theorem \[Main1\] remains true when we consider the Hamiltonian ${H_{\mathrm{ren}}}(P)$ with $\omega$ and $\chi_{\Lambda}$ replaced by $\omega_0(k)=|k|$ and $\chi_{\sigma}^{\Lambda}=\chi_{\Lambda}-\chi_{\sigma}$, where the infrared cutoff $\sigma$ is chosen so that $0<\sigma <\Lambda$. (Note that when $\sigma=0$, we have to take extra care for the infrared problem, see, e.g., [@A.; @Arai; @LoMS; @Sasaki]. We will examine such a case in [@Miyao6].) $\diamondsuit$ In order to explain our achievement, let us introduce the modified Nelson Hamiltonian by $$\begin{aligned} H_{\varrho}(P)=\frac{1}{2} (P-{P_{\mathrm{f}}})^2-g\int_{{\mathbb{R}}^3} dk\frac{\varrho(k)}{\sqrt{\omega(k)}} (a(k)+a(k)^*)+{H_{\mathrm{f}}},\end{aligned}$$ where $\varrho(k)$ is real-valued. Under the assumptions $$\begin{aligned} \omega^{-1/2} \varrho,\ \ \omega^{-1} \varrho\in L^2({\mathbb{R}}^3),\end{aligned}$$ $H_{\varrho}(P)$ is self-adjoint on ${\mathrm{dom}}({P_{\mathrm{f}}}^2)\cap {\mathrm{dom}}({H_{\mathrm{f}}})$, and bounded from below for all $P\in {\mathbb{R}}^3$. In a famous paper [@JFroehlich1], Fr[ö]{}hlich has shown that, if $\varrho(k)>0$ a.e. $k$, then $e^{-\beta H_{\varrho}(P)}$ is positivity improving for all $P\in {\mathbb{R}}^3$ and $\beta >0$ in the Fock representation. His idea has been applied to the polaron problem successfully [@GeLowen; @Moller2; @Spohn2]; In particular, it has been proven in [@Miyao; @Miyao3; @Miyao4] that the semigroup generated by the Fr[ö]{}hlich Hamiltonian without ultraviolet cutoff is positivity improving for all $P\in {\mathbb{R}}^3$. Note that, in [@Sloan; @Sloan2], Sloan has proved that the semigroup generated by the two-dimensional polaron model without ultraviolet cutoff is posivitiy improving for $P=0$; His beautiful method is different from Fr[ö]{}hlich’s approach, and is applicable in the Schr[ö]{}dinger representation. The primary reason for these successes is that no energy renormalization is needed, when we remove the ultraviolet cutoff from the polaron models. In contrast to the polaron problem, the Hamiltonian ${H_{\mathrm{ren}}}(P)$ is defined through an infinite energy renormalization. By this obstacle, Fr[ö]{}hlich’s original method only tells us that $e^{-\beta {H_{\mathrm{ren}}}(P)}$ is postivity preserving for all $P\in {\mathbb{R}}^3$ and $\beta >0$. It has been a long standing problem to prove that $ e^{-\beta {H_{\mathrm{ren}}}(P)} $ is positivity improving for all $P\in {\mathbb{R}}^3$. To overcome this difficulty, we apply operator theoretic correlation inequalities studied in [@MS; @Miyao; @Miyao3; @Miyao4; @Miyao5]. In our previous works on the polaron models [@Miyao; @Miyao3; @Miyao4], we have clarified that this approach is very useful for studies on the semigroup generated by the operator. In the present paper, we further develop this method so that we can get over a difficulty arising from the infinite energy renormalization. For readers’ convenience, we give a brief outline of the proof of Theorem \[Main1\] here. For every $\kappa>0$, let $B_{\kappa}$ be the ball of radius $\kappa$ in ${\mathbb{R}}^3$ centered at the origin. Let ${\mathfrak{F}}^{\le \kappa}$ be the Fock space over $L^2(B_{\kappa})$ and let ${\mathfrak{F}}^{>\kappa}$ be the Fock space over $L^2(B_{\kappa}^c)$, where $B_{\kappa}^c$ is the complement of $B_{\kappa}$. The Fock space ${\mathfrak{F}}$ can be factorized as $$\begin{aligned} {\mathfrak{F}}={\mathfrak{F}}^{\le \kappa}\otimes {\mathfrak{F}}^{>\kappa}. \label{FockFA}\end{aligned}$$ Corresponding to (\[FockFA\]), ${H_{\mathrm{ren}}}(P)$ can be decomposed as $$\begin{aligned} {H_{\mathrm{ren}}}(P)={H_{\mathrm{ren}}}^{\le \kappa}(P) \otimes 1\dot{+} C_{\kappa}\dot{+}1\otimes K_{\kappa}, \label{HDEC}\end{aligned}$$ where $\dot{+}$ indicates the form sum. The local part ${H_{\mathrm{ren}}}^{\le \kappa}(P)$ acts in ${\mathfrak{F}}^{\le \kappa}$, while $K_{\kappa}$ lives in ${\mathfrak{F}}^{>\kappa}$. $C_{\kappa}$ is the cross-term. In Section \[Pf1\], we will prove the following: To show that $e^{-\beta {H_{\mathrm{ren}}}(P)}$ improves the positivity in ${\mathfrak{F}}$, it suffices to show that $e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)}$ improves the positivity in ${\mathfrak{F}}^{\le \kappa}$ and $e^{-\beta K_{\kappa}}$ preserves the positivity in ${\mathfrak{F}}^{>\kappa}$ for all $\kappa>0$. On the other hand, we can apply Fr[ö]{}hlich’s idea to see that $e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)}$ improves the positivity in ${\mathfrak{F}}^{\le \kappa}$. In this way, we obtain Theorem \[Main1\]. The most difficult part in the above is the reduction of the positivity improvingness of $e^{-\beta {H_{\mathrm{ren}}}(P)}$ to the properties of $e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)}$. This procedure can be achieved by extending Faris’ idea in [@Faris] as we will see in Section \[Pf1\]. Path measure methods have been actively studied, and made remarkable progress [@GHL; @MM; @LoMS]. As far as we are aware, this methods can only cover a case where $P=0$; To be precise, it can be proved by a functional integral formula that $e^{-\beta {H_{\mathrm{ren}}}(0)}$ is positivity improving in the [*Schr[ö]{}dinger representation*]{}. Note that this methods work for $P=0$ only. In contrast to this, our methods work for all $P\in {\mathbb{R}}^3$, and are effective in the [*Fock representation*]{}. On the other hand, path measure methods can treat the Hamiltonian ${H_{\mathrm{ren}}}+V$ with an external potential $V:\ {\mathbb{R}}^3\to {\mathbb{R}}$. By using ideas in [@Miyao5], our approach can also cover this case only if $V$ is assumed to be [*ferromagnetic*]{}[^1]; We will discuss this problem in [@Miyao6]. In conclusion, our operator theoretic and path measure methods complement each other and both have specific advantages. Recently, Griesemer and W[ü]{}nsch reported an interesting finding of the domain property of the renormalized Nelson Hamiltonian in [@GW]. Namely, they showed that the domain of the Nelson model satisfies ${\mathrm{dom}}({H_{\mathrm{ren}}}) \cap {\mathrm{dom}}(H_0)=\{0\}$, where $H_0=-\Delta+{H_{\mathrm{f}}}$. Fortunately, this anomalous property unaffects our arguments in the present paper. To be more precise, the point of our proof is the reduction of the problem to the local properties as we mentioned above; this step is essentially based on the algebraic relation (\[HDEC\]), and detailed information on the domain is unnecessary for our proof. The organization of the present paper is as follows: In Section \[SecMono\], we briefly review some basic properties of operator theoretic correlation inequalities. Section \[2ndQuant\] is devoted to study useful properties of the second quantized operators. In Section \[Pf1\], we prove Theorem \[Main1\] by applying operator theoretic correlation inequalities. In Appendx \[AppA\], we give a list of fundamental facts that are used in the main sections. [**Acknowledgments.**]{} I am grateful to Herbert Spohn for comments. I would like to thank an anonymous referee for helpful suggestions that improved the present paper. I would also like to take this opportunity to thank M. Hirokawa and K. R. Ito for useful discussions. This work was partially supported by KAKENHI 18K03315. Operator theoretic correlation inequalities {#SecMono} =========================================== Positivity preserving operators ------------------------------- Let ${\mathfrak{H}}$ be a complex Hilbert space and let ${\mathfrak{P}}$ be a convex cone in ${\mathfrak{H}}$. We say that ${\mathfrak{P}}$ is [*self-dual*]{} if $$\begin{aligned} {\mathfrak{P}}=\{x\in {\mathfrak{H}}\, |\, {\langle}x| y{\rangle}\ge 0\ \forall y\in {\mathfrak{P}}\}.\end{aligned}$$ Henceforth, we always assume that ${\mathfrak{P}}\neq \{0\}$. The following properties of ${\mathfrak{P}}$ are well-known [@Bos; @BR1]: \[BasisSAC\] We have the following: - ${\mathfrak{P}}\cap (-{\mathfrak{P}})=\{0\}$. - There exists a unique involution $j$ in ${\mathfrak{H}}$ such that $jx=x$ for all $x\in {\mathfrak{P}}$. - Each element $x\in {\mathfrak{H}}$ with $jx=x$ has a unique decomposition $x=x_+-x_-$, where $x_+,x_-\in{\mathfrak{P}}$ and ${\langle}x_+| x_-{\rangle}=0$. - ${\mathfrak{H}}$ is linearly spanned by ${\mathfrak{P}}$. <!-- --> - A vector $x$ is said to be [*positive w.r.t. ${\mathfrak{P}}$*]{} if $x\in {\mathfrak{P}}$. We write this as $x \ge 0$ w.r.t. ${\mathfrak{P}}$. - A vector $x \in {\mathfrak{P}}$ is called [*strictly positive w.r.t. ${\mathfrak{P}}$*]{} whenever ${\langle}x| y{\rangle}>0$ for all $y\in {\mathfrak{P}}\backslash \{0\}$. We write this as $x>0 $ w.r.t. ${\mathfrak{P}}$. - Let ${\mathfrak{H}}_{{\mathbb{R}}}=\{x\in {\mathfrak{H}}\, |\, jx=x\}$, where $j$ is given in Proposition \[BasisSAC\]. Let $x, y\in {\mathfrak{H}}_{{\mathbb{R}}}$. If $x-y\in {\mathfrak{P}}$, then we write this as $x\ge y$ w.r.t. ${\mathfrak{P}}$. $\diamondsuit$ [ For each $d\in {\mathbb{N}}$, we set $$\begin{aligned} L^2({\mathbb{R}}^d)_+=\{f\in L^2({\mathbb{R}}^d)\, |\, f(u)\ge 0\ \ \mbox{a.e. $u$} \}.\end{aligned}$$ $L^2({\mathbb{R}}^d)_+$ is a self-dual cone in $L^2({\mathbb{R}}^d)$. $f\ge 0$ w.r.t. $L^2({\mathbb{R}}^d)_+$ if and only if $f(u) \ge 0$ a.e. $u$. On the other hand, $f >0$ w.r.t. $L^2({\mathbb{R}}^d)_+$ if and only if $f(u)>0$ a.e. $u$. $\diamondsuit$ ]{} Let $\mathfrak{V}$ be a dense subspace of ${\mathfrak{H}}$ such that $\mathfrak{V} \cap {\mathfrak{P}}\neq \{0\}$.[^2] Set $$\begin{aligned} \mathscr{L}(\mathfrak{V})=\{\mbox{$A$: linear operator s.t. $\mathfrak{V}\subseteq {\mathrm{dom}}(A)\cap {\mathrm{dom}}(A^*),\ A\mathfrak{V} \subset \mathfrak{V},\ A^*\mathfrak{V} \subset \mathfrak{V}$}\}.\end{aligned}$$ The following lemma is easy to check: \[AuxV\] We have the following: - $\mathscr{L} (\mathfrak{V})$ is a linear space. - If $A, B\in \mathscr{L}(\mathfrak{V})$, then $AB\in \mathscr{L}(\mathfrak{V})$. - If $A\in \mathscr{L}(\mathfrak{V})$, then $A^*\in \mathscr{L}(\mathfrak{V})$. - If $A\in \mathscr{L}(\mathfrak{V})$, then ${\mathrm{dom}}(A) \cap {\mathfrak{P}}\supseteq \mathfrak{V}\cap {\mathfrak{P}}\neq \{0\}$. - If $A\in \mathscr{L}(\mathfrak{V})$, then ${\mathrm{dom}}(A) \cap {\mathfrak{H}}_{{\mathbb{R}}} \supseteq \mathfrak{V}\cap{\mathfrak{H}}_{{\mathbb{R}}} \neq \{0\}$. <!-- --> - Let $A\in \mathscr{L}(\mathfrak{V})$. If $A({\mathrm{dom}}(A) \cap {\mathfrak{P}}) \subseteq {\mathfrak{P}}$, then we write this as $A\unrhd 0$ w.r.t. ${\mathfrak{P}}$. Remark that, by Lemma \[AuxV\] (iv), this definition is meaningful. In this case, we say that $A$ [*preserves the positivity*]{} w.r.t. ${\mathfrak{P}}$. - Let $A, B\in \mathscr{L}(\mathfrak{V})$. Suppose that $A({\mathrm{dom}}(A) \cap {\mathfrak{H}}_{{\mathbb{R}}}) \subseteq{\mathfrak{H}}_{{\mathbb{R}}}$ and $B({\mathrm{dom}}(B) \cap {\mathfrak{H}}_{{\mathbb{R}}}) \subseteq{\mathfrak{H}}_{{\mathbb{R}}}$. If $(A-B)\Big({\mathrm{dom}}(A) \cap {\mathrm{dom}}(B) \cap {\mathfrak{P}}\Big) \subseteq {\mathfrak{P}}$, then we write this as $A\unrhd B$ w.r.t. ${\mathfrak{P}}$. $\diamondsuit$ [ Suppose that $A$ and $B$ are bounded. Then $A\unrhd B$ w.r.t. ${\mathfrak{P}}$ if and only if ${\langle}x|Ay{\rangle}\ge {\langle}x|By{\rangle}$ for all $x, y\in {\mathfrak{P}}$. $\diamondsuit$ ]{} [ Let $F$ be a multiplication operator on $L^2({\mathbb{R}}^d)$ by the function $F(u)$. Assume that $\|F\|_{\infty} <\infty$. If $F(u) \ge 0$ a.e., then $F\unrhd 0$ w.r.t. $L^2({\mathbb{R}}^d)_+$. $\diamondsuit$ ]{} \[PPInq\] Let $A, A_1, A_2, B, B_1, B_2\in \mathscr{L}(\mathfrak{V})$. We have the following: - If $0\unlhd A$ and $0\unlhd B$ w.r.t ${\mathfrak{P}}$, then $0\unlhd A B$ w.r.t. ${\mathfrak{P}}$. - If $0\unlhd A_1\unlhd B_1$ and $0\unlhd A_2\unlhd B_2$ w.r.t. ${\mathfrak{P}}$, then $0\unlhd a A_1+b A_2 \unlhd a B_1+bB_2$ w.r.t. ${\mathfrak{P}}$ for all $a,b\in{\mathbb{R}}_+$. - Suppose that ${\mathfrak{P}}\cap {\mathrm{dom}}(A)$ is dense in ${\mathfrak{P}}$. If $0\unlhd A$ w.r.t. ${\mathfrak{P}}$, then $0\unlhd A^*$ w.r.t. ${\mathfrak{P}}$. [*Proof.*]{} (i) and (ii) are easy to see. \(iii) Let $x\in {\mathrm{dom}}(A) \cap {\mathfrak{P}}$ and let $y\in {\mathrm{dom}}(A^*) \cap {\mathfrak{P}}$. Then we have $$\begin{aligned} {\langle}x|A^*y{\rangle}={\langle}Ax|y{\rangle}\ge 0. \label{xAyP}\end{aligned}$$ Because ${\mathrm{dom}}(A) \cap {\mathfrak{P}}$ is dense in ${\mathfrak{P}}$, (\[xAyP\]) holds true for all $x\in {\mathfrak{P}}$. Thus, $A^*y \ge 0$, which implies that $A^*\unrhd 0$ w.r.t. ${\mathfrak{P}}$. $\Box$\ Let $\mathscr{B}({\mathfrak{H}})$ be the set of all bounded linear operators on ${\mathfrak{H}}$. \[Miura\] Let $A, B, C, D\in \mathscr{B}({\mathfrak{H}})$ and let $a, b\in {\mathbb{R}}$. - If $A \unrhd B \unrhd 0$ and $C\unrhd D \unrhd 0$ w.r.t. ${\mathfrak{P}}$, then $AC\unrhd BD \unrhd 0$ w.r.t. ${\mathfrak{P}}$. - If $A \unrhd 0 $ w.r.t. ${\mathfrak{P}}$, then $A^*\unrhd 0$ w.r.t. ${\mathfrak{P}}$. [*Proof.*]{} (i) By Lemma \[PPInq\] (i), we have $$\begin{aligned} AC-BD=\underbrace{A}_{\unrhd 0}\underbrace{(C-D)}_{\unrhd 0}+ \underbrace{(A-B)}_{\unrhd 0} \underbrace{D}_{\unrhd 0} \unrhd 0\ \ \ \mbox{w.r.t. ${\mathfrak{P}}$}.\end{aligned}$$ \(ii) follows from Lemma \[PPInq\] (iii). $\Box$\ \[WeakCl\] Let $\mathfrak{A}=\{A\in \mathscr{B}({\mathfrak{H}})\, |\, A\unrhd 0 \ \mbox{w.r.t. ${\mathfrak{P}}$}\}$. Then $\mathfrak{A}$ is a weakly closed convex cone. [*Proof.*]{} Let $\{A_n\}_{n=1}^{\infty}$ be a sequence in $\mathfrak{A}$. Assume that $A_n$ weakly converges to $A$. Take $x, y\in {\mathfrak{P}}$ arbitrarily. Because ${\langle}x|A_n y{\rangle}\ge 0$ for all $n\in {\mathbb{N}}$, we have ${\langle}x|Ay{\rangle}\ge 0$, which implies that $A\unrhd 0$ w.r.t. ${\mathfrak{P}}$. Thus, $\mathfrak{A}$ is weakly closed. $\Box$ Positivity improving operators ------------------------------ [ Let $A\in \mathscr{B}({\mathfrak{H}})$. We write $A\rhd 0$ w.r.t. ${\mathfrak{P}}$, if $Ax >0$ w.r.t. ${\mathfrak{P}}$ for all $x\in {\mathfrak{P}}\backslash \{0\}$. In this case, we say that [*$A$ improves the positivity w.r.t. ${\mathfrak{P}}$.*]{} $\diamondsuit$ ]{} The following theorem plays an important role. \[PFF\][(Perron–Frobenius–Faris)]{} Let $A$ be a self-adjoint positive operator on ${\mathfrak{H}}$. Suppose that $0\unlhd e^{-\beta A}$ w.r.t. ${\mathfrak{P}}$ for all $\beta \ge 0$, and that $\inf \mathrm{spec}(A)$ is an eigenvalue. Let $P_A$ be the orthogonal projection onto the closed subspace spanned by eigenvectors associated with $\inf \mathrm{spec}(A)$. Then, the following are equivalent: - $\dim {\mathrm{ran}}P_A=1$ and $P_A\rhd 0$ w.r.t. ${\mathfrak{P}}$. - $0\lhd e^{-\beta A}$ w.r.t. ${\mathfrak{P}}$ for all $\beta >0$. - For each $x, y\in {\mathfrak{P}}\backslash\{0\}$, there exists a $\beta>0$ such that ${\langle}x| e^{-\beta A} y{\rangle}>0$. [*Proof.*]{} See, e.g., [@Faris; @Miyao; @ReSi4]. $\Box$ [ (i) is equivalent to the following: The eigenvalue $\inf \mathrm{spec}(A)$ is simple with a strictly positive eigenvector. $\diamondsuit$ ]{} Second quantized operators {#2ndQuant} ========================== We briefly summarize necessary results concerning the second quantized operators. As to basic definitions, we refer to [@BraRob2] as an accessible text. Basic definitions ----------------- Let ${\mathfrak{H}}$ be a complex Hilbert space. The bosonic Fock space over ${\mathfrak{H}}$ is defined by $$\begin{aligned} {\mathfrak{F}}({\mathfrak{H}})=\Sumoplus {\mathfrak{F}}^{(n)}({\mathfrak{H}}), \ \ \ {\mathfrak{F}}^{(n)}({\mathfrak{H}})={\mathfrak{H}}^{\otimes_{\mathrm{s}}n},\end{aligned}$$ where ${\mathfrak{H}}^{\otimes_{\mathrm{s}}n}$ is the $n$-fold symmetric tensor product of ${\mathfrak{H}}$ with convention ${\mathfrak{H}}^{\otimes_{\mathrm{s}}0}={\mathbb{C}}$. ${\mathfrak{F}}^{(n)}({\mathfrak{H}})$ is called the [*$n$-boson subspace*]{}. A finite particle subspace ${\mathfrak{F}_{\mathrm{fin}}}({\mathfrak{H}})$ is defined by $$\begin{aligned} {\mathfrak{F}_{\mathrm{fin}}}({\mathfrak{H}})=\bigg\{{\varphi}=\Sumoplus {\varphi}_n \in {\mathfrak{F}}({\mathfrak{H}})\, \bigg|\, \mbox{$\exists N\in {\mathbb{N}}_0$ such that ${\varphi}_n=0$ for all $n\ge N$}\bigg\}.\end{aligned}$$ We denote by $a(f)\, (f\in {\mathfrak{H}})$ the annihilation operator on ${\mathfrak{F}}({\mathfrak{H}})$, its adjoint $a(f)^*$, called the creation operator, is defined by $$\begin{aligned} a(f)^*{\varphi}=\sideset{}{^{\oplus}_{n\ge 1}}\sum \sqrt{n}S_n (f\otimes {\varphi}_{n-1})\label{DefCrea}\end{aligned}$$ for ${\varphi}=\sum_{n\ge 0}^{\oplus} {\varphi}_n \in {\mathrm{dom}}(a(f)^*)$, where $S_n$ is the symmetrizer on ${\mathfrak{F}}^{(n)}({\mathfrak{H}})$. The annihilation- and creation operators satisfy the cannonical commutation relations (CCRs) $$\begin{aligned} [a(f), a(g)^*]={\langle}f| g{\rangle}, \ \ [a(f), a(g)]=0=[a(f)^*, a(g)^*]\end{aligned}$$ on ${\mathfrak{F}_{\mathrm{fin}}}({\mathfrak{H}})$. Let $C$ be a contraction operator on ${\mathfrak{H}}$, that is , $\|C\|\le 1$. Then we define a contraction operator $\Gamma(C)$ on ${\mathfrak{F}}({\mathfrak{H}})$ by $$\begin{aligned} \Gamma(C)=\Sumoplus C^{\otimes n}\end{aligned}$$ with $C^{\otimes 0}={1}$, the identity operator. For a self-adjoint operator $A$ on ${\mathfrak{H}}$, let us introduce $$\begin{aligned} {d\Gamma}(A)=0\oplus \sideset{}{^{\oplus}_{n\ge 1}}\sum \sideset{}{_{n\ge k\ge 1}}\sum {1}^{\otimes (k-1)}\otimes A \otimes {1}^{\otimes (n-k)} \end{aligned}$$ acting in ${\mathfrak{F}}({\mathfrak{H}})$. Then ${d\Gamma}(A)$ is essentially self-adjoint. We denote its closure by the same symbol. If $A$ is positive, then one has $$\begin{aligned} \Gamma(e^{-tA})=e^{-t {d\Gamma}(A)},\ \ \ t \ge 0. \label{SemiP2}\end{aligned}$$ The following proposition is well-known. Let $A$ be a positive self-adjoint operator. For each $f\in {\mathrm{dom}}(A^{-1/2})$, we have the following operator inequalities: $$\begin{aligned} a(f)^*a(f)&\le \|A^{-1/2}f\|^2 ({d\Gamma}(A)+{1}),\label{CreAnnInq}\\ a(f)a(f)^*&\le \|A^{-1/2}f\|^2 ({d\Gamma}(A)+{1}),\label{CreAnnInq2}\\ {d\Gamma}(A)+a(f)+a(f)^*&\ge -\|A^{-1/2}f\|^2. \label{VHove}\end{aligned}$$ Fock space over $L^2({\mathbb{R}}^3)$ ------------------------------------- In this study, the bosonic Fock space over $L^2({\mathbb{R}}^3)$ is important. We simply write it as $$\begin{aligned} {\mathfrak{F}}={\mathfrak{F}}(L^2({\mathbb{R}}^3)).\end{aligned}$$ The $n$-boson subspace ${\mathfrak{F}}^{(n)}=L^2({\mathbb{R}}^3)^{\otimes_{\mathrm{s}}n}$ is naturally identified with $ L^2_{\mathrm{sym}}({\mathbb{R}}^{3n}) $. Hence $$\begin{aligned} {\mathfrak{F}}={\mathbb{C}}\oplus \sideset{}{^{\oplus}_{n\ge 1}}\sum L^2_{\mathrm{sym}}({\mathbb{R}}^{3n}). \label{FockId}\end{aligned}$$ The annihilation- and creation operators are symbolically expressed as $$\begin{aligned} a(f)=\int_{{\mathbb{R}}^3}{d}k\, \overline{f(k)}a(k),\ \ a(f)^*=\int_{{\mathbb{R}}^3}{d}k\, f(k) a(k)^*.\end{aligned}$$ If $F$ is a multipilication operator by the function $F(k)$, then ${d\Gamma}(F)$ is formally written as $$\begin{aligned} {d\Gamma}(F)=\int_{{\mathbb{R}}^3}{d}k\, F(k)a(k)^* a(k).\end{aligned}$$ Note that ${d\Gamma}(F) \restriction L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})$ is a mutiplication operator by the function $F(k_1)+\cdots+F(k_n)$. The Fröhlich cone ----------------- Let ${\mathfrak{F}}_+$ be a convex cone defined by (\[FockCone\]). We begin with the following lemma: ${\mathfrak{F}}_+$ is a self-dual cone in ${\mathfrak{F}}$. [*Proof.*]{} It suffices to show that $L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})_+$ is a self-dual cone for all $n\in{\mathbb{N}}_0$. To this end, we set ${\mathfrak{P}}=L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})_+$. It is easy to check that ${\mathfrak{P}}\subseteq {\mathfrak{P}}^{\dagger}$. To prove the converse, we note the following fact: Let $\psi\in L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})$. $\psi\ge 0$ w.r.t. $L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})_+$ if and only if $$\begin{aligned} {\langle}f_1\otimes \cdots \otimes f_n|\psi{\rangle}\ge 0\label{PPEqui} \end{aligned}$$ for all $f_1, \dots, f_n\in L^2({\mathbb{R}}^{3})_+$, where $ ( f_1\otimes \cdots \otimes f_n )(k_1, \dots, k_n)=f_1(k_1)\cdots f_n(k_n) $. But it is easy to prove (\[PPEqui\]) for each $\psi\in {\mathfrak{P}}^{\dagger}$. $\Box$ [The self-dual cone ${\mathfrak{F}}_+$ is called the [*Fröhlich cone*]{}. $\diamondsuit$ ]{} We have the following: - $a(f)$ and $a(f)^*\in \mathscr{L}({\mathfrak{F}_{\mathrm{fin}}})$ for all $f\in L^2({\mathbb{R}}^3)$. - If $F$ is a multiplication operator such that $\|F\|_{\infty} \le 1$, then $\Gamma(F) \in \mathscr{L}({\mathfrak{F}_{\mathrm{fin}}})$. By using the above lemma, we can discuss operator inequalities given in Section \[SecMono\]. The following lemma will be useful. \[ClPos\] Let ${\mathfrak{F}}_{\mathrm{fin}, +}={\mathfrak{F}_{\mathrm{fin}}}\cap {\mathfrak{F}}_+$. Then $ \overline{{\mathfrak{F}}_{\mathrm{fin}, +}}={\mathfrak{F}}_+ $, where the bar indicates the closure in the strong topology. We summarize properties of operators on ${\mathfrak{F}}$ below. All propositions were proven in [@Miyao3]. For reader’s convenience, we will provide proofs. \[PPFockI\] Let $C$ be a contraction operator on $L^2({\mathbb{R}}^3)$. If $C\unrhd 0$ w.r.t. $L^2({\mathbb{R}}^3)_+$, then we have $\Gamma(C)\unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. [*Proof.*]{} Let $f_1, \dots, f_n\in L^2({\mathbb{R}}^3)_+$. Because $C\unrhd 0$ w.r.t. $L^2({\mathbb{R}}^3)_+$, we have $Cf_j \ge 0$ w.r.t. $L^2({\mathbb{R}}^3)_+$, which implies that $Cf_1\otimes \cdots \otimes Cf_n \ge 0$ w.r.t. $L^2({\mathbb{R}}^{3n})_+$. Thus, $$\begin{aligned} {\langle}f_1\otimes \cdots \otimes f_n|C^{\otimes n}\psi{\rangle}= {\langle}C f_1\otimes \cdots \otimes C f_n|\psi{\rangle}\ge 0 \end{aligned}$$ for all $f_1, \dots, f_n\in L^2({\mathbb{R}}^{3})_+$, which implies that $C^{\otimes n} \unrhd 0$ w.r.t. $L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})_+$. $\Box$ \[PPFock3\] Let $B$ be a positive self-adjoint operator. If $e^{- t B}\unrhd 0$ w.r.t. $L^2({\mathbb{R}}^3)_+$ for all $t\ge 0$, then $e^{-t {d\Gamma}(B)}\unrhd 0$ w.r.t. ${\mathfrak{F}}_+$ for all $t\ge 0$. [*Proof.*]{} By (\[SemiP2\]) and Proposition \[PPFockI\], we obtain the desired assertion. $\Box$ \[PPFockII\] If $f \ge 0$ w.r.t. $L^2({\mathbb{R}}^3)_+$, then $a(f)^*\unrhd 0$ and $a(f)\unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. [*Proof*]{}. Let ${\varphi}=\Sumoplus {\varphi}_n\in {\mathfrak{F}}_+\cap {\mathrm{dom}}(a(f)^*)$. By (\[DefCrea\]), we have $$\begin{aligned} \big( a(f)^* {\varphi}\big)_{n+1} (k_1, \dots, k_{n+1})= \frac{1}{\sqrt{n+1}} \sum_{j=1}^{n+1} \underbrace{f(k_j)}_{\ge 0} \underbrace{{\varphi}_n(k_1, \dots, \hat{k}_j, \dots, k_{n+1})}_{\ge 0} \ge 0,\end{aligned}$$ where $\hat{k}_j$ indicates the omission of $k_j$. Thus, $a(f)^*\unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. Because $a(f)=(a(f)^*)^*$, we have $a(f) \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$ by Lemmas \[PPInq\] (iii) and \[ClPos\]. $\Box$ \[ErgoFock\][(Ergodicity)]{} For each $f\in L^2({\mathbb{R}}^3)$, let $\phi(f)$ be a linear operator defined by $$\begin{aligned} \phi(f)=a(f)+a(f)^*.\end{aligned}$$ Note that $\phi(f)$ is essentially self-adjoint. We denote its closure by the same symbol. If $f >0$ w.r.t. $L^2({\mathbb{R}}^3)_+$, that is, $f(k)>0$ a.e. $k$, then $\phi(f)$ is ergodic in the sense that, for any ${\varphi}, \psi\in {\mathfrak{F}}_{\mathrm{fin}, +}\backslash \{0\}$, there exists an $n\in {\mathbb{N}}_0$ such that ${\langle}{\varphi}|\phi(f)^n \psi{\rangle}>0$. [*Proof.*]{} Choose ${\varphi}, \psi\in {\mathfrak{F}}_{\mathrm{fin}, +}\backslash \{0\}$, arbitrarily. We can express ${\varphi}$ and $\psi$ as $$\begin{aligned} {\varphi}=\Sumoplus {\varphi}_n,\ \ \ \psi=\Sumoplus \psi_n.\end{aligned}$$ Because ${\varphi}$ and $\psi$ are non-zero, there exist $p, q\in {\mathbb{N}}_0$ such that ${\varphi}_p \neq 0$ and $\psi_q\neq 0$. Under the identifications $$\begin{aligned} {\varphi}_p=\Sumoplus \delta_{np} {\varphi}_n,\ \ \ \psi_q=\Sumoplus \delta_{nq}\psi_n, \label{FinIdn}\end{aligned}$$ we have ${\varphi}\ge {\varphi}_p$ and $\psi\ge \psi_q$ w.r.t. ${\mathfrak{F}}_+$, where $\delta_{mn}$ is the Kronecker delta. By Proposition \[PPFockII\], we have $$\begin{aligned} {\langle}{\varphi}|\phi(f)^{p+q} \psi{\rangle}\ge {\langle}{\varphi}_p|\phi(f)^{p+q} \psi_q{\rangle}. \label{ErgPf1}\end{aligned}$$ Because $\phi(f)^p\unrhd a(f)^p$ and $\phi(f)^q \unrhd a(f)^q$ w.r.t. ${\mathfrak{F}}_+$, we have $$\begin{aligned} \mbox{the RHS of (\ref{ErgPf1})} \ge {\langle}a(f)^p {\varphi}_p| a(f)^q \psi_q{\rangle}.\label{ErgPf2}\end{aligned}$$ Remark that $$\begin{aligned} a(f)^p{\varphi}_p=\sqrt{p!} {\langle}f^{\otimes p}|{\varphi}_p{\rangle}\Omega,\ \ \ a(f)^q\psi_q=\sqrt{q!} {\langle}f^{\otimes q}|\psi_q{\rangle}\Omega,\end{aligned}$$ where $\Omega=1\oplus 0\oplus0\oplus \cdots$ is the Fock vacuum. Since $ {\langle}f^{\otimes p}|{\varphi}_p{\rangle}>0 $ and $ {\langle}f^{\otimes q}|\psi_q{\rangle}>0 $, we get, by (\[ErgPf1\]) and (\[ErgPf2\]), $$\begin{aligned} {\langle}{\varphi}|\phi(f)^{p+q} \psi{\rangle}\ge \sqrt{p! q!} {\langle}f^{\otimes p}|{\varphi}_p{\rangle}{\langle}f^{\otimes q}|\psi_q{\rangle}>0.\end{aligned}$$ Thus we are done. $\Box$ Local properties ---------------- Let $B_{\kappa}$ be a ball of radius $\kappa$ in ${\mathbb{R}}^3$ centered at the origin and let $\chi_{\kappa}$ be a function on ${\mathbb{R}}^3$ defined by $\chi_{\kappa}(k)=1$ if $k \in B_{\kappa}$ and $\chi_{\kappa}(k)=0$ otherwise. Then as a multiplication operator, $\chi_{\kappa}$ is an orthogonal projection on $L^2({\mathbb{R}}^3)$ and $Q_{\kappa}=\Gamma(\chi_{\kappa})$ is an orthogonal projection on ${\mathfrak{F}}$ as well. We remark the following properties: - If $\kappa_1 \ge \kappa_2$, then $Q_{\kappa_1} \ge Q_{\kappa_2}$. - $Q_{\kappa}$ strongly converges to $1$ as $\kappa\to \infty$. Let us define the local Fock space by $$\begin{aligned} {\mathfrak{F}}^{\le \kappa}=Q_{\kappa}{\mathfrak{F}}.\end{aligned}$$ Since $\chi_{\kappa}L^2({\mathbb{R}}^3)=L^2(B_{\kappa})$, ${\mathfrak{F}}^{\le \kappa}$ can be identified with ${\mathfrak{F}}(L^2(B_{\kappa}))$. In what follows, ${\mathfrak{F}_{\mathrm{fin}}}^{\le \kappa}$ denotes ${\mathfrak{F}_{\mathrm{fin}}}(L^2(B_{\kappa}))$. The following fact will be useful: $$\begin{aligned} {\mathfrak{F}}=\overline{\bigcup_{\kappa\ge 0} {\mathfrak{F}}^{\le \kappa}}.\end{aligned}$$ \[QPP\] For each $\kappa \ge 0$, we set $Q_{\kappa}^{\perp}={1}-Q_{\kappa}$. Then we have the following: - $Q_{\kappa}\unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. - $Q_{\kappa}^{\perp} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. [*Proof.*]{} (i) immediately follows from Proposition \[PPFockI\]. \(ii) Under the identification (\[FockId\]), we see $$\begin{aligned} (Q_{\kappa}{\varphi}_n)(k_1, \dots, k_n)=\Bigg[ \prod_{j=1}^n\chi_{\kappa}(k_j)\Bigg] {\varphi}_n(k_1,\dots, k_n)\end{aligned}$$ for each ${\varphi}_n\in L^2_{\mathrm{sym}}({\mathbb{R}}^{3n})$. Hence $$\begin{aligned} (Q^{\perp}_{\kappa}{\varphi}_n)(k_1,\dots, k_n)=\Bigg\{1-\prod_{j=1}^n \chi_{\kappa}(k_j)\Bigg\} {\varphi}_n(k_1,\dots, k_n). \label{OrthQ}\end{aligned}$$ If ${\varphi}_n(k_1,\dots, k_n)\ge 0$ a.e., then the right hand side of (\[OrthQ\]) is positive for a.e. $k_1, \dots, k_n$ because $1-\prod_{j=1}^n \chi_{\kappa}(k_j)\ge 0$. This means that $Q^{\perp}_{\kappa} \unrhd 0$ w.r.t. ${\mathfrak{F}}_{+}$. $\Box$\ We remark the following: $$\begin{aligned} a(f)Q_{\kappa}&=a(\chi_{\kappa}f)=\int_{|k|\le \kappa}{d}k\, \overline{f(k)} a(k),\\ Q_{\kappa} a(f)^*&=a(\chi_{\kappa}f)^*=\int_{|k|\le \kappa}{d}k\, f(k) a(k)^*,\\ {d\Gamma}(F) Q_{\kappa}&={d\Gamma}(\chi_{\kappa}F)=\int_{|k|\le \kappa}{d}k\, F(k)a(k)^*a(k).\end{aligned}$$ By these facts, we obtain the following proposition. \[CommuL\] We have the following: - $[Q_{\kappa}, a(f)]=Q_{\kappa} a((1-\chi_{\kappa})f)$ on ${\mathfrak{F}_{\mathrm{fin}}}$. - $[Q_{\kappa}, {d\Gamma}(F)]=0$ on ${\mathrm{dom}}({d\Gamma}(F))$. Next let us introduce a natural self-dual cone in ${\mathfrak{F}}^{\le \kappa}$. To this end, define $$\begin{aligned} {\mathfrak{F}}^{\le \kappa}_{n, +}=\big\{ {\varphi}\in L^2_{\mathrm{sym}}(B_{\kappa}^{\times n})\, |\, {\varphi}(k_1,\dots, k_n) \ge 0\ \ a.e. \big\}\end{aligned}$$ with ${\mathfrak{F}}_{0, +}^{\le \kappa}={\mathbb{R}}_+$. Each ${\mathfrak{F}}_{ n, +}^{\le \kappa}$ is a self-dual cone in $L^2(B_{\kappa})^{\otimes_{\mathrm{s}}n}=L^2_{\mathrm{sym}}(B_{\kappa}^{\times n})$. [The [*local Fröhlich cone* ]{} is defined by $$\begin{aligned} {\mathfrak{F}}_{+}^{\le \kappa}=\Sumoplus {\mathfrak{F}}_{n, +}^{\le \kappa}.\end{aligned}$$ ${\mathfrak{F}}_{+}^{\le \kappa}$ is a self-dual cone in ${\mathfrak{F}}^{\le \kappa}$. As before, we define ${\mathfrak{F}}_{\mathrm{fin}, +} ^{\le \kappa}={\mathfrak{F}_{\mathrm{fin}}}^{\le \kappa } \cap {\mathfrak{F}}_+^{\le \kappa}$. Note that $ \overline{{\mathfrak{F}}_{\mathrm{fin}, +} ^{\le \kappa}}={\mathfrak{F}}_+^{\le \kappa} $. $\diamondsuit$ ]{} \[LocalPropErgo\] Propositions \[PPFockI\], \[PPFock3\], \[PPFockII\] and \[ErgoFock\] are still true even if one replaces $L^2({\mathbb{R}}^3)_+$, ${\mathfrak{F}}_+$ and ${\mathfrak{F}}_{\mathrm{fin}, +}$by $L^2(B_{\kappa})_+$, ${\mathfrak{F}}_{ +}^{\le \kappa}$ and ${\mathfrak{F}}_{\mathrm{fin}, +} ^{\le \kappa}$, respectively. Decomposition properties ------------------------ Let $\mathfrak{h}_1$ and $\mathfrak{h}_2$ be complex Hilbert spaces. Remark the following factorization property: $$\begin{aligned} {\mathfrak{F}}(\mathfrak{h}_1\oplus \mathfrak{h}_2)={\mathfrak{F}}(\mathfrak{h}_1)\otimes {\mathfrak{F}}(\mathfrak{h}_2) . \label{GFact}\end{aligned}$$ Corresponding to this, we have the following: - For each $ f\in \mathfrak{h}_1,\ g\in \mathfrak{h}_2 $, $$\begin{aligned} a(f\oplus g)=a(f)\otimes {1}+{1}\otimes a(g). \label{Fac1}\end{aligned}$$ - Let $A$ and $B$ be self-adjoint operators. We have $$\begin{aligned} {d\Gamma}(A\oplus B)=\{{d\Gamma}(A)\otimes {1}+{1}\otimes {d\Gamma}(B)\}^-,\label{Fac2}\end{aligned}$$ where $\{\cdots\}^-$ indicates the closure of $\{\cdots\}$. - Let $C$ and $D$ be contraction operators. We have $$\begin{aligned} \Gamma(C\oplus D)=\Gamma(C)\otimes \Gamma(D). \end{aligned}$$ For each $\kappa >0$, we have the following identification: $$\begin{aligned} L^2({\mathbb{R}}^3)=L^2(B_{\kappa}) \oplus L^2(B_{\kappa}^c), \label{L2Dec}\end{aligned}$$ where $B_{\kappa}^c$ indicates the complement of $B_{\kappa}$. Using (\[GFact\]) and (\[L2Dec\]), we have $$\begin{aligned} {\mathfrak{F}}=&{\mathfrak{F}}^{\le \kappa}\otimes {\mathfrak{F}}^{>\kappa}, \label{FactFock}\end{aligned}$$ where ${\mathfrak{F}}^{>\kappa}={\mathfrak{F}}(L^2(B_{\kappa}^c))$. Thus, we have $$\begin{aligned} {\mathfrak{F}}=&\Sumoplus {\mathfrak{F}}^{\le \kappa}\otimes L^2_{\mathrm{sym}}((B_{\kappa}^c)^{\times n}){\nonumber \\}=&{\mathfrak{F}}^{\le \kappa} \oplus \bigg[ \sideset{}{^{\oplus}_{n\ge 1}}\sum {\mathfrak{F}}^{\le \kappa}\otimes L^2_{\mathrm{sym}}((B_{\kappa}^c)^{\times n}) \bigg], \label{Iden1}\end{aligned}$$ where $ L^2_{\mathrm{sym}}((B_{\kappa}^c)^{\times 0}):={\mathbb{C}}$. The following lemma will be useful. \[QAct\] Let $\psi =\Sumoplus \psi_n(k_1, \dots, k_n)\in {\mathfrak{F}}$. For each $\kappa>0$, we have $$\begin{aligned} Q_{\kappa} \psi=\psi_{\kappa}\otimes \Omega^{>\kappa}, \label{ProjQ}\end{aligned}$$ where $\Omega^{>\kappa}$ is the Fock vacuum in ${\mathfrak{F}}^{>\kappa}$ and $$\begin{aligned} \psi_{\kappa}=\Sumoplus \Bigg[\prod_{\ell=1}^n \chi_{\kappa}(k_{\ell})\Bigg] \psi_n(k_1, \dots, k_n). \label{Defpsik}\end{aligned}$$ A natural self-dual cone in ${\mathfrak{F}}^{>\kappa}$ is given by $$\begin{aligned} {\mathfrak{F}}_+^{>\kappa}=\Sumoplus L^2_{\mathrm{sym}}((B_{\kappa}^c)^{\times n})_+,\end{aligned}$$ where $ L^2_{\mathrm{sym}}((B_{\kappa}^c)^{\times 0})_+:={\mathbb{R}}_+ $. As before, we set ${\mathfrak{F}_{\mathrm{fin}}}^{>\kappa}={\mathfrak{F}_{\mathrm{fin}}}(L^2(B_{\kappa}^c))$ and ${\mathfrak{F}}_{\mathrm{fin}, +}^{>\kappa}={\mathfrak{F}_{\mathrm{fin}}}^{>\kappa} \cap {\mathfrak{F}}_+^{>\kappa}$. Propositions \[PPFockI\], \[PPFock3\], \[PPFockII\] and \[ErgoFock\] are still true even if one replaces $L^2({\mathbb{R}}^3)_+$, ${\mathfrak{F}}_+$ and ${\mathfrak{F}}_{\mathrm{fin}, +}$by $L^2(B_{\kappa}^c)_+$, ${\mathfrak{F}}_{ +}^{> \kappa}$ and ${\mathfrak{F}}_{\mathrm{fin}, +} ^{> \kappa}$, respectively. The self-dual cone ${\mathfrak{F}}_+$ can be expressed as $$\begin{aligned} {\mathfrak{F}}_+={\mathfrak{F}}^{\le \kappa}_+ \oplus \bigg[\sideset{}{^{\oplus}_{n\ge 1}}\sum {\mathfrak{F}}^{\le \kappa}_+\otimes L^2_{\mathrm{sym}}((B_{\kappa}^c)^{\times n})_+\bigg], \label{DSumSC}\end{aligned}$$ where $$\begin{aligned} {\mathfrak{F}}^{\le \kappa}_+\otimes L^2_{\mathrm{sym}}((B_{\kappa}^c)^{\times n})_+ =\Big\{\psi\in {\mathfrak{F}}^{\le \kappa}\otimes L^2_{\mathrm{sym}}((B_{\kappa}^c)^{\times n})\, \Big|\, \psi(k_1, \dots, k_n) \ge0\ \mbox{w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$ a.e.}\Big\}.\end{aligned}$$ \[PCup\] We have the following: - $Q_{\kappa}{\mathfrak{F}}_+={\mathfrak{F}}^{\le \kappa}_+$. - $\displaystyle {\mathfrak{F}}_+=\overline{\bigcup_{\kappa >0} {\mathfrak{F}}_+^{\le \kappa}} $. [*Proof.*]{} (i) This immediately follows from (\[DSumSC\]). \(ii) With the identification $ {\mathfrak{F}}_+^{\le \kappa} = {\mathfrak{F}}_+^{\le \kappa} \oplus \{0\} $, we know that ${\mathfrak{F}}_+\supseteq {\mathfrak{F}}_+^{\le \kappa} $ by (\[DSumSC\]). Hence, $ \displaystyle {\mathfrak{F}}_+\supseteq \overline{\bigcup_{\kappa>0} {\mathfrak{F}}_+^{\le \kappa}} $. Let $\psi\in {\mathfrak{F}}_+$. For each $\kappa>0$, we know that $Q_{\kappa} \psi\in {\mathfrak{F}}_+^{\le \kappa}$ by (\[ProjQ\]). Because $Q_{\kappa}$ strongly converges to ${1}$ as $\kappa\to \infty$, we conclude that $\psi \in \overline{\bigcup_{\kappa>0} {\mathfrak{F}}_+^{\le \kappa}}$. $\Box$ \[TensPP\] Let $\psi\in {\mathfrak{F}}$. The following (i) and (ii) are equivalent: - $\psi\ge 0$ w.r.t. ${\mathfrak{F}}_+$. - ${\langle}\xi \otimes \eta|\psi{\rangle}\ge 0$ for all $\xi\in {\mathfrak{F}}_+^{\le\kappa}$ and $\eta \in {\mathfrak{F}}_+^{>\kappa}.$ [*Proof.*]{} (ii) $\Longrightarrow$ (i): Without loss of generality, we may assume that $\psi\in {\mathfrak{F}_{\mathrm{fin}}}$. Thus, it suffices to consider the case where $\psi=\psi_1\otimes \psi_2$ with $\psi_1\in {\mathfrak{F}_{\mathrm{fin}}}^{\le \kappa}$ and $\psi_2\in {\mathfrak{F}_{\mathrm{fin}}}^{>\kappa}$. Because $ {\langle}\xi\otimes \eta|\psi{\rangle}={\langle}\xi|\psi_1{\rangle}{\langle}\eta|\psi_2{\rangle}\ge 0 $, we can choose $\psi_1$ and $\psi_2$ such that $\psi_1\ge 0$ w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$ and $\psi_2\in {\mathfrak{F}}_+^{>\kappa}$. Thus, we conclude that $\psi\ge 0$ w.r.t. ${\mathfrak{F}}_+$. \(i) $\Longrightarrow$ (ii): By arguments similar to those in the above, it suffices to consider the case $ \psi=\psi_1\otimes \psi_2 $ with $\psi_1\in {\mathfrak{F}}_+^{\le \kappa}$ and $\psi_2\in {\mathfrak{F}}_+^{>\kappa}$. In this case, we easily check that ${\langle}\xi \otimes \eta|\psi{\rangle}\ge 0$ for all $\xi\in {\mathfrak{F}}_+^{\le\kappa}$ and $\eta \in {\mathfrak{F}}_+^{>\kappa}.$ $\Box$ \[TensPPAB\] Let $A\in \mathscr{B}({\mathfrak{F}}^{\le \kappa})$ and $B\in \mathscr{B}({\mathfrak{F}}^{> \kappa})$. If $A\unrhd 0$ w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$ and $B\unrhd 0$ w.r.t. ${\mathfrak{F}}_+^{>\kappa}$, then $A\otimes B\unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. [*Proof.*]{} Let $\xi\in {\mathfrak{F}}_+^{\le \kappa}$ and let $\eta\in {\mathfrak{F}}_+^{>\kappa}$. By the assumption, we have $A^*\xi\ge 0$ and $B^*\eta\ge 0$. Thus, by Lemma \[TensPP\], $$\begin{aligned} {\langle}\xi \otimes \eta|A\otimes B\psi{\rangle}={\langle}(A^* \xi)\otimes (B^*\eta)| \psi{\rangle}\ge 0.\end{aligned}$$ By Lemma \[TensPP\] again, we have $A\otimes B\psi\ge 0$ w.r.t. ${\mathfrak{F}}_+$. $\Box$ Proof of Theorem \[Main1\] {#Pf1} ========================== Decomposition of ${H_{\mathrm{ren}, \Lambda}}(P)$ ------------------------------------------------- In what follows, we always assume that $\kappa <\Lambda$. Let $F$ be a real-valued measurable function on ${\mathbb{R}}^3$. Suppose that $F(k)$ is finite for almost everywhere. Then ${d\Gamma}(F)$ is essentially self-adjoint. For each $\kappa>0$, we set $F^{\le \kappa}=\chi_{\kappa} F$ and $F^{>\kappa}=(1-\chi_{\kappa}) F$. By (\[Fac2\]) and (\[FactFock\]), we have $$\begin{aligned} {d\Gamma}(F)=\{{d\Gamma}(F^{\le \kappa}) \otimes {1}+{1}\otimes {d\Gamma}(F^{>\kappa})\}^{-}.\end{aligned}$$ Keeping this fact in mind, we set $$\begin{aligned} P_{\mathrm{f}, j}^{\le \kappa}={d\Gamma}(k_j \chi_{\kappa}),\ \ P_{\mathrm{f}, j}^{> \kappa}={d\Gamma}(k_j (1-\chi_{\kappa})),\ \ j=1,2,3.\end{aligned}$$ Remark the following formulas: $$\begin{aligned} {d\Gamma}(\omega)&={d\Gamma}(\omega^{\le \kappa}) \otimes 1+1\otimes {d\Gamma}(\omega^{>\kappa}), \label{Dec1}\\ P_{\mathrm{f}, j} &=\{P_{\mathrm{f}, j}^{\le \kappa}\otimes {1}+{1}\otimes P_{\mathrm{f}, j}^{> \kappa}\}^-,\label{Dec2}\\ a(f)&= a(\chi_{\kappa} f)\otimes 1+1\otimes a((1-\chi_{\kappa})f).\label{Dec3}\end{aligned}$$ Let $$\begin{aligned} E^{\Lambda}_{\kappa} =-g^2\int_{{\mathbb{R}}^3} dk \frac{\chi_{\kappa}^{\Lambda}(k)}{\omega(k)\{\omega(k)+k^2/2\}},\ \ \ \chi_{\kappa}^{\Lambda}=\chi_{\Lambda}-\chi_{\kappa}.\end{aligned}$$ Note that $E_{\kappa}^{\Lambda}=E_{\Lambda}-E_{\kappa}$, where $E_{\Lambda}$ is defined by (\[DefE\_k\]), while $E_{\kappa}$ is defined by (\[DefE\_k\]) with $\Lambda$ replaced by $\kappa$. Using (\[Dec1\]), (\[Dec2\]) and (\[Dec3\]), we have $$\begin{aligned} {H_{\mathrm{ren}, \Lambda}}(P) =H_{\mathrm{ren}}^{\le \kappa}(P)\otimes{1}+{1}\otimes K_{\kappa, \Lambda} -(P-{P_{\mathrm{f}}}^{\le \kappa}) \cdot {P_{\mathrm{f}}}^{>\kappa} ,\end{aligned}$$ where $$\begin{aligned} {H_{\mathrm{ren}}}^{\le \kappa}(P)=&\frac{1}{2}(P-{P_{\mathrm{f}}}^{\le\kappa})^2-g \int_{{\mathbb{R}}^3}dk\frac{\chi_{\kappa}(k)}{\sqrt{\omega(k)}} (a(k)+a(k)^*) +{d\Gamma}(\omega^{\le \kappa}) -E_{\kappa},\\ K_{\kappa, \Lambda} =& \frac{1}{2}({P_{\mathrm{f}}}^{>\kappa})^2-g \int_{{\mathbb{R}}^3}dk\frac{\chi^{\Lambda}_{\kappa}(k)}{\sqrt{\omega(k)}} (a(k)+a(k)^*) +{d\Gamma}(\omega^{>\kappa}) -E_{\kappa}^{\Lambda}\end{aligned}$$ and $$\begin{aligned} (P-{P_{\mathrm{f}}}^{\le \kappa}) \cdot {P_{\mathrm{f}}}^{>\kappa}=\sum_{j=1}^3 (P_j-P_{\mathrm{f}, j}^{\le \kappa}) \otimes P_{\mathrm{f}, j}^{>\kappa}.\end{aligned}$$ $e^{-\beta {H_{\mathrm{ren}}}(P)}$ is positivity preserving w.r.t. ${\mathfrak{F}}_+$ -------------------------------------------------------------------------------------- In this subsection, we will show the following proposition. \[GlobalPP\] For all $ P\in {\mathbb{R}}^3 $ and $\beta\ge 0$, we have $e^{-\beta {H_{\mathrm{ren}}}(P)}\unrhd 0$ w.r.t. $\mathfrak{F}_{ +}$. ### Proof of Proposition \[GlobalPP\] \[BasicPP1\] We have the following: - $e^{-\beta {d\Gamma}(\omega)} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$ for all $\beta \ge 0$. - $e^{-\beta (P-{P_{\mathrm{f}}})^2/2} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$ for all $P\in {\mathbb{R}}^3$ and $\beta \ge 0$. [*Proof.*]{} (i) Note that $e^{-\beta \omega} \unrhd 0$ w.r.t. $L^2({\mathbb{R}}^3)_+$ for all $\beta \ge 0$. By Proposition \[PPFock3\], we obtain (i). \(ii) Note that $$\begin{aligned} e^{-\beta (P-{P_{\mathrm{f}}})^2/2}=\Sumoplus e^{-\beta(P-k_1-\cdots-k_n)^2/2}.\end{aligned}$$ Since each multiplication operator $ e^{-\beta(P-k_1-\cdots-k_n)^2/2} $ preserves the positivity w.r.t. $L^2_{\mathrm{sym}} ({\mathbb{R}}^{3n})_+$, we conclude (ii). $\Box$\ \[CPP\] $e^{-\beta {H_{\mathrm{ren}, \Lambda}}(P)} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$ for all $P\in {\mathbb{R}}^3$, $\beta \ge 0$ and $\Lambda>0$. [*Proof.*]{} By Proposition \[PPFockII\] and Lemma \[BasicPP1\], we can apply Proposition \[BasicPertPP\] with $A=\frac{1}{2}(P-{P_{\mathrm{f}}})^2+{d\Gamma}(\omega)$ and $B=-\{a(f)+a(f)^*\},\ \ f=g\frac{\chi_{\Lambda}}{\sqrt{\omega}}$. $\Box$ [ *Proof of Proposition \[GlobalPP\]* ]{} Because $e^{-\beta {H_{\mathrm{ren}, \Lambda}}(P)}$ strongly converges to $e^{-\beta {H_{\mathrm{ren}}}(P)}$, the assertion follows from Proposition \[WeakCl\] and Lemma \[CPP\]. $\Box$ $e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)}$ is positivity improving w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$ -------------------------------------------------------------------------------------------------------------- Our goal here is to prove the following. \[LocalPI\] For all $P\in {\mathbb{R}}^3, \beta >0$ and $\kappa>0$, we have $ e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \rhd 0 $ w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$. ### Proof of Proposition \[LocalPI\] Using arguments similar to those in the proof of Lemma \[BasicPP1\], we have the following. \[BasicRP1\] We have the following: - $e^{-\beta {d\Gamma}(\omega^{\le \kappa })} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$ for all $\beta \ge 0$. - $e^{-\beta (P-{P_{\mathrm{f}}}^{\le \kappa})^2/2} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$ for all $P\in {\mathbb{R}}^3$ and $\beta \ge 0$. \[LocalPP\] For all $P\in {\mathbb{R}}^3, \beta >0$ and $\kappa>0$, we have $ e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P) }\unrhd 0 $ w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$. [*Proof.*]{} By Proposition \[LocalPropErgo\] and Lemma \[BasicRP1\], we can apply Proposition \[BasicPertPP\] with $A=\frac{1}{2}(P-{P_{\mathrm{f}}}^{\le \kappa})^2+{d\Gamma}(\omega^{\le \kappa})$ and $B=-\{a(F)+a(F)^*\},\ \ F=g\frac{\chi_{\kappa}}{\sqrt{\omega}}$. $\Box$ [ *Proof of Proposition \[LocalPI\]* ]{} Let $\displaystyle F=g\frac{\chi_{\kappa}}{\sqrt{\omega}}$. Because $F(k)>0$ on $B_{\kappa}$, $\phi(F)=a(F)+a(F)^*$ is ergodic w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$ by Proposition \[LocalPropErgo\]. Let ${\varphi}, \psi\in {\mathfrak{F}}_+^{\le \kappa} \backslash \{0\}$. We can express ${\varphi}$ and $\psi$ as $ {\varphi}=\Sumoplus {\varphi}_n $ and $\psi=\Sumoplus \psi_n$. Since ${\varphi}$ and $\psi$ are non-zero, there exist $n_1, n_2\in {\mathbb{N}}_0$ such that ${\varphi}_{n_1}\neq 0$ and $\psi_{n_2} \neq 0$. By the identifications similar to (\[FinIdn\]) and the ergodicity of $\phi(F)$, there exists an $\ell\in {\mathbb{N}}_0$ such that $$\begin{aligned} {\langle}{\varphi}_{n_1}|\phi(F)^{\ell} \psi_{n_2}{\rangle}>0. \label{ErgP} \end{aligned}$$ Since ${\varphi}\ge {\varphi}_{n_1}$ and $\psi\ge \psi_{n_2}$ w.r.t. ${\mathfrak{F}}_+^{\le \kappa}$, we have $$\begin{aligned} {\langle}{\varphi}|e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \psi{\rangle}\ge {\langle}{\varphi}_{n_1}|e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \psi_{n_2}{\rangle}\label{Red1}\end{aligned}$$ for all $\beta \ge 0$, by Lemma \[LocalPP\]. Let $H_0=\frac{1}{2}(P-{P_{\mathrm{f}}}^{\le \kappa})^2+{d\Gamma}(\omega^{\le \kappa})$. By the Duhamel formula, we obtain $$\begin{aligned} e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} =\sum_{j=0}^{\ell}D_j+R_{\ell}\ \ \ \mbox{on ${\mathfrak{F}_{\mathrm{fin}}}^{\le \kappa}$},\end{aligned}$$ where $$\begin{aligned} D_j=&\int_0^{t}{d}s_1 \int_0^{t-s_1}{d}s_2 \cdots \int_0^{t-\sum_{i=1}^{j-1}s_i}{d}s_j\times {\nonumber \\}&\times e^{-s_1 H_0} \phi(F)e^{-s_2 H_0}\cdots e^{- s_j H_0} \phi(F)e^{-(t-\sum_{i=1}^{j}s_i)H_0},\\ R_{\ell}=&\int_0^{t}{d}s_1 \int_0^{t-s_1}{d}s_2 \cdots \int_0^{t-\sum_{i=1}^{\ell}s_i}{d}s_{\ell+1} \times {\nonumber \\}& \times e^{-s_1 H_0} \phi(F) e^{-s_2 H_0}\cdots e^{- s_{\ell} H_0} \phi(F) e^{-(t-\sum_{i=1}^{\ell+1}s_i){H_{\mathrm{ren}}}^{\le \kappa}(P)}.\end{aligned}$$ Because $e^{-s H_0} \unrhd 0$ and $\phi(F) \unrhd 0$ w.r.t. ${\mathfrak{F}}^{\le \kappa}_+$, we know that ${\langle}{\varphi}_{n_1}|D_j\psi_{n_2}{\rangle}\ge 0$. Similarly, by Lemma \[LocalPP\], we have ${\langle}{\varphi}_{n_1}|R_{\ell} \psi_{n_2} {\rangle}\ge 0$. Hence, $$\begin{aligned} {\langle}{\varphi}_{n_1}|e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \psi_{n_2}{\rangle}\ge {\langle}{\varphi}_{n_1}|D_{\ell} \psi_{n_2}{\rangle}. \label{Red2}\end{aligned}$$ Let $G(s_1, \dots, s_{\ell})={\langle}{\varphi}_{n_1}| e^{-s_1 H_0} \phi(F)e^{-s_2 H_0}\cdots e^{- s_{\ell} H_0} \phi(F)e^{-(t-\sum_{i=1}^{\ell}s_i)H_0} \psi_{n_2}{\rangle}$. By (\[ErgP\]), we see that $G(0, \dots, 0)>0$. Because $G(s_1, \dots, s_{\ell})$ is positive and continuous, we have $$\begin{aligned} {\langle}{\varphi}_{n_1}|D_{\ell} \psi_{n_2}{\rangle}=\int_0^{t}{d}s_1 \int_0^{t-s_1}{d}s_2 \cdots \int_0^{t-\sum_{i=1}^{\ell-1}s_i}{d}s_{\ell} G(s_1, \dots, s_{\ell})>0.\label{IntPI}\end{aligned}$$ Combining (\[Red1\]), (\[Red2\]) and (\[IntPI\]), we arrive at ${\langle}{\varphi}|e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \psi{\rangle}>0$ for all $\beta >0$. $\Box$ Basic properties of $K_{\kappa, \Lambda}$ ------------------------------------------ In this subsection, we will show the following. \[KExt\] For all $\kappa>0$, there exists a self-adjoint operator $K_{\kappa}$ bounded from below such that - $e^{-\beta K_{\kappa, \Lambda}}$ strongly converges to $e^{-\beta K_{\kappa}}$ for all $\beta \ge 0$, as $\Lambda\to \infty$; - $e^{-\beta K_{\kappa}} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+^{>\kappa}$ for all $\beta \ge 0$. ### Proof of Proposition \[KExt\] (i) We will apply Nelson’s idea [@Nelson]. Choose $K$ such that $\kappa<K<\Lambda$. Let $$\begin{aligned} \beta(k)=g\frac{1-\chi_{K}(k)}{\omega(k)^{1/2}\{ \omega(k)+k^2/2 \}}. \end{aligned}$$ We define an anti-self-adjoint operator $T$ by $$\begin{aligned} T=\{ a(G)-a(G)^*\}^-,\ \ \ G=\beta \chi_{\kappa}^{\Lambda}.\end{aligned}$$ The unitary operator $e^T$ is called the [*Gross transformation*]{}, which was introduced in [@EPGross]. We can check the following (For notational simplicity, we give somewhat formal expressions here.): - $ e^T {P_{\mathrm{f}}}^{>\kappa}e^{-T}={P_{\mathrm{f}}}^{>\kappa}+A+A^* $, where $A=(A_1, A_2, A_3)$ with $A_j=a(k_jG)$. - $e^T a(k)e^{-T}=a(k)+G(k)$. Let $\tilde{K}_{\kappa, \Lambda}=e^{T}K_{\kappa, \Lambda}e^{-T}$. Using the above facts, we obtain the following: $$\begin{aligned} \tilde{K}_{\kappa, \Lambda} =&\frac{1}{2} ({P_{\mathrm{f}}}^{>\kappa})^2+{P_{\mathrm{f}}}^{>\kappa}\cdot A+A^*\cdot{P_{\mathrm{f}}}^{>\kappa} +\frac{1}{2}A^2+\frac{1}{2}A^{*2}+A^*\cdot A{\nonumber \\}&+H_I+{d\Gamma}(\omega^{>\kappa})-E^{K}_{\kappa} , \end{aligned}$$ where $$\begin{aligned} H_I=-g\int_{{\mathbb{R}}^3}dk \frac{\chi_{\kappa}^K(k)}{\sqrt{\omega(k)}}(a(k)+a(k)^*).\end{aligned}$$ We set $$\begin{aligned} \mathcal{J}=\frac{1}{2} ({P_{\mathrm{f}}}^{>\kappa})^2+{d\Gamma}(\omega^{>\kappa}). \end{aligned}$$ Let us define a quadratic form $B_{\Lambda}$ on ${\mathrm{dom}}(\mathcal{J}^{1/2})\times {\mathrm{dom}}(\mathcal{J}^{1/2})$ by $$\begin{aligned} B_{\Lambda}({\varphi}, \psi) =&\sum_{j=1}^3 \Big\{ {\langle}P_{\mathrm{f}, j}^{>\kappa} {\varphi}|A_j \psi{\rangle}+{\langle}A_j{\varphi}|P_{\mathrm{f}, j}^{>\kappa} \psi{\rangle}+\frac{1}{2}{\langle}A_j^*{\varphi}|A_j\psi{\rangle}+\frac{1}{2}{\langle}A_j{\varphi}|A_j^*\psi{\rangle}{\nonumber \\}&+{\langle}A_j{\varphi}|A_j\psi{\rangle}\Big\}+{\langle}{\varphi}|H_I\psi{\rangle}. \label{BLD}\end{aligned}$$ We easily check that $$\begin{aligned} {\langle}{\varphi}|\tilde{K}_{\kappa, \Lambda} \psi{\rangle}= {\langle}\mathcal{J}^{1/2} {\varphi}|\mathcal{J}^{1/2} \psi{\rangle}+B_{\Lambda}({\varphi}, \psi),\ \ {\varphi}, \psi\in {\mathrm{dom}}(\mathcal{J}^{1/2}).\end{aligned}$$ Let $G_{\infty}=\beta (1-\chi_{\kappa})$ and let $A_{\infty}=a(kG_{\infty})$. We define a quadratic form $B_{\infty}$ on ${\mathrm{dom}}(\mathcal{J}^{1/2}) \times {\mathrm{dom}}(\mathcal{J}^{1/2})$ by replacing $A$ with $A_{\infty}$ in (\[BLD\]). \[BInfEst\] Let $C(K)$ be a positive number defined by $$\begin{aligned} C(K)^2 =\int_{{\mathbb{R}}^3} dk \frac{1-\chi_K(k)}{\{\omega(k)+k^2/2\}^2}. \label{DefCK}\end{aligned}$$ For all ${\varepsilon}>0$, there exists a constant $D_{K, {\varepsilon}}>0$ such that $$\begin{aligned} |B_{\infty}({\varphi}, {\varphi})| \le \{ 6C(K)+6C(K)^2+{\varepsilon}\}\|(\mathcal{J}+1)^{1/2} {\varphi}\|^2+D_{K, {\varepsilon}} \|{\varphi}\|^2 \label{BInq}\end{aligned}$$ for all ${\varphi}\in {\mathrm{dom}}(\mathcal{J}^{1/2})$. [*Proof.*]{} Using (\[CreAnnInq\]) and (\[CreAnnInq2\]), we have $ \|A^{\#}_{\infty, j} {\varphi}\| \le \|\omega^{-1/2} k_j G\|\|(\mathcal{J}+1)^{1/2} {\varphi}\| $, where $a^{\#} =a$ or $a^*$. Because $ \|\omega^{-1/2} k_j G\| \le C(K) $, we obtain $$\begin{aligned} \|A^{\#}_{\infty, j} {\varphi}\| \le C(K)\| (\mathcal{J}+1)^{1/2} {\varphi}\|,\ \ {\varphi}\in {\mathrm{dom}}(\mathcal{J}^{1/2}). \label{A1}\end{aligned}$$ On the other hand, we have $$\begin{aligned} \|P_{\mathrm{f}, j}^{> \kappa} {\varphi}\| \le \|(\mathcal{J}+1)^{1/2} {\varphi}\|,\ \ {\varphi}\in {\mathrm{dom}}(\mathcal{J}^{1/2}).\label{P1}\end{aligned}$$ By using (\[A1\]) and (\[P1\]), we can estimate the terms involving $A$ and ${P_{\mathrm{f}}}^{>\kappa}$. In order to estimate ${\langle}{\varphi}|H_I\psi{\rangle}$, we observe, by (\[CreAnnInq\]) and (\[CreAnnInq2\]) again, $$\begin{aligned} |{\langle}{\varphi}|H_I{\varphi}{\rangle}| \le D \|{\varphi}\| \|(\mathcal{J}+1)^{1/2} {\varphi}\|,\end{aligned}$$ where $\displaystyle D=2g \bigg( \int dk \frac{\chi_{\kappa}^K}{\omega^2} \bigg)^{1/2}$. Using $ab \le {\varepsilon}a^2+b^2/4{\varepsilon}$, we obtain $$\begin{aligned} |{\langle}{\varphi}|H_I{\varphi}{\rangle}| \le {\varepsilon}\|(\mathcal{J}+1)^{1/2} {\varphi}\|^2+\frac{D}{4{\varepsilon}} \|{\varphi}\|^2.\end{aligned}$$ Thus we are done. $\Box$\ Choose $K$ sufficiently large as $6C(K)+6C(K)^2<1$. By the KLMN theorem [@ReSi2 Theorem X. 17] and Lemma \[BInfEst\], there exists a unique self-adjoint operator $\tilde{K}_{\kappa}$ such that $$\begin{aligned} {\langle}{\varphi}|\tilde{K}_{\kappa} \psi{\rangle}= {\langle}\mathcal{J}^{1/2} {\varphi}|\mathcal{J}^{1/2} \psi{\rangle}+B_{\infty}({\varphi}, \psi).\end{aligned}$$ Note that $\tilde{K}_{\kappa}$ is bounded from below. \[DiffBL\] We have $$\begin{aligned} |B_{\infty}({\varphi}, {\varphi})-B_{\Lambda}({\varphi}, {\varphi})| \le \Big\{ 6C(\Lambda)+12C(K) C(\Lambda) \Big\} \|(\mathcal{J}+1)^{1/2} {\varphi}\|^2 \label{DiffB}\end{aligned}$$ for all ${\varphi}\in {\mathrm{dom}}(\mathcal{J}^{1/2})$, where $C(K)$ and $C(\Lambda)$ are defiend by (\[DefCK\]). [*Proof*]{}. By (\[CreAnnInq\]) and (\[CreAnnInq2\]), we have $$\begin{aligned} \|(A_{\infty, j}^{\#}-A_j^{\#}) {\varphi}\|& \le \|\omega^{-1/2} k_j \beta (1-\chi_{\kappa} -\chi_{\kappa}^{\Lambda})\| \|(\mathcal{J}+1)^{1/2} {\varphi}\|{\nonumber \\}& \le C(\Lambda) \|(\mathcal{J}+1)^{1/2} {\varphi}\|,\ \ {\varphi}\in {\mathrm{dom}}(\mathcal{J}^{1/2}). \label{DifferenceA}\end{aligned}$$ Using (\[A1\]), (\[P1\]) and (\[DifferenceA\]), we can prove (\[DiffB\]). $\Box$ [*Proof of Theorem \[KExt\] (i)* ]{} Note that $C(\Lambda) \to 0$ as $\Lambda\to \infty$. By Lemma \[DiffBL\] and [@ReSi1 Theorem VIII. 25], $\tilde{K}_{\kappa, \Lambda}$ converges to $\tilde{K}_{\kappa}$ in norm resolvent sense as $\Lambda\to \infty$. Let $T_{\infty}=\{a(G_{\infty})-a(G_{\infty})^*\}^-$. Because $e^T$ strongly converges to $e^{T_{\infty}}$, we obtain the desired result. $\Box$ ### Proof of Proposition \[KExt\] (ii) Using arguments similar to those in the proof of Lemmas \[BasicPP1\] and \[CPP\], we can show the following lemma. \[KPP\] $e^{-\beta K_{\kappa, \Lambda}} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+^{>\kappa}$ for all $\beta \ge 0,\ \kappa>0$ and $\Lambda>0$. [ *Proof of Proposition \[KExt\] (ii)* ]{} By Proposition \[KExt\] (i), $e^{-\beta K_{\kappa, \Lambda}}$ strongly converges to $e^{-\beta K_{\kappa}}$ as $\Lambda\to \infty$. Using Proposition \[WeakCl\] and Lemma \[KPP\], we conclude Proposition \[KExt\] (ii). $\Box$ A key theorem ------------- Let $$\begin{aligned} L_{\kappa}= H_{\mathrm{ren}}^{\le \kappa}(P)\otimes{1}+{1}\otimes K_{\kappa}.\end{aligned}$$ Our purpose in this subsection is to prove the following theorem. \[LocalEq\] The following (i) and (ii) are mutually equivalent: - $e^{-\beta {H_{\mathrm{ren}}}(P)} \rhd 0$ w.r.t. ${\mathfrak{F}}_+$ for all $\beta >0$. - For each ${\varphi}, \psi\in {\mathfrak{F}}_+\backslash \{0\}$, there exist $\beta\ge 0$ and $\kappa>0$ such that ${\langle}{\varphi}|e^{-\beta L_{\kappa}} \psi{\rangle}>0$. ### Proof of Theorem \[LocalEq\] Let $A$ and $B$ be self-adjoint operators, and let $E_A$ and $E_B$ be their spectral measures. Assume that $E_A$ and $E_B$ commute with each other: $E_A(I) E_B(J)=E_B(J)E_A(I)$ for all $I, J\in \mathbb{B}^1$, the Borel sets of ${\mathbb{R}}$. We can decompose $A$ as $A=A_+-A_-$, where $A_+$ and $A_-$ are positive and negative parts of $A$, respectively. Similarly, we have $B=B_+-B_-$. For each $n\in {\mathbb{N}}$, we set $$\begin{aligned} (AB)_{[n]} =& A_+B_++A_-B_--\Big( A_+E_A[0, n] B_- E_B[-n, 0] +A_-E_A[-n, 0]B_+E_B[0, n] \Big).\end{aligned}$$ Note that $E_A[-n, 0]=E_{A_-} [0, n]$ and $E_B[-n, 0]=E_{B_-}[0, n]$. Thus, we have $$\begin{aligned} (AB)_{[n]} &\ge -2n^2, \label{BddAB1}\\ (AB)_{[n]} &\ge (AB)_{[n+1]}. \label{MonoAB}\end{aligned}$$ Similarly, we define $$\begin{aligned} (AB)^{[n]}=A_+E_A[0, n] B_+ E_B[0, n]+A_- E_A[-n, 0] B_- E_B[-n, 0] -(A_+B_-+A_-B_+).\end{aligned}$$ We have $$\begin{aligned} (AB)^{[n]} &\le 2n^2, \label{BddAB2}\\ (AB)^{[n]} &\le (AB)^{[n+1]}. \label{MonoAB2}\end{aligned}$$ For each $\kappa>0$, we define a sequence of self-adjoint operators $\{C_{\kappa, n}^{+}\}_{n=1}^{\infty}$ by $$\begin{aligned} C_{\kappa, n}^+=-\sum_{j=1}^3\Big((P_j-P_{\mathrm{f}, j}^{\le \kappa}) P_{\mathrm{f}, j}^{<\kappa}\Big)_{[n]}.\end{aligned}$$ Similarly, we define $$\begin{aligned} C_{\kappa, n}^-=-\sum_{j=1}^3\Big((P_j-P_{\mathrm{f}, j}^{\le \kappa}) P_{\mathrm{f}, j}^{<\kappa}\Big)^{[n]}.\end{aligned}$$ Let $C_{\kappa}=-(P-{P_{\mathrm{f}}}^{\le \kappa}) \cdot {P_{\mathrm{f}}}^{>\kappa}$. Note that $C_{\kappa, n}^{\pm} {\varphi}$ converges to $C_{\kappa}{\varphi}$ as $n\to \infty$ for each ${\varphi}\in {\mathrm{dom}}(C_{\kappa})$. By (\[BddAB1\]), (\[MonoAB\]), (\[BddAB2\]) and (\[MonoAB2\]), we have $$\begin{aligned} C_{\kappa, n}^+&\le 6n^2, \label{C1}\\ C_{\kappa, n}^+&\le C_{\kappa, n+1}^+, \label{C2}\\ C_{\kappa, n}^- &\ge -6n^2, \label{C3}\\ C_{\kappa, n}^-&\ge C_{\kappa, n+1}^-. \label{C4}\end{aligned}$$ \[PPC\] For all $n\in {\mathbb{N}}$ and $s\ge 0$, we have the following: - $e^{-sC_{\kappa, n}^-} $ is bounded and $e^{-sC_{\kappa, n}^-} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. - $e^{sC_{\kappa, n}^+} $ is bounded and $e^{sC_{\kappa, n}^+} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. [*Proof.*]{} (i) By (\[C3\]), $e^{-s C_{\kappa, n}^-}$ is bounded for all $s\ge 0$. We can express $e^{-sC_{\kappa, n}^-}$ as $$\begin{aligned} e^{-s C_{\kappa, n}^-}=\sideset{}{^{\oplus}_{\ell\ge 0}}\sum F_{\ell}, \label{DirectC}\end{aligned}$$ where $F_{\ell}$ is some multiplication operator on $L^2_{\mathrm{sym}}({\mathbb{R}}^{3\ell})$. We easily see that the function $F_{\ell}$ is positive. Thus, $F_{\ell}\unrhd 0$ w.r.t. $L^2_{\mathrm{sym}}({\mathbb{R}}^{3\ell})_+$ for all $\ell\in {\mathbb{N}}$, which implies (i). Similarly, we can prove (ii). $\Box$ \[PPVan\] Let ${\varphi}, \psi\in {\mathfrak{F}}_+$. - If ${\langle}{\varphi}|\psi{\rangle}=0$, then ${\langle}{\varphi}|e^{-sC_{\kappa, n}^-}\psi{\rangle}=0$ for all $n\in {\mathbb{N}},\ s\ge 0$ and $\kappa>0$. - If ${\langle}{\varphi}|\psi{\rangle}=0$, then ${\langle}{\varphi}|e^{sC_{\kappa, n}^+}\psi{\rangle}=0$ for all $n\in {\mathbb{N}},\ s\ge 0$ and $\kappa>0$. [*Proof.*]{} (i) We can express ${\varphi}$ and $\psi$ as $$\begin{aligned} {\varphi}=\sideset{}{^{\oplus}_{\ell \ge 0}}\sum {\varphi}_{\ell},\ \ \ \psi=\sideset{}{^{\oplus}_{\ell\ge 0}}\sum \psi_{\ell}.\end{aligned}$$ Note that ${\varphi}_{\ell}$ and $\psi_{\ell}$ are positive functions in $L_{\mathrm{sym}}^2({\mathbb{R}}^{3\ell})$. The condition ${\langle}{\varphi}|\psi{\rangle}=0$ is equivalent to the condition ${\langle}{\varphi}_{\ell}|\psi_{\ell}{\rangle}=0$ for all $\ell\in {\mathbb{N}}_0$. Recall the expression (\[DirectC\]). Because $F_{\ell}$ is positive and bounded, we conclude that ${\langle}{\varphi}_{\ell}|F_{\ell}\psi_{\ell}{\rangle}=0$, which implies that $ {\langle}{\varphi}|e^{-sC_{\kappa, n}^-} \psi{\rangle}=\sum_{\ell=0}^{\infty} {\langle}{\varphi}_{\ell}|F_{\ell}\psi_{\ell}{\rangle}=0. $ Similarly, we can prove (ii). $\Box$ \[LKPP\] $e^{-\beta L_{\kappa}} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$ for all $P\in {\mathbb{R}}^3$ and $\beta \ge 0$. [*Proof.*]{} By Propositions \[TensPPAB\], \[LocalPI\] and \[KExt\], we obtain the assertion in the lemma. $\Box$ \[StReC\] We have the following: - $L_{\kappa}\dot{+}C_{\kappa, n}^-$ converges to ${H_{\mathrm{ren}}}(P)$ in strong resolvent sense as $n\to \infty$, where $\dot{+}$ in dicates the form sum. - ${H_{\mathrm{ren}}}(P)\dot{-}C_{\kappa, n}^+$ converges to $L_{\kappa}$ in strong resolvent sense as $n\to \infty$. [*Proof.*]{} (i) Let us define a sequence of closed, positive quadratic form $\{t_n\}_{n=1}^{\infty}$ by $$\begin{aligned} t_n({\varphi}, \psi)={\langle}{\varphi}| \{L_{\kappa}+C_{\kappa, n}^-+ Const.\}\psi{\rangle},\end{aligned}$$ where $Const.$ is chosen such that $t_n$ is uniformly positive. By (\[C4\]), we have $t_1\ge t_2\ge \cdots\ge t_n \ge \cdots$ and $\lim_{n\to \infty}t_n({\varphi}, {\varphi})=t_{\infty}({\varphi}, {\varphi})$, where $t_{\infty}$ is a quadratic form associated with ${H_{\mathrm{ren}}}(P)$. Thus, by [@ReSi1 Theorem S. 16], we obtain (i). Similarly, we can prove (ii) by applying [@ReSi1 Theorem S. 16]. $\Box$ [ *Proof of Theorem \[LocalEq\]* ]{} \(i) $\Longrightarrow$ (ii): We extend the idea in [@Faris]. Let $\psi\in {\mathfrak{F}}_+\backslash \{0\}$. We set $ K(\psi)=\{ {\varphi}\in {\mathfrak{F}}_+\, |\, {\langle}{\varphi}| e^{-\beta L_{\kappa}} \psi{\rangle}=0\ \forall \beta \ge 0\ \forall \kappa>0 \} $. We will show that $K(\psi)=\{0\}$. Let ${\varphi}\in K(\psi)$: ${\langle}{\varphi}|e^{-\beta L_{\kappa}} \psi{\rangle}=0$ for all $\beta \ge 0$ and $\kappa>0$. By Lemma \[PPVan\] (i) and Lemma \[LKPP\], we have ${\langle}e^{-s C_{\kappa, n}^-} {\varphi}|e^{-\beta L_{\kappa}} \psi{\rangle}=0$ for all $n\in {\mathbb{N}},\ s\ge0,\ \beta \ge 0$ and $\kappa>0$, which implies that $e^{-sC_{\kappa, n}^-} K(\psi)\subseteq K(\psi) $. On the other hand, it is easy to check that $e^{-t L_{\kappa}} K(\psi)\subseteq K(\psi)$ for all $t\ge 0$. Hence, $ (e^{-\beta L_{\kappa}/\ell} e^{-\beta C_{\kappa, n}^-/\ell})^{\ell} K(\psi)\subseteq K(\psi) $ for all $\ell\in {\mathbb{N}}$. Taking $\ell\to \infty$, we obtain that $ e^{-\beta (L_{\kappa}\dot{+}C_{\kappa, n}^-)} K(\psi)\subseteq K(\psi) $ for all $n\in {\mathbb{N}}$ and $\beta \ge 0$ by [@ReSi1 Theorem S. 21]. Taking $n\to \infty$, we arrive at $ e^{-\beta {H_{\mathrm{ren}}}(P)} K(\psi) \subseteq K(\psi) $ for all $\beta \ge 0$ by Lemma \[StReC\] (i). Therefore, for each ${\varphi}\in K(\psi)$, it holds that ${\langle}{\varphi}|e^{-\beta{H_{\mathrm{ren}}}(P)} \psi{\rangle}=0$ for all $\beta \ge 0$. By the assumption (i), ${\varphi}$ must be $0$. \(ii) $\Longrightarrow$ (i): We will provide a sketch. For each $\psi\in {\mathfrak{F}}_+\backslash \{0\}$, we set $ J(\psi)=\{ {\varphi}\in {\mathfrak{F}}_+\, |\, {\langle}{\varphi}| e^{-\beta {H_{\mathrm{ren}}}(P)} \psi{\rangle}=0\ \forall \beta \ge 0 \} $. Using arguments similar to those in the previous part, we can show that $ e^{-\beta L_{\kappa}} J(\psi) \subseteq J(\psi) $ for all $\beta \ge 0$ and $\kappa>0$. Thus, for each ${\varphi}\in J(\psi)$, we obtain ${\langle}{\varphi}|e^{-\beta L_{\kappa}} \psi{\rangle}=0$ for all $\beta \ge 0$ and $\kappa>0$. By the assumption (ii), ${\varphi}$ must be $0$, which implies $J(\psi)=\{0\}$. Thus, for each ${\varphi}, \psi\in {\mathfrak{F}}_+\backslash \{0\}$, there exists a $\beta \ge 0$ such that ${\langle}{\varphi}| e^{-\beta {H_{\mathrm{ren}}}(P)} \psi{\rangle}>0$. Applying Theorem \[PFF\], we conclude (i). $\Box$ Completion of proof of Theorem \[Main1\] ---------------------------------------- \[KeyInq\] For all $P\in {\mathbb{R}}^3$ and $ \kappa>0$, we have $$\begin{aligned} e^{-\beta L_{\kappa}} \unrhd {\langle}\Omega^{>\kappa}|e^{-\beta K_{\kappa}} \Omega^{>\kappa} {\rangle}e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \otimes {1}Q_{\kappa}\end{aligned}$$ w.r.t. ${\mathfrak{F}}_+$, where $\Omega^{>\kappa}$ is the Fock vacuum in ${\mathfrak{F}}^{>\kappa}$. [*Proof.*]{} By Proposition \[QPP\], it holds that $Q_{\kappa} \unrhd 0$ and $Q_{\kappa}^{\perp} \unrhd 0$ w.r.t. ${\mathfrak{F}}_+$. Thus, by Lemma \[LKPP\], $$\begin{aligned} e^{-\beta L_{\kappa}} \unrhd Q_{\kappa} e^{-\beta L_{\kappa}} Q_{\kappa}\ \ \mbox{w.r.t. ${\mathfrak{F}}_+$ for all $\beta \ge 0$}.\end{aligned}$$ By Lemma \[QAct\], we have $$\begin{aligned} {\langle}{\varphi}|Q_{\kappa}e^{-\beta L_{\kappa}}Q_{\kappa} \psi{\rangle}=&{\langle}{\varphi}_{\kappa}\otimes \Omega^{>\kappa}|e^{-\beta L_{\kappa}} \psi_{\kappa}\otimes \Omega^{>\kappa}{\rangle}{\nonumber \\}=& {\langle}\Omega^{>\kappa} |e^{-\beta K_{\kappa}} \Omega^{>\kappa}{\rangle}{\langle}{\varphi}_{\kappa}|e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \psi_{\kappa}{\rangle}{\nonumber \\}=&{\langle}\Omega^{>\kappa} |e^{-\beta K_{\kappa}} \Omega^{>\kappa}{\rangle}{\langle}{\varphi}|e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \otimes {1}Q_{\kappa}\psi{\rangle},\end{aligned}$$ which implies that $ Q_{\kappa}e^{-\beta L_{\kappa}}Q_{\kappa} ={\langle}\Omega^{>\kappa}|e^{-\beta K_{\kappa}} \Omega^{>\kappa} {\rangle}e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \otimes {1}Q_{\kappa} $. Here, we used the fact that $Q_{\kappa} e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \otimes 1 Q_{\kappa}=e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \otimes 1 Q_{\kappa}$, which follows from Proposition \[CommuL\]. $\Box$ \[Nenno\] $ {\langle}\Omega^{>\kappa}|e^{-\beta K_{\kappa}} \Omega^{>\kappa}{\rangle}>0 $ for all $\beta\ge 0$ and $\kappa>0$. [*Proof.*]{} Because $\ker(e^{-\beta K_{\kappa}})=\{0\}$ by Proposition \[KExt\], the assertion is easy to check. $\Box$ [*Proof of Theorem \[Main1\]* ]{} Let ${\varphi}, \psi\in {\mathfrak{F}}_+\backslash \{0\}$. Because $Q_{\kappa}$ strongly converges to $1$ as $\kappa\to \infty$, there exists a $\kappa>0$ such that $Q_{\kappa} {\varphi}\neq 0$ and $Q_{\kappa} \psi \neq 0$. By Proposition \[KeyInq\], we have $$\begin{aligned} {\langle}{\varphi}|e^{-\beta L_{\kappa}} \psi{\rangle}\ge {\langle}\Omega^{>\kappa}|e^{-\beta K_{\kappa}} \Omega^{>\kappa} {\rangle}{\langle}{\varphi}| e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \otimes {1}Q_{\kappa} \psi{\rangle}.\label{EqLQ2}\end{aligned}$$ Remark that $$\begin{aligned} {\langle}{\varphi}| e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \otimes {1}Q_{\kappa} \psi{\rangle}={\langle}{\varphi}_{\kappa}|e^{-\beta {H_{\mathrm{ren}}}^{\le \kappa}(P)} \psi_{\kappa}{\rangle}, \label{EqLQ}\end{aligned}$$ where ${\varphi}_{\kappa}$ and $\psi_{\kappa}$ are defined by (\[Defpsik\]). Of course, ${\varphi}_{\kappa} \neq 0$ and $\psi_{\kappa} \neq 0$. By Proposition \[LocalPI\], the right hand side of (\[EqLQ\]) is strictly positive, provided that $\beta>0$. Because $ {\langle}\Omega^{>\kappa}|e^{-\beta K_{\kappa}} \Omega^{>\kappa} {\rangle}>0 $ by Lemma \[Nenno\], we know that the right hand side of (\[EqLQ2\]) is strictly positive. By Theorem \[LocalEq\], we finally conclude that $e^{-\beta {H_{\mathrm{ren}}}(P)} \rhd 0$ w.r.t. ${\mathfrak{F}}_+$ for all $P\in {\mathbb{R}}^3$ and $\beta>0$. $\Box$ A useful proposition {#AppA} ===================== In this appendix, we will review a useful result concerning the operator inequalities introduced in Section \[SecMono\]. \[BasicPertPP\] Let $A$ be a positive self-adjoint operator and let $B$ be a symmetric operator. Assume the following: - $B$ is $A$-bounded with relative bound $a<1$, i.e., ${\mathrm{dom}}(A)\subseteq {\mathrm{dom}}(B)$ and $\|Bx\|\le a \|Ax\|+b\|x\|$ for all $x\in {\mathrm{dom}}(A)$. - $0\unlhd e^{-tA}$ w.r.t. ${\mathfrak{P}}$ for all $t\ge 0$. - $0\unlhd -B$ w.r.t. ${\mathfrak{P}}$. Then $0\unlhd e^{-t(A+B)}$ w.r.t. ${\mathfrak{P}}$ for all $t\ge 0$. [*Proof.*]{} This proposition is already proved in [@Miyao], see also [@Miyao3; @Miyao4; @Miyao5]. For readers’ convenience, we provide a proof. For each ${\varepsilon}>0$, we set $B_{{\varepsilon}}=e^{-{\varepsilon}A} B e^{-{\varepsilon}A}$. By (i) and (iii), $B_{{\varepsilon}}$ is bounded and $-B_{{\varepsilon}} \unrhd 0$ w.r.t. ${\mathfrak{P}}$. Let us consider a self-adjoint operator $C_{{\varepsilon}}=A+B_{{\varepsilon}}$. By the Duhamel formula, we have the following norm convergent expansion: $$\begin{aligned} e^{-t C_{{\varepsilon}}}&=\sum_{n=0}^{\infty}D_n, \label{Duha1}\\ D_n&=\int_{S_n(t)} e^{-s_1 A}(-B_{{\varepsilon}}) e^{-s_2 A}(-B_{{\varepsilon}})\cdots e^{-s_n A} (-B_{{\varepsilon}}) e^{-(\beta-\sum_{j=1}^ns_j)A},\end{aligned}$$ where $\int_{S_n(t)}=\int_0^{\beta}ds_1\int_0^{\beta-s_1}ds_2\cdots \int_0^{\beta-\sum_{j=1}^{n-1}s_j} ds_n$ and $D_0=e^{-t A}$. Since $-B_{{\varepsilon}} \unrhd 0$ and ${\mathrm{e}}^{-t A}\unrhd 0$ w.r.t. ${\mathfrak{P}}$ for all $t \ge 0$, it holds that, by Lemma \[Miura\], $$\begin{aligned} \underbrace{ e^{-s_1 A} }_{\unrhd 0} \underbrace{(-B_{{\varepsilon}})}_{\unrhd 0} \underbrace{ e^{-s_2 A} }_{\unrhd 0}\cdots \underbrace{e^{-s_n A}}_{\unrhd 0} \underbrace{(-B_{{\varepsilon}})}_{\unrhd 0} \underbrace{e^{-(t-\sum_{j=1}^ns_j)A}}_{\unrhd 0} \unrhd 0,\end{aligned}$$ provided that $s_1 \ge 0, \dots, s_n\ge 0$ and $t-s_1-\cdots-s_n\ge 0$. Thus, by Proposition \[WeakCl\], we obtain $D_n\unrhd 0$ w.r.t. ${\mathfrak{P}}$ for all $n\ge 0$. Accordingly, by (\[Duha1\]), we have ${\mathrm{e}}^{-t{C_{{\varepsilon}}}}\unrhd D_{n=0}=e^{-t A} \unrhd 0$ w.r.t. ${\mathfrak{P}}$ for all $t \ge 0$ and ${\varepsilon}\ge 0$. Because $e^{-tC_{{\varepsilon}}}$ strongly converges to $e^{-t(A+B)}$ as ${\varepsilon}\to +0$, we conclude that $e^{-t(A+B)} \unrhd 0$ w.r.t. ${\mathfrak{P}}$ for all $t\ge 0$ by Proposition \[WeakCl\]. $\Box$ [100]{} A. Abdesselam, D. Hasler, Analyticity of the ground state energy for massless Nelson models. Comm. Math. Phys. 310 (2012), 511–536. Z. Ammari, Asymptotic completeness for a renormalized nonrelativistic Hamiltonian in quantum field theory: the Nelson model. Math. Phys. Anal. Geom. 3 (2000), 217–285. A. Arai, Ground state of the massless Nelson model without infrared cutoff in a non-Fock representation. Rev. Math. Phys. 13 (2001), 1075–1094 W. Bös, Direct integrals of selfdual cones and standard forms of von Neumann algebras. Invent. Math. 37 (1976), 241–251. O. Bratteli, D. W. Robinson, [*Operator algebras and quantum statistical mechanics. 1. $C^*$- and $W^*$ algebras, symmetry groups, decomposition of states Second edition.*]{} Texts and Monographs in Physics. 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H. Spohn, [*Dynamics of Charged Particles and Their Radiation Field*]{}, Cambridge University Press, Cambridge, 2004. [^1]: Roughly speaking, we say that $V$ is ferromagnetic if $\hat{V}(k)<0$, where $\hat{V}$ is the Frourier transformation of $V$. [^2]: In concrete applications in Sections \[2ndQuant\] and \[Pf1\], we will see that $\mathfrak{V}$ satisfies a much stronger condition: $\overline{\mathfrak{V} \cap {\mathfrak{P}}}={\mathfrak{P}}$.

--- abstract: | We introduce a special class of knots, called global knots, in $F^2 \times \mathbb{R}$ and we construct new isotopy invariants, called $T$-invariants, for global knots. Some $T$-invariants are of finite type but they cannot be extracted from the generalized Kontsevitch integral (which is consequently not the universal invariant of finite type for the restricted class of global knots). We prove that $T$-invariants separate all global knots of a certain type. As a corollary, we prove the non-invertibility of some links in $S^3$ without making use of the link group. author: - 'T. Fiedler' title: 'Global knot theory in $F^2 \times \mathbb{R}$' --- Introduction and main results ============================= Let $F^2$ be a compact oriented surface with or without boundary, and let $v$ be a Morse-Smale vector field on $F^2$ which is transversal to the boundary $\partial F^2$. (For us, a Morse-Smale vector field is a smooth vector field, having at most isolated non-degenerated singularities, and no limit cycles.) We study oriented knots $K$ in the oriented manifold $F^2 \times \mathbb{R}$. It turns out that there are naturally three types of knot theory, which we call respectively local, global and general. Local knot theory ----------------- Let $F^2$ be the disk $D^2$ or the sphere $S^2$ and let $v$ be a vector field which has only critical points of index +1. Alexander’s theorem says that each knot type (i.e. a knot up to ambient isotopy) has a representative, also called $K$, such that the projection $K \hookrightarrow F^2 \times \mathbb{R} \rightarrow F^2$ is transversal to $v$. Markov’s theorem says that two transversal representatives of the same knot type can be joined by an almost transversal isotopy i.e. an isotopy through transversal knots, besides for a finite number of values of the parameter where a Reidemeister move of type $I$ occurs in a singularity of $v$ (such a move is usually called Markov move). This type of knot theory is traditionally the most studied one and we cannot add anything new here. Global knot theory ------------------ A knot type $K$ is called a [*global knot*]{} if there is a vector field $v$ without critical points or only with critical points of index -1, and there is a representative $K$ such that the projection $K \hookrightarrow F^2 \times \mathbb{R} \rightarrow F^2$ is transversal to $v$. We call such $v$ [*non-elliptic*]{} vector fields. Notice that a knot $K \hookrightarrow D^2 \times \mathbb{R} \hookrightarrow F^2 \times \mathbb{R}$ can never be a global knot. This implies that there is no analogue of Alexander’s theorem here. However, there is an analogue of Markov’s theorem for global knots which is even better than Markov’s theorem: [*If two representatives of the same global knot have projections transversal to the same non-elliptic vector field $v$, then there is an isotopy between them transversal to $v$.*]{} We give a proof in a special case and we indicate the idea of the proof in the general case. Global knots are the main object of our work. They have almost never been studied before, except for the very special case of closed braids. General knot theory ------------------- A knot type $K$ is called a [*general knot*]{} if for each representative $K$ the following property is verified: if $v$ is a vector field such that $K \hookrightarrow F^2 \times \mathbb{R} \rightarrow F^2$ is transversal to $v$, then $v$ has critical points of both indices $+1$ and $-1$. General knots do exist. For example: the Whitehead link seen as a knot in the solid torus. Evidently, if a knot is neither local nor global, then it is general. We prove that in the general setting there is no analogue of Markov’s theorem (in difference to the global setting). More precisely, we give two representatives of some general knot in the solid torus with projections transversal to the same vector field $v$, and we show that they cannot be joined by any transversal isotopy. This indicates that the general case is even much more complicated than the local and the global case. The main achievement of our work is the construction of new isotopy invariants, called [*T-invariants*]{} ($T$ means “transversal”), for global knots. These invariants depend neither on the chosen non-elliptic vector field $v$, nor on the chosen representative of the knot whose projection is transversal to $v$. Hence, $T$-invariants are isotopy invariants in the usual sense. $T$-invariants are defined as “Gauss diagram invariants”. Consequently, their calculation has polynomial complexity with respect to the number of crossings of the knot diagrams. However, not all $T$-invariants are of finite type in the sense of Vassiliev. Moreover, we show that even some $T$-invariants of finite type cannot be extracted from the generalized Kontsevitch integral (see \[A-M-R\]). This comes from the fact that $T$-invariants are well defined only for global knots and not for all knots in $F^2 \times \mathbb{R}$. A knot $K \hookrightarrow F^2 \times \mathbb{R}$ is called a [*solid torus knot*]{} (or a closed braid) if it has a representative whose projection is contained in some annulus $S^1 \times I \hookrightarrow F^2$ (of course, local knots are a special case of solid torus knots). We conjecture that $T$-invariants separate all global knots in $F^2 \times \mathbb{R}$ which are not solid torus knots. We prove this conjecture in the following special case: Let $T^2$ be the torus. An oriented global knot $K \hookrightarrow T^2 \times\mathbb{R}$ is called $\mathbb{Z}/2\mathbb{Z}$-pure if for each crossing of $K$, each of the two oriented loops obtained by splitting the crossing is non-trivial in $H_1(T^2;\mathbb{Z})/\langle [K] \rangle \bigotimes \mathbb{Z}/2\mathbb{Z}\cong \mathbb{Z}/2\mathbb{Z}$ (We consider $K$ as a diagram in $F^2 \times \mathbb{R}$ over $F^2$ and we denote by $[K]$ the homology class represented by $K$. If $K$ is not a solid torus knot then $H_1(T^2;\mathbb{Z})/\langle [K] \rangle \bigotimes \mathbb{Z}/2\mathbb{Z}$ is automatically isomorphic to $\mathbb{Z}/2\mathbb{Z}$.) [*$T$-invariants separate all $\mathbb{Z}/2\mathbb{Z}$-pure global knots in $T^2 \times \mathbb{R}$.*]{} Let $flip: T^2 \times \mathbb{R} \to T^2 \times \mathbb{R}$ be the hyper-elliptic involution on $T^2$ multiplied by the identity on the lines $\mathbb{R}$. An oriented knot $K \hookrightarrow T^2 \times \mathbb{R}$ is called [*invertible*]{} if it is ambient isotopic to $flip(-K)=-flip(K)$. One easily shows that both the generalized HOMFLY-PT and the generalized Kauffman polynomial can never distinguish $K$ from $flip(-K)$. (The polynomials of the cables do not make the distinction either.) On the other hand, we prove the non-invertibility of some global knots in $T^2 \times \mathbb{R}$ using $T$-invariants of finite type. Consequently, these invariants cannot be extracted from the above knot polynomials. Knots in $T^2 \times \mathbb{R}$ are in 1-1 correspondence with ordered 3-component links in $S^3$ containing the Hopf link $H$ as a sublink. Using a $T$-invariant of degree 6 for the knot $K$, we show that the link $L=K \cup H \hookrightarrow S^3$ (see Fig. 1). is not invertible for any chosen orientation on it. Notice that this is the first proof of the non-invertibility of a link with a numerical invariant, which does not make any use of the link group $\pi_1(S^3\setminus $K$;*)$. (Compare with $L$ for approaches which use the link group.) $$\begin{picture}(0,0)\special{psfile=im1.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4506,3244)(264,-2535) \put(2601,-744){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$H$}}} \put(4770,-1204){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K$}}} \put(2510,-2477){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$L=K \cup H$}}} \end{picture}$$ [**Fig. 1**]{} Let us consider the space of all diagrams of a given knot type $K \hookrightarrow F^2 \times \mathbb{R}$. The discriminant is the subspace of all non-generic diagrams. Each path in the space of diagrams which cuts the discriminant only in strata of codimension 1 is a generic isotopy of knots (see \[F\], sect. 1). The construction of $T$-invariants relies on the combination of two different approaches. On one hand, there is the concept of $G$-pure knots and $G$-pure isotopy. A $G$-pure isotopy is an isotopy which does not cut the discriminant in strata with certain homological markings of the crossings. $T$-invariants are, roughly speaking, Gauss diagram formulas which are invariant under $G$-pure isotopy. The problem is now, that even if two knots are isotopic, we cannot grant that there exists a $G$-pure isotopy joining them. At this place, the concept of global knots is introduced. If two isotopic knots are global, then there exists an isotopy through global knots between them. Such an isotopy does not cut the strata of the discriminant depicted in Fig. 2. $$\begin{picture}(0,0)\special{psfile=ima2.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4551,1049)(464,-728) \end{picture}$$ [**Fig. 2**]{} We still do not know wether isotopic $G$-pure global knots can be joined by a $G$-pure isotopy. But for an isotopy through global knots, we have enough control over the “cycles of crossings” (compare with \[F\], sect. 4) to enable us to show that $T$-invariants are actually isotopy invariants! Using $T$-invariants, we show in sect. 9 that there exist isotopic $G$-pure general knots such that 1. there is no $G$-pure isotopy between them 2. there is no isotopy transversal to the vector field $v$ between them Isotopy of global knots ======================= We fix orientations on $F^2$ and $F^2 \times \mathbb{R}$. Let $pr: F^2 \times \mathbb{R} \to F^2$ be the canonical projection, and let $v$ be a non-elliptic vector field. We fix the orientation on the global knot $K \hookrightarrow F^2 \times \mathbb{R}$ in such a way that ($pr(K), v$) induce the given orientation on $F^2$. The natural equivalence relation for global knots is: ambient isotopy through global knots keeping the vector field $v$ fixed. However, we believe that this equivalence relation coincides with the usual ambient isotopy. [*Conjecture*]{}. [*Two global knots with respect to the same vector field $v$ are ambient isotopic if and only if they are ambient isotopic through global knots with respect to $v$.*]{} Remarks: : 1. The conjecture is true in the case of closed braids: this is a consequence of Artin’s theorem. 2. Below we give a proof in the case where $F^2=T^2$. 3. We outline the strategy of the proof in the general case (and we will come back to it in another paper): One can easily prove in a geometrical way that braids are isotopic as tangles if and only if they are isotopics as braids (see e.g. \[P-S\]). This proof can be generalized to all such isotopies of global knots which do not pass through the singularities of $v$. If an isotopy passes through a singularity of $v$, then the resulting knot is no longer transversal to $v$, except for the case where the isotopy passes again through the same singularity but in the opposite direction. This can be proven by constructing 2-disks with piecewise smooth boundary in the trace of the isotopy on $F^2$. The vector field $v$, having only critical points of index -1, can never be transversal to such a disk. Using this fact, one can remove successively these 2-disks in the isotopy. We consider now the case of the torus $T^2$. Let $pr_2: S^1_1 \times S^1_2 \to S^1_2$ be the projection and let $v$ be the unit vector field tangential to the fibres of $pr_2$ (we might perturbate $v$ slightly so that it has no longer any closed orbits). Consequently, a knot $K \hookrightarrow T^2 \times \mathbb{R}$ is global with respect to $v$ if and only if the restriction $pr_2: K \to S^1_2$ is a covering. [*Isotopic global knots with respect to $v$ in $T^2 \times \mathbb{R}$ are isotopic through global knots with respect to $v$.*]{} Artin’s theorem implies that isotopic closed braids are isotopic through closed braids (see e.g. \[M\]). Let $T^2 \times \mathbb{R} \hookrightarrow S^3$ be the standard embedding and let $A_1$ resp. $A_2$ be the cores of the solid tori $S^3 \setminus (T^2 \times \mathbb{R})$. The oriented link $A_1 \cup A_2$ is determined by the following conventions: $lk(A_1, \{*\} \times S^1_2) = lk(A_2, S^1_1 \times \{*\}) = 1$ and $K$ is a global knot in $T^2 \times \mathbb{R}$ with respect to $v$ if and only if $K \cup A_2$ is a closed braid for the natural fibering $S^3 \setminus A_1 \to S^1_2$. Consequently, if $K$ and $K'$ are isotopic global knots then $K \cup A_2$ and $K' \cup A_2$ are closed braids which are isotopic as links. According to Artin’s theorem, they are also isotopic as closed braids. But $A_2$ is just the closure of a 1-string braid and therefore we may assume that it remains fixed in the isotopy of closed braids. This implies that the isotopy of $K$ in $S^3 \setminus (A_1 \cup A_2)$ is an isotopy throughglobal knots ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$. From now on, all the isotopies considered will be isotopies through global knots. [*Basic observation.*]{} [*In an isotopy of global knots, the Reidemeister moves depicted in Fig. 3 can never occur.*]{} $$\begin{picture}(0,0)\special{psfile=im2.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5188,3464)(203,-2864) \end{picture}$$ [**Fig. 3**]{} [**Proof**]{}. Obviously, in the singularities there is always one branch of $K$ which has not the correct orientation (see the middle part of Fig. 3). ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ [*Question*]{}. [*Is there an analogue of Alexander’s theorem (in the right sense) for knots in $T^2 \times \mathbb{R}$?*]{} More precisely, given a diagram $K$, is there an algorithm which either constructs some non-elliptic vector field $v$ and a representative of $K$ transversal to $v$, or otherwise, which proves that $K$ is not a global knot? Construction of $T$-invariants for global knots. ================================================ Let $K_0 \hookrightarrow F^2 \times \mathbb{R}$ be an oriented global knot and let $K_t, t \in [0,1]$ be an isotopy of $K_0$ (through global knots). Let $G$ be a fixed quotient group of $H_1(F^2; \mathbb{Z})$ and let $[K]_G$ be the homology class in $G$ represented by $K=K_0$. [A global knot $K$ is [*$G$-pure*]{} if for each crossing $p$ of $K$ (with respect to the projection $pr$), $[K^+_p]_G \notin \{0, \pm [K]_G \}$. The isotopy $K_t$ is [*$G$-pure*]{} if $K_t$ is $G$-pure for each $t$. A knot type (called $K$ as the knot itself) is [ *$G$-pure*]{} if it has a global representative which is $G$-pure.]{} We remind the definition of the oriented (global) knot $K^+_p$ in Fig. 4 (see also \[F\], sect.  0 and 1). $$\begin{picture}(0,0)\special{psfile=im3.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3894,1698)(439,-1289) \put(1120,-689){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1147,-1232){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K$}}} \put(4316,214){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_p^-$}}} \put(3516,-1231){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_p^+$}}} \end{picture}$$ [**Fig. 4**]{} The [*Gauss diagram*]{} of $K$ is the abstract chord diagram of $pr(K)$, where each chord $p$ is oriented from the undercross to the overcross of the crossing $p$, and it is marked by the writhe $w(p)$ and by the homology class $[K^+_p]_G \in G$. A [*configuration*]{} is a given (abstract) chord diagram with given orientations and homological markings (in $G$) of the chords, but without writhes. A [*Gauss sum of degree $d$ for the knot $K$*]{} (also called a [*Gauss diagram formula*]{}) is a sum which is defined in the following way: Let $D$ be a given configuration of $d$ chords. We consider the integer $$\sum_{D} {\mbox{function (writhes of the crossings of $K$ corresponding to the chords of $D$)}}$$ where $D$ runs over all the subdiagrams of the Gauss diagram of $K$. The function is called the [*weight function*]{} (see also \[F\], sect. 0 and 1). [A [*$G$-pure configuration of degree $m$*]{} is a given configuration $D$ of $m$ oriented chords $(p_1, \dots, p_m)$ with corresponding markings $(a_1, \dots, a_m)$ where each $a_i \in G \setminus \{0, \pm [K]_G \}$, verifying the following 2 conditions:]{} 1. [Let $D$ be represented as a subdiagram $D_0$ of a Gauss diagram of any $G$-pure knot $K_0$, and let $K_t, t \in [0,1]$ be any $G$-pure isotopy of $K_0$ without Reidemeister moves of type $II$ involving one of the crossings $p_i$. Then, $D$ is preserved as a subdiagram of $K_t$ i.e. there is a continuous family $D_t$ of subdiagrams of the Gauss diagrams of $K_t$ and almost each $D_t$ represents $D$.]{} 2. [If $D$ contains a fragment of the type depicted in Fig. 5, then $a_i \not= a_j$.]{} $$\begin{picture}(0,0)\special{psfile=manu20.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2302,2421)(645,-1707) \put(1131,519){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a_i$}}} \put(2406,519){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a_j$}}} \put(1411,-116){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_i$}}} \put(2251,-16){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_j$}}} \end{picture}$$ [**Fig. 5**]{} [*Exemple. $2.1$*]{} $G:= \mathbb{Z}$ and $[K]_G = 0$. Let $a \in \mathbb{Z} \setminus 0$ be fixed. There is exactly one $\mathbb{Z}$-pure configuration of degree 1 which involves the class $a$ (see Fig. 6). There are exactly three $\mathbb{Z}$-pure configurations of degree 2 which involve the class $a$ (see Fig. 7). $$\begin{picture}(0,0)\special{psfile=manu21.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1164,1316)(739,-617) \put(1400,-110){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(1361,534){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture}$$ [**Fig. 6**]{} $$\begin{picture}(0,0)\special{psfile=manu21bis.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4948,1420)(465,-796) \put(821,-156){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(2676,-37){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(796,429){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1351,-796){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3236,-116){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \put(2496,-751){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3196,424){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(4603,-35){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(4436,439){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5071,459){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5176,-91){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \put(1376,-161){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \put(1896,-611){\makebox(0,0)[lb]{\smash{\SetFigFont{14}{16.8}{\rmdefault}{\mddefault}{\updefault},}}} \put(3856,-621){\makebox(0,0)[lb]{\smash{\SetFigFont{14}{16.8}{\rmdefault}{\mddefault}{\updefault},}}} \end{picture}$$ [**Fig. 7**]{} This follows immediately from the fact that a $G$-pure isotopy does not intersect the following strata in the discriminant: $a_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=dr.pstex}\end{picture} } }}^{+(-)}(0|a,0)$, $a_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=br.pstex}\end{picture} } }}^{+(-)}(0|a,-a)$, $a_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=dr.pstex}\end{picture} } }}^{+(-)}(a|a,0)$, $a_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=br.pstex}\end{picture} } }}^{+(-)}(a|0,a)$, (See \[F\], sect. 4.11). In fact, in \[F\] we replaced all triple points of type (0,0)(0,5) by triple points of type (0,0)(0,5) . A closer look to this replacement shows that a Reidemeister move of type $III$ corresponding to (0,0)(0,5) is $G$-pure and a configuration $D$ is invariant under this move if and only if it is invariant under the corresponding move of type (0,0)(0,5) . We illustrate this in Fig. 8. $$\begin{picture}(0,0)\special{psfile=im4.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3584,2021)(465,-1505) \put(1141,-146){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1326,-1505){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1681,-1210){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3538,-1366){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3506,145){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2281,-670){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3171,-290){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1956,-860){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture}$$ [**Fig. 8**]{} The homological markings involved in (0,0)(0,5) are exactly the same as those involved in the corresponding (0,0)(0,5) . Notice that the $G$-pure configuration depicted in the left-hand part of Fig. 9, which appears in the replacement, disappears again. Consequently, the chords $p_1$ and $p_2$ of the above configurations cannot become crossed (as shown in the right-hand part of Fig. 9) in a $G$-pure isotopy. $$\begin{picture}(0,0)\special{psfile=ima9.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3609,1282)(469,-878) \put(656,219){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1291,239){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2116,-711){\makebox(0,0)[lb]{\smash{\SetFigFont{14}{16.8}{\rmdefault}{\mddefault}{\updefault},}}} \put(3696,-183){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \put(3113,-178){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \end{picture}$$ [**Fig.9**]{} [*Example $3.2$*]{} $G := \mathbb{Z}/2\mathbb{Z}$ and $[K]_G=0$. Each chord is marked by the non-trivial element in $\mathbb{Z}/2\mathbb{Z}$. In this case, each configuration which does not contain a fragment as depicted in Fig. 10 $$\begin{picture}(0,0)\special{psfile=ima10.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2357,2242)(610,-1707) \put(1116,-136){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_i$}}} \put(2201,-51){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_j$}}} \end{picture}$$ [**Fig. 10**]{} is a $\mathbb{Z}/2\mathbb{Z}$-pure configuration. Let $D$ be a fixed $G$-pure configuration of degree $m$. Let $\mathcal{D}_i, i \in I$ be a finite set of configurations, each of them with at most $m+n$ oriented chords $(p_1, \dots, p_m, q^{(i)}_1, \dots, q^{(i)}_{n_i})$. Let $n = \max_{i}(n_i)$. All chords have markings in $G \setminus \{0, \pm[K]_G \}$. The given (unordered) chords $(p_1, \dots, p_m)$ form the given $G$-pure subconfiguration $D$ in each $\mathcal{D}_i$. Let $f_i, i \in I$ be functions $$f_i: \underbrace{\mathbb{Z}/2\mathbb{Z} \times \cdots \times \mathbb{Z}/ 2\mathbb{Z}}_{n_i} \rightarrow \mathbb{Z}$$ The linear combination of Gauss diagram formulas $$c(D) := \sum_{\mathcal{D}_i}{f_i(w(q^{(i)}_1), \dots, w(q^{(i)}_{n_i}))}$$ is called a [*$G$-pure class of $D$ of degree at most $n$* ]{} if the following condition holds: Let $D$ be represented as a subdiagram $D_0$ of a Gauss diagram of any $G$-pure knot $K_0$ and let $K_t, t \in [0,1]$ be any $G$-pure isotopy of $K_0$ without Reidemeister moves of type $II$ involving one of the $p_i$. (Hence, there exists a continuous family $D_t$ as in Def. 3.2.). Let $$c(D_t) := \sum_{i} \sum_{\Delta_i}{f_i(w(q^({i})_1, \dots, w_(q^({i})_{n_i}))}$$ where $\Delta_i$ runs through all subdiagrams which represent $\mathcal{D}_i$ in the Gauss diagram of $K_t$, and which contain $D_t$ as the given subconfiguration $D$. The integer $c(D_t)$ is the same for all $t \in [0,1]$ such that the projection $pr: K_t \to F^2$ is generic. (Here, $w(q^{(i)}_j)$ is the writhe of the crossing $q^{(i)}_j$.) [**Remarks**]{}: 1. In the calculation of $c(D_t)$, subdiagrams which coincide up to different numerations of the chords are identified, $$\begin{picture}(0,0)\special{psfile=ima11.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5531,2405)(388,-2164) \put(4295,-935){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(4775,-510){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(4810,-1510){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(1215,-950){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1808,-1298){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(1707,-505){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(3076,-1066){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$=$}}} \end{picture}$$ [**Fig. 11**]{} so that they bring only one term intothe sum (see Fig 11). 2. We say that the class $c(D)$ is of degree $n$ if for each class $c'(D)$ which uses no more than $n-1$ chords, $c(D_t) - c'(D_t)$ is not constant. [*Example $3.3$*]{} $G := \mathbb{Z}$, $[K]_G=0$, $a \in \mathbb{Z} \setminus 0$, $D$, $\mathcal{D}_1$, $\mathcal{D}_2$ are as in Fig. 12, $f_1 \equiv f_2 = w(q_1)w(q_2)$. $$\begin{picture}(0,0)\special{psfile=ima12.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5712,1630)(423,-1232) \put(5609,-1232){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(5136,148){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5654,148){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5525,-827){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(5554,-245){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(5302,-584){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(423,-479){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D=$}}} \put(1531,-509){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1536,233){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2146,-544){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault},}}} \put(4114,-473){\makebox(0,0)[lb]{\smash{\SetFigFont{9}{10.8}{\rmdefault}{\mddefault}{\updefault},}}} \put(2333,-534){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\mathcal{D}_1=$}}} \put(3029,171){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3630,-1161){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3171,-533){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(3667,128){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3601,-768){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(3603,-267){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(4359,-550){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\mathcal{D}_2=$}}} \end{picture}$$ [**Fig. 12**]{} $$c(D) = \sum_{\mathcal{D}_1}{w(q_1)w(q_2)} + \sum_{\mathcal{D}_2}{w(q_1)w(q_2)}$$ is a class of $D$ of degree 2. Indeed, in any $G$-pure isotopy of a knot $K$ such that the crossing $p$ does not disappear, we observe the following: the chord $p$ can get crossed with none of the $q_i, i=1, 2$. The chords $q_1$ and $q_2$ can get crossed together by passing e.g. a stratum of the type $a_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=dr.pstex}\end{picture} } }}^+(a|-a,2a)$. But we count them now in $\mathcal{D}_2$ instead of $\mathcal{D}_1$. Notice that the move depicted in Fig. 13 is again not possible. $$\begin{picture}(0,0)\special{psfile=ima13.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3468,3319)(439,-2841) \put(878,-1016){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1200,-1589){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(727,-209){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1245,-209){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3261,313){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2754,-2144){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(2908,276){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3533,-1034){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3226,-2841){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3606,-1499){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2916,-1402){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2072,-1711){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}or}}} \put(3253,-247){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(3563,-484){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(2663,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1140,-572){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(1186,-1096){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(3330,-1789){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(3645,-2055){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \end{picture}$$ [**Fig. 13**]{} Let e.g. $(\mathcal{D}_1)_t$ be a subdiagram which represents $\mathcal{D}_1$ for a knot $K_t$. If in the isotopy e.g. $(q_1)_t$ disappears (see the left-hand part of Fig. 14), then there is a crossing $(q'_1)_t$ such that the diagram depicted in the right-hand part of Fig. 14 represents also $\mathcal{D}_1$. But for the writhes $w((q_1)_t)=-w((q'_1)_t)$, and hence for $f_1=f_2=w(q_1)w(q_2)$, the contributions of $(q_1)_t$ and $(q'_1)_t$ in $c(D_t)$ cancel out. This shows that $c(D)$ is a class, and calculating examples one easily establishes that it is of degree 3. $$\begin{picture}(0,0)\special{psfile=ima14.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5042,1497)(464,-1280) \put(4510, 52){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5111,-1280){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(5148, 9){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(4352,-615){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_t$}}} \put(4777,-401){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$(q_1')_t$}}} \put(4835,-858){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$(q_2)_t$}}} \put(1192,-679){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$(q_1')_t$}}} \put(554,-671){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$(q_1)_t$}}} \put(2043,-346){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$t$}}} \end{picture}$$ [**Fig. 14**]{} Let $K \hookrightarrow F^2 \times \mathbb{R}$ be a $G$-pure global knot and let $D$ be a $G$-pure configuration of degree $m$. Let $c_i(D), i \in \{1, \dots, k\}$ be a finite collection of $G$-pure classes of $D$ having degree $n_i$ respectively, and let $c_i, i \in \{1, \dots, k\}$ be fixed integers. Let $n = \max_{i}{n_i}$. [The [*$T$-invariant*]{} $T_K(D; c_1(D)=c_1, \dots c_k(D)=c_k)$ for $G$-pure global knots is defined as: $$\sum_{D}{w(p_1) \cdots w(p_m)}$$ Where we sum over all $D$ occuring as subdiagrams of the Gauss diagram of $K$, such that $c_1(D)=c_1, \dots c_k(D)=c_k$. Here, $w(p_i)$ are the writhes of the crossings of $K$ which correspond to the chords of the subdiagram representing $D$. If $n=0$ and $m \not= 0$, then $T_K(D; \emptyset)$ is defined as $$\sum_{D}{w(p_1) \cdots w(p_m)}$$ Where we sum over all $D \subset$ Gauss diagram of $K$. If $m=0$ and $n \not= 0$ then $T_K(\emptyset; c(\emptyset))$ is defined as $$c(\emptyset) = \sum_{i} \sum_{\Delta_i}{f_i(w(q^{(i)}_1, \dots, q^{(i)}_{n_i}))}$$ Where $\Delta_i$ runs through all subdiagrams which represent $\mathcal{D}_i$ in the Gauss diagram of $K$.If $n= m= 0$, i.e. $D= \mathcal{D}_i= \emptyset$, then $T_K(\emptyset, \emptyset)$ is defined as the [*free regular homotopy class*]{} of $pr(K) \subset F^2$. (This is the universal invariant of degree 0.)]{} [The set $\{ c_1(D), \dots, c_k(D) \}$ is called a [*multi-class*]{} of $D$.]{} [**Remark**]{}. If there is no risk of confusion, we will denote shortly by $T_K$ the invariant $T_K(D; c_1(D)= c_1, \dots, c_k(D)= c_k)$. We are now ready to formulate our main result. [*Let $K_0$ and $K_1$ be $G$-pure global knots which are ambient isotopic. Then, for each $T$-invariant for $G$-pure global knots*]{} $$T_{K_0}(D; c_1(D)= c_1, \dots, c_k(D)= c_k)= T_{K_1}(D; c_1(D)= c_1, \dots, c_k(D)= c_k)$$ [**Remark**]{}. The isotopy in the theorem need not to be $G$-pure! [**Proof**]{}. The formal proof is very complicated. We just outline the main steps and let the verification of the details for the reader. It follows from the definition of a class $c(D)$ that $T_K$ is invariant for all such $G$-pure isotopies in which no crossing $p_i$ disappears for a subdiagram (of the Gauss diagram of $K$) representing $D$. Assume now that $D$ is represented by $\{ p_1, \dots, p_i, \dots p_m \}$ and that the move depicted in Fig. 15 occurs in the isotopy: $$\begin{picture}(0,0)\special{psfile=ima15.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4599,650)(441,-271) \put(1668,-271){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_t$}}} \put(576,-136){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_i$}}} \put(1281,-136){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_i'$}}} \end{picture}$$ [**Fig. 15**]{} (Remember that (18,10) (16,8)(-10,4) does not occur in a transversal isotopy.) Then the crossing $p'_i$, with which $p_i$ disappears, verifies: 1. $w(p_i)= -w(p'_i)$ 2. $p'_i \notin \{p_1, \dots, p_i, \dots, p_m \}$. (This follows from 2. in Definition. 3.2.) 3. $\{ p_1, \dots, p'_i, \dots, p_m \}$ represents $D$ ($p_i$ is replaced by $p'_i$). 4. $c(\{ p_1, \dots, p_i, \dots, p_m \} = c(\{ p_1, \dots, p'_i, \dots p_m \}$, (in the right-hand term, $p_i$ is replaced by $p'_i$). Consequently, the contributions of $p_i$ and $p'_i$ in $T_{K_t}$ cancel out. This implies that $T_K$ is invariant for $G$-pure isotopies. Assume now that $K_t, t \in [0,1]$ is an isotopy which is not necessarily $G$-pure (but $K_0$ and $K_1$ are $G$-pure!) The (value of) a class $c(D)$ can change only if two of the crossings among $\{ p_1, \dots, p_m; q^{(i)}_1, \dots, q^{(i)}_n \}$ of a configuration $\mathcal{D}_i$ are involved in a Reidemeister move of type $III$, such that the third crossing involved has the homological marking 0 or $\pm[K]_G$. Performing the isotopy $K_t$, let us watch the traces on $F^2$ of the crossings with markings in $\{0, [K]_G, -[K]_G \}$. These traces form immersed circles, called [*Rudolph diagrams*]{} (see \[F\], sect. 4.11). To each such circle, we associate a family of disks $D_t$ (which are immersed in $F^2$), exactly as in the proof of Theorem. 4.3 in \[F\]. This is possible because in the isotopy $K_t$, there are no Reidemeister moves of type $II$ with opposite directions of the tangencies. Such moves could destroy the disks $D_t$, as shown in Fig. 16. $$\begin{picture}(0,0)\special{psfile=im5.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5707,3415)(386,-2645) \put(5563,-2012){\makebox(0,0)[lb]{\smash{\SetFigFont{17}{20.4}{\rmdefault}{\mddefault}{\updefault}$?$}}} \put(1651,-307){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_t$}}} \put(5552,317){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(5177,605){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(5267, 71){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1131,-1477){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1590,-1855){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(771,-2335){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(3787,-1944){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(2932,-2460){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(4814,-2496){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \end{picture}$$ [**Fig. 16**]{} When Reidemeister moves of type $II$, with equal directions of the tangencies, occur in the isotopy $K_t$, one gets the usual surgeries of the disks, as shown in Fig. 17 and 18. $$\begin{picture}(0,0)\special{psfile=im6.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4605,2831)(43,-2111) \put(1889,174){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_t$}}} \put(3600,-521){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(2855,140){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(4415,118){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1080,-1580){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(2775,-1555){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1870,-2111){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(475,470){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(160,468){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1169,495){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1524,475){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(769,-416){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(135,151){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}(or}}} \put( 43,-83){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\pm [K]_G$)}}} \end{picture}$$ [**Fig. 17**]{} $$\begin{picture}(0,0)\special{psfile=im7.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3826,1277)(453,-861) \put(1331,-236){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_t$}}} \put(3526,-321){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(4051,-311){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(3111,-861){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \end{picture}$$ [**Fig. 18**]{} Remember that in a transversal isotopy, there are no Reidemeister moves of type $I$. Each of the families of disks $D_t$ starts and ends with the empty set because the phenomen explained in Fig. 101 in \[F\] can still not appear. Each disk in $D_t$ has exactly two vertices corresponding both to crossings of type $0$ or $[K]_G$ or $-[K]_G$ which appeared in the underlying circle of Rudolph’s diagram. We distinguish now two cases: - [*The simple case:*]{} Let $r_1, r_2 \in \{ p_1, \dots, p_m; q^{(i)}_1, \dots, q^{(i)}_n \}$ be two crossings of a configuration $\mathcal{D}_i$ used in the definition of $T_K$. We assume that $r_1$ and $r_2$ pass together for the first time $t$ in the isotopy $K_t$ over a vertex of a disk in $D_t$, and neither $r_1$ nor $r_2$ passes over a vertex of [*another*]{} disk in $D_t$. Then, similar arguments to those used in the proof of Theorem. 4.3 in \[F\] show that one of the following three possibilities is realized: 1. $r_1$ and $r_2$ will pass a second time together over the same vertex, but in opposite directions 2. $r_1$ and $r_2$ will pass together also over the second vertex of the disk 3. There exists a third crossing $r'_1$ which has appeared together with $r_1$ by a Reidemeister move of type $II$ (hence, $w(r_1)= -w(r'_1)$, and $r'_1$ can replace $r_1$ in the configuration $\mathcal{D}_i$). Moreover, $r'_1$ will pass together with $r_2$ over a vertex of the disk (compare with the proof of Theorem. 4.3 in \[F\]). (See Fig. 19.) $$\begin{picture}(0,0)\special{psfile=im8.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5186,4907)(324,-4596) \put(885, 34){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \put(1055,-564){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1925,-528){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1630,-993){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(3417,-3749){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(4599,-3762){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(3362,-819){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(4507,-764){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(3676,-179){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1$}}} \put(4952,-3884){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \put(4935,-4178){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1'$}}} \put(3242,-29){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \put(4095,-83){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1'$}}} \put(3376,-4469){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1$}}} \put(627,-2384){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1455,-2093){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1'$}}} \put(601,-2939){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1$}}} \put(962,-3014){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \put(3512,-1454){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \put(3552,-1769){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(4802,-1701){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(4522,-2616){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(4672,-4596){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$D_t$}}} \put(1912,-2799){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(2713,-2811){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}or}}} \put(2717,-2552){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}either}}} \end{picture}$$ [**Fig. 19**]{} Obviously, $T_{K_t}$ does not change for the first two possibilities. If in the third possibility, $r_1$ and $r_2$ enter together into some configuration $\mathcal{D}_i$, then $r_2$ and $r'_1$ (instead of $r_1$) enter also in this configuration $\mathcal{D}_i$. There are two cases to distinguish: - A\) $r_1$ and hence $r'_1$ belong to $\{ p_1, \dots, p_m \} \subset \mathcal{D}_i$. But then, the contributions of $r_1$ and $r'_1$ cancel out in $T_{K_t}$ because the weight function is $w(p_1) \cdots w(p_m)$. - B\) $r_1$ (and hence $r'_1$) belongs to $\{ q^{(i)}_1, \dots, q^{(i)}_n \} \subset \mathcal{D}_i$. But then the definition of a class $c(D)$ implies that the contributions of $r_1$ and $r'_1$ cancel out in $c(D)$, and hence $T_{K_t}$ is again invariant. [*The general case:*]{} During the isotopy $K_t$, $r_1$ and $r_2$ pass together over a vertex of a disk in $D_t$. Let $t_0$ be the smallest value of $t$ for which this occurs. At a time $t_1> t_0$, $r_1$ passes over the vertex of another disk in $D_t$. Notice, that $r_1$ and $r_2$ cannot pass together over the vertex of another disk in $D_t$. We illustrate the general case with an example, shown in Fig. 20. $$\begin{picture}(0,0)\special{psfile=im9.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4487,4329)(446,-3872) \put(1358,262){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_3$}}} \put(766,-286){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \put(833,-1073){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1$}}} \put(2416,-76){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$[K]_G$}}} \put(1276,-811){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(2813,-346){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(3496,-1253){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$[K]_G$}}} \put(2318,-1201){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$S$}}} \put(3421,-1988){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \put(3983,-2768){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_3$}}} \put(3916,-3458){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1$}}} \put(3091,-2805){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$[K]_G$}}} \put(2176,-2355){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$[K]_G$}}} \put(2685,-2551){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1171,-3031){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(2221,-3458){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$S$}}} \put(4594,-811){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_t$}}} \put(1426,-1576){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$(D_1)_t$}}} \put(2948,-1793){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$(D_2)_t$}}} \end{picture}$$ [**Fig. 20**]{} We can assume that the crossing $s$ in Fig. 20 is not of type $0$ or $\pm [K]_G$ because otherwise, the crossings $r_2$ and $r_3$ on the left-hand side would already have passed together over the vertex of a disk in $D_t$. We observe that $r_1$ with $r_2$ on the left-hand side of Fig. 20 form exactly the same configurations as $r_1$ with $r_3$ on the right-hand side of Fig. 20. But the mutual configuration of $r_2$ and $r_3$ has changed (by passing $s$). Notice that we cannot eliminate the disks $(D_1)_t$, $(D_2)_t$ by an isotopy, which is supported in a 3-ball containing just the fragment of $K_t$ drawn in Fig. 20. Nevertheless, we can replace the local piece of the isotopy $K_t$ shown in Fig. 20 by the local piece of a $G$-pure isotopy $K'_t$ shown in Fig. 21. $$\begin{picture}(0,0)\special{psfile=im10.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4727,4555)(206,-4098) \put(4594,-811){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_t'$}}} \put(2243,-3234){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$S$}}} \put(3218,-2716){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \put(2782,-3834){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1$}}} \put(2986,-3317){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_3$}}} \put(1358,262){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_3$}}} \put(2198,-841){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$S$}}} \put(765,-1118){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_1$}}} \put(893,-841){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$r_2$}}} \end{picture}$$ [**Fig. 21**]{} The crossings $r_1, r_2, r_3$ contribute to $T_{K_t}$ on the left-hand-side (resp. right-hand side) of Fig. 20 exactly the same way as they contribute to $T_{K'_t}$ in the left-hand side (resp. right-hand side) of Fig. 21. But the isotopy in Fig. 21 is $G$-pure and this implies, as already proven, that $T_{K'_t}$ is invariant. Consequently, $T_{K_t}$ for the isotopy $K_t$ in fig. 20 is invariant too. Clearly, these arguments can be generalized for the case of more than two disks $D_t$ and several crossings instead of only $s$ in the local piece of the isotopy $K_t$. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ [Let $n= \max_i{n_i}$. The [*degree of $T_K$ as Gauss diagram invariant*]{} is equal to ]{} $m+ n$. This definition is justified by the observation that the complexity of the calculation of $T_K(D; c_1(D)= c_1, \dots, c_k(D)= c_k)$ for knots $K$ is a polynomial of degree $m+ n$ in the number of crossings of $K$. Hence, the invariant $T_K$ is calculated with the same (order of) complexity as a Vassiliev invariant of degree $m+n$. [**Remarks**]{}: 1. Examples 1 and 3 show that the numbers of different $G$-pure configurations $D$ of degree $m$, and of classes $c_i(D)$ of degree $n$ is in general not finite (the configurations depend on the parameter $a \in \mathbb{Z} \setminus 0$). 2. We show in an example in sect. 9 that multi-classes are usefull. As a matter of fact, $T_K(D; c_1(D)= c_1, c_2(D)= c_2)$ contains sometimes more information than $T_K(D; c_1(D)= c_1)$ and $T_K(D; c_2(D)= c_2)$ together. [*If $m= 0$ or $n= 0$, then the invariant $T_K$ is of finite type in the sense of Vassiliev-Gussarov.*]{} We omit the proof of this lemma: it is a straightforward generalization of Oestlund’s proof that the Gauss diagram invariants of Polyak-Viro for knots in $\mathbb{R}^3$ are of finite type (\[P-V\], see also \[F\], sect. 2). Gussarov has proven that each Vassiliev invariant for knots in $\mathbb{R}^3$ can be represented as a Gauss diagram invariant of Polyak-Viro (see \[G-P-V\]). We do not know wether or not this is still true for finite type invariants of global knots. But evidently, each Gauss diagram invariant of finite type which does not use the homological markings $\{0, \pm [K]_G \}$ in $G$ is a $T$-invariant for $m= 0$ (i.e. $D= \emptyset$). [*Let $m> 0$ and $n>0$ and assume that $c(D)$ (from Definition 3.3) is not a Gauss diagram identity, i.e. there exists a knot $K_t$ and a crossing $q$ of $K_t$ such that switching the crossing $q$ changes the value $c(D_t)$ (see \[F\], sect. 4). Then the invariant $T_K$ is not of finite type.*]{} We omit the general proof but show this in an example for a class of degree 1. It is clear that one can find similar examples for any class of degree at least 1. [*Example $3.4$*]{} We fix the system of generators $\{ \alpha, \beta \}$ of $H_1(T^2; \mathbb{Z})$ as shown in Fig. 22. $$\begin{picture}(0,0)\special{psfile=im11.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4005,2040)(961,-2229) \put(4966,-1059){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$T^2$}}} \put(4081,-1666){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(3743,-2109){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\beta$}}} \end{picture}$$ [**Fig. 22**]{} Let $f: T^2 \to S^1$ be a submersion such that the fibers represent $\beta$. The vector field $v$ is defined as the unit tangent vector field to the fibers of $f$. Let us consider the family of knots $K_n, n \in \mathbb{N}$ shown in Fig. 23. $$\begin{picture}(0,0)\special{psfile=im12.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4942,2711)(476,-2327) \put(1486,-1600){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1876,-1637){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1448,-1885){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(1928,-1975){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \put(5093,-1098){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K_n$}}} \put(2558,-1143){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2n$}}} \put(2138,-1720){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2326,-1712){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2963,-1607){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \end{picture}$$ [**Fig. 23**]{} $K_n \hookrightarrow T^2 \times \mathbb{R}$ is a global knot with repect to $v$ and $[K_n]= 3\alpha+ \beta \in H_1(T^2)$. We take as group $G$: $$(H_1(T^2)/\langle [K_n] \rangle) \otimes \mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z}$$ Each knot $K_n$ is $G$-pure, i.e. for each crossing $p$ the loop $pr(K^+_p)$ (as well as the loop $pr(K^-_p)$) represents the non-trivial element in $G \cong \mathbb{Z}/2\mathbb{Z}$. (Thus, each chord is marked by the same element in $G$ and we do not write the marking.) Let us consider the unique $\mathbb{Z}/2\mathbb{Z}$-pure configuration $D$ of degree 1: $$\begin{picture}(0,0)\special{psfile=pag46.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,744)(494,-258) \put(931, 39){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \end{picture}$$ Let us consider the class of $D$ of degree 1 defined by $$c(D)= \sum_{\begin{picture}(0,0)\special{psfile=page47.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(631,578)(564,-175) \put(1021, 59){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(841, 64){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \end{picture} }{w(q)}$$ (One easily verifies that $c(D)$ is indeed a class of $D$, because the chords $p$ and $q$ cannot get crossed in a $\mathbb{Z}/2 \mathbb{Z}$-pure isotopy, and if a new couple of crossings $q_i, i= 1, 2$ appears by a Reidemeister move of type $II$, then $c(D_t)$ remains unchanged.) $K_n$ has the Gauss diagram depicted in Fig. 24. $$\begin{picture}(0,0)\special{psfile=ima24.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2938,2938)(697,-2120) \put(1661,489){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(1466,129){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1126,-611){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(976,-1201){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \put(2451,-66){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(3046,-306){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2501,-1886){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2141,-2041){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1751,-1951){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1386,-1686){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(3396,-866){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}.}}} \put(3401,-916){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}.}}} \put(3406,-971){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}.}}} \end{picture}$$ [**Fig. 24**]{} Consequently, $c(p_1)= -n$, $c(p_2)= 0$ and $c(p_i)= 0$ or $1$ for each crossing $p_i$ of $K_n \setminus \{p_1, p_2 \}$. Therefore, $T_{K_n}(D; c(D)=-n)= +1$ and $T_{K_n}(D; c(D)=r)= 0$ for all $r \notin \{ -n, 0, 1 \}$. Let $K'_n$ be any knot obtained from $K_n$ by changing any of the $2n$ crossings of $K_n \setminus \{ p_1, p_2 \}$. Then $c(p_1)$ becomes strictly bigger than $-n$ and thus, $T_{K'_n}(D; c(D)=-n)= 0$ This shows that no linear combination with coefficients $\pm 1$ of $T_{K_n}(D; c(D)=-n)$ with $T_{K'_n}(D; c(D)=-n)$ can ever be equal to $0$. Moreover, replacing the fragment in $K_n$ depicted in the left-hand side of Fig. 25 by the one depicted in the right-hand side of Fig. 25 does not change the knot $K_n$. Switching any of the $s$ crossings $\{ q_1, \dots, q_s \}$ makes $c(p_1)$ strictly bigger than $-n$. Therefore, $T_K(D; c(D)=-n)$ is of degree at least $2n+ s+ 1$ and hence it is not of finite type. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ $$\begin{picture}(0,0)\special{psfile=ima25.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5439,2777)(409,-2386) \put(1731,-781){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2n$}}} \put(1703,-2386){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2n$}}} \put(4426,-2311){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2s$}}} \put(5386,-1088){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_s$}}} \put(3781,-1058){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(3901,-61){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}by}}} \end{picture}$$ [**Fig. 25**]{} Clearly, $T$-invariants which are not of finite type cannot be extracted from the generalized Kontsevitch integral (which is the universal invariant of finite type). Moreover, if we restrict ourselves to $G$-pure global knots, then the Kontsevitch integral is no longer even the universal invariant of finite type for these knots. [*Exemple $2.5$*]{} Let $a \in G$ and $a \notin \{0, \pm [K]_G \}$. Let $\makebox(50,50){D=} \makebox(50,50){\begin{picture}(0,0)\special{psfile=pag50.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(631,890)(564,-336) \put(758,419){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(810,139){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(999,-31){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \put(936,-336){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$[K]_G-a$}}} \end{picture} }$ (30,50)[ , ]{} $\makebox(50,50){ n=0 }$ $T_K(D; \emptyset)$ is a Gauss diagram invariant of degree 2 for $G$-pure global knots $K \hookrightarrow T^2 \times \mathbb{R}$. One easily verifies that $T_K(D; \emptyset)$ is of degree 2 as a finite type invariant too. If in a (not $G$-pure) isotopy $K_t$, the knot $K$ crosses exactly once a stratum of type e.g. $a^{\mbox{}}_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=dr.pstex}\end{picture} } }}(0|a,-a)$ (see Fig. 26) $$\begin{picture}(0,0)\special{psfile=ima26.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5095,1884)(345,-1225) \put(1166,-894){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(1550,-916){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3665,-129){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(4871,-122){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(4327,-1085){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1680,-252){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \end{picture}$$ [**Fig. 26**]{} then $T_{K_t}(D; \emptyset)$ changes by $\pm 1$. Consequently, $T_K(D; \emptyset)$ is not invariant for [*all*]{} isotopies of $K$ and therefore cannot be extracted from the generalized Kontsevitch integral. [**Remarks**]{}: 1. The above invariant $T_K(D; \emptyset)$ could have been equally considered as an invariant $T_K(\emptyset; c(D))$ with $$c(D)= \sum_{\begin{picture}(0,0)\special{psfile=pag50.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(631,890)(564,-336) \put(758,419){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(810,139){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(999,-31){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \put(936,-336){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$[K]_G-a$}}} \end{picture} }{w(q_1)w(q_2)}$$ 2. Let $n= 0$ and let (50,50)[$D=$]{} (50,50) (0,0) \#1\#2\#3\#4\#5[ @font ]{} (631,956)(564,-336) (723,-336)[(0,0)\[lb\]]{} (1003,455)[(0,0)\[lb\]]{} (50,50)[ or ]{} (50,50)[$D=$]{}(50,50) (0,0) \#1\#2\#3\#4\#5[ @font ]{} (631,710)(564,-175) (1025,400)[(0,0)\[lb\]]{} (720,400)[(0,0)\[lb\]]{} (50,50)[ .]{} The corresponding $T$-invariants $T_K(D; \emptyset)$ are the only other invariants of degree 2 which are of finite type and which cannot be extracted from the generalized Kontsevitch integral for knots in $F^2 \times \mathbb{R}$. More generally, let us consider $T$-invariants of finite type under $G$-pure isotopy from the point of view of the works \[V\], \[K\], \[BN\]. In the case of knots in $\mathbb{R}^3$, the famous theorem of Kontsevitch states that each $\mathbb{C}$-valued functional on the $\mathbb{C}$-vector space of (unmarked) chord diagrams can be integrated to a knot invariant of finite type if it verifies the 1-$T$ and 4-$T$ relations. (It is an easy matter to see that these relations are necessarily verified by each invariant of finite type). Let $\mathcal{K}$ be a fixed free homotopy class of an oriented loop in $F^2$ and let $[K] \in H_1(F^2; \mathbb{Z})$ be the corresponding homology class. Let $M_{\mathcal{K}}$ be the space of all possibly singular knot diagrams $K \hookrightarrow F^2 \times \mathbb{R}$ such that $pr(K)$ represents $\mathcal{K}$. Let $G$ be a fixed quotient group of $H_1(F^2; \mathbb{Z})/\langle [K] \rangle$. Finally, let $M^G_{\mathcal{K}} \hookrightarrow M_{\mathcal{K}}$ be the subspace of all possibly singular $G$-pure diagrams. Here, the marking in $G$ of a double point of $K$ is given by the corresponding marking of the positive resolution (see Fig. 27) $$\begin{picture}(0,0)\special{psfile=ima27.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3781,1104)(626,-2770) \put(1231,-2098){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1268,-2585){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(3953,-2593){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(3983,-2075){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a=[K_p^+] \in G$}}} \put(1628,-1910){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K$}}} \end{picture}$$ [**Fig. 27**]{} (Near each crossing, we write the corresponding marking in $G$.) Evidently, $M_{\mathcal{K}}$ is connected. We do not know wether $M^G_ {\mathcal{K}}$ is always connected or not. However, if we take out the set $\Sigma$ of all singular diagrams in $M^G_{\mathcal{K}}$, then ther are in general different components of $M^G_{\mathcal{K}} \setminus \Sigma$ which represent the same knot type in $F^2 \times \mathbb{R}$ (see sect. 9). Theorem 1 claims that for global knots, $T$-invariants do not depend on the chosen component of $M^G_{\mathcal{K}} \setminus \Sigma$ for a given knot type in $F^2 \times \mathbb{R}$. But let us forget global knots for one moment, and let us consider $G$-pure knots only up to $G$-pure isotopy. Let us have a look at the analogue of the above mentioned relations. Let $\mathcal{A}_G$ be the $\mathbb{C}$-vector space generated by all chord diagrams with homological markings in $G$ of the chords, and with the homotopical marking $\mathcal{K}$ for the whole circle. Finally, let $\mathcal{A}^0_G \hookrightarrow \mathcal{A}_G$ be the subspace generated by the $G$-pure chord diagrams (i.e. there are no chords with marking $0 \in G$). Let $I: \mathcal{A}_G \to \mathbb{C}$ be a functional. We want to integrate it to a knot invariant. - [*$1-T$ relation:*]{} This relation is obtained by going in $M_{\mathcal{K}}$ around a diagram which has a double point (as a singular knot) in a cusp of the projection into $F^2$. See Fig. 28. $$\makebox(10,80){I} \makebox(10,80){\Bigg( } \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima28G.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(512,391)(444,-1707) \put(571,-1707){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \end{picture} } \makebox(10,80){\Bigg) } \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg( } \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima28D.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(512,431)(444,-1742) \put(571,-1742){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \end{picture} } \makebox(10,80){\Bigg) } \makebox(10,80){=} \makebox(10,80){0}$$ [**Fig. 28**]{} - [*$2-T$ relation:*]{} We have this additional relation because, instead of considering only embeddings $S^1 \hookrightarrow F^2 \times \mathbb{R}$, we consider diagrams, i.e. embeddings together with the projection onto $F^2$. Then, the relation is obtained by going in $M_{\mathcal{K}}$ around a diagram which has two double points in an autotangency of the projection (see Fig. 29). $$\makebox(10,80){I} \makebox(10,80){\Bigg( } \makebox(60,80) {\begin{picture}(0,0)\special{psfile=ima29G.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(549,329)(439,-2082) \put(511,-2082){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} }\makebox(10,80){ \Bigg) } \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){ \Bigg( } \makebox(60,80) {\begin{picture}(0,0)\special{psfile=ima29D.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(549,204)(439,-1517) \end{picture}} \makebox(10,80){ \Bigg) } \makebox(10,80){=} \makebox(10,80){0}$$ [**Fig. 29**]{} (The markings in all other crossings or double points of these fragments are determined by the single marking $a \in G$.) There are two cases to consider, which are shown in Fig. 30 and 31. $$\makebox(10,80){\Bigg(} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima301.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(553,357)(443,232) \put(499,232){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(833,246){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima302.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(662,390)(431,-218) \put(516,-207){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(898,-218){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){\Bigg)} \makebox(10,80){-}$$ $$\makebox(10,80){\Bigg(} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima303.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(553,357)(443,232) \put(499,232){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(833,246){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima302.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(662,390)(431,-218) \put(516,-207){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(898,-218){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){\Bigg)} \makebox(10,80){=} \makebox(10,80){0}$$ [**Fig. 30**]{} $$\makebox(10,80){\Bigg(} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima311.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(662,409)(431,-237) \put(887,-218){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(512,-237){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima312.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(553,338)(435,212) \put(507,213){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(852,212){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){\Bigg)} \makebox(10,80){-}$$ $$\makebox(10,80){\Bigg(} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima311.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(662,409)(431,-237) \put(887,-218){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(512,-237){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima314.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(553,365)(443,232) \put(499,232){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(833,246){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){\Bigg)} \makebox(10,80){=} \makebox(10,80){0}$$ [**Fig. 31**]{} - [*$4-T$ relation:*]{} This relation is obtained by going in $M_{\mathcal{K}}$ around a diagram with three double points in a triple point of the projection. For each $a$, $b \in G$, we have the relation shown in Fig. 32. $$\makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(120,80){\begin{picture}(0,0)\special{psfile=ima321.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1470,1245)(116,-921) \put(1051,-598){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$b$}}} \put(400,-604){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(952,-34){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a-b$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(120,80){\begin{picture}(0,0)\special{psfile=ima322.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1556,1116)(199,-624) \put(1108,213){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1004,-348){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$b-a$}}} \put(818,227){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$b$}}} \end{picture} } \makebox(10,80){\Bigg)}$$ $$\makebox(10,80){+} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(120,80){ \begin{picture}(0,0)\special{psfile=ima323.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1510,1304)(1294,-1107) \put(2084,-179){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$b-a$}}} \put(1498,-698){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2255,-805){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$b$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(120,80){ \begin{picture}(0,0)\special{psfile=ima324.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1556,1116)(199,-624) \put(1205,-11){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1034,-341){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$b-a$}}} \put(740,216){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-b$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){=} \makebox(10,80){0}$$ [**Fig. 32**]{} The proofs are completely analogous to the one for knots in $\mathbb{R}^3$ (see also \[Go\] where it is done for knots in the solid torus). Each functional $I$ which can be integrated to a knot invariant verifies $1-T$, $2-T$, $4-T$. [*Question*]{}: [*Can each functional which verifies $1-T$, $2-T$, $4-T$ be integrated to a knot invariant?*]{} [**Remark**]{}. Goryunov \[Go\] has shown that the answer is “yes” in the case of the solid torus. Notice that our chord diagrams are planar and with homological markings. In \[A-M-R\] it is shown that the answer to the above question is “yes” if $\partial F^2 \not= \emptyset$ and if one uses chord diagrams which are immersed in $F^2$ instead of planar chord diagrams with homological markings. But then, it seems to be difficult to find such functionals. Let us consider now functionals $I^0: \mathcal{A}^0_G: \to \mathbb{C}$. Evidently, if $I^0$ can be integrated to a knot invariant under $G$-pure isotopy, then it has to verify the corresponding relations $1-T^0= \emptyset$, $2-T^0$, $4-T^0$ similar to the previous ones but where the marking $0 \in G$ is forbidden. par [*Example 3.6*]{} $$\makebox(50,50){$I^0_1:= T_K \Bigg($}\makebox(50,50){\begin{picture}(0,0)\special{psfile=p52b8.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(631,833)(564,-298) \put(1025,400){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(727,-298){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \end{picture} }\makebox(30,50){; $\emptyset$ }\makebox(10,50){$\Bigg)$}\makebox(10,50){,} \makebox(50,50) {$a\neq 0$}\makebox(10,50){.}$$ $I^0_1$ verifies $1-T$, $2-T$, and $4-T^0$ but it does [*not*]{} always verify $4-T$ (see Fig. 33). $$\makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(120,80){\begin{picture}(0,0)\special{psfile=ima321.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1470,1245)(116,-921) \put(1051,-598){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(400,-604){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(952,-34){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(120,80){\begin{picture}(0,0)\special{psfile=ima322.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1556,1116)(199,-624) \put(1108,213){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1004,-348){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(818,227){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)}$$ $$\makebox(10,80){+} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(120,80){ \begin{picture}(0,0)\special{psfile=ima323.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1510,1304)(1294,-1107) \put(2084,-179){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(1498,-698){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2255,-805){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){I} \makebox(10,80){\Bigg(} \makebox(120,80){ \begin{picture}(0,0)\special{psfile=ima324.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1556,1116)(199,-624) \put(1205,-11){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1034,-341){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(740,216){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \end{picture} } \makebox(10,80){$\Bigg)$} \makebox(10,80){$=$} \makebox(10,80){$-1$}$$ [**Fig. 33**]{} Indeed, the only non-zero contribution to $I^0_1$ from two of the three involved crossings comes from the term shown in Fig. 34, and is equal to $w(p)w(q)= -1$. $$\makebox(10,80){-} \makebox(10,80){\Bigg(} \makebox(10,80){$-I_1^0$} \makebox(10,80){\Bigg(} \makebox(120,80){\begin{picture}(0,0)\special{psfile=ima34.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1556,1116)(199,-624) \put(1205,-11){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(1034,-341){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$0$}}} \put(740,216){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){\Bigg)}$$ [**Fig. 34**]{} [*Exemple $3.7$*]{} $$\makebox(50,50){$I^0_2:= T_K \Bigg($}\makebox(50,50){\begin{picture}(0,0)\special{psfile=pag52DH.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(631,710)(564,-175) \put(1025,400){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(720,400){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} }\makebox(20,50){; }\makebox(50,50){$\emptyset \Bigg)$}$$ with $a \not= 0$ and $a \not= -a$ in $G$. $I^0_2$ verifies $1-T$ and $4-T^0$ but it does [*not*]{} always verify $2-T^0$! Indeed, in case 2, we have the relation shown in Fig. 35. $$\makebox(10,80){\Bigg(} \makebox(10,80){$I_2^0$} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima311.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(662,409)(431,-237) \put(887,-218){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(512,-237){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){$I_2^0$} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima312.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(553,338)(435,212) \put(507,213){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(852,212){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){\Bigg)} \makebox(10,80){-}$$ $$\makebox(10,80){\Bigg(} \makebox(10,80){$I_2^0$} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima311.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(662,409)(431,-237) \put(887,-218){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(512,-237){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){-} \makebox(10,80){$I_2^0$} \makebox(10,80){\Bigg(} \makebox(60,80){\begin{picture}(0,0)\special{psfile=ima314.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(553,365)(443,232) \put(499,232){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(833,246){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} } \makebox(10,80){\Bigg)} \makebox(10,80){\Bigg)} \makebox(10,80){=} \makebox(10,80){$-1$}$$ [**Fig. 35**]{} However, $I^0_2$ is an isotopy invariant for $G$-pure global knots because autotangencies with opposite directions of the branches do not occur in an isotopy through global knots. par [*Question*]{}. [*If a functional $I^0: \mathcal{A}^0_G \to \mathbb{C}$ verifies $2-T^0$ and $4-T^0$, can it be integrated to a knot invariant under $G$-pure isotopy?*]{} Let us consider now Gauss diagram invariants of degree 2 for global knots $K$, which are not of finite type. From now on, $G$ will always be a quotient group of $H_1(F^2; \mathbb{Z})/\langle [K] \rangle$. Hence, a knot is $G$-pure if and only if there is no marking equal to $0$ in $G$. By the above lemmas and the definition of the degree of a Gauss diagram invariant, we must have $m= n= 1$. Consequently, $$\makebox(60,50){$D=$}\makebox(60,50){\begin{picture}(0,0)\special{psfile=pag52M.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,947)(494,-258) \put(931, 39){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(893,524){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture}}$$ for some $a \in G$, $a \not= 0$. $$c_{++}(D)= \sum_{\begin{picture}(0,0)\special{psfile=page52B.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,1131)(494,-428) \put(758,-428){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(826, 29){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(1043,-72){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(953,538){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} }{w(q)} \qquad c_{--}(d)= \sum_{\begin{picture}(0,0)\special{psfile=page53H.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,1033)(494,-383) \put(1002,-383){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(656,485){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1036,104){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(818,119){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \end{picture} }{w(q)} \qquad$$ $$c_{+-}(D)= \sum_{\begin{picture}(0,0)\special{psfile=page53M.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,901)(494,-251) \put(656,485){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1028,480){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1043,-20){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(834,136){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \end{picture} }{w(q)} \qquad c_{-+}(D)= \sum_{\begin{picture}(0,0)\special{psfile=page53B.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,901)(494,-251) \put(656,485){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1028,480){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1056,139){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(833,-17){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \end{picture} }{w(q)}$$ $c_{++}(D)$, $c_{--}(D)$, $c_{+-}(D)$, $c_{-+}(D)$ [*are $G$-pure classes of degree 1 of $D$. Moreover, these are the only $G$-pure classes of degree 1.*]{} [**Proof**]{}. Evidently, $c_{++}$, $c_{--}$, $c_{+-}$, $c_{-+}$ are $G$-pure classes of $D$ because $p$ and $q$ cannot get crossed in a $G$-pure isotopy. Assume now that $p$ and $q$ are crossed. None of the configurations depicted in Fig. 36 $$\begin{picture}(0,0)\special{psfile=ima36.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2688,1241)(281,-615) \put(1973,476){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2675,464){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(447,491){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1171,480){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(508, 78){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1039, 38){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(2025, 78){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(2581, 63){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1523, 22){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}or}}} \end{picture}$$ [**Fig. 36**]{} enters into the class. Indeed, if one of them did, it would be invariant under Reidemeister moves of type $II$. Assume that the configuration in the left-hand side of Fig. 37 $$\begin{picture}(0,0)\special{psfile=ima37.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2688,1241)(281,-615) \put(2675,464){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(447,491){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(508, 78){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1039, 38){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(1171,480){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(1973,476){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(2697,-68){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(2186, 15){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \end{picture}$$ [**Fig. 37**]{} enters into the class. The stratum $a^{\mbox{ }}_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=dr.pstex}\end{picture} } }}(a|-a, 2a)$ of the discriminant forces then the configuration in the right-hand side of Fig. 37 to enter into the class too. But then, the stratum $a^{+}_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=dr.pstex}\end{picture} } }}(a|2a,-a)$ forces the configuration in the left-hand side of Fig. 38 $$\begin{picture}(0,0)\special{psfile=ima38.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2688,1241)(281,-615) \put(447,491){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(508, 78){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(1039, 38){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(1171,480){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2707,-60){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(2161,-101){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(2644,472){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(2000,486){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture}$$ [**Fig. 38**]{} to be also in the class. Now, the stratum $a^{+}_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=br.pstex}\end{picture} } }}(a|2a,-a)$ forces the configuration in the right-hand side of Fig. 38 to be in the class too. Thus, [*all*]{} crossings with marking $-a$ enter into the class. This class is therefore equal to $W_K(-a)$ independently of $p$ (see \[F\] for the definitions of the strata and of $W_K(-a)$). Therefore, the corresponding $T$-invariant would be $W_K(a).W_K(-a)$, which is not new, and is of course of finite type. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ $$\makebox(70,50){$ T_K:= T_K \Bigg( D=$}\makebox(70,50){\begin{picture}(0,0)\special{psfile=pag52M.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,947)(494,-258) \put(893,524){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} }$$ $$\makebox(70,50){ ; $c_{++}(D)= c_1$, $c_{--}(D)= c_2$, $c_{+-}(D)= c_3$, $c_{-+}(D)=c_4\Bigg)$ }$$ [*is the universal $T$-invariant of degree 2 which is not of finite type for $G$-pure global knots.*]{} (Compare with Definition 3.4.) $T_K$ is “universal” means that any other invariant (of degree 2, not of finite type) can be extracted from $T_K$. [**Proof**]{}. The proposition is an immediate consequence of Theorem 1 and Lemma 3.3. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ In sect. 9, we show an application of the above invariant. Let $K_1, K_2 \hookrightarrow F^2 \times \mathbb{R}$ be two global knots with respect to the same non-elliptic vector field $v$ on $F^2$. We assume that $K_i, i= 1, 2$ are not solid torus knots in $F^2 \times \mathbb{R}$. Let $\mathcal{G}$ be the set of all possible quotient groups $G$ of $H_1(F^2, \mathbb{Z})$ such that $K_1$ and $K_2$ have global representatives which are $G$-pure. Let $\mathcal{T}$ be the set of all $T$-invariants of $G$-pure knots with respect to some $G \in \mathcal{G}$. [**Conjecture**]{}: [*If $K_1$ and $K_2$ are not isotopic, then there are $T$-invariants in $\mathcal{T}$ which distinguish them.*]{} [**Remarks**]{}: 1. Remember that the usual invariants of finite type, in particular the free homotopy class of the knot, are a subset of $T$-invariants for $m= 0$. 2. For solid torus knots, the $T$-invariants are nothing but the usual invariants of finite type (extracted from the generalized Kontsevitch integral). Indeed, $H_1(S^1 \times I; \mathbb{Z})/\langle [K] \rangle$ is a finite group $G$. In the Gauss diagram of $K$ with markings $a$, $-a$ in $G$, there are no subdiagrams of the forms depicted in Fig. 39 at all (because $s$ would be a global knot homologous to $0$). $$\begin{picture}(0,0)\special{psfile=ima39.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3861,1337)(458,-896) \put(669,299){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1313,285){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2007,306){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2644,-896){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3994,296){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3338,-895){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3618,-290){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$S$}}} \put(2272,-320){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$S$}}} \put(967,-324){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$S$}}} \end{picture}$$ [**Fig. 39**]{} Hence, there are no specific $T$-invariants with respect to $G$. Therefore, we have to consider a quotient $G'$ of $G$. This means that some $a \not= 0 \in G$ becomes $0$ in $G'$. But, if for a closed braid $K$ and a given $a \in G$, no crossing is of type $[K^+_p]= a$, then it is easily seen that there always exist a crossing $p$ with $[K^+_p]= -a \in G$. But this means that $K$ is never a $G'$-pure global knot and, hence, there are not any specific $T$-invariants. 3. Global solid torus knots are closed braids. They are classified by Artin’s theorem together with Garside’s solution of the conjugacy problem in braid groups. Unfortunately, this solution has exponential complexity. $T$-invariants separate ${\mathbb Z}/ 2 {\mathbb Z}$-pure global knots in $T^2 \times {\mathbb R}$ ================================================================================================== Let $\{ \alpha, \beta \}$ be generators of $H_1(T^2)$ as shown in Fig. 22. It is more convenient to use the non-generic vector field $v$ which is tangent to the fibers of $f$ (see Example 3.4). The difference with a generic vector field (obtained by a small perturbation of $v$) is that for the latter, positive multiples of $\beta$ can be represented by global knots. But in any case, these are solid torus knots and we do not consider them. [A global knot $K \hookrightarrow T^2 \times \mathbb{R}$ with respect to $v$ is called a [*$\mathbb{Z}/2\mathbb{Z}$-pure global knot*]{} if]{}: 1. $H_1(T^2; \mathbb{Z})/\langle [K] \rangle \cong \mathbb{Z}$ 2. $K$ is $\mathbb{Z}/2\mathbb{Z}$-pure for $G= \mathbb{Z}/2\mathbb{Z} \cong (H_1(T^2)/\langle [K] \rangle)/2\mathbb{Z}$ [**Remark**]{}. In particular, condition 1 implies that a $\mathbb{Z}/2\mathbb{Z}$-pure global knot $K$ is a solid torus knot if and only if $K \hookrightarrow T^2$ (i.e. K is a torus knot). We show a typical example of a $\mathbb{Z}/2\mathbb{Z}$-pure global knot in Fig. 40. $[K]= 3\alpha+ \beta$ and $\alpha$ is a generator of $H_1(T^2)/\langle [K] \rangle$. Notice that switching a crossing $p$ does not change the marking of $p$ for a $\mathbb{Z}/2\mathbb{Z}$-pure global knot. Consequently, the property of a global knot to be $\mathbb{Z}/ 2\mathbb{Z}$-pure or not depends only on the underlying curve $pr(K) \hookrightarrow T^2$. $$\begin{picture}(0,0)\special{psfile=im14.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4942,2884)(476,-2500) \put(1919,-1309){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$2\alpha + \beta$}}} \put(1353,-2029){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$2\alpha + \beta$}}} \put(4116,-2455){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$2 \alpha + \beta$}}} \put(3216,-2129){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$2 \alpha + \beta$}}} \put(2477,-1352){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$2\alpha + \beta$}}} \put(2389,-1659){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(2873,-1701){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(3052,-1958){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(4070,-2169){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(1960,-1645){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \end{picture}$$ [**Fig. 40**]{} [*Let $K \hookrightarrow T^2 \times \mathbb{R}$ be a $\mathbb{Z}/ 2\mathbb{Z}$- pure global knot with $c$ crossings. Let $d$ be any natural number not bigger than $c$. Then, the knot $K$ is uniquely determined by the set of all $T$-invariants $T_K$ of finite type, of the degrees $(m= d, n= 0)$ with respect to $G= \mathbb{Z}/2 \mathbb{Z}$.*]{} [**Remark**]{}. The number of such $T$-invariants is finite. Thus, Theorem 2 proves the conjecture in sect. 3, in the case of $\mathbb{Z}/2\mathbb{Z}$-pure global knots in $T^2 \times \mathbb{R}$ and, moreover gives an effective solution to the problem. Notice that we do not need here the $T$-invariants of infinite type. [**Proof**]{}. The proof consists of two steps. [*Step 1:*]{} The Gauss diagram with markings in $G \cong \mathbb{Z}/ 2\mathbb{Z}$ determines $K$. [*Step 2:*]{} The invariants $T_K$ determine the Gauss diagram of $K$. [*Step 1:*]{} For local knots, it is well known and rather evident. But it is not at all obvious for knots in $T^2 \times \mathbb{R}$. By definition, the marking of each crossing of a $\mathbb{Z}/2 \mathbb{Z}$-pure knot is the non-trivial element in $\mathbb{Z}/ 2\mathbb{Z}$. Therefore, we do not write it in Gauss diagrams, configurations, etc …The $T$-invariant of degree $(d_1, d_2)= (0, 0)$ is the free homotopy class of $K$, or (equivalently here), the homology class represented by $K$ (which is a primitive class because $K$ is not a solid torus knot). Consequently, we have to show that the Gauss diagram determines $K$ in its given homology class. [A set of arrows in a Gauss diagram is called a [*bunch of arrows*]{} if their number is even and:]{} 1. they are near to each other (i.e. there are small arcs on $S^1$ between them where no other arrow starts or ends) 2. each two arrows cut in exactly one point 3. the orientation of the arrows is alternating 4. all the arrows have the same writhe (See an example in Fig. 41.) $$\begin{picture}(0,0)\special{psfile=ima41.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4983,2792)(446,-2193) \put(1230,-2045){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(3760,310){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(3760,-2025){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(4235,-2135){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(4710,-1960){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(1577,330){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(4253,443){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(4751,220){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(2441,-75){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$s_1$}}} \end{picture}$$ [**Fig. 41**]{} *After possibly performing Reidemeister moves of type $II$, such that each of them decreases the number of crossings (see left-hand part of Fig. 42), $$\begin{picture}(0,0)\special{psfile=ima42.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4325,446)(85,-369) \end{picture}$$* [**Fig. 42**]{} the Gauss diagram of a $\mathbb{Z}/2\mathbb{Z}$-pure global knot is of the following form: some chord diagram where each chord is replaced by some bunch of arrows. Moreover, there exists a homotopy from $K$ to a torus knot $K'$ which is an isotopy of diagrams besides possibly performing transformations of the type depicted in the right-hand part of Fig. 42. (Of course, $K'$ is determined by the homology class represented by $K$.) [*Example*]{} The knot in Fig. 40 corresponds to the diagram in the right-hand part of Fig. 43. $$\begin{picture}(0,0)\special{psfile=ima43.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3836,1578)(310,-787) \put(2718,245){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3012,437){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3408,377){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3744,362){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2853,-283){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2958,-121){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(3456,-166){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3711,-136){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3735,-325){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3540,-517){\makebox(0,0)[lb]{\smash{\SetFigFont{10}{12.0}{\rmdefault}{\mddefault}{\updefault}$+$}}} \end{picture}$$ [**Fig. 43**]{} Notice that e.g. (40,10) (0,0) \#1\#2\#3\#4\#5[ @font ]{} (290,333)(445,68) can never appear as the underlying chord diagram of a $\mathbb{Z}/2\mathbb{Z}$-pure global knot. (This is an exercise). [**Proof of Lemma $4.1$**]{} We start by eliminating all couples of arrows corresponding to crossings say $q_1$ and $q_2$ as shown in Fig. 44. $$\begin{picture}(0,0)\special{psfile=ima44.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5257,2606)(458,-2228) \put(1590,-2228){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \put(1960,194){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(1165,222){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(1450,-60){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(1184,-395){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1733,-373){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \end{picture}$$ [**Fig. 44**]{} We assume that the arcs $I_1$ and $I_2$ are empty. We have to prove that both arcs $I_1$ and $I_2$ are small or equivalently that $[K^+_{q_1}]= [K^+_{q_2}] \in H_1(T^2; \mathbb{Z})$. Then the above operation corresponds to a Reidemeister move inverse to the one depicted in the left-hand part of Fig. 42. After possibly performing a (global) isotopy of the diagram, we may assume that at least one of the two arcs, say $I_1$, is small (see Fig. 45). $$\begin{picture}(0,0)\special{psfile=im17a.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4942,3698)(476,-3314) \put(2716,-2011){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(2446,-1321){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \end{picture}$$ $$\begin{picture}(0,0)\special{psfile=im17.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5376,2711)(210,-6437) \put(2723,-5625){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \put(2483,-5895){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(3796,-5903){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \end{picture}$$ [**Fig. 45**]{} If the arc $I_2$ cannot be made small at the same time as $I_1$, then we have exactly the situation shown in Fig. 45. But then $w(q_1)= w(q_2)$ in contradiction to our assumption on $q_1$ and $q_2$. We observe now that for a diagram of a $\mathbb{Z}/2\mathbb{Z}$-pure global knot, there are no possible Reidemeister moves of type $III$ at all. Indeed, in $pr(K)$, there cannot be any triangle $$\begin{picture}(0,0)\special{psfile=ima46.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1764,752)(344,-1156) \end{picture}$$ [**Fig. 46**]{} as shown in Fig. 46 because the three crossings $q_i, i=1, 2, 3$ always verify a relation: $[K^+_{q_3}] = [K^+_{q_1}] + [K^+_{q_2}] mod [K]$. Consequently, they could not be all three non-zero in $G \cong \mathbb{Z}/2\mathbb{Z}$. After having performed all possible Reidemeister moves of type $II$ as in the left-hand side of Fig. 42, we obtain a diagram of $K$ which we call [*minimal*]{}. It is characterized by the fact that it does not allow any Reidemeister moves, except for those which strictly increase the number of crossings (i.e. the move inverse to the one in the left-hand side of Fig. 42). [**Claim $1$**]{}. If the minimal diagram is not empty, then it contains always a [*2-gon*]{} (see the left-hand part of Fig. 47). $$\begin{picture}(0,0)\special{psfile=ima47.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4946,2685)(446,-1928) \put(3895,-68){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(3580,-1023){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(4290,-1928){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(4248,562){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \end{picture}$$ [**Fig. 47**]{} Moreover, the following fragment (Fig. 47, right-hand side) of the minimal Gauss diagram of a $\mathbb{Z}/2\mathbb{Z}$-pure knot corresponds always to a 2-gon. Here, $w(q_1)= w(q_2)$ and $I_i, i=1, 2$ are both empty. [**Proof of the claim**]{}. We start with the following observation: Let $D(K)$ be a knot which is obtained from $K$ after performing a Dehn twist of $T^2$. We do not change the vector field $v$. If the Dehn twist is along $\beta$, or positive along $\alpha$, then $D(K)$ is still a global ($\mathbb{Z}/ 2\mathbb{Z}$-pure) knot with respect to $v$ (see Fig. 22). By definition, the two sides of a 2-gon form a loop which is homotopic to $0$ in $T^2$. Let us consider first “fake” 2-gons, i.e. 2-gons in $pr(K) \subset T^2$ such that the corresponding loop is not homotopic to $0$ in $T^2$. Using the above observation, we can restrict our considerations exactly to the two cases shown in Fig. 48. $$\begin{picture}(0,0)\special{psfile=im19.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3155,3979)(436,-3491) \put(2078,-786){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \put(1772,-192){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(2841,-2638){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(2426,-3433){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(3591,-597){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\beta$}}} \put(2116,-2558){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \put(436,323){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}Case $1$}}} \put(466,-1274){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}Case $2$}}} \end{picture}$$ [**Fig.48**]{} $I_1 \cup I_2$ does not cut the rest of the knot $K$. Consequently, in the second case, $K$ is a solid torus knot which is not a torus knot (the minimal diagram is not empty). We do not consider these knots. In the first case, $K$ represents $2\alpha + x\beta, x \in \mathbb{Z}$ and $x$ odd. One easily checks that in the minimal diagram of $K$ there is always some 2-gon. An example is shown in Fig. 49. $$\begin{picture}(0,0)\special{psfile=im20.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3231,1567)(585,-1423) \put(2624,-1187){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(2165,-1191){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(1304,-703){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(1026,-1104){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \end{picture}$$ [**Fig. 49**]{} If we change in the diagram in Fig. 47 exactly one of the crossings $q_1$ or $q_2$ to its inverse, then we obtain a fragment as shown in Fig. 44. We have already proven that the fragment in Fig. 44 corresponds to $$\begin{picture}(0,0)\special{psfile=page69.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(927,389)(85,-312) \end{picture}$$ Consequently, the fragment in Fig. 47 corresponds to $$\begin{picture}(0,0)\special{psfile=page69bis.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(990,395)(418,46) \end{picture}$$ which is a 2-gon. Assume now that the diagram of $K$ (in general position) contains no 2-gons at all. We have already proven that the diagram of $K$ contains no triangles whose three sides form a loop which is contractible in $T^2$. [**Sub-claim**]{}. [*The sides of each $n$-gon in $pr(K) \subset T^2$ form a contractible loop if $n \geq 3$.*]{} Indeed, either we are in the situation analogue to case 2 in Fig. 48, and hence, $K$ is a solid torus knot, or we are in the situation analogue to case 1in Fig. 48. But this is not possible if $n \geq 3$ as shown in Fig. 50: $$\begin{picture}(0,0)\special{psfile=im21.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2762,1011)(699,-1244) \put(1694,-797){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(1544,-1119){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(2732,-883){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_3$}}} \put(3445,-429){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\beta$}}} \put(1821,-455){\makebox(0,0)[lb]{\smash{\SetFigFont{14}{16.8}{\rmdefault}{\mddefault}{\updefault}?}}} \end{picture}$$ [**Fig. 50**]{} The branch of $K$ through $q_1$ and $q_2$ is blocked. This proves the subclaim. The assuption (no 2-gons) together with the subclaim imply that the 4-valent graph $pr(K) \subset T^2$ splits $T^2$ into contractible 4-gons, 5-gons …Let $v_0$ be the number of vertices and $v_1$ be the number of edges of $pr(K)$. Let $v_2$ be the number of components of $T^2 \setminus pr(K)$. Evidently, $v_1= 2v_0$. One has: $v_0- v_1+ v_2= \chi(T^2)= 0$ and hence, $v_0= v_2$. We denote by $\sharp(.)$ the number of (.). $\sharp$(angles)$= 4v_0= 4v_2$. On the other hand, $\sharp$(angles)$= 4\sharp$(4-gons)$+ 5\sharp$(5-gons)$+ \dots$ This implies that $0= \sharp$(5-gons)= $\sharp$(6-gons)$= \dots$ Consequently, $T^2 \setminus pr(K)$ consists only of contractible 4-gons. We take one of them. It has at least two opposite sides which have the same orientation (induced by the orientation of $K$). We add to the 4-gon the two neighbouring 4-gons corresponding to the remaining two sides (see Fig.51). $$\begin{picture}(0,0)\special{psfile=im22.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3177,1003)(438,-1281) \put(1777,-804){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$4-gon$}}} \end{picture}$$ [**Fig. 51**]{} We continue the process and at the end, we obtain an orientable immersed band in $T^2$. But the boundary of the band has two components contradicting the fact that $K$ is a knot. This proves Claim 1. We take now the existing 2-gon and make a homotopy as indicated in Fig. 52. $$\begin{picture}(0,0)\special{psfile=ima52.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4779,1392)(446,-1493) \put(1246,-1298){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K$}}} \put(3916,-1493){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K'$}}} \end{picture}$$ [**Fig. 52**]{} $K'$ is again a $\mathbb{Z}/2\mathbb{Z}$-pure global knot. [**Claim $2$**]{} [*If $K$ was already minimal, then $K'$ is minimal too*]{}. [**Proof of Claim 2**]{} If $K'$ is not minimal, then it contains a fragment as shown in Fig. 53 $$\begin{picture}(0,0) \special{psfile=im23.pstex} \end{picture} \setlength{\unitlength}{4144sp} \begingroup\makeatletter\ifx\SetFigFont\undefined \gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont} \fi\endgroup \begin{picture}(3215,926)(621,-1186) \put(2016,-695){\makebox(0,0)[lb]{\smash{\SetFigFont{17}{20.4}{\rmdefault}{\mddefault}{\updefault}or}}} \end{picture}$$ [**Fig. 53**]{} We have already proven that we can transform $K$ into a torus knot by performing only the operations $$\begin{picture}(0,0)\special{psfile=page74.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5331,431)(85,-342) \put(2574,-195){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}and}}} \end{picture}$$ on the diagram of $K$. Therefore, we may assume that we have eliminated all crossings of $K$ outside of the above fragment. Again, by using appropriate Dehn twists, we can reduce our considerations to the two cases shown in Fig. 54. (We need only $pr(K) \subset T^2$.) $$\begin{picture}(0,0)\special{psfile=im24.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3691,2176)(466,-1861) \put(2141,-1164){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(466,180){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}Case $1$}}} \end{picture}$$ $$\begin{picture}(0,0)\special{psfile=im24bis.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3651,2266)(443,-1936) \put(1848,-1209){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q$}}} \put(443,195){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}Case $2$}}} \end{picture}$$ [**Fig. 54**]{} In case 1, we have $[K^+_q]= 2\alpha$ mod $[K]$ and in case 2, we have $[K^+_q]= 2\alpha+ 2\beta$ mod $[K]$. Consequently, $[K^+_q]= 0 \in G$ and $K$ was not $\mathbb{Z}/2\mathbb{Z}$-pure. This proves Claim 2. By Claim 2, we can reduce the minimal diagram of $K$ to a torus knot by using [*only*]{} the operation $$\begin{picture}(0,0)\special{psfile=page75.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2067,425)(431,-133) \end{picture}$$ Moreover, we have proven that this operation corresponds exactly to the operation in Fig. 55. $$\begin{picture}(0,0)\special{psfile=im25.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2543,1673)(383,-2737) \put(567,-1564){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(627,-1925){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(721,-1220){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(646,-2303){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(383,-2679){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$w(q_1)=w(q_2)$}}} \end{picture}$$ [**Fig. 55**]{} Suppose, that we have a fragment as shown in Fig. 56. $$\begin{picture}(0,0)\special{psfile=im26.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3216,3873)(643,-5158) \put(1433,-2379){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(2048,-1921){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(2461,-2116){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_3$}}} \put(2776,-2536){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_4$}}} \put(2109,-5100){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(2806,-4890){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(1336,-4920){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(1358,-1568){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(1995,-1441){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(2685,-1651){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \end{picture}$$ [**Fig. 56**]{} In fact, we have already shown that this implies automtically $w(q_1)= w(q_2)$, $w(q_3)= w(q_4)$, $w(q_2)= -w(q_3)$, and that $q_2$, $q_3$ can be eliminated by a move $$\begin{picture}(0,0)\special{psfile=page76H.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2063,419)(85,-342) \end{picture}$$ Consequently, repeating the operation $$\begin{picture}(0,0)\special{psfile=page76B.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2821,431)(1518,-342) \end{picture}$$ creates just bunches of arrows and Lemma 4.1 is proven. *Let $K$ be a $\mathbb{Z}/2\mathbb{Z}$-pure global knot.* Then, $K$ is determinated by its Gauss diagram with markings in $G \cong \mathbb{Z}/2\mathbb{Z}$ together with the homology class $[K] \in H_1(T^2; \mathbb{Z})$. [**Proof**]{}. As we have seen in the proof of Lemma 4.1, we can detect all possible moves $$\begin{picture}(0,0)\special{psfile=page76H.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2063,419)(85,-342) \end{picture}$$ with the Gauss diagram. Performing these moves, we obtain the minimal diagram of $K$. By Lemma 4.1, the minimal diagram of $K$ is obtained from the torus knot $K'$ (which is determinated by its homology class), by performing only operations $$\begin{picture}(0,0)\special{psfile=page76B.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2821,431)(1518,-342) \end{picture}$$ on the diagram of $K'$. Each such operation corresponds to a bunch of two arrows. Thus, we only need to show that the resulting knot is completely determined by the place of the bunch in the Gauss diagram, the directions of the arrows and their writhe. Indeed, the operation shown in Fig. 57 $$\begin{picture}(0,0)\special{psfile=ima57.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5160,2767)(548,-2225) \put(4576,-2143){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(1808,197){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \put(1441,-2225){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(3981,-194){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(4011,-1148){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(4386,347){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \end{picture}$$ [**Fig. 57**]{} corresponds to the change in Fig. 58 $$\begin{picture}(0,0)\special{psfile=im27.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5020,1126)(458,-2802) \put(4383,-2029){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \put(4278,-2438){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(1038,-2469){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \put(1028,-1953){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(5478,-2302){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\beta$}}} \end{picture}$$ [**Fig. 58**]{} if $w(q_1)= w(q_2)= +1$, or to the change in Fig. 59 if $w(q_1)= w(q_2)= -1$. $$\begin{picture}(0,0)\special{psfile=im28.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5020,1119)(458,-2794) \put(1038,-2469){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \put(1028,-1953){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(5478,-2302){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\beta$}}} \put(4268,-1958){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_2$}}} \put(4358,-2466){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I_1$}}} \end{picture}$$ [**Fig. 59**]{} Lemma 4.2 is proven. [*Step 2*]{} [*Let $K$ be a $\mathbb{Z}/2\mathbb{Z}$-pure global knot.*]{}\ [*A) Let* ]{} $$\begin{picture}(0,0)\special{psfile=ima60.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1350,1484)(653,-892) \put(1647,-278){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(1141,-233){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(1253,427){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I$}}} \end{picture}$$ [**Fig. 60**]{} [*the diagram in Fig. 60 occur as subdiagram of the minimal diagram of $K$ in such a way that $I= \emptyset$ (i.e. there do not start or end any arrows in $I$). Then $w(q_1)= w(q_2)$.*]{}\ [*Let*]{} $$\begin{picture}(0,0)\special{psfile=ima61.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1377,1484)(629,-892) \put(1647,-278){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(1141,-233){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(1253,427){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I$}}} \put(629,-795){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$J$}}} \end{picture}$$ [**Fig. 61**]{} [*the diagram in Fig. 61 occur as subdiagram of the minimal Gauss diagram of $K$ in such a way that $I= \emptyset$, $J= \emptyset$. Then $w(q_3)= -w(q_1)= -w(q_2)$.*]{} [**Remark**]{}. Lemma 4.1 implies that all three crossings belong to different bunches. Evidently, Lemma 4.3 allows to calculate all the writhes of a minimal diagram if one knows the writhe of [*one*]{} bunch of arrows. [**Proof of Lemma 4.3**]{}. We will prove only A). The proof of B) is similar, and is therefore omitted. Using Lemma 4.1, we can eliminate all crossings of $K$, except of the four crossings shown in Fig. 62. $$\begin{picture}(0,0)\special{psfile=ima62.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(1350,1484)(653,-892) \put(1253,427){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I$}}} \put(1364,142){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(1176, 15){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(1142,-459){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_1$}}} \put(1414,-646){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p_2$}}} \end{picture}$$ [**Fig. 62**]{} We know already that $w(q_1)= w(p_1)$, $w(q_2)= w(p_2)$. After suitable Dehn twists, and after making $I$ small, $K$ is transformed into a knot $K'$ so that one of the possibilities depicted in Fig. 63 is realized. In both cases, $w(q_1)= w(q_2)$. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ $$\begin{picture}(0,0)\special{psfile=im29.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4519,3013)(451,-2613) \put(451,235){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}Case $1$}}} \put(2454,-1723){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(2604,-1499){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(2528,-1250){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1980,-1512){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2566,-2555){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$[K']=3\alpha+ \beta$}}} \end{picture}$$ $$\begin{picture}(0,0)\special{psfile=im29bis.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4549,2678)(496,-2349) \put(2502,-1173){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2812,-1203){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_2$}}} \put(2872,-1393){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$q_1$}}} \put(2807,-1683){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2506,-2291){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$[K']=3\alpha-\beta$}}} \put(496,164){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}Case $2$}}} \end{picture}$$ [**Fig. 63**]{} Let $K$ be the diagram of a $\mathbb{Z}/2\mathbb{Z}$-pure global knot with $c$ crossings and let $K'$ be the corresponding minimal diagram with $c'$ crossings. (We remind that a “diagram” is a knot together with his regular projection into $T^2$.) Each $T$-invariant $T_K$ of degree $(d, 0)$ is $0$ for $d> c'$. Indeed, $K$ is isotopic to $K'$ and in the Gauss diagram of $K'$, there are not any configurations of $d$ arrows. Let $D$ be the Gauss diagram of $K'$, and let $\bar D$ be the Gauss diagram of $K'$, without the writhes. Thus, $\bar D$ is a $\mathbb{Z}/2\mathbb{Z}$-pure configuration of degree $c'$ (see Def. 3.2). By Theorem 1, $$T_K(\bar D; \emptyset):= \sum_{\bar D}w(p_1) \cdots w(p_{c'})$$ is an isotopy invariant of $K$. In each bunch, there is an even number of arrows, and, consequently, $T_K(\bar D; \emptyset)= +1$. By Lemma 4.3, there are only two possibilities for the writhes of $D$. Therefore, the $T$-invariant (of finite type) $T_K(\bar D; \emptyset)$ of degree $(c', 0)$ almost determines $K$: there are at most two knots with the same invariant. Their Gauss diagrams are obtained one from the other by a simultaneous switch of the writhes. Let $K_1$ and $K_2$ be the corresponding knots (we know already that they are determined by their Gauss diagrams). We have to distinguish them by $T$-invariants of smaller degree. Let $p$ be an arrow of $\bar D$ and let $\bar D_p$ be the configuration $\bar D \setminus p$ of degree $c'- 1$. Of course, different $p$ could determine the same configuration $\bar D_p$. Evidently, for each configuration $\bar D_p$, we have $$T_{K_1}(\bar D_p; \emptyset):= \sum_{\bar D_p \subset D(K_1)}\prod_{p_i \in \bar D_p}{w(p_i)}= -T_{K_2}(\bar D_p; \emptyset)$$ Consequently, if $T_K(\bar D_p; \emptyset)\not= 0$ for some $\bar D_p$, then $T_K(\bar D; \emptyset)$ together with $T_K(\bar D_p; \emptyset)$ determine $K$. Assume that for all $p$, $T_K(\bar D_p; \emptyset)= 0$. Evidently, for all couples $(p_1, p_2)$ of arrows in $\bar D$ and the corresponding configurations $\bar D_{(p_1, p_2)}:= \bar D \setminus \{ p_1, p_2 \}$, we have $$T_{K_1}(\bar D_{(p_1, p_2)}; \emptyset)=$$ $$T_{K_2}(\bar D_{(p_1, p_2)}; \emptyset)$$ Therefore, we go on with considering all triples $(p_1, p_2, p_3)$ of arrows in $\bar D$. For the configurations $\bar D_{(p_1, p_2, p_3)}:= \bar D \setminus \{p_1, p_2, p_3 \}$, we have $$T_{K_1}(\bar D_{(p_1, p_2, p_3)}; \emptyset)=$$ $$-T_{K_2}(\bar D_{(p_1, p_2, p_3)}; \emptyset)$$ If again for all triples $(p_1, p_2, p_3)$, one has $T_K(\bar D_{(p_1, p_2, p_3)}; \emptyset)= 0$, then we continue with 5-tuples $(p_1, p_2, p_3, p_4, p_5)$ and so on …At the end, we have either distinguished $K_1$ from $K_2$ or proven that $T_{K_1}(c; \emptyset)= T_{K_2}(c; \emptyset)$ for any $\mathbb{Z}/ 2\mathbb{Z}$-pure configuration $c$ (of course, $T_K(c; \emptyset)= 0$ for all configurations $c$ which are not subconfigurations of $\bar D$). One easily sees that in the latter case, the Gauss diagrams of $K_1$ and $K_2$ are isotopic, and hence, by Lemma 4.2, $K_1$ and $K_2$ are isotopic too. Theorem 2 is proven. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ Non-invertibility of knots in $T^2 \times \mathbb{R}$ ===================================================== Let $flip: T^2 \to T^2$ be the hyper-elliptic involution shown in Fig. 64. $$\begin{picture}(0,0)\special{psfile=im30.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4435,1738)(100,-1649) \put(4370,-645){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\pi$}}} \put(2550,-1360){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\beta$}}} \put(3505,-90){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$T^2$}}} \end{picture}$$ [**Fig. 64**]{} The orientation preserving involution $flip \times id$ on $T^2 \times \mathbb{R}$ will also be called $flip$ for simplicity. $flip$ acts as $-1$ on $H_1(T^2; \mathbb{Z})$. Let $K \hookrightarrow T^2 \times \mathbb{R}$ be any oriented knot, and let $-flip(K)= flip(-K)$ be the knot obtained from $flip(K)$ by reversing its o [$K$ is called [*invertible*]{} if $K$ is ambient isotopic to $-flip(K)$ in $T^2 \times \mathbb{R}$. Otherwise, $K$ is called [*non-invertible*]{}]{}. [**Remarks**]{}. 1. We show in the next section that quantum invariants do not detect non-invertibility. 2. We show in sect. 7 that our notion of invertibility for knots in $T^2 \times{R}$ coincides with the usual notion of invertibility for certain links in $S^3$. 3. The knot $-flip(K)$ is always homotopic to $K$ in $T^2 \times \mathbb{R}$ Let $v$ be our standard vector field on $T^2$ (see sect. 4). Let $K \hookrightarrow T^2 \times \mathbb{R}$ be a (canonically oriented) global knot with respect to $v$. Let $G$ be a quotient group of $H_1(T^2; \mathbb{Z})/\langle [K] \rangle$. We assume that $K$ is a $G$-pure global knot. Let $p$ be a crossing of $K$, and let $p'$ be the corresponding crossing of $-flip(K)$. [*$-flip(K)$ is a $G$-pure global knot with respect to $v$ too. Moreover,*]{} 1. $w(p)= w(p')$ 2. $[K^+_{p'}]= -[K^+_p]$ [*in*]{} $G$ 3. [*Let $D \subset \mathbb{R}^2$ be the Gauss diagram of $K$ without writhes and homological markings. Then, $-flip(D) \subset \mathbb{R}^2$ is obtained from $D$ by a reflection with respect to any line in $\mathbb{R}^2$, followed by the reversion of the orientation of the circle.*]{} [**Proof**]{}. $flip(K)$ is a knot transversal to $v$, but with the wrong orientation. Hence, $-flip(K)$ is a global knot. $flip: T^2 \times \mathbb{R} \to T^2 \times \mathbb{R}$ preserves the orientation and, consequently, $w(p)= w(p')$ for each crossing $p$. The involution $flip$ maps $K^+_p$ to $K^+_{p'}$. Reversing the orientation of $flip(K)$, the knot $-K^-_{p'}$ for $flip(K)$ is mapped to the knot $K^+_{p'}$ for $-flip(K)$. Thus, $[K^+_{p'}]= [K^-_p]=[K]- [K^+_p]$ in $H_1(T^2; \mathbb{Z})$, and hence, $[K^+_{p'}]=-[K^+_p]$ in $G$. If $[K^+_p] \not= 0$ in $G$, then $K$ is $G$-pure. Therefore, if $K$ is $G$-pure, $-flip(K)$ is $G$-pure too. Let $D_K$ be the Gauss diagram of $K$ without writhes and homological markings (in $G$). The circle of $D_K$ is always supposed to be embedded in the standard way in $\mathbb{R}^2: {\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=br.pstex}\end{picture} } }$ $D_{flip(K)}$ is exactly the same Gauss diagram (but the knots $K$ and $flip(K)$ are embedded in different ways), because $flip$ preserves the orientation of the lines $\mathbb{R}$ and, hence, preserves undercrosses and overcrosses. Changing the orientation of $flip(K)$ changes only the orientation of the Gauss diagram $D_{flip(K)}= D_K$. To obtain the standard embedding of the circle in the plane, we only need to perform a reflection with respect to a line in the plane. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ [Let $D$ be any $G$-pure configuration with markings in $G$. The [ *inverse configuration*]{} $\bar D$ is obtained from $D$ by the successive operations:]{} 1. [*a reflection with respect to a line in the plane*]{} 2. [*reversing the orientation of the circle*]{} 3. [*replacing each marking $a \in G$ by $-a \in G$.*]{} [**Remark**]{} The inverse configuration is also a $G$-pure configuration. [*Let $K \hookrightarrow T^2 \times \mathbb{R}$ be a $G$-pure global knot and let $D$ be any $G$-pure configuration. If $K$ is invertible, then for the $T$-invariants (of finite type), the following holds*]{}: $$T_K(D; \emptyset)= T_K(\bar D; \emptyset)$$ [**Proof**]{}. $K$ and $-flip(K)= flip(-K)$ are $G$-pure global knots (with respect to the same $v$), and they are homotopic. Lemma 5.2 follows then immediately from Theorem 1, Lemma 5.1, and the definition of the inverse configuration $\bar D$. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ [**Remark**]{} Lemma 5.2 can be generalized in a straightforward way to the case of general $T$-invariants $T_K(D; c_1(D)= c_1, \dots, c_k(D)= c_k)$ for $G$-pure global knots (see Def. 3.4). In particular, the [*inverse class*]{} $\bar c(\bar D)$ is defined exactly as $c(D)$, replacing $D$ by $\bar D$ and each configuration $\mathcal{D}_i$ by its inverse configuration $\bar \mathcal{D}_i$ (see Def. 3.3 and 5.2). For example, if $K$ is invertible, then $T_K(\emptyset; c(\emptyset))=T_K(\emptyset; \bar c(\emptyset))$ for each $G$-pure class $c(\emptyset)$. [*Let $K \hookrightarrow T^2 \times \mathbb{R}$ be a $\mathbb{Z}/ 2\mathbb{Z}$-pure global knot, let $D$ be the corresponding minimal configuration, and let $D_s \subset D$ be a subconfiguration of highest odd degree such that $T_K(D_s; \emptyset) \not= 0$ (see sect. 4). then, $K$ is invertible if and only if*]{} $$T_K(D; \emptyset)= T_K(\bar D; \emptyset)$$ [*and*]{} $$T_K(D_s; \emptyset)= T_K(\bar D_s; \emptyset)$$ [**Proof**]{}. As shown in the proof of Theorem 2, $T_K(D; \emptyset)$ and $T_K(D_s; \emptyset)$ determine the knot $K$. Lemma 5.3 follows then immediately from Lemma 5.2. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ [*Example 5.1*]{} $$\begin{picture}(0,0)\special{psfile=im31.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4036,2470)(310,-2157) \put(1001,-1924){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1201,-1929){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1361,-1654){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1551,-1634){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1786,-1634){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2016,-1649){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2181,-1174){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2406,-1154){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(4081,-909){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K$}}} \end{picture}$$ [**Fig. 65**]{} The knot shown in Fig. 65 represents $4\alpha+ \beta$ in $H_1(T^2; \mathbb{Z})$ and is a $\mathbb{Z}/2\mathbb{Z}$-pure global knot. Its Gauss diagram is shown in Fig. 66. $$\begin{picture}(0,0)\special{psfile=im32.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2390,2415)(1061,-2161) \put(2244,-23){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2744,-438){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1681,-347){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1264,-1130){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1592,-1863){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2060,-1842){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2531,-1722){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2918,-1410){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \end{picture}$$ [**Fig. 66**]{} Let $D$ be the configuration of degree 6 shown in Fig. 67. $$\begin{picture}(0,0)\special{psfile=im33.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2390,2415)(1061,-2161) \end{picture}$$ [**Fig. 67**]{} Evidently, each configuration which does not contain a subconfiguration as depicted in Fig. 68 $$\begin{picture}(0,0)\special{psfile=ima68.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2234,2558)(3474,-2098) \put(4547,328){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \put(4420,-2049){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\emptyset$}}} \end{picture}$$ [**Fig. 68**]{} is a $\mathbb{Z}/2\mathbb{Z}$-pure configuration (see Def. 3.2). Consequently, $D$ is a $\mathbb{Z}/2\mathbb{Z}$-pure configuration. Using Fig. 66 it takes some seconds to calculate $T_K(D; \emptyset)= -1$. The inverse configuration $\bar D$ is shown in Fig. 69. $$\begin{picture}(0,0)\special{psfile=im34.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2390,2412)(1073,-2140) \end{picture}$$ [**Fig. 69**]{} We see immediately that $\bar D$ does not appear at all in the Gauss diagram of $K$ (the cyclic ordering of the bunches has changed). Therefore, $T_K(\bar D; \emptyset)= 0$, and the knot $K$ is not invertible according to Lemma 5.3. [*Example 5.2*]{} This is a more complicated example. $$\begin{picture}(0,0)\special{psfile=im35.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4229,2616)(929,-2322) \put(3001,-1223){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3501,-1278){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2686,-1623){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2491,-1613){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2211,-1798){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1991,-1798){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3161,-1628){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(3431,-1613){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1776,-2003){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1516,-1968){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(5158,-1483){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K$}}} \end{picture}$$ [**Fig. 70**]{} The knot $K$ drawn in Fig. 70 is a global knot which represents $5\alpha+ \beta$ in $H_1(T^2; \mathbb{Z})$. Let $G:= (H_1(T^2)/\langle [k] \rangle)/3\mathbb{Z} \cong \mathbb{Z}/ 3\mathbb{Z}= \{ 0, a, -a \}$, where the class $a$ is represented by $\alpha$. The Gauss diagram of $K$ is shown in Fig. 71. $$\begin{picture}(0,0)\special{psfile=im36.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2685,2816)(811,-2377) \put(1671,-331){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2156,-11){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2086,274){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(816,-551){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(1221,-926){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1391,-1576){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3066,-1191){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2941,-1526){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2056,-1901){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3016,-791){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2521,-756){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2826,129){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3326,-321){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3496,-751){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3216,-1876){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(811,-1001){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2146,-2377){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2616,-2307){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(811,-1596){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(2586,-161){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \end{picture}$$ [**Fig. 71**]{} Hence, $K$ is $\mathbb{Z}/3\mathbb{Z}$-pure. Let $c(\emptyset)$ be the class of degree 5 shown in Fig. 72 (the weight functions are always the products of the writhes of the 5 crossings). $$\begin{picture}(0,0)\special{psfile=ima72.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5654,4367)(76,-3661) \put(1426,-2521){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1264,-3628){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(811,-3394){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1600,-2662){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(931,-3598){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1168,571){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1435,502){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(973,-479){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1339,-533){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(778,-245){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3379,493){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3538,361){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3190,-569){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(2887,-548){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2671,-236){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(5574,388){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5334,556){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5163,-542){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5109,-533){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(4659,-95){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(1396,-1012){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1570,-1135){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1222,-2071){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(811,-1882){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(913,-2029){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5538,-880){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5253,-775){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(4764,-1582){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(5127,-1891){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(5190,-1894){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3442,-2554){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3280,-3661){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3616,-2695){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2947,-3631){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(2879,-3444){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put( 76, 22){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$c(\emptyset):=$}}} \put(2108, 22){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1966,-1523){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1996,-3136){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(4073,-1373){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(4005,104){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(240,-1478){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(188,-3316){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(4149,-3132){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$ $\ldots$}}} \put(3326,-889){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3164,-1996){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(2711,-1762){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3500,-1030){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2831,-1966){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \end{picture}$$ [**Fig. 72**]{} The only possible strata of triple points in the discriminant for a $\mathbb{Z}/3\mathbb{Z}$-pure isotopy are $a^{\pm}_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=dr.pstex}\end{picture} } }({\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=br.pstex}\end{picture} } })}(a|-a,|-a)$ and $a^{\pm}_{{\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=br.pstex}\end{picture} } }({\makebox(8,8) {\begin{picture}(0,0)(3,5)\special{psfile=br.pstex}\end{picture} } })}(-a|a,|a)$ (see [@F], sect. 1). Therefore, the changings depicted in Fig. 73 are the only possible ones for a couple of crossings in a $\mathbb{Z}/3\mathbb{Z}$-pure isotopy. $$\begin{picture}(0,0)\special{psfile=ima73.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5461,3082)(353,-2378) \put(1606,-1771){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}or}}} \put(3856,-16){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}or}}} \put(638,569){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1126,547){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(2760,554){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(5235,562){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(540,-1231){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1147,-2378){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(3270,-1201){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(2512,-2356){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(4654,-1162){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(5577,-1136){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(3473,554){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(4560,584){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(631,-983){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}if $x \neq y$}}} \end{picture}$$ [**Fig. 73**]{} Here, $x, y \in \{ a, -a \}$. $c(\emptyset)$ is obtained from the configuration shown in the left-hand part of Fig. 74 by applying [*all*]{} possible changings to it. We have shown some of these changings hereabove. Notice that no chord can ever get crossed with the isolated chord, because the part of the configuration shown in the right-hand part of Fig. 74 cannot change at all. $$\begin{picture}(0,0)\special{psfile=ima74.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3428,1427)(454,-721) \put(3316,526){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3583,457){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1168,571){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1435,502){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(973,-479){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1339,-533){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(778,-245){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3426,-585){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture}$$ [**Fig. 74**]{} Consequently, $c(\emptyset)$ is a $\mathbb{Z}/3\mathbb{Z}$-pure class of degree 5 and $T_K(\emptyset; c(\emptyset))$ is an isotopy invariant of $K$. Notice that $p$ is the only arrow in the Gauss diagram of $K$ with marking $a$ and such that there are arrows in $K^+_p$. Using this fact, we easily calculate $T_K(\emptyset; c(\emptyset))= -1$. For the convenience of the reader, we give the Gauss diagram of $flip(-K)$ in Fig. 75. $$\begin{picture}(0,0)\special{psfile=im37.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(2790,2706)(381,-2660) \put(741,-2179){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1011,-1904){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(916,-1404){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(381,-1039){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(686,-679){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(1406,-964){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1011,-1054){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1986,-2229){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1461,-524){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1966,-2655){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(1511,-2660){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2756,-1379){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2626,-1839){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2981,-2094){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3171,-1409){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(2161,-119){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3071,-899){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2471,-869){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2066,-569){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(1396,-219){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \end{picture}$$ [**Fig. 75**]{} Hence, $T_{flip(-K)}(\emptyset; c(\emptyset))= T_K(\emptyset; \bar c( \emptyset))= 0$. (Again, the cyclic order of the two couples of crossed arrows with respect to the isolated arrow $p'$ has changed.) We have proven that $K$ i A remark on quantum invariants for knots in $T^2 \times \mathbb{R}$ =================================================================== Let $L \hookrightarrow T^2 \times \mathbb{R}$ be any oriented link. There are generalized HOMFLY-PT and Kauffman polynomials for $L$ (see e.g.[@H-P]). [*The generalized HOMFLY-PT polynomials of $L$ and of $flip(-L)$ coincide. The generalized Kauffman polynomials of $L$ and of $flip(-L)$ coincide.*]{} [**Proof**]{}.$L$ can be reduced to a linear combination of “initial knots” by using skein relations. These combinations are the same for $flip(-L)$ besides the fact that each “initial knot” $K$ has to be replaced by $flip(-K)$. Consequently, if we find a set of “initial knots” such that for each of them, $K$ is isotopic to $flip(-K)$, then the Lemma follows. As well known, we have to choose a knot $K$ in each free homotopy class of oriented loops in $T^2 \times \mathbb{R}$. Notice that “$flip \circ -$” acts as the identity on $\pi_1(T^2)$. Any primitive class in $H_1(T^2) \cong \pi_1(T^2)$ can be represented by a torus knot which is invariant under “$flip \circ -$”. Let $K \hookrightarrow T^2$. Each class $n[K], n \not= 0$ can be represented as the closure $\hat \beta$ of the braid $\beta= \sigma_1\sigma_2 \cdots \sigma_{n-1}$ in a tubular neighbourhood $V$ of $K \hookrightarrow T^2 \times \mathbb{R}$ which is a solid torus. (Remember that $\sigma_i$ are the standard generators of $B_n$.) We easily see that $flip(- \hat \beta)= \hat \gamma$, where $\gamma= \sigma_{n-1}\sigma_{n-2} \cdots \sigma_2\sigma_1$ in the same (invariant under $flip$) solid torus $V$. But, as well known, $\beta$ is conjugate to $\gamma$ in $V$ and hence, $\hat \beta$ and $\hat \gamma$ are the same knot. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ [**Remarks**]{}. 1. Evidently, Lemma 6.1 is still true if one replaces $L$ by any cable of $L$. 2. Lemma 6.1 implies that the above quantum invariants (and possibly all quantum invariants) can never detect non-invertibility of knots in $T^2 \times \mathbb{R}$. 3. It was already well known that quantum invariants never detect the non-invertibility of links in $S^3$ (see e.g. [@K]). We have shown in sect. 5 that $T$-invariants detect the non-invertibility of knots in $T^2 \times \mathbb{R}$. Thus, these $T$-invariants (of degrees 5 and 6 in the examples) cannot be extracted from the HOMFLY-PT or Kauffman polynomials of the knot or any of its cables. Non-invertibility of links in $S^3$ =================================== Our results about knots in $T^2 \times \mathbb{R}$ can be interpreted as results about certain links in $S^3$. Let $T^2 \times \mathbb{R}$ be the tubular neighbourhood of the standardly embedded torus in $S^3$. Let $T_1$ and $T_2$ be the cores of the corresponding solid tori $S^3 \setminus T^2$. To each knot $K \hookrightarrow T^2 \times \mathbb{R} \hookrightarrow S^3$, we associate the link $K \cup T_1 \cup T_2 \hookrightarrow S^3$. [*Two knots $K, K' \hookrightarrow T^2 \times \mathbb{R}$ are isotopic if and only if the corresponding ordered links $K \cup T_1 \cup T_2$, $K' \cup T_1 \cup T_2 \hookrightarrow S^3$ are isotopic.*]{} [**Proof.**]{} Lemma 1.7 of [@F] implies that the ordered links $K \cup T_1 \cup T_2$ and $K' \cup T_1 \cup T_2$ are isotopic if and only if the ordered links $K \cup T_1$ and $K \cup T_2$ are isotopic in the solid torus $S^3 \setminus T_2$. It is also well known that each isotopy of the solid torus, which is the identity near the boundary and which maps the core of the solid torus to itself, can be isotopically deformed to an isotopy which leaves the core pointwise fixed. ${\mbox{}\nolinebreak\rule{2mm}{2mm}}$ We will use Lemma 7.1 in order to study the invertibility of the link $K \cup T_1 \cup T_2 \hookrightarrow S^3$. Instead of Lemma 7.1, we could use the fact that, there is only one isotopy which inverts the Hopf link $H= T_1 \cup T_2$, up to isotopy of isotopies. Indeed, an isotopy which inverts $H$ inverts also the meridians and longitudes for $T_1$ and $T_2$. Therefore, such an isotopy induces an orientation preserving homeomorphism of the incompressible torus $T^2$ in $S^3 \setminus H$. This homeomorphism acts as $-1$ on $H_1(T^2; \mathbb{Z})$. As the mapping class group of $T^2$ is $SL(2; \mathbb{Z})$, this homeomorphism is isotopic to $flip$. [*Thus, the non-invertibility of the link $L$ in Fig. 1 follows from the non-invertibility of the knot $K$ in Fig. 65.*]{} Indeed, to the knot $K$, we have to add the Hopf link $T_1 \cup T_2$. Notice that the resulting link $L$ is naturally ordered: $K \hookrightarrow S^3$ is not the trivial knot, $lk(K, T_2)= 4$, $lk(K, T_1)= 1$. Hence, if $L$ is invertible, then $L$ is invertible as an ordered link (i.e. respecting the ordering), and one can apply Lemma 7.1. $flip$, seen as an involution on $S^3$, maps simultaneously $T_1$ to $-T_1$ and $T_2$ to $-T_2$. Thus, $L$ is isotopic to $-L$ if and only if $K$ is isotopic to $flip(-K)$ in $T^2 \times \mathbb{R}$. But we have shown that this is not the case, using the $T$-invariant in Example 5.1. $T$-invariants which are not of finite type are usefull too =========================================================== Let $h= (id, -id): T^2 \times \mathbb{R} \to T^2 \times \mathbb{R}$, and let $K \hookrightarrow T^2 \times \mathbb{R}$ be a global knot. Then $h(K)$ is called the [*mirror image*]{} of $K$ and is denoted as usually by $K!$ Clearly, $K!$ is a global knot which is always homotopic to $K$. We give an example of a $\mathbb{Z}/2\mathbb{Z}$-pure global knot $K$ which we distinguish from $K!$. We do this in two ways: first with a $T$-invariant of degree $2$ but which is not of finite type, and then with a $T$-invariant of degree $8$ which is of finite type. We prove moreover that $K$ and $K!$ cannot be distinguished by any Gauss diagram invariant (see [@F]), or by a $T$-invariant of finite type of degree not bigger than 2. [*Hence, $T$-invariants which are not of finite type are sometimes more effective than $T$-invariants of finite type.*]{}\ $$\begin{picture}(0,0)\special{psfile=im38.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(5388,2963)(204,-2834) \put(5318,-1290){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$K$}}} \put(1732,-2479){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1956,-2460){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3712,-2435){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(4110,-2348){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3083,-2024){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(3431,-1900){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2236,-1663){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2435,-1788){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \end{picture}$$ [**Fig. 76**]{} The Gauss diagram of the knot in Fig. 76 is shown in Fig. 77. $$\begin{picture}(0,0)\special{psfile=im39.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(3620,3079)(285,-2684) \put(3698,-2054){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2\alpha+\beta$}}} \put(2002,109){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(2157,239){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(3422, -1){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(3808,-479){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2\alpha+\beta$}}} \put(1441,-645){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2213,-15){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2468, 7){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(1441,-1216){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2671,-923){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(3661,-1223){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2567,-2626){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$\alpha$}}} \put(3105,-106){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$3$}}} \put(3226,-376){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$4$}}} \put(3601,-729){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2910,-1179){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$8$}}} \put(3458,-1014){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$1$}}} \put(2678,-1695){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$7$}}} \put(2588,-1441){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-$}}} \put(2724,-2161){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2$}}} \put(2401,-1966){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$6$}}} \put(1943,-1749){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$5$}}} \put(1756,-2491){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2\alpha+\beta$}}} \put(901,-826){\makebox(0,0)[b]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$2\alpha+\beta$}}} \end{picture}$$ [**Fig. 77**]{} For the convenience of the reader, we have affected numbers to the crossings. We see that there appear only two different homology classes as markings. In particular, $K$ is $\mathbb{Z}/ 2\mathbb{Z}$-pure for $G:= (H_1(T^2)/\langle [K] \rangle)/2 \mathbb{Z}= \{ 0, a \}$. The Gauss diagram of $K!$ is obtained from the one of $K$ by replacing all arrows, writhes and markings by their opposites (but remember that $a= -a$). We start by comparing the invariants of degree 1 (see also [@F], sect. 2.2). $$W_K(\alpha)= W_{K!}(\alpha)= W_K(2\alpha+ \beta)= W_{K!}(2\alpha+ \beta)=0$$ If we see $K$ and $K!$ as knots in $S^3$ using the embedding $T^2 \times \mathbb{R} \hookrightarrow S^3$, then we easily calculate $v_2(K)= v_2(K!)$ for the only Vassiliev invariant of degree 2 (notice that $K!$ is not the mirror image of $K$ in $S^3$ because of the two additional crossings seen in Fig. 76). All the Gauss diagram invariants and $T$-invariants of degree 2 which are of finite type are linear combinations of all possible configurations of degree 2 (see also [@F], sect. 2.4). The weight function is always the product of the two writhes (because of the invariance under Reidemeister moves of type $II$). Therefore, this function is invariant under taking the mirror image. In Fig. 78, we indicate how the configurations change by taking the mirror image. $$\begin{picture}(0,0)\special{psfile=ima78.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4180,4994)(466,-4605) \put(1052,224){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1765,224){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(4397,247){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(3722,269){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(1052,-1553){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1765,-1553){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(4397,-1545){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(3722,-1523){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(1203,-3540){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1691,-3562){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(3783,-3525){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(4271,-3547){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(466, 54){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$I)$}}} \put(481,-1745){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$II)$}}} \put(473,-3755){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$III)$}}} \end{picture}$$ $$\begin{picture}(0,0)\special{psfile=ima78bis.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4112,7400)(443,-7255) \put(458,-163){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$IV)$}}} \put(1185, 10){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1673,-12){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(3735, -5){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(4223,-27){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(4342,-1948){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(3449,-3125){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(1177,-1866){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1784,-3013){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(458,-2233){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$V)$}}} \put(451,-4026){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$VI)$}}} \put(4365,-3771){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(3472,-4948){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1200,-3689){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(1807,-4836){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(443,-6288){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$VII)$}}} \put(4366,-6033){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(3473,-7210){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \put(1201,-5951){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$x$}}} \put(1808,-7098){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$y$}}} \end{picture}$$ [**Fig. 78**]{} In this Figure, $x, y \in \{ \alpha, 2\alpha+ \beta \}$ and $x \not= y$. [**Remarks**]{}. 1. $I)$ can not enter in any invariant because of the invariance under Reidemeister moves of type $II$ (see also Lemma 3.3). 2. $IV)$ is invariant. Thus, if in our example, the left-hand side is equal to the right-hand side for $II)$, $III)$, $V)$, $VI)$, $VII)$, then [*all*]{} invariants of finite type of degree 2 coincide for $K$ and $K!$. We easily calculate the values on both sides and it turns out that they coincide: $II)= +2$, $III)= 0$, $V)= 0$, $VI)= -2$, $VII)= 0$. Let us consider $T$-invariants of infinite type (see Prop. 3.1). Let $$\makebox(10,60){ $D=$ } \makebox(60,60){\begin{picture}(0,0)\special{psfile=pag52M.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,947)(494,-258) \put(931, 39){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(893,524){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} }$$ for $G \cong \mathbb{Z}/2\mathbb{Z}= \{ 0, a \}$. We easily calculate $$T_K(D; c_{++}(D)= +2)= -1$$ ($D$ is the crossing number 1 with the crossing $q$ which is either the crossing number 4 or 5.) $$T_K(D; c_{++}(D)= -1)= +2$$ ($D$ is 4 and 5, $q$ is 1.) $$T_K(D; c_{++}(D)= 0)= -1$$ ($D$ is 2, 3, 6, 7, 8. There are no $q$’s.) For all other $c \in \mathbb{Z}$, $T_K(D; c_{++}(D)= c)= 0$ We keep the numbers for the crossings of $K!$. $$T_{K!}(D; c_{++}(D)= -2)= +1$$ ($D$ is 2, the $q$’s are 3 and 6.) $$T_{K!}(D; c_{++}(D)= +1)= -2$$ ($D$ is 3 and 6, $q$ is 2.) $$T_{K!}(D; c_{++}(D)= 0)= +1$$ and all other $T_{K!}(D; c_{++}(D)= c)= 0$. Consequently, $K$ and $K!$ are not isotopic and we have proven it with an invariant of quadratic complexity. The Gauss diagram of $K$ without the writhes is a $\mathbb{Z}/2\mathbb{Z}$-pure configuration $D$ of degree 8. Clearly, it is different from the corresponding configuration for $K!$. Thus, $K$ and $K!$ are also distinguished by the $T$-invariant of degree 8 of finite type $T_K(D; \emptyset)$. We do not know wether or not there are $G$-pure global knots which can be distinguished [ *only*]{} by $T$-invariants which are not of finite type. But in any case, our example shows that these invariants do it sometimes in a more effective way than the invariants of finite type. $T$-invariants are not well defined for general knots ===================================================== Let $K \hookrightarrow S^3= (\mathbb{R}^2 \times \mathbb{R}) \cup \{ \infty \}$ be a knot and let $m$ be a meridian of $K$. The meridian $m$ is isotopic to $(0 \times \mathbb{R}) \cup \{ \infty \}$ and hence, we can consider $K$ as a knot in $(\mathbb{R}^2 \setminus 0) \times \mathbb{R}$. If two knots $K$, $K'$ are isotopic in $S^3$, then in fact, they are already isotopic in $(\mathbb{R}^2 \setminus 0) \times \mathbb{R}$, where $(0 \times \mathbb{R}) \cup \{ \infty \}$ is a meridian for both knots. We consider the projection $(\mathbb{R}^2 \setminus 0) \times \mathbb{R} \to \mathbb{R}^2 \setminus 0$. Assume that $K$ and $K'$ are isotopic. If they have the same writhe and the same Whitney index, then they are regularly isotopic in $(\mathbb{R}^2 \setminus 0) \times \mathbb{R}$ (see [@F], sect. 2). If we take now the same cable or satellite for two regularly isotopic knots, then the resulting knots are again (regularly) isotopic in $(\mathbb{R}^2 \setminus 0) \times \mathbb{R}$. Let $K$ be the figure-eight knot. As well known, $K$ is isotopic to its mirror image $K!$. As satellite, we take the positive (untwisted) Whitehead double. Consequently, the two knots $W$ and $W'$ shown in Fig. 79 are isotopic in the solid torus $S^3 \setminus m= (\mathbb{R}^2 \setminus 0) \times \mathbb{R}$. $$\begin{picture}(0,0)\special{psfile=ima79.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4287,3420)(769,-3076) \put(1973,-450){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3885,-579){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3466,-1290){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3241,-1516){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3182,-1097){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2948,-1223){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2716,-3076){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(5056,-1928){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$W$}}} \put(1493,-1688){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$m$}}} \put(2619,-1876){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \end{picture}$$ $$\begin{picture}(0,0)\special{psfile=ima79bis.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(4310,3424)(746,-3076) \put(1973,-450){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3885,-579){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(3466,-1290){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(3241,-1516){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(3182,-1097){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$+$}}} \put(2716,-3076){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$-a$}}} \put(5056,-1928){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$W'$}}} \put(1493,-1688){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$m$}}} \put(2948,-1223){\makebox(0,0)[rb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2701,-1853){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture}$$ [**Fig. 79**]{} Let $v$ be a Morse-Smale vector field on $\mathbb{R}^2$, which has a critical point of index 1 in $0= m \cap \mathbb{R}^2$, and such that $v$ is transversal to $pr(W)$. But $pr(W)= pr(W')$ and hence, $v$ is transversal to $pr(W')$ too. Of course, $W$ and $W'$ are [*not*]{} global knots, because $v$ has critical points of index 1 different from 0. Let “$a$” be the generator$$\begin{picture}(0,0)\special{psfile=gener.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(916,988)(1803,-1285) \put(2453,-432){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \put(2325,-977){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$m$}}} \end{picture}$$ of $H_1(S^3 \setminus m; \mathbb{Z})=H_1(\mathbb{R}^2 \setminus 0; \mathbb{Z})$. One has $[W]= [W']= 0$ in $H_1(\mathbb{R}^2 \setminus 0; \mathbb{Z})$ and we easily see that $W$ and $W'$ are $\mathbb{Z}$-pure knots (the markings are shown in Fig. 79 too). It follows from the proof of Theorem 1 that each $T$-invariant $T_W$ 1. is invariant for each isotopy transversal to $v$ and which is not necessarily $\mathbb{Z}$-pure 2. is invariant for each $\mathbb{Z}$-pure isotopy which is not necessarily transversal to $v$. Let $$\makebox(10,60){$D=$} \makebox(60,60){\begin{picture}(0,0)\special{psfile=pag52M.pstex}\end{picture}\setlength{\unitlength}{4144sp}\begingroup\makeatletter\ifx\SetFigFont\undefined\gdef\SetFigFont#1#2#3#4#5{ \reset@font\fontsize{#1}{#2pt} \fontfamily{#3}\fontseries{#4}\fontshape{#5} \selectfont}\fi\endgroup\begin{picture}(772,947)(494,-258) \put(931, 39){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$p$}}} \put(893,524){\makebox(0,0)[lb]{\smash{\SetFigFont{12}{14.4}{\rmdefault}{\mddefault}{\updefault}$a$}}} \end{picture} }$$ and let $c(D)$ be any of the classes of Def. 3.7. We easily calculate $T_W(D; c(D)=c)= T_{W'}(D; c(D)=c)$ for any $c \in \mathbb{Z}$. But e.g. $T_W(D; c_{++}(D)=0, c_{+-}(D)=-1)= +2$ and $T_{W'}(D; c_{++}(D)=0, c_{+-}(D)=-1)= 0$. We have shown above that $W$ and $W'$ are actually isotopic in $(\mathbb{R}^2 \setminus 0) \times \mathbb{R}$. This example has three important consequences: 1. $W$ and $W'$ are transversal to $v$ and they are isotopic. But there is no isotopy transversal to $v$ joining them. 2. $W$ and $W'$ are $\mathbb{Z}$-pure and they are isotopic. But there is no $\mathbb{Z}$-pure isotopy joining them (i.e. there are cycles with marking 0 which cannot be eliminated). 3. Multi-classes contain more information than the classes taken individually. [99]{} J. Andersen, J. Mattes, N. Reshetikhlin: [ *Quantization of the algebra of chord diagrams*]{} Math. Proc. Cambridge Philos. Soc. 124 (1998) J. Birman, X.S. Lin: [*Knot polynomials and Vassiliev invariants*]{} Inv. Math. 111 (1993) D. Bar Natan: [*On the Vassiliev knot invariants*]{} Topology 34 (1995) T. Fiedler: [*Gauss diagram invariants for knots and links*]{} Monography, Kluwer academic publishers (to appear) V. Goryunov: [*Finite order invariants of framed knots in a solid torus and in Arnold’s $\mathcal{J}^+$-theory of plane curves*]{} “Geometry and Physics”, Lect. Notes in Pure and Appl. Math. (1996) M. Gussarov, M. Polyak, O. Viro: [*Finite type invariants of classical and virtual knots*]{} Topology 39 (2000) J. Hoste, J. Przytycki: [*A survey of skein modules of 3-manifolds*]{} Knots 90 (Osaka, 1990) de Gruyter (1992) M. Kontsevitch: [*Vassiliev’s knot invariants*]{} Adv. in Sov. Math. 16 (2) (1993) X.S Lin: [*Finite type link invariants and the non-invertibility of links*]{} Math. Res. Lett. 3 (1996) H. Morton: [*Infinitely many fibred knots having the same Alexander polynomial*]{} Topology 17 (1978) V. Praslov, A. Sossinsky: [*Knots, links, braids and 3-manifolds*]{} Transl. of Math. Monographs 154 AMS (1997) M. Polyak, O. Viro: [*Gauss diagram formulas for Vassiliev invariants*]{} Internat. Math. Res. Notices 11 (1994) V. Vassiliev: [*Cohomology of knot spaces*]{} Theory of Singularities and its Applications Amer. Math. Soc., Providence (1990) Unfortunately, my preprint “New invariants in knot theory” (November 1999) contains some serious errors. I apologize by the reader for this. The present preprint is the result of the correction of these errors. It will be added to the final version of the monography “Gauss diagram invariants for knots and links”. I am very grateful to Séverine for her constant support!

--- abstract: | A family $\T$ of digraphs is a *complete set of obstructions* for a digraph $H$ if for an arbitrary digraph $G$ the existence of a homomorphism from $G$ to $H$ is equivalent to the non-existence of a homomorphism from any member of $\T$ to $G$. A digraph $H$ is said to have *tree duality* if there exists a complete set of obstructions $\T$ consisting of orientations of trees. We show that if $H$ has tree duality, then its arc graph $\delta H$ also has tree duality, and we derive a family of tree obstructions for $\delta H$ from the obstructions for $H$. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions. **Keywords:** constraint satisfaction, tree duality, adjoint functor **2000 Mathematics Subject Classification:** 16B50, 68R10, 18A40, 05C15 author: - | Jan Foniok\ ETH Zurich, Institute for Operations Research\ Rämistrasse 101, 8092 Zurich, Switzerland\ `foniok@math.ethz.ch` - | Claude Tardif\ Royal Military College of Canada\ PO Box 17000, Stn Forces, Kingston, Ontario\ Canada, K7K 7B4\ `Claude.Tardif@rmc.ca` date: 6 May 2009 title: Adjoint functors and tree duality --- Introduction ============ Our primary motivation is the *$H$-colouring problem* (which has become popular under the name *Constraint Satisfaction Problem—CSP*): for a fixed digraph $H$ (a *template*) decide whether an input digraph $G$ admits a homomorphism to $H$. The computational complexity of $H$-colouring depends on the template $H$. For some templates the problem is known to be NP-complete, for others it is tractable (a polynomial-time algorithm exists). Assuming that $\mathrm{P}\ne\mathrm{NP}$, infinitely many complexity classes lie strictly between P and NP [@Lad:StruPTR], but it has been conjectured that $H$-colouring belongs to no such intermediate class for any template $H$ [@FedVar:SNP]. This conjecture has indeed been proved for symmetric templates $H$ [@HelNes:Dicho]. In this paper the focus is on tractable cases. Several conditions are known to imply the existence of a polynomial-time algorithm for $H$-colouring (definitions follow in the next two paragraphs): it is the case if $H$ has a near-unanimity function (nuf), if $H$ has bounded-treewidth duality, if $H$ has tree duality, if $H$ has finite duality (see [@CohJea:CCL; @FedVar:SNP; @HelNesZhu:DualPoly]). Some of the conditions are depicted in the diagram (Fig. \[fig:diag\]). ![The structure of tractable templates[]{data-label="fig:diag"}](sproink.1) A *near-unanimity function* is a homomorphism $f$ from $H^k$ to $H$ with $k\ge3$ such that for all $x,y\in V(H)$ we have $f(x,x,x,\dotsc,x)=f(y,x,x,\dotsc,x)=f(x,y,x,\dotsc,x)= \dotsb=f(x,x,x,\dotsc,y)=x$. The power $H^k$ is the $k$-fold product $H\times H\times\dotsb\times H$ in the category of digraphs and homomorphisms, see [@HelNes:GrH]. A digraph is a tree (has treewidth $k$) if its underlying undirected graph is a tree (has treewidth $k$, respectively). A set $\F$ of digraphs is a *complete set of obstructions* for $H$ if for an arbitrary digraph $G$ there exists a homomorphism from $G$ to $H$ if and only if no $F\in\F$ admits a homomorphism to $G$. A template has *bounded-treewidth duality* if it has a complete set of obstructions with treewidth bounded by a constant; it has *tree duality* if it has a complete set of obstructions consisting of trees; and it has *finite duality* if it has a finite complete set of obstructions. There is a fairly straightforward way to generate templates with finite duality. For an arbitrary tree $T$ there exists a digraph $D(T)$ such that $\{T\}$ is a complete set of obstructions for $D(T)$. The digraph $D(T)$ is unique up to homomorphic equivalence[^1]; it is called the *dual* of $T$. Several explicit constructions are known (see [@F:Diss; @Kom:Phd; @NesTar:Dual; @NesTar:Short]). If $\F$ is a finite set of oriented trees, then the product $D=\prod_{T\in\F}D(T)$ is a template with finite duality and $\F$ is a complete set of obstructions for $D$. This construction yields all digraphs with finite duality [@NesTar:Dual], thus also proving that finite duality implies tree duality. Encouraged by the full description of finite dualities, we aim to provide a construction for some more digraphs with tree duality. To this end we use the arc-graph construction and consider the class $\delta_\pi{\cal C}$ of digraphs generated from finite duals by taking iterated arc graphs and finite Cartesian products. We show that all templates in this class have tree duality. We provide an explicit construction of the resulting tree obstructions, which allows us to show that all the digraphs in $\dpc$ have in fact *bounded-height tree duality*, that is, they have a complete set of obstructions consisting of trees of bounded algebraic height (these are tree obstructions that allow a homomorphism to a fixed directed path). In this context we also prove that the problem of existence of a complete set of obstructions consisting of trees with bounded algebraic height is decidable. The arc-graph construction is a special case of a more general phenomenon: it is a right adjoint in the category of digraphs and homomorphisms. We show in the more general setting of the category of relational structures that right adjoints (characterised by Pultr [@Pul:The-right-adjoints] for all locally presentable categories) preserve tractability of templates and moreover they preserve tree duality and existence of a near-unanimity function. In this case, nevertheless, it remains open to provide a nice general description of complete sets of obstructions. We use some notions and properties of graphs and homomorphisms which the reader can look up in [@HelNes:GrH], as well as some category-theory notions, for which, e.g. [@BarWel:Cat; @Mac:Cat] may be consulted. Arc graphs and tree duality =========================== Let $G=(V,A)$ be a digraph. The *arc graph* of $G$ is the digraph $\delta G =(A,\delta A)$, where $$\delta A=\bigl\{((u,v),(v,w)) : (u,v),(v,w)\in A\bigr\}.$$ Notice that $\delta$ is an endofunctor[^2] in the category of digraphs and homomorphisms. This implies in particular that if $G\to H$, then $\delta G \to \delta H$. (The notation $G\to H$ means that there exists a homomorphism from $G$ to $H$.) If $G$ is a digraph and $\sim$ is an equivalence relation on its vertex set $V(G)$, the *quotient* $G/{\sim}$ is the digraph $(V(G)/{\sim},A)$, where $V(G)/{\sim}$ is the set of all equivalence classes of $\sim$ on $V(G)$, and for $X,Y\in V(G)/{\sim}$ we have $(X,Y)\in A$ if and only if there exist $x\in X$ and $y\in Y$ such that $(x,y)\in A(G)$. Suppose still that $G=(V,A)$ is a digraph. Let $V'=\{o_u,t_u:u\in V\}$ and let $A'=\{(o_u,t_u) : u\in V\}$. Define the relation $\sim_0$ such that $t_u\sim_0 o_v$ if and only if $(u,v)\in A$. Let $\sim$ be the minimal equivalence relation on $V'$ containing $\sim_0$. Set $\delta^{-1}G=(V',A')/{\sim}$. In the following, we use the notation $V'(G)=V'$, $A'(G)=A'$ and $\sim_0$ and $\sim$ for the sets and relations appearing in the definition of $\delta^{-1}$; the precise meaning will be clear from the context. Now $\delta^{-1}$ is also an endofunctor in the category of digraphs. Strictly speaking, it is not an inverse of $\delta$; its name is chosen because of the following property. \[prop:delta\] For any digraphs $G$ and $H$, $$G\to\delta H \qquad\text{if and only if}\qquad \delta^{-1}G\to H.$$ Let $f: G \to \delta H$ be a homomorphism. Then there exist two homomorphisms $o, t: G \to H$ such that $f(u) = (o(u), t(u))$ for all $u \in V(G)$. Define the mapping $\hat g : V'(G) \to V(H)$ by $\hat g(o_u) = o(u)$ and $\hat g (t_u) = t(u)$. If $t_u \sim_0 o_v$, then $(u,v) \in A(G)$, whence $(f(u),f(v)) \in A(\delta H)$ and thus $t(u) = o(v)$. Therefore $\hat g$ is constant on the equivalence classes of $\sim$, and it induces a homomorphism from ${A'(G)/{\sim}} = \delta^{-1}G$ to $H$. Conversely, let $g: \delta^{-1}G \to H$ be a homomorphism. We define $f: V(G) \to V(\delta H)$ by $f(u) = (g(o_u/{\sim}), g(t_u/{\sim}))$. If $(u,v) \in A(G)$, then $t_u/{\sim} = o_v/{\sim}$, whence $(f(u), f(v)) \in A(\delta H)$. Therefore $f$ is a homomorphism. Thus $\delta$ and $\delta^{-1}$ are Galois adjoints[^3] with respect to the ordering by existence of homomorphisms. They are in fact adjoint functors in the category of digraphs and homomorphisms. We return to this topic in Section \[sec:functors\]. For the moment we aim to prove that $\delta$ preserves tree duality. More precisely, from the family ${\cal T}$ of tree obstructions of $H$, we will derive the family $\operatorname{Sproink}({\cal T})$ of tree obstructions of $\delta H$. The *algebraic height* of an oriented tree $T$ is the minimum number of arcs of a directed path to which $T$ maps homomorphically. The algebraic height of every finite oriented tree is well-defined and finite, since every such tree admits a homomorphism to some finite directed path. Thus a tree $T$ is *of height at most one* if its vertex set can be split into two parts $0_T, 1_T$ in such a way that for every arc $(x,y)$ of $T$ we have $x \in 0_T$ and $y \in 1_T$. Note that if the tree $T$ has no arcs, then it has only one vertex and thus one of the sets $0_T$, $1_T$ is empty and the other one is a singleton. Let $T$ be a tree. For every vertex $u$ of $T$, let $F(u)$ be a tree of height at most one. For each arc $e$ of $T$ incident with $u$, let there be a fixed vertex $v(e,F(u))$ in $F(u)$ such that if $u$ is the initial vertex of $e$, then $v(e,F(u))\in 1_{F(u)}$, and if $u$ is the terminal vertex of $e$, then $v(e,F(u))\in 0_{F(u)}$.[^4] A tree $S$ is now constructed by taking all the trees $F(u)$ for all vertices $u$ of $T$, and by identifying the vertex $v(e,F(u))$ with $v(e,F(u'))$ whenever $e=(u,u')$ is an arc of $T$. Any such tree $S$ constructed from $T$ by the above procedure is called a *sproink* of $T$. The set of all sproinks of a tree $T$ is denoted by $\operatorname{Sproink}(T)$. The following lemma asserts that sproinks of obstructions for a template $H$ are indeed obstructions for its arc graph $\delta H$. \[lem:sproink\] Let $T$ be a tree and $H$ a digraph such that $T\notto H$. If $S\in\operatorname{Sproink}(T)$, then $S\notto \delta H$. We prove that $T\to\delta^{-1}S$. Consequently $\delta^{-1}S\notto H$ because $T\notto H$, and therefore $S\notto\delta H$ by Proposition \[prop:delta\]. Thus let $S\in\operatorname{Sproink}(T)$. For a vertex $u$ of $T$, consider the tree $F(u)$, which is a subgraph of $S$. Since $F(u)$ has height at most one, its vertices are partitioned into the sets $0_{F(u)}$ and $1_{F(u)}$. The set $V'(S)$, which appears in the definition of $\delta^{-1}S$, contains $V'(F(u))$ as a subset. If $(x,y)$ is an arc of $F(u)$, then $t_x\sim_0 o_y$. Thus whenever $x\in0_{F(u)}$ and $y\in 1_{F(u)}$, then $t_x\sim o_y$. Hence for any vertex $u$ of $T$ there exists a unique vertex $f(u)$ of $\delta^{-1}S$ that is equal to $t_x/{\sim}$ for all $x\in 0_{F(u)}$ and to $o_y/{\sim}$ for all $y\in 1_{F(u)}$. In this way, we have defined a mapping $f: V(T) \to V(\delta^{-1}S)$. Now assume that $e=(u,v)$ is an arbitrary arc of $T$. Then the vertex $v(e,F(u))$, which belongs to $1_{F(u)}$, has been identified with $v(e,F(v))$, which belongs to $0_{F(v)}$. Let this identified vertex be $x$; it is a vertex of $S$. By definition, $f(u)=o_x/{\sim}$ because $x\in1_{F(u)}$, and $f(v)=t_x/{\sim}$ because $x\in 0_{F(v)}$. Of course $(o_x/{\sim},t_x/{\sim})\in A(\delta^{-1}S)$. Therefore $f: T\to\delta^{-1}S$ is a homomorphism, as we have promised to prove. For a set $\F$ of trees, let $\operatorname{Sproink}(\F)= \bigcup_{T\in\F}\operatorname{Sproink}(T)$. \[thm:spr-thm\] Let $\F$ be a set of trees which is a complete set of obstructions for a template $H$. Then $\operatorname{Sproink}(\F)$ is a complete set of obstructions for $\delta H$. Lemma \[lem:sproink\] implies that $\operatorname{Sproink}(\F)$ is a set of obstructions for $\delta H$. It remains to prove that it is complete, that is whenever $G\notto\delta H$, then there exists some $S\in\operatorname{Sproink}(\F)$ such that $S\to G$. So let $G\notto\delta H$. Thus by Proposition \[prop:delta\] we have $\delta^{-1}G\notto H$. Hence there exists a tree $T\in\F$ such that $T\to \delta^{-1}G$, because $\F$ is a complete set of obstructions for $H$. Consequently it suffices to prove that if $T\to\delta^{-1}G$ then there exists $S\in\operatorname{Sproink}(T)$ such that $S\to G$. Thus assume that $f:T\to\delta^{-1}G$ is a homomorphism. For every $u \in V(T)$, the image $f(u)$ is a $\sim$-equivalence class; put $$\begin{aligned} 1_{u} & = \{ y \in V(G) : o_y \in f(u)\},\\ 0_{u} & = \{ x \in V(G) : t_x \in f(u)\}.\end{aligned}$$ Then $f(u) = 1_{u} \cup 0_u$, and by the definition of $\sim$ as the least equivalence containing $\sim_0$, there exists a tree $F(u)$ of height at most one and a homomorphism $g_u: F(u) \to G$ such that $g_u(0_{F(u)}) = 0_{u}$ and $g_u(1_{F(u)}) = 1_{u}$. For every arc $(u,v)$ of $T$, we have $(f(u), f(v)) \in A(\delta^{-1}G)$ so there exists $x \in V(G)$ such that $o_x \in f(u)$ and $t_x \in f(v)$. We then select $y \in 1_{F(u)}$ and $z \in 0_{F(v)}$ such that $g_u(y) = g_v(z) = x$, and identify them. Proceeding with all such identifications, we construct a tree $S \in \operatorname{Sproink}(T)$ such that $g = \bigcup_{u \in V(T)} g_u : S \to G$ is a well-defined homomorphism. \[cor:arc\] If a digraph $H$ has tree duality, then its arc graph $\delta H$ also has tree duality. Consider $T=\vec P_4$, the directed path with four arcs, and its dual $D=\vec T_4$, the transitive tournament on four vertices. Here $\delta D$ has six vertices, but its core[^5] is the directed path $\vec P_2$ with two arcs. It is well known that a directed graph $G$ admits a homomorphism to $\vec P_2$ if and only if it does not admit a homomorphism from a “thunderbolt”, that is, an oriented path with two forward arcs at the beginning and at the end, and with an odd-length alternating path between them (see Fig. \[fig:thunder\]). Thus the family of all thunderbolts is a complete set of tree obstructions for $\vec P_2$. ![A thunderbolt[]{data-label="fig:thunder"}](sproink.2) Our construction $\operatorname{Sproink}(T)$ gives all obstructions obtained by stacking five trees $L_0$, $L_1$, $L_2$, $L_3$, $L_4$ of height at most one, with one top vertex of $L_i$ identified with one bottom vertex of $L_{i+1}$ for $i = 0, 1, 2, 3$. The example of thunderbolts shows that in fact $L_0$ can be restricted to a single (top) vertex, and $L_4$ can be restricted to a single (bottom) vertex. The same holds for leaves of general trees. Also, $L_1$, $L_2$, $L_3$ can be restricted to *paths* of height one, and it is also true in general that it is sufficient to consider sproinks obtained by replacing vertices by *paths* of height at most one. In fact the name “sproink” is inspired by picturing such a path springing out of every non-leaf of $T$. The results of this section show that we can construct an interesting class of templates with tree duality by repeatedly applying the arc-graph construction to digraphs with finite duality. Moreover, if templates $H_1$, $H_2$, …, $H_k$ all have tree duality, then also their product $H_1\times H_2\times\dotsb\times H_k$ has tree duality as the union of the respective complete sets of obstructions of the factors is a complete set of obstructions for the product. The resulting class of templates is subject to examination in the next section. Finite duality {#sec:fd} ============== Following [@NesTar:Dual], every tree $T$ admits a dual $D(T)$ such that for every digraph $G$, we have $G \rightarrow D(T)$ if and only if $T \notto G$. A digraph $H$ has finite duality if and only if it is homomorphically equivalent to a finite product of duals of trees. In this section, we consider the class $\dpc$, the smallest class of digraphs that contains all duals of trees and is closed under taking arc graphs, finite products and homomorphically equivalent digraphs. It follows from Corollary \[cor:arc\] that all elements of $\dpc$ have tree duality. Moreover we know how to construct a complete set of obstructions for each of these templates, using iterated $\operatorname{Sproink}$ constructions and unions. The question then arises as to how significant the class $\dpc$ is within the class of digraphs with tree duality. It turns out that the digraphs in $\dpc$ have properties that are not shared by all digraphs with tree duality. A digraph $H$ has *bounded-height tree duality* provided there exists a constant $m$ such that $H$ admits a complete set of obstructions consisting of trees of algebraic height at most $m$. \[prop:dpc\] - Every core in $\dpc$ admits a near-unanimity function. - Every member of $\dpc$ has bounded-height tree duality. (i): By Corollary 4.5 of [@Tar:FOD], every structure with finite duality admits a near-unanimity function. Therefore it suffices to show that the class of structures admitting a near-unanimity function is closed under taking cores, finite products and the arc-graph construction. Let $C$ be the core of $H$, $\rho: H \to C$ a retraction and $f:H^k\to H$ a near-unanimity function. Since $C$ is an induced subgraph of $H$, the restriction $\rho \circ f\restrict C^k$ is a near-unanimity function on $C$. Suppose $f_i:H_i^{k_i}\to H_i, i = 1, \ldots, m$ are near-unanimity functions. For $k = \max\{k_i : i = 1, \ldots, m\}$, we define $k$-ary near-unanimity functions $g_i: H_i^k \to H_i$ by $g_i(x_1, \ldots, x_k) = f_i(x_1, \ldots, x_{k_i})$. For $H = \Pi_{i = 1}^m H_i$ we then define a near-unanimity function $g: H^k \to H$ coordinate-wise, by putting $$g((x_{1,1},\ldots, x_{m,1}), \ldots, (x_{1,k}, \ldots, x_{m,k})) = (g_1(x_{1,1},\ldots,x_{1,k}), \ldots, g_m(x_{m,1},\ldots, x_{m,k})).$$ Now suppose that $f: H^k \to H$ is a near-unanimity function. Then $\left ( \delta H \right )^k$ is naturally isomorphic to $\delta(H^k)$, and we define $g: \left ( \delta H \right )^k \to \delta H$ by $$g((u_1,v_1), \ldots, (u_k,v_k)) = (f(u_1,\ldots, u_k),f(v_1,\ldots,v_k)).$$ The fact that $f$ is a homomorphism implies that $g$ is well defined, and $g$ is a homomorphism by the definition of adjacency in $\delta H$. Also, $g$ clearly satisfies the near-unanimity identities, so it is a near-unanimity function on $\delta H$. (ii): The class of digraphs with bounded-height tree duality is obviously preserved by taking cores and finite products. By Theorem \[thm:spr-thm\], if $H$ has a complete set of obstructions consisting of trees of algebraic height at most $k$, then $\delta H$ has a complete set of obstructions consisting of trees of algebraic height at most ${k+1}$, so the class of digraphs with bounded-height tree duality is also preserved by the arc-graph construction. We know a core digraph with tree duality but no near-unanimity function and no bounded-height tree duality. (The example is complicated and out of the scope of this paper, therefore we omit it.) Thus the class $\dpc$ does not capture all core digraphs with tree duality. The problem of generating all structures with tree duality by means of suitable functors applied to structures with finite duality remains nevertheless interesting. Membership in $\dpc$ is not known to be decidable. In the remainder of this section, we show that bounded-height tree duality is decidable. Given a digraph $H$, the *$n$-th crushed cylinder* $H_n^*$ is the quotient $(H^2\times P_n) / {\simeq_n}$, where $P_n$ is the path with arcs $(0,0),(0,1), (1,2), \cdots, (n-1,n), (n,n)$, and $\simeq_n$ is the equivalence defined by $$(u,v,i) \simeq_n (u',v',j) \Leftrightarrow \begin{cases} \text{$i = j = 0$ and $u = u'$}, \\ \text{or $i = j = n$ and $v = v'$}, \\ \text{or $(u,v,i) =(u',v',j)$}. \end{cases}$$ \[thm:123\] For a core digraph $H$ with tree duality, the following are equivalent: - $H$ has bounded-height tree duality, - For some $n$ we have $H_n^* \to H$. - There exists a directed (upward) path from the first projection to the second in $H^{H^2}$. [**(1) $\Rightarrow$ (2):**]{} The two subgraphs obtained from $H_n^*$ by removing the two ends both admit homomorphisms to $H$. Therefore, if a tree obstruction of $H$ admits a homomorphism to $H_n^*$, its image must intersect the two ends hence its algebraic length must be at least $n$. **$\neg{}$(1) $\Rightarrow$ $\neg{}$(2):** Let $T$ be a critical obstruction of $H$ of algebraic length $n+2$. Let $T_0$, $T_n$ be the subgraphs of $T$ obtained by removing the vertices of height $0$ and $n+2$ respectively. Then there exists homomorphisms $f_0: T_0 \to H$ and $f_n: T_n \to H$. Let $h: T \to P_{n+2}$ be the height function of $T$. We define a map $f: T \to H_n^*$ by $$f(u) = \begin{cases} (f_n(u), f_0(u), h(u)-1) / {\simeq_n} & \text{if $h(u) \not \in \{0, n+2\}$,}\\ (f_n(u), f_n(u), 0)/ {\simeq_n}& \text{if $h(u) = 0$,}\\ (f_0(u), f_0(u), n)/ {\simeq_n}& \text{if $h(u) = n+2$.} \end{cases}$$ Let $(u,v)$ be an arc of $T$. Then $h(v) = h(u) + 1$. If $\{h(u), h(v) \} \cap \{ 0, n+2 \} = \emptyset$, we clearly have $(f(u), f(v)) \in A(H_n^*)$. If $h(u) = 0$, then $f(u) = (f_n(u), f_n(u), 0)/ {\simeq_n}$ is an in-neighbour of $(f_n(v), f_n(v), 0)/ {\simeq_n} = (f_n(v), f_0(v), 0)/ {\simeq_n} = f(v)$, and if $h(v) = n+2$, $f(v) = (f_0(v), f_0(v), n)/ {\simeq_n}$ is an out-neighbour of $f(u)$ because $$(f_0(u), f_0(u), n)/ {\simeq_n} = (f_n(u), f_0(u), n)/ {\simeq_n} = f(u).$$ Therefore $f$ is a homomorphism. **(2) $\Leftrightarrow$ (3):** This equivalence follows easily from the definition. The problem whether an input digraph has bounded-height tree duality is decidable. It is decidable whether a digraph has tree duality [@FedVar:SNP] (see Theorem \[thm:u\] below). For a digraph with tree duality, bounded height of the obstructions (the condition (1) of Theorem \[thm:123\]) is equivalent to the condition (3), which involves directed reachability in a finite graph. Hence bounded-height tree duality is decidable. Adjoint functors and generation of tractable templates {#sec:functors} ====================================================== The correspondence of Proposition \[prop:delta\] can be extended to a wide class of functors presented in this section. To illustrate this extension, we first redefine $\delta$ in terms of patterns. Let $P$ be the digraph with vertices $0, 1$ and arc $(0,1)$, and $Q$ the digraph with vertices $0, 1, 2$ and arcs $(0,1), (1,2)$. Furthermore let $q_1, q_2: P \to Q$ be the homomorphisms mapping the arc $(0,1)$ to $(0,1)$ and $(1,2)$ respectively. For an arbitrary digraph $G$, its arc graph $\delta G$ can be described as follows: The vertices of $\delta G$ are the arcs of $G$, that is, the homomorphisms $f: P \to G$. The arcs of $\delta G$ are the couples of consecutive arcs in $G$, that is, the couples $(f_1,f_2)$ such that there exists a homomorphism $g: Q \to G$ satisfying $g \circ q_1 = f_1$ and $g \circ q_2 = f_2$. Thus the functor $\delta$ is generated by the pattern $\{P, (Q,q_1,q_2))\}$ in a way that generalises quite naturally. The rest of this section deals with relational structures. A *vocabulary* is a finite set $\sigma = \{ R_1 , \dotsc , R_m \}$ of relation symbols, each with an arity $r_i$ assigned to it. A $\sigma$-structure is a relational structure $A = \langle \bs A; R_1 (A), \dotsc , R_m (A)\rangle$ where $\bs A$ is a non-empty set called the *universe* of $A$, and $R_i (A)$ is an $r_i$-ary relation on $\bs A$ for each $i$. Homomorphisms of relational structures are relation-preserving mappings between universes; a homomorphism is defined only between structures with the same vocabulary. Cores, trees, quotient structures, etc. can also be defined in the context of relational structures, consult [@LoTar:mosfd] for the details (see also [@Hod:A-shorter-model; @Tar:FOD]). The notions of the constraint satisfaction problem, template, and tree duality also carry over naturally from the setting of digraphs. Let $\sigma$ and $\tau$ be two vocabularies. Let $P$ be a $\sigma$-structure, and for every relation $R$ of $\tau$ of arity $r = a(R)$, let $Q_R$ be a $\sigma$-structure with $r$ fixed homomorphisms $q_{R,i}: P \to Q_R$ for $i = 1, \ldots, r$. Then the family $\{ P \} \cup \{ (Q_R, q_{R,1}, \ldots, q_{R,a(R)}) : R \in \tau \}$ defines a functor $\Psi$ from the category $\mathcal{A}$ of $\sigma$-structures to the category $\mathcal{B}$ of $\tau$-structures as follows. - For a $\sigma$-structure $A$, let $B = \Psi A$ be a $\tau$-structure whose universe is the set of all homomorphisms $f: P \to A$. - For every relation $R$ of $\tau$ of arity $r = a(R)$, let $R(B)$ be the set of $r$-tuples $(f_1, \ldots, f_r)$ such that there exists a homomorphism $g: Q_{R} \to A$ such that for $i = 1, \ldots, r$ we have $g \circ q_{R,i} = f_i$. It was shown by Pultr [@Pul:The-right-adjoints] that functors $\Psi$ defined by means of patterns are right adjoints into a category of relational structures characterised by axioms of a specific type. We exhibit their corresponding left adjoints $\Psi^{-1}$ in the case when both the domain and the range of $\Psi$ is the category of all relational structures with a given vocabulary. For every $\tau$-structure $B$, we define a $\sigma$-structure $\Psi^{-1} B = A/{\sim}$, where - $A$ is a disjoint union of $\sigma$-structures; for every element $x$ of the universe of $B$, $A$ contains a copy $P_x$ of $P$, and for every $R \in \tau$ and $(x_1, \ldots, x_r) \in R(B)$, $A$ contains a copy $Q_{R,(x_1, \ldots, x_r)}$ of $Q_R$. - $\sim$ is the least equivalence which identifies every element $u$ of $P_{x_i}$ with its image $q_{R,i}(u)$ in $Q_{R,(x_1, \ldots, x_r)}$, for every $R \in \tau$, every $(x_1, \ldots, x_r) \in R(B)$ and every $i \in \{1, \ldots, r\}$. \[prop:Psi\] For any $\tau$-structure $B$ and $\sigma$-structure $A$, $$B\to\Psi A \qquad\text{if and only if}\qquad \Psi^{-1} B \to A.$$ Let $h: B\to\Psi A $ be a homomorphism, and put $h(b) = f_b: P\to A$. Then for every $b \in B$, the mapping $f_b$ corresponds to a well-defined homomorphism to $A$ from a copy $P_b$ of $P$. Also, for every $R \in \tau$ and $(b_1, \ldots, b_r) \in R(B)$, we have $(h(b_1), \ldots, h(b_r)) \in R(\Psi A )$, so there exists a homomorphism $g_{(b_1, \ldots, b_r)}: Q_R\to A$ such that $f_{b_i} = g_{(b_1, \ldots, b_r)} \circ q_{R,i}$ for $i = 1, \ldots, r$; the mapping $g_{(b_1, \ldots, b_r)}$ corresponds to a well-defined homomorphism from a copy $Q_{R,(b_1, \ldots, b_r)}$ of $Q_R$ to $A$. Therefore $\bigcup_{b \in B} f_b \cup \bigcup_{\tau} \bigcup_{R(B)} g_{(b_1, \ldots, b_r)}$ corresponds to a well-defined homomorphism $\hat{h}: \bigcup_{b \in B} P_b \cup \bigcup_{\tau} \bigcup_{R(B)} Q_{R,(b_1, \ldots, b_r)} \to A$, such that if $x \sim y$, then $\hat{h}(x) = \hat{h}(y)$. Therefore $\hat{h}$ induces a homomorphism from the quotient structure $\Psi^{-1} B $ to $A$. Conversely, if $h: \Psi^{-1} B \to A$ is a homomorphism, we define a homomorphism $\hat{h}: B\to\Psi A $ by $\hat{h}(b) = f_b$, where $f_b$ corresponds to the restriction of $h$ to the quotient of $P_b$ in $\Psi^{-1} B $. Indeed, if $R \in \tau$ and $(b_1, \ldots, b_r) \in R(B)$, then the restriction of $h$ to the quotient of $Q_{R,(b_1, \ldots b_r)}$ in $\Psi^{-1} B $ corresponds to a homomorphism $g: Q_R \to A$ such that $f_{b_i} = g \circ q_{R,i}$ for $i = 1, \ldots, r$, whence $(\hat{h}(b_1), \ldots, \hat{h}(b_r)) \in R(\Psi A )$. If a $\sigma$-structure $A$ has polynomial CSP, then the $\tau$-structure $\Psi A $ also has polynomial CSP. In fact, Corollary \[cor:arc\] generalises as follows. \[thm:psi\] If a $\sigma$-structure $A$ has tree duality, then the $\tau$-structure $\Psi A $ also has tree duality. We prove Theorem \[thm:psi\] using Feder and Vardi’s characterisation of structures with tree duality. For a $\sigma$-structure $A$, let $\U A$ be the $\sigma$-structure defined as follows. The universe of $\U A$ is the set of all nonempty subsets of $A$, and for $R \in \sigma$ of arity $r$, $R(\U A)$ is the set of all $r$-tuples $(X_1, \ldots, X_{r})$ such that for all $j \in \{1, \ldots, r\}$ and $x_j \in X_j$ there exist $x_k \in X_k, k \in \{1, \ldots, r\}\setminus \{j\}$ such that $(x_1, \ldots, x_{r}) \in R(A)$. \[thm:u\] A structure $A$ has tree duality if and only if there exists a homomorphism from $\U A$ to $A$. Suppose $A$ has tree duality. Then there is a homomorphism $f:\U A\to A$. Let $U=\P(\Psi A )\setminus\{\emptyset\}$ be the universe of $\U\Psi A $ and let $S\in U$. For $p \in P$, define $S_p ={\{f(p): f \in S\}} \in \U A$, and $f_S(p) = f(S_p)$. We claim that $f_S: P \to A$ is a homomorphism. Indeed, for $R \in \sigma$ and $(p_1, \ldots, p_r) \in R(P)$, the $r$-tuples $(f(p_1), \ldots, f(p_r)) \in R(A)$ for all $f \in S$ prove that $(S_{p_1}, \ldots, S_{p_r}) \in R(\U A)$, whence $(f_S(p_1), \ldots, f_S(p_r)) = (f(S_{p_1}), \ldots, f(S_{p_r})) \in R(A)$. Thus we define a map $\hat{f}: \U \Psi A \to \Psi A $ by $\hat{f}(S) = f_S$. We show that it is a homomorphism. For $R \in \tau$ and $(S_1, \ldots, S_r) \in R(\U \Psi A )$, every $f_i \in S_i$, $1 \leq i \leq r$ is contained in an $r$-tuple $(h_1, \ldots, h_r) \in R(\Psi A )$ with $f_j \in S_j$ for $1 \leq j \leq r$ and $h_i = f_i$, whence there exists a homomorphism $g_{(h_1, \ldots, h_r)}: Q_R \to A$ such that $h_j = g_{(h_1, \ldots, h_r)} \circ q_{R,j}$ for $j = 1, \ldots, r$. For $x \in Q$, let $T_x$ be the set of all images $g_{(h_1, \ldots, h_r)}(x) \in A$ (with $(S_1, \ldots, S_r)$ fixed), and $g_{(S_1, \ldots, S_r)}(x) = f(T_x)$. Then $g_{(S_1, \ldots, S_r)} : Q_R \to A$ is a homomorphism, and for $x \in q_{R,j}(P)$ we have $T_x$ = $S_x$ (because they are images of $x$ under restrictions of the same homomorphisms), whence $g_{(S_1, \ldots, S_r)}(x) = f_{S_j}(x)$. Thus $f_{S_j} = g_{(S_1, \ldots, S_r)} \circ q_{R,j}$ for $j = 1, \ldots, r$. Consequently $(f_{S_1}, \ldots, f_{S_r}) = (\hat{f}(S_1), \ldots, \hat{f}(S_r))\in R(\Psi A )$. This shows that $\hat{f}$ is a homomorphism. Unlike the case of the arc-graph construction, we are unable to provide an explicit description of the tree obstructions of $\Psi A $ in terms of those of $A$ for a general right adjoint $\Psi$. However, in isolated cases we can do it, as the following example shows. The endofunctor $\Psi$ on the category of digraphs is defined via the pattern $\{P, (Q, q_1, q_2)\}$, where $P=\vec P_1$ is the one-arc path $u\to v$, $Q=\vec P_3$ is the directed path $0\to1\to2\to3$, the homomorphism $q_1: u\mapsto0,\ v\mapsto 1$, and finally $q_2: u\mapsto 2,\ v\mapsto 3$. Let $T$ be a tree of algebraic height $h$ and consider the unique homomorphism $t$ from $T$ to the directed path $\vec P_h$. The arcs of $T$ are of two kinds: *blue arcs* $A_b(T)=\{(x,y): t(x)=2k,\ t(y)=2k+1 \text{ for some integer $k$}\}$ and *red arcs* $A_r(T)=\{(x,y): t(x)=2k+1,\ t(y)=2k+2 \text{ for some integer $k$}\}$. We define two equivalence relations on the vertices of $T$: $x\sim_b y$ if the (not necessarily directed) path from $x$ to $y$ in $T$ has only blue arcs, and $x\sim_r y$ if the path from $x$ to $y$ in $T$ has only red arcs. Then $T$ has two $\Psi$-Sproinks, namely $T/{\sim_b}$ and $T/{\sim_r}$ with loops removed. For a collection $\T$ of trees, let $\PsiS(\T)$ be the set of all $\Psi$-Sproinks of the trees contained in $\T$. We claim that if $\T$ is a complete set of obstructions for a template $H$, then $\PsiS(\T)$ is a complete set of obstructions for $\Psi H$. To prove it, we follow the idea of the proofs of Lemma \[lem:sproink\] and Theorem \[thm:spr-thm\]. First we prove that $T\to\Psi^{-1}(T/{\sim_b})$. This is not difficult: every blue arc of $T$ was contracted to a vertex of $T/{\sim_b}$ and this vertex was blown up to an arc in $T\to\Psi^{-1}(T/{\sim_b})$. Thus we can map blue arcs to the corresponding blown-up arcs. Red arcs of $T$ are also arcs of $T/{\sim_b}$, and hence we can map each red arc to the arc $(1,2)$ of the corresponding copy of $Q$ in $\Psi^{-1}(T/{\sim_b})$. Clearly such a mapping is a homomorphism. Analogously we show that $T\to\Psi^{-1}T/{\sim_r}$. Finally we want to prove that if $T\to\Psi^{-1}G$, then either $T/{\sim_b}\to G$ or $T/{\sim_r}\to G$. Suppose that $f:T\to\Psi^{-1}G$. Then some arcs of $T$ are mapped by $f$ to arcs corresponding to vertices of $G$ (arcs of copies of $P$), and others are mapped to arcs corresponding to arcs of $G$ (arcs $(1,2)$ of copies of $Q$). Let us call the former v-arcs and the latter a-arcs. It follows from the definition of $\Psi^{-1}$ that either all blue arcs of $T$ are v-arcs and all red arcs of $T$ are a-arcs, or all blue arcs of $T$ are a-arcs and all red arcs of $T$ are v-arcs. In the former case $T/{\sim_b}\to G$, while in the latter case $T/{\sim_r}\to G$. It is notable that in the above example each tree obstruction for $H$ generates finitely many obstructions for $\Psi H $. This is no accident. \[thm:fidu\] Let $\Psi$ be a functor generated by a pattern $\{ P \} \cup \{ (Q_R, q_{R,1}, \ldots, q_{R,a(R)}) : R \in \tau \}$, where for every $R \in \tau$ and $1 \leq i < j \leq a(R)$, the image $q_{R,i}(P)$ is vertex-disjoint from $q_{R,j}(P)$. If a $\sigma$-structure $A$ has finite duality, then the $\tau$-structure $\Psi A $ also has finite duality. The proof uses the characterisation of structures with finite duality of [@Tar:FOD]. The *square* of a $\sigma$-structure $B$ is the structure $B \times B$. It contains the *diagonal* $\Delta_{B \times B} = \{ (b,b) : b \in B\}$. An element $a$ of $B$ is *dominated* by an element $b$ of $B$ if for every $R \in \sigma$, for every $i$ and every $(x_1,\ldots,x_{a(R)}) \in R(B)$ with $x_i = a$, we have $(y_1,\ldots,y_{a(R)}) \in R(B)$ with $y_i = b$ and $y_j = x_j$ for $j \neq i$. A structure $B$ *dismantles to* its induced substructure $C$ if there exists a sequence $x_1,\dots,x_k$ of distinct elements of $B$ such that $B\setminus C = \{x_1,\dots,x_k\}$ and for each $1 \leq i \leq k$ the element $x_i$ is dominated in the structure induced by $C \cup \{x_i,\dots,x_k\}$. The sequence $x_1,\dots,x_k$ is then called a *dismantling sequence* of $B$ on $C$. A structure has finite duality if and only if it has a retract whose square dismantles to its diagonal. Let $A$ be a $\sigma$-structure with finite duality. Without loss of generality, we assume that $A$ is a core, so that $A$ has no proper retracts; thus the square of $A$ dismantles to its diagonal. Let $(x_1,y_1), \ldots, (x_k, y_k)$ be a dismantling sequence of $A \times A$ on $\Delta_{A \times A}$. Then $\Psi(A \times A) \cong \Psi A \times \Psi A $; we want to prove that it dismantles to $\Delta_{\Psi A \times \Psi A } \cong \Psi \Delta_{A \times A} $. For $i \in \{1, \ldots, k\}$, define $X_i$ to be the substructure of $A \times A$ induced by the set $\Delta_{A \times A} \cup \{(x_i,y_i), \ldots, (x_k,y_k)\}$, and let $X_{k+1} = \Delta_{A \times A}$. We will show that $\Psi X_i $ can be dismantled to $\Psi X_{i+1} $, $i = 1, \ldots, k$. Let $b = (b_1,b_2)$ be an element dominating $a = (x_i,y_i)$ in $X_i$. Let $f \in \Psi X_i \setminus \Psi X_{i+1} $, and assume that $f = (f_1, f_2): P \to A \times A$. Then there exists (at least one) $p_0 \in P$ such that $f(p_0) = a$. We define $g = (g_1, g_2): P \to A \times A$ by $g(p_0) = b$ and $g(p) = f(p)$ if $p \neq p_0$. Since $b$ dominates $a$, $g$ is a homomorphism, and obviously $g \in \Psi X_i $. We claim that $g$ dominates $f$. Indeed, for $R \in \tau$ and $(f_1, \ldots, f_{a(R)}) \in R(\Psi X_i )$ such that $f = f_j$, there exists a homomorphism $h: Q_R \to X_i$ such that $f = h \circ q_{R,j}$. Define $h': Q_R \to X_i$ by $h'(q_{R,j}(p_0)) = b$ and $h'(z) = h(z)$ for $z \neq q_{R,j}(p_0)$. Since $b$ dominates $a = h(q_{R,j}(p_0))$, the mapping $h'$ is a homomorphism. By hypothesis, for $\ell \neq j$, the image $q_{R,\ell}(P)$ is disjoint from $q_{R,j}(P)$, whence $f_{\ell} = h' \circ q_{R,\ell}$, while $h' \circ q_{R,j} = g$. Therefore $R(\Psi X_i )$ contains all the $a(R)$-tuples needed to establish the domination of $f$ by $g$. Let $p_1, p_2, \ldots, p_m$ be an enumeration of the elements of $P$. We dismantle $\Psi X_i $ to $\Psi X_{i+1} $ by successively removing the functions $f$ such that $f(p_j) = (x_i,y_i)$ for $j = 1, \ldots, m$. Proceeding in this way for $i = 1, \ldots, k$, we get a dismantling of $\Psi A \times \Psi A \cong \Psi X_1 $ to $\Psi X_{k+1} \cong \Delta_{\Psi A \times \Psi A }$. Therefore $\Psi A $ has finite duality. Perhaps the lack of knowledge of a general construction is natural since there is no restriction on the pattern $\{ P \} \cup \{ (Q_R, q_{R,1}, \ldots, q_{R,a(R)}) : R \in \tau \}$. On the other hand, there are many possible transformations ${\cal T}'$ on a family ${\cal T}$ of tree obstructions, in the style of $\operatorname{Sproink}({\cal T})$. Any such transformation gives rise to a complete set of obstructions to homomorphisms into a structure $H' = \Pi_{T \in {\cal T}'}D_T$; however in general there is no way of guaranteeing that such structure $H'$ is finite, even when ${\cal T}$ is a complete set of obstructions for a finite structure $H$. Concluding comments =================== In this paper we tried to shed more light on the structure of tractable templates with tree duality. Let us turn our attention one more time to Fig. \[fig:diag\]. The grey areas in the diagram are areas that need a closer look in future research. Currently we do not know any digraph with a near-unanimity function and with bounded-height tree duality that could not be generated using right adjoints and products, starting from digraphs with finite duality; it is not clear whether any such “reasonable” class of structures with tree duality can be generated from structures with finite duality with a “reasonable” set of adjoint functors. We have shown here that possession of bounded-height tree duality is decidable. It is natural to ask what its complexity is; in particular, whether it is complete for some class of problems. Equally interesting is the decidability of membership in other classes depicted in Fig. \[fig:diag\]. Tree duality is known to be decidable [@FedVar:SNP], but not known to be in PSPACE. Our decision procedure for bounded-height duality is in PSPACE for graphs with tree duality; this suggests that checking tree duality may be harder than checking bounded height of the obstructions. Furthermore, finite duality is NP-complete [@Tar:FOD]. The decidability of bounded-treewidth duality is unknown, and so is the decidability of a near-unanimity function (see [@Mar:ExNUF] for a related result). The properties of near-unanimity functions proved in the proof of Proposition \[prop:dpc\] (i) in the context of digraphs and the arc-graph construction, also hold in the context of general structures and right adjoints. The proofs carry over naturally. [10]{} M. Barr and C. Wells. . Les Publications CRM, Montr[é]{}al, 3rd edition, 1999. D. Cohen and P. Jeavons. The complexity of constraint languages. In F. Rossi, P. van Beek, and T. Walsh, editors, [*Handbook of Constraint Programming*]{}, volume 2 of [*Foundations of Artificial Intelligence*]{}, chapter 8. Elsevier, 2006. T. Feder and M. Y. Vardi. The computational structure of monotone monadic [SNP]{} and constraint satisfaction: [A]{} study through [D]{}atalog and group theory. , 28(1):57–104, 1999. J. Foniok. . PhD thesis, Charles University, Prague, 2007. P. Hell and J. Ne[š]{}et[ř]{}il. On the complexity of [$H$]{}-coloring. , 48(1):92–110, 1990. P. Hell and J. Ne[š]{}et[ř]{}il. , volume 28 of [*Oxford Lecture Series in Mathematics and Its Applications*]{}. Oxford University Press, 2004. P. Hell, J. Ne[š]{}et[ř]{}il, and X. Zhu. Duality and polynomial testing of tree homomorphisms. , 348(4):1281–1297, 1996. W. Hodges. . Cambridge University Press, 1997. P. Kom[á]{}rek. . PhD thesis, Czechoslovak Academy of Sciences, Prague, 1987. In Czech (Dobr[é]{} charakteristiky ve t[ř]{}[í]{}d[ě]{} orientovan[ý]{}ch graf[u]{}). R. E. Ladner. On the structure of polynomial time reducibility. , 22(1):155–171, 1975. B. Larose, C. Loten, and C. Tardif. A characterisation of first-order constraint satisfaction problems. In [*Proceedings of the 21st IEEE Symposium on Logic in Computer Science (LICS’06)*]{}, pages 201–210. IEEE Computer Society, 2006. C. Loten and C. Tardif. Majority functions on structures with finite duality. , 29(4):979–986, 2008. S. Mac Lane. , volume 5 of [ *Graduate texts in mathematics*]{}. Springer-Verlag, New York Berlin Heidelberg, 2nd edition, 1998. M. Mar[ó]{}ti. The existence of a near-unanimity term in a finite algebra is decidable. Manuscript, 2005. J. Ne[š]{}et[ř]{}il and C. Tardif. Duality theorems for finite structures (characterising gaps and good characterisations). , 80(1):80–97, 2000. J. Ne[š]{}et[ř]{}il and C. Tardif. Short answers to exponentially long questions: Extremal aspects of homomorphism duality. , 19(4):914–920, 2005. A. Pultr. The right adjoints into the categories of relational systems. In [*Reports of the Midwest Category Seminar, IV*]{}, volume 137 of [*Lecture Notes in Mathematics*]{}, pages 100–113, Berlin, 1970. Springer. [^1]: Two digraphs $H$ and $H'$ are *homomorphically equivalent* if there exists a homomorphism from $H$ to $H'$ as well as a homomorphism from $H'$ to $H$. Clearly, if $H$ and $H'$ are homomorphically equivalent, then $H$-colouring and $H'$-colouring are equivalent problems, because $H$ and $H'$ admit homomorphisms from exactly the same digraphs. [^2]: An *endofunctor* is a functor from a category to itself. [^3]: Let $X$ and $Y$ be partially ordered sets. Mappings $\phi:X\to Y$ and $\psi:Y\to X$ are *Galois adjoints* if $\phi(x)\le_{Y}y \,\Leftrightarrow\, x\le_X \psi(y)$ for all elements $x\in X$ and $y\in Y$. [^4]: It follows that if $u$ is neither a source nor a sink of $T$, then both $0_{F(u)}$ and $1_{F(u)}$ are non-empty, and so in this case $F(u)$ is not a single vertex. If $u$ is a source or a sink of $T$, then $F(u)$ may be an arbitrary tree of height at most one. [^5]: The [*core*]{} of a digraph is any of its smallest subgraphs to which it admits a homomorphism. Every digraph $H$ has a unique core $C$ (up to isomorphism), which is moreover the only core homomorphically equivalent to it. In fact, the core $C$ of $H$ is a *retract* of $H$, which means that there exists a homomorphism $\rho:H\to C$ whose restriction on $C$ is the identity mapping (such a homomorphism is called a *retraction*).

--- abstract: 'We present quantum mechanical explanations for unresolved phenomena observed in field-free molecular alignment by a femtosecond laser pulse. Quantum phase analysis of molecular rotational states reveals the physical origin of the following phenomena: strong alignment peaks appear periodically, and the temporal shape of each alignment peak changes in an orderly fashion depending on molecular species; the strongest alignment is not achieved at the first peak; the transition between aligned and anti-aligned states is very fast compared to the time scale of rotational dynamics. These features are understood in a unified way analogous to that describing a carrier-envelope-phase-stabilized mode-locked laser.' author: - Sang Jae Yun - Chul Min Kim - Jongmin Lee - Chang Hee Nam bibliography: - 'MR4.bib' date: 'November 5, 2012' title: 'Quantum Phase Analysis of Field-Free Molecular Alignment' --- Chemical reaction depends on the relative orientation of reactants due to anisotropy in their electronic structure. Likewise, in photochemical processes, the relative orientation between laser polarization and molecular axis is a basic parameter affecting the result. In order to clarify or to control the chemical processes at the most fundamental level, the processes should be examined with aligned molecules. Among several techniques developed so far [@Pirani2001; @Wu1994; @Stapelfeldt2003], the field-free alignment method using an intense femtosecond laser pulse is now widely adopted. Due to the anisotropy of molecular polarizability, the intense laser pulse induces a torque to rotate molecules to the direction of laser polarization. After interacting with the laser pulse, the molecules periodically align to that direction, achieving field-free alignment [@Stapelfeldt2003; @Ortigoso1999]. This method has become popular in revealing molecular structures [@Itatani2004; @Vozzi2011], controlling and tracing chemical reactions [@Larsen1999; @Worner2010], generating high-order harmonics from aligned molecules [@Levesque2007; @Lee2008], and compressing optical pulses [@Bartels2001; @Spanner2003]. The field-free alignment method exhibits unique features in the temporal variation of alignment. A typical temporal structure of the method can be seen in Fig. 1, which is obtained by numerically solving the time-dependent Schrödinger equation (TDSE) for the rotational state of ${{\rm{O}}_{\rm{2}}}$. The degree of alignment can be represented by $\left\langle {\left\langle {{{\cos }^2}\theta } \right\rangle } \right\rangle$, in which $\theta $ is the angle between the laser polarization and the molecular axis and the double bracket means that the value is obtained by Boltzmann-averaging of all $\left\langle {{{\cos }^2}\theta } \right\rangle $ (quantum mechanical expectation value) in a thermal ensemble. The unique features can be summarized as follows. First, it shows a revival structure with a full revival time ${T_{rev}}$ in which multiple fractional revivals occur, showing periodic alignment peaks [@Seideman1999; @Child2003]. Second, each alignment peak has its own temporal profile that changes in an orderly fashion depending on molecular species [@Child2003]. Third, the strongest alignment is not achieved at the first peak [@Leibscher2003]. Fourth, the transition between aligned and anti-aligned states is very fast compared to the time scale of rotational dynamics [@Torres2005], as marked by $\Delta {T_{tr}}$ in Fig. 1. These features have been observed also in experiments [@Bartels2001; @Dooley2003], and many theoretical studies have been made to understand the physical origin of the phenomena [@Seideman1999; @Child2003; @Torres2005; @Seideman2001; @Renard2004; @Renard2005]. Considering that the revival structure is a salient feature of the coherent superposition state of quantum systems, e.g. coherently excited Rydberg atoms [@Leichtle1996], the complex amplitudes of the molecular eigenstates need to be analyzed in detail to explain the features [@Seideman2001; @Renard2004; @Renard2005]. {width="8cm"} In this work we present an analysis of the molecular alignment induced by a femtosecond laser pulse in terms of the quantum phase of molecular rotational states. By numerically solving the TDSE of molecular rotational states, the quantum phase of rotational eigenstates is obtained. With the phase of the eigenstates known, the temporal variation of molecular alignment is described in a way similar to that of mode-locked laser pulses. The results provide the quantum mechanical explanation on all the above features in a unified way, signifying that the quantum phase of rotational states plays a crucial role in understanding the alignment dynamics. To be specific, the alignment of an ${{\rm{O}}_{\rm{2}}}$ molecule, a typical non-polar linear molecule, by a femtosecond laser pulse was considered for analysis. The TDSE for such a case within a rigid rotor model in atomic units is given by [@Stapelfeldt2003; @Lemeshko2010] $$i{d \over {dt}}\left| {\Psi (t)} \right\rangle = {B_0}\left( {{{\bf{J}}^2} - { I \over I_0 }g(t){{\cos }^2}\theta } \right)\left| {\Psi (t)} \right\rangle ,$$ where $B_0$ is the molecular rotational constant, $I$ the cycle-averaged peak intensity, $I_0 \equiv {B_0 c} /\left( 2\pi \Delta \alpha \right)$ the natural intensity scale of the rotational molecular Hamiltonian, and $g(t) = {\sin ^2}\left( {\pi t/\tau } \right)$ the normalized laser-intensity profile. $\Delta \alpha ( \equiv {\alpha _\parallel } - {\alpha _ \bot })$ is the anisotropy of molecular polarizability. ${{\rm{O}}_{\rm{2}}}$ molecule has ${B_0} = 2.856 \times {10^{ - 16}}{~\rm{ erg ~(}}1.437{~\rm{c}}{{\rm{m}}^{ - 1}}{\rm{)}}$ and $\Delta \alpha = 1.12 ~ {\rm{ }}{{\rm{{\AA}}}^3}$ [@NIST], giving a natural rotational time scale $\pi /{B_0} = 11.6~{\rm{ps}}$ and $I_0=1.22 \times 10^{11}{~\rm{W/cm^2}}$. In our calculation, the condition of $\tau=60~{\rm{fs}} = 0.0052 \pi /{B_0}$ and $I = 5 \times {10^{13}}{~\rm{ W/c}}{{\rm{m}}^2} = 410 I_0$ was used. The evolution of molecular rotational states during the laser interaction is investigated by considering an angular momentum eigenstate $\left| {{J_i},{M_i}} \right\rangle $ as an initial state. Since the laser interaction causes sequential Raman-type transitions with the selection rules $\Delta J = 0, \pm 2$ and $\Delta M = 0$ [@Stapelfeldt2003], the rotational state under interaction can be written as $$\left| {\Psi (t)} \right\rangle = \sum\limits_{n = {n_0}}^\infty {\sqrt {{P_{{J_i} + 2n}}(t)} {e^{i{\Phi _{{J_i} + 2n}}(t)}}\left| {{J_i} + 2n,{M_i}} \right\rangle } ,$$ where ${P_{{J_i} + 2n}}$ and ${\Phi _{{J_i} + 2n}}$ are the probability and the quantum phase of $n$-th emerging component, respectively. The emerging components with positive $n$ come from Stokes processes, while those with negative $n$ from anti-Stokes processes. Since ${J_i} + 2n$ cannot be smaller than $\left| {{M_i}} \right|$, the anti-Stokes processes are bound by a non-positive integer ${n_0}$ [@Lemeshko2010]. Inserting Eq. (2) into Eq. (1) gives coupled differential equations of ${P_{{J_i} + 2n}(t)}$ and ${\Phi _{{J_i} + 2n}(t)}$. In obtaining the numerical solution, the Crank-Nicolson method [@Press1992] was used. Figure 2 shows the numerical solution of the rotational state for an initial state $\left| {3,0} \right\rangle $ as an example. {width="8cm"} The quantum phase ${\Phi _{{J_i} + 2n}}$ is the basic element of the alignment dynamics. As seen in Fig. 2 (a), the quantum phase of $n$-th emerging component starts with $\pi \left| n \right|/2$ [@Seideman2001; @Renard2004; @Renard2005]. This phenomenon can be understood by analyzing the mathematical form of Eq. (1). The emerging components are generated through the off-diagonal matrix elements of the interaction Hamiltonian, $-B_0 I g(t){{\cos }^2}\theta / I_0 $, which are always negative as the matrix elements of ${\cos ^2}\theta $ are positive [@Owschimikow2011], and $\Delta \alpha $ is also positive for most linear molecules including ${{\rm{O}}_{\rm{2}}}$. Because of this negativity of the interaction Hamiltonian and the factor $i = \exp (i\pi /2)$ in the left-hand side of Eq. (1), the starting phase of an emerging component must add ${\pi/2}$ to the phase of its parent component. Since a new component arises only through sequential transitions, ${\pi/2}$ is accumulated to its phase whenever a succeeding component emerges. Even when the populating process is of anti-Stokes $\left( {n < 0} \right)$, the phase accumulation is also ${\pi/2}$, not $-{\pi/2}$, because the matrix element of the interaction Hamiltonian is equally negative. Thus, the $n$-th quantum phase starts at the value of $\pi \left| n \right|/2$. Since the alignment is an interference among the rotational eigenstates, not the quantum phase itself but the relative phase determines the dynamics. As the laser interaction goes on, the quantum phase itself varies significantly, but the phase difference of ${\pi/2}$ between neighboring components set at the beginning does not change much because the phase-varying rates are comparable to each other, and the laser duration is very short compared to the time scale of the rotational dynamics. Thus, until the end of the laser pulse, the phase difference between neighboring components is kept to be around ${\pi/2}$ [@Seideman2001; @Renard2004; @Renard2005], as observed in Fig. 2 (a). After the interaction, the molecular wavefunction evolves freely and the dynamics of alignment is determined solely by the molecular state at the end of the interaction (t = 60 fs in Fig. 2). Setting this moment as the new time origin, the field-free evolution of the degree of alignment, i.e. expectation value of ${{{\cos }^2}\theta }$, is given by $$\begin{gathered} \left\langle {{{\cos }^2}\theta } \right\rangle (t) = \sum\limits_{n = {n_0}}^\infty {P_{{J_i} + 2n}^{}C_{{J_i} + 2n,{J_i} + 2n}^{}} \nonumber \\ + \left\{ {\sum\limits_{n = {n_0}}^\infty {{1 \over 2}\sqrt {I_n^{}} } {e^{i\left( {\omega _n^{}t - \varphi _n^{}} \right)}} + {\rm{c}}.{\rm{c}}.} \right\},\end{gathered}$$ where $$\begin{gathered} I_n^{} = 4P_{{J_i} + 2n}^{}P_{{J_i} + 2n + 2}^{}{\left( {C_{{J_i} + 2n,{J_i} + 2n + 2}^{}} \right)^2}, \\ \omega _n^{} = {E_{{J_i} + 2n + 2}} - {E_{{J_i} + 2n}} = {B_0}(8n + 4{J_i} + 6), \\ \varphi _n^{} = \Phi _{{J_i} + 2n + 2}^{} - \Phi _{{J_i} + 2n}^{},\end{gathered}$$ $C_{{J_a},{J_b}}^{} \equiv \left\langle {{J_a},{M_i}} \right|{\cos ^2}\theta \left| {{J_b},{M_i}} \right\rangle $, and ${E_J} \equiv {B_0}J(J + 1)$. Here, $P_{{J_i} + 2n}$ and $\Phi _{{J_i} + 2n}$ are the values evaluated at the end of the laser pulse. Because Eq. (3) is expressed in a complex Fourier series, the time-domain behaviour can be analyzed in the frequency-domain, where the spectral intensity $I_n$, frequency $\omega_n$, and spectral phase $\varphi_n$ are given by Eqs. (4), (5), and (6), respectively. The first term of Eq. (3) is the DC term giving an offset in the time-domain, whereas the second term is the oscillating part having multiple frequencies. Equation (5) expresses an equally spaced frequency comb with a comb spacing of $8{B_0}$. {width="8cm"} The spectral phase ${\varphi _{n}}$ of $\left\langle {{{\cos }^2}\theta } \right\rangle (t)$ in Eq. (3) is the relative phase of two neighboring quantum states, as expressed in Eq. (6). ${\varphi _{n}}$ has nearly binary values close to either ${\pi/2}$ or $ - {\pi/2}$ [@Renard2004; @Renard2005], as shown in Figs. 3 (a)-(c). This is explained by the initial quantum phases $\pi \left| n \right|/2$ of which relative difference is mostly maintained until the end of the laser pulse. Because of this, Stokes processes $\left( {n \ge 0} \right)$ result in ${\varphi _{n}}$ near ${\pi/2}$ whereas anti-Stokes processes $\left( {n < 0} \right)$ generate near $ - {\pi/2}$, making them nearly out of phase. This is the most critical part in understanding the alignment dynamics. In order to obtain experimentally observable alignment dynamics, $\left\langle {{{\cos }^2}\theta } \right\rangle (t)$ should be averaged over an initial thermal ensemble to give $\left\langle {\left\langle {{{\cos }^2}\theta } \right\rangle } \right\rangle (t)$ as shown in Fig. 1. Instead of considering all the initial states at once, it is more systematic to consider only part of them by dividing into groups according to ${M_i}$. Figure 3 illustrates a group with ${M_i} = 0$. For a given ${M_i}$, the lowest-${J_i}$ state generates only Stokes processes so that the spectral phase is nearly constant, ${\pi/2}$, as shown in Fig. 3 (a). Anti-Stokes processes produce low-frequency components and can happen only from initial high-${J_i}$ states of which Boltzmann factors are smaller than that of the lowest-${J_i}$ state. Because of this, after cancelling out the spectral intensity of low-frequency parts due to the opposite phase between Stokes and anti-Stokes processes, the resulting ensemble-averaged phase follows the phase of the Stokes processes as can be seen in Fig. 3 (d). In contrast to the low-frequency parts, the spectral intensity of high-frequency parts add up constructively because only Stokes processes prevail in that region. Consequently, the ensemble-averaged phase of Fig. 3 (d) is formed almost identical to that of Fig. 3 (a). Similarly, other groups having different ${M_i}$ also show the same behavior, nearly constant phase of ${\pi/2}$. Owing to this phase coherence, the resulting spectral intensity from all the initial states is effectively enhanced. This is the main reason for achieving the strong alignment. The resulting spectra of $\left\langle {\left\langle {{{\cos }^2}\theta } \right\rangle } \right\rangle (t)$ from all the initial states are shown in Fig. 4, where almost flat spectral phase and equally spaced frequency comb are observed. By synthesizing the spectral components in Fig. 4, the complex temporal variation in Fig. 1 was obtained. {width="9cm"} The complex temporal dynamics of molecular alignment after the laser pulse can be understood by treating it as a mode-locked signal with almost flat phase. Since an equally spaced frequency comb having a linear spectral phase produces a pulse train in the time-domain [@Weiner2009], it is natural that $\left\langle {\left\langle {{{\cos }^2}\theta } \right\rangle } \right\rangle (t)$ in Fig. 1 also shows pulsed shape like mode-locked laser pulses, exhibiting periodic strong alignment peaks. The temporal profile of each alignment peak in Fig. 1 can be analyzed by using the analogy to the carrier-envelope phase (CEP) of a mode-locked laser pulse. In Fig. 1, the solid line corresponds to the carrier oscillation, while the dashed line the envelope. It is well known that the CEP-sequence is completely determined by the frequency-comb structure [@Jones2000; @Weiner2009]. Let us denote the frequency-comb as ${\omega _m} = m\Delta \omega + \delta \omega $ ($m = $ 1, 2,$ \cdots $), where $\Delta \omega $ is the pulse-repetition rate (multiplied by $2\pi $), and $\delta \omega $ is the offset frequency as shown in Fig. 4. Then, in the time-domain, the increment of CEP from one pulse to the next is $\Delta {\phi _{CEP}} = - 2\pi \cdot \delta \omega /\Delta \omega $ [@Weiner2009]. Since the full revival occurs when the accumulated CEP change reaches $2\pi $, the number of fractional revivals occurring during a full revival period should be $N = 2\pi /\left| {\Delta {\phi _{CEP}}} \right| = \Delta \omega /\left| {\delta \omega } \right|$, and the full revival time should be ${T_{rev}} = 2\pi N/\Delta \omega = 2\pi /\left| {\delta \omega } \right|$. The comb parameters, $\Delta \omega $ and $\delta \omega $, are obtained from Eq. (5) for a given molecular species. Since an ${{\rm{O}}_{\rm{2}}}$ molecule has only odd-$J$ states due to its nuclear spin statistics [@Herzberg1989], $\Delta \omega = 8{B_0}$ and $\delta \omega = 2{B_0}$ as shown in Fig. 4, which results in $\Delta {\phi _{CEP}} = - {\pi/2}$, $N = 4$, and ${T_{rev}} = {\pi / {{B_0}}}$. The CEP of the first alignment peak is always nearly ${\pi/2}$ (sine-like) regardless of molecular species, and it relates to the fact that the strongest alignment is not achieved at the first peak. The first CEP is determined by the temporal phases of all the frequency components at $t=0$. These phases at $t=0$ coincide with the spectral phases as can be deduced from Eq. (3). Since all the spectral phases lie near ${\pi/2}$, the first CEP should also be near ${\pi/2}$. With the results of the last paragraph, the CEP sequence of ${{\rm{O}}_{\rm{2}}}$ molecule should be $\left( {{\pi/2},0, - {\pi/2}, - \pi } \right)$. This explains why the strongest alignment does not occur at the first peak because it occurs when the CEP is $0$. The same consideration gives the CEP sequence $\left( {{\pi/2},\pi , - {\pi/2},0} \right)$ for even-$J$ molecules such as ${\rm{C}}{{\rm{O}}_{\rm{2}}}$ [@Bartels2001] and $\left( {{\pi/2}, - {\pi/2}} \right)$ for all-$J$ molecules such as ${{\rm{N}}_2}$ [@Dooley2003]. Our analysis also provides a detailed explanation on temperature dependence of molecular alignment. It seems intuitively obvious that the degree of alignment decreases with temperature. However, although this phenomenon was partially explained from a classical treatment [@Leibscher2004], no quantum mechanical explanation has been given. Here it can be explained by noting that Stokes and anti-Stokes processes generate opposite spectral phases. At a high temperature, considerable population is at high-${J_i}$ states initially. A higher-${J_i}$ state causes more anti-Stokes processes that diminish the spectral intensity of low-frequency part. The higher the temperature is, the wider the frequency range is diminished. Because a small spectral intensity is equivalent to a low oscillating amplitude in the time-domain, the diminished spectral intensity results in weaker alignment. The fast transition between aligned and anti-aligned states can be explained by considering the center frequency of the frequency comb, ${\omega _c}$ in Fig. 4. ${\omega _c}$ corresponds to the carrier oscillation period with the relation $\Delta {T_{tr}} = {\pi \mathord{\left/ {\vphantom {\pi {{\omega _c}}}} \right. \kern-\nulldelimiterspace} {{\omega _c}}}$. It means that high ${\omega _c}$ is equivalent to fast transition. High ${\omega _c}$ can result from either a high temperature or a strong laser interaction. A high temperature results in a high ${\omega _c}$ with two reasons. First, initially existing high-${J_i}$ states easily generate high-frequency parts because, from Eq. (5), ${\omega _{n}}$ becomes a high frequency with even a small $n$. Second, the spectral intensity of low-frequency part is diminished as explained before. A strong laser field also causes the high ${\omega _c}$ because the strong interaction populates much higher-$J$ states through high-$n$ Stokes transition so that the spectral intensity in the high-frequency region increases. Thus, $\Delta {T_{tr}}$ becomes shorter when the temperature is high or the laser interaction is strong. In Fig. 4, ${\omega _c}$ is $29.8{B_0}$ leading to $\Delta {T_{tr}} = 388~{\rm{ fs}}=0.034 T_{\rm{rev}}$. This fast transition is not surprising because the superposition of the eigenstates of $J = 7$ and $J = 9$ generates ${\omega _c} = 34{B_0}$ leading to a transition time of $341~{\rm{ fs}}~ (=0.030 T_{\rm{rev}})$. The molecular alignment at higher intensity was also examined. For example, at $I = 8 \times {10^{13}}~{\rm{ W/c}}{{\rm{m}}^2} = 655 I_0$, the quantum phase ${\Phi _{{J_i}}}$ showed an abrupt jump due to the redistribution of rotational states during the laser interaction. Due to the phase jump, the ${\pi/2}$ phase-difference between adjacent quantum phases did not hold any more, and the spectral phase of $\left\langle {{{\cos }^2}\theta } \right\rangle (t)$ deviated from that of Figs. 3 (a)-(c). It was, however, found that the Boltzmann-averaging mitigate these effects to make the ensemble-averaged spectral phase to be almost the same as that of Fig. 3 (d). In conclusion, we have presented clear quantum mechanical explanations on the puzzling phenomena observed in the field-free molecular alignment. Analyzing the quantum phase of molecular rotational states, we have shown that $\left\langle {\left\langle {{{\cos }^2}\theta } \right\rangle } \right\rangle (t)$ has almost flat spectral phase and equally spaced frequency comb. All the features of complex alignment dynamics could be explained in an analogous way describing a CEP-stabilized mode-locked laser. The analysis given here can be extended further to more challenging subject such as orientation of polar molecules or three-dimensional alignment of asymmetric top molecules. The clear understanding for alignment dynamics will bring the optimization of experimental conditions and the development of a better method controlling molecular rotational states. This work was supported by the Ministry of Education, Science and Technology of Korea through the National Research Foundation and by the Ministry of Knowledge and Economy of Korea through the Ultrashort Quantum Beam Facility Program.

--- abstract: 'Fully convolutional networks (FCN) has significantly improved the performance of many pixel-labeling tasks, such as semantic segmentation and depth estimation. However, it still remains non-trivial to thoroughly utilize the multi-level convolutional feature maps and boundary information for salient object detection. In this paper, we propose a novel FCN framework to integrate multi-level convolutional features recurrently with the guidance of object boundary information. First, a deep convolutional network is used to extract multi-level feature maps and separately aggregate them into multiple resolutions, which can be used to generate coarse saliency maps. Meanwhile, another boundary information extraction branch is proposed to generate boundary features. Finally, an attention-based feature fusion module is designed to fuse boundary information into salient regions to achieve accurate boundary inference and semantic enhancement. The final saliency maps are the combination of the predicted boundary maps and integrated saliency maps, which are more closer to the ground truths. Experiments and analysis on four large-scale benchmarks verify that our framework achieves new state-of-the-art results.' author: - 'Yunzhi Zhuge, Pingping Zhang, Huchuan Lu' bibliography: - 'IEEEabrv.bib' - 'refs.bib' title: 'Boundary-guided Feature Aggregation Network for Salient Object Detection' --- Salient object detection, Boundary information extraction, Attention, feature fusion. Introduction {#sec:intro} ============ Saliency object detection is a fundamental computer vision task which aims to identify the most eye-catching objects and areas in an image [@Itti1998A][@Achanta2009Frequency] [@Tong2015Salient] [@Wang2015Deep]. In the past two decades, great success has been made in this pixel-labeling task. However, due to several inevitable factors such as cluttered backgrounds or blurred boundaries, it still remains a difficult task to combine all hand-tuned cues in an appropriate way. Recently, deep convolutional neural networks (CNNs) have greatly improved the performances of many computer vision tasks, such as image classification [@Krizhevsky2012ImageNet], semantic segmentation [@FCN] and visual tracking [@Wang2016STCT; @zhang2018non]. With the advantages of fully convolutional networks (FCNs) [@FCN], several FCNs-based attempts have been performed and delivered state-of-the-art performance in predicting saliency maps [@Wang2016Saliency; @Lee2016Deep; @Liu2016DHSNet]. Nonetheless, existing models mainly focus on utilizing high-level features extracted from last convolutional layers. As a result, they are lack of low-level visual information such as object boundary. Thus, these models tend to predict imperfect results with poorly localized object boundaries. In this paper, we propose a novel saliency detection method based on multi-level features and boundary cues. To make full use of the multi-level convolutional features and boundary information, we present a boundary-guided feature aggregating architecture, which simultaneously generates and merges multi-level saliency maps and boundary prediction maps to obtain accurate saliency maps. Our framework has two streams for saliency prediction. In the main stream, we predict saliency maps with the incorporated features maps at different resolutions, and these predicted saliency maps are recursively sent to the refinement stage as the inputs. In another stream, boundary features are extracted through a boundary extraction structure. To utilize the boundary information, we propose an attention-based feature fusion module to integrate two-stream features. Our main contributions are as follows: - We propose a boundary-guided feature aggregation network, termed as BFANet, to utilize multi-level convolutional features and boundary cues for salient object detection. The BFANet first extracts multi-level features, then integrates them into multiple resolutions. The boundary information fusion is performed to enhance these features to generate finer saliency maps. - We propose a boundary extraction network as a branch of the BFANet, which generates boundary feature maps under the supervision of boundary ground truth. Besides, we also introduce an attention-based feature fusion module to produce saliency maps with robust boundary. - Extensive experiments on four large-scale benchmarks have shown that our approach performs favorably against other state-of-the-art methods. Our Proposed Method {#sec:previous work} =================== Our method is mainly motivated by the following facts. First, previous methods usually focus on precisely localizing salient objects, which more or less neglect the sharpness of boundary areas. To enhance the boundary, we propose a two-steam structure to generate saliency maps with accurate object boundaries. Secondly, features of variant scales contribute differently to detection. However, there still exist questions in how to effectively utilize these features. Therefore, we make a feasible attempt and propose an effective attention-based structure to perform the multi-level feature fusion. As shown in Fig. \[diagram\], our proposed framework is composed of three components: Aggregating Feature Extraction Network (AFEN), Boundary Prediction Network (BPN) and Attention-based Feature Fusion Module (AFFM). In the following subsections, we will elaborate these components in detail. Aggregating Feature Extraction Network -------------------------------------- ---------------------------------------------------- ----------------------------------------------------- {width="0.85\linewidth"}   {width="0.16\linewidth"}   ---------------------------------------------------- ----------------------------------------------------- \[sec:AmuletNet for initial prediction\] The residual networks (ResNet) [@He2016Deep] have shown excellent performances in many computer vision tasks. Our feature extraction network is based on the ResNet-101, which extracts multi-level feature maps from raw RGB images for feature integration and saliency prediction. We modify the ResNet-101 and reduce the resolution of features by a factor of 16 comparing to the input image, as shown in the Fig. \[diagram\]. The feature maps are extracted from specified convolutional layers. For the balance of resolution, we adopt the resolution-based feature combination structure (RFC) [@Zhang2017Amulet] to integrate multi-level convolutional features. The RFC structure unifies convolutional features through shrink and extend operations. More specifically, given an input image $\bf{I}$, the integrated feature maps of scale $\tau\in[1,..,5]$ are computed by $$\bf{F}^{\tau} = \it{C}_{m=1}^{\text{4}}( \it{R_{m}^{\tau}}(\bf{F}_{\it{m}}(I);\psi_{\it{m}})), \label{eq:original-bayes}$$ where $\it{R_{m}^{\tau}}({\cdot};\psi_{m})$ represents the reshape operator that expands or shrinks the feature maps by a factor of $\psi_{m}$. $\bf{F}_{\it{m}}$ denotes the $m$-level feature maps. $\it{C}$ is the concatenation operation in channel-wise. The resolution of generated feature maps $\bf{F^{\tau}}$ is $[\frac{W}{2^{\tau}},\frac{H}{2^{\tau}}]$. Besides, to enhance the feature interaction, we adopt the bidirectional information streams [@Zhang2017Amulet], which integrate multi-level features in both bottom-up and top-down directions. Although multi-level features have been extracted and fused in this effective way, there still exists a large gap between predicted saliency maps and ground truth. Due to the defects of down-sampling operations, the predicted saliency maps are blurred or occluded on boundary areas. Boundary Prediction Network {#Boundary Prediction Network} --------------------------- To resolve the blurry boundary problem, we introduce the BPNet, which generates boundary predictions to guide saliency prediction. To generate the boundary labels, we apply the open-source “Canny” algorithm [@Canny] to the binary saliency labels, which usually provide in public saliency benchmarks. An example is shown in Fig. \[boundary\]. The detailed structure of our BPNet is shown at the bottom of Fig. \[diagram\]. We adopt five convolutional blocks of the VGG-16 model [@simonyan2014very] to extract multi-scale boundary features. Given an input image, our BPNet first extracts five scale feature maps $\bf{B}^{\tau}_{\it{f}}$. To progressively merge multi-scale features and enlarge the boundary prediction map, we cascade several Residual Convolution Units (RCU) [@Lin2017RefineNet] on the side-output feature maps. In the scale $\tau$, BPNet generates boundary feature maps $\bf{B}^{\tau}$ by $$\begin{aligned} \bf{B}^{\tau}= \begin{cases} ((\bf{W}^{\tau}{\star}_{s}\bf{B}^{{\tau}+1})\oplus \it{RCU}(\bf{B}_{\it{f}}^{\tau})),\rm{1}\le\it{\tau}<\rm{5}\\ \it{RCU}(\bf{B}_{\it{f}}^{\tau}),\it{\tau}=\rm{5} \end{cases} \label{boundary feature map} \end{aligned}$$ where $\bf{B}^{\it{\tau}}$ and $\bf{B}_{\it{f}}^{\it{\tau}}$ represent boundary feature maps and corresponding convolutional feature maps respectively. ${\star}_{s}$ denotes the deconvolution with a stride $s$ to ensure the same resolution. $\oplus$ is the element-wise addition. Then we apply a $1\times1$ convolution operation on the boundary feature maps of each scale to generate the boundary prediction map $\bf{B}^{\tau}_{\it{p}}$. The comparison of boundary prediction maps is shown in Fig. \[boundary\]. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- {width="0.9\linewidth"}   --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \[boundary\] Attention-based Feature Fusion Module ------------------------------------- [|p[1.8cm]{}&lt;|p[1.3cm]{}&lt;|p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt; p[0.6cm]{}&lt;|]{} Dataset&Metric&Ours&Amulet&UCF&DHS&NLDF&RFCN&DS&DCL&ELD&LEGS&MDF&DRFI&BSCA\ &$\tt{F_{\beta}\uparrow}$&&0.867&0.839&0.872&&0.834&0.825&0.829&0.810&0.785&0.807&0.733&0.705\ &$\tt{MAE}\downarrow$&&&0.078&0.059&0.063&0.107&0.122&0.088&0.079&0.118&0.105&0.164&0.182\ &$\tt{F_{\beta}\uparrow}$&&0.669&0.613&/&&0.626&0.603&0.684&0.611&0.591&0.644&0.550&0.509\ &$\tt{MAE}\downarrow$&&0.090&0.132&/&&0.111&0.120&0.097&0.092&0.133&0.092&0.139&0.190\ &$\tt{F_{\beta}\uparrow}$&&0.709&0.629&0.724&&0.712&0.632&0.714&0.628&0.585&0.673&0.541&0.499\ &$\tt{MAE}\downarrow$&&0.080&0.117&0.067&&0.091&0.090&0.088&0.093&0.138&0.094&0.175&0.197\ &$\tt{F_{\beta}\uparrow}$&&0.861&0.808&0.855&&0.835&0.785&0.853&0.769&0.723&0.801&0.722&0.654\ &$\tt{MAE}\downarrow$&&0.053&0.074&0.053&&0.089&0.078&0.072&0.074&0.119&0.089&0.144&0.175\ \[table\] To efficiently utilize the boundary information and refine saliency maps, we introduce the attention-based feature fusion module (AFFM), which exploits multiple attention cues [@Chen2017SCA]. More specifically, we first reduce the channels of saliency features to the same size of boundary features by shrink dimension. Then we perform a global average pooling on the saliency and boundary features to obtain a feature vector $\textbf{v}=[v_{1},v_{2},...,v_{n}]$ ($n$ is the channel dimension of layers). A spatial softmax operator is applied on the feature vector to generate a normalized weight vector of the feature maps: $$\begin{aligned} {w_{i}}=\frac{e^{v_{i}}}{\sum_{i=1}^{n}e^{v_{i}}}, \label{boundary feature map} \end{aligned}$$ where $w_{i}$ is the weight of channel $i$ and $\sum_{i=1}^{n}w_{i}=1$. Subsequently, the fused feature maps $\bf{F}_{\text{fused}}^{\tau}$ is generated by $$\begin{aligned} \bf{F}_{\it{fused}}^{\tau}=(\bf{w}_{\it{F}}^{\tau}\otimes \bf{F}^{\tau}) \oplus (\bf{w}_{\it{B}}\otimes \bf{B}), \label{equation8} \end{aligned}$$ where $\otimes$ is channel-wise product. $\bf{B}$ denotes the boundary feature maps with the resolution of $256\times256$, which are in accordance with the aggregated saliency feature maps. $\bf{w}_{\it{F}}^{\tau}$ and $\bf{w}_{\it{B}}$ represents the attention weights for saliency features and boundary features respectively. With the fused feature, five paralleled fused prediction modules (FPM) (each of which is composed of a $3\times 3$ convolutional layer and an upsampling layer) are used to predict stage-wise prediction maps. With the stage-wise prediction maps, we add another convolutional layer with a $1\times1$ kernel to predict the final prediction map. Experiments {#sec:Experiments} =========== Training and Testing Datasets {#Dataset} ----------------------------- In the training process, the **DUTS-TR** [@Lijun2017Learning] dataset is chosen as our training dataset, which includes 10, 553 images with accurate pixel-wise annotations. We implement our proposed model based on the Caffe toolbox [@jia2014caffe]. We train and test our method with an NVIDIA 1080 GPU (with 8G memory). The input image is uniformly resized into $256\times256\times3$ pixels and subtracted the ImageNet mean [@deng2009imagenet]. We find our model with this resolution achieves both effectiveness and efficiency. We adopt the sigmoid cross entropy as the loss function for both saliency and boundary prediction. Following previous works [@Zhang2017Amulet; @Zhang2017Learning; @Zhang2018Salient], we train the model until its training loss converges. The weights of FCN backbones are initialized from the VGG-16 [@simonyan2014very] and ResNet-101 [@He2016Deep] models. For other layers, we initialize the weights by the “msra” method. We follow the parameters in [@Zhang2017Amulet] and use the standard SGD method with a batch size 8, momentum 0.9 and weight decay 0.0005. We set the learning rate to 1$\times$e$^{-8}$ and decrease it by 10% after every 10 epoch. Our model has a size of 410 MB and runs around 10 *fps* for saliency inference, which is comparable even faster than most of methods. We evaluate the performance of our method on four large-scale datasets described as follows. **ECSSD** [@Yang2013Saliency] is composed of 1000 images with random objects of different scales. **HKU-IS** [@Li2015Visual] includes 4447 images with fine pixel-wise annotations. Images of this dataset are well chosen to include multiple disconnected salient objects or objects touching the image boundary. **DUTS-TE** [@Lijun2017Learning] has 5019 images with accurate pixel-wise annotations. All images are picked from the ImageNet DET test set and the SUN dataset [@Xiao2010SUN]. **DUT-OMRON** [@Yang2013Saliency] has a total of 5168 high-quality images. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- {width="0.5\linewidth" height="2.6cm"} {width="0.5\linewidth" height="2.6cm"} {width="0.5\linewidth" height="2.6cm"} {width="0.5\linewidth" height="2.6cm"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- \[Fig pr\] Evaluation Metrics. ------------------- To evaluate the performance, we adopt three main metrics, *i.e.*, the PR curves, mean F-measure score and mean absolute error (MAE) [@Borji2015Salient]. Precision-recall (PR) curves can be computed by binarizing the saliency map with a threshold in \[0, 255\] and then comparing the binary maps with the ground truth. In many occasions, both precision and recall are important to measure methods. Therefore, F-measure, which is averaged with precision and recall, is proposed to achieve the overall performance evaluation, $$\begin{aligned} F_{\beta} =\frac{(1+\beta^2)\times Precision\times Recall}{\beta^2\times Precision \times Recall}.\end{aligned}$$ We set $\beta ^2$ to 0.3 to weigh precision more than recall as suggested in [@Achanta2009Frequency; @Borji2015What; @Wang2015Deep]. The above evaluations usually assign high saliency scores to salient pixels, which can be unfair especially for the methods which successfully detect non-salient regions, but miss the detection of salient regions. Therefore, we also adopt the MAE metric [@Borji2015Salient] to measure the average difference between the saliency prediction and the ground truth. $$MAE =\frac{1}{W\times H}\sum_{x=1}^{W}\sum_{y=1}^{H}|S(x,y) - G(x,y)|,$$ where $S$ is the predicted saliency map and $G$ is the binary ground truth mask. It indicates how similar a saliency map is compared to the ground truth. Comparison with Other Methods ----------------------------- ---------------------------------------------- {width="0.94\linewidth"} ---------------------------------------------- We compared our method with other 12 algorithms, including 10 deep learning based algorithms (Amulet[@Zhang2017Amulet], UCF[@Zhang2017Learning], NLDF[@Luo2017Non], DHS[@Liu2016DHSNet], DS[@Li2015DeepSaliency], DCL[@Li2016Deep], ELD[@Lee2016Deep], RFCN[@Wang2016Saliency], LEGS[@Wang2015Deep], MDF[@Li2015Visual]) and 2 conventional algorithms (DRFI[@Jiang2013Salient], BSCA[@Qin2015Saliency]). We compute saliency maps with the original implementations or use them provided by the authors. **Quantitative Results** As shown in Tab. \[table\] and Fig. \[Fig pr\], our model consistently outperforms other methods across all the datasets in terms of all evaluation metrics, which convincingly demonstrates the effectiveness of the proposed method. **Qualitative Results.** Fig. \[Fig saliency\] shows visual comparison between our method and other algorithms. As shown in the 1st row, the foreground is very complex, while our method successfully captures the main components against the competing algorithms. The object in 2nd row is an unusual roof. Many algorithms fail to capture the semantic structure, while our method successfully highlights it. Our proposed method also perform better on images with similar color distribution between foreground and background (3rd row). For disconnected objects (last two rows), our method still performs well, while other algorithms misjudge the interferences. Ablation Studies ---------------- [**[Effects of the boundary branch.]{}**]{} We verify the effectiveness of boundary branch in our framework. The compared models include: (1) With the same ResNet-101, we construct a baseline network, which is composed of encoder-decoder and feature combination structure [@Zhang2017Amulet]. The multi-level features are used to generate stage-wise prediction maps. The prediction map is generated by merging stage-wise prediction maps. (2) We add boundary-stream to the baseline network and use concatenation operations to combine features of both branches for saliency detection. The prediction scheme is similar with baseline network. Different from (1), features of each level incorporate both saliency and boundary information. We name this setting as $Boundary^{+}$. (3) To verify the effectiveness of cascaded RCUs, we implement the boundary-stream approach without RCUs, named $Boundary^{-}$ (4) Finally, AFFM is added to fuse saliency and boundary features to generate attentive features, resulting in our final model $AFFM^{+}$. We perform detailed experiments on two datasets, *i.e.*, ECSSD and DUT-OMRON. The results are shown in Tab. \[table1\]. From quantitative evaluations and qualitative comparisons, one can observe that our proposed components effectively enhance saliency detection performance. Especially, the boundary prediction module improves the MAE with 4%. Qualitative comparisons with different model settings are shown in Fig. \[Fig\_com\]. Compared with the baseline, our boundary-guided prediction is much better in predicting the boundary details. The proposed method indeed produces saliency map with sharp boundaries. The low MAE metric also verifies this fact. ----------------------------------------------------------------------------------------------------------------------------------------- ![Qualitative comparisons with different model settings.[]{data-label="Fig_com"}](duibi.pdf "fig:"){width="0.9\linewidth" height="3cm"} ----------------------------------------------------------------------------------------------------------------------------------------- \[table1\] **Effects of merging multi-scale predictions.** To verify the effects of the multi-scale FPM, we also perform a series experiments on ECSSD dataset. Firstly, we evaluate the results of a single FPM. Then we progressively add more FPMs to obtain the merged multi-scale results. The quantitative results are listed in Tab. \[table3\]. The “$12345$” is our final model. From the results, we can observe that adding more FPMs can integrate more information, thus improving the performances. \[table3\] Conclusion {#sec:Conclusion} ========== In this paper, we propose a boundary-guided aggregating feature fusion network for salient object detection. Different from the methods directly introduce high-level features into shallow layers, our method integrates feature maps into multiple resolutions. The proposed attention-based feature fusion module can effectively refine the results with clear boundaries.

--- abstract: 'We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with Diţă’s construction and generalizes Szöllősi’s method. We extend some known families and present new ones existing in even dimensions. In particular, we find more than 13 millon inequivalent affine families in dimension 32. We also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six and for any family of complex Hadamard matrices existing in every odd dimension.' author: - 'D. Goyeneche' title: | A new method to construct families\ of complex Hadamard matrices in even dimensions --- Keywords: Complex Hadamard matrices, Affine families, Mutually unbiased bases. Introduction ============ In recent years, the complex Hadamard matrices knowledge has exponentially increased. There are many applications to quantum information theory, e.g. they are useful to construct bases of unitary operators, bases of maximally entangled states and unitary depolarisers [@Werner]. Complex Hadamard matrices allow to solve the Mean King Problem [@Vaidman; @Englert; @Klappenecker], to construct error correcting codes [@Heng], to find quantum designs [@Zauner] and also to study spectral sets and Fuglede’s conjecture [@Tao; @Matolcsi; @Kolountzakis; @Kolountzakis2]. Furthermore, they are also useful for constructing some $*$-subalgebras in finite von Neumann algebras [@Popa; @Harpe; @Munemasa; @Haagerup], analyzing bi-unimodular sequences and finding cyclic $n$-roots [@Bjork; @Bjork2] and equiangular lines [@Godsil]. The existence of complex Hadamard matrices in every dimension is assured by the Fourier matrices. However, a complete characterization of inequivalent complex Hadamard matrices is known up to dimension five [@Haagerup]. The complexity of the problem suddenly increases in dimension six. This is not only due to the fact that six is not a prime power number, like what occurs in the mutually unbiased bases problem [@Wootters]. This issue remains open even in lower prime dimensions; e.g. it is open in dimension seven, where a one-parametric family and a few number of single complex Hadamard matrices are known. Also, we do not know if a continuous family exists in dimension eleven. A complete understanding of complex Hadamard matrices could help us to solve the Hadamard conjecture and the mutually unbiased bases problem in non-prime power dimensions. In this work, we present a new method of constructing affine families of complex Hadamard matrices. This method allows us to find families stemming from a particular subset of complex Hadamard matrices existing in even dimensions. This subset includes the Fourier and real Hadamard matrices. This work is organized as follows: In Section II we briefly introduce complex Hadamard matrices. In Section III we present our method to generate affine families in even dimensions. We prove that inequivalent families of complex Hadamard matrices stem from inequivalent real Hadamard matrices when our method is used. We also find two interesting restrictions: *(i)* for any set of four MU bases existing in dimension six and *(ii)* for any family of complex Hadamard matrices existing in every odd dimension. In Section IV we compare our method with existing constructions; i.e. we generalize Szöllősi’s method and we demonstrate our method intersects Diţă’s construction. In Section V we exemplify our method by constructing families stemming from the Fourier matrices in every even dimension. In Section VI we construct families stemming from real Hadamard matrices. We also extend a known family in dimension eight and two families in dimension twelve. Finally, in Section VII we summarize and conclude. Complex Hadamard matrices ========================= In this section, we briefly resume basic properties of complex Hadamard matrices. More detailed explanations can be found in the book of K. Horadam [@Horadam] or in the self contained paper of W. Tadej and K. Życzkowski [@Tadej2]. A square matrix $H$ of size $d$ is called a *complex Hadamard matrix* if its entries are unimodular complex numbers and it has orthogonal columns. The Fourier matrix defined by its entries $$(F_d)_{j,k}=e^{\frac{2\pi i}{d}jk},$$ where $i=\sqrt{-1}$ and $j,k=0,\dots,d-1$, is a privileged example because it represents the *only* construction existing in every dimension $d$ [@Haagerup]. On the other hand, real Hadamard matrices, that is, complex Hadamard matrices having real entries, can only exist in dimensions of the form $d=4k$, where $k=1/2$ or $k$ is an integer positive number. The *Hadamard conjecture* states that real Hadamard matrices exist in all such dimensions and it represents one of the most important open problems in Combinatorics. Currently, the smaller order where a real Hadamard matrix is still unknown is $d=4\times167=668$. Two complex Hadamard matrices are *equivalent* ($H_1\!\sim\! H_2$) if two diagonal matrices $D_1,D_2$ and two permutation matrices $P_1,P_2$ exist, such that $$H_2=D_1P_1H_1P_2D_2.$$ In dimensions two, three and five every complex Hadamard matrix is equivalent to the Fourier matrix and in dimension four a uniparametric family stems from the Fourier matrix. This is the complete characterization of complex Hadamard matrices in $d\leq5$ [@Haagerup]. In dimensions higher than five a complete classification remains open. Recently, it has been proven that several four dimensional families exist in dimension six [@Dita5]. Strangely enough, its expressions are very complicated and none of them can be explicitly written in a single page. A complex Hadamard matrix is *dephased* if every entry of the first row and every entry of the first column are equal to the unity. Given a complex Hadamard matrix $H_1$ it is possible to obtain $H_2\sim H_1$ such that $H_2$ is written in dephased form and, conversely, if $H_1$ and $H_2$ have the same dephased form then $H_2\sim H_1$. However, the dephased form is not unique and we cannot use it to characterize inequivalent complex Hadamard matrices. In the cases of $d=4$ and $d>5$ there exist continuous of inequivalent complex Hadamard matrices. This kind of sets is called a *family*. A family is *affine* [@Tadej2] if there exists a set $H(\mathcal{R})$ stemming from a dephased complex Hadamard matrix $H$, associated with a subspace $\mathcal{R}$ of the real space of $d\times d$ matrices with zeros in the first row and column such that $$H(\mathcal{R})=\{H\circ\exp(iR):R\in\mathcal{R}\}.$$ Here, $R\in\mathbb{R}^{d^2}$ contains $m$ free parameters and generates an $m$-dimensional subspace with basis $R_1\dots,R_m$. We characterize the family with the notation $H(\vec{\xi})$, where $\vec{\xi}$ is an $m$-dimensional real vector. That is $$\label{affine} H(\vec{\xi})=H(R(\vec{\xi}))=H\circ\exp(iR(\vec{\xi})),$$ where $R(\vec{\xi})=\sum_{i=1}^m\xi_iR_i$. The symbol $\circ$ denotes the Hadamard product $$(H_1\circ H_2)_{ij}=(H_1)_{ij}(H_2)_{ij},$$ while $\mathrm{Exp}$ denotes the entrywise exponential function $$(\mathrm{Exp}(H))_{ij}=\exp(H_{ij}).$$ We say an affine family is *maximal* if it is not contained in any larger affine family $H(R')$ stemming from $H$, where $\{R\}\subset \{R'\}$. If a family cannot be written in the form of Eq.(\[affine\]) we say it is *non-affine*. By the other hand, if a complex Hadamard matrix does not belong to a family it is *isolated*. For example, the spectral matrix $S_6$ and every Fourier matrix defined in prime dimensions are isolated [@Tadej]. The transpose of a family is still a family. We say that two families $H_1(\vec{\xi})$ and $H_2(\vec{\nu})$ are *cognate* if the families $H_2(\vec{\nu})$ and $H_1^t(\vec{\xi})$ are equivalent. Here, $t$ denotes matrix transposition. If a family and its transpose determine the same family we say it is *self-cognate*. For example, the one-parametric family $F_4^{(1)}$ is self-cognate. The problem of determining the maximal family stemming from a complex Hadamard matrix is open in dimensions higher than five. Even the problem to find the dimension of the maximal family is still open. The best approach in order to find this number is the *defect* of a complex Hadamard matrix [@Tadej], that is, the dimension of the solution space of the linear system $$\left\{ \begin{array}{rl} R_{0,j}=0,&\hspace{0.5cm}j\in\{1,\dots,d-1\},\\ R_{i,0}=0,&\hspace{0.5cm}i\in\{0,\dots,d-1\},\\ \sum_{k=0}^{d-1}H_{i,k}H_{j,k}^*(R_{i,k}-R_{j,k})=0,&\hspace{0.5cm}0\leq i<j\leq d-1, \end{array} \right.$$ where $R$ is a variable matrix. The defect $\mathbf{d}(H)$ is an upper bound of the dimension of the maximal family stemming from $H$. For example, $\mathbf{d}(H)=0$ implies that $H$ is an isolated matrix but the reciprocal implication is not valid. The defect has been analytically obtained for the Fourier matrices $F_d$ in every dimension [@Tadej]. As a particularly interesting case, it has been proven that $\mathbf{d}(F_d)=0$ when $d$ is prime. Consequently, the Fourier matrices in prime dimensions are isolated. Sometimes, this upper bound is not attained: $\mathbf{d}(F_4)=1$ but $\mathbf{d}(F_2\otimes F_2)=3$, and the maximal affine family existing in dimension four is one-dimensional only [@Haagerup]. Construction of affine families {#construction} =============================== A family of complex Hadamard matrices defines a continuous set of orthogonal bases in $\mathbb{C}^d$ when the parameters of the family are smoothly changed. These bases are given by the columns of such matrices which rotate in a very special way; i.e. orthogonality is preserved and also every entry of every column is restricted to be a unimodular complex number. In this section, we deal with a very particular kind of rotations. The main idea of this work comes from the following question: *Can we define a family of complex Hadamard matrices by introducing\ a parameter in two columns of a single complex Hadamard matrix?* As we will show, this question has only a positive answer for even dimensions. Let $\{\phi_k\}$ be an orthogonal base such that every vector $\phi_k$ defines a column of a $d\times d$ complex Hadamard matrix $H$. Our objective consists in finding two continuous vectors $\phi_a(\xi)$ and $\phi_b(\xi)$ such that: 1. *They are a linear combination of* $\phi_0$ *and* $\phi_1$. 2. *They have unimodular complex entries.* 3. *They are orthogonal.* 4. *The initial conditions* $\phi_a(0)=\phi_0$ *and* $\phi_b(0)=\phi_1$ *hold*. If these conditions are satisfied then the vectors $\{\phi_a(\xi),\phi_b(\xi),\phi_2,\dots,\phi_{d-1}\}$ define a family of complex Hadamard matrices. Let $\{\varphi_k\}$ be the canonical base and let $\{\phi_0,\phi_1\}$ be the first two columns of $H$. Without loosing the generality we can assume that $$\phi_0=\sum_{k=0}^{d-1}\varphi_k,$$ and $$\phi_1=\sum_{k=0}^{d-1}e^{i\alpha_k}\varphi_k,$$ for a given set of unimodular complex numbers $\{e^{i\alpha_k}\}$ restricted to the following condition $$\label{restr2} \sum_{k=0}^{d-1}e^{i\alpha_k}=0.$$ Therefore, proposing a linear combination of $\phi_0$ and $\phi_1$ (C.1) $$\begin{aligned} \label{phi_a} \phi_a(\xi)&=&x(\xi)\phi_0+y(\xi)\phi_1,\nonumber \\ &=&\sum_{k=0}^{d-1}(x(\xi)+y(\xi)e^{i\alpha_k})\varphi_k,\end{aligned}$$ and imposing unimodular entries in the last equation (C.2) we obtain $$\label{restr1} |x(\xi)+y(\xi)e^{i\alpha_k}|=1,$$ for every $k=0,\dots,d-1$. The coupled system of equations given by Eqs.(\[restr2\]) and (\[restr1\]) has a solution if and only if $d$ is an even number and $\phi_1$ is a real vector. In fact, expanding Eq.(\[restr1\]) we obtain $$\label{restr1_2} |x(\xi)|^2+|y(\xi)|^2+2\mathrm{Re}(x^*(\xi)y(\xi)e^{i\alpha_k})=1,$$ for every $k=0,\dots,d-1$. Given that $\phi_a(\xi)$ defined in Eq.(\[phi\_a\]) is normalized, that is $$\label{rest1_3} |x(\xi)|^2+|y(\xi)|^2=1,$$ we have $$\label{rest1_4} \mathrm{Re}(x^*(\xi)y(\xi)e^{i\alpha_k})=0,$$ for every $k=0,\dots,d-1$ and $\xi\in[0,2\pi)$. One way to write the most general solution of Eq.(\[rest1\_3\]) is $$x(\xi)=\cos(\xi)\hspace{0.3cm}\mbox{and}\hspace{0.3cm}y(\xi)=\sin(\xi)e^{i\beta(\xi)},$$ and imposing Eq.(\[rest1\_4\]) we obtain that $$\label{phase1} e^{i\beta(\xi)}e^{i\alpha_k}=(-1)^ki,$$ for every $\xi\in[0,2\pi)$. Therefore, $$\label{sol1} \phi_a(\xi)=\sum_{k=0}^{d-1}e^{i(-1)^k\xi}\varphi_k,$$ up to equivalence. The only pure state that is orthogonal to $\phi_a(\xi)$ (C.3) and it is also a linear combination of $\phi_0$ and $\phi_1$ (C.1) is given by $$\phi_b(\xi)=y(\xi)^*\phi_0-x(\xi)\phi_1.$$ From Eq.(\[phase1\]) without loosing the generality we can choose $$e^{i\beta(\xi)}=i\hspace{0.3cm}\mbox{and}\hspace{0.3cm}e^{i\alpha_k}=(-1)^k,$$ and up to a global sign we obtain $$\label{sol2} \phi_b(\xi)=\sum_{k=0}^{d-1}(-1)^ke^{i(-1)^k\xi}\varphi_k.$$ From the last equation we show that the entries of $\phi_b(\xi)$ are unimodular complex numbers (C.2) and also that the initial conditions imposed in (C.4) hold. Note that the only difference between $\{\phi_0,\phi_1\}$ and $\{\phi_a(\xi),\phi_b(\xi)\}$ is the exponential term $\{e^{i(-1)^k\xi}\}$ appearing in Eqs.(\[sol1\]) and (\[sol2\]). We highlight this is the most general solution up to equivalence. Another interesting consequence arises from Eqs.(\[sol1\]) and (\[sol2\]): *our construction only works in even dimensions*. Otherwise, the vectors $\phi_a(\xi)$ and $\phi_b(\xi)$ are not orthogonal. Before formalizing the above results in a theorem let us define a useful concept. Let $C_A$ and $C_B$ be two columns of a complex Hadamard matrix. We say they are an equivalent to real (ER) pair if $$(C_A^*)_j(C_B)_j=\pm1,$$ for every $j=0,\dots,d-1$. Here, $(C_A)_j$ and $(C_B)_j$ are the $j$th entries of $C_A$ and $C_B$, respectively. The asterisk denotes complex conjugation. For example, the Fourier matrix defined in every even dimension $d$ has $d/2$ ER pairs of columns. Indeed, the $k$th column of $F_d$ is given by $$(F_d)_k=\sum_{l=0}^{d-1}\omega^{lk}\varphi_l,$$ where $\omega=e^{2\pi i/d}$. The $d/2$ ER pairs of $F_d$ are determined by $$\label{ERpairs} \left\{(F_d)_k,(F_d)_{k+\frac{d}{2}}\right\},$$ where $k=0,\dots,\frac{d}{2}-1$. Given that $F_d$ is symmetric it also has $d/2$ ER pairs of rows. The maximal number of ER pairs of columns and rows are denoted by $\eta_c$ and $\eta_r$, respectively. These numbers coincide for the Fourier matrices but they differ in general. For example, the spectral matrix $S_8$ [@Matolcsi2] has $\eta_c=4$ and $\eta_r=0$. We formalize the results found in Eqs.(\[sol1\]) and (\[sol2\]) in the following theorem: \[prop1\] Let $H$ be a complex Hadamard matrix defined in an even dimension $d>2$. If $H$ has $m<d/2$ ER pairs then $H$ belongs to a $m$-dimensional family. If we consider $d/2$ ER pairs of columns one of the $d/2$ parameters generated by our method could be linearly dependent on the others. For example, this occurs for the family stemming from the Fourier matrix in every even dimension. Let us analyze in a separated subsection all consequences of Theorem \[prop1\]. Consequences of Theorem \[prop1\] --------------------------------- Theorem \[prop1\] is the main result of this paper and it has several consequences. We have collected all of them in this subsection in order to have a clear structure of our results. The following corollaries emerge from the above theorem \[nonisolated\] A complex Hadamard matrix having an ER pair of columns or rows is not isolated. \[oddcorol\] A parameter of a family of complex Hadamard matrices defined in every odd dimension cannot appear in only two columns or rows. The proof of these corollaries is trivial from Theorem \[prop1\]. Our intention here is to emphasize that *(i)* two columns of $H$ can contain enough information to affirm that a complex Hadamard matrix *is not* isolated and *(ii)* our construction forbids its extension to every odd dimension. Let us define a particularly interesting case of ER pairs. Let $\{C_1,C_2\}$ and $\{C_3,C_4\}$ be two ER pairs of columns. We say that they are aligned if $$(C_1^*)_k(C_2)_k=(C_3^*)_k(C_4)_k,$$ for every $k=0,\dots,d-1$. The maximal number of aligned pairs of columns and rows is called $\eta_{\bar{c}}$ and $\eta_{\bar{r}}$, respectively. Let us prove that the existence of ER pairs is invariant under equivalence. \[propequi\] Let $H$ and $\tilde{H}$ be two equivalent and dephased complex Hadamard matrices. Let $\{C_1,C_2\}$ be an ER pair of columns of $H$. Then, the corresponding pair $\{\tilde{C_1},\tilde{C_2}\}$ is an ER pair of columns of $\tilde{H}$. Proof: Let $\{C_1,C_2\}$ an ER pair of columns of $H$. Hence, up to equivalence this pair is given by $$C_1=\sum_{k=0}^{d-1}a_k\varphi_k,$$ and $$\label{C2} C_2=\sum_{k=0}^{d-1}(-1)^ka_k\varphi_k.$$ Therefore, we have $$\label{aligned1} (C_1)_k(C_2)_k=(-1)^k.$$ Let $\tilde{H}$ be a complex Hadamard matrix equivalent to $H$. Therefore, there exists unimodular complex numbers $c$ and $b_k$, and an injective function $f:\mathbb{Z}_d\rightarrow\mathbb{Z}_d$ such that $$\label{equi1} \tilde{C_1}=\sum_{k=0}^{d-1}c\,b_{k}\,a_{f(k)}\,\varphi_{k},$$ and $$\label{equi2} \tilde{C_2}=\sum_{k=0}^{d-1}(-1)^{f(0)}\,c\,b_{k} \,(-1)^{f(k)}a_{f(k)}\,\varphi_{k}.$$ The function $f(k)$ and the numbers $b_k$ are related to a permutation operator $P$ and a diagonal unitary operator $D$ applied to $H$, respectively. In order to dephase $\tilde{C_1}$ and $\tilde{C_2}$ we consider $$c=(b_{0}\,a_{f(0)})^*.$$ From Eqs.(\[equi1\]) and (\[equi2\]) we obtain $$\label{aligned2} (\tilde{C_1})^*_k(\tilde{C_2})_k=(-1)^{f(0)}\,(-1)^{f(k)}.$$ Therefore, $\tilde{C_1}$ and $\tilde{C_2}$ are an ER pair of columns.[$\Box$]{} We remark that we could consider $(-1)^{g(k)}$ instead of $(-1)^k$ in Eq.(\[C2\]), where ${g:\mathbb{Z}_d\rightarrow\mathbb{Z}_d}$ is an injective function. Nevertheless, our consideration is general up to equivalence. Let us present the main consequence of this proposition. Let $H$ be a dephased complex Hadamard matrix. Then, the numbers $\eta_c,\eta_r,\eta_{\bar{c}}$ and $\eta_{\bar{r}}$ are invariant under equivalence. Proof: The numbers $\eta_c$ and $\eta_r$ are trivially invariant from Proposition \[propequi\]. On the other hand, $\eta_{\bar{c}}$ is invariant due to the function $f$ appearing in Eq.(\[aligned2\]) which is the same for every ER pair of columns; analogously for $\eta_{\bar{r}}$.[$\Box$]{} In the case of the Fourier matrices we have $$\begin{aligned} \eta_c= \left\{ \begin{array}{c l} \frac{d}{2} &\mbox{if $d$ is even,}\\ 0 &\mbox{if $d$ is odd,} \end{array} \right.\end{aligned}$$ and the same values for $\eta_r,\eta_{\bar{c}}$ and $\eta_{\bar{r}}$. However, for real Hadamard matrices $\eta_c=\eta_r=d/2$ but, it is not possible to guess the functions $\eta_{\bar{c}}$ and $\eta_{\bar{r}}$ *a priori*. This is due to the existence of inequivalent real Hadamard matrices in dimensions $d\geq16$. Let us present some properties of ER pairs in dimension six. A $6\times6$ complex Hadamard matrix $H$ belongs to the family $F_6^{(2)}$ if and only if $\eta_c\neq0$ or $\eta_r\neq0$. Proof: If $H$ belongs to $F_6^{(2)}$ it is trivial to prove that it has 3 ER pairs of columns and rows. This is because the number of ER pairs is the same for any parameter of the family, as we can see from Eqs.(\[sol1\]) and (\[sol2\]). Therefore, $\eta_c=\eta_r=3$ for any member of the family $F_6^{(2)}$. Reciprocally, let us suppose that $H$ is written in a dephased form and $\eta_c\neq0$. Therefore, there exists $\tilde{H}\sim H$ such that it has a real ER pair of columns. Considering that every entry of $\tilde{H}$ is unimodular and every pair of columns is orthogonal we only obtain two free parameters. These parameters generate $F_6^{(2)}$. Analogously for $\eta_r\neq0$.[$\Box$]{} Given a pair of MU bases $\{\mathbb{I},F_6^{(2)}(a,b)\}$ it is not possible to find more than triplets of MU bases for any parameters $(a,b)$ [@Jaming]. From this fact and the last proposition an interesting consequence emerges. \[CorolMUB\] Suppose that four MU bases can be constructed in dimension six, namely $\{\mathbb{I},H_1,H_2,H_3\}$. Then, $\eta_c=\eta_r=0$ for $H_1,H_2$ and $H_3$. This corollary means that if such matrices $H_1,H_2,H_3$ exist, each of them is inequivalent to a complex Hadamard matrix having two real columns. This new result is one of the very few analytically known restrictions for the existence of four MU bases in dimension six. Let us consider ER pairs in dimension four. Here, the most general ER pair of columns can be written in the form $$\begin{aligned} C_1&=&(1,e^{ia},e^{ib},e^{ic}),\\ C_2&=&(1,-e^{ia},e^{ib},-e^{ic}),\end{aligned}$$ for any numbers $a,b,c\in[0,2\pi)$. The $-1$’s can appear in other entries but the above case is general up to equivalence. The inner product between these columns is given by $$\label{pm} \langle C_1,C_2\rangle=1-1+1-1=0.$$ As we can see, the contribution of every pair of entries to the inner product is $\pm1$; this is an exclusive property of ER pairs. Using Eqs.(\[sol1\]) and (\[sol2\]) we construct the continuous pair of ER vectors $$\begin{aligned} C_1(\xi)&=&(e^{i\xi},e^{i(a-\xi)},e^{i(b+\xi)},e^{i(c-\xi)}),\\ C_2(\xi)&=&(e^{i\xi},-e^{i(a-\xi)},e^{i(b+\xi)},-e^{i(c+\xi)}).\end{aligned}$$ Dephasing and considering $\xi\rightarrow-\xi/2$ we find that $$\begin{aligned} C_1(\xi)&=&(1,e^{i(a+\xi)},e^{ib},e^{i(c+\xi)}),\\ C_2(\xi)&=&(1,-e^{i(a+\xi)},e^{ib},-e^{i(c+\xi)})\label{C1C2}.\end{aligned}$$ Note that $\xi$ only appears in entries such that $(C_1)^*_j(C_2)_j=-1$. This is a *mnemonic technique* to construct families from our method. Let us present a proposition regarding families stemming from real Hadamard matrices. \[propHad\] Let $H_1$ and $H_2$ be two inequivalent real Hadamard matrices. Then, the families stemming from them by using our method are inequivalent. Proof: Let $H_1(\vec{\xi}\,)$ be a family stemming from the real Hadamard matrix $H_1$ and constructed from our method. Then, the only way to obtain a real Hadamard matrix into $H_1(\vec{\xi}\,)$ is by considering every entry of $\vec{\xi}$ in the set $\{0,\pi\}$. By the other hand, from Eqs.(\[sol1\]) and (\[sol2\]) it is easy to show that $H_1(\vec{\xi_1}\,)\sim H_1(\vec{\xi_2}\,)$ when $\vec{\xi}_1$ and $\vec{\xi}_2$ have every entry in the set $\{0,\pi\}$. Moreover, every vector contained in an ER pair is changed, at most, in a global sign under these considerations. Therefore, a real Hadamard matrix $H_2\not\sim H_1$ cannot be contained in the family $H_1(\xi\,)$.[$\Box$]{} Despite of Proposition \[propHad\], $H_1\sim H_2$ does not imply that the families stemming from these matrices are equivalent, as we can see in Diţă’s family $D^{(7)}_{12\Sigma}$ [@Dita4]. \[Corol13millon\] In dimension 32 we can generate 13,710,027 inequivalent affine families of complex Hadamard matrices. This corollary is a consequence of the existence of exactly 13,710,027 inequivalent real Hadamard matrices in dimension 32 [@Kharaghania]. Additionally, in dimensions $16, 20, 24$ and $28$ we can construct 5, 3, 60 and 487 inequivalent affine families, respectively. Let us present some general cases where our construction cannot be applied. Let $H$ be a dephased $d\times d$ complex Hadamard matrix such that one of the following conditions hold 1. The entries of $H$ are real in the main diagonal and non-real in other cases. 2. The entries of $H$ are power of roots of the unity $\omega^k$ with $k=0,\dots,d/2-1$, where $\omega=e^{\frac{2\pi i}{d}}$. 3. The entries of $H$ are power of roots of the unity $\omega=e^{\frac{2\pi i}{N}}$, where $N$ is an odd number. Then, our method defined in Theorem \[prop1\] cannot be applied. The proof is trivial because in these cases we do not have ER pairs.[$\Box$]{} For example, we cannot construct a family stemming from the matrices $C_6,D_6,D_{10}$ and $D_{14}$ (see the BTZ catalog [@Bruzda] for explicit expressions of these matrices). Also, this proposition tells us that we cannot construct a family stemming from the isolated matrix $S_6$ and from any isolated matrix recently found by McNulty and Weigert [@McNulty]. Szöllősi and Diţă methods {#Szol_Dita} ========================= In dimension four or higher than five several families of complex Hadamard matrices have been found. Some of them are specific constructions and cannot be extended to other dimensions. A method found by Szöllősi allows us to find a family of complex Hadamard matrices stemming from real Hadamard matrices. This method is stated in Lemma 3.4 [@Szollosi]: **Lemma 3.4** *(Szöllősi)Let $H$ be an arbitrary dephased complex Hadamard matrix of order $d\geq 4$. Suppose that $H$ has a pair of columns, say $u$ and $v$, with the following property: $u_i = v_i$ or $u_i + v_i = 0$ holds for every $i =0,\dots,d-1$. Then, $H$ admits an affine orbit.* This lemma is complemented by Theorem 3.5, which represents the main result of Szöllősi’s paper: **Theorem 3.5** *(Szöllősi)* *Let $H$ be a real Hadamard matrix of order $d\geq12$. Then, $H$ admits an $d/2+1$-parameter affine orbit.* We have noticed that Lemma 3.4 coincides with our Corollary \[nonisolated\]. It is important to realize that the main result found by Szöllősi is Theorem 3.5, which considers families stemming from real Hadamard matrices. This theorem provides the first proof that any real Hadamard matrix defined in $d\geq12$ is not isolated. However, Lemma 3.4 has not been tapped in all its generality. Coincidentally, our method is to do the natural generalization of Szöllősi’s idea to complex Hadamard matrices. The advantage of our method can be appreciated even for constructing families stemming from real Hadamard matrices. In these cases, we can apply our method to ER pairs of rows and columns simultaneously. Consequently, we are able to construct families stemming from real Hadamard matrices having more than $d/2+1$ independent parameters. For example, in dimension eight and twelve our method extends the families found by using Szöllősi’s method, as we will show further along in Section \[RHadamard\]. In order to construct a family of complex Hadamard matrices we can consider Diţă’s construction [@Dita]. This method can be applied to *Diţă type* complex Hadamard matrices: **Diţă type** *A complex Hadamard matrix $H$ of order $d=d_1d_2$ is called Diţă type if there exists complex Hadamard matrices $M$ of order $d_1$ and $N_1,\dots,N_{d_1}$ of order $d_2$ such that $H$ can be cast in the form* $$H= \left( \begin{array}{ccc} m_{11}N_1&\dots&m_{1d_1}N_{d_1}\\ \vdots&&\vdots\\ m_{d_11}N_1&\dots&m_{d_1d_1}N_{d_1} \end{array} \right).$$ The matrices $N_1,\dots,N_{d_1}$ are not necessarily different. **Diţă’s construction** *Let $H$ be a Diţă type complex Hadamard matrix. Then, the following affine family stems from $H$* $$H(\vec{\xi})= \left( \begin{array}{cccc} m_{11}N_1&m_{12}D_2N_2&\dots&m_{1d_1}D_{d_1}N_{d_1}\\ \vdots&&&\vdots\\ m_{d_11}N_1&m_{d_12}D_2N_2&\dots&m_{d_1d_1}D_{d_1}N_{d_1} \end{array} \right),$$ where $D_2,\dots,D_{d_1}$ are diagonal unitary matrices and each of them contains $d_2-1$ free parameters. The total number of free parameters in this construction is $(d_1-1)(d_2-1)+m+n_1+\dots+n_{d_1}$. Here, $m$ and $n_1,\dots,n_{d_1}$ denote the number of free parameters of $M$ and $N_1,\dots,N_{d_1}$, respectively. In the case of $N_1=N_2=\dots=N_{d_1}=N$ we have $H=M\otimes N$ and Diţă type matrices are reduced to Sylvester type [@Sylvester]. Let us show that our method intersects Diţă’s construction. \[Dita\_type\] Let $H$ be a dephased complex Hadamard matrix such that $\eta_{\bar{c}}=\eta_{\bar{r}}=d/2$. Then, $H$ is a Sylvester type. Proof: Suppose that there exists a complex Hadamard matrix such that $\eta_{\bar{c}}=\eta_{\bar{r}}=d/2$. Therefore, permuting rows and columns of $H$ we obtain the following equivalent matrix $$\tilde{H}= \left( \begin{array}{cc} A&B\\ A&-B \end{array} \right),$$ where $A$ and $B$ are $d/2\times d/2$ complex Hadamard matrices. Thus, our method is reduced to a particular case of Diţă’s construction $$\label{Dita_constr} H(\vec{\xi})= \left( \begin{array}{cc} A&D_1B\\ A&-D_1B \end{array} \right)$$ [$\Box$]{} We emphasize that it is not easy to identify Diţă type matrices but it is straightforward to identify ER pairs even in higher dimensions. Fourier families in even dimensions {#Fourier} =================================== In this section, we construct families of complex Hadamard matrices stemming from the Fourier matrices using our method. We analyze even $(d=2k)$ and doubly even $(d=4k)$ dimensions separately because different properties appear. We do not present new results in this section because the Fourier matrices have $\eta_{\bar{c}}=\eta_{\bar{r}}=d/2$ and then our method coincides with Diţă’s construction, as we have noticed in Proposition \[Dita\_type\]. We firstly analyze even dimensions. Fourier families in even dimensions {#fourier-families-in-even-dimensions} ----------------------------------- In dimension four, the Fourier matrix is given by $$\label{F4} F_4= \left( \begin{array}{lccc} 1&1&1&1\\ 1&i&-1&-i\\ 1&-1&1&-1\\ 1&-i&-1&i \end{array} \right).$$ Here we have $\eta_c=\eta_r=2$ but only one independent parameter can be found after dephasing the family. Let $(C_2)_j$ and $(C_4)_j$ be the $j$th entries of the second and the fourth column of $F_4$, respectively. According to our mnemic technique defined after Eq.(\[C1C2\]) we add a phase $e^{i\xi}$ in every entry of $C_1$ and $C_2$ if ${(C_2)_j}^*(C_4)_j=-1$. Consequently, we obtain the following family $$\label{FamilyF4} F_4^{(1)}(\xi)=F_4\circ\exp(iR_{F_4^{(1)}}(\xi)),$$ where, $$R_{F_4^{(1)}}(\xi)= \left( \begin{array}{lccc} \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\\ \bullet&\xi&\bullet&\xi\\ \bullet&\bullet&\bullet&\bullet\\ \bullet&\xi&\bullet&\xi \end{array} \right).$$ Here, the symbol $\bullet$ means zero and $\xi\in[0,2\pi)$. This family is self-cognate and it agrees with the only maximal family existing in dimension four [@Bruzda]. In the same way, we construct a 2-parametric family stemming from the Fourier matrix in dimension six by considering the ER pairs of columns $\{C_2,C_5\}$ and $\{C_3,C_6\}$. That is, $$\label{FamilyF61} F_6^{(2)}(a,b)=F_6\circ\exp(iR^{(2)}_{F_6}(a,b)),$$ and $$\label{FamilyF62} \left(F_6^{(2)}(a,b)\right)^t=F_6\circ\exp\left(i\left(R^{(2)}_{F_6}(a,b)\right)^t\right),$$ where $$R_{F_6^{(2)}}(a,b)= \left( \begin{array}{lccccc} \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\\ \bullet&a&b&\bullet&a&b\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&a&b&\bullet&a&b\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&a&b&\bullet&a&b \end{array} \right).$$ This is the only maximal affine family stemming from the Fourier matrix $F_6$. As we have shown, the Fourier matrices contain $d/2$ ER pairs of columns in every even dimension. Therefore, we can construct the following affine families $$\label{mainsol_a} F_d^{(d/2-1)}(\vec{\xi}\,)=F_d\circ\exp(iR^{(d/2-1)}(\vec{\xi}\,)),$$ and $$\label{mainsol_b} \left(F_d^{(d/2-1)}(\vec{\xi}\,)\right)^t=F_d\circ\exp\left(i\left(R^{(d/2-1)}(\vec{\xi}\,)\right)^t\right),$$ where $$\label{mainsol_c} \left(R^{(d/2-1)}(\vec{\xi}\,)\right)_{i,j}=\left\{ \begin{array}{l} \bullet\mbox{ when $i=0$ or even}\\ \bullet\mbox{ when $i$ is odd and }j=0\,\,\mathrm{mod}\,(d/2)\\ \xi_{[j-1]}\mbox{ otherwise.} \end{array} \right.$$ The above families are not self-cognate. We have noticed from the BTZ catalog [@Bruzda] that in the cases $d=2,4,6,10,14$ our construction agrees with the maximal affine Hadamard family stemming from $F_d$. This motivates us to establish the following conjecture: The maximal affine family of complex Hadamard matrices stemming from the Fourier matrix $F_d$ in dimensions $d=2p$ ($p$ prime) is given by Eqs.(\[mainsol\_a\]) to (\[mainsol\_c\]). Fourier families in doubly even dimensions {#d4k} ------------------------------------------ In the case of dimensions of the form $d=4k,\,k>1$ we can simultaneously apply Theorem \[prop1\] to rows and columns of the Fourier matrices. This procedure increases the number of parameters of the family beyond $d/2-1$. For example, we have obtained the following 5-parametric self-cognate family stemming from $F_8$ $$\label{F8} F_8^{(5)}(a,b,c,d,e)=F_8\circ\exp(iR_{F_8}(a,b,c,d,e)),$$ where $$\label{RF8} R_{F_8}(a,b,c,d,e)= \left( \begin{array}{lccccccc} \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}\\ \vspace{-0.1cm} \bullet&a+d&e&a&\bullet&a+d&e&a\\ \vspace{-0.1cm} \bullet&b&\bullet&b&\bullet&b&\bullet&b\\ \vspace{-0.1cm} \bullet&c+d&e&c&\bullet&c+d&e&c\\ \vspace{-0.1cm} \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \vspace{-0.1cm} \bullet&a+d&e&a&\bullet&a+d&e&a\\ \vspace{-0.1cm} \bullet&b&\bullet&b&\bullet&b&\bullet&b\\ \vspace{-0.1cm} \bullet&c+d&e&c&\bullet&c+d&e&c \end{array} \right).$$ The ER pairs here considered are $\{C_2,C_6\},\{C_3,C_7\},\{C_4,C_8\}$ and $\{R_2,R_6\},\{R_3,R_7\},\{R_4,R_8\}$ ($C=R=\{2,6;3,7;4,8\}$ to abbreviate). The ER pair $\{C_4,C_8\}$ produces a linearly dependent parameter and it has not been considered in Eq.(\[RF8\]). The 5-parametric family given in Eq.(\[F8\]) coincides with the maximal affine family stemming from $F_8$ [@Tadej2]. In the case of $d=12$, the following 9-parametric family can be found $$\label{FamilyF12} F_{12}^{(9)}(a,b,c,d,e,f,g,h,i)=F_{12}\circ\exp(iR_{F_{12}}(a,b,c,d,e,f,g,h,i)),$$ where $$R_{F_{12}}(a,b,c,d,e,f,g,h,i)= \left( \begin{array}{lccccccccccc} \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\\ \bullet&a+f&g&a+h&i&a&\bullet&a+f&g&a+h&i&a\\ \bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b\\ \bullet&c+f&g&c+h&i&c&\bullet&c+f&g&c+h&i&c\\ \bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d\\ \bullet&e+f&g&e+h&i&e&\bullet&e+f&g&e+h&i&e\\ \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\\ \bullet&a+f&g&a+h&i&a&\bullet&a+f&g&a+h&i&a\\ \bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b\\ \bullet&c+f&g&c+h&i&c&\bullet&c+f&g&c+h&i&c\\ \bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d\\ \bullet&e+f&g&e+h&i&e&\bullet&e+f&g&e+h&i&e \end{array} \right).$$ Here, we consider the ER pairs $C=R=\{2,8;3,9;4,10;5,11;6,12\}$, whereas $\{C_6,C_{12}\}$ produces a linearly dependent parameter. This result coincides with the family $F_{12A}^{(9)}$ [@Tadej2] which is self-cognate. A non-affine family stemming from $F_{12}$ has been recently found by Barros and Bengtsson [@Barros]. This family contains the affine families found by Tadej and Życzkowski [@Tadej2]. Finally, we obtain a 13-parametric family stemming from $F_{16}$. That is, $$F_{16}^{(13)}(a,b,c,d,e,f,g,h,i,j,k,l,m)=F_{16}\circ\exp(iR_{F_{16}}(a,b,c,d,e,f,g,h,i,j,k,l,m)),$$ where $R_{F_{16}}(a,b,c,d,e,f,g,h,i,j,k,l,m)$ is given by $$\left( \begin{array}{lccccccccccccccc} \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\\ \bullet&a+h&i&a+j&k&a+l&m&a&\bullet&a+h&i&a+j&k&a+l&m&a\\ \bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b\\ \bullet&c+h&i&c+j&k&c+l&m&c&\bullet&c+h&i&c+j&k&c+l&m&c\\ \bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d\\ \bullet&e+h&i&e+j&k&e+l&m&e&\bullet&e+h&i&e+j&k&e+l&m&e\\ \bullet&f&\bullet&f&\bullet&f&\bullet&f&\bullet&f&\bullet&f&\bullet&f&\bullet&f\\ \bullet&g+h&i&g+j&k&g+l&m&g&\bullet&g+h&i&g+j&k&g+l&m&g\\ \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\\ \bullet&a+h&i&a+j&k&a+l&m&a&\bullet&a+h&i&a+j&k&a+l&m&a\\ \bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b&\bullet&b\\ \bullet&c+h&i&c+j&k&c+l&m&c&\bullet&c+h&i&c+j&k&c+l&m&c\\ \bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d&\bullet&d\\ \bullet&e+h&i&e+j&k&e+l&m&e&\bullet&e+h&i&e+j&k&e+l&m&e\\ \bullet&f&\bullet&f&\bullet&f&\bullet&f&\bullet&f&\bullet&f&\bullet&f&\bullet&f\\ \bullet&g+h&i&g+j&k&g+l&m&g&\bullet&g+h&i&g+j&k&g+l&m&g \end{array} \right).$$ This family is self-cognate. The ER pairs here considered are $C=R=\{2,10;3,11;4,12;5,13;6,14;7,15\}$, whereas $\{C_8,C_{16}\}$ produces a linearly dependent parameter. We remark that the maximal affine family stemming from $F_{16}$ has 17 parameters [@Dita; @Tadej2]. In general, for $d=4k,\,k>1$ we can construct the following families using our method: $$\label{mainsol2_a} F_d(\vec{\alpha},\vec{\beta})=F_d\circ\exp(i(R_1(\vec{\alpha})+R_2(\vec{\beta}))),$$ and $$\label{mainsol2_b} \left(F_d(\vec{\alpha},\vec{\beta})\right)^T=F_d\circ\exp\left(i\left(R_1(\vec{\alpha})+R_2(\vec{\beta})\right)^T\right),$$ where $$\left(R_1(\vec{\alpha})\right)_{i,j}=\left\{ \begin{array}{l} \bullet\mbox{ when $i=0$ or even}\\ \bullet\mbox{ when $i$ is odd and }j=0\,\,\mathrm{mod}\,(d/2)\\ \alpha_{[j-1]}\mbox{ otherwise,} \end{array} \right.$$ and $$R_2(\vec{\beta})=R^T_1(\vec{\beta}).$$ Note that $\vec{\alpha},\vec{\beta}$ can be dependent and $d-2$ is an upper bound for the dimension of the family that we can obtain from $F_d$. We can define a lower bound for the maximal affine family stemming from a complex Hadamard matrix in even dimensions. Let $H$ be a complex Hadamard matrix and $d_{max}(H)$ the dimension of the maximal affine family stemming from $H$. Then, the following lower bound can be established $$\eta_{max}(H)\leq d_{max}(H).$$ where $\eta_{max}$ is the maximal number of linearly independent parameters of a family constructed by using our method. Let us analyze this bound in the case of the Fourier matrices in even dimensions. $d$ $\eta_{max}(F_d)$ $d_{max}(F_d)$ $d$ $\eta_{max}(F_d)$ $d_{max}(F_d)$ ----- ------------------- ---------------- ----- ------------------- ---------------- 2 0 0 10 4 4 4 1 1 12 9 9 6 2 2 14 6 6 8 5 5 16 13 17 As we can see in this table, up to $d=14$ every known affine family stemming from the Fourier matrix can be constructed by using our method. Real Hadamard matrices {#RHadamard} ====================== Fourier matrices have $d/2$ aligned ER pairs of rows and columns for every even dimension. However, for real Hadamard matrices the ER pairs are not necessarily aligned. Also, we have many non-equivalent ways to define the ER pairs. As we will show next, for $d\geq8$ the ER pairs should be intelligently chosen in order to maximize the dimension of a family. In $d=2$ we have an isolated real Hadamard matrix. For $d=4$ every real Hadamard matrix belongs to the one parametric family $F_4^{(1)}(\xi)$ presented in the previous section. Thus, our first interesting case is $d=8$. Here, every real Hadamard matrix is equivalent to $$H_8= \left( \begin{array}{rrrrrrrr} 1&1&1&1&1&1&1&1\\ 1&1&-1&1&-1&-1&1&-1\\ 1&1&1&-1&-1&-1&-1&1\\ 1&-1&1&1&-1&1&-1&-1\\ 1&-1&1&-1&1&-1&1&-1\\ 1&-1&-1&1&1&-1&-1&1\\ 1&1&-1&-1&1&1&-1&-1\\ 1&-1&-1&-1&-1&1&1&1 \end{array} \right).$$ In order to maximize a family obtained from $H_8$, we should choose the ER pairs of rows such that the number of ER pairs of columns is maximal. Let us explicitly construct the family. Applying our method to the rows of $H_8$ and considering $\eta_{\bar{r}}=4$ we obtain, as a first step, the 4-parametric family $$\label{H8} H_8(a,b,c,d)=H_8\circ\exp(iR_{H_8}(a,b,c,d)),$$ where $$\label{RH8} R_{H_8}(a,b,c,d)= \left( \begin{array}{llllllll} \bullet\hspace{0.2cm}&a\hspace{0.2cm}&\bullet\hspace{0.2cm}&a\hspace{0.2cm}&\bullet\hspace{0.2cm}&a\hspace{0.2cm}&\bullet\hspace{0.2cm}&a\\ \bullet&b&\bullet&b&\bullet&b&\bullet&b\\ \bullet&c&\bullet&c&\bullet&c&\bullet&c\\ \bullet&c&\bullet&c&\bullet&c&\bullet&c\\ \bullet&a&\bullet&a&\bullet&a&\bullet&a\\ \bullet&d&\bullet&d&\bullet&d&\bullet&d\\ \bullet&d&\bullet&d&\bullet&d&\bullet&d\\ \bullet&b&\bullet&b&\bullet&b&\bullet&b \end{array} \right).$$ Therefore, we have $9$ inequivalent ways to choose the ER pairs of columns. They are: $$\begin{aligned} C_A&=&\{1,3;5,7;2,4;6,8\},\\ C_B&=&\{1,3;5,7;2,6;4,8\},\\ C_C&=&\{1,3;5,7;2,8;4,6\},\\ C_D&=&\{1,5;3,7;2,4;6,8\},\\ C_E&=&\{1,5;3,7;2,6;4,8\},\\ C_F&=&\{1,5;3,7;2,8;4,6\},\\ C_G&=&\{1,7;3,5;2,4;6,8\},\\ C_H&=&\{1,7;3,5;2,6;4,8\},\\ C_I&=&\{1,7;3,5;2,8;4,6\}.\label{choices}\end{aligned}$$ Let us analyze the case of $C_A$. That is, $$H^{(8)}_{8A}(a,b,c,d,e)=H_8\circ\exp(iR_{H_{8A}}(a,b,c,d,e)),$$ where $$R_{H_{8A}}(a,b,c,d,e)= \left( \begin{array}{lccccccc} \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}\\ \bullet&a+b+d&\bullet&a+b+d&d&a+b+d&d&a+b+d\\ \bullet&a+c+e&\bullet&a+c+e&\bullet&a+c&\bullet&a+c\\ \bullet&a+c+e&\bullet&a+c+e&\bullet&a+c&\bullet&a+c\\ \bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet&\bullet\\ \bullet&a+d+e&\bullet&a+d+e&d&a+d&d&a+d\\ \bullet&a+d+e&\bullet&a+d+e&d&a+d&d&a+d\\ \bullet&a+b+d&\bullet&a+b+d&d&a+b+d&d&a+b+d \end{array} \right).$$ We can considerer $7!!=7\times5\times3\times1=105$ different choices for the ER pairs of rows. The above 9 cases consider only one of these choices. In principle, we do not know how many of these families are inequivalent. Using Diţă’s construction there were found 9 inequivalent 5-parametric families stemming from $H_8$ [@Dita2], which are a particular subset of our solutions here presented. This is very easy to show from Proposition \[Dita\_type\]. In Section \[Szol\_Dita\], we already mentioned that our method generalizes Szöllősi’s method even for real Hadamard matrices. Let us present an example where this is clearly showed. From the real Hadamard matrix $$\label{H12} H_{12}= \left( \begin{array}{rrrrrrrrrrrr} 1&1&1&1&1&1&1&1&1&1&1&1\\ \vspace{0.1cm} 1&1&1&1&1&1&-1&-1&-1&-1&-1&-1\\ \vspace{0.1cm} 1&-1&1&1&-1&-1&1&1&-1&1&-1&-1\\ \vspace{0.1cm} 1&-1&1&-1&1&-1&-1&1&1&-1&1&-1\\ \vspace{0.1cm} 1&-1&-1&1&-1&1&-1&1&1&-1&-1&1\\ \vspace{0.1cm} 1&1&1&-1&-1&-1&1&-1&1&-1&-1&1\\ \vspace{0.1cm} 1&1&-1&-1&1&-1&-1&1&-1&1&-1&1\\ \vspace{0.1cm} 1&1&-1&1&-1&-1&-1&-1&1&1&1&-1\\ \vspace{0.1cm} 1&-1&1&-1&-1&1&-1&-1&-1&1&1&1\\ \vspace{0.1cm} 1&1&-1&-1&-1&1&1&1&-1&-1&1&-1\\ \vspace{0.1cm} 1&-1&-1&-1&1&1&1&-1&1&1&-1&-1\\ \vspace{0.1cm} 1&-1&-1&1&1&-1&1&-1&-1&-1&1&1 \end{array} \right),$$ we find an 8-parametric family from considering the ER pairs given by $C=\{1,6;2,3\}$ and $R=\{1,5;2,11;3,7;4,8;6,12;9,10\}$. That is, $$\label{FamilyF12R} H^{(8)}_{12}(a,b,c,d,e,f,g,h)=H_{12}\circ\exp(iR_{H_{12}}(a,b,c,d,e,f,g,h)),$$ where $R_{H_{12}}(a,b,c,d,e,f,g,h)$ is given by $$\label{FamilyF12R2} \left( \begin{array}{lccccccccccc} \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}\\ \bullet&a+b&a+b&b&a&\bullet&a+b&\bullet&b&a+b&a&\bullet\\ \bullet&a+c\atop+g+h&a+c\atop+g+h&c+h&a+c\atop+h&\bullet&a+c\atop+h&h&h&a+h&a+h&c+h\\ \bullet&a+d\atop+g+h&a+d\atop+g+h&d+h&a+d\atop+h&\bullet&a+h&d+h&h&a+d\atop+h&a+h&h\\ \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}\\ \bullet&a+e\atop+h&a+e\atop+h&e+h&a+e\atop+h&\bullet&a+h&h&e+h&a+h&a+e\atop+h&h\\ \bullet&a+c\atop+g+h&a+c\atop+g+h&c+h&a+c\atop+h&\bullet&a+c\atop+h&h&h&a+h&a+h&c+h\\ \bullet&a+d\atop+g+h&a+d\atop+g+h&d+h&a+d\atop+h&\bullet&a+h&d+h&h&a+d\atop+h&a+h&h\\ \bullet&a+f\atop+g&a+f\atop+g&\bullet&a&\bullet&a+f&f&\bullet&a+f&a&f\\ \bullet&a+f\atop+g&a+f\atop+g&\bullet&a&\bullet&a+f&f&\bullet&a+f&a&f\\ \bullet&a+b&a+b&b&a&\bullet&a+b&\bullet&b&a+b&a&\bullet\\ \bullet&a+e\atop+h&a+e\atop+h&e+h&a+e\atop+h&\bullet&a+h&h&e+h&a+h&a+e+h&h \end{array} \right).$$ We have verified that all the parameters are linearly independent. Considering different combinations of ER pairs of rows and columns we can generate $12!!=46,080$ families but we do not know how many of them are inequivalent. Our family $H^{(8)}_{12}$ is a new result that extends the family $H^{(7)}_{12}$ found by using Szöllősi’s method [@Szollosi]. We have noted that our method can be applied to other complex Hadamard matrices apart from the real and the Fourier matrices. For example, from $H^{(\omega)}_{10}$ [@Dita4] we found the following 5-parametric family $$\label{Hw} H^{(\omega)}_{10}(a,b,c,d,e)=H^{(\omega)}_{10}\circ\exp(iR_{H^{(\omega)}_{10}}(a,b,c,d,e)),$$ where $$H^{(\omega)}_{10}= \left( \begin{array}{rrrrrrrrrr} 1&1&1&1&1&1&1&1&1&1\\ \vspace{-0.1cm} 1&1&1&\omega&\omega^2&-1&-1&1&\omega^2&\omega\\ \vspace{-0.1cm} 1&1&1&\omega^2&\omega&-1&1&-1&\omega&\omega^2\\ \vspace{-0.1cm} 1&\omega&\omega^2&1&1&-1&\omega^2&\omega&-1&1\\ \vspace{-0.1cm} 1&\omega^2&\omega&1&1&-1&\omega&\omega^2&1&-1\\ \vspace{-0.1cm} 1&\omega^2&\omega&1&-1&1&-\omega&-\omega^2&-1&-1\\ \vspace{-0.1cm} 1&-1&1&\omega&\omega^2&1&-1&-1&-\omega^2&-\omega\\ \vspace{-0.1cm} 1&1&-1&\omega^2&\omega&1&-1&-1&-\omega&-\omega^2\\ \vspace{-0.1cm} 1&\omega&\omega^2&-1&1&1&-\omega^2&-\omega&-1&-1\\ \vspace{-0.1cm} 1&-1&-1&-1&-1&-1&1&1&1&1 \end{array} \right),$$ $\omega^2+\omega+1=0$ and $$\label{Romega10} R_{H^{(\omega)}_{10}}(a,b,c,d,e)= \left( \begin{array}{lccccccccc} \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}\\ \bullet&a+b&a&a&a&a+b&\bullet&b&b&b\\ \bullet&a&a+c&a&a&a+c&c&\bullet&c&c\\ \bullet&a&a&a+d&a&a+d&d&d&\bullet&d\\ \bullet&a&a&a&a+e&a+e&e&e&e&\bullet\\ \bullet&a&a&a&a+e&a+e&e&e&e&\bullet\\ \bullet&a+b&a&a&a&a+b&\bullet&b&b&b\\ \bullet&a&a+c&a&a&a+c&c&\bullet&c&c\\ \bullet&a&a&a+d&a&a+d&d&d&\bullet&d\\ \bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm}&\bullet\hspace{0.2cm} \end{array} \right).$$ In this case, we have $\eta_c=5$ and $\eta_r=5$ but they cannot be simultaneously considered in order to obtain more free parameters. It was proven that a 7-parametric family stems from $H^{(\omega)}_{10}$ [@Dita4], and Eq.(\[Hw\]) represents a subset of this family. We have also extended the 7 parametric family $D^{(7)}_{12\Sigma}$ [@Dita3] found by using Diţă’s construction. This extension is obtained by considering the ER pair given by the first and the last row in every subfamily of $D^{(7)}_{12\Sigma}$. This is straightforwardly obtained by inspecting of the family [@Dita3]. We have proven that the new parameter is linearly independent to the rest of the parameters after dephasing the family. We have omitted details here to abbreviate but the twenty subfamilies $D^{(7)}_{12A}$ to $D^{(7)}_{12R}$ are generalized in the same straightforward way. As a last result, we show that the single matrix $D_{12}$ presented by Diţă [@Dita4] $$D_{12}= \left( \begin{array}{rrrrrrrrrrrr} 1&1&1&1&1&1&1&1&1&1&1&1\\ 1&i&i&i&-i&-i&-i&-1&1&1&-1&-1\\ 1&i&i&-i&i&-i&-i&1&-1&-1&1&-1\\ 1&i&-i&i&-i&i&-i&1&-1&-1&-1&1\\ 1&-i&i&-i&i&i&-i&-1&1&-1&-1&1\\ 1&-i&-i&i&i&i&-i&-1&-1&1&1&-1\\ 1&-i&-i&i&i&-i&i&1&1&-1&-1&-1\\ 1&-i&i&i&-i&-i&i&-1&-1&-1&1&1\\ 1&i&-i&-i&i&-i&i&-1&-1&1&-1&1\\ 1&i&-i&-i&-i&i&i&-1&1&-1&1&-1\\ 1&-i&i&-i&-i&i&i&1&-1&1&-1&-1\\ 1&-1&-1&-1&-1&-1&-1&1&1&1&1&1 \end{array} \right),$$ belongs to the intersection of our extension of $D^{(7)}_{12\Sigma}$, namely $D^{(8)}_{12\Sigma}$. That is, $$\label{D12} D_{12}\in\bigcap_{\Gamma=A}^R D^{(8)}_{12\Gamma}.$$ Indeed, every subfamily of $D^{(7)}_{12\Sigma}$ stems from a real Hadamard matrix equivalent to $H_{12}$. All these equivalent real matrices have the same first and last row as $H_{12}$. In order to obtain $D_{12}$ we start by multiplying from the second to the seventh column of $H_{12}$ by $i$ times. Thus, we introduce a parameter $\xi$ by applying our method to the ER pair $\{R_1,R_{12}\}$. In the case of $\xi=-\pi/2$ we obtain $D_{12}$. Analogously for every subfamily $D^{(8)}_{12\Gamma},\,\Gamma=A,\dots,R$. Summary and conclusion ====================== We presented a new method to construct families of complex Hadamard matrices in every even dimension $d>2$ by introducing the concept of *ER pairs*. Let us summarize our results: *(i)* Using our method we have reproduced some previously known results: *We found maximal affine families stemming from the Fourier matrix in $d=4$ (Eq.(\[FamilyF4\])), $6$ (Eqs.(\[FamilyF61\]) and (\[FamilyF62\])), $8$ (Eq.(\[F8\])) and $12$ (Eq.(\[FamilyF12\])). Also, we found families stemming from the Fourier matrix in every even (Eqs.(\[mainsol\_a\]) and (\[mainsol\_b\])) and double even (Eqs.(\[mainsol2\_a\]) and (\[mainsol2\_b\])) dimension.*\ *(ii)* Although our method is defined in even dimensions, we also found a restriction on the distribution of a parameter in families existing in every odd dimension: *Families of complex Hadamard matrices defined in every odd dimension cannot contain a parameter appearing in only two columns or rows (See Corollary \[oddcorol\]).*\ *(iii)* We have generalized Szöllősi’s method for constructing affine families: *Our method increases the number of free independent parameters that can be obtained by using Szöllősi’s method in every even dimension $d\geq12$. (See the beginning of Section \[Szol\_Dita\]).*\ *(iv)* Our method has an intersection with Diţă’s construction: *If a $d\times d$ complex Hadamard matrix has $d/2$ aligned ER pairs of columns or rows then it is Diţă type. (See Proposition \[Dita\_type\]).*\ *(v)* We have constructed several families stemming from $H_8$ in dimension eight: *We found $9\times105=945$ different ways to construct a 5-dimensional family stemming from the real Hadamard matrix $H_{8}$* (See Eq.(\[H8\]) to Eq.(\[choices\])).\ *(vi)* We have extended two families in dimension twelve: *We found 46,080 different ways to generalize the family $H_{12}^{(7)}$ obtained from Szöllősi’s method* (See Eqs. (\[FamilyF12R\])-(\[FamilyF12R2\]) and the paragraph afterwards).\ *The family $D_{12\Sigma}^{(7)}$ obtained from Diţă’s construction was extended to $D_{12\Sigma}^{(8)}$ (See paragraph after Eq.(\[Romega10\])). Also, we have proven that the single matrix $D_{12}$ presented by Diţă belongs to every subfamily of $D_{12\Sigma}^{(8)}$* (See Eq.(\[D12\])).\ *(vii)* We have established a connection between the mutually unbiased (MU) bases problem in dimension six and the ER pairs: *Let $\{\mathbb{I},H_1,H_2,H_3\}$ be a set of four MU bases existing in dimension six. Then, $H_1,H_2$ and $H_3$ do *not* have ER pairs* (See Corollary \[CorolMUB\]).\ And finally, *(viii)* We generated inequivalent affine families stemming from inequivalent real Hadamard matrices. For example: *In dimensions 16, 20, 24, 28 and 32 we can construct 5, 3, 60, 487 and more than 13 millon inequivalent families, respectively* (See Proposition \[propHad\] and Corollary \[Corol13millon\]).\ Our method to construct families considers parameters appearing in pairs of columns or rows of complex Hadamard matrices. This assumption allowed us to construct many families of complex Hadamard matrices in a very easy way. However, our method is not general because the parameters can appear in more than two columns. In fact, several families existing in even dimensions and all families existing in every odd dimension cannot be constructed from our method. This naturally suggests to us to try to generalize the concept of ER pairs. Nevertheless, a general extension to three or four columns seems not easy. We have noted that parameters appearing in three columns can only be shown in the following families of the BTZ catalog: $F^{(2)}_6$, $F^{(4)}_9$, $S^{(5)}_{12}$. This evidence strongly suggests that parameters appearing in exactly three columns are only possible in dimensions of the form $d=3k$. Very interestingly, parameters appearing in four columns are not restricted to doubly even dimensions, as we can see in Petrescu’s family $P^{(1)}_7$. Therefore, a generalization to four columns could lead us to a construction of affine families in every dimension $d\geq6$. We have solved the problem of finding the maximal affine family stemming from a complex Hadamard matrix when the parameters appear in exactly two columns or rows. We hope this method and its generalization to be a useful tool to try to understand the general structure of affine families of complex Hadamard matrices existing in every dimension. Acknowledgments =============== I specially thank to W. Tadej, I. Bengtsson, K. Życzkowski, S. Weigert, P. Diţă, A. Delgado, F. Szöllősi, B. Karlsson and M. Matolcsi for their invaluable comments. Also, I would like to thank to the referee for his many useful comments in order to improve this article. This work is supported by Grants FONDECyT N$^{\text{\underline{o}}}$ 3120066 and MSI P010-30F. [99]{} R. Werner. All teleportation and dense coding schemes, J. Phys. A, 34, (2001), 7081-7094 L. Vaidman, Y. Aharonov and D. Z. Albert, Phys. Rev. Lett 58 1385 (1987). B.-G. Englert and Y. 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P. de la Harpe and V.R.F. Jones, Paires de sous-algebres semi-simples et graphes fortement reguliers, C.R. Acad. Sci. Paris 311, 147-150 (1990). A. Munemasa and Y. Watatani, Orthogonal pairs of $*$-subalgebras and association schemes, C.R. Acad. Sci. Paris 314, 329-331 (1992). U. Haagerup, Ortogonal maximal Abelian $*$-subalgebras of $n\times n$ matrices and cyclic $n$-roots, Operator Algebras and Quantum Field Theory (Rome), Cambridge, MA International Press, (1996), 296-322. G. Björk and R. Fröberg, A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic $n$-roots, J. Symbolic Comp. 12, 329-336 (1991). G. Björck and B. Saffari, New classes of finite unimodular sequences with unimodular Fourier transform. Circulant Hadamard matrices with complex entries, C. R. Acad. Sci., Paris 320 319-24 (1995). C. D. Godsil and A. Roy, Equiangular lines, mutually unbiased bases, and spin models preprint quant-ph/0511004 (2005). W. 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--- author: - | Krishnendu Chatterjee$^\dag$ Laurent Doyen$^{\S}$\ $\strut^\dag$ IST Austria $\strut^\S$ CNRS & LSV, ENS Paris-Saclay, France bibliography: - 'biblio.bib' title: '[**Graph Planning with Expected Finite Horizon**]{}' ---

--- abstract: 'We consider the transport equation driven by the fractional Brownian motion. We study the existence and the uniqueness of the weak solution and, by using the tools of the Malliavin calculus, we prove the existence of the density of the solution and we give Gaussian estimates from above and from below for this density.' author: - | Christian Olivera $^{1,}$[^1] 0.2cm Ciprian A. Tudor $^{2,3,}$ [^2]\ $^{1}$ Departamento de Matemática, Universidade Estadual de Campinas,\ 13.081-970-Campinas-SP-Brazil.\ colivera@ime.unicamp.br\ $^{2}$ Laboratoire Paul Painlevé, Université de Lille 1\ F-59655 Villeneuve d’Ascq, France.\ $^{3}$Academy of Economical Studies, Bucharest, Romania\ tudor@math.univ-lille1.fr title: The density of the solution to the stochastic transport equation with fractional noise --- [*MSC 2010*]{}: Primary 60F05: Secondary 60H05, 91G70. [*Key Words and Phrases*]{}: transport equation, fractional Brownian motion, Malliavin calculus, method of characteristics, existence and estimates of the density. Introduction ============ The purpose of this paper is to study the probability law of the real-valued solution of the following stochastic partial differential equations $$\label{trasportS} \left \{ \begin{array}{lll} du(t, x) + b(t, x) \nabla u(t, x) \ dt + \nabla u(t,x) \circ d B^{H}_{t} + F(t,u) \ dt =0,\\ u(0, x) = u_{0}(x), \end{array} \right .$$ where $B^{H}_{t}=(B^{H_{1}}_{t},...,B^{H_{d}}_{t})$ is a fractional Brownian motion (fBm) in $\mathbb{R}^{d}$ with Hurst parameter $H=(H_{1},..., H_{d}) \in \left[ \frac{1}{2}, 1\right) ^{d}$ and the stochastic integration is understood in the symmetric (Stratonovich) sense. The equation (\[trasportS\]) is usually called the stochastic transport equation and arises as a prototype model in a wide variety of phenomena. Although we introduced (\[trasportS\]) in a general form, we mention that some results will be obtained in dimension one. The stochastic transport equation with standard Brownian noise has been first studied in the celebrated works by Kunita [@Ku3], [@Ku2] and more recently it has been the object of study for many authors. We refer, among many others, to [@CO], [@Fre1], [@FGP2], [@Maurelli], [@MNP], [@Oli]. Our aim is to analyze the stochastic partial equation (\[trasportS\]) when the driving noise is the fractional Brownian motion, including the particular case of the Brownian motion. We will first give, by interpreting the stochastic integral in (\[trasportS\]) as a symmetric integral via regularization in the Russo-Vallois sense [@RV], an existence and uniqueness result for the weak solution to (\[trasportS\]) via the so-called method of characteristics and we express the solution as the initial value applied to the inverse flow generated by the equation of characteristics. This holds, when $H_{i}=\frac{1}{2}, i=1,..,d$ for any dimension $d$ and in dimension $d=1$ if the Hurst parameter is bigger than one half. Using this representation of the solution to (\[trasportS\]), we study the existence and the Gaussian estimates for its density via the analysis of the dynamic of the inverse flow. A classical tools to study the absolute continuity of the law of random variables with respect to the Lebesque measure is the Malliavin calculus. We refer to the monographs [@N] or [@Sans] for various applications of the Malliavin calculus to the existence and smoothness of the density of random variables in general, and of solutions to stochastic equations in particular. We will prove the Malliavin differentiability of the solution to (\[trasport\]) by analyzing the dynamic of the inverse flow generated by the characteristics (\[1\]). Using a result in [@NV] we obtain, in dimension $d=1$ upper and lower Gaussian bounds for the density of the solution to the transport equation. We are also able to find the explicit form of the density in dimension $d\geq 2$ when the driving noise is the standard Brownian motion and the drift is divergence-free (i.e. the divergence of the drift vanishes). We organized our paper as follows. In Section 2 we recall the existence and uniqueness results for the solution to the transport equation driven by the standard Brownian motion. In Section 3, we analyze the weak solution to the transport equation when the noise is the fBm, via the method of characteristics. In Section 4 we study the Malliavin differentiability of the solution to the equation of characteristics and this will be applied in Section 4 to obtain the existence and the Gaussian estimates for the solution to the transport equation. In Section 6 we obtain an explicit formula for the density when the noise is the Wiener process and the drift is divergence-free. Stochastic transport equation driven by standard Brownian motion ================================================================ Throughout the paper, we will fix a probability space $(\Omega, \mathcal{F}, P)$ and a $d$-dimensional Wiener process $(B_{t}) _{t \in [0, T]}$ on this probability space. We will denote by $(\mathcal {F}_{t}) _{t\in [0,T]}$ the filtration generated by $B$. We will start by recalling some known facts on the solution to the transport equation driven by a standard Wiener process in $\mathbb{R} ^{d}$. The equation (\[trasportS\]) is interpreted in the strong sense, as the following stochastic integral equation $$u(t,x)=u_{0}(x)-\int_{0}^{t}b(s,x)\nabla u(s,x)\ ds -\sum_{i=0}^{d}\int_{0}^{t} \partial_{x_i} u(s,x)\circ dB_{s}^{i} \label{transintegral} - \int_{0}^{t} F(t,u) \ ds$$ for $t\in [0,T]$ and $x\in \mathbb{R} ^{d}$. The solution to (\[trasportS\]) is related with the so-called equation of characteristics. That is, for $0\leq s\leq t$ and $x\in\mathbb{R}^{d}$, consider the following stochastic differential equation in $\mathbb{R}^{d}$ $$\label{11}X_{s,t}(x)= x + \int_{s}^{t} b(r, X_{s,r}(x)) \ dr + B_{t}-B_{s}.$$ and denote by $X_{t}(x): = X_{0,t}(x), t\in [0,T], x\in \mathbb{R} ^{d}$. For $m \in {\mathbb{N}}$ and $0< \alpha < 1$, let us assume the following hypothesis on $b$: $$\label{REGULCLASS} b\in L^{1}((0,T); C_{b}^{m,\alpha}(\mathbb{R}^{d}))$$ where $C^{m,\alpha}(\mathbb{R}^{d})$ denotes the class of functions of class $C ^{m}$ on $\mathbb{R}^{d}$ such that the last derivative is Hölder continuous of order $\alpha$. Let us recall the definition of the stochastic flow (see e.g. [@Ku]). A stochastic flow is a family of maps $(\Phi _{s,t} : \mathbb{R} ^{d} \to \mathbb{R} ^{d} ) _{ 0\leq s\leq t\leq T}$ such that - $\lim _{t\to s_{+} } \Phi_{s,t} (x) =x$ for every $x\in \mathbb{R} ^{d}$. - $\Phi _{u,t} \circ \Phi _{s,u} = \Phi _{s,t} $ if $0\leq s\leq u \leq t$. Note that in [@Ku] the some measurability is also required in the definition of the flow but, since we are working later in the paper with non-semimartingales, we will omit it. It is well known that under conditions (\[REGULCLASS\]), $X_{s,t}(x)$ is a stochastic flow of $C^{m}$-diffeomorphism (see for example [@Ku2] and [@Ku]). Moreover, the inverse $Y_{s,t}(x):=X_{s,t}^{-1}(x)$ satisfies the following backward stochastic differential equation $$\label{itoassBac}Y_{s,t}(x)= x - \int_{s}^{t} b(r, Y_{r,t}(x)) \ dr - (B_{t}-B_{s}).$$ for every $0\leq s\leq t\leq T$, see [@FGP2] or [@Ku2] pp. 234. In order to get the solution of (\[trasportS\]) via the stochastic characteristic method we considerer the following ordinary differential equation $$\label{deter} Z_{t}(r)= r + \int_{0}^{t} F(s, Z_{s}(r)) \ ds .$$ We have the following representation of the solution to the transport equation in terms of the inital data and of the inverse flow (\[itoassBac\]). We refer to e.g. [@Ku] or [@Chow], Section 3 for the proof. \[lemaexis\] Assume (\[REGULCLASS\]) for $m\geq3$ and let $u_{0}\in C^{m,\delta}(\mathbb{R}^{d}), F \in L^{\infty}((0,T); C ^{m}_{b}(\mathbb{R} ^{d})) $. Then the Cauchy problem (\[transintegral\]) has a unique solution $u(t,.)$ for $0\leq t\leq T$ such that it is a $C^{m}$-semimartingale which can be represented as $$u(t,x)=Z_{t}(u_{0}(X_{t}^{-1}(x))), \hskip0.3cm t \in [0,T], x\in \mathbb{R} ^{d}$$ where $Z$ is the unique solution to (\[deter\]) and $X_{t} ^{-1} = X_{0,t} ^{-1}= Y_{0,t}$ for every $t\in [0,T]$. The weak solution of the transport equation driven by fractional Brownian motion ================================================================================ We discuss in this section the existence, uniqueness and the representation of the solution to the standard equation driven by a fractional Brownian motion with Hurst parameter bigger than one half. We refer to the last section (the Appendix) for the basic properties of this process. We will restrict throughout this section to the case $d=1$ and and we will use the concept of weak solution. We explain at the end of this section (see Remark \[19a-2\]) why we need to assume these restrictions. Consider the following one-dimensional Cauchy problem: given an initial-data $u_0$, find $u(t,x;\omega) \in {\mathbb{R}}$, satisfying $$\label{trasport} \left \{ \begin{aligned} &\partial_t u(t, x;\omega) + \Big(\, \partial_x u(t, x;\omega) \, \big(b(t, x) + \frac{d B ^{H}_{t}}{dt}(\omega)\big ) \Big)= 0, \\[5pt] &u|_{t=0}= u_{0}, \end{aligned} \right .$$ with $T>0$, $\big( (t,x) \in U_T, \omega \in \Omega \big)$, where $U_T= [0,T] \times {\mathbb{R}}$, and $b:[0,T]+ \times {\mathbb{R}}\to {\mathbb{R}}$ is a given vector field. The noise $B ^{H}$ is a fractional Brownian motion with Hurst parameter $H>\frac{1}{2}$ and the stochastic integral in (\[trasport\]) will be understood in the symmetric sense via regularization [@RV] or [@RV1]. The fBm $B ^{H}$ is related to the Brownian moption $B$ via (\[BH\]). Let us first recall the notion of weak solution to (\[trasport\]). \[defisolu\] A stochastic process $u\in L^{\infty}(\Omega\times[0, T]\times \mathbb{R})$ is called a weak $L^{p}-$solution of the Cauchy problem , when for any $\varphi \in C_c^{\infty}({\mathbb{R}})$, $\int _{\mathbb{R}} u(t, x)\varphi(x)dx$ is an adapted real value proces which has a continuous modification, finite covariation, and for all $t \in [0,T]$, we have $P$-almost surely $$\begin{aligned} \int_{\mathbb{R}} u(t,x) \varphi(x) dx &=& \int_{\mathbb{R}} u_{0}(x) \varphi(x) \ dx +\int_{0}^{t} \!\! \int_{\mathbb{R}} u(s,x) \ b(s,x) \partial_{x} \varphi(x) \ dx ds\nonumber \\ &&\; + \int_{0}^{t} \!\! \int_{\mathbb{R}} u(s,x) \, b'(s,x) \, \varphi(x) \ dx ds \nonumber \\ &&\; + \int_{0}^{t} \!\! \int_{\mathbb{R}} u(s,x) \ \partial_{x} \varphi(x) \ dx d^{\circ}B ^{H} _s.\label{DISTINTSTR}\end{aligned}$$ where $b'(s,x)$ denotes the derivative of $b(s,x)$ with respect to the variable $x$. At this point, we need to recall the definition of the symmetric integral $d^{\circ} B ^{H}$ that appears in (\[DISTINTSTR\]). assume $(X_t)_{ t\geq 0} $ is a continuous process and $(Y_t)_{ t\geq 0}$ is a process with paths in $L_{loc}^{1}(\mathbb{R}^{+})$, i.e. for any $ b > 0$, $ \int_{0}^{b}|Y_t| dt <\infty$ a.s. The generalized stochastic integrals (forward, backward and symmetric) are defined through a regularization procedure see [@RV], [@RV1]. That is, let $ I^{0}(\epsilon, Y, dX)$ be the $\varepsilon-$symmetric integral $$I^{0}(\epsilon, Y, dX)=\int_{0}^{t} Y_{s} \frac{(X_{s+\epsilon}-X_{s-\epsilon})}{2\epsilon} ds \ t \geq 0.$$ The symmetric integral $\int_{0}^{t} Y d^{\circ} X$ is defined as $$\int_{0}^{t} Y d^{\circ} X: = \lim_{\epsilon\rightarrow 0}I^{0}(\varepsilon, Y, dX)(t),$$ for every $t\in [0,T]$, provided the limit exist ucp (uniformly on compacts in probability). Similarly to Lemma \[lemaexis\], we also have a representation formula for the weak solution in terms of the initial condition $u_0$ and the (inverse) stochastic flow associated to SDE (\[1\]). \[repre\] Assume that $b \in L^{\infty}((0,T); C_b^{1}(\mathbb{R}^{d}))$. Then there exists a $C^{1} (\mathbb{R}) $ stochastic flow of diffeomorhism $X_{s,t}, 0\leq s\leq t\leq T$ that satisfies $$\label{1} X_{s,t} (x)= x+ \int_{s}^{t} b(u, X_{s,u} (x)) du + B ^{H} _{t}-B ^{H} _{s}$$ for every $x\in \mathbb{R}^{d}$. Moreover, if $d=1$, given $u_{0}\in L^{\infty}(\mathbb{R})$, the stochastic process $$\label{19a-1}u(t, x):= u_0(X_{t}^{-1}(x)), \hskip0.5cm t\in [0,T], x\in \mathbb{R}$$ is the unique weak $L^{\infty}-$ solution of the Cauchy problem , where $X_{t}:= X_{0,t}$ for every $t\in [0,T]$. [**Proof:** ]{} We will proceed in several steps: first we show that (\[1\]) is a diffeomorphism flow, then we prove the uniquennes of the $L^{\infty}$ weak solution to (\[trasport\]) and then we show that (\[19a-1\]) satisfies the transport equation (\[trasport\]). Let us first show that (\[1\]) generates a flow of diffeomorphism. By doing the linear transformation $$Z_{s,t}=X_{s,t}(x)- ( B_{t}^{H}-B_{s}^{H} )$$ we deduce that the equation (\[1\]) is equivalent to the random equation $$\label{itoassequevalent} Z_{s,t}(x)= x + \int_{s}^{t} b(r, Z_{s,r}(x) + B_{r}^{H}-B_{s}^{H} ) \ dr$$ for $0 \leq s\leq t\leq T$. From the classical theory for ordinary differential equations (see e.g. [@AMR]) we have that $Z_{s,t}(x)$ with $ 0\leq s\leq t\leq T$ is a $C^{1}(\mathbb{R}^{d}) $ diffeomorphism flow. Thus we deduce that $X_{s, t}(x)$ is a $C^{1}(\mathbb{R}^{d}) $ diffeomorphism flow. In a second step, we will show that the transport equation with fBm noise admits a unique $L^{\infty}$ weak solution. By linearity we have to show that a weak $L^{\infty}-$solution with initial condition $u_{0}=0$ vanishes identically. Applying the Itô-Ventzel for the symmetric integral formula (see Proposition 9 of [@FlandRusso]) to $F(y)=\int u(t,x) \varphi(x-y) \ dx $ (which depends on $\omega$), we obtain that $$\begin{aligned} \int _{\mathbb{R}} u(t,x) \varphi(x-B^{H}_{t}) dx &=& \int_{0}^{t} \int _{\mathbb{R}} b(s,x) \partial_x \varphi(x-B ^{H} _{s}) u(s,x) dx ds \nonumber\\ &&+ \int_{0}^{t} \int _{\mathbb{R}} b'(s,x) \varphi(x-B ^{H} _{s}) u(s,x) dx ds \nonumber \\ && + \int_{0}^{t} \int _{\mathbb{R}} u(s,x) \partial_x \varphi(x-B ^{H}_{s}) dx d^{\circ} B ^{H}_{s}\nonumber \\ && + \int_{0}^{t} \int _{\mathbb{R}} u(s,x) \partial_y [ \varphi(x-B^{H}_{s})] dx d^{\circ} B ^{H}_{s} \label{19a-4}.\end{aligned}$$ We observe that $\partial_y [ \varphi(x-B^{H}_{s})]=- \partial_x \varphi(x-B ^{H}_{s})$. Thus the process $$V(t,x): =u(t,x+ {B ^{H}_{t}})$$ verifies $$\begin{aligned} \int _{\mathbb{R}}V(t,x) \varphi(x) dx &=& \int_{0}^{t} \int _{\mathbb{R}} b(s,x+B ^{H}_{s}) \partial_x \varphi(x) V(s,x)dx ds\\ && + \int_{0}^{t} \int _{\mathbb{R}} b'(s,x+B ^{H}_{s}) \varphi(x) V(s,x) dx ds.\end{aligned}$$ Let $\phi_{\varepsilon}$ be a standard mollifier and let $V_{\varepsilon}(t,x): =V(t,.)\ast \phi_{\varepsilon}$. Then it holds $$\begin{aligned} \int _{\mathbb{R}} V(t,z) \phi_{\varepsilon}(x-z) dz&=& \int_{0}^{t} \int _{\mathbb{R}} V(s,z) \, b(s,z+B ^{H}_{s}) \, \partial_z \phi_{\varepsilon}(x-z) dz ds \\ &&+ \int_{0}^{t} \int _{\mathbb{R}} u(s,z) b'(s,z+B ^{H}_{s}) \, \phi_{\varepsilon}(x-z) dz ds \end{aligned}$$ From an algebraic convenient manipulatio we get $$\frac{dV_{\varepsilon}}{dt}-b(t,x-B ^{H}_{t}) \partial_x V_{\varepsilon}=\mathcal{R}_{\varepsilon}(b,u)$$ where $ \mathcal{R}_{\varepsilon}(b,u)$ is the commutator defined as $$\mathcal{R}_{\varepsilon}(b,u)=(b\partial _{x}) (\phi_{\varepsilon}\ast u )- \phi_{\varepsilon}\ast((b\partial _{x})u).$$ Since $b(s,x+B^{H}_{s})$ belongs a.s to $ L^{\infty} ((0,T); C ^{1}_{b} (\mathbb{R} ))$ then by the Commuting Lemma (see Lemma II.1 of [@DL]), the process $V_{\varepsilon}(t,x)=V(t,.)\ast \phi_{\varepsilon}$ satisfies $$\lim_{\varepsilon \rightarrow 0}\frac{dV_{\varepsilon}}{dt}-b(t,x-B ^{H}_{t}) \partial_x V_{\varepsilon}=0 \mbox{ a.s. in }\ L^{1}([0,T], L^{1}_{loc}(\mathbb{R})).$$ We deduce that if $\beta\in C^{1}(\mathbb{R})$ and $\beta^{\prime}$ is bounded, then $$\label{norma} \frac{d\beta(V)}{dt}-b(t,x-B ^{H}_{t}) \partial _{x} \beta(V)=0.$$ Now, by Theorem II. 2 of [@DL], we define for each $M \in [0,\infty)$ the function $\beta_M(t)=(|t| \wedge M )^p$ and obtain that $$\frac{d}{dt}\int \beta_M(V(t,x))dx \leq C \int \beta_M(V(t,x))dx.$$ Taking expectation we have that $$\frac{d}{dt}\int \mathbb{E}(\beta_M(V(t,x)))dx \leq C \int \mathbb{E}(\beta_M(V(t,x)))dx.$$ From Gronwall Lemma we conclude that $\beta_M(V(t,x))=0$. Thus $u=0$. Let us finally show that (\[19a-1\]) satisfies (\[trasport\]). We have that, by the change of variables $X_{t} ^{-1}(y)=x$ $$\label{PUSHFORWARD} \int_{\mathbb{R}} u_0(X_t^{-1}) (y) \; \varphi(y) dy =\int_{{\mathbb{R}}^d} u_0(x) \; X'_t (x)\varphi(X_t (x) ) dx ,$$ for each $t \in [0,T]$, where $X' _{t}(x)$ denotes the derivative with respect to $x$ of $X_{t}(x)$. Notice that $X'_{t}(x)= 1+ \int_{0} ^{t} b'(s, X_{s}(x)) X '_{s}(x)ds$ for every $t\in [0,T], x\in \mathbb{R}$. By applying Itô’s formula (see [@RV1], [@RV]) to the product $$X'_t (x) \varphi(X_t (x))$$ and using the fact that $B^{H}$ has zero quadratic variation when $H>\frac{1}{2}$ we obtain that $$\begin{aligned} \int_{\mathbb{R}} u_0 (X_t^{-1}(x)) \varphi(x) dx &=& \int_{\mathbb{R}} u_0(x) dx +\int_{0}^{t} \int_{\mathbb{R}} u_0(x) b(s,X_s(x)) X'_t (x) \cdot \varphi ' (X_s (x)) dx ds \\ &+& \int_{0}^{t}\int_{\mathbb{R}} u_0(x) b'(s,X_s (x) ) X'_t (x) \varphi '(X_s(x) ) dx ds \nonumber \\ &+& \int_{0}^{t} \int_{\mathbb{R}} u_0(x) X'_t (x) \varphi '(X_s(x)) dy d^{\circ}B_{s}^{H}. \label{19a-3}\end{aligned}$$ Note that the Itô formula in [@RV1] guarantees the existence of the symmetric stochastic integrals in (\[19a-3\]) above. Now, by the change variable $ y=X_t (x)$ we have that $$\begin{aligned} && \int_{\mathbb{R}} u_0(X_t^{-1}(x)) \varphi(x) dx = \int_{\mathbb{R}} u_0(x) dx + \int_{0}^{t} \int_{\mathbb{R}} u_0(X_s^{-1}(x)) b(s,y) \cdot \varphi '(y) dy ds \\ && +\int_{0}^{t} \int_{\mathbb{R}} u_0(X_s^{-1}(x)) b'(s,y) \varphi'(y) dy ds \\ &&+ \int_{0}^{t} \int_{\mathbb{R}} u_0(X_s^{-1}(x)) \; \varphi '(y) dy d^{\circ} B_{s}^{H}.\end{aligned}$$ From this we conclude que $u(t,x)=u_0(X_t^{-1}(x))$ is a weak solution of (\[trasport\]). Its adaptedness is a consequence of (\[BH\]). Thus the unique solution to (\[trasport\]) is $u(t,x)=u_0(X_t^{-1}(x))$ for every $t\in [0,T]$ and for every $x\in \mathbb{R}$. \[19a-2\] - We need to restrict to the situation $d=1$ in order to get the existence of the symmetric integral in (\[19a-3\]) or (\[19a-4\]). Here we also used the hypothesis $H> \frac{1}{2}$ that ensures that there is not a second derivative term in the Itô formula. - The uniqueness of the weak solution can be obtined with weaker assumption on the drift $b$ by following the proof of Theorem 3.1 in [@CO]. Fractional Brownian flow ======================== In this section we will analyze the properties of the stochastic flow generated by the fractional Brownian motion. We will call it the fractional Brownian flow in the sequel. Fix $d\geq 1$ and let $ B^{H}= ( B ^{H_{1}}, B^{H_{2}}, \ldots, B^{H_{d}})$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H= (H_{1}, H_{2}, \ldots , H_{d}) \in (0,1) ^{d}$. Recall (see Theorem \[repre\]) that if $b \in L^{\infty}((0,T), C^{1}_{b} (\mathbb{R} ^{d}))$, (\[1\]) generates a $C^{1}$-stochastic flow of diffeomorphism. We next describe the dynamic of the inverse flow of (\[1\]). Let $b \in L^{\infty}((0,T), C^{1}_{b} (\mathbb{R} ^{d}))$ and denote, for every $0\leq s\leq t\leq T $ and for every $x\in \mathbb{R} ^{d}$ $$\label{y} Y_{s,t} (x)= X_{s, t} ^{-1}(x)$$ the inverse of the stochastic flow given by (\[1\]). Then the inverse flow satisfies the backward stochastic equation $$\label{back} Y_{s,t}(x)= x-\int_{s}^{t} b(r, Y_{r,t} )dr -( B^{H}_{t} -B^{H}_{s})$$ for every $x\in \mathbb{R} ^{d}.$ [**Proof:** ]{} It follows from Kunita [@Ku]. Indeed, Lemms 6.2, page 235 in [@Ku] says that for any continuous function in two variables $g$ we have $$\int_{s}^{t} g(r, X_{s,r}(y)) dr | _{y=X_{s,t}^{-1} (x) } = \int_{s}^{t} g(r, X_{r,t}^{-1} (x))dr$$ and it suffices to apply the above identity to (\[1\]). We need the following useful lemma. \[6a-5\] Let us introduce the notation, for $t\in [0,T]$ and $x\in \mathbb{R} ^{d}$, $$\label{r} R_{t,x} (u)= Y_{ t-u, t} (x), \hskip0.5cm \mbox{ if } u \in [0,t].$$ Then we have, for every $t\in [0,T], u\in [0,t] $ and $x\in \mathbb{R} ^{d}$ $$\label{5a-1} R_{t,x}(u)= x- \int_{0} ^{u} b(a, R_{t, x} (a) )da -( B ^{H}_{t}- B ^{H} _{t-u}).$$ [**Proof:** ]{} In (\[y\]) we use the change of notation $u=t-s$ and we get for every $y\in \mathbb{R} ^{d}$, $$R_{t,y}(u)= y- \int_{t-u}^{t} b( r, Y _{r,t} (x))dr -(B_{t}^{H}- B_{t-u}^{H})$$ and then, with the change of variables $a=t-r$ in the integral $dr$, we can writye $$R_{t,y}(u)= y-\int_{0} ^{u} b(a, Y_{t,y}(a) )da -( B ^{H}_{t}- B ^{H}_{t-u})$$ with $R_{u,y}(u)=y$. As a consequence of the above Lemma \[6a-5\], we get the uniqueness of solution to the backward equation (\[back\]) satisfied by the inverse flow. If $ (\tilde{Y} _{s,t} ) _{0\leq s\leq t\leq T}$ is another two parameter process that satisfies (\[y\]) with $\tilde{Y} _{s,s}(x)=x$ and $b $ is Lipschitz in $x$ uniformy with respect to $t$, then $\tilde{Y}_{s,t} (x)= Y_{s,t}(x) $ for every $0\leq s\leq t$ and for every $x\in \mathbb{R} ^{d}$. [**Proof:** ]{} If $\tilde{Y}$ satisfies (\[y\]), then, if we denote $\tilde{R}_{t,x}(u)=\tilde{ Y}_{t-u,t}(x) $, we get from Lemma that $\tilde{R}$ satisfies (\[r\]) and the Gronwall lemma and the Lipschitz assumption on the drift $b$ imply the conclusion. We denote by $D$ the Malliavin derivative with respect with the fBm $ B^{H}$ (see the Appendix). \[5a-3\] Assume $b \in L^{\infty}((0,T), C^{1}_{b} (\mathbb{R} ^{d}))$ and let $X_{s,t}$ be given by (\[1\]). Then, for every $0\leq s\leq t \leq T$ and for every $x\in \mathbb{R}^{d}$, the components of inverse flow $Y^{i}_{s,t} $ ($1\leq i\leq d$) are Malliavin differentiable and for every $\alpha \in [s, t]$ $$D_{\alpha} Y^{i}_{s,t}(x)=-\int_{s}^{t} \sum_{j=1} ^{d} \frac{\partial b^{i}}{\partial x_{j}}(r, Y_{r,t}) D_{\alpha} Y ^{j} _{r,t} (x) dr -1$$ and $D_{\alpha} Y^{i}_{s,t}(x)=0$ if $\alpha \notin [s,t]$. We denoted by $b^{i}$ ($1\leq i\leq d$) the components of the vector mapping $b$. [**Proof:** ]{} It suffices to show that the random variable $R_{t,x}(u)$ defined by (\[r\]) is Malliavin differentiable for any $x\in \mathbb{R} ^{d}$ and for every $0\leq u\leq t\leq T$. We will give the sketsch of the proof which follows by a routine fix point argument. Fix $x\in \mathbb{R} ^{d}, t \in [0,T]$ and define the iterations $$R^{(0)} _{t,x}(u)=x, \mbox{ for every } u\in [0,t]$$ and for $n\geq 1$, $$R_{t,x} ^{(n)}(u)= x-\int_{0} ^{u} b(a, R^{(n-1)}_{t, x} (a) )da -( B ^{H}_{t}- B ^{H} _{t-u}).$$ By induction, we can prove by standard arguments (see e.g. [@N], Theorem 2.2.1) that for every $p\geq 1$ $$\sup_{0\leq u\leq t} \mathbf{E} \left| R ^{(n) }_{t,x} (u)\right| ^{p} <\infty,$$ $$R^{(n), j} _{t,x}(u) \in \mathbb{D} ^{1, \infty}, \hskip0.5cm j=1,.., d,$$ and $$\sup _{n \geq 1}\sup_{ \alpha \in [0,T]} \mathbf{E} \left| D_{\alpha} R^{(n), j} _{t,x}(u) \right| ^{p} < \infty$$ where $ R^{(n), j} _{t,x}(u) $ denotes the $j$ th component of $ R^{(n)} _{t,x}(u) .$ Moreover, the sequence of random variables $ (R^{n} _{t,x}(u)) _{n \geq 1}$ converges in $L^{p}$ to $R_{t,x}(u)$ which is the unique solution to (\[5a-1\]). It follows from Lemma 1.2.3 in [@N] that $R _{t,x}(u)$ belongs to $ \mathbb{D} ^{1,\infty}$. Note that, when the noise is the standard Brownian motion, the Malliavin differentiability of $Y$ is also claimed in [@MNP]. Existence and Gaussian bounds for the density of the solution to the transport equation in dimension one ======================================================================================================== In this section we will assume that $d=1$. On the other hand, the results in these section (except Theorem \[6a-3\]) will hold for every $H\in (0,1)$. We also mention that we will use the notation $c,C..$ for generic positive constants that may vary from line to line. From Proposition \[5a-3\] we immediately obtain the explicit expression for the Malliavin derivative of the inverse flow. If $b \in L^{\infty}((0,T), C^{1}_{b} (\mathbb{R} ^{d}))$ and $Y_{s,t}$ is defined by (\[y\]), we have for every $\alpha$ and for every $0\leq s\leq t\leq T$ $$\label{5a-4} D_{\alpha} Y _{s,t} (x)=- 1_{[s,t]} (\alpha )e ^{- \int_{s} ^{\alpha} b'(r, Y_{r,t}) dr}$$ with $b'(t,x)$ the derivative of $b(t,x)$ with respect to $x$. [**Proof:** ]{} For every $\alpha $, we have $$\begin{aligned} D_{\alpha} Y_{s,t} (x) &=& -\int_{s}^{t} b'(s_{1}, Y_{s_{1}, t} (x)) D_{\alpha } Y _{s_{1}, t}(x) ds_{1}- 1_{[s,t] } (\alpha) \end{aligned}$$ and by iterating the above relation we can write, for every $0\leq s\leq t\leq T$ and for evey $\alpha \in [0,T]$, $$\begin{aligned} D_{\alpha} Y_{s,t} (x) &=& -1_{[s,t] } (\alpha ) \sum_{n\geq 0} (-1) ^{n} \int_{s}^{\alpha} ds_{1} \int_{s_{1} }^{\alpha} ds_{2} ...\int_{s_{n-1} }^{\alpha} ds_{n}\\ &&\times b'(s_{1}, Y_{s_{1}, t} (x))b'(s_{2}, Y_{s_{2}, t} (x))..b'(s_{n}, Y_{s_{n}, t} (x))\\ &=& -1_{[s,t] } (\alpha ) \sum_{n\geq 0}\frac{ (-1) ^{n} }{n!} \left( \int_{s} ^{t} dr b'(r, Y_{r,t}(x) )\right) ^{n} \\ &=& - 1_{[s,t]} (\alpha )e ^{- \int_{s} ^{\alpha} b'(r, Y_{r,t}) dr}.\end{aligned}$$ The main tool in order to obtain the Gaussian estimates for the density of the solution to the trasport equation is the following result given in [@NV]. \[6a-4\] I If $ F \in \mathbb{F} ^{1,2}$, let $$g_{F}(F)= \int_{0} ^{\infty} d\theta e^{-\theta} \mathbf{E} \left[ \mathbf{E} '\left( \langle DF, \widetilde{DF} \rangle _{\mathcal{H}} | F \right) \right]$$ where for any random variable $X$, we denoted $$\tilde{X}(\omega , \omega ') = X ( e ^{-\theta }w + \sqrt{1-e^{-2\theta} }\omega ').$$ Here $\tilde{X}$ is defined on a product probability space $\left( \Omega \times \Omega ', {\cal{F}} \otimes {\cal{F}}, P\times P'\right)$ and ${\mathbf E}'$ denotes the expectation with respect to the probability measure $P'$. If there exists two constants $\gamma _{min}$ and $\gamma _{max}$ such that almost surely $$0\leq \gamma_{ min} \leq g_{F}(F) \leq \gamma _{max}$$ then $F$ admits a density $\rho$. Moreover, for every $z\in \mathbb{R}$, $$\frac{ \mathbf{E} \vert F -\mathbf{E} F \vert }{2\gamma ^{2}_{max} }e ^{ -\frac{(z-\mathbf{E} F) ^{2}}{2\gamma^{2} _{min} } }\leq \rho(z) \leq \frac{ \mathbf{E} \vert F -\mathbf{E} F \vert }{2\gamma^{2} _{min} }e ^{ -\frac{(z-\mathbf{E} F) ^{2}}{2\gamma ^{2} _{max} }}$$ To apply the above result, we need to controll the Malliavin derivative of the inverse flow. This will be done in the next result. Notice that a similar method has been used in e.g. [@AB], [@BKT] or [@NQ] for various types of stochastic equations. In the sequel $\mathcal{H}$ denotes the canonical Hilbert space associated to the fractional Brownian motion (see the Appendix). \[6a-3\] Assume $H>\frac{1}{2}$ and $b \in L^{\infty}((0,T); C^{1}_{b}(\mathbb{R}))$. Then there exist two positive constants $c<C$ such that for every $t\in [0,T]$ and for every $x \in \mathbb{R } $ $$\label{6a-2} ct^{2H} \leq \langle D Y_{0,t}(x), \widetilde{ D Y_{0,t}(x)} \rangle _{\mathcal{ H}} \leq Ct ^{2H }$$ where $Y_{0,t}$ is given by (\[back\]). [**Proof:** ]{} Assume $H=\frac{1}{2}$. Then $H= L ^{2} ([0,T]) $ and $$\langle D Y_{0,t}(x), \widetilde{ D Y_{0,t}(x)} \rangle _{\mathcal{ H}}= \int_{0} ^{t} d\alpha e ^{- \int_{0} ^{\alpha} b'(r, Y_{r,t}(x)) dr} e ^{- \int_{0} ^{\alpha} b'(r, \widetilde{Y_{r,t}(x)}) dr}$$ and since $$\label{6a-1} e^{-T\Vert b'\Vert _{\infty} } \leq e ^{- \int_{s} ^{\alpha} b'(r, Y_{r,t}) dr}\leq e ^{T\Vert b'\Vert _{\infty}}$$ (and a similar bound holds for the tilde process) we obtain $$ct \leq \langle D Y_{0,t}(x), \widetilde{ D Y_{0,t}(x)} \rangle _{\mathcal{ H}} \leq Ct$$ with two positive constant $c$ and $C$. Assume $H> \frac{1}{2}$. Then by (\[27i-1\]) $$\langle D Y_{0,t}(x), \widetilde{ D Y_{0,t}(x)} \rangle _{\mathcal{ H}} = \alpha _{H}\int_{0} ^{t} d\alpha \int_{0} ^{t} d\beta e ^{- \int_{0} ^{\alpha} b'(r, Y_{r,t}(x)) dr}e ^{- \int_{0} ^{\beta } b'(r, \widetilde{Y_{r,t}(x)}) dr}\vert \alpha - \beta \vert ^{2H-2}$$ and inequality (\[6a-1\]) implies that $$c\int_{0} ^{t} d\alpha \int_{0} ^{t} d\beta \vert \alpha - \beta \vert ^{2H-2} \leq \langle D Y_{0,t}(x), \widetilde{ D Y_{0,t}(x)} \rangle _{\mathcal{ H}} \leq C\int_{0} ^{t} d\alpha \int_{0} ^{t} d\beta \vert \alpha - \beta \vert ^{2H-2}$$ which immediately gives (\[6a-2\]). Assume $H< \frac{1}{2}$. Then Proposition \[6a-1\] in [@BKT] implies the lower bound in (\[6a-2\]). Concerning the upper bound, it suffices again to follow [@BKT], Section 3.4 and to note that for every $\alpha, \beta \in [0,T]$ with $\alpha >\beta$ we have $$\vert D_{\alpha } Y _{0, t} (x) -D_{\beta} Y _{0,t} (x) \vert \leq e ^{- \int_{0} ^{\beta } b'(r, Y_{r,t}) dr}\left| e ^{- \int_{\beta } ^{\alpha} b'(r, Y_{r,t}) dr}-1 \right| \leq c(\alpha -\beta)$$ and the same bound holds for the $\widetilde{Y_{0,t}}$. 0.3cm Denote by $m: =\mathbf{E} u(t,x)$ (it satisfies a parabolic equation, see e.g. [@FGP2]). We are ready to state our main result. Let $u(t,x)$ be the solution to the transport equation (\[trasport\]). Assume that $u_{0} \in C^{1}(\mathbb{R})$ such that there exist $0<c<C$ with $c\leq u_{0}'(x) <C$ for every $x\in \mathbb{R}$ and $b\in L^{\infty}((0,T); C^{1}_{b}(\mathbb{R}))$. Then, for every $t\in [0,T]$ and for every $x\in \mathbb{R}$, the random variable $u(t,x)$ is Malliavin differentiable. Moreover $u(t,x)$ admits a density $\rho _{u(t,x) }$ and there exist two positive constants $c_{1}, c_{2}$ such that $$\label{5a-5} \frac{\mathbf{E} \vert u(t,x)- m\vert }{2c_{1} t^{2H}}e ^{-\frac{(y-m) ^{2}}{2c_{2} t^{2H}}}\leq \rho_{ u(t,x)} \leq \frac{\mathbf{E} \vert u(t,x)- m\vert }{2c_{2} t^{2H}}e ^{-\frac{(y-m) ^{2}}{2c_{1} t^{2H}}}$$ [**Proof:** ]{} Since by Theorem \[repre\], $u(t,x)= u_{0} (Y_{0,t}(x))$, we get the Malliavin differentiability of $u(t,x)$ from Proposition \[5a-3\] and the chain rule for the Malliavin derivative (see e.g. [@N]). Moreover, the chain rule implies $$D_{\alpha} u(t,x)= u_{0} '( Y_{0,t} (x)) D_{\alpha } Y_{0,t} (x)$$ and thus $$\langle D u(t,x) , \widetilde{Du(t,x)} \rangle _{ \mathcal{H}}= u_{0} '( Y_{0,t} (x)) \widetilde{ u_{0} '( Y_{0,t} (x))} \langle D Y_{0,t}(x), \widetilde{ D Y_{0,t}(x)} \rangle _{\mathcal{ H}} .$$ By Proposition \[6a-3\] and the asumption $u_{0} \in C^{1}_{b}$, there exists two strictly positive constants $c<C$ such that $$ct^{2H} \leq \langle D u(t,x) , \widetilde{Du(t,x)} \rangle _{ \mathcal{H}}\leq C t^{2H}$$ for every $t\in [0,T]$ and for every $x \in \mathbb{R}$. Now, Proposition \[6a-4\], point 2. implies that, if $F=u(t,x)$ then $$c_{1} t ^{2H} \leq g_{F}(F) \leq c_{2} t^{2H}$$ and Proposition \[6a-4\], point 1. gives the conclusion. Explicit expression of the density when the noise is the Brownian motion in $\mathbb{R} ^{d}$ ============================================================================================= We obtained above the existence and Gaussian estimate for the solution to the transport equation in dimension 1 and for $H\geq \frac{1}{2}$. In this section, we will assume $d\geq 2$, $H=\frac{1}{2}$, that is, the transport equation is driven a standard Brownian motion in $\mathbb{R} ^{d}$. We obtain the followin explicit expression for the density of the solution when the divergence of the drift $b$ vanishes. \[um\] Assume $d\geq 2$and let $u_{0}$ be a $ C^{m,\delta}(\mathbb{R}^{d}) $ diffeomorphism. Assume (\[REGULCLASS\]) for $m\geq3$. Moreover, suppose that $$\label{DIVB} div \, b= 0 .$$ Fix $t \in [0,T]$ and $x\in\mathbb{R}^{d} $. Then the law of the solution of (\[trasportS\]), has a density $\widetilde{\rho}$ with respect to the Lebesgue measure. Moreover the density $\widetilde{\rho}$ admits the representation $$\label{9a-2} \widetilde{\rho}= Ju_{0}( Z_t^{-1}(y)) JZ_{t} \rho(u_{0}^{-1}( Z_t^{-1}(y) ),t,x)$$ where $ \rho$ denotes the density of the solution to (\[11\]). [**Proof:** ]{} Let $u(t,x)$ solution of the SPDE (\[trasportS\] ). By Lemma \[lemaexis\] we have that $u(t,x)$ has the representation $$u(t,x)=Z_t(u_{0}(X_{t}^{-1}(x))).$$ Let $\phi_{\varepsilon}$ be a standard mollifier and consider a smooth function $\varphi \in C_c^{\infty}( \mathbb{R}^{d} )$. Then $$\begin{aligned} \mathbf{E} [\varphi(u(t,x))]&=& \mathbf{E} [\varphi(Z_t(u_{0}(X_{t}^{-1}(x))) )]\\ &=& \mathbf{E} \ [ \lim_{\epsilon\rightarrow 0 } \int_{\mathbb{R}^{d}} \phi_{\varepsilon}(y-x) \varphi(Z_t(u_{0}(X_{t}^{-1}(y))) ) dy ] \\ &=& \lim_{\epsilon\rightarrow 0 } \mathbf{E}\ [ \int_{\mathbb{R}^{d}} \phi_{\varepsilon}(y-x) \varphi(Z_t(u_{0}(X_{t}^{-1}(y))) ) dy ].\end{aligned}$$ The assumption (\[DIVB\]) implies that $JX_t=1$, where $JX_t$ denote of the Jacobian map of $X_t$. By doing one more time a chamge of variable, we can write $$\begin{aligned} \mathbf{E} [\varphi(u(t,x))]&=& \lim_{\epsilon\rightarrow 0 } \mathbf{E} [ \int_{\mathbb{R}^{d}} \phi_{\varepsilon}(y-x) \varphi(Z_t(u_{0}(X_{t}^{-1}(y))) ) dy ]\nonumber \\ &=& \lim_{\epsilon\rightarrow 0 } \mathbf{E} [ \int_{\mathbb{R}^{d}} \phi_{\varepsilon}(X_t(y)-x) \varphi(Z_t(u_{0}(y)) ) dy ] \nonumber \\ &=& \lim_{\epsilon\rightarrow 0 } \int_{\mathbb{R}^{d}} \mathbf{E} [ \phi_{\varepsilon}(X_t(y)-x) ] \varphi(Z_t(u_{0}(y)) ) dy ] \label{9a-1}\end{aligned}$$ The random variable $X_{t}(x) $ admits a density $\rho$ in any dimension $d$. This is an easy consequence of equation (\[1\]) (see e.g. [@N]). Therefore, (\[9a-1\]) becomes $$\begin{aligned} \mathbf{E} [\varphi(U(t,x))]&=& \lim_{\epsilon\rightarrow 0 } \int_{\mathbb{R}^{d}} \mathbf{E} [ \phi_{\varepsilon}(X_t(y)-x) ] \varphi(Z_t(u_{0}(y)) ) dy ]\\ &=& \lim_{\epsilon\rightarrow 0 } \int_{\mathbb{R}^{d}} \int_{\mathbb{R}^{d}} \phi_{\varepsilon}(u-x) \rho(u,t,y) du \ \varphi(Z_t(u_{0}(y)) ) dy \end{aligned}$$ and by calculating the limit above when $\varepsilon \to 0$ we get $$\mathbf{E} [\varphi(u(t,x))]= \int_{\mathbb{R}^{d}} \rho(y,t,x) \varphi(Z_t(u_{0}(y)) ) dy .$$ Finally, the making succesively the changes of variables $w=u_{0} (y)$ and $y= Z_{t}(w)$ we obtain $$\begin{aligned} \mathbf{E} [\varphi(u(t,x))]&=& \int_{\mathbb{R}^{d}} \ Ju_{0} \ \rho(u_{0}^{-1}(w),t,x) \ \varphi(Z_t(w) ) dw \\ &=& \int_{\mathbb{R}^{d}} \ Ju_{0}( Z_t^{-1}(y)) \ JZ_{t} \ \rho(u_{0}^{-1}( Z_t^{-1}(y) ),t,x) \ \varphi(y) \ dy\end{aligned}$$ and thus relation (\[9a-2\]) is obtained. The assumption $\div b=0$ can be interpreted as follows (see [@L]): in fluid mechanics or more generally in continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the velocity of the fluid. This is equivalent to the condition that the divergence of the fluid velocity is zero. Appendix ======== We present here some basic element on the fractional Brownian motion and on the Malliavin calculus. Fractional Brownian motion -------------------------- Consider $(B_{t}^{H})_{t\in\lbrack0,T]}$ a fractional Brownian motion with Hurst parameter $H\in(0,1)$. Recall that is it a centered Gaussian process with covariance function $$\label{cov} \mathbf{E} B ^{H}_{t} B ^{H}_{s} :=R_{H}(t,s)= \frac{1}{2} ( t^{2H} + s^{2H}- \vert t-s\vert ^{2H}, \hskip0.5cm s,t \in [0,T].$$ The fractional Brownian motion can be also defined as the only self-similar Gaussian process with stationary increments. Denote by ${\mathcal{H}}$ its canonical Hilbert space . If $H=\frac{1}{2}$ then $B^{\frac{1}{2}}$ is the standard Brownian motion (Wiener process) $W$ and in this case ${\mathcal{H}}=L^{2}([0,T])$. Otherwise $\mathcal{H}$ is the Hilbert space on $[0,T]$ extending the set of indicator function $\mathbf{1}_{[0,T]}, t\in [0,T]$ (by linearity and closure under the inner product) the rule $$\left\langle \mathbf{1}_{[0,s]};\mathbf{1}_{[0,t]}\right\rangle _{\mathcal{H}}=R_{H}\left( s,t\right) :=2^{-1}\left( s^{2H}+t^{2H}-\left\vert t-s\right\vert ^{2H}\right) .$$ The followings facts will be needed in the sequel (we refer to [@PiTa1] or [@N] for their proofs): [$\bullet$ ]{} If $H>\frac{1}{2}$, the elements of ${\cal{H}}$ may be not functions but distributions; it is therefore more practical to work with subspaces of ${\cal{H}}$ that are sets of functions. Such a subspace is $$\begin{aligned} \left| {\cal{H}}\right| &=&\left \{ f:[0,T]\to \mathbb{R} \,\,\Big | \int _{0}^{T} \int_{0}^{T} \vert f(u)\vert \vert f(v)\vert \vert u-v\vert ^{2H-2} dvdu <\infty \right \}.\end{aligned}$$ Then $\left| {\cal{H}}\right|$ is a strict subspace of $ {\cal{H}}$ and we actually have the inclusions $$\begin{aligned} \label{inclu1} L^{2}([0,T]) \subset L^{\frac{1}{H} } ([0,T]) \subset \left| {\cal{H}}\right| \subset {\cal{H}}.\end{aligned}$$ [$\bullet$ ]{} The space $\left| {\cal{H}}\right|$ is not complete with respect to the norm $\Vert \cdot \Vert _{{\cal{H}}}$ but it is a Banach space with respect to the norm $$\begin{aligned} \Vert f\Vert ^{2}_{\left| {\cal{H}}\right| }&=&\int _{0}^{T} \int_{0}^{T} \vert f(u)\vert \vert f(v)\vert \vert u-v\vert ^{2H-2} dvdu .\end{aligned}$$ [$\bullet$ ]{} If $H>\frac{1}{2}$ and $f,g$ are two elements in the space $\left| {\cal{H}}\right|$, their scalar product in ${\cal{H}}$ can be expressed by $$\label{27i-1} \langle f,g\rangle _{{\cal{H}}}=\alpha _{H} \int_{0} ^{T} \int_{0} ^{T} dudv \vert u-v\vert ^{2H-2} f(u) g(v)$$ where $\alpha _{H}= H(2H-1)$. [$\bullet$ ]{} when $H<\frac{1}{2}$ then the canonical Hilbert space is a space of functions. We have $$C^{\gamma } \subset \mathcal{H} \subset L ^{2} ([0,T])$$ for all $\gamma > \frac{1}{2}-H$ where $C^{\gamma}$ denotes the class of Hölder continuous functions of order $\gamma$. [$\bullet$ ]{} The fBm admits a representation as Wiener integral of the form $$B ^{H}_{t}=\int_{0}^{t}K_{H}(t,s)dW_{s}, \label{BH}$$ where $W=\{W_{t},t\in T\}$ is a Wiener process, and $K_{ H}(t,s)$ is the kernel $$K_{H}( t,s)=d_{H}\left( t-s\right) ^{H-\frac{1}{2}}+s^{H-\frac {1}{2}}F_{1}\left( \frac{t}{s}\right) , \label{for1}$$ $d_{H}$ being a constant and $$F_{1}\left( z\right) =d_{H}\left( \frac{1}{2}-H\right) \int _{0}^{z-1}\theta^{H-\frac{3}{2}}\left( 1-\left( \theta+1\right) ^{H-\frac{1}{2}}\right) d\theta.$$ If $H>\frac{1}{2}$, the kernel $K_{H}$ has the simpler expression $$\label{K} K_{H}(t,s)= c_{H} s^{\frac{1}{2}-H} \int _{s}^{t} (u-s)^{H-\frac{3}{2}} u^{H-\frac{1}{2}} du$$ where $t>s$ and $c_{H} =\left( \frac{ H(H-1) }{\beta( 2-2H, H-\frac{1}{2}) } \right) ^{\frac{1}{2}}.$ A a $d$ dimensional fractional Brownian motion $B^{H}= (B^{H_{1}}, \ldots, B ^{H_{d}}$ with Hurst parameter $H=(H_{1}, \ldots , H_{d} ) \in (0,1) ^{d}$ is a centered Gaussian process in $\mathbb{R} ^{d} $ with independent components and the covariance of the $i$th component is given by $$R_{H_{i}}(t,s)=\mathbf{E} B^{H_{i}}_{t}B^{H_{i}} _{s} =\frac{1}{2} ( t^{2H_{i}}+ s^{2H_{i}}-\vert t-s\vert ^{2H_{i}})$$ for every $1\leq i\leq d$. The Malliavin derivative ------------------------ Here we describe the elements from the Malliavin calculus that we need in the paper. We refer [@N] for a more complete exposition. Consider ${\mathcal{H}}$ a real separable Hilbert space and $(B (\varphi), \varphi\in{\mathcal{H}})$ an isonormal Gaussian process on a probability space $(\Omega, {\cal{A}}, P)$, which is a centered Gaussian family of random variables such that $\mathbf{E}\left( B(\varphi) B(\psi) \right) = \langle\varphi, \psi\rangle_{{\mathcal{H}}}$. We denote by $D$ the Malliavin derivative operator that acts on smooth functions of the form $F=g(B(\varphi _{1}), \ldots , B(\varphi_{n}))$ ($g$ is a smooth function with compact support and $\varphi_{i} \in {{\cal{H}}}, i=1,...,n$) $$DF=\sum_{i=1}^{n}\frac{\partial g}{\partial x_{i}}(B(\varphi _{1}), \ldots , B(\varphi_{n}))\varphi_{i}.$$ It can be checked that the operator $D$ is closable from $\mathcal{S}$ (the space of smooth functionals as above) into $ L^{2}(\Omega; \mathcal{H})$ and it can be extended to the space $\mathbb{D} ^{1,p}$ which is the closure of $\mathcal{S}$ with respect to the norm $$\Vert F\Vert _{1,p} ^{p} = \mathbf{E} F ^{p} + \mathbf{E} \Vert DF\Vert _{\mathcal{H}} ^{p}.$$ We denote by $ \mathbb{D} ^{k, \infty}:= \cap _{p\geq } \mathbb{D} ^{k,p}$ for every $k\geq 1$. In our paper, $\mathcal{H}$ will be the canonical Hilbert space associated with the fractional Brownian motion, as defined in the previous paragraph. [99]{} vPotential Anal. 38(2), 573-587. Applied Mathematical Sciences, 75. Springer-Verlag, New York. Preprint. Random Operators and Stochastic Equations, 21(2), 125-134. Stochastic Partial Differential Equations, Chapman Hall/CRC, 2007. , Invent. Math., 98, 511–547. F Fedrizzi , F. Flandoli. [*Noise prevents singularities in linear transport equations*]{}, Journal of Functional Analysis, 264, 1329-1354, 2013. F. Flandoli, M. Gubinelli, E. Priola, 2010. [*Well-posedness of the transport equation by stochastic perturbation*]{}, Invent. Math., 180(1): 1-53 Ann. Probab. 30(1), 270-292. Ecole d’été de probabilités de Saint-Flour, XII—1982, 143–303, Lecture Notes in Math., 1097, Springer, Berlin. , in Proceedings of the Taniguchi International Symposium on Stochastic Analysis, North-Holland Mathematical Library, 249-269. Stochastic flows and stochastic differential equations, Cambridge University Press, 1990. Lecture Series in Mathematics and its applications, 3, Oxford University Press. , Comptes Rendus Mathematique, 349, 11-12, 669–672. Preprint. Electronic Journal of Probability, 14, paper 78, 2287-2309. Stochastic Process. Appl. 119(11), 3914-3938. Nonlinear Anal. 96, 211-215. Probab. Theory Rel. Fileds, 97(3), 403-421. Séminaire de Probabilités XL, Lecture Notes in Mathematics 1899, 147-186. Fundamental Sciences, EPFL Press, Lausanne (2005). [^1]: Supported by FAPESP 2012/18739-0 [^2]: Supported by the CNCS grant PN-II-ID-PCCE-2011-2-0015. Associate member of the team Samos, Université de Panthéon-Sorbonne Paris 1

--- abstract: 'I review recent and less recent work on globular cluster systems in early-type galaxies. Explaining their properties and possible assembly scenarios, touches on a variety of astrophysical topics from cluster formation itself to galaxy formation and evolution and even details of observational techniques. The spectacular cluster systems of central galaxies in galaxy clusters may owe their richness to a plethora of less spectacular galaxies and their star formation processes. It seems that dwarf galaxies occupy a particularly important role.' author: - Tom Richtler bibliography: - 'utrecht.bib' title: 'Globular cluster systems of early-type galaxies - do we understand them?' --- Introduction - globular clusters and dwarf galaxies =================================================== Talking about globular cluster systems with the intention to reach an audience of non-experts is a particular challenge because often, a specific discovery or result is sparkling brighter inside the cosmos of experts than outside, where the horizon is wider and sparks dim easily. On the other hand, the variety of astrophysical problems relevant to star cluster research (see the reviews of @brodie06, @harris10, and do not miss @kissler09) joins a large and explosive congregation, where sparks can have tremendous effects. The formation of globular clusters (GCs) is yet to be understood in detail (as is usually the case with dissipative processes in astrophysics), but it is not a mystery. We see GCs today forming in a variety of galaxy types, most spectacularly in starbursts, triggered by galaxy interactions (e.g. @schweizer09). The origin of the most massive star cluster known, with a mass of $\mathrm 8 \times 10^7 M_\odot$ [@maraston04], W3 in the interacting galaxy NGC 7252, can be convincingly related to the starburst, which occurred 0.5 Gyr ago. Such massive objects may form by rapid merging of star cluster complexes [@fellhauer05; @kissler06]. Also in “normal” spiral galaxies, the formation of massive star clusters is supported by a high star-formation rate [@larsen00]. Integrating the cosmic star formation rate (e.g. @hopkins04), one finds that about 65% of all stellar mass has been formed before z=1 (age 8.8 Gyr for a standard cosmology). In this epoch, the star formation rate was much higher than it is today, so most GCs in the Universe are old, but do not represent by nature the oldest populations. This “cosmic” argument does not apply to individual galaxies and many intermediate-age GCs have been found in early-type galaxies (e.g. @puzia05).\ Messier 92 is one of the best investigated Galactic GCs. Its age can currently be constrained to be $11 \pm 1.5$ Gyr [@dicecco10] ($1.6 < z < 5$), and it is one of the metal-poorest clusters. An iron abundance of \[Fe/H\]=-2.3 [@kraft03] and a mass of $1.5\times10^5$ $M_\odot$ mean a total iron mass of only 0.8$M_\odot$, a mass, which can be produced by a few SNII supernovae. That there is no detectable star-to-star variation of the iron abundance (which is not the case for other elements, e.g. @angelou12), permits the conclusion that M92 is not self-enriched and that it has formed in an already well-mixed environment. This environment cannot have been very massive since there is no significant field population with this iron abundance. If we call it a “dwarf galaxy” then we are close to the scenario proposed by @searle78 who called these entities “protogalactic fragments”. Dwarf galaxies donate GCs to the Milky Way. This becomes manifest through the Sagittarius stream [@ibata01; @siegel11] and other candidates (e.g. @pawlowski12 and references therein). Moreover, the “young halo clusters” trace the plane of Milky Way satellites [@yoon02; @kroupa05; @keller12]. @forbes10 estimate that an appreciable fraction of the Galactic GC system has been accreted through dwarf galaxies. If accretion plays an important role for the assembly of the metal-poor part of relatively isolated spiral galaxies, what role does it occupy in really dense environments? The richness of globular cluster systems ======================================== The richest globular cluster systems are not so rich ---------------------------------------------------- A popular quantitative measure for the richness of a globular cluster system (GCS) is the “specific frequency” $S_N$, which has been defined by [@harris81] as $S_N = N_{GC} 10^{0.4 (M_V+15)}$, where $N_{GC}$ is the total number of GCs and $M_V$ the host galaxy’s absolute V-magnitude. We find the richest GCSs in terms of GC number around central galaxies in galaxy clusters, the nearest being M87 (Virgo) [@tamura06; @harris09c; @strader11], NGC 1399 (Fornax) [@kissler99; @richtler04; @dirsch03a; @schuberth10], NGC 3311 (Hydra I) [@wehner08; @richtler11]. $S_N$-values for these galaxies, which can host more than 10000 GCs, are somewhat uncertain, not so much for the number of GCs, but because $M_V$ can be easily underestimated for these galaxies with very extended stellar halos. The case of NGC 3311 is illustrative, because this galaxy had the reputation of showing a particularly high $S_N$, e.g. @mclaughlin95 quoted $S_N = 15 \pm 6$. @wehner08 found 16500 GCs within a radius of 150 kpc and adopted $M_V= -22.8$, thus $S_N = 12.5$, the uncertainties still permitting an extreme lower limit of $S_N \approx 9$. But integrating the V-luminosity profile of @richtler11 out to the same radius results in $M_V = -24$ and $S_N = 4.1$, which is a normal value for giant elliptical galaxies. A similar point has been made for NGC 1399 [@ostrov98; @dirsch03a]. Therefore there is no hard evidence that $S_N$-values for central giant ellipticals are dramatically higher than for normal ellipticals. Intuition tells us that accretion of dwarf galaxies for ellipticals in galaxy clusters should be even more important than for spiral galaxies (@cote98 formulated this beyond intuition; see also @hilker99). Clear evidence, e.g. in the form of streams, is only just now emerging, for example in the cases of M87 [@romanowski12] or NGC 3311 [@arnaboldi12]. The halos of these central galaxies have been built up by long-term accretion of material from the cluster environment, a process which is still on-going. Strong evidence for a significant growth of massive elliptical galaxies since z=2 (10.3 Gyr) has been provided by @vandokkum10. They find that outside a core with a size of about 5kpc, elliptical galaxies increased their mass by a factor of 4 within the last 10 Gyr. This happens predominantly through minor and dry mergers [@tal12]. Obviously, GCSs should share the same fate. For NGC 1399, kinematical data indicate that many GCs cannot have formed [*in situ*]{}. One finds in the GCS of NGC 1399 objects, which by their high radial velocities must reach apocentric distances of 500 kpc or even greater [@richtler04; @schuberth10]. These few objects near their pericenters must trace a much larger (unknown) population with high space velocities, but low radial velocities. Their orbital velocities result from potential differences that exist within the Fornax cluster rather than within NGC 1399, so one may call them “intracluster GCs” [@kissler99]. The poorest globular cluster systems can be the richest {#sec:dwarfs} ------------------------------------------------------- The GCSs of early-type dwarf galaxies are as interesting as those of giant ellipticals. The highest specific frequency known is that of the Fornax dwarf spheroidal with 5 GCs ($S_N\approx30$). Are dwarf galaxies for some reason more efficient in forming GCs? @miller07 indeed found a trend of increasing $S_N$ with decreasing brightness of the host galaxy for dwarf ellipticals in Fornax, Virgo and the Leo group. The largest data base in this respect is the HST/ACS Virgo survey [@peng08], in which about 100 early-type Virgo galaxies have been imaged down to a magnitude of $M_V \approx -16$. This work does not support a clear relation between $S_N$ and host galaxy brightness, but dwarf galaxies fill a larger $S_N$-interval than giant ellipticals. Probably a key finding is that dwarf galaxies with large clustercentric distances consistently show low $S_N$-values, while high $S_N$-values are found among the (projected) inner dwarf galaxy population, i.e. among those galaxies with a higher probability of interactions, which may have triggered star-bursts. The phenomenon of “bimodality” =============================== The metallicity distribution of Galactic GCs is “bimodal”, i.e. well represented by two Gaussians, with metal-rich and metal-poor GCs being the bulge and the halo clusters, respectively [@zinn85]. Do we find a similar fundamental structure in the GCSs of elliptical galaxies? In an influential paper, @ashman92 explored the idea that GCs form in mergers and interactions of galaxies (see their introduction for the history of this concept) and hypothesized that the metallicity distribution of GCs in elliptical galaxies should be bimodal. Adopting the merger paradigm for elliptical galaxies, metal-poor clusters are the old GCs of the pre-merger components, while metal-rich clusters form in starbursts triggered by gas-rich mergers of spiral galaxies. This [*prediction*]{} of bimodality in the [*color*]{} distribution of GCs has indeed been found in many elliptical galaxies (e.g. @zepf93 [@whitmore95; @gebhardt99; @larsen01; @kundu01; @peng06; @harris09a; @harris09b]). Studies in the Washington photometric system showed this bimodality particularly well (e.g. @geisler96 [@dirsch03a; @dirsch03b; @bassino06]): it can be characterized by two Gaussians with peaks at C-R=1.35 (the “blue” peak) and C-R=1.75 (the “red” peak) (these peaks are not found in Fig.\[fig:n1316\_utrecht\]!). The blue peak is remarkably constant among the investigated galaxies, while the red peak gets slightly redder with increasing host galaxy brightness (see also @larsen00). However, this color bimodality does not apply to the brightest clusters in some GCSs, which avoid very red and very blue colors. The HST/ACS surveys in Virgo and the Fornax confirm this with a much larger database [@peng06; @peng08; @jordan09]. It turns out that bimodal color distributions are mainly a signature of bright host galaxies, which does not come as a surprise: early-type galaxies follow a well-defined color-magnitude relation (e.g. @smithcastelli08 [@misgeld08]) and one does not expect GCs to be redder than the host galaxy itself. A striking difference between the blue and red clusters is that the mean color of the red subpopulation strongly correlates with the host galaxy luminosity and thus with its color, while the mean color of blue clusters is more or less constant. Blue and red GC populations, however, differ in more than only their colors. As shown in many papers, the radial number density profiles of red clusters are steeper and resemble more the light profile of their host galaxies. Accordingly, they show a lower velocity dispersion than the blue clusters. For NGC 1399, Fig.20 of @schuberth10 demonstrates that the velocity dispersion exhibits a sudden change between red and blue clusters, not a smooth transition. Do these bimodal color distributions reflect bimodal metallicity distributions? There had been some claims that a non-linearity of the color-metallicity relation, in combination with a considerable scatter around this relation, can produce a bimodal color distribution, even if the underlying metallicity distribution is not bimodal [@richtler06; @yoon06]. More recent work strengthens this point. @yoon11a [@yoon11b] derive GC metallicity distributions from a non-linear color-metallicity relation and show that the inferred metallicity distributions are rather single-peaked (I caution that their example NGC 4374 is a multiple SNIa host galaxy, see more remarks below). A further complicating point is that their color-metallicity relation becomes so insensitive to metallicity for clusters metal-poorer than about \[Fe/H\]=-1.5, that to infer a metallicity distribution from a color distribution seems to be only meaningful if color and metallicity are related without any generic scatter. But this is not the case, because metal-poor Galactic GCs having the same metallicity can show quite different CMDs (Fig.\[fig:BIcolor\]). Therefore, the reconstructed metallicity distribution for metal-poor clusters might be considered as a mean distribution without saying much about an individual object. However, a weak metallicity-brightness relation for metal-poor clusters, as described by @harris09b, is expected, because massive clusters have a higher probability to originate from more massive host galaxies with an overall higher metallicity. ![Colour-metallicity relations for Galactic GCs with reddenings less than E(B-V)=0.15, using B-I data from @harris96 and C-T1 data from @harris77. The non-linearity is clearly seen. For clusters metal-poorer than about \[Fe/H\]=-1.5, the color becomes a very bad proxy for the metallicity, more so for B-I than for C-T1. Note, however, the large deviations for some clusters. Also note that these colors, measured with diaphragms, refer only to the innermost parts of GCs. []{data-label="fig:BIcolor"}](metalcolorwash.pdf){width="70.00000%"} If IR-bands are included, bimodality may vanish almost completely, as shown by @blakeslee12 for NGC 1399, and @chies11 [@chies12] for a sample of 17 early-type galaxies. Galaxies with very pronounced blue peaks in g-z are NGC 4374 and NGC 4526. These galaxies hosted three and two SNIa events, respectively. Therefore, one may assume significant intermediate-age populations to be present, and probably also intermediate-age GCs, and the color is not anymore a pure metallicity indicator. More SNIa host galaxies in their sample are NGC 4382, NGC 4621, and NGC 4649. The striking deficiency of red clusters in NGC 4660 (which one also finds in the sample of @peng06) is remarkable. There may be more individuality among GCSs than previously thought. In those cases, where metallicities of larger GC samples have been derived from integrated spectra (not only of early-type galaxies!) [@foster10; @foster11; @caldwell11], the metallicity distributions appear unimodal with one broad peak around \[Fe/H\]=-1. Once more: giant and dwarf galaxies =================================== The question to what degree $S_N$-values of galaxies reflect the efficiency of GC formation is difficult to answer (relating GC masses to host galaxy mass as do e.g. @peng08 [@georgiev10] would be more physical, but observationally more difficult to determine). Although the $S_N$-values of central galaxies may not be as high as previously thought, they are still higher than elliptical galaxies of lower luminosity. The ACS Virgo survey [@peng08] reveals a shallow minimum around $M_V \approx$ -20, but without a well-defined relation for the dwarf or for the giant regime, although many dwarf galaxies as faint as $M_V = -16$ show $S_N$-values rivalling or exceeding typical values for giant galaxies. Including fainter galaxies from @lotz04 lets the trend of increasing $S_N$ with decreasing luminosity appears clearer: values higher than 10 are normal, particularly for nucleated dwarf galaxies. This is also visible in the compilation of @georgiev10. They embed the variation of $S_N$-values in the context of galaxy formation, inspired by @dekel06 (see @forbes05 for an earlier account). In brief, star formation in low mass galaxies is regulated by stellar feedback, in high mass galaxies by virial shocks, which in both regimes leads to a suppression of field star formation, and favors star clusters. One notes that this interpretation implies an alternative view: here it is a generic property of star formation processes in giant ellipticals, which produces the rich GCSs, not the infall of less massive galaxies with intrinsically higher $S_N$-values. It may be, however, difficult to defend this view in front of all evidence for the role of accretion. Starbursts in dwarf galaxies might hold the key for a proper understanding. Can metallicity itself be a parameter for efficient GC formation? At low metallicity, the energy and momentum input into the interstellar medium via radiation-driven stellar winds is reduced [@kudritzki02], leading to a higher star formation efficiency [@dib11]. @glover12a find that molecules are not necessarily a prerequisite for star formation, but that dust is an important ingredient for cooling processes. Star formation occurs in cold gas at higher temperatures, and Jeans masses are increased [@glover12b]. The formation of dense substructures in a collapsing cloud is efficiently suppressed at low metallicities, because turbulence cannot create clumps as efficiently as in high-metallicity clouds [@glover12b]. This may lead to a more coherent star formation inside a star-forming cloud. These are all factors which support the dynamical stability of star-forming clouds, and plausibly favor the formation of compact and coherent structures, which lead to the extremely clustered star formation for example in Blue Compact Galaxies (e.g. @adamo11). Globular clusters outside galaxies ================================== “Cosmological” formation of GCs, i.e. GCs embedded in a dark halo, is interesting to imagine, but difficult to prove. GCs apparently exist outside galaxies, however, this does not mean that they were formed outside galaxies. The objects with high radial velocities around central galaxies spend most of their orbital life far way from the cluster center. But there are also GC populations in galaxy clusters without a central galaxy. @west11 found in a population of intergalactic GCs in Abell 1185, mostly metal-poor. A recent survey with HST/ACS of the Coma cluster uncovered a huge population of GCs, filling the entire cluster core [@peng11]. The authors estimate a total of about 47000 GCs out to a radius of 570 kpc. Dissolution of dwarf galaxies and tidal stripping from more massive galaxies might both contribute to create this largest GCS in the local Universe. Isolated elliptical galaxies ============================ Suspecting dwarf galaxies as donators of metal-poor GCs, it would be interesting to compare cluster galaxies with isolated elliptical galaxies. Unfortunately, the data are sparse. After the compilation of @spitler08, no new work on isolated ellipticals has been reported, leaving NGC 720 [@kissler96] and NGC 821 [@spitler08] as prototypes. Both present relatively poor cluster systems. However, many “isolated ellipticals” exhibit tidally disturbed features and may be in fact late mergers [@tal09], including NGC 720 that does not exhibit obvious morphological peculiarities, but strong population gradients [@rembold05]. NGC 1316 - cluster formation in a late merger ============================================= A galaxy whose brightness is dominated by intermediate-age populations [@kuntschner00], is the merger remnant NGC 1316 (Fornax A) in the outskirts of the Fornax galaxy cluster. It might illustrate the processes that were in action during the youth of giant ellipticals to form metal-rich clusters (see @richtler12a for references). It experienced a major starburst about 2-3 Gyr ago that produced many massive star clusters, the brightest one (114 in the list of @goudfrooij01a) having a mass of the order $2\times10^7 M_\odot$. Note that @brodie11 would not call it an “Ultra Compact Dwarf”, because its effective radius is smaller than 10 pc [@goudfrooij12]. The imprint of the 2 Gyr starburst is a well defined peak in the color distribution of GCs [@richtler12a] (Fig.\[fig:n1316\_utrecht\]). Cluster ages from integrated spectra have been determined only for a few of the brightest clusters [@goudfrooij01b], but the colors (with the assumption of solar metallicity for all bright clusters) fit well to these ages. Star formation continued after this starburst and a second peak corresponds to 0.8 Gyr (which still has to be confirmed by spectroscopy). We find GCs with ages as young as 0.5 Gyr. Including fainter clusters probably samples older objects and lets these peaks largely disappear. A very interesting observation is that the brightest clusters seem to avoid the systemic radial velocity of NGC 1316 by showing large negative offsets up to 500 km/s. This indicates elongated orbits and a population of massive clusters far away from the center, which still has to be identified. The starburst seem to have happened in a very early stage of the merger with the merger components still separated with high relative velocities. Very remarkable is the object SH2, discovered by @schweizer80. Rather than as a normal HII-region, it appears as an ensemble of star clusters with ages around 0.1 Gyr [@richtler12b]. Its exact nature still awaits investigation, but a plausible hypothesis is that of an infalling dwarf galaxy, having recently experienced a starburst. In this case, one would expect low metallicities and it would constitute an example, how metal-poor clusters are donated to a GCS. ![ Left panel: color histogram of 178 confirmed GCs in NGC 1316 (Richtler et al., in preparation). The ages assume solar metallicity. Right panel: radial velocities vs. R-magnitudes for the same clusters augmented by a sample from @goudfrooij01b (crosses). Note, how the bright cluster population avoids the systemic velocity (dashed line) and how the velocity dispersion increases with fainter magnitudes.[]{data-label="fig:n1316_utrecht"}](n1316_utrecht.pdf){width="80.00000%"} Inventory ========= A cautiously positive answer to the title question seems to be appropriate. A description of a global picture (which generously ignores details) can be: giant elliptical galaxies assembled their metal-poor GCs by the accretion of dwarf galaxies. The metal-rich GCs formed in early starbursts, triggered by gas-rich mergers. Whether colors alone can provide an adequate description of the substructure of an GCS, has become doubtful. With higher precision of kinematical data, more substructures in GCSs will be detected and perhaps single merger events can be identified. The key to all that is the physics of star cluster formation in starbursts, whether in dwarf galaxies or in massive systems. Modern simulations of galaxy mergers now resolve star cluster scales (see @hopkins12 and references therein) and will probably provide the physical understanding.

--- abstract: 'This work investigates the scattering coefficients for inverse medium scattering problems. It shows some fundamental properties of the coefficients such as symmetry and tensorial properties. The relationship between the scattering coefficients and the far-field pattern is also derived. Furthermore, the sensitivity of the scattering coefficients with respect to changes in the permittivity and permeability distributions is investigated. In the linearized case, explicit formulas for reconstructing permittivity and permeability distributions from the scattering coefficients is proposed. They relate the exponentially ill-posed character of the inverse medium scattering problem at a fixed frequency to the exponential decay of the scattering coefficients. Moreover, they show the stability of the reconstruction from multifrequency measurements. This provides a new direction for solving inverse medium scattering problems.' author: - 'Habib Ammari[^1]' - 'Yat Tin Chow[^2]' - 'Jun Zou[^3]' title: The Concept of Heterogeneous Scattering Coefficients and Its Application in Inverse Medium Scattering --- [Mathematics Subject Classification (MSC2000): 35R30, 35B30]{} [Keywords: inverse medium scattering, scattering coefficients, heteregeneous inclusions, far-field measurements, sensitivity, reconstruction algorithm]{} Introduction ============ In this work we will be concerned with the following transverse magnetic polarized wave scattering problem u + \^2 u = 0 & \^2 , \[scattering1\] where $\mu, \varepsilon > 0 $ are the respective permittivity and permeability coefficients of the medium. We consider an inhomogeneous medium $\Omega$ contained inside a homogeneous background medium, and assume that $\Omega$ is an open bounded connected domain with a $\mathcal{C}^{1,\alpha}$ boundary for some $0< \alpha<1$. Let $\nu$ denote the outward normal vector at $\partial \Omega$, and $\mu_0, \varepsilon_0 > 0 $ be the medium coefficients of the homogeneous background medium. Suppose that $\mu, \varepsilon \in L^\infty$ and $\mu - \mu_0$ and $\varepsilon - \varepsilon_0$ are supported in $\Omega$. Moreover, there exist positive constants $\underline{\mu}$ and $\underline{\varepsilon}$ such that $\mu(x)\geq \underline{\mu}$ and $\varepsilon(x) \geq \underline{\varepsilon}$ in $\Omega$. Under these settings, we can write the equation (\[scattering1\]) as follows: u + \^2 u = 0 & ,\ u + k\_0\^2 u = 0 & \^2 \\ ,\ u\^+ = u\^[-]{} & ,\ = & .\ \[scattering2\] Here and throughout this paper, the superscripts $\pm$ indicate the limits from outside and inside of $\Omega$, respectively, and $\partial/\partial \nu$ denotes the normal derivative. We shall complement the system (\[scattering2\]) by the physical outgoing Sommerfeld radiation condition: (u - u\_0) - i k\_0 (u - u\_0) = O(|x|\^[-]{})& |x| . \[sommerfield\] where $k_0 = \omega \sqrt{\mu_0 \varepsilon_0} $ is the wavenumber and $u_0$ is an incident field, solving the homogeneous Helmholtz equation $ (\Delta + k_0^2) u_0 = 0$ in $\mathbb{R}^d$. The solution $u$ to the system (\[scattering2\]) and (\[sommerfield\]) represents the total field due to the scattering from the inclusion $\Omega$ corresponding to the incident field $u_0$. The notion of scattering coefficients was previously studied for homogeneous electromagnetic inclusions [@homoscattering] (see also [@lim]) in order to enhance near-cloaking. The purpose of this paper is twofold. We first introduce the concept of inhomogeneous scattering coefficients and investigate some of their important properties and their sensitivity with respect to changes in the physical parameters. Then we make use of this new concept for solving the inverse medium scattering problem and understanding the associated fundamental issues of stability and resolution. The inhomogeneous scattering coefficients can be obtained from the far-field data by a least-squares method [@tran]. Explicit reconstruction formulas of the inhomogeneous electromagnetic parameters from the scattering coefficients at a fixed frequency or at multiple frequencies are derived in the linearized case. These formulas show that the exponentially ill-posed characteristics of the inverse medium scattering problem at a fixed frequency [@gallagher; @isakov1; @john] is due to the exponential decay of the scattering coefficients. Moreover, they clearly indicate the stability of the reconstruction from multifrequency measurements [@bao2; @bao3; @isakov2; @isakov3]. Based on the decay property of the inhomogeneous scattering coefficients, a resolution analysis analogous to the one in [@foundations] can be easily derived. The resolving power, i.e., the number of scattering coefficients which can be stably reconstructed from the far-field measurements, can be expressed in terms of the signal-to-noise ratio in the far-field measurements. The scattering coefficient based approach introduced in this paper is a new promising direction for solving the long-standing inverse scattering problem with heterogeneous inclusions. It could be combined with the continuation method developed in [@bao1; @coifman] for achieving a good resolution and stability for the image reconstruction. For the sake of simplicity, we shall restrict ourselves to the scattering problem in two dimensions, but all the results and analysis hold true also for three dimensions. The paper is organized as follows. In section \[sec2\] we introduce the notion of inhomogeneous scattering coefficients. Section \[sec3\] provides integral representations of the scattering coefficients and shows their exponential decay. This property is the root cause of the exponentially ill-posed character of the inverse medium scattering problem. In section \[sec4\] we prove that the scattering coefficients are nothing else but the Fourier coefficients of the far-field pattern, then derive transformation formulas for the scattering coefficients under rigid transformations and scaling in section \[sec5\]. In section \[sec6\] we provide a sensitivity analysis with respect to the electromagnetic parameters for the scattering coefficients. In section \[sec7\] we derive new reconstruction formulas from the scattering coefficients at one frequency and at multiple frequencies as well. A few concluding remarks are given in section \[sec8\]. Appendix \[appendixA\] is to construct a Neumann function for the inhomogeneous Helmholtz equation on a bounded domain. Appendices \[appendixB\] and \[appendixC\] are to show the existence of some functions used in the derivation of the explicit reconstruction formulas in the linearized case. Integral representation and scattering coefficients {#sec2} =================================================== In this section we define the scattering coefficients of inhomogeneous inclusions. The idea of scattering coefficients for inclusions with homogeneous permittivity and permeability was initially introduced in [@homoscattering]. We extend this idea and define such a notion for inhomogeneous inclusions following the idea in [@nonhomopolarization; @homoscattering]. We first derive the fundamental representation of the solution $u$ to the system (\[scattering2\])-(\[sommerfield\]). For $k_0>0$, let $\Phi_{k_0}$ be the fundamental solution to the Helmholtz operator $\Delta + k_0^2$ in two dimensions satisfying $$(\Delta + k_0^2)\Phi_{k_0}(x) = \delta_0(x)$$ subject to the outgoing Sommerfeld radiation condition: $$\f{\partial}{\partial r}\Phi_{k_0} - i k_0 \Phi_{k_0} = O(|x|^{-\f{3}{2}}) \quad \text{ as } |x| \rightarrow \infty \, .$$ Then $\Phi_{k_0}$ is given by \_[k\_0]{} (x) = - H\^[(1)]{}\_0(k\_0|x|) , \[fundamental\] where $H^{(1)}_0$ is the Hankel function of the first kind of order zero. We can easily deduce from Green’s formula that if $u$ is the solution to (\[scattering2\])-(\[sommerfield\]), then we have for $x \in \mathbb{R}^2 \backslash \overline{\Omega}$ that (u - u\_0)(x) &=& \_ \_[k\_0]{} (x-y) (y) d (y) - \_ (u- u\_0)\^+ (y) d (y)\ &=& \_ \_[k\_0]{} (x-y) (y) d (y) - \_ u\^+ (y) d (y)\ &=& \_ () \_[k\_0]{} (x-y) (y) d (y) - \_ u\^- (y) d (y) , \[scattered1\] where the second equality holds since $u_0$ satisfies the homogeneous Helmholtz equation. Let $g = \f{1}{\mu} \f{\p u^{-}}{\p \nu} $. Then we define the Neumann-to-Dirichlet (NtD) map $\Lambda_{\mu,\varepsilon}$: $H^{-\f{1}{2}}(\p \Omega) \to H^{\f{1}{2}}(\p \Omega)$ such that for any $g\in H^{-\f{1}{2}}(\p \Omega)$, $u=\Lambda_{\mu,\varepsilon}g\in H^{\f{1}{2}}(\p \Omega)$ is the trace of the solution to the following system: u + \^2 u = 0 ; = g . \[interior\_pde\] We remark that $\Lambda_{\mu_0,\varepsilon_0}$ is well-defined if $\omega \sqrt{\mu_0 \varepsilon_0}$ is not a Neumann eigenvalue of $-\Delta$ on $\Omega$. For general distributions $\varepsilon$ and $\mu$, in order to ensure the well-posedeness of $\Lambda_{\mu,\varepsilon}$, one should assume, throughout this paper, that $0$ is not a Neumann eiganvalue of $\nabla \cdot (1/\mu) \nabla + \omega^2 \varepsilon$ in $\Omega$. With this definition of $\Lambda_{\mu,\varepsilon}$, we have $\Lambda_{\mu,\varepsilon} [g] = u^{-}$ and $\f{1}{\mu_0}\Lambda_{\mu_0,\varepsilon_0} [\f{\p\Phi_{k_0}}{\p \nu}] = \Phi_{k_0} $ on $\p \Om$. We can therefore rewrite (\[scattered1\]) as (u - u\_0)(x) &=& \_ \_0 \_[k\_0]{} (x-y) g(y) d (y) - \_ \_0 \_[\_0,\_0]{}\^[-1]{} \[\_[k\_0]{}\](x-y)\_[,]{} \[g\] (y) d (y) . \[scattered2\] One can check that $\Lambda_{\mu,\varepsilon}$ is self-adjoint under the duality pair $\langle \cdot, \cdot \rangle_{H^{-\f{1}{2}},H^{\f{1}{2}}}$ on $\p \Omega$. So, we can further write (u - u\_0)(x) = \_0 \_ \_[k\_0]{} (x-y) \_[\_0,\_0]{}\^[-1]{} ( \_[\_0,\_0]{} - \_[,]{} ) \[g\] (y) d (y) , x \^d \\. \[ingegral1\] We now use Graf’s addition formula [@Watson] to derive an asymptotic expression of $u - u_0$ as $|x| \rightarrow \infty$. For the fundamental solution (\[fundamental\]), we recall the Graf’s addition formula for $|x| > |y|$: H\^[(1)]{}\_0 (k\_0 |x-y|) = \_[n]{} H\^[(1)]{}\_n (k\_0 |x|) e\^[i n \_x]{} J\_n (k\_0 |y|) e\^[-i n \_y]{} , \[graf\] where $x$ is in polar coordinate $(|x|,\theta_x)$, and the same for $y$. Now we define (u\_0)\_m (y) := J\_m (k\_0 |y|) e\^[i m \_y]{}, \[incidence\] and let $u_m$ to be the total field corresponding to the incident field $(u_0)_m$, namely the solution to (\[scattering2\])-(\[sommerfield\]) with the incident field $u_0$ replaced by $(u_0)_m$. If we write g\_m := , then for any incident field $u_0$ admitting the expansion u\_0 (y) = \_[m]{} a\_m J\_m (k\_0 |y|) e\^[i m \_y]{} , we have g = = \_[m]{} a\_m g\_m . \[gm\] Putting (\[graf\]) and (\[gm\]) into (\[ingegral1\]), we get the following asymptotic formula as $|x| \rightarrow \infty$: & &(u - u\_0)(x)\ &=& - \_[m,n]{} \_ a\_m H\^[(1)]{}\_n (k\_0 |x|) J\_n (k\_0 |y|) e\^[i n (\_x - \_y )]{} \_[\_0,\_0]{}\^[-1]{} ( \_[\_0,\_0]{} - \_[,]{} ) \[g\_m\] (y) d (y) . \[ingegral2\] This motivates us to introduce the following definition. The scattering coefficients $\{W_{nm}\}_{m,n \in \mathbb{Z}}$ at frequency $\omega$ of the inhomogeneous scatterer $\Omega$ with the permittivity and permeability distributions $\varepsilon, \mu$ are defined by W\_[nm]{} = W\_[nm]{} \[, , , \] := \_0 \_ J\_n (k\_0 |y|) e\^[-i n \_y]{} \_[\_0,\_0]{}\^[-1]{} ( \_[\_0,\_0]{} - \_[,]{} ) \[g\_m\] (y) d (y). \[eq:w\_nm\] With this definition and the derivations above, we immediately come to the following integral representation theorem from (\[ingegral2\]). For an incident field of the form $u_0 (y) = \sum_{m\in \mathbb{Z}} a_m J_m (k_0 |y|) e^{i m \theta_y}$, the total field $u$ (i.e., the solution of (\[scattering2\])-(\[sommerfield\])) has the following asymptotic representation: (u - u\_0)(x) = - \_[m,n]{} a\_m H\^[(1)]{}\_n (k\_0 |x|) e\^[i n \_x]{} W\_[nm]{} |x| . \[eq:ufromwnm\] Representation and decay property of scattering coefficients {#sec3} ============================================================ In this section we would like to represent the scattering coefficients using layer potentials and study their decay properties. In order to do this, we first introduce the Neumann function of the Helmholtz equation and the single and double layer potentials. Let $N_{\mu,\varepsilon} (x,y)$ be the fundamental solution to the problem (\[interior\_pde\]), i.e., for each fixed $z \in \Omega$, $N_{\mu,\varepsilon} (\cdot , z)$ is the solution to N\_[,]{} (, z) + \^2 N\_[,]{} (, z) = - \_[z]{} () ; N\_[,]{} (, z) = 0 . \[Neumann\] Let $\mathcal{N}_{\mu,\varepsilon} [g] (x) := \int_{\partial \Omega} N_{\mu,\varepsilon} (x , y) g(y) d \sigma(y)$ for $x \in \Omega$. Then we can see that $\mathcal{N}_{\mu,\varepsilon} [g] (x)$ is the solution to (\[interior\_pde\]), and that \_[,]{} \[g\] (x) = \_[,]{} \[g\] (x) , x , \[def\_ntd\] by noting the relation (cf. [@book]) $$\f{1}{\mu} \f{\partial}{\partial \nu} \mathcal{N}_{\mu,\varepsilon}[g]= g \quad \mbox{on } \partial \Omega\,.$$ Let $\mathcal{S}_{k_0} [\phi]$ and $\mathcal{D}_{k_0} [\phi]$ be the following single and double layer potentials on $\p \Om$: \_[k\_0]{} \[\](x) = \_ \_[k\_0]{}(x-y) (y) d (y), x \^2, and \_[k\_0]{} \[\](x) = \_ (x-y) (y) d (y), x \^2 . Then the layer potentials $\mathcal{S}_{k_0}$ and $\mathcal{D}_{k_0}$ satisfy the following jump conditions: ( \_[k\_0]{}\[\] )\^ = ( I + \^\*\_[k\_0, ]{} )\[\], ( \_[k\_0]{}\[\] )\^ = ( I + \_[k\_0, ]{} ) \[\], \[jump\_condition\] where $\mathcal{K}_{k_0, \Omega}$ is the boundary integral operator defined by $$\mathcal{K}_{k_0, \Omega} [\phi](x) = \int_{\partial \Omega} \f{\p \Phi_{k_0}}{\p \nu_y}(x-y) \phi(y) d \sigma (y)$$ and $ \mathcal{K}^*_{k_0, \Omega}$ is the $L^2$ adjoint of $ \mathcal{K}_{k_0, \Omega}$ with $L^2$ being equipped with the real inner product. Note that $ \f{1}{2} I + \mathcal{K}^*_{k_0, \Omega}$ is invertible if $k_0^2$ is not a Dirichlet eigenvalue of $-\Delta$ on $\Omega$; see [@eigenpaper; @book]. From (\[ingegral1\]) and the transmission conditions (\[scattering2\]), we can see that the solution $u$ to (\[scattering2\])-(\[sommerfield\]) can be represented as u(x) = u\_0 (x) + \_0 \_[k\_0]{} \[\] x \^d \\ ; u(x) = \_[,]{} \[\] x \[total\_u\] for some density pair $(\phi, \psi) \in L^2(\partial \Omega) \times L^2(\partial \Omega)$ which satisfies $$u_0 = \Lambda_{\mu,\varepsilon}[\psi] - \mu_0 \mathcal{S}_{k_0} [\phi] \quad \m{and} {\quad}\f{1}{\mu_0} \f{\p u_0}{\p \nu} = - ( \f{1}{2} I + \mathcal{K}^*_{k_0, \Omega} ) [\phi] + \psi {\quad}\m{on} {\quad}\partial \Omega\,.$$ If we define A := - \_0 \_[k\_0]{} & \_[,]{}\ - ( I + \^\*\_[k\_0, ]{} ) & I , \[operator\_A\] then we can write $(\phi,\psi)$ as the solution to the following equation A \ = u\_0\ ,\[potential\] and show the following result. The operator $A: L^2(\partial \Omega) \times L^2(\partial \Omega) \rightarrow L^2(\partial \Omega) \times L^2(\partial \Omega)$ is invertible. Let $(\phi,\psi) \in L^2(\partial \Omega) \times L^2(\partial \Omega)$ be such that $ A \begin{pmatrix} \phi \\ \psi \end{pmatrix} = 0$. Let $u$ be defined by $$u = \left\{\begin{array}{l} \mathcal{N}_{\mu,\varepsilon} [\psi] \quad \text{ in } \Omega,\\ \mu_0 \mathcal{S}_{k_0} [\phi] \quad \text{ in } \mathbb{R}^d \backslash \overline{\Omega}. \end{array} \right.$$ From the jump conditions $$\left\{ \begin{array}{l} \mu_0 \mathcal{S}_{k_0} [\phi] = \mathcal{N}_{\mu,\varepsilon} [\psi] \quad \text{ on } \partial \Omega, \\ \mu_0 ( \f{1}{2} I + \mathcal{K}^*_{k_0, \Omega} ) = \frac{\partial}{\partial \nu} \mathcal{N}_{\mu,\varepsilon} [\psi] = \mu_0 \psi \quad \text{ on } \partial \Omega, \end{array} \right.$$ one can see that $u$ satisfies the Helmholtz equation (\[scattering1\]) together with the outgoing Sommerfeld radiation condition: u - i k\_0 u = O(|x|\^[-]{})& |x| . \[sommerfieldo\] Uniqueness of a solution to (\[scattering1\]) subject to the Sommerfeld radiation condition (\[sommerfieldo\]) shows that $u=0$ in $\mathbb{R}^d$. Then, since $k_0^2$ is not a Dirichlet eigenvalue of $-\Delta$ on $\Omega$, we have $\phi=0$, hence $\psi=0$ as well. This shows the injectivity of $A$. Next, since $\f{1}{\mu_0} \Phi_{k_0}(|x-y|)$ and $N_{\mu,\varepsilon}(x,y)$ have the same singularity type (i.e., of logarithmic type) as $|x-y| \rightarrow 0$ [@singular] (see Appendix \[appendixA\]) and $\mathcal{K}^*_{k_0, \Omega}$ is a compact operator on $L^2(\partial \Omega)$, it follows that $A$ is a compact perturbation of the invertible operator on $ L^2(\partial \Omega) \times L^2(\partial \Omega)$ which is given by $$\begin{pmatrix} - \mu_0 \mathcal{S}_{k_0} & \mu_0 \mathcal{S}_{k_0} \\ - \frac{1}{2} I & I \end{pmatrix} \,.$$ Therefore, Fredholm alternative holds and injectivity of $A$ shows its invertibility. We define $(\phi_m, \psi_m)$ as the pair of solution to the above equation (\[potential\]) corresponding to the incident field $u_0 (y) = (u_0)_m (y) := J_m (k_0 |y|) e^{i m \theta_y} $ defined as in (\[incidence\]), then $W_{nm}$ can be simply expressed as W\_[nm]{} = \_0 \_ J\_n (k\_0 |y|) e\^[ - i n \_y]{} \_m (y) d (y) = \_0 (u\_0)\_n, \_m \_[L\^2()]{} . \[inner\] Using this expression, we can derive the decay property of scattering coefficients. Again from the fact that the functions $\f{1}{\mu_0} \Phi_{k_0}(|x-y|)$ and $N_{\mu,\varepsilon}(x,y)$ have the same logarithmic type singularity as $|x-y| \rightarrow 0$ [@singular], we obtain from (\[potential\]) that ||\_m||\_[L\^2()]{} + ||\_m||\_[L\^2()]{} C ( ||(u\_0)\_m ||\_[L\^2()]{} + || (u\_0)\_m ||\_[L\^2()]{} ). Using the asymptotic behavior of the Bessel function $J_m$ [@handbook], J\_m (t) / ()\^[|m|]{} 1 \[decayhaha\] as $m \rightarrow \infty$, we have ||(u\_0)\_n ||\_[L\^2()]{} ||\_m ||\_[L\^2()]{} for some constants $C_1$ and $C_2$. Therefore, we deduce from (\[inner\]) that $$|W_{nm}| = | \mu_0 \langle (u_0)_m, \phi_m \rangle_{L^2(\partial \Omega)} | \leq ||(u_0)_n ||_{L^2(\partial \Omega)} ||\phi_m ||_{L^2(\partial \Omega)} \leq \f{C^{|m|+|n|}}{|m|^{|m|} |n|^{|n|}}$$ for some constant $C$, leading to the following theorem. There exists a constant $C$ depending on $(\mu,\varepsilon,\omega)$ such that |W\_[nm]{}| n, m . \[decay\_conclusion\] Far-field pattern {#sec4} ================= In this section we shall derive the far-field pattern of the scattered field in terms of the scattering coefficients. We consider the incident field $u_0$ as a plane wave of the form $u_0 = e^{i k_0 \xi \cdot x}$ with $\xi$ being on the unit circle. We recall the Fourier mode $(u_0)_m (y) := J_m (k_0 |y|) e^{i m \theta_y}$ in (\[incidence\]), and the solution pair $(\phi_m, \psi_m)$ to (\[potential\]) corresponding to the incident field $(u_0)_m$. Then by the well-known Jacobi-Anger decomposition, we have the following decomposition of the plane wave in terms of $(u_0)_m$: u\_0 = e\^[i k\_0 x]{} = \_[m]{} e\^[i m ( - \_)]{} J\_m (k\_0 |x|) e\^[i m \_x]{} = \_[m]{} e\^[i m ( - \_)]{} (u\_0)\_m , ł[eq:u0]{} where $\xi=(\cos \theta_\xi, \sin\theta_\xi)$ and $x=|x|(\cos \theta_x, \sin \theta_x)$. Let $(\phi, \psi)$ be the solution pair to (\[potential\]) corresponding to the incident field $u_0 = e^{i k_0 \xi \cdot x}$, then using (\[eq:u0\]) and the principle of superposition we have = \_[m]{} e\^[i m ( - \_)]{} \_m = \_[m]{} e\^[i m ( - \_)]{} \_m . It follows directly from (\[total\_u\]) that u - e\^[i k\_0 x]{} = \_0 \_ \_[k\_0]{} (x-y) (y) d (y) = \_0 \_[m]{} e\^[i m ( - \_)]{} \_ \_[k\_0]{} (x-y) \_m(y) d (y). \[representation\_plane\] In order to derive the far-field pattern from expression (\[representation\_plane\]), we consider the asymptotic expansion of $\Phi_{k_0} (x-y)$ as $|x| \rightarrow \infty$. Noting the expression (\[fundamental\]) of $\Phi_{k_0}$ and the two approximations that H\^[(1)]{}\_0 (t) = ( e\^[i (t - )]{} + O(t\^[-1]{}) ) = e\^[i (t - )]{} + O(t\^[-]{}) t and $|x-y| = |x| - |y| \cos(\theta_x - \theta_y) + O(|x|^{-1})$ as $|x| \rightarrow \infty$, we arrive at the following asymptotic expansion of $\Phi_{k_0} (x-y)$: \_[k\_0]{} (x-y) = e\^[- i ]{} e\^[i k\_0 (|x| - |y| (\_x - \_y)) ]{} + O(|x|\^[-]{}) |x| . \[fundamental2\] Substituting this into (\[representation\_plane\]) yields u - e\^[i k\_0 x]{} = -i e\^[- i ]{} \_[m]{} e\^[i m ( - \_)]{} \_ e\^[- i k\_0 |y| (\_x - \_y)]{} \_m(y) d (y) + O(|x|\^[-]{}) , \[farfield1\] from which and the Jacobi-Anger identity e\^[- i k\_0 |y| (\_x - \_y)]{} = \_n J\_n( k\_0 |y|) e\^[- i n (\_y + )]{} e\^[i n \_x]{} \[representation\_plane2\] it follows that $$u (x)- e^{i k_0 \xi \cdot x} = -i e^{- i \f{\pi}{4}} \f{\mu_0 e^{i k_0 |x|} }{\sqrt{8 \pi k_0 |x|}} \sum_{m,n \in \mathbb{Z}} i^{(m-n)} e^{- i m \theta_\xi } e^{i n \theta_x} \int_{\p \Omega} J_n( k_0 |y|) e^{- i n \theta_y } \phi_m(y) d \sigma (y) + O(|x|^{-\f{3}{2}}) \, .$$ Comparing this expression with the representation of $W_{nm}$ in (\[inner\]), we infer that u(x) - e\^[i k\_0 x]{} = -i e\^[- i ]{} \_[m,n ]{} i\^[(m-n)]{} e\^[- i m \_]{} e\^[i n \_x]{} W\_[n m]{} + O(|x|\^[-]{}) . \[farfield\_exp\] This motivates us with the following definition of the far-field pattern. Consider the total field $u$ satisfying (\[scattering2\])-(\[sommerfield\]) with the incident field $u_0(x) = e^{i k_0 \xi \cdot x}$. Then the far-field pattern $A_{\infty} [\varepsilon, \mu, \omega] (\theta_\xi, \theta_x)$ is defined by \[defA\] u(x) - e\^[i k\_0 x]{} = -i e\^[- i ]{} A\_ \[, , \] (\_, \_x) + O(|x|\^[-]{}) |x| . By comparing (\[defA\]) with (\[farfield\_exp\]) we come to the following theorem. Let $\theta_\xi$ and $\theta_x$ be respectively the incident and the scattered direction. Then the far-field pattern $A_{\infty} [\varepsilon, \mu, \omega] (\theta_\xi, \theta_x)$ defined by (\[defA\]) can be expressed in the explicit form: A\_ \[, , \] (\_, \_x) = \_[m,n ]{} i\^[(m-n)]{} e\^[- i m \_]{} e\^[i n \_x]{} W\_[n m]{} \[, , \]. \[farfield\_def\] It is easy to see that the bounds in (\[decay\_conclusion\]) ensure the converges of the above series uniformly with respect to $\theta_\xi$ and $\theta_x$, so $A_{\infty} [\varepsilon, \mu, \omega]$ is well-defined. Moreover, one can see that reconstructing the scattering coefficients from the far-field pattern is an exponentially ill-posed problem if the measurements of $ A_{\infty}$ are corrupted with noise. Transformation rules and properties of scattering coefficients {#sec5} ============================================================== In this section, we derive more properties, including some transformation rules for the scattering coefficients. To do so, we first represent the scattering coefficients in terms of an exterior NtD map. For any $g \in H^{-\f{1}{2}}(\p\Om)$, the action of the exterior NtD map $\Lambda_{\mu_0,\varepsilon_0}^{e}: H^{-\f{1}{2}} (\p\Om)\to H^{\f{1}{2}} (\p\Om)$ is defined by the trace $u=\Lambda_{\mu_0,\varepsilon_0}^{e}g \in H^{\f{1}{2}} (\p\Om)$ of the solution $u$ to the system: u + \_0 \^2 u = 0 & \^d \\ ,\ = g & ,\ u - i k\_0 u = O(|x|\^[-]{})& |x| . \[exterior\_pde\] With the help of the exterior NtD map $\Lambda_{\mu_0,\varepsilon_0}^{e}$, we can derive some new representation of the scattering coefficients. Let $(u_0)_n$ and the scattering coefficients $W_{nm}$ be defined as in (\[incidence\]) and (\[eq:w\_nm\]), respectively, and let $\Lambda_{\mu,\varepsilon}$ and $\Lambda_{\mu_0,\varepsilon_0}^{e}$ be the interior and exterior NtD maps. Then the scattering coefficients $W_{nm}$ can be expressed as W\_[nm]{} = (u\_0)\_n, [A]{}\_[,]{}(u\_0)\_m \_[L\^2()]{} n, m , \[wmn\_bilinear\] where the operator ${\cal A}_{\mu,\varepsilon}$ is given by \_[,]{} := \_0 \_[\_0,\_0]{}\^[-1]{} (\_[\_0,\_0]{} - \_[,]{}) (\_[,]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{} (\_[\_0,\_0]{} - \^[e]{}\_[\_0,\_0]{}) \_[\_0,\_0]{}\^[-1]{} . \[long\_operator\] For a given incident field $u_0$, let $(\phi,\psi) \in L^2(\p \Om) \times L^2(\p \Om)$ be the density pair that solves (\[potential\]). Then it follows from the jump conditions of the layer potentials in (\[jump\_condition\]) that &=& + ( - I + \^\*\_[k\_0, ]{} ) \[\] + = + ( \_[k\_0]{}\[\] )\^[-]{} + , \[process1\] &=& ( I + \^\*\_[k\_0, ]{} ) \[\] + = ( \_[k\_0]{}\[\] )\^[+]{} + . \[process2\] By directly applying the interior and exterior NtD operators to (\[process1\]) and (\[process2\]), we obtain $$\begin{cases} \Lambda_{\mu_0,\varepsilon_0}^{e} [\psi] & = \mu_0 \mathcal{S}_{k_0}[\phi] + \f{1}{\mu_0} \Lambda_{\mu_0,\varepsilon_0}^{e} \left[\f{\p u_0}{\p \nu} \right] \, , \\ \Lambda_{\mu_0,\varepsilon_0}[\psi] & = \Lambda_{\mu_0,\varepsilon_0}[\phi] + \mu_0 \mathcal{S}_{k_0}[\phi] + u_0 \, , \\ \Lambda_{\mu,\varepsilon}[\psi] & = u_0 + \mu_0 \mathcal{S}_{k_0} [\phi] \, , \end{cases}$$ which combines to give (\_[,]{} - \_[\_0,\_0]{}\^[e]{} ) \[\] & = ( \_[\_0,\_0]{}- \_[\_0,\_0]{}\^[e]{}) = ( \_[\_0,\_0]{}- \_[\_0,\_0]{}\^[e]{}) \_[\_0,\_0]{}\^[-1]{} ,\ (\_[\_0,\_0]{} - \_[,]{})\[\]& = \_[\_0,\_0]{}\[\] . \[process3\] Substituting the first equation in (\[process3\]) into the second, we readily get $$\phi = \Lambda_{\mu_0,\varepsilon_0}^{-1}(\Lambda_{\mu_0,\varepsilon_0} - \Lambda_{\mu,\varepsilon})[\psi] = \Lambda_{\mu_0,\varepsilon_0}^{-1}(\Lambda_{\mu_0,\varepsilon_0} - \Lambda_{\mu,\varepsilon})(\Lambda_{\mu,\varepsilon} - \Lambda_{\mu_0,\varepsilon_0}^{e} )^{-1} ( \Lambda_{\mu_0,\varepsilon_0}- \Lambda_{\mu_0,\varepsilon_0}^{e}) \Lambda_{\mu_0,\varepsilon_0}^{-1} \left[ u_0 \right].$$ In particular, if $(\phi_m,\psi_m) \in L^2(\p \Om) \times L^2(\p \Om)$ be the density pair that satisfies (\[potential\]) corresponding to the incident field $u_0 (y) = (u_0)_m (y) := J_m (k_0 |y|) e^{i m \theta_y} $ as in (\[incidence\]), then $\phi_m$ satisfies \_m = \_[\_0,\_0]{}\^[-1]{}(\_[\_0,\_0]{} - \_[,]{})(\_[,]{} - \_[\_0,\_0]{}\^[e]{} )\^[-1]{} ( \_[\_0,\_0]{}- \_[\_0,\_0]{}\^[e]{}) \_[\_0,\_0]{}\^[-1]{} = \_[,]{} (u\_0)\_m. \[potential\_property\] Substituting (\[potential\_property\]) into (\[inner\]), we conclude that $$W_{nm} = \mu_0 \langle (u_0)_n, \phi_m \rangle_{L^2(\partial \Omega)} = \langle (u_0)_n, {\cal A}_{\mu,\varepsilon}(u_0)_m \rangle_{L^2(\partial \Omega)}\,.$$ With the representations (\[inner\]) and (\[wmn\_bilinear\]), we can derive some special transformation rules for the scattering coefficients. \[coll\_four\] The scattering coefficients $\{W_{nm}\}_{n, m \in \mathbb{Z}}$ in (\[eq:w\_nm\]) meet the following transformation rules: 1. $ W_{nm} [\varepsilon, \mu, \omega, \Omega] = \overline{ W_{mn} [\varepsilon, \mu, \omega, \Omega]} $; 2. $ W_{nm} [\varepsilon, \mu, \omega, e^{i \theta} \Omega] = e^{i (m-n)\theta} W_{nm} [\varepsilon, \mu, \omega, \Omega] $  for all $\theta \in [0, 2 \pi]$; 3. $ W_{nm} [\varepsilon, \mu, \omega, s \Omega] = W_{nm} [\varepsilon, \mu, s \omega, \Omega] $  for all $s >0 $; 4. $ W_{nm} [\varepsilon, \mu, \omega, \Omega + z ]= \sum_{l,l \in \mathbb{Z}} \overline{(u_0)_{p}(z)}(u_0)_{l}(z) W_{n-p,m-l} [\varepsilon, \mu, \omega, \Omega ]$  for all $z \in \mathbb{R}^2$, where we identify the spaces before and after translation, rotation and scaling by the natural isomorphism, e.g., $H^s (\partial \Omega) \cong H^s (e^{i \theta} \partial \Omega)$. We start with the first result in Corollary\[coll\_four\]. From representation (\[wmn\_bilinear\]) of $W_{nm}$, it suffices to show that the operator ${\cal A}_{\mu,\varepsilon}$ defined in (\[long\_operator\]) is self-adjoint. To do this, we utilize the following identity for any operators $A$, $B$, $C$ such that $A-C$ and $B-C$ are invertible: (A-C)\^[-1]{} - (B-C)\^[-1]{} = (A-C)\^[-1]{} (B-A) (B-C)\^[-1]{} = (B-C)\^[-1]{} (B-A) (A-C)\^[-1]{} . \[operator\_identity\] Using this we can write (\_[\_0,\_0]{}-\_[\_0,\_0]{}\^e)\^[-1]{} - (\_[,]{}-\_[\_0,\_0]{}\^e)\^[-1]{} = (\_[,]{}-\_[\_0,\_0]{}\^e)\^[-1]{} (\_[,]{}-\_[\_0,\_0]{}) (\_[\_0,\_0]{}-\_[\_0,\_0]{}\^e)\^[-1]{}. \[operator\_identity2\] Substituting (\[operator\_identity2\]) into (\[long\_operator\]), we get && [A]{}\_[,]{}\ &=& \_0 \_[\_0,\_0]{}\^[-1]{} (\_[\_0,\_0]{} - \_[,]{}) (\_[,]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{} (\_[\_0,\_0]{} - \^[e]{}\_[\_0,\_0]{}) \_[\_0,\_0]{}\^[-1]{}\ &=& \_0 \_[\_0,\_0]{}\^[-1]{} (\_[\_0,\_0]{} - \_[,]{}) \_[\_0,\_0]{}\^[-1]{} +\_0 \_[\_0,\_0]{}\^[-1]{} ( \_[,]{} - \_[\_0,\_0]{}) (\_[,]{}-\_[\_0,\_0]{}\^e)\^[-1]{} (\_[,]{}-\_[\_0,\_0]{}) \_[\_0,\_0]{}\^[-1]{}. Now the self-adjointness of ${\cal A}_{\mu,\varepsilon}$ is a consequence of the self-adjointness of $\Lambda_{\mu_0,\varepsilon_0}$, $\Lambda_{\mu,\varepsilon}$ and $\Lambda_{\mu_0,\varepsilon_0}^e$. To see the second result in Corollary\[coll\_four\], we consider the change of coordinates from $(|y|, \theta_y)$ to $(|\widetilde{y}|, \widetilde{\theta_y})$, with $\widetilde{\theta_y} + \theta = \theta_y$ and $|\widetilde{y}| = |y|$. It follows from definition (\[incidence\]) that $(u_0)_m (y)= J_m (k_0|y|) e^{i m \theta_y} = J_m (k_0|\widetilde{y}|) e^{i m (\widetilde{\theta_y} + \theta)} = (u_0)_m (\widetilde{y}) e^{i m \theta}$. Let $\widetilde{u_m}(\widetilde{y})$ be the solution to \_ ( \_ u() ) + \^2 () u() = 0 & \^2 ,\ (u - u\_0)() - i k\_0 (u - u\_0)() = O(||\^[-]{})& |x| . with the incident field $u_0(\widetilde{y}) = (u_0)_m (\widetilde{y})$, and let $u_m(y)$ be the solution to (\[scattering1\])-(\[sommerfield\]) with the incident field $u_0(y)=(u_0)_m (y) = (u_0)_m (\widetilde{y}) e^{i m \theta}$. Then we can see that $u_m(y)$ is actually $u_m(y) = \widetilde{u_m}(\widetilde{y}) e^{i m \theta}$. Therefore we observe that the density pair $(\psi_m,\phi_m)$ satisfying (\[potential\]) with incident field $u_0 (y) = (u_0)_m (y)$ and electromagnetic parameters $\mu(y)$ and $\varepsilon(y)$ actually has the form $ (\psi_m(y), \phi_m(y))= (\widetilde{\phi}_m (\widetilde{y}), \widetilde{\phi}_m (\widetilde{y}) )e^{i m \theta} $, where $(\widetilde{\psi}_m,\widetilde{\phi}_m)$ satisfies (\[potential\]) with incident field $(u_0)_m (\widetilde{y})$ and parameters $\mu(\widetilde{y})$, $\varepsilon(\widetilde{y})$. Hence we derive from (\[inner\]) that W\_[nm]{}\[, , , e\^[i ]{} \] &=& \_0 \_[e\^[i]{} ]{} J\_n(k\_0|y|) e\^[-i n \_y]{} \_m d (y)\ &=& \_0 \_[e\^[i]{} ]{} J\_n(k\_0||) e\^[-i n ( +)]{} (y)() e\^[i m ]{} d (y)\ &=& e\^[i (m-n) ]{} \_0 \_[e\^[i ]{} ]{} J\_n(k\_0||) e\^[-i n ]{} () d ()\ &=& e\^[i (m-n)]{} W\_[nm]{} \[, , , \]. This proves the second result in Corollary\[coll\_four\]. Next for the third result in Corollary\[coll\_four\], we consider the change of coordinates from $(|y|, \theta_y)$ to $(|\widetilde{y}|, \widetilde{\theta_y})$, with $\widetilde{\theta_y} = \theta_y$ and $s|\widetilde{y}| = |y|$. We know from (\[incidence\]) that $(u_0)_m (y)= J_m (k_0|y|) e^{i m \theta_y} = J_m (s|\widetilde{y}|) e^{i m \widetilde{\theta_y})}$. Let $\widetilde{u_m}(\widetilde{y})$ be the solution to the following system \_ ( \_ u() ) + (s)\^2 () u() = 0 & \^2 ,\ (u - u\_0)() - i k\_0 (u - u\_0)() = O(||\^[-]{})& |x| with the incident field $u_0(\widetilde{y}) = (u_0)_m (\widetilde{y})$, then it is easy to see that the solution $u_m(y)$ to the system (\[scattering1\])-(\[sommerfield\]) with the incident field $u_0(y)=(u_0)_m (y) = (u_0)_m (s \widetilde{y})$ takes the form $u_m(y) = \widetilde{u_m}( s \widetilde{y})$. With this, we observe that the density pair $(\psi_m,\phi_m)$ satisfying (\[potential\]) with incident field $u_0 (y) = (u_0)_m (y)$ and parameters $\mu(y)$ and $\varepsilon(y)$ is given by $(\psi_m(y), \phi_m(y) )= (\widetilde{\psi}_m ( s \widetilde{y}), \widetilde{\phi}_m ( s \widetilde{y}))/s$ with $(\widetilde{\psi}_m,\widetilde{\phi}_m)$ satisfying (\[potential\]) with incident field $(u_0)_m (\widetilde{y})$ and parameters $\mu(\widetilde{y})$, $\varepsilon(\widetilde{y})$. This comes from the fact that $\f{\p u_m^{-}}{\p \nu_y} = \f{\p \widetilde{u_m}^- }{\p \nu_{\widetilde{y}}} ( s \widetilde{y}) /s= \widetilde{\psi}_m ( s \widetilde{y})/s$, by comparing (\[process3\]) with (\[eq:w\_nm\]) and (\[inner\]). Now the desired third result in Corollary\[coll\_four\] follows from the straightforward derivations: W\_[nm]{} \[, , , s \] &=& \_0 \_[s ]{} J\_n(k\_0|y|) e\^[-i n \_y]{} \_m(y) d (y)\ &=& \_0 \_[s ]{} J\_n(k\_0 s||) e\^[-i n \_y]{} \_m(s ||) d (y)\ &=& \_0 \_ J\_n(k\_0 s||) e\^[-i n \_y]{} \_m(s ||) d ()\ &=& W\_[nm]{} \[, , s , \]. Finally we come to derive the last relation in Corollary\[coll\_four\]. To do so, we consider the change of coordinates from $(|y|, \theta_y)$ to $(|\widetilde{y}|, \widetilde{\theta_y})$ that has point $z$ as the origin. Then the definition of $(u_0)_m$ in (\[incidence\]) and the Graf’s addition formula (\[graf\]) allow us to write $$(u_0)_m = J_m (k_0|y|) e^{i m \theta_y} = \sum_{a \in \mathbb{Z}} J_a (k_0|z|) e^{i m \theta_z} J_{m-a}(k_0|\widetilde{y}|) e^{i (m-a) \widetilde{\theta_y}}.$$ By the linearity of operator $A$ in (\[operator\_A\]), the density pair $(\psi_m,\phi_m)$ satisfying (\[potential\]) with the incident field $u_0 (y) = (u_0)_m (y)$ can be expressed in the form $(\psi_m, \phi_m)= \sum_{a \in \mathbb{Z}} J_a (k_0|z|) e^{i m \theta_z} (\widetilde{\psi}_{m-a} (\widetilde{y}), \widetilde{\phi}_{m-a} (\widetilde{y}))$, where $(\widetilde{\psi}_m,\widetilde{\phi}_m)$ satisfies (\[potential\]) with the incident field $(u_0)_m (\widetilde{y})$. With these preparations, the last result in Corollary\[coll\_four\] follows readily from the following derivations: &&W\_[nm]{} \[, , , + z \]\ &=& \_0 \_[+ z]{} J\_n(k\_0|y|) e\^[-i n \_y]{} \_m (y) d (y)\ &=& \_0 \_[b]{} J\_a (k\_0|z|) e\^[i m \_z]{} \_[+ z]{} J\_[n-b]{}(k\_0||) e\^[-i (n-b) ]{} \_m (y) d (y)\ &=& \_0 \_[a,b]{} J\_b (k\_0|z|) e\^[- i m \_z]{} J\_a (k\_0|z|) e\^[i m \_z]{} \_ J\_[n-b]{}(k\_0||) e\^[- i (n-b) ]{} \_[m-a]{} (,) d ()\ &=& \_[a,b]{} (u\_0)\_[a]{}(z) W\_[n-b,m-a]{} \[, , , \]. We end this section with one more representation of $W_{nm}$. Let $(u_0)_m$ be defined as in (\[incidence\]) and $u_m$ be the solution to (\[scattering1\])-(\[sommerfield\]) with the incident field $(u_0)_m$. Then the scattering coefficients in (\[eq:w\_nm\]) admits the following representation for any $n, m \in \mathbb{Z}$: W\_[nm]{} = \^2 \_0 \_ (\_0(y)-(y)) (y) u\_m(y) d (y) + \_0 \_ (-) (y) u\_m(y) d (y) . \[new\_representation\_Wnm\] Let $(\psi_m,\phi_m)$ be the density pair $(\psi_m,\phi_m)$ that satisfies (\[potential\]) with the incident field $u_0 (y) = (u_0)_m (y)$. Then it follows directly from (\[inner\]), (\[jump\_condition\]) and (\[potential\]) that W\_[nm]{}& = & \_0 \_ (y) \_m (y) d (y)\ & = & \_0 \_ (y) d (y)\ & = & \_0 \_ (y) ( \_m(y) - (y) ) d (y) - \_0\_ (y) (y) d (y). Using Green’s identity and (\[potential\]), we can further derive W\_[nm]{} & = & \_0 \_ (y) ( \_m(y) - (y) ) d (y) - \_0 \_ (y) \_[k\_0]{}\[\_m\](y) d (y)\ & = & \_0 \_ (y) ( \_m- ) d (y) - \_ (y) (\_[,]{}\[\_m\] - (u\_0)\_m) d (y)\ & = & \_0 \_ (y) \_m(y) d (y) - \_ (y) \_[,]{}\[\_m\](y) d (y). Now the desired representation of $W_{nm}$ follows from (\[total\_u\]) and (\[def\_ntd\]), the comparison of (\[process3\]) with (\[eq:w\_nm\]) and (\[inner\]), and the Green’s identity: W\_[nm]{} & = &\_0 \_ (y) \_m(y) d (y) - \_ (y) u\_m(y) d (y)\ & = & \_0 \_ (y) d (y) - \_ (y) u\_m(y) d (y)\ & = & \^2 \_0 \_ (\_0-) (y) u\_m(y) d (y) + \_0 \_ (-) (y) u\_m(y) d (y). Sensitivity analysis {#sec6} ==================== In this section, we shall investigate the sensitivity of the scattering coefficients with respect to the changes in the permittivity and permeability distributions. This will provide us with perturbation formulas for evaluating the gradients that are needed in numerical minimization algorithms for reconstructing the permittivity and permeability distributions. We study a perturbation of $W_{nm}$ for $n,m \in \mathbb{Z}$ with respect to a change of $(\mu, \varepsilon)$. More specifically, we consider the difference $W_{nm}^{\delta} - W_{nm}$ between W\_[nm]{}\^ : = W\_[nm]{} W\_[nm]{} := W\_[nm]{}\[ , , , \] \[perturbed\] in terms of the differences $\varepsilon^\delta - \varepsilon$ and ${1}/{\mu^{\delta}} - {1}/{\mu}$, where $(\mu,\varepsilon)$ and $(\mu^\delta,\varepsilon^\delta)$ are two different sets of electromagnetic parameters. In the subsequent analysis, we shall often write := { || \^- ||\^2\_[L\^()]{} + || - ||\^2\_[L\^()]{}}\^[1/2]{}. \[orderepsilon\] We first note that if $\widehat{\varepsilon}$ is small enough, then the NtD map $ \Lambda_{\mu^\delta,\varepsilon^\delta}$ is well defined provided that $\Lambda_{\mu,\varepsilon}$ is well defined. This follows from the theory of collectively compact operators; see [@anselone; @book]. Next we show the following expression for the difference $W_{nm}^{\delta} - W_{nm}$. For all $n,m \in \mathbb{Z}$, the difference $W_{nm}^{\delta} - W_{nm}$ can be represented in terms of the interior and exterior NtD maps $\Lambda_{\mu,\varepsilon}$ and $\Lambda_{\mu_0,\varepsilon_0}^{e}$ as follows: W\_[nm]{}\^ - W\_[nm]{} & = &\_0 \_ (y) (\_[,]{} - \_[\^,\^]{}) \[\_m\^\] (y) d (y) , \[B\_bilinear\_useful\] where $\psi_n$ and $\psi_m^\delta$ are given by \_n &=& (\_[,]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{} (\_[\_0,\_0]{} - \^e\_[\_0,\_0]{}) \_[\_0,\_0]{}\^[-1]{} (u\_0)\_n , \[recall1\]\ \_m\^&=& (\_[\^,\^]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{} (\_[\_0,\_0]{} - \^e\_[\_0,\_0]{}) \_[\_0,\_0]{}\^[-1]{} (u\_0)\_m . \[recall2\] Using the identity (\[operator\_identity\]) we can write (\_[\^,\^]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{} - (\_[,]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{} = (\_[,]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{} (\_[,]{}-\_[\^,\^]{}) (\_[\^,\^]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{} , \[identity\_00\] which enables us to derive &&(\_[\_0,\_0]{} - \_[\^,\^]{}) (\_[\^,\^]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{} - (\_[\_0,\_0]{} - \_[,]{}) (\_[,]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{}\ &=& (\_[,]{} - \_[\^,\^]{}) (\_[\^,\^]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{} + (\_[\_0,\_0]{} - \_[,]{})\ &=& (\_[,]{}-\_[\^,\^]{}) (\_[\^,\^]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{}\ &=& (\_[\_0,\_0]{}-\^[e]{}\_[\_0,\_0]{}) (\_[,]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{} (\_[,]{}-\_[\^,\^]{}) (\_[\^,\^]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{} . \[identity\_01\] It follows directly from (\[identity\_01\]) and definition (\[long\_operator\]) for the operators ${\cal A}_{\mu,\varepsilon}$ and ${\cal A}_{\mu^\delta,\varepsilon^\delta}$ that [&&[A]{}\_[\^,\^]{} - [A]{}\_[,]{}\ &=&\_0 \_[\_0,\_0]{}\^[-1]{} { (\_[\_0,\_0]{} - \_[\^,\^]{}) (\_[\^,\^]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{} - (\_[\_0,\_0]{} - \_[,]{}) (\_[,]{} - \^[e]{}\_[\_0,\_0]{})\^[-1]{} } (\_[\_0,\_0]{} - \^[e]{}\_[\_0,\_0]{}) \_[\_0,\_0]{}\^[-1]{}\ &=&\_0 \_[\_0,\_0]{}\^[-1]{} (\_[\_0,\_0]{}-\^[e]{}\_[\_0,\_0]{}) (\_[,]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{} (\_[,]{}-\_[\^,\^]{}) (\_[\^,\^]{}-\^[e]{}\_[\_0,\_0]{})\^[-1]{} (\_[\_0,\_0]{} - \^[e]{}\_[\_0,\_0]{}) \_[\_0,\_0]{}\^[-1]{} . ]{}Now identity (\[B\_bilinear\_useful\]) is a consequence of the above relation and the representation (\[wmn\_bilinear\]) for $W_{nm}$ and $W_{nm}^{\delta}$, $$W_{nm}^{\delta} - W_{nm} = \langle (u_0)_n, \left({\cal A}_{\mu^\delta,\varepsilon^\delta} - {\cal A}_{\mu,\varepsilon} \right) (u_0)_m \rangle_{L^2(\partial \Omega)} = \langle \psi_n, \left(\Lambda_{\mu,\varepsilon} - \Lambda_{\mu^\delta,\varepsilon^\delta}\right) [\psi_m^\delta] \rangle_{L^2(\partial \Omega)} \, ,$$ where $\langle , \rangle$ denotes the complex inner product on $L^2(\partial \Omega)$. The following identity will be useful for the subsequent analysis. \[quadratic\] For the solutions $u_i$ ($i = 1,2$) to the two systems ( u\_i )+ \^2 \_i u\_i = 0 ; = g , \[systems\_purb\] the following identity holds & & \_ (\_[\_2,\_2]{} - \_[\_1,\_1]{}) \[g\] d\ & = & \_ ( - ) ( -| (u\_1 - u\_2) |\^2 + | u\_1 |\^2 + | u\_2 |\^2)\ & & - \^2 \_ ( \_1 - \_2 ) ( -| u\_1 - u\_2 |\^2 + | u\_1 |\^2 + | u\_2 |\^2) dx . \[quadraticpolarization\] It follows easily from (\[systems\_purb\]) and integration by parts that \_ (\_[\_1,\_1]{})\[g\] d &=& \_ ( |u\_1|\^2 - \^2 \_1 |u\_1|\^2 ) dx , \[101\]\ \_ (\_[\_2,\_2]{})\[g\] d &=& \_ ( u\_2 - \^2 \_1 u\_2 ) dx , \[102\]\ \_ (\_[\_2,\_2]{})\[g\] d &=& \_ ( |u\_2|\^2 - \^2 \_2 |u\_2|\^2 ) dx , \[103\]\ \_ u\_2 d &=& \_ ( |u\_2|\^2 - \^2 \_1 |u\_2|\^2 ) dx . \[104\] Combining (\[101\])-(\[104\]) yields & & \_ | (u\_2 - u\_1) |\^2 dx - \^2 \_ \_1 | u\_2 - u\_1 |\^2 dx + \_ ( - ) | u\_2 |\^2 dx - \^2 \_ ( \_2 - \_1 ) | u\_2 |\^2 dx\ &=& \_ (\_[\_1,\_1]{})\[g\] d - 2 \_ (\_[\_2,\_2]{})\[g\] d + \_ u\_2 d + \_ (\_[\_2,\_2]{})\[g\] d - \_ u\_2 d\ &=& \_ (\_[\_1,\_1]{} - \_[\_2,\_2]{}) \[g\] d , which gives the identity & & \_ | (u\_2 - u\_1) |\^2 dx - \^2 \_ \_1 | u\_2 - u\_1 |\^2 dx + \_ ( - ) | u\_2 |\^2 dx - \^2 \_ ( \_2 - \_1 ) | u\_2 |\^2 dx\ &=& \_ (\_[\_1,\_1]{} - \_[\_2,\_2]{}) \[g\] d . \[201\] Swapping $u_1$ and $u_2$ in the above identity implies & & \_ | (u\_1 - u\_2) |\^2 dx - \^2 \_ \_2 | u\_1 - u\_2 |\^2 dx + \_ ( - ) | u\_1 |\^2 dx - \^2 \_ ( \_1 - \_2 ) | u\_1 |\^2 dx\ &=& \_ (\_[\_2,\_2]{} - \_[\_1,\_1]{}) \[g\] d . \[202\] Now (\[quadraticpolarization\]) follows by subtracting (\[201\]) from (\[202\]). By the same arguments as those in [@abboud] (see also [@stability]), we can derive the following estimate. The difference between the interior NtD maps $\Lambda_{\mu,\varepsilon}$ and $\Lambda_{\mu^\delta,\varepsilon^\delta}$ can be represented in terms of the differences between two sets of electromagnetic parameters $(\mu,\varepsilon)$ and $(\mu^\delta,\varepsilon^\delta)$: ||\_[\^,\^]{} - \_[,]{} || C ( ||\^- ||\_[L\^()]{} + || - ||\_[L\^()]{} ) . \[ineq1\] Now we can further our analysis on the difference $W_{nm}^{\delta} - W_{nm}$ in terms of $\varepsilon^\delta - \varepsilon$ and ${1}/{\mu^{\delta}} - {1}/{\mu}$ using (\[B\_bilinear\_useful\]) and (\[ineq1\]). Recalling $\psi_n$ and $\psi_m^\delta$ from (\[recall1\]) and (\[recall2\]), we can define the solutions $u_m, u_m^\gamma, u_n^\delta$ and $u_n^{\delta \gamma}$ to the following four systems: && u\_m + \^2 u\_m = 0 ; u\_m = \_m ;\ && u\_m\^+ \^\^2 u\_m\^= 0 ; u\_m\^= \_m ;\ && u\_n\^ + \^2 u\_n\^= 0 ; u\_n\^ = \^\_n ;\ && u\_n\^+ \^\^2 u\_n\^= 0 ; u\_n\^= \^\_n . Noting from (\[total\_u\]) that $\psi_n$ and $\psi_m^\delta$ are the density functions in the Neumann potential along $\partial \Omega$ with coefficients $(\mu,\varepsilon)$ and $(\mu^\delta,\varepsilon^\delta)$ respectively, the solutions $u_m$ and $u_n^\delta$ solve (\[scattering1\])-(\[sommerfield\]) with coefficients $(\mu,\varepsilon)$ and $(\mu^\delta,\varepsilon^\delta)$ and the incident field $(u_0)_m$ and $(u_0)_n$ defined as in (\[incidence\]). For convenience, we introduce a bilinear form: B(p , q) := \_ (\_[,]{} - \_[\^,\^]{}) \[q\] d p, q H\^[-]{}(). \[def\_B\_bilinear\] Then (\[quadraticpolarization\]) gives us an explicit expression of $B(g ,g)$ for $g \in H^{-\f{1}{2}}(\p \Om)$. By (\[B\_bilinear\_useful\]), the difference $W_{nm}^\delta - W_{nm}$ can be split using the bilinear form $B$ as W\_[nm]{}\^- W\_[nm]{} &=& \_0 B(\_m , \^\_n)\ &=&\ && +\ &:=& + , \[hahaintegral\] where $\I$ and $\II$ are given by &:=& B(\_m + \^\_n, \_m + \^\_n) - B(\_m , \_m )- B(\^\_n, \^\_n) ,\ &:=& B(\_m + i \^\_n, \_m + i \^\_n) - B(\_m , \_m ) - B(\^\_n, \^\_n) . By direct calculations, we get the following expression of the term $\I$: &=& B(\_m + \^\_n, \_m + \^\_n) - B(\_m , \_m ) - B(\^\_n, \^\_n)\ &=& \_ ( - ) ( -| (u\_m + u\_n\^ - u\_m\^- u\_n\^) |\^2 + | (u\_m + u\_n\^) |\^2 + | (u\_m\^+ u\_n\^) |\^2) dx\ && - \^2 \_ (\^- ) ( -| u\_m + u\_n\^ - u\_m\^- u\_n\^|\^2 + | u\_m + u\_n\^ |\^2 + | u\_m\^+ u\_n\^|\^2) dx\ && -\ && - . \[hahareal0\] From (\[ineq1\]), we get ||u\_m - u\_m\^||\^2\_[H\^1()]{} = O ( ) ||u\_n\^ - u\_n\^||\^2\_[H\^1()]{} = O ( ) , \[haharealbound\] where $\widehat{\varepsilon}$ is defined as in (\[orderepsilon\]). Then using (\[haharealbound\]), we further the estimate of the term $\I$ as follows: [& = & \_ ( - ) ( | (u\_m + u\_n\^) |\^2 - | u\_m |\^2 - | u\_n |\^2) dx\ &&- \^2 \_ (\^- ) ( | u\_m + u\_n\^|\^2 - | u\_m |\^2 - | u\_n |\^2) dx + O (\^2 )\ & = & 2 + O (\^2 ). \[hahareal\] ]{} Similarly, we can derive the following estimate for the term $\II$: & = & B(\_m + i \^\_n, \_m + i \^\_n) - B(\_m , \_m ) - B(\^\_n, \^\_n)\ & = & 2 + O (\^2 ) . \[hahaimag\] Substituting (\[hahareal\]) and (\[hahaimag\]) in (\[hahaintegral\]) gives W\_[nm]{}\^- W\_[nm]{} &=& \_0 \_ ( - ) u\_m dx - \_0 \^2 \_ (\^- ) u\_m dx + O (\^2 ) . \[perturbation\_final0\] Furthermore, we have from (\[eq:ufromwnm\]) that for all $m \in \mathbb{Z}$, u\_m - (u\_0)\_m = - \_[m]{} H\^[(1)]{}\_n (k\_0 |x|) e\^[i n \_x]{} W\_[nm]{} , u\_m\^- (u\_0)\_m = - \_[m]{} H\^[(1)]{}\_n (k\_0 |x|) e\^[i n \_x]{} W\_[nm]{}\^ , \[eq:ufromwnm\_diff2\] subtracting the first one from the second in (\[eq:ufromwnm\_diff2\]) gives u\_m\^= u\_m - \_[m]{} H\^[(1)]{}\_n (k\_0 |x|) e\^[i n \_x]{} ( W\_[nm]{}\^- W\_[nm]{}). \[eq:ufromwnm\_diff3\] Now replacing $u_m^\delta$ in (\[perturbation\_final0\]) by (\[eq:ufromwnm\_diff3\]), we arrive at the following theorem. \[perturbation\] Assume $(\mu,\varepsilon)$ and $(\mu^\delta,\varepsilon^\delta)$ are two different sets of electromagnetic parameters, and $W_{nm}$ and $W_{nm}^{\delta}$ are defined as in (\[perturbed\]). Let $\widehat{\varepsilon}$ be defined as in (\[orderepsilon\]). For any $m \in \mathbb{Z}$, let $u_m$ be the solution to (\[scattering1\])-(\[sommerfield\]) with the coefficients $(\mu,\varepsilon)$ and the incident field $(u_0)_m$ in (\[incidence\]). Then the following estimate holds for any $n, m \in \mathbb{Z}$: W\_[nm]{}\^- W\_[nm]{} = \_0 \_ ( - ) u\_m - \^2 \_0 \_ (\^- ) u\_m + O (\^2 ) . The above formula provides a sensitivity analysis in terms of electromagnetic parameters $(\mu,\varepsilon)$ for arbitrary medium domains $\Om$. In order to derive reconstruction formulas for $\mu$ and $\varepsilon$ from the scattering coefficients, we shall achieve more explicit and detailed sensitivity analysis and representation formulas for scattering coefficients $W_{nm}$ when the medium domains are of some special geometry. This is our focus in the next section. Explicit reconstruction formulas in the linearized case {#sec7} ======================================================= For a given $\widehat{\varepsilon} > 0$, consider $\mu, \varepsilon$ such that $(|| \varepsilon - \varepsilon_0 ||_{L^\infty(\Omega)}^2 + \big|\big|{\mu}^{-1} - {\mu_0^{-1}}\big|\big|^2_{L^\infty(\Omega)})^{1/2} = \widehat{\varepsilon}$. Then it follows from (\[new\_representation\_Wnm\]) and the definition of $(u_0)_m$ in (\[incidence\]) that W\_[nm]{} & =& \_0 \_ ( - ) ( J\_n(k\_0|y|) e\^[- i n \_y]{} ) ( J\_m(k\_0|y|) e\^[i m \_y]{} ) dy\ & & - \^2 \_0 \_ ( (y) - \_0 ) J\_n(k\_0|y|) J\_m(k\_0|y|) e\^[ i (m - n) \_y]{} dy + O (\^2 ) . ł[eq:wnm]{} Now for all $n \neq 0$, we have by direct computing \_[x\_1]{} (J\_n(k r) e\^[i n ]{}) &=& (\_[r]{} - \_ ) (J\_n(k r) e\^[i n ]{})\ &=& ( J\_[n-1]{}(k r) - J\_[n+1]{}(k r)) e\^[i n ]{} - J\_n(k r) e\^[i n ]{} ,\ \_[x\_2]{} (J\_n(k r) e\^[i n ]{}) &=& (\_[r]{} + \_ ) (J\_n(k r) e\^[i n ]{})\ &=& ( J\_[n-1]{}(k r) - J\_[n+1]{}(k r)) e\^[i n ]{} + J\_n(k r) e\^[i n ]{} , which implies the explicit expression for the gradient term in (\[eq:wnm\]) for $n ,m \neq 0$: [&& ( J\_n(k\_0|y|) e\^[- i n \_y]{} ) ( J\_m(k\_0|y|) e\^[i m \_y]{} )\ & = & e\^[i (m-n) \_y]{} . ]{} For $n = 0$, we have $J_0' = - J_1$ and \_[x\_1]{} (J\_0(k r)) = - k ( J\_1 (k r)) , \_[x\_2]{} (J\_0(k r)) = - k ( J\_1 (k r)) , which yields the following explicit expressions for the gradient term for $n = 0$ or $m =0$: [( J\_0(k\_0|y|) ) ( J\_m(k\_0|y|) e\^[i m \_y]{} ) & = & e\^[i m \_y]{} ,\ ( J\_n(k\_0|y|) e\^[-i n \_y]{} ) ( J\_0(k\_0|y|) ) & = & e\^[ -i n \_y]{} ,\ ( J\_0(k\_0|y|) ) ( J\_0(k\_0|y|) ) & = & k\_0\^2 ( J\_[1]{}(k\_0 |y|) )\^2 . ]{} These explicit formulas lead us to the following corollary. ł[cor:wnm]{} Let $(\mu,\varepsilon)$ be a pair of electromagnetic parameters in $\Omega$, and $\widehat{\varepsilon}= (|| \varepsilon - \varepsilon_0 ||^2_{L^\infty} + \big|\big|{\mu}^{-1} - {\mu_0^{-1}}\big|\big|^2_{L^\infty})^{1/2} $. Then the scattering coefficients $W_{nm}[ \, \varepsilon, \mu, \omega, \Omega \,]$ admit the following expansions: W\_[nm]{} & =& \_ ( - )( J\_[n-1]{}(k\_0 |y|) - J\_[n+1]{}(k\_0 |y|)) ( J\_[m-1]{}(k\_0 |y|) - J\_[m+1]{}(k\_0 |y|)) e\^[i (m-n) \_y]{} dy\ & & + \_0 n m \_ ( - ) J\_n(k\_0 |y|)J\_m(k\_0 |y|) e\^[i (m-n) \_y]{} dy\ & & - \^2 \_0 \_ ( (y) - \_0 ) J\_n(k\_0|y|) J\_m(k\_0|y|) e\^[ i (m - n) \_y]{} dy + O (\^2 ) (   n,m0) \[wnm\_explicit\]\ W\_[00]{} & =& \_0 k\_0\^2 \_ ( - ) ( J\_[1]{}(k\_0 |y|) )\^2 dy - \^2 \_0 \_ ( (y) - \_0 ) ( J\_[0]{}(k\_0 |y|) )\^2 dy + O (\^2 ) \[wnm\_explicit3\]\ W\_[n0]{} & =&- \_ ( - ) ( J\_[n-1]{}(k\_0 |y|) - J\_[n+1]{}(k\_0 |y|)) ( J\_[1]{}(k\_0 |y|) ) e\^[ -i n \_y]{} dy\ & & - \^2 \_0 \_ ( (y) - \_0 ) J\_n(k\_0|y|) J\_0(k\_0|y|) e\^[ -i n \_y]{} dy + O (\^2 ) (   n0). \[wnm\_explicit2\] By means of the asymptotic behavior (\[decayhaha\]) and the estimates in Corollary\[cor:wnm\], we obtain the following estimate for all $n, m \in \mathbb{Z}$: | W\_[nm]{} | & & || - ||\_[L\^()]{}\ & & + \_0 || - ||\_[L\^()]{} + \^2 \_0 || - \_0 ||\_[L\^()]{} + C \^2 . ł[eq:estimatew]{} Moreover, by comparing (\[wnm\_explicit\]) with (\[decayhaha\]) for large $m$ and $n$, we can see that the two integrals with the term $({\mu}^{-1} - {\mu_0^{-1}})$ dominate. This suggests that we may separate the effect of $({\mu}^{-1} - {\mu_0^{-1}})$ and $\varepsilon - \varepsilon_0$ on $W_{nm}$ and recover $\mu$ and $\varepsilon$ alternatively: First use the scattering coefficients $W_{nm}$ for large $m,n$ to recover $\mu$, then use the scattering coefficients $W_{nm}$ for small $m,n$ to recover $\varepsilon$. Furthermore, with the integral expression (\[wnm\_explicit\]) we may work out each term explicitly for some special domains, e.g., $\Omega = B_R(0)$. For simplicity, we will present our detailed derivations and calculations for the special case with $\mu = \mu_0$ but $\varepsilon \neq \varepsilon_0$ in the remainder of this section, though most of the conclusions can be extended to the general case with $\mu \neq \mu_0$ and $\varepsilon \neq \varepsilon_0$. It is easy to see for the special domain $\Omega = B_R(0)$ and the special case with $\mu = \mu_0$ but $\varepsilon \neq \varepsilon_0$ that $W_{nm}$ are simplified to be W\_[nm]{} &=& - \^2 \_0 \_[0]{}\^[R]{} \_[0]{}\^[2 ]{} ((y) - \_0) J\_n(k\_0r\_y) J\_m(k\_0r\_y) e\^[ i (m - n) \_y]{} r\_y dr\_y d \_y + O (\^2 ) , \[wnm\_epsilon\] where $y = (r_y, \theta_y)$ is the polar coordinate. Radially symmetric case ----------------------- In this subsection we derive formulas to recover the electromagnetic parameter $\varepsilon$ from the scattering coefficients $W_{nm}$ in the case with $\mu = \mu_0$, but $\varepsilon \neq \varepsilon_0$ with $\varepsilon$ being radially symmetric in $\Omega = B_R(0)$. We shall write $\widehat{\varepsilon} := || \varepsilon - \varepsilon_0||_{L^\infty(\Omega)} $, and $\varepsilon(y) = \varepsilon(r_y)$. It is straightforward to see from (\[wnm\_epsilon\]) that W\_[nm]{} = - 2 \^2 \_0 \_[0]{}\^[R]{} ((r\_y) - \_0) \[J\_n(k\_0r\_y)\]\^2 r\_y dr\_y + O (\^2 )   m = n   and   O (\^2 )   m n. \[wnm\_epsilon\_radial\] It follows readily from (\[wnm\_explicit\]), (\[wnm\_explicit2\]) and (\[wnm\_explicit3\]) that the same conclusion as in (\[wnm\_epsilon\_radial\]) for $m \neq n$ can be obtained for the more general case when $\mu \neq \mu_0$ and $\varepsilon \neq \varepsilon_0$, provided that both $\mu$ and $\varepsilon$ are radial symmetric in $\Omega = B_R(0)$. In the later part of this subsection, we shall establish an explicit formula for computing the electromagnetic parameter $\varepsilon$ in terms of the scattering coefficients $W_{nn}(k) := W_{nn}[ \, \varepsilon, \mu, \omega(k), \Omega \, ]$, where $ \omega(k)= {k}/{\sqrt{\varepsilon_0 \mu_0}} \label{frequency} $ is the frequency depending on $k \in \mathbb{R}^+$. For the sake of convenience, we define the following coefficient \_[n]{}\^[(0)]{}:= \_0\^ dk. \[H\_def1\] Using the following orthogonality of the Bessel functions $\{ J_{n}( r k) \}_{r >0}$ for a given $n \in \mathbb{Z}$: \^\_[0]{} J\_[n]{}(r k) J\_[n]{}(r’ k) k dk = r, r’ &gt;0 , \[orthogonal\] we obtain from (\[wnm\_epsilon\]) and (\[frequency\]) that \_[n]{}\^[(0)]{} = \_0\^ dk &=& - \_[0]{}\^[R]{} ((r\_y) - \_0) ( \_0\^J\_n(k r\_y) J\_n(k r\_y)k dk ) r\_y dr\_y + O (\^2 )\ &=& - \_[0]{}\^[R]{} ((r\_y) - \_0) dr\_y + O (\^2 ) , which gives the average of $ \varepsilon(r_y) - \varepsilon_0 $ along the radial direction. Next, we shall extend the above observation to obtain more information about $\varepsilon$. This motivates us with the following definition. For $n \in \mathbb{Z}$, let $W_{nn}(k) := W_{nn}[ \, \varepsilon, \mu, \omega(k), \Omega \, ]$ be defined as in (\[eq:w\_nm\]) with $\omega(k)= {k}/{\sqrt{\varepsilon_0 \mu_0}}$. For $l, n \in \mathbb{Z}$ and $l \geq 0$, let $g^{(l)}_{n}(k)$ be functions such that \_0\^g\^[(l)]{}\_[n]{}(k) J\_n(k r) J\_n(k r) k\^2 dk = r\^[l-1]{} r &gt; 0. \[moment\] Then we define the coefficients $\mathcal{H}_{n}^{(l)}$ by \_[n]{}\^[(l)]{}:= \_0\^ g\^[(l)]{}\_[n]{}(k) W\_[nn]{}(k) dk   l, n ,  l 0. \[H\_def2\] We will show the existence of functions $ g^{(l)}_{n}$ satisfying (\[moment\]) and derive their explicit expressions in Appendix \[appendixB\]. We see from the orthogonality relation (\[orthogonal\]) that $g^{(0)}_{n}(k) = {1}/{k}$. Thus the definition of $\mathcal{H}_{n}^{(l)}$ in (\[H\_def2\]) is consistent with (\[H\_def1\]) for $l = 0$. With this definition, we are able to recover the $l$-th moment of $\varepsilon(r_y) - \varepsilon_0$ from the scattering coefficients $W_{nn}(k)$ measured at different wavenumber $k$ but for one fixed $n \in \mathbb{Z}$. Putting (\[wnm\_epsilon\]), (\[moment\]) into (\[H\_def2\]), we get \_[n]{}\^[(l)]{}= \_0\^ g\^[(l)]{}\_[n]{}(k) W\_[nn]{}(k) dk &=& - \_[0]{}\^[R]{} ((r\_y) - \_0) ( \_0\^g\^[(l)]{}\_[n]{}(k) J\_n(k r\_y) J\_n(k r\_y) k\^2 dk ) r\_y dr\_y + O (\^2 )\ &=& - \_[0]{}\^[R]{} r\_y\^[l]{} ((r\_y) - \_0) dr\_y + O (\^2 ) . By this relation, the electromagnetic coefficient $\varepsilon$ can be reconstructed explicitly. \[recover1\] Let $\Omega = B_R(0)$ be the disk of center $0$ and radius $R$. Let $(\mu,\varepsilon)$ be the pair of electromagnetic parameters in $\Omega$ and $(\mu_0,\varepsilon_0)$ be the parameters of the homogeneous background. Assume that the parameters satisfy $\mu=\mu_0$ and $\varepsilon$ is radially symmetric, i.e., $\varepsilon(y) = \varepsilon(r_y)$, and $\widehat{\varepsilon} = || \varepsilon - \varepsilon_0||_{L^\infty(\Omega)}$. Then the coefficients $\mathcal{H}_{n}^{(l)}$ defined in (\[H\_def2\]) satisfy the following relationship for $l, n \in \mathbb{Z}$ and $l \geq 0$, \_[n]{}\^[(l)]{} = - \_[0]{}\^[R]{} r\_y\^[l]{} ((r\_y) - \_0) dr\_y + O (\^2 ) . \[Mellin\] For $\alpha \in \mathbb{Z}$, the $\alpha$-th Fourier coefficient $\mathfrak{F}_{r_y}\left[\varepsilon(r_y) - \varepsilon_0\right]( \alpha )$ of $\varepsilon(r_y) - \varepsilon_0$ can be written explicitly by \_[r\_y]{}( ) = - \_[l=0]{}\^ \_[n]{}\^[(l)]{} + O (\^2 ) \[recover\_05\] for a fixed $n \in \mathbb{Z}$, and the electromagnetic coefficient $\varepsilon$ can be explicitly expressed as, for a fixed $n \in \mathbb{Z}$, (- \_0)(r\_y) = - \_[ =-]{}\^ \_[l=0]{}\^ e\^[i r\_y ]{} \_[n]{}\^[(l)]{} + O (\^2 ) . \[recover\_formula1\] We remark that, with (\[recover\_formula1\]), we are able to reconstruct $\varepsilon$ from a set of scattering coefficients $\{ W_{nn}(k) | k \in \mathbb{R}^+ \}$ for all wavenumbers $k$ but with only a fixed $n \in \mathbb{Z}$ . Choosing $n$ small yields a stable reconstruction of $\varepsilon$ from far-field patterns at frequencies $k \in [0, k_{\mathrm{max}}]$ by approximating $\mathcal{H}_n^{(l)}$ with $\int_0^{k_{\mathrm{max}}} g^{(l)}_{n}(k) W_{nn}(k) \,dk$ and truncating the infinite sums in (\[recover\_formula1\]). Angularly symmetric case ------------------------ In this subsection we would like to recover the electromagnetic parameter $\varepsilon$ from the scattering coefficients $W_{nm}$ for the special domain $\Omega = B_R(0)$ and the special case when $\mu = \mu_0$ and the electromagnetic coefficient $\varepsilon$ only depends on $\theta_y$, i.e., $\varepsilon(y) = \varepsilon(\theta_y)$. Directly from (\[wnm\_epsilon\]), we have, for $n,m \in \mathbb{Z}$, W\_[nm]{} &=& - \^2 \_0 ( \_0\^[2 ]{} ((\_y) - \_0) e\^[i(m-n) \_y]{} d \_y ) ( \_[0]{}\^[R]{} J\_n(k\_0r\_y) J\_m(k\_0r\_y) r\_y dr\_y ) + O (\^2 )\ &=& - \^2 \_0 C\_[k\_0]{} (m,n) \_0\^[2 ]{} ((\_y) - \_0) e\^[i(m-n) \_y]{} d \_y + O (\^2 ) . \[wnm\_epsilon\_angular\] where $C_{k_0} (m,n)$ is given by C\_[k\_0]{} (m,n) & : =& \_[0]{}\^[R]{} J\_n(k\_0r\_y) J\_m(k\_0r\_y) r\_y dr\_y , n,m . We can see that for $n,m \in \mathbb{Z}$, $C_{k_0} (m,n)$ actually satisfies C\_[k\_0]{} (m,n) & : =& \_[0]{}\^[R]{} J\_n(k\_0r\_y) J\_m(k\_0r\_y) r\_y dr\_y\ & =& \_[0]{}\^[2 ]{} \_[0]{}\^[2 ]{} e\^[- i ( n + m ) ]{} d d\ & =& \_[0]{}\^[2 ]{} \_[0]{}\^[2 ]{} e\^[- i ( n + m ) ]{} d d\ & =& \_[,]{} (n,m) , \[coefficient\_fourier\] where $\mathfrak{F}_{\theta,\phi}$ stands for the Fourier coefficient in both arguments $\theta$ and $\phi$. Formula (\[coefficient\_fourier\]) indicates that the coefficients $C_{k_0} (m,n)$, $m,n \in \mathbb{Z}$, can be approximated via FFT or calculated explicitly. From (\[wnm\_epsilon\_angular\]), we can obtain the Fourier coefficients $\mathfrak{F}_{\theta_y} \left[\varepsilon(\theta_y) - \varepsilon_0\right]$ of $\varepsilon(\theta_y) - \varepsilon_0$ as follows: \_[\_y]{} (n-m) = - + O (\^2 ) , for all $n,m \in \mathbb{Z}$. Thus we have the following corollary. \[recover2\] Let $\Omega = B_R(0)$ and $\widehat{\varepsilon} := || \varepsilon - \varepsilon_0||_{L^\infty(\Omega)}$, and the same assumptions be assumed for $(\mu,\varepsilon)$ and $(\mu_0,\varepsilon_0)$ as in Corollary\[recover1\], except that the radial symmetry of $\varepsilon$ is now replaced by the angular symmetry, i.e., $\varepsilon(y) = \varepsilon(\theta_y)$. Then for all $n,m \in \mathbb{Z}$, the scattering coefficients $W_{nm}$ defined in (\[eq:w\_nm\]) satisfy the following relationship with the Fourier coefficients of $\varepsilon(\theta_y) - \varepsilon_0$: \_[\_y]{} (n-m) = - + O (\^2 ) . \[expill\] Let $\{ (n_{l}, m_{l}) \}_{l \in \mathbb{Z}} \subset \mathbb{Z} \times \mathbb{Z}$ be such that $n_{l}-m_{l} = l$ for all $l \in \mathbb{Z}$. Then the electromagnetic coefficient $\varepsilon$ can be explicitly expressed by (- \_0)(\_y) = - \_[l=-]{}\^ e\^[i 2 l \_y]{} + O (\^2 ) . \[recover\_formula2\] We can see from (\[recover\_formula2\]) that in order to recover the electromagnetic coefficient $\varepsilon$ in the angular symmetric case, we only need to know $\{ W_{n_{l} m_{l}} \}_{l \in \mathbb{Z}}$ where $\{ (n_{l}, m_{l}) \}_{l \in \mathbb{Z}} \subset \mathbb{Z} \times \mathbb{Z}$ is such that $n_{l}-m_{l} = l$ for $l \in \mathbb{Z}$. So we do not necessarily require all the scattering coefficients $W_{nm}$ to recover $\varepsilon$, instead we may choose $\{ W_{n_{l} m_{l}} \}_{l \in \mathbb{Z}}$ of any particular $\{ (n_{l}, m_{l}) \}_{l \in \mathbb{Z}}$, for instance we may fix $n_{l} = 0$. Truncating the sum in (\[recover\_formula2\]) up to $N$ gives a stable reconstruction formula (for the low-frequency part) with an angular resolution limit depending on $N$. Higher is $N$ better is the angular resolution. When $W_{nm}$ are corrupted by noise, $N$ can be computed as a function of the signal to noise ratio in the measurements. General case ------------ In this subsection, we try to derive formulas to recover the parameter $\varepsilon$ from the set of scattering coefficients $\{W_{nm}(k)|n,m \in \mathbb{Z}, k \in \mathbb{R}^+\}$, where $W_{nm}(k) := W_{nm}[ \, \varepsilon, \mu, \omega(k), \Omega \, ]$ is defined in (\[eq:w\_nm\]) with $\omega(k)$ satisfying (\[frequency\]) when $\Omega = B_R(0)$, $\mu = \mu_0$, without any assumption on the parameter $\varepsilon$. We would like to combine the ideas in the proofs of (\[recover\_formula1\]) and (\[recover\_formula2\]) to get a more general result. Now we start with a general $\varepsilon$ which admits the Fourier expansion: (r\_y,\_y) - \_0 = \_[ ]{} \_[\_y]{} () e\^[i \_y]{}, \[fourier\_epsilon\] where $\mathfrak{F}_{\theta_y} \left[\varepsilon(r_y,\theta_y) - \varepsilon_0 \right](\alpha)$ is the $\alpha$-th Fourier coefficient with respect to $\theta_y$ fixing $r_y$. Then we plug the expansion (\[fourier\_epsilon\]) into (\[wnm\_epsilon\]) to get W\_[nm]{} &=& - 2 \^2 \_0 \_[0]{}\^[R]{} \_[\_y]{} (n-m) J\_n(k\_0r\_y) J\_m(k\_0r\_y) r\_y dr\_y + O (\^2 ) . \[finaldev\] Following the definition of $\mathcal{H}_{n}^{(l)}$ in (\[H\_def2\]), we define a generalized coefficient $\mathcal{H}^{(l)}_{nm}$ below. For $n, m \in \mathbb{Z}$, let $W_{nm}(k) := W_{nm}[ \, \varepsilon, \mu, \omega(k), \Omega \, ]$ be defined as in (\[eq:w\_nm\]) where $\omega(k)$ is defined as in (\[frequency\]). For $l, n, m \in \mathbb{Z}$ and $l \geq 0$, let $g^{(l)}_{nm}(k)$ be functions such that \_0\^g\^[(l)]{}\_[nm]{}(k) J\_n(k r) J\_m(k r) k\^2 dk = r\^[l-1]{} , \[moment2\] for any $r > 0$. Then the coefficients $\mathcal{H}_{nm}^{(l)}$ are defined as, for $l, n, m \in \mathbb{Z}$ and $l \geq 0$, \_[nm]{}\^[(l)]{}:= \_0\^g\^[(l)]{}\_[nm]{}(k) W\_[nm]{}(k) dk . \[H\_def3\] We refer to Appendix \[appendixC\] for the existence of functions $g^{(l)}_{nm}$ satisfying (\[moment2\]). With this definition, we are able to recover, for all $n,m\in \mathbb{Z}$, the $l$-th moment of the Fourier coefficients $\mathfrak{F}_{\theta_y} \left[ \varepsilon(r_y,\theta_y) - \varepsilon_0 \right](n-m)$ with respect to $r_y$ from the scattering coefficients $W_{nm}(k)$ measured at different frequencies $k$ . Actually, we have, putting (\[frequency\]), (\[moment2\]) and (\[finaldev\]) into (\[H\_def2\]), \_[nm]{}\^[(l)]{}&=& \_0\^ g\^[(l)]{}\_[nm]{}(k) W\_[nm]{}(k) dk\ &=& - \_[0]{}\^[R]{} \_[\_y]{} (n-m) ( \_0\^g\^[(l)]{}\_[nm]{}(k) J\_n(k r\_y) J\_m(k r\_y) k\^2 dk ) r\_y dr\_y + O (\^2 )\ &=& - \_[0]{}\^[R]{} r\_y\^[l]{} \_[\_y]{} (n-m) dr\_y + O (\^2 ) , for all $n,m\in \mathbb{Z}$. Therefore, similar to (\[recover\_05\]), we get, for all $n,m,\alpha \in \mathbb{Z}$, \_[r\_y,\_y]{} (, n-m) = - \_[l=0]{}\^ \_[nm]{}\^[(l)]{} + O (\^2 ) . Fixing a set $\{ (n_{p}, m_{p}) \}_{p \in \mathbb{Z}} \subset \mathbb{Z} \times \mathbb{Z}$ such that $n_{p}-m_{p} = p$ for $p \in \mathbb{Z}$, we are able to recover $\varepsilon - \varepsilon_0$ explicitly expressed as - \_0 = - \_[ =-]{}\^ \_[ p =-]{}\^ \_[l=0]{}\^ e\^[ i ]{} \_[n\_[p]{}m\_[p]{}]{}\^[(l)]{} + O (\^2 ) . \[recover3\] Let $\Omega = B_R(0)$ and $\widehat{\varepsilon} := || \varepsilon - \varepsilon_0||_{L^\infty(\Omega)}$, and the same assumptions be assumed for $(\mu,\varepsilon)$ and $(\mu_0,\varepsilon_0)$ as in Corollary\[recover1\], except that the radial symmetry of $\varepsilon$ is now replaced by the Fourier expansion (\[fourier\_epsilon\]). Then for $l, n, m \in \mathbb{Z}$ and $l \geq 0$, the coefficients $ \mathcal{H}_{nm}^{(l)}$ defined in (\[H\_def3\]) satisfy the following relationship: \_[nm]{}\^[(l)]{} = - \_[0]{}\^[R]{} r\_y\^[l]{} \_[\_y]{} (n-m) dr\_y + O (\^2 ) . \[Mellin2\] Moreover, for all $n,m,\alpha \in \mathbb{Z}$, the $(\alpha , n-m)$-th Fourier coefficient of $\varepsilon - \varepsilon_0$ can be written explicitly by \_[r\_y,\_y]{} (, n-m) = - \_[l=0]{}\^ \_[nm]{}\^[(l)]{} + O (\^2 ) . \[recover\_25\] Let $\{ (n_{p}, m_{p}) \}_{p \in \mathbb{Z}} \subset \mathbb{Z} \times \mathbb{Z}$ be such that $n_{p}-m_{p} = p$ for all $p \in \mathbb{Z}$, then the electromagnetic coefficient $\varepsilon$ can be explicitly expressed by (- \_0)(r\_y, \_y) = - \_[ =-]{}\^ \_[ p =-]{}\^ \_[l=0]{}\^ e\^[ i ]{} \_[n\_[p]{}m\_[p]{}]{}\^[(l)]{} + O (\^2 ) . \[recover\_formula3\] We remark that expression (\[recover\_formula3\]) generalizes (\[recover\_formula1\]) and (\[recover\_formula2\]). Moreover, similar to observations in previous subsections, we can see that in order to recover the electromagnetic coefficient $\varepsilon$, we only need to know $\{ W_{n_{p} m_{p}} (k) | p \in \mathbb{Z}, k \in \mathbb{R}^+ \}$ where $\{ (n_{p}, m_{p}) \}_{p \in \mathbb{Z}} \subset \mathbb{Z} \times \mathbb{Z}$ is such that $n_{p}-m_{p} = p$ for $p \in \mathbb{Z}$. Therefore, we may choose a particular choice $\{ (n_{p}, m_{p}) \}_{p \in \mathbb{Z}}$, for instance we can let $n_{p} = 0$. This tells us that we are able to recover $\varepsilon$ with incomplete data of the scattering coefficients. As pointed out earlier, we may truncate the series in (\[recover\_formula3\]) and approximate $\mathcal{H}_{n_{p}m_{p}}^{(l)}$ by $\int_0^{k_{\mathrm{max}}} g^{(l)}_{n_{p}m_{p}}(k) W_{n_{p}m_{p}}(k) \,dk$. Concluding remarks {#sec8} ================== In this paper we have introduced the concept of scattering coefficients for inverse medium scattering problems in heterogeneous media, and established important properties (such as symmetry and tensorial properties) of the scattering coefficients as well as their various representations in terms of the NtD maps. An important relationship between the scattering coefficients and the far-field pattern is also derived. Furthermore, the sensitivity of the scattering coefficients with respect to the changes in the permittivity and permeability distributions is explored, which enables us to derive explicit reconstruction formulas for the permittivity and permeability parameters in the linearized case. These formulas show on one hand the stability of the reconstruction from multifrequency measurements, and on the other hand, the exponential instability of the reconstruction from far-field measurements at a fixed frequency. The scattering coefficient based approach introduced in this work is a new promising direction for solving the long-standing inverse scattering problem with heterogeneous inclusions. They can be combined with some existing methods such as the continuation method [@bao1; @bao2; @bao3; @coifman] to improve the stability and the resolution of the reconstructed images. Construction of the Neumann function {#appendixA} ==================================== In this section we construct the Neumann function $N_{\mu,\varepsilon}$ associated with -L u := u + \^2 u \[pde\] in $\Omega$, which is an open connected domain with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$ for $d=2,3$. We shall also estimate its singularity. Again, we assume that $0$ is not a Neumann eigenvalue of $L$ on $\Omega$. In order to show the existence of the Neumann function, we first consider the following problem: given $f \in \mathcal{C}^{\infty}_c (\Omega)$, find $u \in H^1(\Omega)$ such that u + \^2 u = f ; = 0 . \[systemtwo\] By the well-known De Giorgi-Nash-Moser Theorem [@moser] for the $L^\infty$ coefficient and Sobelov embedding, we have for $R>0$ such that $B_{R} \subset \Omega$ that || u ||\_[L\^(B\_[R/2]{})]{} & & C (R\^[1-]{} || u ||\_[L\^ ()]{} + R\^2 ||f||\_[L\^(B\_[R]{})]{} )\ & & C (R\^[1-]{} || u ||\_[H\^1()]{} + R\^2 ||f||\_[L\^(B\_[R]{})]{} ) . \[ineq1b\] On the other hand, one can prove using the same argument as in [@abboud] that for all $f \in \mathcal{C}^{\infty}_c (\Omega)$, there exists a unique $u \in H^1(\Omega)$ satisfying (\[systemtwo\]) such that || u ||\_[H\^1()]{} C || f ||\_[H\^1()]{} . \[ineq2\] Therefore, combining (\[ineq1b\]) and (\[ineq2\]), we have || u ||\_[L\^(B\_[R/2]{})]{} & & C (R\^[1-]{} || f ||\_[H\^1()]{} + R\^2 ||f||\_[L\^(B\_[R]{})]{} ) . \[ineq3\] Now consider $f \in \mathcal{C}^{\infty}_c (\Omega)$ such that the support of $f$ is contained in $B_{R} \subset \Omega$ for some $R$. Then for any $\phi \in H^1(\Omega)$, we deduce by the Hölder inequality and Sobelov embedding that | \_ f dx | || f ||\_[L\^ (B\_[R]{})]{} || ||\_[L\^ ()]{} C || f ||\_[L\^ (B\_[R]{})]{} || ||\_[H\^1()]{} C R\^ ||f||\_[L\^(B\_[R]{})]{} || ||\_[H\^1()]{} . \[ineq3point5\] For all $u \in H^1(\Omega), \Delta u \in L^2(\Omega)$ with $\f{\partial u }{\partial \nu} = 0$, the following Poincaré-type inequality can be shown by contradiction ||u||\_[H\^1()]{}\^2 C |L u , u \_[L\^2()]{}| . \[ineq10\] Setting $\phi = u \in H^1(\Omega)$ in (\[ineq3point5\]) and combining it with (\[ineq10\]), we get ||u||\_[H\^1()]{}\^2 C |L u , u \_[L\^2()]{}| = C | \_ f u dx | C R\^ ||f||\_[L\^(B\_[R]{})]{} || u ||\_[H\^1()]{} , which gives ||u||\_[H\^1()]{} C R\^ ||f||\_[L\^(B\_[R]{})]{} . \[ineq4\] Therefore, combining (\[ineq1b\]) and (\[ineq4\]), we have || u ||\_[L\^(B\_[R/2]{})]{} & & C R\^2 ||f||\_[L\^(B\_[R]{})]{} . \[ineq5\] This inequality plays a key role in proving the existence of the Neumann function and and establishing its estimate. Now we are ready to construct a Neumann function for the system (\[pde\]), following the technique in [@singular]. Fix a function $\varphi \in \mathcal{C}^{\infty}_c (B_1(0))$ and $0 \leq \varphi \leq 2$ such that $ \int_{B_1(0)} \varphi dx = 1 $. Let $y \in \Omega$ be fixed. For any $\varepsilon >0$, we define \_[,y]{} (x) = \^[-d]{} () . Let $N^{\varepsilon}(\cdot,y) \in H^1(\Omega)$ be the “averaged Neumann function” such that it satisfies (\[systemtwo\]) with $f = \varphi_{\varepsilon,y} $, then we immediately have from (\[ineq5\]) that for all $\varepsilon \leq \f{R}{2}$, || N\^(,y) ||\_[L\^(B\_[R/2]{})]{} & & C R\^2 ||\_[,y]{}||\_[L\^(B\_[R]{})]{} . \[ineq6\] This $L^\infty$ estimate for $N^{\varepsilon}(\cdot,y)$ can be further improved later. It is worth mentioning that we have the $H^1$ estimate for $N^{\varepsilon}(\cdot,y)$ by the Hölder inequality and Sobelov embedding. Indeed for all $\phi \in H^1(\Omega)$, we have | \_ \_[,y]{} dx | || \_[,y]{} ||\_[L\^ (B\_)]{} || ||\_[H\^1()]{} C \^ || ||\_[H\^1()]{} . \[ineq6point5\] Setting $\phi = N^{\varepsilon}(\cdot,y) \in H^1(\Omega)$ in (\[ineq6point5\]) and combining it with (\[ineq10\]), we yield ||N\^(,y)||\_[H\^1()]{}\^2 C |L N\^(,y) , N\^(,y) \_[L\^2()]{}| = C | \_ \_[,y]{} N\^(,y) dx | C \^ || N\^(,y) ||\_[H\^1()]{} , which gives || N\^(,y) ||\_[H\^[1]{}()]{} C \^ . \[ineq7\] Now we come back to the $L^\infty$ bound for $N^{\varepsilon}(\cdot,y)$. For all $\varepsilon \leq \f{R}{2}$, $R \leq d_y$ where $d_y$ is the distance between $y$ and $\partial \Omega$, \_ f N\^(,y) dx = - \_ L u N\^(,y) dx = - \_ u L N\^(,y) dx = - \_ u \_[,y]{} dx , hence we have from $ \int_{B_1(0)} \varphi dx = 1 $, $\varphi \geq 0$ and (\[ineq5\]) that | \_ f N\^(,y) dx | || \_[,y]{} ||\_[L\^1(B\_[R/2]{})]{} || u ||\_[L\^(B\_[R/2]{})]{} = || u ||\_[L\^(B\_[R/2]{})]{} C R\^2 ||f||\_[L\^(B\_[R]{})]{} . Therefore, by duality, we have the $L^1$ estimate for $N^{\varepsilon}(\cdot,y)$ with $\varepsilon \leq \f{R}{2}$ and $R \leq d_y$ as follows || N\^(,y) ||\_[L\^1(B\_[R]{})]{} & & C R\^2 . \[ineq8\] We wish to use De Giorgi-Nash-Moser theorem once again to get a sharp $L^\infty$ estimate for $N^{\varepsilon}(\cdot,y)$ from (\[ineq8\]) following an idea in [@singular]. Indeed, for any $x \in \Omega$ such that $0 < |x-y| < {d_y}/{2}$, take $R : = {2|x-y|}/{3}$. Note that if $\varepsilon < {R}/{2}$, then $ N^{\varepsilon}(\cdot,y) \in H^{1}(B_R(x))$ satisfies $-L N^{\varepsilon}(\cdot,y) = 0$ in $B_R(x)$. For $r \leq \f{R}{3}$, we derive by the De Giorgi-Nash-Moser theorem for the $L^\infty$ coefficient, we get that |N\^(x,y)| || N\^(,y) ||\_[L\^(B\_r(x))]{} C r\^[-d]{} || N\^(,y) ||\_[L\^1(B\_[r]{}(x))]{} C r\^[-d]{} || N\^(,y) ||\_[L\^1(B\_[3r]{}(y))]{} C r\^[2-d]{} . \[ineq8b\] Therefore we recover the result in [@singular] for our operator $L$: for any $x,y \in \Omega$ satisfying $0 < |x-y| < {/d_y}{2}$, we have |N\^(x,y)| C |x-y|\^[2-d]{} 0 . \[combineineq1\] Moreover it follows directly from (\[ineq9\]) that for all $\epsilon < \f{r}{6}$, || N\^(,y) ||\_[L\^ (\\B\_r(y))]{} C r\^, \[ineq12\] while for $\epsilon \geq \f{r}{6}$, || N\^(,y) ||\_[L\^ (\\B\_r(y))]{} C || N\^(,y) ||\_[H\^1()]{} C r\^. \[ineq13\] Now the combination (\[ineq12\]) with (\[ineq13\]) yields || N\^(,y) ||\_[L\^(\\B\_r(y))]{} C r\^ , r (0 ,) , &gt; 0 . \[combineineq2\] On the other hand, the following estimate comes from (\[combineineq1\]) and (\[combineineq2\]) for all $r \in \left(0 ,d_y\right)$ that || N\^(,y) ||\_[L\^(\\B\_r(y))]{} + || N\^(,y) ||\_[L\^2(\\B\_r(y))]{} C r\^ , &gt; 0 . \[totalcombineineq\] With this estimate (\[totalcombineineq\]), we can readily derive the following estimate for $r \in \left(0 ,d_y\right)$ by following the same argument as in [@singular]: || N\^(,y) ||\_[L\^p(B\_r(y))]{} C r\^[2 - d + ]{} , &gt; 0 , p \[1 , ) ,\ \[totalcombineineq2\] || N\^(,y) ||\_[L\^p(B\_r(y))]{} C r\^[1 - d + ]{} , &gt; 0 , p \[1 , ) . \[totalcombineineq3\] Now the same argument as in [@singular] will ensure the existence of a sequence $\{\varepsilon_n\}_{n=1}^\infty$ going to zero and a function $N(\cdot,y)$ such that $N^{\varepsilon_n}(\cdot,y)$ converges to $N(\cdot,y)$ weakly in $W^{1,p}(B_r(y))$ for $1 < p < \f{d}{d-1}$ and weakly in $H^1(\Omega \backslash B_r(y))$ for all $r \in \left(0 , d_y \right)$. It is then routine (see [@singular]) to get an estimate of $N(\cdot,y)$ from (\[totalcombineineq\]) for all $r \in \left(0 , d_y\right)$, || N(,y) ||\_[L\^(\\B\_r(y))]{} + || N(,y) ||\_[L\^2(\\B\_r(y))]{} C r\^ , \[totalcombineineqlimit\] and from (\[totalcombineineq2\]) and (\[totalcombineineq3\]) for all $r \in \left(0 , d_y\right)$ that || N(,y) ||\_[L\^p(B\_r(y))]{} C r\^[2 - d + ]{} , p \[1 , ) ,\ \[totalcombineineq2limit\] || N(,y) ||\_[L\^p(B\_r(y))]{} C r\^[1 - d + ]{} , p \[1 , ) . \[totalcombineineq3limit\] Our section ends with the pointwise estimate for $N(x,y)$ by using De Giorgi-Nash-Moser theorem once again. For any $x \in \Omega$ such that $0 < |x-y| < \f{d_y}{2}$, take $R : = \f{2|x-y|}{3}$. From (\[totalcombineineqlimit\]) we have $N(\cdot,y)\in H^1(B_R(x))$ satisfying $-L N(\cdot,y)=0$ in $B_R(x)$. Then by De Giorgi-Nash-Moser theorem for the $L^\infty$ coefficient, we can deduce the following estimate with the same technique as in (\[ineq8b\]), |N(x,y)| C r\^[-d]{} || N(,y) ||\_[L\^1(B\_[r]{}(x))]{} C r\^[-d]{} || N(,y) ||\_[L\^1(B\_[3r]{}(y))]{} C |x-y|\^[2-d]{} . \[totalcombineineq4limit\] This gives the estimate of the singularity type as $x$ approaches to $y$. Existence of functions $g^{(l)}_n$ {#appendixB} ================================== In this section we wish to show the existence of functions $g^{(l)}_n$ satisfying (\[moment\]) for all $l, n \in \mathbb{N}$ and provide their explicit expressions. From the fact that \^2 = \_0\^[/2]{} J\_[2n]{} (2 kr ) d \[integral\] for all $n \in \mathbb{N}$, we substitute (\[integral\]) into (\[moment\]) to get, for all $l , n \in \mathbb{N}$, that \_0\^[/2]{} \_0\^g\^[(l)]{}\_[n]{}(k) J\_[2n]{} (2 kr ) k\^2 dk d = r\^[l-1]{} r &gt; 0 , \[moment2b\] Recall the following orthogonal relationship for Hankel functions \_0\^J\_[2n]{} (kr)J\_[2n]{}(k’r ) r dr = . \[ortho\] for all $k , k' > 0$ and $n \in \mathbb{N}$,. Now, for $l, n \in \mathbb{N}$, consider the Hankel tranform of $r^{l-1}$ of order $2n$ at $p >0$, \[\_[2n]{} (r\^[l-1]{}) \](p)&:=& \_0\^ r\^[l-1]{} J\_[2n]{}(r p ) r dr . By a change of variables, we have \[\_[2n]{} (r\^[l-1]{}) \](p)&=& \_0\^ r\^[l-1]{} J\_[2n]{}(r p ) r dr\ &=& \_0\^[/2]{} \_0\^ g\^[(l)]{}\_[n]{}(k) ( \_0\^J\_[2n]{} (2 kr )J\_[2n]{}(r p ) r dr ) k\^2 dk d\ &=& \_0\^ \_[= 0]{}\^[= /2]{} k g\^[(l)]{}\_[n]{}(k) d ( 2 k ) dk\ &=& \_0\^ \_[0]{}\^[2k]{} k g\^[(l)]{}\_[n]{}(k) d l dk . \[eqil1\] From orthogonality relation (\[ortho\]), we get that from (\[eqil1\]) that \[\_[2n]{} (r\^[l-1]{}) \](p) &=& \_0\^ \_[{p&lt;2k}]{}(k) dk\ &=& \_\^ dk . \[eqil2\] Therefore, for $p >0$, we have - 2p \[\_[2n]{} (r\^[l-1]{}) \](2 p) &=& - \_[p]{}\^ dk . \[eqil3\] Now we recall that the Abel transform of an integrable function $f(r)$ defined on $r\in (0,\infty)$ is as follows F(y) := \[ (f)\](y) := 2 \_[y]{}\^ dr , y (0,) , \[abel\] whenever the above integral is well-defined. If $f(r) = O(\f{1}{r})$ as $r \rightarrow \infty$, then its inverse Abel transform is well-defined and $f$ satisfies the following f(r) = \[\^[-1]{} (F)\](r) := - \_[r]{}\^ dy , r (0,) . \[invabel\] Comparing (\[invabel\]) and (\[eqil3\]), we can see that, for all $l , n \in \mathbb{N}$, the functions G\^[(l)]{}\_[n]{}(p) : = - 2p \[\_[2n]{} (r\^[l-1]{}) \](2 p) , p (0,) \[defGin\] are nothing but the inverse Abel transform of a primitive function of $k^2 g^{(l)}_{n}(k)$. Therefore, applying Abel transform to both sides of the equation (\[eqil3\]) and then differentiating with respect to the argument of the function, we get (k) = k\^2 g\^[(l)]{}\_[n]{}(k) . \[eqil4\] Consequently, we have the following explicit expression for $g^{(l)}_{n}$ g\^[(l)]{}\_[n]{}(k) = (k) , k (0,), \[answer\] where $G^{(l)}_{n}$ is defined as in (\[defGin\]). One can see by direct substitution of (\[answer\]) back into (\[moment\]) that the functions $g^{(l)}_{n}$ defined as (\[answer\]) satisfy equation (\[moment\]). Therefore, we have shown existence of functions satisfying (\[moment\]). Existence of functions $g^{(l)}_{nm}$ {#appendixC} ===================================== In this section we show the existence of functions $g^{(l)}_{nm}$ for $l,n,m \in \mathbb{Z}$ and $l\geq 0$ which satisfies (\[moment2\]), namely the integral equation \_0\^g\^[(l)]{}\_[nm]{}(k) J\_n(k r) J\_m(k r) k\^2 dk = r\^[l-1]{} r &gt; 0 . \[generalmoment\] For this purpose, we would like to first investigate the following integral, which will be useful in the subsequent discussion. For $n,m \in \mathbb{N}$ and $p \in \mathbb{C}$ such that $m + n > Re(p)>0$, we consider the following integral, A\_[nm]{}(p) := \_0\^J\_n(x)J\_m(x)x\^[-p]{} dx, p , m + n &gt; Re(p)&gt;0 . \[integralAnm\] We observe that the function $A_{nm}: \{p \in \mathbb{C}\,: \, m + n > Re(p)>0 \}\rightarrow \mathbb{C}$ is a holomorphic function on the strip $\{p \in \mathbb{C}\,: \, a<Re(p)<b \}$ for some $a, b \in \mathbb{R}$ such that $a < b$. This comes from the fact that for $n,m \in \mathbb{N}$ and $p \in \mathbb{C}$ such that $m + n > Re(p)>0$, the integral $A_{nm}(p)$ defined in (\[integralAnm\]) can be expressed in the following form, A\_[nm]{}(p)=\_0\^J\_m(x)J\_n(x)x\^[-p]{} dx = \[integral\_explicit\] Now given $a, b \in \mathbb{R}$ and $s \in \mathbb{C}$ such that $a <Re(s)<b$, we recall the definition of the Mellin tranform of an integrable function $f(r)$ defined for $r\in (0,\infty)$: (y) := \_0\^r\^[s-1]{} f(r) dr , a &lt;Re(s)&lt;b whenever the above integral is well-defined. With $a,b \in \mathbb{R}$, we write the function $\zeta_{a,b}$ as \_[a,b]{}(x) = x\^[-a]{} 0 &lt; x 1 , x\^[-b]{} 1 &lt; x &lt; . We define the linear space $\mu_{a,b}(0,\infty)$ as the space of all infinitely smooth compactly supported complex valued functions $\phi \in C_c^\infty(0,\infty)$ for which || ||\_[k,\_[a,b]{},K]{} := \_[K]{} |\_[a,b]{}(x) x\^[k+1]{} D\_x\^k (x)| is finite for all $k \in \mathbb{N}$ and for any compact set $K \Subset (0,\infty)$. Consider an increasing sequence of compact sets $\{K_n \Subset (0,\infty)\}_{n\in \mathbb{N}}$ such that $\bigcup_{n\in \mathbb{N}} K_n = (0,\infty)$, the countable norms $|| \cdot ||_{k,\zeta_{a,b},K_n} \, , k,n \in \mathbb{N}$ gives a topology on $\mu_{a,b}(0,\infty)$ such that $\mu_{a,b}(0,\infty)$ becomes a complete locally convex space. We define the dual of $\mu_{a,b}(0,\infty)$, $\mu_{a,b}'(0,\infty)$, and equip it with the weak topology. With these definitions at hand, the Mellin transform can be naturally extended to the space $\mu_{a,b}'(0,\infty)$, see [@alomari; @Zemanian] for more details. We denote the generalized Mellin transform also as $\mathcal{M}$. Now from (\[generalmoment\]) and (\[integralAnm\]), we have for all $l,n,m \in \mathbb{Z}$ with $l\geq 0$ and $p\in \mathbb{C}$ such that $Re(p)>l$, (l-p)&=& \_0\^\_0\^g\^[(l)]{}\_[nm]{}(k) J\_n(k r) J\_m(k r) r\^[-p]{} k\^2 dk dr\ &=& \_0\^g\^[(l)]{}\_[nm]{} (k) ( \_[r=0]{}\^J\_n(k r) J\_m(k r) (kr)\^[-p]{} d(kr) ) k\^[p + 1]{} dk\ &=& \_0\^g\^[(l)]{}\_[nm]{} (k) ( \_[0]{}\^J\_n(r) J\_m(r) r\^[-p]{} d r ) k\^[p + 1]{} dk\ &=& A\_[nm]{}(p) (p+2) , where $A_{nm}(p)$ is known explicitly as (\[integral\_explicit\]). Therefore we get, for all $l,n,m \in \mathbb{Z}$ with $l\geq 0$ and $p\in \mathbb{C}$ such that $Re(p)>l$, (p+2) = , then the existence of $g^{(l)}_{nm}$ is ensured by the Mellin inverse transform. [11]{} T. Abboud and H. 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Part II: the Helmholtz equation*, Comm. Math. Phys. 317 (2013), pp. 485-502. H. Ammari, M.P. Tran, and H. Wang, Shape identification and classification in echolocation, preprint, arxiv:1308.5625, (2013). P.M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1971. G. Bao, Y. Chen, and F. Ma, *Regularity and stability for the scattering map of a linearized inverse medium problem*, J. Math. Anal. Appl. 247 (2000), pp. 255-271. G. Bao, J. Junshan, and F. Triki, *A multi-frequency inverse source problem*, J. Diff. Equat. 249 (2010), pp. 3443-3465. G. Bao, J. Junshan, and F. Triki, *Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data*, Contemp. Math. 548 (2011), pp. 45-60. R. Coifman, M. Goldberg, T. Hrycak, M. Israeli, and V. Rokhlin, *An improved operator expansion algorithm for direct and inverse scattering computations*, Waves Random Media 9 (1999), pp. 441-457. V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006. V. Isakov, *Increased stability in the Cauchy problem for some elliptic equations*, in: C. Bardos, A. Fursikov (Eds.), Instability in Models Connected with Fluid Flows. I, in: International Math. Series, vol. 6, Springer, 2008, pp. 339-362. V. Isakov and S. Kindermann, *Subspaces of stability in the Cauchy problem for the Helmholtz equation*, Methods Appl. Anal. 18 (2011), pp. 1-29. F. John, *Continuous dependence on data for solutions of partial differential equations with a prescribed bound*, Comm. Pure Appl. Math. 13 (1960), pp. 551-585. J. Moser, *A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations*, Comm. Pure Appl. Math. 13 (1960), pp. 457-468. G. N. Watson, [Theory of Bessel Functions]{}, 2nd edition, Cambridge University Press (1944). A.H. Zemanian, [Generalized integral transformation]{}, Dover Publications, Inc., New York(1968). [^1]: Department of Mathematics and Applications, Ecole Normale Sup$\acute{\text{e}}$rieure, 45 Rue d’Ulm, 75005 Paris, France. The work of this author was supported by ERC Advanced Grant Project MULTIMOD–267184. (habib.ammari@ens.fr). [^2]: Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong (ytchow@math.cuhk.edu.hk). [^3]: Department of Mathematics, Chinese University of Hong Kong, Shatin, N.T., Hong Kong. The work of this author was substantially supported by Hong Kong RGC grants (projects 405513 and 404611). (zou@math.cuhk.edu.hk).

--- abstract: | Incompressibility is established for three-dimensional and two-dimensional deformations of an anisotropic linearly elastic material, as conditions to be satisfied by the elastic compliances. These conditions make it straightforward to derive results for incompressible materials from those established for the compressible materials. As an illustration, the explicit secular equation is obtained for surface waves in incompressible monoclinic materials with the symmetry plane at $x_3=0$. This equation also covers the case of incompressible orthotropic materials. The displacements and stresses for surface waves are often expressed in terms of the elastic stiffnesses, which can be unbounded in the incompressible limit. An alternative formalism in terms of the elastic compliances presented recently by Ting is employed so that surface wave solutions in the incompressible limit can be obtained. A different formalism, also by Ting, is employed to study the solutions to two-dimensional elastostatic problems. In the special case of incompressible monoclinic material with the symmetry plane at $x_3=0$, one of the three Barnett-Lothe tensors $\mathbf{S}$ vanishes while the other two tensors $\mathbf{H}$ and $\mathbf{L}$ are the inverse of each other. Moreover, $\mathbf{H}$ and $\mathbf{L}$ are diagonal with the first two diagonal elements being identical. An interesting physical phenomenon deduced from this property is that there is no interpenetration of the interface crack surface in an incompressible bimaterial. When only the inplane deformation is considered, it is shown that the image force due to a line dislocation in a half-space or in a bimaterial depends only on the magnitude, not on the direction, of the Burgers vector. author: - 'Michel Destrade, Paul A. Martin, Tom C.T. Ting' title: 'The incompressible limit in linear anisotropic elasticity, with application to surface waves and elastostatics' --- Introduction ============ Linear anisotropic elasticity is characterized by two material constants, which can be taken as the shear modulus $\mu$ and Poisson’s ratio $\nu$. These constants satisfy $\mu>0$ and $-1<\nu<\halft$. The incompressible limit is $\nu\to\halft$. To see why this is so, we write down Hooke’s law, relating the stress components $\sigma_{ij}$ to the strain components $\epsilon_{kl}$ as $$\sigma_{ij}=\mu\left( \frac{2\nu}{1-2\nu}\delta_{ij}\epsilon_{kk} +2\epsilon_{ij}\right).$$ In the above, is the Kronecker delta and repeated indices imply summation. Contracting, we obtain $$\epsilon_{ii}= \frac{1-2\nu}{2\mu(1+\nu)}\sigma_{ii} =\frac{\nu}{\lambda(1+\nu)}\sigma_{ii}, \label{EPSIG}$$ where $\lambda$ is a Lamé constant. If the material is incompressible, $\epsilon_{ii}=0$ for every possible deformation, whence $_1$ gives $\nu=\halft$. Let us now turn to linear [*anisotropic*]{} elasticity, and consider the corresponding incompressible limit. For such materials, we have $\sigma_{ij} = C_{ijks} \epsilon_{ks}$, where the $C$’s are the elastic stiffnesses. In the special case of isotropy, the non-trivial stiffnesses are $C_{1111} = C_{2222} = C_{3333}=\lambda +2 \mu$, $C_{1122} = C_{1133}=C_{2233}=\lambda$ and $C_{1212}=C_{1313}=C_{2323}=\mu$. From $_2$, the incompressible limit corresponds to $\lambda\to\infty$. This suggests that, in general, some of the stiffnesses will become unbounded in the incompressible limit, and therefore it will be safer to work with the coefficients of the elastic *compliance* matrix $ \mathbf{s}$ rather than with those of the elastic stiffness matrix $ \mathbf{C}$. This is so because $ \mathbf{s}$ is the inverse of $ \mathbf{C}$, and possible infinite components of $ \mathbf{C}$ will simply correspond to some components (or combination of components) of $ \mathbf{s}$ being equal to zero. In order to consider incompressible linearly elastic anisotropic materials directly, some authors have modified the stress-strain law by introducing a hydrostatic pressure $P$, as $\sigma_{ij} = - P \delta_{ij} + C_{ijks} \epsilon_{ks}$. Incompressibility is then imposed by supplementing the condition $\epsilon_{ii}=0$. Although formally acceptable, and supported by similar considerations in finite elasticity, this approach is risky as it may lead to potentially meaningless results, when the stiffnesses appear in their final expressions. For example, consider some recent developments in the theory of surface waves in linear anisotropic elastic materials. For compressible materials the secular equation was obtained explicitly for monoclinic materials with the symmetry plane at $x_3=0$ [@Dest01a; @Ting01]. At the same time, some attention has been given to the consideration of interface waves in anisotropic materials which are *incompressible* (see for instance [@NaSo99] or [@Dest01c], and the references therein). In this paper we show that results obtained in the general (compressible) case can be easily specialized to the incompressible case, simply by imposing the conditions for incompressibility on the elastic compliances, without having to introduce an arbitrary pressure. We adopt the following plan for the paper. In Section 2 we recall the three-dimensional stress-strain laws of linear anisotropic elasticity, and establish that the constraint of incompressibility yields simple mathematical conditions, which are written for the elastic compliances $s_{\alpha \beta}$. Unlike the case of isotropic elastic materials, the conditions of incompressibility are different for *two*-dimensional deformations. These conditions are established in Section 3 and written for the *reduced elastic compliances* $s'_{\alpha \beta}$. In both Sections, a necessary and sufficient condition for the strain energy density to be positive semidefinite is presented. Then we show in Section 4 how simple it is to deduce an explicit secular equation for surface waves in a monoclinic material with the symmetry plane at $x_3 = 0$ for the incompressible case from that for the compressible case. The secular equation is only a part of the surface wave solution. In the literature, the stresses and displacements for surface waves in an anisotropic elastic material are expressed in terms of the elastic stiffnesses, as briefly summarized in Section 5. These expressions have to be converted to ones for the reduced elastic compliances. This has been done by Ting [@Ting01] and is outlined in Section 6. The conversion presented in Section 6 does not apply to elastostatics. A different formulation, again by Ting [@Ting99], is reviewed in Section 7. In Section 8 we consider the special case of incompressible monoclinic materials with the symmetry plane at $x_3=0$ under a static loading. Interesting physical phenomena are discovered due to the incompressibility of the material. Incompressibility for three-dimensional deformations ==================================================== When the displacement $\mathbf{u}$ in an anisotropic linear elastic material depends on the three material coordinates $x_1$, $x_2$, $x_3$, the deformation is three-dimensional. The relation between the strains $\epsilon_\alpha$ and the stresses $\sigma_\alpha$ in the contracted notation [@Voig10] is $$\label{strain-stress3D} \epsilon_\alpha = s_{\alpha \beta} \sigma_\beta,$$ where $s_{\alpha \beta}$ are the elastic compliances. In particular, for isotropic materials, we have, $$\nonumber \mathbf{s} = \frac{1}{2\mu(1+\nu)} \begin{bmatrix} 1 & & & & & \\ -\nu & 1 & & & & \\ -\nu & -\nu & 1 & & & \\ 0 & 0 & 0& 2(1+\nu) & & \\ 0 & 0 & 0 & 0 & 2(1+\nu) & \\ 0 & 0 & 0 & 0& 0 & 2(1+\nu) \end{bmatrix}. \label{isocom}$$ In an *incompressible* material the vanishing of the volume change is given by $$\epsilon_1 + \epsilon_2 + \epsilon_3 = \sum_{\beta=1}^{6}\left( \sum_{\alpha=1}^{3} s_{\alpha \beta} \right) \sigma_\beta = 0.$$ If this is to hold for any stresses we must have $$\label{incompr3D} \sum_{\alpha=1}^{3} s_{\alpha \beta} =0, \quad \text{ for } \beta = 1,2,3,4,5,6.$$ There are six conditions for incompressibility. When the material is isotropic, is trivially satisfied for $\beta = 4,5,6$ while for $\beta = 1,2,3$ it recovers the condition that $\nu = \halft$. Now we show that is *structurally invariant* [@Ting01b]. If holds for a coordinate system $x_j$, it holds for any other coordinate system $x^*_i$ obtained from $x_j$ by an orthogonal transformation $\mbox{\boldmath $\Omega$}$, say. Let $$\label{rot} x^*_i = \Omega_{ij} x_j, \quad \Omega_{ik} \Omega_{jk} = \delta_{ij} = \Omega_{ki}\Omega_{kj}$$ In the four-index tensor notation, the elastic compliances $s_{ijks}$ referred to the rotated coordinate system $x^*_i$ become $$s^*_{ijks} = \Omega_{ip} \Omega_{jq} \Omega_{kr} \Omega_{st} s_{pqrt}.$$ By contracting $i = j$ and using $_3$, this equation yields $$\label{rotContr} s^*_{iiks} = \Omega_{kr} \Omega_{st} s_{pprt}.$$ However, in the four-index tensor notation is $s_{pprt}=0$. Equation then gives $s^*_{iiks}=0$. This completes the proof. The constraint says that the first three rows of the $6 \times 6$ matrix $\mathbf{s}$ are linearly dependent. This means that $\mathbf{s}$ is singular, and that the rank of $\mathbf{s}$ is at most five. We assume that the rank is five, because that is the case for isotropic materials. The strain energy density cannot be negative for an incompressible material. Hence $\mathbf{s}$ must be positive semidefinite. The rank of $\mathbf{s}$ being five implies that there exists a $5 \times 5$ submatrix that is non-singular. According to a theorem presented in [@Hohn65], a necessary and sufficient condition for the matrix $\mathbf{s}$ of rank five to be positive semidefinite is that the five leading principal minors of the non-singular submatrix be positive. It means that this non-singular submatrix must be positive definite. To apply the theorem we write the matrix $\mathbf{s}$ satisfying the constraint in the form $$\label{s6x6} \mathbf{s} = \begin{bmatrix} s_{22} + 2s_{23} + s_{33} & & & & & \\ -(s_{22} + s_{23}) & s_{22} & & & & \\ -(s_{23} + s_{33}) & s_{23} & s_{33} & & & \\ -(s_{24} + s_{34}) & s_{24} & s_{34} & s_{44} & & \\ -(s_{25} + s_{35}) & s_{25} & s_{35} & s_{45} & s_{55} & \\ -(s_{26} + s_{36}) & s_{26} & s_{36} & s_{46} & s_{56} & s_{66} \end{bmatrix}.$$ Only the lower triangle of the matrix is shown since it is symmetric. The $5 \times 5$ submatrix on the lower right corner of $\mathbf{s}$ can be prescribed arbitrarily and the elements in the first column (and hence the first row) of $\mathbf{s}$ are then determined. We will therefore take the $5 \times 5$ submatrix on the lower right corner of $\mathbf{s}$ to be non-singular. Before we write down the leading principal minors of this submatrix, we introduce the following notation for the minors of $\mathbf{s}$ . Let $s(n_1, \ldots, n_k | m_1, \ldots, m_k)$ be the $k \times k$ minor of the matrix $s_{\alpha \beta}$, the elements of which belong to the rows of $s_{\alpha \beta}$ numbered $n_1, \ldots n_k$ and columns numbered $m_1, \ldots m_k$, $1 \le k \le 6$. A principal minor is $s(n_1, \ldots, n_k | n_1, \ldots, n_k)$, which is written as $s(n_1, \ldots, n_k)$ for simplicity. If the leading principal minors are taken from the lower right corner of the submatrix, a necessary and sufficient condition for the matrix $\mathbf{s}$ to be positive semidefinite is $$\label{iff1} s_{66} > 0, \quad s(5,6) > 0, \quad s(4,5,6) > 0, \quad s(3,4,5,6) > 0, \quad s(2,3,4,5,6) > 0.$$ If they are taken from the top left corner of the submatrix, we have $$\label{iff2} s_{22} > 0, \quad s(2,3) > 0, \quad s(2,3,4) > 0, \quad s(2,3,4,5) > 0, \quad s(2,3,4,5,6) > 0.$$ Equation or is the necessary and sufficient condition for the matrix $\mathbf{s}$ to be positive semidefinite. The first two inequalities in are the necessary and sufficient conditions for the $3 \times 3$ submatrix on the top left corner of the matrix $\mathbf{s}$ to be positive semidefinite. When the three equations for $\beta = 1,2,3$ in are solved for $s_{12}, s_{23}, s_{31}$, we have $$\label{s12s23s31} s_{12} = \halft (s_{33} - s_{11} - s_{22}), \quad s_{23} = \halft (s_{11} - s_{22} - s_{33}), \quad s_{31} = \halft (s_{22} - s_{33} - s_{11}).$$ Hence $s_{11}, s_{22}, s_{33}$ are all we need to prescribe the $3 \times 3$ submatrix. The $s_{11}, s_{22}, s_{33}$ are, respectively, $1/E_1, 1/E_2, 1/E_3$, where $E_i$ are the Young’s moduli. With the $s_{23}$ given in , the second inequality in is $$\label{2ndIneq} s(2,3) - s_{22} s_{33} - \textstyle{\frac{1}{4}} (s_{11} - s_{22} - s_{33})^2>0.$$ Since $s_{22}>0$, equation tells us that $s_{33}>0$. Equation can then be written as $$[(\sqrt{s_{22}} + \sqrt{s_{33}})^2 - s_{11}] [s_{11} - (\sqrt{s_{22}} - \sqrt{s_{33}})^2] > 0.$$ It tells us that $s_{11}>0$. This is rewritten in a form symmetric with respect to $s_{11}, s_{22}, s_{33}$ as $$\label{symm} (U+V+W)(U+V-W)(V+W-U)(W+U-V)>0,$$ where $U = \sqrt{s_{11}}, V = \sqrt{s_{22}}, W = \sqrt{s_{33}}$. Scott [@Scot00] obtained the same inequality, involving the area modulus of elasticity. From Hero’s formula, the left hand side of is, after taking the square root and dividing the result by 4, the area of a triangle whose three sides are $U, V, W$. Thus $\sqrt{s_{11}}, \sqrt{s_{22}}, \sqrt{s_{33}}$ must form a triangle with a nonzero area for the $3 \times 3$ submatrix to be positive semidefinite. Another geometrical interpretation of the constraint on $s_{11}, s_{22}, s_{33}$ can be made by noticing that is equivalent to $$V+W > U > |V-W|.$$ In a rectangular coordinate system $U, V, W$, the point ($U, V, W$) is inside a triangular cone (or pyramid) in the space $U>0, V>0, W>0$. The three edges of the cone lie on the three coordinate planes making an equal angle ($\pi/4$) with the coordinate axes. When the material is compressible, Zheng and Chen [@ZhCh00] employed the notation $$n_i = \frac{-s_{jk}}{s_{jj}s_{kk}} = \sqrt{\frac{E_k}{E_j}} \nu_{jk} = \sqrt{\frac{E_j}{E_k}} \nu_{kj},$$ where $\nu_{ij}$ are Poisson’s ratios and $\{i, j, k\}$ is a cyclic permutation of $\{1,2,3\}$. The condition for the $3 \times 3$ submatrix to be positive definite is $|n_i| <1$, ($i=1,2,3$) and $$\label{zhongi} n_1^2 + n_2^2 + n_3^2 + 2 n_1 n_2 n_3 <1.$$ The geometry of the solid represented by resembles that of a Chinese delicacy called *Zongzi*. For an incompressible material, ($n_1, n_2, n_3$) lies on the surface of a *Zongzi*. Equation can be written in a symmetric form as $$s(2,3) = \halft (s_{11}s_{22} + s_{22}s_{33} + s_{33}s_{11}) - \textstyle{\frac{1}{4}} (s_{11} + s_{22} + s_{33})^2 >0.$$ Hence the three $2 \times 2$ minors $s(2,3), s(3,1), s(1,2)$ are identical. Incompressibility for two-dimensional deformations ================================================== When the displacement $\mathbf{u}$ depends on $x_1, x_2$, but not on $x_3$, the deformation is two-dimensional. In this case $ \epsilon_3 = u_{3,3} = 0$, and is replaced by $$\label{strain-stress2D} \epsilon_\alpha = s'_{\alpha \beta} \sigma_\beta,$$ where $$\label{reduced} s'_{\alpha \beta} = s_{\alpha \beta} - \frac{ s_{\alpha 3} s_{3 \beta}}{s_{33}},$$ are the reduced elastic compliances [@Lekh63]. It should be noted that $s'_{\alpha 3} = s'_{3 \alpha} = 0$. With , the incompressibility condition $\epsilon_1 + \epsilon_2 = 0$ yields $$\label{incompr2D} s'_{1 \beta} + s'_{2 \beta} =0, \quad \text{ for } \beta = 1,2,4,5,6.$$ When the material is isotropic, is trivially satisfied for $\beta =4,5,6$, while for $\beta = 1,2$, it recovers the condition that $\nu = \halft$. Under a rotation of the coordinate system about the $x_3$-axis, Ting [@Ting01b] has shown that the following relations for the elastic stiffnesses $C_{\alpha \beta}$ in the contracted notation are *structurally invariant*: $$\label{strucInv1} C_{16} + C_{26} = C_{11} - C_{22} =0, \quad C_{14} + C_{24} = C_{15} + C_{25} =0.$$ They are called Type 1A and 4A, respectively. He pointed out that applies also to $s'_{\alpha \beta}$. Following his derivation it can be shown that $$\label{strucInv2} C_{11} + C_{12} = C_{12} + C_{22} = C_{16} + C_{26} = 0$$ is structurally invariant, and that it applies to $s'_{\alpha \beta}$. Thus the incompressibility condition is *structurally invariant* under rotation of the coordinate axes about the $x_3$-axis. The reduced elastic compliance matrix that satisfies has the structure $$\mathbf{s'} = \begin{bmatrix} s'_{22} & & & & \\ -s'_{22} & s'_{22} & & & \\ -s'_{24} & s'_{24} & s'_{44} & & \\ -s'_{25} & s'_{25} & s'_{45} & s'_{55} & \\ -s'_{26} & s'_{26} & s'_{46} & s'_{56} & s'_{66} \ \end{bmatrix}.$$ The matrix $\mathbf{s'}$ must be positive semidefinite. A necessary and sufficient condition for the matrix $\mathbf{s'}$ to be positive semidefinite is that the four leading principal minors of the $4 \times 4$ submatrix on the lower right corner of $\mathbf{s'}$ be positive. If the leading principal minors are taken from the lower right corner of the submatrix, a necessary and sufficient condition for $\mathbf{s'}$ to be positive semidefinite is $$s'_{66} > 0, \quad s'(5,6) > 0, \quad s'(4,5,6) > 0, \quad s'(2,4,5,6) > 0.$$ If they are taken from the top left corner of the submatrix, we have $$s'_{22} > 0, \quad s'(2,4) > 0, \quad s'(2,4,5) > 0, \quad s'(2,4,5,6) > 0.$$ Using , equation can be rewritten as $$s_{1\beta} + s_{2 \beta} + w s_{3 \beta} = 0, \quad w = -(s_{13} + s_{23})/s_{33}.$$ It is an identity when $\beta=3$. An elastic compliance matrix that satisfies this equation has the structure $$\label{s5x5} \mathbf{s} = \begin{bmatrix} s_{22} + 2s_{23} + w^2 s_{33} & & & & & \\ -(s_{22} + w s_{23}) & s_{22} & & & & \\ -(s_{23} + w s_{33}) & s_{23} & s_{33} & & & \\ -(s_{24} + w s_{34}) & s_{24} & s_{34} & s_{44} & & \\ -(s_{25} + w s_{35}) & s_{25} & s_{35} & s_{45} & s_{55} & \\ -(s_{26} + w s_{36}) & s_{26} & s_{36} & s_{46} & s_{56} & s_{66} \end{bmatrix}.$$ where $w$ is arbitrary. It reduces to when $w=1$. Thus incompressibility in three-dimensional deformations implies incompressibility in two-dimensional deformations, but the converse need not hold. A necessary and sufficient condition for the matrix $\mathbf{s}$ in to be positive semidefinite is identical to the one in or . It should be noted that or does not involve $w$. If the matrix $\mathbf{s}$ in is positive semidefinite for any $w$, then $s_{11}$ and $s(1,2)$, which can be computed easily, should be non-negative for any $w$. It can be shown that $$s_{11} = (w \sqrt{s_{33}} + \frac{s_{23}}{\sqrt{s_{33}}})^2 + \frac{s(2,3)}{s_{33}}, \quad s(1,2) = w^2 s(2,3),$$ so that $s_{11}$ and $s(1,2)$ are indeed non-negative for any $w$. When $w = 0$, $s(1,2) = 0$ but the rank of the $3 \times 3$ submatrix on the top left corner of the matrix $\mathbf{s}$ is two for any $w$. Secular equation for surface waves in incompressible monoclinic materials ========================================================================= The interest for considering incompressibility for surface waves in linear anisotropic elasticity is threefold. From a *historical* perspective, it must be remembered that Rayleigh, the initiator of the theoretical study of elastic surface waves, did treat the case of an incompressible linearly isotropic elastic half-space [@Rayl85]. Although some literature can be found on the subject of surface waves in incompressible, finitely elastic, stress-induced anisotropic half-spaces [@Flav63; @Will73; @DoOg90; @Chad97], very few papers are placed within the counterpart context of linearly elastic, anisotropic half-spaces, subject to the internal constraint of incompressibility. Second, from an *experimental* point of view, it is accepted [@NaSo99; @NaSo97; @SoNa99; @Sutc92; @GuGu99] that certain elastic materials may be modeled as incompressible, linearly elastic, anisotropic materials. According to Nair and Sotiropoulos [@NaSo97], such is the case for “polymer Kratons, thermoplastic elastomers, rubber composites when low frequency waves are considered to justify the assumption of material inhomogeneity, etc”. Third, the *theoretical* aspect of incompressibility in linear anisotropic elasticity has not been addressed in this context, and it is important to derive the secular equation in terms of the compliances rather than in terms of the stiffnesses. Here attention is turned to surface waves propagating with speed $v$ in the direction of the $x_1$-axis in the half-space $x_2>0$. The material is monoclinic with the symmetry plane at $x_3=0$. In the general (compressible) case the secular equation for the surface wave has been obtained explicitly by Destrade [@Dest01a] using the method of first integrals introduced by Mozhaev [@Mozh95], and by Ting [@Ting01] using a modified Stroh [@Stro62] formalism. Letting $X = \rho v^2$ where $\rho$ is the mass density, the secular equation is $$\begin{gathered} \label{secularCompr} [\eta - (1+r_2)X] \{(\eta - X) [ (\eta - X)(n_{66}X -1) + r_6^2 X] + X^2[ (\eta - X)n_{22} + r_2^2] \} \\ + 2 r_6 X^2 (\eta - X) [(\eta - X) n_{26} + r_2 r_6] = 0.\end{gathered}$$ It is a quartic in X. In , (see [@Ting01]) $$\begin{aligned} \label{constants} & \eta = \frac{1}{s'_{11}}, \quad r_2 = -\frac{s'_{12}}{s'_{11}}, \quad r_6 = -\frac{s'_{16}}{s'_{11}}, \nonumber \\ & n_{66} = \frac{s'(1,6)}{s'_{11}}, \quad n_{26} = \frac{s'(1,2|1,6)}{s'_{11}}, \quad n_{22} = \frac{s'(1,2)}{s'_{11}}.\end{aligned}$$ The incompressible case was first studied by Nair and Sotiropoulos [@NaSo99], although they did not establish the secular equation explicitly. The secular equation for incompressible materials can be deduced directly from by imposing the incompressibility conditions $s'_{2 \beta} = -s'_{1 \beta}$. The $r_2, n_{26}, n_{22}$ in simplify to $$r_2 =1, \quad n_{26} = 0, \quad n_{22} = 0,$$ and the secular equation reduces to $$(\eta - 2X) [(\eta - X)^2 (n_{66}X -1) + X^2] + r_6^2 \eta X(\eta - X) = 0.$$ It can be written in a non-dimensional form as $$\begin{aligned} & (1- 2 \xi)[(1-\xi)^2 (\kappa \xi - 1) + \xi^2] + r_6^2 \xi (1 - \xi) = 0, \\ & \xi = X/\eta = \rho v^2/s'_{11}, \quad \kappa =n_{66}/s'_{11}.\end{aligned}$$ For incompressible orthotropic materials for which $s'_{16}=0$, the secular equation further simplifies to, since $(1- 2 \xi) \ne 0$, $$(1-\xi)^2 (1 - \kappa \xi) = \xi^2, \quad \kappa = s'_{66}/s'_{11}.$$ This cubic in $\xi$ has a more compact and satisfying form than that obtained in terms of the stiffnesses [@Dest01c] which, as stressed in the Introduction, are not easily defined for incompressible anisotropic materials. The secular equation is only a part of the surface wave solution. A complete solution requires the computation of the displacements and stresses. This is discussed next. The Stroh formalism for steady state motion =========================================== In a fixed rectangular coordinate system $x_i$ ($i=1,2,3$) the stress-strain law and the equations of motion are $$\begin{aligned} \label{stress-strain} & \sigma_{ij} = C_{ijks} u_{k,s}, \\ \label{motion} & C_{ijks} u_{k,sj} = \rho \ddot{u}_i.\end{aligned}$$ in which the dot stands for differentiation with time $t$. Consider a steady state motion with the steady wave speed $v$ propagating in the direction of the $x_1$-axis. A solution for the displacement vector $\mathbf{u}$ of can be written as [@Stro62] $$\label{displVector} \mathbf{u} = \mathbf{a} f(z), \quad z = x_1 - vt + p x_2,$$ in which $f$ is an arbitrary function of $z$, and $p$ and $\mathbf{a}$ satisfy the eigenrelation $$\begin{aligned} \label{eigen1} & \mbox{\boldmath $\Gamma$} \mathbf{a} = \mathbf{0}, \\ & \mbox{\boldmath $\Gamma$} = \mathbf{Q} - X \mathbf{I} + p( \mathbf{R} + \mathbf{R}^T) + p^2 \mathbf{T}, \\ & X = \rho v^2.\end{aligned}$$ In the above the superscript $T$ stands for the transpose, $ \mathbf{I}$ is the unit matrix, and $ \mathbf{Q}, \mathbf{R}, \mathbf{T}$ are $3 \times 3$ matrices whose elements are $$Q_{ik} = C_{i1k1} , \quad R_{ik} = C_{i1k2} , \quad T_{ik} = C_{i2k2}.$$ The matrices $\mathbf{Q}$ and $\mathbf{T}$ are symmetric and so is the matrix $\mbox{\boldmath $\Gamma$}$. Introducing the new vector $ \mathbf{b}$ defined by $$\label{b} \mathbf{b} = ( \mathbf{R}^T + p \mathbf{T}) \mathbf{a} = -[ p^{-1} (\mathbf{Q} - X \mathbf{I}) + \mathbf{R}] \mathbf{a},$$ in which the second equality follows from , the stress determined from can be written as $$\sigma_{i1} = - \phi_{i,2} - \rho v \dot{u}_i, \quad \sigma_{i2} = \phi_{i,1}.$$ The $\phi_i$ ($i=1,2,3$) are the components of the stress function vector $$\label{stressVector} \mbox{\boldmath $\phi$} = \mathbf{b} f(z).$$ There are six eigenvalues $p_\alpha$ and six Stroh eigenvectors $ \mathbf{a}_\alpha$ and $ \mathbf{b}_\alpha$($\alpha = 1,2,\ldots,6$). When $p_\alpha$ are complex, they consist of three pairs of complex conjugates. If $p_1, p_2, p_3$ are the eigenvalues with a positive imaginary part, the remaining three eigenvalues are the complex conjugates of $p_1, p_2, p_3$. Assuming that $p_1, p_2, p_3$ are distinct, the general solution obtained from superposing three solutions of and associated with $p_1, p_2, p_3$ can be written in matrix notation as $$\label{u-phi} \mathbf{u} = \mathbf{A} <f(z_*)> \mathbf{q}, \quad \mbox{\boldmath $\phi$} = \mathbf{B} <f(z_*)> \mathbf{q},$$ where $ \mathbf{q}$ is an arbitrary constant vector and $$\begin{aligned} & \mathbf{A} = [ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3], \quad \mathbf{B} = [ \mathbf{b}_1, \mathbf{b}_2, \mathbf{b}_3], \nonumber \\ & <f(z_*)> = \text{Diag} [ f(z_1), f(z_2), f(z_3)], \label{AB}\\ & z_\alpha = x_1 - vt + p_\alpha x_2. \nonumber\end{aligned}$$ For surface waves in the half-space $x_2 \ge 0$, the function $f(z)$ is chosen as $$f(z) = e^{ikz},$$ where $k$ is the real wave number. Since the imaginary parts of $p_1, p_2, p_3$ are positive, $_1$ assures us that $\mathbf{u} \to \mathbf{0}$ as $x_2 \to \infty$. The surface traction at $x_2=0$ vanishes if $\mbox{\boldmath $\phi$} = \mathbf{0}$ at $x_2=0$, i.e., $$\mathbf{B} \mathbf{q} = \mathbf{0}.$$ This has a nontrivial solution for $\mathbf{q}$ when the determinant of $\mathbf{B}$ vanishes, i.e., $$\label{B=0} | \mathbf{B} | = 0.$$ This is the secular equation for $v$. For a monoclinic material with the symmetry plane at $x_3=0$, leads to . The displacement $\mathbf{u}$ and the stress function vector $\mbox{\boldmath $\phi$}$ given in require the computation of the eigenvalues $p_\alpha$ and the eigenvectors $ \mathbf{a}_\alpha$ and $ \mathbf{b}_\alpha$ ($\alpha=1,2,3$). They are provided by and which are in terms of the elastic stiffnesses. They are not suitable for taking the incompressible limit. A different expression in terms of the reduced elastic compliances is needed. This is presented next. Steady state motion for incompressible materials ================================================ The two equations in can be written in a standard eigenrelation as [@InTo69; @BaLo73; @ChSm77] $$\begin{aligned} \label{Nxi} & \mathbf{N}\mbox{\boldmath $\xi$} = p \mbox{\boldmath $\xi$}, \\ & \mathbf{N} = \begin{bmatrix} \mathbf{N}_1 & \mathbf{N}_2 \\ \mathbf{N}_3 + X \mathbf{I} & \mathbf{N}_1^T \end{bmatrix}, \quad \mbox{\boldmath $\xi$} = \begin{bmatrix} \mathbf{a} \\ \mathbf{b} \end{bmatrix}, \\ & \mathbf{N}_1 = -\mathbf{T}^{-1} \mathbf{R}^T, \quad \mathbf{N}_2 = \mathbf{T}^{-1}, \quad \mathbf{N}_3 = \mathbf{R}\mathbf{T}^{-1}\mathbf{R}^T - \mathbf{Q}.\end{aligned}$$ It was shown in [@Ting88] that $ \mathbf{N}_1, \mathbf{N}_2, \mathbf{N}_3$ have the structure $$\label{eigen2} -\mathbf{N}_1 = \begin{bmatrix} r_6 & 1 & s_6 \\ r_2 & 0 & s_2 \\ r_4 & 0 & s_4 \end{bmatrix}, \mathbf{N}_2 = \begin{bmatrix} n_{66} & n_{26} & n_{46} \\ n_{26} & n_{22} & n_{24} \\ n_{46} & n_{24} & n_{44} \end{bmatrix}, -\mathbf{N}_3 = \begin{bmatrix} m_{55} & 0 & -m_{15} \\ 0 & 0 & 0 \\ -m_{15} & 0 & m_{11} \end{bmatrix}.$$ An explicit expression of the elements of $ \mathbf{N}_1, \mathbf{N}_2, \mathbf{N}_3$ was given in [@ChSm77] (see also [@Ting96 p. 167] in terms of the reduced elastic compliances and in [@BaCh90] in terms of the elastic stiffnesses. The expressions in term of the reduced elastic compliances are $$\begin{aligned} & r_\alpha = \frac{1}{\Delta} s'(1,5|5,\alpha), \quad s_\alpha = \frac{1}{\Delta} s'(1,5|\alpha,1), \nonumber \\ & n_{\alpha \beta} = \frac{1}{\Delta} s'(\alpha,1,5|\beta,1,5), \quad m_{\alpha \beta} = \frac{1}{\Delta} s'_{\alpha \beta}, \quad \Delta = s'(1,5).\end{aligned}$$ Since $s'_{2\beta} = - s'_{1\beta}$ for incompressible materials, it can be shown that $$r_2 =1, \quad s_2 =0, \quad n_{26}=n_{22}=n_{24}=0.$$ Thus, for incompressible materials, the matrices $ \mathbf{N}_1$ and $\mathbf{N}_2$ have the simpler expressions (see also Chadwick [@Chad97]), $$-\mathbf{N}_1 = \begin{bmatrix} r_6 & 1 & s_6 \\ 1 & 0 & 0 \\ r_4 & 0 & s_4 \end{bmatrix}, \mathbf{N}_2 = \begin{bmatrix} n_{66} & 0 & n_{46} \\ 0 & 0 & 0 \\ n_{46} & 0 & n_{44} \end{bmatrix}.$$ Equation consists of six scalar equations. The second and the fifth equations provide the identities $$\label{a1a2b1b2} a_1 + p a_2 =0, \quad b_1 + p b_2 = X a_2.$$ The first identity could have been deduced by inserting the solution into the condition of incompressibility $$\label{incompCond} \epsilon_1 + \epsilon_2 = u_{1,1} + u_{2,2} = 0.$$ With $ \mathbf{N}_1, \mathbf{N}_2, \mathbf{N}_3$ expressed in terms of $s'_{\alpha\beta}$, equation can be employed to compute the eigenvalues $p$ and the eigenvectors $\mathbf{a}$ and $\mathbf{b}$. Equation consists of two equations, $$\label{eigen3} (\mathbf{N}_1 - p \mathbf{I}) \mathbf{a} + \mathbf{N}_2 \mathbf{b} = \mathbf{0}, \quad (\mathbf{N}_3 + X\mathbf{I}) \mathbf{a} + (\mathbf{N}_1^T - p \mathbf{I})\mathbf{b} = \mathbf{0}.$$ Assuming that $(\mathbf{N}_3 + X\mathbf{I})$ in not singular, $_2$ can be solved for $\mathbf{a}$ and $_1$ can be written as [@Ting01] $$\begin{aligned} \label{eigen4} & \widehat{\mbox{\boldmath $\Gamma$}} \mathbf{b} = \mathbf{0}, \\ & \widehat{\mbox{\boldmath $\Gamma$}} = \widehat{\mathbf{Q}} + p( \widehat{\mathbf{R}} + \widehat{\mathbf{R}}^T) + p^2 \widehat{\mathbf{T}}.\end{aligned}$$ In the above, $$\label{ThatRhatQhat} \widehat{\mathbf{T}}= (\mathbf{N}_3 + X\mathbf{I})^{-1}, \quad \widehat{\mathbf{R}} = \mathbf{N}_1 \widehat{\mathbf{T}}, \quad \widehat{\mathbf{Q}} = \mathbf{N}_1 \widehat{\mathbf{T}}\mathbf{N}_1^T + \mathbf{N}_2.$$ The matrices $\widehat{\mathbf{T}}$ and $\widehat{\mathbf{Q}}$ are symmetric, so is $\widehat{\mbox{\boldmath $\Gamma$}}$. Equation provides the eigenvalue $p$ and the eigenvector $\mathbf{b}$. The eigenvector $\mathbf{a}$ obtained from $_2$ is, using and , $$\label{a} \mathbf{a} = -(\widehat{\mathbf{R}}^T +p \widehat{\mathbf{T}})\mathbf{b} = (p^{-1} \widehat{\mathbf{Q}} + \widehat{\mathbf{R}})\mathbf{b}.$$ We have thus presented equations for computing the eigenvalues $p$ and the eigenvectors $ \mathbf{a}$ and $\mathbf{b}$ needed for the surface wave solution in terms of $s'_{\alpha\beta}$. The surface wave solution for an incompressible material is then complete. Elastostatics for incompressible materials ========================================== The solutions and remain valid for elastostatics if we set $v=0$. The derivation in - also holds for elastostatics if we let $X=0$. However, the derivation from to is not valid for elastostatics because $(\mathbf{N}_3 + X\mathbf{I})$ is singular when $X=0$. A different approach is needed to find $ \mathbf{a}$ and $\mathbf{b}$ in terms of $s'_{\alpha\beta}$. A modified Lekhnitskii formalism in the style of Stroh was proposed by Ting [@Ting99] in which the vector $\mathbf{b}$ satisfies the eigenrelation (see also [@BaCh90; @BaKi97]) $$\label{eigenStatics} \begin{bmatrix} 1 & -p & 0 \\ 0 & l_4 & -l_3 \\ 0 & -l_3 & l_2 \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \mathbf{0}.$$ In the above $$\begin{aligned} \label{l2l3l4} & l_2 = s'_{55} p^2 - 2 s'_{45}p + s'_{44}, \nonumber \\ & l_3 = s'_{15} p^3 - (s'_{14} + s'_{56})p^2 + (s'_{25} + s'_{46})p - s'_{24}, \\ & l_4 = s'_{11} p^4 - 2s'_{16} p^3 + (s'_{66} + 2s'_{12})p^2 - 2s'_{26}p + s'_{22}. \nonumber\end{aligned}$$ From the eigenvalues $p$ are computed from the sextic equation $$\label{sexticStatics} l_2 l_4 - l_3 l_3 =0,$$ originally given by Lekhnitskii [@Lekh63]. The vector $\mathbf{a}$ is [@Ting99] $$\label{aStatics} \mathbf{a}= \begin{bmatrix} g_1 & -h_1 \\ p^{-1}g_2 & -p^{-1}h_2 \\ g_5 & h_5 \end{bmatrix} \begin{bmatrix} b_2 \\ b_3 \end{bmatrix},$$ in which $$\label{g_alpha-h_alpha} g_\alpha = s'_{\alpha 1} p^2 - s'_{\alpha 6} p + s'_{\alpha 2}, \quad h_\alpha = s'_{\alpha 5} p - s'_{\alpha 4}.$$ We have thus the eigenvalues $p$ and the eigenvectors $\mathbf{a}$ and $\mathbf{b}$ all in terms of $s'_{\alpha\beta}$. When the material is incompressible, $s'_{2 \beta} = - s'_{1 \beta}$ and the $l_3, l_4$ in simplify to $$\begin{aligned} \label{l3l4Incomp} & l_3 = (s'_{15} p - s'_{14})(p^2 -1) - s'_{56}p^2 + s'_{46}p, \nonumber \\ & l_4 = s'_{11}(p^2 -1)^2 - 2s'_{16} p(p^2 -1) + s'_{66}p^2.\end{aligned}$$ Also, gives $$g_2 = -g_1, \quad h_2= -h_1.$$ Equation can then be written as $$\label{aStaticsIncomp} \mathbf{a}= \begin{bmatrix} g_1 & -h_1 \\ -p^{-1}g_1 & p^{-1}h_1 \\ g_5 & h_5 \end{bmatrix} \begin{bmatrix} b_2 \\ b_3 \end{bmatrix}.$$ The $a_1, a_2$ computed from indeed satisfy the identity $_1$. In the next section we study the special case of incompressible monoclinic materials with the symmetry plane at $x_3=0$. Monoclinic materials with the symmetry plane at $x_3=0$ ======================================================= When the material is monoclinic with the symmetry plane at $x_3=0$, $l_3$ vanishes identically so that the sextic equation leads to $l_2=0$ or $l_4=0$. If the material is incompressible, $l_4$ is given by and we have $$\label{p} (p - p^{-1})^2 - 2 \alpha (p - p^{-1}) + \beta =0,$$ where $$\alpha = s'_{16}/s'_{11}, \quad \beta = s'_{66}/s'_{11}.$$ Since $p_1, p_2$ are the roots of with a positive imaginary part, gives $$\label{roots} p - p^{-1} = \alpha + i \gamma,$$ in which $$\gamma = \sqrt{\beta - \alpha^2} = \sqrt{s'(1,6)}/s'_{11}.$$ Equation tells us that $$\label{p1p2} p_1 + p_2 = \alpha + i \gamma, \quad p_1 p_2 = -1.$$ We also obtain an explicit expression of $p_1, p_2$ as $$p_1, p_2 = \frac{\alpha + i \gamma}{2} \pm \sqrt{(\frac{\alpha + i \gamma}{2})^2 +1}.$$ The three Barnett-Lothe [@BaLo73] tensors $\mathbf{S}, \mathbf{H}, \mathbf{L}$ appear often in the solutions to anisotropic elasticity problems. They are real. Explicit expressions of $\mathbf{S}, \mathbf{H}, \mathbf{L}$ for monoclinic materials with the symmetry plane at $x_3=0$ have been presented in [@Ting96 p.174]. Specializing to incompressible materials using leads to $$\begin{aligned} \label{SHL} & \mathbf{S} = \mathbf{0}, \quad \mathbf{H} = \mathbf{L}^{-1} = \text{Diag} [\gamma s'_{11}, \gamma s'_{11}, 1/\mu],\\ & \mu = [s'(4,5)]^\halft.\end{aligned}$$ The quantity $\mu$ is the shear modulus when the material is isotropic. The structure of $\mathbf{S}, \mathbf{H}, \mathbf{L}$ in provides the following interesting results in elastostatics for incompressible materials. The order of the stress singularity at an interfacial crack tip in a bimaterial consisting of two dissimilar materials bonded together is not a complex number when $\mathbf{S} \mathbf{L}^{-1}$ in the two materials are identical. In this case, the physically unrealistic interpenetration of the crack surface displacement does not occur (see, for example, [@Ting96 p.144]). For a bimaterial for which both materials are incompressible, $\mathbf{S} = \mathbf{0}$ according to . Hence $\mathbf{S} \mathbf{L}^{-1}$ vanishes in both materials. Therefore there is no interpenetration of the crack surface when the material is incompressible and monoclinic with the symmetry plane at $x_3=0$. The inplane displacement and the antiplane displacement for a monoclinic material with the symmetry plane at $x_3=0$ are uncoupled [@Stro62]. We can therefore consider the inplane and antiplane deformations separately. Consider the inplane deformation. The Barnett-Lothe tensors now require only the $2 \times 2$ matrix located at the top left corner of $\mathbf{S}, \mathbf{H}, \mathbf{L}$. From we have $$\label{SHLincomp} \mathbf{S} = \mathbf{0}, \quad \mathbf{H} = \mathbf{L}^{-1} = \gamma s'_{11} \mathbf{I},$$ where $ \mathbf{I}$ is the $2 \times 2$ identity matrix. Consider now an infinite monoclinic material subject to a line of concentrated force $\mathbf{f}$ and a line of dislocation with Burgers vector $ \widehat{\mathbf{b}}$ applied along the $x_3$-axis. The strain energy in the annual region bounded by the two radii $r_2>r_1$ can be shown to be $$\label{strainEnergyCompr} \frac{1}{4 \pi} \ln (\frac{r_2}{r_1}) ( \mathbf{f}^T \mathbf{Hf} + \widehat{\mathbf{b}}^T \mathbf{L} \widehat{\mathbf{b}}),$$ for a compressible material [@Ting96 p.249]. When the material is incompressible and when the vectors $\mathbf{f}$ and $ \widehat{\mathbf{b}}$ lie on the $x_3=0$ plane, use of in yields $$\frac{1}{4 \pi} \ln (\frac{r_2}{r_1}) [ \gamma s'_{11} |\mathbf{f}| + (\gamma s'_{11})^{-1} |\widehat{\mathbf{b}}|].$$ This strain energy depends only on the magnitudes, not the directions, of the vectors $\mathbf{f}$ and $ \widehat{\mathbf{b}}$. Consider next a half-space with a traction-free boundary surface subject to a line dislocation with Burgers vector $ \widehat{\mathbf{b}}$ in the half-space [@Ting96 pp. 264-265]. When the material is incompressible it can be shown that, by virtue of , the image force that is attracted to the free-surface depends on the magnitude, not on the direction, of the Burgers vector $ \widehat{\mathbf{b}}$. Likewise, if the boundary surface is a rigid surface [@TiBa93], the image force that is repelled by the rigid surface depends on the magnitude, not on the direction, of the Burgers vector $ \widehat{\mathbf{b}}$. Moreover, the magnitude of the repel force is identical to the attracted force when the boundary is a free-surface. The same result applies to a line dislocation in a bimaterial that consists of two dissimilar materials bonded together [@Ting96 p. 286]. When the material is incompressible, the image force that is attracted to or repelled by the interface depends on the magnitude, not on the direction, of the Burgers vector. 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--- address: | Max-Planck-Institute for Physics (Werner Heisenberg Institute), Föhringer Ring 6\ 80805 München, Germany\ E-mail: bracinik@mppmu.mpg.de\ For H1 and ZEUS collaborations author: - Juraj Bracinik title: Jets in Photoproduction and in the Transition Region to DIS at the HERA collider --- Introduction ============ The photon is probably the best known elementary particle. It is a quantum of the gauge field, and as such it is considered to be massless, charge-less and to couple point-like. While interactions of photons with leptons are described by QED with high precision, interactions of photons with hadrons still bring surprises. This is caused by the fact, that in the same way, as the photon can fluctuate into an electron-positron pair, it can fluctuate into a pair of quark and anti-quark, which interact strongly. The photon then behaves as a hadron. In Leading Order (LO), it is possible to distinguish between direct processes (the photon interacts electromagnetically) and resolved processes (the photon fluctuates into a partonic system, of which one of the partons then interacts). Beyond LO this classification becomes ambiguous. Theoretical description of photon-proton interactions ===================================================== Perturbative QCD calculations which aim to describe interactions of photons with protons use as an input parton density functions of the proton (obtained from global fits to DIS and hadron collision data) and of the photon (extracted from data on $\gamma \gamma$ collisions). Depending on the way perturbative QCD is used, there are two groups of approaches. Leading order plus Parton Shower (LO+PS) models combine LO matrix elements with parton showers re-summing the leading logarithmic contributions from all orders.[@h1_dijets_herwig] Hadronization is included using a QCD motivated phenomenological model and predictions are directly compared to data on hadron level. Next-to-leading order (NLO) calculations use matrix elements up to a fixed order in $\alpha_S$ (for most processes up to $\alpha_S^2$).[@h1_incl_frixione_ridolfi; @zeus_incl_klasen; @zeus_dijets_graudenz] They provide predictions on parton level. Before comparing to data, NLO predictions are corrected for hadronization effects. These are estimated using LO+PS models discussed above. Experimental conditions ======================= Depending on how events are selected, we distinguish between tagged photoproduction (scattered electron is measured in downstream calorimeter, $Q^2 \le 10^{-2} \; {\rm GeV}^2$), un-tagged photoproduction (electron is not observed in main detector, $Q^2 \le 1 \; {\rm GeV}^2$) and low $Q^2$ region (electron is measured in main detector, $Q^2 \ge 2 \; {\rm GeV}^2$). The distribution of the center-of-mass energy of the $\gamma p$ system depends on the exact event selection; at HERA it extends up to $280 \; {\rm GeV}$. The results are presented in the hadronic CMS (center-of-mass frame of $\gamma p$ system), for jet finding, an inclusive $k_{t}$ algorithm is used[@ktclus] in the same frame. Before being compared to theory, data are corrected to hadron level, i.e. acceptance and effects due to the detector and the reconstruction software are corrected for. These corrections are calculated using LO+PS models together with a detailed detector simulation. Inclusive jets ============== To check our understanding of jet production in photoproduction one can simply count the number of jets as a function of their transverse energy. The cross section of inclusive jets in the pseudo-rapidity range $-1 < \eta < 2.5$ has been measured by both H1 (Fig. \[fig:incl\_et\]) and ZEUS.[@h1_inclusive; @zeus_inclusive] One can see an excellent agreement between the NLO calculation and the data. Agreement extends down to low value of $E_T$ ($5 \; {\rm GeV}$), where hadronization corrections (including effects of the underlying event) become significant. The dominant experimental error is coming from the energy scale uncertainty, which is smaller then the renormalization scale uncertainty of the theory. Dijets in photoproduction and at low $Q^2$ DIS ============================================== A measurement of inclusive jet production has clear advantages. Theoretical predictions are “safe”, the measurement is least restrictive in phase space and offers good statistical precision. On the other hand, dijets allow to construct more differential quantities, for example $x_{\gamma}^{obs}$.[^1] At parton level in LO this variable corresponds to the fractional photon energy of the parton “in” the photon entering the hard subprocess. Higher orders, hadronization and detector resolution smear-out the correlation, but still one expects for direct processes $x_{\gamma}^{obs}$ to be close to one and significantly smaller than one for resolved processes. In photoproduction ($Q^2 = 0$) a photon behaves part of the time as a hadron, which is manifested by the presence of its resolved part. On the other hand, in DIS, at high enough $Q^2$, the photon is point-like. What happens in the transition region? ZEUS has presented the cross section on dijets in photoproduction and DIS (Fig. \[fig:dijets\_zeus\_nlo\] left) in the hadronic CMS, in the pseudo-rapidity range $-3 < \eta^* < 0$ and for $E_T> 7.5 \; (6.5) \; {\rm GeV}$.[@zeus_dijets] While in photoproduction the cross section of dijet production is in agreement with NLO,[@h1_incl_frixione_ridolfi] in DIS however, the NLO assuming only pointlike photon[@zeus_dijets_graudenz], underestimates the cross section. Requiring $x_{\gamma}^{obs} > 0.75 $($x_{\gamma}^{obs} < 0.75 $), it is possible to enhance direct (resolved) processes. The sample with predominantly direct processes is well described by NLO, while a discrepancy is observed at small $x_{\gamma}^{obs}$. The same data are shown in Fig. \[fig:dijets\_zeus\_nlo\] (right side) in the form of the ratio $R=\sigma(x_{\gamma}^{obs} < 0.75)/\sigma(x_{\gamma}^{obs} > 0.75)$. In this ratio correlated experimental and theoretical uncertainties partly cancel. We can see that the discrepancy between data and direct NLO extends up to rather high $Q^2$ values and is most remarkable at low $E_T$. A change of scale (using $Q^2$ instead of $Q^2 + E_T^2$), improves the agreement at low $Q^2$, but not at higher $Q^2$ values. This result may be a hint that higher orders are needed in the perturbative calculation. At low $Q^2$, it is possible to include them effectively using the concept of resolved virtual photons. H1 has measured the triple differential cross section of dijets as a function of $Q^2$, $E_T$ and $x_{\gamma}$ in DIS at low $Q^2$ in the region $-2.5 < \eta^* < 0$ and $E_T > 7 \; (5) \; {\rm GeV}$.[@h1_dijets] The comparison of the data with NLO shows that NLO underestimates the cross section, the discrepancy being most clearly visible for low $Q^2$, $E_T$ and $x_{\gamma}$. The inclusion of a resolved transverse virtual photon component in NLO reduces the discrepancy, but agreement with data is still not perfect. The same data are compared to the LO+PS model of HERWIG[@h1_dijets_herwig] in Fig. \[fig:dijets\_h1\_nlo\]. Again, direct processes alone underestimate the data and inclusion of transverse resolved photons improves the agreement. Including, in addition a contribution from resolved longitudinal photons yields an even better description.. An alternative approach to modeling $\gamma p$ interactions is represented by the CASCADE model,[@h1_dijets_cascade2] which is based on the CCFM evolution, with angle ordering instead of $k_T$ ordering of the radiated gluons and using un-integrated parton densities of the proton. This model is in reasonable (but not perfect) agreement with the data. This is remarkable, as CASCADE does not use any concept of a resolved photon; low $x_{\gamma}^{obs}$ events are produced by different evolution from the proton side. Study of color dynamics in three jet events in photoproduction ============================================================== Three jet events are interesting as they feel the triple gluon vertex. It would be nice to find three-jet observables sensitive to the structure of the gauge group behind the strong interaction. ZEUS has measured cross sections of three-jet production as a function of angles $\theta_H$, $\alpha_{23}$ and $\beta_{KSW}$. Jets with $-1 < \eta < 2.5$ and $E_T > 14 \; {\rm GeV} $ were selected.[@zeus_threejets] They are ordered according to their $E_T$. Then $\theta_H$ is defined as the angle between the plane defined by the beam axis and jet $1$ and the plane defined by jets $2$ and $3$, $\alpha_{23}$ is the angle between jets $2$ and $3$. If predominantly direct events are selected ($x_{\gamma} > 0.7$), we have in LO four terms (see Fig. \[fig:threejets\_diagrams\]). Detailed analysis shows that cross sections plotted as a function of $\theta_H$, $\alpha_{23}$ and $\beta_{KSW}$ are sensitive to the presence of the triple gluon vertex. The shape of term B (triple gluon vertex in quark induced events) is different from the other three terms. [@zeus_threejets] A comparison of data with LO predictions[@zeus_incl_klasen] is shown in Fig. \[fig:threejets\_zeus\]. There is very good agreement with SU(3) expectations. In addition, it is possible to rule out several exotic possibilities (like $SU(N)$ for large $N$, or $C_F=0$). Current precision does not allow to distinguish between SU(3) and Abelian case, because SU(3) predicts only $10 \% $ probability for events with a triple gluon vertex in quark induced events. Summary ======= Measured cross sections for inclusive jets in photoproduction are in excellent agreement with NLO. Study of dijets at low $Q^2$ show discrepancies between data and NLO, indicating the need for higher orders in the perturbative expansion. Inclusion of resolved longitudinal virtual photon component in low $Q^2$ DIS improves the description of the data. Three-jet events are sensitive to the triple gluon vertex, allowing to study the gauge structure of the strong interaction. The data are in agreement with LO QCD predictions. Current precision does not allow to discriminate between SU(3) and the Abelian case. [99]{} G.Marchesini et al., [*Comp. Phys. Com.*]{} [**67**]{} (1992) 465. S.Frixione, [*Nucl. Phys.*]{} B [**507**]{} (1997) 315; S.Frixione, G.Ridolfi, [*Nucl. Phys.*]{} B [**507**]{} (1997) 295. M.Klasen, T.Kleinwort, G.Kramer, [*Eur. Phys. J. Direct*]{} C [**1**]{} (1998) 1. D.Graudenz, hep-ph/9710244, 1997. S.D.Ellis, D.E.Soper, [*Phys. Rev.*]{} D [**48**]{}, 3160 (1993). H.Jung, [*Comp. Phys. Com.*]{} [**143**]{} (2002) 100. C.Adloff et al. \[H1 col.\], [*Eur. Phys. J.*]{} C [**29**]{} (2003) 497. A.Aktas et al. \[H1 col.\], [*Eur. Phys. J.*]{} C [**37**]{} (2004) 141. S. Chekanov et al. \[ZEUS col.\], [*Phys. Lett.* ]{} B [**560**]{} (2003) 7. S. Chekanov et al. \[ZEUS col.\], [*Eur. Phys. J.*]{} C [**35**]{} (2004) 487. ZEUS col., ABS 5-0271 submitted to this conference. [^1]: defined as $x_{\gamma}^{obs} = \sum_{jets} (E_j^* - p_z^*) / \sum_{hadrons} (E_j^* - p_z^*)$

--- abstract: | Throughout this Abstract, $G$ is a topological Abelian group and $\widehat{G}$ is the space of continuous homomorphisms from $G$ into $\mathbb T$ in the compact-open topology. A dense subgroup $D$ of $G$ [ *determines*]{} $G$ if the (necessarily continuous) surjective isomorphism $\widehat{G}\twoheadrightarrow\widehat{D}$ given by $h\mapsto h|D$ is a homeomorphism, and $G$ is [*determined*]{} if each dense subgroup of $G$ determines $G$. The principal result in this area, obtained independently by [L. Au]{}[ß]{}[enhofer]{} and [M. J. Chasco]{}, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup $D_i$ determines $G_i$ with $G_i$ compact, then $\oplus_i\,D_i$ determines $\Pi_i\,G_i$. In particular, if each $G_i$ is compact then $\oplus_i\,G_i$ determines $\Pi_i\,G_i$. 3. Let $G$ be a locally bounded group and let $G^+$ denote $G$ with its Bohr topology. Then $G$ is determined if and only if ${G^+}$ is determined. 4. Let $\non(\mathcal{N})$ be the least cardinal $\kappa$ such that some $X \subseteq \TT$ of cardinality $\kappa$ has positive outer measure. No compact $G$ with $w(G)\geq\non(\mathcal{N})$ is determined; thus if $\non(\mathcal{N})=\aleph_1$ (in particular if CH holds), an infinite compact group $G$ is determined if and only if $w(G)=\omega$. Question. Is there in ZFC a cardinal $\kappa$ such that a compact group $G$ is determined if and only if $w(G)<\kappa$? Is $\kappa=\non(\mathcal{N})$? $\kappa=\aleph_1$? address: - | W. W. Comfort\ Department of Mathematics, Wesleyan University\ Middletown, CT 06459 - | S. U. Raczkowski and F. Javier Trigos-Arrieta\ Department of Mathematics, California State University, Bakersfield\ 9011 Stockdale Highway\ Bakersfield, CA, 93311-1099 author: - 'W. W. Comfort' - 'S. U. Raczkowski' - 'F. Javier Trigos-Arrieta' title: Concerning the dual group of a dense subgroup --- (474,66)(0,0) (0,66)(1,0)[40]{}[(0,-1)[24]{}]{} (43,65)(1,-1)[24]{}[(0,-1)[40]{}]{} (1,39)(1,-1)[40]{}[(1,0)[24]{}]{} (70,2)(1,1)[24]{}[(0,1)[40]{}]{} (72,0)(1,1)[24]{}[(1,0)[40]{}]{} (97,66)(1,0)[40]{}[(0,-1)[40]{}]{} (143,66)[(0,0)\[tl\][Proceedings of the Ninth Prague Topological Symposium]{}]{} (143,50)[(0,0)\[tl\][Contributed papers from the symposium held in]{}]{} (143,34)[(0,0)\[tl\][Prague, Czech Republic, August 19–25, 2001]{}]{} [^1] [^2] [^3] Terminology, Notation and Preliminaries ======================================= For $X$ a set and $\kappa$ a cardinal, we write $[X]^\kappa=\{A\subseteq X:|A|=\kappa\}$. For each space $X=(X,\sT)$ we write $$\sK(X):=\{K\subseteq X:K \mbox{ is $\sT$-compact}\}.$$ All groups considered here, whether or not equipped with a topology, are Abelian groups written additively. The identity of a group $G$ is denoted $0$ or $0_G$, and the torsion subgroup of $G$ is denoted $\tor(G)$. The reals, rationals, and integers are denoted $\RR$, $\QQ$, and $\ZZ$, respectively, and the “unit circle” group $\TT$ is the group $(-\frac{1}{2},\frac{1}{2}]$ with addition mod $1$. Except when we specify otherwise, these groups carry their usual metrizable topology. The symbol $\PP$ denotes the set of positive prime integers. The set of homomorphisms $h:G\rightarrow\TT$, a group under pointwise operation, is denoted $\Hom(G,\TT)$. For a subgroup $A$ of $\Hom(G,\TT)$ we denote by $(G,\sT_A)$ the group $G$ with the topology induced by $A$. Evidently $(G,\sT_A)$ is a Hausdorff topological group if and only if $A$ separates points of $G$. The topology $\sT_A$ is the coarsest topology on $G$ for which the homomorphism $e_A:G\rightarrow\TT^A$ given by $(e_A(x))_h=h(x)$ ($x\in G$, $h\in A$) is continuous. When $G=(G,\sT)$ is a topological group, the set of $\sT$-continuous functions in $\Hom(G,\TT)$ is a subgroup of $\Hom(G,\TT)$ denoted $\widehat{G}$ or $\widehat{(G,\sT)}$; in this case the topology $\sT_{\widehat{G}}$ is the [*Bohr topology*]{} associated with $\sT$, and $(G,\sT_{\widehat{G}})$ is denoted $G^+$ or $(G,\sT)^+$. When $\widehat{(G,\sT)}$ separates points we say that $G$ is a [*maximally almost periodic*]{} group and we write $G=(G,\sT)\in{\bf MAP}$. Whether or not $(G,\sT)\in{\bf MAP}$, the closure of $e[G]$ in $\TT^{\widehat{G}}$, denoted $b(G)$ or $b(G,\sT)$, is the [*Bohr compactification*]{} of $(G,\sT)$. The Bohr compactification $b(G)$ of a topological group $G$ is characterized by the condition that each continuous homomorphism from $G$ into a compact Hausdorff group extends continuously to a homomorphism from $b(G)$. From this and the uniform continuity of continuous homomorphisms it follows that if $D$ is a dense subgroup of $G$ then $b(D)=b(G)$. It is conventional to suppress mention of the function $e_{\widehat{G}}$ and to write simply $\widehat{G}=\widehat{G^+}$. When $G\in{\bf MAP}$ we write $G^+\subseteq b(G)\subseteq\TT^{\widehat{G}}$, the inclusions being both algebraic and topological. A group $G$ with its discrete topology is denoted $G_d$. For notational convenience, and following van Douwen [@vdii], for every (Abelian) group $G$ we write $G^\#=(G_d)^+\subseteq b(G_d)$. A subset $S$ of a topological group $G$ is said to be [*bounded*]{} in $G$ if for every nonempty open $V\subseteq G$ there is finite $F\subseteq G$ such that $S\subseteq F+V$; $G$ is [*locally bounded*]{} \[resp., [*totally bounded*]{}\] if some nonempty open subset of $G$ is bounded \[resp., $G$ itself is bounded\]. It is a theorem of Weil [@weili] that each locally bounded group $G$ embeds as a dense topological subgroup of a locally compact group $W(G)$, unique in the obvious sense; the group $W(G)$ is compact if and only if $G$ is totally bounded. We denote by [**LCA**]{} \[resp., **LBA**\] the class of locally compact \[resp., locally bounded\] Hausdorff Abelian groups. The relation ${\bf LCA}\subseteq{\bf MAP}$ is a well known consequence of the Gel$'$fand-Raĭkov Theorem (cf. [@hri 22.17]); since each subgroup $S\subseteq G\in{\bf MAP}$ clearly satisfies $S\in{\bf MAP}$, we have in fact the relations ${\bf LCA}\subseteq\textbf{LBA}\subseteq{\bf MAP}$. Let $S$ be a subgroup of $G\in$ [*[**LBA**]{}*]{}. Then - $S$ is [*dual-embedded*]{} in $G$ in the sense that each $h\in\widehat{S}$ extends to an element of $\widehat{G}$; - if $h\in\widehat{S}$ and $x\in G\backslash\overline{S}^G$, the extension $k\in\widehat{G}$ of $h$ may be chosen so that $k(x)\neq0$. It follows that for each subgroup $S$ of a group $G\in\textbf{LBA}$ the topology of $S^+$ coincides with the topology inherited by $S$ from $G^+$. This validates the following notational convention. For $S\subseteq G\in{\bf LBA}$, $S$ not necessarily a subgroup of $G$, we denote by $S^+$ the set $S$ with the topology inherited from $G^+$. When $G$ is discrete, so that $G^+=G^\#$, we write $S^\#$ in place of $S^+$ when $S\subseteq G$. (Glicksberg [@glickii]). Let $K\subseteq G\in{\bf LBA}$. Then $K\in\sK(G)$ if and only if $K^+\in\sK(G^+)$. Hence if $K\in\sK(G)$, then $K$ and $K^+$ are homeomorphic. (Flor [@flor]. See also Reid [@reid]). Let $G\in{\bf LBA}$ and let $x_n\rightarrow p\in b(G)=b(W(G))$ with each $x_n\in G^+\subseteq(W(G))^+\subseteq b(G)$. Then - $p\in (W(G))^+$, and - not only $x_n\rightarrow p$ in $(W(G))^+\subseteq b(G)$ but also $x_n\rightarrow p$ in $W(G)$. Strictly speaking, the papers cited above in connection with Theorems 0.2 and 0.3 deal with groups $G\in{\bf LCA}$. Our modest generalization to the case $G\in{\bf LBA}$ is justified by 0.2 and 0.3 as originally given and by these facts about $G\in{\bf LBA}$: - $G$ is a (dense) topological subgroup of $W(G)\in{\bf LCA}$; - $G^+$ is a (dense) topological subgroup of $(W(G))^+$; and - $b(G)=b(W(G))$. In what follows, groups of the form $\widehat{G}$ will be given the [*compact-open*]{} topology. This is defined as usual: the family $$\{U(K,\epsilon):K\in\sK(G), \epsilon>0\}$$ is a base at $0\in\widehat{G}$, where for $A\subseteq G$ one writes $$U(A,\epsilon)=\{h\in\widehat{G}:x\in A\Rightarrow|h(x)|<\epsilon\}.$$ We have noted already that for $G\in{\bf MAP}$ the groups $\widehat{G}$ and $\widehat{G^+}$ are identical; that is, $\widehat{G}=\widehat{G^+}$ as groups. Our principal interest in Theorem 0.2 is that for $G\in{\bf LBA}$ it gives a topological consequence, as follows. Let $G\in{\bf LBA}$. Then $\widehat{G}=\widehat{G^+}$ as topological groups. That is, the compact-open topology on $\widehat{G}$ determined by $\sK(G)$ coincides with the compact-open topology on $\widehat{G}$ determined by $\sK(G^+)$. We acknowledge with thanks several helpful conversations relating to §6 with these mathematicians: Adam Fieldsteel, Michael Hru[š]{}[á]{}k, and Stevo Todor[č]{}evi[ć]{}. The Groups $\widehat{G}$ for $G$ Metrizable =========================================== If $D$ is a dense subgroup of an Abelian topological group $G=(G,\sT)$ then every $h\in\widehat{D}$ extends (uniquely) to an element of $\widehat{G}$; and of course, each $h\in\widehat{G}$ satisfies $h|D\in\widehat{D}$. Accordingly, abusing notation slightly, we have $\widehat{D}=\widehat{G}$ as groups. Since groups of the form $\widehat{G}$ carry the compact-open topology, it is natural to inquire whether the identity $\widehat{G}=\widehat{D}$ is topological as well as algebraic. Informally: do $\sK(G)$ and $\sK(D)$ induce the same topology on the set $\widehat{G}=\widehat{D}$? The question provokes this definition. Let $G$ be an Abelian topological group. - Let $D$ be a dense subgroup of $G$. Then $D$ [*determines*]{} $G$ (alternatively: $G$ is [*determined*]{} by $D$) if $\widehat{G}=\widehat{D}$ as topological groups. - $G$ is [*determined*]{} if every dense subgroup of $G$ determines $G$. <!-- --> - It is a theorem of Kaplan [@kaplan48 2.9] (cf. Banasczyzk [@banaszcz 1.3] and Raczkowski [@raczphd §3.1] for alternative treatments, and Au[ß]{}enhofer [@Au98] and [@Au99 3.4] for a generalization) that for each $G$ the family $\{U(K,\frac{1}{4}):K\in\sK(G)\}$ is basic at $0\in\widehat{G}$. (For notational simplicity, henceforth we write $U(K):=U(K,\frac{1}{4})\subseteq\widehat{G}$ for $K\in\sK(G)$.) Thus the condition that a group $G$ is determined by its dense subgroup $D$ reduces to (i.e., is equivalent to) the condition that $\sK(D)$ is cofinal in $\sK(G)$ in the sense that for each $K\in\sK(G)$ there is $E\in\sK(D)$ such that $U(E)\subseteq U(K)$. - Let $D$ and $S$ be dense subgroups of a topological group $G$ such that $D\subseteq S\subseteq G$. Then since $\sK(D)\subseteq\sK(S)\subseteq\sK(G)$, $D$ determines $G$ if and only if $D$ determines $S$ and $S$ determines $G$. In particular, a dense subgroup of a determined group is determined. - The principal theorem in this corner of mathematics is the following result, obtained independently by Außenhofer [@Au99 4.3] and Chasco [@chasco]. This is the point of departure of the present inquiry. Every metrizable, Abelian group is determined. Is every topological group determined? Is every [**MAP**]{} group determined? Are there nonmetrizable, determined groups? Is every closed (or, open) subgroup of a determined group itself determined? Is the class of determined groups closed under passage to continuous homomorphisms? Continuous isomorphisms? The formation of products? These are some of the questions we address. Determined Groups: $G$ vs. $G^+$ ================================ Let $D$ be a subgroup of $G\in{\bf LBA}$. Then $D$ is dense in $G$ if and only if $D^+$ is dense in $G^+$. Let $D$ be a subgroup of $G\in{\bf LBA}$. Then $D$ determines $G$ if and only if $D^+$ determines $G^+$. Let $G\in{\bf LBA}$. Then $G$ is determined if and only if $G^+$ is determined. Let $G$ be an [**LBA**]{} group such that $G^+$ determines $b(G)$. Then - $G$ is totally bounded (and hence $G=G^+$); and - if also $G\in{\bf LCA}$ then $G$ is compact (and hence $G=G^+=b(G)$). Let $G\in{\bf LBA}$. Then $b(G)$ is determined if and only if $W(G)$ is compact and determined; in this case $W(G)=b(G)$. Let $G\in{\bf LCA}$. If $G$ is noncompact then $G^+$ does not determine $b(G)$ (and hence $b(G)$ is not determined). Let $G$ be a closed subgroup of a product of [**LBA**]{} groups. Then a dense subgroup $D$ of $G$ determines $G$ if and only $D^+$ determines $G^+$. Thus $G$ is determined if and only if $G^+$ is determined. Determined Groups: Some Examples ================================ There are totally bounded, nonmetrizable, determined groups. Let $G$ be an arbitrary determined [**LBA**]{} group such that $G$ is not totally bounded. (Appealing to Theorem 1.3, one might choose $G\in\{\ZZ,\QQ,\RR\}$.) That $G^+$ is as required follows from three facts: - $G^+$ is determined (Theorem 2.3); - a group with a dense metrizable subgroup is itself metrizable [@Bo2-66 Prop. IX §2.1.1]; - $b(G)$ is not metrizable. A nondetermined group may have a dense, determined subgroup. The image of a nondetermined group under a continuous homomorphism may be determined. We see in Theorem 4.8 below that compact groups of weight $\geq\cc$ are nondetermined. Each such group maps by a continuous homomorphism onto either the group $\TT$ or a group of the form $(\ZZ(p))^\omega$ ($p\in\PP$) [@comfremusiv], and such groups are determined by Theorem 1.3. Obviously an [**LBA**]{} group with no proper dense subgroup is vacuously determined. We mention three classes of such groups. - Discrete groups. - Groups of the form $G^\#=(G_d)^+$. (It is well known [@comfsaks 2.1] that every subgroup of such a group is closed.) - ${\bf LCA}$ groups of the type given by Rajagopalan and Subrahmanian [@rajsub]. Specifically, let $\kappa\geq\omega$, fix $p\in\PP$, and topologize the group $G:=(\ZZ(p^\infty))^\kappa$ so that its subgroup $H:=(\ZZ(p))^\kappa$ in its usual compact topology is open-and-closed in $G$. <!-- --> - A determined group may contain a nondetermined open-and-closed subgroup. - There are non-totally bounded, nonmetrizable, determined [ **LBA**]{} groups. Although compact groups of the form $K^\kappa$ with $\kappa\geq\cc$ are not determined, we see in Corollary 3.11 below that such groups do contain nontrivial determining subgroups. Let $\{G_i:i\in I\}$ be a set of groups, let $S_i\subseteq G_i$, and let $p\in G:=\Pi_{i\in I}\,G_i$. Then - $s(p)=\{i\in I:p_i\neq0_i\}$; - $\oplus_{i\in I}\,G_i=\{x\in G:|s(x)|<\omega\}$; and - $\oplus_{i\in I}\,S_i=(\Pi_{i\in I}\,S_i)\cap(\oplus_{i\in I}\,G_i)$. In this context we often identify $S_i$ with the subset $S_i\times\{0_{I\backslash\{i\}}\}$ of $G$. In particular we write $G_i\subseteq G$ and we identify $\widehat{G_i}$ with $\{h|G_i:h\in\widehat{G}\}$. We use the following property to find some determining subgroups of certain (nondetermined) products. A topological group $G$ has the [*cofinally zero*]{} property if for all $K\in\sK(G)$ there is $F\in\sK(G)$ such that every $h\in U(F)$ satisfies $h|K\equiv0$. We record two classes of groups with the cofinally zero property. - $G$ is a determining subgroup of a compact Abelian group. (There is $F\in\sK(G)$ such that $U(F)=\{0\}$, so each $h\in U(F)$ satisfies $h|K\equiv0$ for all $K\in\sK(G)$.) - $G$ is a torsion group of bounded order. (Given $K\in\sK(G)$, let $n>4$ satisfy $nx=0$ for all $x\in G$ and use Remark 1.2(a) to choose $F\in\sK(G)$ such that $U(F)\subseteq U(K,\frac{1}{n})$.) Let $\{G_i:i\in I\}$ be a set of [**LBA**]{} groups with the cofinally zero property and let $G=\Pi_{i\in I}\,G_i$. If $D_i$ is a dense, determining subgroup of $G_i$, then $D:=\oplus_{i\in I}\,D_i$ determines $G$. Let $\{G_i:i\in I\}$ be a set of determined [**LBA**]{} groups with the cofinally zero property and let $G=\Pi_{i\in I}\,G_i$. If $D_i$ is a dense subgroup of $G_i$, then $\oplus_{i\in I}\,D_i$ determines $G$. Let $\{G_i:i\in I\}$ be a set of compact Hausdorff groups and let $G=\Pi_{i\in I}\,G_i$. Then $\oplus_{i\in I}\,G_i$ determines $G$. \[3.12\] The image under a continuous homomorphism of a compact determined group is determined. It is easily checked that if a locally compact space $X$ is $\sigma$-compact then it is [*hemicompact*]{}, i.e., some countable subfamily $\{K_n:n<\omega\}$ of $\sK(X)$ is cofinal in $\sK(X)$ in the sense that for each $K\in\sK(X)$ there is $n<\omega$ such that $K\subseteq K_n$. It follows that if an [**LCA**]{} group $G$ is $\sigma$-compact (equivalently: Lindelöf) then $w(\widehat{G})\leq\omega$, so $\widehat{G}$ in this case is determined by Theorem 1.3. Nondetermined groups: Some Examples =================================== The principal result of this section is that compact Abelian groups of weight $\geq\cc$ are nondetermined. Let $G$ be an [**LBA**]{} group with a proper dense subgroup $D$ such that each $K\in\sK(D)$ satisfies either - $K$ is finite or - $\langle K\rangle$ is closed in $D$. Then $D$ does not determine $G$. We noted in Theorem 2.4(b) that if $G^+$ determines $b(G)$ with $G\in{\bf LCA}$, then $G$ is compact (in fact $G=G^+=b(G)$). Lemma 4.1 allows a more direct proof in the case that $G$ is discrete. Let $G$ be an infinite Abelian group. Then $G^\#$ does not determine $b(G_d)$. ([@comfrossi]). Let $G$ be an Abelian group. - If $A$ is a point-separating subgroup of $\Hom(G,\TT)$, then $(G,\sT_A)$ is a totally bounded, Hausdorff topological group with $\widehat{(G,\sT_A)}=A$; - for every totally bounded Hausdorff topological group topology $\sT$ on $G$ the subgroup $A:=\widehat{(G,\sT)}$ of $\Hom(G,\sT)$ is point-separating and satisfies $\sT=\sT_A$. It is easily checked that for each Abelian group $G$ the set $\Hom(G,\TT)$ is closed in the compact space $\TT^G$. Thus $\Hom(G,\TT)$, like every Hausdorff (locally) compact group, carries a Haar measure. Our convention here is that Haar measure is complete, so in particular every subset of a measurable set of measure $0$ is itself measurable (and of measure $0$). Concerning Haar measure $\lambda$ on a [**LCA**]{} group $G$ we appeal frequently to the [*Steinhaus-Weil Theorem*]{}: [*If $S\subseteq G$ is $\lambda$-measurable and $\lambda(S)>0$, then the difference set $S-S:=\{x-y:x,y\in S\}$ contains a nonempty open subset of $G$; thus $S$, if a subgroup of $G$, is open in $G$.*]{} ([@comftrigwu 3.10]). Let $G$ be an Abelian group, let $\{x_n:n<\omega\}$ be a faithfully index sequence in $G$, and let $$S:=\{h\in\Hom(G,\TT):h(x_n)\rightarrow0\in\TT\}.$$ Let $\lambda$ be the Haar measure of $\Hom(G,\TT)$. Then $S$ is a $\lambda$-measurable subgroup of $\Hom(G,\TT)$, with $\lambda(S)=0$. Let $X$ be a compact Hausdorff space such that $|X|<2^{\aleph_1}$. Then - ([@hajjuh76]) $X$ contains a closed, countably infinite subspace; and - $X$ contains a nontrivial convergent sequence. Let $G$ be an Abelian group such that $|G|<2^{\aleph_1}$ and let $A$ be a dense subgroup of $\Hom(G,\TT)$ such that either - $A$ is non-Haar measurable, or - $A$ is Haar measurable, with $\lambda(A)>0$. Then $(G,\sT_A)$ does not determine $W(G,\sT_A)$. Let $G$ be a compact, Abelian group such that $w(G)\geq\cc$. Then $G$ is not determined. (Outline) Step 1. $\TT^\cc$ is not determined. \[Proof. There is a nonmeasurable subgroup $A$ of $\TT$ algebraically of the form $\oplus_\cc\,\ZZ$. Apply Theorem 4.7(i) to the (dense) subgroup $e_A(\ZZ)$ of $\TT^A=\TT^\cc$.\] Step 2. $F^\cc$ is not determined ($F$ a finite Abelian group, $|F|>1$). \[Proof. We have $b((\oplus_\omega\,F)_d)=F^\cc$, so Corollary 4.2 applies.\] Step 3. There is a continuous isomorphism $\phi:G\twoheadrightarrow K^\cc$ with either $K=\TT$ or $K=F$ as in Step 2, so Theorem \[3.12\] applies. \[CH\] Let $G$ be a compact Abelian group. Then $G$ is determined if and only if $G$ is metrizable. \[CH\] Let $\{G_i:i\in I\}$ be a set of compact Abelian groups with each $|G_i|>1$, and let $G=\Pi_{i\in I}\,G_i$. Then $G$ is determined if and only if $|I|\leq\omega$ and each $G_i$ is determined. We close this section with an example indicating that the intersection of dense, determining subgroups may be dense and nondetermining. There are dense, determining subgroups $D_i$ ($i=0,1$) of $\TT^\cc$ such that $D_0\cap D_1$ is dense in $\TT^\cc$ and does not determine $\TT^\cc$. Let $Z$ be a dense, cyclic, nondetermining subgroup of $\TT^\cc$, let $A_i$ ($i=0,1$) be dense, torsion subgroups of $\TT$ such that $A_0\cap A_1=\{0_\TT\}$, and set $D_i:=Z+\oplus_\cc\,A_i\subseteq\TT^\cc$ ($i=0,1$). Then $A_i$ determines $\TT$ by Theorem 1.3 so $\oplus_\cc\,A_i$ determines $\TT^\cc$ by Lemma 3.9, so $D_i$ determines $\TT^\cc$ ($i=0,1$); but the dense subgroup $Z=D_0\cap D_1$ of $\TT^\cc$ does not determine $\TT^\cc$. Concerning Topological Linear Spaces ==================================== Let $\kappa$ be a cardinal number and denote by $l^1_\kappa$ the space of real $\kappa$-sequences $x=\{x_\xi:\xi<\kappa\}$ such that $||x||_1:=\sum_{\xi<\kappa}|x_\xi|<\infty$. The additive topological group $l_\kappa^1$ respects compactness (cf. Remus and Trigos-Arrieta [@remustrig97a]). The group $\widehat{(l_\kappa^1)^+}$ is not discrete, so the Weil completion $W((l_\kappa^1)^+)$ is another example of a compact nondetermined group. A topological group $G$ is [*(group) reflective*]{} if the evaluation mapping $\Omega_G:G\rightarrow \widehat{\widehat{G}}$ defined by $\Omega_G(x)(h):=h(x)$ for $x\in G$, $h\in\widehat{G}$ is a topological isomorphism of $G$ onto $\widehat{\widehat G}$. Let $G$ be a noncomplete, reflective group and let $R(G)$ be its Raĭkov completion. Then $R(G)\in\bf MAP$ and $G$ does not determine $R(G)$. Example 5.4 [*infra*]{} illustrates Theorem 5.2. A reflexive locally convex vector space (LCS) in which every closed bounded subset is compact is called a [*Montel space*]{}. Reflexivity and boundedness (Schaefer[@shaf:86] §I.5, §IV.5) are meant here in the sense of topological vector spaces. By a [*Montel group*]{} we mean the underlying (additive) topological group of a Montel space. Since by definition these are reflexive LCS, Montel groups are reflective as proven by Smith [@smith52]. Kōmura [@komura64] and Amemiya and Kōmura [@amemiyakomura] construct by induction three different noncomplete Montel spaces, the completion of each being a “big product” of copies of $\RR$, and one of them being exactly $\RR^\cc$. These groups indicate that Theorem 5.2 is not vacuous. One of the groups constructed in [@amemiyakomura] is separable. Thus in particular, again by Theorem 5.2, we see that $\RR^\cc$ has a countable dense subgroup which does not determine $\RR^\cc$. The remarks above show again that the property of being determined is not $\cc$-productive. Cardinals $\kappa$ Such That $\omega<\kappa\leq\cc$ =================================================== It is well known (cf. for example [@kunen80 2.18] or [@cies 8.2.4]) that under Martin’s Axiom \[MA\] every cardinal $\kappa$ with $\omega\leq\kappa<\cc$ satisfies $2^\kappa=\cc$. In particular under MA $+\neg$CH it follows from Theorem 4.6(b) that every compact Hausdorff space $X$ such that $|X|< 2^{\aleph_1}=\cc$ contains a nontrivial convergent sequence. Malykhin and [Š]{}apirovski[ĭ]{} [@malysap73] have achieved a nontrivial extension of this result: Under MA, every compact Hausdorff space $X$ with $|X|\leq\cc$ contains a nontrivial convergent sequence. \[MA\] Let $G$ be a group with $|G|\leq 2^{\omega}$, and let $A$ be a dense nonmeasurable subgroup of $\widehat{G_d}$. Then every compact subset of $(G,\sT_A)$ is finite, so its completion $W(G,\sT_A)$ is not determined. If we denote by $\lambda_G$ the (completed) Haar measure on a LCA group $G$, let $\lambda^*_{G}$ stand for the associated outer measure. The existence of a nonmeasurable subset $X$ of $\TT$ (with $|X|=\cc$) is well known, so the case $\kappa=\cc$ of the following theorem generalizes the statement in Step 1 of the proof of Theorem 4.8. Let $\omega < \kappa\leq\cc$. If there is $X\in [\TT]^\kappa$ such that $\lambda_\TT^*(X)>0$, then there is a nonmeasurable, free Abelian subgroup $A$ of $\TT$ algebraically of the form $A=\oplus_\kappa\,\ZZ$. Responding to a question on a closely related matter, Stevo Todorčević [@To01] proposed and proved the above result for $\kappa=\aleph_1$. His proof additionally yields that $X\backslash\tor(\TT)$ can be broken into $\omega$-many pairwise disjoint independent sets, each of cardinality $\aleph_1$. For torsion groups of prime order, we obtain the following. Let $F$ be a finite group of prime order $p$, and let $\kappa_1, \kappa_2$ be infinite cardinals such that $\kappa_1\leq2^{\kappa_2}$. If there is $X\in [F^{\kappa_2}]^{\kappa_1}$ such that $\lambda^*_{F^{\kappa_2}}(X)>0$, then there is a nonmeasurable subgroup $A$ of $F^{\kappa_2}$ algebraically of the form $A=\oplus_{\kappa_1} F$. For an ideal $\sI$ of subsets of a set $S$ we write as usual $${\mathrm{non}}(\sI)=\min\{|Y|:Y\subseteq S,~Y\notin\sI\}.$$ Let $F$ be a finite group ($|F|>1$), let $\sN(\TT)$ and $\sN(F^\omega)$ denote the $\sigma$-algebra of $\lambda_\TT$- and $\lambda_{F^\omega}$-measurable sets of measure zero. As with any two compact metric spaces of equal cardinality equipped with atomless (“continuous”) probability measures, the spaces $\TT$ and $F^\omega$ are Borel-isomorphic in the sense that there is a bijection $\phi:\TT\twoheadrightarrow F^\omega$ such that the associated bijection $\overline{\phi}:\sP(\TT)\twoheadrightarrow\sP(F^\omega)$ carries the Borel algebra $\sB(\TT)$ onto the Borel algebra $\sB(F^\omega)$ in such a way that $\lambda_{F^\omega}(\overline{\phi}(B))=\lambda_\TT(B)$ for each $B\in\sB(\TT)$. (See [@kechris 17.41] or [@sriva98 3.4.23] for a proof of this “Borel isomorphism Theorem for measures”.) The cardinals $\non(\sN(\TT))$ and $\non(\sN(F^\omega))$ are equal. We write $$\non(\sN):=\non(\sN(\TT))=\non(\sN(F^\omega)),$$ a definition justified by Lemma 6.5. Let $G$ be a compact Abelian group such that $w(G)\geq\non(\sN)$. Then $G$ is nondetermined. Questions ========= Is there a compact group $G$ with a countable dense subgroup $D$ such that $w(G)>\omega$ and $D$ determines $G$? If $\{G_i:i\in I\}$ is a set of topological Abelian groups and $D_i$ is a dense determining subgroup of $G_i$, must $\oplus_{i\in I}\,D_i$ determine $\Pi_{i\in I}\,G_i$? In particular, does $\oplus_{i\in I}\,G_i$ determine $\Pi_{i\in I}\,G_i$? In particular, does $\oplus_\cc\,\RR$ determine $\RR^\cc$? Consider the following cardinals: - $\mm_\TT :=$ the least cardinal $\kappa$ such that $\TT^\kappa$ is nondetermined; - $\mm_{f\exists}$ \[resp., $\mm_{f\forall}] :=$ the least cardinal $\kappa$ such that some \[resp., each\] finite group $F$ has $F^\kappa$ nondetermined; - $\mm_{c\exists}$ \[resp., $\mm_{c\forall}] :=$ the least cardinal $\kappa$ such that some \[resp., each\] compact abelian group of weight $\kappa$ is nondetermined; - $\mm_{p\exists}$ \[resp., $\mm_{p\forall}] :=$ the least cardinal $\kappa$ such that some \[resp., each\] product of $\kappa$-many compact determined groups is nondetermined. It follows from Theorems 1.3 and 6.6 that each $\mm_x$, with the possible exception of $\mm_{p\exists}$, satisfies $\aleph_1 \leq \mm_x\leq\non(\sN)$. Further if $\non(\sN)=\aleph_1$, then all seven cardinals $\mm_x$ are equal to $\aleph_1$. The condition $\non(\sN)=\aleph_1$ is clearly consistent with CH, and it has been shown to be consistent as well with $\neg$CH (see for example [@bartjudah], [@fremlin84] and [@jechii Example 1, page 568]), so in particular there are models of ZFC + $\neg$CH in which every compact (Abelian) group $G$ satisfies: $G$ is determined if and only if $G$ is metrizable. (Without appealing to the cardinal $\non(\sN)$, Michael Hru[š]{}[á]{}k [@Hr00] in informal conversation suggested the existence of models of ZFC + $\neg$CH in which $\{0,1\}^{\aleph_1}$ is nondetermined.) Are the various cardinal numbers $\mm_x$ equal in ZFC? Are they equal to one of the familiar “small cardinals” conventionally noted in the Cicho[ń]{} diagram (cf. [@bartjudah], [@vaughan90])? Is each $\mm_x=\non(\sN)$? Is each $\mm_x=\aleph_1$? Is each $\cf(\mm_x)>\omega$? We know of no models of ZFC in which $\TT^{\aleph_1}$, or some group of the form $F^{\aleph_1}$ ($F$ finite, $|F|>1$), is determined, so we are forced to consider the possibility that the following questions have an affirmative answer. Are the following (equivalent) statements theorems of ZFC? - The group $\TT^{\aleph_1}$ and groups of the form $F^{\aleph_1}$ ($F$ finite, $|F|>1$) are nondetermined. - A compact abelian group $G$ is determined if and only if $G$ is metrizable. The following question is suggested by those above. Is there in ZFC a cardinal $\kappa$ such that a compact group $G$ is determined if and only if $w(G)<\kappa$? Is it consistent with ZFC that $\mm_{p\exists}=2$? Is it consistent with ZFC that $\mm_{p\exists}=\omega$? Question 7.7 has analogues in the context of groups which are not assumed to be compact, as follows. In ZFC alone or in augmented axiom systems: Is the product of finitely many determined groups necessarily determined? 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--- abstract: 'In a multitime hybrid differential game with mechanical work payoff, the multitime upper value function and the multitime lower value function are viscosity solutions of original PDEs of type Hamilton-Jacobi-Isaacs.' author: - 'Constantin Udrişte, Elena-Laura Otobîcu, Ionel Ţevy' title: Viscosity solution PDEs in hybrid games with mechanical work payoff --- \[section\] \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Definition]{} \[theorem\][Remark]{} [**MSC2010**]{}: multitime hybrid differential games; multitime viscosity solution; multitime dynamic programming. [**Keywords**]{}: 49L20, 91A23, 49L25, 35F21. Multitime lower or upper value function ======================================= All variables and functions must satisfy suitable conditions (for example, see [@[8]]). We analyze a [*multitime hybrid differential game, with two teams of players*]{}, whose [*Bolza payoff*]{} is the sum between a path independent curvilinear integral (mechanical work) and a function of the final event (the terminal cost, penalty term) and whose evolution PDE is an [*m-flow*]{}. The optimal control problem is: [*Find $$\min_{v(\cdot)\in V}\max_{u(\cdot)\in U} J(u(\cdot),v(\cdot))=\int_{\Gamma_{0T}} L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))ds^\alpha+g(x(T)),$$ subject to the Cauchy problem $$\frac{\partial x^i}{\partial s^\alpha}(s)=X^i_\alpha(s,x(s),u_\alpha(s),v_\alpha(s)),$$ $$x(0)=x_0, \,\,s\in \Omega_{0T}\subset \mathbb{R}_+^m, \,\,x\in \mathbb{R}^n,$$*]{} where $i=1,...,n$; $\alpha =1,...,m$; $u=(u^a)$, $a=1,...,p$, $v=(v^b)$, $b=1,...,q$ are the controls. To simplify, suppose that the curve $\Gamma_{0T}$ is an increasing curve in the multitime interval $\Omega_{0T}$. We vary the starting multitime and the initial point. We obtain a larger family of similar multitime problems containing the functional $$J_{x,t}(u(\cdot),v(\cdot))=\int_{\Gamma_{tT}} L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))ds^\alpha+g(x(T)),$$ and the evolution constraint (Cauchy problem for first order PDEs system) $$\frac{\partial x^i}{\partial s^\alpha}(s)=X^i_\alpha(s,x(s),u_\alpha(s),v_\alpha(s)),\,\, x(t)=x, \,\,s\in \Omega_{tT}\subset \mathbb{R}_+^m,\,\, x\in \mathbb{R}^n.$$ Let $\Psi$ and $\Phi$ be suitable strategies of the two equips of players. \(i) The function $$m(t,x)=\min_{\Psi\in \mathcal{V}} \max_{u(\cdot)\in U} J_{t,x}( u(\cdot),\Psi[u](\cdot))$$ is called the multitime lower value function. \(ii) The function $$M(t,x)=\max_{\Phi\in \mathcal{U}} \min_{v(\cdot)\in V} J_{t,x}(\Phi[v](\cdot),v(\cdot))$$ is called the multitime upper value function. The papers [@[1]]-[@[4]], [@[12]] refer to viscosity solutions of Hamilton-Jacobi-Isaacs equations. To understand the multitime optimal control and our recent results see the papers [@[5]]-[@[11]]. Viscosity solutions of\ multitime upper/lower PDEs ========================== The key original idea is that the multitime upper value function or the multitime lower value function are solutions of PDEs, defined in the next Theorem. Our PDEs contain some implicit assumptions and are valid under certain conditions which are defined and analyzed for multitime hybrid differential games. \(i) The multitime upper value function $M(t,x)$ is the viscosity solutions of the multitime upper PDE $$\frac{\partial M}{\partial t^\alpha}(t,x)+\min_{v_\alpha\in \mathcal{V}} \max_{u_\alpha\in \mathcal{U}} \left\lbrace \frac{\partial M}{\partial x^i}(t,x) X_\alpha^i(t,x,u_\alpha,v_\alpha)+L_\alpha(t,x,u_\alpha,v_\alpha)\right\rbrace =0,$$ which satisfies the terminal condition $M(T,x)=g(x).$ \(ii) The multitime lower value function $m(t,x)$ is the viscosity solution of the multitime lower PDE $$\frac{\partial m}{\partial t^\alpha}(t,x)+\max_{u_\alpha \in \mathcal{U}} \min_{v_\alpha \in \mathcal{V}} \left\lbrace \frac{\partial m}{\partial x^i}(t,x) X_\alpha^i(t,x,u_\alpha,v_\alpha)+L_\alpha(t,x,u_\alpha,v_\alpha)\right\rbrace =0,$$ which satisfies the terminal condition $m(T,x)=g(x).$ We introduce the so-called upper and lower Hamiltonian defined respectively by $$H^+_\alpha(t,x,p)=\min_{v_\alpha\in \mathcal{V}} \max_{u_\alpha \in \mathcal{U}}\lbrace p_i(t) X_\alpha^i(t,x,u_\alpha,v_\alpha)+L_\alpha(t,x,u_\alpha,v_\alpha)\rbrace,$$ $$H^-_\alpha(t,x,p)=\max_{u_\alpha\in \mathcal{U}} \min_{v_\alpha\in \mathcal{V}}\lbrace p_i(t) X_\alpha^i(t,x,u_\alpha,v_\alpha)+L_\alpha(t,x,u_\alpha,v_\alpha)\rbrace.$$ We prove only the first statement. For $s\in \Omega_{tt+h},$ we use the Cauchy problem $$\frac{\partial x^i}{\partial s^\alpha}(s)=X^i_\alpha(s,x(s),u_\alpha(s),v_\alpha(s)),$$ $$x(t)=x,\,\, s\in \Omega_{tt+h}\subset \mathbb{R}_+^m,\,\, x\in \mathbb{R}^n$$ and the cost functional (mechanical work) $$J_{t,x}(u(\cdot),v(\cdot))=\int_{\Gamma_{tt+h}} L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))ds^\alpha.$$ For $s\in \Omega_{tT}\setminus \Omega_{tt+h},$ the cost is $M(t+h,x(t+h))$. Consequently, $$J_{t,x}(u(\cdot),v(\cdot))=\int_{\Gamma_{tt+h}} L_\alpha (s,x(s),u_\alpha(s),v_\alpha(s))ds^\alpha+M(t+h,x(t+h)),$$ with $M(t,x)\geq M(t+h,x(t+h))$, because $M(t,x)$ is the greatest cost. Thus we have the multitime dynamic programming optimality condition $$M(t,x)=\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in V(t)}\bigg\{\int_{\Gamma_{tt+h}} L_\alpha (s,x(s),\Phi [v_\alpha](s),v_\alpha(s))ds^\alpha +M(t+h,x(t+h))\bigg\}.$$ Let $(\omega)\in C^1(\Omega_{0T}\times \mathbb{R}^n)$ be a generating vector field. We analyse two cases: Suppose $M-\omega$ attains a local maximum at $(t,x)\in \Omega_{0T}\times \mathbb{R}^n.$ We must prove the inequality $$\frac{\partial \omega}{\partial t^\alpha}(t,x)+H^+_{\alpha}\left( t,x,\frac{\partial \omega}{\partial x^i}(t,x)\right) \geq 0.\eqno(1)$$ For that, we suppose the contrary $$\frac{\partial \omega}{\partial t^\alpha}(t,x)+H^+_{\alpha}\left( t,x,\frac{\partial \omega}{\partial x^i}(t,x)\right) \leq -\theta_\alpha <0,$$ for each $\alpha=\overline{1,m}$ and for some constant 1-form $\theta_\alpha>0.$ Let $h=(h^\alpha),$ with $h^\alpha>0.$ We use the [*Fundamental Lemma*]{} in the next Section. This implies that, for each sufficiently small $\Vert h\Vert$ and all $\omega\in \mathcal{A}(t),$ the relation $$\int_{\Gamma_{tt+h}} \bigg(L_\alpha(s,x(s),\Phi[v_\alpha](s),v_\alpha(s)) +\frac{\partial \omega}{\partial x^i} X_\alpha^i(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))+\frac{\partial \omega}{\partial s^\alpha}\bigg) ds^\alpha \leq -\frac{h^{\alpha}\theta_{\alpha}}{2}$$ holds for $v_\alpha\in \mathcal{V}(t).$ Thus $$\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in V(t)} \bigg\{\int_{\Gamma_{tt+h}} \left(L_\alpha(s,x(s),\Phi[v_\alpha](s),v_\alpha(s)) +\frac{\partial \omega}{\partial x^i} X_\alpha^i(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))\right.$$ $$\hspace{5cm}\left.+\frac{\partial \omega}{\partial s^\alpha}\right)ds^\alpha \bigg\}\leq -\frac{h^{\alpha}\theta_{\alpha}}{2},\eqno(2)$$ with $x(\cdot)$ solution of the previous Cauchy problem. Because $M-\omega$ has a local maximum at the point $(t,x),$ we have $$M(t,x)-\omega(t,x)\geq M(t+h,x(t+h))-\omega(t+h,x(t+h)).$$ The multitime dynamic programming optimality condition and by the local maximum definition, we can write $$M(t,x)-M(t+h,x(t+h))=\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in \mathcal{V}(t)} \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),\Phi [v_\alpha](s),v_\alpha(s))ds^\alpha.$$ Consequently, we have $$\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in \mathcal{V}(t)} \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),\Phi [v_\alpha](s),v_\alpha(s))ds^\alpha \geq \omega(t,x)-\omega(t+h,x(t+h))$$ or $$\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in \mathcal{V}(t)}\int_{\Gamma_{tt+h}} L_\alpha (s,x(s),\Phi [v_\alpha](s),v_\alpha(s))ds^\alpha +\omega(t+h,x(t+h))-\omega(t,x)\geq 0.\eqno(3)$$ On the other hand, $$\omega(t+h,x(t+h))-\omega(t,x)= \int_{\Gamma_{tt+h}} d\omega=\int_{\Gamma_{tt+h}} D_\alpha\omega\,\, ds^\alpha$$ $$=\int_{\Gamma_{tt+h}} \left( \frac{\partial \omega}{\partial x^i} X_\alpha^i(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))+\frac{\partial \omega}{\partial s^\alpha}\right) ds^\alpha.$$ So, the relation $(3)$ contradicts the relation $(2)$. Suppose $M-\omega$ attains a local minimum at $(t,x)\in \Omega_{0T}\times \mathbb{R}^n.$ We must prove that $$\frac{\partial \omega}{\partial t^\alpha}(t,x)+H^+_{\alpha}(t,x,\frac{\partial \omega}{\partial x^i}(t,x))\leq 0, \eqno(4).$$ To do this, we suppose the contrary $$\frac{\partial \omega}{\partial t^\alpha}(t,x)+H^+_{\alpha}(t,x,\frac{\partial \omega}{\partial x^i}(t,x))\geq \theta_\alpha >0,$$ for each $\alpha=\overline{1,m}$ and for some constant 1-form $\theta_\alpha>0.$ Let $h=(h^\alpha),$ with $h^\alpha>0.$ We use the [*Fundamental Lemma*]{} in the next Section. This implies that, for each sufficiently small $\Vert h\Vert$ and all $\omega\in \mathcal{A}(t),$ the relation $$\int_{\Gamma_{tt+h}} \bigg( L_\alpha(s,x(s),\Phi[v_\alpha](s),v_\alpha(s)) +\frac{\partial \omega}{\partial x^i} X_\alpha^i(s,x(s),\Phi[v_\alpha](s),v_\alpha(s)) +\frac{\partial \omega}{\partial s^\alpha}\bigg) ds^\alpha\geq\frac{h^\alpha\theta_\alpha}{2}$$ holds for $v_\alpha\in \mathcal{V}(t).$ Thus $$\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in \mathcal{V}(t)} \bigg\{ \int_{\Gamma_{tt+h}} \left( L_\alpha(s,x(s),\Phi[v_\alpha](s),v_\alpha(s)) +\frac{\partial \omega}{\partial x^i} X_\alpha^i(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))\right.$$ $$\hspace{5cm}\left.+\frac{\partial \omega}{\partial s^\alpha}\right) ds^\alpha\bigg\}\geq\frac{h^\alpha\theta_\alpha}{2}.\eqno(5)$$ Because $M-\omega$ has a local minimum at the point $(t,x),$ we have $$M(t,x)-\omega(t,x)\leq M(t+h,x(t+h))-\omega(t+h,x(t+h)),$$ where $x(\cdot)$ is the solution of the previous Cauchy problem. By the multitime dynamic programming optimality condition and by the local minimum definition, we can write $$M(t,x)-M(t+h,x(t+h))=\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in \mathcal{V}(t)} \left\lbrace \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),\Phi [v_\alpha](s),v_\alpha(s))ds^\alpha\right\rbrace.$$ Using the inequality $$M(t,x)-M(t+h,x(t+h))\leq \omega(t,x)-\omega(t+h,x(t+h)),$$ we find $$\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in \mathcal{V}(t)} \left\lbrace\int_{\Gamma_{tt+h}} L_\alpha (s,x(s),\Phi [v_\alpha](s),v_\alpha(s))ds^\alpha \right\rbrace\leq \omega(t,x)-\omega(t+h,x(t+h))$$ and $$\max_{\Phi\in \mathcal{A}(t)}\min_{v_\alpha\in \mathcal{V}(t)}\bigg\{ \int_{\Gamma_{tt+h}} L_\alpha (s,x(s),\Phi [v_\alpha](s),v_\alpha(s))ds^\alpha\bigg\} +\omega(t+h,x(t+h))-\omega(t,x)\leq 0.\eqno(6)$$ On the other hand, $$\omega(t+h,x(t+h))-\omega(t,x)= \int_{\Gamma_{tt+h}} d\omega=\int_{\Gamma_{tt+h}} D_\alpha\omega \,\,ds^\alpha$$ $$=\int_{\Gamma_{tt+h}} \bigg(\frac{\partial \omega}{\partial x^i} X_\alpha^i(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))+\frac{\partial \omega}{\partial s^\alpha}\bigg)ds^\alpha.$$ That is why the relation $(6)$ contradicts the relation $(5).$ Fundamental contradict Lemma ============================ The short proofs in the previous section are based on an interesting Lemma. \[l-1\] Let $\omega\in C^1(\Omega_{0T}\times \mathbb{R}^n)$. (i)If $M-\omega$ attains a local maximum at $(t_0,x_0)\in \Omega_{0T}\times \mathbb{R}^n$ and $$\omega_{t^{\alpha}}(t_0,x_0)+ H^+_\alpha\left(t_0,x_0,\frac{\partial\omega}{\partial x^i}(t_0,x_0)\right)\leq - \theta_\alpha <0,$$ then, for all vectors $h=(h^\alpha)$, with sufficiently small $||h||$, there exists a control $v=(v_\alpha)\in {\mathcal V}(t_0)$ such that the relation $(2)$ holds for all strategies $\Phi\in {\mathcal A}(t_0)$. \(ii) If $M-\omega$ attains a local minimum at $(t_0,x_0)\in \Omega_{0T}\times \mathbb{R}^n$ and $$\omega_{t^{\alpha}}(t_0,x_0)+ H^+_\alpha\left(t_0,x_0,\frac{\partial\omega}{\partial x^i}(t_0,x_0)\right)\geq \theta_\alpha >0,$$ then, for all vectors $h=(h^\alpha)$, with sufficiently small $||h||$, there exists a control $u=(u_\alpha)\in {\mathcal U}(t_0)$ such that the relation $(5)$ holds for all strategies $\Psi\in {\mathcal B}(t_0)$. We introduce the 1-form $\Lambda$ of components $$\Lambda_\alpha = L_\alpha(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))+\frac{\partial \omega}{\partial x^i} X_\alpha^i(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))+\frac{\partial \omega}{\partial s^\alpha}.$$ \(i) By hypothesis $$\min_{v\in {\mathcal V}}\,\, \max_{u\in {\mathcal U}} \,\Lambda_\alpha (t_0,x_0,u_\alpha,v_\alpha)\leq -\theta_\alpha <0.$$ Consequently there exists some control $v^*\in {\mathcal V}$ such that $$\max_{u\in {\mathcal U}} \, \Lambda_\alpha (t_0,x_0,u_\alpha,v^*_\alpha)\leq -\theta_\alpha,$$ for each $\alpha=\overline{1,m}.$ On the other hand, the uniform continuity of the 1-form $\Lambda=(\Lambda_\alpha)$ implies $$\max_{u\in {\mathcal U}} \, \Lambda_\alpha (t_0,x(s),u_\alpha,v^*_\alpha)\leq - \frac{1}{2}\,\theta_\alpha$$ provided $s\in \Omega_{t_0t_0+h}$, for any small $||h||>0$, and $x(\cdot)$ is solution of PDE on $\Omega_{t_0t_0+h}$, for any $u(\cdot),v(\cdot)$, with initial condition $x(t_0)=x_0$. It follows that, for the control $v(\cdot)=v^*$ and for any strategy $\Phi\in {\mathcal A}(t_0)$, we have $$L_\alpha(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))+\frac{\partial \omega}{\partial x^i} X_\alpha^i(s,x(s),\Phi[v_\alpha](s),v_\alpha(s))+\frac{\partial \omega}{\partial t^\alpha}\leq \frac{-\theta_\alpha}{2}$$ for $s\in \Omega_{t_0t_0+h}$. Taking the curvilinear integral along an increasing curve $\Gamma_{t_0t_0+h}$, we obtain the relation $(2)$. \(ii) The inequality in the Lemma reads $$\min_{v\in {\mathcal V}}\,\, \max_{u\in {\mathcal U}} \,\Lambda_\alpha (t_0,x_0,u_\alpha,v_\alpha)\geq \theta_\alpha >0.$$ Consequently, for each control $v\in {\mathcal V}$ there exists a control $u=u(v)\in U$ such that $$\Lambda_\alpha (t_0,x_0,u_\alpha,v_\alpha)\geq \theta_\alpha.$$ The uniform continuity of the 1-form $\Lambda$ implies $$\Lambda_\alpha (t_0,x_0,u_\alpha,\xi_\alpha)\geq \frac{3}{4}\,\theta_\alpha,\,\,\forall \xi\in B(v,r)\cap V\, \hbox{and some}\,\, r=r(v)>0.$$ Due to compactness of ${\mathcal V}$, there exists finitely many distinct points $$v_1,...,v_n \in {\mathcal V}; \,u_1,..., u_n\in {\mathcal U}$$ and the numbers $r_1,...,r_n >0$ such that ${\mathcal V}\subset \bigcup_{i=1}^{n} B(v_i,r_i)$ and $$\Lambda_\alpha (t_0,x_0,u_i,\xi)\geq \frac{3}{4}\,\theta_\alpha,\,\,\forall \xi\in B(v_i,r_i).$$ Define $$\psi:{\mathcal V}\to {\mathcal U},\, \psi(v)=u_k\,\, \hbox{if}\,\, v\in B(u_k,r_k)\setminus \bigcup_{i=1}^{k-1} B(u_i,r_i),\,k=\overline{1,n}.$$ In this way, $$\Lambda_\alpha (t_0,x_0,\psi(v_\alpha),v_\alpha)\geq \frac{3}{4}\,\theta_\alpha, \forall v\in {\mathcal V}.$$ Again, the uniform continuity of the 1-form $\Lambda$ and a sufficiently small $||h||>0$ give $$\Lambda_\alpha (s,x(s),\psi(v_\alpha),v_\alpha)\geq \frac{1}{2}\,\theta_\alpha, \forall v\in {\mathcal V},\, s\in \Omega_{t_0t_0+h},$$ and any solution $x(\cdot)$ of PDE on $\Omega_{t_0t_0+h}$, for any $u(\cdot), v(\cdot)$ and with initial condition $x(t_0)=x_0$. Now define a new strategy $$\Psi\in {\mathcal B}(t_0),\,\,\Psi[v_\alpha](s)=\psi(v_\alpha(s)),\, \forall v\in {\mathcal V}(t_0),\,s\in \Omega_{t_0t_0+h}.$$ Finally, for each $\alpha,$ we have the inequality $$\Lambda_\alpha (s,x(s),\Psi[v_\alpha](s),v_\alpha(s))\geq \frac{1}{2}\,\theta_\alpha, \forall \,s\in \Omega_{t_0t_0+h},$$ and taking the curvilinear integral along an increasing curve $\Gamma_{t_0t_0+h}$, we find the result in Lemma. [50]{} M. G. Crandall, L. C. Evans and P. L. Lions, *Some properties of viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc., 282, 2, (1984), 487-502. M. G. Crandall and P. L. Lions, *Viscosity solutions of Hamilton-Jacobi equations*, Trans. Amer. Math. Soc., 277, (1983), 1-42. E. N. Barron, L. C. Evans, R. Jensen, *Viscosity solutions of Isaacs’equations and differential games with Lipschitz controls*, Journal of Differential Equations, 53, (1984), 213-233. P. E. Souganidis, *Existence of viscosity solutions of Hamilton-Jacobi equations*, Journal of Differential Equations, 56, (1985), 345-390. C. Udrişte, *Multi-time controllability, observability and bang-bang principle*, J. Optim. Theory Appl., 138, 1, (2008), 141-157. C. Udrişte, *Equivalence of multitime optimal control problems*, Balkan J. Geom. Appl. 15, 1, (2010), 155-162. C. Udrişte, *Simplified multitime maximum principle*, Balkan J. Geom. Appl. 14, 1, (2009), 102-119. C. Udrişte, I. Ţevy, *Multitime dynamic programming for curvilinear integral actions*, J. Optim. Theory and Appl., 146, (2010), 189-207. C. Udrişte, A. Bejenaru, *Multitime optimal control with area integral costs on boundary*, Balkan J. Geom. Appl., 16, 2, (2011), 138-154 C. Udrişte, *Multitime maximum principle for curvilinear integral cost*, Balkan J. Geom. Appl., 16, 1, (2011), 128-149. C. Udrişte, I. Ţevy, *Multitime dynamic programming for multiple integral actions*, J. Glob. Optim., 51, 2, (2011), 345-360. C. Udrişte, I. Ţevy, E.-L. Otob\^ icu, [*Viscosity solutions of divergence type PDEs associated to multitime hybrid games*]{}, U.P.B. Sci. Bull., Series A, Vol..., Iss. ..., 2016. A. Davini, M. Zavidovique, *On the (non) existence of viscosity solutions of multi-time Hamilton-Jacobi equations,* Preprint (2013), [http://www.math.jussieu.fr/zavidovique/articles/NonCommuting\ Nov. 2013.pdf]{} Constantin Udrişte, Elena-Laura Otobîcu, Ionel Ţevy\ University [Politehnica]{} of Bucharest, Faculty of Applied Sciences,\ Department of Mathematics and Informatics,\ Splaiul Independentei 313, RO-060042, Bucharest, Romania.\ E-mail: [udriste@mathem.pub.ro]{}; [laura.otobicu@gmail.com]{}; [vascatevy@yahoo.fr]{}

--- abstract: 'The trace norm is widely used in multi-task learning as it can discover low-rank structures among tasks in terms of model parameters. Nowadays, with the emerging of big datasets and the popularity of deep learning techniques, tensor trace norms have been used for deep multi-task models. However, existing tensor trace norms cannot discover all the low-rank structures and they require users to manually determine the importance of their components. To solve those two issues together, in this paper, we propose a Generalized Tensor Trace Norm (GTTN). The GTTN is defined as a convex combination of matrix trace norms of all possible tensor flattenings and hence it can discover all the possible low-rank structures. In the induced objective function, we will learn combination coefficients in the GTTN to automatically determine the importance. Experiments on real-world datasets demonstrate the effectiveness of the proposed GTTN.' bibliography: - 'GTTN.bib' --- Introduction ============ Given multiple related learning tasks, multi-task learning [@caruana97; @zy17b] aims to exploit useful information contained in them to help improve the performance of all the tasks. Multi-task learning has been applied to many application areas, including computer vision, natural language processing, speech recognition and so on. Over past decades, many multi-task learning models have been devised to learn such useful information shared by all the tasks. As reviewed in [@zy17b], multi-task learning models can be categorized into six classes, including the feature transformation approach [@aep06; @msgh16], feature selection approach [@otj06; @lpz09; @ls12], low-rank approach [@ptjy10; @hz16; @yh17], task clustering approach [@xlck07; @jbv08; @kd12; @hz15a], task relation learning approach [@bcw07; @zy10a; @lcwy17; @zwy18], and decomposition approach [@jrsr10; @cly10; @zw13; @hz15b]. Among those approaches, the low-rank approach is effective to identify low-rank model parameters. When model parameters of a task can be organized in a vector corresponding to for example binary classification tasks or regression tasks on vectorized data, the matrix trace norm or its variants is used as a regularizer on the parameter matrix, each of whose columns stores parameters for a task, to identify the low-rank structure among tasks. Nowadays with the collection of complex data and the popularity of deep learning techniques, each data point can be represented as a tensor (e.g., images) and each learning task becomes complex, e.g., multi-class classification tasks. In this case, the parameters of all the tasks are stored in a tensor, making the matrix trace norm not applicable, and instead tensor trace norms [@rabp13; @wst14; @yh17] are used to learn low-rank parameters in the parameter tensor for multi-task learning. Different from the matrix trace norm which has a unique definition, the tensor trace norm has many variants as the tensor rank has multiple definitions. Here we focus on overlapped tensor trace norms which equals the sum of the matrix trace norm of several tensor flattenings of the tensor. An overlapped tensor trace norm relies on the way to do the tensor flattening. For example, the Tucker trace norm [@tucker66] conducts the tensor flattening along each axis in the tensor and the Tensor-Train (TT) trace norm [@oseledets11] does it along successive axes starting from the first one. There are two limitation in the existing tensor trace norms. Firstly, for a $p$-way tensor, we can see that there are $2^p-2$ possible tensor flattenings but existing overlapped tensor trace norms only utilize a subset of them, making them fail to capture all the low-rank structures in the parameter tensor. Another limitation of existing tensor trace norms is that all the tensor flattenings used in a tensor trace norm are assumed to be equally important, which is suboptimal to the performance. In this paper, to overcome the two aforementioned limitations of existing overlapped tensor trace norms, we propose a Generalized Tensor Trace Norm (GTTN). The GTTN exploits all possible tensor flattenings and it is defined as the convex sum of matrix trace norms of all possible tensor flattenings. In this way, the GTTN can capture all the low-rank structures in the parameter tensor and hance overcome the first limitation. Moreover, to alleviate the second limitation, we treat combination coefficients in the GTTN as variables and propose an objective function to learn them from data. Another advantage of learning combination coefficients is that it can show the importance of some axes, which can improve the interpretability of the learning model and give us insights for the problem under investigation. To obtain a full understanding of the GTTN, we study properties of the proposed GTTN. For example, the number of tensor flattenings with distinct matrix trace norms is proved to be $2^{p-1}-1$ and so when $p\le 5$ we encountered in most problems, such number is not so large that the computational complexity is comparable to existing tensor trace norms. We also analyze the dual norm of the GTTN and give a generalization bound. Extensive experiments on real-world datasets demonstrate the effectiveness of the proposed GTTN. Existing Tensor Trace Norms =========================== In multi-task learning, trace norms are widely used as the regularization to learn a low-rank structure among model parameters of all the tasks as minimizing the trace norm will enforce some singular values to approach zero. When both a data point and model parameters of a task are represented in vectorized forms in regression tasks or binary classification tasks, the matrix trace norm can be used and it is defined as $\|\mathbf{W}\|_*=\sum_{i}\sigma_i(\mathbf{W})$ with each column of the parameter matrix $\mathbf{W}$ storing the parameter vector of the corresponding task and $\sigma_i(\mathbf{W})$ denoting the $i$th largest singular value of $\mathbf{W}$. Regularizing $\mathbf{W}$ with $\|\mathbf{W}\|_*$ will make $\mathbf{W}$ tend to be low-rank, which leads to the linear dependency among parameter vectors of different tasks and reflects the relatedness among tasks in terms of model parameters. Nowadays, the data such as images can be represented in a matrix or tensor form in the raw representation (e.g., pixel-based representation) and transformed representation after for example convolutional operations. Moreover, each task becomes more complex, for example, a multi-class classification task. In those cases, parameters of all the tasks can be organized as a $p$-way tensor ($p\ge 3$), e.g., $\mathcal{W}\in\mathbb{R}^{d_1\times\ldots\times d_p}$. That is, for multi-class classification tasks, when $p$ equals 3, $d_1$ denotes the number of hidden units in the last hidden layer, $d_2$ can represent the number of classes, and $d_3$ can be the number of tasks. In such cases, the matrix trace norm is no longer applicable and instead tensor trace norms are investigated. According to [@ts13], tensor trace norms can be classified into two categories, including overlapped tensor trace norms and latent tensor trace norms. An overlapped tensor trace norm transforms a tensor into matrices in different ways and compute the sum of the matrix trace norm of different transformed matrices. A latent tensor trace norm decomposes the tensor into multiple latent tensors and then compute the sum of the matrix trace norm of matrices which are transformed from the latent tensors. Deep multi-task learning mainly uses the overlapped tensor trace norm, which is the focus of our study. As reviewed in [@yh17], three tensor trace norms belonging to the overlapped tensor trace norm are used in deep multi-task learning, including the Tucker trace norm, TT trace norm, and Last Axis Flattening (LAF) trace norm. In the following, we will review those three tensor trace norms. Tucker Trace Norm ----------------- Based on the Tucker decomposition [@tucker66], the Tucker trace norm for a tensor $\mathcal{W}\in\mathbb{R}^{d_1\times\ldots\times d_p}$ can be defined as $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*=\sum_{i=1}^p\alpha_i\|\mathcal{W}_{(i)}\|_*,\label{def_Tucker_trace_norm}$$ where $[p]$ denotes a set of positive integers no larger than $p$, $\mathrm{permute}(\mathcal{W},\mathbf{s})$ permutes the tensor $\mathcal{W}$ along axis indices in $\mathbf{s}$ that is a permutation of $[p]$, $\mathrm{reshape}(\mathcal{W},\mathbf{a})$ reshapes the tensor $\mathcal{W}$ with the new size stored in a vector $\mathbf{a}$, $\mathcal{W}_{(i)}:=\mathrm{reshape}(\mathrm{permute}(\mathcal{W},[i,1,\ldots,i-1,i+1,\ldots,p]),[d_i,\prod_{j\ne i}d_j])$ is the mode-$i$ tensor flattening to transform $\mathcal{W}$ to a matrix along the $i$th axis, and $\alpha_i$ denotes the weight for the model-$i$ flattening. To control the scale of $\{\alpha_i\}$, here $\{\alpha_i\}$ are required to satisfy that $\alpha_i\ge 0$ and $\sum_{i=1}^p\alpha_i=1$. Based on Eq. (\[def\_Tucker\_trace\_norm\]), we can see that the Tucker trace norm is a convex combination of matrix trace norms of tensor flattening along each axis, where $\alpha_i$ controls the importance of the mode-$i$ tensor flattening. Without a priori information, different tensor flattenings are usually assumed to have equal importance by setting $\alpha_i$ to be $\frac{1}{p}$. Besides being used in deep multi-task learning, the Tucker trace norm has been adopted in multilinear multi-task learning [@rabp13; @wst14] which assumes the existence of multi-modal structures contained in multi-task learning problems. TT Trace Norm ------------- Based on the tensor-train decomposition [@oseledets11], the TT trace norm for a tensor $\mathcal{W}\in\mathbb{R}^{d_1\times\ldots\times d_p}$ can be defined as $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*=\sum_{i=1}^{p-1}\alpha_i\|\mathcal{W}_{[i]}\|_*,\label{def_TT_trace_norm}$$ where $\mathcal{W}_{[i]}=\mathrm{reshape}(\mathcal{W},[\prod_{j=1}^id_j,\prod_{j=i+1}^pd_j])$ and $\alpha_i$ denotes a nonnegative weight. Different from the mode-$i$ tensor flattening $\mathcal{W}_{(i)}$, $\mathcal{W}_{[i]}$ unfolds the tensor to a matrix along the first $i$ axes. Similar to the Tucker trace norm, $\{\alpha_i\}$ are assumed to satisfy that $\alpha_i\ge 0$ and $\sum_{i=1}^{p-1}\alpha_i=1$. Usually $\alpha_i$ is set by users to be $\frac{1}{p-1}$ if there is no additional information for the importance of each term in Eq. (\[def\_TT\_trace\_norm\]). LAF Trace Norm -------------- The LAF trace norm for a tensor $\mathcal{W}\in\mathbb{R}^{d_1\times\ldots\times d_p}$ can be defined as $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*=\|\mathcal{W}_{(p)}\|_*.\label{def_LAF_trace_norm}$$ The last axis in $\mathcal{W}$ is the task axis and hence the LAF trace norm is equivalent to place the matrix trace norm on $\mathcal{W}_{(p)}$ each of whose rows stores model parameters of each task. Compared with the Tucker trace norm in Eq. (\[def\_Tucker\_trace\_norm\]), the LAF trace norm can be viewed as a special case of the Tucker trace norm where $\alpha_p$ equals 1 and other $\alpha_i$’s ($i\ne p$) are equal to 0. Given a tensor trace norm, the objective function of a deep multi-task model can be formulated as[^1] $$\min_{\bm{\Theta}}\sum_{i=1}^m\frac{1}{n_i}\sum_{j=1}^{n_i}l(f_i(\mathbf{x}^i_j;\bm{\Theta}),y^i_j)+\lambda{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*, \label{obj_DMTL}$$ where $m$ denotes the number of tasks, $n_i$ denotes the number of training data points in the $i$th task, $\mathbf{x}^i_j$ denotes the $j$th data point in the $i$th task, $y^i_j$ denotes the label of $\mathbf{x}^i_j$, $f_i(\cdot;\bm{\Theta})$ denotes a learning function for the $i$th task given a deep multi-task neural network parameterized by $\bm{\Theta}$, $l(\cdot,\cdot)$ denotes a loss function such as the cross-entropy loss for classification tasks and the square loss for regression tasks, $\mathcal{W}$ denotes a part of $\bm{\Theta}$ that is regularized by a tensor trace norm, and $\lambda$ is a regularization parameter. In problem (\[obj\_DMTL\]), the tensor trace norm can be the Tucker trace norm, or the TT trace norm, or the LAF trace norm. Generalized Tensor Trace Norm ============================= In this section, we first analyze existing tensor trace norms and then present the proposed generalized tensor trace norm as well as the optimization and generalization bound. Analysis on Existing Tensor Trace Norms --------------------------------------- As introduced in the previous section, we can see that overlapped tensor trace norms rely on different ways of tensor flattening. For example, the Tucker trace norm reshapes the tensor along each axis and the LAF trace norm focuses on the last axis, while the TT trace norm reshapes the tensor by combining the first several axes. Given the physical meaning of each axis, the LAF trace norm only considers the inter-task low-rank structure among tasks, but differently both the Tucker and TT trace norms consider not only the inter-task low-rank structure among tasks but also the intra-task low-rank structure among, for example, features. In this sense, the Tucker and TT trace norms seems to be superior to the LAF trace norm. For overlapped tensor trace norms like the Tucker and TT trace norms, there are two important issues. 1) How to choose the ways of tensor flattening? 2) How to determine the importance of of different ways of tensor flattening? For the first issue, the Tucker trace norm chooses to reshape along each axis while the TT trace norm combines the first several axes together to do the tensor flattening. Different ways of tensor flattening encode the belief on the existence of the low-rank structure in $\mathcal{W}$. So the Tucker trace norm assumes that the low-rank structure exists in each axis while the TT trace norm considers the combinations of the first several axes have low-rank structure. However, those models may fail when such assumptions do not hold. For the second issue, current models usually assume the equal importance of different ways of tensor flattening, which is reflected in the equal value of $\{\alpha_i\}$. Intuitively, different ways of tensor flattening should have different degrees in terms of the low-rank structure and hence $\{\alpha_i\}$ should be different from each other. In this sense, $\{\alpha_i\}$ with an equal value incur the suboptimal performance. GTTN ---- To solve the above two issues together, we propose the generalized tensor trace norm. For the first issue, since for most problems we do not know which ways of tensor flattening are helpful to learn the low-rank structure, we can try all possible ways of tensor flattening. To mathematically define this, we define $\mathcal{W}_{\{\mathbf{s}\}}$ as $$\mathcal{W}_{\{\mathbf{s}\}}=\mathrm{reshape}\Big(\mathrm{permute}(\mathcal{W},[\mathbf{s},\neg\mathbf{s}]), \Big[\prod_{i\in\mathbf{s}}d_i,\prod_{j\in\neg\mathbf{s}}d_j\Big]\Big),$$ where $\mathbf{s}$ is a nonempty subset of $[p]$ (i.e., $\mathbf{s}\subset[p]$) and $\neg\mathbf{s}$ denotes the complement of $\mathbf{s}$ with respect to $[p]$ (i.e., $\neg\mathbf{s}=[p]-\mathbf{s}$). So $\mathcal{W}_{\{\mathbf{s}\}}$ is a tensor flattening to a matrix with a dimension corresponding to axis indices in $\mathbf{s}$ and the other to axis indices in $\neg\mathbf{s}$. When $\mathbf{s}=\{i\}$ contains only one element, $\mathcal{W}_{\{\mathbf{s}\}}$ becomes $\mathcal{W}_{(i)}$, the mode-$i$ tensor flattening used in the Tucker trace norm. When $\mathbf{s}=[i]$, $\mathcal{W}_{\{\mathbf{s}\}}$ becomes $\mathcal{W}_{[i]}$ that is used in the TT trace norm. Moreover, this new tensor flattening can be viewed as a generalization of $\mathcal{W}_{(i)}$ and $\mathcal{W}_{[i]}$ as $\mathbf{s}$ can contain more than one element, which is more general than $\mathcal{W}_{(i)}$, and it does not require that elements in $\mathbf{s}$ should be successive integers from 1, which is more general than $\mathcal{W}_{[i]}$. As we aim to consider all possible ways of tensor flattening, similar to the Tucker and TT trace norms, we define the GTTN as $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*=\sum_{\mathbf{s}\subset[p],\mathbf{s}\ne\emptyset}\alpha_{\mathbf{s}}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*,\label{GTTN}$$ where $\mathbf{s}$ is also used as a subscript to index the corresponding weight for $\|\mathcal{W}_{\{\mathbf{s}\}}\|_*$, $\bm{\alpha}$ denotes the set of $\alpha_{\mathbf{s}}$’s, $\mathcal{C}_{\bm{\alpha}}=\{\bm{\alpha}|\alpha_{\mathbf{s}}\ge 0$ and $\sum_{\mathbf{s}\subset[p]}\alpha_{\mathbf{s}}=1\}$ defines a constraint set for $\bm{\alpha}$. Then based on the GTTN, we can solve the first issue to some extent as it can discover all the low-rank structures by considering all possible ways of tensor flattening with appropriate settings of $\bm{\alpha}$. In Figure \[fig\_GTTN\], we show the difference among the Tucker trace norm, TT trace norm, LAF trace norm and GTTN for a four-way tensor at the top. In the bottom of Figure \[fig\_GTTN\], we can see that there are seven possible tensor flattenings. The Tucker trace norm uses $\mathcal{W}_{\{1\}}$, $\mathcal{W}_{\{2\}}$, $\mathcal{W}_{\{3\}}$, and $\mathcal{W}_{\{4\}}$. The TT trace norm relies on $\mathcal{W}_{\{1\}}$, $\mathcal{W}_{\{1,2\}}$, and $\mathcal{W}_{\{1,2,3\}}$. The LAF trace norm only contains $\mathcal{W}_{\{4\}}$. The calculation of the GTTN is based on all the seven tensor flattenings. From this example, we can see that the union of tensor flattenings used in the Tucker, TT, and LAF trace norms cannot cover all the possible ones and the GTTN utilizes some additional tensor flattening (e.g., $\mathcal{W}_{\{1,3\}}$ and $\mathcal{W}_{\{1,4\}}$). In this sense, the GTTN can discover more low-rank structures than existing tensor trace norms. -0.05in ![Comparison among the Tucker trace norm, TT trace norm, LAF trace norm, and GTTN. At the top, there is a four-way tensor where each cube is a slice along the last axis. The 7 matrices denotes all the possible tensor flattenings. If a tenor flattening is used by the Tucker trace norm, it will have an orange rectangle. If a tenor flattening is used by the TT trace norm, it will have a red rectangle. If a tenor flattening is used by the LAF trace norm, it will have a blue rectangle. If a tenor flattening is used by the GTTN, it will have a purple rectangle.[]{data-label="fig_GTTN"}](flatenning.pdf){width="1\linewidth"} -0.2in -0.15in For the number of distinct summands in the right-hand side of Eq. (\[GTTN\]), we have the following theorem.[^2] \[theorem\_num\_summands\] The right-hand side of Eq. (\[GTTN\]) has $2^{p-1}-1$ distinct summands. As shown in the proof of Theorem \[theorem\_num\_summands\], $\mathcal{W}_{\{\mathbf{s}\}}$ and $\mathcal{W}_{\{\neg\mathbf{s}\}}$ are transpose matrices to each other with equal matrix trace norm and we can eliminate one of them to reduce the computational cost. For notational simplicity, we do not explicitly do the elimination in the formulation but in computation, we did do that. In problems we encounter, $p$ is at most $5$ and so the GTTN has at most 15 distinct summands. So the number of distinct summands are not so large, making the optimization efficient. Similar to the Tucker and TT trace norms, GTTN defined in Eq. (\[GTTN\]) still faces the second issue. Here to solve the second issue, we view $\bm{\alpha}$ as variables to be optimized and based on Eq. (\[GTTN\]), the objective function of a deep multi-task model based on GTTN is formulated as $$\min_{\bm{\Theta},\bm{\alpha}\in\mathcal{C}_{\bm{\alpha}}}\sum_{i=1}^m\frac{1}{n_i}\sum_{j=1}^{n_i}l(f_i(\mathbf{x}^i_j;\bm{\Theta}),y^i_j) +\lambda{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*. \label{obj_GTTN}$$ Compared with problem (\[obj\_DMTL\]), we can see two differences. Firstly, the regularization terms in two problems are different. Secondly, problem (\[obj\_GTTN\]) treat $\bm{\alpha}$ as variables to be optimized but the corresponding entities are constants which are set by users. In the following theorem, we can simplify problem (\[obj\_GTTN\]) by eliminating $\bm{\alpha}$. \[theorem\_obj\_GTTN\_reformulation\] Problem (\[obj\_GTTN\]) is equivalent to $$\min_{\bm{\Theta}}\sum_{i=1}^m\frac{1}{n_i}\sum_{j=1}^{n_i}l(f_i(\mathbf{x}^i_j;\bm{\Theta}),y^i_j) +\lambda\min_{{\mathbf{s}\subset[p]\atop\mathbf{s}\ne\emptyset}}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*,\label{obj_GTTN_2}$$ According to problem (\[obj\_GTTN\_2\]), learning $\bm{\alpha}$ will tend to choosing a tensor flattening with the minimal matrix trace norm. Optimization ------------ Even though problem (\[obj\_GTTN\_2\]) is equivalent to problem (\[obj\_GTTN\]), in numerical optimization, we choose problem (\[obj\_GTTN\]) as the objective function to be optimized. One reason is that problem (\[obj\_GTTN\_2\]), which involves the minimum of matrix trace norms, is more complicated than problem (\[obj\_GTTN\]) to be optimized. Another reason is that the learned $\bm{\alpha}$ in problem (\[obj\_GTTN\]) can visualize the importance of each tensor flattening, which can improve the interpretability of the learning model. Since problem (\[obj\_GTTN\]) is designed for deep neural networks, the Stochastic Gradient Descent (SGD) technique is the first choice for optimization. However, problem (\[obj\_GTTN\]) is a constrained optimization problem, making SGD techniques not directly applicable. The constraints in problem (\[obj\_GTTN\]) constrain $\bm{\alpha}$ to form a $(p-1)$-dimensional simplex. To convert problem (\[obj\_GTTN\]) to an unconstrained problem that can be optimized by SGD, we reparameterize each $\alpha_{\mathbf{s}}$ as $$\alpha_{\mathbf{s}}=\frac{\exp\{\beta_{\mathbf{s}}\}}{\sum_{\mathbf{t}\subset[p],\mathbf{t}\ne\emptyset}\exp\{\beta_{\mathbf{t}}\}}.$$ With such reparameterization, problem (\[obj\_GTTN\]) can be reformulated as [$$\min_{\bm{\Theta},\bm{\beta}}\sum_{i=1}^m\frac{1}{n_i}\sum_{j=1}^{n_i}l(f_i(\mathbf{x}^i_j;\bm{\Theta}),y^i_j) +\frac{\lambda\sum_{\mathbf{s}\subset[p]\atop\mathbf{s}\ne\emptyset}\exp\{\beta_{\mathbf{s}}\}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*} {\sum_{\mathbf{t}\subset[p],\mathbf{t}\ne\emptyset}\exp\{\beta_{\mathbf{t}}\}}.\label{obj_GTTN_3}$$ ]{}For each parameter $\theta\in\bm{\Theta}-\mathcal{W}$, its gradient can be computed based on the first term in the objective function of problem (\[obj\_GTTN\_3\]). For each $\beta_{\mathbf{s}}$, its gradient can be computed as $$\begin{aligned} \frac{\partial h}{\partial \beta_{\mathbf{s}}}&=& -\frac{\lambda\exp\{\beta_{\mathbf{s}}\}\sum_{\mathbf{t}\subset[p]\atop\mathbf{t}\ne\emptyset} \exp\{\beta_{\mathbf{t}}\}\|\mathcal{W}_{\{\mathbf{t}\}}\|_*} {\left(\sum_{\mathbf{t}\subset[p],\mathbf{t}\ne\emptyset}\exp\{\beta_{\mathbf{t}}\}\right)^2}\\ &&+\frac{\lambda\exp\{\beta_{\mathbf{s}}\}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*} {\sum_{\mathbf{t}\subset[p],\mathbf{t}\ne\emptyset}\exp\{\beta_{\mathbf{t}}\}}.\end{aligned}$$ For $\mathcal{W}$, the computation of its gradient comes from both terms in the objective function of problem (\[obj\_GTTN\_3\]). The first term is the conventional training loss and the second term involves the matrix trace norm which is non-differentiable. According to [@watson92], we can compute the subgradient instead, that is, $\frac{\partial \|\mathbf{X}\|_*}{\partial \mathbf{X}}=\mathbf{U}\mathbf{V}^T$ where $\mathbf{X}=\mathbf{U}\bm{\Sigma}\mathbf{V}^T$ denotes the singular value decomposition of a matrix $\mathbf{X}$. Generalization Bound -------------------- For the GTTN defined in Eq. (\[GTTN\]), we can derive its dual norm in the following theorem. \[theorem\_GTTN\_dual\_norm\] The dual norm of the GTTN defined in Eq. (\[GTTN\]) is defined as $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^{\star}}= \min_{\sum_{\mathbf{s}\ne\emptyset\atop\mathbf{s}\subset[p]}\alpha_{\mathbf{s}}\mathcal{Y}^{(\mathbf{s})}=\mathcal{W}} \max_{\mathbf{s}\ne\emptyset\atop\mathbf{s}\subset[p]}\|\mathcal{Y}^{(\mathbf{s})}_{\{\mathbf{s}\}}\|_{\infty},$$ where $\mathcal{Y}^{(\mathbf{s})}$ is a variable indexed by $\mathbf{s}$ and $\|\cdot\|_{\infty}$ denotes the spectral norm of a matrix that is equal to the maximum singular value. Without loss of generality, here we assume $\bm{\Theta}=\mathcal{W}$ which can simplify the analysis. We rewrite problem (\[obj\_GTTN\]) into an equivalent formulation as $$\min_{\mathcal{W}}\sum_{i=1}^m\frac{1}{n_i}\sum_{j=1}^{n_i}l(f_i(\mathbf{x}^i_j;\mathcal{W}),y^i_j)\ \mathrm{s.t.}\ {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*\le\gamma,\label{obj_GTTN_4}$$ where $\bm{\alpha}$ is assumed to be fixed to show its impact to the bound. Here each data point is a tensor and binary classification tasks are considered,[^3] implying that $\mathcal{W}\in\mathbb{R}^{d_1\times\ldots\times d_{p-1}\times m}$ and $\mathbf{x}^i_j\in\mathbb{R}^{d_1\times\ldots\times d_{p-1}}$. The learning function for each task is a linear function defined as $f_i(\mathbf{x};\mathcal{W})=\langle\mathcal{W}_i,\mathbf{x}\rangle$, where $\langle\cdot,\cdot\rangle$ denotes the inner product between two tensors with equal size and $\mathcal{W}_i$ denotes the $i$th slice along the last axis which is the task axis. For simplicity, different tasks are assumed to have the same number of data points, i.e., $n_i$ equals $n_0$ for $i=1,\ldots,m$. It is very easy to extend our analysis to general settings. The generalization loss for all the tasks is defined as $L(\mathcal{W})=\frac{1}{m}\sum_{i=1}^m\mathbb{E}_{(\mathbf{x},y)\sim\mathcal{D}_i}[l(f_i(\mathbf{x};\mathcal{W}),y)]$, where $\mathcal{D}_i$ denotes the underlying data distribution for the $i$th task and $\mathbb{E}[\cdot]$ denotes the expectation. The empirical loss for all the tasks is defined as $\hat{L}(\mathcal{W})=\frac{1}{m}\sum_{i=1}^m\frac{1}{n_i}\sum_{j=1}^{n_i}l(f_i(\mathbf{x}^i_j;\mathcal{W}),y^i_j)$. We assume the loss function $l(\cdot,\cdot)$ has values in $[0,1]$ and it is Lipschitz with respect to the first input argument with a Lipschitz constant $\rho$. Each training data $\mathbf{x}^i_j$ is assumed to satisfy $\langle\mathbf{x}^i_j,\mathbf{x}^i_j\rangle\le 1$. To characterize correlations between features, we assume that $\mathbf{C}_{\mathbf{s}}=\mathbb{E}[(\mathbf{x}^i_j)_{\{\mathbf{s}\}}(\mathbf{x}^i_j)_{\{\mathbf{s}\}}^T] \preceq\frac{\kappa}{d}\mathbf{I}$ for any $\mathbf{s}\ne\emptyset$ and $\mathbf{s}\subset[p-1]$, where $\mathbf{A}\preceq\mathbf{B}$ means that $\mathbf{B}-\mathbf{A}$ is a positive semidefinite matrix, $d=\prod_{i\in[p-1]}d_i$, and $\mathbf{I}$ denotes an identity matrix with an appropriate size. For problem (\[obj\_GTTN\_4\]), we can derive a generalization bound in the following theorem. \[theorem\_generalization\_bound\] For the solution $\hat{\mathcal{W}}$ of problem (\[obj\_GTTN\_4\]) and $\delta>0$, with probability at least $1-\delta$, we have $$\begin{aligned} L(\hat{\mathcal{W}})\le& \hat{L}(\hat{\mathcal{W}})+\frac{2\rho\gamma C}{mn_0}\min_{\mathbf{s}\ne\emptyset\atop\mathbf{s}\subset[p]}\left(\frac{\kappa m\sqrt{\ln d_{\mathbf{s}}}}{\alpha_{\mathbf{s}}n_0d}+\frac{\ln d_{\mathbf{s}}}{\alpha_{\mathbf{s}}n_0}\right)\\ &+\sqrt{\frac{2}{m}\ln\frac{1}{\delta}}.\end{aligned}$$ According to Theorem \[theorem\_generalization\_bound\], we can see that each $\alpha_{\mathbf{s}}$ can be used to weigh the second term which is related to the model complexity. Experiments =========== In this section, we conduct empirical studies for the proposed GTTN. Experimental Settings --------------------- ### Datasets **ImageCLEF dataset**. This dataset contains 12 common categories shared by 4 tasks: Caltech-256, ImageNet ILSVRC 2012, Pascal VOC 2012, and Bing. Totally, there are about 2,400 images in all the tasks. **Office-Caltech dataset**. This dataset consists of 4 tasks and 2,533 images in total. One task consists of data from 10 common categories shared in the Caltech-256 dataset, and the other three tasks consist of data from the Office dataset whose images are collected from 3 distinct domains/tasks, e.g., Amazon, Webcam and DSLR. **Office-31 dataset**. This dataset contains 31 categories from Amazon, webcam, and DSLR. Totally, there are 4,110 images in all the tasks. **Office-Home dataset**. This dataset contains images from 4 domains/tasks, which are artistic images, clip art, product images, and real-world images. Each task contains images from 65 object categories collected in the office and home settings. There are about 15,500 images in all the tasks. ### Baselines We compare the GTTN method with various competitors, including the deep multi-task learning (DMTL) method where different tasks share the first several layers as the common feature representation, the Tucker trace norm method (denoted by Tucker), the TT trace norm method (denoted by TT), LAF trace norm method (denoted by LAF), LAF Tensor Factorisation method (denoted by LAF-TF) [@yh17a]. ### Implementation details We employ the Vgg19 network [@sz15] to extract features for image data by using the output of the pool5 layer and fc7 layer, respectively, for all the models in comparison. After that, if the pool5 layer is used, the feature representation extracted is a 3-way $7 \times 7 \times 512 $ tensor and all the multi-task learning models adopt a five-layer architecture where the three hidden layers are used to transform along each mode of the input with the ReLU activation function and they have 6, 6, 256 hidden units, respectively. Otherwise, if the fc7 layer is used, all the multi-task learning models adopt a two-layer fully-connected architecture with the ReLU activation function and 1024 hidden units, where the first layer is shared by all the tasks. The architecture used is illustrated in Figure \[fig\_architecture\]. To see the effect of training size on the performance, we vary the training proportion from 50% to 70% at an interval of 10%. The performance measure is the classification accuracy. Each experimental setting will repeat 5 times and we report the average performance as well as the standard deviation. For all the baseline methods, we follow their original model selection procedures. The regularization parameter $\lambda$ that controls the trade-off between the training cross-entropy loss and the regularization term is set by 0.25 and 0.65, respectively, for all the 6 methods to test the sensitivity of the performance with respect to to $\lambda$. In addition, we use Adam with the learning rate varying as $\eta = \frac{0.02}{1 + p}$, where $p$ is the number of the iteration and we adopt mini-batch SGD with $\text{batch\_size} = 16$. Experimental Results -------------------- -0.1in ![The architecture used by all the multi-task learning models in comparison for experiments.[]{data-label="fig_architecture"}](model.pdf){width="\linewidth"} -0.25in -0.1in -0.03in ![Performance on the ImageCLEF dataset with $\lambda=0.25$.[]{data-label="image-CLEF-25"}](image-CLEF-lambda-25.pdf){width="1\linewidth"} -0.2in -0.05in ![Performance on the ImageCLEF dataset with $\lambda=0.65$.[]{data-label="image-CLEF-65"}](image-CLEF-lambda-65.pdf){width="1\linewidth"} -0.2in -0.05in ![Performance on the Office-Caltech10 dataset with $\lambda=0.25$.[]{data-label="Office-caltech10-25"}](office-caltech10-lambda-25.pdf){width="1\linewidth"} -0.2in -0.05in ![Performance on the Office-Caltech10 dataset with $\lambda=0.65$.[]{data-label="Office-caltech10-65"}](Office-caltech10-lambda-65.pdf){width="1\linewidth"} -0.2in -0.05in ![Office-31 ($\lambda=0.25$)[]{data-label="Office-31-25"}](Office-31-lambda-25.pdf){width="1\linewidth"} -0.2in -0.1in ![Performance on the Office-31 dataset with $\lambda=0.65$.[]{data-label="Office-31-65"}](Office-31-lambda-65.pdf){width="1\linewidth"} -0.2in -0.05in ![Performance on the Office-Home dataset with $\lambda=0.25$.[]{data-label="Office-Home-25"}](OfficeHomeDataset-lambda-25.pdf){width="1\linewidth"} -0.2in -0.05in ![Performance on the Office-Home dataset with $\lambda=0.65$.[]{data-label="Office-Home-65"}](OfficeHomeDataset_lambda-65.pdf){width="1\linewidth"} -0.2in The experimental results are reported in Figures \[image-CLEF-25\]-\[Office-Home-65\] based on different feature extractors (i.e., pool5 or fc7) and different regularization parameters (i.e., 0.25 or 0.65). Since the output of the fc7 layer is in a vectorized representation, the model parameter $\mathcal{W}$ is a 3-way tensor. In this case, we can see that the Tucker trace norm possesses three tensor flattenings, the TT trace norm utilizes two tensor flattenings, and the GTTN also has three tensor flattenings. So in this case, both the GTTN and Tucker trace norm utilize all the possible tensor flattenings with the only difference that the GTTN learns the combination coefficients $\bm{\alpha}$ but the Tucker trace norm manually sets them to be identical. According to the results, we can see the GTTN outperforms the Tucker trace norm in most cases, which verifies that learning $\bm{\alpha}$ is better than fixing it. When using the pool5 layer as the feature extractor, the feature representation is in a 3-way tensor, making the parameter $\mathcal{W}$ a 5-way tensor. In this case, we can see that the GTTN method performs significantly better than other baseline methods. This is mainly because the GTTN utilizes more tensor flattenings than other baseline models and hence it may discover more low-rank structures. Analysis on Learned $\bm{\alpha}$ --------------------------------- Tables \[pool5-25\] and \[pool5-65\] show the learned $\bm{\alpha}$ of GTTN based on the pool5 layer when $\lambda$ takes the value of 0.25 and 0.65, respectively. In this case, the parameter $\mathcal{W}$ is a 5-way tensor and hence the GTTN contains 15 different flattenings, including $\mathcal{W}_{\{1\}}$, $\mathcal{W}_{\{2\}}$, $\mathcal{W}_{\{3\}}$, $\mathcal{W}_{\{4\}}$, $\mathcal{W}_{\{5\}}$, $\mathcal{W}_{\{1,2\}}$, $\mathcal{W}_{\{1,3\}}$, $\mathcal{W}_{\{1,4\}}$, $\mathcal{W}_{\{2,3\}}$, $\mathcal{W}_{\{2,4\}}$, $\mathcal{W}_{\{3,4\}}$, $\mathcal{W}_{\{1,2,3\}}$, $\mathcal{W}_{\{1,2,4\}}$, $\mathcal{W}_{\{1,3,4\}}$, and $\mathcal{W}_{\{2,3,4\}}$, which correspond to each component of $\bm{\alpha}$ in Tables \[pool5-25\] and \[pool5-65\]. According to the results, we can see that different tensor flattenings have varying weights. Similarly, Tables \[fc7-25\] and \[fc7-65\] show the learned $\bm{\alpha}$ of GTTN based on the fc7 layer when $\lambda$= 0.25 and $\lambda$= 0.65, respectively. In this case, the parameter $\mathcal{W}$ is a 3-way tensor, which contains 3 different flattenings by GTTN method, i.e., $\mathcal{W}_{\{1\}}$, $\mathcal{W}_{\{2\}}$ $\mathcal{W}_{\{1,2\}}$. We can notice that the weight of ${W}_{\{1,2\}}$ is among the maximum in most settings, which may imply that the combination of the first two axes is very important. ----------------------------- -------------------------------------- ------------------------------------- ------------------------------------- \*[**ImageCLEF** ]{} 0.0736, **0.0799** , 0.0789, 0.0548, 0.0674, 0.0724, 0.0688, 0.0620, 0.0757, 0.0668, 0.0699, 0.0610,   0.0724, 0.0780, 0.0592, 0.0741, 0.0691, 0.0799, 0.0630, **0.0823**, 0.0683, **0.0819**, 0.0608, 0.0718,   0.0529, 0.0526, 0.0470, 0.0613, 0.0603, 0.0541, 0.0531, 0.0661, 0.0629, 0.0502, 0.0542, 0.0792,   0.0699, 0.0745, 0.0709 0.0727, 0.0657, 0.0632 0.0610, 0.0678, 0.0686 \*[**Office-Caltech10** ]{} 0.0627, 0.0739, 0.0709, 0.0604, 0.0722, 0.0676, 0.0783, 0.0482, 0.0697, 0.0762, 0.0883, 0.0497,   0.0707, 0.0667, 0.0564, 0.0705, 0.0690, 0.0725, 0.0597, 0.0705, **0.0901**, 0.0837, 0.0536, 0.0685,   0.0610, 0.0564, 0.0476, **0.0876** , 0.0583, 0.0503, 0.0584, 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------------------------------------- ------------------------------------- \[pool5-25\] -0.1in ----------------------------- ------------------------------------- -------------------------------------- ------------------------------------- \*[**ImageCLEF** ]{} 0.0672, 0.0666, 0.0695, 0.0523, 0.0688, 0.0739, **0.0808**, 0.0602, **0.0821**, 0.0795, 0.0705, 0.0549,   0.0712, 0.0690, 0.0670, 0.0791, 0.0687, 0.0680, 0.0563, 0.0726, 0.0741, 0.0787, 0.0528, 0.0682,   0.0563, 0.0675, 0.0521, 0.0664, 0.0515, 0.0507, 0.0590, 0.0754, 0.0595, 0.0494, 0.0463, 0.0579,   **0.0809**, 0.0713, 0.0637 0.0678, 0.0763, 0.0698 0.0704, 0.0743, 0.0814 \*[**Office-Caltech10** ]{} 0.0662, 0.0746, 0.0760, 0.0545, 0.0681, 0.0648, **0.0863**, 0.0500, 0.0665, 0.0730, 0.0682, 0.0613,   0.0596, 0.0737, 0.0566, **0.0792**, 0.0711, 0.0731, 0.0495, 0.0667, 0.0749, **0.0866**, 0.0453, 0.0857,   0.0600, 0.0618, 0.0564, 0.0646, 0.0518, 0.0528, 0.0604, 0.0722, 0.0566, 0.0492, 0.0505, 0.0750,   0.0740, 0.0715, 0.0713 0.0721, 0.0768, 0.0841 0.0612, 0.0686, 0.0773 \*[ **Office-31** ]{} 0.0874, 0.0772, **0.0910**, 0.0562, **0.0833**, 0.0806, 0.0811, 0.0571, 0.0680, 0.0736, 0.0788, 0.0574,   0.0684, 0.0806, 0.0509, 0.0726, 0.0767, 0.0694, 0.0602, 0.0617, 0.0720, 0.0700, 0.0547, 0.0732,   0.0518, 0.0514, 0.0621, 0.0539, 0.0651, 0.0575, 0.0686, 0.0553, 0.0535, 0.0548, 0.0622, 0.0663,   0.0557, 0.0642, 0.0767 0.0541, 0.0700, 0.0593 0.0588, 0.0763, **0.0804** \*[**Office-Home** ]{} 0.0687, 0.0672, 0.0780, 0.0619, 0.0673, 0.0810, 0.0668, 0.0497, **0.0907**, 0.0834, 0.0835, 0.0492,   0.0731, 0.0786, 0.0480, **0.0798**, 0.0820, 0.0791, 0.0492, **0.0892** , 0.0773, 0.0819, 0.0466, 0.0852,   0.0523, 0.0572, 0.0749, 0.0633, 0.0589, 0.0517, 0.0819, 0.056, 0.0515, 0.0432, 0.0751, 0.0522,   0.0591, 0.0651, 0.0730 0.0524, 0.0484, 0.0865 0.0523, 0.0523, 0.0755 ----------------------------- ------------------------------------- -------------------------------------- ------------------------------------- \[pool5-65\] -0.1in -- ---------------------------- ---------------------------- ---------------------------- -- 0.3861, 0.2246, **0.3893** 0.3825, 0.2336, **0.3839** 0.3718, 0.2154, **0.4128** **0.3911**, 0.2246, 0.3843 **0.3953**, 0.2152, 0.3895 **0.3984**, 0.2302,0.3714 0.3186, 0.2507, **0.4307** 0.3041, 0.2787, **0.4170** 0.2662, 0.2864, **0.4474** 0.3162, 0.2750, **0.4088** 0.2901, 0.2724, **0.4374** 0.3057, 0.2630, **0.4313** -- ---------------------------- ---------------------------- ---------------------------- -- \[fc7-25\] -0.1in -- ---------------------------- ---------------------------- ---------------------------- -- 0.2992, 0.2834, **0.4173** 0.3216, 0.2753, **0.4029** 0.3229, 0.2908, **0.3861** **0.4052**, 0.2244, 0.3704 0.3759, 0.2462, **0.3779** 0.3871, 0.2106, **0.4023** 0.3609, 0.2184, **0.4207** **0.3926**, 0.2415, 0.3658 0.3279, 0.2399, **0.4322** 0.2789, 0.2944, **0.4267** 0.3113, 0.2618, **0.4269** 0.2746, 0.2672, **0.4582** -- ---------------------------- ---------------------------- ---------------------------- -- \[fc7-65\] Conclusion ========== In this paper, we devise a generalized tensor trace norm to capture all the low-rank structures in a parameter tensor used in deep multi-task learning and identify the importance of each structure. We analyze properties of the proposed GTTN, including its dual norm and generalization bound. Empirical studies show that it outperforms state-of-the-art counterparts and the learned combination coefficients can give us more understanding of the problem studied. As a future work, we are interested in extending the idea of GTTN to study tensor Schatten norms. Appendix {#appendix .unnumbered} ======== Proof for Theorem \[theorem\_num\_summands\] {#proof-for-theorem-theorem_num_summands .unnumbered} -------------------------------------------- [**Proof**]{}. For a valid $\|\mathcal{W}_{\{\mathbf{s}\}}\|_*$, it is required that $\mathbf{s}$ and $\neg\mathbf{s}$ should not be empty, implying that $\mathbf{s}\ne \emptyset$ and $\mathbf{s}\ne [p]$. So the total number of valid summands in the right-hand side of Eq. (\[GTTN\]) is $2^p-2$. Based on the definition of $\mathcal{W}_{\{\mathbf{s}\}}$, we can see that $\mathcal{W}_{\{\mathbf{s}\}}$ is equal to the transpose $\mathcal{W}_{\{\neg\mathbf{s}\}}$, making $\|\mathcal{W}_{\{\mathbf{s}\}}\|_*=\|\mathcal{W}_{\{\neg\mathbf{s}\}}\|_*$. So for $\|\mathcal{W}_{\{\mathbf{s}\}}\|_*$, there will always be an equivalent $\|\mathcal{W}_{\{\neg\mathbf{s}\}}\|_*$, leading to $2^{p-1}-1$ distinct summands in the right-hand side of Eq. (\[GTTN\]). $\Box$ Proof for Theorem \[theorem\_obj\_GTTN\_reformulation\] {#proof-for-theorem-theorem_obj_gttn_reformulation .unnumbered} ------------------------------------------------------- [**Proof**]{}. Based on Eq. (\[GTTN\]), we rewrite problem (\[obj\_GTTN\]) as $$\min_{\bm{\Theta},\bm{\alpha}\in\mathcal{C}_{\bm{\alpha}}}\sum_{i=1}^m\frac{1}{n_i}\sum_{j=1}^{n_i}l(f_i(\mathbf{x}^i_j;\bm{\Theta}),y^i_j) +\lambda\sum_{\mathbf{s}\subset[p]\atop\mathbf{s}\ne\emptyset}\alpha_{\mathbf{s}}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*,$$ which is equivalent to $$\min_{\bm{\Theta}}\sum_{i=1}^m\frac{1}{n_i}\sum_{j=1}^{n_i}l(f_i(\mathbf{x}^i_j;\bm{\Theta}),y^i_j) +\lambda\min_{\bm{\alpha}\in\mathcal{C}_{\bm{\alpha}}} \sum_{\mathbf{s}\subset[p]\atop\mathbf{s}\ne\emptyset}\alpha_{\mathbf{s}}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*.$$ So we just need to prove that $$\min_{\bm{\alpha}\in\mathcal{C}_{\bm{\alpha}}} \sum_{\mathbf{s}\subset[p]\atop\mathbf{s}\ne\emptyset}\alpha_{\mathbf{s}}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*= \min_{{\mathbf{s}\subset[p]\atop\mathbf{s}\ne\emptyset}}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*.$$ The optimization problem in the left-hand side of the above equation is a linear programming problem with to $\bm{\alpha}$. It is easy to show that $\sum_{\mathbf{s}\subset[p]}\alpha_{\mathbf{s}}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*\ge \min_{\mathbf{s}\subset[p]\atop\mathbf{s}\ne\emptyset}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*$ for $\bm{\alpha}\in\mathcal{C}_{\bm{\alpha}}$, where the equality holds when the corresponding coefficient for $\min_{\mathbf{s}\subset[p]\atop\mathbf{s}\ne\emptyset}\|\mathcal{W}_{\{\mathbf{s}\}}\|_*$ equals 1 and other coefficients equals 0. Then we reach the conclusion.$\Box$ Proof for Theorem \[theorem\_GTTN\_dual\_norm\] {#proof-for-theorem-theorem_gttn_dual_norm .unnumbered} ----------------------------------------------- [**Proof**]{}. We define a linear operator $\Phi(\mathcal{W})=[\mathrm{vec}(\alpha_{\{[1]\}}\mathcal{W}_{\{[1]\}});\ldots;\alpha_{\{[2:p]\}}\mathrm{vec}(\mathcal{W}_{\{[2:p]\}})]$, where $\mathrm{vec}(\cdot)$ denotes the columnwise concatenation of a matrix and $[i:j]$ denotes a set of successively integers for $i$ to $j$. We define the $q$ norm as $$\|\mathbf{y}\|_q=\sum_i\|\mathcal{Y}^{(\pi(i))}_{\{\pi(i)\}}\|_*,$$ where $\mathcal{Y}^{(\pi(i))}_{\{\pi(i)\}}$ denotes the inverse vectorization of a subvector $\mathbf{z}_{(i-1)*N+1:kN}$ of $\mathbf{z}$ into a $\prod_{j\in\pi(i)}p_j\times\prod_{j\in\neg\pi(i)}p_j$ matrix where $N=\prod_{j=1}^pd_j$ and $\pi(i)$ transforms an index $i$ into a subset of $[p]$. Based on the definition of the dual norm, we have $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^{\star}}=\sup\langle\mathcal{W},\mathcal{X}\rangle\ \mathrm{s.t.}\ {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{X} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*}\le 1,$$ where $\langle\cdot,\cdot\rangle$ denotes the inner product between two tensors with equal size. Since this maximization problem satisfies the Slater condition, the strong duality holds. Thus, due to Fenchel duality theorem, we have [$$\sup_{\mathcal{X}}(\langle\mathcal{W},\mathcal{X}\rangle-\delta({{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{X} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*}\le 1))=\inf_{\mathbf{y}}(\delta(-\Phi^T(\mathbf{y})+\mathcal{X})+\|\mathbf{y}\|_{q^\star}),$$ ]{}where $\delta(C)$ is an indicator function of condition $C$ and it outputs 0 when $C$ is true and otherwise $\infty$. Since the dual norm of the trace norm is the spectral norm, we reach the conclusion.$\Box$ Proof for Theorem \[theorem\_generalization\_bound\] {#proof-for-theorem-theorem_generalization_bound .unnumbered} ---------------------------------------------------- Before presenting the proof for Theorem \[theorem\_generalization\_bound\], we first prove the following theorem. \[theorem\_M\_upperbound\] $\sigma^i_j$, a Rademacher variable, is an uniform $\{\pm 1\}$-valued random variable, and $\mathcal{M}$ is a $d_1\times\ldots\times d_{p-1}\times d_p$ tensor with $\mathcal{M}_i=\sum_{j=1}^{n_0}\frac{1}{n_0}\sigma^i_j\mathbf{x}^i_j$, where $d_p$ equals $m$. Then we have $$\mathbb{E}[{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{M} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^\star}]\le \min_{\mathbf{s}\ne\emptyset\atop\mathbf{s}\subset[p]}\frac{C}{\alpha_{\mathbf{s}}}\left(\frac{\kappa m}{n_0d}\sqrt{\ln d_{\mathbf{s}}}+\frac{\ln d_{\mathbf{s}}}{n_0}\right).$$ where $d_{\mathbf{s}}=\prod_{i\in\mathbf{s}}d_i+\prod_{j\in\neg\mathbf{s}}d_j$, $C$ is an absolute constant, [**Proof**]{}. We define $d_p=m$. According to Theorem \[theorem\_GTTN\_dual\_norm\], we have $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{M} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^\star}=\min_{\sum_{\mathbf{s}\ne\emptyset\atop\mathbf{s}\subset[p]}\alpha_{\mathbf{s}}\mathcal{Y}^{(\mathbf{s})}=\mathcal{M}} \max_{\mathbf{s}\ne\emptyset\atop\mathbf{s}\subset[p]}\|\mathcal{Y}^{(\mathbf{s})}_{\{\mathbf{s}\}}\|_{\infty}$$ Since for each $\mathbf{s}$ we can make $\alpha_{\mathbf{s}}\mathcal{Y}^{\mathbf{s}}$ equal to $\mathcal{M}$, we have $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{M} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^\star}\le \frac{1}{\alpha_{\mathbf{s}}}\|\mathcal{M}_{\{\mathbf{s}\}}\|_{\infty}\ \forall \mathbf{s}\ne\emptyset,\ \mathbf{s}\subset[p],$$ which implies that $${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{M} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^\star}\le \min_{\mathbf{s}}\frac{1}{\alpha_{\mathbf{s}}}\|\mathcal{M}_{\{\mathbf{s}\}}\|_{\infty}.$$ So we can get $$\begin{aligned} \mathbb{E}[{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{M} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^\star}] \le&\mathbb{E}\left[\min_{\mathbf{s}}\frac{1}{\alpha_{\mathbf{s}}}\|\mathcal{M}_{\{\mathbf{s}\}}\|_{\infty}\right]\\ \le&\min_{\mathbf{s}}\mathbb{E}\left[\frac{1}{\alpha_{\mathbf{s}}}\|\mathcal{M}_{\{\mathbf{s}\}}\|_{\infty}\right].\end{aligned}$$ Based on Theorem 6.1 in [@tropp12], we can upper-bound each expectation as $$\begin{aligned} \mathbb{E}\left[\|\mathcal{M}_{\{\mathbf{s}\}}\|_{\infty}\right]\le C(\sigma_{\mathbf{s}}\sqrt{\ln d_{\mathbf{s}}}+\psi_{\mathbf{s}}\ln d_{\mathbf{s}}),\end{aligned}$$ where $\mathcal{Z}^{i,j}$ is a $d_1\times\ldots\times d_{p-1}\times d_p$ zero tensor with only the $i$th slice along the last axis equal to $\frac{1}{n_0}\sigma^i_j\mathbf{x}^i_j$, $\psi_{\mathbf{s}}$ needs to satisfy $\psi_{\mathbf{s}}\ge\|\mathcal{Z}^{i,j}_{\{\mathbf{s}\}}\|_{\infty}$, and [$$\begin{aligned} &\sigma_{\mathbf{s}}^2\\ =&\max\Big( \big\|\sum_{i=1}^{m}\sum_{j=1}^{n_0}\mathbb{E}\big[\mathcal{Z}^{i,j}_{\{\mathbf{s}\}}(\mathcal{Z}^{i,j}_{\{\mathbf{s}\}})^T\big]\big\|_{\infty}, \big\|\sum_{i=1}^{m}\sum_{j=1}^{n_0}\mathbb{E}\big[(\mathcal{Z}^{i,j}_{\{\mathbf{s}\}})^T\mathcal{Z}^{i,j}_{\{\mathbf{s}\}}\big]\big\|_{\infty} \Big).\end{aligned}$$ ]{}As the Frobenius norm of a matrix is larger than its spectral norm, $\|\mathcal{Z}^{i,j}_{\{\mathbf{s}\}}\|_{\infty}\le \frac{1}{n_0}$ and we simply set $\psi_{\mathbf{s}}=\frac{1}{n_0}$. For $\sigma_{\mathbf{s}}$, we have $$\mathbb{E}\Big[\sum_{j=1}^{n_0}\mathcal{Z}^{i,j}_{\{\mathbf{s}\}}(\mathcal{Z}^{i,j}_{\{\mathbf{s}\}})^T\Big]=\frac{1}{n_0} \mathbf{C}_{\mathbf{s}-\{p\}}\preceq\frac{\kappa}{n_0d}\mathbf{I},$$ implying that $$\left\|\sum_{i=1}^{m}\sum_{j=1}^{n_0}\mathbb{E}\big[\mathcal{Z}^{i,j}_{\{\mathbf{s}\}}(\mathcal{Z}^{i,j}_{\{\mathbf{s}\}})^T\big]\right\|_{\infty} \le\frac{\kappa m}{n_0d}.$$ Similarly, we have $$\mathbb{E}\Big[\sum_{j=1}^{n_0}(\mathcal{Z}^{i,j}_{\{\mathbf{s}\}})^T\mathcal{Z}^{i,j}_{\{\mathbf{s}\}}\Big] =\mathrm{diag}\left(\frac{\mathrm{tr}(\mathbf{C}_{\mathbf{s}-\{p\}})}{n_0}\right)\preceq\frac{\kappa}{n_0d}\mathbf{I},$$ where $\mathrm{tr}(\cdot)$ denotes the trace of a matrix and $\mathrm{diag}(\cdot)$ converts a vector or scalar to a diagonal matrix. This inequality implies $$\left\|\sum_{i=1}^{m}\sum_{j=1}^{n_0}\mathbb{E}\big[\mathcal{Z}^{i,j}_{\{\mathbf{s}\}}(\mathcal{Z}^{i,j}_{\{\mathbf{s}\}})^T\big]\right\|_{\infty} \le\frac{\kappa m}{n_0d}.$$ By combining the above inequalities, we reach the conclusion.$\Box$ Then we can prove Theorem \[theorem\_generalization\_bound\] as follows. [**Proof**]{}. By following [@bm02], we have $$\begin{aligned} L(\hat{\mathcal{W}})&\le& \hat{L}(\hat{\mathcal{W}})+\sup_{{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*\le\gamma}\left\{L(\mathcal{W})-\hat{L}(\mathcal{W})\right\}\\ &=&\hat{L}(\hat{\mathcal{W}})+\sup_{{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*\le\gamma}\left\{\mathbb{E}[\hat{L}(\mathcal{W})]-\hat{L}(\mathcal{W})\right\}.\end{aligned}$$ When each pair of the training data $(\mathbf{x}^i_j,y^i_j)$ changes, the random variable $\sup_{{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*\le\gamma}\left\{\mathbb{E}[\hat{L}(\mathcal{W})]-\hat{L}(\mathcal{W})\right\}$ can change by no more than $\frac{2}{mn_0}$ due to the boundedness of the loss function $l(\cdot,\cdot)$. Then by McDiarmid’s inequality, we can get [$$\begin{aligned} &P\left(\sup_{\mathcal{W}\in\mathcal{C}}\left\{\mathbb{E}[\hat{L}(\mathcal{W})]-\hat{L}(\mathcal{W})\right\} -\mathbb{E}\left[\sup_{\mathcal{W}\in\mathcal{C}}\left\{\mathbb{E}[\hat{L}(\mathcal{W})]-\hat{L}(\mathcal{W})\right\}\right]\ge t\right)\\ &\le \exp\left\{-\frac{t^2mn_0}{2}\right\},\end{aligned}$$ ]{}where $P(\cdot)$ denotes the probability and $\mathcal{C}=\{\mathcal{W}|{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{W} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_*\le\gamma\}$. This inequality implies that with probability at least $1-\delta$, [$$\begin{aligned} \sup_{\mathcal{W}\in\mathcal{C}}\left\{\mathbb{E}[\hat{L}(\mathcal{W})]-\hat{L}(\mathcal{W})\right\} \le&\mathbb{E}\left[\sup_{\mathcal{W}\in\mathcal{C}}\left\{\mathbb{E}[\hat{L}(\mathcal{W})]-\hat{L}(\mathcal{W})\right\}\right]\\ &+\sqrt{\frac{2}{mn_0}\ln\frac{1}{\delta}}.\end{aligned}$$ ]{}Based on the the property of the Rademacher complexity, we have $$\begin{aligned} &\mathbb{E}\left[\sup_{\mathcal{W}\in\mathcal{C}}\left\{\mathbb{E}[\hat{L}(\mathcal{W})]-\hat{L}(\mathcal{W})\right\}\right]\\\le& 2\rho\mathbb{E}\left[\sup_{\mathcal{W}\in\mathcal{C}}\left\{\frac{1}{mn_0}\sum_{i=1}^m\sum_{j=1}^{n_0}\sigma^i_jf_i(\mathbf{x}^i_j)\right\}\right].\end{aligned}$$ Then based on the definition of $\mathcal{M}$ and the Hölder’s inequality, we have $$\sup_{\mathcal{W}\in\mathcal{C}}\left\{\frac{1}{mn_0}\sum_{i=1}^m\sum_{j=1}^{n_0}\sigma^i_jf_i(\mathbf{x}^i_j)\right\} \le\frac{\gamma}{m}{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{M} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^\star}.$$ By combining the above inequalities, with probability at least $1-\delta$, we have $$\begin{aligned} L(\hat{\mathcal{W}})&\le& \hat{L}(\hat{\mathcal{W}})+\frac{2\rho\gamma}{m}\mathbb{E}[{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert \mathcal{M} \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{*^\star}]+\sqrt{\frac{2}{mn_0}\ln\frac{1}{\delta}}.\end{aligned}$$ Then by incorporating Theorem \[theorem\_M\_upperbound\] into this inequality, we reach the conclusion.$\Box$ [^1]: Here for simplicity, we assume the tensor trace norm regularization is placed on only one $\mathcal{W}$. This formulation can easily be extended to multiple $\mathcal{W}$’s with the tensor trace norm regularization. [^2]: All the proofs are put in the appendix. [^3]: The analysis is easy to extend to regression tasks and multi-class classification tasks.

--- abstract: | We discuss the stability properties of the solutions of the general nonlinear [Schr[ö]{}dinger]{}  equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time (${\mathcal P}{\mathcal T}$) symmetric superpotential $W(x)$ that we considered earlier \[Kevrekidis [*et al.*]{} Phys. Rev. E **92**, 042901 (2015)\]. In particular we consider the nonlinear partial differential equation $ \{ i \, \partial_t + \partial_x^2 - V^{-}(x) + | \psi(x,t) |^{2\kappa} \} \, \psi(x,t) = 0 \>, $ for arbitrary nonlinearity parameter $\kappa$. We study the bound state solutions when $V^{-}(x) = (1/4- b^2) \sech^2(x)$, which can be derived from two different superpotentials $W(x)$, one of which is complex and ${\mathcal P}{\mathcal T}$ symmetric. Using Derrick’s theorem, as well as a time dependent variational approximation, we derive exact analytic results for the domain of stability of the trapped solution as a function of the depth $b^2$ of the external potential. We compare the regime of stability found from these analytic approaches with a numerical linear stability analysis using a variant of the Vakhitov-Kolokolov (V–K) stability criterion. The numerical results of applying the V-K condition give the *same* answer for the domain of stability as the analytic result obtained from applying Derrick’s theorem. Our main result is that for $\kappa>2$ a *new* regime of stability for the exact solutions appears as long as $b > b_{\text{crit}}$, where $b_{\text{crit}}$ is a function of the nonlinearity parameter $\kappa$. In the absence of the potential the related solitary wave solutions of the NLSE are *unstable* for $\kappa>2$. author: - Fred Cooper - Avinash Khare - Andrew Comech - Bogdan Mihaila - 'John F. Dawson' - Avadh Saxena date: ',  EDT' title: 'Stability of exact solutions of the nonlinear [Schr[ö]{}dinger]{} equation in an external potential having supersymmetry and parity-time symmetry' --- \[s:Intro\]Introduction ======================= The topic of Parity-Time (${\mathcal{PT}}$) symmetry and its relevance for physical applications on the one hand, as well as its mathematical structure on the other, have drawn considerable attention from both the physics and the mathematics community. Originally the proposal of Bender and his collaborators [@Bender_review; @special-issues] towards the study of such systems was made as an alternative to the postulate of Hermiticity in quantum mechanics. In view of the formal similarity of the [Schr[ö]{}dinger]{} equation with Maxwell’s equations in the paraxial approximation, it was realized that such ${\mathcal{PT}}$ invariant systems can in fact be experimentally realized in optics [@review; @Muga; @PT_periodic; @experiment]. Subsequently, these efforts motivated experiments in several other areas including ${\mathcal{PT}}$ invariant electronic circuits [@tsampikos_recent; @tsampikos_review], mechanical circuits [@pt_mech], and whispering-gallery microcavities [@pt_whisper]. Concurrently, the notion of supersymmetry (SUSY) originally espoused in high-energy physics has also been realized in optics [@heinr1]. The key idea is that from a given potential one can obtain a SUSY partner potential with both potentials possessing the same spectrum (with exception of possibly one eigenvalue) [@Gendenshtein; @susy]. An interplay of SUSY with ${\mathcal{PT}}$ symmetry is expected to be quite rich and is indeed useful in achieving transparent as well as one-way reflectionless complex optical potentials [@bagchi; @ahmed; @midya]. In a previous paper [@pt1] we explored the interplay between ${\mathcal{PT}}$ symmetry, SUSY and nonlinearity. In Ref. [@pt1] we derived the exact solutions of the general nonlinear [Schr[ö]{}dinger]{} equation (NLSE) with arbitrary nonlinearity in 1+1 dimensions when in an external potential given by a shape invariant [@Gendenshtein; @avinashfred] supersymmetric and ${\mathcal{PT}}$ symmetric complex potential. In particular, we considered the nonlinear partial differential equation $$\label{eqn1} \qty{ i \, \partial_t + \partial_x^2 - V^\pm(x) + | \psi(x,t) |^{2\kappa} } \, \psi(x,t) = 0 \>,$$ for arbitrary nonlinearity parameter $\kappa$, with $$\label{eqn1x} V^{\pm}(x) = W_1^2(x)\mp W_1'(x) \>,$$ and the partner potentials arise from the superpotential $$\label{eqn1y} W_1(x) = \qty( m - 1/2 ) \, \tanh{x} - i b \, \sech{x} \>,$$ giving rise to \[VpVm\] $$\begin{aligned} V^{+}(x) &= \qty( -b^2 - m^2 + 1/4 ) \, \sech^2(x) \label{eqn7a} \\ & \quad - 2 i \, m \, b \, \sech(x) \, \tanh(x), \notag \\ V^{-}(x) &= \qty( - b^2 - (m-1)^2 + 1/4 ) \, \sech^2(x) \label{eqn8a} \\ & \quad - 2 i \, \qty( m - 1 ) \, b \, \sech(x) \, \tanh(x) \>. \notag\end{aligned}$$ For $m=1$, the *complex* potential $V^{+}(x)$ has the same spectrum, apart from the ground state, as the *real* potential $V^{-}(x)$ and we used this fact in our numerical study of the stability of the bound state solutions of the NLSE in the presence of $V^{+}(x)$ (see Ref. [@pt1]). To complete our study of this system of nonlinear [Schr[ö]{}dinger]{} equations in ${\mathcal{PT}}$ symmetric SUSY external potentials, we will study the stability properties of the bound state solutions of NLSE in the presence of the external real SUSY partner potential $V^{-}(x)$, and compare the stability regime of these solutions, which depends on the parameters $(b, \kappa)$, to the stability regime of the related *solitary* wave solutions to the NLSE in the absence of the external potential, which depend only on the parameter $\kappa$. Because the NLSE in the presence of $V^{-}(x)$ is a Hamiltonian dynamical system, we can use variational methods to study the stability of the solutions when they undergo certain small deformations. We will compare the results of this type of analysis with a linear stability analysis based on the V–K stability criterion [@comech; @stab1]. In our previous paper [@pt1] we determined the exact solutions of the equation for $m=1$ and for $V^{+}(x)$, which was complex. We studied numerically the stability properties of these solutions using linear stability analysis. We found some unusual results for the stability which depended on the value of $b$. In that paper, because of the complexity of the potential, the energy was not conserved and a Hamiltonian formulation of the problem was not possible. However for the partner potential $V^{-}(x)$, when $m=1$, the potential is real. We note that $V^{-}(x)$ has the symmetry $b \leftrightarrow m-1$, so that we can obtain $V^{-}(x)$ from two *different* superpotentials $W_1(x)$ and $W_2(x) $ that have the ${\mathcal P}{\mathcal T}$ symmetric forms at arbitrary $m$: \[SSW1W2\] $$\begin{aligned} W_1(x) &= \qty( m - 1/2 ) \, \tanh(x) - i b \, \sech(x) \>, \label{SSW1} \\ W_2(x) &= ( b + 1/2 ) \, \tanh(x) - i (m-1) \sech(x) \>. \label{SSW2}\end{aligned}$$ In particular, when $m=1$ we can determine the spectrum of bound states of the linear [Schr[ö]{}dinger]{} equation with potential $V^{-}(x)$ using the *real* superpotential $W_2(x) = \qty( b + 1/2 ) \, \tanh{x}$ and the real shape invariant sequence of partner potentials. When $m=1$ the first superpotential $W_1(x)$ has the *complex* partner potential $V^{+}(x)$ which we studied in [@pt1] , whereas $W_2(x)$ is the usual *real* shape invariant SUSY potential. Both superpotentials yield the same real potential $ V^{-}(x) = -(b^2 - 1/4) \sech^2(x)$. Because this potential is real, one can use variational methods to study the stability of the exact solutions to the NLSE in the potential $V^{-}(x)$. We will consider both Derrick’s theorem [@ref:derrick] as well as a time dependent variational approximation [@var1; @var2] to study the stability of the exact solutions. Because of the similarity of the solutions to those of the NLSE equation, we are able to use a variant of the V–K stability criterion to study spectral stability of the solutions. The results of the V–K analysis agree with the new regime of stability found from Derrick’s theorem and the time dependent variational approach. The latter approach allows us to obtain the frequency of small oscillations of the perturbed solutions. We are able to obtain exact analytic results because we can formulate the problem in terms of Hamilton’s action principle. The Euler-Lagrange equations lead to the dynamical equations which have a conserved Hamiltonian. This is in sharp contrast with the stability analysis for the solutions in the presence of $V^{+}(x)$ which had to be done numerically. The latter system is dissipative in nature. This paper is structured as follows. In Sec. \[s:SSModel\] we review the non-hermitian SUSY model that we studied in Ref. [@pt1]. In Sec. \[s:HamPrinc\] we consider Hamilton’s principle for the NLSE in the real external potential $V^{-}(x)$. Section \[s:Derrick\] describes the application of Derrick’s theorem for determining the domain of stability of the solutions. Here we determine analytically a new domain of stability for the solutions, when compared to the solutions in the absence of an external potential. We find that for $b > b_{\text{crit}}$ a new domain of stability exists. Sec. \[s:collective\] contains a collective coordinate approach which allows one to study dynamically the blowup or collapse of the solution as well as small oscillations around the exact solution when slightly perturbed. By setting the frequency to zero we obtain analytically a domain of stability that agrees with the result of Derrick’s theorem. In Sec. \[NS.s:LinearStability\] we provide the details of a linear stability analysis based on the V-K stability criterion, which leads to identical conclusions as Derrick’s theorem. Section \[s:conclude\] contains a summary of our main results. \[s:SSModel\]A Linear Non-Hermitian Supersymmetric Model ======================================================== We are motivated by the $\mathcal{PT}$ symmetric SUSY superpotential $$\label{eqn6} W_1(x) = \qty( m - 1/2 ) \, \tanh(x) - i b \, \sech(x) \>,$$ which gives rise to the supersymmetric partner potentials given by Eqs. . In what follows we will specialize to the case $m=1$. In that case, $V^{-}(x)$ is the well known P[ö]{}schl-Teller potential [@poschl; @LL]. The relevant bound state eigenvalues assume an extremely simple form as $$\label{eqn9} E_n^{(-)} = -\frac{1}{4} \, \qty[ \, 2 b - 2 n - 1 \, ]^2 \>.$$ Such bound state eigenvalues only exist when $n < b - 1/2$. We notice that for the ground state (n=0) to exist requires $b > 1/2$. The existence of a first excited state (n=1) requires $b > 3/2$. We will find that the stability of the NLSE solutions in this external potential will depend on the depth of the well, which will lead to a critical value of $b = b_{\text{crit}}$, above which the solutions are stable. In what follows we will be concerned with the properties of the NLSE in the presence of the external potential centered at $x=0$, $$\label{Vmdef} V^{-}(x) = - \qty( b^2 - 1/4 ) \, \sech^2(x) \qc b > 1/2 \>.$$ In particular we are interested in the bound state solutions of $$\label{eqn1-II} \qty{ i \, \partial_t + \partial_x^2 - V^{-}(x) + | \psi(x,t) |^{2\kappa} } \, \psi(x,t) = 0 \> .$$ If we assume a solution of the form $$\label{eqn-form} \psi(x,t) = A \sech^\alpha(x) \, \exp{-i \omega t} \>,$$ it is easy to show that an exact solution is given by: $$\label{exact} \psi(x,t) = A(b,\kappa) \, \sech^{1/\kappa}(x) \, e^{-i \omega t} \>,$$ where $\omega = - 1/\kappa^2$ and $$\label{Abound} A^{2\kappa}(b,\kappa) = ( 1/\kappa + 1/2)^2 - b^2 \>.$$ For the complex potential $V^{+}(x)$, the amplitude of the solution is given by [@pt1] $$A^{2\kappa}(b,\kappa) = \frac{ \qty( ( 1/\kappa + 1/2)^2 - m^2 ) \qty( ( 1/\kappa + 1/2)^2 - b^2 ) } { \qty( 1/\kappa + 1/2 )^2 } \>,$$ so that there are two separate regimes where $A$ is real. In contrast, for $V^{-}(x)$ there is only one regime for an attractive $V^{-}(x)$ where $A$ is real, namely $$\label{brange} b_{\text{min}} = 1/2 \leq b \leq b_{\text{max}} = (1/\kappa + 1/2) \>.$$ The analysis of the stability of the solutions for $V^{+}(x)$ in Ref. [@pt1] showed a very complicated pattern. Even for $\kappa=1$ there is a regime of instability as a function of $b$ for the nodeless solution. For $\kappa=3$ all solutions in that case found analytically and numerically were unstable. Only for $\kappa < 2/3$ were the solutions stable. In contrast to that analysis where each value of $\kappa$ had to be investigated separately, in the case of the real potential $V^{-}(x)$, we are able to address the stability question for all $\kappa$ using Derrick’s theorem as well as the V–K stability criterion. The mass $M$ of the bound state for the case $V^{-}(x)$ is given by $$\begin{aligned} M &= \int |\psi(x,t)|^2 \dd{x} \label{Mbound} \\ &= \frac{ \sqrt{\pi} \, \qty[ (1/2 + 1/\kappa)^2 - b^2 ]^{1/\kappa} \, \Gamma[1/\kappa] } { \Gamma[1/2 + 1/\kappa] } \>. \notag\end{aligned}$$ If we turn off the external potential by setting $b \to 1/2$, $A^{\kappa}[1/2,\kappa] \to (1/\kappa + 1/2)$ and the mass $M$ of the bound state goes to the mass of the solitary wave solutions, $$\label{Msol} M \to \frac{ \sqrt{\pi} \, \qty[ (\kappa + 1)/\kappa^2 ]^{1/\kappa} \, \Gamma[1/\kappa] } { \Gamma[1/2 + 1/\kappa] } \>.$$ \[s:HamPrinc\]Hamilton’s Principle of Least Action for the NLSE in an external potential ======================================================================================== Let us first discuss Hamilton’s principle of least action for the usual NLSE without a confining potential. The NLSE with arbitrary nonlinearity in 1+1 dimensions is given by $$\label{ham.e:1} \qty{ i \, \partial_t + \partial_x^2 + g \, | \psi(x,t) |^{2\kappa} } \, \psi(x,t) = 0 \>.$$ The second term causes diffusion and the third term attraction and the competition allows for solitary wave blowup which depends on $\kappa$. Here $g$ can be scaled out of the equation by letting $$\psi(x,t) \mapsto g^{-1/( 2\kappa)}\, \psi(x,t) \>,$$ so that the linear equation for the rescaled equation is obtained in the limit $\psi(x,t) \rightarrow 0$. While the solitary waves are stable for $\kappa < 2$, for $\kappa = 2$ there is a critical mass $M$ necessary for blowup to occur, where the width of solitary wave goes to zero. For $\kappa > 2$, blowup occurs in a finite amount of time. The classical action for the NLSE is $\Gamma[\psi,\psi^{\ast}] = \int L[\psi,\psi^{\ast}] \dd{t}$, where the Lagrangian $L[\psi,\psi^{\ast}]$ is given by $$\begin{aligned} L[\psi,\psi^{\ast}] &= \int \dd{x} \bigl \{ \, i \, \qty[ \psi^{\ast} (\partial_t \psi) - (\partial_t \psi^{\ast} ) \psi ] / 2 \label{LagNLSE} \\ & \qquad + ( \partial_x \psi^{\ast} )( \partial_x \psi) - | \psi |^{2(\kappa+1)} / ( \kappa + 1 ) \, \bigr \} \>. \notag\end{aligned}$$ The NLSE follows from the Hamilton’s principle of least action, $\delta \Gamma / \delta \psi = 0$ and $\delta \Gamma / \delta \psi^{\ast} = 0$, which leads to Eq.  with $g \to 1$. Multiplying this equation by $\psi^{\ast}(x,t)$ and subtracting its complex conjugate, it is easy to prove that the mass $M$, defined by $M = \int |\psi(x,t)|^2 \dd{x}$, is conserved. We now want to add a *real* SUSY potential to the NLSE. We will consider the addition of $V^{-}(x)$ given in Eq.  so that the equation of motion is now given by $$\label{motion2} \qty{ i \, \partial_t + \partial_x^2 - V^{-}(x) + | \psi(x,t) |^{2\kappa} } \, \psi(x,t) = 0 \>.$$ The action which leads to Eq.  is given by $\Gamma[\psi,\psi^{\ast}] = \int L[\psi,\psi^{\ast}] \dd{t}$ where $$\begin{aligned} L[\psi,\psi^{\ast}] &= \int \dd{x} i \, \qty[ \psi^{\ast} (\partial_t \psi) - (\partial_t \psi^{\ast} ) \psi ]/2 - H[\psi,\psi^{\ast}] \>, \notag \\ H[\psi,\psi^{\ast}] &= \int \dd{x} \bigl \{ \, ( \partial_x \psi^{\ast} )( \partial_x \psi) - | \psi |^{2(\kappa+1)} / (\kappa + 1) \notag \\ & \qquad\qquad + \psi^{\ast} \, V^{-}(x) \, \psi \, \bigr \} \>. \label{Hdef}\end{aligned}$$ \[s:Derrick\]Derrick’s theorem ============================== Derrick’s theorem [@ref:derrick] states that for a Hamiltonian dynamical system, for a solitary wave solution to be stable it must be stable to changes in scale transformation $x \mapsto \beta x$ when we keep the mass of the solitary wave fixed. That is the Hamiltonian needs to be a minimum in $\beta$ space. First let us look at the case of the NLSE without an external potential: Derrick’s method is based on whether a scale transformation which keeps the mass $M$ invariant, raises or lowers the energy of a solitary wave. For the NLSE with Hamiltonian $$\begin{aligned} H &= \int \dd{x} \qty[ ( \partial_x \psi^\ast ) \, ( \partial_x \psi ) - | \psi |^{2(\kappa+1)} / (\kappa + 1) ] \\ & \equiv H_1 - H_2 \>, \notag\end{aligned}$$ where both $H_1$ and $H_2$ are positive definite. A static solitary wave solution can be written as $$\psi(x,t) = r(x) \, e^{- i \omega t} \>.$$ The exact solution has the property that it minimizes the Hamiltonian subject to the constraint of fixed mass as a function of a stretching factor $\beta$. This can be seen by studying a variational approach as done in  [@variational], or by directly studying the effect of a scale transformation that respects conservation of mass. In the latter approach, which generalizes the method used by Derrick  [@ref:derrick], we let $x \mapsto \beta x$, and consider the stretched wave function, $$\label{stretchpsi} \psi_{\beta}(x,t) = \beta^{1/2} r(\beta x) \, e^{ - i \omega t} \>,$$ so that $$M = \int \dd{x} \abs{ \psi_\beta(x,t) }^2 = \int \dd{x} \abs{ \psi(x,t) }^2$$ is preserved by the transformation. Defining $H_\beta$ as the value of $H$ for the stretched solution $\psi_\beta(x,t)$, one finds that $\partial H_{\beta} / \partial \beta |_{\beta=1} = 0$ is consistent with the equations of motion. The stable solutions must then also satisfy: $$\label{d2Hbeta-dbeta2} \pdv[2]{H_{\beta}}{\beta} \ge 0 \>.$$ If we write $H$ in terms of the two positive definite pieces $H_1$, $H_2$, then $$\label{H1H2defs} H_{\beta} = \beta^2 \, H_1 - \beta^{\kappa} \, H_2 \>,$$ we find $$\label{H1andH2} \eval{ \pdv{H_{\beta}}{\beta} }_{\beta=1} = 2 \, H_1 - \kappa \, H_2 = 0 \>,$$ so that $H_1 = (\kappa / 2 ) \, H_2$. This result is consistent with the equations of motion. In fact for the NLSE the exact solution has $r(x) = A \, \sech^{1/\kappa}(x)$ where $A^{2\kappa}= (\kappa + 1)/\kappa^2$. One finds then using $$\int_{-\infty}^\infty \dd{x} \sech^r(x) = \frac{\sqrt{\pi} \, \, \Gamma[ r/2] }{\Gamma[1/2+r/2]} \>,$$ that \[H1andH2eval\] $$\begin{aligned} H_1 &= \frac{ \sqrt{\pi} \, \qty[ (\kappa + 1)/\kappa^2 ]^{1/\kappa} \Gamma[1/\kappa] } { 2 \kappa^2 \, \Gamma[3/2 + 1/\kappa] } \>, \label{H1-val} \\ H_2 &= \frac{ \sqrt{\pi} \, \qty[ (\kappa + 1)/\kappa^2 ]^{1/\kappa} \Gamma[1/\kappa] } { \kappa^3 \, \Gamma[3/2 + 1/\kappa] } \>, \label{H2-val} \end{aligned}$$ so that the exact solution is indeed a minimum of the Hamiltonian with respect to scale transformations, with $H_1 = (\kappa/2) \, H_2$. The second derivative is given by $$\label{d2Hbeta} \pdv[2]{H_{\beta}}{\beta} = 2 \, H_1 - \kappa(\kappa - 1) \, \beta^{\kappa - 2} \, H_2 \>,$$ which when evaluated at the stationary point yields $$\label{d2H2stable} \pdv[2]{H_{\beta}}{\beta} = 2 \, (2 - \kappa) \, H_1 \ge 0 \>,$$ for stability. This result indicates that solutions are unstable to changes in the width, compatible with the conserved mass, when $\kappa > 2$. The case $\kappa=2$ is a marginal case where it is known that blowup occurs at a critical mass (see for example Ref. [@var2]). The result found above for the NLSE has also been found by various other methods such as linear stability analysis and using strict inequalities. Numerical simulations (see Ref. [@Rose:p]) have been done for the critical case $\kappa=2$ showing that blowup (self-focusing) occurs when the mass $M > 2.72$. For $\kappa >2$ a variety of analytic and numerical methods have been used to study the nature of the blowup at finite time [@kevrekedis]. \[s:VKcriterion\]Linear Stability and the Vakhitov–Kolokokov criterion ---------------------------------------------------------------------- In the case of the nonlinear [Schr[ö]{}dinger]{} equation, one can perform a linear stability analysis of the exact solutions. Namely one lets $$\label{nlse.e:1} \psi(x,t) = \qty[ \psi_\omega(x) + r(x,t) ] \, e^{-i \omega t} \>,$$ and linearizes the NLSE to find an equation for $r(x,t)$ to first order, $$\label{nlse.e:2} \partial_t \, r(x,t) = A_{\omega} \, r(x,t) \>,$$ and studies the eigenvalues of the differential operator $A_\omega$. If the spectrum of $A_\omega$ is imaginary, then the solutions are spectrally stable. V–K showed [@stab1] that when the spectrum is purely imaginary $\dd M(\omega)/\dd \omega < 0$. Also they showed that when $\dd M(\omega)/\dd \omega > 0$, there is a real positive eigenvalue so that there is a linear instability. For the NLSE, there is a class of solutions with arbitrary nonlinearity parameter $\kappa$. Namely \[nlse.e:3\] $$\begin{aligned} \psi_{\omega}(x,t) &= A(\kappa,\beta) \, \sech^{1/\kappa}(\beta x) \, e^{-i \omega t} \>, \label{nlse.e:3-a} \\ A^{2\kappa}(\kappa,\beta) &= \beta^2 ( \kappa + 1 ) /\kappa^2 \qc \omega = - \beta^2 / \kappa^2 \>. \label{nlse.e:3-b}\end{aligned}$$ When we do not have an external potential, we know explicitly how the mass changes when we change $\omega$ at fixed $\kappa$. That is $$\begin{aligned} M(\omega) &= \int_{-\infty}^{+\infty} \!\!\! \dd{x} \abs{ \psi_{\omega}(x,t) }^2 = A^2(\beta,\kappa) \, C_1(\kappa) / \beta \label{nlse.e:4} \\ &= \frac{ \sqrt{\pi} \, \qty[ - \omega / (\kappa + 1) ]^{1/\kappa} \Gamma[1/\kappa] } { \kappa \, \sqrt{-\omega} \, \Gamma[1/2 + 1/\kappa] } \>, \notag \end{aligned}$$ where $$\label{nlse.e:5} C_1(\kappa) = \int_{-\infty}^{+\infty} \!\!\! \dd{x} \sech^{2/\kappa}(x) \>.$$ We find $$\label{nlse.e:6} \dv{M}{\omega} = a_1 \, (\kappa - 2) \qc a_1 > 0 \>.$$ Thus for $\kappa > 2$ the solitary waves are unstable. This agrees with the result of Derrick’s theorem. When we have an external potential, we will need to determine the solutions numerically as we change $\omega$. This will be accomplished in Sec. \[NS.s:LinearStability\]. \[ss:Derrick-external\]Adding an external potential --------------------------------------------------- So now let us look at our situation when we have in addition the real external potential: $$\label{Dext.e:1} V^{-}(x) = - (b^2 - 1/4) \sech ^2(x) \>.$$ The exact solution to the NLSE in the presence of $V^{-}(x)$ is given by Eq. . This solution is similar in form to the usual solution to the NLSE except this nodeless solution is pinned to the potential so that there is no translational invariance. When $b=1/2$ this solution goes over to a particular solution of the NLSE with width parameter $\beta=1$. Under the scale transformation $x \to \beta x$, the stretched solution which preserves the mass $M$ is given by: \[Dext.e:2\] $$\begin{aligned} \psi_{\beta}(x,t) &= A(b,\kappa) \, \beta^{1/2} \, \sech^{1/\kappa}(\beta x) \, e^{-i \omega t} \>, \label{Dext.e:2-a} \\ A^{2\kappa}(b,\kappa) &= \qty( 1/2 + 1/\kappa )^2 - b^2 \qc \omega = - 1 / \kappa^2 \>. \label{Dext.e:2-b}\end{aligned}$$ The stretched wave function $\psi_{\beta}(x,t)$ is no longer an exact solution. The stretched Hamiltonian for the external potential case is now given by $$\label{Dext.e:3} H_{\beta} = \beta^2 H_1(b,\kappa) - \beta^{\kappa} \, H_2(b,\kappa) + H_3(b,\kappa,\beta) \>,$$ where $$\begin{aligned} H_1(b,\kappa) &= A^2(b,\kappa) \, f_1(\kappa) , \\ f_1(\kappa) &= \frac{ \sqrt{\pi} \, \Gamma[1/\kappa] } { 2 \kappa^2 \, \Gamma[3/2+1/\kappa] } \>, \end{aligned}$$ and $$\begin{aligned} H_2(b,\kappa) &= A^{2(\kappa+1)}(b,\kappa) \, f_2(\kappa) , \\ f_2(\kappa) &= \frac{ \sqrt{\pi} \, \Gamma[1+1/\kappa] } { (\kappa + 1) \, \Gamma[3/2+1/\kappa] } \>, \end{aligned}$$ with $A(b,\kappa)$ now given by , and $$\begin{aligned} &H_3(b,\kappa,\beta) = \int_{-\infty}^{+\infty} \!\!\!\! \psi^{\ast}_{\beta}(x,t) \, V^{-}(x) \, \psi_{\beta}(x,t) \dd{x} \\ & = \qty(1/4 - b^2) \, A^2(b,\kappa) \, \int_{-\infty}^{+\infty} \!\!\!\!\! \beta \, \sech^{2/\kappa}(\beta x) \, \sech^2(x) \dd{x} \>.\end{aligned}$$ Thus, we find $$\begin{aligned} &\eval{ \pdv{H_3}{\beta} }_{\beta=1} \!\!\!\! = \qty(1/4 - b^2) \, A^2(b,\kappa) \\ & \! \times \int_{-\infty}^{+\infty} \!\! [\, \sech^{2/\kappa + 2}(x) - 2 x \, \sech(x) \, \sech^{2/\kappa + 3}(x) / \kappa \,] \dd{x} \>.\end{aligned}$$ Using the identity, $$\begin{aligned} &\pdv{\sech^{2+2/\kappa}( \lambda x )}{\lambda} \label{Dext.e:4} \\ & \quad = - (2/\kappa + 2) \, x \, \sinh(\lambda x) \, \sech^{2/\kappa + 3}( \lambda x) \>, \notag\end{aligned}$$ we obtain $$\label{Dext.e:5} \eval{ \pdv{H_3}{\beta} }_{\beta=1} \!\!\! = - A^{2}(b,\kappa) \, \frac{ \sqrt{\pi} \, (1/4 - b^2 ) \, \Gamma[1/\kappa] } { (\kappa + 1) \, \Gamma[3/2+1/\kappa] } \>.$$ As in the case when $V^{-}(x) = 0$, we again find $$\label{Dext.e:6} \eval{ \pdv{H_{\beta}}{\beta} }_{\beta=1} \!\!\! = 0$$ for our exact solution. So the stretched solution is again an extremum of $H_\beta$ with $M$ kept fixed. For the second derivative we have $$\label{Dext.e:7} \eval{ \pdv[2]{H_{\beta}}{\beta} }_{\beta=1} \!\!\! = 2 \, H_1 - \kappa ( \kappa - 1 ) \eval{ \pdv[2]{H_{3}}{\beta} }_{\beta=1} \>,$$ where $$\label{Dext.e:8} \eval{ \pdv[2]{H_{3}}{\beta} }_{\beta=1} \!\!\! = (1/4 - b^2 ) \, A^{2}(b,\kappa) \, \qty[ I_1 + I_2 + I_3 ] / \kappa \>,$$ with $$\begin{aligned} I_1 &= - \int \dd{x} 2 x^2 \, \sech^{2/\kappa + 2}(x) \>, \\ I_2 &= \int \dd{x} 2 ( 2/\kappa + 1 ) \, x^2 \, \sinh^2(x) \, \sech^{2/\kappa + 4}(x) \>, \\ I_3 &= - \int \dd{x} 4 x \, \sinh(x) \, \sech^{2/\kappa + 3}(x) \>.\end{aligned}$$ We can again evaluate these integrals using the first identity Eq.  and the identity: $$\begin{aligned} &\pdv[2]{\sech^{2+2/\kappa}( \lambda x )}{\lambda} \label{Dext.e:9} \\ & \quad = (2/\kappa +2)(2/\kappa + 3) \, x^2 \sinh^2(\lambda x) \, \sech^{2/\kappa + 4}(\lambda x) \notag \\ & \qquad\qquad - (2/\kappa + 2) \, x^2 \, \sech^{2/\kappa + 2}(\lambda x) \>. \notag\end{aligned}$$ We will also need the following hypergeometric function: $$\begin{aligned} &ug(\kappa) = \int \dd{x} x^2 \, \sech^{2/\kappa + 2}(x) = \frac{2^{(\kappa + 2)/\kappa} \kappa^3}{(\kappa+1)^3} \label{Dext.e:10} \\ & \times {}_4F_3( \, 1+1/\kappa, 1+1/\kappa, 1+1/\kappa, 2+2/\kappa; \notag \\ & \qquad\qquad 2+1/\kappa, 2+1/\kappa, 2+1/\kappa; -1 \, ) \>. \notag\end{aligned}$$ Using these results, Eq.  gives $$\begin{aligned} \eval{ \pdv[2]{H_{3}}{\beta} }_{\beta=1} \!\!\!\!\! &= (1/4 - b^2 ) \, A^{2}(b,\kappa) \, f_3(\kappa) \>, \label{Dext.e:11} \\ f_3(\kappa) &= - \qty[ \frac{4 \, ug(\kappa)}{2+3\kappa} + \frac{4 \sqrt{\pi} \, \kappa \, \Gamma[1+1/\kappa]} {(\kappa+1)(3\kappa+2) \, \Gamma[3/2+1/\kappa] } ] . \notag\end{aligned}$$ The critical value is determined from: $$\begin{aligned} \eval{ \pdv[2]{H_{\beta}}{\beta} }_{\beta=1} \!\!\! &= A^{2}(b,\kappa) \, \bigl [ \, 2 \, f_1(\kappa) \label{Dext.e:12} \\ & \hspace{-2em} - \kappa ( \kappa + 1 ) \, A^{2\kappa}(b,\kappa) \, f_2(\kappa) + ( 1/4 - b^2 ) \, f_3(\kappa) \, \bigr ] \>. \notag\end{aligned}$$ Solving for the critical value of $b^2$, we find $$\label{Dext.e:13} b_{\text{crit}}^2 = \frac{ \qty( \kappa^3 + 3 \kappa^2 - 8 \kappa - 4 ) \, f_2(\kappa) - \kappa \, f_3(\kappa) } { 4 \kappa^2 ( \kappa - 1 ) \, f_2(\kappa) - 4 \kappa \, f_3(\kappa) ) } .$$ The result of calculating the second derivative at $\beta=1$ and setting it equal to zero is that the domain of stability is now as follows: for $\kappa < 2$ and all $b$ in the range $1/2 < b < b_{\text{max}} = 1/\kappa + 1/2$, the solution is stable, as it was for the solitary wave solutions of the NLSE. Here $b=1/2$ corresponds to no external potential. When $\kappa > 2$ the solitary wave solutions of the NLSE were *unstable*. Instead, in the presence of the confining potential, a new domain of stability occurs when $\kappa > 2$ as long as $b_{\text{crit}} < b < b_{max}$, where $b_{\text{crit}}$ is given by Eq. . We see this in the result for $\kappa = 2.1$ shown in Fig. \[NS.f:dHdbeta\]. In Fig. \[NS.f:bvskappa\], we show both $b_{\text{crit}}(\kappa)$ and $b_{\text{max}}(\kappa)$ as a function of $\kappa$. The region between $b=1/2$ and $b_{\text{crit}}(\kappa)$ is unstable. As we will show in Sec. \[NS.s:LinearStability\], this analytic result for $b_{\text{crit}}$ given by Eq.  is confirmed by our linear stability analysis. Just as there is a critical mass for instability in the NLSE at $\kappa=2$, for $ \kappa >2$ we can interpret the critical value of $b$ in terms of a critical mass which depends on $\kappa$ above which the solution is unstable. Since from Eq. , we have $$\label{PF.e:Mbound-II} M = \frac{ \sqrt{\pi} \, \qty[ (1/2 + 1/\kappa)^2 - b^2 ]^{1/\kappa} \, \Gamma[1/\kappa]} { \Gamma[1/2 + 1/\kappa] } \>,$$ we see that the mass $M$ *decreases* as we increase $b$ at fixed $\kappa$. So as we go from the unstable case $b=1/2$ (no potential) and increase $b$ we decrease the mass until we reach an $M_{\text{crit}}$ below which the solution is stable. Finally we reach the curve $M=0$ which corresponds to $b=b_{\text{max}}[\kappa] = 1/2 + 1/\kappa$. The different regimes are shown in Fig. \[NS.f:Mcrit\]. In the lightly shaded regime the solutions are unstable. The maximum value of the mass is given by the case $b=1/2$, when there is no longer a stabilizing potential. The interval $b_{\text{crit}} < b < b_{\text{max}}$ corresponds to the regime $0 < M < M_{\text{crit}}$. This is the stable regime denoted by the darker shaded area in Fig. \[NS.f:Mcrit\]. \[s:collective\]Collective coordinate approach for studying perturbations to the exact solution =============================================================================================== In order to follow the time evolution of a slightly perturbed solitary wave or bound solution to a Hamiltonian dynamical system, without solving numerically the time dependent partial differential equations for $\psi(x,t)$, one can introduce time-dependent collective coordinates assuming that the general shape of the original solution is maintained apart from the height, width, and position, etc. This will allow us to see whether these parameters just oscillate around the original values or whether the parameters grow or decrease in time. When instabilities are seen in the variational results, it suggests that the exact solutions are also unstable. Unlike Derrick’s theorem when applied to the NLSE, the collective coordinate method can be applied to the special case $\kappa=2$. It also gives an approximate description of wave function blow-up or collapse in the unstable regime, and oscillation of the perturbed solution in the stable regime. In the next section, we first apply this approach when there is no external potential. \[ss:blowup\]Self-similar analysis of blowup and critical mass for the NLSE --------------------------------------------------------------------------- Let us remind ourselves of the collective coordinate variational approach to blow-up for the NLSE [@var1; @var2] with no external potential. Using this method, we found previously that when $\kappa = 2$, there is a critical value of the mass required before blowup could take place. Derrick’s theorem has nothing to say about the stability of the solitary wave solution for this case. To make the collective coordinate approach concrete, we assume self-similar solutions of the form: $$\begin{aligned} \label{var.e:1} \psi(x,t) &= A(t) \, f[ \beta(t) y(t) ] \\ & \qquad \times \exp[ i \qty( v \, y(t) / 2 + \Lambda(t) \, y^2(t) - \omega t ) ] \>. \notag\end{aligned}$$ Here $\Lambda(t)$, $A(t)$, and $\beta(t)$ are arbitrary real functions of time alone, and $y(t) = x - q(t)$. For no external potential translation invariance gives $q(t) = v_0 t$. In particular at $t = 0$ and $v_0 = 0$, we will start with the exact solution of the form $\psi(x,0) = A \, \sech^{1/\kappa}(\beta x)$ and assume that this solution just changes during the time evolution in amplitude and width. With this assumption one can derive the dynamical equations for $A(t)$ and $\beta(t)$ from Hamilton’s principle of least action with the Lagrangian given in Eq. . Noether’s theorem yields three conservation laws: conservation of probability, conservation of momentum, and conservation of energy. Conservation of probability gives “mass” conservation: $$\label{var.e:2} M = \int_{-\infty}^{\infty } \!\!\! \dd{x} \abs{\psi(x,t)}^2 = \frac{A^2(t)}{\beta(t)} \, \int_{-\infty}^{\infty } \!\!\! \dd{z} f^2(z) \>,$$ and allows one to rewrite $A(t)$ in terms of the conserved mass $M$, the width parameter $\beta(t)$, and a constant $C_1$ whose value depends on $f(z)$. Thus, $$\label{var.e:3} A^2(t) = \frac {M \beta(t)}{C_1} \qc C_1 = \int_{-\infty}^{\infty } \!\!\! \dd{z} f^2(z) \>.$$ We will therefore keep $M$ in our definition of $A(t)$ since it will be a relevant parameter when $\kappa=2$. For $f(z) = \sech^{1/\kappa}(z)$, one obtains $$\label{var.e:4} C_1 = \frac{\sqrt{\pi } \, \Gamma[ 1/\kappa ]}{\Gamma[1/2+1/\kappa]} \>.$$ Setting $\beta(t) = 1/G(t)$, in terms of the new collective coordinates $\qty[G,\Lambda]$, the Lagrangian is given by $$\label{var.e:5} L[G,\Lambda] = K[G,\Lambda] - H[G,\Lambda] \>,$$ where \[var.e:6\] $$\begin{aligned} \frac{K[G,\Lambda]}{M} &= \frac{i}{2 M} \int \dd{x} \qty[ \psi^{\ast} (\partial_t \psi) - (\partial_t \psi^{\ast} ) \psi ] \notag \\ &= \frac{1}{2} \, v^2 + \omega - \dot{\Lambda} \, G^2 \, \frac{C_2}{C_1} \>, \label{var.e:6a} \\ \frac{H[G,\Lambda]}{M} &= - \frac{1}{M} \int \dd{x} \qty[ ( \partial_x \psi^{\ast} )( \partial_x \psi) + | \psi |^{2(\kappa+1)} / ( \kappa + 1 ) ] \notag \\ & \hspace{-3em} = \frac{v^2}{4} + \frac{C_3}{C_1} \, \frac{1}{G^2} + 4 \Lambda^2 \, \frac{C_2}{C_1} \, G^2 - \frac{1}{(\kappa + 1)} \, \frac{C_4}{C_1} \, \qty( \frac{M}{C_1 G} )^{\kappa} \>, \label{var.e:6b}\end{aligned}$$ where \[var.e:7\] $$\begin{aligned} C_2 &= \int_{-\infty}^{\infty } \!\!\! \dd{z} z^2 f^2(z) \label{var.e:7a} \\ &= 2^{(2/\kappa - 1)} \kappa^3 \, {}_4F_3( \, 1/\kappa, 1/\kappa, 1/\kappa, 2/\kappa; \notag \\ & \qquad\qquad 1+1/\kappa, 1+1/\kappa, 1+1/\kappa; -1 \, ) \>, \notag \\ C_3 &= \int_{-\infty}^{\infty } \!\!\! \dd{z} \qty( f'(z) )^2 \label{var.e:7b} \\ &= \frac{\sqrt{\pi} \, \Gamma[ 1+1/\kappa ]}{2 \kappa \, \Gamma[3/2+1/\kappa]} \>, \notag \\ C_4 &= \int_{-\infty}^{\infty } \!\!\! \dd{z} f^{(2 \kappa + 2)}(z) \label{var.e:7c} \\ &= \frac{\sqrt{\pi} \, \Gamma[ 1+1/\kappa ]}{2 \kappa \, \Gamma[3/2+1/\kappa]} = 2 \kappa \, C_3 \>. \notag\end{aligned}$$ Collecting terms from and , the Lagrangian is given by $$\begin{aligned} \frac{L[G,\Lambda]}{M} &= \frac{1}{4} \, v^2 + \omega - \dot{\Lambda} \, G^2 \, \frac{C_2}{C_1} - \frac{C_3}{C_1} \, \frac{1}{G^2} \label{var.e:8} \\ & \qquad - 4 \Lambda^2 \, \frac{C_2}{C_1} \, G^2 + \frac{1}{(\kappa + 1)} \, \frac{C_4}{C_1} \, \qty( \frac{M}{C_1 G} )^{\kappa} \>. \notag\end{aligned}$$ From the Euler-Lagrange equations we obtain the second order differential equation for $G$, $$\label{var.e:9} \ddot{G} = 4 \, \frac{C_3}{C_2} \, \frac{1}{G^3} - \frac{4 \, \kappa^2}{(\kappa + 1)} \, \frac{ C_3}{C_2 G} \, \qty( \frac{M}{C_1 G} )^{\kappa} \>,$$ and the relation $\Lambda = \dot{G}/(2G)$. In solving these equations, we will use for the mass $M$, when we are not at the critical value $\kappa=2$, the expression for the mass for the solitary wave solution given by Eq. . If we do this, we can rewrite Eq.  as $$\label {gddot2} \ddot G = 4 \frac{C_3}{C_2} \frac{1}{ G^3 } - \frac{C_3}{C_2 } \frac{4}{G^{\kappa+1} } \>.$$ One notices that for $G[0]=1$, $\ddot G =0$, as it must for an exact solution. We see that to get $G \rightarrow 0$ when $\kappa=2$ we need to have $M > M^{\star}$ where $M^{\star}$ is the value of the mass for the exact solution. So initial conditions with a mass greater than this are necessary to see blow up at $\kappa=2$. By multiplying both sides of by $\dot G$ and integrating with respect to time we obtain a first integral of the second order differential equation, which up to a multiplicative factor is the same as setting the conserved Hamiltonian divided by the mass $M$ to a constant $E$. This gives $$\label{var.e:10} E = \frac{C_2}{C_1} \frac{ \dot G ^2}{4} + \frac{C_3}{C_1} \frac{1}{ G^2 } - \frac{1}{(\kappa+1)} \frac{2 \kappa C_3}{C_1} \, \qty( \frac{M}{C_1 G} )^\kappa \>.$$ We notice that at the critical value of $\kappa =2$, the last two terms both go like $1/G^2$. Self-focusing occurs when the width can go to zero. Since $\dot G^2 $ needs to be positive, this means that at $\kappa=2$, the mass has to be greater than $M^\star$ for $G$ to be able to go to zero. We find [@NLDE] $$\label{var.e:11} M^\star = \sqrt{ \frac{3 C_1^2} {4} } = \frac{\pi}{2}\sqrt{3} = 2.7207 \dotsb \>,$$ provided we use the exact solution (which is a zero-energy solution) for $\kappa=2$, namely $f=\sech^{1/2}(z)$. This agrees well with numerical estimates of the critical mass [@Rose:p] and is slightly lower than the variational estimate obtained earlier by Cooper [*et al.*]{} [@variational] using a post-Gaussian trial wave functions instead of a trial wave function based on the exact solution. For $\kappa \neq 2$, if we use the mass of the exact solitary wave solution, the energy conservation equation simplifies to $$E = \frac{C_2}{C_1} \, \frac{ \dot G ^2}{4}+\frac{C_3}{C_1}\, \qty( \frac{1}{ G^2 } - \frac{2}{\kappa G^\kappa} )$$ In the supercritical case when $G \to 0$, we have $$\label{var.e:12} \frac{C_2}{C_1} \, \frac{ \dot G ^2}{4} = \frac{1}{(\kappa+1)} \frac{2 \kappa C_3}{C_1} \, \qty( \frac{M }{C_1 G} )^\kappa \>.$$ This “mean-field” result was obtained earlier in Refs. [@var2; @variational]. To show the difference between the stability at $\kappa=3/2$ and $\kappa=5/2$, we have solved Eq.  for the initial conditions $G(0)=0.001 $, $\dot{G}(0) = 0$, with the results shown in Fig. \[f:NLSE-3-5\]. For small oscillations we can assume $$\label{var.e:13} G(t) = 1 + \epsilon \, g(t) \>,$$ from which we obtain the equation, $$\begin{gathered} \label{var.e:14} \ddot{g} + \omega^2 \, g = 0 \>, \\ \omega^2 = \qty( C_3 / C_2 ) \, \qty[ 12 - 4 \kappa^2 \qty( M/C_1 )^\kappa ] \>. \notag\end{gathered}$$ Setting $\omega=0$ in Eq. , leads to the same criterion for the critical mass when $\kappa=2$. The same equation gives the frequency of small oscillations when $\kappa < 2$. For $\kappa = 3/2$, the predicted period of oscillation is $T=2 \pi/\omega = 12.5998$ in good agreement with Fig. \[f:NLSE-3-5\]a. ![\[f:NLSE-3-5\] $G(t)$ from Eq.  for (a) $\kappa = 3/2$ and (b) $\kappa = 5/2$. The latter case corresponds to “blowup".](NLSE-blowup-v2.pdf){width="0.9\columnwidth"} ![\[f:Vfit\] Potential fit for $\kappa = 3/2$. The solid line (blue online) is the exact derivative of the potential from Eq. , the dashed line (red online) is the fitted function $F(a,b,c,d)$ of Eq. .](dVdG-3-2-NL-v3.pdf){width="0.9\columnwidth"} \[s:varNLSEexternal\]Adding an external potential ------------------------------------------------- Now we would like to see how this argument is modified when we add the external potential $V^{-}(x)$. In this case the exact solution is “pinned” to the origin. The Lagrangian is again given by Eq.  with the addition of the potential term: $$\begin{aligned} \label{var.e:15} H'[G] / M &= \int_{-\infty}^{+\infty} \!\!\! \dd{x} \psi^{\ast}(x,t) \, V^{-}(x) \, \psi(x,t) / M \\ &= - (b^2 - 1/4)\, {\mathcal{V}}[G,\kappa] / C_1 \>, \notag\end{aligned}$$ where $$\begin{aligned} \label{var.e:16} {\mathcal{V}}[G,\kappa] &= \int_{-\infty}^{+\infty} \!\!\! \dd{y} \sech^{2/\kappa}(y) \, \sech^{2}(G y) \>, \\ &\xrightarrow{G \to 1} \frac{\sqrt{\pi} \, \Gamma[1+1/\kappa]}{\Gamma[3/2+1/\kappa]} \>. \notag\end{aligned}$$ The Lagrangian now becomes: $$\begin{aligned} &\frac{L[G,\Lambda]}{M} = \frac{1}{4} \, v^2 + \omega - \dot{\Lambda} \, G^2 \, \frac{C_2}{C_1} - \frac{C_3}{C_1} \, \frac{1}{G^2} - 4 \Lambda^2 \, \frac{C_2}{C_1} \, G^2 \notag \\ & \> + \frac{1}{(\kappa + 1)} \, \frac{C_4}{C_1} \, \qty( \frac{M}{C_1 G} )^{\kappa} + \frac{(b^2 - 1/4)}{C_1} \, {\mathcal{V}}[G,\kappa] \>. \label{var.e:17}\end{aligned}$$ The Euler-Lagrange equations now give $$\begin{aligned} \label{var.e:18} \ddot{G} &= 4 \, \frac{C_3}{C_2} \, \frac{1}{G^3} - \frac{4 \, \kappa^2}{(\kappa + 1)} \, \frac{ C_3}{C_2 G} \, \qty( \frac{M}{C_1 G} )^{\kappa} \\ & \hspace{2em} + \frac{2 \, (b^2 - 1/4)}{C_1} \, \pdv{{\mathcal{V}}[G,\kappa]}{G} \>, \notag\end{aligned}$$ where $$\begin{aligned} \pdv{{\mathcal{V}}[G,\kappa]}{G} &= - 2 \int_{-\infty}^{+\infty} \!\!\! \dd{y} y \, \sech(Gy) \, \sech^{3}(G y) \, \sech^{2/\kappa}(y) \notag \\ &\xrightarrow{G \to 1} - \frac{\kappa}{\kappa + 1} \, \frac{\sqrt{\pi} \, \Gamma[1+1/\kappa]}{\Gamma[3/2+1/\kappa]} \>. \label{var.e:19}\end{aligned}$$ To solve this equation numerically we fit the numerical values of the integral in by a function of the form: $$\label{var.e:20} F(a,b,c,d) = a \, e^{-d \, G} \sech^b(G) \tanh (c \, G) \>.$$ Using Mathematica, one obtains an extremely accurate 4-parameter fit. For example, the result of this fit for $\kappa = 3/2$ is shown in Fig. \[f:Vfit\] for $a=4.290$, $b=-1.528$, $c=-2.214$, and $d=2.222$. Different fit parameters are used for each value of $\kappa$. Plots of the solutions $G(t)$ of Eq.  for different values of $b$ and for $\kappa = 3/2$, $2$, $2.1$, and $5/2$ are shown in Fig. \[f:Gt\]. ![\[f:Gt\] The solid (blue online), dotted (green online), and dashed (red online) lines are the solutions $G(t)$ of Eq.  for $\kappa = 3/2$, $2$, $2.1$, and $5/2$, and (a) $b^2 = 0.25,0.75,1.3611$, for $\kappa = 3/2$, (b) $b^2 = 0.25,0.5,1$, for $\kappa = 2$, (c) $b^2 = 0.25,0.5,0.95$, for $\kappa = 2.1$, and (d) $b^2 = 0.25,0.5,0.81$, for $\kappa = 5/2$.](Gt-kappa-NL.pdf){width="0.87\columnwidth"} We can study analytically the stability of the solutions in this variational approximation by linearizing Eq.  around the exact solution $G=1$, $$\label{var.e:21} G(t) = 1 + \epsilon \, g(t) \>.$$ To evaluate the effect of the external potential on the small oscillation equation we just need to know that: $$\begin{aligned} \label{var.e:22} &y \, \sech(Gy) \, \sech^{3}(G y) = y \, \sech(y) \, \sech^{3}(y) \\ & \quad - \epsilon \, g(t) \, y^2 \qty[ 2 \sech^2(y) - 3 \sech^4(y) ] + \order{ \epsilon^2 } \notag \>. \end{aligned}$$ Substitution of this expansion into gives $$\begin{gathered} \label{var.e:22a} \ddot{g} + \omega^2 \, g = 0 \>, \\ \omega^2(\kappa,b) = (C_3/C_2) \, \qty[ 12 - 4 \kappa^2 \, \qty( b_{\text{max}}^2(\kappa) - b^2 ) ] \notag \\ \hspace{3em} - 4 \, ( b^2 - 1/4 ) \, \qty[ 2 \, ug(\kappa) - 3 \, ug_2(\kappa) ] / C_2 \notag \>, \end{gathered}$$ with $b_{\text{max}}(\kappa) = 1/2 + 1/\kappa$, where $ug(\kappa)$ is given by Eq.  and where $$\label{var.e:23} ug_2(\kappa) = \int_{-\infty}^{+\infty} \!\!\! \dd{y} y^2 \, \sech^{2/\kappa + 4}(y) \>.$$ The collective coordinate method allows one to approximately calculate the small oscillation frequency as well as the time evolution of the system using Eq. . In Fig. \[f:Gt\] we assumed $\epsilon = 0.01, g(0)= -1 , \dot g = 0$. The relevant values of $b_{\text{crit}}$ are $1/2, 0.525, 0.5785$ for $\kappa = 2, 2.1, 2.5$. For $\kappa=3/2$ we get oscillation for the entire range from $b=1/2$ to $b=b_{max}$ as seen in Fig. \[f:Gt\]a. As predicted, for $\kappa=2$, once we get above $b^2=1/4$, which is the case with no potential, then the solution is stable as seen in Fig. \[f:Gt\]b. For $\kappa=2.1$, once we get above $b=b_{\text{crit}}$, then the solution is stable as seen in Fig. \[f:Gt\]c. For $\kappa=5/2$ we get similar results to $\kappa= 2.1$, as seen in Fig. \[f:Gt\]d. In the stable regime, the oscillation periods are accurately predicted by Eq. . Setting $\omega^2(\kappa,b)=0$, determines the critical value of $b$ at a given $\kappa$ below which the solutions are unstable for $\kappa > 2$. The expression for $b_{\text{crit}}$ obtained this way is identical to the expression for $b_{\text{crit}}$ obtained from Derrick’s theorem in Eq.  and shown in Fig. \[NS.f:bvskappa\]. In the domain of instability one finds that if we look at initial conditions where $G(0)=1, (g(0)=0)$ and $\epsilon=0.01$, $\dot g = \pm 1$, then for the minus sign one gets “blow up” ($G \rightarrow 0$), and for the plus sign we get collapse of the solution ($G \rightarrow \infty$). In Fig. \[collapse\], we give an example of collapse when $\kappa = 3$ and we are in the unstable regime. A first integral of the second order differential equation resulting from the Lagrange’s equation for $G$ can be obtained by setting the conserved Hamiltonian to a constant $E$. One then has $$\begin{aligned} E &= \frac{C_3}{C_1} \frac{1}{ G^2 } + 4 \Lambda^2 \, \frac{C_2}{C_1} \, G^2 - \frac{2 \kappa}{(\kappa+1)}\frac{C_3}{C_1} \qty( \frac{M }{C_1 G})^\kappa \notag \\ & \hspace{2em} - \frac{( b^2 - 1/4 )}{C_1 } \, {\mathcal{V}}[G,\kappa] \>. \label{Econs}\end{aligned}$$ From the energy conservation equation we can see immediately that at $\kappa=2$ the exact solution we found does not blow up. This is for two reasons: first, when the width parameter $ G \rightarrow 0$, then ${\mathcal{V}}[G,\kappa]$ becomes a constant independent of $G$ and therefore the potential does not affect the small $G$ behavior of the differential equation; secondly the mass of the exact solution depends now on $b$ and $\kappa$ and it is lower than the critical mass needed for blowup. That is, the mass of the bound solution is given by: $$\begin{aligned} M &= A^2[\kappa,b] \, C_1[\kappa] = \frac{\sqrt{\pi} \, \qty( b_{\text{max}}^2(\kappa) - b^2 )^{1/\kappa} \, \Gamma[1/\kappa] }{ \Gamma[1/2 + 1/\kappa] } \notag \\ & \hspace{2em} \xrightarrow{\kappa \to 2} \pi \, \sqrt{1 - b^2} \>. \label{Mkappav} \end{aligned}$$ The maximum value of this occurs when the external potential goes to zero at $b=1/2$. When $b > 1/2$, the mass of the exact solution is *always* less than $M^{\star}$, so that these solutions are always stable when $\kappa=2$. For the NLSE with no external potential, when the stability depends on the mass of the initial wave function at $\kappa=2$, the critical value is that of the exact solitary wave solution. See also Fig. \[NS.f:Mcrit\] and the discussion thereof. \[NS.s:LinearStability\]Linear Stability ======================================== Let us perform the linear stability analysis of the solitary wave solutions $\phi_{\omega}(x) \, e^{-i\omega t}$ to the nonlinear [Schr[ö]{}dinger]{} equation in the external potential. We take a perturbed solitary wave solution in the form $\psi(x,t)= [\, \phi_{\omega}(x)+r(x,t) \, ] \, e^{-i\omega t}$ and consider the linearized equation on $R(x,t) = \qty[ \, \Re{r(x,t)},\Im{r(x,t)} \,]$, $$\partial_t R(x,t) = {\mathcal{A}}(\omega) \, R(x,t) \>.$$ If the spectrum of ${\mathcal{A}}(\omega)$ has eigenvalues with positive real part, then the corresponding solitary wave is called linearly unstable; otherwise, it is called spectrally stable. In general, the spectral stability does not imply nonlinear stability, but for the nodeless solutions to the nonlinear [Schr[ö]{}dinger]{} equation one can use the Lyapunov-type approach to prove the orbital stability; see e.g. Ref. [@MR901236]. The equation we are solving is $$\label{nls-potential} \qty{ i \, \partial_t + \partial_x^2 - V^{-}(x) + | \psi(x,t) |^{2\kappa} } \, \psi(x,t) = 0 \>,$$ with $$\label{def-V} V^{-}(x) = -\qty(b^2-1/4) \, \sech^2{x} \>.$$ We are interested in the stability of the solitary wave solution $\psi_{\omega,b}(x,t) = \phi_{\omega,b}(x) \, e^{-i \omega t}$ to , with the amplitude $\phi_{\omega,b}(x)$ satisfying $$\label{stationary-eq} \omega \, \phi_{\omega,b}(x) = \qty[ - \partial_x^2 + V\sp{-}(x) - \abs{\phi_{\omega,b}(x)}^{2\kappa} ] \, \phi_{\omega,b}(x)\>.$$ For $\omega = \omega_{\kappa} = -1/\kappa^2$, one has the explicit expression $$\label{PF.e:psiphi-def} \phi_{\omega_{\kappa},b}(x) = \qty[ b_{\text{max}}^2(\kappa) - b^2 ]^{1/(2\kappa)} \, \sech^{1/\kappa}{x} \>,$$ with $b_{\text{max}}(\kappa) = 1/2 + 1/\kappa$. We will perform the spectral analysis of the linearization operator following the V–K approach [@stab1]. We consider the perturbation of the solitary wave, $\psi(x,t) = \qty[ \, \phi_{\omega,b}(x)+r(x,t) \, ] \, e^{-i\omega t}$, with $r(x,t)=u(x,t)+iv(x,t)$, and with $u(x,t)$ and $v(x,t)$ real. The linearized equation on $u(x,t)$ and $v(x,t)$ is given by $$\begin{aligned} \partial_t \begin{pmatrix}u\\v\end{pmatrix} &= \mathcal{A}(\omega,b)\begin{pmatrix}u\\v\end{pmatrix} \label{def-operator-A} \\ &= \begin{pmatrix}0&L_{-}(\omega,b)\\-L_{+}(\omega,b)&0\end{pmatrix} \, \begin{pmatrix}u\\v\end{pmatrix} \>, \notag\end{aligned}$$ where the self-adjoint operators $L_\pm(\omega,b)$ are given by \[Lpmdefs\] $$\begin{aligned} L_{-}(\omega,b) &= - \partial^2_x + V^{-}(x) - \omega - |\phi\sb{\omega,b}|^{2\kappa} \>, \label{Lpmdefs-a} \\ L_{+}(\omega,b) &= - \partial^2_x + V^{-}(x) - \omega - \qty( 2\kappa+1 ) \, |\phi\sb{\omega,b}|^{2\kappa} \>. \label{Lpmdefs-b}\end{aligned}$$ The stationary equation satisfied by $\phi_{\omega,b}$ and its derivative with respect to $\omega$ give the relations $$\label{lm-lp} L_{-}(\omega,b)\,\phi_{\omega,b} = 0 \qc L_{+}(\omega,b) \, \partial_{\omega} \phi_{\omega,b} = \phi_{\omega,b} \>.$$ We need to perform the spectral analysis of $L\sb{-}(\omega,b)$ and $L\sb{+}(\omega,b)$. We start with reviewing the V–K approach from [@comech] for the case $b=1/2$, when $V^{-}(x)\equiv 0$. For a given value $\kappa > 0$, let $$\begin{aligned} \label{def-varphi} \varphi_\omega(x) &:= \phi\sb{\omega,1/2}(x) \\ &= (\kappa+1)^{1/(2\kappa)} \, |\omega|^{1/(2\kappa)} \, \sech^{1/\kappa}(\kappa x\sqrt{|\omega|}) \notag\end{aligned}$$ be the profile of a solitary wave for the case when $V^{-}(x)=0$ (when $b=1/2$). By the V–K theory, the linearization at $\varphi_\omega$ is such that $$L_{-}(\omega) = L_{-}(\omega,1/2) = - \partial^2_x - \omega - |\varphi_\omega|^{2\kappa}$$ has a simple eigenvalue $\lambda = 0$ as its smallest eigenvalue, corresponding to the eigenfunction $\varphi_\omega(x)$, while $$L_{+}(\omega)= L_{+}(\omega,1/2) = - \partial^2_x - \omega - \qty( 2 \kappa + 1) \,|\varphi_\omega|^{2\kappa}$$ has one simple negative eigenvalue on the subspace of even functions, and a simple eigenvalue at $\lambda=0$ on the subspace of odd functions corresponding to the eigenfunction $\partial_x \varphi_{\omega}(x)$. For any nonzero eigenvalue $\lambda \in \sigma_{p}({\mathcal{A}}(\omega,1/2))$ of the linearization operator from , one has the relation $\lambda^2\psi = -L_{-}(\omega)L_{+}(\omega) \, \psi$ with nonzero $\psi$. Being in the range of $L_{-}$, which is self-adjoint, $\psi$ is orthogonal to the null space of $L_{-}(\omega)$; this allows us to arrive at $$\label{lambda-square} \lambda^2 \, \expval{\psi,L_{-}(\omega)^{-1}\psi} = - \expval{\psi,L_{+}(\omega)\psi} \>,$$ hence $\lambda^2 \in \mathbb{R}$. Thus, the linear instability could only be caused by a positive eigenvalue of $\mathcal{A}(\omega,1/2)$. From , one can see that one could have $\lambda>0$ if the right-hand side of becomes positive for some $\psi$ orthogonal to the kernel of $L_{-}(\omega)$; in other words, if the minimization problem $$\label{mp} \mu = \inf_{\substack{ \expval{\psi,\varphi_{\omega} } = 0 , \\ \expval{\psi,\psi} = 1 }} \expval{ \psi,L_{+}(\omega)\psi }$$ gives a negative value of $\mu$. By [@stab1], finding the minimum of under constraints $\expval{\psi,\psi} = 1$ and $\expval{\psi,\varphi_{\omega}} = 0$ leads to the relation $$\label{lm} L_{+}(\omega) \, \psi = \mu \, \psi + \nu \, \varphi\sb{\omega} \>,$$ with $\mu,\,\nu$ Lagrange multipliers; pairing the above with $\psi$ shows that $\mu$ in and is the same. Writing $\psi = (L_{+}(\omega)-\mu)^{-1} \, \nu \, \varphi_{\omega}$ and taking into account that $\expval{\psi,\varphi_{\omega}} = 0$, we see that we need to analyze the location of the first root of the V–K function $$\label{vk-function} f(z) = \expval{ \varphi_{\omega}, (L_{+}(\omega) - z)^{-1} \, \varphi_{\omega} } \>,$$ which is defined for $z$ in the resolvent set of the operator $L\sb{+}(\omega)$ restricted onto the subspace of even functions. This domain includes the interval $(z_0,z_2)$, where $z_0 < 0$ is the smallest negative eigenvalue of $L_{+}$ and $z_2 > 0$ is the next eigenvalue of $L_{+}$, on the subspace of even functions. Since clearly $f'(z) > 0$ for $z \in (z_0,z_2)$, one has $f(\mu) = 0$ at some $\mu \in (z_0,z_2)$, $ \mu > 0$ (hence stability) if and only if $f(0) <0 $, which leads to $\expval{ \varphi\sb{\omega},L\sb{+}(\omega)^{-1}\varphi\sb{\omega} } < 0$, and, using , we arrive at the V–K stability condition $$\label{vk-condition} \dv{\omega} \expval{ \varphi_{\omega},\varphi_{\omega} } < 0 \>.$$ An elementary computation based on shows that is satisfied (for all $\omega<0$) if and only if $\kappa \in (0,2)$. The left-hand side of becomes identically zero for $\kappa = 2$ and becomes positive for $\kappa > 2$ (again, for all $\omega<0$). Now let us consider $L\sb\pm(\omega,b)$ with $b > 1/2$ and $\omega = \omega_{\kappa} = -1/\kappa^2$. As in the case of no potential, one has $L_{-}(\omega,b)\ge 0$, with $\lambda = 0$ a simple eigenvalue corresponding to the eigenfunction $\phi\sb{\omega,b}$. At $\omega = \omega_\kappa$, one has $$|\phi_{\omega,b}(x)|^{2\kappa} - V^{-}(x) = |\varphi_\omega(x)|^{2\kappa}$$ and $$L_{-}(\omega,b) = - \partial^2_x - \omega - |\varphi\sb{\omega}(x)|^{2\kappa} = L_{-}(\omega,1/2) \>.$$ We note that $$L_{+}(\omega,b) = L_{-}(\omega,b) - 2\kappa \, |\phi_{\omega,b}(x)|^{2\kappa} < L_{-}(\omega,b) \>,$$ hence the smallest eigenvalue $z_0(\omega,b)$ of $L_{+}(\omega,b)$ (assumed on the subspace of even functions) is negative. At $\omega = \omega_{\kappa} = - 1/\kappa^2$, one has $$\label{lp-kappa} L_{+}(\omega_{\kappa},b) = L_{+}(\omega_{\kappa},1/2) + 2\kappa \, (b^2-1/4) \, \sech^{2}{x} \>,$$ hence for $b > b'$ and $b,\,b' \in (1/2,b_{\text{max}}(\kappa))$, $$\label{lp-greater} L_{+}(\omega_{\kappa},b) > L_{+}(\omega_{\kappa},b') \>.$$ Just as in the case $b=1/2$ which we considered above, the linear instability takes place when the minimization problem $$\label{mp-b} \mu = \inf_{\substack{ \expval{\psi,\varphi_{\omega} } = 0 , \\ \expval{\psi,\psi} = 1 }} \expval{ \psi,L_{+}(\omega,b)\psi }$$ gives a negative value of $\mu$. As we already pointed out in the case $b=1/2$, one has $\mu > 0$ for $\kappa \in (0,2)$ (equivalently, $\varphi_{\omega}$ are linearly stable), and $\mu=0$ for $\kappa=2$. Due to , one then also has $\mu > 0$ for $\kappa\in (0,2)$, $\omega = -1/\kappa^2$, $b \in [\, 1/2,b_{\text{max}}(\kappa) \,)$ and for $\kappa=2$, $\omega=-1/\kappa^2$, $b \in (\, 1/2,b_{\text{max}}(\kappa) \,)$. Thus, for these values of $\kappa$ and $b$, the solitary waves $\phi\sb{\omega,b}e^{-\omega t}$ are spectrally stable. For $\kappa>2$, the story is different: while $\mu$ in is negative for $b=1/2$ corresponding to the linear instability of $\varphi\sb\omega(x) e^{-i\omega t}$, $\mu$ could become positive if $b$ exceeds some critical value $b_{\text{crit}}(\kappa)$: $$\partial_{\omega} \! \expval{ \phi\sb{\omega,b},\phi\sb{\omega,b} } < 0 \qc\! b \in (b_{\text{crit}}(\kappa),b_{\text{max}}(\kappa)) \qc\! \omega=-1/\kappa^2.$$ Numerically, we proceed as follows. We pick $\kappa>2$ and use the shooting method to construct a solitary wave $\phi_{\omega,b}$ and find the critical value $b_{\text{crit}}>1/2$ above which $\partial_\omega \expval{\phi_{\omega,b},\phi_{\omega,b} }|_{(\omega_{\kappa},b)}$ becomes negative \[that is, when $b = b_{\text{crit}}(\kappa)$, a positive eigenvalue from the spectrum of $\mathcal{A}(\omega,b)$ collides with a negative eigenvalue, and they produce a pair of purely imaginary eigenvalues; for $b \in (b_{\text{crit}}(\kappa), b_{\text{max}}(\kappa))$, spectral stability takes place\]. This gives us the critical values $b_{\text{crit}}$ [vs.]{} $\kappa$ in agreement with Fig. \[NS.f:bvskappa\]. We find remarkably that Derrick’s theorem and the V–K spectral analysis of stability give identical results. The same result for the stability regime was also obtained by setting the oscillation frequency for small oscillations around the exact solution to zero using the time dependent variational method. In distinction with the case without a potential, in the presence of the external potential $V^{-}(x)$ the results of the stability analysis are much more interesting because of the additional $b$ dependence of the exact solution. For $\kappa >2$ it is possible to interpret the results of V-K and Derrick’s theorem in terms of a critical mass $M_{crit}$ below which the solution is stable, or equally in terms of a critical depth $b$ for the confining potential above which the solution is stable. \[s:conclude\]Conclusions ========================= In this paper we studied the stability of the exact solution of the NLSE in a real P[ö]{}schl-Teller potential which is the SUSY partner of a *complex* $\mathcal{PT}$ symmetric potential studied previously [@pt1]. Unlike the previous problem which required detailed numerical analysis for every value of the nonlinearity parameter $\kappa$, the real external potential problem here results in a Hamiltonian dynamical system amenable to several variational approaches to the stability problem, such as Derrick’s theorem [@ref:derrick], V–K theory [@stab1], and a time dependent variational approach. Using these methods we were able to show that for $\kappa >2$ the pinned solution has a region of stability that was not available to the solitary wave solution of the NLSE without an external potential. The latter solutions are known to blow up in a finite time interval when perturbed appropriately. The analytic result for the re-entry regime of stability found using Derrick’s theorem was corroborated by a numerical study of spectral stability based on the V–K theory. This result is different from the result found numerically for the stability of the solution for the complex SUSY partner external potential $V^{+}(x)$. The analysis of the stability of the solutions for $V^{+}(x)$ in [@pt1] showed a very complicated pattern. Even for $\kappa=1$ there is a regime of instability as a function of $b$ for the nodeless solution. At $\kappa=3$ all the solutions found for $V^{+}(x)$, were unstable due to oscillatory instabilities. Only for $\kappa < 2/3$ were the solutions stable. In contrast, for the $V^{-}(x)$ potential we are able to address the stability question for all $\kappa$ *analytically* and show that the effect of the external potential is to introduce a new domain of stability for all $\kappa > 2$, when compared to the stability of the related solitary wave solutions in the absence of an external potential. The stability properties of the solutions of the NLSE in the presence of the partner potentials $V^{\pm}(x)$ are quite different from one another due to the dissipative versus conservative nature of these potentials. F.C. would like to thank the Santa Fe Institute and the Center for Nonlinear Studies at Los Alamos National Laboratory for their hospitality. A.K. is grateful to Indian National Science Academy (INSA) for awarding him INSA Senior Scientist position at Savitribai Phule Pune University, Pune, India. The research of A.C. was carried out at the Institute for Information Transmission Problems, Russian Academy of Sciences at the expense of the Russian Foundation for Sciences (Project 14-50-00150). B.M. and J.F.D. would like to thank the Santa Fe Institute for their hospitality. B.M. acknowledges support from the National Science Foundation through its employee IR/D program. The work of A.S. was supported by the U.S. Department of Energy. [99]{} C. M. Bender, Rep. Prog. 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--- abstract: 'We perform a theoretical investigation into the classical and quantum dynamics of an optical field in a cavity containing a moving membrane (“membrane-in-the-middle” set-up). Our approach is based on the Maxwell wave equation, and complements previous studies based on an effective Hamiltonian. The analysis shows that for slowly moving and weakly reflective membranes the dynamics can be approximated by unitary, first-order-in-time evolution given by an effective Schrödinger-like equation with a Hamiltonian that does not depend on the membrane speed. This approximate theory is the one typically adopted in cavity optomechanics and we develop a criterion for its validity. However, in more general situations the full second-order wave equation predicts light dynamics which do not conserve energy, giving rise to parametric amplification (or reduction) that is forbidden under first order dynamics and can be considered to be the classical counterpart of the dynamical Casimir effect. The case of a membrane moving at constant velocity can be mapped onto the Landau-Zener problem but with additional terms responsible for field amplification. Furthermore, the nature of the adiabatic regime is rather different from the ordinary Schrödinger case, since mode amplitudes need not be constant even when there are no transitions between them. The Landau-Zener problem for a field is therefore richer than in the standard single-particle case. We use the work-energy theorem applied to the radiation pressure on the membrane as a self-consistency check for our solutions of the wave equation and as a tool to gain an intuitive understanding of energy pumped into/out of the light field by the motion of the membrane.' author: - 'F. Hasan' - 'D. H. J. O’Dell' title: 'Parametric amplification of light in a cavity with a moving dielectric membrane: Landau-Zener problem for the Maxwell field ' --- Introduction {#sec:intro} ============ Most textbooks on quantum optics (see, e.g. [@Heitler; @Loudon; @Walls+Milburn; @Mandel+Wolf]) begin with Maxwell’s equations and use them to obtain a wave equation for the field which is second order in time and space. The normal modes of this equation behave like independent harmonic oscillators and can be quantized by the methods of ordinary non-relativistic quantum mechanics. In this way, the quantum dynamics of the electromagnetic field is shown to be governed by the Schrödinger equation which is first order in time and hence unitary (the lack of Lorentz invariance in Schrödinger’s equation should not worry us because normal modes separate time and space [@Tong]). This standard procedure breaks down in the presence of moving mirrors or dielectrics because there are no normal modes in time dependent systems. Our mission in this paper is to study the nature of the dynamics, especially adiabaticity and parametric amplification, for an optical field in the presence of a moving dielectric in a cavity. In the absence of true normal modes we use time-evolving modes which become coupled, an approach inspired by the papers of C. K. Law [@Law1; @Law2; @Law3]. We are primarily interested in classical fields, however, we are naturally led to a comparison with the quantum case because under certain approximations the time-evolving classical modes obey first-order equations which are mathematically analogous to Schrödinger equations. The differences between first and second order wave equations have been previously studied in the context of the Klein-Gordon equation where it is known that the wave function cannot be interpreted as a probability amplitude, in contrast to that of the Schrödinger equation [@Greiner]. Indeed, the Klein-Gordon equation does not provide a consistent description of a single particle precisely because it allows particle creation and annihilation (the Klein-Gordon equation does, however, correctly describe the normal modes of a free spinless quantum field). Similarly, in the dynamical Casimir effect (DCE) pairs of photons are generated from the vacuum by a moving mirror [@Dodonov09; @Dalvit10], and here we study the classical analogue of this phenomenon in the form of parametric amplification. A well known form of the DCE is Davies-Fulling-DeWitt radiation [@Davies75; @Fulling76; @Dewitt] generated in response to the uniform acceleration of a single mirror in free space. It is related to the Unruh effect [@Unruh], and therefore ultimately to Hawking radiation [@Hawking]. The DCE in a cavity with a moving end mirror was first investigated by Moore in 1970 [@Moore]. If the mirror is oscillated at twice the frequency of a cavity mode the condition for parametric resonance is fulfilled and the effect is exponentially enhanced [@Dodonov95; @Dalvit99; @Plunien; @Schaller]. Still, the effect is tiny and various schemes have been devised to enhance or mimic it. When a gas or semiconductor is ionized to produce a plasma the refractive index can drop to near zero in a picosecond [@Yablonovitch; @Manko], and when the ionization is produced by a periodically pulsed laser the result can be a rapidly oscillating plasma mirror [@Lozovik; @Crocce; @Braggio05]. Similarly, a coherently pumped $\chi^{(2)}$ nonlinear crystal forms an optical parametric oscillator whose nonlinear susceptibility oscillates at optical frequencies [@Lambrecht10]. The first system to successfully observe the DCE operated in the microwave regime and used a superconducting circuit made of a coplanar transmission line, the effective length of which can be changed at frequencies exceeding 10 GHz by modulating the inductance [@Wilson; @Johansson]. The interaction of light with a moving dielectric is a rich problem whose history goes back at least as far as the investigations carried out by Fresnel [@Fresnel] and Fizeau [@Fizeau] in the 19th century. It has close connections to the theory of special relativity, and, in the case of nonuniform motion, to general relativity [@PiwnickiLeonhardt]. An active modern area of research that involves moving dielectrics is the field of optomechanics [@Kippenberg+Vahala07; @AspelmeyerKippenbergMarquardt], where light and mechanical oscillators are coupled through radiation pressure. The prototypical system consists of a cavity made of two mirrors, one of which is mounted on a spring. When pumped by a laser, the optical field that builds up inside the cavity can displace the mobile mirror by radiation pressure. Such a set-up was realized in 1983 by Dorsel *et al* [@dorsel83] who observed a lengthening of the cavity. The dynamic version of this effect, where the mirror position and light field amplitude oscillate, can be used to heat or cool the mirror motion, as first demonstrated by Braginsky and co-workers in experiments with microwave cavities [@braginsky67; @braginsky70] in the 1960s. The past decade has seen renewed theoretical [@KippenbergRae; @GirvinMarquardt; @AspelmeyerGenes; @FreegardeXuereb] and experimental [@AspelmeyerZeilinger; @arcizet06; @Schliesser06; @VahalaRokhsari; @KarraiFavero; @Schliesser08; @thompson08; @HarrisJayich; @HarrisZwickl; @Painter1; @Groblacher09; @Park09; @Schliesser09; @Rocheleau10; @Sankey2010; @KippenbergRiviere; @Chan11; @LehnertRegal1; @HarrisLee] activity in optomechanics, with one of the principal aims being to laser cool a mechanical object towards its quantum ground state. In particular, the experiment [@Chan11] achieved a sub-single phonon occupancy of a nanomechanical oscillator. Optomechanical systems have now been realized in diverse physical media including ultrahigh-Q microtoroids [@VahalaRokhsari], mirrors attached to cantilevers [@KarraiFavero; @AspelmeyerZeilinger], optomechanical crystals [@Painter1], mechanical oscillators in microwave and optical cavities [@LehnertRegal1], cold atom clouds [@murch08; @brennecke08], hybrid atom-membrane optomechanics [@Treutlein1; @MeystreBariani], as well as the ‘membrane-in-the-middle’ cavities [@thompson08; @HarrisJayich; @HarrisZwickl; @Sankey2010; @HarrisLee] that will be the focus of this paper. Radiation pressure and its quantum fluctuations (shot noise) on mirrors also turn out to be significant issues in high precision optical interferometers, like those designed to detect gravitational waves [@Braginsky+Manukin; @braginsky01; @braginsky02; @corbitt06]. In this paper we investigate the dynamics of light stored in a ‘membrane-in-the-middle’ type optical cavity, as depicted schematically in Figure \[fig:DoubleCavityPic\]. This arrangement has been realized in a series of experiments by the Yale group [@thompson08; @HarrisJayich; @HarrisZwickl; @Sankey2010; @HarrisLee], and is made of two highly-reflective end mirrors between which a thin moveable membrane (slab of SiN dielectric approximately 50 nm thick) is suspended, forming two subcavities. Light is transmitted between the two cavities at a rate determined by the membrane reflectivity: when its reflectivity is high the membrane strongly alters the optical mode structure of the cavity producing a network of avoided crossings as a function of membrane displacement (see Figure \[fig:AvoidedCrossing\]). The quadratic form of the mode structure at an avoided crossing lends itself to a quantum non-demolition measurement of the membrane’s energy and hence a fundamental demonstration of the quantization of the energy of a mechanical oscillator, something which is not possible with linear coupling [@Sankey2010; @HarrisLee]. Nonclassical correlations between two mechanical modes in such membranes has also been demonstrated experimentally [@Patil15; @Vengalatorre15]. The small gaps between the optical modes at avoided crossings in a membrane-in-the-middle cavity mean that such systems have a fundamentally multi-mode character. This has led other authors [@HarrisPhotonShuttle], as well as us [@NickPaper], to suggest that Landau-Zener type physics might be relevant to the optical dynamics caused by membrane motion. The celebrated Landau-Zener problem is one of the few exactly solvable problems in time-dependent quantum mechanics and provides a paradigm for analyzing the dynamical control of quantum systems, including the breakdown of adiabatic transfer between states. Applying this to the electromagnetic field where photon number is not conserved is one of the main themes of this paper. In our previous paper [@NickPaper] we showed how to approximately map the dynamics of two interacting *classical* optical fields obeying the Maxwell wave equation in the membrane-in-the-middle cavity system onto the *mathematics* of the Landau-Zener model, and hence how to analyze the efficiency of light transfer from one subcavity to the other by moving the membrane. Such deterministic transfer of light between cavities is a basic element of a quantum network [@Kimble], and has technological significance for cavity-QED realizations of quantum information processing [@Roadmap]. The optomechanical interaction between a mirror and a cavity mode of frequency $\omega_{\mathrm{cav}}$ arises from radiation pressure and is usually written [@AspelmeyerKippenbergMarquardt; @HarrisJayich; @HarrisPhotonShuttle; @MarquardtHeinrich2] $$\begin{aligned} \hat{H}_{\mathrm{optomech}} & = & \hbar \omega_{\mathrm{cav}}(x) \hat{a}^{\dag} \hat{a} \\ & = & \hbar \left(\omega_{\mathrm{cav}} + x \frac{ \partial \omega_{\mathrm{cav}}}{\partial x} + x^2 \frac{ \partial^2 \omega_{\mathrm{cav}}}{\partial x^2} \ldots \right) \hat{a}^{\dag} \hat{a} \nonumber \label{eq:Hoptomech}\end{aligned}$$ where $x$ is the mirror displacement from equilibrium and $\hat{a}^{\dag} \hat{a}$ gives the number of photons in the cavity mode. This interaction depends parametrically on mirror displacement through the dependence of the mode frequency $\omega_{\mathrm{cav}}$ on $x$. For small displacements in comparison to the optical wavelength it is sufficient to expand $\omega_{\mathrm{cav}}$ as shown; for the case of a single cavity with a mobile end mirror only the linear term is needed, but in a double cavity near an avoided crossing this vanishes and the leading term is quadratic. In currently experimentally accessible regimes this Hamiltonian gives an excellent description. Nevertheless, $H_{\mathrm{optomech}}$ as written above has *no dependence on the mirror’s speed* (for either the exact or expanded form). This means that the DCE is excluded which is unsatisfactory form a purely theoretical standpoint. A more complete Hamiltonian for the double cavity which does include the DCE has been derived by Law [@Law3] and will be discussed in Section \[sec:DCE\]. However, rather than using a Hamiltonian, our approach here will be based on the Maxwell wave equation. A similar approach to the one we take has recently been used by Castaños and Weder [@WederCastanos] to describe a single cavity with a mobile end mirror; they combine Maxwell’s wave equation for the light with Newton’s equation for the mobile mirror. We, on the other hand, give the membrane a prescribed trajectory in order to make full contact with the Landau-Zener problem. The DCE is a phenomenon that originates in quantum zero-point fluctuations of the vacuum [@Jaekel], and so does not occur in the classical field description. Nevertheless, there are classical analogues to the creation/annihilation of photons in the form of parametric amplification/reduction of pre-existing classical fields by time-dependent cavity boundaries [@Askaryan], however even this is absent in the standard Landau-Zener model applied previously to the membrane-in-the-middle system in references [@HarrisPhotonShuttle; @NickPaper] because it obeys unitary time evolution. The approximate mapping from the Maxwell wave equation to an effective Schrödinger-like wave equation has the unintended consequence of treating classical field amplitudes as though they were probability amplitudes; whereas the sum of the squares of classical field amplitudes is proportional to the total energy and is not constrained to be constant in a time-dependent cavity, the sum of the squares of probability amplitudes is fixed at unity even for time-dependent Hamiltonians. The second-order-in-time nature of the Maxwell wave equation thus allows for a richer dynamical behaviour than is present in the standard Landau-Zener problem. We shall see that the second-order-in-time dynamics includes a type of evolution which is adiabatic in the sense that there is no transfer (scattering) of light between modes and yet the magnitudes can still change due to parametric amplification/reduction, something which cannot occur in unitary evolution. In this paper we shall specifically investigate how such “beyond Landau-Zener” phenomena depend on membrane speed and reflectivity. Nonunitary effects such as parametric amplification of the cavity field are negligible in standard optomechanical experiments but interesting from a fundamental perspective. In order to evaluate the prospects of observing them it is important to know what membrane speeds can be achieved. One way to move the membrane in a prescribed motion such as a Landau-Zener sweep is to use a piezoelectric motor (as is used, for example, to stabilize cavities against vibrations [@Ottl]). The maximum speed would then be around 10 m/s. However, much greater effective speeds can be achieved without moving the membrane at all, but by instead filling the subcavities with dielectrics whose indices of refraction can be changed independently in time, thereby changing their relative optical lengths, see Appendix C of [@NickPaper]. Ultrafast electro-optical control of the refractive index allows the effective optical length of a cavity to be changed on time scales shorter than 10 ps [@LipsonPreble; @LipsonDong; @Reed; @Yanik; @LipsonXu07]. This control can be achieved by using a laser to excite a plasma of free charge in the dielectric, similar to the original proposal in reference [@Yablonovitch] mentioned above. Related effects can also be generated electrically [@LipsonXu05]. In this way we estimate that effective membrane speeds of $20,000$ m/s are achievable. A very important conceptual and practical difference between the ‘membrane-in-the-middle’ setup considered here and the original Davis-Fulling-DeWitt moving mirror proposals, as well as Moore’s moving cavity end mirror, is that in the latter cases a perfectly reflective mirror moves, whereas in the former case a dielectric membrane of finite reflectivity moves. While a perfect mirror is in any case an idealization, when it moves it leads to pathologies in the theory as recognized by Moore [@Moore]. According to Barton and Eberlein [@Barton93] “In essence, the displacement of a perfectly reflecting surface forces the description of the quantized field out of the original Hilbert space and into another.” In other words, the creation and annihilation operators for the field for two different mirror positions cannot be defined in the same Hilbert space. Ways around this problem include only working with dielectrics with finite refractive indices, like in references [@Barton93] and [@Salamone94], or to use Law’s [@Law1; @Law2; @Law3] effective Hamiltonian approach which does not even attempt to describe the true interactions between the field and the charges and currents in the mirror but rather imposes the zero boundary condition at the mirror by hand and then works with the photon operators associated with the continuously evolving ‘instantaneous’ modes. In this paper we adopt a hybrid approach in which the only moving element is a dielectric with a finite refractive index plus we impose the zero boundary condition at the stationary end mirrors. ![Schematic of the amplitude of light in a double cavity with perfectly reflective end mirrors and a partially transmissive, moveable central membrane.[]{data-label="fig:DoubleCavityPic"}](DoubleCavitypap){width="0.9\columnwidth"} The plan for this paper is as follows. In Section \[sec:waveequation\] we derive the wave equation obeyed by the electric field in the presence of a moving dielectric in the non-relativistic regime. In Section \[sec:setup\] we solve it for its normal modes in the case of a static membrane. Section \[sec:adiabaticbasisEOM\] considers the case where the membrane is moving and derives the general equations of motion for the optical field by expanding it over the instantaneous normal modes, i.e. an adiabatic basis that continuously evolves. An important special case that plays a central role in this paper is that of two modes interacting at an avoided crossing: in Section \[sec:CavityEnergy\] we give expressions for the energy of such a dichromatic field. In Section \[sec:adiabatictheorem\] we give the results of numerically solving the general equations of motion for a membrane moving at a constant velocity which gives a Landau-Zener type sweep of the field through an avoided crossing. In Section \[sec:LocalModeDynamics\] we solve the same problem but in the local (diabatic) basis where approximations can be more readily made (and which we verify numerically). These approximations include neglecting the time-dependence of the mode functions and also reducing the second-order-in-time wave equation to one which is first order. We obtain, in Section \[sec:ApproximationCondition\], an analytic criterion for when this first order reduction is valid in terms of the basic parameters of membrane reflectivity and speed. Section \[sec:RadiationPressure\] gives a physical explanation for the change in energy in the cavity as the membrane moves in terms of the work done by the radiation pressure on the membrane. Throughout this paper we try to compare and contrast the classical description of an optical field with the quantum case. This approach culminates in Section \[sec:DCE\] where we give a detailed quantum description and connect it to the classical field dynamics. We give our conclusions in Section \[sec:conclusion\]. We have also included four appendices which contain details excluded from the main text; Appendix \[app:NonrelativisticApproximation\] discusses relativistic corrections to the wave equation in the presence of a moving dielectric membrane, Appendix \[app:initialcondition\] derives the rather subtle initial conditions for the field that we use for numerically integrating the equations of motion, Appendix \[app:quantumEOM\] gives the derivation of the quantum equations of motion and Appendix \[app:analyticexpressions\] sketches the calculation of the various coefficients that appear in the quantum Hamiltonian, as well as giving estimates of their magnitudes in contemporary experiments. Wave equation in the presence of a moving dielectric {#sec:waveequation} ==================================================== In their most general form the four Maxwell equations read $$\begin{aligned} \nabla \cdot \mathbf{D} & = & \rho_{f} \\ \nabla \cdot \mathbf{B} & = & 0 \\ \nabla \times \mathbf{E} & = & - \frac{\partial \mathbf{B}}{\partial t} \label{eq:faraday} \\ \nabla \times \mathbf{H} & = & \mathbf{J}_{f} + \frac{\partial \mathbf{D}}{\partial t} \label{eq:ampere}\end{aligned}$$ where $\mathbf{D}$ is the displacement field, $\mathbf{E}$ is the electric field, $\mathbf{H}$ is the magnetizing field and $\mathbf{B}$ is the magnetic field. $\rho_{f}$ and $\mathbf{J}_{f}$ are the free charge and free current, respectively, which exist in the end mirrors but not in the dielectric which is assumed to only contain bound charge and polarization current. In this paper we follow Law’s [@Law1; @Law2; @Law3] approach, where the electromagnetic field is set to zero at the surfaces of the end mirrors by hand. This *effective* theory avoids us having to deal explicitly with the complicated interaction between the fields and $\mathbf{J}_{f}$ and $\rho_{f}$ in the end mirrors, and we can therefore set these source terms to zero everywhere. We also assume that the electromagnetic properties of the dielectric are linear and isotropic so that they obey the constitutive relations $\mathbf{D}=\epsilon \mathbf{E}$ and $\mathbf{B}=\mu \mathbf{H}$. In fact, we will only consider the case of a non-magnetic dielectric and hence $\mu(x,t) \rightarrow \mu_{0}$, where $\mu_{0}$ is the permeability of free space. Substituting $\mathbf{B}=\mu_{0} \mathbf{H}$ into Faraday’s law Eq. (\[eq:faraday\]), taking the curl of both sides, then taking the time derivative of Ampère’s law Eq. (\[eq:ampere\]), and combining the two equations gives $-\nabla \times \left( \nabla \times \mathbf{E} \right) = \mu_{0} \partial^2 \mathbf{D} / \partial t^2$. We can use the standard vector identity $\nabla \times \nabla \times \mathbf{E} = \nabla (\nabla \cdot \mathbf{E})-\nabla^2 \mathbf{E}$ to re-write the left hand side, but unlike the vacuum case, we do not have $\nabla \cdot \mathbf{E} = 0$ everywhere. Rather, because $\rho_{f}=0$, we have $\nabla \cdot (\epsilon \mathbf{E})=0$ and hence $\epsilon (\nabla \cdot \mathbf{E})+ \mathbf{E} \cdot \nabla \epsilon=0$. Thus, the electric field satisfies $$\begin{aligned} \nabla^2 \mathbf{E} + \nabla \left( \mathbf{E} \cdot \nabla \log \epsilon \right) = \mu_{0} \frac{\partial^2 \epsilon \mathbf{E}}{\partial t^2} . \label{eq:firstwaveequation}\end{aligned}$$ In fact, the second term on the left hand side vanishes identically in the situations we shall consider in this paper where the dielectric function only varies along the cavity axis, whereas the electric field is polarized transversally to this. Because we shall only consider a single polarization we are in essence using the scalar field model [@Barton93; @Salamone94; @Calucci92] in one dimension. In order to analyse the right hand side of Eq. (\[eq:firstwaveequation\]) we need a model for the dielectric function of a moving membrane. If we assume a gaussian profile of the form $$\epsilon_{\mathrm{membrane}}=\alpha \epsilon_{0} \frac{\exp[-(x-vt)^2/w^2]}{\sqrt{\pi} w}$$ where $\epsilon_{0}$ is the permittivity of free space and $w$, $v$ and $\alpha$ characterise the membrane’s thickness, velocity and dielectric strength, respectively, we find that the rate of change of the dielectric properties obey $$\frac{\partial \epsilon_{\mathrm{membrane}}}{\partial t} \le \alpha \epsilon_{0} \sqrt{\frac{2}{\pi}} \frac{v}{w^2} \exp[-1/2] \label{eq:depsilondt}$$ where the right hand side has been evaluated at the point $x-vt=w/\sqrt{2}$ where the gaussian changes most rapidly. For the ‘velocity’ we put $v=5000$ m/s, which is a typical value used in this paper (although velocities up to $v=20,000$ m/s are considered), and guided by the Yale experiments [@HarrisJayich; @HarrisZwickl] we set $w=50$ nm. In order to estimate the membrane’s reflectivity $R$, and hence the value of $\alpha$, we note that when the membrane is much thinner than the wavelength of light, as is the case when $w=50$ nm, we can let $w \rightarrow 0$. In this limit the gaussian reduces to a $\delta$-function and $$R=\frac{k^2 \alpha^2}{4+k^2 \alpha^2} . \label{eq:reflectivity}$$ For example, taking $\lambda=2 \pi /k = 785 $ nm and $\alpha= 1.7 \times 10^{-6}$ m gives a membrane reflectivity of 98%. In fact the $\delta$-function approximation can also be used for thicker membranes provided resonances are avoided where a significant amount of electromagnetic energy is concentrated inside the dielectric [@NickPaper]. Inserting the above numbers into Eq. (\[eq:depsilondt\]) gives the estimate $\partial \epsilon_{\mathrm{membrane}}/ \partial t \lesssim 10^{12} \epsilon_{0}$. The factor $10^{12}$ s$^{-1}$ should be compared with the optical frequency $\omega_{\mathrm{optical}}= \mathcal{O}[10^{15}]$ s$^{-1}$ which characterizes the time-dependence of the electric field. This means that one can reasonably ignore the time derivatives of $\epsilon$ on the right hand side of Eq. (\[eq:firstwaveequation\]) and adopt the standard wave equation but with a space- and time-dependent dielectric function: $$\begin{aligned} \nabla^2 \mathbf{E} - \mu_{0} \epsilon(x,t) \frac{\partial^2 \mathbf{E}}{\partial t^2} =0 . \label{eq:timedependentMaxwellwave}\end{aligned}$$ Inside a dielectric light does not travel at the same speed as in vacuum and this means that the above equation is not relativistically invariant and hence is subject to relativistic corrections when the membrane moves [@PiwnickiLeonhardt]. However, as shown in Appendix \[app:NonrelativisticApproximation\], these corrections turn out to be small for the speeds we consider here and will be neglected. In fact, because we model the dielectric by a $\delta$-function, strictly speaking there is no light inside the medium and the membrane only acts as a boundary condition, somewhat like that due to the end mirrors. It can then be argued that any relativistic corrections due to the medium vanish identically. Static membrane {#sec:setup} =============== Our treatment of the dynamics of light in a double cavity is based upon finding the normal modes of the field, and these depend on the position of the membrane. While normal modes only exist for a stationary membrane, which is the focus of the present section, when interpreted as the instantaneous modes at each position of the membrane they can be used as a complete and orthogonal basis for the moving membrane case to be discussed in subsequent sections. As above, the membrane is taken to be a thin piece of dielectric material whose spatial profile is modelled by a $\delta$-function. It can transmit light, in contrast to the two end mirrors, which are assumed to be perfectly reflective. Once an initial optical field is established in the double cavity the external pump fields are presumed to be turned off and losses are neglected. The dynamics of light in the stationary version of this model were studied by Lang *et al* [@Lang73] in 1973 in the context of modelling lasers as open systems. One of the subcavities represented the laser cavity and the other, which was much longer, represented the outside world. More recently, the dynamic version of the model has been used by Linington and Garraway [@Linington08; @liningtonthesis] to study dissipation control in cavities with moving end mirrors, and Castaños and Weder [@WederCastanos] have used it to find the classical dynamics of a thin end mirror. When choosing a coordinate system it is convenient to pretend that the membrane is always located at the origin and the end mirrors are at $x=-L_1$ and $L_2$. The total length of the cavity is $L=L_1+L_2$ and the distance of the membrane from the center is $\Delta L/2$, where $\Delta L = L_1 - L_2$ is the difference in length between the two subcavities so that $L_{1/2} = (L \pm \Delta L)/2$. Thus, we write the dielectric function of the double cavity as $$\epsilon(x,\Delta L)= \begin{cases} \epsilon_{0}[1+\alpha\delta(x)], & -L_1<x<L_2 \\ \infty, & x>L_2,\ x<-L_1 . \end{cases}$$ We emphasize that despite this choice of coordinate system, the physical situation we are describing is one in which the membrane is mobile and the end mirrors are fixed. ![The wavenumbers of the normal modes inside a double cavity form a network of avoided crossings when plotted as a function of the difference in length between the two subcavities. The total length of the double cavity is $L=100 \mu$m. The red dashed lines correspond to a perfectly reflective central membrane ($\alpha \rightarrow \infty$). The green solid lines correspond to a membrane of reflectivity $98 \%$ (i.e. $\alpha= 1.7 \times10^{-6}$ m), the magenta, small dotted lines correspond to a membrane reflectivity of $91 \%$ (i.e. $\alpha=8.0 \times 10^{-7}$m), the blue dashed dotted lines correspond to $61 \%$ (i.e. $\alpha=3.1 \times 10^{-7}$m), and the larger, black dotted lines correspond to a membrane reflectivity of $28 \%$ (i.e. $\alpha=1.6 \times 10^{-7}$m). All curves except the red curve have avoided crossings. The gap at the avoided crossing ($2\Delta$) goes down as the reflectivity is increased.[]{data-label="fig:AvoidedCrossing"}](AvoidedCrossing2){width="1\columnwidth"} We take the mirrors to lie in the y-z plane and to be translatable along the x-axis, and consider the case where the electric and magnetic fields are polarized along the $z$ and $y$ axes, respectively. In terms of the vector potential $\mathbf{A}=A(x,t)\hat{\mathbf{z}}$, we have $\mathbf{E}(x,t)=E(x,t) \hat{\mathbf{z}}=-({\partial}_t A)\hat{\mathbf{z}}$ and $\mathbf{B}(x,t)=B(x,t)\hat{\mathbf{y}}=-({\partial}_x A)\hat{\mathbf{y}}$. The Maxwell wave equation then takes the form $$\frac{{\partial}^{2}E(x,t)}{{\partial}x^{2}}-\mu_{0}\epsilon(x,\Delta L)\frac{{\partial}^{2}E(x,t)}{{\partial}t^{2}}=0. \label{eq:StationaryMaxwell}$$ The method for solving this equation in terms of normal modes is well known. However, we shall go through it carefully here as a reference for the moving membrane case we tackle in the rest of this paper. To this end we perform a separation of variables, by putting $E(x,t)=C(t)U(x)$, which gives the two equations $$\begin{aligned} && \frac{d^{2}U}{d x^{2}}+k^2 \frac{\epsilon(x, \Delta L)}{\epsilon_{0}} U=0 \label{eq:Utimeindependent} \\ && \frac{d^{2}C}{d t^{2}}+\omega^2 C=0 \label{eq:classicalSHO}\end{aligned}$$ where $\omega^2= c^2 k^2$ is the separation constant and $c=1/\sqrt{\epsilon_{0} \mu_{0}}$ is the speed of light in vacuum. The solutions to Eq. (\[eq:Utimeindependent\]) that obey the boundary conditions $E(x=-L_{1},t)=E(x=L_{2},t)=0$ due to the end mirrors are the global modes of the entire double cavity $$\begin{aligned} & U_m(x,\Delta L) = \label{eq:globalmodes} \\ & \begin{cases} A_{m}(\Delta L)\sin[k_{m}(\Delta L)(x+L_{1}(\Delta L))], & -L_{1}\le x\le0 \\ B_{m}(\Delta L)\sin[k_{m}(\Delta L)(x-L_{2}(\Delta L))], & 0\le x\le L_{2} . \nonumber \end{cases}\end{aligned}$$ The allowed wavenumbers $k_{m}$ satisfy [@Lang73] $$\cos(2k_{m}\Delta L)-\cos(k_{m}L)=2\frac{\sin(k_{m}L)}{\alpha k_{m}} \, \label{eq:evaleqn}$$ where $m$ is an integer that labels them. Both $k_m$ and $U_m$ depend parametrically on $\Delta L$; when Eq. (\[eq:evaleqn\]) is solved as a function of membrane displacement the result is a network of avoided crossings as shown in Fig. \[fig:AvoidedCrossing\]. An important property of the mode functions is that they are orthogonal in the Sturm-Liouville sense. If in addition we impose normalization they obey $$\frac{1}{\epsilon_{0}}\int_{-L_{1}}^{L_{2}} \, \epsilon(x,\Delta L) \, U_{l}(x,\Delta L) \, U_{m}(x,\Delta L) \, \mathrm{dx}=\delta_{lm}. \label{eq:orthonormal}$$ The time dependence of the field is determined by Eq. (\[eq:classicalSHO\]) which is the equation of motion for a harmonic oscillator. Factorizing it as $(-i \partial_{t}-\omega_{m})(i\partial_t-\omega_{m})C_{m}=0$ we see that there are two solutions of the form $C_{m}^{\pm}(t) = c_{\pm} \exp[\pm i \omega_{m} t]$. The electric field is a linear combination of $C_{m}^{\pm}(t)$ and must be real. We can therefore put $$E(x,t)= \sum_{m} \left[ C_{m}^{+}(t) + C_{m}^{-}(t) \right] \, U_m(x,\Delta L) \ , \label{eq:staticEfield}$$ where the constants $c_{\pm}$ are complex conjugates of each other. Thus, $C_{m}^{+}(t)=[C_{m}^{-}(t)]^{\ast}$, and in this sense the harmonic oscillator equation can be replaced by the single first order equation $(i\partial_t-\omega_{m})C_{m}=0$. Although the harmonic oscillator equation is second order, and hence its solution requires two arbitrary constants (an amplitude and a phase), there is no loss of information in going over to a first order equation because $C_{m}$ is now a complex number specified by two real numbers. In the quantum theory $C_{m}(t)$ and $C_{m}^{\ast}(t)$ become lowering and raising operators, respectively, that obey the Heisenberg equations of motion: $$\begin{aligned} \left(i\frac{\partial }{\partial t}- \omega_{m} \right) \hat{C}_{m}(t) & = & 0 \\ \left( -i\frac{\partial}{\partial t} - \omega_{m} \right) \hat{C}_{m}^{\dag}(t) & = & 0 .\end{aligned}$$ These equations are not independent: there is really a single equation and its hermitian conjugate. The electric field in Eq. (\[eq:staticEfield\]) becomes an operator proportional to $\hat{C}_{m}^{\dag}(t) + \hat{C}_{m}(t)$ which can be recognized as the position operator (up to constant factors). It is of considerable significance that the quantum equations of motion (and also the classical ones in this case) are first order in time as this ensures that the commutator $[\hat{C}_{m}(t), \hat{C}_{n}^{\dag}(t)]=\delta_{mn}$ is preserved under the dynamics. This quantization procedure becomes problematic in the presence of a moving membrane because then the dielectric function depends on time and prevents a separation of variables i.e. there are no normal modes. Quantization in this situation will be discussed in Section \[sec:DCE\]. In this paper we focus on the dynamics near an avoided crossing, and hence parameterize the two relevant eigenfrequencies as $$\omega_{2/1}(\Delta L)=\omega_{\mathrm{av}}\pm\sqrt{\Delta^{2}+\Gamma^2 (\Delta L)} \label{eq:EvenOddfrequencies}$$ where $\Delta$ is half the separation between the two frequencies at the avoided crossing, $\omega_\mathrm{av}$ is their average, and $\Gamma \equiv\sqrt{\gamma} \, \Delta L$ varies linearly with the membrane’s displacement from the avoided crossing. In [@NickPaper] we showed that for the $\delta$-function membrane model \[and for optical frequencies where $\omega = \mathcal{O} (10^{15})$ s$^{-1}$\] that $$\begin{aligned} \omega_{0} & \equiv & \frac{2cn\pi}{L} = \omega_{\mathrm{av}} - \Delta \approx \omega_{\mathrm{av}} \\ \Delta & = & \frac{\omega_{0}}{2}\frac{1}{1+\frac{\omega_{0}^{2}L\alpha}{4c^{2}}} \approx \frac{2 c^2}{\omega_{0} L \alpha} \label{eq:deltadef} \\ \gamma & = & \frac{\alpha \, \Delta \, \omega_{0}^{3}}{2Lc^{2}} \approx \frac{\omega_{0}^2}{L^2} \label{eq:gammadef} \end{aligned}$$ where $n$ denotes the $n^\mathrm{th}$ pair of modes as counted up from the fundamental mode in a cavity with a perfectly centered and perfectly reflective membrane. For a chosen avoided crossing, the mode corresponding to the lower branch is labelled by the subscript $1$, while that forming the upper branch is labelled by the subscript $2$. When the mirror is perfectly centered the electric field mode functions are either symmetric or antisymmetric. The antisymmetric modes correspond to the lower eigenfrequency ($\omega_1$) of the avoided crossing, while the symmetric state corresponds to the higher eigenfrequency ($\omega_2$). This is in contrast [@NickPaper] to the case of material particles governed by the Schrödinger equation where the scenario is reversed, i.e. the state with the lower eigenvalue is symmetric. An alternative basis to the global modes is provided by the local modes $$\begin{array}{c} \phi_{L}(x,\Delta L)=-\sin \theta \ U_{2}(x,\Delta L)+\cos \theta \ U_{1}(x,\Delta L)\\ \phi_{R}(x,\Delta L)=\cos \theta \ U_{2}(x,\Delta L)+\sin \theta \ U_{1}(x,\Delta L) \end{array} \label{eq:localmodes}$$ where $$\sin\theta=-\sqrt{\frac{1}{2}-\frac{\Gamma(\Delta L)}{2\sqrt{\Delta^{2}+\Gamma(\Delta L)^{2}}}} \label{eq:sindefn}$$ and $$\cos\theta=\sqrt{\frac{1}{2}+\frac{\Gamma(\Delta L)}{2\sqrt{\Delta^{2}+\Gamma(\Delta L)^{2}}}} \ , \label{eq:cosdefn}$$ see Appendix D of [@NickPaper] for a derivation. The local modes are localized in the left ($\phi_{L}$) and right ($\phi_{R}$) subcavities. Although this localization is not perfect, it becomes strong even for moderate membrane reflectivities. The orthonormality of the global modes is inherited by the local modes so that $$\frac{1}{\epsilon_{0}}\int_{-L_{1}}^{L_{2}}\epsilon(x,\Delta L)\phi_{i}(x,\Delta L)\phi_{j}(x,\Delta L)\mathrm{dx}=\delta_{ij}.$$ where $\{i,j\}=\{L,R\}$. The usefulness of the local basis, when used for dynamics near an avoided crossing, will become apparent in Section \[sec:LocalModeDynamics\]. From hence forth the global basis will be referred to as the *adiabatic* basis and the local basis as the *diabatic* basis. This terminology is borrowed from the Landau-Zener problem where the energies of the diabatic states cross linearly as a function time whereas the adiabatic states have an avoided crossing with a minimum gap of $2 \Delta$. The differences between the diabatic and the adiabatic modes are most stark at the (avoided) crossing; far from the (avoided) crossing they become equal to each other. One note of caution: as explained in Appendix D in reference [@NickPaper] the diabatic modes are *not* the same as the perfectly uncoupled modes when the two sides of the cavity are independent except in the limit $\alpha \rightarrow \infty$. Moving membrane {#sec:adiabaticbasisEOM} =============== In this section we derive the equations of motion describing the time evolution of light in a double cavity with a moving membrane. Following Linington [@liningtonthesis], we write the evolving electric field in the instantaneous eigenbasis (adiabatic basis) and find differential equations that are second order in time for the corresponding amplitudes. These equations of motion, given in Eq. (\[eq:LiningtonEquation\]) below, will be referred to as the *adiabatic second order equations* (ASOE) and provide us with the most accurate description of the dynamics (they do not assume any adiabatic approximation). The results predicted by the ASOE are the benchmark against which we compare the validity of the approximate dynamics given by the *diabatic second order equations* (DSOE) and the *diabatic first order equations* (DFOE) which will be introduced later. The adiabatic modes for any instantaneous position of the membrane form a complete basis and we can expand the electric field in terms of them $$E(x,t)= \sum_{n} c_{n}(t)\exp \left\{ -i\int^t_{t_0}\omega_n(t^{\prime})\mathrm{dt^{\prime}} \right\} U_n(x,t) \label{eq:globalmodeansatz}$$ where the instantaneous mode functions $U_n(x,t)$ at time $t$ are specified in Eq. (\[eq:globalmodes\]) and the time-dependent coefficients $c_{n}(t)$ are in general complex numbers. Although we have not made it explicit, it is understood that the physical electric field is given by the real part of Eq. (\[eq:globalmodeansatz\]). Substituting equation (\[eq:globalmodeansatz\]) into (\[eq:timedependentMaxwellwave\]), one finds [@liningtonthesis] $$\begin{gathered} \underset{n}{\sum} \left[\underbrace{-2i\omega_n \frac{{\partial}}{{\partial}t} (c_n(t)U_n(x,t))}_\mathrm{1} + \underbrace{\frac{{\partial}^2}{{\partial}t^2} (c_n(t)U_n(x,t))}_\textrm{2} \right. \\ \left. \underbrace{-i\frac{{\partial}\omega_n(t)}{{\partial}t}c_n(t)U_n(x,t)}_\textrm{3}\right]\exp\left[-i\int_{t_0}^t\omega_n(t^\prime)\mathrm{dt}^\prime\right] = 0. \label{eq:LiningtonEquation0}\end{gathered}$$ Term 1 is by far the dominant one due to the very large optical frequency prefactor. In the slow membrane regime term 2 is small while term 3 is much smaller still because the adiabatic mode can change more significantly in comparison to the rate of change of the optical frequency near an avoided crossing. Right at the avoided crossing, the frequencies are at a maximum or a minimum and hence their rate of change is zero. The relative magnitude of all these terms is analyzed in greater detail in reference [@liningtonthesis]. In particular, for faster membrane speeds terms 2 and 3 can become of similar magnitude. By projecting out the $m^{\mathrm{th}}$ amplitude using the orthnormality of adiabatic modes, we find from Eq. (\[eq:LiningtonEquation0\]) that the amplitudes corresponding to the adiabatic basis satisfy the *ASOE* [@liningtonthesis] $$\begin{aligned} \ddot{c}_{m}(t)- & i\dot\omega_{m}(t)c_{m}(t)-2i\omega_{m}(t)\dot{c}_{m}(t) +\underset{n}{\sum}\left\{ [2\dot{c}_{n}(t)- \right. \nonumber \\ & \left. 2i\omega_{n}(t)c_{n}(t)]P_{mn}(t)+c_{n}(t)Q_{mn}(t)\right\}=0 . \label{eq:LiningtonEquation}\end{aligned}$$ In these equations $$\begin{aligned} \theta_{mn}(t) & \equiv & \int_{t_{0}}^{t}[\omega_{m}(t^{\prime})-\omega_{n}(t^{\prime})]\,\mathrm{dt}^{\prime} \nonumber \\ P_{mn}(t) & \equiv & e^{i\theta_{mn}(t)} \int_{-L_{1}}^{L_{2}}\frac{\epsilon(x,t)}{\epsilon_{0}}U_{m}(x,t)\frac{\partial U_{n}(x,t)}{\partial t} \,\mathrm{dx} \label{eq:Pdefinition} \nonumber \\ Q_{mn}(t) & \equiv & e^{i\theta_{mn}(t)}\int_{-L_{1}}^{L_{2}}\frac{\epsilon(x,t)}{ \epsilon_{0}}U_{m}(x,t)\frac{\partial^{2}U_{n}(x,t)}{\partial t^{2}} \,\mathrm{dx}. \nonumber\end{aligned}$$ The integrals $P_{mn}(t)$ and $Q_{mn}(t)$ depend on the motion of the membrane through the time-dependence of the adiabatic mode functions $U_n(x,t)$. If the membrane is stationary $P_{mn}$ and $Q_{mn}$ vanish and there is no coupling between the different adiabatic modes. The coupled differential equations Eq.  (\[eq:LiningtonEquation\]) are second order in time and we therefore need to specify two conditions at the initial time $t_{0}$ in order to solve them. We choose $c_m(t_0)$ and $\dot{c}_m(t_0)$. However, while $c_m(t_0)$ can be found for any choice of the initial field configuration by projecting it over the expansion given in Eq. (\[eq:globalmodeansatz\]), it is not so obvious what to choose for $\dot{c}_m(t_0)$. In particular, if we assume that for times $t<t_{0}$ the membrane is stationary then we show in Appendix \[app:initialcondition\] that the correct initial condition for the time derivatives of the coefficients is $\dot{c}_m(t_0)=-\sum_{n} P_{mn}(t_0)c_n(t_0)$. Energy of a dichromatic field {#sec:CavityEnergy} ============================= A key quantity in our analysis of the dynamics is the instantaneous energy of the electromagnetic field $$\begin{aligned} \mathcal{E} & =\frac{1}{2}\int_{\mathcal{V}}\left[\epsilon(x,t)|E(x,t)|^{2}+\mu_{0}|H(x,t)|^{2}\right]\mathrm{dV} \nonumber \\ & =\frac{\mathcal{A}}{2}\int\left[\epsilon(x,t)|E(x,t)|^{2}+\mu_{0}|H(x,t)|^{2}\right]\mathrm{dx} \label{eq:energygeneral}\end{aligned}$$ where $H(x,t)=B(x,t)/\mu_0$, $\mathcal{A}$ is the area of the mode functions, and $\mathcal{V}$ is the volume of the cavity. Note that the vanishing volume of the $\delta$-function membrane means that there is no contribution from it. In this paper we are interested in the field dynamics when passing through an avoided crossing where attention can be restricted to just two modes. We therefore consider a dichromatic field in the adiabatic basis with frequencies $\omega_{1}$ and $\omega_{2}$. The total electric field can then be written as $$\begin{gathered} E(x,t)=c_{1}(t)\exp[-i\theta_{1}(t)]U_{1}(x,t) + \\ c_{2}(t)\exp[-i\theta_{2}(t)]U_{2}(x,t)\end{gathered}$$ where $U_m(x,t)$ is defined in Eq. (\[eq:globalmodes\]) and $$\theta_{m}(t)=\int_{t_{0}}^{t}\omega_{m}(t^{\prime})\mathrm{dt}^{\prime}.$$ Hence, the energy per unit area becomes $$\frac{\mathcal{E}}{\mathcal{A}}=\frac{\epsilon_{0}}{2}\left\{ |c_{1}(t)|^{2}+|c_{2}(t)|^{2}\right\} +\frac{\mu_{0}}{2}\int_{-L_{1}}^{L_{2}}|H(x,t)|^{2}\mathrm{dx}.$$ Assuming that, as usual, the magnetic field makes a contribution to the energy equal to that of the electric field, we arrive at the following expression for the total energy per unit area: $$\frac{\mathcal{E}}{\mathcal{A}}=\epsilon_{0}\left\{ |c_{1}(t)|^{2}+|c_{2}(t)|^{2}\right\} . \label{eq:lightenergy}$$ In time-independent situations the Hamiltonian gives the energy of a system. However, this is not necessarily true in time-dependent systems where the Hamiltonian still plays the role of the generator of dynamics but need not coincide with the energy. Reference [@schutzhold98] proves that Eq. (\[eq:energygeneral\]) is the correct expression for the instantaneous energy even in time-dependent situations. Although our approach to finding the dynamics in this paper is based upon the wave equation rather than the Hamiltonian, we shall have occasion to derive the Hamiltonian in Sec. \[sec:DCE\] and will find that it contains extra velocity-dependent terms not present in Eq. (\[eq:energygeneral\]). Field dynamics while traversing an avoided crossing {#sec:adiabatictheorem} =================================================== In this section we apply the full ASOE derived in Section \[sec:adiabaticbasisEOM\] to the case of a Landau-Zener style sweep of the membrane through an avoided crossing. The Landau-Zener problem is a rare example of where the time-dependent Schrödinger equation can be solved exactly and the adiabaticity of the motion evaluated analytically. The Schrödinger equation for the Landau-Zener problem is $$\begin{aligned} i \frac{d}{dt} & \left( \begin{array}{c} a_{L}\\ a_{R} \end{array} \right) = H_{\mathrm{LZ}}(t) \left( \begin{array}{c} a_{L}\\ a_{R} \end{array} \right) \label{eq:SchrodLZ}\end{aligned}$$ where, in the notation used in this paper, the Landau-Zener Hamiltonian takes the form $$H_{\mathrm{LZ}}(t) \equiv \left( \begin{matrix} \omega_{\mathrm{av}}+ \Gamma(t) & \Delta \\ \Delta & \omega_{\mathrm{av}}- \Gamma(t) \end{matrix} \right) . \label{eq:HLZ}$$ It describes the case where two *diabatic* levels cross linearly in time and in the double cavity system this corresponds to a membrane moving at constant velocity $v$. Given that the membrane displacement is $\Delta L/2$, and that $\Gamma(t) \equiv \sqrt{\gamma} \, \Delta L(t)$ \[see Eq. (\[eq:EvenOddfrequencies\]) and the definitions given below it\], for a Landau-Zener sweep we must put $$\Gamma(t)=\sqrt{\gamma} \, \Delta L(t)= \sqrt{\gamma} 2 v t.$$ The diabatic levels cross at $t=0$ and have a constant coupling given by $\Delta$. In the *adiabatic* basis the same Schrödinger equation becomes $$\begin{aligned} i \frac{d}{dt} & \left( \begin{array}{c} c_{2}\\ c_{1} \end{array} \right) = \left( \begin{matrix} \omega_{2}(t) & 0 \\ 0 & \omega_{1}(t) \end{matrix} \right) \left( \begin{array}{c} c_{2}\\ c_{1} \end{array} \right)\end{aligned}$$ where $\omega_{2/1}(t)=\omega_{\mathrm{av}} \pm \sqrt{\Delta^2+\Gamma^{2}(t)}$. There is an avoided crossing between the two adiabatic states with a gap of $2 \Delta$ at $t=0$. If the system starts in one of the adiabatic states at $t=-\infty$ the probability that it has made a transition to the other adiabatic state by $t=+\infty$ is given by [@Landau; @Zener; @Stenholm] $$P_{\mathrm{LZ}}=\exp \left[- \pi \Delta^2/ (2 v \sqrt{\gamma}) \right]. \label{eq:P_LZ}$$ The process becomes more adiabatic as the velocity $v$ is reduced; the population transfer approaches zero exponentially fast in $1/v$. ![Dynamics of an initially empty mode when traversing an avoided crossing at five different speeds. We simulated the field dynamics using the ASOEs given in Eq. (\[eq:adiabaticLinington\]) in the two-level approximation near an avoided crossing. $c_{2}$ is the amplitude associated with the upper adiabatic mode, where the initial condition is $c_1=1$ and $c_2=0$. According to Eq. (\[eq:lightenergy\]), $\vert c_{n} \vert^2$ is proportional to the electromagnetic energy of the $n^{\mathrm{th}}$ mode. We see that as the membrane speed goes down, the energy pumped into the initially unpopulated mode tends to zero. Parameters: membrane reflectivity $98\%$ (i.e. $\alpha=1.5 \times 10^{-6}$m); length of double cavity $100 \mu$m; maximum membrane displacement $\Delta L/2 = \pm 1 \times 10^{-7}$m. The adiabatic modes shown are those with $n=128$, where we label the modes in terms of the wavenumbers for a perfectly reflecting membrane for which $k_{n}=2 \pi n /(L\pm\Delta L)$. These perfectly localized modes come in pairs that are degenerate at $\Delta L=0$. []{data-label="fig:AdiabaticCriteriaSingleMode2"}](AdiabaticCriteriaSingleMode2){width="\linewidth"} ![Dynamics of an initially excited mode when traversing an avoided crossing at different speeds as calculated using the ASOEs. This figure is for exactly the same setup as Fig. \[fig:AdiabaticCriteriaSingleMode2\] except here we plot the results for the lower mode. We see that as the membrane speed is decreased the energy of this mode is not conserved but has a slight upward curve. Combined with Fig. \[fig:AdiabaticCriteriaSingleMode2\], this tells us that whilst the slowly moving membrane limit is sufficient to avoid nonadiabatic transitions, energy is not conserved.[]{data-label="fig:AdiabaticCriteriaSingleMode"}](AdiabaticCriteriaSingleMode){width="\linewidth"} We should not expect the Landau-Zener theory to apply to the classical electromagnetic field because the latter does not obey the Schrödinger equation. Nevertheless, as we shall see, there are regimes where we can map the passage of the electromagnetic field through an avoided crossing onto the Landau-Zener problem. In particular, we find that decreasing the membrane speed is a sufficient criteria for achieving adiabaticity in the Maxwell wave equation in the sense of vanishing transfer between adiabatic modes. However, contrary to the Schrödinger case, we find that even at very slow membrane speeds we do not conserve the sum $\vert c_{1}(t) \vert^2+\vert c_{2}(t) \vert^2$. In quantum mechanics the coefficients $c_{n}(t)$ are probability amplitudes and the sum of their squares represents the total probability which is conserved under the unitary evolution provided by the Schrödinger equation. The same is not true in the Maxwell case where, as we saw in Section \[sec:CavityEnergy\], the sum of the squares represents the total energy which is in general not conserved when an external parameter is varied. Physically, the electromagnetic field interacts with the membrane via radiation pressure and as a result energy can be transferred back and forth between the field and the external agent moving the membrane. There is always radiation pressure on the membrane (except right at an avoided crossing) and therefore some energy is pumped into/out of the system regardless of how slowly the membrane is being moved. This is a fundamental difference between adiabaticity in the Schrödinger and Maxwell wave equations. We consider the situation where the membrane moves at constant speed $v$ from position $x=-L_0$ to $L_0$ over the time $t=-T_0$ to $T_0$. The displacement of the membrane from the center is given by $\Delta L/2$, $$\frac{\Delta L(t)}{2}= \frac{L_0}{T_0}t$$ and $v=L_0/T_0$. We investigate the effects of varying the speed by fixing $L_0$ and changing $T_0$. It is useful to introduce the scaled time variable $$\tau=\frac{t}{T_0}=\lambda t,\: -1\le\tau\le1$$ i.e. $\lambda = \frac{1}{T_0}$. In terms of these variables the ASOE given in Eq. (\[eq:LiningtonEquation\]) become $$\begin{aligned} \frac{dc_{m}}{d\tau} = -\frac{d\omega_{m}}{d\tau}\frac{c_m}{2\omega_{m}}- & \frac{i\lambda}{2\omega_{m}}\frac{d^2 c_{m}}{d\tau^2}-\underset{n}{\sum}\left\{ \left[\frac{i\lambda}{\omega_{m}}\frac{d c_{n}}{d\tau}+\right.\right. \nonumber \\ & \left.\left.\frac{\omega_{n}}{\omega_{m}}c_{n}\right]\bar{P}_{mn}+\frac{i\lambda}{2\omega_{m}}c_{n}\bar{Q}_{mn}\right\} \label{eq:adiabaticLinington}\end{aligned}$$ where $$\begin{aligned} \bar{\theta}_{mn} & \equiv & \frac{1}{\lambda}\int_{-1}^{\tau}\left[\omega_{m}(\tau^{\prime})-\omega_{n}(\tau^{\prime})\right]d\tau^{\prime} \nonumber \\ \bar{P}_{mn} & \equiv & e^{i\bar{\theta}_{mn}} \int_{-L_{1}}^{L_{2}}\frac{\epsilon(\tau,x)}{\epsilon_{0}}U_{m}(\tau,x){\partial}_{\tau}U_{n}(\tau,x)dx\nonumber \\ \bar{Q}_{mn} & \equiv & e^{i\bar{\theta}_{mn}} \int_{-L_{1}}^{L_{2}}\frac{\epsilon(\tau,x)}{\epsilon_{0}}U_{m}(\tau,x){\partial}_{\tau}^{2}U_{n}(\tau,x)dx. \nonumber\end{aligned}$$ Let us assume that a single mode $c_m$ is initially populated and all other modes are empty. When $T_0 \rightarrow \infty$, we have $\lambda \rightarrow 0$ and the factors $\bar{P}_{mn}$ and $\bar{Q}_{mn}$ for $n \neq m$ approach zero due to the presence of the phase term which oscillates infinitely rapidly in that limit. However, the diagonal terms $\bar{P}_{mm}$ and $\bar{Q}_{mm}$ have no such phase term and are generally non-zero. The term $(d\omega_m/d\tau)(c_m/2\omega_m)$ in Eq. (\[eq:adiabaticLinington\]) is related to the slope of the frequency and is non-zero everywhere except exactly at the avoided crossing. Since the first term and the diagonal part of the fourth term of the right hand side of Eq. (\[eq:adiabaticLinington\]) do not approach zero in the slow membrane limit, $d c_m/d\tau$ does not approach zero. Meanwhile, the rates of change of all initially unpopulated states do approach zero in the slow membrane limit because the first and fourth terms in Eq. (\[eq:adiabaticLinington\]) depend on the mode population. This indicates that all the initially empty modes continue to remain empty in the slow membrane limit despite the fact that the initially occupied mode can in general change its amplitude no matter how slowly the membrane is moved. ![The fractional change in total energy of the system as it traverses an avoided crossing as calculated using the ASOEs. Here $\mathcal{E}_{0}$ is the initial energy and we use the same parameters as in Figs. \[fig:AdiabaticCriteriaSingleMode2\] and \[fig:AdiabaticCriteriaSingleMode\]. The plot shows that even at very slow membrane speeds the energy change as a function of time does not vanish but instead tends to a limiting curve. Hence, even though $v \rightarrow 0$, we find $\sum_{n} \vert c_{n}(\tau) \vert^2 \neq \mathrm{constant}$, confirming that adiabaticity does not imply energy conservation.[]{data-label="fig:AdiabaticityEnergy"}](AdiabaticCriteriaSingleModeWorkEnergy){width="\linewidth"} This analysis of the equations of motion is supported by the numerical results shown in Figs. \[fig:AdiabaticCriteriaSingleMode2\],\[fig:AdiabaticCriteriaSingleMode\], and \[fig:AdiabaticityEnergy\] where we plot dynamics for a pair of adiabatic modes as they traverse an avoided crossing at various speeds. At higher speeds energy is removed from the initially excited mode and transferred to the initially empty mode as can be seen from the almost perfect mirror symmetry of the $\vert c_{1} \vert^2$ and $\vert c_{1} \vert^2$ curves about the midpoint $\vert c_{1} \vert^2=\vert c_{1} \vert^2=0.5 $. This is the type of behaviour we would expect in the standard Landau-Zener problem with the Schrödinger equation. And at very low speeds we see from Fig. \[fig:AdiabaticCriteriaSingleMode2\] that the amplitude of the initially empty mode remains zero indicating adiabatic evolution, as expected. However, in Fig. \[fig:AdiabaticCriteriaSingleMode\] we see that at low speeds the various curves for the initially excited mode converge towards a limiting curve where there is a finite change in energy of the mode. To make this point clearer we plot the change in total energy of the system in Fig. \[fig:AdiabaticityEnergy\]. To be precise, we plot the change in energy divided by the initial energy $|c_1|^2+|c_2|^2-1=\Delta \mathcal{E}_{\mathrm{ASOE}}/\mathcal{E}_0$, and as one can see no matter how slowly the membrane is moved the energy pumped into the system converges to a curve that always lies above the zero axis. We also note that the slow speed limiting curve is symmetric about the avoided crossing at $\tau=0$, indicating that whatever energy is pumped in when approaching the avoided crossing is pumped out as it recedes. However, at higher speeds there is a noticeable net energy gain by the electromagnetic field. Dynamics in the diabatic basis {#sec:LocalModeDynamics} ============================== ![Dynamics on traversing an avoided crossing as seen in the diabatic basis, with the field initially localized on the right. Membrane speed: $5000 \ \mathrm{m s}^{-1}$. All other parameters are the same as in Fig. \[fig:AdiabaticCriteriaSingleMode2\]. The results were calculated using both the ASOE and DSOE schemes with the two sets of curves lying right on top of each other.[]{data-label="fig:FullvsFirstOrder"}](FullvsPartialphi5000){width="\linewidth"} In this Section we first obtain the second-order-in-time equations of motion in the diabatic basis (DSOE) and then approximate them to first-order-in-time equations (DFOE). So far we have worked in the adiabatic basis which corresponds to the instantaneous normal modes of the double cavity. One feature of this basis is that as an avoided crossing is traversed the two mode functions involved radically change their structure by exchanging the sides upon which they are principally localized, see Fig. 4 in [@NickPaper]. Conversely, the expansion amplitudes $c_{m}(t)$ in the adiabatic basis experience only exponentially small changes in the slow membrane regime. The opposite is true for the diabatic basis where the mode functions hardly change but there is a large change in the amplitudes. The diabatic basis is advantageous for making analytic calculations because to a good approximation we can ignore the time dependence of the mode functions and focus all our attention on the amplitudes, a fact Zener points out in his original paper [@Zener]. We shall confirm this property below. Assuming as before that the membrane motion is restricted to be in the vicinity of an avoided crossing, we employ the two-level approximation and let $$E(x,t)=a_L(t)\phi_L(x,t)+a_R(t)\phi_R(x,t). \label{eq:localmodeelectricfield}$$ Substitution into the Maxwell wave equation given in Eq. (\[eq:timedependentMaxwellwave\]), and neglecting the terms $\dot{\phi}_{L/R}$ and $\ddot{\phi}_{L/R}$, yields $$a_L(t){\phi}^{\prime \prime}_L+a_R(t){\phi}^{\prime \prime}_R=\mu_{0}\epsilon(x,t)[\ddot{a}_L(t)\phi_L+\ddot{a}_R(t)\phi_R] \label{eq:PartialSecondOrder0}$$ where the dots indicate time derivatives and the dashes spatial derivatives. The diabatic modes are not normal modes of the double cavity and so even for a stationary membrane the light oscillates back and forth between the left and right modes in a fashion analogous to the Rabi oscillations of a two-level atom interacting with a single mode field. The combined effect of this intrinsic oscillation and the moving membrane leads to a much larger rate of change of the diabatic amplitudes compared to the adiabatic amplitudes. ![A comparison of the fractional change in energy calculated using the ASOE and DSOE for the same avoided crossing dynamics as shown in Fig. \[fig:FullvsFirstOrder\] except that here we also vary the membrane speed. We see that the order of magnitude of difference between ASOE and DSOE is of the order of $1 \times 10^{-5}$. Here, $\Delta \mathcal{E}_{\mathrm{ASOE}}/\mathcal{E}_0$ is generated by Eq. (\[eq:LiningtonEquation\]) and $\Delta \mathcal{E}_{\mathrm{DFOE}}/\mathcal{E}_0$ is generated by Eq. (\[eq:PartialSecondOrder\]). Although a speed of $20,000$ ms$^{-1}$ seems very high, such effective speeds can be achieved by changing the background index of refractions rather than physically moving the mirror.[]{data-label="fig:FullvsPartialReflectivity98"}](FullvsPartial){width="\linewidth"} ![A comparison of the fractional change in energy calculated using the ASOE and DSOE for the same avoided crossing dynamics as shown in Fig. \[fig:FullvsFirstOrder\] except that here we also vary the membrane reflectivity. The results lead us to the same conclusion as in Fig. \[fig:FullvsPartialReflectivity98\]. Here, $\Delta \mathcal{E}_{\mathrm{ASOE}}/\mathcal{E}_0$ is generated by equation (\[eq:LiningtonEquation\]) and $\Delta \mathcal{E}_{\mathrm{DFOE}}/\mathcal{E}_0$ is generated by equation (\[eq:PartialSecondOrder\]).[]{data-label="fig:FullvsPartialSpeed5000"}](FullvsPartialEOMspeed5000){width="\linewidth"} In order to find the spatial derivatives of the diabatic modes in Eq. (\[eq:PartialSecondOrder0\]), we express each diabatic mode in the adiabatic basis whose second derivatives we know in terms of the Sturm-Liouville relationship given in Eq. (\[eq:Utimeindependent\]), ${\partial}_x^2U_m(x,\Delta L)= - (\epsilon(x,\Delta L) / \epsilon_0) k_m^2 U_m(x,\Delta L)$, and then convert back to the diabatic basis. In matrix form, we find [@NickPaper] $$\begin{aligned} - & \left( \begin{array}{c} \ddot{a}_{L}\\ \ddot{a}_{R} \end{array} \right) = \\ & \left( \begin{matrix} \omega^2_2 \cos^2\theta+\omega_1^2 \sin^2\theta & (\omega_1^2-\omega_2^2) \cos\theta\sin\theta \\ (\omega_1^2-\omega_2^2) \cos\theta\sin\theta & \omega_1^2 \cos^2\theta+\omega_2^2 \sin^2\theta \end{matrix} \right) \nonumber \left( \begin{array}{c} a_{L}\\ a_{R} \end{array} \right)\end{aligned}$$ or $$\left( \frac{d^2}{dt^2} + M_{\mathrm{DSOE}} \right) \left( \begin{array}{c}a_{L}\\ a_{R} \end{array} \right) =0 \label{eq:PartialSecondOrder}$$ where $$M_{\mathrm{DSOE}}= \left( \begin{matrix} \left[\omega_{\mathrm{av}}+\Gamma(t)\right]^{2}+\Delta^{2} & 2\Delta \, \omega_{\mathrm{av}} \\ 2\Delta \, \omega_{\mathrm{av}} & \left[\omega_{\mathrm{av}}-\Gamma(t)\right]^{2}+\Delta^{2} \end{matrix} \right) \label{eq:DSOEmatrix}$$ and we have made use of the identities $$\begin{aligned} && \sin \theta \cos \theta \left( \omega_{1}^{2} - \omega_{2}^2 \right) = 2 \Delta \omega_{\mathrm{av}} \\ && \omega_{1}^2 \cos^{2} \theta + \omega_{2}^2 \sin^{2} \theta = (\omega_{\mathrm{av}}-\Gamma )^2 + \Delta^2 \\ && \omega_{2}^2 \cos^{2} \theta + \omega_{1}^2 \sin^{2} \theta = (\omega_{\mathrm{av}}+\Gamma)^2 + \Delta^2 .\end{aligned}$$ We refer to Eq. (\[eq:PartialSecondOrder\]) as the *diabatic second order equations* (DSOE). ![This figure shows the trend of agreement between DSOE and DFOE as we vary the membrane reflectivity for the same avoided crossing dynamics as shown in Fig. \[fig:FullvsFirstOrder\]. For first order dynamics, $\Delta \mathcal{E}_{\mathrm{DFOE}}/\mathcal{E}_0$ has to be identically zero, while for second order dynamics it is generally nonzero. Hence the difference of this quantity from zero can be used to quantify the validity of the first order model. As reflectivity goes up, the first order approximation becomes less valid. []{data-label="fig:PartialvsFirstSpeed5000"}](PartialSecondvsFirstspeed5000){width="\linewidth"} The DSOE are strongly reminiscent of the second order in time harmonic oscillator equation given in Eq. (\[eq:classicalSHO\]) for the static membrane, albeit in the present case there are two modes involved. This begs the question as to whether the DSOE can be factorized and reduced to a first order equation like the harmonic oscillator equation can. To this end we note that $M_{\mathrm{DSOE}}=H_{\mathrm{LZ}}^2$, and thus it is tempting to write Eq. (\[eq:PartialSecondOrder\]) as $$\left(i\frac{d}{dt} - H_{\mathrm{LZ}} \right) \left(- i\frac{d}{dt} -H_{\mathrm{LZ}} \right) \left( \begin{array}{c}a_{L}\\ a_{R} \end{array} \right) =0 \ \mbox{?} \label{eq:DSOEFactorize}$$ This factorization is correct in the time-independent case, but due to the time-dependence of $H_{\mathrm{LZ}}$, when the left hand side of Eq. (\[eq:DSOEFactorize\]) is expanded there is an extra term $-i \dot{H}_{\mathrm{LZ}}$ not present in the DSOE given in Eq. (\[eq:PartialSecondOrder\]). ![This figure shows the trend of agreement between DSOE and DFOE as we vary the membrane speed for the same avoided crossing dynamics as shown in Fig. \[fig:FullvsFirstOrder\]. As our intuition would suggest, the first order approximation becomes less valid for higher speeds. []{data-label="fig:PartialvsFirstReflectivity98"}](PartialSecondvsFirstchangingvelocity){width="\linewidth"} Although the DSOE cannot be exactly reduced to first-order-in-time equations, a first order approximation can be derived as we now show. We start by transforming the left/right mode amplitudes $$a_{L/R}=\tilde{a}_{L/R}\exp\left\{ -i\int_{t_{0}}^{t}\beta_{L/R}(t^{\prime})dt^{\prime}\right\} \label{eq:DFOEansatz}$$ where $$\beta_{L/R}(t) \equiv \sqrt{(\Gamma(t)\pm\omega_{\mathrm{av}})^{2}+\Delta^{2}}. \label{eq:betadef}$$ Substituting for the new variables removes the fast oscillations $$\begin{aligned} \ddot{a}_{L/R}= & \left\{\ddot{\tilde{a}}_{L/R}-2i\beta_{L/R}\dot{\tilde{a}}_{L/R}-i\dot{\beta}_{L/R}\tilde{a}_{L/R}- \right.\nonumber \\ & \left. \beta_{L/R}^{2}\tilde{a}_{L/R}\right\} \exp\left\{ -i\int_{t_{0}}^{t}\beta_{L/R}(t^{\prime})dt^{\prime}\right\} . \label{eq:SecondtoFirstOrder}\end{aligned}$$ Intuitively, we expect that during a slow sweep the first and third terms on the right hand side will be small and hence we shall ignore them. We will check the validity of these assumptions numerically below (and in Section \[sec:ApproximationCondition\] we derive an analytic criterion for the validity of the first order approximation). We have that $$i\dot{\tilde{a}}_{L/R}=\frac{\omega_{\mathrm{av}}\Delta}{\beta_{L/R}}\tilde{a}_{R/L}\exp\left\{ \pm i\int_{t_{0}}^{t}[\beta_{L}-\beta_{R}]dt^{\prime}\right\} .$$ Assuming $\omega_{\mathrm{av}}$ is very large we can put $$\beta_{L/R}(t)\approx\omega_{\mathrm{av}}\left\{ 1\pm\frac{\Gamma(t)}{\omega_{\mathrm{av}}}+\frac{1}{2}\frac{\Delta^{2}}{\omega_{\mathrm{av}}^{2}}\right\}.$$ Hence, $\beta_{L}-\beta_{R}\approx2\Gamma$ and $\omega_{\mathrm{av}}/\beta_{L/R}\approx1$, giving $$i\dot{\tilde{a}}_{L/R}=\Delta \, \tilde{a}_{R/L}\exp\left\{ \pm2i\int_{t_{0}}^{t}\Gamma(t^{\prime})dt^{\prime}\right\}.$$ Changing the variables back to $a_{L/R}$, we finally obtain $$i\left(\begin{array}{c} \dot{a}_{L}\\ \dot{a}_{R} \end{array}\right)=\left( \begin{matrix} \omega_{\mathrm{av}}+\Gamma(t) & \Delta \\ \Delta & \omega_{\mathrm{av}}-\Gamma(t) \end{matrix} \right)\left(\begin{array}{c} a_{L}\\ a_{R} \end{array}\right) . \label{eq:PartialFirstOrder}$$ In the case that the membrane moves at a constant speed, so that $\Gamma(t)$ is linear in time, these are exactly the Landau-Zener equations \[see Eq. (\[eq:SchrodLZ\])\]. We refer to Eq. (\[eq:PartialFirstOrder\]) as the *diabatic first order equations* (DFOE). The results of numerically simulating the dynamics using the ASOE and DSOE schemes are compared in Figs. \[fig:FullvsFirstOrder\], \[fig:FullvsPartialReflectivity98\], and \[fig:FullvsPartialSpeed5000\]. In each case the light is initially located on the right side of the cavity and the membrane is moved from left to right. When the calculation is done using the ASOE we can still plot the results in the diabatic basis by switching basis using Eq. (\[eq:localmodes\]). From Fig. \[fig:FullvsFirstOrder\] we see that the results using the ASOE and DSOE lie on top of each other so that their differences are small relative to the order of magnitude of the amplitudes themselves. Hence, for these parameters we are safe in ignoring the time-dependence of the diabatic mode functions. To get a closer look at the differences, we compute the change in energy relative to its initial value $\Delta \mathcal{E}/\mathcal{E}_0$ for the two sets of equations of motion. From Figs. \[fig:FullvsPartialReflectivity98\] and \[fig:FullvsPartialSpeed5000\], we see that as long as the membrane motion is close to the avoided crossing, the difference is of the order of $10^{-5}$ even for speeds as high as $20,000$ ms$^{-1}$. Let us now check the validity of the first-order-in-time approximation embodied in the DFOE approach, i.e. how good of an approximation it is to ignore the first and the third terms in Eq. (\[eq:SecondtoFirstOrder\]). In Figs. \[fig:PartialvsFirstSpeed5000\] and \[fig:PartialvsFirstReflectivity98\] we compare the relative change in energy with time using the DSOE and the DFOE. The difference between the first and second order models is directly related to the energy pumped into and out of the system because the first order dynamics preserves $\sum_{n} \vert c_{n} \vert^2$, meaning that $\Delta \mathcal{E}_{\mathrm{DFOE}}$ is identically zero. We can see that for increasing reflectivity and speed, the first order approximation becomes less valid. Nevertheless, in the optical frequency regime it is a very good approximation as the discrepancy is only of the order of $10^{-3}$. This number also shows that ignoring the time dependence of the diabatic mode functions is a much smaller effect than the first order reduction of the DSOE to the DFOE. The finding that the first-order-in-time approximation becomes less valid as reflectivity is increased appears to be in contradiction to the results in our previous paper [@NickPaper]. In particular, Fig. 7 of [@NickPaper] shows that the approximation becomes better as the coupling $\Delta$ is decreased. This might be interpreted (erroneously) as saying that reflectivity should be increased for a better match. That paper showed that the first order equations of motion depend on a single dimensionless parameter $v \sqrt{\gamma} / \Delta^2$, a result which is consistent with the Landau-Zener transition probability given in Eq. (\[eq:P\_LZ\]), whereas the second order equations of motion depend additionally upon the dimensionless quantity $\Delta/\omega_{\mathrm{av}}$. Therefore, a comparison of the two dynamics where $v \sqrt{\gamma} / \Delta^2$ is held constant but $\Delta/ \omega_{\mathrm{av}}$ is varied should agree in the limit $\Delta/ \omega_{\mathrm{av}} \rightarrow 0$. This is correct. However, holding $v \sqrt{\gamma} / \Delta^2$ constant and reducing $\Delta$ implies that the speed $v$ must also be decreasing, ensuring that the first order dynamics becomes a better approximation as higher order time derivatives present in the corrections become smaller. Such a comparison of the dynamics is not a good test of the role of reflectivity because it is not just the reflectivity that is varied. In this paper, and in particular in Fig. \[fig:PartialvsFirstSpeed5000\], we study a different situation: we fix the initial and final mirror positions and then sweep through the avoided crossing at a fixed speed while varying the reflectivity. Analytic criterion for validity of first order dynamics {#sec:ApproximationCondition} ======================================================= In the previous section we presented numerical evidence showing that the second order Maxwell wave equation can, in certain regimes, be approximated by a first-order-in-time Schrödinger-like equation. In particular, we saw in Figs. \[fig:PartialvsFirstSpeed5000\] and \[fig:PartialvsFirstReflectivity98\] that the approximation became better when the membrane reflectivity and speed are low. However, apart form dropping higher derivatives, it is not clear where in the derivation of the first order equation Eq. (\[eq:PartialFirstOrder\]) the restriction to small reflectivities or speeds came in. Let us develop a criterion that allows us to evaluate when the first order approximation is valid depending upon the mirror reflectivity and speed. ![A plot of the analytical condition given in Eq. (\[eq:ratio\]) for the validity of the DFOE as a function of membrane reflectivity and speed. When $r$ is large the DFOE is a good approximation. It can be clearly seen that r becomes large at small speeds. The dependence on reflectivity is much gentler, but there is still a discernible monotonic increase in $r$ as the reflectivity is reduced. These trends are in agreement with the numerical results shown in previous sections, showing that the DFOE becomes a better approximation at low membrane speed and reflectivity.[]{data-label="eq:ratio condition"}](RatioCondition5){width="\linewidth"} Comparing Eqns. (\[eq:PartialSecondOrder\]) and (\[eq:DSOEFactorize\]), we see that the first order approximation is equivalent to solving the equation $$\frac{d^2}{dt^2} \left( \begin{array}{c}a_{L}\\ a_{R} \end{array} \right) = \left( - H_{\mathrm{LZ}}^2 + i \dot{H}_{\mathrm{LZ}} \right) \left( \begin{array}{c}a_{L}\\ a_{R} \end{array} \right) \label{eq:DerivativeOfFirsOrder}$$ which differs from the true equation by the term $$i \dot{H}_{\mathrm{LZ}} = i\left( \begin{matrix} \dot{\Gamma} & 0 \\ 0 & -\dot{\Gamma} \end{matrix} \right).$$ Thus, for the DFOE to be a valid approximation to the DSOE, we require that the ratio $r \equiv ||H_{\mathrm{LZ}}^2||^2/||\dot{H}_{\mathrm{LZ}}||^2$ be large, i.e.  $||H_{\mathrm{LZ}}^2||$ be much larger than $||\dot{H}_{\mathrm{LZ}}||$. The symbol $||\cdot||$ represents the norm of the matrix given by the square root of the sum of each matrix element squared [@SpenceFriedberg]. Substituting in $H_{\mathrm{LZ}}$ and $\dot{H}_{\mathrm{LZ}}$, the ratio is given by $$\begin{aligned} r = & &\frac{8\Delta^2\omega_{\mathrm{av}}^2+(\left[\omega_{\mathrm{av}}+\Gamma\right]^2+\Delta^2)^2+(\left[\omega_{\mathrm{av}}-\Gamma\right]^2+\Delta^2)^2}{2\dot{\Gamma}^2} \\ =& & \frac{(\gamma^{2}\Delta L^{4}+\Delta^{4}+\omega_{\mathrm{av}}^{4})+6\omega_{\mathrm{av}}^{2}(\Delta^{2}+\gamma\Delta L^{2})+2\gamma\Delta^{2}\Delta L^{2}}{\gamma\dot{\Delta L}^{2}} \label{eq:ratio}\end{aligned}$$ where the second line follows from putting $\Gamma=\sqrt{\gamma} \, \Delta L$ and simplifying. The role of the optical frequency and mirror speed in the validity of the DFOE is quite clear from this expression for $r$: increasing $\omega_{\mathrm{av}}$ and decreasing $\dot{\Delta L}$ contribute to increasing $r$. What is not as obvious is the role of the reflectivity which according to Eqs. (\[eq:deltadef\]) and (\[eq:gammadef\]) appears in the terms $\Delta$ and $\gamma$ through their dependence upon $\alpha$. Intuitively, it seems that a higher reflectivity causes the membrane to perturb the field more and should therefore should lead to a breakdown of the first order approximation. That this is indeed the case can be demonstrated by differentiating $r$ with respect to $\alpha$, from which we find that the derivative is always negative showing that $r$ monotonically decreases as $\alpha$ (and hence $R$) increases. The dependence of $r$ on reflectivity and speed are shown in Fig. \[eq:ratio condition\]. A further pictorial explanation can be found in the structure of the frequencies $\omega_{2/1}$ near an avoided crossing as shown in Fig. \[fig:AvoidedCrossing\]. One can see that as the central membrane reflectivity approaches unity the avoided crossing curves become steeper (asymptotically approaching the diabatic frequencies given by the red dashed curves) and change very rapidly at the avoided crossing itself as $\Delta$ shrinks. This implies a faster change of the frequencies \[and quantities such as $\beta$ given in Eq. (\[eq:betadef\])\] with membrane position and hence that second order derivatives become more important in this limit. Radiation Pressure and the Work-Energy Theorem {#sec:RadiationPressure} ============================================== In this section we attempt to give a more physical explanation for the change in energy of the electromagnetic field seen in the second-order-in-time descriptions of the dynamics (ASOE and DSOE). By applying the work-energy theorem $\Delta \mathcal{E} = \mathcal{W} = \int \mathbf{F} \cdot \mathrm{d}\mathbf{x}$, we show that the radiation pressure exerted by the field on the membrane fully accounts for the changes in electromagnetic energy we have computed in Sections \[sec:adiabatictheorem\] and \[sec:LocalModeDynamics\]. This also provides a self consistency check on our numerical simulations. Starting from the Maxwell stress tensor, we carefully derive the radiation pressure of light in the two mode approximation near an avoided crossing. We show that the radiation pressure obtained by simply adding the pressures due to each adiabatic mode \[$U_{1/2}(x,\Delta L)$\] individually leads to erroneous results and is not equivalent to the radiation pressure applied by the net electric field that includes interference. The effect of radiation pressure can in fact be seen in Figs. \[fig:PartialvsFirstSpeed5000\] and \[fig:PartialvsFirstReflectivity98\] where the light is initially localized on the right side of the cavity and the membrane is moved from left to right at a constant speed. The radiation pressure pushes against the membrane, and hence, to maintain a constant speed, we need to apply a force equal in magnitude to the radiation force, but in the opposite direction. Therefore, positive work is done by the membrane on the optical field and the latter’s energy will increase. Furthermore, one can see in Figs. \[fig:PartialvsFirstSpeed5000\] and \[fig:PartialvsFirstReflectivity98\] that the energy pumped in reaches a maximum value. This occurs at the point where the light intensities on the left and right sides of the membrane are equal and the radiation pressure cancels, as can be seen in Fig. \[fig:RadPress1\] which plots the radiation pressure corresponding to the various curves in Fig. \[fig:PartialvsFirstSpeed5000\]. Past this point the light intensity is greater on the left and the radiation pressure points in the same direction as the membrane motion which means that light does work upon the membrane. An external force has to be applied in the opposite direction to the membrane motion in order to maintain a constant speed. The force due to radiation pressure on some region of volume $\mathcal{V}$ and surface area $\mathcal{S}$ is given by [@Griffiths] $$\mathbf{F}=\int_{\mathcal{S}}\overleftrightarrow{\mathbf{T}}\cdot \mathrm{d}\mathbf{a}-\frac{\partial}{\partial t}\int_{\mathcal{V}}\epsilon(\mathbf{r})\mathbf{E}\times\mathbf{B} \, \mathrm{dV} \label{eq:radiationpressure}$$ where $\overleftrightarrow{\mathbf{T}}$ is the Maxwell stress tensor defined by $$T_{ij}\equiv\epsilon_{0} \left(E_{i}E_{j}-\frac{1}{2}\delta_{ij}E^{2}\right)+\frac{1}{\mu_{0}}\left(B_{i}B_{j}-\frac{1}{2}\delta_{ij}B^{2}\right).$$ We note in passing that the second term on the right hand side of Eq. (\[eq:radiationpressure\]) is responsible for the difference between the Abraham and Minkowski expressions for the momentum of light in a medium [@Hinds05]. We shall neglect it here because in the $\delta$-function membrane model the volume $\mathcal{V}$ is vanishingly small. The first term, on the other hand, depends on the surface $\mathcal{S}$ of the membrane interfaces and this does not vanish. Since the only non-zero components of the electromagnetic field are $E_z$ and $B_{y}$, the stress tensor is purely diagonal. Furthermore, we only require the force along the $x$-axis and thus the only component of $\overleftrightarrow{T}$ we need is $T_{xx}$ which is given by $$T_{xx}=-\frac{\epsilon_{0}}{2}E_{z}^{2}-\frac{1}{2\mu_{0}}B_{y}^{2}.$$ Hence, the force on the membrane is $$F=\int_{\mathcal{S}}T_{xx}\mathrm{d}a_{x}=\mathcal{A}\left\{ -\frac{\epsilon_{0}}{2}E_{z}^{2}-\frac{1}{2\mu_{0}}B_{y}^{2}\right\} _{\mathrm{left}}^{\mathrm{right}}$$ where $\mathcal{A}$ is the transverse area of the cavity mode at the membrane and the limits are evaluated at the left and right interfaces of the membrane. It is useful to first picture this for the case of a membrane of finite width and then take the limit as the width shrinks to zero. The radiation pressure is therefore $$\begin{aligned} \mathcal{P}=\frac{F}{\mathcal{A}} & =\left\{ -\frac{\epsilon_{0}}{2}E_{z}^{2}-\frac{1}{2\mu_{0}}B_{y}^{2}\right\} _{\mathrm{left}}^{\mathrm{right}} \nonumber \\ & =-\frac{1}{2\mu_{0}}\left\{ B_{y}^{2}\right\} _{\mathrm{left}}^{\mathrm{right}}\end{aligned}$$ and is simply proportional to the difference of the magnetic field intensity between the two sides. The electric field does not contribute because it is continuous at the membrane interface. By contrast, the magnetic field is discontinuous because it is related to the spatial derivative of the electric field, and for a $\delta$-function dielectric $\partial_{x} E$ has finite jump across it. With this expression for the radiation pressure in hand, the work-energy theorem predicts that the change in the energy of the electromagnetic field will be $$\frac{\Delta \mathcal{E}(\tau)}{\mathcal{A}}=-v\int_{-1}^{\tau} \mathcal{P}(\tau^{\prime}) \, \mathrm{d}\tau^{\prime} \label{eq:WorkI}$$ where $\tau$ is defined in Sec. \[sec:adiabatictheorem\] and the negative sign recognizes the fact that we need the work done on the field by the membrane rather than vice versa. Once the magnetic field has been computed, the radiation pressure interpretation of the physical mechanism behind the energy change can be verified by comparing it against Eq. (\[eq:lightenergy\]) which gives $$\frac{\Delta \mathcal{E}(\tau)}{\mathcal{A}}=\epsilon_{0}\left\{ |a_{1}(\tau)|^{2}+|a_{2}(\tau)|^{2}-1\right\} \label{eq:energyformula}$$ assuming that we pick the initial amplitude sum to be $1$. ![Comparison of the change in energy $\Delta \mathcal{E}$ of the optical field (solid blue curves) with the work done $\mathcal{W}$ by radiation pressure on the membrane (red dash-dot curves) during passage through an avoided crossing. Both quantities are in units of the initial energy $\mathcal{E}_{0}$. Eq. (\[eq:energyformula\]) is used to calculate $\Delta \mathcal{E}$ and the radiation pressure is obtained by summing the contributions given by Eq. (\[eq:pressuremodem\]) for the two modes separately. The agreement is good but breaks down at higher membrane reflectivities. The membrane speed is 5000 ms$^{-1}$. The same mode amplitudes $\{c_{1}(t),c_{2}(t)\}$ were used for both sets of curves and were calculated using the ASOE. As usual, the light is initially localized on the right hand side of the membrane. []{data-label="fig:WorkEnergyII"}](WorkEnergyForceIISpeed5000){width="\linewidth"} ![Same as Fig. \[fig:WorkEnergyII\], except that the radiation pressure is calculated using Eq. (\[eq:WorkII\]) which includes interference between the contributions from each mode. As can be seen, the agreement is excellent at all reflectivities.[]{data-label="fig:WorkEnergyI"}](WorkEnergyForceISpeed5000){width="\linewidth"} The magnetic field entering the expression for radiation pressure can be obtained from the electric field using Maxwell’s equations. The electric field due to the $m^{\mathrm{th}}$ mode is $$\mathbf{E}_{m}(x,t)=c_{m}(t) U_{m}(x,t) \exp[-i\theta_{m}(t)] \, \hat{\mathbf{z}}$$ which gives rise to the displacement field $\mathbf{D}(x,t)= \epsilon(x,t) \mathbf{E}(x,t)$. According to the Maxwell equation $$\nabla\times\mathbf{H}=\frac{{\partial}\mathbf{D}}{{\partial}t}$$ the magnetizing field satisfies $$\begin{gathered} {\partial}_{x}H_{m}(x,t)\hat{\mathbf{z}} = \bigg\{-i \omega_{m}(t) \epsilon(x,t) c_{m}(t) U_{m}(x,t) \\ + \frac{{\partial}}{{\partial}t}[\epsilon(x,t) c_{m}(t)U_{m}(x,t)]\bigg\} \exp[-i\theta_{m}(t)] \hat{\mathbf{z}} . \label{eq:Hfieldderivative}\end{gathered}$$ Given the large magnitude of the optical frequency $\omega_{m}$, we can to a very good approximation ignore the second term on the right hand side. Incorporating the boundary conditions at the end mirrors, the solutions to Eq. (\[eq:Hfieldderivative\]) can be written $$H_{m}(x,t)=i c \epsilon_0 c_{m}(t) \exp[-i\theta_{m}(t)]G_{m}(x, \Delta L)$$ where $$G_{m}(x,\Delta L)= \begin{cases} A_{m}\cos[k_{m}(x+L_{1})], & -L_{1} \le x \le 0\\ B_{m}\cos[k_{m}(x-L_{2})], & 0 \le x \le L_{2} \end{cases}$$ \[compare with  Eq. (\[eq:globalmodes\]), in particular, the amplitudes $A_{m}$ and $B_{m}$ are the same as for the electric field modes $U_{m}(x,\Delta L)$\]. The radiation pressure on the membrane located at $x=0$ due to a monochromatic field of frequency $\omega_{m}$ is, therefore, $$\begin{aligned} \mathcal{P}_{m} &= -\frac{\mu_{0}}{2}\left\{ \vert H_{m} \vert^{2}\right\} _\mathrm{left}^\mathrm{right} \nonumber \\ &=-\frac{\epsilon_{0}}{2} \vert c_{m}(t) \vert^2 \left\{B_{m}^{2}\cos^{2}(k_{m}L_{2}) - A_{m}^{2}\cos^{2}(k_{m}L_{1}) \right\} . \label{eq:pressuremodem}\end{aligned}$$ ![Evolution of the radiation pressure on the central membrane during passage through an avoided crossing for four different membrane reflectivities. The membrane speed is held constant at 5000 ms$^{-1}$ and the radiation pressure is calculated using (\[eq:WorkII\]). The maximum radiation pressure is greater at larger membrane reflectivities, however, at $98 \%$ reflectivity the radiation pressure exhibits oscillatory behaviour due to the non-adiabatic nature of the optical dynamics.[]{data-label="fig:RadPress1"}](RadiationPress1){width="\linewidth"} ![Same as Fig. \[fig:RadPress1\] except that here we fix the membrane reflectivity at $98 \%$ but choose five different membrane speeds. The highest mirror speed leads to non-adiabatic optical dynamics and consequently an oscillatory behaviour of the radiation pressure. This figure corresponds to the optical dynamics shown in Figs. \[fig:AdiabaticCriteriaSingleMode2\] and \[fig:AdiabaticCriteriaSingleMode\].[]{data-label="fig:RadPress2"}](RadiationPress2){width="\linewidth"} In Fig. \[fig:WorkEnergyII\] we compare the results of the radiation pressure calculation with the exact result given in Eq. (\[eq:energyformula\]). The total radiation pressure is taken to be the sum of that due to each monochromatic light field separately. We see that the agreement is excellent for the lower reflectivity cases, but there are noticeable differences between the $98 \%$ reflectivity curves. This is because we are not including interference between the two modes involved in the avoided crossing. Rather of summing up the forces due to individual frequencies of light, let us instead find the force due to the net electromagnetic field. The total electric field is $$\begin{gathered} \mathbf{E}(x,t)= \left\{ c_{1}(t)\exp[-i\theta_{1}(t)]U_{1}(x,t)+ \right. \\ \left. c_{2}(t)\exp[-i\theta_{2}(t)]U_{2}(x,t) \right\} \hat{\mathbf{z}}.\end{gathered}$$ Following the analogous steps as the single mode case, we find that the magnetizing field due to two modes is $$\begin{gathered} H(x,t) = i c \epsilon_0 \left\{ c_{1}(t)\exp[-i\theta_{1}(t)]G_{1}(x,\Delta L)+ \right. \\ \left. c_{2}(t)\exp[-i\theta_{2}(t)]G_{2}(x, \Delta L)\right\}\end{gathered}$$ and therefore the radiation pressure is given by $$\begin{gathered} \mathcal{P} =-\bigg\{ \vert c_{1}(t) \vert^2 \left( B_{1}^{2}(t) \cos^{2}[k_{1} L_{2} ]-A_{1}^{2}(t) \cos^{2}[k_{1} L_{1} ] \right) \\ +\vert c_{2}(t) \vert^2 \left( B_{2}^{2}(t) \cos^{2}[k_{2} L_{2} ]-A_{2}^{2}(t) \cos^{2}[k_{2} L_{1} ] \right) \\ +2 \Re \left[ c_{1}^{\ast}(t) c_{2}(t) e^{i \theta_{12}} \right] \left( B_{1}(t)B_{2}(t) \cos [k_{1} L_{2}]\cos[k_{2} L_{2} ] \right. \\ - \left. A_{1}(t)A_{2}(t) \cos [k_{1} L_{1}]\cos[k_{2} L_{1} ] \right) \bigg\} \frac{\epsilon_{0}}{2} . \label{eq:WorkII}\end{gathered}$$ The cross terms on the third and fourth lines are not included in Fig. \[fig:WorkEnergyII\], but are included in Fig. \[fig:WorkEnergyI\] where we find perfect agreement with the general result given in Eq. (\[eq:energyformula\]). It is instructive to plot the radiation pressure itself during passage through an avoided crossing, and this is done in Figs. \[fig:RadPress1\] and \[fig:RadPress2\] where we now exclusively use the more accurate form for the radiation pressure given in Eq. (\[eq:WorkII\]). Initially the light is localized on the right of the membrane producing a radiation pressure in the $-x$ direction; on the other side of the avoided crossing the light has swapped sides and so the radiation pressure reverses direction. In Fig.  \[fig:RadPress1\] the effect of changing the reflectivity of the membrane is shown. As expected, the maximum radiation pressure increases with reflectivity and thus it is possible to do more work on the optical field in the regime of high reflectivity. In Fig.  \[fig:RadPress2\] we see the effect of varying the membrane speed. The pressure curves do not all pass through zero at the same point, there being a slight lag at higher speeds. Perhaps the most striking feature of both Figs. \[fig:RadPress1\] and \[fig:RadPress2\] is that at higher reflectivities and speeds the radiation pressure develops oscillations. If the transfer is adiabatic then light is smoothly transferred from one side of the cavity to the other with the radiation pressure monotonically reversing direction. However, non-adiabatic passage means that not all the light is transferred to the other side. Instead, the system is left in an “excited” state with a certain fraction of the light sloshing back and forth between the two sides of the cavity leading to an oscillatory radiation pressure. Quantization {#sec:DCE} ============ Although the focus of this paper is on classical fields, in this section we review the quantum version of the problem in order to better understand the connection between the two. In Section \[sec:setup\] we discussed quantization for the case of a static membrane and showed how the second-order-in-time wave equation became a first order Heisenberg equation of motion for the field operator $\hat{C}(t)$. The dynamic membrane case is more involved and our approach here, which makes use of Dirac’s canonical quantization method, is adapted from Law’s treatment of a single cavity with a moving end mirror [@Law1] (in reference [@Law3] Cheung and Law treat the problem of the double cavity but they consider the membrane position and momentum as dynamical variables to be included in the Hamiltonian rather than following a prescribed motion as we do here). As for the static case, the quantum operators in the Heisenberg representation can be constructed from the solutions to the classical wave equation. It is more usual to work with the vector potential $A(x,t)$ than the electric field; the former satisfies the wave equation $$\frac{\partial^2 A}{\partial x^2}-\frac{\partial}{\partial t} \left[ \mu_{0} \epsilon(x,t) \frac{\partial A}{\partial t} \right]=0 \label{eq:waveeqnforA}$$ with the same boundary conditions at the end mirrors as the electric field. Unlike in the static case, it is necessary to introduce an auxiliary variable $\pi(x,t) \equiv \epsilon(x,t) \partial A(x,t)/ \partial t$ which is the ‘momentum’ conjugate to $A(x,t)$. Canonical quantization is achieved by imposing the commutation relation $[ \hat{A}(x,t), \hat{\pi}(x',t)]=i \hbar \delta(x-x')/\mathcal{A}$, where $\mathcal{A}$ is the area of the mode functions. A separation of variables can be achieved by expanding $\hat{A}(x,t)$ and $\hat{\pi}(x,t)$ over the adiabatic modes with time-dependent amplitudes $\hat{Q}_{n}(t)$ and $\hat{P}_{n}(t)$: $$\begin{aligned} \hat{A}(x,t) & = & \frac{1}{\sqrt{\epsilon_{0}}} \sum_{n} \hat{Q}_{n} (t) U_n(x,t) \label{eq:Aexpansion} \\ \hat{\pi}(x,t) & = & \frac{\epsilon(x,t)}{\sqrt{\epsilon_{0}}} \sum_{n} \hat{P}_{n} (t) U_n(x,t). \label{eq:piexpansion}\end{aligned}$$ $\hat{Q}_{n}(t)$ and $\hat{P}_{n}(t)$ play roles analogous to the canonical position and momentum variables of the harmonic oscillator and indeed they obey the canonical commutator $[\hat{Q}_{m}(t),\hat{P}_{n}(t)]=i (\hbar/\mathcal{A}) \delta_{m,n}$, a relation which is inherited from that between $\hat{A}(x,t)$ and $\hat{\pi}(x,t)$. In the static case $\hat{P}_{n}(t)$ can be eliminated in favor of $\hat{Q}_{n}(t)$, however, the fact that the separation into space- and time-dependent variables is not complete in the dynamic case (because the mode functions also depend on time), introduces an extra coupling between $\hat{Q}_{n}(t)$ and $\hat{P}_{n}(t)$ that prevents a description purely in terms of $\hat{Q}_{n}(t)$ and hence in terms of a single first-order-in-time equation for $\hat{Q}_{n}(t)$. Explicit expressions for $\hat{Q}_{n}(t)$ and $\hat{P}_{n}(t)$ can be found by inverting the above equations by using the orthonormality of the mode functions \[Eq. (\[eq:orthonormal\])\] giving $$\begin{aligned} \hat{Q}_{n}(t) & = & \frac{1}{\sqrt{\epsilon_{0}}} \int_{-L_{1}}^{L_{2}} \epsilon(x,t) \hat{A}(x,t) U_{n}(x,t) \ d x \label{eq:Qn} \\ \hat{P}_{n}(t) & = & \frac{1}{\sqrt{\epsilon_{0}}} \int_{-L_{1}}^{L_{2}} \hat{\pi}(x,t) U_{n}(x,t) \ d x. \label{eq:Pn}\end{aligned}$$ Equations of motion for $\hat{Q}_{n}(t)$ and $\hat{P}_{n}(t)$ are obtained by taking time derivatives of these expressions (details are given in Appendix \[app:quantumEOM\]) $$\begin{aligned} \frac{d \hat{Q}_n(t)}{dt} & = & \hat{P}_{n}(t)- \sum_{m} G_{nm} (t) \hat{Q}_{m} (t) \label{eq:dQdt} \\ \frac{d\hat{P}_n(t)}{dt} & = & - \omega_{n}^{2}(t) \hat{Q}_{n}(t)+ \sum_{m} G_{mn} (t) \hat{P}_{m} (t)\label{eq:dPdt}\end{aligned}$$ where $G_{nm}(t)= \dot{q} g_{nm}(q) $ and $$g_{nm}(q) = \int_{-L_{1}}^{L_{2}} \frac{\epsilon(x,q)}{\epsilon_{0}} U_{n}(x,q) \frac{\partial U_{m}(x,q)}{\partial q} dx . \label{eq:gdefinition}$$ To keep this expression compact we have introduced the symbol $q$ for the membrane displacement $\Delta L/2$. It is clear that the membrane motion introduces coupling between $\hat{Q}_{n}(t)$ and $\hat{P}_{n}(t)$ that is absent in the static case. The coupling is governed by $G_{nm}(t)$ and is directly proportional to the velocity of the membrane $v= \dot{q}$. By integrating the coupled equations of motion Eqns. (\[eq:dQdt\]) and ($\ref{eq:dPdt}$) forward in time the quantum dynamics of the electromagnetic field can be calculated from given initial conditions. However, in order to gain physical insight it is useful to find the corresponding Hamiltonian, i.e. the Hamiltonian that gives $d\hat{Q}_{n}/dt$ and $d\hat{P}_{n}/dt$ as its equations of motion via the Heisenberg equation $i \hbar d\hat{O}/dt=[\hat{O},\hat{H}]$, where $\hat{O}$ stands for either $\hat{Q}$ or $\hat{P}$. One can verify that the Hamiltonian that does the trick is [@Law1] $$\begin{aligned} \hat{H} & = &\frac{1}{2}\sum_{n} \bigg[ \hat{P}_{n}^2+\omega_{n}^{2}(t)\hat{Q}_{n}^2 -G_{nn}(t) \left(\hat{P}_{n} \hat{Q}_{n}+\hat{Q}_{n}\hat{P}_{n}\right) \bigg] \nonumber \\ && -\sum_{m \neq n} G_{m,n}(t) \hat{P}_{m} \hat{Q}_{n} \ . \label{eq:QuantumHamiltonian1}\end{aligned}$$ The first two terms describe a harmonic oscillator with a parametrically driven frequency. The third term introduces correlations in phase space that produce a “squeezing effect” [@schutzhold98] and the final term introduces further correlations that have been called the “acceleration effect” [@schutzhold98], even though a constant velocity is enough (there is no acceleration in the particular cases we have considered in the earlier sections of this paper). The squeezing and acceleration effects give rise to field dynamics such as parametric amplification and transfer of excitations between modes. Indeed, the squeezing term in the Hamiltonian corresponds to that of a degenerate parametric amplifier [@gardiner]. Parametric amplification/reduction is most clearly seen in the Hamiltonian if it is expressed in terms of the annihilation and creation operators defined as $$\begin{aligned} \hat{C}_{n}(t) \equiv \frac{1}{\sqrt{2 \hbar \omega_{n}(t)}}\left( \omega_n(t)\hat{Q}_n(t)+i \hat{P}_{n}(t) \right) \label{eq:annihilationdef} \\ \hat{C}^{\dag}_{n}(t) \equiv \frac{1}{\sqrt{2 \hbar \omega_{n}(t)}}\left( \omega_n(t)\hat{Q}_n(t)-i \hat{P}_{n}(t) \right) \label{eq:creationdef}\end{aligned}$$ which annihilate and create photons in the $n^{\mathrm{th}}$ adiabatic mode. To find the Hamiltonian we proceed similarly to before by taking time derivatives of the expressions for $\hat{C}_{n}$ and $\hat{C}^{\dag}_{n}$ to obtain their equations of motion in terms of $d\hat{Q}_{n}/dt$ and $d\hat{P}_{n}/dt$ whose expressions are already known and then inferring the Hamiltonian that generates them. The resulting Hamiltonian is [@Law1] $$\hat{H} = \sum_n \hbar \omega_n(t) \hat{C}_{n}^{\dag} \hat{C}_{n} -\frac{i \hbar \dot{q}}{2} \hat{\Lambda}(q) \label{eq:QuantumHamiltonian2}$$ where $$\begin{aligned} \hat{\Lambda}(q) & = & \sum_{n} \left\{ g_{nn}(q)-\frac{1}{2 \omega_{n}} \frac{\partial \omega_{n}}{\partial q}\right\} \left[ \left(\hat{C}^{\dag}_{n}\right)^2 - \hat{C}_{n}^{2} \right] \label{eq:Lambda1} \\ && +\sum_{m \neq n} \sqrt{\frac{\omega_{m}(q)}{\omega_{n}(q)}} g_{mn}(q) \left( \hat{C}^{\dag}_{m} \hat{C}^{\dag}_{n} + \hat{C}^{\dag}_{m} \hat{C}_{n} - \mathrm{h.c.} \right) \nonumber\end{aligned}$$ in which h.c. stands for hermitian conjugate. $ \hat{\Lambda}(q)$ contains the terms responsible for non-adiabatic transfer (scattering of photons between modes) and also amplification/reduction processes. Due to its prefactor of $\dot{q}$, these terms arise purely as a result of membrane motion. The diagonal terms in $\hat{\Lambda}(t)$ give rise to single-mode squeezing by creating and annihilating photons in pairs in the same mode, whereas the off-diagonal ($m \neq n$) terms give rise both to two-mode squeezing ($ \hat{C}^{\dag}_{m} \hat{C}^{\dag}_{n} -$ h.c. terms) where pairs of photons are created and annihilated in different modes, and scattering between modes ($ \hat{C}^{\dag}_{m} \hat{C}_{n} - $ h.c. terms). We saw the classical analogues of these processes in Sections \[sec:adiabatictheorem\], \[sec:LocalModeDynamics\] and \[sec:RadiationPressure\] where we found both the transfer of field energy between modes and the amplification/reduction of the total energy in both modes even in the absence of transfer. However, unlike in the classical case, in quantum mechanics photons can be created from the vacuum, and this is the DCE. In the two-mode case the Hamiltonian can be written out explicitly. In order to obtain analytic expressions for the coefficients one can approximate the adiabatic mode functions by superpositions of modes perfectly localized in either the left or right sides of the cavity, as detailed in Appendix \[app:analyticexpressions\]. One finds $$\begin{aligned} g_{11} & = & g_{22}=0 \\ g_{12} & = & -g_{21}= -\frac{d \Gamma (q)}{dq} \frac{\Gamma(q) \Delta}{2 (\Gamma^2(q)+\Delta^2)^{3/2}}\end{aligned}$$ where $\Gamma(q)= 2\sqrt{\gamma}q \approx 2 (\omega_{\mathrm{av}}/L) q$ so that $d \Gamma/ dq \approx 2 \omega_{\mathrm{av}}/L$. In addition, at this level of approximation $$\frac{1}{\omega_{1}}\frac{d \omega_{1}}{ dq} \approx -\frac{1}{\omega_{2}}\frac{d \omega_{2}}{ dq} = \frac{4 \omega_{\mathrm{av}}}{L \sqrt{\Delta^2+\Gamma^2(q)}}\frac{q}{L}$$ and $$\begin{aligned} \sqrt{\frac{\omega_{1}(q)}{\omega_{2}(q)}} \approx 1- \frac{\sqrt{\Delta^2+\Gamma^2(q)}}{\omega_{\mathrm{av}}} \\ \sqrt{\frac{\omega_{2}(q)}{\omega_{1}(q)}} \approx 1+ \frac{\sqrt{\Delta^2+\Gamma^2(q)}}{\omega_{\mathrm{av}}}.\end{aligned}$$ As shown in Appendix \[app:analyticexpressions\], the corrections to unity in these latter two expressions are necessary to consistently keep terms of the same magnitude as $(1/\omega_{1}) d \omega_{1}/dq$. The two-mode quantum Hamiltonian in the adiabatic basis then takes the form $$\hat{H}=\hbar \omega_{1}(t) \hat{C}_{1}^{\dag}\hat{C}_{1} +\hbar \omega_{2}(t) \hat{C}_{2}^{\dag}\hat{C}_{2} -\frac{i \hbar \dot{q}}{2} \hat{\Lambda}(q) \label{eq:QuantumHamiltonian3}$$ where $$\begin{aligned} \hat{\Lambda}(q) & = & \frac{1}{2 \omega_{1}}\frac{d \omega_{1}}{dq} \left( \hat{C}_{2}^{\dag} \hat{C}_{2}^{\dag} - \hat{C}_{2} \hat{C}_{2} + \hat{C}_{1} \hat{C}_{1} - \hat{C}_{1}^{\dag} \hat{C}_{1}^{\dag} \right) \nonumber \\ & & + 2 g_{21}\bigg\{ \left( \hat{C}_{2}^{\dag} \hat{C}_{1} - \hat{C}_{1}^{\dag} \hat{C}_{2} \right) \nonumber \\ && \quad +\frac{\sqrt{\Delta^2+\Gamma^2(q)}}{\omega_{\mathrm{av}}} \left(\hat{C}_{2}^{\dag}\hat{C}_{1}^{\dag}- \hat{C}_{2}\hat{C}_{1} \right)\bigg\}. \label{eq:Lambda3}\end{aligned}$$ We note that the squeezing terms \[that appear on the first and third lines of $\hat{\Lambda}(q)$\] are weaker by a factor of $\sim \Delta/\omega_{\mathrm{av}}$ than the intermode transfer terms \[that appear on the second line of $\hat{\Lambda}(q)$\]. Finally, in order to compare the quantum field Hamiltonian with that of Landau-Zener problem, let us re-write it in the diabatic basis. This can be done by rotating the operators (in the Schrödinger representation) as $$\begin{aligned} \hat{C}_{1} & = & \sin \theta \ \hat{a}_{R}+ \cos \theta \ \hat{a}_{L} \\ \hat{C}_{2} & = & \cos \theta \ \hat{a}_{R} - \sin \theta \ \hat{a}_{L}\end{aligned}$$ where $\sin \theta$ and $\cos \theta$ are defined in Eqns. (\[eq:sindefn\]) and (\[eq:cosdefn\]). Making use of the following exact results: $$\begin{aligned} && \cos^2 \theta - \sin^{2} \theta = \frac{\Gamma(q)}{\sqrt{\Delta^2+\Gamma^2(q)}} \\ && \cos \theta \sin \theta = \frac{\Delta}{2\sqrt{\Delta^2+\Gamma^2(q)}} \\ && \cos \theta \sin \theta \left( \omega_{1}-\omega_{2}\right)= \Delta \\ && \omega_{2} \cos^2 \theta + \omega_{1} \sin^{2} \theta = \omega_{\mathrm{av}} + \Gamma(q) \\ && \omega_{2} \sin^2 \theta + \omega_{1} \cos^{2} \theta = \omega_{\mathrm{av}} - \Gamma(q) \ ,\end{aligned}$$ we obtain the Hamiltonian $$\begin{aligned} \hat{H} & = & \hbar \left\{\omega_{\mathrm{av}} +\Gamma(q)\right\} \hat{a}_{R}^{\dag}\hat{a}_{R} +\hbar\left\{ \omega_{\mathrm{av}}-\Gamma(q)\right\} \hat{a}_{L}^{\dag}\hat{a}_{L} \nonumber \\ && +\hbar \Delta (\hat{a}^{\dag}_{R} \hat{a}_{L}+\hat{a}^{\dag}_{L}\hat{a}_{R}) -\frac{i \hbar \dot{q}}{2} \hat{\Lambda}(q) \label{eq:QuantumHamiltonian4}\end{aligned}$$ where this time $$\begin{aligned} && \hat{\Lambda}(q) = 2 g_{21} \left( \hat{a}_{R}^{\dag} \hat{a}_{L} - \hat{a}_{L}^{\dag} \hat{a}_{R} \right) \nonumber \\ && + \bigg\{ \frac{1}{2 \omega_{1}}\frac{d \omega_{1}}{dq} \frac{2 \Delta}{\sqrt{\Delta^2+\Gamma^2(q)}} -2 g_{21} \frac{\Gamma}{\omega_{\mathrm{av}}} \bigg\} \nonumber \\ && \quad \times \left( \hat{a}_{R} \hat{a}_{L} - \hat{a}^{\dag}_{R} \hat{a}^{\dag}_{L} \right) \nonumber \\ && + \bigg\{ \frac{1}{2 \omega_{1}}\frac{d \omega_{1}}{dq} \frac{\Gamma(q)}{\sqrt{\Delta^2+\Gamma^2(q)}} + g_{21} \frac{\Delta}{ \omega_{\mathrm{av}}} \bigg\} \nonumber \\ && \quad \times \left( \hat{a}_{R}^{\dag} \hat{a}_{R}^{\dag} - \hat{a}_{R} \hat{a}_{R} + \hat{a}_{L} \hat{a}_{L} - \hat{a}_{L}^{\dag} \hat{a}_{L}^{\dag} \right) . \label{eq:Lambda4}\end{aligned}$$ The first part of the Hamiltonian \[everything except $\hat{\Lambda}(q)$\] is independent of the membrane velocity and conserves total photon number. It has the structure of a many-particle version of the Landau-Zener problem: The diagonal terms feature the diabatic energies $ \hbar \{\omega_{\mathrm{av}} \pm \Gamma(q)\}$ that vary linearly with $q$, and the off-diagonal term gives the constant photon transfer rate $\Delta$ between the two diabatic modes. $\hat{\Lambda}(q)$ contains the “beyond Landau-Zener” effects including photon pair creation and annihilation in the form of both single and two mode squeezing, and also (photon number conserving) intermode transfer (the first line). Current treatments of the Landau-Zener (“Photon Shuttle”) problem in the optomechanical literature [@HarrisPhotonShuttle] do not include pair creation and annihilation as these effects are expected to be tiny in present experimental setups; even the dominant term in $\hat{\Lambda}(q)$ is an intermode transfer term, albeit a velocity dependent one. Using the experimental numbers given in Reference [@Sankey2010] we can estimate (see Appendix \[app:analyticexpressions\]) that at membrane velocities of $10$ m/s, and at displacements of the order of half way to the next avoided crossing, this term would give a comparable contribution to that of the static membrane transfer rate $\Delta= 2 \pi \times 0.1$ MHz. The equations of motion that arise from the two-mode Hamiltonian in the adiabatic basis given in Eqns. (\[eq:QuantumHamiltonian3\]) and (\[eq:Lambda3\]) are the quantum equivalents of our ASOE derived in Section \[sec:adiabaticbasisEOM\]. One might guess, therefore, that the equations of motion that arise from the two-mode Hamiltonian in the diabatic basis given in Eqns. (\[eq:QuantumHamiltonian4\]) and (\[eq:Lambda4\]) would be the quantum equivalents of the DSOE derived in Section \[sec:LocalModeDynamics\]. However, this is not quite true because when deriving the DSOE we made the approximation of ignoring the time-dependence of the diabatic mode functions on the grounds that in the single particle Landau-Zener problem this is a much smaller effect than the change in amplitudes. Nevertheless, we saw in Section \[sec:LocalModeDynamics\] that energy is not conserved by the DSOE and this can be attributed to the fact that they are second order equations in time that are not trivially first order equations that have been differentiated a second time (as shown in Section \[sec:ApproximationCondition\]) which would conserve energy like the DFOE. Comparing with the quantum Hamiltonian in the diabatic basis, if the time-dependence of the mode functions is ignored then $g_{nm}=0$, but there is still a contribution to photon generation coming from $(1/\omega_{1}) d \omega_{1}/dq$ in $\hat{\Lambda}(q)$. We shall not numerically solve the quantum field equations found in this section, but will leave that to a future publication. Rather, our purpose has been to understand the structure of the quantum theory in comparison to the classical one. Summary and Conclusions {#sec:conclusion} ======================= In this paper we have examined the Landau-Zener problem in the context of an optical field whose modes undergo an avoided crossing. It can therefore be viewed as a study of adiabaticity for fields satisfying the Maxwell wave equation and is related to generalizations of the Landau-Zener theory to the many-particle case in condensed matter physics contexts [@Anglin03; @Tomadin08; @Altland08; @Itin09; @Oka10; @Chen11; @Polkovnikov11; @Qian13]. By comparing the effects of successive approximations, such as ignoring the time-dependence of the modes in the diabatic basis and reducing the Maxwell wave equation to an effective Schrödinger equation, we have emphasized some significant differences to the original Landau-Zener problem which is posed in terms of the (true) single-particle Schrödinger wave equation. In the diabatic basis (whose modes are *not* instantaneous normal modes) almost all the time-evolution occurs in the coefficients as opposed to the mode functions such that the time-evolution of the latter can be ignored. However, reducing the second order Maxwell wave equation to a first order effective Schrödinger equation turns out to be a more severe approximation, at least conceptually, because it prevents changes in the energy of the field associated with parametric amplification (and reduction) that may be considered as classical analogues of the DCE. The Maxwell wave equation therefore allows for a type of evolution unfamiliar from the single-particle case but which becomes particularly evident in the regime of a slowly moving membrane where the non-adiabatic transfer between the modes switches off (like in the single-particle case) and yet the total energy (i.e. photon population) can change. Furthermore, the energy dependence on membrane position does not vanish as the membrane velocity vanishes but tends to a fixed function that depends only on the membrane reflectivity. This type of behaviour was explained in Section \[sec:RadiationPressure\], both qualitatively and quantitatively, by looking at the work done by the radiation pressure on the membrane, and this never vanishes except right at the centre of the avoided crossing. An analytic criterion \[given in Eq. (\[eq:ratio\])\] can be derived which predicts when beyond single-particle effects become important. Apart from the expected role of the membrane velocity, i.e. faster membranes cause more amplification/reduction, the criterion depends on the reflectivity. A more reflective membrane perturbs the modes more, giving a sharper change in the adiabatic mode frequencies as the membrane passes through an avoided crossing. The criterion predicting when the single-particle picture breaks down is obtained by examining when the Maxwell wave equation can be factorized into a product of two effective Schrödinger equations (which are Hermitian conjugates of each other). The factorization is exact for a static membrane but is approximate in the presence of a moving membrane, as shown in Section \[sec:LocalModeDynamics\]. This raises the question of what exactly is the connection between the effective Schrödinger equation used to describe the classical field and the true quantum field description? The answer is rather little, at least in the moving membrane case. The effective Schrödinger equation obtained in this paper is nothing more than an approximation to a classical field equation, and the classical field amplitude that obeys it has no interpretation in terms of a probability amplitude even though it happens to be a complex number in our treatment (the real part gives the physical electric field). Furthermore, there is only a single Schrödinger equation for each mode \[the 2x2 matrix equation given in Eq. (\[eq:PartialFirstOrder\]) is for two modes\]. In the true quantum field description, as given in Section \[sec:DCE\], each mode is described by two canonical coordinates, $\hat{Q}$ and $\hat{P}$, whose first order equations of motion \[Eqns. (\[eq:dQdt\]) and (\[eq:dPdt\])\] only take on the harmonic oscillator form in the limit of a stationary membrane. Only in this limit can $\hat{P}$ be eliminated to obtain the second order in time equation of motion purely in terms of $\hat{Q}$ which is that of a free harmonic oscillator. Converting the canonical coordinates to annihilation and creation operators leads to a Hamiltonian with two pieces: one piece \[Eq. (\[eq:QuantumHamiltonian4\])\] which is straightforward generalization of the single-particle Landau-Zener hamiltonian to the many-particle case, and a second ‘beyond Landau-Zener’ piece \[Eq.  (\[eq:Lambda4\])\] which depends linearly on the membrane velocity and includes the terms responsible for pair creation and annihilation. The evolution of the quantum field obeys the true Schrödinger equation $$i \hbar \frac{\partial \vert \Psi (t) \rangle}{\partial t}= \hat{H}(t) \vert \Psi (t) \rangle \label{eq:trueschrodinger}$$ where $\hat{H}(t)$ can be any one of the Hamiltonians given in Section \[sec:DCE\] and $\vert \Psi (t) \rangle$ is the state vector in Fock space describing the occupation of the various modes by photons. Coming back to the connection to Klein-Gordon equation mentioned in the Introduction, it is known that in the time-independent case it can be exactly reformulated in terms of two coupled Schrödinger equations (see p19 of Reference [@Greiner]), as is to be expected in general for a second order equation. The solutions to each Schrödinger equation individually satisfy the Klein-Gordon equation. In the same time-independent regime the Maxwell wave equation can be exactly reformulated in terms of a single Schrödinger equation (for each mode)—see Sections \[sec:setup\] and \[sec:LocalModeDynamics\]. The difference arises because the Klein-Gordon equation describes a massive field which is in general complex whereas the Maxwell field is real: this means that the Klein-Gordon field excitations include particles and antiparticles whereas in the Maxwell case the photon is massless and is its own antiparticle. Of course, the Maxwell field can have two different polarizations (whereas the Klein-Gordon field is spinless) although we have not made use of this possibility in this work since we assumed a single linear polarization. A close analogy exists between the non-relativistic limit of the Klein-Gordon equation and the effective Schrödinger equation given in Eq. (\[eq:PartialFirstOrder\]) that forms the DFOE approximation used in this paper. Substituting the ansatz $\psi(r,t)=\phi(r,t) \exp[-i m c^2 t/\hbar]$ into the Klein-Gordon equation, where $m$ is the rest mass, the non-relativistic limit is obtained by assuming that the rest mass energy $mc^2$ greatly exceeds the kinetic energy, i.e. $\vert i \hbar \partial \phi /\partial t \vert \ll mc^2 \phi$ (see p7 of Reference [@Greiner]). Thus, second order time derivatives of $\phi$ can be neglected and this leads directly to Schrödinger’s equation for a single massive particle as an approximation to the Klein-Gordon equation. The non-relativistic ansatz should be compared with that introduced in Eq. (\[eq:DFOEansatz\]) which reduces the second order Maxwell wave equation encapsulated in the DSOE to the first order Schrödinger-like DFOE. In both cases the exponential accounts for the dominant time-dependence: this arises from the rest mass energy in the Klein-Gordon case, and in the Maxwell case from the quantities $ \sqrt{(\Gamma(t)\pm\omega_{\mathrm{av}})^{2}+\Delta^{2}}$ given in Eq. (\[eq:betadef\]), i.e.  the diagonal terms of the DSOE given in Eq. (\[eq:DSOEmatrix\]). Also, second order time derivatives are likewise ignored in order to obtain the DFOE. Just as Schrödinger’s equation knows nothing about antiparticles and, indeed, conserves particle number, the Schrödinger-like DFOE knows nothing about parametric amplification of the Maxwell field. Electric field in a moving dielectric {#app:NonrelativisticApproximation} ====================================== As predicted by Fresnel in 1818 [@Fresnel] and observed by Fizeau in 1851 [@Fizeau], the apparent refractive index of a medium depends upon its velocity. This effect is in principle present in the moving membrane studied in this paper, and we shall therefore make a rough estimate of the size of the effect. Inside a stationary dielectric with a uniform refractive index $n_{r}$ the electric field obeys the wave equation $$\frac{{\partial}^2 E} {{\partial}x^2}-\frac{n_{r}^2}{c^{2}}\frac{{\partial}^{2}{E}}{{\partial}t^{2}}=0.$$ Now consider a dielectric moving with velocity $\mathbf{v}$ in the laboratory. In order to find the transformed wave equation we follow [@PiwnickiLeonhardt] and first rewrite the above wave equation as $$\frac{{\partial}^2 E} {{\partial}x^2}-\frac{1}{c^{2}}\frac{{\partial}^{2}{E}}{{\partial}t^{2}}-\frac{n_{r}^2-1}{c^{2}}\frac{{\partial}^{2}{E}}{{\partial}t^{2}}=0.$$ The first two terms form an invariant combination under Lorentz transformation. However, the third term is not invariant and to first order in $\vert \mathbf{v} \vert/c$ the time derivative transforms as ${\partial}/{\partial}t \rightarrow {\partial}/{\partial}t + \mathbf{v} \cdot \nabla $. Therefore, to this order of approximation, the electric field in the dielectric satisfies $$\frac{{\partial}^2 E} {{\partial}x^2}-\frac{n_{r}^2}{c^{2}}\frac{{\partial}^{2}{E}}{{\partial}t^{2}}-2 \frac{n_{r}^{2}-1}{c^{2}} \mathbf{v} \cdot \nabla \frac{{\partial}{E}}{{\partial}t}=0. \label{eq:RelMaxEq}$$ when viewed from the laboratory frame. The highest membrane velocity considered in this paper is $20,000$ ms$^{-1}$, and the highest membrane reflectivity is $98 \%$ for a wavenumber $k=8 \times 10^6$ m$^{-1}$. Using Eq. (\[eq:reflectivity\]) for the reflectivity, we find that this implies that the $\delta$-membrane dielectric coefficient takes the value $\alpha=1.7 \times 10^{-6}$ m. Assuming a membrane of width $w=50$ nm, we can use the relation $\alpha = 2 w n_{r}^2$ derived in Appendix B in reference [@NickPaper] between $\alpha$ and the refractive index to obtain $n_{r} \approx 4$. Armed with the refractive index, and assuming $E(x,t)=E_{0} \exp{\left [i(kx-\omega t)\right ]}$, we can compare the order of magnitude of each term in the transformed wave equation Eq. (\[eq:RelMaxEq\]). We have $\frac{{\partial}^2 E} {{\partial}x^2} \sim k^2$ ; $\frac{n^2}{c^{2}}\frac{{\partial}^{2}{E}}{{\partial}t^{2}} \sim n^2 k^2=16k^2$ ; $v \frac{n^{2}-1}{c^{2}}\frac{{\partial}}{{\partial}x}\frac{{\partial}{E}}{{\partial}t} \sim \frac{v}{c}(n^2-1)k^2=0.001k^2$. We conclude that for the velocities considered in this paper the motion of the membrane only introduces a modification three orders of magnitude smaller than the standard static membrane effect and will therefore be neglected. Initial conditions for the electric field in the adiabatic basis. {#app:initialcondition} ================================================================= In this appendix we find an expression for $\dot{c}_{m}(t_{0})$, where $c_{m}(t)$ is the $m^{\mathrm{th}}$ expansion coefficient of the electric field in the adiabatic basis \[Eq. (\[eq:globalmodeansatz\])\] that is quoted at the end of Section \[sec:adiabaticbasisEOM\]. Our approach is adapted from that given in Appendix F.2 in Reference [@liningtonthesis]. We start from the two Maxwell equations $\nabla \times \mathbf{E}= -{\partial}\mathbf{B}/ {\partial}t$ and $\nabla \times \mathbf{H} = {\partial}\mathbf{D} / {\partial}t$ and put $\mathbf{B}(\mathbf{r},t)=\mu_{0} \mathbf{H}(\mathbf{r},t)$ and $\mathbf{D}(\mathbf{r},t)=\epsilon(\mathbf{r},t) \mathbf{E}(\mathbf{r},t)$, where $\epsilon(\mathbf{r},t)$ is the time and space dependent dielectric function appropriate to the double cavity \[$n_{r}(\mathbf{r},t)= c \sqrt{\epsilon(\mathbf{r},t) \mu_{0} }$ is the refractive index\]. Under the physically reasonable assumption that the time evolution of the dielectric function is much smaller than the optical frequency that determines the time evolution of the electric field, the second Maxwell equation becomes $\nabla \times \mathbf{B} \approx \epsilon \mu_{0} {\partial}\mathbf{E} / {\partial}t$. In our one-dimensional system the two Maxwell equations take the forms ${\partial}E / {\partial}x = {\partial}B / {\partial}t$ and ${\partial}B / {\partial}x = \epsilon(x,t) \mu_{0} {\partial}E / {\partial}t$, respectively. The key assumption we now make is that for $t<t_{0}$ the membrane is stationary $\epsilon(x,t) \rightarrow \epsilon(x)$. This means that the adiabatic mode functions and frequencies for $t<t_{0}$ are time independent. Next we expand the electric and magnetic field amplitudes over the adiabatic basis as $$\begin{aligned} E(x,t<t_{0}) & = & \sum_{n} c_{n} U_n(x) e^{ -i\omega_n t} \\ B(x,t<t_{0}) & = & \frac{i}{c} \sum_{n} c_{n} V_n(x) e^{ -i \omega_n t} \label{eq:EandB}\end{aligned}$$ where we note that the expansion coefficients are the same for both fields and that $\omega_{n}=c k_{n}$. We have also introduced $V_{n}(x)$ as the adiabatic mode functions for the magnetic field. Due to the fact that the membrane is assumed to be stationary, the adiabatic modes are not merely instantaneous eigenmodes like in the moving membrane case but are true normal modes of the double cavity that are independent of one another. This implies that the Maxwell equations must be satisfied for each mode individually and allows us to determine the relationship between the $U_{n}$ and $V_{n}$ mode functions as $$\begin{aligned} \frac{{\partial}U_{n}(x)}{{\partial}x } & = & k_{n} V_{n}(x) \label{eq:dUdx} \\ \frac{{\partial}V_{n}(x)}{{\partial}x } & = & - n_{r}^2(x) \, k_{n} U_{n}(x) . \label{eq:dVdx}\end{aligned}$$ The second of these equations can be used to express the gradient of the total magnetic field in terms of the electric field mode functions $U_{n}$ $$\frac{{\partial}B}{{\partial}x} = - \frac{i}{c} n_{r}^2(x) \sum_{n} c_{n} k_{n} U_{n} (x) e^{ -i \omega_n t} .$$ We now consider times infinitesimally greater than $t_{0}$ when the membrane starts moving. Inserting the above result for ${\partial}B/{\partial}x$ into ${\partial}B / {\partial}x = \epsilon(x,t) \mu_{0} {\partial}E / {\partial}t$ and introducing the time-dependence of all quantities gives $$\begin{aligned} - i & \underset{n}{\sum} & c_{n}(t) \omega_{n}(t) U_{n} (x,t) e^{ -i\int^t_{t_0}\omega_n(t^{\prime})\mathrm{dt^{\prime}} } \\ \nonumber & = & \frac{{\partial}}{{\partial}t} \left\{ \sum_{n} c_{n}(t) U_{n}(x,t) e^{ -i\int^t_{t_0}\omega_n(t^{\prime})\mathrm{dt^{\prime}} } \right\}\end{aligned}$$ which simplifies to $$\sum_{n} \frac{{\partial}}{{\partial}t} \bigg\{ c_{n}(t) U_{n}(x,t) \bigg\} e^{ -i\int^t_{t_0}\omega_n(t^{\prime})\mathrm{dt^{\prime}} } = 0 . \label{eq:initialcondition1}$$ We emphasize that this result is only valid for $t \approx t_{0}$ since in order to derive it we assumed the results given in Eqns. (\[eq:dUdx\]) and (\[eq:dVdx\]) which rely on the time independence of the normal modes. Projecting out the $m^{\mathrm{th}}$ coefficient using the orthonormality of the mode functions, we can express the relation given in Eq. (\[eq:initialcondition1\]) at the initial time $t=t_{0}$ as $$\dot{c}_{m}(t_{0})=- \sum_{n} P_{mn}(t_{0}) c_{n}(t_{0})$$ where the function $P_{mn}(t)$ is defined in Eq. (\[eq:Pdefinition\]). This fixes $\dot{c}_{m}(t_{0})$ for any particular choice of the initial coefficients $c_{n}(t_{0})$. Derivation of the quantum equations of motion {#app:quantumEOM} ============================================= In this appendix we give the derivation of Eqns. (\[eq:dQdt\]) and (\[eq:dPdt\]) which are the equations of the motion for the “position” $\hat{Q}_{n}$ and “momentum” $\hat{P}_{n}$ operators for the field modes that appear in Section \[sec:DCE\]. The derivation begins by taking the time derivatives of Eqns. (\[eq:Qn\]) and (\[eq:Pn\]) for $\hat{Q}_{n}$ and $\hat{P}_{n}$, respectively. Taking the $\hat{Q}_{n}$ case first we have $$\begin{aligned} \frac{d \hat{Q}_{n}}{dt} & = & \frac{1}{\sqrt{\epsilon_{0}}} \int_{-L_{1}}^{L_{2}} dx \bigg[\frac{\partial \epsilon(x,t)}{\partial t} \hat{A}(x,t) U_{n}(x,t) \\ && + \epsilon(x,t) \frac{\partial \hat{A}(x,t)}{\partial t} U_{n}(x,t) + \epsilon(x,t) \hat{A}(x,t) \frac{\partial U_{n}(x,t)}{\partial t} \bigg] \nonumber \\ & = & \frac{1}{\sqrt{\epsilon_{0}}} \int_{-L_{1}}^{L_{2}} dx \bigg[ \frac{\partial \epsilon(x,t)}{\partial t} \hat{A}(x,t) U_{n}(x,t) \nonumber \\ &&+ \hat{\pi}(x,t) U_{n}(x,t) + \epsilon(x,t) \hat{A}(x,t) \frac{\partial U_{n}(x,t)}{\partial t} \bigg] \\ &=& \hat{P}_{n}(t) - \sum_{m} G_{nm}(t) \hat{Q}_{m}(t)\end{aligned}$$ which is the result given in the main text. In going from the first equality to the second we used the definition $\pi(x,t) \equiv \epsilon(x,t) \partial A(x,t)/ \partial t$ which in turn gives $ \hat{P}_{n}(t)$ on the last line when we use the expression given in Eq. (\[eq:Pn\]) for $ \hat{P}_{n}(t)$. We also replaced $\hat{A}(x,t)$ in the other two terms by its expansion over $\hat{Q}_{m}(t)U_{m}(x,t)$ given in Eq. (\[eq:Aexpansion\]) : $$\begin{aligned} && \int_{-L_{1}}^{L_{2}} \frac{dx}{\sqrt{\epsilon_{0}}} \bigg[\frac{\partial \epsilon(x,t)}{\partial t} \hat{A}(x,t) U_{n}(x,t) + \epsilon(x,t) \hat{A}(x,t) \frac{\partial U_{n}(x,t)}{\partial t} \bigg] \nonumber \\ && = \sum_{m} \hat{Q}_{m}(t) \int_{-L_{1}}^{L_{2}} dx \bigg[\frac{\partial}{\partial t} \frac{\epsilon(x,t)}{\epsilon_{0}} U_{m}(x,t) U_{n}(x,t) \nonumber \\ && \quad + \frac{\epsilon(x,t)}{\epsilon_{0}} U_{m}(x,t) \frac{ \partial U_{n}(x,t)}{\partial t} \bigg] \\ && = - \sum_{m}\hat{Q}_{m}(t) \int_{-L_{1}}^{L_{2}} dx \frac{\epsilon(x,t)}{\epsilon_{0}} \frac{\partial U_{m}(x,t)}{\partial t} U_{n}(x,t) \\ && = - \sum_{m} G_{nm} (t)\hat{Q}_{m}(t) \end{aligned}$$ where $G_{nm}(t)=\dot{q} g_{nm}(t)$ and $g_{nm}(t)$ is defined in Eq. (\[eq:gdefinition\]). In going from the first equality to the second equality in this expression we made use of a relation obtained by differentiating the orthonormalization condition Eq. (\[eq:orthonormal\]) with respect to time : $$\frac{\partial}{\partial t} \int_{-L_{1}}^{L_{2}} dx \frac{\epsilon(x,t)}{\epsilon_{0}} U_{m}(x,t)U_n(x,t) = 0.$$ The equation of motion for $\hat{P}_{n}(t)$ is obtained similarly; differentiating Eq. (\[eq:Pn\]) with respect to time yields $$\frac{d \hat{P}_{n}}{dt} = \int_{-L_{1}}^{L_{2}} \frac{dx}{\sqrt{\epsilon_{0}}} \bigg[ \frac{\partial \hat{\pi}(x,t)}{\partial t} U_{n}(x,t)+ \hat{\pi}(x,t) \frac{\partial U_{n}(x,t)}{\partial t} \bigg]. \label{eq:dPdtappendix}$$ The first term can be reexpressed in terms of $\hat{A}(x,t)$ by using the wave equation (\[eq:waveeqnforA\]) to write $$\frac{\partial \hat{\pi}(x,t)}{\partial t} = \frac{1}{\mu_{0}} \frac{\partial^2 \hat{A}(x,t)}{\partial x^2}$$ and replacing $\hat{A}(x,t)$ by its expansion over $\hat{Q}_{m}(t)U_{m}(x,t)$ as given in Eq. (\[eq:Aexpansion\]) gives $$\begin{aligned} \int_{-L_{1}}^{L_{2}} && \frac{dx}{\sqrt{\epsilon_{0}}} \frac{\partial \hat{\pi}(x,t)}{\partial t} U_{n} (x,t) \nonumber \\ && = \sum_{m} \hat{Q}_{m}(t) \int_{-L_{1}}^{L_{2}} \frac{dx}{\mu_{0} \epsilon_{0}} \frac{\partial^2 U_{m}(x,t)}{\partial x^2} U_{n}(x,t) \nonumber \\ && = - \sum_{m} \hat{Q}_{m}(t) \omega_{m}^{2}(t). \label{eq:1stterm}\end{aligned}$$ In the last step we used the time-independent wave equation (\[eq:Utimeindependent\]) satisfied instantaneously by the adiabatic mode functions $U_{m}(x,t)$ to remove the second spatial derivative, leaving an integral corresponding to the orthonormality condition Eq. (\[eq:orthonormal\]). The second term in Eq. (\[eq:dPdtappendix\]) is treated by substituting the expansion of $\hat{\pi}(x,t)$ over $\hat{P}_{m} U_{m}(x,t)$ as given in Eq. (\[eq:piexpansion\]) to give $$\begin{aligned} \int_{-L_{1}}^{L_{2}} && \frac{dx}{\sqrt{\epsilon_{0}}} \hat{\pi}(x,t) \frac{\partial U_{n}(x,t)}{\partial t} \nonumber \\ && = \sum_{m} \hat{P}_{m}(t) \int_{-L_{1}}^{L_{2}} dx \frac{\epsilon(x,t)}{\epsilon_{0}} U_{m}(x,t) \frac{\partial U_{n}(x,t)}{\partial t} \nonumber \\ && = \sum_{m} \hat{P}_{m}(t) G_{mn}(t). \label{eq:2ndterm}\end{aligned}$$ The sum of Eqns. (\[eq:1stterm\]) and (\[eq:2ndterm\]) give the expression for $d \hat{P}_{n}/dt $ quoted in Eq. (\[eq:dPdt\]) in the main part of the paper. Analytic expressions and orders of magnitude for coefficients in the quantum Hamiltonian {#app:analyticexpressions} ======================================================================================== In this appendix we outline the calculation of the coefficients $(1/\omega_{1}) d \omega_{1}/dq$, $(1/\omega_{2}) d \omega_{2}/dq$, $\omega_{1}/\omega_{2}$, $g_{11}$, $g_{22}$, $g_{12}$, and $g_{21}$, that appear in the two-mode quantum Hamiltonians given in Eqns. (\[eq:Lambda3\]) and (\[eq:Lambda4\]). We first consider $(1/\omega_{1}) d \omega_{1}/dq$, where $\omega_{1}= \omega_{\mathrm{av}}-\sqrt{\Delta^2+\Gamma^2(q)}$. Noting that $\Gamma=2 \sqrt{\gamma} q \approx 2 (\omega_{\mathrm{av}}/L) q$ the derivative can be taken. When dividing by $\omega_{1}$ we make the assumption that $ \omega_{\mathrm{av}} \gg \sqrt{\Delta^2+\Gamma^2(q)}$ (recall that $ \omega_{\mathrm{av}}$ is assumed to be an optical frequency $\approx 2 \pi \times 10^{15}$ Hz, whereas the gap $\Delta$ at an avoided crossing, which gives the order of magnitude for $\sqrt{\Delta^2+\Gamma^2(q)}$, is assumed to be tiny in comparison; in experiments $\Delta$ ranges from $2 \pi \times 1 $ GHz [@thompson08] to $2 \pi \times 0.1 $ MHz [@Sankey2010].) Thus we have that $$\begin{aligned} \frac{1}{\omega_{1}} \frac{d \omega_{1}}{dq} & \approx & -\frac{4 \gamma q}{\sqrt{\Delta^2+4 \gamma^2 q^2}} \times \frac{1}{\omega_{\mathrm{av}}} \nonumber \\ & \approx & -\frac{4 \omega_{\mathrm{av}} q/L^2}{\sqrt{\Delta^2+4 \omega^2 q^2/L^2}}\end{aligned}$$ where to obtain the second line we put $\gamma \approx \omega_{\mathrm{av}}^2/L^2 $, see Eq. (\[eq:gammadef\]). Within the same set of approximations, $(1/\omega_{2}) d \omega_{2}/dq$ takes exactly the same magnitude but is of opposite sign. This makes intuitive sense because after an avoided crossing one mode bends down ($\omega_{1}$) and the other bends up ($\omega_{2}$). We can thus replace all instances of the one coefficient by the (negative) of the other. Let us also estimate the magnitude of $(1/\omega_{1}) d \omega_{1}/dq$. In the vicinity of an avoided crossing we can replace $\sqrt{\Delta^2+\Gamma^2(q)}$ by $\Delta$ and thus $$\frac{1}{2 \omega_{1}} \frac{d \omega_{1}}{dq} \sim \mathcal{O} \left( -\frac{2}{L} \frac{\omega_{\mathrm{av}}}{\Delta} \frac{q}{L} \right)$$ which varies linearly with the membrane displacement $\Delta L = 2q$. In the experiment by Thompson *et al* [@thompson08], the total length of the double cavity was $L=6.7$ cm, $\Delta=2 \pi \times 1 $ GHz, and $\omega_{\mathrm{av}}\approx \omega_{\mathrm{laser}}= 10^{15}$ rad/s. Inputting these numbers we find $(1/\omega_{1}) d \omega_{1}/dq \sim 2 \times 10^6 \times (q/L) $ m$^{-1}$. The distance the membrane needs to travel to go between two avoided crossings is $(q/L) \approx c \pi/(2 L \omega_{\mathrm{av}}) \approx 7 \times 10^{-6} $ and so this sets an upper limit on the magnitude of $(q/L)$ we are interested in. Thus, as the membrane travels from one avoided crossing to halfway to the next one $(1/\omega_{1}) d \omega_{1}/dq$ varies in magnitude from 0 to 10 m$^{-1}$. This number depends on $1/L^2$ and so in smaller cavities it would grow accordingly. The basic approximation underlying our calculation of $g_{ij} \equiv(1/\epsilon_{0})\int_{-L_{1}}^{L_2} dx \epsilon(x,q) U_{i}(x,q)\partial U_{j}(x,q)/\partial q$, is to assume that we can expand the adiabatic modes in terms of mode functions which are perfectly localized on the left or right side of the membrane: $$\begin{aligned} \phi_{L}^{(0)} & = & \sqrt{\frac{2}{L_{1}}} \sin \left[ n \pi \left(x/L_{1}+1 \right) \right], \ -L_{1} \le x \le 0 \\ \phi_{R}^{(0)} & = & \sqrt{\frac{2}{L_{2}}} \sin \left[ n \pi \left(x/L_{2}+1 \right) \right], \ 0 \le x \le L_{2}. \end{aligned}$$ These modes in general differ from the diabatic modes which only equal these expressions in the limit $\Delta \rightarrow 0$. Nevertheless, as $\Delta$ is decreased one finds that these rapidly become excellent approximations for the diabatic modes, the corrections being exponentially small. Expanding the adiabatic modes as $$\begin{aligned} U_{1}= \sin \theta \ \phi_{R}^{(0)} + \cos \theta \ \phi_{L}^{(0)} \\ U_{2}= \cos \theta \ \phi_{R}^{(0)} - \sin \theta \ \phi_{L}^{(0)} \end{aligned}$$ where $\sin \theta$ and $\cos \theta$ are given, as usual, by Eqns. (\[eq:sindefn\]) and (\[eq:cosdefn\]), we can obtain analytic results for $g_{11}$, $g_{22}$, $g_{12}$, and $g_{21}$. One finds that $$g_{11}=g_{22}= \cos \theta \frac{d}{dq} \cos \theta + \sin \theta \frac{d}{dq} \sin \theta =0$$ and $$\begin{aligned} g_{12}=-g_{21} &=& \sin \theta \frac{d}{dq} \cos \theta - \cos \theta \frac{d}{dq} \sin \theta \nonumber \\ && = - \frac{d\Gamma(q)}{dq} \frac{\Gamma(q)\Delta}{2(\Delta^2+\Gamma^{2}(q))^{3/2}} \nonumber \\ && \approx - \frac{\omega_{\mathrm{av}}}{L} \frac{\Gamma(q)\Delta}{(\Delta^2+\Gamma^{2}(q))^{3/2}} . \end{aligned}$$ To obtain an order of magnitude estimate for $g_{12}$ we make the same assumptions as for $(1/\omega_{1}) d \omega_{1}/dq$ above and find $$g_{12} \sim \mathcal{O} \left( -\frac{2}{L} \left(\frac{\omega_{\mathrm{av}}}{\Delta}\right)^2 \frac{q}{L} \right)$$ which is a factor of $\omega_{\mathrm{av}}/\Delta \approx 10^5 $ *bigger* than $(1/\omega_{1}) d \omega_{1}/dq$. Finally, we need the factors $\sqrt{\omega_{1}/\omega_{2}}$ and $\sqrt{\omega_{2}/\omega_{1}}$ which multiply $g_{12}$ and $g_{21}$, respectively, in the main Hamiltonian given in Eqns. (\[eq:QuantumHamiltonian2\]) and (\[eq:Lambda1\]). We have $$\begin{aligned} \sqrt{\frac{\omega_{2}}{\omega_{1}}} & = &\sqrt{\frac{\omega_{\mathrm{av}}+\sqrt{\Delta^2+\Gamma^2}}{\omega_{\mathrm{av}}-\sqrt{\Delta^2+\Gamma^2}}} \nonumber \\ & = & 1 + \frac{\sqrt{\Delta^2+\Gamma^2}}{\omega_{\mathrm{av}}}+\frac{1}{2}\left( \frac{\sqrt{\Delta^2+\Gamma^2}}{\omega_{\mathrm{av}}} \right)^2 + \cdots \end{aligned}$$ and $$\begin{aligned} \sqrt{\frac{\omega_{1}}{\omega_{2}}} & = &\sqrt{\frac{\omega_{\mathrm{av}}-\sqrt{\Delta^2+\Gamma^2}}{\omega_{\mathrm{av}}+\sqrt{\Delta^2+\Gamma^2}}} \nonumber \\ & = & 1 - \frac{\sqrt{\Delta^2+\Gamma^2}}{\omega_{\mathrm{av}}}+\frac{1}{2}\left( \frac{\sqrt{\Delta^2+\Gamma^2}}{\omega_{\mathrm{av}}} \right)^2 + \cdots \end{aligned}$$ The corrections to unity, in powers of $\sqrt{\Delta^2+\Gamma^2}/\omega_{\mathrm{av}}$, are small. However, the first correction must be retained to be consistent with other terms involving$(1/\omega_{1}) d \omega_{1}/dq$ which is a factor $\Delta/\omega_{\mathrm{av}}$ smaller than $g_{12}$ and $g_{21}$. [8]{}

--- abstract: 'Excited states in $^{14}$O have been investigated both experimentally and theoretically. Experimentally, these states were produced via neutron-knockout reactions with a fast $^{15}$O beam and the invariant-mass technique was employed to isolate the 1$p$ and 2$p$ decay channels and determine their branching ratios. The spectrum of excited states was also calculated with the Shell Model Embedded in the Continuum that treats bound and scattering states in a unified model. By comparing energies, widths and decay branching patterns, spin and parity assignments for all experimentally observed levels below 8 MeV are made. This includes the location of the second 2$^{+}$ state that we find is in near degeneracy with the third 0$^{+}$ state. An interesting case of sequential 2$p$ decay through a pair of degenerate $^{13}$N excited states with opposite parities was found where the interference between the two sequential decay pathways produces an unusual relative-angle distribution between the emitted protons.' author: - 'R.J. Charity' - 'K.W. Brown' - 'J. Oko[ł]{}owicz' - 'M. P[ł]{}oszajczak' - 'J.M. Elson' - 'W. Reviol' - 'L.G. Sobotka' - 'W.W. Buhro' - 'Z. Chajecki' - 'W.G. Lynch' - 'J. Manfredi' - 'R. Shane' - 'R.H. Showalter' - 'M.B. Tsang' - 'D. Weisshaar' - 'J.R. Winkelbauer' - 'S. Bedoor' - 'A.H. Wuosmaa' title: 'Invariant-mass spectroscopy of $^{14}$O excited states' --- INTRODUCTION ============ The past decade has been one in which nuclear structure theory has seriously attacked the treatment of the effects of open channels. Clearly this advance is required to splice structure and reaction theories. Thus far, the lessons from this effort are many and include an explanation of several levels close to decay thresholds [@smec2; @smec3; @oko2016; @oko2018] and the demonstration of the enhanced mixing of standard shell-model eigenstates when the coupling to the continua is allowed [@smec; @smec2; @smec3]. Both of these features can destroy touchstones, like mirror symmetry, employed by nuclear physicists for decades. The present work contains a combined experimental - theoretical study of $^{14}$O. All excited states of this nucleus can decay by 1$p$ emission and 2$p$ emission becomes possible above 6.6 MeV of excitation for which several excited states of $^{13}$N are possible intermediates. The branching between these possible decay routes is accessible by invariant-mass spectroscopy, the experimental tool employed here. Comparing energies, widths and branching information to Shell Model Embedded in the Continuum (SMEC) calculations provide evidence that the second 2$^{+}$ state has an almost degenerate partner for which only the third 0$^{+}$ state is fully consistent with the SMEC calculations. Having made this assignment, there is little ambiguity as to the assignment of the third 2$^{+}$ state and assignments to other high-lying levels are considered. EXPERIMENTAL METHODS ==================== The experiment was performed at the Coupled Cyclotron Facility at the National Superconducting Cyclotron Laboratory at Michigan State University. Details have been published elsewhere [@Brown:2014; @Brown:2015; @Brown:2017]. Briefly, a mixed $^{17}$Ne ($E/A$=62.9 MeV, 11%, 1.6$\times$10$^{4}$ pps) and $^{15}$O($E/A$=52.1 MeV, 89%) secondary beam was produced from a primary $^{20}$Ne beam ($E/A$=150 MeV, 175 pnA). This secondary beam impinged on a 1-mm-thick $^9$Be target and charged particles produced in the subsequent reactions were detected in the High Resolution Array (HiRA) [@Wallace:2007]. In this experiment, the array consisted of fourteen $E$-$\Delta E$ telescopes which were arranged around the beam axis and subtended polar angles from 2$^\circ$ to 13.9$^\circ$. $^{13}$N and $^{12}$C fragments were only identified in the inner two telescopes where their yield was greatest. The CsI(Tl) $E$ detectors in each telescope were calibrated using $E/A$=55 and 75-MeV proton and $E/A$=55 and 75-MeV $N$=$Z$ cocktail beams. The $^{13}$N calibration was constructed from the $^{14}$N calibrations. The relative locations of the target and each HiRA telescope were determined accurately using a coordinate measurement machine arm. Using the same experimental data, we have examined narrow one- and two-proton resonances in $^{12-15}$N, $^{13}$O and $^{15}$O whose centroids are well constrained. The invariant-mass peaks were found to be less than 10 keV from their tabulated values [@Solove:1991] and so we use 10 keV as the systematic uncertainty in extracting excitation energies. The $\gamma$-ray array CAESAR [@Weisshaar:2010] surrounded the target to detect any $\gamma$ rays in coincidence with the detected charged particles. For this experiment, the array consisted of 158 CsI(Na) crystals covering polar angles between 57.5$^\circ$ and 142.4$^{\circ}$ in the laboratory, with complete azimuthal coverage. The only use of this array in this study was to verify that the invariant-mass peaks obtained with the detected 2$p$+$^{12}$C channel involved decays to the ground state of $^{12}$C rather than the $\gamma$-decaying first-excited state. For the normalization of cross sections, the number of beam particles was determined by counting using a thin plastic-scintillator foil placed in the focal point of the A1900 separator. The loss in the beam flux in its transportation to the target and the relative contribution of each beam species was determined by temporarily placing a CsI(Tl) detector in the target position. As we are only interested in relative yields, the uncertainties in the cross sections quoted in this work are statistical only. However, based on past experience [@Charity:2019] we expect an overall $\pm$15% systematic uncertainty. Simulations {#sec:sim} =========== In order to extract intrinsic widths and cross sections, it is important to understand the experimental invariant-mass resolution and the detection efficiency. To explore this, we have performed Monte Carlo simulations including the detector geometry and their energy and position resolutions. The major contribution to the energy resolution comes from the energy loss and small-angle scattering of the decay fragments as they leave the target which are calculated in the simulations using Refs. [@Ziegler:1985; @Anne:1988]. The reaction is assumed to occur at a random depth within the $^9$Be target. The primary angular distribution of $^{14}$O$^*$ fragments produced in the reactions is chosen so as to give “detected” secondary distributions in the simulations that are consistent with the experimental distributions. Figure \[fig:res\](a) shows the predicted energy resolution as a function of the emission angle $\theta_{C}$ of the remaining heavy “core” fragment in the $^{14}$O$^*$ center-of-mass frame for a reaction induced by a $^{15}$O beam particle. Here zero degrees is the beam axis. Results are shown for simulations of 1$p$ and 2$p$ decay where the excitation energies are chosen to match the 2$^{+}_{1}$ and 2$^{+}_{3}$ states of $^{14}$O [@Solove:1991], respectively. The protons are assumed to be emitted isotropically in the $^{14}$O$^*$ center-of-mass and the 2$p$ decay is treated as sequential emission though the 5/2$^{+}$, third-excited state of $^{13}$N. The resolution, as measured by the FWHM of the response function, shows a similarly strong dependence on $\theta_{C}$ for both simulations. This behavior is due to the use of a thick target. ![Simulated invariant-mass resolution for 1$p$ and 2$p$ emissions from $^{14}$O. Panel(a) shows the dependence of the resolution on the emission angle $\theta_C$ of the heavy core in the $^{14}$O$^*$ center-of-mass frame. Dependence of the (b) resolution and (c) detection efficiency on the $^{14}$O$^*$ excitation energy for all and only transverse emissions of the core. Note that the efficiency for transverse emissions has been multiplied by a factor of 4 for display purposes.[]{data-label="fig:res"}](res.pdf) In analyzing both the experimental and simulated events, the energies of the detected fragments are increased to account for their average energy loss in the target material. These increases are calculated assuming the reaction occurred in the center of the target, but in reality this could be anywhere between the front and back of the target. For a 1$p$ decay, the differential velocity loss in the target material is $\approx$5 times larger for the $^{13}$N fragment than for the emitted proton [@Ziegler:1985]. Thus the uncertainty in the velocity of the heavy fragment is most important in determining the resolution. For transverse decay, [*i.e.*]{}, where the $p$-$^{13}$N relative velocity vector is perpendicular to the beam axis, this uncertainty in the $^{13}$N laboratory velocity acts perpendicularly to the relative velocity and thus only affects its value in second order. The final resolution is dominated by small-angle scattering of the fragments in the target material. However for longitudinal decay, the uncertainty due to the energy loss acts in first order while that from the small-angle scattering acts in second order. The observed angular dependence therefore reflects the relatively larger importance of the energy-loss contribution over the small-angle-scattering contribution to the final resolution. Clearly, improved resolution can be obtained from selecting events associated with transverse emission. From here on we will define these as events with $\left|\cos\theta_{C}\right|<$0.2. Figures \[fig:res\](b) and \[fig:res\](c) shows the simulated resolution and detection efficiency as a function of excitation energy. Results are shown for all “detected” events and those in our transverse gate. The improved resolution from the transverse gate comes at the cost of about a factor of 4 reduction in the efficiency. The resolution is best close to the threshold and deteriorates with increasing excitation energy. The rapid increase in the FWHM at the higher excitation energies for events without any gate, is due to the loss of transverse events, [*i.e.*]{} the efficiency for transverse emission drops as the typical opening angles of the fragments in the laboratory frame become larger than the angular acceptance of the detector. Apart from the improved resolution, there are two other reasons to use the transverse events to fit the excitation energies and widths. First, the determination of the experimental resolution is more certain for the transverse events. In determining an average resolution for all detected events, one needs to know the relative number of events as a function $\theta_{C}$. This dependents on the intrinsic $\theta_{C}$ distribution and the relative detection efficiency as a function of $\theta_{C}$. The latter depends of the assumed primary-fragment angular distribution and the placement of the detectors. By using only a small range of $\theta_{C}$ values these uncertainties are eliminated. Second, the effect of errors in the CsI(Tl) calibration for the heavy fragments are minimized. For transverse emission such errors, like the energy-loss correction, act perpendicular to the relative velocity and thus only contribute in 2^nd^ order. EXPERIMENTAL RESULTS ==================== Values of the excitation energy, widths, and cross sections for $^{14}$O levels determined in this work are listed in Table \[tab:level\] and Fig. \[fig:level\] displays the energy level diagram and observed decay branches. Excited $^{14}$O fragments were produced from reactions induced with both the $^{15}$O and $^{17}$Ne beam particles. This is illustrated from the distribution of the center-of-mass velocity of the detected $p$+$^{13}$N and 2$p$+$^{12}$C events which is shown in Fig. \[fig:vel\]. Both distributions contain two peaks, with their maxima close to the velocities of the two beam species. Note that the low-energy particle-identification thresholds of the CsI(Tl) detectors severely restricts events below $\approx$8.5 cm/ns. We have also looked for peaks in the $\alpha$+$^{10}$C invariant-mass distribution as a number of clustered states are predicted just above the threshold for this channel [@Baba:2019]. However, no resolved states were observed.  [ccccccc]{} $E^*$ & $\Gamma$ & $J^{\pi}$& $\sigma$($p_0$) & $\sigma$($p_1)$ & $\sigma$($p_2+p_3$) & $E^*_{eval.}$\ [\[]{}MeV\] & \[keV\] & & \[mb\] & \[mb\] & \[mb\] & \[MeV\]\ 5.164(2) & & 1$^-_1$ & 0.25(2) & & & 5.173(10)\ 6.285(2) & &3$^-_1$ & 0.37(14) & & & 6.272(10)\ 6.585(1) & $<$25 & 2$^{+}_{1}$ & 3.46(5) & & & 6.590(10)\ 7.669(53) & $<$128 & $\mathbf{(0^+_3)}$ & &\ 7.768& 76& 2$^{+}_{2}$ & & & & 7.768(10)\ 8.787(13) & 182(32) & $\mathbf{(1^-_2)}$ & $<$1.0 & 0.16(5) & 0.37(12)& 8.720(40)\ 9.755(10) & 229(51) & $\mathbf{(2^{+}_{3})}$ & $<$0.7 & 0.21(2) & 3.11(36) & 9.715(20)\ 11.195(30) & $<$220 &$\mathbf{(2^+_4)}$ & $<$3.4 & $<$0.25 & 0.78(35) & 11.240(50)\ \[tab:level\] ![Laboratory velocity distribution of the parent $^{14}$O$^*$ fragments reconstructed from the detected $p$+$^{13}$N and 2$p$+$^{12}$C events. Values of the velocities of the $^{15}$O and $^{17}$Ne beam particles are indicated for comparison. Gates $G17$, $G15_{peak}$, and $G15_{low}$ used to examine the $p$+$^{13}$N events are indicated. []{data-label="fig:vel"}](vel.pdf) $p$+$^{13}$N Events {#sec:p13N} ------------------- ### Excitation-energy distributions Excitation-energy distributions for the $p$+$^{13}$N channel determined with the invariant-mass technique are shown in Fig. \[fig:p13N\] for three gates on the parent $^{14}$O$^*$ velocity. As there are no particle-bound excited states in $^{13}$N, all single-proton decays are to the ground-state of $^{13}$N. The improved resolution for transverse events (solid red histograms) is easily seen compared to those for all events (dotted blue histograms). The reconstructed $^{14}$O excitation-energy distribution for events induced with the $^{17}$Ne beam (gate $G17$) is displayed in Fig. \[fig:p13N\](a). The transverse-emission spectrum shows the largest number of resolved peaks and these can be identified by the arrows in this figure at the energies listed in the most recent evaluation [@Solove:1991] and for the $J$=0$^-_1$ and 2$^{-}_{1}$ states from [@Teranishi:2007; @Wang:2008]. The mechanism for producing $^{14}$O states from $^{17}$Ne probably involves the knockout of one to three nucleons producing states which sequential decay to $^{14}$O$^*$. ![$^{14}$O excitation-energy distributions from detected $p$+$^{13}$N events. Solid red histograms are where the $^{13}$N fragment is emitted transversely from the parent $^{14}$O$^*$ system while the dotted blue histograms are for all events scaled by the factor 0.3. (a) gated on the $^{17}$Ne beam (gate $G17$), (b,c) on the $^{15}$O beam. (b) is for $^{14}$O velocity gate $G15_{peak}$ around the peak, while (c) is for the $G15_{low}$ gate containing the low-velocity tail. The arrows in (a) show the locations of states listed in the most recent evaluation [@Solove:1991] and from [@Teranishi:2007; @Wang:2008].[]{data-label="fig:p13N"}](Ex_p13Na.pdf) We expect single-neutron knockout to be the most important mechanism with the $^{15}$O beam especially for $^{14}$O$^*$ velocities near to the $^{15}$O beam velocity. The neutron configuration of $^{15}$O is well described by a hole in the $p$ shell and thus $p$-shell knockout will produce $J^{\pi}$=0$^+$, 1$^+$, and 2$^+$ states. The excitation-energy spectrum gated around the $^{15}$O beam velocity (gate $G15_{peak}$ in Fig. \[fig:vel\]) shown in Fig. \[fig:p13N\](b) is dominated by a peak at the first 2$^+$ energy and a contribution from a peak near the energy of the second 2$^+$ state is also visible. The lowest-energy peak is the $1^-_{1}$ state which could be produced from knockout of a deeply bound 0$s_{1/2}$ neutron. Also present is a low-energy shoulder on the 2$^+_1$ peak which can be attributed to the 3$^-_1$ state. This cannot be the result of simple neutron knockout reaction, and suggests more complicated multi-step processes are contributing. Such multi-step processes are expected to be more dominant at lower $^{14}$O velocities as verified in Fig. \[fig:p13N\](c) which is gated on the low-velocity tail of the $^{15}$O-induced events , [*i.e.*]{} gate $G15_{low}$ in Fig. \[fig:vel\]. Here we see the relative contribution of the 1$^-_1$ and 3$^-_1$ peaks are enhanced significantly. We will concentrate on fitting the centroids, widths and cross sections determined with the $^{15}$O beam as the statistics are better. For the yields, we have fit the combined $G15_{low}$ and $G15_{peak}$ gates. However, in fitting of the centroids and widths of the 2$^+_1$ and 2$^{+}_2$ peaks, we have just used the $G15_{peak}$ gate as contamination from the other peaks, such as the 3$^-_1$, is minimal. Figure \[fig:fitp13N\] shows a fit to this combined $G15_{low}+G15_{peak}$ spectrum using Breit-Wigner line shapes and a smooth background (dashed curve). Backgrounds are typical needed in fitting invariant-mass spectra and, in general, represent contributions from non-resonant breakup and unresolved wide states. However in this case, we do not expect any unresolved wide states in this interval of excitation energy. Our Monte Carlo simulations were used to incorporate the experimental resolution and the fitted contribution from individual levels are shown by the dotted curves. This fit includes peaks for the 1$^-_1$, 0$^+_2$, 3$^{-}_1$, 2$^{+}_{1}$, 2$^{-}_{1}$, and 2$^{+}_2$ states. Other than the highest-energy peak which will be discussed later, only the 1$^-_1$, 3$^-_1$ and 2$^+_1$ peaks have significant yields in the fits. The fitted centroids of these levels in Table \[tab:level\] are consistent with the evaluated values [@Solove:1991] within our statistical and systematic uncertainties. ![Fit to the excitation-energy distribution for transverse $p$+$^{13}$N events with the combined $G15_{peak}$ and $G15_{low}$ gates. Individual curves (dotted) are shown for each state included in the fit and the fitted background distribution is shown as the dashed blue curve. Arrows indicate the locations of states listed in the most recent evaluation [@Solove:1991] and from [@Teranishi:2007; @Wang:2008]. []{data-label="fig:fitp13N"}](fitp13N.pdf) The intrinsic width of the 2$^{+}_{1}$ state is of interest as this state sits very close to the 2$p$ threshold. In such cases, a collective state can be formed which carries many features of the nearby particle-emission threshold [@smec2; @smec3]. A well know case is $^{8}$Be$_{g.s.}$ which has strong 2$\alpha$ correlation in its wavefunction due to the presence of the nearby 2$\alpha$ decay threshold. In the present case, the wavefunction should have strong 2$p$ character thus giving it a small decay width for the 1$p$ channel. We found the fitted width be consistent with zero with an upper limit of 25 keV at the 3$\sigma$ level. This is consistent with the more restrictive limit of $<$5 keV from [@Negret:2005]. For the peak at the energy of the 2$^+_2$ state, a good fit was obtained with a single peak with a centroid of $E^*$=7.723(2) MeV and an intrinsic width of $\Gamma$=128(18) keV (solid red curve in Fig. \[fig:2plus2\](a)\]. However, the latter is almost a factor of two larger than the value of $\Gamma$=76(10) keV listed in the most recent evaluation [@Solove:1991]. More recent measurements give similarly small values; 63(16) keV [@Teranishi:2007] and 62(10) keV [@Wang:2008]. We therefore conclude that this peak is either not the 2$^+_2$ state or more likely a doublet of the $2^+_2$ and a previously unknown $^{14}$O state. To further highlight this point, Fig. \[fig:2plus2\](a) also shows a calculation (dot-dashed magenta curve) with the centroid fixed to the 2$^{+}_{2}$ evaluated value and the width fixed to the weighted average of the previously reported values. This calculated peak is shifted to higher excitation energies than the experimental peak and does not reproduce the data very well. ![Fits to the peak near the location of the known 2$^+_2$ state. The solid red curve in (a) is a fit with a single Breit-Wigner intrinsic line shape where the centroid and width are allowed to vary. The dot-dashed magenta curve was obtained when the centroid and width are constrained to the known values for the 2$^+_2$ state. (b) shows a fit assuming the observed peak is a doublet. The individual peaks are indicated by the dashed lines. The higher-energy peak has its centroid and width constrained to the values for the 2$^+_2$ state. The smooth dashed blue curve in both panels is the fitted background.[]{data-label="fig:2plus2"}](2plus2.pdf) Figure \[fig:2plus2\](b) shows a fit to this peak with a doublet of states. The higher-energy member of the doublet has its centroid and width constrained to the 2$^+_2$ values \[same as for the dot-dashed curve in Fig. \[fig:2plus2\](a)\] while the width of the low-energy member is assumed to be zero. As long as the width of this new state is about the same as the 2$^+_2$ level or smaller, rather similar fits are produced. The maximum width of this new state is $\Gamma$=128 keV, i.e. the value obtained from fitting this peak as a singlet. ### Parent velocity distribution {#sec:pCVcm} To explore the production mechanism for the observed states, Fig. \[fig:Vcm\] displays reconstructed $^{14}$O$^*$ velocity spectra gated on some of the different peaks in the excitation spectra. For the 2$^+_1$ peak, we have employed a narrow gate to preclude the possibility of any significant contribution from the neighboring 3$^-_1$ peak. Using the simulated resolution, we then use this yield to subtract the relevant contribution from this state under the 3$^-_1$ peak. Velocity distributions for the peaks at the 2$^+_1$ and 2$^+_2$ energies are shown as the solid curves in the Fig. \[fig:Vcm\](a) while distribution for the 1$^-_1$ and 3$^-_1$ peaks are shown as the solid curves in Fig. \[fig:Vcm\](b), The former show very sharp peaks at the $^{15}$O beam velocity expected for single-neutron knockout from a valance state. No dependence of the velocity distribution for the peak at the 2$^+_2$ energy was observed as we scanned the energy across the peak. If this peak is a doublet, then the unknown member also appears to be produced by the knockout of a valence $p$-shell neutron and thus is $J^{\pi}$=0$^+$, 1$^+$, or 2$^{+}$. Indeed the 0$^{+}_{3}$ state is a candidate for this state based on theoretical consideration (see later). The 1$^-_1$ and 3$^-_1$ distributions have broader peaks in this region and the 3$^-_1$ distribution has a flat top, while the 1$^-_1$ is peaked below the $^{15}$O beam velocity. The knockout of a deeply bound $s_{1/2}$ neutron should produce a broader peak which could explain the result for the 1$^-_1$ peak, but as mentioned before, the population mechanism for the 3$^-_1$ peak is not clear. ![Reconstructed $^{14}$O$^*$ parent velocity distributions in the laboratory frame determined for the indicated peaks. For reference, the two beam velocities are shown by the arrows.[]{data-label="fig:Vcm"}](Vcm2.pdf) 2$p$+$^{12}$C Events {#sec:pp12C} -------------------- ### Excitation-energy distribution The reconstructed excitation-energy distribution for the 2$p$+$^{12}$C events where the $^{12}$C fragment is emitted transversely is shown in Fig. \[fig:pp12C\](a). The peaks observed in this spectrum as associated with 2$p$ decay to the ground state of $^{12}$C. While a small 4.44-MeV $\gamma$-ray yield associated with the decay of the first-excited state of $^{12}$C is present in the coincident $\gamma$-ray spectrum, this yield is not correlated with the invariant-mass peaks and is thus associated with background. This is probably from non-resonant breakup reactions. ![$^{14}$O excitation-energy distributions from detected 2$p$+$^{12}$C events. (a) Distribution for all events where the $^{12}$C fragment is emitted transversely. (b) Distribution of all events for the gate on the $J^{\pi}$=1/2$^-$ intermediate state of $^{13}$N. (c) Distribution of transversely-emitted events for the gate on the $J^{\pi}$=3/2$^-$ and 5/2$^+$ intermediate states of $^{13}$N. Curves shows fits to these distribution with the fitted background given by the dashed blue curves and the contribution from the individual states is also shown. The arrows in (a) show the energies of $^{14}$O levels in the most recent evaluation [@Solove:1991]. The labels associated with these indicate the spin and parity assigned in this evaluation when known. The level associated with the question mark was not considered firmly established in the evaluation.[]{data-label="fig:pp12C"}](Ex_pp12C.pdf) This invariant-mass spectrum is dominated by a peak at 9.78 MeV which in the most recent evaluation is tentatively assigned as the 2$^+_{3}$ level. We also observe a small peak just above the 2$p$ threshold near the energy of the $J^{\pi}$=2$^+_2$ state. The tabulations also tentatively assign a state at 8.72 MeV that corresponds to the small peak \[labeled “?” in Fig. \[fig:pp12C\](a)\] on the low-energy side of the main 2$^+_3$ peak. Clearly we have confirmed the existence of this state. In addition there is a structure above the 2$^+_3$ peak which probably has contributions from the 10.89 and 11.2-MeV states listed in the tabulations. No peaks associated with branches of these levels to 1$p$ channel were observed, possibly due to the low efficiency and poor energy resolution at high-excitation-energy for this channel (Sec. \[sec:sim\]). However, Table \[tab:level\] provides upper limits to their 1$p$ cross section. As shown in Fig. \[fig:level\], the sequential 2$p$ decay of these observed states can pass through either the $J^{\pi}$=1/2$^+$ first-excited state or the 3/2$^-$, 5/2$^+$ doublet. The latter are separated by 45 keV and, as their intrinsic widths are 62 and 47 keV, respectively, cannot be resolved and hence can be considered degenerate. Information on the 2$p$ decay mechanism can be gleaned from Jacobi 2-dimensional correlation plots. Consider the schematic velocity vectors in Fig. \[fig:jacy\] for two-proton decay where we have designated the protons as “1” and “2”. The relative velocity vector $\bm{V}_{p-core}$ between the core and proton “2” can be used to calculate a relative energy $E_{p-core}$. If $\bm{V}_{1}$ is the velocity vector between proton “1” and the center-of-mass of the other two particles, then $\theta_k$ is the angle between $\bm{V}_1$ and $\bm{V}_{p-core}$. In the standard Jacobi Y representation, the $x$ axis is $E_{p-core}/E_{T}$ where $E_T$ is the total decay energy and the $y$ axis is $\cos\theta_k$. For $\cos\theta_k\approx$-1, the relative angle between the protons is small, while for $\cos\theta_k$=1, they are emitted back-to-back in the $^{14}$O$^*$ center-of-mass frame. For each event, the two-dimensional Jacobi Y histogram is incremented twice, using coordinates calculated by assigning each of the two detected protons to be proton “2” in Fig. \[fig:jacy\] and the other to be proton “1”. Experimental correlation plots are shown for the 8.79 and 9.78-MeV states in Figs. \[fig:Jac\](a) and \[fig:Jac\](b), respectively. ![Schematic showing the definition of the angle $\theta_k$ used in the Jacobi Y correlation plots. $\bm{V}_{p-core}$ is the relative velocity between proton “2” and the core while $\bm{V}_1$ is the relative velocity between proton “1” and the center of mass of the other two particles. Also shown is the relationship of $\theta_k$ to the relative angle $\theta_{pp}$. []{data-label="fig:jacy"}](jacy.pdf) ![Jacobi Y correlation plots for the 2$p$ decay of the 8.79-MeV and 9.78-MeV peak in the 2$p$+$^{12}$C excitation energy distributions. (a) and (c) are from experimental data while (b) and (d) are the corresponding simulated distributions. Curves in (b) and (d) identify the ridge tops associated with different decays. Solid curves are for decay through the 3/2$^-$, 5/2$^+$ doublet, while dotted curves are for decays through the 1/2$^+$ singlet.[]{data-label="fig:Jac"}](Jac.pdf) A signature of a sequential 2$p$ decay through a single intermediate state is the presence of two separated ridges. One ridge should be vertical associated with the second emitted proton. Here $E_{p-core}$ is determined from the invariant mass of the intermediate state and thus independent of $\theta_k$. The other ridge, corresponding to the first emitted proton, should be tilted, as $E_{p-core}$ in this case depends on the recoil momentum imparted to the core from the second emitted proton and is thus $\theta_k$ dependent. Neither of the experimental Jacobi Y plots in Figs. \[fig:Jac\](a) and \[fig:Jac\](c) show two cleanly separated ridges, but they can still be understood as sequential. The correlation plots in Figs. \[fig:Jac\](b) and \[fig:Jac\](d) show corresponding simulated results obtained assuming sequential decays through both the 1/2$^-$ singlet and 3/2$^-$, 5/2$^+$ doublet of states in $^{13}$N with the relative intensities adjusted to reproduce the experimental correlations. For the 8.79-MeV state, one can discern a total of four ridges in Fig. \[fig:Jac\](a). To help locate these ridges we have drawn curves along the ridge tops in the simulated distribution of Fig. \[fig:Jac\](b). Dashed-curves marking the ridges for decay through the 1/2$^+$ $^{13}$N state and the solid curves for decay through the 3/2$^-$, 5/2$^+$ doublet. On the other hand for the 9.78-MeV state, we observe one very intense ridge in Fig. \[fig:Jac\](c). This structure can be traced in the simulations \[Fig. \[fig:Jac\](d)\] to the two overlapping ridges from the decay through the 3/2$^-$, 5/2$^+$ doublet, [*i.e.*]{} at this excitation energy the two proton energies are approximately equal. A very close examination of the experimental spectrum in Fig. \[fig:Jac\](a) reveals another pair of faint ridges associated with decay through the 1/2$^+$ intermediate $^{13}$N state \[short dashed curves in Fig. \[fig:Jac\](d)\]. It would be useful to gate on $E_{p-core}$ values for the different intermediate states and project out the $^{14}$O invariant-mass spectra. However for the 8.79-MeV peak in Fig. \[fig:Jac\](a), ridges from the two sequential decay paths overlap at $\cos\theta_{k}\approx$1. To avoid this problem we have further restricted the events to $\cos\theta_k<$-0.2. With this extra condition, the excitation energy spectra gated on the 1/2$^-$ singlet and 3/2$^-$, 5/2$^+$ doublet are displayed in Fig. \[fig:pp12C\](b) and \[fig:pp12C\](c), respectively. The latter is for transverse emission, but due to smaller number of events, the former has no gate on the emission direction of the core. These spectra show that the 2$p$ decay path of the 2$^+_2$ level and/or its doublet partner passes only through the 1/2$^-$ singlet (as dictated by energy conservation), while the 8.79 and 9.78-MeV states have contributions from both decay paths. The structures above the 9.78-MeV state appear to be predominately associated with the doublet of intermediate states. To quantify these findings we have fitted these spectra with Breit-Wigner line shapes for each $^{14}$O level considered and use the Monte Carlo simulations to introduce the experimental resolution and apply exactly the same gates as employed for the experimental data in order to extract reliable relative yields for the two decay branches. There is one exception, the lowest-energy peak which is close to threshold may have an asymmetric line shape for this decay branch. We take a limiting $R$-matrix approximation assuming the total width is dominated by decay to the ground state of $^{13}$N and thus can be described by a Breit-Wigner form $B(E^*)$. Therefore, the line shape for the small branch to the $^{13}$N excited state is $P_\ell(E^*-S_p) B(E^*)$ where the first term is the $R$-matrix barrier penetration factor. In the fit, this line shape was fixed assuming decay from the 2$^+_2$ state only, with the Breit-Wigner parameters set to the tabulated values for this state and the penetration factor calculated for $\ell$=1 with a channel radius of $a$=4.85 fm. This allows for a good representation of the low-energy data in Fig. \[fig:pp12C\](b), but we cannot rule out contributions from the proposed doublet partner of the 2$^{+}_2$ state. The fitted yields for all decay branches are listed in Table \[tab:level\]. For the lowest-energy peak, we list only the yield up to $E^*$=8.3 MeV as our approximate line shape has a very long high-energy tail and much of the total yield is associated with this. In the fit, the structure above $E^*$=10.3 MeV is not entirely clear. We have included a moderately narrow peak at $E^*\approx$11.2 MeV to account for the rapidly drop off in yield at this energy. In addition we have added a very wide state between this and the 2$^+_3$ level. However its parameters are not well constrained and similar fits could be obtained by replacing this wide level with some overlapping narrower peaks. ### Parent velocity distribution {#sec:2pCVcm} The reconstructed $^{14}$O$^*$ velocity distributions for the 9.78 and 8.79 peaks are plotted in Fig. \[fig:Vcm\] as the dashed curves. The 9.78-MeV velocity distribution is sharply peaked at the $^{15}$O beam velocity similar to the 2$^+_1$ level and the doublet at the 2$^+_2$ energy, suggesting that is it also produced from the knockout of a valence $p$-shell neutron and hence is restricted to $J^{\pi}$=0$^+$, 1$^+$, or $2^+$. This is consistent with its tentative assignment of $J^{\pi}$=2$^+$ [@Solove:1991; @Negret:2006]. The 8.79-MeV distribution has a broader peak in this velocity region and it is more similar in shape to the 1$^-_1$ peak in the $p$+$^{13}$N channel, possibly from the knockout of a $s_{1/2}$ neutron or some more complex process. ### Distributions of relative angle between protons Information about the spin and parity of a sequential 2$p$ emitter can be gleaned from the distributions of relative angles $\theta_{pp}$ between the protons [@Charity:2009]. The $\theta_{pp}$ distribution for the 9.78 MeV state from the $^{15}$O beam obtained by gating only on the intense ridge in Fig. \[fig:Jac\](c) ($^{13}$N intermediate state is either the 3/2$^-$ or 5/2$^+$) is plotted in Fig. \[fig:thetapp\] as the black circular data points. No gate on transversely-emitted cores is applied here as its affect on the resolution of the relative angle is minor. This distribution has been corrected for the experimental acceptance as determined by our Monte Carlo simulations. In principle $\theta_{pp}$ is just 180$^{\circ}$-$\theta_{k}$ when $\theta_k$ in Fig. \[fig:jacy\] is calculated when proton “2” is the second emitted proton. However in this case, we cannot distinguish which is the first and second emitted protons as they have similar kinetic energies. Therefore both ways of calculating $\theta_k$ are included in Fig. \[fig:thetapp\]. This has little effect as these two ways of calculating $\theta_k$ give very similar values. To further investigate this, and the effect of the detector resolution, the fitted distribution (as described below) is introduced into the Monte Carlo simulations and the simulated events analyzed in the same manor as the experimental data. The distribution obtained from this procedure was found to deviate by, at most, 3% from the primary distribution used as input into the simulations. This small deviation was used as a correction to generate the red square data points. ![The black circular data points give the distribution of relative-angle $\theta_{pp}$ between the two emitted protons in the sequential decay of the 9.78-MeV state. The dashed curves shows a calculation for isotropic proton emission including the Coulomb final-state interaction between the protons. The red square data points have been corrected to remove this final-state interaction and also the effects of the detector resolution. The solid curve is a fit with a quartic function in $\cos\theta_{pp}$.[]{data-label="fig:thetapp"}](PP2.pdf) Classically this distribution should be symmetric about $\theta_{pp}$=90$^{\circ}$ ($\cos\theta_{pp}$=0). If the orbital angular momentum vectors of the two protons are aligned, then we expect a “U” shaped distribution with a minimum at $\theta_{pp}$=90$^{\circ}$, whereas if these vectors are perpendicular then the distribution will peak at 90$^{\circ}$. If one or both of the protons are emitted isotropically, or the two spin vectors have no angular correlations, then the distribution should be flat. A full quantum treatment can be found in Refs. [@Frauenfelder:1953; @Biendenham:1953]. The experimental distribution does not follow any of these three possibilities and is quite remarkable having a distinct asymmetry about $\theta_{pp}$=90$^{\circ}$ ($\cos\theta_{pp}$=0). There are two ways to introduce an asymmetry about $\theta_{pp}$=90$^{\circ}$. First, there is the possibility of final-state interactions (FSI) between the two protons. A sizable effect was observed for the 2$p$ decay of the 2^nd^ excited-state of $^{17}$Ne [@Charity:2018] where the second proton is emitted with a larger velocity than the first and thus, if emitted in the same direction, can catch up. Coulomb final-state interaction between the two protons resulted in a depletion of events near $\theta_{pp}$=0 compared to $\theta_{pp}$=180$^{\circ}$. From fitting the magnitude of this effect, the intrinsic width of $^{16}$F$_{g.s}$ could be extracted and was found to be consistent with the value obtained from directly measuring the width of the peak line shape [@Charity:2018]. Here, the decay width of the two possible intermediate states are larger (62 and 47 keV) than the value of 23 keV for the $^{16}$F$_{g.s.}$. However both protons are emitted with almost the same velocity so the second proton will not catch up to the first. To estimate the magnitude of these interactions, we have used the classical calculation described in [@Charity:2018]. The predicted $\theta_{pp}$ distributions, assuming each proton is emitted isotropically is shown as the dashed black curve in Fig. \[fig:thetapp\] for decay through the 5/2$^+$ intermediate state. The result for the 3/2$^-$ intermediate state is not that different. The magnitude of the final-state interaction is not large enough to explain the observed asymmetry. The red data points in Fig. \[fig:thetapp\] also contain a correction to remove the effects of the final-state interactions based on this calculation. This is largely for display purposes as subsequently we do not fit the region of small $\theta_{pp}$ ($\cos\theta_{pp}>$0.8) where the effect of the FSI are significant The second way to produce the asymmetry relies on the unusual occurrence in this decay that there are two possible intermediate states in $^{13}$N which are degenerate (3/2$^-$ and 5/2$^+$) and thus we cannot distinguish energetically whether the decay is through only one of these or there are decay branches though both of them. If the latter is true, then there can be interference between these two decay paths. In addition as the parity of the two degenerate states are opposite, this interference term will be asymmetric. In general if there are two decay paths “a” and “b” with amplitudes $\alpha$ and $\beta$, then the relative angular distribution will be [@Biendenham:1953]. $$W(\theta_{pp}) = \lvert \alpha \rvert^2 W_{a}(\theta_{pp}) + \lvert \beta \rvert^2 W_{b}(\theta_{pp}) + (\alpha \beta^* + \alpha^* \beta) W_{ab}(\theta_{pp})$$ where $W_a(\theta_{pp})$ and $W_{b}(\theta_{pp})$ are the distributions for decay purely through the “a” and “b” paths, respectively, and $W_{ab}(\theta_{pp})$ is the interference term. Normally one considers a parent state with mixed a configuration that has decay paths through the same intermediate state. For example if the spin of this state is 2$^+$, then we can consider both $s_{1/2}$ and $d_{5/2}$ decays to the 5/2$^+$ intermediate state. The interference term is such cases will be symmetric about $\theta_{pp}$=90$^{\circ}$. However the interference between a $p_{1/2}$ decay to the 3/2$^-$ state and a $s_{1/2}$ decay to the 5/2$^+$ state will be asymmetric. We have considered the possibility of $p_{1/2}$ and $p_{3/2}$ decays to the 3/2$^-$ intermediate state and $s_{1/2}$, $d_{3/2}$, and $d_{5/2}$ decays to the 5/2$^+$ intermediate state. This gives 10 possible interference terms, six of which are asymmetric. The asymmetric terms can either enhance or suppress the small relative angles depending on the relative phase of their two components. With all these contributions, the final relative-angle distribution is quartic in $\cos\theta_{pp}$. The best quartic fit is shown as the solid red curve in Fig. \[fig:thetapp\] which reproduces the experimental data quite well. With five possible decay branches and their phases we have nine parameters (one of the phases can be set to zero as only relative phases are important). However as the final distribution is a quartic function of $\cos\theta_{pp}$ with only five parameters, it is not surprising that our best fit can be reproduced by many different sets of branching ratios and phases. There are however some restrictions on the branching ratios. The presence of a $\cos\theta_{pp}^{4}$ term requires a decay branch where the two emitted protons have $d_{5/2}$ character or $\ell\geq$3. We will ignore the latter possibility. With the presence of this $d_{5/2}$-$d_{5/2}$ decay path, we obtain $\cos^4\theta_{pp}$ terms from the distribution for pure $d_{5/2}$-$d_{5/2}$ decay and from its interference with the $d_{3/2}$-$d_{5/2}$ decay path. The magnitude of the fitted $\cos^4\theta_{pp}$ coefficient constrains the branching ratio by $d$-wave emission to the 5/2$^+$ state in $^{13}$N to be at least 13.2% of the total branching ratio to the doublet. Of course there must be decay paths to both members of the doublet in order to get the asymmetric interference terms. Consider two extremes. Firstly, if the decay to the 5/2$^+$ intermediate state dominates, then we find there must be at least a 3.8% decay branch to the 3/2$^-$ state. On the other hand if decay to the 3/2$^-$ state dominates, then we require at least a 15.5% decay branch to the 5/2$^+$ intermediate state. Clearly it only takes a small admixture of one of the decay branches to the other to produce a significant asymmetric distribution. Therefore our observation of this asymmetry is not surprising. Theoretical results {#sec:theory} =================== To describe spectra of $^{14}$O resonances and their properties, we used the Shell Model Embedded in the Continuum (SMEC) which describes resonances in the framework of the shell model for open quantum systems [@smec; @smec1; @smec11; @smec2; @smec3]. Recent SMEC applications included the study of correlations and clustering in near-threshold states [@smec2; @smec3; @oko2016; @oko2018] and the change of the continuum coupling strength for like and unlike nucleons in exotic nuclei [@Charity:2018]. Here, the scattering environment is provided by the one-proton decay channels. The Shell Model (SM) states are coupled to the environment of one-proton decay channels through the energy-dependent effective Hamiltonian: $${\cal H}(E)=H_{{\cal Q}_0{\cal Q}_0}+W_{{\cal Q}_0{\cal Q}_0}(E), \label{eq21}$$ where ${\cal Q}_{0}$ and ${\cal Q}_{1}$ denote the two orthogonal subspaces of Hilbert space containing 0 and 1 particle in the scattering continuum, respectively. $H_{{\cal Q}_0{\cal Q}_0}$ stands for the standard SM Hamiltonian and $W_{{\cal Q}_0{\cal Q}_0}(E)$: $$W_{{\cal Q}_0{\cal Q}_0}(E)=H_{{\cal Q}_0{\cal Q}_1}G_{{\cal Q}_1}^{(+)}(E)H_{{\cal Q}_1{\cal Q}_0}, \label{eqop4}$$ is the energy-dependent continuum coupling term, where $G_{{\cal Q}_1}^{(+)}(E)$ is the one-nucleon Green’s function and ${H}_{{Q}_0,{Q}_1}$, ${H}_{{Q}_1{Q}_0}$ are the coupling terms between the subspaces ${\cal Q}_{0}$ and ${\cal Q}_{1}$. $E$ stands for a scattering energy of particle in the continuum and the energy scale is defined by the lowest one-proton decay threshold. Each decay-channel state in $^{14}$O is defined by the coupling of one proton in the scattering continuum of $^{13}$N in a given SM state. Internal dynamics of the ${\cal Q}_{0}$ system include couplings to the environment of decay channels and is given by the energy-dependent effective Hamiltonian ${\cal H}(E)$. Each decay threshold is associated with a non-analytic point of the scattering matrix. The coupling of different SM eigenfunctions to the same decay channel induces a mixing among the SM eigenfunction which reflects the nature of the decay threshold. This continuum-induced configuration mixing, which consists of the Hermitian principal-value integral describing virtual continuum excitations and the anti-Hermitian residuum that represents the irreversible decay out of the internal space ${\cal Q}_{0}$, can radically change the structure of near-threshold SM states. Unitarity in a multichannel system implies that the external mixing of SM eigenfunctions changes whenever a new channel opens up. Hence, the eigenfunctions depend on the energy window imposed theoretically, [*i.e.*]{} on the chosen set of decay channels, and vary with the increasing total energy of the system, [*i.e.*]{} with the variation of the environment of scattering states and decay channels. The effective Hamiltonian ${\cal H}(E)$ is Hermitian for energies below the lowest particle-emission threshold and non-Hermitian above it. Consequently, the energy-dependent solutions of ${\cal H}(E)$ are real for bound states ($E<0$) and complex $\tilde{E}_i - (1/2) i \tilde{\Gamma}_i$ in the continuum ($E>0$). In general, due to the energy dependence of $\tilde{E}_i$ and $\tilde{\Gamma}_i$, the line shape of the resonance differs from a Breit–Wigner shape, even without any interferences and far from the decay thresholds. The energy and the width of resonance states are determined by the fixed-point conditions [@smec]: $$\begin{aligned} E_i &=& {\tilde E}_i(E=E_i) \nonumber \\ \Gamma_i &=& {\tilde \Gamma}_i (E=E_i). \label{eqfp}\end{aligned}$$ The solution of these equations sets the energy scale for states with the quantum numbers $J$, $\pi$ of state $i$. In practice, to have the same scale for the whole spectrum, an energy reference is defined by a single representative state, [*e.g.*]{} the ground state of the nucleus. However, to compare parameters of a physical resonance (widths, branching ratios) with experimental data, we solve the fixed-point equations (\[eqfp\]) for each resonance studied at the experimental energy of this resonance relative to the elastic-channel threshold. The SMEC Hamiltonian in this work consists of the monopole-based SM interaction (referred to as YSOX [@yuan2012]) in the full $psd$ model space plus the Wigner-Bartlett contact interaction [@Shalit:1963]: $$V_{12}=V_0 \left[ \alpha + \beta P^{\sigma}_{12} \right] \delta\langle\bf{r}_1-\bf{r}_2\rangle ~ \ , \label{WB}$$ for the coupling between SM states and the decay channels, where $\alpha + \beta = 1$ and $P^{\sigma}_{12}$ is the spin exchange operator. The SM eigenstates have good isospin but the continuum-coupling term $W_{{\cal Q}_0{\cal Q}_0}(E)$ in ${\cal H}(E)$ breaks the isospin conservation due to different radial wave functions for protons and neutrons, and different proton and neutron separation energies. $J^{\pi}$ $E^{*}_{\rm exp}$ $\Gamma_{\rm exp}$ $\Gamma_{\rm th}$ ----------- ------------------- -------------------- ------------------- $1^-_1$ 5.164(12) 38(2) 57.5 $0^-_1$ 5.71(14) 400(44) 280 $0^+_2$ 5.931(10) $<$12 22 $3^-_1$ 6.285(12) 37.7(17) 655 $2^+_1$ 6.585(11) $<$25 0.1 $2^-_1$ 6.764(10) 96(5) 964 $2^+_2$ 7.768(10) 68(6) 8.39 $2^+_3$ 9.755(10) 229(51) 563 : \[table\_spectra\] The widths of known $^{14}$O resonances calculated in the SMEC using the modified YSOX interaction and the Wigner-Bartlett continuum coupling with strength $V_0 = -350$ MeV$\cdot$fm$^3$. Fixed-point equations (\[eqfp\]) are solved for each resonance at energy $E$ equal to its experimental energy relative to the elastic-channel threshold. Comparsions to experimental width $\Gamma_{\rm exp}$ are made. (For more details, see the text.) The radial single-particle wave functions (in ${\cal Q}_0$) and the scattering wave functions (in ${\cal Q}_1$) are generated by the Woods-Saxon (WS) potential which includes spin-orbit and Coulomb parts. The radius and diffuseness of the WS potential are $R_0=1.27 A^{1/3}$ fm and $a=0.67$ fm, respectively. The spin-orbit potential is $V_{\rm SO}=6.4$ MeV, and the Coulomb part is calculated for a uniformly charged sphere with radius $R_0$. The depth of the central part for protons is adjusted to reproduce the measured proton separation energy ($S_p$=4.627 MeV) for the $p_{1/2}$ single-particle state . Similarly, the depth of the potential for neutrons is chosen to reproduce the measured neutron separation energy ($S_n$=23.179 MeV) for the $p_{3/2}$ single-particle state. The original YSOX interaction fails to reproduce energies of the low-lying unnatural parity states in $^{14}$O. Spectrum of these states was greatly improved if the cross-shell $T=1$ matrix elements: $\langle 0p_{1/2}1s_{1/2};J^{\pi}=0^-|V|0p_{1/2}1s_{1/2};J^{\pi}=0^-\rangle$ and $\langle 0p_{1/2}1s_{1/2};J^{\pi}=0^-|V|0p_{1/2}1s_{1/2};J^{\pi}=1^-\rangle$ were made smaller by 1 MeV and 2 MeV, respectively. This change implies a modification of the $T=1$ monopole term ${\cal M}^{T=1}(0p_{1/2} 1s_{1/2})$ which becomes -1.067 MeV, as compared to +0.683 MeV in the original YSOX interaction. No change was made in the $T=0$ monopole, ${\cal M}^{T=0}(0p_{1/2} 1s_{1/2}) = -2.842$ MeV. The continuum-coupling constant of the Wigner-Bartlett interaction $V_0 = -350$ MeV$\cdot$fm$^3$ has been chosen to obtain an overall satisfactory reproduction of the spectrum of proton resonances in $^{14}$O. The spin-exchange parameter $\alpha$ ($\beta=1-\alpha$) in the coupling between SM states and the decay channels \[Eq. (\[WB\])\] has a standard value of $\alpha = 0.73$ [@smec]. The SMEC results are compared with the experimental excitation energies in Fig. \[fig:level\]. In this calculation, for each $J^{\pi}$ separately, the SM states are mixed via the coupling to 9 channels, including the elastic channel $[{^{13}}$N($1/2^-) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$ and 8 inelastic channels $[{^{13}}$N($K^{\pi}) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$ which correspond to the excited states of $^{13}$N: $K^{\pi}=1/2^+_1$, $3/2^-_1$, $5/2^+_1$, $5/2^+_2$, $3/2^+_1$, $7/2^+_1$, $5/2^-_1$, and $3/2^+_2$. The fixed-point equation in the SMEC calculation is solved for the ground state $0^+_1$. With this choice, the SMEC and experimental ground-state energy have the same origin of the energy scale. The agreement between experimental and calculated spectrum is quite remarkable. The one conspicuous failure of these calculations is the over prediction of the one-proton emission widths of the higher-spin unnatural-parity resonances, $3^-_1$ and $2^-_1$, (see Table \[table\_spectra\]). The experimental results suggest that the $2^+_2$ resonance coincides with another resonance of unknown spin and parity. This resonance doublet decays mainly to the $1/2^-$ ground state $^{13}$N with a small decay branch to the first-excited state $1/2^+$. The SMEC predicts that the partner of the $2^+_2$ resonance could be $0^+_3$ resonance. This assignment is also favored by the observed parent velocity distribution of the doublet (see Secs. \[sec:pCVcm\] and \[sec:2pCVcm\]). The dependence of the $2^+_2$ and $0^+_3$ eigenvalues on the continuum-coupling strength is shown in Fig. \[fig\_1add\]. Below $V_0 \approx -330$ MeV$\cdot$fm$^3$ these eigenvalues are nearly degenerate \[panel (a)\]. The calculated (total) decay widths of the $2^+_2$ and $0^+_3$ resonances are changing smoothly with increasing continuum coupling \[panel (b)\] and at $V_0 = -350$ MeV$\cdot$fm$^3$ are 8.4 keV and 129 keV, respectively. The branching ratios exhibit a weak dependence on the continuum-coupling strength. For $V_0 = -350$ MeV$\cdot$fm$^3$, $\Gamma(2^+_2 \rightarrow 1/2^-)/\Gamma(\rm tot)= 0.25$, whereas $\Gamma(0^+_3 \rightarrow 1/2^-)/\Gamma(\rm tot)= 0.73$. Hence, the narrow $2^+_2$ resonance is predicted to decay mainly to the first-excited state $1/2^+$, while the broader $0^+_3$ resonance decays predominantly to the ground state of $^{13}$N. ![ Results for the $2^+_2$ and $0^+_3$ levels in the SMEC as a function of the continuum coupling strength $V_0$. Panel (a) shows the eigenenergies and panel (b) the widths of SMEC eigenstates which form a doublet of resonances in the interval $-450 \leq V_0 \leq -330$ MeV$\cdot$fm$^3$. The dotted vertical line shows the value of $V_0 = -350$ MeV$\cdot$fm$^3$ which gives an overall reasonable reproduction of the resonances reported in this work, see Fig. \[fig:level\][]{data-label="fig_1add"}](Efxp-w-w-mag2p2-0p3_0p1_vs_V0.pdf "fig:"){width="1.00\linewidth"}\ -.1truecm ![ Results for the $2^+_2$ and $0^+_3$ levels in the SMEC as a function of the continuum coupling strength $V_0$. Panel (a) shows the eigenenergies and panel (b) the widths of SMEC eigenstates which form a doublet of resonances in the interval $-450 \leq V_0 \leq -330$ MeV$\cdot$fm$^3$. The dotted vertical line shows the value of $V_0 = -350$ MeV$\cdot$fm$^3$ which gives an overall reasonable reproduction of the resonances reported in this work, see Fig. \[fig:level\][]{data-label="fig_1add"}](Gfxp-w-w_2p2-0p3_0p1_vs_V0.pdf "fig:"){width="1.0\linewidth"} The present work confirms the existence of a resonance at 8.787(13) MeV with a width of $\Gamma = 182(32)$ keV which is listed as tentative in the most recent evaluation. This resonance decays both to the $3/2^-_1$, $5/2^+_1$ doublet and to the $1/2^+_1$ resonance (more to the former than the latter), and while the peak associated with decay to the ground state was not observed, we place an upper limit of 65% for this branch (Table \[tab:level\]). The SMEC calculations predict $1^-_2$, $2^-_2$, and $3^+_1$ resonances in this energy region which have not been connected to experimental levels. Their predicted decay widths are 200 keV, 87 keV, and 744 keV, respectively. The predicted ground-state branch for the $2^-_2$ resonance (81%) largely exceeds the experimental upper limit. The predicted branching fractions for the $1^-_2$ resonance to the ground state, $1/2^+_1$ first-excited state and to the $3/2^-_1$, $5/2^+_1$ doublet are: 50%, 9% and 41%, respectively. Finally, the $3^+_1$ resonance is predicted to decay mainly to $1/2^+_1$ first-excited state (89%) and the remaining fraction goes to the $3/2^-_1$, $5/2^+_1$ doublet. As compared to the experiment, this resonance has both inverted decay fractions and significantly larger total decay width. Thus the predicted energy, width and branching fractions for the $1^-_2$ state are consistent with those for the 8.787 MeV resonance. As this cannot be said for the other two candidates, this assignment is made. A resonance at 11.195(30) MeV, with a width $\Gamma < 220$ keV was observed to decay to the $3/2^-_1$, $5/2^+_1$ pair. No peak associated with decay to the ground state was observed, however again due to poor efficiency for this decay, we can only constrain this branching ratio to $<$81% (see Table \[tab:level\]). In the vicinity of this resonance, the SMEC calculations predict $2^+_4$, $3^-_2$, and $3^+_2$ resonances with total decay width $\Gamma(\rm tot)$ equal 268 keV, 1700 keV, and 103 keV, respectively. The $2^+_4$ state is predicted to decay mainly to the $3/2^-_1$, $5/2^+_1$ doublet with a branching ratio: $\Gamma(2^+_4 \rightarrow 1/2^-)/\Gamma(\rm tot) = 0.7$. The remaining flux goes to the $1/2^+_1$ first-excited state. The $3^-_2$ resonance is predicted to decay to the doublet with a probability 98% (the remaining flux in this decay goes to the ground state) but its very-large predicted width removes it as a suitable candidate. Finally, the $3^+_2$ resonance decays mainly to the first-excited state $1/2^+_1$ with a probability 60% and the remaining flux goes to the doublet, [*i.e.*]{} branching fractions inverted from what is observed. For the observed resonance at 11.195(30) MeV, $2^+_4$ is the most suitable assignment (best overall agreement with energy, decay width, branching ratios), however $3^+_2$ cannot be excluded because despite its inverted decay fractions, it has a suitable decay width. As the present calculations do not include coupling to the $\alpha$+$^{10}$C channel the above assignment is tentative. Configuration mixing in resonance spectra ----------------------------------------- Continuum induced mixing of SM eigenstates is strong if there are avoided crossings of SMEC eigenstates [@smec; @oko2009]. These crossings can be conveniently studied by calculating energy trajectories of the double poles of the scattering matrix, the so-called exceptional points (EPs) [@zirn83], of the effective Hamiltonian with the complex-extended continuum coupling strength $V_0$. The connection of EPs to avoided crossings, spectral properties [@hei91; @Dukelsky:2009], and associated geometric phases have been discussed in simple models in considerable detail [@heis98; @heis00; @Dembowski:2001; @Dembowski:2003; @Keck:2003]. Their manifestation in scattering experiments has been considered in [@oko2009]. Since the effective Hamiltonian is energy dependent, the essential information about configuration mixing is contained in the trajectories of coalescing eigenvalues $E_{i_1}(E) = E_{i_2}(E)$, the so-called exceptional threads (ETs), for a complex value of the continuum coupling strength $V_0$. EPs correspond to common roots of the equations: $$\frac{\partial^{(\nu)}}{\partial {\cal E}} {\rm det}\left[{\cal H}\left(E;V_0\right) -{\cal E}I\right] = 0,~~~\nu=0,1. \label{discr}$$ Single-root solutions of Eq. (\[discr\]) correspond to EPs associated with either decaying or capturing states. The maximum number of such roots is: $M_{\rm max} = 2n(n - 1)$, where $n$ is the number of states of given angular momentum $J$ and parity $\pi$. The factor 2 in the expression on $M_{\rm max}$ comes from the symmetry of solutions of ${\cal H}(E)$ with respect to the transformation $V_0 \rightarrow -V_0$. This symmetry is broken above the lowest particle-emission threshold but it rarely happens that the ET change from a physical value $V_0<0$ to an unphysical value for $V_0$. Below the first particle-emission threshold, half of all ETs correspond to the poles with the asymptotic of a decaying state whereas the other half has the capturing state asymptotic. In the continuum, this symmetry is broken, and at higher excitation energies, the analytic continuation of a decaying pole in the coupling constant may become the capturing pole, or [*vice versa*]{}. It should be stressed that both decaying and capturing poles influence the configuration mixing in physical SMEC wave functions. It has been shown [@smec2; @smec3] that ETs exhibit generic features that are fairly independent of both the continuum-coupling strength and the detailed nature of the coupling matrix elements. This makes ETs particularly suitable for the investigation of the susceptibility of the eigenfunctions to the features of the coupled multichannel network in the whole domain of the continuum-coupling strength. ### $2^+$ resonances The upper panel of Fig. \[fig\_2a\] shows the spectrum of four $2^+$ eigenvalues of the SMEC Hamiltonian plotted as a function of the (real) continuum coupling constant $V_0$. For all $V_0$, the fixed-point equation is solved at the ground-state $0^+_1$. The $2^+_1$ and $2^+_2$ eigenenergies decrease monotonically with increasing $V_0$. However the $2^+_3$ and $2^+_4$ eigenvalues have an avoided crossing near $V_0 \approx -280$ MeV$\cdot$fm$^3$ corresponding to $E^* \approx 9.7$ MeV. The evolution of the $2^+_3$ and $2^+_4$ resonance widths as a function of $V_0$ is shown in the lower panel of Fig. \[fig\_2a\]. For lower values of $|V_0|$, one can see a significantly different dependence of the $2^+_3$ and $2^+_4$ eigenvalues to the continuum coupling. With increasing $|V_0|$, at first the width of $2^+_4$ resonance is both much larger and grows much faster than the width of $2^+_3$ resonance until $V_0\approx -250$ MeV$\cdot$fm$^3$. The width of the $2^+_3$ resonance then decreases and the two widths coincide for $V_0 \approx -280$ MeV$\cdot$fm$^3$. This indicates that the $2^+_3$ and $2^+_4$ eigenvalues are close to a $2^+$ EP. For even more negative coupling strengths, the width of higher energy $2^+_4$ resonance again increases rapidly overtaking that of the $2^+_3$ resonance at $V_0 \approx -425$ MeV$\cdot$fm$^3$. ![The SMEC spectrum of $2^+_i$ eigenstates ($i=1,2,3,4$) in $^{14}$O is shown as a function of the (real) continuum coupling strength $V_0$. Panel (a) presents the eigenenergies and panel (b) the widths of $2^+_3$ and $2^+_4$ SMEC eigenstates, the states that are involved in the avoided crossing at $V_0 \approx -280$ MeV$\cdot$fm$^3$, see vertical arrows. The dotted vertical line shows the value of $V_0 = -350$ MeV$\cdot$fm$^3$ which gives an overall reasonable reproduction of the resonances reported in this work. For more details, see the text.[]{data-label="fig_2a"}](Efxp-w-w-mag2p_0p1_vs_V0.pdf "fig:"){width="1.00\linewidth"}\ -.4truecm ![The SMEC spectrum of $2^+_i$ eigenstates ($i=1,2,3,4$) in $^{14}$O is shown as a function of the (real) continuum coupling strength $V_0$. Panel (a) presents the eigenenergies and panel (b) the widths of $2^+_3$ and $2^+_4$ SMEC eigenstates, the states that are involved in the avoided crossing at $V_0 \approx -280$ MeV$\cdot$fm$^3$, see vertical arrows. The dotted vertical line shows the value of $V_0 = -350$ MeV$\cdot$fm$^3$ which gives an overall reasonable reproduction of the resonances reported in this work. For more details, see the text.[]{data-label="fig_2a"}](Gfxp-w-w_2p_0p1_vs_V0.pdf "fig:"){width="1.00\linewidth"} Figure \[fig\_2ab\] shows the behavior of partial decay rates of the $2^+_3$ and $2^+_4$ resonances. For small values of $|V_0|$, the decay properties of $2^+_3$ and $2^+_4$ eigenvalues are quite different: the $2^+_3$ resonance decays mainly to the first-excited state $1/2^+_1$ \[panel (b)\] whereas $2^+_4$ decays to the $3/2^-_1$ - $5/2^+_1$ doublet \[panel (a)\]. The decay fractions remain approximately the same for the two resonances from the avoided crossing ($V_0 \approx -280$ MeV$\cdot$fm$^3$) until $V_0 \simeq -350$ MeV$\cdot$fm$^3$ and then diverge for even larger $|V_0|$. ![The branching fractions, as a function of the continuum-coupling strength $V_0$, for the one-proton decay of $2^+_3$ and $2^+_4$ resonances either to (a) the doublet of resonances $3/2^-_1$ - $5/2^+_1$ or (b) the first-excited state $1/2^+_1$ in $^{13}$N. The arrows show the location of the avoided crossing and the dotted vertical line is at $V_0 = -350$ MeV$\cdot$fm$^3$, the value of the $V_0$ that gives a satisfactory reproduction of the observed $^{14}$O spectrum.[]{data-label="fig_2ab"}](Rdfxp-w-w_2p_0p1_vs_V0.pdf "fig:"){width="1.00\linewidth"}\ -.3truecm ![The branching fractions, as a function of the continuum-coupling strength $V_0$, for the one-proton decay of $2^+_3$ and $2^+_4$ resonances either to (a) the doublet of resonances $3/2^-_1$ - $5/2^+_1$ or (b) the first-excited state $1/2^+_1$ in $^{13}$N. The arrows show the location of the avoided crossing and the dotted vertical line is at $V_0 = -350$ MeV$\cdot$fm$^3$, the value of the $V_0$ that gives a satisfactory reproduction of the observed $^{14}$O spectrum.[]{data-label="fig_2ab"}](R2fxp-w-w_2p_0p1_vs_V0.pdf){width="1.00\linewidth"} Figure \[fig\_2b\] displays two exceptional threads (ETs) in the complex plane ${\cal R}e(V_0)$ - ${\cal I}m(V_0)$ that are both in the physical region, close to the real axis, and in the energy region relevant for the mixing of the $2^+_3$ and $2^+_4$ resonances. The ET shown by the thick red curve, which crosses the real axis at $V_0 = -262$ MeV$\cdot$fm$^3$, is responsible for the avoided crossing (see Fig. \[fig\_2a\]). It should be noted that in this presentation, the ET dynamics is independent of the reference state chosen for solving the fixed-point equation (\[eqfp\]). ![Exceptional threads for $2^+$ eigenstates of SMEC in $^{14}$O are shown as a function of the real and imaginary parts of the continuum coupling strength $V_0$. Negative (positive) imaginary values of $V_0$ correspond to outgoing, [*i.e.*]{} decaying, (ingoing, [*i.e.*]{} capturing) asymptotics. Different points on these ETs correspond to different excitation energies. The filled circle points to the threshold energy of the lowest decay channel (the elastic reaction channel). Open circles denote excitation energies at which subsequent (inelastic) channels open. The open double circle corresponds to the opening of two nearly-degenerate decay channels at 8.129 MeV and 8.174 MeV. Arrows indicate the direction of increasing excitation energy along each ET.[]{data-label="fig_2b"}](realx-w_2p3-4_14O.pdf "fig:"){width="1.00\linewidth"}\ Each point along the ET corresponds to a different scattering energy $E$. For energies below the lowest proton decay threshold (the elastic-channel threshold: $[{^{13}}$N($1/2^-_1) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$) both ETs shown in Fig. \[fig\_2b\] are straight lines with reflection symmetry with respect to the real axis. This symmetry is broken above the first decay threshold. Different circles show the points (energies) at which various one-proton decay channels open. The filled circle denotes the lowest-energy decay threshold. The open double circle shows the nearly degenerate second and third decay channels, $[{^{13}}$N($3/2^{-}_1) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$ and $[{^{13}}$N($5/2^{+}_1) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$, which open at 8.129 MeV and 8.174 MeV, respectively. The ET depicted by the thick red curve, which at low excitation energies corresponds to the succession of double poles of the $S$ matrix with the decaying asymptotic \[${\cal I}m(V_0) < 0$\], crosses the real axis at $E^* \approx 10.35$ MeV ($V_0 = -262$ MeV$\cdot$fm$^3$) close to $E^*=10.991$ MeV, where the $[{^{13}}$N($5/2^+_2) \otimes {\rm p}({\ell_j})]^{J^{\pi}}$ inelastic channel opens. This ET dynamics explains the nature of the avoided crossing at $E^* \approx 9.7$ MeV (see Fig. \[fig\_2a\]) as due to the proximity of a $J^{\pi}=2^+$ EP with significant $\ell=0$ coupling to the $[{^{13}}$N($5/2^+_2) \otimes {\rm p}({\ell_j})]^{J^{\pi}}$ channel. The loop seen in the ET dynamics is due to the opening of higher-energy decay channels. The ET depicted by the green line corresponds to the double poles with capturing asymptotics ([*i.e.*]{} above real axis). This ET evolves smoothly with the excitation energy and remains a close, but irrelevant, spectator to the configuration mixing of the $2^+_3$ and $2^+_4$ eigenvalues. ![ Exceptional threads for $2^+$ eigenstates are displayed in the ${\cal I}m(V_0)$ - $E$ plane. The thick red curve is the exceptional thread responsible for the avoided crossing.[]{data-label="fig_2c"}](realx-w-E_14O.pdf){width="1.00\linewidth"} Another representation of $2^+$ ETs, in the region of small ${\cal I}m(V_0)$, is shown in Fig. \[fig\_2c\]. In the plane of excitation energy $E^*$ and ${\cal I}m(V_0)$ one observes that with increasing excitation energy (and thus higher density of decay channels), more and more ETs cross the physical real axis. Close to the crossing points, the influence of the double-pole singularity on the configuration mixing of SM eigenvalues increases and hence the movement of SMEC eigenvalues is strongest. In this high excitation-energy region, one peculiar ET (shown with the blue curve) actually crosses the real-$V_0$ axis twice. While this pattern is quite complex, there is only one ET, depicted with the thick red curve, that appears at the excitation energy sufficiently close to the $2^+_3$ and $2^+_4$ eigenvalues to be relevant for their avoided crossing. ### $1^-$ resonances Figure \[fig\_xa\] shows the spectrum of three $1^-$ eigenvalues of the SMEC plotted as a function of the (real) continuum coupling constant $V_0$. In this case there no avoided crossings. With increasing $|V_0|$, the $1^-_1$, $1^-_2$, and $1^-_3$ energies increase gradually. The total one-proton decay widths and branching ratios (to different $^{13}$N states) also vary smoothly with $V_0$. ![SMEC spectrum of $1^-$ eigenstates in $^{14}$O is shown as a function of the continuum coupling strength $V_0$. For more details see the caption of Fig. \[fig\_2a\] and the discussion in text.[]{data-label="fig_xa"}](Efxp-w-w-mag1m_0p1_vs_V0.pdf){width="1.00\linewidth"} ![ Exceptional threads for $1^-$ eigenstates of SMEC in $^{14}$O are shown as a function of the real and imaginary parts of the continuum coupling strength $V_0$. The circles on each curve denote where the excitation energies corresponding to the thresholds for the elastic (solid) and inelastic (open) channels are located. Arrows indicate the direction of increasing excitation energy along each ET.[]{data-label="fig_xb"}](realx-w-mod_1m1-2_14O.pdf "fig:"){width="1.00\linewidth"}\ This weak configuration mixing among $1^-$ states finds an explanation in the pattern of the most relevant ETs in the ${\cal R}e(V_0)$ - ${\cal I}m(V_0)$ plane (see Figure \[fig\_xb\]). To follow their evolution, it is necessary to consider both ${\cal R}e(V_0) < 0$ and ${\cal R}e(V_0) > 0$ half-planes. Below the first decay threshold, the ET corresponding to the poles having a decaying asymptotic is shown as the piece of the short red curve below the filled circle, with the filled circle representing its value at threshold. Its symmetric counterpart for ${\cal R}e(V_0) > 0$, having the capturing asymptotics, is given by the blue curve above its filled circle. These two sets of double poles change rather weakly with increasing excitation energy, showing a turning near the threshold of the first inelastic channel. These poles are deep inside of the complex plane and hence cannot influence the configuration mixing of $1^-$ eigenstates. The double-poles which have capturing asymptotics above the first particle threshold are shown by the small piece of the long green curve above its filled circle at ${\cal R}e(V_0) < 0$. Its symmetry partner is given by the small piece of the long magenta curve below its filled circle at ${\cal R}e(V_0) > 0$. Their evolution with excitation energy is rapid and just above the first decay threshold (filled circles), the ET with the capturing asymptotic (green curve) crosses the ${\cal R}e(V_0) = 0$ axis and moves into the right half-plane \[${\cal R}e(V_0) > 0]$ without changing its asymptotics. Its symmetry partner (magenta curve) makes a move in the opposite direction, entering the half-plane ${\cal R}e(V_0) < 0$. This evolution happens deep in the complex plane for $0.5<|{\cal I}m(V_0)|<1.1$ GeV$\cdot$fm$^3$. Above the channel doublet: $[{^{13}}$N($3/2^{-}_1) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$, $[{^{13}}$N($5/2^{+}_1) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$, both ETs make a rapid turn passing through the real axis \[${\cal I}m(V_0) = 0$\] and change their asymptotics. These crossings appear well above a physical range of the continuum coupling constants (${\cal R}e(V_0) \approx 1.5$ GeV$\cdot$fm$^3$) and thus have no influence on the spectra. After opening of all decay channels both of these ETs (depicted by the green and magenta lines) return to their original quadrant of the ${\cal R}e(V_0)$ - ${\cal I}m(V_0)$ plane. ### $0^+$ resonances Figure \[fig\_4a\] shows the spectrum of three $0^+$ eigenvalues as a function of the $V_0$. The consequence of the fixed-point equations being solved for the ground state (for each $V_0$) is that the eigenenergy of the $0^+_1$ state does not change with $V_0$. We find that the $0^+_3$ stays approximately constant through the relevant $V_0$ range. This means that correlation energy that results from the coupling to the continuum is the same for $0^+_1$ (the ground state) and $0^+_3$. The same cannot be said of $0^+_2$. This state drifts up with increasing $|V_0|$, relative to the others, as the (negative) correlation energy is less. For very large $|V_0|$, an avoiding crossing (and thus mixing) of the $0^+_2$ and $0^+_3$ eigenvalues is approached. This crossing (at $V_0 \approx -450$ MeV$\cdot$fm$^3$) corresponds to $E^* \approx 7$ MeV. ![The SMEC spectrum of $0^+$ eigenstates in $^{14}$O is shown as a function of the continuum coupling strength $V_0$. For more details see the caption of Fig. \[fig\_2a\] and the discussion in text.[]{data-label="fig_4a"}](Efxp-w-w-mag0p_0p1_vs_V0.pdf){width="1.00\linewidth"} ![ Exceptional threads for $0^+$ eigenstates of the SMEC in $^{14}$O are shown as a function of the real and imaginary parts of the continuum coupling strength $V_0$. The circles on each curve denote where the excitation energies corresponding to the thresholds for the elastic (solid) and inelastic (open) channels are located. Arrows indicate the direction of increasing excitation energy along each ET.[]{data-label="fig_4b"}](realx-w_0p2-3_14O.pdf){width="1.00\linewidth"} Figure \[fig\_4b\] displays the two relevant ETs for this avoided crossing. In this case the ETs are both close to the real axis and have both decaying and capturing asymptotics. The dynamics of these ETs is caused by the $\ell=0$ coupling to the decay channel $[{^{13}}$N($1/2^{+}_1) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$ which opens at 6.991 MeV. The ET shown in red, crosses the real axis at $V_0 \approx -288.4$ MeV$\cdot$fm$^3$ as a result of opening of higher lying channels: $[{^{13}}$N($K^{\pi}) \otimes {\rm p}(\ell_j)]^{J^{\pi}}$ with $K^{\pi}=$$5/2^+_2$, $3/2^+_1$, $7/2^+_1$, $5/2^-_1$, and $3/2^+_2$. The excitation energy associated with the value of $V_0$ when this ET passes through the physical region near the real axis is very high ($E^* = 12.65$ MeV), well above where these states actually exist. Thus, as in the case of the $1^-$ states, strong mixing due to the proton continuum is not expected for the $0^+$ states. Nevertheless, in the relevant range of the continuum coupling strength, the $0^+_1$ and $0^+_3$ resonances drift down fastest relative to other resonances. In particular, the eigenstate $0^+_3$ gains $\approx 1.5$ MeV in correlation energy with respect to $0^+_2$ state at $V_0 = -350$ MeV$\cdot$fm$^3$. Discussion ========== The role of the continuum is important for the $^{14}$O states investigated. Specifically, the exceptional points strongly influence the spectrum and structure of low-energy resonances. Fortunately the continuum coupling acts differently on each of the $\{J^{\pi}\}$ resonance sets and, within a given set, acts selectively on certain states depending on the location of branch points ([*i.e.*]{} decay thresholds) and double poles of the scattering matrix. The latter do not vary in a systematic way from one nucleus to another. From one point of view this poses a tremendous challenge for the microscopic nuclear theory vis-a-vis the microscopic determination of effective nucleon-nucleon interaction. From another point of view, with data that are sufficiently discriminatory, the continuum coupling constant can be fixed (for a given nucleus). The case studied here $^{14}$O seems to just such a case. Figure \[fig:level\] presents the experimental and the SMEC results side by side. A search through $V_0$ for a doublet partner for $2^+_2$, finds that only the third $0^+$ state is a serious candidate. One of the two states in the doublet must have a branching ratio to the 1/2$^+$ $^{13}$N excited state of greater than 2.5% to produce the observed yield in the 2$p$+$^{12}$C channel. At the implied coupling strength (which gives $2^+_2$ and $0^+_3$ degeneracy), the $2^+_3$ level coincides with the observed peak at 9.755(10) MeV. Its branching ratio to the negative-parity $^{13}$N ground state is at most 17%. The SMEC prediction for this branch is 1%. Most of the remaining decay strength is to the 3/2$^-$, 5/2$^+$ doublet, with decays to the 1/2$^+$ level accounting for only 6.3(9)%. The SMEC prediction for this branch is $\approx 20\%$. The 2$^+\rightarrow$ 5/2$^+$ decay path produces a $\cos^4(\theta_{pp})$ term in the $\theta_{pp}$ distribution as observed in the data. The observed structure at 8.787(13) MeV must be $1^-$ as the other unmatched resonances in this energy region: $2^-_2$ and $3^+_1$, exhibit decay patterns which disagree with the experimental findings. Several spin-parity possibilities exist for the highest-excitation-energy peak observed in the present work \[11.195(30) MeV\] with the best candidate being the 2$^+_4$ level, but as it is above the alpha threshold (for which the continuum coupling is not considered in this work) there is some uncertainty. This work does not undertake a study of the mirror of $^{14}$O, [*i.e.*]{} $^{14}$C. The nucleon and alpha thresholds are shifted up (in going to $^{14}$C) by $\approx 3.54$ MeV and $\approx 1.89$ MeV, making the continuum coupling problem wholly different. However the present work does highlight one very interesting issue in the mirror comparison. The position of $0^+_2$ is $\approx 0.68$ MeV higher in $^{14}$C than in $^{14}$O while $0^+_3$ (with our assignment in $^{14}$O) is shifted up $\approx 2.0$ MeV. Not having done the calculations for $^{14}$C, and not having the capability to include coupling to the $\alpha$ channel, we cannot provide an answer for these significant and largely different shifts. However this subject cannot be dropped without making some comments. There are two mechanisms contributing to these relative shifts. The first mechanism is the well known Thomas-Ehrmann effect which is related to the difference of Coulomb energies in mirror states and hence, is of a geometrical nature. Its relevance is primarily limited to states with significant $s$-orbit contribution as these can expand when a relevant decay threshold is passed in one of the mirror partners. The second mechanism results from the coupling of SM states to decay channels. This continuum coupling may provide both strong energy shifts and collective modifications of the involved wave functions depending on the nature of the matrix elements and a distance from the decay threshold. In our study of $^{14}$O, the continuum coupling prevails for the $0^+$ states. Indeed the coupling to continuum in $^{14}$O shifts down $0^+_3$ relative to $0^+_2$ by $\approx 1.5$ MeV at $V_0 = -350$ MeV$\cdot$fm$^3$. On the other hand, the contribution of the $s^2$ configurations are $<1\%$, $\approx 59\%$, $\approx 10\%$ for the $0^+_1$, $0^+_2$, and $0^+_3$ states, respectively. Therefore without an additional source of mixing, the strong mirror shift between the $0^+_3$ states of $^{14}$C and $^{14}$O cannot be induced by the standard Thomas-Ehrmann effect. CONCLUSIONS =========== The A = 14 isobaric chain contains many fascinating issues. One of these is the perturbation of the $^{14}$O shell-model spectrum by open decay channels. Only the ground state is particle bound and proton decay to the first four states of $^{13}$N are the only open channels until the $\alpha + ^{10}$C channel opens at 10.12 MeV. This work is a first attempt to study this nuclide, in this excitation widow, with insight gained from invariant-mass measurements combined with the SMEC, a continuum-cognizant shell model. Experimentally, $^{14}$O excited states are produced from knockout reactions with a fast $^{15}$O beam. The decay produced from 1$p$ and 2$p$ decay are detected with the HiRA array and used to construct invariant mass-distributions. From the momentum correlations observed in the 2$p$ decays, sequential decay pathways through a number of possible $^{13}$N intermediate states can be inferred and branching ratios extracted. The model study, in and of itself, indicated that exceptional points lurk in the Hilbert space, making this type of project challenging. This experimental information on energies, widths and branching ratios when compared to the same information from the calculations, provides discrimination between the possible spin and parities. We found that the previously known 2$^+_2$ state forms a doublet with a previously unknown state. If the strength of the coupling of shell-model states to the continuum is varied, one finds that the only possible partner is the 0$^+_3$ level. Also the assignment of 2$^+_3$ becomes clear leaving the only observed state between the second and third 2$^+$ states to have a 1$^-$ assignment. Thus all observed levels between the proton and alpha thresholds are assigned. The decay of the 2$^+_3$ excited state demonstrates an interesting example of interference in sequential 2$p$ decay. The sequential decay paths for this state are largely through a pair of degenerate states in $^{13}$N of opposite parity (3/2$^-$ and 5/2$^{+}$). The interference between these two decays paths results in a $p$-$p$ relative angle ($\theta_{pp}$) distribution which is no longer symmetric about $\theta_{pp}$=90$^\circ$, the expectation for most sequential 2$p$ decays. Finally, our assignment of the $0^+_3$ state in $^{14}$O motivates a detailed comparison between $^{14}$O and its mirror $^{14}$C with models that can include coupling to all relevant open channels. This work was supported by the U.S. Department of Energy, Division of Nuclear Physics under grant No. DE-FG02-87ER-40316, No. DE-FG02-04ER-41320, and No. DE-SC0014552 and by the National Science Foundation under Grant No. PHY-0606007 and by the COPIN and COPIGAL French-Polish scientific exchange programs. K.W.B. was supported by a National Science Foundation Graduate Fellowship under Grant No. DGE-1143954 and J.M. was supported by a Department of Energy National Nuclear Security Administration Stewardship Science Graduate Fellowship under cooperative Agreement Number DE-NA0002135. 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--- author: - | Alessio Pomponio[^1]\ \ SISSA, via Beirut 2/4\ I-34014 Trieste\ [*pomponio@sissa.it*]{} date: title: '**Singularly perturbed Neumann problems with potentials**' --- Introduction ============ In this paper we study the following problem: $$\label{EQe} \left\{ \begin{array}[c]{ll} -{\varepsilon}^2 \operatorname{div}\left(J(x)\nabla u\right)+V(x)u=u^p & \text{in }{\Omega}, \\ \dfrac{{\partial}u}{{\partial}\nu}=0 & \text{on }{\partial}{\Omega}, \end{array} \right.$$ where ${\Omega}$ is a smooth bounded domain with external normal $\nu$, $N{\geqslant}3$, $1<p<(N+2)/(N-2)$, $J\colon{{\mathbb{R}^N}}\to {\mathbb{R}}$ and $V\colon{{\mathbb{R}^N}}\to {\mathbb{R}}$ are $C^2$ functions. When $J\equiv 1$ and $V\equiv 1$, then becomes $$\label{EQN} \left\{ \begin{array}[c]{ll} -{\varepsilon}^2 \varDelta u +u=u^p & \text{in }{\Omega}, \\ \dfrac{{\partial}u}{{\partial}\nu}=0 & \text{on }\partial{\Omega}. \end{array} \right.$$ Such a problem was intensively studied in several works. For example, Ni & Takagi, in [@NT; @NT2], show that, for ${\varepsilon}$ sufficiently small, there exists a solution $u_{\varepsilon}$ of which concentrates in a point $Q_{\varepsilon}\in {\partial}{\Omega}$ and $H(Q_{\varepsilon}) \to \max_{{\partial}{\Omega}} H$, here $H$ denotes the mean curvature of ${\partial}{\Omega}$. Moreover in [@Li], using the Liapunov-Schmidt reduction, Li constructs solutions with single peak and multi-peaks on ${\partial}{\Omega}$ located near any stable critical points of $H$. Since the publication of [@NT; @NT2], there have been many works on spike-layer solutions of , see for example [@DY; @dPFW; @GPW; @G; @GW; @W2] and references therein. What happens in presence of potentials $J$ and $V$? In this paper we try to give an answer to this question and we will show that, for the existence of concentrating solutions, one has to check if at least one between $J$ and $V$ is not constant on ${\partial}{\Omega}$. In this case the concentration point is determined by $J$ and $V$ only. In the other case the concentration point is determined by an interplay among the derivatives of $J$ and $V$ calculated on ${\partial}{\Omega}$ and the mean curvature $H$. On $J$ and $V$ we will do the following assumptions: (J) : $J\in C^2 ({\Omega}, {\mathbb{R}})$, $J$ and $D^2 J$ are bounded; moreover, $$J(x){\geqslant}C>0 \quad \textrm{for all } x\in{\Omega};$$ (V) : $V\in C^2 ({\Omega}, {\mathbb{R}})$, $V$ and $D^2 V$ are bounded; moreover, $$V(x){\geqslant}C>0 \quad \textrm{for all } x\in{\Omega}.$$ Let us introduce an auxiliary function which will play a crucial rôle in the study of . Let ${\Gamma}\colon {\partial}{\Omega}\to {\mathbb{R}}$ be a function so defined: $$\label{eq:Gamma} {\Gamma}(Q)=V(Q)^{\frac{p+1}{p-1}- \frac{N}{2}} J(Q)^{\frac{N}{2}}.$$ Let us observe that by [**(J)**]{} and [**(V)**]{}, ${\Gamma}$ is well defined. Our first result is: \[th1\] Let $Q_0 \in {\partial}{\Omega}$. Suppose [**(J)**]{} and [**(V)**]{}. There exists ${\varepsilon}_0>0$ such that if $0<{\varepsilon}<{\varepsilon}_0$, then possesses a solution $u_{\varepsilon}$ which concentrates in $Q_{\varepsilon}$ with $Q_{\varepsilon}\to Q_0$, as ${\varepsilon}\to 0$, provided that one of the two following conditions holds: $(a)$ : $Q_0$ is a non-degenerate critical point of ${\Gamma}$; $(b)$ : $Q_0$ is an isolated local strict minimum or maximum of ${\Gamma}$. Hence, if $J$ and $V$ are not constant on the boundary ${\partial}{\Omega}$, the concentration phenomena depend only by $J$ and $V$ and not by the mean curvature $H$. Our second result deals with the other case and, more precisely, we will show that, if $J$ and $V$ (and so also ${\Gamma}$) are constant on the boundary, then the concentration phenomena are due by another auxiliary function which depends on the derivatives of $J$ and $V$ on the boundary and by the mean curvature $H$. Let $\bar {\Sigma}\colon {\partial}{\Omega}\to {\mathbb{R}}$ be the function so defined: $$\begin{gathered} \label{eq:Sigmabar} \bar {\Sigma}(Q)\equiv\ k_1 \int_{{\mathbb{R}}^-_{\nu(Q)}} J'(Q)[x] \left|\left({\nabla }\bar U \right)\!\left(k_2 x \right)\right|^2 d x \\ +k_3 \int_{{\mathbb{R}}^-_{\nu(Q)}} V'(Q)[x] \left[ \bar U \!\!\left(k_2 x\right)\right]^2 d x -k_4 H(Q),\end{gathered}$$ where $\bar U$ is the unique solution of $$\left\{ \begin{array} [c]{lll} -\varDelta \bar U+ \bar U= \bar U^p & \text{in }{\mathbb{R}}^{N}, \\ \bar U>0 & \text{in }{\mathbb{R}}^{N}, \\ \bar U(0)=\max_{{\mathbb{R}}^{N}}\bar U, \end{array} \right.$$ $\nu(Q)$ is the outer normal in $Q$ at ${\Omega}$, $${\mathbb{R}}^-_{\nu(Q)} \equiv \left\{ x\in{{\mathbb{R}^N}}: x \cdot \nu(Q) {\leqslant}0\right\},$$ and, for $i=1, \ldots, 4$, $k_i$ are constants which depend only on $J$ and $V$ and not on $Q$ (see Remark \[re:Sigma\] for an explicit formula). Our second result is: \[th2\] Suppose [**(J)**]{} and [**(V)**]{} with $J$ and $V$ constant on the boundary ${\partial}{\Omega}$. Let $Q_0 \in {\partial}{\Omega}$ be an isolated local strict minimum or maximum of $\bar {\Sigma}$. There exists ${\varepsilon}_0>0$ such that if $0<{\varepsilon}<{\varepsilon}_0$, then possesses a solution $u_{\varepsilon}$ which concentrates in $Q_{\varepsilon}$ with $Q_{\varepsilon}\to Q_0$, as ${\varepsilon}\to 0$. Suppose that $J\equiv 1$ and fix any $Q_0 \in {\partial}{\Omega}$. For $k\in {\mathbb{N}}$, let $V_k$ be a bounded smooth function constantly equal to $1$ on the ${\partial}{\Omega}$ and in the whole ${\Omega}$, except a little ball tangent at ${\partial}{\Omega}$ in $Q_0$, with ${\nabla }V_k (Q_0)=-k \nu(Q_0)$ (see figure 1). {height="4.8cm"}\ It is easy to see that, outside a little neighborhood of $Q_0$ in ${\partial}{\Omega}$, we have $$\bar {\Sigma}(Q)= -C_1 H(Q),$$ while $$\bar {\Sigma}(Q_0)= -C_1 H(Q_0) + k C_2,$$ where $$\begin{aligned} C_1 &=& \frac{1}{2} \bar{B} + \left(\frac{1}{2}-\frac{1}{p+1}\right) \bar{A}, \\ C_2 &=& - \frac{1}{2} \int_{\{ \nu(Q_0) \cdot x{\leqslant}0 \}} \nu(Q_0) \cdot x \;\bar U^2 d x.\end{aligned}$$ Since $C_2>0$, we can choose $k\gg 1$ such that $Q_0$ is the absolute maximum point for $\bar {\Sigma}$ and hence there exists a solution concentrating at $Q_0$. Theorem \[th1\] will be proved as a particular case of two multiplicity results in Section 6, where we will prove also Theorem \[th2\]. The proof of the theorems relies on a finite dimensional reduction, precisely on the perturbation technique developed in [@AB; @ABC; @AMS]. In Section 2 we give some preliminary lemmas and some estimates which will be useful in Section 3 and Section 4, where we perform the Liapunov-Schmidt reduction, and in Section 5, where we make the asymptotic expansion of the finite dimensional functional. Finally we mention that problem , but with the Dirichlet boundary conditions, is studied by the author and by S. Secchi in [@PS], where we show that there are solutions which concentrate in minima of an auxiliary function, which depends only on $J$ and $V$. [**Acknowledgments**]{} The author wishes to thank Professor Antonio Ambrosetti and Professor Andrea Malchiodi for suggesting the problem and for useful discussions. [**Notation**]{} - ${\mathbb{R}}^N_+\equiv \left\{ (x_1, \ldots, x_N)\in{{\mathbb{R}^N}}: x_N > 0\right\}$. - If $\mu \in {{\mathbb{R}^N}}$, then ${\mathbb{R}}^-_\mu \equiv \left\{ x\in{{\mathbb{R}^N}}: x \cdot \mu {\leqslant}0\right\}$, where with $x \cdot \mu$ we denote the scalar product in ${{\mathbb{R}^N}}$ between $x$ and $\mu$. - If $r>0$ and $x_0 \in {{\mathbb{R}^N}}$, $B_r (x_0)\equiv \left\{ x\in{{\mathbb{R}^N}}: |x- x_0| <r \right\}$. We denote with $B_r$ the ball of radius $r$ centered in the origin. - If $u \colon {{\mathbb{R}^N}}\to {\mathbb{R}}$ and $P\in {{\mathbb{R}^N}}$, we set $u_P \equiv u(\cdot - P)$. - If $U^Q$ is the function defined in , when there is no misunderstanding, we will often write $U$ instead of $U^Q$. Moreover if $P=Q/{\varepsilon}$, then $U_P \equiv U^Q(\cdot -P)$. - If $Q\in {\partial}{\Omega}$, we denote with $\nu(Q)$ the outer normal in $Q$ at ${\Omega}$ and with $H(Q)$ the mean curvature of ${\partial}{\Omega}$ in $Q$. - If ${\varepsilon}>0$, we set ${\Omega_\varepsilon}\equiv {\Omega}/{\varepsilon}\equiv\{x\in {{\mathbb{R}^N}}: {\varepsilon}x \in {\Omega}\}$. - We denote with $\|\cdot \|$ and with $(\cdot \mid \cdot)$ respectively the norm and the scalar product of $H^1({\Omega_\varepsilon})$. While we denote with $\|\cdot \|_+$ and with $(\cdot \mid \cdot)_+$ respectively the norm and the scalar product of $H^1({{\mathbb{R}^N_+}})$. - If $P \in {\partial}{\Omega}_{\varepsilon}$, we set ${\partial}_{P_i} \equiv \frac{{\partial}}{{\partial}e_i}$, where $\left\{ e_1,\ldots,e_{N-1}\right\}$ is an orthonormal basis of $T_P(\partial{\Omega}_{\varepsilon})$. Analogously, if $Q\in {\partial}{\Omega}$, we set ${\partial}_{Q_i} \equiv \frac{{\partial}}{{\partial}{\tilde e}_i}$, where $\left\{ {\tilde e}_1,\ldots,{\tilde e}_{N-1}\right\}$ is an orthonormal basis of $T_Q({\partial}{\Omega})$. Preliminary lemmas and some estimates ===================================== First of all we perform the change of variable $x \mapsto {\varepsilon}x$ and so problem becomes $$\label{EQ} \left\{ \begin{array}[c]{ll} -\operatorname{div}\left(J({\varepsilon}x)\nabla u\right)+V({\varepsilon}x)u=u^p & \text{in }{\Omega}_{\varepsilon}, \\ \dfrac{{\partial}u}{{\partial}\nu}=0 & \text{on }{\partial}{\Omega}_{\varepsilon}, \end{array} \right.$$ where ${\Omega}_{\varepsilon}={\varepsilon}^{-1}{\Omega}$. Of course if $u$ is a solution of , then $u(\cdot/{\varepsilon})$ is a solution of . Solutions of (\[EQ\]) are critical points $u\in H^1({\Omega}_{\varepsilon})$ of $$f_{\varepsilon}(u)= \frac{1}{2}\int_{{\Omega_\varepsilon}} J({\varepsilon}x)|\nabla u|^2 dx+ \frac{1}{2}\int_{{\Omega_\varepsilon}} V({\varepsilon}x)u^2dx -\frac{1}{p+1}\int_{{\Omega_\varepsilon}} |u|^{p+1}.$$ The solutions of (\[EQ\]) will be found near a $U^Q$, the unique solution of $$\left\{ \begin{array} [c]{lll} -J(Q)\varDelta u+ V(Q) u= u^p & \text{in }{\mathbb{R}}^{N},\\ u>0 & \text{in }{\mathbb{R}}^{N},\\ u(0)=\max_{{\mathbb{R}}^{N}} u, & \end{array} \right.$$ for an appropriate choice of $Q \in {\partial}{\Omega}$. It is easy to see that $$\label{eq:UQ} U^Q(x)=V(Q)^{\frac{1}{p-1}}\,\bar U\left(x \sqrt{V(Q)/J(Q)} \right),$$ where $\bar U$ is the unique solution of $$\left\{ \begin{array} [c]{lll} -\varDelta \bar U+ \bar U= \bar U^p & \text{in }{\mathbb{R}}^{N}, \\ \bar U>0 & \text{in }{\mathbb{R}}^{N}, \\ \bar U(0)=\max_{{\mathbb{R}}^{N}}\bar U, \end{array} \right.$$ which is radially symmetric and decays exponentially at infinity with its derivatives. We remark that $U^Q$ is a solution also of the “problem to infinity”: $$\label{eq:Q} \left\{ \begin{array} [c]{lll} -J(Q) \varDelta u+ V(Q) u= u^p & \text{in }{\mathbb{R}}^{N}_+, \\ \dfrac{{\partial}u}{{\partial}\nu}=0 & \text{on } {\partial}{\mathbb{R}}^N_+. \end{array} \right.$$ The solutions of (\[eq:Q\]) are critical points of the functional defined on $H^1({{\mathbb{R}^N_+}})$ $$\label{eq:F} F^{Q}(u)= \frac{1}{2}J(Q)\int_{{\mathbb{R}}^N_+}|\nabla u|^2 + \frac{1}{2}V(Q)\int_{{\mathbb{R}}^N_+}u^2 -\frac{1}{p+1}\int_{{{\mathbb{R}^N_+}}}|u|^{p+1}.$$ We recall that we will often write $U$ instead of $U^Q$. If $P={\varepsilon}^{-1}Q \in {\partial}{\Omega_\varepsilon}$, we set $U_P \equiv U^Q(\cdot - P)$ and $$Z^{\varepsilon}\equiv \{ U_P : P\in {\partial}{\Omega_\varepsilon}\}.$$ \[lem:nf\] For all $Q \in {\partial}{\Omega}$ and for all ${\varepsilon}$ sufficiently small, if $P=Q/{\varepsilon}\in {\partial}{\Omega_\varepsilon}$, then $$\label{eq:nf} \|\nabla f_{\varepsilon}(U_P)\|=O({\varepsilon}).$$ $$\begin{gathered} (\nabla f_{\varepsilon}(U_P) \mid v) = \int_{{\Omega_\varepsilon}} J({\varepsilon}x) \nabla U_P \cdot {\nabla }v +\int_{{\Omega_\varepsilon}} V({\varepsilon}x) U_P v -\int_{{\Omega_\varepsilon}} U_P^p v \\ =\int_{\frac{{\Omega}-Q}{{\varepsilon}}} J({\varepsilon}x +Q) \nabla U \cdot {\nabla }v_{-P} +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} V({\varepsilon}x +Q) U v_{-P} -\int_{\frac{{\Omega}-Q}{{\varepsilon}}} U^p v_{-P} \\ =\int_{\frac{{\Omega}-Q}{{\varepsilon}}} J(Q) \nabla U \cdot {\nabla }v_{-P} +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} V(Q) U v_{-P} -\int_{\frac{{\Omega}-Q}{{\varepsilon}}} U^p v_{-P} \\ +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} (J({\varepsilon}x +Q)-J(Q)) \nabla U \cdot {\nabla }v_{-P} +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} (V({\varepsilon}x +Q)-V(Q)) U v_{-P} \\ =\int_{\frac{{\Omega}-Q}{{\varepsilon}}} \left[-J(Q) \varDelta U + V(Q) U - U^p \right] v_{-P} +J(Q) \int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}U_P}{{\partial}\nu} v \\ +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} (J({\varepsilon}x +Q)-J(Q)) \nabla U \cdot {\nabla }v_{-P} +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} (V({\varepsilon}x +Q)-V(Q)) U v_{-P}.\end{gathered}$$ Hence, since $U\equiv U^Q$ is solution of , we get $$\begin{gathered} \label{eq:nf2} (\nabla f_{\varepsilon}(U_P) \mid v) = J(Q) \int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}U_P}{{\partial}\nu} v +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} (J({\varepsilon}x +Q)-J(Q)) \nabla U \cdot {\nabla }v_{-P} \\ +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} (V({\varepsilon}x +Q)-V(Q)) U v_{-P}.\end{gathered}$$ Let us estimate the first of these three terms: $$\begin{gathered} \left|J(Q) \int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}U_P}{{\partial}\nu} v \right| {\leqslant}C \|v\|_{L^2({\partial}{\Omega_\varepsilon})} \left( \int_{{\partial}{\Omega_\varepsilon}}\left|\frac{{\partial}U_P}{{\partial}\nu}\right|^2\right)^{1/2}.\end{gathered}$$ First of all, we observe that there exist ${\varepsilon}_0>0$ and $C>0$ such that, for all ${\varepsilon}\in(0,{\varepsilon}_0)$ and for all $v\in H^1({\Omega_\varepsilon})$, we have $$\|v\|_{L^2({\partial}{\Omega_\varepsilon})}{\leqslant}C \|v\|_{H^1({\Omega_\varepsilon})}.$$ Moreover, after making a translation and rotation, we can assume that $Q$ coincides with the origin $\cal O$ and that part of ${\partial}{\Omega}$ is given by $x_N=\psi(x')=\frac 12 \sum^{N-1}_{i=1}\l_i x_i^2+O(|x'|^3)$ for $|x'| < \mu$, where $\mu$ is some constant depending only on ${\Omega}$. Then for $|y'|<\mu/{\varepsilon}$, the corresponding part of ${\partial}{\Omega_\varepsilon}$ is given by $y_N=\Psi (y')= {\varepsilon}^{-1} \psi({\varepsilon}y')= \frac{{\varepsilon}}{2} \sum^{N-1}_{i=1}\l_i y_i^2+O({\varepsilon}^2 |y'|^3)$. Then it is easy to see that $$\frac{{\partial}U}{{\partial}\nu}(y', \Psi(y')) ={\varepsilon}\left[\sum_{i=1}^{N-1} \l_i y_i \frac{{\partial}U}{{\partial}y_i}(y',0) -\frac 12 \frac{{\partial}^2 U}{{\partial}y_N^2}(y',0) \sum_{i=1}^{N-1} \l_i y_i^2\right] +O({\varepsilon}^2).$$ Let us observe that by the exponential decay of $U$ and of its derivatives, we get: $$\begin{gathered} \int_{{\partial}\tilde{{\Omega}}_{\varepsilon}}\!\!\left|\frac{{\partial}U}{{\partial}\nu}\right|^2 \!\!\!= \! {\varepsilon}^2 \!\int_{{\partial}\tilde{{\Omega}}_{\varepsilon}}\!\! \left[\sum_{i=1}^{N-1} \l_i y_i \frac{{\partial}U}{{\partial}y_i}(y',0) -\frac 12 \frac{{\partial}^2 U}{{\partial}y_N^2}(y',0) \sum_{i=1}^{N-1} \l_i y_i^2\right]^2 \!\!\!\!+\!o({\varepsilon}^2) \!=\!O({\varepsilon}^2),\end{gathered}$$ where ${\partial}\tilde{{\Omega}}_{\varepsilon}\equiv {\partial}{\Omega_\varepsilon}\cap B_{{\varepsilon}^{-1/2}}$. Therefore $$\label{eq:deU} \left( \int_{{\partial}{\Omega_\varepsilon}}\left|\frac{{\partial}U}{{\partial}\nu}\right|^2\right)^{1/2} =\left( \int_{{\partial}{\Omega_\varepsilon}\cap B_{{\varepsilon}^{-1/2}}}\left|\frac{{\partial}U}{{\partial}\nu}\right|^2\right)^{1/2} +o({\varepsilon})=O({\varepsilon}).$$ Let us calculate the second term of . We start observing that, from the assumption $D^2 J$ bounded, we infer that $$|J({\varepsilon}x +Q)-J(Q)| {\leqslant}{\varepsilon}|J'(Q)| |x| + c_1 {\varepsilon}^2 |x|^2,$$ and so, using again the exponential decay of $U$ and of its derivatives, $$\begin{gathered} \int_{\frac{{\Omega}-Q}{{\varepsilon}}} \!\!(J({\varepsilon}x +Q)-J(Q)) \nabla U \!\! \cdot \!\!{\nabla }v_{-P} {\leqslant}\|v\| \!\left(\! \int_{\frac{{\Omega}-Q}{{\varepsilon}}}\!\! |J({\varepsilon}x +Q)-J(Q)|^2 |\nabla U|^2 \!\right)^{1/2} \nonumber \\ {\leqslant}c_2 \|v\| \left[\int_{{\mathbb{R}^N_+}}{\varepsilon}^2 |J'(Q)|^2 |x|^2|\nabla U|^4 + \int_{{\mathbb{R}^N_+}}{\varepsilon}^4 |x|^4 |\nabla U|^4 \right]^{1/2} =O({\varepsilon})\|v\|. \label{eq:restoJ} \end{gathered}$$ Analogously, we can say that: $$\label{eq:restoV} \int_{\frac{{\Omega}-Q}{{\varepsilon}}} (V({\varepsilon}x +Q)-V(Q)) U v_{-P} =O({\varepsilon})\|v\|.$$ Now the conclusion follows immediately by , , and . We here present some useful estimates that will be used in the sequel. \[lemma1.2\] Let $P=Q/{\varepsilon}\in {\partial}{\Omega}_{\varepsilon}$. Then we have: $$\label{eq:1.4} \int_{{\Omega}_{\varepsilon}} U_P^{p+1}= \int_{{\mathbb{R}}^N_+}\left(U^Q\right)^{p+1} -{\varepsilon}\frac{H(Q)}{2} \int_{{\mathbb{R}}^{N-1}} \left[U^Q(y',0)\right]^{p+1} |y'|^2 d y'+o({\varepsilon}),$$ $$\label{eq:1.5} \int_{{\partial}{\Omega}_{\varepsilon}} \frac{{\partial}U_P}{{\partial}\nu} U_P =-{\varepsilon}\frac{(N-1)H(Q)}{4} \int_{{\mathbb{R}}^{N-1}} \left[U^Q(y',0)\right]^2 d y'+o({\varepsilon}),$$ $$\begin{gathered} \label{eq:1.6} J(Q) \int_{{\Omega}_{\varepsilon}} |{\nabla }U_P|^2 +V(Q) \int_{{\Omega}_{\varepsilon}} U_P^2 \\ =\int_{{\mathbb{R}}^N_+}\left(U^Q\right)^{p+1} -{\varepsilon}\frac{H(Q)}{2}\int_{{\mathbb{R}}^{N-1}} \left[U^Q(y',0)\right]^{p+1} |y'|^2 d y' \\ -{\varepsilon}J(Q) \frac{(N-1)H(Q)}{4} \int_{{\mathbb{R}}^{N-1}} \left[U^Q(y',0)\right]^2 d y'+o({\varepsilon}), \end{gathered}$$ $$\label{eq:J} \int_{{\Omega}_{\varepsilon}} J({\varepsilon}x)|\nabla U_P|^2= J(Q) \int_{{\Omega_\varepsilon}}|\nabla U_P|^2 +{\varepsilon}\int_{{\mathbb{R}}^-_{\nu(Q)}}J'(Q)[x] |\nabla U^Q|^2 +o({\varepsilon}),$$ $$\label{eq:V} \int_{{\Omega}_{\varepsilon}} V({\varepsilon}x) U_P^2= V(Q) \int_{{\Omega_\varepsilon}} U_P^2 +{\varepsilon}\int_{{\mathbb{R}}^-_{\nu(Q)}} V'(Q)[x] \left(U^Q\right)^2 +o({\varepsilon}).$$ Moreover, we have $$\label{eq:1.8} \int_{{\Omega}_{\varepsilon}} U_P^p \; {\partial}_{P_i} U_P= {\varepsilon}\frac{1}{p+1} \bar C {\partial}_{Q_i} {\Gamma}(Q) + o({\varepsilon}),$$ $$\label{eq:1.7} {\partial}_{P_i} \left[ J(Q) \!\int_{{\Omega}_{\varepsilon}} \!|{\nabla }U_P|^2 +V(Q) \!\int_{{\Omega}_{\varepsilon}}\! U_P^2 \right] = {\varepsilon}\bar C {\partial}_{Q_i} {\Gamma}(Q) + o({\varepsilon}).$$ where $\bar C= \int_{{\mathbb{R}^N_+}}\bar U^{p+1}$ and ${\Gamma}$ is defined in . The first two formulas can be proved repeating the arguments of Lemma 1.2 of [@Li]. Equation follows easily by and observing that $$J(Q) \int_{{\Omega}_{\varepsilon}} |{\nabla }U_P|^2 +V(Q) \int_{{\Omega}_{\varepsilon}} U_P^2= \int_{{\Omega_\varepsilon}}U_P^{p+1} +J(Q)\int_{{\partial}{\Omega_\varepsilon}}\frac{{\partial}U_P}{{\partial}\nu} U_P.$$ Let us prove . Arguing as in the proof of , we infer: $$\begin{gathered} \int_{{\Omega}_{\varepsilon}} J({\varepsilon}x)|\nabla U_P|^2= \int_{\frac{{\Omega}-Q}{{\varepsilon}}} J({\varepsilon}x +Q )|\nabla U^Q|^2 \\ =J(Q)\int_{\frac{{\Omega}-Q}{{\varepsilon}}} |\nabla U^Q|^2 +{\varepsilon}\int_{\frac{{\Omega}-Q}{{\varepsilon}}} J'(Q)[x]|\nabla U^Q|^2 +o({\varepsilon}) \\ =J(Q)\int_{{\Omega}_{\varepsilon}} |{\nabla }U_P|^2 +{\varepsilon}\int_{{\mathbb{R}}^-_{\nu(Q)}} J'(Q)[x]|\nabla U^Q|^2 +o({\varepsilon}).\end{gathered}$$ We can prove equation repeating the arguments of . Since $$\int_{{\Omega}_{\varepsilon}} U_P^p \; {\partial}_{P_i} U_P= \frac{1}{p+1} {\partial}_{P_i} \int_{{\Omega}_{\varepsilon}} U_P^{p+1},$$ equations and follow easily because, as observed by [@Li], the error terms $O({\varepsilon})$ in and become of order $o({\varepsilon})$ after applying ${\partial}_{P_i}$ to them. Invertibility of $D^2 f_{\varepsilon}$ on $\left(T_{U_P}Z^{\varepsilon}\right)^\perp$ ===================================================================================== In this section we will show that $D^2 f_{\varepsilon}$ is invertible on $\left(T_{U_P}Z^{\varepsilon}\right)^\perp$, where $T_{U_P} Z^{\varepsilon}$ denotes the tangent space to $Z^{\varepsilon}$ at $U_P$. Let $L_{{\varepsilon},Q}:(T_{U_P}Z^{\varepsilon})^\perp\to (T_{U_P}Z^{\varepsilon})^\perp$ denote the operator defined by setting $(L_{{\varepsilon},Q}v \mid w)= D^2 f_{\varepsilon}(U_P)[v,w]$. \[lem:inv\] There exists $C>0$ such that for ${\varepsilon}$ small enough one has that $$\label{eq:inv} |(L_{{\varepsilon},Q}v \mid v)|{\geqslant}C \|v\|^{2},\qquad \forall\; v\in(T_{U_P}Z^{{\varepsilon}})^{\perp}.$$ By , if we set $\a(Q)=V(Q)^{\frac{1}{p-1}}$ and $\b(Q)=\sqrt{V(Q)/J(Q)}$, we have that $U^Q(x)=\a(Q) \bar U(\b(Q) x)$. Therefore, we have: $$\begin{gathered} {\partial}_{P_i} U^Q(x-P) = {\partial}_{P_i} \left[\a({\varepsilon}P) \bar U(\b({\varepsilon}P)(x-P))\right] = \\ {\varepsilon}{\partial}_{P_i} \a({\varepsilon}P) U^Q(\b({\varepsilon}P)(x-P)) \!+\! {\varepsilon}\a({\varepsilon}P) {\partial}_{P_i} \b({\varepsilon}P) {\nabla }U^Q(\b({\varepsilon}P)(x-P))\cdot (x-P) \\ -\a({\varepsilon}P) \b({\varepsilon}P) ({\partial}_{x_i} U^Q)(\b({\varepsilon}P)(x-P)).\end{gathered}$$ Hence $$\label{eq:de_iU} {\partial}_{P_i} U^Q (x-P)=-{\partial}_{x_i} U^Q(x-P)+O({\varepsilon}).$$ For simplicity, we can assume that $Q={\varepsilon}P$ is the origin $\cal O$. Following [@Li], without loss of generality, we assume that $Q={\varepsilon}P$ is the origin $\cal O$, $x_N$ is the tangent plane of ${\partial}{\Omega}$ at $Q$ and $\nu(Q)=(0,\ldots, 0, -1)$. We also assume that part of ${\partial}{\Omega}$ is given by $x_N=\psi(x')=\frac 12 \sum^{N-1}_{i=1}\l_i x_i^2+O(|x'|^3)$ for $|x'| < \mu$, where $\mu$ is some constant depending only on ${\Omega}$. Then for $|y'|<\mu/{\varepsilon}$, the corresponding part of ${\partial}{\Omega_\varepsilon}$ is given by $y_N=\Psi (y')= {\varepsilon}^{-1} \psi({\varepsilon}y')= \frac{{\varepsilon}}{2} \sum^{N-1}_{i=1}\l_i y_i^2+O({\varepsilon}^2 |y'|^3)$. We recall that $T_{U^{\cal O}} Z^{\varepsilon}= {\rm span}_{H^1({\Omega_\varepsilon})} \{{\partial}_{P_1} U^{\cal O}, \ldots, {\partial}_{P_{N-1}}U^{\cal O} \}$. We set $$\begin{aligned} {\cal V}_{\varepsilon}&=& {\rm span}_{H^1({\Omega_\varepsilon})} \{U^{\cal O}, {\partial}_{x_1} U^{\cal O}, \ldots, {\partial}_{x_{N-1}}U^{\cal O} \}, \\ {\cal V}_+ &=& {\rm span}_{H^1({{\mathbb{R}^N_+}})} \{U^{\cal O}, {\partial}_{x_1} U^{\cal O}, \ldots, {\partial}_{x_{N-1}}U^{\cal O} \}.\end{aligned}$$ By it suffices to prove (\[eq:inv\]) for all $v\in {\rm span}\{U^{\cal O},\phi\}$, where $\phi$ is orthogonal to ${\cal V}_{\varepsilon}$. Precisely we shall prove that there exist $C_{1},C_{2}>0$ such that, for all ${\varepsilon}>0$ small enough, one has: $$\begin{aligned} (L_{{\varepsilon},\cal O}U^{\cal O} \mid U^{\cal O})& {\leqslant}& - C_{1}< 0. \label{eq:neg} \\ (L_{{\varepsilon},\cal O}\phi \mid \phi)&{\geqslant}& C_{2} \|\phi\|^2. \label{eq:claim}\end{aligned}$$ The proof of (\[eq:neg\]) follows easily from the fact that $U^{\cal O}$ is a Mountain Pass critical point of $F^{\cal O}$ and so from the fact that there exists $c_0>0$ such that, for all ${\varepsilon}>0$ small enough, one finds: $$D^2 F^{\cal O}(U^{\cal O})[U^{\cal O},U^{\cal O}] < -c_0< 0.$$ Indeed, arguing as in the proof of Lemma \[eq:nf\] (see and ) and by and , we have: $$\begin{gathered} (L_{{\varepsilon}, \cal O} U^{\cal O} \mid U^{\cal O})= \int_{{\Omega_\varepsilon}} J({\varepsilon}x) |\nabla U^{\cal O}|^2 + \int_{{\Omega_\varepsilon}} V({\varepsilon}x) (U^{\cal O})^2 - p \int_{{\Omega_\varepsilon}} (U^{\cal O})^{p+1} \\ =J({\cal O})\int_{{\Omega_\varepsilon}} |\nabla U^{\cal O}|^2 + V({\cal O}) \int_{{\Omega_\varepsilon}} (U^{\cal O})^2 - p \int_{{\Omega_\varepsilon}} (U^{\cal O})^{p+1}+O({\varepsilon}) \\ =D^2 F^{\cal O}(U^{\cal O})[U^{\cal O},U^{\cal O}]+O({\varepsilon})< -c_0+O({\varepsilon})<-C_1.\end{gathered}$$ Let us prove (\[eq:claim\]). As before, the fact that $U^{\cal O}$ is a Mountain Pass critical point of $F^{\cal O}$ implies that $$\label{eq:D2F+} D^2 F^{\cal O}(U^{\cal O})[\tilde \phi,\tilde \phi]>c_1 \|\tilde \phi\|^2_+ \quad \forall \tilde \phi \perp {\cal V}_+.$$ Let us consider a smooth function $\chi_{1}:{{\mathbb{R}^N}}\to {\mathbb{R}}$ such that $$\chi_{1}(x) = 1, \quad \hbox{ for } |x| {\leqslant}{\varepsilon}^{-1/8}; \qquad \chi_{1}(x) = 0, \quad \hbox{ for } |x| {\geqslant}2 {\varepsilon}^{-1/8};$$ $$|\nabla \chi_{1}(x)| {\leqslant}2{\varepsilon}^{1/8}, \quad \hbox{ for } {\varepsilon}^{-1/8} {\leqslant}|x| {\leqslant}2 {\varepsilon}^{-1/8}.$$ We also set $ \chi_{2}(x)=1-\chi_{1}(x)$. Given $\phi \perp {\cal V}_{\varepsilon}$, let us consider the functions $$\phi_{i}(x)=\chi_{i}(x)\phi(x),\quad i=1,2.$$ If $Q\neq {\cal O}$, then we would take $$\phi_{i}(x)=\chi_{i}(x-P)\phi(x),\quad i=1,2.$$ With calculations similar to those of [@AMS], we have $$\label{eq:phi} \| \phi \|^2 = \| \phi_1 \|^2 + \| \phi_2 \|^2 + \underbrace{2\int_{{\mathbb{R}^N}}\chi_{1}\chi_{2}(\phi^{2}+|\nabla \phi|^{2})}_{I_\phi} + O({\varepsilon}^{1/8})\| \phi \|^2.$$ We need to evaluate the three terms in the equation below: $$\label{eq:L} (L_{{\varepsilon},{\cal O}}\phi \mid \phi)= (L_{{\varepsilon},{\cal O}}\phi_{1} \mid \phi_{1})+ (L_{{\varepsilon},{\cal O}}\phi_{2} \mid \phi_{2})+ 2(L_{{\varepsilon},{\cal O}}\phi_{1} \mid \phi_{2}).$$ Let us start with $(L_{{\varepsilon},{\cal O}}\phi_{1} \mid \phi_{1})$. Let $\eta=\eta_{\varepsilon}$ a smooth cutoff function satisfying $$\eta(y) = 1, \quad \hbox{ for } |y| {\leqslant}{\varepsilon}^{-1/4}; \qquad \eta(y) = 0, \quad \hbox{ for } |y| {\geqslant}2 {\varepsilon}^{-1/4};$$ $$|\nabla \eta(y)| {\leqslant}2{\varepsilon}^{1/4}, \quad \hbox{ for } {\varepsilon}^{-1/4} {\leqslant}|y| {\leqslant}2 {\varepsilon}^{-1/4}.$$ Now we will straighten ${\partial}{\Omega_\varepsilon}$ in the following way: let $\Phi \colon {{\mathbb{R}^N_+}}\cap B_{{\varepsilon}^{-1/2}} \to {\Omega_\varepsilon}$ be a function so defined: $$\Phi(y',y_N)=(y', y_N + \Psi(y')).$$ We observe that: $$D \Phi (y)= \left( \begin{array}{ccc|c} 1 & & & 0 \\ & \ddots & & \vdots \\ & & 1 & 0 \\ \hline & \nabla_{y'} \Psi(y') & & 1 \end{array} \right).$$ Let us defined ${\tilde{\phi_1}}\in H^1({{\mathbb{R}^N_+}})$ as: $${\tilde{\phi_1}}(y)= \left\{ \begin{array}{lll} \phi_1(\Phi(y))\, \eta(y) & \quad {\rm if }\; |y| {\leqslant}{\varepsilon}^{-1/2}, \\ 0 & \quad {\rm if }\; |y|>{\varepsilon}^{-1/2}. \end{array} \right.$$ We get: $$\begin{gathered} \int_{{{\mathbb{R}^N_+}}}|\nabla {\tilde{\phi_1}}|^2 = \int_{{{\mathbb{R}^N_+}}\cap B_{2 {\varepsilon}^{-1/4}}}\left| \nabla \left[ \phi_1(\Phi(y)) \right]\right|^2 d y \\ =\int_{{{\mathbb{R}^N_+}}\cap B_{2 {\varepsilon}^{-1/4}}} \sum_{i=1}^{N-1}\left| \frac{{\partial}\phi_1}{{\partial}x_i}(\Phi) + {\varepsilon}\l_i y_i \frac{{\partial}\phi_1}{{\partial}x_N}(\Phi)\right|^2 +\left|\frac{{\partial}\phi_1}{{\partial}x_N}(\Phi)\right|^2 +o({\varepsilon})\|\phi\|^2 \\ =\int_{{{\mathbb{R}^N_+}}\cap B_{2 {\varepsilon}^{-1/4}}}|(\nabla \phi_1)(\Phi)|^2 + O({\varepsilon}^{7/8})\|\phi\|^2= \int_{{\Omega_\varepsilon}}|\nabla \phi_1|^2 + O({\varepsilon}^{7/8})\|\phi\|^2.\end{gathered}$$ Analogously, we have: $$\int_{{{\mathbb{R}^N_+}}} |{\tilde{\phi_1}}|^2 = \int_{{\Omega_\varepsilon}}|\phi_1|^2,$$ and so $$\|{\tilde{\phi_1}}\|^2_+=\|\phi_1\|^2+ O({\varepsilon}^{7/8})\|\phi\|^2.$$ Let us now evaluate $(L_{{\varepsilon},{\cal O}}\phi_{1}|\phi_{1})$: $$\begin{gathered} (L_{{\varepsilon},{\cal O}}\phi_{1} \mid \phi_{1})= \int_{{\Omega_\varepsilon}} J({\varepsilon}x) |\nabla \phi_1|^2 +\int_{{\Omega_\varepsilon}} V({\varepsilon}x) \phi_1^2 -p \int_{{\Omega_\varepsilon}} (U^{\cal O})^{p-1} \phi_1^2 \\ =J({\cal O})\int_{{\Omega_\varepsilon}} |\nabla \phi_1|^2 +V({\cal O})\int_{{\Omega_\varepsilon}} \phi_1^2 -p \int_{{\Omega_\varepsilon}} (U^{\cal O})^{p-1} \phi_1^2 \\ +{\varepsilon}\int_{{\Omega_\varepsilon}} J'({\cal O})[x] |\nabla \phi_1|^2 +{\varepsilon}\int_{{\Omega_\varepsilon}} V'({\cal O})[x] \phi_1^2 +o({\varepsilon})\|\phi\|^2 \\ =J({\cal O})\int_{{\Omega_\varepsilon}} |\nabla \phi_1|^2 +V({\cal O})\int_{{\Omega_\varepsilon}} \phi_1^2 -p \int_{{\Omega_\varepsilon}} (U^{\cal O})^{p-1} \phi_1^2 +O({\varepsilon}^{7/8})\|\phi\|^2 \\ =J({\cal O})\int_{{{\mathbb{R}^N_+}}} |\nabla {\tilde{\phi_1}}|^2 +V({\cal O})\int_{{{\mathbb{R}^N_+}}} {\tilde{\phi_1}}^2 -p \int_{{{\mathbb{R}^N_+}}} [U^{\cal O}(\Phi)]^{p-1} {\tilde{\phi_1}}^2 +O({\varepsilon}^{7/8})\|\phi\|^2 \\ =D^2 F^{\cal O}(U^{\cal O})[{\tilde{\phi_1}},{\tilde{\phi_1}}] -p \int_{{{\mathbb{R}^N_+}}} \left( [U^{\cal O}(\Phi)]^{p-1}-(U^{\cal O})^{p-1}\right) {\tilde{\phi_1}}^2 +O({\varepsilon}^{7/8})\|\phi\|^2.\end{gathered}$$ We have: $$\begin{gathered} \left|\int_{{{\mathbb{R}^N_+}}} \left( [U^{\cal O}(\Phi)]^{p-1}-(U^{\cal O})^{p-1}\right) {\tilde{\phi_1}}^2 \right| {\leqslant}C \int_{{{\mathbb{R}^N_+}}} |\Psi(y')| {\tilde{\phi_1}}^2 \\ = O({\varepsilon}^{3/4})\|{\tilde{\phi_1}}\|^2 =O({\varepsilon}^{3/4})\|\phi\|^2.\end{gathered}$$ Therefore, we have that $$\label{eq:L11} (L_{{\varepsilon},{\cal O}}\phi_{1} \mid \phi_{1})=D^2 F^{\cal O}(U^{\cal O})[{\tilde{\phi_1}},{\tilde{\phi_1}}] + O({\varepsilon}^{3/4})\|\phi\|^2.$$ We can write ${\tilde{\phi_1}}=\xi + \zeta$, where $\xi \in {\cal V}_+$ and $\zeta \perp {\cal V}_+$. More precisely $$\xi=({\tilde{\phi_1}}\mid U^{\cal O})_+ \, U^{\cal O} \|U^{\cal O}\|^{-2}_+ +\sum_{i=1}^{N-1}({\tilde{\phi_1}}\mid {\partial}_{P_i} U^{\cal O})_+ \, {\partial}_{P_i} U^{\cal O} \|{\partial}_{P_i} U^{\cal O}\|^{-2}_+.$$ Let us calculate $({\tilde{\phi_1}}| U^{\cal O})_+$. $$\begin{gathered} ({\tilde{\phi_1}}\mid U^{\cal O})_+ =\int_{{{\mathbb{R}^N_+}}} \nabla {\tilde{\phi_1}}\cdot \nabla U^{\cal O} +\int_{{{\mathbb{R}^N_+}}} {\tilde{\phi_1}}U^{\cal O} \\ =\int_{{{\mathbb{R}^N_+}}\cap B_{2 {\varepsilon}^{-1/4}}}\nabla \left[ \phi_1(\Phi(y)) \right]\cdot \nabla U^{\cal O} +\int_{{{\mathbb{R}^N_+}}\cap B_{2 {\varepsilon}^{-1/4}}} \phi_1(\Phi(y))\, U^{\cal O} \\ =\int_{{{\mathbb{R}^N_+}}\cap B_{2 {\varepsilon}^{-1/4}}} \!\!\!\!\!\!\!\left[(\nabla \phi_1)(\Phi)\cdot \nabla U^{\cal O} + \phi_1(\Phi)\, U^{\cal O}\right] \!+\!{\varepsilon}\! \sum_{i=1}^{N-1}\int_{{{\mathbb{R}^N_+}}\cap B_{2 {\varepsilon}^{-1/4}}} \!\!\!\!\!\!\!\! \l_i y_i \frac{{\partial}\phi_1}{{\partial}x_N}(\Phi) \frac{{\partial}U^{\cal O}}{{\partial}x_i} \\ =\int_{{\Omega_\varepsilon}} \nabla \phi_1 \cdot \nabla U^{\cal O}(\Phi^{-1}) +\int_{{\Omega_\varepsilon}} \phi_1 U^{\cal O}(\Phi^{-1}) +O({\varepsilon}^{7/8})\|\phi \|^2 \\ =\int_{{\Omega_\varepsilon}} \nabla \phi_1 \cdot \nabla U^{\cal O} +\int_{{\Omega_\varepsilon}} \phi_1 U^{\cal O} +O({\varepsilon}^{3/4})\|\phi \| =O({\varepsilon}^{3/4})\|\phi \|.\end{gathered}$$ In an analogous way, we can prove also that $({\tilde{\phi_1}}\mid {\partial}_{P_i} U^{\cal O})_+=O({\varepsilon}^{3/4})\|\phi \|$, and so $$\begin{aligned} \|\xi\|_+ &=&O({\varepsilon}^{3/4})\|\phi \|, \label{eq:xi} \\ \|\zeta\|_+ &=&\|\phi_1 \| + O({\varepsilon}^{3/4})\|\phi \|. \label{eq:zeta}\end{aligned}$$ Let us estimate $D^2 F^{\cal O}(U^{\cal O})[{\tilde{\phi_1}},{\tilde{\phi_1}}]$. We get: $$\label{eq:D2F} D^2 F^{\cal O}(U^{\cal O})[{\tilde{\phi_1}},{\tilde{\phi_1}}] =D^2 F^{\cal O}(U^{\cal O})[\zeta,\zeta] +2 D^2 F^{\cal O}(U^{\cal O})[\zeta,\xi] +D^2 F^{\cal O}(U^{\cal O})[\xi,\xi].$$ By and , we know that $$D^2 F^{\cal O}(U^{\cal O})[\zeta,\zeta]>c_1 \|\zeta\|^2_+ =c_1 \|\phi_1 \|^2 + O({\varepsilon}^{3/4})\|\phi \|^2,$$ while, by and straightforward calculations, we have $$\begin{aligned} D^2 F^{\cal O}(U^{\cal O})[\zeta,\xi] &=& O({\varepsilon}^{3/4})\|\phi \|^2, \\ D^2 F^{\cal O}(U^{\cal O})[\xi,\xi] &=& O({\varepsilon}^{3/2})\|\phi \|^2.\end{aligned}$$ By these estimates, and , we can say that $$\label{eq:L11-fin} (L_{{\varepsilon},{\cal O}}\phi_{1} \mid \phi_{1})>c_1 \|\phi_1 \|^2 + O({\varepsilon}^{3/4})\|\phi\|^2.$$ Using the definition of $\chi_i$ and the exponential decay of $U^{\cal O}$, we easily get $$\begin{aligned} (L_{{\varepsilon},{\cal O}}\phi_{2} \mid \phi_{2}) & {\geqslant}& c_2 \|\phi_{2}\|^{2}+o({\varepsilon})\|\phi\|^{2}, \label{eq:L22} \\ (L_{{\varepsilon},{\cal O}}\phi_{1} \mid \phi_{2}) & {\geqslant}& c_3 I_\phi + O({\varepsilon}^{1/8})\|\phi\|^{2}, \label{eq:L12}\end{aligned}$$ where $I_\phi$ is defined in . Therefore by , , , and recalling we get $$(L_{{\varepsilon},{\cal O}}\phi \mid \phi){\geqslant}c_4 \|\phi\|^{2}+ O({\varepsilon}^{1/8})\|\phi\|^{2}.$$ This completes the proof of the lemma. The finite dimensional reduction ================================ \[lem:w\] For ${\varepsilon}>0$ small enough, there exists a unique $w=w({\varepsilon}, Q)\in (T_{U_P} Z^{\varepsilon})^{\perp}$ such that $\nabla f_{\varepsilon}(U_P + w)\in T_{U_P} Z$. Such a $w({\varepsilon},Q)$ is of class $C^{2}$, resp. $C^{1,p-1}$, with respect to $Q$, provided that $p{\geqslant}2$, resp. $1<p<2$. Moreover, the functional ${{\cal A}}_{\varepsilon}(Q)=f_{\varepsilon}(U_{Q/{\varepsilon}} +w({\varepsilon},Q))$ has the same regularity of $w$ and satisfies: $$\nabla {{\cal A}}_{\varepsilon}(Q_0)=0 \quad \Longleftrightarrow \quad \nabla f_{\varepsilon}\left(U_{Q_0/{\varepsilon}}+w({\varepsilon},Q_0)\right)=0.$$ Let ${{\cal P}}={{\cal P}}_{{\varepsilon}, Q}$ denote the projection onto $(T_{U_P} Z^{\varepsilon})^\perp$. We want to find a solution $w\in (T_{U_P} Z^{\varepsilon})^{\perp}$ of the equation ${{\cal P}}\nabla f_{\varepsilon}(U_P +w)=0$. One has that $\nabla f_{\varepsilon}(U_P+w)= \nabla f_{\varepsilon}(U_P)+D^2 f_{\varepsilon}(U_P)[w]+R(U_P,w)$ with $\|R(U_P,w)\|=o(\|w\|)$, uniformly with respect to $U_P$. Therefore, our equation is: $$\label{eq:eq-w} L_{{\varepsilon},Q}w + {{\cal P}}\nabla f_{\varepsilon}(U_P)+{{\cal P}}R(U_P,w)=0.$$ According to Lemma \[lem:inv\], this is equivalent to $$w = N_{{\varepsilon},Q}(w), \quad \mbox{where}\quad N_{{\varepsilon},Q}(w)=-L_{{\varepsilon},Q}\left( {{\cal P}}\nabla f_{\varepsilon}(U_P)+{{\cal P}}R(U_P,w)\right).$$ By it follows that $$\label{eq:N} \|N_{{\varepsilon},Q}(w)\| = O({\varepsilon}) + o(\|w\|).$$ Then one readily checks that $N_{{\varepsilon},Q}$ is a contraction on some ball in $(T_{U_P} Z^{\varepsilon})^{\perp}$ provided that ${\varepsilon}>0$ is small enough. Then there exists a unique $w$ such that $w=N_{{\varepsilon},Q}(w)$. Let us point out that we cannot use the Implicit Function Theorem to find $w({\varepsilon},Q)$, because the map $({\varepsilon},u)\mapsto {{\cal P}}\nabla f_{\varepsilon}(u)$ fails to be $C^2$. However, fixed ${\varepsilon}>0$ small, we can apply the Implicit Function Theorem to the map $(Q,w)\mapsto {{\cal P}}\nabla f_{\varepsilon}(U_P + w)$. Then, in particular, the function $w({\varepsilon},Q)$ turns out to be of class $C^1$ with respect to $Q$. Finally, it is a standard argument, see [@AB; @ABC], to check that the critical points of ${{\cal A}}_{\varepsilon}(Q)=f_{\varepsilon}(U_P+w)$ give rise to critical points of $f_{\varepsilon}$. \[rem:w\] From (\[eq:N\]) it immediately follows that: $$\label{eq:w} \|w\|=O({\varepsilon}).$$ For future references, it is convenient to estimate the derivative ${\partial}_{P_i} w$. \[lem:Dw\] If $\gamma=\min\{1,p-1\}$, then, for $i=1, \ldots, N-1$, one has that: $$\label{eq:Dw} \|{\partial}_{P_i} w\|=O({\varepsilon}^\gamma).$$ We will set $h(U_P,w)=(U_P+w)^p -U_P^p -p U_P^{p-1}w$. With these notations, and recalling that $L_{{\varepsilon},Q}w = -\operatorname{div}(J({\varepsilon}x) {\nabla }w) +V({\varepsilon}x)w -p U_P^{p-1}w$, it follows that, for all $v\in (T_{U_P} Z^{\varepsilon})^{\perp}$, since $w$ satisfies , then: $$\begin{gathered} \int_{{\Omega_\varepsilon}} J({\varepsilon}x) {\nabla }U_P \cdot {\nabla }v +\int_{{\Omega_\varepsilon}} V({\varepsilon}x) U_P v -\int_{{\Omega_\varepsilon}} U_P^p v \\ +\!\int_{{\Omega_\varepsilon}} \!\!J({\varepsilon}x) {\nabla }w \cdot {\nabla }v +\!\int_{{\Omega_\varepsilon}} \!\!V({\varepsilon}x) w v -p\!\int_{{\Omega_\varepsilon}}\!\! U_P^{p-1} w v - \!\int_{{\Omega_\varepsilon}}\!\! h(U_P,w)v =0.\end{gathered}$$ Hence ${\partial}_{P_i} w$ verifies: $$\begin{gathered} \int_{{\Omega_\varepsilon}} J({\varepsilon}x) {\nabla }({\partial}_{P_i} U_P) \cdot {\nabla }v +\int_{{\Omega_\varepsilon}} V({\varepsilon}x) ({\partial}_{P_i} U_P) v -p \int_{{\Omega_\varepsilon}} U_P^{p-1}({\partial}_{P_i} U_P) v \nonumber \\ +\int_{{\Omega_\varepsilon}} J({\varepsilon}x) {\nabla }({\partial}_{P_i} w) \cdot {\nabla }v +\int_{{\Omega_\varepsilon}} V({\varepsilon}x) ({\partial}_{P_i} w) v -p\int_{{\Omega_\varepsilon}} U_P^{p-1} ({\partial}_{P_i} w) v \nonumber \\ -p(p-1)\int_{{\Omega_\varepsilon}} U_P^{p-2}({\partial}_{P_i} U_P)w v - \int_{{\Omega_\varepsilon}} \left[h_{U_P}({\partial}_{P_i} U_P) + h_w ({\partial}_{P_i} w) \right]v =0. \label{eq:dw}\end{gathered}$$ Let us set $L'=L_{{\varepsilon},Q}-h_w $. Then can be written as $$\begin{gathered} (L' ({\partial}_{P_i} w)\mid v) =p(p-1)\int_{{\Omega_\varepsilon}} U_P^{p-2}({\partial}_{P_i} U_P)w v + \int_{{\Omega_\varepsilon}} h_{U_P}({\partial}_{P_i} U_P) v \nonumber \\ -\int_{{\Omega_\varepsilon}} J({\varepsilon}x) {\nabla }({\partial}_{P_i} U_P) \cdot {\nabla }v -\int_{{\Omega_\varepsilon}} V({\varepsilon}x) ({\partial}_{P_i} U_P) v +p \int_{{\Omega_\varepsilon}} U_P^{p-1}({\partial}_{P_i} U_P) v. \label{eq:L'}\end{gathered}$$ It is easy to see that $$\label{eq:L'1} \left| p(p-1)\int_{{\Omega_\varepsilon}} U_P^{p-2}({\partial}_{P_i} U_P)w v \right| {\leqslant}c_1 \|w\|\|v\|$$ and, if $\gamma = \min\{1,p-1\}$, $$\label{eq:L'2} \left| \int_{{\Omega_\varepsilon}} h_{U_P}({\partial}_{P_i} U_P) v \right| {\leqslant}c_2 \|w\|^\gamma \|v\|.$$ Let us study the second line of . We recall that often we will write $U$ instead of $U^Q$. Reasoning as in the proof of Lemma \[eq:nf\] (see and ), we infer: $$\begin{gathered} I\equiv \int_{{\Omega_\varepsilon}} J({\varepsilon}x) {\nabla }({\partial}_{P_i} U_P) \cdot {\nabla }v +\int_{{\Omega_\varepsilon}} V({\varepsilon}x) ({\partial}_{P_i} U_P) v -p \int_{{\Omega_\varepsilon}} U_P^{p-1}({\partial}_{P_i} U_P) v \\ =\int_{\frac{{\Omega}-Q}{{\varepsilon}}} J(Q) {\nabla }({\partial}_{P_i} U) \cdot {\nabla }v_{-P} +\int_{\frac{{\Omega}-Q}{{\varepsilon}}} V(Q) ({\partial}_{P_i} U) v_{-P} \\ +{\varepsilon}\int_{{\Omega_\varepsilon}} J'(Q)[x-P] {\nabla }({\partial}_{P_i} U_P) \cdot {\nabla }v +{\varepsilon}\int_{{\Omega_\varepsilon}} V'(Q)[x-P] ({\partial}_{P_i} U_P) v \\ -p \int_{{\Omega_\varepsilon}} U_P^{p-1}({\partial}_{P_i} U_P) v +O({\varepsilon}) \|v \|.\end{gathered}$$ Suppose, for simplicity, $Q$ coincides with the origin $\cal O$ and that part of ${\partial}{\Omega}$ is given by $x_N=\psi(x')=\frac 12 \sum^{N-1}_{i=1}\l_i x_i^2+O(|x'|^3)$ for $|x'| < \mu$, where $\mu$ is some constant depending only on ${\Omega}$. Then for $|y'|<\mu/{\varepsilon}$, the corresponding part of ${\partial}{\Omega_\varepsilon}$ is given by $y_N=\Psi (y')= {\varepsilon}^{-1} \psi({\varepsilon}y')= \frac{{\varepsilon}}{2} \sum^{N-1}_{i=1}\l_i y_i^2+O({\varepsilon}^2 |y'|^3)$. Since by ${\partial}_{P_i} U_P = -{\partial}_{x_i} U_P+O({\varepsilon})$, by integration by parts, we get: $$\begin{aligned} {\varepsilon}\int_{{\Omega_\varepsilon}} J'(Q)[x-P] {\nabla }({\partial}_{P_i} U_P)\! \cdot \!{\nabla }v &=& {\varepsilon}\int_{{\Omega_\varepsilon}} {\partial}_{Q_i} J(Q) {\nabla }U_P \cdot {\nabla }v +O({\varepsilon})\|v\|, \\ {\varepsilon}\int_{{\Omega_\varepsilon}} V'(Q)[x-P] ({\partial}_{P_i} U_P) v &=& {\varepsilon}\int_{{\Omega_\varepsilon}} {\partial}_{Q_i} V(Q) U_P v +O({\varepsilon})\|v\|.\end{aligned}$$ Hence $$\begin{gathered} I =\int_{{\Omega_\varepsilon}} J(Q) {\nabla }({\partial}_{P_i} U_P) \cdot {\nabla }v +{\varepsilon}\int_{{\Omega_\varepsilon}} {\partial}_{Q_i} J(Q) {\nabla }U_P \cdot {\nabla }v \\ +\int_{{\Omega_\varepsilon}} V(Q) ({\partial}_{P_i} U_P) v +{\varepsilon}\int_{{\Omega_\varepsilon}} {\partial}_{Q_i} V(Q) U_P v -p \int_{{\Omega_\varepsilon}} U_P^{p-1}({\partial}_{P_i} U_P) v +O({\varepsilon}) \|v \|.\end{gathered}$$ Being $U=U^Q$ solution of , we have that $$-J(Q) \varDelta ({\partial}_{P_i} U) -{\varepsilon}{\partial}_{Q_i}J(Q) \varDelta U +V(Q) ({\partial}_{P_i} U) +{\varepsilon}{\partial}_{Q_i} V(Q) U -p U^{p-1}({\partial}_{P_i} U)=0$$ and so $$\begin{gathered} I=J(Q) \int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}}{{\partial}\nu} ({\partial}_{P_i} U_P) v +{\varepsilon}{\partial}_{Q_i} J(Q) \int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}U_P}{{\partial}\nu} v +O({\varepsilon}) \|v \|. \end{gathered}$$ Arguing again as in the proof of Lemma \[eq:nf\] (see ), we can prove that $$\left|J(Q) \int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}}{{\partial}\nu} ({\partial}_{P_i} U_P) v +{\varepsilon}{\partial}_{Q_i} J(Q) \int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}U_P}{{\partial}\nu} v\right| = O({\varepsilon}) \|v \|.$$ Hence $$\label{eq:I} I=O({\varepsilon}^{3/4}) \|v \|.$$ Putting together , , and , we find $$|(L'({\partial}w_i)\mid v)|=\left(c_3 \|w\|^{\gamma}+ O({\varepsilon}) \right)\|v \|.$$ Since $h_w\to 0$ as $w \to 0$, the operator $L'$, likewise $L$, is invertible for ${\varepsilon}>0$ small and therefore one finds $$\|{\partial}_{P_i} w\|{\leqslant}c_4 \|w\|^{\gamma}+ O({\varepsilon}).$$ Finally, by Remark \[rem:w\], the Lemma follows. The finite dimensional functional ================================= \[th:sviluppo\] Let $Q \in {\partial}{\Omega}$ and $P=Q/{\varepsilon}\in {\partial}{\Omega_\varepsilon}$. Suppose [**(J)**]{} and [**(V)**]{}. Then, for ${\varepsilon}$ sufficiently small, we get: $$\label{eq:A} {{\cal A}}_{\varepsilon}(Q)= f_{\varepsilon}(U_P + w({\varepsilon},Q)) = c_0 {\Gamma}(Q) +{\varepsilon}{\Sigma}(Q) +o({\varepsilon}),$$ where ${\Gamma}$ is the auxiliary functions introduced in , $$c_0\equiv\left(\frac{1}{2}-\frac{1}{p+1}\right)\int_{{{\mathbb{R}^N_+}}} \bar U^{p+1},$$ and ${\Sigma}\colon {\partial}{\Omega}\to {\mathbb{R}}$ is so defined: $$\begin{gathered} \label{eq:Sigma} {\Sigma}(Q)\equiv \frac{1}{2}\int_{{\mathbb{R}}^-_{\nu(Q)}} \!\!\!\!J'(Q)[x] |\nabla U^Q|^2 d x +\frac{1}{2}\int_{{\mathbb{R}}^-_{\nu(Q)}}\!\!\!\!V'(Q)[x] \left(U^Q\right)^2 d x \\ -\frac{1}{2} \bar{B}^Q J(Q) H(Q) -\left(\frac{1}{2}-\frac{1}{p+1}\right) \bar{A}^Q H(Q),\end{gathered}$$ with $$\begin{aligned} \bar{A}^Q &\equiv& \frac 12 \int_{{\mathbb{R}}^{N-1}}\left[U^Q(x',0)\right]^{p+1} |x'|^2 d x', \\ \bar{B}^Q &\equiv& \frac{(N-1)}{4} \int_{{\mathbb{R}}^{N-1}}\left[U^Q(x',0)\right]^2 d x.\end{aligned}$$ Moreover, for all $i=1,\ldots,N-1$, we get: $$\label{eq:DA} {\partial}_{P_i} {{\cal A}}_{\varepsilon}(Q)= {\varepsilon}c_0 {\partial}_{Q_i} {\Gamma}(Q)+o({\varepsilon}).$$ In the sequel, to be short, we will often write $w$ instead of $w({\varepsilon},Q)$. It is always understood that ${\varepsilon}$ is taken in such a way that all the results discussed previously hold. First of all, reasoning as in the proofs of and and by , we can observe that $$\begin{aligned} \int_{{\Omega_\varepsilon}} J({\varepsilon}x)\nabla U_P \cdot \nabla w &=& J(Q) \int_{{\Omega_\varepsilon}} \nabla U_P \cdot \nabla w + o({\varepsilon}), \label{eq:J2} \\ \int_{{\Omega_\varepsilon}} V({\varepsilon}x)U_P \,w &=& V(Q) \int_{{\Omega_\varepsilon}} U_P \,w +o({\varepsilon}). \label{eq:V2}\end{aligned}$$ We have: $$\begin{gathered} {{\cal A}}_{\varepsilon}(Q) = f_{\varepsilon}(U_P + w({\varepsilon},Q)) \\ =\frac{1}{2}\int_{{\Omega_\varepsilon}}\!\! J({\varepsilon}x) |\nabla (U_P +w)|^2 +\frac{1}{2}\int_{{\Omega_\varepsilon}}\!\! V({\varepsilon}x)(U_P+w)^2 -\frac{1}{p+1}\int_{{\Omega_\varepsilon}}\! (U_P+w)^{p+1}\end{gathered}$$ \[by \] $$\begin{gathered} =\frac{1}{2}\int_{{\Omega_\varepsilon}} J({\varepsilon}x) |\nabla U_P|^2 +\frac{1}{2}\int_{{\Omega_\varepsilon}} V({\varepsilon}x)U_P^2 -\frac{1}{2}\int_{{\Omega_\varepsilon}} U_P^{p+1} \\ +\int_{{\Omega_\varepsilon}} J({\varepsilon}x)\nabla U_P \cdot \nabla w +\int_{{\Omega_\varepsilon}} V({\varepsilon}x)U_P \,w -\int_{{\Omega_\varepsilon}} U_P^p \,w +\left(\frac{1}{2}-\frac{1}{p+1}\right) \int_{{\Omega_\varepsilon}} \!\!U_P^{p+1} \\ -\frac{1}{p+1}\int_{{\Omega_\varepsilon}}\!\left[ (U_P+w)^{p+1} -U_P^{p+1}-(p+1) U_P^p \,w\right] +o({\varepsilon})=\end{gathered}$$ \[by , , , and and with our notations\] $$\begin{gathered} =\frac{1}{2} \int_{{{\mathbb{R}^N_+}}} U^{p+1} -\frac{{\varepsilon}}{2} \bar{A}^Q H(Q) -\frac{{\varepsilon}}{2} \bar{B}^Q J(Q) H(Q) +\frac{{\varepsilon}}{2}\int_{{\mathbb{R}}^-_{\nu(Q)}}\!\!\!\!J'(Q)[x] |\nabla U|^2 \\ +\frac{{\varepsilon}}{2}\int_{{\mathbb{R}}^-_{\nu(Q)}}V'(Q)[x] U^2 -\frac{1}{2}\int_{{{\mathbb{R}^N_+}}} U^{p+1} +\frac{{\varepsilon}}{2} \bar{A}^Q H(Q) \\ +J(Q) \int_{{\Omega_\varepsilon}} \nabla U_P \cdot \nabla w +V(Q) \int_{{\Omega_\varepsilon}} U_P \,w -\int_{{\Omega_\varepsilon}} U_P^p \,w \\ +\left(\frac{1}{2}-\frac{1}{p+1}\right) \int_{{{\mathbb{R}^N_+}}} U^{p+1} -{\varepsilon}\left(\frac{1}{2}-\frac{1}{p+1}\right) \bar{A}^Q H(Q) +o({\varepsilon}).\end{gathered}$$ From the fact that $U$ is solution of , we infer $$\begin{gathered} J(Q) \int_{{\Omega_\varepsilon}} \nabla U_P \cdot \nabla w +V(Q) \int_{{\Omega_\varepsilon}} U_P \,w -\int_{{\Omega_\varepsilon}} U_P^p \,w \\ =\int_{{\Omega_\varepsilon}} \left[ -J(Q) \varDelta U_P + V(Q)U_P - U_P^p \right] w +J(Q)\int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}U_P}{{\partial}\nu} w \\ =J(Q)\int_{{\partial}{\Omega_\varepsilon}} \frac{{\partial}U_P}{{\partial}\nu} w =o({\varepsilon}).\end{gathered}$$ By these considerations we can say that $$\begin{gathered} {{\cal A}}_{\varepsilon}(Q) = \left(\frac{1}{2}-\frac{1}{p+1}\right) \int_{{{\mathbb{R}^N_+}}} U^{p+1} \\ +{\varepsilon}\Bigg[ \frac{1}{2}\int_{{\mathbb{R}}^-_{\nu(Q)}} J'(Q)[x] |\nabla U|^2 +\frac{1}{2}\int_{{\mathbb{R}}^-_{\nu(Q)}}V'(Q)[x] U^2 \\ -\frac{1}{2} \bar{B}^Q J(Q) H(Q) -\left(\frac{1}{2}-\frac{1}{p+1}\right) \bar{A}^Q H(Q) \Bigg] +o({\varepsilon}).\end{gathered}$$ Now the conclusion of the first part of the theorem follows observing that, since by $$U^Q (x)=V(Q)^{\frac{1}{p-1}}\,\bar U\left(x \sqrt{V(Q)/J(Q)} \right),$$ then $$\int_{{{\mathbb{R}^N_+}}} U^{p+1}= V(Q)^{\frac{p+1}{p-1}-\frac{N}{2}}J(Q)^{\frac{N}{2}} \int_{{{\mathbb{R}^N_+}}} \bar U^{p+1}.$$ Let us prove now the estimate on the derivatives of ${{\cal A}}_{\varepsilon}$. First of all, we observe that by and by , we infer that $$\left|{\nabla }f_{\varepsilon}(U_P)[{\partial}_{P_i} w]\right| =O({\varepsilon}^{1+\g}),$$ and so, by and , we have: $$\begin{gathered} {\partial}_{P_i} {{\cal A}}_{\varepsilon}(Q) ={\nabla }f_{\varepsilon}(U_P +w)[{\partial}_{P_i} U_P +{\partial}_{P_i} w]= {\nabla }f_{\varepsilon}(U_P +w)[{\partial}_{P_i} U_P]+ O({\varepsilon}^{1+\g}) \\ ={\nabla }f_{\varepsilon}(U_P)[{\partial}_{P_i} U_P] +D^2 f_{\varepsilon}(U_P) [w, {\partial}_{P_i} U_P] \\ +\left({\nabla }f_{\varepsilon}(U_P +w) -{\nabla }f_{\varepsilon}(U_P) -D^2 f_{\varepsilon}(U_P)[w] \right) [{\partial}_{P_i} U_P] + O({\varepsilon}^{1+\g}).\end{gathered}$$ But $$\| {\nabla }f_{\varepsilon}(U_P +w) -{\nabla }f_{\varepsilon}(U_P) -D^2 f_{\varepsilon}(U_P)[w] \| = o(\|w \|) =o({\varepsilon})$$ and, moreover, by also $D^2 f_{\varepsilon}(U_P) [w, {\partial}_{P_i} U_P] =O({\varepsilon}^{1+\g})$, therefore $$\label{eq:DA-1} {\partial}_{P_i} {{\cal A}}_{\varepsilon}(Q) ={\nabla }f_{\varepsilon}(U_P)[{\partial}_{P_i} U_P] + O({\varepsilon}^{1+\g}).$$ Let us calculate ${\nabla }f_{\varepsilon}(U_P)[{\partial}_{P_i} U_P]$. $$\begin{gathered} {\nabla }f_{\varepsilon}(U_P)[{\partial}_{P_i} U_P] \!=\!\!\int_{{\Omega_\varepsilon}}\!\!\!\! J({\varepsilon}x) {\nabla }U_P\! \cdot \!{\nabla }({\partial}_{P_i} U_P) \!+\!\!\int_{{\Omega_\varepsilon}}\!\!\!\! V({\varepsilon}x) U_P ({\partial}_{P_i} U_P) \!-\!\!\int_{{\Omega_\varepsilon}}\!\!\!\! U_P^p ({\partial}_{P_i} U_P) \\ =J(Q) \!\int_{{\Omega_\varepsilon}}\!\! {\nabla }U_P \cdot {\nabla }({\partial}_{P_i} U_P) +V(Q) \!\int_{{\Omega_\varepsilon}} \!\!U_P ({\partial}_{P_i} U_P) \\ +{\varepsilon}\!\!\int_{{\mathbb{R}}^-_{\nu(Q)}}\!\!\!\!\!\!\!\!J'(Q)[x] {\nabla }U\! \cdot \!{\nabla }({\partial}_{P_i} U)\! +\!{\varepsilon}\!\! \int_{{\mathbb{R}}^-_{\nu(Q)}}\!\!\!\!\!\!\!\!V'(Q)[x] U ({\partial}_{P_i} U)\! -\!\!\int_{{\Omega_\varepsilon}}\!\!\!\! U_P^p ({\partial}_{P_i} U_P) \!+\!o({\varepsilon}).\end{gathered}$$ Suppose, for simplicity, $Q$ coincides with the origin $\cal O$ and that part of ${\partial}{\Omega}$ is given by $x_N=\psi(x')=\frac 12 \sum^{N-1}_{i=1}\l_i x_i^2+O(|x'|^3)$ for $|x'| < \mu$, where $\mu$ is some constant depending only on ${\Omega}$. Then for $|y'|<\mu/{\varepsilon}$, the corresponding part of ${\partial}{\Omega_\varepsilon}$ is given by $y_N=\Psi (y')= {\varepsilon}^{-1} \psi({\varepsilon}y')= \frac{{\varepsilon}}{2} \sum^{N-1}_{i=1}\l_i y_i^2+O({\varepsilon}^2 |y'|^3)$. Since by ${\partial}_{P_i} U_P = -{\partial}_{x_i} U_P +O({\varepsilon})$, by integration by parts, we get: $$\begin{aligned} \int_{{\mathbb{R}}^-_{\nu(Q)}}\!\!\!J'(Q)[x] {\nabla }U\! \cdot \!{\nabla }({\partial}_{P_i} U) &=&\frac 12 \int_{{\mathbb{R}}^-_{\nu(Q)}}\! {\partial}_{Q_i} J(Q) |{\nabla }U|^2, \\ \int_{{\mathbb{R}}^-_{\nu(Q)}}\!V'(Q)[x] U ({\partial}_{P_i} U) &=&\frac 12 \int_{{\mathbb{R}}^-_{\nu(Q)}}\! {\partial}_{Q_i} V(Q) U^2.\end{aligned}$$ Therefore we infer $${\nabla }f_{\varepsilon}(U_P)[{\partial}_{P_i} U_P] =\frac 12 {\partial}_{P_i}\!\!\left[ J(Q) \!\!\int_{{\Omega_\varepsilon}}\!\!|{\nabla }U_P|^2\! +\!V(Q) \!\!\int_{{\Omega_\varepsilon}} \!\!U_P^2 \right] -\int_{{\Omega_\varepsilon}} \!\!U_P^p \; ({\partial}_{P_i} U_P)+o({\varepsilon}),$$ and so, by and , $${\nabla }f_{\varepsilon}(U_P)[{\partial}_{P_i} U_P] = {\varepsilon}\left[\left( \frac 12 - \frac{1}{p+1}\right) \int_{{\mathbb{R}^N_+}}\bar U^{p+1}\right] {\partial}_{Q_i} {\Gamma}(Q) = {\varepsilon}c_0 {\partial}_{Q_i} {\Gamma}(Q)+o({\varepsilon}).$$ By this equation and by , follows immediately. \[re:Gamma\] Let us observe that by and , for ${\varepsilon}$ sufficiently small, we have $$\label{eq:C1} \|{{\cal A}}_{\varepsilon}- c_0 {\Gamma}\|_{C^1({\partial}{\Omega})}=O({\varepsilon}).$$ \[re:Sigma\] By , it is easy to see that, if $J$ and $V$ are constant on the boundary ${\partial}{\Omega}$, then $\bar {\Sigma}$, defined in , coincides with ${\Sigma}$, defined in with the following definitions: $$\begin{array}{lll} C_J \equiv J_{|_{{\partial}{\Omega}}},& \quad&C_V \equiv V_{|_{{\partial}{\Omega}}}, \\ k_1 \equiv \frac{(C_V)^{\frac{p+1}{p-1}}}{2 C_J},& &k_2 \equiv \sqrt{C_V/C_J}, \\ k_3 \equiv \frac{(C_V)^{\frac{2}{p-1}}}{2}, & &k_4 \equiv -\frac{1}{2} \bar{B} C_J -\left(\frac{1}{2}-\frac{1}{p+1}\right) \bar{A}, \end{array}$$ where $$\begin{aligned} \bar{A} &\equiv& \frac {(C_V)^{\frac{p+1}{p-1}}}{2} \int_{{\mathbb{R}}^{N-1}}\left[\bar U \left(x'\sqrt{C_V/C_J},0 \right)\right]^{p+1} |x'|^2 d x', \\ \bar{B} &\equiv& \frac{(N-1)(C_V)^{\frac{2}{p-1}}}{4} \int_{{\mathbb{R}}^{N-1}}\left[\bar U \left(x'\sqrt{C_V/C_J},0 \right)\right]^2 d x'. \end{aligned}$$ Proofs of Theorem \[th1\] and Theorem \[th2\] ============================================= In this section we will state and prove two multiplicity results for whose Theorem \[th1\] is a particular case. Finally we will prove also Theorem \[th2\]. Let us start introducing a topological invariant related to Conley theory. Let $M$ be a subset of ${{\mathbb{R}^N}}$, $M\ne \emptyset$. The [*cup long*]{} $l(M)$ of $M$ is defined by $$l(M)=1+\sup\{k\in {\mathbb{N}}\mid \exists\, \a_{1},\ldots,\a_{k}\in \check{H}^{*}(M)\setminus 1, \,\a_{1}\cup\ldots\cup\a_{k}\ne 0\}.$$ If no such class exists, we set $l(M)=1$. Here $\check{H}^{*}(M)$ is the Alexander cohomology of $M$ with real coefficients and $\cup$ denotes the cup product. Let us recall Theorem 6.4 in Chapter II of [@C]. \[th:Chang\] Let $N$ a Hilbert-Riemannian manifold. Let $g \in C^2(N)$ and let $M\subset N$ be a smooth compact nondegenerate manifold of critical points of $g$. Let $U$ be a neighborhood of $M$ and let $h \in C^1(N)$. Then, if $\|g-h\|_{C^1(\bar{U})}$ is sufficiently small, the function $g$ possesses at least $l(M)$ critical points in $U$. Let us suppose that ${\Gamma}$ has a smooth manifold of critical points $M$. We say that $M$ is nondegenerate (for ${\Gamma}$) if every $x\in M$ is a nondegenerate critical point of ${\Gamma}_{|M^{\perp}}$. The Morse index of $M$ is, by definition, the Morse index of any $x\in M$, as critical point of ${\Gamma}_{|M^{\perp}}$. We now can state our first multiplicity result. \[th:lc\] Let [**(J)**]{} and [**(V)**]{} hold and suppose ${\Gamma}$ has a nondegenerate smooth manifold of critical points $M\subset {\partial}{\Omega}$. There exists ${\varepsilon}_0>0$ such that if $0<{\varepsilon}<{\varepsilon}_0$, then has at least $l(M)$ solutions that concentrate near points of $M$. Fix a $\delta$-neighborhood $M_\delta$ of $M$ such that the only critical points of ${\Gamma}$ in $M_\delta$ are those in $M$. We will take $U=M_\delta$. For ${\varepsilon}$ sufficiently small, by and Theorem \[th:Chang\], ${{\cal A}}_{\varepsilon}$ possesses at least $l(M)$ critical points, which are solutions of by Lemma \[lem:w\]. Let $Q_{\varepsilon}\in M$ be one of these critical points, then $u_{\varepsilon}^{Q_{\varepsilon}}=U_{Q_{\varepsilon}/{\varepsilon}}+w({\varepsilon}, Q_{\varepsilon})$ is a solution of . Therefore $$u_{\varepsilon}^{Q_{\varepsilon}}(x/{\varepsilon})\simeq U_{Q_{\varepsilon}/{\varepsilon}}(x/{\varepsilon})= U^{Q_{\varepsilon}}\left(\frac{x-Q_{\varepsilon}}{{\varepsilon}} \right)$$ is a solution of . Moreover, when we deal with local minima (resp. maxima) of ${\Gamma}$, the preceding results can be improved because the number of positive solutions of can be estimated by means of the category and $M$ does not need to be a manifold. \[th:cat\] Let [**(J)**]{} and [**(V)**]{} hold and suppose ${\Gamma}$ has a compact set $X \subset {\partial}{\Omega}$ where ${\Gamma}$ achieves a strict local minimum (resp. maximum), in the sense that there exist $\delta>0$ and a $\d$-neighborhood $X_{\delta }\subset {\partial}{\Omega}$ of $X$ such that $$b\equiv \inf\{{\Gamma}(Q) : Q\in \partial X_{\d}\}> a\equiv {\Gamma}_{|_X}, \quad \left({\rm resp. }\; \sup\{{\Gamma}(Q) : Q\in \partial X_{\d}\}< {\Gamma}_{|_X} \right).$$ Then there exists ${\varepsilon}_0>0$ such that has at least $\operatorname{cat}(X,X_\d)$ solutions that concentrate near points of $X_{\d}$, provided ${\varepsilon}\in (0,{\varepsilon}_0)$. Here $\operatorname{cat}(X,X_\d)$ denotes the Lusternik-Schnirelman category of $X$ with respect to $X_\d$. We will treat only the case of minima, being the other one similar. We set $Y=\{Q\in X_{\d} :{{\cal A}}_{{\varepsilon}}(Q){\leqslant}c_0 (a+b)/2\}$. By it follows that there exists ${\varepsilon}_0>0$ such that $$\label{eq:X} X \subset Y \subset X_{\d},$$ provided ${\varepsilon}\in (0,{\varepsilon}_0)$. Moreover, if $Q \in \partial X_{\d}$ then ${\Gamma}(Q){\geqslant}b$ and hence $${{\cal A}}_{{\varepsilon}}(Q){\geqslant}c_0 {\Gamma}(Q) + O({\varepsilon}) {\geqslant}c_0 b + O({\varepsilon}).$$ On the other side, if $Q\in Y$ then ${{\cal A}}_{{\varepsilon}}(Q){\leqslant}c_0 (a+ b)/2$. Hence, for ${\varepsilon}$ small, $Y$ cannot meet $\partial X_{\d}$ and this readily implies that $Y$ is compact. Then ${{\cal A}}_{{\varepsilon}}$ possesses at least $\operatorname{cat}(Y,X_{\d})$ critical points in $ X_{\d}$. Using (\[eq:X\]) and the properties of the category one gets $$\operatorname{cat}(Y,Y){\geqslant}\operatorname{cat}(X,X_{\d}),$$ and the result follows. Let us observe that the (a) of Theorem \[th1\] is a particular case of Theorem \[th:lc\] while the (b) of Theorem \[th1\] is a particular case of Theorem \[th:cat\]. Let us now prove Theorem \[th2\]. Let $Q$ be a minimum point of $\bar {\Sigma}$ (the other case is similar) and let ${\Lambda}\subset {\partial}{\Omega}$ be a compact neighborhood of $Q$ such that $$\min_{\Lambda}\bar {\Sigma}<\min_{{\partial}{\Lambda}}\bar {\Sigma}.$$ By and Remark \[re:Sigma\], it is easy to see that for ${\varepsilon}$ sufficiently small, there results: $$\min_{{\Lambda}} {{\cal A}}_{\varepsilon}<\min_{{\partial}{\Lambda}}{{\cal A}}_{\varepsilon}.$$ Hence, ${{\cal A}}_{\varepsilon}$ possesses a critical point $Q_{\varepsilon}$ in ${\Lambda}$. 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--- abstract: 'Although the radio emission from most quasars appears to be associated with star forming activity in the host galaxy, about ten percent of optically selected quasars have very luminous relativistic jets apparently powered by a SMBH which is located at the base of the jet. When these jets are pointed close to the line of sight their apparent luminosity is enhanced by Doppler boosting and appears highly variable. High resolution radio interferometry shows directly the outflow of relativistic plasma jets from the SMBH. Apparent transverse velocities in these so-called “blazars” are typically about 7c but reach as much as 50c indicating true velocities within one percent of the speed of light. The jets appear to be collimated and accelerated in regions as much as a hundred parsecs downstream from the SMBH. Measurements made with Earth to space interferometers indicate apparent brightness temperatures of $\sim 10^{14}$ K or more. This is well in excess of the limits imposed by inverse Compton cooling. The modest Doppler factors deduced from the observed ejection speeds appear to be inadequate to explain the high observed brightness temperatures in terms of relativistic boosting.' --- Why Radio? ============ Historically the first speculations about the existence of active galactic nuclei (AGN) and super massive black holes (SMBHs) came from the huge energy requirements implied by the discovery of distant powerful radio galaxies and quasars. Today, radio observations remain crucial to understanding the role of SMBHs in astrophysics. Only at radio wavelengths is it possible to image the region immediately surrounding the SMBH central engine and the relativistic jets which apparently originate with the SMBH. Typical resolution obtained with Very Long Baseline Interferometer (VLBI) observations at centimeter wavelengths is of the order of 0.001 arccsecond (1 milliarcsec). Thus, for nearby sources such as those located in the Virgo cluster, a linear resolution of 1 milliarcsec corresponds to only about 0.1 parsec or about 100 Schwartzchild radii for the SMBH located in the nucleus of M87. For $z \sim 1$ the resolution for Earth-based VLBI is $\sim $ 10 kpc. However, using an Earth to space interferometry at $\sim$ 1 cm or Earth-based systems at millimeter wavelengths, the resolution is improved by more than another order of magnitude. Finally, we note that at radio wavelengths, there is no obscuration, even close to the SMBH. However, not all radio sources are due to AGN; and not all AGN and SMBHs are radio sources. Nearly all observed extragalactic radio emission is probably due to synchrotron radiation from ultra relativistic electron with energies $\sim$ 10 GeV moving in weak magnetic fields with $B\sim 10^{-5}$ to $10^{-4}$ Gauss. The high energy electrons are thought to be accelerated in one of two ways; either by a central engine associated with accretion onto a SMBH in elliptical galaxies or in quasars, or by supernovae following massive star formation (starbursts) in the nucleus of early type galaxies. Unfortunately, both processes are often referred to as AGN, and this has led to considerable confusion in the literature. Above $\sim 1$ mJy, the radio source number-flux density relation is dominated by sources driven by SMBHs. These more powerful sources are characterized by extended radio lobes, and by highly beamed relativistic jets extending from a few parsecs to hundreds of kiloparsecs from the SMBH. At microJy levels there is an increasing contribution from star formation related activity rather than from a SMBH. However, there is an uncomfortably large spread in the observed microJy source count even among different observers using the same instrument, the VLA, in the same field. Most likely these discrepancies are the result of systematic errors in the reported flux densities due to uncertainties in corrections for the effects of resolution [@CCF12]. A recent complexity comes from the ARCADE 2 balloon measurement of a 3 GHz sky brightness of $54 \pm 6$ K which is about 5 sigma above that expected from known radio sources, suggesting a possible population of previously unrecognized weak sub microJansky sources [@F11 (Fixen et al. 2011)]. However, deep 3 GHz VLA observations showed no evidence for any source population greater than about 30 nanoJy. Any population of weaker sources that could produce the excess sky brightness would need to have a sky density greater than 6 x $10^4$ per ster, or 60 times greater than the density of the faintest (mag 29) galaxies in the Hubble Ultra Deep Field [@CCF12 (Condon et al. 2012)]. Thus if the excess background temperature observed by ARCADE is real and due to discrete sources, these sources cannot be associated with any known galaxy population. It will be important to verify that the ARCADE 2 results were not contaminated by unrecognized Galactic radio emission. Radio emission due to AGN can usually be distinguished from that due to star formation in a variety of ways. $\bullet$  [**Morphology:**]{} Star formation sources may have dimensions of the order of a few tenths of an arcsecond or a few kiloparsecs at cosmological distances while SMBH driven AGN sources are typically very small, of the order of 0.001 arcsec (10 pc) or less, and are coincident with the galaxy nucleus or QSO. Quasars and AGN powered by SMBHs are often variable on time scales as short as days with corresponding changes in their morphology indicating highly collimated outflows with apparent superluminal velocity. SMBH driven AGN may also contain extended lobes tens or hundreds of kiloparsecs distant from the compact nucleus, and sometimes show optical and radio jets joining the nucleus and radio lobes. $\bullet$ [**Radio Spectra:**]{} Star forming sources and the extended radio lobes of AGN generally have steep radio spectra. Due to synchrotron self absorption, the compact sources generally have flat or even inverted spectra. $\bullet$ [**Brightness Temperature:**]{} Star forming sources mostly have measured brightness temperature up to $\sim 10^6$ K while the compact flat spectrum sources associated with AGN have brightness temperature $10^{11-12}$ K or more. $\bullet$ [**Radio Luminosity:**]{} Star forming regions typically have a radio luminosity close to $10^{22-23}$ W/Hz and follow the well known correlation between radio and FIR luminosity [@C92]. Radio galaxies and quasars driven by SMBHs may be $10^{4-5}$ times more luminous so their radio luminosity greatly exceeds that expected from the radio/FIR relation characteristic of star forming regions. $\bullet$ [**X and $\gamma$-ray emission:**]{} Star forming regions are only weak x-ray sources with typical luminosity $\sim 10^{42}$ ergs/sec while SMBH driven AGN can be strong x-ray, $\gamma$-ray, and TeV sources. $\bullet$ [**Host Galaxies:**]{} SMBH driven radio sources are located in the nuclei of elliptical galaxies or are associated with quasars which themselves are thought to be the bright nuclei of elliptical galaxies that greatly outshine their host galaxy. Low (optical) luminosity AGN are typically found in early type spiral (often classified as Seyfert) galaxies. Radio emission from star forming regions is typically associated with spiral galaxies but may also be found in the host galaxies of radio quiet quasars (see Section 4). Early Evidence for AGN and SMBHs ================================ Perhaps the first suggestions that the nuclei of galaxies may contain more than just stars came from Sir James Jeans in 1929 who remarked in his book on Astronomy and Cosmogony [@2014IAUS..304...78K (Jeans 1929)],\ > The centres of the nebulae are of the nature of singular points at which matter is poured into our universe from some other and entirely spatial dimension so that to a denizen of our universe, they appear as points at which matter is being continuously created. The modern understanding of the important role of galactic nuclei probably began with the famous paper by Karl Seyfert (1943) who reported on his study of broad strong emission lines in the nucleus of seven spiral nebulae. Interestingly, although Seyfert’s name ultimately became attached to the broad category of spiral galaxies with active nuclei, his 1943 paper received no citations until 1951, and apparently went unnoticed until Baade and Minkowski [@BM54 (1954)] drew attention to the similarity of the Cygnus A radio source spectrum with that of the galaxies studied by Seyfert. Not until the 1949 Nature paper by Bolton, Stanley, and Slee [@BSS49 (1949)] did astronomers finally recognize the vast energy requirements of radio galaxies. Bolton et al. had identified three of the strongest discrete radio sources with the Crab Nebula, M87, and NGC 5128, Until that time the discrete radio sources were widely thought to be associated with galactic stars. This was understandable, as Karl Jansky and Grote Reber had observed radio emission from the Milky Way. The Milky Way is composed of stars, so it was natural to assume that the discrete radio sources had a stellar origin. Bolton et al. understood the importance of their identification of the Taurus A radio source with the Crab Nebula which was widely recognized as the remnant of the 1054 supernova reported by Chinese observers. BSS correctly identified two other strong sources with M87 and NGC 5128, but realizing that if they were extragalactic, their absolute radio luminosity would need to be a million times more luminous than that of the Crab Nebula, they argued that “NGC 5128 and NGC 4486 (M87) have not been resolved into stars, so there is little direct evidence that they are true galaxies.” So they concluded that they are within our own Galaxy. Indeed their paper carried the title “Positions of Three Discrete Radio Sources of Galactic Radio Frequency Radiation." John Bolton later argued that he really did understand that M87 and NGC 5128 were very luminous radio sources, but that he was concerned that that in view of their apparent extraordinary radio luminosity, Nature might not publish their paper. The following years saw the identification of more radio galaxies, and the changed paradigm which had previously considered all discrete radio sources to be stellar to one with most high latitude sources were assumed to be extragalactic. The energy requirements were exacerbated in 1951 with the identification of Cygnus A, the second strongest radio source in the sky with a magnitude 18 galaxy at what was then considered a high redshift of 0.056 and a corresponding radio luminosity about $10^3$ times more luminous than M87 and NGC 5128 [@BM54 (Baade and Minkowski 1954)]. The total energy contained in relativistic particles and magnetic fields in the radio lobes of Cygnus A and other powerful radio galaxies was estimated to be at least $10^{60-61}$ ergs [@B59]. Hoyle, Fowler, Burbidge and Burbidge [@HFBB64 (1964)] were apparently the first to call attention to gravitational collapse as a possible energy source to power radio galaxies. By the middle of 1960, many radio sources had been identified with galaxies having red shifts up to 0.24 [@B60]. Typically the optical counterpart of strong radio sources was identified with an elliptical galaxy that was the brightest member of a cluster. In 1960, Rudolph Minkowski [@M60 (1960)] identified 3C 295 with a mag 20 galaxy at z=0.46. 3C 295 is about ten times smaller than Cygnus A and ten times more distant consistent with the idea that the smallest radio sources might be path finders to finding very distant galaxies. But a few months later Caltech radio astronomers identified the first of several very small sources with what appeared to be galactic stars, thus raising questions about the extragalactic nature of other small diameter radio sources. The First Quasars ================= While searching for ever more distant radio galaxies, Caltech radio astronomers John Bolton and Tom Matthews identified 3C 48 with an apparent stellar object. At the 107th meting of the American Astronomical Society held in New York in December 1960, Allan Sandage [@S60] reported the discovery of “The First True Radio Star." Before he left to return to Australia, John Bolton [@B90] speculated that 3C 48 had a high redshift of 0.37, but was apparently dissuaded by Jesse Greenstein and Ira Bowen on the grounds that there was a 3 or 4 Angstrom discrepancy among the corresponding rest wavelengths. In a subsequent analysis of the complex emission line spectrum, Jesse Greenstein [@G62] interpreted the 3C 48 spectrum in terms of emission lines from highly ionized states of rare earth elements. He briefly considered a possible redshift of 0.37, but quickly dismissed the possibility that 3C 48 was extragalactic. Nearly two years would pass, and other compact radio sources would be identified as galactic stars before a series of lunar occultations would lead to the identification of 3C 273 with a star like object at a redshift of 0.16 and the immediate realization that 3C 48 was also extragalactic with a redshift of 0.37 leading to the recognition of quasi stellar radio sources or “quasars” as the extremely bright nuclei of galaxies. The apparent high radio as well as optical luminosity of quasars, coupled with their very small dimensions presented a further challenge to understanding the source of energy and how this energy is converted to relativistic particles and magnetic fields. Radio Loud and Radio Quiet Quasars ================================== The following years led to the identification of more quasars at ever larger redshifts and the suggestion that quasars are powered by accretion onto super massive black holes (SMBH) with masses up to $10^9$ solar masses or more [@LB69]. Generally, the identified quasars had a significant UV excess compared with stars, so due to the redshift of their spectrum, they appeared blue on photographic plates facilitating their identification with radio sources with even modest position accuracy. In 1965, Sandage noted that the density of blue stellar objects on the sky was some thousand time greater than that of 3C radio sources. Sandage argued that what he called “quasi stellar galaxies” are related to quasars, except that they are not strong radio sources. But, his paper was widely attacked, perhaps in part because of the perceived irregular treatment by the Astrophsyical Journal. Sandage’s paper was received on May 15, 1965 at the Astrophysical Journal, but S. Chandrasekar, the ApJ editor was apparently so impressed by Sandage’s claim for a “New Constituent of Universe" that he held up publication of the Journal, and Sandage’s paper appeared in the May 15 issue. Tom Kinman [@K65] along with Lynds and Villere [@LV65] argued that most of Sandage’s Blue Stellar Objects were only blue galactic stars, while Fritz Zwicky [@Z65] pointed out that he had previously called attention to this phenomena, and he later accused Sandage of “one of the most astounding feats of plagiarism” [@Z71]. As it turned out, most of Sandage’s Blue Stellar Objects were just that, “blue stellar objects," and only some ten percent of optically identified quasars are strong radio sources. But, it has now been more than half a century since we have divided quasars into the two classes of radio loud and radio quiet quasars, and it has still not been clear if there are two distinct populations or rather whether the radio loud population is merely the extreme end of a continuous distribution of radio luminosity. Proponents of each interpretation claim that the other interpretation is due to selection effects. Many of the previous investigations designed to distinguish between radio loud and radio quiet quasars were limited by contamination from low luminosity AGN with absolute optical magnitudes greater than -23, biased samples based on radio rather than optical selection criteria, and inadequate sensitivity to detect radio emission from most of the radio quiet population. In an attempt to overcome these limitations, Kimball et al. [@K11] observed 179 quasars selected from the SDSS. All of the these quasars were within the redshift range 0.2 to 0.3 and were brighter than $M_{i}$ = -23 so were genuine quasars that presumably contained a SMBH. The observations were made with the Jansky Very Large Array at 6 GHz reaching an rms noise of 6 $\mu$Jy. All but about 6 quasars were detected as radio sources with an observed radio luminosity sharply peaked between $10^{22}$ and $10^{23}$ Watts/Hz characteristic of the radio luminosity typically observed from star forming galaxies. About ten percent of the SDSS sample are strong radio sources with radio luminosities ranging up to $10^{27}$ W/Hz. Kimball et al. concluded that the radio emission from radio quiet quasars is due to star formation in the host galaxy. Similar conclusions were reached by Padovani et al. (2011, 2014) based on the identification and classification of the microJy radio sources found in a deep VLA survey of the Extended Chandra Deep Field South. Based on radio, optical, IR, and X-ray data, Padovani et al. concluded that the microJy radio emission from AGN, like that of galaxies, is powered primarily by starbursts, and not the SMBHs which powers the AGN. Condon et al. [@C13] argue that these starbursts are fueled by the same gas that flows into the SMBH that powers the quasar and thus accounts for the co-evolution of star formation and SMBHs. Jet Kinematics and Relativistic Beaming ======================================= Shortly after the recognition of quasars, radio source observations in both the Soviet Union [@S65] and the U.S. [@D65] demonstrated variability on time scales of months or less. This presented a problem. Causality arguments suggested linear dimensions, d $\leq$ c$\tau$ where c is the speed of light and $\tau$ the characteristic time scale of the observed variability. Knowing the quasar redshift and corresponding distance puts a limit to the angular size which for many variable sources was only $\sim 10^{-5}$ arcseconds and the corresponding lower limit to the brightness temperature which appeared to be significantly in excess of the inverse Compton limit of $\sim 10^{11.5}$ K For most variable sources, the apparent violation of the inverse Compton limit is now understood in terms of relativistic beaming. Due to relativistic effects, we observe apparent jet speeds, luminosities, and brightness temperatures which are related to the corresponding intrinsic quantities in the AGN rest frame through the Doppler factor, $\delta$, the Lorentz factor, $\gamma$, and the jet orientation, $\theta$, with respect to the line of sight [@C07]. The apparent velocity transverse to the line of sight, $\beta_\mathrm{app}$, the apparent luminosity, $L$, the apparent brightness temperature, $T_\mathrm{app}$ and the Doppler factor, $\delta$, can be calculated from the Lorentz factor, $\gamma$, $\theta$, and the intrinsic luminosity, $L_o$. The apparent transverse velocity $\beta_\mathrm{app}$ is given by $$\beta_\mathrm{app} = \frac{\beta\sin\theta}{1-\beta\cos\theta}\,, \label{eq:beta_app}$$ where $\beta = v/c$ For small values of $\theta$, because the radiating source is almost catching up with its own radiation, equation 5.1 shows that the apparent transverse can exceed the speed of light, which is commonly referred to as “superluminal motion.” The apparent luminosity, $L$, is given by $$L = L_o \delta^n\,, \label{eq:lum}$$ where the Doppler factor, $\delta$, is $$\delta = \gamma^{-1}(1-\beta\cos\theta)^{-1}\,, \label{eq:delta}$$ and where $L_o$ is the luminosity that would be measured by an observer in the AGN frame. The quantity $n$ depends on the geometry and spectral index and is typically in the range between 2 and 3. The Lorentz factor, $\gamma$, is given by $$\gamma = {(1-\beta^2)}^{-{1/2}}. \label{eq:lorentz}$$ Quasars or AGN with highly Doppler boosted relativistic jets pointed nearly along the line-of-sight are often referred to as “blazars.” Blazars are characterized by rapid flux density variability, apparent superluminal motion, and strong x-ray and $\gamma$-ray emission. High resolution observations of blazars provide unique insight to the process by which relativistic jets are accelerated and collimated in the region close to the SMBH. We want to understand: $\bullet$ How and where is the relativistic beam accelerated and collimated into narrow jets? Are there accelerations or decelerations? Do all parts of the jet move at the same speed? $\bullet$ What causes the curvature of jets? Does the flow follow a curved trajectory or is the motion ballistic and characteristic of a rotating nozzle? $\bullet$ Does the observed apparent velocity reflect the true bulk velocity of motion? What determines the jet velocity? Is the velocity related to other properties such as radio, optical, x, or $\gamma$-ray luminosity? $\bullet$ What is the maximum observed brightness temperature? Does it exceed the inverse Compton limit? $\bullet$ What is the energy production mechanism? $\bullet$ What can we learn from radio observations about the nature of the SMBH? Very Long Baseline observations made since 1971 have confirmed the apparent superluminal motion expected from highly relativistic bulk motion. Since 1995, the NRAO Very Long Baseline Array (VLBA) has been used to study the motions of a large sample of quasars and AGN at 7mm by a group from Boston University [@M12] and by the international MOJAVE group [@K04; @L09; @H09; @LAA13; @H14]. More detailed information may be found on the respective web sites: Boston University 7 mm program: [**http://www.bu.edu/blazars/VLBAproject.html**]{} MOJAVE 2 cm program: [**: //www.physics.purdue.edu/ mlister/MOJAVE/**]{} The results of these programs may be summarized as follows. $\bullet$ Radio loud quasars and AGN show highly relativistic bulk motion with a broad distribution of apparent velocities. In general, the jets appear one sided, probably due to differential Doppler boosting so that the approaching jet appears much brighter than the receding one. Each jet appears to have its own characteristic velocity but there is an appreciable spread in the apparent velocity of the different features within a given jet. The typical apparent velocity, $\beta_{app} \sim 8$ corresponding to an intrinsic value of $\beta \sim 0.99$. The maximum observed apparent velocity, $\beta \sim 50$ corresponds to an intrinsic value of $\beta \sim 0.999$. The parent jet population is mostly only mildly relativistic, but is under represented in flux density limited samples due to the effect of Doppler boosting. $\bullet$ The jets with the fasted apparent velocities have the highest apparent luminosity, likely reflecting a correlation between intrinsic speed and intrinsic luminosity rather than simply being the result of Doppler boosting (Cohen et al. 2007). $\bullet$ Apparent inward motions are uncommon and are likely the result of a feature moving outward along a curved trajectory approaches the line-of-sight so that the apparent separation from the jet base transverse to the line-of-sight appears to decrease with time. $\bullet$ Individual jet features may show both apparent accelerations and decelerations. Both the apaprent speed and direction of motion may change with time, but changes in speed are more common than changes in direction, indicating real changes in the Lorentz factor as features propagate down the jet. In general the apparent speed is greater further down the jet, so that acceleration must take place at distances at least up to $\sim$ 100 pc from the base of the jet (Homan et al. 2009, 2014). $\bullet$ Many jets show a curved structure and in a few cases there is evidence of an oscillatory behavior. Sometimes the flow appears to follow pre-existing channels; other times the flow appears ballistic as from a rotating nozzle, perhaps due to precession possibly resulting from a binary black hole pair (Lister et al. 2013). $\bullet$ In some cases the direction of ejection appears to vary within a well defined cone forming what appears to be an edge brightened jet (Lister et al. 2013) such as shown by the jet in the nearby radio galaxy M87 where there is sufficient linear resolution to resolve the jet transverse to its structure [@K07]. $\bullet$ There appears to be a relation between radio and gamma ray emission. There is statistical evidence that radio outbursts follow a $\gamma$-ray event by $\sim$ 1 month, but it has been difficult to convincingly establish a one to one correlation between individual radio and $\gamma$-ray events [@P10]. Brightness Temperature Issues ============================= As described above, inverse Compton scattering limits the maximum observed brightness temperature, T $ < \sim 10^{11.5}$. At the inverse Compton limit, the energy contained in relativistic particles greatly exceeds that in the magnetic field which is perhaps not unreasonable in a very young source. If the particle and magnetic energies are in equilibrium, then the corresponding brightness temperature is only $\sim 10^{10.5}$. The observed brightness temperature may be calculated from [@K06] $$T_b = \frac{2\ ln 2}{k\pi}\frac{S\lambda^2}{\theta^2} ~K = 1.4\times10^9S\frac{\lambda^2}{\theta^2}(1+z)~K\,, \label{eq:Tb}$$ where k is the Boltzman constant, $\theta$ the angular size in milliarcsec, S is the flux density in Janskys, and $\lambda$ the wavelength in cm. The resolution of a radio interferometer, is given by the ratio of the observing wavelength, to the interferometer baseline, D; or $\theta$ = $\lambda$/D. Putting this back into eqn. 6.1 gives $$T_b = 80SD^2(1+z)~K\,, \label{eq:Tb}$$ so the maximum brightness temperature that may be measured depends only on the flux density and baseline length, and is independent of wavelength. For ground based observations with a maximum baseline of $\sim 8,000$ km, the highest brightness temperatures which can be reached are $\sim 10^{13}$ K. Recent observations with the Russian RadioAstron space VLBI satellite have suggested lower limits to brightness temperatures of 3C 273 and other sources $\sim 10^{14}$ K (Kellermann et al. 2014, Kovalev 2014), or at least two to three orders of magnitude greater than the limit set by inverse Compton cooling. Several explanations are possible [@KIK14]. 1\. For a relativistically beamed source the apparent brightness temperature is boosted by a factor $\delta$. To explain the high observed brightness temperatures in this way would require Doppler factors, $\delta \sim$ $10^2$ to $10^3$. But typical observed values of $\delta \sim \gamma \sim 10$ with maximum observed values $\sim 50$, and for 3C 273, $\gamma \sim 15$ (Lister et al. 2013). Possibly the bulk flow which is related to the Doppler boosting might be much greater than the pattern flow observed by the VLBA, but Cohen et al. (2007) have shown that this is unlikely. 2\. The observed emission may be coherent such as observed in pulsars or the Sun including possible stimulated synchrotron emission. 3\) The radio emission might be the result of synchrotron emission from protons rather than electrons which would enhance the upper limit to the brightness temperature by about the ratio of the proton to electron mass or more than a factor of 1000. However, proton synchrotron radiation would require a magnetic field strength more than $10^6$ times stronger than needed for electron synchrotron radiation of the same strength at the same wavelength. 4\) There may be a continuous acceleration of relativistic particles which balances the energy losses due to inverse Compton cooling. Summary and Issues ================== About ten percent of quasars and bright elliptical galaxies are strong radio sources with radio luminosity, $P_r > 10^{23}~W/Hz$ and are thought to be driven by accretion onto a SMBH. The weaker radio sources, $P_r < 10^{23}~W/Hz$ are mostly due to star formation in the host galaxy. The observed properties of radio jets can be interpreted in terms of a highly relativistic outflow from a central engine driven by a SMBH of up to $10^9$ solar masses. But, many questions remain. $\bullet$ Most quasars and AGN are not strong radio sources. Why are only $\sim$ 10% of quasars strong radio sources, although all quasars presumably contain a SMBH to account for their extraordinary optical luminosity. $\bullet$ How do SMBHs generate relativistic jets? $\bullet$ How are the jets confined and shaped as they propagate away from the SMBH? What are the relative roles of velocity shear, hydrodynamic turbulence, shocks, and plasma instabilities in shaping the form and kinematics of relativistic jets? $\bullet$ Why do only some jets produce $\gamma$-rays? How and where are the $\gamma$-rays produced? 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--- abstract: | We derive sharp regularity for viscosity solutions of an inhomogeneous infinity Laplace equation across the free boundary, when the right hand side does not change sign and satisfies a certain growth condition. We prove geometric regularity estimates for solutions and conclude that once the source term is comparable to a homogeneous function, then the free boundary is a porous set and hence, has zero Lebesgue measure. Additionally, we derive a Liouville type theorem. When near the origin the right hand side grows not faster than third degree homogeneous function, we show that if a non-negative viscosity solution vanishes at a point, then it has to vanish everywhere. **Keywords:** Infinity Laplacian, regularity, dead-core problems, porosity.\ **AMS Subject Classification (2010):** 35B09, 35B53, 35B65, 35R35. address: - 'Instituto Federal de Educação, Ciência e Tecnologia do Rio Grande do Sul, Canoas, Brazil' - 'CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal' author: - 'Nicolau M.L. Diehl' - Rafayel Teymurazyan title: | Reaction-diffusion equations\ for the infinity Laplacian --- Introduction {#s1} ============ Reaction-diffusion equations arise naturally when modeling certain phenomena in biological, chemical and physical systems. In this paper we study reaction-diffusion equations for infinity Laplacian, which despite being too degenerate to realistically represent a physical diffusion process, has been studied in the framework of optimization and free boundary problems (see, for example, [@ATU19], [@RST17], [@RT12; @RTU15; @TU17], just to cite a few). More precisely, we establish regularity and geometric properties of solutions of the problem $$\label{1.1} \Delta_\infty u=f(u) \quad \text{in} \quad \Omega,$$ where $\Omega\subset\R^n$, $f\in C(\R_+)$ and $$\label{1.2} 0\le f(\delta t)\leq M\delta^\gamma f(t),$$ with $M>0$, $\gamma\in[0,3)$, $t>0$ bounded, and $\delta>0$ small enough. Additionally, we assume that $$\label{comparison} f \text{ is non-decreasing }.$$ Here, $\R_+$ is the set of non-negative numbers, and the infinity Laplacian is defined as follows: $$\Delta_\infty u(x):=\sum_{i,j=1}^nu_{x_i}u_{x_j}u_{x_ix_j},$$ with $u_{x_i}=\partial u/\partial x_i$. Note that the continuity of $f$ provides that $f(u)$ is bounded once $u$ is bounded. Note also that is quite general in the sense that it needs to hold only for $\delta$ close to zero. For example, it holds for functions that are homogeneous of degree $\gamma$. Condition is needed to guarantee the comparison principle. Solutions of are understood in the viscosity sense according to the following definition: \[d1.1\] A function $u\in C(\Omega)$ is called a viscosity super-solution (resp. sub-solution) of , and written as $\Delta_\infty u\le f(u)$ (resp. $\ge$), if for every $\phi\in C^2(\Omega)$ such that $u-\phi$ has a local minimum at $x_0\in\Omega$, with $\phi(x_0)=u(x_0)$, we have $$\Delta_\infty\phi(x_0)\leq f(\phi(x_0)).\quad \textrm{(resp. $\geq$)}$$ A function $u$ is called a viscosity solution if it is both a viscosity super-solution and a viscosity sub-solution. The infinity Laplace operator is related to the absolutely minimizing Lipschitz extension problem: for a given Lipschitz function on the boundary of a bounded domain, find its extension inside the domain in a way that has the minimal Lipschitz constant, [@A67]. It is known (see [@J93]) that such function $u$ has to be an infinity harmonic one, i.e. $\Delta_\infty u=0$ (in the viscosity sense). The regularity issue of infinity harmonic functions received extensive attention over the years. As was shown in [@ES08], the infinity harmonic functions in the plane are $C^{1,\alpha}$, for a small $\alpha$ (it is conjectured that the optimal regularity is $C^{1,\frac{1}{3}}$). In higher dimensions infinity harmonic functions are known to be everywhere differentiable (see [@ES11]). As for the inhomogeneous case of $\Delta_\infty u=f$, it is known that the Dirichlet problem has a unique viscosity solution, provided $f$ does not change sign (see [@LW08]). Moreover, as was shown in [@L14], for bounded right hand side, the Lipschitz estimate and everywhere differentiability of solutions remain true. The case of $f$ not being bounded away from zero, mainly, when $f=u_+^\gamma$, where $u_+:=\max\left(u,0\right)$ and $\gamma\in[0,3)$ is a constant, was studied in [@ALT16] (dead-core problem). The authors show that for such right hand side (strong absorbtion) across the free boundary $\partial\{u>0\}$ non-negative viscosity solutions are of class $C^{\frac{4}{3-\gamma}}$. The denominator $3-\gamma$ is related to the degree of homogeneity of the operator, which is three, i.e., $\Delta_\infty(Cu)=C^3\Delta_\infty u$, for any constant $C$. Note that for $\gamma\in(0,3)$ this regularity is more than the conjectured $C^{1,\frac{1}{3}}$, i.e., we obtain higher regularity across the free boundary. This result allows to establish Hausdorff dimension estimate for the free boundary $\partial\{u>0\}$ and conclude that it has Lebesgue measure zero. We extend these results for the source term $f$ satisfying . In particular, it includes equations with the right hand side $$f(t)=e^t-1\,\,\,\textrm{ and }\,\,\,f(t)=\log(t^2+1)$$ among others (see Section \[s7\] for more examples). In fact, our results are true in a broader context, when allowing “coefficients” in the right hand side, that is, when in one has $f=f(x,u)$, as long as $f(x,u)$ satisfies as a function of $u$ and is continuous (and bounded) as a function of $x$ (see Section \[s7\]). For simplicity, we restrict ourselves to the case of $f(x,u)=f(u)$. Our strategy is the following: by means of a flattening argument, we show that across the free boundary $\partial\{u>0\}\cap\Omega$ non-negative viscosity solutions of are of class $C^{\frac{4}{3-\gamma}}$, when holds. When the source term is comparable to a homogeneous function of degree $\gamma$, this result is sharp in the sense that across the free boundary non-negative viscosity solutions grow exactly as $r^{\frac{4}{3-\gamma}}$ in the ball of radius $r$. We also analyze the borderline (critical) case, that is, when $\gamma=3$ (which is also the degree of the homogeneity of the infinity Laplacian). Unlike [@ALT16], $f$ is not given explicitly, which makes it harder to construct a barrier function - needed for our analysis. Nevertheless, we are able to show that in this case has a viscosity sub-solution whose gradient has modulus separated from zero. We use this function to build up a suitable barrier to conclude that if a viscosity solution vanishes at a point, it has to vanish everywhere. Our results remain true when the right hand side has some “bounded coefficients” (see Remark \[r7.1\]). For simplicity we restrict ourselves with the right hand side “without coefficients”. The paper is organized as follows: in Section \[s2\], we prove an auxiliary result (flattening solutions) (Lemma \[l2.2\]), which we use in Section \[s3\] to derive the main regularity result (Theorem \[t3.1\]), and as a consequence, in Section \[Liouville\], we obtain Liouville type theorems (Theorem \[t3.2\] and Theorem \[t3.3\]). In Section \[s4\], we prove several geometric measure estimates (Theorem \[t4.1\] (non-degeneracy) and Corollary \[c4.1\] (porosity)), and conclude that the free boundary has Lebesgue measure zero (Corollary \[c4.2\]). In Section \[s5\], when $\gamma=3$, we show that the only non-negative viscosity solution that has zero, is the function that is identically zero (Theorem \[t5.1\]). Finally, in Section \[s7\] we bring some examples of source terms for which our results are true. Preliminaries {#s2} ============= In this section we list some preliminaries, as well as prove an auxiliary lemma for future reference. We start by the comparison principle, the proof of which can be found in [@CIL92; @LW08]. \[l2.1\] Let $u$, $v\in C(\overline{\Omega})$ be such that $$\Delta_\infty u-f(u)\le0, \,\,\,\Delta_\infty v-f(v)\ge0\,\,\textrm{ in }\,\,\Omega$$ in the viscosity sense, and $f$ satisfy or $\inf f>0$. If $u\geq v$ on $\partial\Omega$, then $u\geq v$ in $\Omega$. The comparison principle, together with Perron’s method leads to the following result (for the proof we refer the reader to [@CIL92], for example). In fact, existence of solutions can be shown even without directly applying the comparison principle, as it was done, for example, in [@RTU19 Theorem 3.1]. \[t2.1\] If $\Omega\subset\R^n$ is bounded and $\varphi\in C(\partial\Omega)$ is a non-negative function, then there is a unique and non-negative function $u$ that solves the Dirichlet problem $$\label{2.1} \left\{ \begin{aligned} \Delta_\infty u&=f(u) \text { in } \Omega, \\ u&=\varphi\text { on } \partial\Omega \end{aligned}\right.$$ in the viscosity sense. The following auxiliary lemma is a variant of the flatness improvement technique introduced in [@ALT16; @T13; @T16] to study the regularity properties of solutions of dead-core problems. \[l2.2\] Let $g\in L^\infty(B_1)\cap C(B_1)$ be a non-negative function such that $$\|g\|_\infty\le\max\{1,M\}\sup_{[0,\|u\|_\infty]}f,$$ where $f$ and $M$ are as in . For any given $\mu >0 $ there exists a constant $\kappa _{\mu}=\kappa(\mu,n)>0$ such that if in $B_1$ a continuous functions $v$, which vanishes at the origin and $v\in[0,1]$, satisfies, in viscosity sense, $$\Delta_\infty v - \kappa_\mu^4g(v) = 0$$\ for $0<\kappa\leq \kappa_{\mu}$, then\ $$\sup_{B_{1/2}} v \leq \mu.$$ We argue by contradiction assuming that there exist $\mu^* >0$, $\{v_i\}_{i\in \N}$ and $\{\kappa_i\}_{i\in \N}$ with $v_i(0)=0$, $0\leq v_i\leq1$, in $B_1$ satisfying in viscosity sense to $$\Delta_\infty v_i-\kappa_i^4 g(v_i)= 0$$\ where $\kappa_i = \text{o}(1)$, while $$\label{2.2} \sup_{B_{1/2}} v_i > \mu^*.$$\ By local Lipschitz regularity (see [@L14 Corollary 2], for example), the sequence $\{v_i\}_{i\in \N}$ is pre-compact in the $C^{0,1}(B_{3/4})$. Hence, by Arzelà-Ascoli theorem, $v_i$ converges (up to a subsequence) to a function $v_\infty$ locally uniformly in $B_{2/3}$. Moreover, $v_\infty(0)=0, \; 0\leq v_\infty \leq 1 $ and $\Delta_\infty v_\infty = 0.$ The maximum principle for the infinity harmonic functions then yields $v\equiv 0$, which contradicts to once $i$ is big enough. The following definition is for future reference. \[d1.2\] A function $u$ is called an entire solution, if it is a viscosity solution of in $\R^n$. We close this section by reminding the notion of porosity. \[porosity\] The set $E\subset\R^n$ is called porous with porosity $\sigma$, if there is $R>0$ such that $\forall x\in E$ and $\forall r\in (0,R)$ there exists $y\in\R^n$ such that $$B_{\sigma r}(y)\subset B_{r}(x)\setminus E.$$ A porous set of porosity $\sigma$ has Hausdorff dimension not exceeding $n-c\sigma^n$, where $c>0$ is a constant depending only on dimension. In particular, a porous set has Lebesgue measure zero (see [@Z88], for instance). Regularity across the free boundary {#s3} =================================== In this section we make use of Lemma \[l2.2\] and derive regularity result for viscosity solutions of across the free boundary $\partial\{u>0\}$. \[t3.1\] If $u$ is a non-negative viscosity solution of , where $f$ satisfies , and $x_0\in\partial \{u>0\}\cap\Omega$, then there exists a constant $C>0$, depending only on $\gamma$, $\|u\|_\infty$ and $\dist (x_0 , \partial \Omega)$, such that $$u(x) \leq C |x - x_0|^{\frac{4}{3-\gamma}}$$ for $x \in \{u>0\} $ near $x_0$. The idea is to use an iteration argument and carefully choose sequence of functions that allows to make use of the Lemma \[l2.2\]. Observe that without loss of generality, we may assume that $x_0=0$ and $B_1\subset\Omega$. For $\mu =2^{-\frac{4}{3-\gamma}}$, let now $\kappa_{\mu} >0$ be as in Lemma \[l2.2\]. We then construct the first member of the sequence by setting $$w_0(x):= \tau u(\rho x) \quad \text{in} \quad B_1,$$ where $$\tau:= \min \left\{1, \|u\|_\infty^{-1} \right\}\,\,\,\textrm{ and }\,\,\,\rho :=\kappa_\mu \tau^{-\frac{3-\gamma}{4}}.$$ Note that $\tau^3\rho^4=\kappa_\mu^4\tau^\gamma$, $w_0(0)=0$ and in $w_0\in[0,1]$. Since $u$ is a viscosity solution of , then $$\Delta_\infty w_0(x) - \tau^3\rho^4 f(\tau^{-1}w_0(x))=0$$ or, equivalently, $$\label{3.1} \Delta_\infty w_0(x)-\kappa_\mu^4\tau^{ \gamma} f(\tau^{-1}w_0(x)) = 0.$$ Since $\tau\le1$, then $g(w_0):=\tau^\gamma f(\tau^{-1}w_0)\le f(u(\rho x))\le\displaystyle\sup_{[0,\|u\|_\infty]}f$. From Lemma \[l2.2\], we obtain $$\sup_{B_{1/2}} w_0 \leq 2^{-\frac{4}{3-\gamma}}.$$ For $i\in \N $, we then define $$w_i(x) := 2^{ -\frac{4}{3-\gamma}}w_{i-1}(2^{-1}x).$$ and observe that $w_i(0)=0$, $w_i\in[0,1]$ and $w_i$ satisfies $$\Delta_\infty w_i(x)=\kappa_\mu^4 2^{\frac{4\gamma}{3-\gamma}i}\tau^\gamma f\left(\tau^{-1}2^{-\frac{4}{3-\gamma}i}w_i(x)\right).$$ Using , for $i$ big we estimate $$2^{\frac{4\gamma}{3-\gamma}i}\tau^\gamma f\left(\tau^{-1}2^{-\frac{4}{3-\gamma}i}w_i(x)\right)\le M\tau^\gamma f\left(\tau^{-1}w_i(x)\right)\le M\displaystyle\sup_{[0,\|u\|_\infty]}f.$$ Once again applying Lemma \[l2.2\], one gets $$\sup_{B_{1/2}} w_i \leq 2^{-\frac{4}{3-\gamma}},$$ or in other terms, $$\sup_{B_{1/4}} w_{i-1} \leq 2^{-2 \frac{4}{3-\gamma}}.$$ Continuing this way, for $w_0$ we obtain $$\label{3.2} \sup_{B_{2^{-i}}} w_0 \leq 2^{-i\frac{4}{3-\gamma}}.$$ Next, for a fixed $0<r\leq \frac{\rho}{2}$, by choosing $i\in \N$ such that $$2^{-(i+1)} < \frac{r}{\rho} \leq 2^{-i},$$ and using , we estimate $$\begin{split} \sup_{B_{r}} u &\leq \sup_{B_{\rho2^{-i}}} u = \tau^{-1}\sup_{B_{\rho2^{-i}}} w_0\\ &\leq\tau^{-1} 2^{-i\frac{4}{3-\gamma}} = 2^{\frac{4}{3-\gamma}}\tau^{-1}2^{-(i+1)\frac{4}{3-\gamma}}\\ & \leq \left(\tau^{-1}2\rho^{-1}\right)^{\frac{4}{3-\gamma}}r^{\frac{4}{3-\gamma}}\\ &=Cr^{\frac{4}{3-\gamma}}. \end{split}$$ Geometrically Theorem \[t3.1\] means that no matter how “bad” the function $u$ is in $\{u>0\}$, it touches the free boundary $\partial\{u>0\}$ smoothly. In other words, a non-negative viscosity solution of may have cusp singularities in its positivity set, and yet it is smooth near its free boundary. Liouville type results {#Liouville} ====================== Despite the regularity information being available only across the free boundary, it is enough to derive the following Liouville type theorem. \[t3.2\] If $u$ is an entire solution, holds and $u(x_0)=0$ for a $x_0\in\R^n$ with $$\label{3.3} u(x) = o\left(|x|^{\frac{4}{3-\gamma}}\right),\,\,\,\textrm{ as }\,\,\,|x| \rightarrow \infty,$$ then $u \equiv0$. Without loss of generality we may assume that $x_0=0$. For $k\in\mathbb{N}$, set $$u_k(x):= k^{\frac{-4}{3-\gamma}}u(kx),\quad x\in B_1,$$ where $B_1$ is the ball of radius one centered at the origin. Note that $ u_k(0)=0 $. Since $u$ is an entire solution, for $x\in B_1$ one has $$\Delta_\infty u_k(x)=k^{\frac{-4 \gamma}{3-\gamma}} f\left(k^{\frac{4}{3-\gamma}} u_{k}(x)\right).$$ Note that the right hand side of the last equation satisfies . From Theorem \[t3.1\], we then deduce that if $x_k\in\overline{B}_r$ is such that $$u_k(x_k)=\sup_{\overline{B}_r} u_k,$$ where $r>0$ is small, then in $B_r$ one has $$\label{3.4} \|u_k \|_\infty\to0,\,\,\,\textrm{ as }\,\,\,k\to\infty.$$ In fact, if $|kx_k|$ remains bounded as $k\to\infty$, then applying Theorem \[t3.1\] to $u_k$ we obtain $$\label{3.5} u_k (x_k) \le C_k|x_k|^{\frac{4}{3-\gamma}},$$ where $C_k>0$ and $C_k\to0$. This implies that $u(kx_k)$ remains bounded as $k\to\infty$, and therefore $u_k(x_k)\to0$, as $k\to\infty$, and is true. It remains true also in the case when $|kx_k|\to\infty$, as $k\to\infty$, since then from we get $$u_k(x_k) \leq |kx_k|^{-\frac{4}{3-\gamma}}k^{-\frac{4}{3-\gamma}}\to0,\,\,\,\textrm{ as }\,\,\,k\to\infty.$$ Now, if there exists $y\in\R^n$ such that $u(y)>0$, by choosing $k\in\mathbb{N}$ large enough so $y\in B_{kr}$ and using and , we estimate $$\dfrac{u(y)}{|y|^{\frac{4}{3-\gamma}}} \leq \sup_{B_{kr}} \dfrac{u(x)}{|x|^{\frac{4}{3-\gamma}}}=\sup_{B_r} \dfrac{u_k(x)}{|x|^{\frac{4}{3-\gamma}}}\leq \dfrac{u(y)}{2|y|^{\frac{4}{3-\gamma}}},$$ which is a contradiction. In fact, once the comparison principle holds, the condition can be weakened in the following sense (Theorem \[t3.3\] below). Let $x_0\in\R^n$ and $r>0$ be fixed, and let $u\ge0$ be the unique solution of in $B_r(x_0)$ with $\varphi\equiv\alpha_r>0$ constant, guaranteed by Theorem \[t2.1\]. Note that $u$ is a viscosity sub-solution of $$\label{3.7} \left\{ \begin{aligned} \Delta_\infty v &= \lambda v_+^{\gamma} &&\textrm{ in } B_r(x_0),\\ v&=\alpha_r &&\textrm{ on } \partial B_r(x_0), \end{aligned} \right.$$ where $$\label{3.6} \lambda:=M^{-1}\beta^{-\gamma}f(\beta),$$ and $\beta>\|u\|_\infty$ is a constant big enough so holds. Then the condition can be weakened and substituted by $$\label{3.8} \limsup_{|x| \rightarrow \infty} \dfrac{u(x)}{|x-x_0|^{\frac{4}{3-\gamma}}} < \left( \lambda \frac{(3-\gamma)^4}{64(1+\gamma)} \right)^{\frac{1}{3-\gamma}},$$ where $\lambda$ is defined by , and Theorem \[t3.2\] can be improved to the following variant (see Theorem \[t3.3\] below). The choice of the right hand side of comes from the explicit structure of the unique solution of , which, as observed in [@ALT16], is given by $$\label{3.9} v(x):=\Upsilon\left(|x-x_0|-r + \left(\frac{\alpha_r}{\Upsilon}\right)^{\frac{3-\gamma}{4}} \right)^{\frac{4}{3-\gamma}}_+,$$ where $$\label{3.10} \Upsilon:= \left(\lambda \frac{(3-\gamma)^4}{64(1+\gamma)}\right)^{\frac{1}{3-\gamma}}.$$ \[t3.3\] Let , hold. If $u$ is an entire solution and satisfies , then $u\equiv0$. Once $r>0$ is large enough, then guarantees, with $\Upsilon>0$ defined by , $$\sup_{\partial B_r} \dfrac{u(x)}{r^{\frac{4}{3-\gamma}}} \leq \theta\Upsilon,$$ for some $\theta < 1$. On the other hand, using , one has that the unique solution of , with $\alpha_r=\displaystyle\sup_{\partial B_r(x_0)}u$, given by , is a viscosity sub-solution of . The comparison principle, Lemma \[l2.1\], then implies that $u\le v$ in $B_r(x_0)$. Letting $r\to\infty$, we conclude that $u\equiv0$. \[r3.1\] As can be seen from , the plateau of $v$, i.e., the set $\{v=0\}$, is the ball $\overline{B}_{R}(x_0) $, where $$0<R:=r - \left(\frac{\alpha_r}{\Upsilon}\right)^{\frac{3-\gamma}{4}}.$$ Since $0\le u\le v$, the plateau of $u$ contains the $\overline{B}_R(x_0)$. \[r3.2\] Note that the inequality has to be strict. For example, if $$w(x):=\Upsilon|x-x_0|^{\frac{4}{3-\gamma}}$$ then $$\limsup_{|x| \rightarrow \infty} \dfrac{w(x)}{|x-x_0|^{\frac{4}{3-\gamma}}}=\Upsilon,$$ but $w$ is not identically zero. Non-degeneracy and porosity {#s4} =========================== In this section we show that once $$\label{5.1} f(\delta t)\ge N\delta^\gamma f(t)\ge0,$$ with $N>0$, $\gamma\in[0,3)$, $t>0$ bounded, and $\delta>0$ small enough, then across the free boundary non-negative viscosity solutions of grow exactly as $r^\frac{4}{3-\gamma}$ in the ball $B_r$, for $r>0$ small enough. As a consequence, we conclude that the touching ground surface is a porous set, which implies that it has Hausdorff dimension less than $n$, and so its Lebesgue measure is zero (see [@Z88]). We start by the following non-degeneracy theorem. \[t4.1\] Let hold. Let also $f$ satisfy or $\inf f>0$. If $u$ is a non-negative viscosity solution of , then there exists a universal constant $c>0$, depending only on dimension and $\gamma$, such that $$\sup_{B_{r}(x_0)} u \ge cr^{\frac{4}{3-\gamma}},$$ where $x_0\in\overline{\{u>0\}}\cap\Omega$ and $0<r<\operatorname{dist}(x_0, \partial\Omega)$. Since $u$ is continuous, it is enough to prove the theorem for points $x_0\in\{u>0\}\cap\Omega$. Set $$v(x):=c|x-x_0|^{\frac{4}{3-\gamma}},$$ with a constant $c\in(0,\Upsilon)$, where $\Upsilon>0$ is defined by . Using , direct computation reveals that the choice of $c$ makes $v$ a viscosity super-solution of in $B_r(x_0)$, where $r>0$ is such that $B_r(x_0)\subset\Omega$. If $v\ge u$ on $\partial B_r(x_0)$, then the comparison principle, Lemma \[l2.1\], would imply $v\ge u$ in $B_r(x_0)$, contradicting to the fact that $0=v(x_0)<u(x_0)$. Hence, there is a point $y\in\partial B_r(x_0)$ such that $v(y)<u(y)$. We then estimate $$\sup_{B_r(x_0)} u \geq u(y) \ge v(y) =cr^{\frac{4}{3-\gamma}}.$$ As a consequence, we obtain that the free boundary is a porous set, therefore it has Hausdorff dimension strictly less than $n$, hence its Lebesgue measure is zero. Note that from one has $f(0)=0$, so to use the comparison principle, Lemma \[l2.1\], it is enough to assume that $f$ is non-decreasing, that is, holds. \[c4.1\] Let , and hold. If $u$ is a bounded non-negative viscosity solution of , then $\partial\{u>0\}$ is a porous set. Let $x\in\partial\{u>0\}$ and $y\in\overline{B}_r(x)$ be such that $$u(y)=\sup_{B_{r}(x)}u.$$ By Theorem \[t4.1\], $u(y)\ge cr^{\frac{4}{3-\gamma}}$. On the other hand, Theorem \[t3.1\] provides $$u(y)\le C\left[d(y)\right]^{\frac{4}{3-\gamma}},$$ where $d(y):=\text{dist}\left({y,\partial\{u>0\}}\right)$. Therefore, $$\left(\frac{c}{C}\right)^{\frac{3-\gamma}{4}}r\le d(y).$$ Hence, if $\sigma:=\frac{1}{2}\left(\frac{c}{C}\right)^{\frac{3-\gamma}{4}}$, one has $$B_{2\sigma r}(y)\subset\{u>0\}.$$ We now choose $\xi\in(0,1)$ such that for the point $z:=\xi y+(1-\xi)x$ we have $|y-z|=\sigma r$. Then $$B_{\sigma r}(z)\subset B_{{2\sigma}r}(y)\cap B_r(x).$$ Moreover, we have $$B_{2\sigma r}(y)\cap B_r(x)\subset \{u>0\},$$ which together with the previous inclusion implies $$B_{\sigma r}(z)\subset B_{2\sigma r}(y)\cap B_r(x)\subset B_r(x)\setminus\partial\{u>0\},$$ that is, the set $\partial\{u>0\}$ is porous with porosity $\sigma$. \[c4.2\] If , , hold, and $u$ is a viscosity solution of , then Lebesgue measure of the set $\partial\{u>0\}$ is zero. The borderline case {#s5} =================== Although, in general, one cannot expect more than $C^{1,\alpha}$ regularity for viscosity solutions of , Theorem \[t3.1\] provides higher and higher regularity across the free boundary, as $\gamma\in[0,3)$ gets closer to 3. In this section we analyze the limit case of $\gamma=3$. The scaling property of the operator plays an essential role here, as $\gamma=3$ is also the degree of homogeneity of the infinity Laplacian, meaning that $\Delta_\infty(Cu)=C^3\Delta_\infty u$, for any constant $C$. Observe that Theorem \[t3.1\] cannot be applied directly, since the estimates deteriorate as $\gamma\to3$. Thus, in this section is substituted by $$\label{6.1} 0\le f(\delta t)\le M\delta^3f(t),$$ with $M>0$, $t>0$ bounded and $\delta>0$ small. Our first observation states as follows. \[l5.1\] If $u$ is a non-negative viscosity solution of , where $f$ satisfies , then its every zero is of infinite order. This is a consequence of Theorem \[t3.1\]. To see that it is enough to rewrite , for $\gamma=3$, as $$f(\delta t)\leq M_\delta\delta^{3-\beta}f(t),$$ where $M_\delta:=M\delta^\beta$ and $\beta>0$. An application of Theorem \[t3.1\] with $M=M_\delta$ leads to the conclusion that if $u(z)=0$ for $z\in\Omega$, then $D^nu(z)=0$, $\forall n\in\mathbb{N}$. Furthermore, we show that if a non-negative viscosity solution of vanishes at a point, then it must vanish everywhere. For $f\equiv0$ this follows from the Harnack inequality. The particular case, when $f$ is homogeneous of degree three, that is, $f(t)=M t^3$, was studied in [@ALT16], where by means of a suitable barrier function, was concluded that if non-negative viscosity solution vanishes in an inner point, then it has to vanish everywhere. Unlike [@ALT16], our function $f$ is not given explicitly, which makes the construction of a suitable barrier function more complicated. Observe that once holds, then $f(0)=0$, hence $\inf f=0$, so to use the comparison principle, one needs to assume that $f$ is non-decreasing. \[t5.1\] Let $u$ be a non-negative viscosity solution of , where $f$ satisfies and . If $\{u=0\}\cap\Omega\neq\emptyset$, then $u\equiv0$. We argue by contradiction, assuming that there is $x\in\Omega$ such that $u(x)=0$, but $u(y)>0$ for a point $y\in\Omega$. Without loss of generality we may assume that $$r:=\dist\left(y,\{u=0\}\right)<\frac{1}{10}\dist\left(y,\partial\Omega\right).$$ We aim to construct a sub-solution of which stays below $u$ on $\partial B_r(y)$. Let $w$ be an infinity sub-harmonic function in $B_r(y)$ such that $|\nabla w|\ge\eta$ for $\eta\ge0$ constant to be chosen later. Such function can be built up as a limit, as $p\to\infty$, of $p$-super-harmonic functions with modulus of gradient separated from zero by $\eta$. We refer the reader for details to [@J93]. Now if $g$ is a smooth function and $v=g(w)$, direct computation reveals that $$\Delta_{\infty} v=\left[g'(w)\right]^3\Delta_{\infty} w+\left[g^{\prime}(w)\right]^{2} g^{\prime \prime}(w)|\nabla w|^{4}.$$ Thus, for $g(t)=e^t+t$, $$\label{5.2} \Delta_{\infty} v\ge\left[g^{\prime}(w)\right]^{2} g^{\prime \prime}(w)|\nabla w|^{4},$$ since $g^\prime\ge1$ and $\Delta_\infty w\ge0$. Also $g^{\prime\prime}\ge e^{-\|w\|_\infty}>0$, and yields (recall that $|\nabla w|\ge\eta$) $$\label{5.3} \Delta_\infty v\ge\mu\eta,$$ where $\mu:=e^{-\|w\|_\infty}>0$. Choosing $$\eta>\frac{M}{\mu}\max_{[0,\|v\|_\infty]}f,$$ from we obtain $$\Delta_\infty v-Mf(v)\ge\Delta_\infty v-\mu\eta\ge0,$$ i.e., $v$ is a sub-solution of . The latter together with gives, for any small constant $\delta>0$, $$\Delta_\infty\left(\delta v\right)-f\left(\delta v\right)\ge\delta^3\left(\Delta_\infty v-Mf(v)\right)\ge0,$$ that is, the function $\delta v$ is also a sub-solution of . We choose $\delta>0$ small enough to guarantee $$\delta v(x)\le u(x),\,\,\,x\in\partial B_r(y),$$ and by the comparison principle, Lemma \[l2.1\], $$\label{5.4} \delta v(x)\le u(x),\,\,\,x\in B_r(y).$$ Observe, that writing as $$f(\delta t)\le M\delta\delta^2f(t),$$ and applying Theorem \[t3.1\] with $\widetilde{M}=M\delta$, we arrive at $$\label{5.5} \sup_{B_d(z)}u\le Cd^4,$$ where $z\in\partial B_r(y)\cap\partial\{u>0\}$, and $d>0$ is small. In fact, we choose $0<d<\left(\frac{\delta\eta}{4C}\right)^\frac{1}{3}$. Using the fact that $|\nabla v|=g^\prime|\nabla w|\ge\eta$, recalling and and the choice of $d$, we estimate $$\delta\eta d\le\sup_{B_d(z)}\delta|v(x)-v(z)|\le\sup_{B_d(z)}\delta v\le\sup_{B_d(z)}u\le Cd^4\le\frac{1}{4}\delta\eta d,$$ which is a contradiction. Examples and beyond {#s7} =================== We close the paper with some examples of the source term, for which our results are valid. We start with the following remark. \[r7.1\] The results in this paper remain true, without changes in the proofs, when in the right hand side has continuous (bounded) coefficients, i.e., $$\Delta_\infty u=f(x,u),$$ as long as $f(x,u)$ satisfies (and , when needed) as a function of $u$. Note that $f\equiv0$ satisfies , , , therefore, our results resemble those for the non-negative infinity harmonic functions. Also, the results are true when in one has $f(t)=t^\gamma$, $t\ge0$, $\gamma\in[0,3)$ (studied in [@ALT16]). In view of Remark \[r7.1\], we can allow some coefficients too, as long as they remain bounded, that is, functions of the type $f(x,t)=g(x)t^\gamma$, where $g$ is a non-negative continuous, bounded function. When in the last example $\gamma=0$, i.e., the source term depends only on $x$, $f(x,t)=g(x)$, across the free boundary one has $C^{\frac{4}{3}}$ regularity, once $g(x)\ge0$ is continuous and bounded. Furthermore, the non-degeneracy result is true, and hence the free boundary is a porous set and has zero Lebesgue measure. An example of the source term, not constructed via power functions is $f(t):=e^t-1$, which satisfies with $M=1$ and $\gamma=0$, since $e^{\delta t}<e^t$, for $\delta>0$ small and $t>0$. Hence, applying Theorem \[t3.1\] to $$\Delta_\infty u=e^u-1,$$ we conclude that non-negative viscosity solutions of the above equation are $C^\frac{4}{3}$ near the free boundary $\partial\{u>0\}$. Moreover, if $u$ is an entire solution of the last equation, which vanishes at a point and $$u(x)=o\left(|x|^\frac{4}{3}\right),\,\,\,\textrm{ as }\,\,\,|x|\rightarrow\infty,$$ then Theorem \[t3.2\] implies that it has to be identically zero. Same conclusion can be made when in the source term is $f(t):=\log (t^2+1)$. Of course, Theorem \[t3.1\] can be applied also for any linear combination (with continuous, bounded coefficients) of the above source terms. In fact, Theorem \[t3.1\] is true for any non-negative continuous function, which is non-decreasing around zero and vanishes at the origin. We point out that (as well as and ) is required to hold only around zero, so Theorem \[t3.1\] is true also for source terms that are any of the above examples around zero and can be anything outside, while remaining non-negative and continuous (and in case of Theorem \[t4.1\] and its consequences, also non-decreasing). We finish with an application of Theorem \[t5.1\]. Let $u$ be a non-negative viscosity solution of $$\label{7.1} \Delta_\infty u=\log\left(1+u^3\right).$$ Since $f(t)=\log(1+t^3)$ satisfies with $M=1$ and $\gamma=0$, Theorem \[t3.2\] implies $C^{\frac{4}{3}}$ regularity of $u$ near the touching ground. On the other hand, the function $f(t)$ can be written as $$\log\left(1+t^3\right)=t^3\frac{\log\left(1+t^3\right)}{t^3}:=t^3g(t).$$ Set $g(0)=1$. Then $g$ is continuous, bounded ($0\le g\le1$) function, and the non-decreasing function $f(t)=g(t)t^3$ satisfies with $M=1$. Applying Theorem \[t5.1\], we conclude that if $u$ is a non-negative viscosity solution of , which is zero at a point, then it must be identically zero. [**Acknowledgments.**]{} NMLD was partially supported by Instituto Federal de Educação, Ciência e Tecnologia do Rio Grande do Sul. NMLD thanks the Analysis group at Centre for Mathematics of the University of Coimbra (CMUC) for fostering a pleasant and productive scientific atmosphere during his postdoctoral program. RT was partially supported by FCT – Fundação para a Ciência e a Tecnologia, I.P., through projects PTDC/MAT-PUR/28686/2017 and UTAP-EXPL/MAT/0017/2017, and by CMUC – UID/MAT/00324/2013, funded by the Portuguese government through FCT and co-funded by the European Regional Development Fund through Partnership Agreement PT2020. [99]{} G. 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--- abstract: 'Magnetic Resonance (MR) Imaging and Computed Tomography (CT) are the primary diagnostic imaging modalities quite frequently used for surgical planning and analysis. A general problem with medical imaging is that the acquisition process is quite expensive and time-consuming. Deep learning techniques like generative adversarial networks (GANs) can help us to leverage the possibility of an image to image translation between multiple imaging modalities, which in turn helps in saving time and cost. These techniques will help to conduct surgical planning under CT with the feedback of MRI information. While previous studies have shown paired and unpaired image synthesis from MR to CT, image synthesis from CT to MR still remains a challenge, since it involves the addition of extra tissue information. In this manuscript, we have implemented two different variations of Generative Adversarial Networks exploiting the cycling consistency and structural similarity between both CT and MR image modalities on a pelvis dataset, thus facilitating a bidirectional exchange of content and style between these image modalities. The proposed GANs translate the input medical images by different mechanisms, and hence generated images not only appears realistic but also performs well across various comparison metrics, and these images have also been cross verified with a radiologist. The radiologist verification has shown that slight variations in generated MR and CT images may not be exactly the same as their true counterpart but it can be used for medical purposes.' author: - Vismay Agrawal - Avinash Kori - Vikas Kumar Anand - Ganapathy Krishnamurthi bibliography: - 'report.bib' title: Structurally aware bidirectional unpaired image to image translation between CT and MR --- Introduction ============ Computer Tomography (CT) and Magnetic Resonance (MR) Imaging are the two most frequently used imaging modalities for medical diagnosis and surgical planning. While CT provides contrast information for bone studies, MR imaging provides information for soft tissue contrast. In this manuscript we propose a Generative Adversarial Networks (GANs) [@goodfellow2014generative] based technique to convert CT images to realistic looking MR images. As the physics of acquisition in the case of MR and CT differ significantly, developing techniques for direct transformation of images from MR space to CT space is not a trivial problem. But recent advancements in data-driven modeling, specifically GANs have provided realistic looking results on image to image translation by separating content and style from an image. We make use of similar techniques to separate style and content from MR and CT images and interchange them. To overcome the challenges in obtaining paired data, we conducted our experiments along the lines of cycleGAN [@zhu2017unpaired] based approach, where the models were trained on unpaired data to maintain the cyclic consistency between different imaging modalities. The use of unpaired data drastically increases the amount of input-output pairs used in translational networks. It also assures that the network is not overfitting itself on the given data but is learning the visual properties of different image modalities. We have also played with loss functions and have shown that using structural similarity in loss function can generate realistically looking images which can be used in looking at bone structure and fractures in a better way. These findings were also cross verified by a radiologist. Previously, Conditional GANs [@isola2017image] have been used with U-net [@Ronneberger2015UNetCN] based architecture as a generator and convolutional PatchGAN classifier as a discriminator to learn the mapping from input images to output images. In some instances, CNNs and GANs have also been implemented in MR to CT translation [@kaiser2019mri] [@nie2017medical]. cycleGANs have been used to perform a bidirectional unpaired image to image translation between Cardiac CT and MR data [@chartsias2017adversarial] and it has been shown that the generated data can be used to increase the accuracy of the network in segmentation tasks. Wolterink et al. [@wolterink2017deep] has shown that cycleGAN trained on unpaired data was able to outperform the model trained on paired data for MR to CT translation. Jin et al. [@jin2019deep] study was focused on CT to MR translation which consisted of dual cycle-consistency loss using paired and unpaired training data, but due to the use of paired data in the training process, the method can’t be used in the situation with a lack of paired data. With all these, we can see that although many research work has been done on brain and cardiac medical scans, there aren’t many successful works with pelvis scans. Also, we have observed that most of the research was focused on obtaining CT images from their MR counterpart, with little or no details about bi-directional translations. Materials and Methods ===================== Data ---- Our data consisted of human pelvis unpaired (taken from multiple patients) MR and CT 3D volumes. T2 weighted images were obtained from local hospital (Apollo Speciality Hospital, Chennai, TamilNadu, India) with appropriate ethical clearance which were acquired using a 1.5 Tesla system (Achieva, Philips Medical System), while CT volumes we extracted from Liver Tumour Segmentation (LiTS) challenge database [@LITS], under the guidance of expert radiologists. In total, we have about 55 MR and CT volumes, average with about 30 slices in MR and about 80 slices in CT volumes. We extracted axial slices from these MR and CT volumes to train the network. The data split used in training and testing is shown in table \[Data\]. Training Set Test Set Volume Dimension ----------- -------------- ---------- ------------------ -- CT Volume 50 5 (512, 512, 80) MR Volume 50 5 (768, 768, 30) : Dataset Details[]{data-label="Data"} Pre-processing of data ---------------------- As the number of slices was different in MR and CT, to maintain an equal number of slices we resampled all volumes to have 80 slices on an average. Each image slice was then normalized using the min-max normalization method. The obtained image was resized to the dimension (286, 286) using bicubic interpolation and subsequently, it was cropped to a dimension (256, 256) from a random location. Data augmentation was performed by flipping and rotating the input image at random. Transformer Network ------------------- Our pipeline consisted of two identical Generator networks for CT to MR and MR to CT image to image translation, i.e. $G_{CT}: I_{MR} \rightarrow{} I_{CT}$ and $G_{MR}: I_{CT} \rightarrow{} I_{MR} $. During training, one of the networks performs the required translation and the other one tries to undo the same. We have used ResNet [@he2016deep] as a generator network with 9 resblocks. For each generator, we have a corresponding discriminator network, $D_{CT}$ and $D_{MR}$, which aims to discriminate between translated and original, CT and MR images respectively. The pipeline of our network is given in Fig. \[pipeline\] Loss function ------------- In the two proposed networks, the first one (cycleGAN) uses loss function as proposed in [@Yang2018UnpairedCycleGAN] and in the second network (cycleGAN-SSIM) we have included the structural similarity loss function [@wang2004image] on top of all the other losses. The loss functions can be divided into multiple parts as stated below. ### GAN Loss $$\begin{aligned} \mathcal{L}_{GAN} (D_{MR}, G_{MR}, I_{CT})= \ & \mathbb{E}_{I_{CT} \sim Data_{CT}}[(D_{MR}(G_{MR}(I_{CT})) - 1)^2] \\ & + \mathbb{E}_{I_{MR} \sim Data_{MR}}[(D_{CT}(G_{CT}(I_{MR})) - 1)^2]\end{aligned}$$ The GAN loss aims to minimize the possibility of the translated image being recognized as a fake image by the discriminator. ### Cycle Loss $$\begin{aligned} \mathcal{L}_{cycle} = \ & \mathbb{E}_{I_{MR} \sim Data_{MR}}[||G_{MR}(G_{CT}(I_{MR})) - I_{MR})||_1] \\ & + \mathbb{E}_{I_{CT} \sim Data_{CT}}[||G_{CT}(G_{MR}(I_{CT})) - I_{CT})||_1]\end{aligned}$$ The cycle loss function ensures the cyclic consistency between the input and generated images. ### Identity Loss $$\begin{aligned} \mathcal{L}_{identity} = \ & \mathbb{E}_{I_{CT} \sim Data_{CT}}[||(G_{CT}(I_{CT})) - I_{CT})||_1] \\ & + \mathbb{E}_{I_{MR} \sim Data_{MR}}[||(G_{MR}(I_{MR})) - I_{MR})||_1]\end{aligned}$$ The root of the identity loss lies in the fact that the generator should be aware of the input image modality, and the generator shouldn’t translate the input image if it corresponds to same imaging modality as the output of the generator. ### Structural Similarity (SSIM) Loss $$\begin{aligned} \mu_{MR/CT} &= \mathbb{E}_{I_{MR/CT} \sim Data_{MR/CT}}[G_{CT/MR}(I_{MR/CT})] \\ \sigma_{MR/CT} &= \mathbb{E}_{I_{MR/CT} \sim Data_{MR/CT}}[(G_{CT/MR}(I_{MR/CT}))^2]\\ % \mu_{CT} &= \mathbb{E}_{I_{CT} \sim Data_{CT}}[G_{MR}(I_{CT})] \\ % \sigma_{CT} &= \mathbb{E}_{I_{CT} \sim Data_{CT}}[(G_{MR}(I_{CT}))^2]\\ \sigma_{MR, CT} &= \mathbb{E}_{I_{CT} \sim Data_{CT}, I_{MR} \sim Data_{MR}}[G_{MR}(I_{CT})G_{CT}(I_{MR})]\\ \mathcal{L}_{SSIM} &= 1 - \ \frac{(\mu_{CT}\mu_{MR} + c_1)(2\sigma_{MR,CT} + c_2)}{(\mu_{MR}^2 + \mu_{CT}^2 + c_1)(\sigma_{MR}^2 + \sigma_{CT}^2 + c_2)}\end{aligned}$$ Here, we chose $c_1$ and $c_2$ to be 0.0001 and 0.009 respectively. SSIM helps in maintaining the structural integrity between inter-domain images. ### Generator Net Loss $$\begin{aligned} \mathcal{L}_{G_{net}} = \ & \mathcal{L}_{GAN} + \lambda_{cyc} * \mathcal{L}_{cycle} + \lambda_{id} * \mathcal{L}_{identity} + \lambda_{ssim} * \mathcal{L}_{ssim} \end{aligned}$$ Net generator loss is defined as a superposition of all the aforementioned losses namely, GAN loss, cycle loss, identity and SSIM loss. ### Discriminator Loss $$\begin{aligned} \mathcal{L}_{dis_{MR/CT}} = &\ \mathbb{E}_{I_{CT/MR} \sim Data_{CT/MR}}[\ (D_{MR/CT}(I_{CT/MR}) - 1)^2\ ] \\ &+ \mathbb{E}_{I_{CT/MR} \sim Data_{CT/MR}}[\ (D_{MR/CT}(G_{MR/CT}(I_{CT/MR}))^2\ ] % \\ % \mathcal{L}_{dis_{CT}} = &\ \mathbb{E}_{I_{MR} \sim Data_{MR}}[\ (D_{CT}(I_{MR}) - 1)^2\ ] \\ % &+ \mathbb{E}_{I_{MR} \sim Data_{MR}}[\ (D_{CT}(G_{CT}(I_{MR}))^2\ ]\end{aligned}$$ As our aim is to predict label 1 for a real imaging modality and label 0 for a translated one, the loss function is the distance between the prediction and the real label. Comparison Metrics ------------------ Quantitative analysis of the generated images was performed using FID index (Fréchet Inception Distance) [@gan_metric], MI (mutual information) [@gan_metric] and SSIM index [@gan_metric]. ### FID In case of FID, pretrained densenet121 ($FID(.)$) is considered as base network, and feature embedding for real MRI/CT and corresponding fake MRI/CT is obtained, the distance between these two embeddings are observed. If the distribution of generator matched with the distribution of real image the distance between feature embeddings will reduce and FID score will increase. $$\begin{aligned} {FID}_{CT} = &\ \mathbb{E}_{I_{CT} \sim Data_{CT}, I_{MR} \sim Data_{MR}}<FID(G_{CT}(I_{MR})), FID(I_{CT})>\end{aligned}$$ where $ <.> $ indicates inner product between two vectors. FID for MR can be calculated similarly. ### SSIM To quantitatively measure the structural integrity of the generated images we made use structural similarity as a metric. SSIM between generated CT/MR and input MR/CT provides us with an information how structurally close the given pair of generated MR and CT are. $$\begin{aligned} {SSIM} &= \frac{(\mu_{CT}\mu_{MR} + c_1)(2\sigma_{MR,CT} + c_2)}{(\mu_{MR}^2 + \mu_{CT}^2 + c_1)(\sigma_{MR}^2 + \sigma_{CT}^2 + c_2)}\end{aligned}$$ ### MI To measure the textural similarity between generated MR/CT with actual MR/CT we made use of mutual information as a metric. $$\label{MI} MI(G, R) = \mathbf{E}(P_{GR}(G,R)) \times log(\frac{P_{GR}(G,R)}{P_{G}(G)P_{R}(R)})$$ where $P_{GR}(G, R)$ denotes joint distribution between generated and real distribution, $\mathbf{E}$ denotes expectation value and $P_G(G), P_R(R)$ denotes marginals. ### Pixel wise accuracy (pixacc) To gauge the performance of transformation network in generating recovered MR/ CT using real MR/ CT images, cosine similarity between them was used as a metric, as stated below. $$\label{pixacc} pixacc_{MR/CT} = \mathbb{E}_{I_{CT} \sim Data_{CT}, I_{MR} \sim Data_{MR}}\frac{I_{MR/CT} \cdot G_{MR/CT}(I_{CT/MR})} {||I_{MR/CT}||\cdot ||G_{MR/CT}(I_{CT/MR})||}$$ Results and Discussion ====================== The performance metrics were being calculated for 200 slices and the average results are shown in Table \[scores\]. The loss plots between cycleGAN and cycleGAN-SSIM, given in Fig \[loss\] clearly show the lower convergence in case of using structural similarity loss for both generator and discriminator, we can conclude that the network was able to exploit the structural consistencies between different image modalities to boost its learning rate. --------------- ------- ------- ------- -------- ------- ------- ------- -------- Model FID SSIM MI pixacc FID SSIM MI pixacc cycleGAN 0.193 0.408 0.273 0.986 0.177 0.562 0.332 0.994 cycleGAN-SSIM 0.200 0.416 0.290 0.988 0.184 0.562 0.336 0.995 --------------- ------- ------- ------- -------- ------- ------- ------- -------- : Mean comparison metrics []{data-label="scores"} We can see that although all the results were higher in the case of the network trained with SSIM loss function, there aren’t any significant differences in their scores. The use of structural similarity loss function was preserving information about the edges of the generated images, especially in the case of CT to MR translation (Fig \[CT2MRexamples\]), wherein visual inspection shows enhanced bone details, and relatively higher structural similarity and mutual information scores for cycleGAN-SSIM model. The MR to CT translation (Fig \[MR2CTexampls\]) had similar SSIM scores for both the models, this denotes that cycleGAN was learning to structural information in CT images. In such a case addition of the SSIM loss function was just acting as a catalyst to increase the learning rate of the model. High pixacc scores reflect the fact that both the models generate realistically looking images and such synthesised images are suitable for use in medical diagnosis. MR to CT translation had higher pixacc scores than CT to MR translation because MR has more tissue contrasts and such information is missing in CT, thus it isn’t possible for the network to completely recover the lost information in latter transformation. According to the reviews by the radiologist, in comparison to cycleGAN-SSIM model, the images generated by cycleGAN were more similar to real and CT MR images. Radiologist pointed out that MR images generated by cycleGAN model would be better for routine diagnosis, whereas images generated by cycleGAN-SSIM model would be better for special cases of diagnosis. In cycleGAN-SSIM model, due to preservation in textural details of input CT image content, its MR counterpart contained contrast feature of MR image with blended with structural information of CT images. Such images can be used in parallel with CT images for diagnosis of bone fractures and also in imaging the lung to identify tiny nodules and calcification. \ \ \ \ \ \ Conclusion and Future work ========================== In this work, we have shown that the cycleGAN can be used in medical image translation tasks. Moreover, the use of structural similarity loss with cycleGAN can boost the learning rate of GAN, images generated by such model are more structurally sound and hence gives a better diagnosis of fractures. In future, we plan to conduct similar experiments with entire 3D volumes, i.e 3D style and content separation from MRI and CT volumes. We would also like to extend this work to identify tiny nodule and calcification in the lungs, as pointed by the radiologist.

--- abstract: 'C4 plants, such as maize, concentrate carbon dioxide in a specialized compartment surrounding the veins of their leaves to improve the efficiency of carbon dioxide assimilation. Nonlinear relationships between carbon dioxide and oxygen levels and reaction rates are key to their physiology but cannot be handled with standard techniques of constraint-based metabolic modeling. We demonstrate that incorporating these relationships as constraints on reaction rates and solving the resulting nonlinear optimization problem yields realistic predictions of the response of C4 systems to environmental and biochemical perturbations. Using a new genome-scale reconstruction of maize metabolism, we build an 18000-reaction, nonlinearly constrained model describing mesophyll and bundle sheath cells in 15 segments of the developing maize leaf, interacting via metabolite exchange, and use RNA-seq and enzyme activity measurements to predict spatial variation in metabolic state by a novel method that optimizes correlation between fluxes and expression data. Though such correlations are known to be weak in general, here the predicted fluxes achieve high correlation with the data, successfully capture the experimentally observed base-to-tip transition between carbon-importing tissue and carbon-exporting tissue, and include a nonzero growth rate, in contrast to prior results from similar methods in other systems. We suggest that developmental gradients may be particularly suited to the inference of metabolic fluxes from expression data.' address: 'Laboratory of Atomic and Solid State Physics/Institute of Biotechnology Cornell UniversityIthaca, NY' author: - Eli Bogart - 'Christopher R. Myers' bibliography: - 'multiscale\_c4.bib' title: 'Multiscale metabolic modeling of C4 plants: connecting nonlinear genome-scale models to leaf-scale metabolism in developing maize leaves' --- Introduction {#introduction .unnumbered} ============ C4 photosynthesis is an anatomical and biochemical system which improves the efficiency of carbon dioxide assimilation in plant leaves by restricting the carbon-fixing enzyme Rubisco to specialized bundle sheath compartments surrounding the veins, where a high-CO~2~ environment is maintained that favors CO~2~ over $\text{O}_2$ in their competition for Rubisco active sites, thus suppressing photorespiration [@vonCaemmerer2003]. C4 plants are geographically and phylogenetically diverse, and represent the descendants of over 60 independent evolutionary origins of the system [@Sage2011]. They include major crop plants such as maize, sugarcane and sorghum as well as many weeds and, relative to non-C4 (C3) plants, typically show improved nitrogen and water use efficiencies [@Brown1999]. The agricultural and ecological significance of the C4 system and its remarkable convergent evolution have made it the object of intense study. The core biochemical pathways are now generally understood [@Kanai1999] but many areas of active research remain, including the genetic regulation of the C4 system [@Hibberd2010], the importance of particular components of the system to its function (e.g., [@Studer2014]), the significance of inter-specific variations in C4 biochemistry [@Furbank2011], details of the process of C4 evolution, [@Sage2004; @Christin2009; @Griffiths2013; @Heckmann2013; @Way2014] and the prospect of increasing yields of C3 crops by artificially introducing C4 functionality to those species [@Covshoff2012; @vonCaemmerer2012]. Computational and mathematical modeling is a proven approach to gaining insight into C4 photosynthesis and will play an important role in addressing these questions. High-level nonlinear models of photosynthetic physiology [@VonCaemmerer2000] relating enzyme activities, light and atmospheric CO~2~ levels, and the rates of CO~2~ assimilation by leaves have been widely applied to infer biochemical properties from macroscopic experiments and explore the responses of C4 plants under varying conditions. More recently, detailed kinetic models have been used to explore the optimal allocation of resources to enzymes in an NADP-ME type C4 plant [@Wang2014] and the relationship between the three decarboxylation types [@Wang2014a]. Large-scale constraint-based metabolic models offer particular advantages for the investigation of connections between the C4 system and a plant’s metabolism more broadly (for example, partitioning of nonphotosynthetic functions between mesophyll and bundle sheath, or the evolutionary recruitment of nonphotosynthetic reactions into the C4 cycle) and for interpreting high-throughput experimental data from C4 systems. Photosynthesis is difficult to describe, however, using the standard approach of flux balance analysis (FBA), which predicts reaction rates $v_1, v_2, \ldots v_N$ in a metabolic network by optimizing a biologically relevant function of the rates subject to the requirement that the system reach an internal steady state, $$\label{FBA} \begin{aligned} & \max_{(v_1, v_2, \ldots, v_N) \in \mathbb R^N} & & f(\mathbf v) \\ & \text{s.t.} & & S\cdot\mathbf v = \mathbf 0,\\ \end{aligned}$$ where the stoichiometry matrix $S$ is determined by the network structure [@Orth2010]. The relationship between the rate $v_c$ of carbon fixation by Rubisco and the rate $v_o$ of the Rubisco oxygenase reaction depends nonlinearly on the ratio of the local oxygen and carbon dioxide concentrations (here expressed as equivalent partial pressures), $$\label{vo_vc_ratio} \frac{v_o}{v_c} = \frac {1}{S_R} \frac{P_{O2}}{P_{CO2}}$$ where $S_R$ is the specificity of Rubisco for CO~2~ over O~2~. In the C4 case, the CO~2~ level in the bundle sheath compartment is itself a function of the rates of the reactions of the C4 carbon concentration system and the rate of diffusion of CO~2~ back to the mesophyll. With the addition of (\[vo\_vc\_ratio\]), the problem (\[FBA\]) becomes nonlinear and cannot be solved with typical FBA tools; instead (as the problem is also nonconvex [@Boyd2004]), a general-purpose nonlinear programming algorithm is required to numerically solve it. Prior constraint-based models of plant metabolism have typically ignored the constraint (\[vo\_vc\_ratio\]) or assumed the oxygen and carbon dioxide levels $P_{O2}$ and $P_{CO2}$ are known and fixed $v_o/v_c$ accordingly [@deOliveiraDalMolin2010; @Saha2011]. While this approach is suitable for mature C4 leaves under many conditions, where $v_o/v_c$ is approximately zero, it may break down in some of the most important targets for simulation: developing tissue, mutants, and C3-C4 intermediate species, where $P_{CO2}$ in the bundle sheath compartment is not necessarily high. In other recent work, a high-level physiological model was used to determine $v_o$, $v_c$, and other key reaction rates given a few parameters, which were then fixed in order to solve eq. (\[FBA\])[@Heckmann2013] . This method yields realistic solutions, but its application is limited by the lack of a way to set the necessary phenomenological parameters (e.g., the maximum rate of PEP regeneration in the C4 cycle) based on lower-level, per-gene data (e.g., from transcriptomics or experiments on single-gene mutants). Here, we treat the problem in a more general way by incorporating the nonlinear constraint (\[vo\_vc\_ratio\]) directly into the optimization problem (\[FBA\]) and solving the resulting nonlinear program numerically with the IPOPT package [@Wachter2006], using a new computational interface that we have developed, which allows rapid, interactive development of nonlinearly-constrained FBA problems from metabolic models specified in SBML format [@Hucka2003]. Using a new genome-scale reconstruction of the metabolic network of *Zea mays*, developed with particular attention to photosynthesis and related processes, we confirm that this approach can reproduce the nonlinear responses of well-validated, high-level physiological models of C4 photosynthesis [@VonCaemmerer2000], while also providing detailed predictions of fluxes throughout the network. Finally, we combine the results of enzyme assay measurements and multiple RNA-seq experiments and apply a new method to infer the metabolic state at points along a developing maize leaf (Fig. \[schematicfig\]a) using a model of mesophyll and bundle sheath tissue in fifteen segments of the leaf, interacting through vascular transport of sucrose, glycine, and glutathione. We compare our results to radiolabeling experiments. ![[**Maize plant and models.**]{} (a) Nine-day-old maize plant (image from [@Li2010]). (b) Organization of the two-cell-type metabolic model, showing compartmentalization and exchanges across mesophyll and bundle sheath cell boundaries. (c) Combined 121-compartment model for leaf 3 at the developmental stage shown in (a). Fifteen identical copies of the model shown in (b) represent 1-cm segments from base to tip.[]{data-label="schematicfig"}](Fig1.pdf){width="125.00000%"} Results {#results .unnumbered} ======= Metabolic reconstruction of *Zea mays* {#metabolic-reconstruction-of-zea-mays .unnumbered} -------------------------------------- A novel genome-scale metabolic model was generated from version 4.0 of the CornCyc metabolic pathway database [@CornCyc] and is presented in two forms. The comprehensive reconstruction involves [2720]{} reactions among [2725]{} chemical species, and incorporates CornCyc predictions for the function of 5204 maize genes, with 2064 reactions associated with at least one gene. A high-confidence subset of the model, excluding many reactions not associated with manually curated pathways or lacking computationally predicted gene assignments as well as all reactions which could not achieve nonzero flux in FBA calculations, involves [635]{} reactions among [603]{} species, with 469 reactions associated with a total of 2140 genes. Both the comprehensive and high-confidence models can simulate the production of all major maize biomass constituents (including amino acids, nucleic acids, fatty acids and lipids, cellulose and hemicellulose, starch, other carbohydrates, and lignins, as well as chlorophyll) under either heterotrophic or photoautotrophic conditions and include chloroplast, mitochondrion, and peroxisome compartments, with key reactions of photosynthesis (including a detailed representation of the light reactions), photorespiration, the NADP-ME C4 cycle, and mitochondrial respiration localized appropriately. Gene associations for reactions present in more than one subcellular compartment have been refined based on the results of subcellular proteomics experiments and computational predictions (as collected by the Plant Proteomics Database, [@Sun2009]) to assign genes to reactions in appropriate compartments. A model for interacting mesophyll and bundle sheath tissue in the leaf was created by combining two copies of the high-confidence model, with transport reactions to represent oxygen and CO~2~ diffusion and metabolite transport through the plasmodesmata, and restricting exchange reactions appropriately (nutrient uptake from the vascular system to the bundle sheath, and gas exchange with the intercellular airspace to the mesophyll). A schematic of the two-cell model is shown in Fig. \[schematicfig\]b. Both single-cell versions of the model and the two-cell model, designated iEB5204, iEB2140, and iEB2140x2 respectively (based on the primary author’s initials and number of genes included, according to the established naming convention [@Reed2003]), are available in SBML format (-.) Nonlinear flux-balance analysis {#nonlinear-flux-balance-analysis .unnumbered} ------------------------------- To solve nonlinear optimization problems incorporating the constraints discussed above, we developed a Python package which – given a model in SBML format, arbitrary nonlinear constraints, a (potentially nonlinear) objective function, and all needed parameter values – infers the conventional FBA constraints of eq. (\[FBA\]) from the structure of the network, automatically generates Python code to evaluate the objective function, all constraint functions, and their first and second derivatives, and calls IPOPT through the pyipopt interface [@pyipopt]. Source code for the package is available in and online (<http://github.com/ebogart/fluxtools>). The software has been used to successfully solve nonlinear FBA problems with over [84000]{} variables and [62000]{} constraints. Figure \[comparisonfig\] demonstrates that, as expected, optimizing the rate of CO~2~ assimilation in the two-cell-type model with nonlinear kinetic constraints \[eqs. (\[rubisco\_kinetics\]), (\[pepc\_kinetics\]), (\[leakiness\])\] produces predictions consistent with the results of the physiological model of [@VonCaemmerer2000]. Note that the effective value of one macroscopic physiological parameter may be governed by many microscopic parameters in the genome-scale model. In the figure, the effective maximum PEP regeneration rate $V_{pr}$ is controlled by the maximum rate of three decarboxylase reactions in the bundle sheath compartment, but with an appropriate choice of parameter values any of at least 10 reactions of the C4 system could become the rate-limiting step in PEP regeneration, and in the calculations below, expression levels for any of the 42 genes associated with these reactions () could influence the net PEP regeneration rate. ![[**$\text{CO}_2$ assimilation rates ($A$) predicted by the C4 photosynthesis model of [@VonCaemmerer2000], solid lines, and the present nonlinear genome-scale model (markers) maximizing $\text{CO}_2$ assimilation with equivalent parameters.**]{} Left, $A$ vs mesophyll $\text{CO}_2$ levels with varying PEPC levels (top to bottom, $V_{p,\text{max}}=$ 110, 90, 70, 50, and 30 [molm^-2^s^-1^]{}). Right, $A$ vs total maximum activity of all bundle sheath decarboxylase enzymes (equivalent to the maximum PEP regeneration rate $V_{pr}$) at varying Rubisco levels (top to bottom, $V_{c,\text{max}}=$ 70, 60, 50, 40, and 30 [molm^-2^s^-1^]{}). Other parameters as in Table 4.1 of [@VonCaemmerer2000], except with nonphotorespiratory respiration rates $r_d=r_m=0$.[]{data-label="comparisonfig"}](Fig2.pdf){width="\textwidth"} Flux predictions in the developing leaf based on multiple data channels {#flux-predictions-in-the-developing-leaf-based-on-multiple-data-channels .unnumbered} ----------------------------------------------------------------------- Maize leaves display a developmental gradient along the base-to-tip direction, with young cells in the immature base and fully differentiated cells at the tip [@Li2010; @Nelson2011]. To explore variations in metabolic state along this axis, we combined the RNA-seq datasets of Wang et al. [@fifteensegment] and Tausta et al. [@Tausta2014] to estimate expression levels (as FPKM) for 39634 genes in the mesophyll and bundle sheath cells at 15 points, representing 1 cm segments of the third leaf of a 9-day-old maize plant, which includes a full gradient of developmental stages. The combined dataset provides expression information for 920 reactions in the two-cell model (460 each in mesophyll and bundle sheath cells). A whole-leaf metabolic model, iEB2140x2x15, was created from fifteen copies of the two-cell model, each representing a 1-cm segment, interacting through the exchange of sucrose, glycine, and glutathione through a common compartment representing the phloem. The resulting 121-compartment model, Fig. \[schematicfig\]c, involves [18780]{} reactions among [16575]{} metabolites. Subject to the requirements that reaction rates in each of the 15 segments obey both the FBA steady-state constraints (eq. \[FBA\]) and the nonlinear constraints governing Rubisco kinetics (eqs. \[rubisco\_kinetics\], \[leakiness\], and \[pepc\_kinetics\], presented in detail below) we determined the set of rates $v_{ij}$ for each reaction $i$ at each segment $j$ which were most consistent with the base-to-tip variation in the gene expression data, by optimizing the objective function $$\label{fitting} F(v) = \sum_{i=0}^{N_r}\sum_{j=1}^{15} \frac{\left(e^{s_i}\left|v_{ij}\right|-d_{ij}\right)^2} {\delta^2_{ij}} + \alpha \sum_{i=0}^{N_r} s_i^2$$ where $N_r=920$ is the number of reactions associated with at least one gene present in the expression data, $d_{ij}$ and $\delta_{ij}$ are the expression data and associated experimental uncertainty for reaction $i$ at leaf segment $j$, and $s_i$ is an optimizable scale factor associated with reaction $i$. Effectively, this calculation – similar to the method of Lee et al. [@Lee2012] or FALCON [@Barker2014] – performs a constrained least-squares fit of the fluxes to the expression data. Allowing the scale factors $s_i$ to vary emphasizes agreement between fluxes and data in their trend along the developmental gradient, rather than in their absolute value: if the data associated with reaction $R_i$ has average value 100 FPKM, a solution in which $R_i$ has mean flux 10 [molm^-2^s^-1^]{} but correlates well with the data can achieve (with appropriate choice of scale factor) a lower cost than a solution in which $R_i$ has mean flux 100 [molm^-2^s^-1^]{} but is anticorrelated. The penalty term $\alpha \sum s_i^2$ favors solutions in which, generally, reactions with larger associated expression data carry higher fluxes. The parameter $\alpha$ controlling the tradeoff between these criteria was set arbitrarily to 1.0 in the work presented here. We require $s_a = s_b$ if reactions $a$ and $b$ are mesophyll and bundle sheath instances of the same reaction. To constrain the overall scale of the fluxes and further improve accuracy, we incorporated enzyme activity assay data from [@fifteensegment] for seventeen enzymes (including Rubisco and PEPC) along the 15 leaf segments as additional constraints on the optimization problem, requiring for each enzyme $k$ and segment $j$ $$\label{enzymeequation} E_{jk} \geq \left| v_{k1}\right| + \ldots + \left| v_{kn}\right|$$ where $E_{jk}$ is the measured maximal activity of the enzyme at that segment and the sum on the right hand side includes all the reactions which represent enzyme $k$ in the mesophyll, bundle sheath, and subcompartments of those cells if applicable. Solving the optimization problem yielded predictions for reaction rates and other variables (). Upper and lower bounds on selected variables () were determined through flux variability analysis (FVA) [@Mahadevan2003], allowing the objective function to increase by 0.1% from its optimal value. ### Predicted source-sink transition {#predicted-source-sink-transition .unnumbered} As shown in Fig. \[sourcesinkfig\], in the outer, more photosynthetically developed, portion of the leaf, our optimal fit predicts net CO~2~ uptake, with most of the assimilated carbon incorporated into sucrose and exported to the phloem. Near the base of the leaf, sucrose is predicted to be imported from the phloem and used to drive a high rate of biomass production, with some concomitant net release of CO~2~ to the atmosphere by respiration. ![[**Source-sink transition along the leaf as predicted by optimizing the agreement between fluxes in the nonlinear model and RNA-seq data.**]{} (a) Predicted rates of exchange of carbon with the atmosphere and phloem along the leaf. (b) Experimental observation of the source-sink transition, reproduced from [@Li2010]. Upper image, photograph of leaf 3; middle image, autoradiograph of leaf 3 after feeding ^14^CO~2~ to leaf 2; lower image, autoradiograph of leaf 3 after feeding ^14^CO~2~ to the tip of leaf 3. (c) Total biomass production in the best-fitting solution. In panels a and c, dotted lines indicate minimum and maximum predicted rates consistent with an objective function value no more than 0.1% worse than the optimum.[]{data-label="sourcesinkfig"}](Fig3.pdf) This transition between a carbon-exporting source region and a carbon-importing sink region is well known, and the predicted transition point between the two, approximately 6 cm above the base of the leaf, can be compared to the ^14^C-labeling results of Li et al. [@Li2010] in the same experimental conditions. Fig. \[sourcesinkfig\]b shows the location of labeled carbon in leaf 3 after feeding labeled CO~2~ to leaf 2 (center image) or leaf 3 (bottom image). Li et al. [@Li2010] identified the sink region as the lowest 4 cm of the leaf; the transition is not perfectly sharp and quantitative comparison of exchange fluxes is not possible, but the nonlinear FBA results appear to slightly overestimate the size of the sink region. Agreement might be improved under a different assumption about net sucrose import or export by leaf 3 (here, we have assumed that the import visible in the center image is exactly balanced by the export suggested by the high density of labeled carbon at the absolute base in the lower image.) The net rate of CO~2~ assimilation predicted in the outer, most mature leaf segments, 8-11 [molm^-2^s^-1^]{}, is lower than that typically measured in more mature maize plants (e.g., rates of 20-30 [molm^-2^s^-1^]{} in 22-day-old wild-type plants under comparable conditions [@Studer2014]), but photosynthetic capacity may still be increasing even in these segments. In addition to sucrose, glycine and glutathione are predicted to be exported from the source region through the phloem and reimported by the sink region, consistent with our expectations that nitrogen and sulfur reduction will occur preferentially in the photosynthesizing region (). Note that this behavior emerges from the data even though there is no explicit requirement in the model that net phloem transport occur in a basipetal direction. ### Predicted C4 system function {#predicted-c4-system-function .unnumbered} Figure \[c4fig\] shows predicted rates of key reactions of the C4 system and CO~2~ and O~2~ levels in the bundle sheath. As expected, the model predicts that a C4 cycle will operate in the source region of the leaf, elevating the CO~2~ level in the bundle sheath. The CO~2~ level is also elevated in the source region; this is an immediate consequence of respiration in the bundle sheath and eq. (\[leakiness\]). It may be overestimated here because we have assumed a constant value for the bundle sheath CO~2~ conductivity (as measured by Bellasio et al. [@Bellasio2014]); in fact, gene expression associated with synthesis of the diffusion-resistant suberin layer between bundle sheath and mesophyll peaks at 4 cm above the leaf base [@fifteensegment], so $g_s$ is presumably higher below that point. ![[**Operation of the C4 system in the best-fitting solution.**]{} (a) Rates of carboxylation by PEPC in the mesophyll and Rubisco in the mesophyll and bundle sheath. (b) Rates of $\text{CO}_2$ release by PEP carboxykinase and chloroplastic NADP-malic enzyme in the bundle sheath. (c) Transport of 3-phosphoglycerate and glyceraldehyde 3-phosphate from bundle sheath to mesophyll (or the reverse, where negative) and glyceraldehyde 3-phosphate dehydrogenation rate in the mesophyll chloroplast, showing the involvement of the mesophyll in the reductive steps of the Calvin cycle throughout the source region. (d) Oxygen and carbon dioxide levels in the bundle sheath. Straight lines show mesophyll levels. Throughout, dotted lines indicate minimum and maximum predicted values consistent with an objective function value no more than 0.1% worse than the optimum.](Fig4.pdf){width="125.00000%"} 5.0cm \[c4fig\] In the Calvin cycle, most reactions are predicted to be bundle-sheath specific, but the reductive phase is active in both cells, with approximately half the 3-phosphoglycerate produced in the bundle sheath transported to the mesophyll and returned as dihydroxyacetone phosphate (Fig. \[c4fig\]c); this is a known aspect of NADP-ME C4 metabolism connected to reduced photosystem II activity in the bundle sheath cells [@Hatch1987], which is also predicted here (). Consistent with conclusions drawn independently from the transcriptomic data, as well as proteomic data from the same system [@Li2010; @Majeran2010; @fifteensegment], the model does not predict a C3-like metabolic state as a developmental intermediate stage. As expected in maize [@Wingler1999a], a significant role for phosphoenolpyruvate carboxykinase (PEPCK) as a decarboxylating enzyme operating in the bundle sheath in parallel with NADP-ME is predicted (Fig. \[c4fig\]b). While the predictions are generally consistent with the standard view of the C4 system in maize, there are minor discrepancies. In the mesophyll, our calculations predict that malate production occurs in the mitochondrion, rather than the chloroplast. In both mesophyll and bundle sheath, phosphoenolpyruvate is formed by pyruvate-orthophosphate dikinase (PPDK) in the chloroplast at a higher rate than necessary to sustain the C4 cycle; the excess is converted again to pyruvate by pyruvate kinase in the cytoplasm, with the resulting ATP consumed by the model’s generic ATPase reaction. Finally, in the bundle sheath, a modest rate of PEPC activity is predicted, recapturing CO~2~ only to have it released again by the decarboxylases (S3 Figure). Further refinement of the associations of genes to reactions in the model might resolve some of these discrepancies. ### Global agreement between fluxes and data {#global-agreement-between-fluxes-and-data .unnumbered} Figure \[costfig\] summarizes overall properties of the predicted fluxes. It is not clear why agreement between data and predicted fluxes is poorer at the base, as shown in Fig. \[costfig\]a. As discussed below, the cell-type-specific RNA-seq data from Tausta et al. [@Tausta2014] does not extend below the fourth segment from the base of the leaf; at the base we have assumed expression levels for all genes are equal in mesophyll and bundle sheath. Though proteomics experiments on the same system [@Majeran2010] generally found limited cell-type specificity at the leaf base, this assumption is likely an oversimplification, and could limit the ability of the algorithm to find a flux prediction consistent with the data there. ![[**Agreement between RNA-seq data and predicted fluxes.**]{} (a) Contribution of each segment to the objective function (eq. (\[fitting\]), excluding costs associated with scale factors). (b) Cumulative histogram of Pearson correlations between data and predicted fluxes for all reactions. (c) Predicted fluxes versus expression data at the tip of the leaf (blue, raw fluxes; red, after rescaling each flux $v_i$ by the optimal factor $e^{s_i}$ of eq. (\[fitting\])). Some outliers with very low predicted flux are not shown. (d) Relationship between RNA-seq and proteomics measurements for 506 proteins in the 14th segment from the base, redrawn from the data of [@Ponnala2014]. NSAF, normalized spectral abundance factor. []{data-label="costfig"}](Fig6.pdf){width="125.00000%"} 5.0cm For most reactions, the correlation between the base-to-tip expression pattern and the base-to-tip trend in predicted flux is high. The cumulative histogram in Fig. \[costfig\]b shows that the Pearson correlation $r>0.92$ for more than half of the reactions in the model with associated expression data. Differences in expression levels between different reactions, however, correlate only weakly with the differences in fluxes between those reactions, as shown for segment 15 in Fig. \[costfig\]c (blue circles). After rescaling fluxes by the optimal per-reaction scale factors, a clear relationship emerges (Fig. \[costfig\]c, red circles), confirming that the scale factors are functioning as intended. Of course we should not expect a perfect correlation between data on transcript levels and predicted fluxes through associated reactions. The limited correlation between fluxes and expression data across different reactions presumably follows, in part, from the imperfect correlation between expression data and protein abundance across different genes, as illustrated in Fig. \[costfig\]d with data from the same experimental system [@Ponnala2014], as well as from the different catalytic capabilities of different enzymes, posttranslational regulation, differences in substrate availability, etc. ### Reconciling expression data and network structure {#reconciling-expression-data-and-network-structure .unnumbered} Figure \[pathwayfig\] illustrates the operation of the fitting algorithm in detail, using two regions of the metabolic network with simple structure as examples. ![[**Comparison of RNA-seq data to predicted fluxes for a linear pathway and around a metabolic branch point.**]{} Upper panels, chlorophyllide a synthesis in the mesophyll; lower panels, production of arogenate in the bundle sheath by prephenate transaminase and its consumption by arogenate dehydrogenase and arogenate dehydratase. Left, aggregate RNA-seq data and experimental standard deviations for each reaction rescaled by a uniform factor (see text). Right, same data and errors further rescaled by reaction-specific optimal factors ($e^{-s_i}$, in the variables of eq. \[fitting\]) to best match data with predicted fluxes (solid circles). Fluxes are equal for all reactions of the linear pathway (1, uroporphyrinogen decarboxylase, 2, coproporphyrinogen oxidase, 3, protoporphyrinogen oxidase, 4, magnesium chelatase, 5, magnesium protoporphyrin IX methyltransferase, 6, magnesium protoporphyrin IX monomethyl ester cyclase, 7, divinyl chlorophyllide a 8-vinyl-reductase, 8, protochlorophyllide reductase.) Error bars represent standard deviations of expression measurements across multiple replicates.[]{data-label="pathwayfig"}](Fig5.pdf){width="125.00000%"} 5.0cm In Fig. \[pathwayfig\]a, expression data for eight reactions of the pathway leading to chlorophyllide a are shown. Expression levels for the different reactions at any point on the leaf may span an order of magnitude or more, but the FBA steady-state assumption requires the rates of all reactions in this unbranched[^1] pathway to be equal at each point. Applying the optimal rescaling determined for each reaction’s expression data, shown in panel b, allows the flux prediction for the pathway (solid dots) to achieve reasonable agreement with the data. (Note that data for reaction 4 cannot be further scaled down because of the lower limit $\exp (-5)$ on its scale factor $\exp (s_4)$, imposed for technical reasons.) Figure \[pathwayfig\]c shows data for a three-reaction branch point in aromatic amino acid synthesis. To balance production and consumption of arogenate, the prephenate transaminase flux must equal the sum of the fluxes through arogenate dehydrogenase (to tyrosine) and arogenate dehydratase (to phenylalanine) but expression is consistently lower for the transaminase than the other enzymes. After rescaling (Fig. \[pathwayfig\]d), the data agree well with the stoichiometrically consistent flux predictions (solid dots). The predicted ratio of dehydrogenase to dehydratase flux reflects data for downstream reactions. ### Comparison to other methods for integrating RNA-seq data {#comparison-to-other-methods-for-integrating-rna-seq-data .unnumbered} shows predictions that result when the scale factors $s_i$ of eq. (\[fitting\]) are fixed to zero. The source-sink transition is apparent but the C4 cycle operates at lower levels, the example pathways of Fig. \[pathwayfig\] (and a number of others) show little or no activity, and predicted fluxes along the leaf are not as tightly correlated with their associated expression data. shows the metabolic state predicted by applying the expression data for each reaction as an upper bound on the absolute value of the reaction rate as in the E-Flux method [@Colijn2009] to the fifteen-segment model with the same RNA-seq data. The C4 system is predicted to operate, but no source-sink transition is apparent, and typical data-predicted flux correlations are poor. Imposing a realistic biomass composition restores the source-sink transition and somewhat improves correlation between data and fluxes (). Fluxes predicted by E-Flux are generally smaller than those predicted by the least-squares method, with or without per-reaction scale factors. compares the fluxes predicted at the tip by optimizing agreement with the data through the non-biological objective function (eq. \[fitting\]), fluxes predicted at the tip with an explicit biological objective function (maximizing CO~2~ assimilation) constrained by the experimental data in the E-Flux method, and fluxes predicted in an FBA calculation which ignores the data entirely (minimizing total flux while achieving the same CO~2~ assimilation rate as predicted at the tip by the least-squares method.) Both data-integration methods lead to predictions very different from the unconstrained FBA calculation. Discussion {#discussion .unnumbered} ========== Reconstruction {#reconstruction .unnumbered} -------------- Our model is the fourth published genome-scale metabolic reconstruction of the major crop plant *Zea mays*, and the first such reconstruction developed solely from maize data sources, rather than as a direct or indirect adaptation of the *Arabidopsis thaliana* model AraGEM [@deOliveiraDalMolin2010]. Direct reaction-to-reaction comparison of iEB5204 with C4GEM [@GomesdeOliveiraDalMolin2010], iRS1563 [@Saha2011], and its successor model [@Simons2014a] is difficult because those models use a naming scheme for compounds and reactions ultimately based on KEGG [@Kanehisa2014; @Kanehisa2000] while this model, like its parent database, uses the nomenclature of MetaCyc and the BioCyc database collection. The models are broadly similar in size and biological scope. As published, C4GEM included [1588]{} reactions associated with [11623]{} maize genes; iRS1563, [1985]{} reactions associated with [1563]{} genes; the model of Simons et al. [@Simons2014a], [3892]{} unique reactions and [5824]{} genes; and iEB5204, [2720]{} reactions with [5204]{} genes. All models can simulate the production of similar sets of basic biomass constituents (including amino acids, carbohydrates, nucleic acids, lipids and fatty acids, and cell wall components) under photosynthetic and non-photosynthetic conditions and include key reactions of the C4 cycle. The model of Simons et al. [@Simons2014a] also offers extensive coverage of secondary metabolism. However, the present model has several advantages which make it particularly suitable for integration with transcriptomics data: Gene associations : The gene associations included in iEB5204 are those presented in CornCyc [@CornCyc], which are generated by the PMN Ensemble Enzyme Prediction Pipeline (E2P2)[@E2P2], a homology-based protein sequence annotation algorithm trained on a reference dataset of experimentally validated enzyme sequences. The E2P2 approach is more comprehensive and scalable than the development procedures of the previous maize reconstructions (which involve, for example, obtaining gene associations by transferring annotations from Arabidopsis genes to their best maize BLAST hits and manually selecting annotations for remaining maize genes from among BLAST hits in other species.) The entire set of gene associations in the FBA model may be readily updated based on improvements in the E2P2 prediction algorithm. High-confidence submodel : In developing the fitting algorithm we found that, to obtain plausible metabolic state predictions, a conservative reconstruction was preferable to a comprehensive one. For example, early tests with the comprehensive version of the model suggested that the fitting algorithm often found low-cost solutions involving high fluxes through reactions which, on investigation, we determined were unlikely to be active in maize. Because of the model’s connection to the CornCyc database, it was straightforward to create a reduced, high-confidence version of the model by preferentially excluding reactions not included in any manually curated plant metabolic pathway, even if candidate associated genes had been identified computationally, leading to more realistic results. Reproducibility : In an effort to improve the reusability of the model and encourage its application to other data sets, we have provided the full source code ( and ) for all calculations presented here, as has been recommended (see, e.g., [@Sandve2013]). Previous reconstructions do offer two features absent from this model: gene associations for intracellular transport reactions, and gene associations which take into account the structure of protein complexes. Both should be considered in future work. In agreement with [@Latendresse2012], we found that building the model starting from a metabolic pathway database was considerably more straightforward than the standard process of *de novo* reconstruction [@Thiele2010]. Reasonable effort was still required to bring the model to a functional state by identifying reactions or pathways present in the CornCyc database which could not be handled automatically by the Pathway Tools export facility (for example, because they involved polymerization, or could not be checked automatically for conservation violations) and determining how to represent them appropriately in the FBA model. The model construction process here could readily be adapted to generate metabolic models describing any of the more than 30 crop and model plant species for which Pathway Tools-based metabolic pathway databases [@Karp2010] have been developed by the Plant Metabolic Network [@PMNOverview], Sol Genomics Network [@Fernandez-Pozo2014], Gramene [@Monaco2014], and others (e.g.,[@Urbanczyk-Wochniak2007; @Naithani2014; @Jung2014]) allowing the present data-fitting method to be applied to RNA-seq data from those organisms. The level of model development effort required and quality of fit results will vary depending on the extent of curation of the pathway database and quality of the gene function annotations. Nonlinear optimization {#nonlinear-optimization .unnumbered} ---------------------- In contrast to the linear and convex optimization methods employed in nearly all prior constraint-based modeling work, general constrained nonlinear optimization algorithms typically require more effort from the user (who might be required to supply functions which evaluate the first and second derivatives of all constraints with respect to all variables in the problem). They are slower, are more sensitive to choices of starting point and problem formulation, are not guaranteed to converge to an optimal point even if one exists, and, when they do converge to an optimum, cannot guarantee that it is globally optimal. The software package we present allows the rapid and effective development of metabolic models with nonlinear constraints despite these complications. All necessary derivatives of constraint functions are taken analytically, and Python code to evaluate them is automatically generated. A model in SBML format may be imported, nonlinear constraints added and removed, and the problem repeatedly solved to test various design choices, solver options, and initial points, all within an interactive session, with a minimum of initial investment of effort in programming. In the present case, agreement between nonlinear FBA calculations maximizing growth and the predictions of classical physiological models confirmed that the true, globally optimal CO~2~ assimilation rate was found successfully. For the data-fitting calculations, where the true optimal cost is not known, we cannot exclude the possibility that there exist other optimal solutions, qualitatively distinct from the flux distributions and quasi-optimal regions presented above, with equivalent or lower costs. In practice, we encountered occasional cases in which reaction or pathway fluxes were initially predicted to be zero even when associated with nonzero data, despite the existence of a superior alternative solution with nonzero predicted fluxes. A step to detect and correct these situations was incorporated into the fitting algorithm. Many future applications for the software are possible. Our approach to Rubisco kinetics may easily be extended to other models of C4 metabolism or, more generally, to any FBA calculation in a photosynthetic organism where the CO~2~ level at the Rubisco active site, and thus the Rubisco oxygenation/carboxylation ratio, is not known *a priori*. A recent genome-scale metabolic reconstruction of the model alga *Chlamydomonas reinhardtii*, for example, was identified by the authors as being deficient in describing algal metabolism under low CO~2~ conditions due to the fact that the Rubisco carboxylase and oxygenase fluxes were treated as independent and not competitive, as we have done here [@Chang2011]. Ensuring that rates of Rubisco oxygenation, Rubisco carboxylation, and PEPC carboxylation are consistent with our knowledge of their kinetics is a special case of the more general problem of integrating kinetic and constraint-based modeling, to which diverse approaches have been proposed (e.g., [@Mahadevan2002; @Smallbone2007; @Jamshidi2010; @Feng2012; @Cotten2013; @Chowdhury2014]). To our knowledge, no prior work has simply imposed kinetic laws as additional, nonlinear constraints in the ordinary FBA optimization problem. Our results demonstrate the potential of this approach in systems where the kinetics of a few well-understood reactions are crucial. It remains to be seen how many kinetic laws may be incorporated in this way at once, and to what extent their introduction usefully constrains the space of possible steady-state flux distributions even when relevant kinetic parameters are not known (but instead are treated as optimizable variables, an approach with connections to ensemble kinetic modeling [@Tan2011]). Nonlinear constraints may also be of use in enforcing thermodynamic realizability of flux distributions, and relaxing requirements of linearity or convexity may stimulate the development of novel objective functions – either for data integration purposes, as here, or as alternatives to growth-rate maximization. Data fitting {#data-fitting .unnumbered} ------------ The expression of a gene encoding a metabolic enzyme need not correlate with the rate of the reaction that enzyme catalyzes. The relationship between transcription and degradation of mRNA and control of flux is indirect, mediated by protein translation, folding, and degradation, complex formation, posttranslational modification, allosteric regulation, and substrate availability. Indeed, as reviewed by [@Hoppe2012], experimentally observed correlations among RNA-seq or microarray data (each itself an imperfect proxy for mRNA abundance or transcription rate), protein abundance, enzyme activity, and fluxes are variable and often weak. For example, RNA-seq and quantitative proteomic data obtained from maize leaves at the same developmental stage studied here, harvested simultaneously from plants grown together, showed Pearson correlation approximately 0.6 across the entire dataset, but some significantly lower values were found when correlations were restricted to genes of particular functional classes, and measured mRNA/protein ratios for individual genes varied up to 10-fold along the gradient [@Ponnala2014]. A subset of this data is shown in Fig. \[costfig\]d. The most comprehensive study of the issue in plants so far [@temp_schwender] found so little agreement between RNA-seq and 13C-MFA data from embryos of two *Brassica napus* accessions that the authors concluded the inference of central metabolic fluxes from transcriptomics is, in general, impossible. In this light, it is not surprising that methods for integrating transcriptomic data with metabolic models to predict reaction rates have met with limited success. Machado and Herrgård [@Machado2014a] reviewed 18 such methods and assessed the performance of seven of them on three test datasets from *E. coli* and *Saccharomyces cerevisiae* where experimentally measured intracellular and extracellular fluxes were available for comparison. None of the methods consistently outperformed parsimonious FBA simulations which completely ignored transcriptomic data. In contrast, in the present work the use of transcriptomic data (and a limited number of enzyme activity measurements) allowed the correct prediction of a metabolic transition from the base of the leaf to the tip, which could not have been expected based on FBA calculations alone: without such data, all points along the gradient would be identical, and the biomass-production-maximizing solution would be the same at each. The predicted position of the source-sink transition is not perfectly accurate, and the overall performance of the model cannot be evaluated until the predicted reaction rates are compared to detailed experimental flux measurements. Nonetheless, the results are encouraging. We offer two explanations for this apparent success. First, the metabolic transition between the heterotrophic sink region at the base and the photoautotrophic source region at the tip is particularly dramatic, involving a large number of reactions which are effectively absent in one region but carry high fluxes in the other [@Li2010]; so long as even a slight correlation between transcript levels and fluxes exists, such a reconfiguration should be apparent from expression data. Second, although the developing maize leaf is biologically more complex than microbial growth experiments, the relationship between expression levels and fluxes may be actually be closer in the leaf. Leaf development is a stereotyped, frequently repeated, relatively slow, one-way process, in which the precise sequence of events is subject to evolutionary optimization. Coordination of transcription with required fluxes will lead to efficient use of resources. In contrast, the test cases of [@Machado2014a] involve microbial responses to varying environmental conditions and under- and over-expression mutations. Environmental responses must be rapid, flexible and reversible – criteria a complex, scripted transcriptional response may not satisfy – while transcriptional responses to novel mutations, by definition, cannot have been evolutionarily optimized. This hypothesis could be tested by evaluating performance of the present method on RNA-seq data from mutant maize plants, or plants subject to environmental challenges. We note also that methods that did not constrain or optimize the growth rate predicted zero growth rates in almost all the test cases studied by Machado and Herrgård [@Machado2014a]. The present method also does not constrain or optimize the growth rate but consistently does predict nonzero growth as reflected in nonzero biomass production (whether with a flexible biomass composition was used, as above, or a fixed biomass composition, as in and ). The whole-leaf model {#the-whole-leaf-model .unnumbered} -------------------- Large-scale metabolic models of interacting cells of multiple types first appeared in 2010, with C4GEM [@GomesdeOliveiraDalMolin2010] and a model of human neurons interacting with their surrounding astrocytes [@Lewis2010]. Many more complex multicellular FBA models have since appeared, including studies of the metabolism of interacting communities of microbial species in diverse natural environments or artificial co-cultures [@Salimi2010; @Zhuang2011; @Zomorrodi2012; @Zengler2012; @Khandelwal2013; @Chiu2014; @Zomorrodi2014] (also [@Stolyar2007] at a smaller scale) and of the metabolic capacities of host animals and their symbionts [@Bordbar2010] or parasites [@Heinken2013]. In plants, diurnal variation in C3 and CAM plant metabolism has been simulated with a model which represents different phases of the diurnal cycle with different abstract compartments, with transport reactions representing accumulation of metabolites over time [@Cheung2014]. In the most direct antecedent of the present work, Grafahrend-Belau and coauthors developed a multiscale model of barley metabolism [@Grafahrend-Belau2013] which represented leaf, stem, and seed organs as subcompartments of a whole-plant FBA model, with nutrients exchanged through the phloem. Combining the FBA model with a high-level dynamic model of plant metabolism allowed them to predict changes in metabolism over time, including the transition between a biomass-producing sink state and a fructan-remobilizing source state in the stem late in the plant’s life cycle. The whole-leaf model presented here occupies an intermediate position between prior C4 models, with single mesophyll and bundle sheath cells, and multi-organ whole-plant models such as [@Grafahrend-Belau2013]. It represents the first attempt to model spatial variations in metabolic state within a single organ, allowing the study of developmental transitions in leaf metabolism by incorporating data from more and less differentiated cells at a single point in time, rather than modeling development dynamically. Other interacting cell models incorporate *a priori* qualitative differences in the metabolic capabilities of their components (e.g., leaf, stem, and seed, or neurons and astrocytes). In contrast in the work presented here, in order to allow the metabolic differences between any two adjacent points to be purely quantitative, the same metabolic network must be used for all points. This simplifies the process of model creation but implies that meaningful predictions of spatial variation depend entirely on the integration of (spatially resolved) experimental data. The ability of the model to capture the experimentally observed shift from sink to source tissue along the developmental gradient based on RNA-seq and enzyme activity measurements shows that this may be done successfully with high-resolution -omics data and careful model construction. Methods {#methods .unnumbered} ======= Reconstruction process {#reconstruction-process .unnumbered} ---------------------- A local copy of CornCyc 4.0 [@CornCyc] was obtained from the Plant Metabolic Network and a draft metabolic model was created using the MetaFlux module of Pathway Tools 17.0 [@Latendresse2012]. The resulting model, including reaction reversibility information, was converted to SBML format and iteratively revised, as described in detail in , until all desired biomass components could be produced under both heterotrophic and photosynthetic conditions and realistic mitochondrial respiration and photorespiration could operate. An overall biomass reaction was adapted from iRS1563 [@Saha2011] with minor modifications to components and stoichiometry, as detailed in . To allow calculations with flexible biomass composition, individual sink reactions were added for most species participating in the biomass reaction, as well as several relevant species (including chlorophyll) not originally included in the iRS1563 biomass equation. Core metabolic pathways were assigned appropriately to subcellular compartments (e.g., the TCA cycle and mitochondrial electron transport chain to the mitochondrion; the light reactions of photosynthesis, the Calvin cycle, and some reactions of the C4 cycle to the chloroplast; and some reactions of the photorespiratory pathway to the peroxisome) and the intracellular transport reactions necessary for their operation were added. The model was thoroughly tested for consistency and conservation violations, confirming that no species could be created without net mass input or destroyed without net mass output (except species representing light, which can be consumed to drive futile cycles.) The base metabolic model iEB5204 is provided in SBML format as . Gene association rules for reactions with associated genes in CornCyc are provided following COBRA conventions [@Becker2007]. Additional annotations give the record in the CornCyc database associated with each reaction and species, where applicable. To produce the higher-confidence version of the reconstruction, iEB2140 (), reactions in the base model which were not associated with any identified metabolic pathway in CornCyc, and those for which no genes for a catalyzing enzyme had been identified by computational function prediction, were removed from the model if their removal did not prevent photosynthesis, photorespiration, or the production of any biomass component. Then, all reactions which could not achieve nonzero steady-state rates were removed. Mesophyll-bundle sheath model {#mesophyll-bundle-sheath-model .unnumbered} ----------------------------- A model for leaf tissue () was created by taking two copies of the high-confidence model, representing mesophyll and bundle sheath cells, and adding reactions representing transport through the plasmodesmata which connect the cytoplasmic spaces of adjacent cells. Though in principle most small molecules can cross the plasmodesmata by diffusion [@Weiner1988], unrealistic concentration gradients may be required to drive high diffusive fluxes, and processes other than simple diffusion may play a role in the rapid exchanges which do occur [@Sowinski2008]. Given this uncertainty we conservatively restricted such transport to species known or expected to be exchanged between cell types (under at least some circumstances); a complete list is given in . Net import or export of metabolites from the system was limited to the mesophyll, for gases exchanged with the intercellular airspace, or the bundle sheath, for soluble metabolites exchanged with the leaf’s vascular system. Reactions were not otherwise restricted *a priori* to a particular cell type. To facilitate integration with cell-type-specific RNA data, gene associations in this model are tagged with the relevant cell type, e.g. ‘bs\_GRMZM2G039273’ vs ‘ms\_GRMZM2G039273’. Leaf gradient model {#leaf-gradient-model .unnumbered} ------------------- The choice of phloem transport metabolites (other than sucrose) is a compromise. Glycine is the most abundant amino acid in maize phloem [@Ohshima1990], and glutathione is a putative phloem sulfur transport compound [@Bourgis1999], but many other amino acids are present in the phloem sap, and other compounds (e.g., S-methyl-methionine [@Bourgis1999]) may play roles in phloem sulfur transport. However, we found that the available data did not adequately constrain rates of phloem transport if multiple transport species of each type were allowed, resulting in high rates of transport from the base towards the tip, against the direction of bulk flow in the phloem. For simplicity, export of metabolites from the leaf to the rest of the plant through the phloem was neglected and net import of sucrose was not allowed. Each segment was taken to have the same total area, so that a 1 [molm^-2^s^-1^]{} rate of sucrose loading in one segment exactly balanced a 1 [molm^-2^s^-1^]{} rate of sucrose unloading in another segment. Note that the whole-leaf model is constructed dynamically within the data-fitting code, rather than being loaded from an SBML file. Physiological constraints {#physiological-constraints .unnumbered} ------------------------- Rubisco carboxylase and oxygenase rates $v_c$ and $v_o$ in mesophyll and bundle sheath chloroplasts were constrained to obey Michaelis-Menten kinetic laws with competitive inhibition, $$\label{rubisco_kinetics} \begin{aligned} v_c &=\frac {v_{c,\text{max}} \left[\text{CO}_2\right]} {\left[\text{CO}_2\right] + k_c \left( 1 + \frac{\left[\text{O}_2\right]}{k_o}\right)}\\ v_o &=\frac {v_{o,\text{max}} \left[\text{O}_2\right]} {\left[\text{O}_2\right] + k_o \left( 1 + \frac{\left[\text{CO}_2\right]}{k_c}\right)},\\ \end{aligned}$$ and the relationship $v_{o,\text{max}}/v_{c,\text{max}}= k_C/(k_O\cdot S_R)$ was imposed, from which eq. (\[vo\_vc\_ratio\]) follows [@VonCaemmerer2000]. The Michaelis-Menten constants for oxygen and carbon dioxide $k_C$ and $k_O$ and the Rubisco specificity $S_R$ were set to values typical of C4 species: $k_C$, 650 molmol^-1^; $k_O$, 450 mmolmol^-1^; $S_R$, [2590]{} [@VonCaemmerer2000]. The rate of PEP carboxylation in the mesophyll was bounded above by an appropriate kinetic law, $$\label{pepc_kinetics} v_p = \frac {v_{p,\text{max}} \left[\text{CO}_2\right]} {k_{C,p} + \left[\text{CO}_2\right]}$$ with $0 \leq v_{p,\text{active}} \leq v_{p,\text{max}} $ and an appropriate $k_{C,p}$ ([80]{}mmolmol^-1^, [@VonCaemmerer2000]). The parameters $v_{p\text{max}}$ and $v_{c,\text{max}}$ representing the total amount of Rubisco and PEPC available may be fixed to permit comparison to models parameterized in those terms or allowed to vary. Rates of oxygen and carbon dioxide diffusion from the bundle sheath to the mesophyll, $L$ and $L_O$, were constrained to obey the relationship $$\label{leakiness} \begin{aligned} L &= g_{BS}\left(\text{CO}_{2,BS} - \text{CO}_{2,ME}\right) \\ L_O &= g_{BS,O}\left(\text{O}_{2,BS} - \text{O}_{2,ME}\right) \end{aligned}$$ with $g_{BS,O}=0.047g_{BS}$ [@VonCaemmerer2000]. All simulations used the bundle sheath CO~2~ conductivity measured by [@Bellasio2014] for maize plants grown under high light, $1.03 \pm 0.18$ [molm^-2^s^-1^]{}. While $g_{BS}$ undoubtedly varies along the developmental gradient, its deviation from this value (measured in fully-expanded leaves, 3-4 weeks after planting) is likely greatest below the region of high suberin synthesis identified 4 cm from the leaf base [@fifteensegment]; as the C4 cycle was not predicted to operate at high rates in this region, the impact of this discrepancy should be limited. Resistance to CO~2~ diffusion from the intercellular airspace to the mesophyll cells was neglected; ref. [@Kromdijk2010] reported $g_m\approx 1$ mmolm^-2^s^-1^ in maize under a variety of conditions, suggesting the mesophyll and intercellular CO~2~ levels would differ only slightly at the rates of CO~2~ assimilation and release dealt with here. Similarly, all intracellular compartments were taken to have equal CO~2~ concentrations. Optimization calculations {#optimization-calculations .unnumbered} ------------------------- The nonlinear modeling package uses the libsbml python bindings to read SBML files [@Bornstein2008] and an internal representation of SBML models derived from the SloppyCell package [@SloppyCell; @Myers2007]. IPOPT calculations used version 3.11.8 with the linear solver ma97 from the HSL Mathematical Software Library [@HSL]. Where not specified, convergence tolerance was $10^{-5}$, or $10^{-4}$ in FVA calculations. To solve purely linear problems (e.g., to test the production of biomass species during the reconstruction process, where nonlinear constraints were not used) the GNU Linear Programming Kit, version 4.47 [@glpk], was called through a Python interface [@pyglpk]. Comparison with other models {#comparison-with-other-models .unnumbered} ---------------------------- Python code used to calculate the predictions of the models of von Caemmerer [@VonCaemmerer2000] for comparison with nonlinear optimization results is provided in . Integrating biochemical and RNA-seq data {#integrating-biochemical-and-rna-seq-data .unnumbered} ---------------------------------------- ### RNA-seq datasets {#rnadatasets .unnumbered} To obtain mesophyll- and bundle-sheath-specific expression levels at 15 points, we combined the non-tissue-type-specific data of Wang et al. [@fifteensegment], measured at 1-cm spatial resolution, with the tissue-specific data of Tausta et al. [@Tausta2014] obtained by using laser capture microdissection (LCM) – measured 4 cm, 8 cm and 13 cm from the leaf base (the upper three highlighted positions in Fig. \[sourcesinkfig\]b). This integration was achieved by determining for each gene at each of those points with LCM data the ratio of the average RPKM in the mesophyll ($M$) to the sum of the average RPKM values for mesophyll and bundle sheath ($M+B$); furthermore, we assumed that the $M/(M+B)$ ratio at the leaf base was 0.5 (based on the proteomic experiments of Majeran et al. [@Majeran2010], which showed only limited mesophyll-bundle sheath specificity there), and linearly interpolating to estimate $M/(M+B)$ ratios at all 15 points. For very weakly expressed genes, we did not impose cell-type specificity: where the sum of mesophyll and bundle sheath RPKM in the LCM data was less than 0.1, we assumed $M/(M+B)=0.5$. We then divided the mean whole-leaf FPKM measurement at each point into mesophyll and bundle sheath portions according to these ratios. To associate expression data with a reaction, data for its associated genes were summed, dividing the data for a gene associated with multiple reactions in the model equally among them. The uncertainties $\delta_{ij}$ in the objective function (eq. (\[fitting\])) were estimated in an ad hoc way by splitting the standard deviations of the FPKM values over multiple experimental replicates according to the $M/(M+B)$ ratios and then summing the uncertainties for all genes associated with a particular reaction, imposing a minimum relative error of 0.05 and a minimum absolute uncertainty corresponding to 7.5 FPKM. To globally rescale the expression data to be comparable to expected flux values, data for PEPC and Rubisco were compared to the enzyme activity measurements discussed below and a simple linear regression performed, yielding a conversion factor of 204 FPKM $\approx$ 1 [molm^-2^s^-1^]{} for these enzymes. All expression data were divided by this factor before solving the optimization problem. ### Enzyme activity measurements {#enzyme-activity-measurements .unnumbered} Enzyme activities constrained by measurements in [@fifteensegment] were alanine aminotransferse, aspartate aminotransferase, fructose bisphosphate aldolase, glyceraldehyde 3-phosphate dehydrogenase (NADPH), glyceraldehyde 3-phosphate dehydrogenase (NADH), glutamate dehydrogenase (NADH), malate dehydrogenase (NADH), malate dehydrogenase (NADPH), PEPC, phosphofructokinase, phosphoglucomutase, phosphoglucose isomerase, phosphoglycerokinase, Rubisco, transketolase, triose phosphate isomerase, and UDP-glucose pyrophosphorylase. For Rubisco and PEPC, enzyme data constrained the sum of the variable kinetic parameters $v_{c,\text{max}}$ and $v_{p,\text{max}}$ in mesophyll and bundle sheath compartments, rather than the sum of the associated fluxes. Enzyme data in nanomole per minute per gram fresh weight was converted to micromole per second per square meter of leaf surface area assuming a fresh weight of 150 gm^-2^. ### Handling reversible reactions {#handling-reversible-reactions .unnumbered} The objective function (eq. (\[fitting\])) optimizes the agreement between the absolute value of the flux through each reaction with its data, but IPOPT requires a twice continuously differentiable objective function. We use a reformulation $F'$ representing each absolute value $|v_{ij}|$ as the product of the flux and a parameter $\sigma_{ij}$ representing its sign: $$\label{fitting_no_abs} F'(v) = \sum_{i=0}^{N_r}\sum_{j=1}^{15} \frac{\left(e^{s_i}\sigma_{ij}v_{ij}-d_{ij}\right)^2} {\delta^2_{ij}} + \alpha \sum_{i=0}^{N_r} s_i^2$$ Similarly, the enzyme activity data constraint, eq. (\[enzymeequation\]), was rewritten to replace absolute values in this way. Reaction rates with positive (negative) sign parameter were required to take values greater than a small negative (less than a small positive) tolerance, typically 1.0. Choosing the $\sigma_{ij}$ to optimize $F'$ is a very large scale mixed-integer nonlinear programming problem. We arrive at an approximate solution using a heuristic method similar in spirit to that of [@Lee2012], with three steps. 1. The subproblems representing each segment of the leaf are solved separately, with all scales $s_{i}$ set to zero and modest upper and lower bounds on the reactions representing nutrient exchange with the phloem. Within each segment, a sign for the reversible reaction $r_1$ with the highest associated expression data is chosen by first setting its sign $\sigma_1$ to $+1$, finding the minimum-flux best-fitting flux distribution $\mathbf{v}^+$ ignoring the costs associated with all other reversible reactions (but including costs associated with all irreversible reactions), then finding the cost $c^+$ of the best-fitting flux distribution $\mathbf{v'}^+$ considering the costs of the reversible reactions with nonzero fluxes in $\mathbf{v}^+$ (temporarily setting their signs according to their values in that case.) A cost $c^{-}$ is determined analogously after setting the sign $\sigma_1$ to $-1$ , and if $c^{-}<c^{+}$, $\sigma_1=-1$ is chosen; otherwise, $\sigma_1=+1$. Then the reversible reaction with the second-highest expression data $r_2$ is treated in the same way, considering $r_1$ to be irreversible. 2. When signs for all reversible reactions have been chosen at a segment, a final best-fitting flux distribution given those signs is determined. Then the full optimization problem, combining all fifteen segments, is solved with the chosen sign parameters fixed, using those flux distributions to provide a nearly-feasible initial guess. 3. The sign-choice process in each subproblem is then solved again, fixing the scale factors $s_{i}$ and rates of metabolite exchange with the phloem to those determined in the full problem. If no signs change, or if the new signs do not decrease the objective function value, fitting stops; otherwise, step 2 is repeated. 4. Finally, for each reaction $j$ with nonzero data and maximum absolute flux less than ${0.0001}$ at any point in the leaf model, a lower bound of $-0.99d_i$ is imposed on the term $\left(e^{s_i}\sigma_{ij}v_{ij}-d_{ij}\right)$ in the objective function, for $i=1,\ldots,15$, and the full fifteen-segment optimization problem is solved again. The final step addresses the observation that the optimization process occasionally converged to a solution in which a few reactions with associated data were predicted to have zero flux when a better solution with nonzero flux existed. In some cases (e.g. the $s_i=0$ case shown in ) this step did not lead to an overall reduction in the objective function and was omitted. Steps 1 and 3 take between one and eight hours per segment using an AMD Opteron 6272 and may be easily parallelized across up to 15 processors. Step 2 may take up to 2 hours in the first iteration but is often faster in later iterations, when the initial guess is closer to the optimum. Typically the procedure stops after 4-5 iterations, requiring about 24 total hours of wall time using 15 processors. ### Special cases {#special-cases .unnumbered} The Rubisco oxygenase, Rubisco carboxylase, and mesophyll PEPC fluxes are excluded from the objective function. Instead, terms are added comparing the transcriptomic data for those enzymes to the variables which explicitly represent their activity level: for Rubisco, $v_{c,\text{max}}$ in mesophyll and bundle sheath compartments, and for PEPC, $v_{p\text{max}}$ in the mesophyll. Scale factors for the mesophyll and bundle sheath Rubisco activities are not constrained to be equal. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by National Science Foundation grant IOS-1127017 and a grant to the International Rice Research Institute from the Bill and Melinda Gates Foundation. The authors thank Tom Brutnell, Lin Wang, Lori Tausta, Qi Sun, Zehong Ding, and Tim Nelson for data and comments, Sue Rhee, Kate Dreher, and Peifen Zhang for helpful discussion of our use of CornCyc, and Lei Huang and Brandon Barker for discussions of metabolic modeling. [ @text@text]{} Supporting Information {#supporting-information .unnumbered} ====================== Outline {#S9_Appendix} ======= This appendix describes the of creation of a metabolic model for maize from CornCyc. It covers the creation of an SBML model with exchange and biomass reactions and limited subcellular compartmentalization which can successfully simulate the production of many biomass components and photosynthetic carbon dioxide assimilation, the adaptation of the biomass equation from iRS1563, some considerations in the process of expanding the model to describe interacting mesophyll and bundle sheath compartments, and some modifications made in response to preliminary fitting results. Sections \[export\] through \[sbml\] explain in detail the process of constructing the underlying metabolic model at the one-cell level. Section \[refinement\] discusses in detail changes made to gene associations based on early data fitting results. Section \[biomass\] describes changes to the iRS1563 biomass equation. Section \[plasmodesmata\] discusses plasmodesmatal transport in the two-cell model. Filenames referred to are in the `model_development` subdirectory of the project source code (see S15 Protocol.) Exporting the CornCyc FBA model from Pathway Tools {#export} ================================================== CornCyc 4.0 [@CornCyc] was obtained from the Plant Metabolic Network and upgraded from from Pathway Tools 16.5 to 17.0 locally. The frame `PWY-561` was removed from the database because otherwise some of the reactions of that pathway were excluded from the FBA export, apparently due to a bug. A simple FBA problem was solved using the Pathway Tools FBA functionality [@PathwayToolsManual], producing an output file which includes all reactions in the FBA model Pathway Tools generates internally, both those which are active in the solution to the FBA problem and those which are not. Note that this list of reactions is distinct from the list of reactions in the database itself; the Pathway Tools software prepares this set of reactions through an extensive process of excluding reactions which are unbalanced or otherwise undesirable while expanding reactions with classes of compounds as products or reactants into sets of possible specific instantiations which respect conservation of mass [@Latendresse2012]. Working with the Pathway Tools FBA reaction set (rather than, e.g, an SBML export of the CornCyc database) allows us take advantage of this pre-processing; however, it comes at the cost of needing to reintroduce into the FBA model many reactions which are present in the CornCyc database but are excluded from the FBA export for one reason or another. Reaction data was extracted from the FBA output file, and reactions were translated to refer to species by their CornCyc frame ID (to allow easy reference to the database and comparison with previous work, and avoid possible ambiguities.) Reactions were then added and removed from the model as described below. Discarding reactions ==================== Polymerization reactions ------------------------ Pathway Tools attempts to include an expanded representation of certain polymerization reactions in the exported FBA model, but this function is considered experimental [@PathwayToolsManual]; these reactions were ignored. Note that some reactions representing polymer growth were added manually later in the process. ATPases ------- We removed all reactions from CornCyc which have the effective stoichiometry > ` {ADP: 1.0, ATP: -1.0, PROTON: 1.0, WATER: -1.0, |Pi|: 1.0} ` There are nine such reactions: - `RXN-11109`, - `3.6.4.6-RXN`, - `RXN-11135`, - `RXN0-1061`, - `ADENOSINETRIPHOSPHATASE-RXN`, - `3.6.4.4-RXN`, - `3.6.4.9-RXN`, - `3.6.4.5-RXN`, - `3.6.4.3-RX`N all treated as reversible by the Pathway Tools export procedure. Typically these are simplified representations of the metabolic effect of enzymes whose complete function is outside the scope of the database, as, for example, EC 3.6.4.3, the microtubule-severing ATPase. \[MaintenanceATPase\] In their place, we added a single generic ATPase reaction to represent cellular maintenance costs, etc., with no associated genes. Reactions involving generic electron donors and acceptors --------------------------------------------------------- Numerous reactions in the database are written with generic representations of electron carrier species (‘a reduced electron acceptor’, ‘an oxidized electron acceptor’). Most of these reactions are outside the areas of emphasis of the model (e.g., brassinosteroid biosynthesis), have no curated pathway assignment, or also appear in forms which do specify the electron carrier species (e.g., the generic nitrate reductase reaction, `NITRATEREDUCT-RXN`, vs `NITRATE-REDUCTASE-NADH-RXN`,) and so could be safely neglected. A small set of exceptions identified in early drafts included reactions of fatty acid synthesis, handled as discussed below, and proline dehydrogenase, `RXN-821`, catalyzed by a mitochondrial-membrane-bound flavoprotein which donates electrons directly to the mitochondrial electron transport chain [@Elthon1982]. Because we have not thoroughly compartmentalized amino acid metabolism, we implemented this reaction as donating electrons to NAD^+^ instead. Duplicates ---------- A number of other reactions were removed because they appeared to be exact (possibly unintentional) duplicates, down to gene associations, of other reactions in the database; or because they were being replaced by modified forms as discussed below. These are given in `reactions_to_remove.txt`. Non-metabolic reactions ----------------------- A number of reactions present in CornCyc were removed because the database indicated, e.g. through the Enzyme Commision summary for the relevant EC number, that they were primarily involved in extrametabolic functions (e.g., cell movement, regulation). These included the GTPases `RXN-5462`, `3.6.5.2-RXN`, and `3.6.5.5-RXN`. Glucose-6-phospate ------------------ In the reduced model (discussed below) only one reaction, myo-inositol-1-phosphate synthase, consumes the generic glucose-6-phosphate species, rather than alpha-G6P or beta-G6P. To ensure that this reaction was appropriately connected to other G6P producing and consuming reactions we manually split it into two instances, one for alpha-G6P and one for beta-G6P. UDP-glucose ----------- For apparently all reactions in CornCyc involving UDP-glucose, the instantiation procedure produced one version involving generic UDP-D-glucose and one version involving UDP-alpha-D-glucose, the only child of the UDP-D-glucose class. UDP-alpha-D-glucose participated in almost no reactions other than these instantiations (in the reduced model, described below, only one: UDP-sulfoquinovose synthase, EC 3.13.1.1). As such there is little to distinguish the generic and specific versions of the reactions, which add complexity to the model and degeneracy to optimization predictions without providing significant information about the function of the system, so we removed the specific versions and changed the UDP-sulfoquinovose synthase to act on a generic UDP-D-glucose substrate. Minor revisions to achieve basic functionality ============================================== Mitochondrial electron transport chain -------------------------------------- The CornCyc representation of the mitochondrial electron transport pathway (`PWY-3781`, plus the mitochondrial ATPase (`ATPSYN-RXN`, EC 3.6.3.14)) was adjusted. Some reactions excluded from the initial Pathway Tools export because the balance state of reactions involving cytochrome C could not be determined were readded manually; ubiquinones/ubiquinols were uniformly represented as ubiquinone-8/ubiquinol-8, and compartments were assigned to reactants and products to properly represent the transport of protons between the mitochondrial matrix and the mitochondrial intermembrane space. In CornCyc, as in MetaCyc and other related databases, transport of protons across the membrane is represented explicitly for complex I but not for complex III and complex IV; in agreement with the standard description of mitochondrial electron transport (see, e.g., [@Brownleader]) proton transport was added to these reactions with a stoichiometry of 2 H+/e- for complex III and 1 H+/e- for complex IV. The stoichiometry of complex IV was further adjusted to include the H+ from the mitochondrial matrix that binds to oxygen to form water. Photosynthesis: light reactions ------------------------------- Similarly, some modifications were made to the light reactions of photosynthesis (`PWY-101`). Reactions involving plastocyanins were not exported and were added manually; a chloroplastic ATP synthase and a reaction describing cyclic electron transport around PS I were added; and the stoichiometry of proton transport was adjusted in accordance with recent literature, assuming a Q cycle and ratio of 14 H+/3 ATP for the chloroplast ATP synthase [@Allen2003]. Reduction of oxygen to superoxide at photosystem I (the Mehler reaction) was added to allow flux through the pathways of chloroplastic reactive oxygen species detoxification: superoxide dismutase and the ascorbate-glutathione cycle, including a reaction representing the direct, non-enzymatic reduction of monodehydroascorbate by ferredoxin [@Asada1999; @FoyerHarbinsonChapter]. Key reactions in biomass component production and nutrient uptake ----------------------------------------------------------------- Several components of biomass required either manual adjustment of reactions from the database or the addition of abstract synthesis reactions summarizing the behavior of pathways which could not easily be represented in more detail. ### Starch Starch synthase (`GLYCOGENSYN-RXN`) is not exported from CornCyc by default (it is a polymerization reaction, and marked as unbalanced in the PGDB); it was added manually in a form that produces the equivalent of one 1,4-alpha-D-glucan subunit. The starch branching enzyme EC 2.4.1.18 (`RXN-7710`) is not exported from CornCyc by default (one reactant, starch, has an unspecified structure); it was added manually as $$\text{a 1,4-alpha-D-glucan subunit} \to \text{an amylopectin subunit}$$ Note that this stoichiometry is not intended to suggest that the branching enzyme introduces branches at each subunit. CornCyc provides a detailed reconstruction of the reactions of starch degradation (`PWY-6724`) which is by nature difficult to convert to a form suitable for FBA calculations, as many of the stoichiometry coefficients are undefined. To incorporate the effects of the glucan-water and phosophoglucan-water dikinases, for example, we would need to specify how many glucosyl residues must be phosphorylated (and then dephosphorylated) to produce “an exposed unphosphorylated, unbranched malto-oligosaccharide tail on amylopectin” of a given length; modeling the release of maltose from that tail would require an estimate of the typical unbranched length of such tails, etc. Rather than estimate average values for these parameters, we divide the reactions of the pathway into two types: those which condition starch for depolymerization , and actual depolymerization reactions. The first class (the dikinases above plus isoamylase) share the abstract stoichiometry $$\text{a starch subunit} \to \text{an exposed starch subunit}$$ (neglecting any ATP costs), while the second class (beta amylase and disporportionating enzyme) convert exposed starch subunits to sugars appropriately. The beta-maltose releasing reactions of the starch degradation pathway in CornCyc have no associated genes. We temporarily associated these reactions with the beta amylase record in the database (`RXN-1827`, EC 3.2.1.2) pending further review. During transient starch degradation, beta-maltose and glucose are exported into the cytosol, where maltose is split, releasing one glucose molecule and donating one glucosyl residue to a cytosolic heteroglycan, from which it may be released in turn as glucose-1-phosphate [@Fettke2009]. In Arabidopsis, specific enzymes (DPE2 and PHS2) are known to be implicated in this process [@Streb2012]. In simulations with this CornCyc-based FBA model we find the typical mode of breakdown of cytosolic maltose is to alpha-D-glucose and alpha-D-glucose-1-phosphate via `AMYLOMALT-RXN`, $$\text{maltotriose} + \text{beta-maltose} \to \text{maltotetraose} + \text{beta-D-glucose}$$ and `RXN0-5182`, $$\text{maltotetraose} + \text{phosphate} \to \text{maltotriose} + \text{alpha-D-glucose-1-phosphate}$$ effectively the standard pathway but with maltotriose/maltotetraose playing the role of the cytosolic heteroglycan pool. This approximation leads to a reasonable effective stoichiometry but it is possible that the genes associated with these reactions do not accurately represent the genes involved in the true underlying process; we have not systematically looked for maize counterparts of the Arabidopsis genes, for example. ### Cellulose The UDP-forming cellulose synthase, EC 2.4.1.12, is not exported from CornCyc by default (it is a polymerization reaction, and marked as unbalanced in the PGDB); it was added manually in a form that produces the equivalent of one subunit. ### Hemicellulose Similarly, the following hemicellulose polymerization reactions were added manually: - 1,4-beta-D-xylan synthase, EC 2.4.2.24, - reactions `RXN-9093` (EC 2.4.2.-) and `RXN-9094` (EC 2.4.1-), representing the addition of arabinose and glucuronate to xylan to form arabinoxylan and glucuronoxylan respectively (note that the corresponding subunits notionally consist of one xylan subunit plus arabionose/glucuronate), - glucomannan synthase, EC 2.4.1.32, - `RXN-9461` (EC 2.4.2.39), representing the addition of xylose to a glucan (as implemented, cellulose) to form xyloglucan (again, the corresponding effective subunit corresponds to one glucan subunit plus xylose)– note this representation ignores the previous step in CornCyc’s xyloglucan biosynthesis pathway, xyloglycan 4-glucosyltransferase (EC 2.4.1.168). In addition to these explicit descriptions of hemicellulose formation from CornCyc, we added generic reactions representing the donation of the following sugar residues from activated donor molecules to unspecified generic polysaccharides: - arabinose (from UDP-L-arabinose) - galactose (from GDP-L-galactose) - galacturonate (from UDP-D-galacturonate) - glucose (from UDP-glucose) - glucuronate (from UDP-D-glucuronate) - mannose (from GDP-alpha-D-mannose) - xylose (from UDP-alpha-D-xylose) These reactions allow the model to represent flux of these sugars towards hemicelluloses or other polysaccharides without explicit synthesis pathways in CornCyc, or the construction of a hemicellulose term in the biomass equation in terms of the overall composition of hemicellulose without reference to specific synthesis reactions, as in our adaptation of the biomass reaction of iRS1563 (see the biomass reaction discussion, below.) ### Miscellaneous cell wall components The following additional cell wall comoponent production reactions from CornCyc were added manually: - `2.4.1.43-RXN`, representing the formation of homogalacturonan from galacturonate - `RXN-9589` (EC 2.4.2.41), representing the addition of xylose to homogalacturonan to form xylogalacturonan (note the resulting xylogalacturonan subunit notionally consists of one galacturonate plus xylose) - `13-BETA-GLUCAN-SYNTHASE-RXN` (EC 2.4.1.12), representing the formation of callose from glucose. Suberin production is not represented in CornCyc in detail but pathways for the synthesis of three key precursors, N-feruloyltyramine, octadecenedioate, and docosanediotate, are provided. Sinks for N-feruloyltyramine and octadecenedioate were added to the model to represent the flow of material towards suberin production; docosanedioate was neglected because no genes are associated with the reactions of its synthesis pathway. N-feruloyltyramine may be produced from trans-caffeate via either ferulate or caffeoyl-CoA; the branch through ferulate was initially dropped from the reduced version of the model used for data analysis because it relies on trans-feruloyl-CoA synthase, EC 6.2.1.34, which has no associated genes, but it was preserved in subsequent versions of the model because high expression levels for caffeate O-methyltransferase suggest this branch is indeed active. (In CornCyc, the tyramine N-feruloyltransferase that produces N-feruloyltyramine from feruloyl-CoA could also catalyze the production of other hydroxycinnamic acid tyramine amides (cinnamoyltyramide, sinapoyltyramide, p-coumaroyl-tyramine) but we have neglected these for now.) ### Fatty acids and lipids Plant fatty acid and lipid biosynthesis is rich in complexity (see, e.g., [@Li-Beisson2013]), and attempting to describe it in the FBA model at the level of detail at which it is currently understood would require a daunting number of reactions among the species representing the combinations of lipid head groups and acyl chains. Though CornCyc presents some pathways of lipid metabolism at such a high resolution, we have adopted a simplified approach which aims to include enough detail to allow the model to: - predict based on RNA-seq data the total flow of biomass into fatty acids and lipids - coarsely predict differences in the types of lipids and fatty acids produced, based on RNA-seq data - approximately preserve the iRS1563 biomass equation. The model describes in detail the sequence of reactions by which fatty acids up to lengths of 16 and 18 are synthesized in the chloroplast (though currently these reactions occur in the cytoplasmic compartment!), and the formation of oleate (as oleoyl-ACP) by the stearoyl-ACP desaturase (`PWY-5156`; [@Ohlrogge1995; @Li-Beisson2013]). In practice, these fatty acids may then enter the ‘prokaryotic’ pathway of glycerolipid synthesis in the chloroplast or leave the chloroplast and enter the ‘eukaryotic’ pathway of glycerolipid synthesis in the endoplasmic reticulum, with further desaturation of the acyl chains occurring after their incorporation into lipids. We simplify this process by effectively decoupling the synthesis of different types of lipids (as distinguished by head groups) from the desaturation of their associated acyl chains. Reactions from lipid synthesis pathways are implemented as if all lipid species had one 16:0 and one 18:1 acyl chain, by implementing the glycerol-3-phosphate O-acyltransferase and 1-acylglycerol-3-phosphate O-acyltransferase reactions (RXN-10462 and 1-ACYLGLYCEROL-3-P-ACYLTRANSFER-RXN), written in the database with generic acyl-acp substrates, with oleoyl-ACP and palmitoyl-ACP as substrates respectively. (This corresponds to the prokaryotic pathway; in the eukaryotic pathway oleoyl-CoA and palmitoyl-CoA would supply the acyl groups for diacylglycerol formation instead [@Li-Beisson2013]. However the same genes are associated with the reactions of diacylglycerol synthesis in the two pathways (`PWY-5667`; `PWY0-1319`) in CornCyc and so they cannot be distinguished based on expression data alone; we have chosen one arbitrarily.) This supply of diacylglycerol is sufficient to allow, without further modification to the CornCyc FBA export, the synthesis of a variety of lipids, including: - phosphatidylcholine, phosphatidylethanolamine, phosphatidylglycerol, phosphatidylinositol; - sulfoquinovosyldiacylglycerol. UDP-glucose epimerase is exported from CornCyc in the UDP-glucose-producing direction by default; we allowed it to run in the reverse direction as well, consistent with literature evidence [@Dormann1998; @Brenda5.1.3.2], which allowed the production of mono- and digalactosyldiacylglycerol. In sphingolipid metabolism, dihydrosphingosine, 4-hydroxysphinganine and sphinganine 1-phosphate may be produced, and sink reactions were added for them. Production of the ceramides and their derivatives would require the choice of a particular fatty acid source for the sphinganine acyltransferase, written by default with the generic substrate ‘a long-chain acyl-coA’; per the CornCyc description page for `PWY-5129`, in leaf sphingolipids C20 to C26 fatty acids are typical. Currently, the FBA model lacks a detailed implementation of production of very long chain fatty acids by elongation (a generic representation is present in CornCyc), so no supply of C20-26 fatty acids is available. We have deferred this issue to future work. Separately, we model the desaturation of oleate to linoleate and linolenate and palmitate to palmitoleate. These (along with palmitate and stearate) are the fatty acid components of the iRS1563 biomass reaction, which originally incorporated them as triglycerides; our modified biomass equation consumes free fatty acids, rather than attempt to specify the precise ratios in which they are to be found in different lipid species in the leaf. The CornCyc pathways for linoleate and linolenate produce them as lipid linoleoyl groups and lipid linolenoyl groups respectively, incorporated in generic lipid molecules; to allow these reactions to balance, and to provide linoleate and linolenate for the biomass reaction, we added lipases which release free linoleate/linolenate from the lipid linoleoyl and lipid linolenoyl groups, regenerating the pool of generic ‘lipid’ species (which participate only in the linoleate pathway, within the FBA model.) Note, however, that other reactions within the model but outside the indicated synthesis pathways are capable of producing linoleate and linolenate as well. CornCyc includes no complete pathway for the production of palmitoleic acid; as there is experimental evidence it is produced in maize leaves (see the discussion of the biomass equation, below) we introduced the acyl-ACP $\Delta$9-desaturase reaction from the palmitoleate biosynthesis pathway of AraCyc (`RXN-8389`, 1.14.99.-), producing palmitoleoyl-ACP from palmitoyl-ACP [@AraCycPWY-5366], which restores this functionality (in combination with the palmitoleoyl-ACP hydrolase, `RXN-9550`, which is present in CornCyc.) Note that there is some evidence that the stearoyl-ACP desaturase enzyme may also catalyze this reaction [@Gibson1993]. The oleoyl-acyl carrier protein hydrolase (EC 3.1.2.14) from CornCyc is unbalanced with respect to hydrogen; a version with an additional proton on the right hand side was added manually. The $\Delta$9-desaturase and the desaturases producing linoleate and linolenate (`RXN-9667` and `RXN-9669`) were written originally with generic electron donor and acceptor species. Initial review of the extensive literature on plant fatty acid desaturation suggests that the electron source for desaturases depends on their location within the cell, with chloroplastic desaturases accepting electrons from ferredoxin while desaturases in the endoplasmic reticulum accept electrons from NADH via cytochrome b5 or fused cytochrome domains (see, eg, [@Sperling1995; @Harwood1996; @Shanklin1998].) As discriminating between chloroplastic and extrachloroplastic fatty acid desaturation is not a high priority for the model, NADH was used as the sole electron donor for all three of these reactions. The ferredoxin-dependent stearoyl-ACP desaturase `RXN-7903`, not exported from the database by default because it is marked as unbalanced, was added in a form adjusted for hydrogen and charge balance. Ferredoxin-NADP oxidoreductase was made reversible to ensure NADPH can drive this reaction in the dark, as is observed [@Shanklin1998]. ### Nucleic acids polymerization Reactions representing the pyrophosphate-releasing incorporation of (d)NTPs into RNA and DNA were added and associated with the DNA-directed DNA polymerase and DNA-directed RNA polymerase reactions in the database. (In each case, it is assumed that all nucleotides occur with equal frequency.) Ascorbate-glutathione cycle --------------------------- To allow the NADPH-monodehydroascorbate reductase reaction to function in the cycle as curated, we split the L-ascorbate peroxidase reaction (EC 1.11.1.11) into its two subreactions, which by default are not exported in the FBA problem. Gamma-glutamyl cycle -------------------- The gamma-glutamyltransferase was lumped together with `GAMMA-GLUTAMYLCYCLOTRANSFERASE-RXN`, originally written in terms of the instanceless class ‘`L-2-AMINO-ACID`’ which appeared in no other stoichiometries in the FBA export, and the dipeptidase `RXN-6622`, which is the only reaction that can consume the cysteinylglycine product of the gamma-glutamyltransferase, forming a combined reaction which can carry flux. The combined reaction retained the gene associations of the gamma-glutamyltransferase, as the other two reactions have no associated genes. Methionine synthesis from homocysteine -------------------------------------- The methionine synthase reaction of CornCyc’s methionine biosynthesis pathway, `HOMOCYSMET-RXN`, EC 2.1.1.14, specifically requires 5-methyltetrahydropteryltri-L-glutamate as a cofactor. Polyglutamylation of folates is present in CornCyc in an abstract representation (with tetrahydrofolate synthase catalyzing the addition of a glutamyl group to a 5-methyltetrahydropteryl with $n$ glutamyl groups); we have not converted this into an explicit representation in the FBA model. Instead, `HOMOCYSMETB12-RXN`, EC 2.1.1.13, acts to produce methionine from homocysteine; the effects of this possible inaccuracy on the behavior of the rest of the network should be limited. Basic import and export ----------------------- The following species are given overall import/export reactions: - `WATER` - `CARBON-DIOXIDE` - `OXYGEN-MOLECULE` - `PROTON` - `NITRATE` - `SULFATE` - `|Pi|` - `|Light|` - `MG+2` These reactions exchange species inside the cell with species in meaningfully labeled compartments where possible (eg, oxygen and CO~2~ are exchanged with the intercellular air space, mineral nutrients with the xylem, etc.) In addition, to facilitate exchange among compartments in the whole-leaf model, a number of exchanges with a phloem compartment were set up: these included sucrose, glycine (as a representative of the amino acids detected in maize phloem sap by Ohshima et al [@Ohshima1990],) and the potential phloem sulfur transport compound glutathione [@Bourgis1999]. Note that these reactions should be inactive, or restricted to the exporting direction only, when not modeling transport within the leaf (except for sucrose, where a free supply should be allowed in heterotrophic conditions.) Defining the biomass components ------------------------------- Two types of biomass reactions are added to the model: - Sinks for individual species, for simulations (e.g, fits to RNAseq data) where the relative rates of production of different components are unknown. The species given such sinks are listed in `biomass_components.txt`. - A set of reactions producing a combined biomass species, made up of assorted components in fixed proportions, for simulations where the maximum rate of production of biomass is of interest, and an approximately realistic biomass composition needs to be enforced directly. These reactions were taken with minor modifications from [@Saha2011]; their adaptation is described below and they are listed in are listed in `adapted_irs1563_biomass.txt`. To conceptually and practically separate these types of biomass reactions, which in general should not both be active in any one calculation, the biomass species they produce are located within two separate abstract biomass compartments in the SBML model. In general, the biomass sink reactions have no gene associations, but an exception was made for the twenty reactions representing incorporation of amino acids into protein, which inherit the gene associations of the corresponding tRNA ligase reactions in CornCyc. (In principle these could be distinguished from sink reactions representing the expansion of free amino acid pools as cells grow and divide, but we have ignored this issue for now.) Note that, to support the adapted iRS1563 biomass equation, a reaction representing the production of free galactose from GDP-L-galactose was introduced (otherwise, release of galactose from UDP-galactose was catalyzed by two reactions in the pathways of indole-3-acetyl-ester conjugate biosynthesis and indole-3-acetate activation, likely not a major route for carbohydrate production.) Free galactose is not included in the individual biomass species used for data fitting. Compartmentalization ==================== Approaches differ to the subcellular compartmentalization in FBA models of eukaryotes, ranging from the assignment of compartments to a few key pathways known to function primarily outside the cytosol, as in the mitochondrial and chloroplastic “modules” of AraMeta [@Poolman2009] and RiceMeta [@Poolman2013] to the extremely comprehensive, data-driven approach of [@Mintz-Oron2012]. Here, we did not attempt to comprehensively assign reactions to their proper compartments; instead, we started with a modular approach similar to [@Poolman2013] in which some core metabolic pathways were compartmentalized (in our case, the TCA cycle and mitochondrial electron transport chain in the mitochondrion, the light reactions of photosynthesis, Calvin cycle, and some reactions of the C4 and photorespiratory pathways in the chlorophyll, and some reactions of the photorespiratory pathway in the peroxisome, with transport reactions added as necessary.) We then refined the compartment assignments of other reactions and pathways as needed to permit key metabolic functions and compartmentalize a limited number of additional reactions whose incorrect assignment to the cytosol we judged particularly likely to lead to misleading results. More details on individual compartmentalization choices and transport reactions are given below. Intracellular transport ----------------------- Sources (beyond those detailed below) informing the addition of intracellular transport reactions in the model included the transport reactions present in AraMeta [@Poolman2009], reviews of photorespiratory metabolism with attention to compartmentalization [@Reumann2006; @Foyer2009], a review of chloroplast transporters [@Weber2011], and a review of transport processes in C4 photosynthesis [@Brautigam2011]. In most cases we have not tried to reflect the mechanisms of the transport systems, where those are known, in any detail (exceptions include the triose phosphate-phosphate and PEP-phosphate transporters across the chloroplast envelope), nor have we associated genes with the transporters, even when they are known. Future work should pay greater attention to this aspect of the system. Photorespiratory pathway ------------------------ Following [@DouceHeldt] we assumed that reducing power was supplied to the peroxisome through an oxaloacetate-malate shuttle and NAD(H)-dependent malate dehydrogenase, and added an oxaloacetate-malate antiporter and a copy of `MALATE-DEH-RXN` to the peroxisome. Reactions of the pathway were localized following [@Reumann2006] and [@Foyer2009]. Note that glycine decarboxylase was assigned exclusively to the mitochondrion, while serine hydroxymethyltransferase was present in both the mitochondrion and the cytoplasm, where it plays a role in one-carbon metabolism [@Hanson2001]. Various ferredoxin-consuming pathways ------------------------------------- The model includes several pathways or reactions (e.g., sulfite and nitrite reduction and the chlorophyll cycle) which rely on ferredoxins for reducing power, and are localized to the chloroplast, where, in the light, reduced ferredoxins may be supplied by the photosynthetic electron transport chain. Rather than assign the reactions of these pathways to compartments appropriately, we added a reaction exchanging reduced ferredoxins and oxidized ferredoxins across the chloroplast boundary to supply ferredoxin-driven pathways in the cytosol. We emphasize that this is a convenient simplification and is not intended to represent a realistic mechanism. Ascorbate production -------------------- The L-galactonolactone dehydrogenase responsible for the final step of the ascorbate production pathway in CornCyc reduces cytochrome C and has been experimentally localized to the mitochondrial inner membrane, with its catalytic site facing outwards, into the intermembrane space [@Bartoli2000]. As the outer membrane is generally permeable to small molecules we have treated this reaction as acting directly on cytoplasmic galactonolactone and ascorbate. A sink for ascorbate as a biomass component was added, as it is found in substantial quantities in leaves (see, e.g., [@Foyer1983; @Smirnoff1996].) Ascorbate-glutathione cycle --------------------------- This cycle is present in multiple cellular compartments; in the model we included only cytosolic and chloroplastic instances (of which only the chloroplastic was ultimately expected to be relevant, as there was no supply of superoxides in the cytosol.) Note that none of the genes associated with monodehydroascorbate reductase could be assigned to the chloroplast under the rules described below: two had curated location in the peroxisome while `GRMZM2G320307` had no curated location and TargetP prediction of mitochondrial (`GRMZM2G320307_P01`) and cytoplasmic (`GRMZM2G320307_P02`, `GRMZM2G320307_P03`) locations. Reduction of monodehydroascorbate may also proceed non-enzymatically (see above) so this (enzymatic) reaction was removed from the chloroplast in favor of direct reduction by ferredoxin. Gene associations for compartmentalized reactions ================================================= Where a reaction was present in more than one compartment– that is, when two or more reactions in different compartments were associated with the same reaction record in CornCyc– we examined the genes associated with those reactions in CornCyc and assigned them to the instance of the reaction in the most appropriate compartment, as far as possible. Where the Plant Proteome Database [@Sun2009] provided manually curated location assignments for genes, those were used; otherwise, we used automatic location predictions by TargetP [@Emanuelsson2000] or in some cases referred to the gene’s annotation (both also provided by PPDB.) In general we assumed the appropriate location for a gene product was the cytoplasmic compartment absent a specific prediction of localization in the chloroplast, mitochondrion, or peroxisome. Where proteins were predicted to occur in a compartment where an no instance of a particular reaction was present, those gene associations were generally dropped from the model. When a gene was associated with a reaction in more than one compartment and also a reaction present in only one compartment, in general the association with the reaction in only one compartment was dropped, except for reactions which we believed based on literature evidence (including comments in CornCyc and PPDB) were assigned to the cytoplasmic compartment only because our compartmentalization process was incomplete. Some details on the judgment calls made in this process are provided in the comments to the file `gra_overrides.txt`; we comment here on a few unusual cases. NADH dehydrogenases ------------------- Cyclic electron transport around Photosystem I may occur through the chloroplast NADH dehydrogenase complex or an alternate pathway which in Arabidopsis involves PGR5 [@Munekage2004; @Shikanai2007]. In C3 plants the PGR5-dependent pathway may play the major role in tuning the photosynthetic ATP/NADPH ratio, while the NADH dehydrogenase pathway is implicated in stress responses [@Shikanai2007]. In contrast, in C4 plants the expression of the chloroplast NADH-dehydrogenase appears to correlate with photosynthetic ATP demand, while PGR5 expression does not, suggesting it is the NADH-dehydrogenase CET pathway which allows increased the increased ATP production required by the C4 system [@Takabayashi2005]. Thus, genes associated in CornCyc with the NADH dehydrogenase reaction for which a chloroplast location was predicted were reassociated with the model’s cyclic electron transport reaction (despite the fact that our somewhat abstract cyclic electron transport reaction may not accurately represent the biochemistry of the NADH-dependent pathway.) Pyruvate dehydrogenases ----------------------- In practice, pyruvate dehydrogenase complexes are found in the mitochondrion and chloroplast, but here we have not fully compartmentalized the chloroplastic pyruvate dehydrogenase and the pathways it supplies, instead leaving it in the cytosol. Thus, genes associated with the reactions of the complex with predicted chloroplast localization were associated instead with the cytosolic version. Genes with no curated or predicted location were left associated with both forms (splitting their expression data between them, in the fitting process.) Testing and consistency checking ================================ The compartmentalized single-cell model was checked in detail for conservation violations by testing the feasibility of net production or consumption of a unit of each internal species with all external transport and biomass sink reactions suppressed. Where such production was found feasible, the reactions involved were carefully inspected and stoichiometry coefficients adjusted to restore balance if necessary. In practice, this led only to the correction of erroneous reactions added by hand; as expected, no balance issues were found with reactions exported from CornCyc. In the final version, no such unrealistic processes are possible in the model under normal conditions. (Note that the species representing light input may be consumed in isolation, but the use of light energy to drive a futile cycle is not unrealistic, though we have not examined the details of the process found by the consistency checker in any detail.) Of course, demonstrating that no such production/consumption is feasible does not guarantee that all reactions in the model are properly balanced. Testing also verified that all individual biomass sink reactions, and the combined biomass reaction, could proceed at nonzero rates. SBML export {#sbml} =========== Component names --------------- SBML distinguishes a component’s name from its ID. Reactions and species in the SBML model were given name attributes according to the by calling the Pathway Tools `get_name_string` function on the frames in the database from which they derive, if any. The IDs of the SBML components were derived from the frame handles, replacing special characters with underscores as necessary to conform to the SBML sID standard. Note that for some reactions in CornCyc, the result of `get_name_string` is an EC number different from the EC number indicated by the label of the frame (e.g, `2.7.1.133-RXN`, for which ‘EC 2.7.1.159’ is returned.) The frame in CornCyc (if any) from which each reaction in the SBML model is ultimately derived is preserved as a comment in the reaction’s Notes element, to resolve any ambiguity. Gene annotations ---------------- Each reaction in the FBA model associated with a particular parent frame in CornCyc was given an association rule that combined all genes associated with that reaction in CornCyc, as well as all genes associated with all generic reactions of which the parent reaction is a specific form, in a logical ‘or’ relationship, stored in the reaction’s Notes element per the COBRA standard. Model refinement {#refinement} ================ Phosphoribulokinase ------------------- In early attempts to fit the model to the leaf gradient data, high costs were associated with the mesophyll phosphoribulokinase reaction in the source tissue when the bundle sheath CO~2~ level was high. We noted that in CornCyc 4.0 several genes were associated with both PRK and glyceraldehyde-3-phosphate dehydrogenase. To clarify the role of these genes we referred to annotations in the Plant Proteome Database [@Sun2009] and best hits in the Conserved Domain Database ([@Marchler-Bauer2004], accessed through NCBI.) Of the eight genes associated with PRK in CornCyc, three (`GRMZM2G039723`, `GRMZM2G337113`, `GRMZM2G162845`) appeared to encode GAPDH enzymes (per PPDB annotations and the presence of `Gp_dh_N` and `Gp_dh_C` domains), three (`GRMZM2G162529, GRMZM2G463280, GRMZM2G026024`) appeared encode to encode genuine phosphoribulokinases (per PPDB annotations and the presence of PRK domains), and two appeared to encode CP12-type regulatory proteins, with no obvious evidence for any individual protein sharing more than one of these roles. The regulatory role of CP12 does involve forming a complex with PRK and GAPDH, but this reduces, rather than enhancing or enabling, their individual activities [@Lopez-Calcagno2014]. We removed the PRK associations of the GAPDH and CP12 genes from our model. PPDB assigned these three GAPDH genes to a plastidic location based on experimental evidence, so we associated them with those reactions exclusively (removing associations with the cytosolic instances of EC 1.2.1.13 and/or EC 1.2.1.12.) Biomass equation {#biomass} ================ We developed a biomass equation following that used in [@Saha2011]. Our calculations are based on supplementary file S4 of that paper[^2], in particular sheet 2, ‘Biomass\_rxn’. That sheet derives a biomass equation corresponding to the production of one gram of plant dry weight, based on literature data on biomass composition; the description is divided into subreactions forming (e.g.) ‘nitrogenous compounds’, ‘lignin’, etc., which then participate in an overall biomass reaction.) The units of the stoichiometric coefficients are mmol. We have adopted most of the biomass composition assumptions of Saha et al wholesale, with gratitude for their efforts in compiling this data from the literature. However, we have made some minor adjustments, resulting in a different overall stoichiometry for biomass production. Fatty acids ----------- Saha et al represent the total lipid/fatty acid contribution to biomass as a pool of triglycerides in proportions apparently based on a maize oil measurement and thus probably reflective of seed triglyceride composition. We substitute measurements of the fatty acid content of mature maize leaf membrane lipids [@Rizov2000] and write a biomass sub-reaction which consumes the relevant free fatty acids (rather than their derivatives in the form of triacylglycerols, membrane lipids, etc.,) as shown in Table \[fatty\_acid\]. Fatty acid CornCyc compound mol. wt. (g/mol) mole fraction ------------- ------------------ ------------------ --------------- palmitic 255.42 0.104 palmitoleic 253.4 0.056 stearic 283.47 0.011 oleic 281.46 0.044 linoleic 279.44 0.132 linolenic 277.43 0.646 : Fatty acid proportions in biomass.[]{data-label="fatty_acid"} \ Weighting the molecular weights by the mole fractions, we find one mole of fatty acid in appropriate proportions weighs 272.4 g. Dividing the mole fractions by the overall molar weight and multiplying coefficients by 1000 to convert to millimoles, we arrive at the final equation: > 0.382 `PALMITATE` + 0.206 `CPD-9245` + 0.04 `STEARIC_ACID` + 0.162 `OLEATE_CPD` + 0.485 `LINOLEIC_ACID` + 2.372 `LINOLENIC_ACID` = where the left-hand side represents 1 g. Fractions add to less than 1.0 because we ignore trace (mole fraction $\leq$ 0.01) amounts of C14:0 and C20:0 fatty acids. Note that the leaf fatty acid composition is known to change along the developmental gradient, so specifying any single composition is an approximation; see [@Leech1973]. Hemicellulose ------------- We adopted the hemicellulose production reaction as is, using the species added to the model for this purpose, ‘`polysaccharide_[sugar]_unit`’. The resulting equation is: > 0.548 `polysaccharide_arabinose_unit` +\ > 1.248 `polysaccharide_xylose_unit ` +\ > 0.301 `polysaccharide_mannose_unit ` +\ > 0.144 `polysaccharide_galactose_unit` +\ > 3.254 `polysaccharide_glucose_unit` +\ > 0.166 `polysaccharide_galacturonate_unit` +\ > 0.166 `polysaccharide_glucuronate_unit` = `hemicellulose_biomass`. Total carbohydrates ------------------- We recalculated the stoichiometries of the carbohydrate-producing reaction to account for the differing molecular weight of our representation of cellulose (‘`CELLULOSE_monomer_equivalent`’, effectively a glucose molecule), account for the fact that one unit of hemicellulose represents one gram, not one (milli)mole, and express pectin in terms of `polysaccharide_galacturonate_unit`, reflecting a belief that UDP is released in the formation of pectin from UDP-D-galacturonate, rather than retained in the polymer [@CornCycPWY-1061]. It is not clear what form the ‘mannose’ referred to by Penning de Vries et al should be assumed to take, as free mannose is not found in plants under most circumstances (see, e.g., [@Herold1977; @Schnarrenberger1990; @PlantCycMANNCAT-PWY].) Here we somewhat arbitrarily choose mannose-6-phopshate. Table \[carbohydrates\] shows the calculation, resulting in the equation: > 0.067 `RIBOSE` + 0.278 `GLC` + 0.111 `FRU` + 0.039 `MANNOSE-6P` + 0.056 `GALACTOSE` + 0.146 `SUCROSE` + 2.220 `CELLULOSE_monomer_equivalent` + 0.400 `hemicellulose_biomass` + 0.259 `polysaccharide_galacturonate_unit` = `carbohydrates_biomass`. Organic acids ------------- We adopt this reaction as is. In the terminology of our model, the resulting equation is: > 0.556 `OXALATE` + 0.676 `GLYOX` + 1.515 `OXALACETIC_ACID` + 0.746 `MAL` + 1.562 `CIT` + 1.724 `CIS-ACONITATE` = `organic_acids_biomass`. Protein and free amino acids ---------------------------- We adopt these reactions as is. In the terminology of our model, the resulting equations are: > 1.15 `L-ALPHA-ALANINE` + 0.0959 `ARG` + 0.414 `L-ASPARTATE` + 0.0313 `CYS` + 1.53 `GLT` + 0.0445 `GLY` + 0.0915 `HIS` + 0.465 `ILE` + 1.51 `LEU` + 5.71e-05 `LYS` + 0.123 `MET` + 0.314 `PHE` + 0.762 `PRO` + 0.612 `SER` + 0.175 `THR` + 0.00409 `TRP` + 0.244 `TYR` + 0.25 `VAL` = `protein_biomass` and > 0.624 `L-ALPHA-ALANINE` + 0.319 `ARG` + 0.418 `L-ASPARTATE` + 0.231 `CYS` + 0.378 `GLT` + 0.740 `GLY` + 0.358 `HIS` + 0.424 `ILE` + 0.424 `LEU` + 0.380 `LYS` + 0.373 `MET` + 0.337 `PHE` + 0.483 `PRO` + 0.529 `SER` + 0.467 `THR` + 0.272 `TRP` + 0.307 `TYR` + 0.475 `VAL` = `free_aa_biomass`. Lignin ------ We adopt this reaction as is. In the terminology of our model, the resulting equation is: > 2.221 `COUMARYL-ALCOHOL` + 1.851 `CONIFERYL-ALCOHOL` + 1.587 `SINAPYL-ALCOHOL` = `lignin_biomass`. Nucleic acids ------------- We adopt this reaction as is (though note that, as discussed above, nucleotide triphosphates are not necessarily the appropriate best representation for polymerized nucleic acids). In the terminology of our model, the resulting equation is: > 0.247 `ATP` + 0.239 `GTP` + 0.259 `CTP` + 0.258 `UTP` + 0.255 `DATP` + 0.247 `DGTP` + 0.268 `DCTP` + 0.259 `TTP` = `nucleic_acids_biomass`. Nitrogenous compounds --------------------- We use the same nitrogenous compound weight fraction breakdown, but recalculate the stoichiometric coefficients accounting for the fact that the protein biomass, free amino acid biomass, and nucleotide biomass species each represent one gram, so that the appropriate stoichiometric coefficients of those species for the production of one total gram of nitrogenous compounds are simply the weight fractions; see Table \[nitrogenous\]. Component Species in model unit wt (mg) wt fraction units/g product --------------- ------------------ -------------- ------------- ----------------- Amino acids 1000.000 0.100 0.100 Proteins 1000.000 0.870 0.870 Nucleic acids 1000.000 0.030 0.030 : Nitrogenous biomass breakdown.[]{data-label="nitrogenous"} The resulting equation is > 0.100 `free_aa_biomass` + 0.870 `protein_biomass` + 0.030 `nucleic_acids_biomass` = `nitrogenous_biomass`. Inorganic materials ------------------- We ignore these entirely, as they play no other role in the model. (Note that even in iRS1563 the two species involved, potassium and chloride, participate only in source and sink reactions.) Total biomass reaction ---------------------- We drop the inorganic materials term (note that weight fractions now add to 0.95) and recalculate the stoichiometric coefficients, accounting for the fact that the component biomass subspecies each represent one gram; see Table \[total\]. The final equation is > 0.230 `nitrogenous_biomass` + 0.565 `carbohydrates_biomass` + 0.025 `fatty_acids_biomass` + 0.080 `lignin_biomass` + 0.050 `organic_acids_biomass` = `total_biomass`. Saha et al additionally incorporate an ATP cost in their overall biomass reaction, based on that used in an earlier Arabidopsis model (AraGEM [@GomesdeOliveiraDalMolin2010]) Combining this ATP hydrolysis with a sink of total biomass, we arrive at the overall equation for biomass production and growth (`CombinedBiomassReaction`): > 1.0 `total_biomass` + 30.0 `ATP` + 30.0 `WATER` = 30.0 `ADP` + 30.0 `Pi` + 30.0 `PROTON` Protonation ----------- Throughout, note that the molecular weights of species in our model may differ somewhat from those used in the iRS1563 table because of differing assumptions about protonation. The practical consequences of this difference should be limited. Oxalate ------- Early drafts of the model could not produce oxalate. CornCyc indicates its production as resulting only from ascorbic acid catabolism with concomitant production of L-threonate. Recent reviews suggest this is the primary pathway of oxalate production in plant species which form calcium oxalate crystals, with the threonate ultimately being oxidized to tartrate [@Franceschi95; @Franceschi2005; @Debolt2007], though the pathways of production of soluble oxalate are less clear [@Franceschi2005]. We found little immediate evidence that tartrate (or threonate) is formed in maize leaves at levels comparable to that of oxalate, or of pathways which could further metabolize the tartrate. Of the three reactions in iRS1563 which could produce oxalate, only one has an associated gene: oxalate carboxylase (oxalate = formate + CO2); KEGG R00522 (EC4.1.1.2). The gene, ‘ACG37538’, may correspond to `GRMZM2G103512`, whose best Arabidopsis hit is `AT1G09560.1` (germin-like protein 5); it may thus be more likely to be an oxalate-consuming oxidase [@Lane1993] than an oxalate carboxylase, though no function was computationally predicted for `GRMZM2G103512` in CornCyc. We decided the available information did not allow us to accurately model oxalate production in maize. However, to retain the iRS1563 biomass equation and ensure that mass and elemental balance was preserved, we allowed production of oxalate from oxaloacetate by oxaloacetase (EC 3.7.1.1; PlantCyc `OXALOACETASE-RXN`, [@PlantCycOXALOACETASE-RXN]). This simple reaction has been observed in fungi [@Hayaishi1956] but is considered unlikely to be widespread in plants [@Franceschi2005]. Plasmodesmatal transport reactions {#plasmodesmata} ================================== Species allowed to be exchanged between cell types through the plasmodesmata included: - carbon dioxide and oxygen; - known C4 cycle metabolites alanine, aspartate, malate, PEP, and pyruvate; - the Calvin cycle intermediates glyceraldehyde 3-phosphate and 3-phosphoglycerate; - photorespiratory metabolites glycerate, glycolate, serine, and glycine; - nutrients sucrose, phosphate, nitrate, ammonia, sulfate and magnesium; - glutamate and 2-ketoglutarate; - and cysteine and glutathione [@Burgener1998]. The inclusion of compounds involved in NAD-ME C4 or C3-C4 intermediate photorespiratory carbon concentrating mechanism is not meant to suggest such a system is necessarily active in maize but merely reflects our knowledge that significant transport of those species between mesophyll and bundle sheath can occur under at least some circumstances. {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} [max width=1.25,center]{} reaction name in model associated genes --------------------------------- --------------------------------------------------- ------------------ malate dehydrogenase (NADP) `MALATE_DEHYDROGENASE_NADP__RXN_chloroplast` 1 alanine aminotransferase `ALANINE_AMINOTRANSFERASE_RXN` 10 aspartate aminotransferase `ASPAMINOTRANS_RXN` 7 NAD-malic enzyme `EC_1_1_1_39` 2 NADP-malic enzyme (cytosol) `MALIC_NADP_RXN` 4 NADP-malic enzyme (chloroplast) `MALIC_NADP_RXN_chloroplast` 2 PEPCK `PEPCARBOXYKIN_RXN` 6 PPDK `PYRUVATEORTHOPHOSPHATE_DIKINASE_RXN_chloroplast` 2 adenylate kinase `ADENYL_KIN_RXN_chloroplast` 6 pyrophosphatase `INORGPYROPHOSPHAT_RXN_chloroplast` 2 0.25cm {width="\textwidth"} Index to additional supporting information files {#index-to-additional-supporting-information-files .unnumbered} ================================================ Except as noted, these are available as arXiv ancillary files. S11 Model {#S11_Model .unnumbered} --------- [**iEB5204 in SBML format.**]{} S12 Model {#S12_Model .unnumbered} --------- [**iEB2140 in SBML format.**]{} S13 Model {#S13_Model .unnumbered} --------- [**iEB2140x2 in SBML format.**]{} S14 Protocol {#S14_Protocol .unnumbered} ------------ [**Source code for the nonlinear constraint-based modeling package fluxtools.**]{} Available at <http://github.com/ebogart/fluxtools>. S15 Protocol {#S15_Protocol .unnumbered} ------------ [**Source code and input files for the calculations discussed above.**]{} Available at <http://github.com/ebogart/multiscale_c4_source>. S16 Table {#S16_Table .unnumbered} --------- [**Predicted variable values along the leaf gradient.**]{} S17 Table {#S17_Table .unnumbered} --------- [**Upper and lower bounds on predicted values of selected variables along the leaf gradient, from FVA calculations.**]{} [^1]: The branch leading to heme production is not included in the reconstruction. [^2]: Specifically, `journal.pone.0021784.s004.xls`, as downloaded from the PLoS One web site 20 November 2013

--- author: - 'T. Kramer' - 'M. Läuter' date: Received title: ' Outgassing induced acceleration of comet 67P/Churyumov-Gerasimenko ' --- Introduction ============ Studying the non-gravitational acceleration of comets provides important insights into the sublimation of cometary ices. Earth-bound astrometry allows one to determine the orbital corrections arising from the sublimation of volatiles. Non-gravitational accelerations are often invoked to explain the orbital evolution of comets and even interstellar objects entering the solar system ([@Whipple1950; @Marsden1973; @Micheli2018; @Sekanina2019]). As pointed out by [@Yeomans2004] it is therefore of interest to compare widely used models with the precise data returned from spacecraft missions. For comet 67P [@Krolikowska2003] provided an assessment of the Marsden parameters within the asymmetric outgassing model of [@Yeomans1989] before the arrival of Rosetta at the comet. The determination of accurate parameters from Earth bound telescopic observations requires to monitor the position of the comet at several apparitions. The situation is different for the orbit of 67P as observed by Rosetta. Rosetta accompanied 67P for more than two years and provided measurements of the three-dimensional position vector with an accuracy better than 100 km (see [@Godard2017]). The discrepancy between Earth bound orbit prediction and the position where Rosetta encountered 67P $590$ days before perihelion requires to adjust the Marsden-type orbit by $2000$ km. From this it can be estimated that Rosetta provided an up-to twenty times higher accuracy compared to previous orbit determinations. In conjunction with the precisely known rotation state of 67P we perform an attribution of the actual orbital changes to the sublimation activity across the nucleus. To account for the observed gas release of 67P requires to extend the non-gravitational acceleration models developed by Whipple, Marsden, and Sekanina. To this end we introduce a Fourier decomposition of the sublimation induced force and establish a general expression connecting the diurnal variations of the sublimation rate with the orbital evolution. This formalism simplifies the analysis by eliminating intermediate angles and emphasizes the lack of proportionality between total production rate and non-gravitational acceleration. Orbit changes by non-gravitational acceleration =============================================== The equation of motion for the position vector $\vec{r}$ of the cometary nucleus is described by contributions of solar system gravitational acceleration $\vec{a}_\mathrm{G}$ and the additional non-gravitational acceleration $\vec{a}_\mathrm{NG}$ (see e.g. [@Yeomans2004]) $$\label{eq:orbit} \ddot{\vec{r}} = \vec{a}_\mathrm{G} + \vec{a}_\mathrm{NG}.$$ In general this equation holds in any inertial system but we considered it in the Earth’s equatorial system. The gravitational part $\vec{a}_\mathrm{G} = \vec{a}_\mathrm{G}(\vec{r},t)$ is evaluated at the momentary position vector and takes into account the acceleration due to all solar system planets and the Earth Moon, Pluto, Vesta, and Ceres. Additional corrections due to relativistic effects are ignored here, since they play only a minor role for 67P. Relativistic corrections could be added for other objects moving faster around the sun. The non-gravitational part is affected by the sublimation of volatiles on the nucleus across the surface. For given initial conditions $\vec{r}(t_0)$, $\dot{\vec{r}}(t_0)$ and a suitable model of the non-gravitational acceleration, the orbit of a comet can be integrated with high precision. Marsden-type orbits {#ssec:marsden} ------------------- To understand the origin of the non-gravitational acceleration $\vec{a}_\mathrm{NG}$ it is helpful to consider different reference frames. Starting from the icy snowball model introduced by [@Whipple1950; @Whipple1951] to account for sublimation processes, Marsden and co-authors developed a powerful parametrization for the non-gravitational acceleration in a series of papers [@Marsden1968; @Marsden1969; @Marsden1970; @Marsden1971; @Marsden1972; @Marsden1973]. Marsden expressed $\vec{a}_\mathrm{NG}$ with respect to three right-handed orthogonal unit vectors with $\vec{e}_\mathrm{r}$ pointing from the sun to the nucleus, $\vec{e}_\mathrm{n}$ directed along the orbital angular momentum perpendicular to the orbital plane, and $\vec{e}_\mathrm{t}$ being orthogonal to both, $\vec{e}_\mathrm{r}$ and $\vec{e}_\mathrm{n}$. By comparing 14 cometary orbits [@Marsden1973] derived the following model for the non-gravitational acceleration $$\label{eq:marsden} \vec{a}_{\mathrm{NG},\text{Marsden}} =g(r') (A_1 \vec{e}_r+A_2\vec{e}_t+A_3\vec{e}_n),$$ with the constant parameters $A_1$, $A_2$, $A_3$ and the empirical activity function $$\label{eq:gr} g(r')=\alpha {\left(\frac{r}{r_0}\right)}^{-m}{\left(1+{\left(\frac{r}{r_0}\right)}^{n}\right)}^{-k}.$$ The solar distance $r'=r(t-\Delta t)$ includes a time-shift $\Delta t$ introduced by [@Yeomans1989] to account for asymmetries of the activity with respect to the perihelion as studied by [@Sekanina1988]. As we discuss later, $g(r)$ is not directly proportional to the sublimation flux as originally stated by [@Marsden1973], Eq. (4). The Marsden parameters provide an excellent, albeit empirical description of the non-gravitational acceleration. The integration of a Marsden-type orbit proceeds by solving Eq.  with the non-gravitational acceleration Eq. . ![ Upper panel: Euclidean error norm of various orbital solutions with respect to the multi-arc ESOC reference. Lower panel: Residuals with respect to the Earth bound range of the orbital solutions. $\mathbf{r}_0$ is the purely gravitational solution, $\mathbf{r}_\text{Marsden}$ a Marsden type orbit, and $\mathbf{r}_{\rm obs}$ the best fit reconstructed orbit for the non-gravitational acceleration shown in Fig. \[fig:angpqr\]. The shaded band indicates the variation across $31$ initial positions for the Marsden solution. []{data-label="fig:errmarsden"}](fig_1a.pdf "fig:"){width="0.99\linewidth"} ![ Upper panel: Euclidean error norm of various orbital solutions with respect to the multi-arc ESOC reference. Lower panel: Residuals with respect to the Earth bound range of the orbital solutions. $\mathbf{r}_0$ is the purely gravitational solution, $\mathbf{r}_\text{Marsden}$ a Marsden type orbit, and $\mathbf{r}_{\rm obs}$ the best fit reconstructed orbit for the non-gravitational acceleration shown in Fig. \[fig:angpqr\]. The shaded band indicates the variation across $31$ initial positions for the Marsden solution. []{data-label="fig:errmarsden"}](fig_1b.pdf "fig:"){width="0.99\linewidth"} Orbit determination based on spacecraft data -------------------------------------------- To move beyond the Marsden model requires more detailed data from spacecraft missions or radar observations about the magnitude and direction of the observed non-gravitational acceleration in space. For 67P the Rosetta mission provides a two years data set for both, the rotation-axis orientation (see [@Kramer2019]) and the orbital evolution, but required an iterative process to obtain the three-dimensional non-gravitational acceleration with high precision. We started with the the multi-arc orbital solution from the flight dynamics team at ESOC, which is available as SPICE kernel (CORB\_DV\_257\_03\_\_\_T19\_00345.BSP) and is returned from the Horizons system as position for 67P during the Rosetta mission. In the following we refer to it as $\vec{r}_\text{ESOC}(t)$. Before and after the Rosetta mission, Horizons returns the solution 67P/K154/2 based on Marsden parameters with a discontinuous jump by $2000$ km into the Rosetta period. Besides this discontinuity in Horizons, further discontinuities in the ESOC position vector $\vec{r}_\text{ESOC}(t)$ during the Rosetta mission exist, which do not allow one to obtain the acceleration by a second derivative of the position vector (see the discussion by [@Attree2019]). An accurate estimate of the non-gravitational acceleration is tied to finding the best possible initial condition during a time with negligible cometary activity. We performed an iterative orbit refinement to identify the initial conditions which minimize the error norm with respect to the ESOC data in the time period $(-400,-200)$ days from perihelion (see Table \[tab:ICorbit\]). During this initial search only the gravitational acceleration was considered, thus we evaluated the cometary orbits with Eq.  with the setup $\vec{a}_\mathrm{NG}=0$. In the next step we added Marsden-type non-gravitational accelerations with the parameters used by Horizons (see Table \[tab:A123Horizons\] and the parameter estimation by [@Krolikowska2003]) and we solved Eqs. , to obtain $\vec{r}_\text{Marsden}(t)$. We extended the integration to the years 1959-2022 (limited by two Jupiter encounters) and verified that our initial conditions lead to orbital solutions within $4400$ km with respect to the Horizons solution. We compared our results with the ones obtained by submitting the corresponding osculating elements to the Advanced Horizons Asteroid & Comet SPK web interface [@Horizons2019]. For the given initial conditions our Marsden-type solution (Fig. \[fig:errmarsden\]) is considerably closer to $\vec{r}_\text{ESOC}$ than the Marsden solution given by [@Attree2019]. [@Attree2019] reported a Marsden solution with a difference in Earth bound range $|\vec{r}_\text{ESOC}-\vec{r}_\text{Earth}|-|\vec{r}_\text{Marsden}-\vec{r}_\text{Earth}|$ of $1500$ km $400$ days after perihelion, while our Marsden solution at that time differs only by $50$ km. This highlights the necessity to perform a throughout statistical ensemble analysis of initial conditions to validate the non-gravitational part of the orbital acceleration. In the final iteration we fitted the remaining difference vector $\Delta \vec{r}(t)=\vec{r}_\text{ESOC}-\vec{r}_\text{Marsden}$ to a combination of exponentials and polynomials up to fourth order, which provided (upon adding it to the Marsden solution) a differentiable representation of the observed orbit $$%\label{eq:robs} \vec{r}_{\rm obs}(t)=\vec{r}_\text{Marsden}(t)+\Delta \vec{r}_\text{fit}(t)$$ (see Fig. \[fig:errmarsden\]). Only after these iterative steps the non-gravitational acceleration was obtained from $$\label{eq:ngobs} \vec{a}_{\mathrm{NG},\mathrm{obs}}=\ddot{\mathbf{r}}_\mathrm{obs}(t) -\vec{a}_{\mathrm{G}}(\mathbf{r}_\mathrm{obs}(t)).$$ The validation of the second order derivative of the initially noisy position vector required a careful analysis of errors introduced by the fit. We have repeated the entire analysis by applying the smoothing and differentiation filter introduced by [@Savitzky1964] with the corrections by [@Steinier1972] to the larger difference vector $\vec{r}_\text{ESOC}-\vec{r}_0$, where $\vec{r}_0$ includes only gravitational forces. The ten times larger distance increases the fit uncertainties but leads to qualitative similar results in the period $\pm 100$ days from perihelion. Remaining systematic errors are discussed in Sect. \[sec:physprop\]. ![Non-gravitational acceleration of 67P in the Earth equatorial frame. Upper panel: Observed $\vec{a}_{\mathrm{NG},\mathrm{obs}}$. The shaded band indicates the variation across the set of initial conditions. Lower panel: Marsden model $\vec{a}_{\mathrm{NG},\mathrm{Marsden}}$. The parameters are from Tables \[tab:ICorbit\], \[tab:A123Horizons\]. []{data-label="fig:angpqr"}](fig_2a.pdf "fig:"){width="0.99\linewidth"}\ ![Non-gravitational acceleration of 67P in the Earth equatorial frame. Upper panel: Observed $\vec{a}_{\mathrm{NG},\mathrm{obs}}$. The shaded band indicates the variation across the set of initial conditions. Lower panel: Marsden model $\vec{a}_{\mathrm{NG},\mathrm{Marsden}}$. The parameters are from Tables \[tab:ICorbit\], \[tab:A123Horizons\]. []{data-label="fig:angpqr"}](fig_2b.pdf "fig:"){width="0.99\linewidth"} The resulting non-gravitational acceleration is shown in Fig. \[fig:angpqr\]. In combination with the initial condition (Table \[tab:ICorbit\]), the retrieved $\vec{a}_{\mathrm{NG},\mathrm{obs}}$ reproduced the multi-arc solution $\vec{r}_\text{ESOC}$ for $\pm400$ days around perihelion with an mean error of $21$ km, mainly caused by the nonphysical jumps (see black curve in Fig. \[fig:errmarsden\]). At most times the error is around 10 km ($\approx 5$ cometary radii). To test the sensitivity of the orbit with respect to the initial conditions, we investigated an ensemble of 1000 nearby initial conditions and verified that no further improvement is obtained. Out of the 1000 initial conditions we identified $31$ orbits with a mean error $<22$ km in the $\pm400$ days interval around perihelion, which all originate from a phase space volume extending about $3$ km around the initial position listed in Table \[tab:ICorbit\] and with $<10^{-5}$ variations in the velocities. In the following we show results for this set of $31$ solutions in the form of shaded bands to estimate the uncertainties of derived quantities from the orbits. quantity value unit ----------- ------------------------- ------ $x$ $+1.332184491942$ au $y$ $-2.770636585665$ au $z$ $-1.613532002605$ au $\dot{x}$ $0.0042333005604361959$ au/d $\dot{y}$ $0.0074132927293021402$ au/d $\dot{z}$ $0.0034828287065240359$ au/d : Representative initial conditions for the orbit integration at JDB $2456897.71970$ ($\approx-350$ days from perihelion), mean equator and equinox of the Earth J2000 frame. Additional $30$ nearby initial conditions have been identified which differ on average less than $22$ km over the $(-400,400)$ days interval from the partly nonphysical ESOC orbit. []{data-label="tab:ICorbit"} quantity value unit ------------ ---------------------------------- ---------- $A_1$ $1.066669896245\times 10^{-9}$ au/d$^2$ $A_2$ $-3.689152188599\times 10^{-11}$ au/d$^2$ $A_3$ $2.483436092734\times 10^{-10}$ au/d$^2$ $\Delta t$ $35.07142$ d : Non-gravitational acceleration parameters used by Horizons in conjunction with the standard parameters $\alpha=0.1112620426$, $k=4.6142$, $m=2.15$, $n=5.093$, $r_0=2.808$ for $g(r)$ in Eq. (\[eq:gr\]). []{data-label="tab:A123Horizons"} Transformation of the non-gravitational acceleration to the cometary body {#sec:trafo} ========================================================================= Next we connected the observed non-gravitational acceleration to the cometary activity on the surface. We assume that the nucleus is not in tumbling rotation and that the rotation period and axis orientation are fixed during a single cometary rotation. For 67P this is an excellent approximation since the orientation over $800$ days changed only by $0.5^\circ$ and the comet rotation period $T_{\rm rot}$ decreased by $21$ minutes from $12.4$ h $300$ days before the 2015 perihelion (see [@Godard2017; @Kramer2019]). The Marsden non-gravitational acceleration given by Eq.  is restricted to the direction dictated by the time-independent linear combination $A_1,A_2,A_3$ of the co-moving basis. This rigid link ignores the physical properties of the nucleus, in particular the rotation axis orientation and rotation period encoded in the angular velocity vector $\vec{\omega}$. For cometary activity driven by the solar illumination on the nucleus the $A_1$, $A_2$, $A_3$ components are no longer time independent. [@Sekanina1967] studied the arising temporal variation of the Marsden parameters under the assumption of a fixed orientation of the rotation axis. Sekanina introduced a coordinate system that takes into account the obliquity of the comet equatorial plane with the orbital plane to study the illumination conditions of the subsolar point during the orbital motion. This approach was further extended by [@Whipple1979], [@Sekanina1984] and [@Sitarski1990] to time-dependent Marsden parameters, including precession models with a changing rotation axis and associated oblateness of the nucleus. Before the rotation state of 67P was known, [@Krolikowska2003] applied different models to 67P including forced precession solutions. For 67P a detailed shape model is available from [@Preusker2017] and the changes of the rotation state are known (see [@Jorda2016; @Kramer2019]). 67P showed a very repetitive diurnal pattern of gas and dust in the coma across the entire illuminated nucleus, indicating a very regular and periodically repeating outgassing [@Kramer2016; @Kramer2017; @Kramer2018; @Lauter2019]. Of particular interest is the surface activity with respect to the subsolar point in the body frame. For a given rotation vector $\vec{\omega}$ and position vector $\vec{r}$ of the nucleus the rotation matrix $\tens{R}_{\mathrm{com\rightarrow equ}}$ is the transformation from the cometary equatorial frame (without nucleus rotation) to the Earth’s equatorial frame, $$\label{eq:comtrans} \tens{R}_{\mathrm{com\rightarrow equ}} = \left( \frac{\vec{h}}{|\vec{h}|}, -\frac{\vec{\omega}\times\vec{r}}{|\vec{\omega}\times\vec{r}|}, \frac{\vec{\omega}}{|\vec{\omega}|} \right),$$ with $\vec{h} = -(\vec{\omega}\times\vec{r})\times\vec{\omega}$. It puts the sun at a fixed subsolar longitude. This construction is very similar to Sekanina’s system (both share the basis vector $\vec{\omega}/|\vec{\omega}|$). Any arising force (observed or modeled) from cometary activity $\vec{F}^\mathrm{equ} = \tens{R}_{\mathrm{com\rightarrow equ}} R_{\rm z}(-\omega t)\vec{F}^{\rm bf}$ is expanded in a Fourier series with respect to the subsolar longitude $$\label{eq:FBF} \vec{F}^{\rm bf}(\lambda_\sun)= \begin{pmatrix} D_{x0}-D_{x1} \sin\lambda_\sun-D_{x2}\cos\lambda_\sun+\ldots\\ D_{y0}-D_{y1} \sin\lambda_\sun-D_{y2}\cos\lambda_\sun+\ldots\\ D_{z0}-D_{z1} \sin\lambda_\sun-D_{z2}\cos\lambda_\sun+\ldots \end{pmatrix} .$$ The Fourier coefficients $D$, in principle defined for each single rotation, are slowly varying functions with the orbital positions around the sun. This expression encompasses comets with few active regions (see [@Jewitt1997] for a simple model of a rectangular shaped comet) as well as globally active ones. The Fourier representation facilitates the rotational averaging across one rotation period ($\lambda_\sun(t)=-\omega t$ for 67P) in the inertial system $$\label{eq:Fav} \langle\vec{F}^\mathrm{equ}\rangle = \tens{R}_{\mathrm{com\rightarrow equ}} \int_0^{T_{\rm rot}} \frac{{\rm d}t}{T_{\rm rot}} \tens{R}_{\rm z}(-\omega t) \vec{F}^{\rm bf}(-\omega t)$$ with the rotation matrix around the $z$ axis $\tens{R}_\mathrm{z}$ in the notation of [@Montenbruck2000]. The final expression for the non-gravitational acceleration acting on the orbit of a comet with mass $M$ ($10^{13}$ kg for 67P) is given by inserting Eq. (\[eq:FBF\]) into Eq. (\[eq:Fav\]) $$\label{eq:FF} \vec{a}_{\mathrm{NG},\mathrm{obs}}=\frac{\langle\vec{F}^{\rm equ}\rangle}{M} = \frac{1}{M} \tens{R}_{\mathrm{com\rightarrow equ}} \begin{pmatrix} D_1 \\ D_2 \\ D_3 \end{pmatrix}$$ with the three linear combinations remaining from the complete Fourier expansion $$\begin{aligned} \begin{pmatrix} D_1 \\ D_2 \\ D_3 \end{pmatrix} &=& \begin{pmatrix} (-D_{y1}-D_{x2})/2 \\ (+D_{x1}-D_{y2})/2 \\ D_{z0} \end{pmatrix}\\\nonumber &=&\sqrt{D_1^2+D_2^2+D_3^2} \begin{pmatrix} \cos \phi_D \cos \lambda_D \\ \cos \phi_D \sin \lambda_D \\ \sin \phi_D \end{pmatrix} .\end{aligned}$$ The parameters $D_1$, $D_2$, $D_3$ are the force coefficients with respect to the cometary equatorial frame (represented by the transformation in Eq. ). The link from Eq.  to the commonly used Sekanina angles is established by expressing $$\label{eq:commont} \tens{R}_{\mathrm{com}\rightarrow\mathrm{equ}} = (\vec{P},\vec{Q},\vec{R}) \tens{R}_\mathrm{z}(\Phi) \tens{R}_\mathrm{x}(-I) \tens{R}_\mathrm{z}(\pi-\theta_0)$$ in terms of the orbital vectors $\vec{P}$, $\vec{Q}$, $\vec{R}$, the obliquity $I$ of the orbit plane to the equator of the comet, the argument of the subsolar meridian at perihelion $\Phi$, and the longitude of the subsolar meridian from the ascending node of the orbit plane on the equator $\theta_0$. The Marsden basis vectors in Sect. \[ssec:marsden\] have the representation $$\label{eq:marmont} (\vec{e}_\mathrm{r}, \vec{e}_\mathrm{t}, \vec{e}_\mathrm{n}) = (\vec{P},\vec{Q},\vec{R}) \tens{R}_\mathrm{z}(-\nu)$$ with the true anomaly $\nu$. The vector with longitude $\lambda_D$ and latitude $\phi_D$ in the cometary equatorial frame transforms to the components $U_\mathrm{r}, U_\mathrm{t}, U_\mathrm{n}$ for the Marsden basis by the relation $$\begin{pmatrix} U_\mathrm{r} \\ U_\mathrm{t} \\ U_\mathrm{n} \end{pmatrix} = (\vec{e}_\mathrm{r}, \vec{e}_\mathrm{t}, \vec{e}_\mathrm{n})^{-1} \tens{R}_{\mathrm{com}\rightarrow\mathrm{equ}} \begin{pmatrix} \cos\phi_D \cos\lambda_D \\ \cos\phi_D \sin\lambda_D \\ \sin \phi_D \end{pmatrix}.$$ Setting $\theta + \pi = \lambda_D$ and $\phi = - \phi_D$ yields Eq. (2) in [@Sekanina1981]. Assuming additionally $-\phi_D$ to be the latitude of the subsolar point this relation simplifies to the classical Eq. (4) in [@Sekanina1981]. Physical properties from non-gravitational acceleration {#sec:physprop} ======================================================= The parameters $D_1$, $D_2$, $D_3$ and with it the non-gravitational acceleration arise from the diurnally averaged activity along the spin axis and the amplitudes in the equatorial plane of the comet. This is reflected in the observed data once decomposed in the cometary equatorial frame in terms of spherical coordinates, see Fig. \[fig:drobust\]. The longitude $\lambda_D$ and latitude $\phi_D$ denote the direction of the acceleration with respect to the subsolar point. Around perihelion the observed acceleration points towards the north in accordance with the subsolar latitude $\phi_{\rm sun}$ around $-50^\circ$, while at the two equinox crossings the acceleration lies in the equatorial plane. Up to a shift towards more southern latitudes the seasonal variation of the subsolar latitude is reflected in the observations. This provides a confirmation of the validity of Sekanina’s approach (see Sect. \[sec:trafo\]) with respect to the seasonal component. The determination of the diurnal lag angle with respect to the solar illumination shows larger uncertainties with no lag discernible up to perihelion. After perihelion the acceleration vector lags behind (in time) with respect to the momentary anti-solar direction up to $50^\circ$. This lag disappears around 140 days after perihelion, when the coma gets more and more CO$_2$ dominated (see [@Lauter2019]). The lag cannot be explained by a forced precession model, since the rotation state changes of 67P are small. In principle a varying surface activity across the nucleus can lead to a lag angle if local surface normal directions do not point towards the sun (see [@Samarasinha1996] for an illustration). [@Davidsson2004] studied the variations of the non-gravitational force with respect to varying activity patterns across an ellipsoidal nucleus (see their Fig. 6). They find small directional changes of the non-gravitational acceleration. Coma observations of 67P indicate a very repetitive gas and dust release across the entire illuminated surface, see [@Kramer2018]. This is inline with the observation that sublimation models of 67P are not capable of reproducing the reported orbit $\vec{r}_{\mathrm{ESOC}}$ within an Earth-bound range error $<10$ km. For instance, the model of [@Attree2019] results in a deviation of $140$ km with respect to the observed Earth bound range $150$ days after perihelion. Besides the shape of the nucleus, another possible cause of an diurnal lag angle is a delay between the maximum illumination and the highest gas release in some areas. From the observation of sunset jets on 67P [@Shi2016] obtain thermal delays of peak surface and sublimation temperatures of about $1-2$ h (corresponding to a rotation of $29^\circ-58^\circ$) in the Ma’at region in April 2015. Their calculations are based on a thermal inertia of $50$ Wm$^{-2}$K$^{-1}$s$^{1/2}$. At perihelion and at other locations on the surface, the thermal delays might differ from these values, or effects can cancel. Finally, the observed lag angle could be affected by an (unknown) systematic error of the ESOC orbit, which cannot be detected by the present analysis. A conclusive attribution of the observed lag angle to a physical process requires further modelling.\ The initial motivation for the study of non-gravitational acceleration was to deduce the total production flux of a comet. The total sublimation flux $Q(r)$ is approximately given by the product of the average gas velocity $v_{\rm gas,av}$ with the magnitude $a_{\mathrm{NG}}$ of the non-gravitational acceleration divided by the cometary mass $M$ $$\label{eq:aq} a_{\mathrm{NG},\mathrm{Marsden}}=\sqrt{A_1^2+A_2^2+A_3^2}\; g(r)\approx Q(r) v_{\rm gas,av}/M$$ The Fourier decomposition of the total force allows one to clarify the relation between sublimation flux and non-gravitational acceleration after factoring out the average gas velocity $v_{\rm av}$: $$\begin{aligned} \label{eq:Mav1} Q&=& \int_0^{T_{\rm rot}} \frac{{\rm d}t}{T_{\rm rot}} \int_{\text{surf}}\!\!\!{\rm d}S\; \frac{|\vec{F}(\vec{r}_{\rm bf},t)|}{|\vec{v}_{\rm gas}(\vec{r}_{\rm bf},t)|} \\\nonumber &\approx& \int_0^{T_{\rm rot}} \frac{{\rm d}t}{T_{\rm rot}} \left(\int_{\text{surf}}\!\!\!\!\!{\rm d}S\; |\vec{F}(\vec{r}_{\rm bf},t)|\right) {\left(\int_{\text{surf}}\!\!\!\!\!{\rm d}S\; |\vec{v}_{\rm gas}(\vec{r}_{\rm bf},t)|\right)}^{-1}.\end{aligned}$$ Next, we applied the triangle inequality to obtain a lower bound for the sublimation flux $$Q \ge \int_0^{T_{\rm rot}} \!\!\!\!\! \frac{{\rm d}t}{T_{\rm rot}\; v_{\rm gas,av}} \left| \vec{F}^{\rm bf}(-\omega t)\right|,\; v_{\rm gas,av}=\int_{\text{surf}}\!\!\!\!\!{\rm d}S\; |\vec{v}_{\rm gas}(\vec{r}_{\rm bf},t)| .$$ Inserting Eq. (\[eq:FBF\]) for the force components and neglecting faster oscillating terms allowed us to perform the integration analytically $$\label{eq:Mav} Q\approx\frac{4 \sqrt{\Delta}}{2\pi v_{\rm gas,av}} {\rm E}\left( \frac{-D_{x1}^2-D_{y1}^2-D_{z1}^2+D_{x2}^2+D_{y2}^2+D_{z2}^2}{\Delta} \right),$$ where $\Delta=D_{x0}^2+D_{y0}^2+D_{z0}^2+D_{x2}^2+D_{y2}^2+D_{z2}^2$, ${\rm E}(x)$ denotes the complete elliptic integral. The result shows that the magnitude of the non-gravitational acceleration contains additional force components besides the $D_1$, $D_2$, $D_3$ components, which lead to minor corrections. An entirely independent determination of the gas production for 67P has been performed by [@Lauter2019] by analyzing the in-situ data of the Double Focusing Mass Spectrometer (DFMS) of the Rosetta Orbiter Spectrometer for Ion and Neutral Analysis (ROSINA). The non-gravitational acceleration by Eqs. (\[eq:aq\]) shows a remarkable agreement to the ROSINA derived water production $Q_{\rm H2O}(r)$ of 67P from [@Lauter2019], Fig. \[fig:drobust\], with gas velocity $v_{\rm gas,av}=480$ m/s. ![ Observed non-gravitational acceleration in the cometary equatorial frame, magnitude (upper panel) and corresponding latitude and longitude of the direction of the sublimation force (lower panel). The latitude $\phi_D$ is correlated with the anti-solar latitude ($-\phi_{\rm sun}$) (black line in the lower panel), while the longitude correlates with the night terminator $\lambda_D=180^\circ$. The shaded band indicates the variation across the set of initial conditions. The red graph in the upper panel shows the ROSINA derived water production of 67P from [@Lauter2019]. []{data-label="fig:drobust"}](fig_3a.pdf "fig:"){width="0.99\linewidth"} ![ Observed non-gravitational acceleration in the cometary equatorial frame, magnitude (upper panel) and corresponding latitude and longitude of the direction of the sublimation force (lower panel). The latitude $\phi_D$ is correlated with the anti-solar latitude ($-\phi_{\rm sun}$) (black line in the lower panel), while the longitude correlates with the night terminator $\lambda_D=180^\circ$. The shaded band indicates the variation across the set of initial conditions. The red graph in the upper panel shows the ROSINA derived water production of 67P from [@Lauter2019]. []{data-label="fig:drobust"}](fig_3b.pdf "fig:"){width="0.9\linewidth"} Conclusions =========== The Rosetta mission to comet 67P provided the unique opportunity to retrieve the non-gravitational acceleration of a comet independently from Earth bound observations. In conjunction with the known rotation state the position data derived from Rosetta allowed us to verify commonly invoked assumptions about the non-gravitational acceleration. We have shown the sensitivity of the orbit reconstruction to the initial conditions and identified a phase-space volume 350 days before perihelion that leads to orbital solutions following the reported ESOC positions within a mean deviation of $22$ km. This close match allowed us to extract the three-dimensional non-gravitational acceleration and to relate it to the activity on the nucleus. Using a Fourier series we have decomposed the non-gravitational acceleration into the averaged outgassing along the rotation axis and the amplitudes of the outgassing along the equatorial plane of the comet. A similar analysis and Fourier theory has been carried out by [@Kramer2019] for the rotation state of 67P. We provided error bounds for all derived quantities based on an extensive analysis of initial conditions for the orbital integration. We find up to perihelion no clear signal of a lag angle between illumination and force direction, while at later times deviations from the instantaneous illumination become apparent. The seasonal effect of the solar illumination on the non-gravitational acceleration is reflected by a strong correlation of the subsolar latitude and the sublimation force direction. The agreement of the observed non-gravitational acceleration with the seasonal illumination conditions demonstrates that the non-gravitational acceleration can be explained by the water-ice distribution on the entire nucleus. The derivation of the diurnal lag carries considerably larger errors at times later than 100 days after perihelion, but points to a shift toward larger delay times 50-100 days after perihelion. The non-gravitational acceleration alone is most indicative about active areas in terms of a zonal mean. Due to the lack of longitudinal variations the detection of local active areas across the surface can not be expected. This is in contrast to the torque affecting the rotation axis orientation and rotation period: both react sensitively to local activity variations ([@Kramer2019]). Finally, we have shown that solely based on the analysis of the Rosetta orbit a close agreement with the in-situ measurements of the gas coma and production of 67P exists. This connection allows one to relate Earth bound astrometry and production estimates with accurate in-situ measurements and models of cometary activity, a prerequisite to advance non-gravitational acceleration models. The authors acknowledge the North-German Supercomputing Alliance (HLRN) for providing computing time on the Cray XC40. We thank the anonymous referee for helpful comments. 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J. E. 1964, Analytical Chemistry, 36, 1627 Sekanina, Z. 1967, Bulletin of the Astronomical Institute of Czechoslovakia, 18, 15 Sekanina, Z. 1981, Annual Review of Earth and Planetary Sciences, 9, 113 Sekanina, Z. 1984, The Astronomical Journal, 89, 1573 Sekanina, Z. 1988, The Astronomical Journal, 96, 1455 Sekanina, Z. 2019, arXiv:1901.08704 \[astro-ph\] \[\] Shi, X., Hu, X., Sierks, H., [et al.]{} 2016, Astronomy & Astrophysics, 586, A7 Sitarski, G. 1990, Acta Astronomica, 40, 405 Steinier, J., Termonia, Y., & Deltour, J. 1972, Analytical Chemistry, 44, 1906 Whipple, F. & Sekanina, Z. 1979, Astronomical Journal, 84, 1894 Whipple, F. L. 1950, The Astrophysical Journal, 111, 375 Whipple, F. L. 1951, Astrophysical Journal, 113, 464 Yeomans, D. K. & Chodas, P. W. 1989, The Astronomical Journal, 98, 1083 Yeomans, D. K., Chodas, P. W., Sitarski, G., Szutowicz, S., & Królikowska, M. 2004, in Comets [[II]{}]{} ([The University of Arizona Press]{}), 16

--- abstract: 'In this letter we discuss new soft theorems for the Goldstone boson amplitudes with non-vanishing soft limits. The standard argument is that the non-linearly realized shift symmetry leads to the vanishing of scattering amplitudes in the soft limit, known as the Alder zero. This statement involves certain assumptions of the absence of cubic vertices and the absence of linear terms in the transformations of fields. For theories which fail to satisfy these conditions, we derive a new soft theorem which involves certain linear combinations of lower point amplitudes, generalizing the Adler zero statement. We provide an explicit example of $SU(N)/SU(N-1)$ sigma model which was also recently studied in the context of $U(1)$ fibrated models. The soft theorem can be then used as an input into the modified soft recursion relations for the reconstruction of all tree-level amplitudes.' author: - Karol Kampf - Jiri Novotny - Mikhail Shifman - Jaroslav Trnka title: New Soft Theorems for Goldstone Boson Amplitudes --- =1 Introduction ============ In this paper we connect two different topics which have been intensively studied in last few years: soft limits of scattering amplitudes in effective field theories, and the $U(1)$ fibrated CP$(N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1)$ sigma models. The tree-level S-matrix in these models exhibit a very special behavior in the soft limit which gives rise to the new type of soft theorems, distinct from the usual Adler zero. [**Sigma models**]{}: The $U(1)$ fibrated CP$(N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1)$ models represent a class of sigma models interpolating between CP$(N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1)$ and $S^{2N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1}$ target spaces [@tu1; @tu2; @tu3]. These models correspond to the cosets $\left[ \left( SU(N)/SU(N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1)\times U(1)\right) \right] \times U(1) $. For brevity in the following we refer to these models as $SU(N)/SU(N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1)$. The above class contains an extremely interesting example of $N=2$, including CP(1) and $S^{3}$ models, both being integrable and exactly solvable in two spacetime dimensions [@tu3; @tu4; @tu5; @W2]. The algebraic form of the interpolating Lagrangian is $$\mathcal{L}=\frac{1}{2\lambda ^{2}} \Bigl\{ \Bigr[ \sum_{a=1,2,3} J_{\mu }^{a}J^{a \mu} \Bigr] -\kappa J_{\mu }^{3}J^{3 \mu }\Bigr\}\,, \label{cl1}$$ where the current $J_{\mu }$ is defined as $$J_{\mu }=U^{\dagger }\partial _{\mu }U\equiv 2\mathrm{i}\,\sum J_{\mu }^{a}T^{a}\,, \quad J_{\mu }^{a}=-\mathrm{i}\,\mathrm{Tr}\,(J_{\mu }T^{a})\,. \label{cl2}$$ Here $U$ is an arbitrary $x$-dependent matrix, $U(x)\in SU(2)$, the generators are proportional to the Pauli matrices, $T^{a}=\frac 12 \tau ^{a}$, and $\kappa $ is a numerical deformation parameter. If $\kappa =1$ the theory is equivalent to the CP(1) model, while at $\kappa =0$ it reduces to the $SU(2)\times SU(2)/SU(2)$ Principal Chiral Model (PCM) whose target space is $S^{3}$. For arbitrary $N$ we can extend (\[cl1\]) as follows: $$\mathcal{L}=\frac{1}{2\lambda ^{2}} \sum_{a=1}^{2N-2}\Bigl[(J^{N^{2}-2N+a})^{2} +\frac{1-\kappa}{N}(J^{N^{2}-1})^{2}\Bigr]\,. \label{general Lagrangian}$$ [**Scattering amplitudes**]{}: Recently, there has been a huge progress in new methods for the calculation of on-shell scattering amplitudes in QFTs. While most work has been focused on gauge theory and gravity, especially with maximal supersymmetry, new surprising results have been obtained in the case of effective field theories (EFT). The general approach is to fix the amplitude uniquely by imposing certain sets of constraints. The universal example is a tree-level factorization on poles, $$\lim_{P^{2}\rightarrow 0}A_{n}=\sum\frac{A_{L}A_{R}}{P^{2}}\,, \label{fact}$$ where the sum runs over internal states. The set of all factorizations is enough to completely specify tree-level S-matrix in a large class of QFTs, called [*on-shell constructible*]{}, including gauge theories or gravity, and it can be then calculated using the recursion relations [@recur]. This does not apply to EFTs due to the presence of unfixed contact terms with no poles, which originate from higher-dimensional operators in the Lagrangian. In [@Cheung:2015ota] it was shown that when the amplitude vanishes for one of the momenta going to zero, we can impose this information as a constraint and use *soft recursion relations* for on-shell reconstruction. This singles out a set of exceptional EFTs where all coefficients in the Lagrangian are fixed by the requirement of a certain degree of vanishing, $A_{n}=\mathcal{O}(p^{\sigma })$, in the soft limit [@Cheung:2014dqa; @Cheung:2016drk; @Cheung:2018oki; @Elvang:2018dco; @Cheung:2015ota; @Low:2019ynd; @unique]. The primary example is the PCM describing the spontaneous symmetry breaking $SU(N) \times SU(N) \rightarrow SU(N)$. It has been known since 1970s [@Susskind:1970gf] that the requirement of the vanishing soft limit of amplitudes, known also as the *Adler zero*, on any two-derivate theory specifies NLSM as a unique solution. In [@NLSM] it was found that in this model the group part of tree-level amplitudes can be stripped, similar to Yang-Mills amplitudes, dramatically simplifying the calculations. Adler zero ========== First we review the standard textbook derivation of the Adler zero for amplitudes of Nambu-Goldstone bosons (NGB). We start with the theory for the single NGB corresponding to the spontaneous breaking of one-parameter continuous symmetry. The NGB couples to the associated Noether current $N^{\mu }(x)$ with a strength parametrized by the decay constant $F$, $$\langle 0|N^{\mu }(x)|\phi (p)\rangle =-\mathrm{i}p^{\mu }Fe^{-\mathrm{i} p\cdot x} \,. \label{mat1}$$ The matrix element of this current between physical states has a pole for $p^{2}\rightarrow 0$, and the residue corresponds to the scattering amplitude for the NGB emission. For the element between out state $\langle \alpha |$ and in state $|\beta \rangle $ we get $$\langle \alpha |N^{\mu }(0)|\beta \rangle =F\frac{p^{\mu }}{p^{2}} A_{n}(\alpha +\phi (p),\beta )+R^{\mu }(p). \label{mat3}$$ Here $A_{n}(\alpha +\phi (p),\beta )$ is the on-shell amplitude which involves emission of the state $\phi $ with momentum $p$, where $p^{\mu }=P_{\beta }^{\mu }-P_{\alpha }^{\mu }$ is the difference between incoming and outgoing momenta, and $R^{\mu }(p)$ is the regular function for $p^{2}\rightarrow 0$. Due to the conservation of the current we have $p_{\mu }\langle \alpha |N^{\mu }(0)|\beta \rangle =0$ and therefore, $$A_{n}(\alpha +\phi (p),\beta )=-\frac{1}{F}p_{\mu }R^{\mu }(p)\,. \label{mat4}$$ Suppose that $R^{\mu }(p)$ is regular also in the limit $p\rightarrow 0$. This is an additional assumptions which does not follow automatically from the standard polology. Then the amplitude $A_{n}$ vanishes if the NGB momentum is soft, $$\lim_{p\rightarrow 0}A_{n}(\alpha +\phi (p),\beta )=0\,.$$ This is the statement of the Adler zero. The same argument applies to the theory with multiple Goldstone bosons. To summarize, we have the nonperturbative Adler zero provided the matrix element $\langle \alpha |N^{\mu }(0)|\beta \rangle $ of the Noether current corresponding to the spontaneous symmetry breaking *has no other singularity for $p\rightarrow 0$ besides the NGB pole*. Therefore, the violation of the Adler zero is possible only when there are additional singularities in the matrix element of the Noether current. This is achieved in the case when the Noether current can be inserted into the external lines of the amplitude $A_{n}(\alpha ,\beta )$, i.e. when there are quadratic terms in the expansion of the operator $N^{\mu }$ in the elementary fields. There are two sources of these quadratic terms: 1. The presence of cubic vertices in the Lagrangian; 2. The presence of linear terms in the nonlinearly realized symmetry transformation corresponding to the Noether current $N^{\mu }$. Schematically, $$\delta \phi =a+b\phi +\mathcal{O}(\phi ^{2}),\qquad b\neq 0.$$ These two conditions are *not* sufficient: even when at least one of the above conditions is satisfied, the theory can still have the Adler zero – a more detailed analysis is needed. Note that the cubic vertices can be always removed by means of field redefinitions, as there are no on-shell three-point amplitudes (apart from $\phi ^{3}$ theory). In such a case the presence of the linear term in $\delta \phi $ is crucial. Note that e.g. in the PCM parametrized by the Lagrangian $$\mathcal{L}=F^{2}\mathrm{Tr}(\partial ^{\mu }U^{\dagger })(\partial _{\mu }U),\quad U=e^{\frac{\mathrm{i}}{F}\phi },\;\phi =\phi ^{a}T^{a}\,, \label{NLSM}$$ where $U\in SU(N)$ transforms under the general element $(V_{R},V_{L})$ of the chiral group $SU(N)\times SU(N)$ as $$U\rightarrow V_{R}UV_{L}^{-1}\,, \label{chiral}$$ there are no cubic vertices, and the matrix $\phi$ of $N^{2}-1$ scalar fields transforms under the axial transformation $V_{L}=V_{R}^{-1}=1+\mathrm{i}\alpha ^{a}T^{a}\equiv 1+\mathrm{i}\alpha $ (with $\alpha$ infinitesimal) as $$\delta _{\alpha }\phi =2F\alpha -\frac{1}{6F}\{\alpha ,\phi ^{2}\}+\frac{1}{3F}\phi \alpha \phi +O\left( \phi ^{3},\alpha ^{2}\right) .$$ The linear term is absent, and consequently the theory has the Adler zero. New soft theorem ================ Let us assume a general two-derivative Lagrangian for $N$ fields $\left\{ \phi _{I}\right\} _{I=1}^{N}$ with a cubic vertex, $$\mathcal{L}=\frac{1}{2}\partial_\mu \phi _{I}\partial^\mu \phi _{I}+ \frac{1}{2}K_{IJK} \partial_\mu \phi _{I} \partial^\mu \phi _{J}\phi _{K}+ \mathcal{O}(\phi ^{4}) \label{Lagr2}$$ with sum over repeating indices tacitly assumed. Let the transformation of the fields corresponding to spontaneously broken symmetry contain, besides the constant term, also a linear term, $$\delta ^{J}\phi _{I}=F_{I}^{J}+\sum_{K=1}^{N}C_{IK}^{J}\phi _{K}+\mathcal{O}(\phi ^{2}). \label{inv}$$ The invariance of (\[Lagr2\]) under the symmetry (\[inv\]) requires non-trivial constraints between all coefficients, namely, $$B_{IK}^{J}\equiv C_{IK}^{J}+\frac{1}{2}\sum_{L=1}^{N}K_{IKL}F_{L}^{J}$$ must be antisymmetric, $B_{IK}^{J}=-B_{KI}^{J}$. The Noether current $N_{\mu }^{J}$ contains a quadratic term in the field expansion $$N_{\mu }^{J}=\sum_{I=1}^{N}F_{I}^{J}\partial _{\mu }\phi _{I}+\sum_{L,K=1}^{N}\mathcal{K}_{LK}^{J}\phi _{K}\partial _{\mu }\phi _{L}+ \mathcal{O}(\phi ^{3})\,,$$ where $\mathcal{K}_{IK}^{J}$ depend on both parameters $C$ and $K$ $$\mathcal{K}_{IK}^{J}=C_{IK}^{J}+\sum_{M=1}^{N}F_{M}^{J}K_{MIK}.$$ At the tree-level the matrix element $\langle \alpha |N_{\mu }^{J}|\beta \rangle $ has additional singular terms from inserting the current into external legs. The remainder $R_{\mu }^{J}$ is not regular for $p\rightarrow 0$, hence the soft limit of $p^{\mu }R_{\mu }^{J}$ is non-zero and reduces to $$\lim_{p\rightarrow 0}p^{\mu }R_{\mu }^{J}=-\sum_{L\in \alpha \cup \beta }\sum_{K=1}^{N}\mathcal{C}_{LK}^{J}A_{n-1}^{K,L}\left( \alpha ,\beta \right),$$ where the $A_{n-1}^{K,L}(\alpha ,\beta )$ is the $(n{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1)$pt amplitude, the particle $\phi _{L}\left( p_{L}\right) $ is omitted and is replaced by particle $\phi _{K}\left( p_{L}\right) $ with momentum $p_{L}$. The sum over $L$ is over the indices of all the particles in the in and out states. Therefore the soft theorem has the form $$\lim_{p\rightarrow 0}\sum_{I=1}^{N}F_{I}^{J}A_{n}(\alpha +\phi _{I}(p),\beta )=\sum_{I\in \alpha \cup \beta }\sum_{K=1}^{N}\mathcal{C}_{IK}^{J}A_{n-1}^{K,I}\left( \alpha ,\beta \right) . \label{soft}$$ Here the coefficient function $\mathcal{C}_{IK}^{J}$ is related to the original parameters in the Lagrangian and transformation as $$\mathcal{C}_{IK}^{J}=B_{IK}^{J}+\frac{1}{2} \sum_{M=1}^{N}F_{M}^{J}(K_{MIK}-K_{MKI})=-\mathcal{C}_{KI}^{J}. \label{coef2}$$ However, since the on-shell amplitudes are invariant with respect to redefinition of the fields of the form $\phi _{I}=\phi _{I}^{\prime }+O\left( \phi^{\prime\,2}\right) $, the constants $\mathcal{C}_{IK}^{J}$ do not depend on such a reparametrization of the Lagrangian. Note that several conditions must be satisfied in order to get non-zero right hand side of (\[soft\]): 1. The coefficients $\mathcal{C}_{IK}^{J}$ must be non-zero, i.e. no cancellation between parameters in (\[Lagr2\]), (\[inv\]) occurs. 2. The theory needs to have both even and odd amplitudes, as the amplitudes on the right hand side have $(n\mathrm{\rule[2.4pt]{6pt}{0.65pt}}1)$ external legs. Most sigma models do have only even point amplitudes and therefore, they preserve the Adler zero. Example of the sigma model ========================== As an explicit example we consider a theory of two types of NGB fields: a vector of multiple complex scalar fields $\Phi _{I}^{+}$, $I=1,\dots,N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1$, and a single real scalar $\chi $. We use the parametrization $$\hat{u}=\left( \begin{array}{c} \frac{\Phi ^{+}}{F} \\ \sqrt{1-\frac{\Phi ^{-}\cdot \Phi ^{+}}{F^{2}}} \end{array}\right) \,,$$ where $\Phi ^{+}=(\phi _{1}^{+},\phi _{2}^{+},\dots ,\phi _{N-1}^{+})^{T}$, $ \Phi ^{-}=\left[ \Phi ^{+}\right] ^{\dagger }$ and $\cdot $ stands for the contraction over the $I$ index. The Lagrangian of the model is $$\begin{aligned} \mathcal{L}& =\frac{(\partial \chi )^{2}}{2}{\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt}}\,F^{2}(\partial ^{\mu }\hat{u}^{\dagger }\!\cdot\! \partial _{\mu }\hat{u}){\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt}}\, \frac{\mathrm{i}F_{0}}{2}\,\partial ^{\mu }\chi (\partial _{\mu }\hat{u}^{\dagger }\cdot u\,{{\rm\rule[2.4pt]{6pt}{0.65pt}}}\,\hat{u}^{\dagger }\cdot \partial _{\mu }\hat{u}) \nonumber \\ & -\Bigl( F^{2}\,\mathrm{\rule[2.4pt]{6pt}{0.65pt}}\,\frac{F_{0}^{2}}{2} \Bigr) (\hat{u}^{\dagger }\cdot \partial ^{\mu }\hat{u})(\partial _{\mu } \hat{u}^{\dagger }\cdot \hat{u}). \label{Lagr1}\end{aligned}$$ It has two coupling constants $F$, $F_{0}$ which play the role of the decay constants of the NGB $\phi _{I}^{+}$ and $\chi $ respectively. The model described by (\[Lagr1\]) is a different parametrization of the $SU(N)/SU(N-1)$ non-linear sigma model (\[general Lagrangian\]). The relation with the original couplings is $$F_{0}=\frac{1}{\lambda }\left( 1-\kappa \right) ^{1/2},~~~~F=\frac{1}{\sqrt{2}\lambda }.$$ Let us briefly summarize limiting cases of our model (for details and discussion see [@future]). The limit $\kappa \rightarrow 1$ gives $ F_{0}\rightarrow 0$ and $\chi $ decouples: we get CP$(N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1)$ model. The case $ \lambda \rightarrow 0$ with $1-\kappa =O(\lambda ^{2})$ means $F\rightarrow \infty $ , $F_{0}$ finite and the theory is free. The limit $\kappa \rightarrow 0$, $\lambda $ fixed means $F_{0}=\sqrt{2}F$ which gives $ O(2N)/O(2N{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1)$ model. Note that the model (\[Lagr1\]) satisfies the first condition for the Adler zero violation as it involves the cubic term $$\mathcal{L}\ni \mathrm{i}\frac{F_{0}}{2F^{2}}\partial ^{\mu }\chi \,(\partial _{\mu }\phi _{I}^{-}\cdot \phi _{I}^{+}-\partial _{\mu }\phi _{I}^{-}\cdot \phi _{I}^{+})\,.$$ The Lagrangian is derivatively coupled in the $\chi $ field, and it is therefore trivially invariant under the shift symmetry $$\label{eqshiftchi} \delta \chi = a \,.$$ Since the cubic vertices can be eliminated by the reparametrization $\Phi ^{\pm }=\Phi ^{\pm \prime }\exp \bigl( \pm i\frac{F_{0}}{2F}\chi \bigr)$, which does not spoil this property, all scattering amplitudes have the vanishing soft limit at $p_{\chi }\rightarrow 0$, i.e. for $\chi $ the Adler zero is valid. After this reparametrization, the Lagrangian is also invariant under a more complicated transformation involving the linear terms, $$\begin{aligned} & \delta \chi =\frac{F_{0}}{2F^{2}}\left( a_{I}^{-}\cdot \phi _{I}^{+}+a_{I}^{+}\cdot \phi _{I}^{-}\right) +\mathcal{O}((\chi ,\phi ^{\pm })^{2})\,, \nonumber \\ & \delta \phi _{I}^{\pm }=\mp \mathrm{i}a_{I}^{\pm }\Bigl( 1\mp \frac{F_{0} } {2F^{2}}\chi \Bigr) +\mathcal{O}((\chi ,\phi ^{\pm })^{2})\,, \label{sym}\end{aligned}$$ where we introduced shift parameters $a_{I}^{\pm }$. Note that the symmetry mixes the single scalar field $\chi$ and multiple scalars $\phi _{I}^{\pm }$. Calculating $\mathcal{C}_{IK}^{J}$ in (\[coef2\]) we learn that $\mathcal{C}_{IK}^{J}$ is non-zero. Furthermore the model involves both odd and even amplitudes, and therefore, the scattering amplitude does not vanish when the momentum of one of $\phi _{I}^{\pm }$ is taken soft. Because of the form of the Lagrangian (\[Lagr1\]) the only allowed amplitudes have the same number of $\phi ^{+}$ and $\phi ^{-}$ fields. If we think about $\phi ^{\pm }$ as charged scalars, this just stands for charge conservation. Let us consider now the scattering amplitude of $2n$ fields $\phi _{I}^{\pm }$ and $m$ fields $\chi$, with total $M=2n+m$ external legs, $$\includegraphics[scale=0.38]{pic1}$$ $$\begin{aligned} & A_{M}(\{\phi _{I_{i}}^{+}\},\{\phi _{J_{j}}^{-}\},\{\chi \})\equiv \label{ampl} \\ & A(\phi _{I_{1}}^{+}(p_{1}){\scalebox{0.9}{$\dots$}}\phi _{I_{n}}^{+}(p_{n}),\phi _{J_{1}}^{-}(q_{1}){\scalebox{0.9}{$\dots$}}\phi _{J_{n}}^{-}(q_{n}),\chi (k_{1}){\scalebox{0.9}{$\dots$}}\chi (k_{m})). \nonumber\end{aligned}$$ The soft theorem when the $p_{1}\rightarrow 0$ then reads $$\lim_{p_{1}\rightarrow 0}A_{M}=\frac{\mathrm{i}F_{0}}{2F^{2}} \sum_{i=1}^{m}A_{M-1}^{(i)}-\frac{\mathrm{i}F_{0}}{2F^{2}} \sum_{j=1}^{n}\delta _{I_{1}J_{j}}A_{M-1}^{(j)}\,, \label{soft2}$$ where the lower point amplitudes are defined as follows: $$\begin{aligned} & A_{M-1}^{(i)}\equiv A(\phi _{I_{1}}^{+}(k_{i}){\scalebox{0.9}{$\dots$}}\phi _{I_{n}}^{+},\{\phi _{J_{j}}^{-}\},\chi (k_{1}){\scalebox{0.9}{$\dots$}}\widehat{\chi (k_{i})} {\scalebox{0.9}{$\dots$}}\chi (k_{m})) \nonumber \\ & A_{M-1}^{(j)}\equiv A(\phi _{I_{2}}^{+}{\scalebox{0.9}{$\dots$}}\phi _{I_{n}}^{+},\phi _{J_{1}}^{-}{\scalebox{0.9}{$\dots$}}\widehat{\phi _{J_{j}}^{-}(q_{j})}{\scalebox{0.9}{$\dots$}}\phi _{J_{n}}^{-},\chi (q_{j}),\{\chi \}) . \nonumber\end{aligned}$$ In the first case, $A_{M-1}^{(i)}$, we start with $A_{M}$ defined in (\[ampl\]) and remove particle $\chi (k_{i})$, then we replace the particle $\phi _{I_{1}}^{+}(p_{1})$ by $\phi _{I_{1}}^{+}(k_{i})$, i.e. just replace momenta keeping the quantum numbers the same, and finally sum over all particles $\chi (k_{i})$ which are removed. In the case of $A_{M-1}^{(j)}$ we remove particle $\phi _{I_{1}}^{+}$ completely as well as $\phi _{J_{j}}^{-}$, and add a new single scalar particle $\chi (q_{j})$ with the momentum of removed $\phi ^{-}$ particle. Graphically we have (left picture corresponds to $A^{(i)}$ while the right for $A^{(j)}$) $$\includegraphics[scale=0.32]{pic2}$$ where the red color stands for removed legs and blue for the legs added. For $q_{1}\rightarrow 0$ the soft theorem is the same except the overall sign on the right hand side of (\[soft2\]). As discussed earlier any amplitude vanishes for $k_{j}\rightarrow 0$. In the following, we focus now on the $N=2$ case which describes only three fields: $\phi ^{\pm }$, $\chi $. To check the soft theorem we first calculate all non-vanishing 4pt amplitudes, $$\begin{aligned} & A_{4}(\phi _{1}^{+},\phi _{2}^{+},\phi _{3}^{-},\phi _{4}^{-})= \frac{1}{4F^{4}}(3F_{0}^{2}-8F^{2})s_{12}\,, \nonumber \\ & A_{4}(\phi _{1}^{+},\phi _{2}^{-},\chi _{3},\chi _{4})=\frac{F_{0}^{2}}{4F^{4}}s_{12}\,,\end{aligned}$$ where $s_{ij}=(p_i+p_j)^2$ and we used the notation $\phi _{1}^{+}\equiv \phi ^{+}(p_{1})$ etc., for simplicity. There is only one non-trivial 5pt amplitude, $$A_{5}(\phi _{1}^{+},\phi _{2}^{+},\phi _{3}^{-},\phi _{4}^{-},\chi _{5})=\frac{\mathrm{i}F_{0}}{F^{6}}\Bigl( F^{2}\,{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}\,\frac{F_{0}^{2}}{2}\Bigr)(s_{12}{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}s_{34}).\! \label{amp5}$$ The soft theorem (\[soft2\]) for $p_{1}\rightarrow 0$ predicts, $$\begin{aligned} \lim_{p_{1}\rightarrow 0}A_{5}& =\frac{\mathrm{i}F_{0}}{2F^{2}}A_{4}(\phi _{5}^{+},\phi _{2}^{+},\phi _{3}^{-},\phi _{4}^{-}) \label{softT} \nonumber \\ & -\frac{\mathrm{i}F_{0}}{2F^{2}}\left[ A_{4}(\phi _{2}^{+},\phi _{3}^{-},\chi _{4},\chi _{5}){\hspace{0.5pt}\text{{\small+}}\hspace{-0.5pt}}\,A_{4}(\phi _{2}^{+},\phi _{4}^{-},\chi _{3},\chi _{5})\right] \nonumber \\ &=-\frac{\mathrm{i}F_{0}}{F^{6}}\Bigl( F^{2}\,{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}\,\frac{F_{0}^{2}}{2}\Bigr)s_{34}\,,\end{aligned}$$ in agreement with the direct calculation (\[amp5\]). Amplitude reconstruction ======================== The knowledge of the soft theorem (\[soft2\]) can be used as an input in the modified version of the soft recursion relations introduced in [@Cheung:2015ota]. We start with the momentum shift where all but two particles are shifted in the way that allows to access the soft limit, $$\begin{aligned} & \hat{p_{i}}=p_{i}(1-a_{i}z)p_{i},\quad i=1,\dots n{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}2, \label{shift1} \\ & \hat{p_{j}}=p_{j}+zq_{j},\quad \quad \,\,\,\,\,j=n{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}1,n, \label{shift2}\end{aligned}$$ where the parameters $a_{i}$ and vectors $q_{j}$ must preserve on-shell conditions and momentum conservation. For this shift any scattering amplitude scales like $A_{n}(z)=\mathcal{O}(z^{2})$, just based on the momentum counting. Then we consider a residue theorem for the meromorphic function $F_{n}(z)$, $$F_{n}(z)\equiv \frac{A_{n}(z)}{z\prod_{i}(1-a_{i}z)}\,. \label{GRT}$$ We need at least three factors of $(1-a_{i}z)$ in the denominator to have vanishing residue at $z\rightarrow \infty$, i.e. $$\lim_{R\to\infty}\oint_{|z|=R} dz\,F_n(z) = 0 \,.$$ We can then express the residue at $z=0$, the original amplitude $A_{n}$, as the sum of all other residues $$A_{n}=-\sum_{k}\mathrm{Res}_{z=z_{k}}\,F_n(z)-\sum_{i} \mathrm{Res}_{z=\frac{1}{a_{i}}}F_n(z)\,.$$ The first sum on the right hand side refers to factorization poles from $A_{n}(z)$, each term is equal to the product of corresponding lower point amplitudes. The second sum is over the soft limit poles when one of the $\hat{p_{j}}\rightarrow 0$. In [@Cheung:2015ota] we considered only theories with vanishing soft limits, i.e. the second sum never contributed, but now the contribution is non-zero and it is given by (\[soft\]). As an example, we will reconstruct the 5pt amplitude from $N=2$ model, $A_{5}(\phi _{1}^{+},\phi _{2}^{+},\phi _{3}^{-},\phi _{4}^{-},\chi )$. We shift legs 1,2,5 as (\[shift1\]) and 3,4 as (\[shift2\]). The amplitude does not have any factorization poles, and the only poles of $F_5(z)$ are soft poles. As the shifted amplitude vanishes for $\hat{p_{5}} \rightarrow 0$ the only contributions come from $\hat{p_{1}}$ or $\hat{p_{2}} \rightarrow 0$ soft limits. The residue at $z=1/a_1$ then reads $$\mathrm{Res}_{z=\frac{1}{a_{1}}}F_5(z)=- \frac{\widehat{A_{5}}|_{z= 1/a_{1}}}{(1-a_{2}/a_{1})(1-a_{5}/a_{1})}\,. \label{res}$$ The value of the shifted amplitude $\widehat{A_{5}}|_{z=1/a_{1}}$ can be obtained from the soft theorem (\[softT\]) by considering the shifted kinematics, $$\hat{p_{1}}=0,\quad \hat{p_{2}}=\Bigr( \frac{a_{1}-a_{2}}{a_{1}}\Bigl) p_{2},\quad \hat{p_{5}}=\Bigr( \frac{a_{1}-a_{5}}{a_{1}}\Bigr) p_{5} .$$ Plugging the result into (\[res\]) we get $$\mathrm{Res}_{z=\frac{1}{a_{1}}}F_5(z)=\frac{\mathrm{i}F_{0}}{F^{6}}\Bigl( F^{2}\,{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}\,\frac{F_{0}^{2}}{2}\Bigr) s_{25} .\label{t1}$$ Similarly, the residue at the pole $z=\frac{1}{a_{2}}$ for $\hat{p_{2}}=0$ gives $$\mathrm{Res}_{z=\frac{1}{a_{2}}}F_5(z)=\frac{\mathrm{i}F_{0}}{F^{6}}\Bigl( F^{2}\,{{{\rm\rule[2.4pt]{6pt}{0.65pt}}}}\,\frac{F_{0}^{2}}{2}\Bigr) s_{15},\label{t2}$$ and after using the momentum conservation the sum of (\[t1\]), (\[t2\]) reproduces the formula (\[amp5\]). Uniqueness of the model ======================= In the last part we turn the procedure around, and will reconstruct our non-linear sigma model for $N=2$ as a unique theory which satisfies soft theorem of the type (\[soft\]). Following the logic of [@Cheung:2014dqa] we start with the ansatz for the amplitude of three types of scalar fields $\phi^{\pm },\chi$ in terms of kinematical invariants and impose the soft theorem of the general type $$\lim_{p_1\rightarrow 0}A_{n}=\sum_{i}c_{i}A_{n-1}^{(i)} \label{soft3}$$ as a constraint. If the right-hand side is zero we deal with the standard Adler theorem (for more details see [@Cheung:2016drk]). To go beyond the standard situation we demand a non-zero right-hand side when shifting charged particles, and keep the Adler zero only for the neutral $\chi$. We went up to the 7pt amplitudes to check that the unique answer is our model, $U(1)$-fibrated CP$(1)$, and the general $c_i$ constants are set in accordance with (\[soft2\]). The natural question is if there are more theories of this type for more than three scalar fields beyond our explicit example (\[Lagr1\]). This is an open question, and we believe that this procedure is a very useful tool to address the problem and potentially find new theories with non-trivial soft theorems. In principle, we can also look at amplitudes for theories with only two types of scalar fields. In the upcoming work [@future] we will prove that for any such theory, under the assumption that the soft theorem (\[soft\]) with $F_{I}^{J}=F\delta _{I}^{J}\neq 0$ is valid, and assuming non-vanishing 4pt amplitude, all the odd-particle amplitudes have to vanish. Therefore all Goldstone boson amplitudes must necessarily have the Adler zero. This supports the statement that the only non-linear sigma model for two scalars are CP$(1)=O(3)/O(2)$ and $O(1,2)/O(2)$. Conclusion ========== In this letter we found a new soft theorem for the Goldstone boson amplitudes. Using the example of $SU(N)/SU(N-1)$ non-linear sigma models, we showed that generically the amplitudes do not vanish in the soft limit but rather reduce to a recursion. Explicit expressions are presented in the simplest $N=2$ case which describes a pair of charged NGBs and a single neutral NGB. We proved that this theory can be uniquely fixed from the tree-level $S$-matrix if we impose the soft theorem as a constraint. Consequently, we derived the recursion relations to reconstruct all tree-level amplitudes. Our work opens new avenues in studying NLSMs, and more generally EFTs using non-vanishing soft limits of scattering amplitudes. In [@future] we will generalize this work, and use the soft theorems as the theoretical tool to explore larger space of theories based on properties of their scattering amplitudes. 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--- abstract: 'During the last few years of his life, Ramanujan had adamantly tried to invert the modular invariant. Subsequent efforts failed until May 30, 2011 when an explicit closed formula for an inverse was presented at the CCRAS (Moscow, Russia). This very formula, along with some special values of the modular invariant, is given in this paper.' author: - 'S. Adlaj[^1]' date: 'May 30, 2011' title: An inverse of the modular invariant --- In a previous paper [@A], a justification for defining an *essential elliptic function* was made. Yet, enabling an inversion of the modular invariant is, perhaps, even more convincing. We shall not elaborate upon describing previous attempts for inverting the modular invariant aside from mentioning two typical references [@B; @C]. The first reference provides a glimpse upon Ramanujan latest efforts, whereas the appendix of the second concludes with a well-known expression for a point $\tau$ in the fundamental domain as a ratio of hypergeometric functions, thereby linking $\tau$ with an intermediate variable $\lambda$. Formula (3.3), in the same paper, yields the modular invariant $j$ as a (well-known) fractional transformation of $\lambda$, of degree 6. We point out this transformation so as to suggest that verifying a formula for an inverse of the modular invariant is as straightforward as verifying a root of a given hexic. An inversion of the modular invariant is afforded via successively composing the functions $$k_0(x) = \frac{i \hspace{.04cm} G \left( \sqrt{1 - x^2} \ \right)}{G(x)}, \ k_1(x) = \frac{\sqrt{x + 4} - \sqrt{x}}{2}, \ k_2(x) = \frac{3}{2} \left( \frac{x}{k_3(x)} + k_3(x) \right) - 1,$$ where $$k_3(x) = \sqrt[3]{\sqrt{x^2 - x^3} - x}$$ and $G(x)$ is the arithmetic-geometric mean of $1$ and $x$. In other words, the function $$k = k_0 \circ k_1 \circ k_2$$ is an inverse of the modular invariant, which (we need not point out) is not single-valued. An ascending sequence of special values of the modular invariant on the boundary of the fundamental domain and the {#an-ascending-sequence-of-special-values-of-the-modular-invariant-on-the-boundary-of-the-fundamental-domain-and-the .unnumbered} ------------------------------------------------------------------------------------------------------------------- $$j \left( 1/2 + 2 \sqrt{- 1} \ \right) = \frac{\left( \displaystyle \frac{4}{3} \left( 1 - 2 \left( 33 + 24 \sqrt{2} - 4 \sqrt{140 + 99 \sqrt{2}} \ \right)^2 \right)^2 - 1 \right)^3}{\left( 1 - 2 \left( 33 + 24 \sqrt{2} - 4 \sqrt{140 + 99 \sqrt{2}} \ \right)^2 \right)^2 - 1} <$$ $$< j \left( 1/2 + \sqrt{- 3} \ \right) = \frac{\left( \displaystyle 12 \left( 555 + 16 \left( 20 \sqrt{3} - \left( 8 / \sqrt{3} + 5 \right) \sqrt{ 26 + 15 \sqrt{3}} \ \right) \right)^2 - 1 \right)^3}{9 \left( 555 + 16 \left( 20 \sqrt{3} - \left( 8 / \sqrt{3} + 5 \right) \sqrt{ 26 + 15 \sqrt{3}} \ \right) \right)^2 - 1} <$$ $$< j \left( 1/2 + \sqrt{- 2} \ \right) = \frac{\left( \displaystyle \frac{4}{3} \left( 1 - 2 \left( 5 + 4 \sqrt{2} - 2 \sqrt{2 \left(7 + 5 \sqrt{2} \ \right)} \ \right)^2 \right)^2 - 1 \right)^3}{\left( 1 - 2 \left( 5 + 4 \sqrt{2} - 2 \sqrt{2 \left( 7 + 5 \sqrt{2} \ \right) } \ \right)^2 \right)^2 - 1} <$$ $$< j \left( 1/2 + \sqrt{- 1} \ \right) = \left( 181 - 19 \left( \frac{3}{\sqrt{2}} \right)^3 \right)^3 < j \left( \frac{1 + \sqrt{- 3}}{2} \ \right) = 0 <$$ $$< j \left( \frac{1 + 2\sqrt{- 2}}{3} \ \right) = \left( \frac{5 \left( 19 - 13\sqrt{2} \ \right)}{6} \right)^3 < j \left( \frac{1 + 4 \sqrt{- 3}}{7} \ \right) =$$ $$\frac{\left( \displaystyle 12 \left( 555 - 16 \left( 20 \sqrt{3} - \left( 8 / \sqrt{3} - 5 \right) \sqrt{ 26 - 15 \sqrt{3}} \ \right) \right)^2 - 1 \right)^3}{9 \left( 555 - 16 \left( 20 \sqrt{3} - \left( 8 / \sqrt{3} - 5 \right) \sqrt{ 26 - 15 \sqrt{3}} \ \right) \right)^2 - 1} < j \left( \sqrt{- 1} \ \right) = 1 <$$ &lt;$$< j \left( \frac{2}{\sqrt{-3}} \ \right) = \frac{375 \left( 35010 - 20213 \sqrt{3} \ \right)}{16} < j \left( \sqrt{- 2} \ \right) = \left( \frac{5}{3} \right)^3 < j \left( \sqrt{- 3} \ \right) = \frac{125}{4} <$$ $$< j \left( 2 \sqrt{- 1} \ \right) = \left( \frac{11}{2} \right)^3 < j \left( \frac{4}{\sqrt{- 3}} \ \right) =$$ $$= \frac{\left( \displaystyle 12 \left( 555 - 16 \left( 20 \sqrt{3} + \left( 8 / \sqrt{3} - 5 \right) \sqrt{ 26 - 15 \sqrt{3}} \ \right) \right)^2 - 1 \right)^3}{9 \left( 555 - 16 \left( 20 \sqrt{3} + \left( 8 / \sqrt{3} - 5 \right) \sqrt{ 26 - 15 \sqrt{3}} \ \right) \right)^2 - 1} < j \left( 2\sqrt{- 2} \ \right) =$$ $$= \left( \frac{5 \left( 19 + 13\sqrt{2} \ \right)}{6} \right)^3 < j \left( 2\sqrt{- 3} \ \right) = \frac{375 \left( 35010 + 20213 \sqrt{3} \ \right)}{16} < j \left( 4 \sqrt{- 1} \ \right) =$$ $$= \left( 181 + 19 \left( \frac{3}{\sqrt{2}} \right)^3 \right)^3 < j \left( 4 \sqrt{- 2} \right) = \frac{\left( \displaystyle \frac{4}{3} \left( 1 - 2 \left( 5 + 4 \sqrt{2} + 2 \sqrt{ 2 \left( 7 + 5 \sqrt{2} \ \right) } \right)^2 \right)^2 - 1 \right)^3}{\left( 1 - 2 \left( 5 + 4 \sqrt{2} + 2 \sqrt{2 \left( 7 + 5 \sqrt{2} \ \right)} \right)^2 \right)^2 - 1} <$$ $$< j \left( 4 \sqrt{- 3} \right) = \frac{\left( \displaystyle 12 \left( 555 + 16 \left( 20 \sqrt{3} + \left( 8 / \sqrt{3} + 5 \right) \sqrt{ 26 + 15 \sqrt{3}} \ \right) \right)^2 - 1 \right)^3}{9 \left( 555 + 16 \left( 20 \sqrt{3} + \left( 8 / \sqrt{3} + 5 \right) \sqrt{ 26 + 15 \sqrt{3}} \ \right) \right)^2 - 1} <$$ $$< j \left( 8 \sqrt{- 1} \right) = \frac{\left( \displaystyle \frac{4}{3} \left( 1 - 2 \left( 33 + 24 \sqrt{2} + 4 \sqrt{140 + 99 \sqrt{2}} \ \right)^2 \right)^2 - 1 \right)^3}{\left( 1 - 2 \left( 33 + 24 \sqrt{2} + 4 \sqrt{140 + 99 \sqrt{2}} \ \right)^2 \right)^2 - 1}.$$ (500,520)(-2,0) (210,616) (310,300) (310,260) (310,210) (210,136) (225,-20) (-10,-20) (450,-20) (105,-20) (128,-20) (310,-20) (328,-20) (116,-60)[The fundamental domain with some points,]{} (48,-80)[at which the value of the modualr invariant is calculated, being marked]{} {width="160mm"} [9]{} S. Adlaj. Eighth lattice points // arXiv:1110.1743v1\[math.NT\]. B. Berndt & H. Chan. Ramanujan and the modular j-invariant // Canad. Math. Bull. Vol. 42 (4), pp. 427–440. K. Vogeler & M. Flohr B. Pure Gauge SU(2) Seiberg-Witten Theory and Modular Forms // arXiv:hep-th/0607142v2. [^1]: *e-mail: SemjonAdlaj@gmail.com*

--- author: - Kalyani Bagri - Ranjeev Misra - Anjali Rao - 'J. S. Yadav' - 'S. K. Pandey' bibliography: - 'ref1.bib' date: 'Received  2016 month day; accepted  20016  month day' title: 'Systematic Analysis of Low/Hard State RXTE Spectra of GX 339-4 to Constrain the Geometry of the System ' --- Introduction {#sect:intro} ============ Black hole binaries are some of the highly variable astronomical objects observed in the X-ray sky and their variability is revealed in both spectral and timing characteristics. The occasional presence of Quasi Periodic Oscillations (QPOs), variation of their centroid frequency, and rare occurrence of high frequency QPOs etc. are some of the manifestations of variability observed in timing properties. The variability in spectral behavior can be described in terms of spectral states exhibiting different shapes, which can be interpreted as being due to varying relative contribution from two or more spectral components. The two major components observed in the spectrum include a multi-temperature thermal component ([@mitsuda84; @shakura73]) originating from an accretion disk and a power law tail due to inverse Comptonized radiation from the corona. The thermal component is mainly present towards softer X-ray bands ($\leq$10 keV) and it dominates the overall spectrum when a system is in the soft spectral states. The copious amount of soft seed photons originating from the disk results in a steep spectrum with a photon index $\Gamma \geq$2. However, the strength of the thermal emission from the disk weakens during the hard spectral state and the spectrum is dominated by a hard power law component that extends towards the softer energies as well. The weak thermal component implies a smaller number of soft seed photons resulting in a relatively flat spectrum with a photon index of $\Gamma <$2. The transient black hole binaries undergo a series of spectral state transitions during an outburst in a systematic manner, which is often depicted as Q-shaped track in hardness-intensity diagrams [@fender09]. These diagrams have shown that a transient black hole binary is observed in the hard state when it enters the outburst, followed by transition to the hard intermediate state (HIMS), soft intermediate state (SIMS) and soft state [@belloni11; @fender09; @homanbelloni05]. The order is reversed as the outburst begins to decline and a system may show a few excursions to different spectral states [see @belloni11; @fender09; @homanbelloni05]. However, the overall scheme of state transitions remains the same for transient black hole binaries. On the other hand, the persistent black hole binaries have not shown a predictable trend of state transitions as in the case of transient black hole binaries, for example, Cyg X-1 remains in the hard state most of the time with occasional excursions to the soft state [@grinberg]. The seed photon starved hard spectral state is generally understood in terms of an accretion disk truncated far from the central black hole and the inner region being replaced by the hot inner flow which is sometimes modeled as an Advection Dominated Accretion Flow [ADAF; @narayanyi95]. The truncation of disk may be described by the disk evaporation model, which was first proposed by [@meyer2K] and later extended by [@liu02] and [@meyermeyer03]. The model considers a continuous evaporation of material from the disk, which feeds the corona. [@qiaoliu09] calculated the evaporation rate as a function of disk radius and showed that it is maximum at a certain radius for a given value of the viscosity parameter. The model predicts that the truncation of disk will occur at a radius where the accretion rate is equal to the maximum evaporation rate and the inner disk will survive the evaporation only when the accretion rate is higher than the maximum evaporation rate. The model successfully explains the finding of a truncated accretion disk [e.g. @plant14; @plant15; @tomsick09 etc.] as a consequence of the low mass accretion rate during the hard state, and an extended disk approaching the Innermost Stable Circular Orbit (ISCO) owing to the higher mass accretion rate in the soft state. It also provides a framework to the models addressing the steady jet emission during the hard states (see [@belloni10] and [@done07] for detailed reviews). The estimation of the inner disk radius using the observational data is crucial to understanding the truncation of accretion disk, which can be measured by modeling of Fe line profile and disk continuum. If the accretion disk is truncated during the hard states, the Fe emission line should be narrow and symmetric as the relativistic effects due to gravitational field of the black hole are weaker on the disk material. However, in soft states, the line should be broadened and skewed by the strong general relativistic effects in the inner region. Therefore, finding a broad Fe line in hard state serves as a strong evidence towards the extension of the disk in the inner regions and hence the violation of the truncated disk model. In such a scenario, a seed photon starved system during the hard state remains a puzzle. In several reports, authors have not only found a broad and skewed Fe line in the hard state, but also estimated the spin parameter by modeling the line profile and found a value of spin parameter consistent with the results obtained from the study of continuum and reflection component. [@miniutti04] studied the three BeppoSAX observations of a black hole candidate XTE J1650-500 during its 2001-02 outburst and perhaps detected the source in the power law dominated hard state ($\Gamma \sim$1.8) in one of the observations. They find a broad and strongly relativistic Fe emission line in the spectrum. The modeling of the line revealed the presence of the disk extending to $\sim$1.34 $R_g$, which led them to suggest the presence of a Kerr black hole in the system. The extension of the accretion disk to the inner region in another black hole binary Swift J1753.5-0127 is shown by [@miller06] during the decline of its 2005 outburst, wherein the inner disk was found at or close to the ISCO by modeling the continuum with a number of models. Their results show that the disk can be present in the inner regions during the hard state at very low luminosities down to $L_{X} \simeq$ 0.003 $L_{Edd}$. The same set of observations belonging to the hard state were further studied by [@reis09-395] to find the spin parameter and inclination angle using a model that included a power law and reflection component. The innermost emitting region was shown to extend close to $\sim$3.1$R_g$ and a spin parameter of $\sim$0.76 was reported by assuming that the inner edge of the disk is at the ISCO. Another result presenting the violation of the truncated disk model is shown in [@reis09l52] wherein the authors extracted the spectrum of XTE J1118+480 in its hard state using Chandra and RXTE and found the presence of thermal component with disk temperature of $\sim$ 0.21 keV, which suggested that the emission is originating from an accretion disk extending close to the radius of marginal stability. XTE J1817-330 was studied by [@rykoff07] using Swift observations during its 2006 outburst. They used the disk continuum model and found the inner disk radius to be consistent with the ISCO. They showed that the luminosity follows the relation $L_X \propto T^4$ roughly during the decline of the outburst, which led them to suggest the presence of a geometrically stable disk in the inner regions at accretion rate as low as 0.001 $L_{Edd}$. [@reis10] studied eight black hole binaries in their hard spectral state. The modeling of the disk continuum revealed that the luminosity in all the systems is consistent with the relation $L_X \propto T^4$ down to $\sim$ 5$\times$10$^{-4} L_{Edd}$. The six sources showed truncation radius not larger than 10 $R_g$ and the Fe line detected in four of the black hole binaries at luminosities down to 1.5$\times$10$^{-3} L_{Edd}$ excluded a truncated disk. Other studies on the finding of the inner disk radius close to the ISCO during low/hard state include [@reynolds10] and [@reynoldsmiller13]. Hence, there are good number of evidence suggesting that the disk may not be truncated in the hard spectral state. While several results show the violation of the truncation model, there are also number of results that support the truncation of the disk in the hard state. [@tomsick09] have reported the detection of Fe line in the hard state of GX 339-4, which is seen at low luminosities of $\sim$0.14$\%$ $L_{Edd}$ using Suzaku and RXTE observations. The truncated disk scenario is supported by their results wherein the inner disk radius is shown to increase by a factor of $>$27 as compared to the value found when the source was bright. Although, the inner disk radius is shown to be dependent on the inclination angle (i.e. $R_{in} >$35 $R_g$ at $i$=0$^\circ$ and $R_{in} >$175 $R_g$ at $i$=30$^\circ$), the results provide a direct evidence for the absence of the inner disk at low luminosities. The detection of Fe line in the hard state of GX 339-4 is reported by [@shidatsu11] and an inner disk radius of $\sim$13.3 $R_g$ is estimated by modeling its profile. Their results indicate that the accretion disk evolves inward as luminosity increases in the range $\sim$0.001 $< L_X/L_{Edd} < \sim $0.02 when the source remains in the hard state. The hints of disk recession can be found in [@petrucci14], wherein the spectral results of five Suzaku observations taken during the decline of 2010-2011 outburst are presented. An inner disk radius of $<$10-30 $R_g$ is found in the first two observations, however it remains unconstrained for the latter observations due to low statistics. [@plant14] have presented a very detailed study of reflection component observed in the joint spectral fitting of PCA and HEXTE spectra for the three outbursts of GX 339-4. Their results support the truncation of the inner disk during the hard state, and a decrease in the coronal height in the soft state. Truncation of disk in GX 339-4 in the hard state is again corroborated by [@plant15] with the spectral studies performed using XMM-Newton and Suzaku observations. [@kole14] studied GX 339-4 with XMM-Newton data in its hard state, and used both Fe line emission and disc continuum methods to measure the inner radius of the accretion disk. They estimated the effects of instrumental and modeling uncertainties and showed that both the methods provide results consistent with the truncated disc model. In addition, there are several reports where authors reanalyzed the results suggesting violation of truncation of disk and reclaimed the finding of truncated disk. A reanalysis of BeppoSAX observation of GX 339-4 by [@donegier06] confirmed the finding of a broad Fe line caused by the extreme relativistic effects previously reported by [@miller02] and [@miniutti04]. However, the authors reinstated the truncation of the inner disk by showing that the relativistic smearing can be significantly reduced by considering resonance Fe K line absorption from an outflowing disk wind. [@donetrigo10] reanalyzed XMM-Newton data of GX 339-4 studied by [@miller06-653] and [@reis08] which claimed the detection of broad Fe line in the hard state. A detailed reanalysis showed that MOS data of XMM-Newton is heavily piled-up and a broad Fe line is an artifact of the same. The spectrum extracted with PN timing mode data of the same observation revealed a narrow line consistent with the truncation of disk in hard state. Other reports showing the truncation of accretion disk in the hard state for various black hole binaries include [@basak16], [@rao15], [@yuannarayan14], [@cabanac09], and [@gierlinski08]. The Fe line emission is just one feature of the reflection component detected in black hole binaries. Another major feature of this component is the broad hump that appear at $\sim$10-30 keV. The reflection hump appearing towards higher energies, if not modeled properly, can give rise to an artificial hardening to an intrinsically soft spectrum. Therefore, it may be possible that reflection can be the reason behind hardening of an originally soft spectrum. It implies that the system is not seed photon starved and hence the finding of a broad Fe line can also be reconciled. In order to test the hypothesis, we have studied the black hole binary GX 339-4, which is already an object in the debate of truncation of accretion disk in hard state [e.g. @plant14]. It is a stellar mass galactic black hole binary harboring a low-mass donor and a confirmed black hole with a mass of 5.8 $\pm$ 0.5 $M_\circ$ [@hynes03] and distance of $>$6 kpc [@hynes04]. GX 339-4 has shown multiple outbursts in the past, which have been regularly monitored with RXTE. We have studied 4 outbursts of the object during 2002-2011 and in particularly its spectra. Our motivation is to fit the spectra with and without reflection component, in order to understand the effect the component has on the photon index. In the next section we will discuss the observations and the results found in the present work are discussed in the sections 2 and 3 respectively. Observations and Spectral Fitting {#sect:data reduction} ================================= This work presents a spectral analysis of pointed observations of the black hole binary [GX 339-4]{} with Proportional Counter Array [PCA; @jahoda96] onboard Rossi X-ray Timing Explorer (RXTE) during 2002-2011. We have studied a total of 1160 pointed observations available on the High Energy Astrophysics Science Analysis Archive (HEASARC) covering the four outbursts (2002-03, 2004-05, 2006-07, 2010-11) of the transient object. We study Standard-2 spectra from PCA data, and the spectral fitting was performed in the energy range of 3-20 keV using the spectral fitting package [XSPEC]{} version 12.8.2. All spectral parameters are presented with a confidence interval of 90% unless otherwise mentioned. The spectra studied here belong to the different spectral states of the four outbursts, which is manifested by their variable shapes. The spectral components have varying relative strengths, indicating the changing geometry and evolution of the physical processes in the system. Therefore, the spectra require different spectral models and we follow a scheme of spectral fitting where we begin with the simplest model of a power law absorbed by the interstellar medium and increase the complexity of the model by including the thermal emission from the accretion disk, Fe K$\alpha$ emission and reflection continuum. All the models are listed in the Table (1) and a description of the fitting procedure is given below. We begin the spectral fitting of all the spectra with the simplest model of an absorbed power law listed as M1 in the Table (1) and mention the number of spectra explained with the model with alphabets A, B, C and D belonging to 2002-03, 2004-05, 2006-07 and 2010-11 outbursts respectively. The interstellar absorption is modeled with the model [wabs]{} [@morrison83] available in [XSPEC]{} and the absorption column is fixed at 3.74 x 10$^{21}$ cm$^2$. It is found that a total of 261 (\[A\]66; \[B\]82; \[C\]96, \[D\]17) spectra provide a good fit ($\Delta \chi^2 <$ 1.2) with this model. Since there is no soft disk component seen in this class of spectra, it is expected that the observations providing a good fit with this model belong to the hard spectral state. Fig (\[fluxtime\]) shows the variation of flux as a function of time and it can be seen that all the spectra falling in this class appear towards the lower flux values shown in red. It is clear that the remaining spectra which are not well explained with the model M1 include other spectral components and the spectral model needs to be modified to account for the additional components. Therefore, the remaining spectra are fitted with the canonical model of a black hole binary consisting of a soft thermal emission from a disk and a non-thermal component originating from the corona. The disk component is modeled with a multicolor disk black body model [DISKBB]{} [@mitsuda84] and the non-thermal component is modeled with [POWERLAW]{}. It is found that none of the remaining spectra are explained with the model and exhibit a larger residual close to 6 keV. Therefore, we include [GAUSSIAN]{} in the model with the centroid energy fixed at 6.4 keV to account for the Fe line emission from the disk. This model [WABS\*(DISKBB+POWERLAW+GAUSSIAN)]{} is listed as M2 in Table (1). The width of Gaussian is allowed as a free parameter. A total of 301 (\[A\]61; \[B\]81; \[C\]38, \[D\]121) spectra are explained with this model giving reduced $\Delta \chi^2\leq$1.2. As a next step in the spectral analysis, we allowed both the centroid energy and the width of Gaussian as free parameters and we name the model as M3 in Table (1). The model resulted in 210(\[A\]31; \[B\]89; \[C\]52; \[D\]48) spectra giving reduced $\Delta \chi^2\leq$1.2. Fig (\[fluxtime\]) shows the flux values for this model with pink color. It was found that some of the remaining spectra showed improvement in the spectral fitting when a narrow Gaussian line was included in the model in addition to the broad Gaussian. This resulted in a model M4 [WABS\*(DISKBB+POWERLAW+GAUSSIAN+GAUSSIAN)]{}. The model provides a good fit to 200 (\[A\]51; \[B\]52; \[C\]51, \[D\]46) spectra. For the spectral fitting of remaining spectra, the column density is allowed as a free parameter and this model M5 provides a good fit to 47 (\[A\]16; \[B\]5; \[C\]2;\[D\]24) spectra. The remaining spectra are not explained with any of the above mentioned models resulting in higher $\Delta \chi^2$ values. Model systematic errors were introduced for these spectra and systematic errors of 0.5%, 1%, 2% and 3% allow to obtain a good fit for the 87(\[A\]25; \[B\]4; \[C\]28, \[D\]30), 33 (\[A\]3; \[B\]5; \[C\]14, \[D\]11), 15 (\[C\]10, \[D\]5 and 6(\[B\]1; \[C\]4;\[D\]1) spectra with model M5. Table (1) summarizes the list of models, free parameters, systematic error and the number of spectra fitted. There are a number of free parameters in the spectral models and we focused on the photon index and width of Fe line, in particular, for this study as discussed in the next section. [c l l c ]{} Model & & Free Parameters & No.of spectra\ $M_1$& wabs\*powerlaw & $\Gamma$, N$_{pl}$& 261\ $M_2$& wabs(diskbb+powerlaw+gaussian) & T$_{in}$, N$_{dbb}$, $\Gamma$, N$_{pl}$, $\sigma$, N$_{gau}$ & 301\ $M_3$& wabs(diskbb+powerlaw+gaussian) & T$_{in}$, N$_{dbb}$, $\Gamma$, N$_{pl}$, E$_{gau}$, $\sigma$, N$_{gau}$ & 210\ $M_4$& wabs(diskbb+gaussian+powerlaw+gaussian) & T$_{in}$, N$_{dbb}$, $\Gamma$, N$_{pl}$, N$_{gau1}$, E$_{gau2}$, $\sigma_2$, N$_{gau2}$ & 200\ $M_5$& wabs(diskbb+gaussian+powerlaw+gaussian) & T$_{in}$, N$_{dbb}$, $\Gamma$, N$_{pl}$, E$_{gau1}$, N$_{gau1}$, E$_{gau2}$, $\sigma_2$, N$_{gau2}$, $N_H$ & 47\ &          0.5% SE & & 87\ &          1.0% SE & & 33\ &          2.0% SE & & 15\ &          3.0% SE & & 6\ \ \ \ \ \ ![The plot shows the total flux in 3-20 keV energy range as a function of time for the four outbursts during 2002-11. The spectra providing good fit with model M1, M2, M3, M4 and, M5 are shown with red, green, black, cyan, and pink respectively.[]{data-label="fluxtime"}](ms0172_fig1.eps){width="8cm" height="12cm"} Results and Discussion {#sect:discussion} ====================== The scheme of spectral fitting discussed above is expected to separate the hard spectra from the softer ones. It is expected that the hard spectra would appear towards the lower flux values. This is justified by the flux values shown in Fig (\[fluxtime\]), which presents the 2-10 keV flux as a function of time for the four outbursts. It is seen that a majority of the spectra explained with the model M1 appear towards lower flux values, however, a few of the spectra appear towards higher flux as well. The spectra explained with model M2, M3, M4 and M5 are shown in green, black, cyan, and pink respectively. It is noticeable that the spectra explained with model M4 and M5 appear towards higher flux in the plot. The standard model of accretion disk describe the thermal emission from the disk in terms of blackbody radiation emitted by the annuli of different radii, integrated over the inner and outer edge of the disk. The inner edge of the disk may extend down to the ISCO at the maximum beyond which no stable orbits are allowed. The model, however does not account for the non-thermal emission commonly seen in the spectrum of black hole binaries. The overall spectrum of black hole binaries including thermal and non-thermal components is explained in terms of a geometrically thin and optically thick accretion disk along with geometrically thick and optically thin hot inner flow [@narayanyi95]. The hot inner flow intercepts a fraction of soft seed photons from the accretion disk which is inverse Compton scattered. The photons escaping directly from the hot flow give rise to the non-thermal component in the spectrum. A fraction of Comptonized radiation is reflected by the accretion disk resulting in an additional characteristic spectrum. The reflection spectrum mainly consists of a broad continuum called Compton hump at $\sim$30 keV and an Fe K$\alpha$ line at 6.4 keV. In the soft spectral state, the accretion rate is high and the disk extends in the inner regions approaching the ISCO. The effects of general and special relativity are prominent on the emission from the inner region resulting in a broad and skewed Fe line. On the other hand, the accretion disk is expected to be truncated far away from the black hole during the hard spectral state, where the effects of relativity are weaker and a narrow Fe line is expected to be observed. Therefore, the presence of a broad and skewed Fe line during hard state presents a violation of the truncated disk scenario. The disk extending close to the ISCO during hard state also does not explain the weaker disk component generally seen in black hole binaries. The spectra belonging to the hard state with broader Fe line need to be investigated in detail in order to understand the discrepancy. We study the Hardness-Intensity diagram for the four outbursts during 2002-11. For the sake of clarity, we represent the spectra providing good fit with model M1, M2, M3, M4 and M5 are shown with red, green, black, cyan, and pink respectively in Fig \[HID\]. While the spectra with $\Gamma<1.8$ and $\sigma>1.5$ keV are shown by blue color with different symbols. These spectra are fitted with models (M3, M4 and M5) are indicated as m3, m4 and m5 in Fig \[HID\]. The hard state spectrum is defined as a thermal Comptonization component with photon index $\Gamma$ $\sim$ 1.8, a weak disk emission and a moderate reflection component. We selected the spectra with photon index $<$1.8 and broad Fe line with FWHM $>$1.5 keV and found a total of 76 \[A:20; B:19; C:21; D:16\] spectra matching these criteria. If we interpret the broad Fe line as a result of the relativistic effects in the vicinity of the black hole, these spectra present a case where the truncation of the disk stands violated. Therefore, we focus our further investigation mainly on this set of spectra. Fig (\[fluxtimeredblue\]), which is the same plot as Fig (\[fluxtime\]) shows all the 76 cases marked as blue points with almost all of them appearing towards higher flux values. The evolution of Fe line width and photon index with time is shown in Fig (\[sigwidthtime\]) for the first outburst. The two occurrences of spectra with $\Gamma<1.8$ and $\sigma>1.5$ keV can be seen in Fig (\[sigwidthtime\]). ![Hardness Intensity Diagram for the four outbursts during 2002-11. The spectra providing good fit with model M1, M2, M3, M4 and, M5 are shown with red, green, black, cyan, and pink respectively. The spectra with $\Gamma<1.8$ and $\sigma>1.5$ keV are shown with blue.[]{data-label="HID"}](ms0172_fig2.eps){width="8cm" height="12cm"} The higher value of soft energy flux and the finding of broad Fe line will be consistent if these spectra belong to soft spectral state with $\Gamma >$1.8. Therefore, a hypothesis is proposed that the observed spectra with broad Fe line actually belong to the soft state, however the hardening of spectra may artificially be resulted by the reflection hump that appear towards the higher energies. Therefore, we have added reflection component [reflionx]{} to the 76 spectra from the four outbursts. The change in the power law index is studied before and after adding the reflection component in the model. We found that the error bars of photon index are either very large or zero for the 40 spectra. So we have provided the spectra for the 36 observations only where the parameter is constrained. Fig (\[beforeafter\]) shows the two values plotted against each other with the green line representing the same value of power law index before and after adding the reflection. The points lying above and below this line show the softening and hardening of spectra respectively. For some of the spectra, the photon index after adding reflection was found to be higher than the best-fit value obtained without reflection component. However, the photon index with reflection component was found to be $>$2 only for two spectra and the remaining spectra continued to provide photon index $<$2. Therefore, the occurrence of hard spectrum and broad Fe line is not understood in the light of contribution from the reflection component. One of the possible solution is to model the spectrum with two Comptonization models [see @yamada; @basak17], where a soft Comptonization component is considered in addition to the main Comptonization component. The model used by [@basak17] to study the hard state spectrum of Cyg X-1 consists of reflection from both Comptonization components. The soft Comptonization component models the soft excess towards lower energies in their work. The model yields the best-fit with $R_{in}$=13-20 $R_g$, indicating a truncated accretion disk in hard state. ![Total flux in 3-20 keV energy range as a function of time for the four outbursts during 2002-11. The observations with $\Gamma$ $<$ 1.8 and $\sigma$ $>$ 1.5 keV are shown with blue points and remaining observations with red. It is noticeable that the blue points appear towards the higher flux values.[]{data-label="fluxtimeredblue"}](ms0172_fig3.eps){width="8cm" height="12cm"} ![The plot shows the evolution of Fe line width (top panel) and photon index (bottom panel) with time for 2002-03 outburst. The observations with $\Gamma$ $<$ 1.8 and $\sigma$ $>$ 1.5 keV are shown with blue points and remaining observations with red.[]{data-label="sigwidthtime"}](ms0172_fig4.eps){width="8cm" height="12cm"} ![The values of photon index with reflection versus the photon index without reflection for spectra with $\Gamma$ $<$ 1.8 and $\sigma$ $>$ 1.5 keV from all the four outbursts. The plot includes the values of photon indices from 36 spectra where the parameter is constrained before and after the addition of reflection component. The green line represent the same value of photon index before and after adding the reflection.[]{data-label="beforeafter"}](ms0172_fig5.eps){width="13cm" height="9cm"} Conclusions =========== The two main features of the reflection component observed in the spectrum of black hole binaries include an Fe emission line at 6.4 keV and broad Compton hump at $\sim$30 keV. The reflection hump appearing towards higher energies can give rise to an artificial hardening to an intrinsically soft spectrum. In order to test the hypothesis, we studied spectra from the four outbursts of [GX 339-4]{} using RXTE/PCA data observed between 2002 to 2011. We studied the spectra with different models of increasing complexity. We particularly shortlisted those cases where a broad Fe line ($\sigma >$1.5 keV) was observed in the hard state with $\Gamma <$1.8. These spectra are refitted by adding reflection component and the values of the photon index before and after adding the reflection were compared. It is found that the addition of reflection component in the spectral fitting results in the higher values of photon indices in few cases. In fact, two spectra with photon index $<$1.8 showed photon index $>$2 after addition of reflection component. However, there are several spectra where the photon index remains $\lesssim$1.8. Therefore, the results show that the reflection component does not completely explain the puzzle about the existence of broad Fe line in the hard spectral state. \[lastpage\]

--- abstract: 'In this paper we show how to approximate the transition density of a CARMA(p, q) model driven by means of a time changed Brownian Motion based on the Gauss-Laguerre quadrature. We then provide an analytical formula for option prices when the log price follows a CARMA(p, q) model. We also propose an estimation procedure based on the approximated likelihood density.' author: - 'Lorenzo Mercuri, Andrea Perchiazzo, Edit Rroji' title: 'Finite Mixture Approximation of CARMA(p,q) Models' --- Introduction ============ The aim of this paper is to provide a simple approximation procedure for the transition density of a Continuous Autoregressive Moving Average Model driven by a Time Changed Brownian Motion. The Continuous Autoregressive Moving Average (CARMA hereafter) model with guassian transition density was first introduced in [@doob1944] as a continuous counterpart of the well known ARMA process defined in discrete time. Recently this model has gained a significant attention in literature due to the relaxation of the gaussianity assumption. A Lévy CARMA model has been proposed in [@Brockwell2001] and the associated marginal distribution is allowed to be skewed and fat-tailed. These features increase the appealing of these processes especially in modeling financial time series \[see for examples [@Brockwell2011; @Iacus2015] and references therein\]. Indeed, in CARMA(p,q) models it is possible to work directly with market data without being forced of considering an equally spaced time grid necessary in discrete-time models like for example in ARMA(p,q) models. The CARMA(p,q) process can be seen as a generalization of the Ornstein-Uhlenbeck process (OU). The OU process is not sufficiently flexible for financial applications since its autocorrelation function shows a monotonic decreasing (negative exponential) behaviour. In this context, the CARMA(p,q) model seems to be useful as it is able to capture a more complex shape for the dependence structure as discussed in [@Brockwell2004]. The nice statistical and mathematical properties make this class of continuous time models very suitable for modeling commodities [@NUALART2000; @BENTH2014392], interest rates [@andresen2014carma], mortality intensity [@Hitaj2019], spot electricity prices [@garcia2011estimation] and temperature [@benth2007putting]. In order to apply the CARMA model on real data, for the evaluation of derivatives on commodities and/or for the evaluation of insurance contracts, it is necessary to know the transition density of the process. In the case of a CARMA(p,q) model where the driving noise is a Brownian motion, the transition density is Gaussian. Therefore, an estimation procedure \[see [@Tomasson2015] for details\] can be obtained directly combining the Gaussian likelihood function with the Kalman Filter while for the pricing of financial/insurance contracts we have to compute just the expected value of a transformation of a Gaussian random variable. We refer for instance to the pricing formula for options on futures derived in [@PASCHKE20102742] where the log-spot price is a gaussian CARMA(p,q) process. Similar results are obtained for interest rate derivatives \[see [@andresen2014carma] for details\]. The main contribution of this paper is to propose a finite mixture of normals that approximates the transition density of a Time Changed Brownian Motion CARMA(p,q) process (TCBm-CARMA hereafter). This approximation increases the appealing of the CARMA model in practical applications since, as a finite mixture of normals, it has a level of computational complexity similar to the gaussian CARMA for estimation on real data and for evaluation of financial and insurance contracts. The choice of a Time Changed Brownian Motion (TCBm) as a driving noise increases also the ability of the CARMA to capture the statistical features of data. In the case of the TCBm-CARMA, our results generalize in a straightforward manner the estimation procedure in [@Iacus2015] based on the Quasi-Gaussian Likelihood (QGMLE) contrast function \[see [@yoshida2011polynomial; @Masuda2013] and reference therein for a complete discussion of the QGMLE procedure\]. Indeed we do not need a two step procedure but we are able to estimate autoregressive, moving average and Lévy measure parameters at the same time. Pricing formulas of financial contracts are again simple linear convex combinations of gaussian pricing formulas. For instance for options written on futures we have a convex linear combination of pricing formulas in [@PASCHKE20102742]. The construction of our approximated transition density for a TCBm-CARMA(p,q) model is based on two main components: the dyadic Riemann sum approximation of a stochastic integral \[see [@attal2003quantum] for a complete discussion\] and the Gauss-Laguerre quadrature \[see [@abramowitz70a] for more details\]. The main idea behind this approach is to approximate the distribution associated to the subordinator process at unitary time with a discrete random variable where the realizations are the zeros of the Laguerre polynomial with a fixed order and the corresponding probability is obtained using the Gauss-Laguerre quadrature. Based on our knowledge the first authors that applied this approach in two different situation are [@Madan2013] for a option pricing purpose and [@LOREGIAN2012217] for the estimation of the Variance Gamma distribution using the EM-algorithm proposed by [@Dempster77maximumlikelihood]. Several authors, recently have used the Laguerre polynomials to derive approximated closed formulas for the pricing of financial contracts \[see [@Belle2019] and reference therein\] and insurance contracts \[see [@ZHANG2019329] and reference therein\] for some specific exponential Lévy processes. A comparison of some numerical techniques including the Gauss-Laguerre quadrature for pricing derivatives under an exponential Variance Gamma process has been presented in [@Aguilar2020]. The paper is organized as follows. Section \[finite-approximation-of-the-density-of-a-normal-variance-mean-mixture\] reviews the Gauss-Laguerre approximation for a Normal Variance Mean Mixture random variable. In Section \[LCARMA\] we extend the Gauss-Laguerre approximation to the case of the transition density of a TCBm-CARMA(p,q) model and we propose an estimation method that maximizes the approximated likelihood function. In Section \[option-pricing-in-a-luxe9vy-carmapq-model.\] we discuss how to apply our approximated density in the evaluation of a transformation of the exponential TCBm-CARMA(p,q) model. In particular we derive specific formulas for the futures term structure and for option prices on futures. Section \[conclusion\] concludes the paper. Finite Approximation of the Density of a Normal Variance Mean Mixture ===================================================================== First we recall the formal definition of a Normal Variance Mean Mixture discussed in [@Barndorff97]. A random variable $Y$ is a Normal Variance Mean Mixture if we have: $$Y = \mu + \theta \Lambda + \sigma\sqrt{\Lambda}Z \label{1}$$ $Z\sim N\left(0,1\right)$. $\Lambda$ is a continuous positive random variable with an exponentially slowly density function $f\left(u\right)$ defined as: $$f\left(u\right)=e^{-\varphi_+u}u^{\lambda-1}L_{\theta}\left(u\right)\mathbbm{1}_{\left\{u\geq0\right\}}, \label{2}$$ $\varphi_+\geq0$, $L_{\theta}\left(u\right):\left[0,+\infty\right)\rightarrow\left[0,+\infty\right)$ function with slowly variation, i.e.: $$\lim_{u\rightarrow+\infty}\frac{L\left(\alpha u\right)}{L\left(u\right)}=1.$$ In order to construct a discrete version of the random variable $\Lambda$, we use the Gauss-Laguerre quadrature. Let $f\left(x\right)$ be a function with support $\left[0,+\infty\right)$ such that $$\int_0^{+\infty} f\left(x\right)e^{-x}\mbox{d}x<+\infty,$$ we have the follwing approximation: $$\int_0^{+\infty} f\left(x\right)e^{-x}\mbox{d}x\approx\sum_{i=1}^{m}w\left(k_i\right)f\left(k_i\right). \label{approx:Laguerre}$$ $k_i$ is the $i$-th root of the Laguerre polynomial $L_m\left(k_i\right)$ and the weights $w\left(k_i\right), \ i=1,\ldots,m$ are: $$w\left(k_i\right) = \frac{k_i}{\left(m+1\right)^2L^2_{m+1}\left(k_i\right)}.$$ We start from the moment generating function of the random variable $\Lambda$: $$\mathbb{E}\left(e^{c\Lambda}\right)=\int_0^{+\infty}e^{cu}e^{-\varphi_+u}u^{\lambda-1}L_{\theta}\left(u\right)\mbox{d}u. \label{mgf}$$ Posing $k=\varphi_+u$ in , we get: $$\mathbb{E}\left(e^{c\Lambda}\right)=\int_0^{+\infty}e^{-k}e^{c\frac{k}{\varphi}}\left(\frac{k}{\varphi}\right)^{\lambda-1}L_{\theta}\left(\frac{k}{\varphi}\right)\frac{\mbox{d}k}{\varphi}=\int_0^{+\infty}e^{-k}\frac{e^{c\frac{k}{\varphi}}}{k}\left(\frac{k}{\varphi}\right)^{\lambda}L_{\theta}\left(\frac{k}{\varphi}\right)\mbox{d}k.$$ Applying the formula in , we have: $$\mathbb{E}\left(e^{c\Lambda}\right)\approx \sum_{i=1}^m e^{c\left(\frac{k_i}{\varphi_+}\right)}\frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right).$$ It is to worth noting that $$\sum_{i=1}^m \frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)\approx\int_0^{+\infty}\frac{e^{-k}}{k}\left(\frac{k}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k}{\varphi_+}\right)\mbox{d}k.$$ Using the substitution $u=\frac{k}{\varphi_+}$ we get: $$\sum_{i=1}^m \frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)\approx\int_0^{+\infty}e^{-\varphi_+u}u^{\lambda-1}L_\theta\left(u\right)\mbox{d}u=1,$$ therefore we have: $$\mathbb{E}\left(e^{c\Lambda}\right)\approx \sum_{i=1}^m e^{c\left(\frac{k_i}{\varphi_+}\right)}\frac{\frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)}{\sum_{i=1}^m \frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)}. \label{eq:MgfApprox}$$ The right hand side of the equation can be seen as the moment generating function of a positive random variable $\Lambda_m$ with a finite support and defined as: $$\Lambda_m=\left\{ \begin{array}{lcl} u_1=\frac{k_1}{\varphi_+}& & \mathbb{P}\left(u_1\right)=\frac{\frac{w\left(k_1\right)}{k_1}\left(\frac{k_1}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_1}{\varphi_+}\right)}{\sum_{i=1}^m \frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)}\\ \vdots & & \vdots\\ u_i=\frac{k_i}{\varphi_+}& & \mathbb{P}\left(u_i\right)=\frac{\frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)}{\sum_{i=1}^m \frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)}\\ \vdots & & \vdots\\ u_m=\frac{k_m}{\varphi_+}& & \mathbb{P}\left(u_m\right)=\frac{\frac{w\left(k_n\right)}{k_m}\left(\frac{k_m}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_m}{\varphi_+}\right)}{\sum_{i=1}^m \frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)}\\ \end{array} \right. . \label{eq:approxMixRVNVMM}$$ The next step is to consider a sequence of random variables $Y_m$ defined as: $$Y_m=\mu+\theta \Lambda_m+\sqrt{\Lambda_m}Z,$$ with $Z\sim N\left(0,1\right)$ independent of $\Lambda_m$. For any $m$ the density of $Y_m$ is a finite mixture of normal with the following form: $$f_{Y_m}\left(y\right)=\sum_{i=1}^m\phi(y,\mu_0+\mu u_i,\sigma^2 u_i)\mathbb{P}\left(u_i\right) \label{approxDens}$$ where $\phi(x,a,b)$ is a normal density at point $x$ with mean $a$ and variance $b$. Using the definition of $\Lambda_m$ $$f_{Y_m}\left(y\right)=\sum_{i=1}^m\phi\left(y,\mu+\theta \frac{k_i}{\varphi_+};\sigma^2 \frac{k_i}{\varphi_+}\right)\frac{\frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)}{\sum_{i=1}^m \frac{w\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right)} \label{eq:approximated}$$ Applying the Gauss-Laguerre quadrature we get: $$f_{Y_m}\left(y\right)\stackrel{m\rightarrow+\infty}{\longrightarrow}\int_0^{+\infty}\phi\left(y,\mu_0+\mu \frac{k}{\varphi_+}; \sigma^2 \frac{k}{\varphi_+}\right)\frac{e^{-k}}{k}\left(\frac{k}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k}{\varphi_+}\right)\mbox{d}k.$$ Substituting $u=\frac{k}{\varphi_+}$, we have: $$f_{Y_m}\left(y\right)\stackrel{m\rightarrow+\infty}{\longrightarrow}\int_0^{+\infty}\phi\left(y,\mu_0+\mu u;\sigma^2 u\right)e^{-\varphi_+u}u^{\lambda-1}L_{\theta}\left(u\right)\mbox{d}u.$$ The right-hand side is the density of the random variable in . Observe that approximation discussed here can be applied in three wide applied distributions: Variance Gamma, Normal Inverse Gaussian, Generalized Hyperbolic. In all cases, the density of the mixing random variable belongs to the class defined in . Indeed we obtain the density of a Gamma random variable with shape $\alpha$ and rate $\beta$ parameters posing the following condition: $$\varphi_+=\beta,\ \lambda = \alpha, \ L_{\left(\alpha,\beta\right)}\left(u\right)=\frac{\beta^{\alpha}}{\Gamma\left(\alpha\right)},$$ therefore the density in approximate the density of a Variance Gamma random variable.The density of an Inverse Gaussian IG$\left(a,b\right)$ can be obtained from by posing: $$\varphi_+=\frac{b^2}{2},\ \lambda=-\frac12, \ L_{a,b}\left(u\right)=\left[\frac{a}{\sqrt{2\pi}}\right]e^{ab-\frac{a^2}{2x}}.$$ In this case we obtain an approximation of the Normal Inverse Gaussian density using .The Generalized Inverse Gaussian density with $a>0$, $b>0$ and $p\in\mathbb{R}$ is a special case of when: $$\varphi_+=\frac{\alpha}{2},\ \lambda=p, \ L_{a,b,p}\left(u\right)=\frac{\left(\frac{a}{b}\right)^{\frac{p}{2}}}{2K_p\left(\sqrt{ab}\right)}e^{-\frac{b}{2u}}$$ where $K_p\left(x\right)$ is a modified Bessel function of the second kind. Using we approximate the density of a Generalized Hyperbolic distribution. Figure \[fig:Fig1\] shows the behavior of the analytic and approximated moment generating functions for the Gamma, Variance Gamma, Inverse gaussian, Normal Inverse Gaussian model. To generate the approximated moment generating function we use $m=40$. ![\[fig:Fig1\] Comparison between theoretical and approximated moment generating function for a $\Gamma\left(1,1\right)$, the corresponding symmetric Variance Gamma centered in zero, a IG$\left(1,1\right)$ and its associated symmetric Normal Inverse Gaussian centered in zero.](Fig1-1.pdf){width="50.00000%"} In Appendix \[EMderivation\] derive the Expectation Maximization algorithm for the approximated density in . Lévy CARMA(p,q) model. {#LCARMA} ====================== In this section, we review the main features of Lévy CARMA(p,q) models. The CARMA model, firstly introduced by [@doob1944] as a generalization in continuous time setup of the Gaussian ARMA model, has recently gained a rapid development in different areas due to the substitution of the Brownian Motion with a general Lévy process as driving noise \[see [@Brockwell2001] for a discussion of a CARMA model driven by a Lévy process with finite second order moments\]. The formal definition of a Lévy CARMA(p,q) model $Y_t$ with $p > q \geq 0$ is based on the continuous version of the state-space representation of an autoregressive moving average-ARMA(p,q) model: $$Y_t =\mathbf{b}^{\top}X_t \label{eqCar}$$ where $X_t$ satisfies: $$\mbox{d}X_t = \mathbf{A}X_{t-}\mbox{d}t+\mathbf{e}\mbox{d}Z_t. \label{eq:CarSDE}$$ $\left\{Z_{t}\right\}_{t\geq0}$ is a Lévy process. The matrix $\mathbf{A}$ with dimension $p\times p$ is defined as: $$\mathbf{A}=\left[ \begin{array}{ccccc} \\ 0 & 1 & 0 & \ldots & 0\\ 0 & 0 & 1 & \ldots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \ldots & 1\\ -a_p & -a_{p-1} & -a_{p-2} & \ldots & -a_1\\ \end{array} \right]_{p\times p}.$$ The vectors $\mathbf{e}$ and $\mathbf{b}$ with dimension $p\times 1$ are defined as follows: $$\mathbf{e}=\left[0,0,\ldots,1\right]^\top$$ $$\mathbf{e}=\left[b_0,0,\ldots,b_{p-1}\right]^\top$$ where $b_{q+1}=\ldots=b_{p-1}=0$. Given the initial point $X_s$, the solution of thee Eq. is: $$X_t= e^{\mathbf{A}\left(t-s\right)}X_s+\int_{0}^{+\infty}e^{\mathbf{A}\left(t-s\right)}\mbox{d}Z_u, \ \forall t>s,$$ where $e^\mathbf{A}=\underset{h=0}{\stackrel{+\infty}{\sum}}\frac{1}{h!}\mathbf{A}^h$.We report in the following the scale property of a CARMA(p,q) process. This property introduces a constraint between the Lévy measure parameters and the moving average vector $\mathbb{b}$. Indeed it is possible to introduce a new Lévy process $L_t$ defined as: $$L_t=\frac{1}{a}Z_t, \ a>0.$$ We also define the state process $X^{\prime}_t$ as: $$X^{\prime}_t =\frac{1}{a}X_t$$ and a new moving average vector $\tilde{\mathbf{b}}=a\mathbf{b}$, the CARMA(p,q) process in can be written equivalently as: $$Y_t= \tilde{\mathbf{b}}^\top X^{\prime}_t$$ where $X^{\prime}_t$ satisfies the following Stochastic Differential Equations: $$\mbox{d}X_t^{\prime}=A X_{t-}^{\prime}\mbox{d}t+\mathbf{e}\mbox{d}L_t.$$ As reported in [@Brockwell2011], under the assumption that all eigenvalues $\lambda_1,\ldots,\lambda_p$ of matrix $\mathbf{A}$ are distinct and their real part is negative, we can write the CARMA(p,q) model as a summation of a finite number of continuous autoregressive models of order 1, i.e. CAR(1) models. Therefore: $$Y_t = \mathbf{b}^\top e^{\mathbf{A}\left(t-s\right)} X_s +\int_0^{+\infty}\underset{i=1}{\stackrel{p}\sum}\left[\alpha\left(\lambda_i\right)e^{\lambda_i\left(t-u\right)}\right]\mathbb{I}_{s\leq u\leq t}\mbox{d}Z_u \label{expr:Sol}$$ with $\alpha\left(z\right)=\frac{b\left(z\right)}{a^{\prime}\left(z\right)}$ where $a\left(z\right)$ and $b\left(z\right)$ are polynomial functions defined as: $$a\left(z\right)=z^p+a_1z^{p-1}+\ldots+a_p,$$ $$b\left(z\right)=b_0+b_1z+\ldots+b_{p-1}z^{p-1}.$$ Under the additional requirement of the existence of a cumulant generating function for $Z_1$, the conditional moment generating function of a CARMA(p,q) model $Y_t$ given the information at time $s<t$ is obtained: $$\mathbb{E}_s\left[e^{cY_t}\right]=e^{c\mathbf{b}^\top e^{\mathbf{A}\left(t-s\right)X_s}}\exp\left[\int_{s}^{t}\kappa\left(c\underset{i=1}{\stackrel{p}{\sum}}\left[\alpha\left(\lambda_i\right)e^{\lambda_i\left(t-u\right)}\right]\right)\mbox{d}u\right] \label{mgfCARMA}$$ with $\kappa\left(c\right)=\ln \mathbb{E}\left[e^{cZ_1}\right]<+\infty.$ Once the state variable $X_s$ is filtered from observable data, from a theoretical point of view, the result in can be used to compute the transition density from time $s$ to time $t$ by means of the Fourier Transform because the characteristic function is obtained from the moment generating function evaluated at $iu$.In order to get an estimate of the state variable from the observed data $Y_{t_0},Y_{t_1},\ldots, Y_{t_i},\ldots$, it is possible to use the approach discussed in [@Brockwell2011] and recently implemented in [@Iacus2015]. As first step, the vector $\hat{X}_{q,t}$ containing the first $q-1$ components of the state process $X_t$ can be written in terms of $Y_{t-1}$ as follows: $$\mbox{d}\hat{X}_{q,t} = \mathbf{B}\hat{X}_{q,t-}\mbox{d}t+\mathbf{e}Y_{t-1}\mbox{d}t \label{eq:StochFilterBrock}$$ where $$\mathbf{B}=\left[ \begin{array}{ccccc} \\ 0 & 1 & 0 & \ldots & 0\\ 0 & 0 & 1 & \ldots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \ldots & 1\\ -b_0 & -b_1 & -b_{2} & \ldots & -b_{q-1}\\ \end{array} \right]_{p\times p}$$ and $$\mathbf{e}_q=\left[0,\ldots,0,1\right]^{\top}.$$ The remaining $p-q$ components of $X_t$ are obtained from the higher order derivatives of the first component $X_{0,t}$ in the state vector, i.e.: $X_t$ with respect to time: $$X_{j,t}=\frac{\partial^{j-1}X_{0,t}}{\left(\partial t\right)^{j-1}}, j = q,\ldots,p-1.$$ Combining the approach in [@Brockwell2011] with the result in , it is possible to introduce an estimation procedure of the Lévy CARMA(p,q) model based on the Maximum Likelihood method. This procedure requires the numerical evaluation of two integrals, the first in the definition of the moment generating function and the second in the inversion formula of the characteristic function. In this section, we show that in the case of a Time Changed Brownian motion, we can can approximate the density using the Laguerre polynomials overcomung the numerical integration problems that arise in the standard approach. We start considering the case of the Ornstein Uhlenbeck that does not require the estimation of the state process then we move to the general CARMA(p,q) model. Estimation of an Ornstein Uhlenbeck driven by a Time Changed Brownian Motion. {#Est:OUdrivenTCBM} ----------------------------------------------------------------------------- Let $\left(\Omega, \mathcal{F}, \mathbb{F}, \mathcal{P}\right)$ be a filtered probability space where $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\geq0}$ is a filtration, the process $Y_t$ is a Time Changed Brownian Ornstein-Uhlenbeck (TCBm-OU hereafter) $Y_t$ satisfies the following stochastic differential equation: $$\mbox{d}Y_t=-aY_{t-}\mbox{d}t+\mbox{d}W_{\Lambda_t}, \ Y_{t_0}=y_0. \label{eq:OU}$$ where $W_{\Lambda_t}$ is a Brownian Motion stopped by the subordinator process $\Lambda_t$. The solution of the SDE in is: $$Y_t = y_0e^{-a\left(t-t_0\right)}+\int_{t_0}^{t}e^{-a\left(t-u\right)}\mbox{d}W_{\Lambda_{u}}.$$ It is worth noting that the distribution at time 1 of the process $W_{\Lambda_t}$ is a Normal Variance Mean Mixture centered in zero. Defining the $\sigma$-field $\mathcal{G}_{t_0,t}=\sigma\left(\mathcal{F}_{t_0} \cup \sigma\left(\left\{\Lambda_u\right\}_{u\leq t}\right)\right)$ with $t_0<t$, we have: $$W_{\Lambda_t}-W_{\Lambda_{t_0}}\left|\mathcal{G}_{t_0,t}\right.\sim N\left(0,\Lambda_t-\Lambda_{t_0}\right).$$ The $\sigma$-field $\mathcal{G}_{t_0,t}$ is crucial for the construction of the approximated transition density of the TCBm-OU process. Given the information associated to the $\sigma$-field $\mathcal{G}_{t_0,t}$, the conditional distribution for $Y_t$ becomes[^1]: $$Y_t\left|\mathcal{G}_{t_0,t}\right.\sim N\left(y_0e^{-a\left(t-t_0\right)}, \int_{t_0}^{t}e^{-2a\left(t-u\right)}\mbox{d}\Lambda_u\right). \label{eq:CondDist}$$ Let us define $V_{t_0}^{t}$ as: $$V_{t_0}^{t}=\int_{t_0}^{t}e^{-2a\left(t-u\right)}\mbox{d}\Lambda_{u}. \label{eq:newV}$$ We can approximate the integral in with a left Riemann sum as follows: $$V_{t_0}^{t}\approx V_{t_0}^{t}\left(n\right)= \underset{k=0}{\stackrel{\left[2^n\left(t-t_0\right)\right]-1}{\sum}}e^{-2a\left(t-t_0-k2^{-n}\right)}\left(\Lambda_{t_0+\left(k+1\right)2^{-n}}-\Lambda_{t_0+k2^{-n}}\right). \label{eq:ApproxV}$$ The increments $\Lambda_{t_0+\left(k+1\right)2^{-n}}-\Lambda_{t_0+k2^{-n}}$ in have a density of the shape in . Therefore we can approximate these densities using the Laguerre polynomials. To this aim, we first introduce a discrete random variable $\mathcal{U}_k$:$$\mathcal{U}_k=\left\{ \begin{array}{lll} u_1 & & \mathbb{P}\left(u_1\right)\\ \vdots & & \vdots\\ u_m & & \mathbb{P}\left(u_m\right) \end{array} \right.$$that approximates the $k-th$ increment $\Lambda_{t_0+\left(k+1\right)2^{-n}}-\Lambda_{t_0+k2^{-n}}$. The random variable $V_{t_0}^{t}\left(n\right)$ can be approximated introducing the new random variable $V_{t_0}^{t}\left(n,m\right)$ defined using dyadic Riemann sums reads: $$V_{t_0}^{t}\left(n,m\right)=\left\{ \begin{array}{lll} \underset{k=0}{\stackrel{\left[2^n\left(t-t_0\right)\right]-1}{\sum}} e^{-2a\left(t-t_0-k2^{-n}\right)}u_1 & \left[2^n\left(t-t_0\right)\right]-1,0,\ldots,0 & \mathbb{P}^{\left[2^n\left(t-t_0\right)\right]-1}\left(u_1\right)\\ \vdots & & \vdots\\ \underset{k=0}{\stackrel{\left[2^n\left(t-t_0\right)\right]-1}{\sum}} e^{-2a\left(t-t_0-k2^{-n}\right)}u_k & n_1,\ldots,n_m & \underset{i=1}{\stackrel{m}{\prod}}\mathbb{P}^{n_i}\left(u_i\right)\\ \vdots & & \vdots\\ \underset{k=0}{\stackrel{\left[2^n\left(t-t_0\right)\right]-1}{\sum}} e^{-2a\left(t-t_0-k2^{-n}\right)}u_m & 0,\ldots,0,\left[2^n\left(t-t_0\right)\right]-1 & \mathbb{P}^{\left[2^n\left(t-t_0\right)\right]-1}\left(u_m\right) \end{array} \right. . \label{ApproxVOUTCBmAP}$$ Observe that the random variable $V_{t_0}^{t}\left(n,m\right)$ has $m^{\left[2^n\left(t-t_0\right)\right]-1}$ realizations. Denoting $V_{t_0}^{t}\left(n,m,i\right)$ the i$-th$ realization of the random variable $V_{t_0}^{t}\left(n,m\right)$ and $\mathbb{P}\left[V_{t_0}^{t}\left(n,m,i\right)\right]$ its probability, we obtain the following approximated density: $$f_{Y_t\left|\mathcal{F}_{t_0}\right.}\left(y\right)=\sum_{i=1}^{m^{\left[2^n\left(t-t_0\right)\right]-1}} \phi\left(y,y_0e^{-a\left(t-t_0\right)},V_{t_0}^{t}\left(n,m,i\right)\right)\mathbb{P}\left[V_{t_0}^{t}\left(n,m,i\right)\right]. \label{eq:ApproxDens}$$ To check the accuracy of this approximation, we compare the theoretical moment generating function of an Ornstein-Uhlenbeck driven by a Variance Gamma model obtained through the result in [@Hitaj2019] with the moment generating function of the finite mixture of normals with density .Figure \[fig:Fig2\] reports a graphical comparison of the theoretical and the approximated moment generating function of a VG-CAR(1) with $a = 0.25$, $t=\frac14$ and $t_0$. The interval $\left[t_0,t\right)$ has been divided into subintervals of length $\Delta t =2^{-6}\approx0.01562$ and fixing $m=2$ we get 65536 realizations of the random variable $V_{t_0}^{t}\left(n,m\right)$. ![\[fig:Fig2\] Comparison of theoretical and approximated moment generating function for a VG-CAR(1) model](Fig2-1.pdf){width="50.00000%"} The result in can be used to construct a Maximum Likelihood Estimation procedure. In the following we perform a simulation and estimation study for the VG-CAR(1) model. As benchmark we use the Quasi-Gaussian Likelihood method extended to the SDE driven by a standardized Lévy noise introduced in [@Masuda2013]. We perform the following steps: 1. We simulate a sample for a VG-CAR(1) model where $a=0.25$ while the distribution at time 1 of the subordinator process is $\Gamma\left(1,1\right)$. In the simulation we use the Euler-Maruyama method with a frequency $\Delta t = 0.01$. 2. We get a new trajectory by subsampling the data obtained at the previous point with a lower frequency, i.e. $\Delta t = 1$. 3. We estimate the parameters, using the data obtained in step 2, by maximizing the log-likelihood constructed using the Laguerre approximation. ![Sample path of a VG-OU process.\[Fig:SampleOU\]](unnamed-chunk-3-1.pdf){width="50.00000%"} ## b a Shape ## 0.2226184 0.9900000 --------- # YUIMA ESTIMATION QMLE BASED ON MASUDA ## 0.2394667 1.0822139 1.0501550 # Estimation Based on Gauss Laguerre Quadrature ## 0.2400000 1.0000000 1.0000000 # TRUE PARAMETERS Estimation of a Gaussian CARMA(p,q) model. ------------------------------------------ In this section we review the literature for the estimation methods of CARMA(p,q) model driven by a Brownian Motion. As discussed in [@Tomasson2015], we have two different approaches for the estimation of a Gaussian CARMA process. The first is based on the frequency domain representation of the CARMA process. The estimated parameters are obtained by minimizing a distance between theoretical $f\left(\omega\right)$ and empirical $\hat{f}\left(\omega\right)$ spectral density, for instance: $$\underset{a_1,\ldots, a_p \ \ \ b_1,\ldots,b_q}{\text{argmin}}\int_{-\infty}^{+\infty}\left\{\ln\left[f\left(\omega\right)\right]+\frac{\hat{f}\left(\omega\right)}{f\left(\omega\right)}\right\}\mbox{d}\omega$$ where $$f\left(\omega\right)=\frac{\mathbf{b}\left(i\omega\right)\mathbf{b}\left(-i\omega\right)}{2\pi\mathbf{a}\left(i\omega\right)\mathbf{b}\left(-i\omega\right)}.$$ The alternative estimation approach is based on the time domain representation of the CARMA process. In this case, the unobservable state process can be extrapolated using the Kalman filter therefore we get the estimates for the model parameters by maximizing the loglikelihood function or minimizing the least-squares error. A detailed description of the Kalman filter and the construction of the gaussian loglikelihood function can be found in [@Iacus2015]. Estimation of a Lévy CARMA(p,q) model driven by a Time Changed Brownian Motion. {#estimation-of-a-luxe9vy-carmapq-model-driven-by-a-time-changed-brownian-motion.} ------------------------------------------------------------------------------- Here we discuss how to estimate the CARMA(p,q) model when the driving noise is a Time Changed Brownian Motion. In this case we propose two alternatives. The first approach combines the Kalman Filter with the approximation transition density of the CARMA(p,q) process while the second use the methodology for recovering noise with the estimation method discussed for the Normal Variance Mean Mixture. ### Lévy CARMA estimation using the approximated transition density {#luxe9vy-carma-estimation-using-the-approximated-transition-density} In order to obtain an approximated transition density for a CARMA(p, q) process we first need to determine the conditional mean and the conditional variance of the state process $X_t$ given the information contained in the $\sigma$-field $\mathcal{G}_{t_0,t}$ and the state process at $X_{t_0}$ defined respectively as: $$\mathbb{E}\left[X_t\left|\mathcal{G}_{t_0,t},X_{t_0}\right.\right]=e^{\mathbf{A}\left(t-t_0\right)}X_{t_0}.$$ $$\mathbb{V}\text{ar}\left[X_t\left|\mathcal{G}_{t_0,t},X_{t_0}\right.\right]=\int_{t_0}^{t}e^{\mathbf{A}\left(t-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(t-u\right)}\mbox{d}\Lambda_u.$$ Therefore the transition density of the CARMA(p,q) model $Y_t$ given $\mathcal{G}_{t_0,t}$ and $X_{t_0}$ is: $$Y_t\left|\left(\mathcal{G}_{t_0,t},X_{t_0}\right.\right)\sim N\left(\mathbf{b}^\top e^{\mathbf{A}\left(t-t_0\right)}X_{t_0}, \int_{t_0}^{t}\mathbf{b}e^{\mathbf{A}\left(t-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(t-u\right)}\mathbf{b}^\top \mbox{d}\Lambda_u \right)$$ Defining the quantity $V_{t_0}^{t}=\int_{t_0}^{t}\mathbf{b}e^{\mathbf{A}\left(t-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(t-u\right)}\mathbf{b}^\top \mbox{d}\Lambda_u$, the transition density of the CARMA(p,q) process $Y_t$ given $X_{t_0}$ can be written in the following form: $$f_{Y_t\left|X_{t_0}\right.}\left(y\right)=\int_{0}^{+\infty} \varphi\left(y;\mathbf{b}e^{\mathbf{A}\left(t-t_0\right)}X_{t_0},v\right)g_{V_{t_0}^{t}}\left(v\right)\mbox{d}v, \label{eq:RealDensityCarma}$$ where $\varphi\left(y,\mu,\sigma^2\right)$ is a normal density with mean $\mu$ and variance $\sigma^2$; $g_{V_{t_0}^t}\left(v\right)$ is the density of $V_{t_0}^{t}$. As done in Section \[Est:OUdrivenTCBM\], we approximate the integral in $V_{t_0}^t$ with a left Reimann sum and we have: $$V_{t_0}^t\approx V_{t_0}^{t}\left(n,m\right)=\left\{ \begin{array}{lll} \underset{k=0}{\stackrel{\left[2^n\left(t-t_0\right)\right]-1}{\sum}} \mathbf{b}e^{\mathbf{A}\left(t-t_0-k2^{-n}\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(t-t_0-k2^{-n}\right)}\mathbf{b}^\top u_1 & \left[2^n\left(t-t_0\right)\right]-1,0,\ldots,0 & \mathbb{P}^{\left[2^n\left(t-t_0\right)\right]-1}\left(u_1\right)\\ \vdots & &\vdots\\ \underset{k=0}{\stackrel{\left[2^n\left(t-t_0\right)\right]-1}{\sum}} \mathbf{b}e^{\mathbf{A}\left(t-t_0-k2^{-n}\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(t-t_0-k2^{-n}\right)}\mathbf{b}^\top u_k & n_1,\ldots,n_m & \underset{i=1}{\stackrel{m}{\prod}}\mathbb{P}^{n_i}\left(u_i\right)\\ \vdots & & \vdots\\ \underset{k=0}{\stackrel{\left[2^n\left(t-t_0\right)\right]-1}{\sum}} \mathbf{b}e^{\mathbf{A}\left(t-t_0-k2^{-n}\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(t-t_0-k2^{-n}\right)}\mathbf{b}^\top u_m & 0,\ldots,0,\left[2^n\left(t-t_0\right)\right]-1 & \mathbb{P}^{\left[2^n\left(t-t_0\right)\right]-1}\left(u_m\right) \end{array} \right. , \label{eq:ApproxVarCar}$$ Thus $f_{Y_t\left|X_{t_0}\right.}\left(y\right)$ can be approximated with the finite mixture density function $\hat{f}_{Y_{t}\left|X_{t_0}\right.}\left(y\right)$ that reads: $$\hat{f}_{Y_{t}\left|X_{t_0}\right.}\left(y\right)=\sum_{i=1}^{m^{\left[2^n\left(t-t_0\right)\right]-1}} \phi\left(y,\mathbf{b}^\top e^{\mathbf{A}\left(t-t_0\right)}X_{t_0},V_{t_0}^{t}\left(n,m,i\right)\right)\mathbb{P}\left[V_{t_0}^{t}\left(n,m,i\right)\right], \label{eq:ApproxDensCarma}$$ where $V_{t_0}^t\left(n,m,i\right)$ denotes the $i-th$ realization of the random variable $V_{t_0}^{t}\left(n,m\right)$ in .For the approximated loglikelihood fuction $\hat{\mathcal{L}}\left(\theta\right)$ we need to infer the state process $X_t$. From the estimated process $\hat{X}_t$, we can determine the optimal value for the parameter vector $\theta$ solving the following optimization problem $$\theta=\text{argmax} \sum_{i=1}^{N}\ln\left[\hat{f}_{Y_{t_i}\left|\hat{X}_{t_{i-1}}\right.}\left(y_{t_i}\right)\right].$$ In this paper we consider two alternatives for the estimation of the state process $X_t$: the Kalman Filter and the filtering approach discussed in Section \[LCARMA\] and proposed in [@Brockwell2011]. In the following table we compare the GQMLE approach discussed in [@Iacus2015] for a General Lévy CARMA(p,q) model and our approaches. The labels `GL-HF` and `GL-HFKF` denote the Maximum Likelihood estimation method based on our approximated transition density, the only difference is related to the method for filtering the state process from the observable data. In `GL-HF` case, the estimated state process $\left\{\hat{X}_{t}\right\}_{t\geq0}$ is obtained using the dynamic in \[see [@Brockwell2011] for more information\] while in `GL-HFKF` case the standard Kalman Filter is used. ## a1 a2 b0 b1 Shape Scale ## 1.35000000 0.05000000 0.20000000 1.00000000 1.00000000 1.00000000 # True Parameters ## 1.38164866 0.04634073 0.18808589 0.99993332 1.15596369 1.00265283 # GL-HF ## 1.31162953 0.04494326 0.19126241 0.98709469 1.12985742 1.01003225 # GL-HFKF ## 1.35175679 0.04813515 0.18653096 1.03154645 ---------- ---------- # GQMLE Option Pricing in a Lévy CARMA(p,q) model. {#option-pricing-in-a-luxe9vy-carmapq-model.} ========================================== In this section we discuss, using the approximated transition density, how to evaluate the expected value of the transformation $g\left(X_T\right)$ where $X_T$ can be a Normal Variance Mean Mixture or a CARMA with a Time Changed Brownian Motion driving noise. In the Normal Variance Mean Mixture case we discuss also the behaviour of the error term while in the second case we analyze it by a comparison with the Monte Carlo simulation. The result here can be applied to extend the option pricing formula for options on futures contracts proposed in [@PASCHKE20102742] for the gaussian CARMA model. This approach can be used also for the evaluation of the term structure of futures. Normal Variance Mean Mixture {#OptNVMM} ---------------------------- Starting from the formal definition of Normal Variance Mean Mixture in , we define the sequence of function $\mathsf{E}\left[g\left(X_T^{m}\right)\left|\mathcal{F}_{0}\right.\right]$ as following: $$\sum_{i=1}^{m}\mathsf{E}\left[g\left(\mu+\theta \Lambda_m+\sqrt{\Lambda_m}Z\right)\left|\mathcal{F}_0,\Lambda_m=u_i\right.\right]\mathbb{P}\left(u_i\right) \label{eq:SectN1}$$ where $\Lambda_m$ and $\mathbb{P}\left(u_i\right)$ are defined in . The quantity $\left[g\left(\mu+\theta \Lambda_n+\sqrt{\Lambda_n}Z\right)\left|\mathcal{F}_0,\Lambda_n=u_i\right.\right]$ is the expectation of a gaussian distribution with mean $\mu+\theta \Lambda_n$ and variance $\Lambda_n$. The formulas proposed in this section can be applied for the evaluation of the contingent claim when the underlying is a transformation of a Time Change Brownian Motion. In the next section we show a comparison of our approach with a Monte Carlo simulation when the log price is a Variance Gamma process and the function $g$ is the final payoff of a European Call Option. ### Simulation Comparison Figure \[fig:FigOptionPriVar1\] shows the behaviour of a European Call option price for varying value of $n$ in the Gauss-Laguerre approximation approach. In this example the model parameters are $r=0$, $\theta=-0.5$, $\alpha=1$, $\beta=1$, underlying price $S_0=1$ and time to maturity $T=1$. ![Comparison of prices obtained using Monte Carlo simulation and the Laguerre Option pricing formula for an ATM European Call option. \[fig:FigOptionPriVar1\]](FigOptionPriVar1-1.pdf){width="50.00000%"} We analyze also the behaviour of the approximation for different strike levels in Figure \[fig:FigOptionPriVar2\] and for varying Time to maturity in Figure \[fig:FigOptionPriVar3\]. In the latter it is important to satisfy the condition $\alpha T\geq1$ otherwise we need to use the Generalized Gauss Laguerre approximation due to the presence of a no negligible singularity in the Mixing Gamma random Variable at point zero. ![Comparison of price obtained using Monte Carlo simulation and the Laguerre Option pricing formula for different levels of strike price.\[fig:FigOptionPriVar2\]](FigOptionPriVar2-1.pdf){width="50.00000%"} ![Comparison of prices obtained using Monte Carlo simulation and the Laguerre Option pricing formula for different levels of strike price.\[fig:FigOptionPriVar3\] ](FigOptionPriVar3-1.pdf){width="50.00000%"} Time Changed CARMA process -------------------------- We discuss here how to extend the general result in Section \[OptNVMM\] for the Time Changed Brownian Motion to the TCBm-CARMA process. The main idea is to use the approximation of $V^t_{t_0}$ introduced in Equation . The general pricing formula of the final payoff $g\left(Y_T\right)$ can be derived following the same steps as in the previous section. The resulting formula reads: $$\mathsf{E}\left[g\left(Y_{T}\right)\left|\mathcal{F}_{t_0}\right.\right]=\sum_{k=1}^{m^{\left[2^n\left(T-t_0\right)\right]-1}}\mathsf{E}\left[g\left(Y_{T}^{m,n}\right)\left|\mathcal{F}_{t_0},V_{t_0}^{T}=V_{t_0}^{T}\left(m,n,k\right)\right.\right]\mathbb{P}\left(V_{t_0}^{T}\left(m,n,k\right)\right), \label{mainResCarmaPricing}$$ where $Y_{T}^{m,n}\left|\mathcal{F}_{t_0},V_{t_0}^{T}=V_{t_0}^{T}\left(m,n,k\right)\right.\sim N\left(b^{\top}e^{\mathbf{A}\left(T-t_0\right)}X_{t_0},V_{t_0}^{T}\left(m,n\right)\right)$. This result can easily find applications in different financial modeling topics such as the construction of futures term structure, option pricing of bond pricing under the hypothesis that the dynamics of the underlying follows a Time Change CARMA model. ### Futures Term Structure with a TCBm CARMA(p,q) model In the filtered probability space we assume that it exists an equivalent martingale measure $\mathbb{Q}\sim\mathbb{P}$ exists. We also assume that the price $S_t$ of the commodity asset follows an exponential TCBm-CARMA(p,q) model under the measure $\mathbb{Q}$ defined as: $$S_t = S_{t_0} e^{Y_{t}},$$ where $Y_t$ is a CARMA(p,q) model described in Section \[LCARMA\]; the driving noise in a Time Change Brownian motion i.e. $$L_t=W_{\Lambda_t}$$ where $W_t$ is a Brownian Motion and $\Lambda_t$ is an independent subordinator process with cumulant generating function $k_\Lambda\left(u\right)$ defined as: $$k_\Lambda\left(u\right):=\ln\left[\mathsf{E}\left(e^{u \Lambda_1}\right)\right].$$ Arbitrage theory is based on the assumption that price of a future should be equal to the expected value of the price at maturity under the risk neutral measure $\mathbb{Q}$. Therefore, the log future price with maturity $T\geq t_0$ can be written as: $$\label{q-measure} \ln F^T_{t_0} = \ln \mathsf{E}^{\mathbb{Q}}\left[ S_{T}\left|\mathcal{F}_{t_0}\right.\right]$$ Defining the $\sigma$-field $\mathcal{G}^{t}_{t_0}=\sigma\left(\mathcal{F}_{t_0} \cup \sigma\left(\left\{\Lambda_u\right\}_{u\leq t}\right)\right)$ with $t\geq t_0$ we have: $$W_{\Lambda_t}-W_{\Lambda_{t_0}}\left|\mathcal{G}^t_{t_0}\right.\sim N\left(0,\Lambda_t-\Lambda_{t_0}\right).$$ Using the iterative property of the conditional expected value, equation can be rewritten as: $$\label{q-measure exp} \ln F^T_{t_0} = \ln \mathsf{E}^{\mathbb{Q}}\left[\mathsf{E}^{\mathbb{Q}}\left( S_{T}\right|\mathcal{G}^{T}_{t_0})|\mathcal{F}_{t_0}\right].$$ It is worth to notice that the random variable $\ln S_{T}\left|\mathcal{G}^{T}_{t_0}\right.$ is normally distributed. Therefore, we have that: $$\mathsf{E}^\mathbb{Q}\left(S_{T}\right|\mathcal{G}^{T}_{t_0}) = \exp\left(\ln S_{t_{0}} + \mathsf{E}^{\mathbb{Q}}\left[\ln S_{T}|\mathcal{G}^{T}_{t_0}\right] + \frac{1}{2} \mathsf{VAR}^\mathbb{Q}\left[\ln S_{T}|\mathcal{G}^{T}_{t_0}\right]\right).$$ Then: $$\label{second step} \ln F^T_{t_0} = \ln \mathsf{E}^{\mathbb{Q}}_{t}\left[ e^{\ln S_{t_{0}}} e^{\mathsf{E}^\mathbb{Q}\left[\ln S_{T}|\mathcal{G}^{T}_{t_0}\right] + \frac{1}{2} \mathsf{VAR}^\mathbb{Q}\left[\ln S_{T}|\mathcal{G}^{T}_{t_0}\right]}|\mathcal{F}_{t_0}\right],$$ and rearranging: $$\label{third step} \ln F^T_{t_0} = \ln S_{t_{0}} + \ln \left[\mathsf{E}^{\mathbb{Q}}\left(e^{\mathsf{E}^\mathbb{Q}\left[\ln S_{T}|\mathcal{G}^{T}_{t_0}\right] + \frac{1}{2} \mathsf{VAR}^\mathbb{Q}\left[\ln S_{T}|\mathcal{G}^{T}_{t_0}\right]}\right)|\mathcal{F}_{t_0}\right].$$ At this stage, it is possible to introduce the conditional transition density of a CARMA(p,q) model driven by a Time Changed Brownian Motion $Y_T$ given $\mathcal{G}^{T}_{t_0}$ as: $$Y_T\left| \mathcal{G}^T_{t_0}\right.\sim N\left(\mathbf{b}^\top e^{\mathbf{A}\left(T-t_0\right)}X_{t_0}, \int_{t_0}^{T}\mathbf{b}^{\top}e^{\mathbf{A}\left(T-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(T-u\right)}\mathbf{b} \mbox{d}\Lambda_u \right)$$ Given this result, we obtain: $$\label{fourth step} \ln F^{T}_{t_0}= \ln S_{t_{0}} + \ln \left[\mathsf{E}^{\mathbb{Q}}\left(e^{\mathbf{b}^\top e^{\mathbf{A}\left(T-t_0\right)}X_{t_{0}} + \frac{1}{2} \int_{t_0}^{T}\mathbf{b}^{\top}e^{\mathbf{A}\left(T-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(T-u\right)}\mathbf{b} \mbox{d}\Lambda_u}|\mathcal{F}_{t_0}\right)\right].$$ Simplifying: $$\label{fifth step} \ln F^T_{t_0} = \ln S_{t_{0}} + \mathbf{b}^\top e^{\mathbf{A}\left(T-t_0\right)}X_{t_0} + \ln \left[\mathsf{E}^{\mathbb{Q}}\left(e^{\frac{1}{2} \int_{t_0}^{T}\mathbf{b}^{\top}e^{\mathbf{A}\left(T-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(T-u\right)}\mathbf{b} \mbox{d}\Lambda_u}|\mathcal{F}_{t_0}\right)\right].$$ We use the following theorem proposed in [@Eberlein1999] in order to evaluate the expected value in . \[EberlRaibleProp\] Let $\Lambda_{t}$ be a subordinator process with cumulant generating function $k_\Lambda\left(u\right)$ and $f\left(u\right):\left[0,+\infty\right)\rightarrow\mathbb{C}$ be a complex left continuous function such that $\left|\mathsf{Re}\left(f\right)\right|\leq M$ then: $$\label{EbRaiEq} \mathsf{\mathsf{E}}\left[\exp\left(\int_{0}^{+\infty}f\left(u\right)\mbox{d}\Lambda_{u}\right)\right]=\exp\left(\int_{0}^{+\infty} k_{\Lambda}\left(f\left(u\right)\right)\mbox{d}u\right).$$ Using the above theorem and the following property of the cumulant function $$k_{\Lambda}\left(u \mathbbm{1}_{A}\right)=\mathbbm{1}_{A} k_{\Lambda}\left(u \right)$$ we obtain the final result $$\ln F^{T}_{t_0} = \ln S_{t_{0}} + \mathbf{b}^\top e^{\mathbf{A}\left(T-t_0\right)}X_{t_0} + \int_{t_0}^{T} k_{\Lambda} \left(\frac{1}{2}\mathbf{b}^\top e^{\mathbf{A}\left(T-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(T-u\right)}\mathbf{b} \right) \mbox{d}u . \label{sixthstep}$$ The approximated transition density of the TCBm-CARMA(p,q) model gives the possibility of evaluating the formulas in in a easy way. By applying the general result in we get the following approximation: $$\ln F^{T}_{t_0}\left(m,n\right)= \ln S_{t_{0}} + \mathbf{b}^\top e^{\mathbf{A}\left(T-t_0\right)}X_{t_0} +\ln \sum_{k=1}^{m^{\left[2^{n}\left(T-t_0\right)\right]-1}} e^{\frac12 V_{t_0}^{T}\left(m,n,k\right)}\mathbb{P}\left(V_{t_0}^{T}\left(m,n,k\right)\right)$$ A numerical comparison of the approximated formula with the pricing results obtained through Monte Carlo simulation is reported below. The MC value is evaluated using 10.000 simulated trajectories of a symmetric VG-CARMA(2,1) model with autoregressive parameters $a_1=1.4$ $a_2=0.5$, moving average parameters $b_0=0.2$ $b_1=1$ and Gamma subordinator process $\left(\Lambda_{t}\right)_{t\geq0}$ with shape parameter $\alpha=1$ and scale parameter $\beta=1$. The simulated trajectories are obtained using the Euler discretization scheme for a Lévy CARMA(p,q) model as described in [@Iacus2015] on a regular grid with $\Delta t=\frac{T}{200}$ where $T$ is the maturity of the Future.It is to worth to observe that since we have that $\alpha T<1$, we can use the Generalized Gauss Laguerre Quadrature to avoid numerical issues that may arise due to the singularity at point 0. See Table \[ComOptFuture\] for the futures term structure and Figures \[fut1m\]-\[fut4m\] for an analysis based on the number of points $m$ used in the approximation. T Lag. MC Ub Lb ---------------- --------- --------- --------- --------- $\frac{1}{12}$ 1.04697 1.04918 1.06285 1.03550 $\frac{2}{12}$ 1.08293 1.08248 1.09719 1.06778 $\frac{3}{12}$ 1.12130 1.12183 1.15005 1.09361 $\frac{4}{12}$ 1.14691 1.14367 1.16442 1.12292 : Pricing results for a future contract using MC and the approximated formula based on the Gauss-Laguerre quadrature.\[ComOptFuture\] ![Future price with maturity 1 month.\[fut1m\]](FuturesT1MCARMAVG21.pdf){width="50.00000%"} ![Future price with maturity 2 months.\[fut2m\]](FuturesT2MCARMAVG21.pdf){width="50.00000%"} ![Future price with maturity 3 months.\[fut3m\]](FuturesT3MCARMAVG21.pdf){width="50.00000%"} ![Future Price with Maturity 4 Months.\[fut4m\]](FuturesT4MCARMAVG21.pdf){width="50.00000%"} ### Futures Option Pricing formula in a TCBm CARMA(p,q) model Here we discuss how to modify our general result in order to extend the result about the Futures option prices in [@PASCHKE20102742] for a Gaussian CARMA(p,q) model to the TCBm-CARMA(p,q) model. Here we do not consider here the non-stationary factor $Z_t$ in equation (7) of [@PASCHKE20102742] but we assume that the log price is simply CARMA(p,q) model with gaussian innovations. We highlight the fact that extension to the ABM-CARMA(p,q) model proposed in [@PASCHKE20102742] is also straightforward in our context. In [@PASCHKE20102742] model the futures log Price has the following form: $$\ln F\left(t,T\right) = \mathbf{b}^{\top}\mathcal{A}\left(t,T\right)X_t+\frac12 \mathbf{b}^{\top}\mathcal{B}\left(t,T\right)\mathbf{b}$$ where $$\mathcal{A}\left(t,T\right)=e^{\mathbf{A}\left(T-t\right)}$$ $$\mathcal{B}\left(t,T\right)=\int_{t}^{T}e^{\mathbf{A}\left(T-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}\left(T-u\right)}\mbox{d}u.$$ If we want to evaluate a European Call Option on the Futures price, we have to consider three points in time: time $t$ the day where we evaluate the contract derivative, time $T_0>t$ the maturity of the option contract and time $T_F>T_0$ the maturity of the underlying future contract. The price of the call option at time $t$ can be obtained using no arbitrage arguments as follows: $$C_t=e^{-r\left(T_0-t\right)}\mathsf{E}^{\mathbb{Q}}\left[\left[F\left(T_{0},T_{F}\right)-K\right]_+\left|\mathcal{F}_t\right.\right].$$ If the state process $\left(X_{t}\right)_{t\geq0}$ is driven by a Brownian Motion, the price is analytic and reads as follows: $$C_t=e^{-r\left(T_0-t\right)}\left[F\left(t,T_F\right)\Phi\left(d_1\right)-K\Phi\left(d_2\right)\right]$$ where $$d_{1,2}=\frac{\ln\left(\frac{F\left(t,T_F\right)}{K}\right)\pm \frac12 \sigma^2\left(t,T_0,T_F\right)}{\sigma\left(t,T_0,T_F\right)}.$$ The forward integrated variance is defined as: $$\sigma^2\left(t,T_0,T_F\right)=\mathbf{b}^{\top}\left[\int_{t}^{T_0}e^{\mathbf{A}\left(T_F-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(T_F-u\right)}\mbox{d}u\right]\mathbf{b}.$$ To extend in our setup this result we use the sigma field $\mathcal{G}_{t}^{T_F}$ therefore if the case of a TCBm-CARMA(p,q) model we have: $$C_t=e^{-r\left(T_0-t\right)}\mathsf{E}\left[ \mathsf{E}\left[\left(F\left(T_0,T_F\right)-K\right)_+\left|\mathcal{G}_{t}^{T_F}\right.\right]\left|\mathcal{F}_t\right.\right]$$ The internal expectation under $\mathcal{G}_{t}^{T_F}$ is exactly the formula in [@PASCHKE20102742] for a fixed value of the integrated Variance: $$\mathsf{E}\left[\left(F\left(T_0,T_F\right)-K\right)_+\left|\mathcal{G}_{t}^{T_F}\right.\right]=\mathsf{E}\left[\left(F\left(T_0,T_F\right)-K\right)_+\left|\mathcal{F}_{t}, \sigma^2\left(t,T_0,T_F\right)\right.\right]$$ where $$\sigma^2\left(t,T_0,T_F\right)\left|\mathcal{G}_{t}^{T_F}\right.= \mathbf{b}^{\top}\left[\int_{t}^{T_0}e^{\mathbf{A}\left(T_F-u\right)}\mathbf{e}\mathbf{e}^{\top}e^{\mathbf{A}^\top\left(T_F-u\right)}\mbox{d}\Lambda_u\right]\mathbf{b}.$$ The conditional mean becomes: $$\mathsf{E}\left[\left(F\left(T_0,T_F\right)-K\right)_+\left|\mathcal{F}_{t}, \sigma^2\left(t,T_0,T_F\right)=\sigma^2\right.\right]=F\left(t,T_F\right)\Phi\left(d_{1,\sigma^2}\right)-K\Phi\left(d_{2,\sigma^2}\right).$$ The Gauss-Laguerre quadrature can be used to construct the random variable $\sigma^2_{m,n}\left(t,T_0,T_F\right)$ following the same approach in . The generic $k^{th}$ realization of the random variable $\sigma^2_{m,n}\left(t,T_0,T_F\right)$ has this form: $$\sigma^2_{m,n,k}\left(t,T_0,T_F\right) = \underset{k=0}{\stackrel{\left[2^n\left(T_0-t\right)\right]-1}{\sum}} \mathbf{b}^\top e^{\mathbf{A}\left(T_{F}-t-k2^{-n}\right)} \mathbf{e}\mathbf{e}^{\top} e^{\mathbf{A}^\top\left(T_{F}-t-k2^{-n}\right)}\mathbf{b} u_k \label{eq:FIV}$$ with probability $$\mathbb{P}\left(\sigma^2_{m,n,k}\left(t,T_0,T_F\right)\right)= \underset{i=1}{\stackrel{m}{\prod}}\mathbb{P}^{n_i}\left(u_i\right)$$where $n_i$ is the times that the realization $u_{i}$ appears in the trajectory of the approximated subordinators and we have this constraint: $$\sum_{i=1}^{m}n_i=\left[2^n\left(T_0-t\right)\right]-1.$$ Now the pricing formula has the same representation in where instead of the random variable $V_{t_0}^{T}\left(n,m\right)$ that can be seen as an approximation of the spot integrated variance we have the Gauss Laguerre approximation of the Forward Integrated Variance which realization are in . The same result can be applied in a straightforward manner to the case of the European Put price when the underlying is a Future contract. Indeed it is worth to notice the construction proposed in this paper implies a Law convergence consequently the convergence of the formulas in for the TCBm-CARMA(p,q) model and in for the Time Changed Brownian motion is ensured when the function $g$ is a bounded continuous function while for a lower-semi continuous function bounded from below only a lower bound can be established. Therefore the convergence behavior is clear in the case of the put option prices and to avoid issues due to this fact we perform the following steps. We first use the Gauss-Laguerre approximation scheme for the Put option price. Then we obtain the corresponding Call price using the put-call parity formula. We report in the following Tables and figures the comparison between the Gauss-Laguerre and MC prices for different call option prices. {width="50.00000%"} [@ ccccc]{}\ \ K & Gauss L & MC & UB & LB\ \ $0.50000$ & $0.59129$ & $0.59324$ & $0.60810$ & $0.57838$\ $0.55263$ & $0.53957$ & $0.54151$ & $0.55635$ & $0.52668$\ $0.60526$ & $0.48803$ & $0.48998$ & $0.50478$ & $0.47517$\ $0.65789$ & $0.43668$ & $0.43863$ & $0.45340$ & $0.42385$\ $0.71053$ & $0.38564$ & $0.38759$ & $0.40233$ & $0.37285$\ $0.76316$ & $0.33489$ & $0.33684$ & $0.35154$ & $0.32213$\ $0.81579$ & $0.28465$ & $0.28659$ & $0.30126$ & $0.27193$\ $0.86842$ & $0.23499$ & $0.23694$ & $0.25156$ & $0.22231$\ $0.92105$ & $0.18615$ & $0.18810$ & $0.20268$ & $0.17352$\ $0.97368$ & $0.13859$ & $0.14053$ & $0.15507$ & $0.12599$\ $1.02632$ & $0.09339$ & $0.09534$ & $0.10983$ & $0.08084$\ $1.07895$ & $0.06907$ & $0.07101$ & $0.08539$ & $0.05663$\ $1.13158$ & $0.06190$ & $0.06384$ & $0.07806$ & $0.04963$\ $1.18421$ & $0.05653$ & $0.05847$ & $0.07253$ & $0.04441$\ $1.23684$ & $0.05218$ & $0.05413$ & $0.06804$ & $0.04022$\ $1.28947$ & $0.04850$ & $0.05044$ & $0.06421$ & $0.03667$\ $1.34211$ & $0.04544$ & $0.04739$ & $0.06102$ & $0.03376$\ $1.39474$ & $0.04282$ & $0.04477$ & $0.05827$ & $0.03127$\ $1.44737$ & $0.04050$ & $0.04245$ & $0.05582$ & $0.02907$\ $1.50000$ & $0.03842$ & $0.04037$ & $0.05362$ & $0.02711$\ \ {width="50.00000%" height="\textheight"} [@ ccccc]{}\ \ K & Gauss L & MC & UB & LB\ \ $0.50000$ & $0.62469$ & $0.63412$ & $0.66088$ & $0.60736$\ $0.55263$ & $0.57347$ & $0.58291$ & $0.60964$ & $0.55617$\ $0.60526$ & $0.52263$ & $0.53207$ & $0.55877$ & $0.50536$\ $0.65789$ & $0.47229$ & $0.48173$ & $0.50840$ & $0.45506$\ $0.71053$ & $0.42261$ & $0.43204$ & $0.45868$ & $0.40541$\ $0.76316$ & $0.37354$ & $0.38298$ & $0.40957$ & $0.35639$\ $0.81579$ & $0.32525$ & $0.33468$ & $0.36123$ & $0.30814$\ $0.86842$ & $0.27804$ & $0.28747$ & $0.31397$ & $0.26098$\ $0.92105$ & $0.23225$ & $0.24168$ & $0.26813$ & $0.21524$\ $0.97368$ & $0.18861$ & $0.19805$ & $0.22444$ & $0.17166$\ $1.02632$ & $0.14833$ & $0.15776$ & $0.18409$ & $0.13144$\ $1.07895$ & $0.12303$ & $0.13247$ & $0.15868$ & $0.10626$\ $1.13158$ & $0.11087$ & $0.12030$ & $0.14637$ & $0.09424$\ $1.18421$ & $0.10179$ & $0.11123$ & $0.13714$ & $0.08531$\ $1.23684$ & $0.09453$ & $0.10396$ & $0.12973$ & $0.07820$\ $1.28947$ & $0.08843$ & $0.09786$ & $0.12349$ & $0.07223$\ $1.34211$ & $0.08315$ & $0.09259$ & $0.11808$ & $0.06709$\ $1.39474$ & $0.07855$ & $0.08799$ & $0.11335$ & $0.06263$\ $1.44737$ & $0.07451$ & $0.08394$ & $0.10918$ & $0.05871$\ $1.50000$ & $0.07091$ & $0.08035$ & $0.10546$ & $0.05523$\ \ {width="50.00000%" height="\textheight"} [@ ccccc]{}\ \ Strike & $Price^{Laguerre}$ & MC-mid & MC-lwb & MC-upb\ \ $0.50000$ & $0.61870$ & $0.61249$ & $0.62156$ & $0.60342$\ $0.55263$ & $0.56662$ & $0.56042$ & $0.56946$ & $0.55138$\ $0.60526$ & $0.51475$ & $0.50854$ & $0.51755$ & $0.49953$\ $0.65789$ & $0.46311$ & $0.45690$ & $0.46586$ & $0.44793$\ $0.71053$ & $0.41178$ & $0.40557$ & $0.41449$ & $0.39665$\ $0.76316$ & $0.36088$ & $0.35467$ & $0.36353$ & $0.34581$\ $0.81579$ & $0.31031$ & $0.30410$ & $0.31291$ & $0.29530$\ $0.86842$ & $0.26029$ & $0.25409$ & $0.26282$ & $0.24535$\ $0.92105$ & $0.21092$ & $0.20471$ & $0.21339$ & $0.19604$\ $0.97368$ & $0.16243$ & $0.15623$ & $0.16483$ & $0.14762$\ $1.02632$ & $0.11522$ & $0.10902$ & $0.11756$ & $0.10047$\ $1.07895$ & $0.07070$ & $0.06449$ & $0.07297$ & $0.05601$\ $1.13158$ & $0.05450$ & $0.04829$ & $0.05660$ & $0.03999$\ $1.18421$ & $0.04791$ & $0.04170$ & $0.04982$ & $0.03359$\ $1.23684$ & $0.04311$ & $0.03691$ & $0.04484$ & $0.02897$\ $1.28947$ & $0.03929$ & $0.03308$ & $0.04084$ & $0.02531$\ $1.34211$ & $0.03623$ & $0.03002$ & $0.03763$ & $0.02241$\ $1.39474$ & $0.03369$ & $0.02748$ & $0.03494$ & $0.02001$\ $1.44737$ & $0.03147$ & $0.02526$ & $0.03259$ & $0.01794$\ $1.50000$ & $0.02956$ & $0.02335$ & $0.03054$ & $0.01615$\ \ Conclusion ========== In this paper we propose an approximation procedure for the evaluation of the transition density of a TCBm-CARMA(p,q) process that resultsto be a finite mixture of normals. Exploiting this structure we obtain a simple estimation procedure and pricing formulas for financial contracts whose value depend only on the value of the underlying at maturity modelled as an exponential TCBm-CARMA(p,q). A possible extension of our proposed approximation methodology to the pricing of path dependent contracts may be based on the result in [@Hieber2012] for the evaluation of the first passage time for a Time Changed Brownian Motion. Indeed the process $V_{t_0}^{t}$ has the same structure of a subordinator while the TCBm-CARMA can be seen as a TCBm where the random time is the process $V_{t_0}^{t}$. This could also give us the possibility to extend our approach to the evaluation of the density function for the time-until death variable that is necessary for the evaluation of contracts with minimimum guaranteed death benefit. [100]{} M. Abramowitz and I. Stegun. . , [New York]{}, 1970. J.-P. Aguilar. Some pricing tools for the variance gamma model. , 12 2020. 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As a first step we determine the complete-data log-likelihood function defined as: $$\begin{aligned} \mathcal{L}^{\star}\left(\mu_0,\mu, \sigma, \varphi_+, \lambda,\theta\right)&=&\sum_{t=1}^T\ln\left[\phi\left(y_t;\mu_0+\mu U_t; \sigma^2 U_t\right)\mathbb{P}\left(U_t,\varphi_+,\lambda,\theta\right)\right]\nonumber\\ &=&\sum_{t=1}^T\sum_{i=1}^{m}D_{t,i}\ln\left[\phi\left(y_t;\mu_0+\mu u_i; \sigma^2 u_i\right)\mathbb{P}\left(u_i,\varphi_+,\lambda,\theta\right)\right]\end{aligned}$$ where $D_{t,i}$ assumes value 1 when $U_t=u_i$ and $0$ otherwise. Following the seminal work of [@Dempster77maximumlikelihood], we perform the Expectation-step (E-step henceforth) evaluating the conditional distribution of the variables $\left\{U_t\right\}_{t=1,\ldots,T}$ given the observed data. Applying the Bayes’ theorem we have: $$\mathbb{P}\left(U_t=u_i\left|y_t,\Theta_{h-1}\right.\right)=\frac{\phi\left(y_t;\mu_{0,h-1}+\mu_{h-1} u_i; \sigma^2_{h-1} u_i\right)\mathbb{P}\left(u_i,\varphi_{+,h-1},\lambda_{h-1},\theta_{h-1}\right)}{\underset{i=1}{\stackrel{m}{\sum}}\phi\left(y_t;\mu_{0,h-1}+\mu_{h-1} u_i; \sigma^2_{h-1} u_i\right)\mathbb{P}\left(u_i,\varphi_{+,h-1},\lambda_{h-1},\theta_{h-1}\right)}$$ where $\Theta_{h-1} =\left(\mu_{0,h-1},\mu_{h-1}, \sigma^2_{h-1},\varphi_{+,h-1}, \lambda_{h-1},\theta_{h-1}\right)$. The E-step consists of computing the conditional expectation of $\mathcal{L}^{\star}\left(\mu_0,\mu,\varphi_+, \lambda,\theta\right)$ in the following way: $$\mathbb{E}\left[\mathcal{L}^{\star}\left(\mu_{0,h},\mu_{h}, \sigma_{h},\varphi_{+,h}, \lambda_{h},\theta_{h}\right)\right]=\sum_{i=1}^{m}\sum_{t=1}^T\ln\left[\phi\left(y_t;\mu_{0,h}+\mu_{h} u_i; \sigma^2_{h} u_i\right)\mathbb{P}\left(u_i,\varphi_{+,h},\lambda_{h},\theta_{h}\right)\right]\mathbb{P}\left(U_t=u_i\left|y_t,\Theta_{h-1}\right.\right).$$ Recalling that $u_i=\frac{k_i}{\varphi_+}$ we get: $$\begin{aligned} \mathbb{E}\left[\mathcal{L}^{\star}\left(\mu_{0,h},\mu_{h},\sigma_{h},\varphi_{+,h}, \lambda,\theta\right)\right]&=&\sum_{i=1}^{m}\sum_{t=1}^T\ln\left[\phi\left(y_t;\mu_{0,h}+\mu_{h} \frac{k_i}{\varphi_{+,h}}; \sigma^2_h\frac{k_i}{\varphi_{+,h}}\right)\mathbb{P}\left(\frac{k_i}{\varphi_{+,h}},\varphi_{+,h},\lambda_{h},\theta_{h}\right)\right]\mathbb{P}\left(U_t=\frac{k_i}{\varphi_{+,h-1}}\left|y_t,\Theta_{h-1}\right.\right)\nonumber\\ &=&\sum_{i=1}^{m}\sum_{t=1}^T\ln\left[\phi\left(y_t;\mu_{0,h}+\mu_{h} \frac{k_i}{\varphi_{+,h}}; \sigma^2_h\frac{k_i}{\varphi_{+,h}}\right)\right]\mathbb{P}\left(U_t=\frac{k_i}{\varphi_{+,h-1}}\left|y_t,\Theta_{h-1}\right.\right)\nonumber\\ &+&\sum_{i=1}^{m}\sum_{t=1}^T\ln\left[\mathbb{P}\left(\frac{k_i}{\varphi_{+,h}},\varphi_{+,h},\lambda_{h},\theta_{h}\right)\right]\mathbb{P}\left(U_t=\frac{k_i}{\varphi_{+,h-1}}\left|y_t,\Theta_{h-1}\right.\right) \label{quant2}\end{aligned}$$ The Maximization-step (M-step henceforth) is based on the maximization of the quantity in , i.e.: $$\left(\mu_{0,h},\mu_{h},\sigma_{h},\varphi_{+,h}, \lambda_{h},\theta_{h}\right)= \underset{ \begin{array}{c} \scriptsize{\mu_{0,h},\mu_{h},\sigma_{h}} \\ \scriptsize{\varphi_{+,h}, \lambda_{h},\theta_{h}} \end{array} }{\text{argmax }} \mathbb{E}\left[\mathcal{L}^{\star}\left(\mu_{0,h},\mu_{h},\sigma_{h},\varphi_{+,h}, \lambda_{h},\theta_{h}\right)\right] \label{Prob1}$$ Using the following parametrization: $$\left\{ \begin{array}{l} \mu=\tilde{\mu}\varphi_+\\ \sigma=\tilde{\sigma}\sqrt{\varphi_+} \end{array} \right. .$$ The problem in becomes: $$\underset{ \begin{array}{c} \scriptsize{\mu_{0,h},\mu_{h},\sigma_{h}} \\ \scriptsize{\varphi_{+,h}, \lambda_{h},\theta_{h}} \end{array} }{\text{argmax }} \sum_{i=1}^{m}\sum_{t=1}^T\ln\left[\phi\left(y_t;\mu_{0,h}+\tilde{\mu}_{h} k_i; \tilde{\sigma}^2_h k_i\right)\right]\mathbb{P}\left(U_t=\frac{k_i}{\varphi_{+,h-1}}\left|y_t,\Theta_{h-1}\right.\right) + \sum_{i=1}^{m}\sum_{t=1}^T\ln\left[\mathbb{P}\left(\frac{k_i}{\varphi_{+,h}},\varphi_{+,h},\lambda_{h},\theta_{h}\right)\right]\mathbb{P}\left(U_t=\frac{k_i}{\varphi_{+,h-1}}\left|y_t,\Theta_{h-1}\right.\right)$$ that can be split as follows: $$\underset{ \scriptsize{\mu_{0,h},\mu_{h},\sigma_{h}} }{\text{argmax }} \mathbb{H}_1\left(\mu_{0,h},\mu_{h},\sigma_{h}\right):=\sum_{i=1}^{m}\sum_{t=1}^T\ln\left[\phi\left(y_t;\mu_{0,h}+\tilde{\mu}_{h} k_i; \tilde{\sigma}^2_h k_i\right)\right]\mathbb{P}\left(U_t=\frac{k_i}{\varphi_{+,h-1}}\left|y_t,\Theta_{h-1}\right.\right) \label{Prob:1a}$$ $$\underset{ \scriptsize{\varphi_{+,h}, \lambda_{h},\theta_{h}} }{\text{argmax }} \mathbb{H}_2\left(\varphi_{+,h}, \lambda_{h},\theta_{h}\right):=\sum_{i=1}^{m}\sum_{t=1}^T\ln\left[\mathbb{P}\left(\frac{k_i}{\varphi_{+,h}},\varphi_{+,h},\lambda_{h},\theta_{h}\right)\right]\mathbb{P}\left(U_t=\frac{k_i}{\varphi_{+,h-1}}\left|y_t,\Theta_{h-1}\right.\right) \label{Prob:2a}$$ Gauss Laguerre Quadrature ------------------------- In this section we review some results about the Gauss-Laguerre quadrature necessary to understand the behavior of our approximation scheme. We refer to [@Rabinowitz1967; @Uspensky1928; @abramowitz70a] for a complete discussion about this quadrature. Let $f\left(x\right)$ be a continuous function on the support $\left[0,+\infty\right)$ and let the integral $\int_{0}^{+\infty}f\left(x\right)e^{-x}\mbox{d}x<+\infty$ be finite with $f$ be $2m$ differentiable. Then we have: $$\int_{0}^{+\infty}e^{-x}f\left(x\right)\mbox{d}x=\sum_{i=1}^m\omega\left(u_i\right)f\left(u_i\right)+ \mathcal{R}_{m}$$ where $$\mathcal{R}_{m}=\frac{\left(m!\right)^2}{\left(2m\right)!}f^{\left(2m\right)}\left(\epsilon\right), \ \ \epsilon \in (0,+\infty).$$ Generalized Gauss Laguerre Quadrature ------------------------------------- The Generalized Gauss-Laguerre quadrature can be applied in the presence of non negligible singularity at $x=0$. Following [@Rabinowitz1967], let $f\left(x\right)$ be a non-negative continuous function such that $\omega\left(x\right)f\left(x\right)$ is a monotonically non negative not increasing in $\left(0,+\infty\right)$ where $\omega\left(x\right)= x^{\alpha}e^{-x}, \ \alpha>-1$, $$f\left(x\right)\leq \frac{e^{x}}{x^{\alpha+1+\rho}}$$ for some $\rho>0$ then, if the function $f\left(x\right)$ is $2n$ differentiable, the Generalized Gauss-Laguerre quadrature has the following form: $$\int_{0}^{+\infty} \omega\left(x\right)f\left(x\right)\mbox{d}x=\sum_{i=1}^{m}\omega\left(u_i\right)f\left(u_i\right)+\mathcal{R}_{m},$$ with $\omega\left(u_i\right)=\frac{\Gamma\left(m+\alpha\right)u_i}{m!\left(m+\alpha\right)\left[L^{\alpha}_{m-1}\left(u_i\right)\right]^2}$ and $L_m^{\alpha}\left(x\right)$ is the generalized Laguerre polynomial. The residual term $\mathcal{R}_{m}$ can be written as: $$\mathcal{R}_m=\frac{m!\Gamma\left(m+\alpha+1\right)}{\left(2m\right)!}f^{\left(2m\right)}\left(\epsilon\right),\ \epsilon\in\left(0,+\infty\right).$$ A standard example where it is necessary to use the Generalized Gauss-Laguerre quadrature is the numerical evaluation of the moment generating function of a Gamma random variable with shape parameter $\alpha \in \left(0,1\right)$. The usage of the Generalized Gauss Laguerre is due to the fact that, in this case, we have a singularity at $x=0$; the requirements described in this section can be easily checked and the error term can be evaluated due to smooth condition of the exponential function. For the case of $\alpha \geq 1$ the standard Gauss Laguerre quadrature described in the previous section can be easily applied. Error computation in the option pricing formula in the case of NVMM {#erroroptNVMM} ------------------------------------------------------------------- It is worth to notice that the formula in can be written as: $$\sum_{i=1}^{m}\mathsf{E}\left[g\left(\mu+\theta \Lambda_m+\sqrt{\Lambda_m}Z\right)\left|\mathcal{F}_0,\Lambda_m=u_i\right.\right]\mathbb{P}\left(u_i\right)=\frac{A_m}{B_m}$$ where: $$A_m=\sum_{i=1}^{m}\mathsf{E}\left[g\left(\mu+\theta \Lambda_m+\sqrt{\Lambda_m}Z\right)\left|\mathcal{F}_0,\Lambda_m=u_i\right.\right]\frac{\omega\left(k_i\right)}{k_i}\left(\frac{k_i}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_i}{\varphi_+}\right), \ k_i=u_i\varphi_+$$ and $$B_m=\sum_{j=1}^m\frac{\omega\left(k_j\right)}{k_{j}}\left(\frac{k_j}{\varphi_+}\right)^{\lambda}L_{\theta}\left(\frac{k_j}{\varphi_+}\right).$$ We analyze the term $A_m$ as $m\rightarrow+\infty$, by Gauss - Laguerre Quadrature we have: $$\lim_{m\rightarrow +\infty}A_m=\int_{0}^{+\infty}\mathsf{E}\left[g\left(\mu \Lambda +\theta \Lambda_T+\sqrt{\Lambda_T}Z \right)\left|\mathcal{F}_0, \Lambda_T=k\right.\right]\left(\frac{k}{\varphi_+}\right)^{\lambda}\frac{L_{\theta}\left(k/\varphi_+\right)}{k}\mbox{d}k, \label{limit:A_n}$$ where the integral in the right hand is exactly the expectation of the function $g\left(Y_{T}\right)$ where $Y_T$ is a normal variance mean mixture (it is enough to solve the integral using the substitution $\frac{k}{\varphi_+}=u$). Denoting with $A$ the integral in , we have the following result due to the standard Gauss-Laguerre quadrature: $$A= A_m+\mathcal{R}_{m}\left(A_m\right)$$ where the remaining term has the following form: $$\mathcal{R}_{m}\left(A_m\right)= \frac{\left(m!\right)^2}{\left(2m\right)!} \partial^{2m}\left[\mathsf{E}\left[g\left(\mu T +\theta \Lambda_T+\sqrt{\Lambda_T}Z \right)\left|\mathcal{F}_0, \Lambda_T=\epsilon\right.\right]\left(\frac{\epsilon}{\varphi_+}\right)^{\lambda}\frac{L_{\theta}\left(\epsilon/\varphi_+\right)}{\epsilon}\right], \ \epsilon \in\left(0,+\infty\right).$$ A discussion about the behaviour of the remaining term $\mathcal{R}_{m}\left(A_m\right)$ can be found in [@lubinsky1983geometric]. The author proved, under mild conditions, the geometric convergence for a Gauss-Laguerre quadrature for a function that can be written as a power series \[see [@Mastroianni1994; @XIANG2012434] for a complete discussion and generalizations\]. We analyze the behaviour of term $B_m$ that: $$\lim_{m\rightarrow+\infty}B_m=\int_0^{+\infty}\left(\frac{k}{\varphi_+}\right)^{\lambda}\frac{L_{\theta}\left(k/\varphi_+\right)}{k}\mbox{d}k. \label{limit:B_n}$$ Using the substitution $u=\frac{k}{\varphi_+}$, the integral is equal to one because the integrand function is the density in . Denoting with $B$ the integral in we have $$B = B_m + \mathcal{R}_{m}\left(B_m\right).$$ The remaining term $\mathcal{R}_{m}\left(B_m\right)$ has the following form: $$\mathcal{R}_{m}\left(B_m\right)=\frac{\left(m!\right)^2}{\left(2m\right)!}\partial^{2m}\left[\left(\frac{\epsilon}{\varphi_+}\right)^{\lambda}\frac{L_{\theta}\left(\epsilon/\varphi_+\right)}{\epsilon}\right], \ \epsilon \in \left(0,+\infty\right)$$ We are now able to establish the error term behaviour of our approximation approach for the normal variance mean mixture. The result presented here holds when we have a no negligible singularity at $x=0$ but the result for this type approximation can easily to generalize to case of the singularity at $x=0$ using the Generalized Gauss-Laguerre quadrature. We define the error term $\mathcal{R}_m$ as: $$\begin{aligned} \mathcal{R}_m &:=& \mathsf{E}\left[g\left(Y_T\right)\left|\mathcal{F}_0\right.\right]-\mathsf{E}\left[g\left(Y^{m}_T\right)\left|\mathcal{F}_0\right.\right] \nonumber \\ &=& \frac{\mathcal{A}_m+\mathcal{R}_{m}\left(\mathcal{A}_m\right)}{\mathcal{B}_m+\mathcal{R}_{m}\left(\mathcal{B}_m\right)}-\frac{\mathcal{A}_m}{\mathcal{B}_m} \nonumber \\ &=&\frac{\mathcal{A}_m+\mathcal{R}_{m}\left(\mathcal{A}_m\right)}{\mathcal{B}_m+\mathcal{R}_{m}\left(\mathcal{B}_m\right)}-\frac{\mathcal{A}_m}{\mathcal{B}_m+\mathcal{R}_{m}\left(\mathcal{B}_m\right)}+\frac{\mathcal{A}_m}{\mathcal{B}_m+\mathcal{R}_{m}\left(\mathcal{B}_m\right)}-\frac{\mathcal{A}_m}{\mathcal{B}_m} \nonumber \\\end{aligned}$$ Noting that $\mathcal{R}_{m}\left(\mathcal{B}_m\right)+\mathcal{B}_m=1$, we have $$\mathcal{R}_m = \mathcal{R}_{m}\left(\mathcal{A}_m\right)-\frac{\mathcal{A}_m}{\mathcal{B}_m}\mathcal{R}_{m}\left(\mathcal{B}_m\right)$$ Therefore $$\left|\mathcal{R}_m\right|\leq \left|\mathcal{R}_{m}\left(\mathcal{A}_m\right)\right|+\left|\frac{\mathcal{A}_m}{\mathcal{B}_m}\right|\left|\mathcal{R}_{m}\left(\mathcal{B}_m\right)\right|.$$ [^1]: Using the result in and the interated expected value, we obtain the moment generating function of a TCBm-OU process. : $$\begin{aligned} \mathbb{E}_{\mathcal{F}_{t_0}}\left[\mathbb{E}\left[e^{cY_t}\left|\mathcal{G}_{t_0,t}\right.\right]\right]&=&e^{cy_0e^{-a\left(t-t_0\right)}}\mathbb{E}_{\mathcal{F}_{t_0}}\left[e^{\frac{c^2}{2}\int_{t_0}^te^{-2a\left(t-u\right)}\mbox{d}\Lambda_u}\right]\nonumber\\ &=& e^{cy_0e^{-a\left(t-t_0\right)}+\int_{t_0}^t \kappa_{\Lambda}\left(\frac{c^2}{2}e^{-2a\left(t-u\right)}\right)\mbox{d}u}.\end{aligned}$$ where $\kappa_{\Lambda}\left(u\right)=\ln\left[\mathbb{E}\left(e^{u{\Lambda}_1}\right)\right]$. The quantity $e^{cy_0e^{-a\left(t-t_0\right)}+\int_{t_0}^t \kappa_{\Lambda}\left(\frac{c^2}{2}e^{-2a\left(t-u\right)}\right)\mbox{d}u}$ is the moment generating function of an TCBm-OU process and it can be alternatively obtained applying the result in [@Eberlein1999].

--- abstract: 'This paper presents an Iterated Tabu Search algorithm (denoted by ITS-PUCC) for solving the problem of Packing Unequal Circles in a Circle. The algorithm exploits the continuous and combinatorial nature of the unequal circles packing problem. It uses a continuous local optimization method to generate locally optimal packings. Meanwhile, it builds a neighborhood structure on the set of local minimum via two appropriate perturbation moves and integrates two combinatorial optimization methods, Tabu Search and Iterated Local Search, to systematically search for good local minima. Computational experiments on two sets of widely-used test instances prove its effectiveness and efficiency. For the first set of 46 instances coming from the famous circle packing contest and the second set of 24 instances widely used in the literature, the algorithm is able to discover respectively 14 and 16 better solutions than the previous best-known records.' address: - 'School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan, 430074, China' - 'Department of Mathematics, Simon Fraser University Surrey, Central City, 250-13450 102nd AV, Surrey, British Columbia, V3T 0A3, Canada ' author: - Tao Ye - Wenqi Huang - Zhipeng Lü title: Iterated Tabu Search Algorithm for Packing Unequal Circles in a Circle --- Packing ,Circle packing ,Global optimization ,Tabu search ,Iterated local search Introduction ============ Given $n$ circles and a container of predetermined shape, the circle packing problem is concerned with a dense packing solution, which can pack all the circles into the smallest container without overlap. The circle packing problem is a well-known challenge in discrete and computational geometry, and it arises in various real-world applications in the field of packing, cutting, container loading, communication networks and facility layout [@castillo2008]. In the field of global optimization, the circle packing problem is a natural and challenging test bed for evaluating various global optimization methods. This paper focuses on solving a classic circle packing problem, the Packing Unequal Circles in a Circle (PUCC) problem. As indicated in previous papers [@addis2008u; @grosso2010; @hifi2009], the PUCC problem has an interesting and important characteristic that it has a both continuous and combinatorial nature. It has continuous nature because the position of each circle is chosen in $R^2$. The combinatorial nature is due to the following two facts: (1) A packing pattern is composed of $n$ circles, and shifting a circle to a different place would produce a new packing pattern; (2) The circles have different radiuses, and swapping the positions of two different circles may result in a new packing pattern. In this paper, we pay special attention to the continuous and combinatorial characteristic of the PUCC problem. We propose an algorithm which integrates two kinds of optimization techniques: A continuous local optimization procedure which minimizes overlaps between circles and produces locally optimal packing patterns, and an Iterated Tabu Search (ITS) procedure which exploits the combinatorial nature of the problem and intelligently uses two appropriate perturbation moves to search for globally optimal packing patterns. The proposed algorithm is assessed on two sets of widely used test instances, showing its effectiveness and efficiency. For the first set of 46 instances coming from the famous circle packing contest, the algorithm is able to discover 14 better solutions than the previous best-known records. For the second set of 24 instances widely used in the literature, the algorithm can improve 16 best-known solutions in a reasonable time. The rest of this paper is organized as follows. Section 2 briefly reviews the most related literature. Section 3 formulates the PUCC problem. Section 4 describes the details of the proposed algorithm. Section 5 assesses the performance of the algorithm through extensive computational experiments. Section 6 analyzes some key ingredients of the algorithm to understand the source of its performance. Finally, Section 7 concludes this paper and proposes some suggestions for future work. Related Literature ================== Over the last few decades, the circle packing problem has received considerable attention in the literature. The simplest and most widely studied cases are the packing of equal circles in a square or in a circle. Though researchers have spent significant effort on the two problems, only a few packings (up to tens of circles) have been proved to be optimal by purely analytical methods and computer-aided proving methods [@szabo2007; @graham1998]. A second category of research aims at finding the best possible packings without optimality proofs. Following this spirit, various heuristic approaches have been proposed, including: Billiard simulation [@graham1998], minimization of energy function [@nurmela1997], nonlinear programming approaches [@birgin2005; @birgin2010], Population Basin Hopping method [@addis2008e; @grosso2010], formulation space search heuristic algorithm [@beasley2011], quasi-physical global optimization method [@huang2011], greedy vacancy search method [@huang2010] and so on. With these approaches, best-known packings for up to thousands of circles have been found, which are reported and continuously updated on the Packomania website [@specht2012]. There are also a number of papers devoted to the unequal circle packing problem. Most previous papers on the unequal circle packing problem can be classified into two categories: Constructive approaches and global optimization approaches. The constructive approaches build a packing by successively placing a circle into the container. These approaches usually include two important components: A placement heuristic, which determines several candidate positions for a new circle in the container, and a tree search strategy, which controls the tree search process and avoids exhaustive enumeration of the solution space. The widely used placement heuristics include the principle of Best Local Position (BLP) [@hifi2004; @hifi2007; @hifi2008; @akeb2010a] and the Maximum Hole Degree (MHD) rule [@huang2005; @huang2006; @lv2008; @akeb2009]. The tree search strategies include the self look-ahead search strategy [@huang2005; @huang2006], Pruned-Enriched-Rosenbluth Method (PERM) [@lv2008], beam search algorithm [@akeb2009] and the hybrid beam search looking-ahead algorithm [@akeb2010a]. The global optimization approaches formulate the unequal circle packing problem as a mathematical programming problem, then the task becomes to find the global minimum of a mathematical model. These kind of approaches include the quasi-physical quasi-human algorithm by @wang2002, the Tabu Search and Simulated Annealing hybrid approach [@zhang2005], the Population Basin Hopping algorithm [@addis2008u; @grosso2007; @grosso2010], the GP-TS algorithm by @huang2012a, the Iterated Local Search algorithm by @huang2012b, the Formulation Search Space algorithm by @beasley2012 and the Iterated Tabu Search algorithm by [@fu2013] for the circular open dimension problem. For the circle packing problem, there also exist many important literature not mentioned here. Interested readers are referred to the review articles by @castillo2008 and @hifi2009, the book by @szabo2007 and the Packomania website [@specht2012]. Problem Formulation =================== Given $n$ disks, each having radius $r_i$ $(i=1, 2, \cdots, n)$, the PUCC problem consists in finding a dense packing solution, which can pack all $n$ disks into the smallest circular container of radius $R$ without overlap. We designate the container center as the origin of the cartesian coordinate system and locate disk $i$ $(i=1,2,\dots,n)$ by the coordinate position of its center $(x_i, y_i)$. The PUCC problem can be formulated as: $$\begin{aligned} minimize \quad R , \quad s.t. : \nonumber \\ \sqrt{x_i^2 + y_i^2} + r_i \leq R \\ \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2} \geq r_i + r_j\end{aligned}$$ where $ i,j = 1,2,\cdots,n; i \neq j$. Eq.(1) ensures that each disk is completely in the container and Eq.(2) guarantees that no overlap exists between any two disks. Note that, this problem can also be formulated in other ways, see for example @birgin2005 and @grosso2010. A packing solution is described by two variables: The radius of the container $R$ and the **packing pattern** denoted by the positions of all $n$ disks $ X= (x_1, y_1,x_2, y_2, \cdots, x_n, y_n) $. The infeasibility of a packing can be caused by two kinds of overlaps: Overlaps between two disks and overlaps between a disk and the exterior of the container. We define the overlapping depth between disks $i$ and $j$ $(i,j=1,2,\cdots,n;i\neq j )$ as: $$\label{eq:oij} o_{ij}= max \ \{ 0 , r_i + r_j - \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2} \} .$$ and the overlapping depth between disk $i$ $( i=1,2,\cdots,n )$ and the exterior of the container as: $$\label{eq:oxi} o_{0i}= max \ \{ 0, \sqrt{x_i^2 + y_i^2}+ r_i - R \} .$$ Adding all squares of overlapping depth together, we get a penalty function measuring overlaps of a packing $$\label{eq:exs} E(X, R) = \sum_{i=0}^{n-1}{\sum_{j=i+1}^{n} o_{ij}^2} .$$ Thus, a packing $(X, R)$ is feasible (non-overlapping) if and only if $E(X, R)=0$. Sometimes, we fix the radius of the container at a constant value $R$ and the penalty function becomes $$\label{eq:exs0} E_{R}(X) = E(X, R).$$ Note that, finding a packing pattern $X$ with $E_R(X)=0$ corresponds to solving the following circle-packing decision problem [@birgin2005]: Given a circular container with fixed radius $R$, find out a feasible pattern $X$ which can pack all the circles into the container without overlap. Our original PUCC problem aims to find the smallest container of radius $R^*$ and a corresponding non-overlapping packing pattern $X$. In practice, the PUCC problem can be solved as a serial of circle-packing decision problems with descending $R$ [@huang2010]. The main steps are as follows: (1) Let $\overline{R}$ be an upper bound of $R^*$. Initialize $\overline{R}$ with a relatively large number such that all circles can be easily packed into the container of radius $\overline{R}$ without overlap. (2) Set $R \leftarrow \overline{R}$ and launch an algorithm to find a feasible $X$ with $E_R(X) = 0$ (i.e., to solve the corresponding circle-packing decision problem). (3) Tighten the packing $(X, R)$, i.e, to minimize $R$ while keeping $X$ basically unchanged. This step can be achieved using various approaches, like the simple bisection method described in [@huang2011], the simple penalty method described in [@huang2010], the standard local optimization solver SNOPT adopted by [@addis2008u] and the more sophisticated augmented Lagrangian method [@andreani2007; @birgin2008; @birgin2009]. After this step, we can usually obtain a better (at least not worse) packing $(X', R')$. (4) Set $\overline{R} \leftarrow R'$ and go to step 2. The loop of steps 2-4 is ended until a certain termination criterion (like time limit) is satisfied. In the rest of this paper, we will first introduce an Iterated Tabu Search algorithm to solve the circle-packing decision problem, and then use it to search for dense packing solutions for the PUCC problem in the computational experiments section. Iterated Tabu Search Algorithm {#sec:ITS} ============================== This section describes the Iterated Tabu Search (ITS) procedure for solving the circle-packing decision problem. As indicated in Section 3, this problem can be transformed to an unconstraint global optimization problem: $$minimize \ \ \ E_R(X).$$ This subproblem is very difficult because there exist enormous local minima in the solution space. @grosso2010 have shown that, even for the equal circle packing problem, the number of local minima tends to increase very quickly with the number of circles. For the more complex unequal circle packing problem, it is very possible that the number of local minima will be significantly larger. The main rationale behind the ITS procedure is as follows: (1) Each local minimum of $E_R(X)$ corresponds to a packing pattern of $n$ disks in the container. (2) If we perturb the current local minimizer $X$ by swapping the positions of two different disks (or shifting the position of one disk) and then call the LBFGS procedure to minimize $E_R(X)$, we can obtain a new local minimizer. (3) By systematically using the two perturbation moves, swap and shift, we can obtain a set of neighboring local minima from the current local minima. Furthermore, we can build a neighborhood structure on the set of local minima of $E_R(X)$. (4) Since there is a neighborhood structure, some Stochastic Local Search methods [@slsbook], such as Tabu Search [@tabusearch] and Iterated Local Search [@ils], can be employed to search for good local minima. The outline of the ITS procedure is given in Algorithm \[alg:its\]. The procedure performs searches on the set of local minima of $E_R(X)$ and follows an Iterated Local Search schema. In Algorithm \[alg:its\], we run the ITS procedure in a multi-start fashion. At each run, the algorithm starts from a randomly generated local minimum (steps 2-3). It goes through the $SwapTabuSearch$ procedure (step 4) and reaches a swap-optimal local minimum (which will be defined in the next section). Then the search explores the solution space by repeatedly escaping from local optima traps (step 6) and moving to another local optimum (step 7). This process is repeated until the best-found solution has not been improved during the last $PerturbDepth$ iterations. \[alg:its\] The SwapTabuSearch Procedure {#sec:tabusearch} ---------------------------- In the $SwapTabuSearch$ procedure, we build a neighborhood structure on the set of local minima of $E_R(X)$. A **swap move** performed on a packing pattern $X$ is defined as swapping the positions of two disks with different radiuses and then locally minimizing $E_R(X)$. For two local minima of $E_R(X)$, $X$ and $X'$, we say $X'$ is a **neighbor** of $X$ if and only if $X'$ can be reached by performing a swap move on $X$. The **neighborhood** of $X$ is a set containing all the neighbors of $X$. We use $E_R(X)$ as the evaluation function, and a local minimum $X$ of $E_R(X)$ is called a **swap-optimal** local minimum if it has better solution quality than all its neighbors (or it cannot be improved via any swap move). Totally, there are $n*(n-1)$ possible swap moves for a packing pattern with $n$ disks. However, for efficiency purposes, a restricted neighborhood is used in this paper. We first sort the disks in a nondecreasing order w.r.t. their radius values, such that for disks $i$ and $j $, $r_i \leq r_j$ if $i < j$. A swap move can only be performed on a pair of disks with neighboring radius values. That is to say, disk $i$ can only exchange positions with disks $i-1$ and $i+1$. Then, there are in total $n-1$ swap moves and a local minimum $X$ has at most $n-1$ different neighbors. The $SwapTabuSearch$ procedure follows a Tabu Search strategy. At the beginning of the search, the tabu list is empty and all swap moves are admissible. At each step, the algorithm chooses a best admissible move which leads to the best nontabu solution. The aspiration criterion is used such that a tabu move can be selected if it generates a solution that is better than the best-found solution. Once a move is selected, it is declared tabu for the next $TabuTenure$ steps. The procedure is repeated until the best-found solution has not been improved with the last $TabuDepth$ steps. The sketch of the $SwapTabuSearch$ procedure is presented in Algorithm \[alg:SwapTabuSearch\]. \[alg:SwapTabuSearch\] the best-found solution $X^*$ The ShiftPerturb Procedure {#sec:perturb} -------------------------- In the $ShiftPerturb$ procedure, the algorithm escapes a local optimum by a series of shift moves. A shift move performed on a packing pattern $X$ is defined as shifting the position of a randomly chosen disk to a random place in the container and then locally minimizing $E_R(X)$. The number of times the shift move is performed is controlled by a parameter $PerturbStrength$. As pointed out in previous research [@ils], the perturbation strength is very important for Iterated Local Search. If it is too weak, the local search may undo the perturbation and the search will be confined in a small area of the solution space. On the contrary, if the perturbation is too strong, the Iterated Local Search will behave like random restart, leading to poor performance. After preliminary computational tests, we choose the value of $PerturbStrength$ to be a random integer from $[1, n/8]$. Performance Assessment ====================== In this section, we assess the performance of the proposed algorithm through computational experiments on two sets of widely-used test instances. We also compare the results of our algorithm with some state-of-the-art algorithms in the literature. Experimental Protocol --------------------- The algorithm is programmed in C++ and complied using GNU G++. All computational experiments are carried out on a personal computer with 4Gb memory and a 2.8GHz AMD Phenom II X6 1055T CPU. Table \[tbl:parameter\] gives the settings of the four important parameters of the algorithm. Note that all the computational results are obtained without special tuning of the parameters, i.e., all the parameters used in the algorithm are fixed for all the tested instances. Test Instances {#sec:instances} -------------- Two sets of test problems are considered, in total constituting 70 instances. The first set comes from the famous circle packing contest (see <http://www.recmath.org/contest/CirclePacking/index.php>). This contest started on October 2005 and ended on January 2006. During this period, the participants were invited to propose densest packing solutions to pack $n(n=5,6,\dots,50)$ circles, each having radius $r_i = i (i=1, 2,\dots, n)$ into the smallest containing circle without overlap. 155 groups from 32 countries took part in the contest and submitted a total of 27490 tentative solutions. After the contest, these results were further improved respectively by @muller2009, Eckard Specht [@specht2012], Zhanghua Fu et al. [@specht2012]. Currently, all the best-known records are published and continuously updated on the Packomania website. The second set of instances consists of 24 problem instances first presented by [@huang2005]. These instances are frequently used in the literature by many authors, see for example [@huang2006; @akeb2009; @akeb2010a]. The size of these instances ranges from $n=10$ to $60$. A detailed description of these instances can be found in [@huang2005]. Computational results on the circle packing contest instances ------------------------------------------------------------- For the circle packing contest instances, researchers usually pay more attention on the solution quality. Especially during the contest, people mostly focus on finding better solutions than the best-known records and rarely consider the computational resource used. After researchers have solved these instances using various approaches and large amount of computational resource, this set of instances now becomes a challenging benchmark to test the *discovery capability* [@grosso2007] of a new algorithm. Therefore, our first experiment concentrates on searching for high-quality solutions. For each run of each instance, we usually set the time limit to 24 hours, run the algorithm multiple times and record the best-found solutions. Table \[tbl:ccin\_result\] gives the computational results. Column 1 lists the best-known records on the Packomania website. Columns 2-4 respectively report the solution difference between some top reference results and the best-known records. These include: the best results found by [@addis2008u] (who is the champion of the circle packing contest) using PBH algorithm, the best records obtained by all the participants in the contest, the best results found by [@muller2009] using Simulated Annealing (SA) algorithm. Column 5 gives the solution difference between our results and the best-known records. The results indicated in bold are better than the best-known ones. Table \[tbl:ccin\_result\] omits the results for $n=5, 6, \dots, 20$, because our results and all the reference results are the same on these instances. Note that, our program generates solutions with a maximum error on the distances of $10^{-9}$. We have sent all the improved results to Eckard Specht. Using his own local optimization solver, he has processed our results to a high precision ($10^{-28}$) and published them on the Packomania website. Table \[tbl:compare\] summarizes the comparison of our results with the reference results. The rows *better*, *equal* and *worse* respectively denote the number of instances for which the proposed algorithm gets solutions that are better, equal and worse than each reference result. Table \[tbl:compare\] shows that the proposed algorithm is able to discover a number of better solutions than the previous best reference results, demonstrating its efficacy in finding high-quality solutions. In fact, we also tested the proposed algorithm on the larger instances of $n=51, 52, \dots, 100$. Some preliminary experiments show that the algorithm can improve almost all previous best-known results. Interested readers can refer to the Packomania website. All the reference algorithms in Table \[tbl:ccin\_result\] concentrate on finding high-quality solutions and do not reveal their computational statistics. In order to further evaluate the proposed algorithm in terms of search efficiency, we conduct additional experiments to compare the proposed algorithm with two recently published algorithms in a time-equalized basis. For each instance of $n=5,6, \dots, 32$, we set the maximum time limit to 10000 seconds. We record the best-found solution and the elapsed time when it is first detected by the algorithm. To reduce the impact of randomness, each instance is independently solved for 10 times. Table \[tbl:contest2\] gives the computational results. Columns 2-3, 4-5 respectively list the best-found solution and the needed computing time of TS/NP algorithm and FSS algorithm. Columns 2 and 3 are extracted from [@aimudahka2010] where the algorithm ran on a computer with a Pentium IV, 2.66 Ghz CPU and 512Mb RAM. Columns 4 and 5 are extracted from [@beasley2012]. Their experiments were done on a computer with a Intel(R) Core(TM) i5-2500 3.30 GHz CPU and 4.00 GB RAM. Columns 6-8 give the computational statistics of our algorithm, including the best-found solution, the number of hit times and the averaged computing time to detect the best-found solution. Columns 6-8 show that, for all the 28 instances, the proposed algorithm can reach (or improve) the previous best-known records listed in Table \[tbl:ccin\_result\] within the given time limit. Especially for $n\leq 25$, the algorithm can robustly detect the best-known records in a short time. When compared with the two reference algorithms, one observes that the proposed algorithm can usually find better solutions within the time limit. These results provide evidence of the search efficiency of ITS-PUCC algorithm. Computational results on the NR instances ----------------------------------------- This section tests the proposed algorithm on the 24 NR instances. For each instance, we set the time limit to 10000 seconds, and record the best-found solution and the elapsed time when it is first detected by the algorithm. Each instance is solved for 10 times from different randomly generated starting points. The computational results are presented in Table \[tbl:NR\]. Column 1 gives the instance name. Columns 2-3, 4-5, 6-7, 8-9, respectively present the best-found solution and the needed computing time of A1.5 Algorithm in [@huang2006], Beam Search (BS) algorithm in [@akeb2009], Algorithm 2 in [@akeb2010b] and GP-TS algorithm in [@huang2012a]. Columns 10-12 give the computational statistics of our algorithm, including the best-found solution, the number of hit times and the averaged computing time for detecting the best-found solution. In experiments, our program generates solutions with a maximum error on the distance of $10^{-9}$. However, in order to keep consistent with previous papers, we report in Table \[tbl:NR\] the results with 4 significant digits after the decimal point. Table \[tbl:NR\] demonstrates that, for all the tested 24 instances, the proposed algorithm can find 16 better solutions than the best results found by the references algorithms (as indicated in bold in the table). For the other 8 instances, it can reach the best-known solutions efficiently and robustly. These results further provide evidence of the competitiveness of the proposed algorithm. Algorithm Analysis ================== In this section, we turn our attention to analyzing the two most important ingredients of the proposed algorithm: the $SwapTabuSearch$ procedure and the $ShiftPerturb$ procedure. Analysis of The SwapTabuSearch Procedure ---------------------------------------- The $SwapTabuSearch$ procedure is a key component of the proposed algorithm, which enables the algorithm to intelligently examines the neighboring packing patterns through swap moves. In order to make sure the Tabu Search strategy makes a meaningful contribution, we conduct experiments to compare the Tabu Search strategy with a simple local search strategy called Steepest Descent [@slsbook]. For comparison, we use the same neighborhood structure as described in Section \[sec:tabusearch\] and implement the Steepest Descent strategy as follows. At each iteration, the search examines each neighbor of the current solution and find out the best neighbor with the least objective value $E_R$. If the best neighbor $X'$ is better than the current solution $X$, i.e., $E_R(X') \leq E_R(X)$, then the search moves to $X'$ and proceeds to the next iteration; otherwise the search stops and declares reaching a local minimum. A representative instance *NR15-2* is chosen as our test bed. This instance is nontrivial. Though many previous papers have tested it, only few state-of-the-art algorithms, like Beam Search [@hifi2008], PBH[@addis2008u; @grosso2010], SA[@muller2009] can obtain the optimal packing pattern. We set the radius of container $R$ to the best-known value, randomly generate initial $X$ and call both algorithms to minimize $E_R(X)$. We run both algorithms 1000 times from different randomly generated starting points and record in Table \[tbl:ts\_sd\] respectively the best-found solution (Column 2), the average solution quality (Column 3), the average number of search steps for each local search (Column 4) and the average elapsed time for each local search (Column 5). From Table \[tbl:ts\_sd\], we observe that, the Tabu Search strategy shows clear advantage over Steepest Descent strategy. Each time, the Tabu Search strategy can find the global minimum from a randomly generated starting point, while the Steepest Descent strategy fails for all 1000 runs. In fact, we try to run the Steepest Descent strategy from 100000 randomly generated starting points, it still cannot find the global minimum. The main reason for the difference is that, with the Steepest Descent strategy, the search is easily trapped in poor local minimum. As shown in Table \[tbl:ts\_sd\], the average number of search step for each local search is only 7. However, with the Tabu Search strategy, the search can escape from low-quality local minimum trap and proceed to explore the neighboring area. Figure \[fig:sd\_ts\] shows a typical search trajectory of Tabu Search, compared with the search trajectory of Multistart Steepest Descent. In Figure \[fig:sd\_ts\], both algorithms start from the same initial solution, a packing pattern with $E_R = 8.77845794$. After 7 search steps, both of them encounter a local minimum with $E_R = 1.37297106 $. At this time, the Steepest Descent strategy is trapped, the search has to proceed from a new randomly generated initial solution. However, with the Tabu Search strategy, the search is able to escape from the local minimum with $E_R = 1.3729106$, proceed to examine the neighboring area, and finally find the global minimum at the 362th search step. Search Strategy Best-found solution Average solution quality Search steps Time (s) ------------------ --------------------- -------------------------- -------------- ---------- Tabu Search 0.000000 0.000000 1269 2 Steepest Descent 0.092194 2.085742 7 0 : Computational statistics of Tabu Search strategy and Steepest Descent strategy from 1000 randomly generated initial packings[]{data-label="tbl:ts_sd"} ![Comparison of search trajectories between Multistart Steepest Descent and Tabu Search []{data-label="fig:sd_ts"}](cmp-trajectory.eps){width="4in"} These experiments reveal that, the Tabu Search strategy helps to perform an intensified examination around the incumbent packing pattern and makes possible discovering those hidden good solutions. The same experiments have been performed on several other instances, leading to similar observation. Analysis of the ShiftPerturb Procedure -------------------------------------- In order to verify the effectiveness of the $ShiftPerturb$ procedure, we conduct experiments to compare the proposed ITS algorithm with a Multistart Tabu Search algorithm. In the Multistart Tabu Search algorithm, when the $SwapTabuSearch$ procedure finishes, the search proceeds from a new randomly generated initial solution. The parameter setting of $SwapTabuSearch$ is the same as listed in Table \[tbl:parameter\]. We test the Multistart Tabu Search algorithm on the 28 circle packing instances with $5 \leq n \leq 32$. Each instance is solved for 10 times. The time limit for each run is also set to 10000 seconds. The computational results show clear advantage of ITS algorithm over Multistart Tabu Search algorithm. For the instances of $5 \leq n\leq 24$, the Multistart Tabu Search algorithm can also detect the best-known records, but with lower success rates and relatively longer time. Nevertheless, for each instance of $25 \leq n \leq 32$, the Multistart Tabu Search algorithm fails to detect the best-known solution for all the 10 runs within the given time limit. We conjecture the superior of ITS over Multistart Tabu Search may be explained from the following two aspects. First, the Iterated Local Search framework helps the search to perform a more intensified examination around the incumbent solution, making it possible to repeatedly discover better solutions. This is supported by our observations from computational experiments that, with the ITS algorithm, the search can usually generate a sequence of local minima with descending objective value. The final solution obtained by one run of ITS is usually much better than that found by the first run of $SwapTabuSearch$. Second, the shift move in the $ShiftPerturb$ procedure is complementary to the swap move, enabling the search to reach some packing patterns which are hard to detect only through swap moves. Conclusion and Future Work ========================== In this paper, we have presented a heuristic global optimization algorithm for solving the unequal circle packing problem. The proposed algorithm uses a continuous local optimization method to generate locally optimal packings and integrates two combinatorial optimization methods, Tabu Search and Iterated Local Search, to systematically search for good local minima. The efficiency and effectiveness of the algorithm have been demonstrated by computational experiments on two sets of widely used test instances. For the 46 challenging circle packing contest instances and the 24 widely-used NR instances, the algorithm can respectively improve 14 and 16 previous best-known records in a reasonable time. There are two main directions for future research. On the one hand, the presented algorithm can be further improved by incorporating other advanced strategies. Possible improvements include the following: First, reduce the solution space by first ignoring several smaller disks and only looking for optimal packing pattern of the remaining larger disks. The smaller disks can be inserted into the holes after the larger disks have been placed into the container. This strategy was proposed in [@addis2008u] and had proved to be very useful. Second, test other Stochastic Local Search methods, such as Simulated Annealing used in [@muller2009], Variable Neighborhood Search [@slsbook] and so on. Third, the proposed algorithm is a single-solution based method. It is possible to strengthen the robustness of the algorithm by employing some population-based methods, like the Population Basin Hopping method proposed in [@grosso2007]. On the other hand, the ideas behind the proposed algorithm can also be applied to other hard global optimization problems. Many real-world global optimization problems, such as the cluster optimization problem in computational chemistry [@blj2011] and the protein folding problem in computational biology [@huanglv2005], have the same characteristics as the unequal circle packing problem, i.e., they have both a continuous and combinatorial nature. For these kinds of problems, it is possible to build a neighborhood structure on the set of local minima via appropriate perturbation moves, and then to employ some advanced combinatorial optimization methods to systematically search for good local minima. Acknowledgement {#acknowledgement .unnumbered} =============== We thank the anonymous reviewers whose detailed and valuable suggestions have significantly improved the manuscript. We thank Eckard Specht for processing our data and publishing it on the Packomania website. This work was supported by National Natural Science Foundation of China (Grant No. 61100144, 61262011). 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--- abstract: 'We show that if a link $L$ has a closed $n$-braid representative admitting non-degenerate exchange move, an exchange move that does not obviously preserve the conjugacy class, $L$ has infinitely many non-conjugate closed $n$-braid representatives.' address: 'Department of Mathematics, Kyoto University, Kyoto 606-8502, JAPAN' author: - Tetsuya Ito title: 'A non-degenerate exchange move always produces infinitely many non-conjugate braids ' --- Let $B_n$ be the braid group with standard generators $\sigma_1,\ldots,\sigma_{n-1}$. We denote the closure of a braid $\beta \in B_n$ by $\widehat{\beta} \subset S^{3}$. For an oriented link $L \subset S^{3}$ let $\mathbf{Br}_n(L)= \{\beta \in B_n \: | \:\widehat{\beta}=L\}$ be the set of $n$-braids whose closures are $L$. The set $\mathbf{Br}_n(L)$ may contain infinitely many mutually non-conjugate braids. However, Birman and Menasco proved a remarkable (non)finiteness theorem [@BM]: $\mathbf{Br}_n(L)$ modulo *exchange move* $\alpha \sigma_{n-1}\beta \sigma_{n-1}^{-1} \leftrightarrow \alpha \sigma_{n-1}^{-1}\beta \sigma_{n-1}$ ($\alpha,\beta \in B_{n-1}$) has only finitely many conjugacy classes. In particular, when $\mathbf{Br}_n(L)$ does not contain a braid admitting exchange move, $\mathbf{Br}_n(L)$ contains only finitely many conjugacy classes. Then we ask the converse[^1]: *Does $\mathbf{Br}_n(L)$ contain infinitely many mutually non-conjugate braids if $\mathbf{Br}_n(L)$ contains a braid admitting exchange move ?* This question was studied in [@SS; @Sh; @St1; @St2] where it was shown that under some additional and technical assumptions, iterations of exchange move indeed produce infinitely many non-conjugate braids. In this note we give a simpler and shorter proof of infiniteness under the weakest assumption. We use a formulation of iterations of exchange moves following [@SS]: We say that an $n$-braid $\beta$ *admits an exchange move* if one can write $\beta=AB$ for $A \in \langle \sigma_1^{\pm 1},\ldots,\sigma_{n-2}^{\pm 1}\rangle$ and $B \in \langle \sigma_2^{\pm 1},\ldots,\sigma_{n-1}^{\pm 1}\rangle$. For $k \in {\mathbb{Z}}$ and an $n$-braid $\beta=AB$ admitting an exchange move, the *$k$-iterated exchange move* of $\beta=AB$ is the braid ${\mathrm{ex}}^{k}(\beta)= A\tau^{k}B\tau^{-k}$ where $\tau=(\sigma_2\cdots \sigma_{n-2})^{n-2} \in B_n$. We say that an (iterated) exchange move is *degenerate* if $A \tau = \tau A$ or $B \tau = \tau B$. Otherwise, an (iterated) exchange move is *non-degenerate*. A $k$-iterated exchange move is attained by exchange move $|k|$ times so the closures of $\beta$ and ${\mathrm{ex}}^k(\beta)$ represent the same link. A degenerate exchange move is an exchange move that obviously preserves the conjugacy classes. Our main theorem shows that, except this trivial cases, iterated exchange moves *always* produce infinitely many mutually non-conjugate braids. We identify the braid group $B_n$ with the mapping class group $MCG(D_n)$ of the $n$-punctured disk $D_n$. Let ${\mathrm{ent}}(\beta)$ be the topological entropy of $\beta$, the infimum of the topological entropy of homeomorphism that represents $\beta$. \[theorem:main\] If $\beta \in \mathbf{Br}_n(L)$ admits a non-degenerate exchange move, then the set $\{{\mathrm{ent}}({\mathrm{ex}}^k(\beta)) \: | \: k \in {\mathbb{Z}}\}$ is unbounded. In particular, the set $\{{\mathrm{ex}}^{k}(\beta) \: | \: k \in {\mathbb{Z}}\} \subset \mathbf{Br}_n(L) $ contains infinitely many distinct conjugacy classes. Let $S$ be a closed orientable surface minus finitely many disks and puncture points in its interior. A simple closed curve $c$ in $S$ is *essential* if $c$ is not boundary parallel nor surrounding single puncture. We denote by $T_c$ the Dehn twist along $c$. A family of essential simple closed curves $\{c_1,\ldots,c_N\}$ *fills* $S$ if $\min i(c,c_i)\neq 0$ for any essential simple closed curve $c$, where $i(c,c')$ denotes the geometric intersection number. Our proof is based on the following theorem of Fathi [@Fa Theorem 7.9]. \[theorem:Fathi\] Let $f \in MCG(S)$ and $c_1,\ldots,c_N$ be essential simple closed curves in $S$. Assume that 1. The set of curves $\{c_1,\ldots, c_N\}$ fills $S$. 2. $i(c_i,c_{i+1})\neq 0$ for $i=1,\ldots,N-1$ and $i(c_{N},c_1)\neq 0$. Then for given $R>0$, there is $k=k(R)>0$ such that $ T_{c_1}^{n_{1}}T_{c_{2}}^{n_{2}}\cdots T_{c_N}^{n_N}f $ is pseudo-Anosov whose dilation is $>R$ whenever $|n_i|>k$ for all $i$. Although the statements and assumptions of Theorem \[theorem:Fathi\] are bit different, Theorem \[theorem:Fathi\] follows from the proof of [@Fa Theorem 7.9]; First, the assumptions (i) and (ii) allow us to apply an interpolation inequality [@Fa Theorem 7.4]. Second, to get dilatation bound we take a choice of $\varepsilon>0$ in page 149 of the proof of [@Fa Theorem 7.9] as $\varepsilon=(K^{2}R^{2l-1})^{-1}$ instead of $(2K^{2})^{-1}$ in the original argument. Then the same argument gives the desired dilatation bound. The braid $\tau$ in iterated exchange move corresponds to the Dehn twist $T_{c}$ along the simple closed curve $c$ surrounding $2$nd,...,($n-2$)th punctures. The non-degeneracy assumption is equivalent to saying that $A(c)\neq c$ and $B(c) \neq c$. For $i>0$ let $c_{2i}=\beta^{i-1}(A(c))=(AB)^{i-1}A(c), c_{2i-1}=\beta^{i-1}(c)=(AB)^{i}(c)$. Then for $N>0$, ${\mathrm{ex}}^{k}(\beta)^{N} = T_{c_1}^{k}T_{c_2}^{-k}\cdots T_{c_{2N-1}}^{k}T_{c_{2N}}^{-k}\beta^{N}$. We equip a complete hyperbolic metric (of finite area) on $D_n$ and take $c_i$ as a closed geodesic. Let $S$ be the minimum complete geodesic subsurface of $D_n$ that contains $\{c_1,c_2,c_3,c_4,\ldots\}=\{c,A(c), \beta(c),\beta(A(c)),\ldots\}$. Here the minimum means the minimum with respect to inclusions. Then for sufficiently large $M$ the set of curves $\{c_1,c_2,\ldots,c_{M}\}$ fills $S$. By non-degeneracy assumption, $S$ contains $\partial D_n$ as its boundary. \[claim:A\] $\beta$ preserves the subsurface $S$ setwise. In particular, the restriction ${\mathrm{ex}}^{k}(\beta)|_S \in MCG(S)$ is well-defined for all $k$. If $S=D_n$ then it is obvious so we assume that $S \neq D_n$. Then $C=\partial S \setminus \partial D_n$ is a non-empty multi curve. If $\beta(S) \neq S$ then $\beta^{-1}(S) \neq S$ and $i(\beta^{-1}(C),C) \neq 0 $. Since $\{c_1,c_2,\ldots,c_{M}\}$ fills $S$ this means $i(\beta^{-1}(C),c_i)=i(C,\beta(c_i)) = i(C,c_{i+2})\neq 0$ for some $i$, this is a contradiction since $i(C,c_{i+2})=0$ by definition. \[claim:B\] There exists $N\geq M$ such that $i(c_{1},c_{2N})\neq 0$. Assume to the contrary that $i(c_{1},c_{2N}) =0$ for all $N\geq M$. Let $S'$ be the minimum geodesic subsurface of $S$ that contains $\{c_{2M},c_{2(M+1)},\ldots\}=\{c_{2M},\beta(c_{2M}),\beta^{2}(c_{2M}),\ldots\}$. By the same argument as Claim \[claim:A\], $\beta$ preserves $S'$, and $i(c,c')=0$ for every curve $c' \subset S'$. Then $\beta^{-2(M-1)}(c_{2M}) = A(c) \subset S'$ so $i(c,A(c))=0$. This contradicts with non-degeneracy assumption. By non-degeneracy assumption $i(c_{i},c_{i+1})\neq 0$ for every $i>0$. Thus by Claim \[claim:B\] there is $N>0$ such that such $${\mathrm{ex}}^{k}(\beta)^{N}|_S = T_{c_1}^{k}T_{c_2}^{-k}\cdots T_{c_{2N-1}}^{k}T_{c_{2N}}^{-k}\beta^{N}|_S \in MCG(S)$$ satisfies the assumptions of Theorem \[theorem:Fathi\]. Hence for any given $R>0$, whenever $|k|$ is sufficiently large, ${\mathrm{ex}}^{k}(\beta)^{N}|_S$ is pseudo-Anosov whose dilatation $\lambda({\mathrm{ex}}^{k}(\beta)^{N}|_S)$ is $>R$. Since $$\begin{aligned} {\mathrm{ent}}( {\mathrm{ex}}^{k}(\beta)) & = \frac{{\mathrm{ent}}( {\mathrm{ex}}^{k}(\beta)^{N} )}{N} \geq \frac{{\mathrm{ent}}( {\mathrm{ex}}^{k}(\beta)^{N}|_S)}{N} = \frac{\log \lambda({\mathrm{ex}}^{k}(\beta)^{N}|_S)}{N} \geq \frac{\log R}{N} \end{aligned}$$ the set $\{{\mathrm{ent}}({\mathrm{ex}}^{k}(\beta)) \: | \: k \in {\mathbb{Z}}\}$ is unbounded. Our proof shows that as $k$ increase, a non-degenerate $k$-iterated exchange move increases the entropy, the complexity of dynamics, as long as $k$ is sufficiently large. Since the core of Birman-Menasco’s proof of (non)finiteness theorem is to reduce the complexity (the number of singular points) of braid foliation of Seifert surface, it is natural to expect relations between the entropy and braid foliation. If a braid $\beta$ is obtained from $\beta'$ by an exchange move reducing the complexity of braid foliation, then ${\mathrm{ent}}(\beta) \leq {\mathrm{ent}}(\beta')$ ? Acknowledgement {#acknowledgement .unnumbered} =============== The author is partially supported by JSPS KAKENHI Grant Numbers19K03490, 16H02145. This research was supported in part by funding from the Simons Foundation and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program. [1]{} J. Birman and W. Menasco, [*Studying links via closed braids VI: a non-finiteness theorem*]{} Pacific J. Math, **156** 1992, 265–285. A. Fathi, [*Dehn twists and pseudo-Anosov diffeomorphisms,*]{} Invent. Math., **87** (1987), 129-151. R. Shinjo, [*Non-conjugate braids whose closures result in the same knot,*]{} J. Knot Theory Ramifications [**19**]{} (2010), 117–124. R. Shinjo and A. Stoimenow [*Exchange moves and non-conjugate braid represenattives of knots,*]{} Nagoya Math J. to appear. A. Stoimenow, [*On non-conjugate braids with the same closure link,*]{} J. Geom. [**96**]{} (2009), 179–186. A. Stoimenow, [*Non-conjugate braids with the same closure link from density of representations,*]{} J. Math. Pures Appl. (9) [**94**]{} (2010), 470–496. [^1]: We need to be bit careful to formulate the problem since some exchange moves are ‘trivial’ in the sense that they obviously yield conjugate braids.

--- abstract: 'We study the size-dependent exciton fine structure in monolayer black phosphorus quantum dots (BPQDs) deposited on different substrates (isolated, Si and SiO$_2$) using a combination of tight-binding method to calculate the single-particle states, and the configuration interaction formalism to determine the excitonic spectrum. We demonstrate that the substrate plays a dramatic role on the excitonic gaps and excitonic spectrum of the QDs. For reasonably high dielectric constants ($\varepsilon_{sub} \sim \varepsilon_{Si} = 11.7 \varepsilon_0$), the excitonic gap can be described by a single power law $E_X(R) = E_X^{(bulk)} + C/R^{\gamma}$. For low dielectric constants $\varepsilon_{sub} \leq \varepsilon_{SiO_2} = 3.9 \varepsilon_0$, the size dependence of the excitonic gaps requires the sum of two power laws $E_X(R) = E_g^{(bulk)} + A/ R^{n} - B/R^{m}$ to describe both strong and weak quantum confinement regimes, where $A$, $B$, $C$, $\gamma$, $n$, and $m$ are substrate-dependent parameters. We also predict that the exciton lifetimes exhibit a strong temperature dependence, ranging between 2-8 ns (Si substrate) and 3-11 ns (SiO$_2$ substrate) for QDs up 10 nm in size.' author: - 'J. S. de Sousa$^1$, M. A. Lino$^{1,2}$, D. R. da Costa$^1$, A. Chaves$^1$, J. M. Pereira Jr.$^1$, G. A. Farias$^1$' title: Substrate effects on the exciton fine structure of black phosphorus quantum dots --- Introduction ============ Black phosphorus (BP) has recently become one of the most attractive two-dimensional materials due to its physical and chemical properties. BP exhibits a highly anisotropic band structure with large direct band gap of the order of 2 eV [@ref4; @ref5; @ref6; @ref7; @ref8; @chaves2016], high carrier mobilities [@xia2014; @castellanos2015; @rudenko2014; @rudenko2015; @cakir2014; @pereira2015], nonlinear optical response [@lu2015], magneto-optical Hall effect [@tahir2015], integer quantum Hall effect [@li2016], and thermoelectricity [@zhang2014]. All these properties make BP a strong candidate for the development of optical and electronic applications. It was recently shown that BP exhibits, depending on the substrate, very large exciton binding energies that can withstand large in-plane electric fields, giving rise to excited excitonic states [@chaves2015]. Two recent reports on the optical properties of bulk monolayer BP deposited on quartz and Si substrates, by Zhang *et al.* and Li *et al.*, reported photoluminescence peaks at 1.67 eV and 1.73 eV, respectively [@zhang2016; @li2017]. The assumption that the difference in the peaks energies of both measurements is caused by the substrate is the main motivation of this work. BP quantum dots (BPQDs) have also been already produced. For example, Sofer *et al.* produced BPQDs with few layers with average size of 15 nm [@ref12]. Sun *et al.* synthesized BPQDs as small as $2.6\pm1.8$ nm of diameter and $1.5\pm0.6$ nm of thickness with a wet exfoliation method [@ref13]. Zhang *et al.* also fabricated BPQDs by wet exfoliation, obtaining BPQDs with lateral sizes of $4.9\pm1.6$ nm and thicknesses of $1.9\pm0.9$ nm [@ref11]. Xu *et al.* produced BPQDs with average size of $2.1\pm0.9$ nm in large scale by solvothermal synthesis [@ref14]. From theoretical point of view, BP and their nanostructures have also been intensively investigated. Rudenko *et al.* developed a tight-binding (TB) parameterization for mono and bilayer BP that has become the basis for the theoretical investigation of several BP structures [@rudenko2015; @rudenko2014]. Pereira *et al.* derived a continuum model to describe the band structure of BP, departing from the paremeterization of Rudenko *et al.*. They also investigated the Landau levels in the mono and bilayer BP [@pereira2015]. de Sousa *et al.* proposed new types of boundary conditions for BP nanoribbons with different edge types to be used in theoretical modelling of BP nanostructures within the continuum model [@desousa2016]. Zhang *et al.* investigated the electronics properties of BPQDs with different geometries under the effect of external magnetic fields [@zhang2015]. Lino *et al.* studied the additional energy spectrum of small BPQDs, and demonstrated the feasibility of observing Coulomb blocked effects in BPQDs at room temperature [@lino2017]. Substrate effects on the electronic properties of monolayer BP have been investigated by Mogulkoc *et al.* [@mogulkoc2016]. They have reported the broadening of the single-particle gap and renormalization of the effective masses of monolayer BP due to the interaction between carriers in BP and substrate polarons. In particular, the single-particle gap broadening can be of the order of 30 meV for BP deposited on SiO$_2$. In this work, we calculate the size-dependent excitonic fine structure of monolayer (ML) circular BPQDs using a combination of a TB method to calculate single-particle states, and the configuration interaction (CI) formalism to determine the excitonic fine structure. We aim to understand the effect of different substrates on the exciton fine structure of the BPQDs. This paper is organized as follows. The Theoretical background to calculated single-particle and excitonic states, as well as the optical properties are described in Section \[sec:model\]. Our results are presented in Section \[sec:results\], and discussed in Section \[sec:discussion\]. \[sec:model\] Theoretical background ==================================== Nearly circular BPQDs were formed by generating a large ML-BP sheet with armchair (zigzag) direction aligned to the $x$ ($y$) axis, and the atoms outside a given radius (measured with respect to center of mass of the large sheet) were disregarded, resulting in QDs with $C_4$ symmetry. Our choice for the dot shape is based on samples produced by exfoliation who exhibits no uniform edges. Some examples of BPQDs studied in this work are depicted in Figures \[fig:BPQDs\](a)-\[fig:BPQDs\](c). The energy spectrum of the BPQDs was calculated by solving Schrödinger equation represented in a linear combination of atomic orbital (LCAO) basis, such that the effective Hamiltonian reads $$\hat{H} = \sum_i \epsilon_i |i\rangle\langle i | + \sum_{i,j} t_{i,j} |i\rangle\langle j |.$$ The generalized index $i = \{\vec{R}_i,\alpha,\nu\}$ represents the orbital $\nu$ of the atomic species $\alpha$ at the atomic site $\vec{R}_i$. $\epsilon_i$ represents the onsite energy of the i-th site, and $t_{i,j}$ represents the hopping parameter between i-th and j-th sites. Since all atoms are identical, the onsite energies only provide an energy reference to the energy spectrum (we adopted $\epsilon_i = 0$ eV). As for the hopping parameters and lattice constants, we adopted the $10$ hopping paramenter TB model of Rudenko *et al.* [@rudenko2015]. Excitonic spectra ----------------- The excitons wavefunctions $\Psi_\lambda $ are expressed as linear combination of single substitution Slater determinants $\Phi_{v,c}$ [@franceschetti1999; @desousa2012] $$\Psi_\lambda (\vec{r}_e,\vec{r}_h)=\sum_{v}^ {N_v}\sum_{c}^ {N_c}C_{v,c}^{\lambda} \Phi_{v,c},$$ where $\lambda$ denotes the exciton quantum numbers, $N_v$ ($N_c$) corresponds to the number of valence (conduction) states included in the expansion. The determinants $\Phi_{v,c}$ are obtained from the ground state (GS) Slater determinant $\Phi_0$ by promoting one electron from the valence band state $\psi_v$ to the conduction band state $\psi_c$: $$\begin{aligned} \Phi_0(\vec{r}_1,...,\vec{r}_N)= \mathcal{A}[\psi_1(\vec{r}_1),...,\psi_v(\vec{r}_v),...\psi_N(\vec{r}_N)], \\ \Phi_{v,c}(\vec{r}_1,...,\vec{r}_N)= \mathcal{A}[\psi_1(\vec{r}_1),...,\psi_c(\vec{r}_v),...\psi_N(\vec{r}_N)] ,\end{aligned}$$ where $N$ is the number of electrons in the system, and $\mathcal{A}$ is the anti-symmetrisation operator. In the $\{ \Phi_{v,c} \}$ basis, the excitonic spectrum is obtained by solving the following effective Schrödinger equation: $$\begin{aligned} \label{eq:excfine} \sum_{v',c'}^{N_v,N_c}\hspace{-0.2cm}\left[(\epsilon_c\hspace{-0.05cm}-\hspace{-0.05cm}\epsilon_v\hspace{-0.05cm}-\hspace{-0.05cm}E_\lambda)\delta_{v,v'}\delta_{c,c'}\hspace{-0.05cm}-\hspace{-0.05cm}J_{vc,v'c'}\hspace{-0.05cm}+\hspace{-0.05cm}K_{vc,v'c'}\right] C_{v',c'}^{\lambda}\hspace{-0.05cm}=\hspace{-0.05cm}0.\nonumber\\\end{aligned}$$ $\epsilon_{c,v}$ represents the single-particle energy states in the conduction and valence bands, respectively. For simplicity, spin effects in both single-particle and excitonic spectra are reserved for future studies. The quantities $J_{vc,v'c'}$ and $K_{vc,v'c'}$ represent the direct Coulomb and exchange energies: $$\begin{aligned} \label{eq:eb} J_{vc,v'c'}\hspace{-0.15cm}=\hspace{-0.15cm}\int\hspace{-0.25cm}\int\hspace{-0.15cm}\psi_{v}^{*}(\vec{r}_2)\psi_{c}^{*}(\vec{r}_1)V(|\vec{r}_1\hspace{-0.1cm}-\hspace{-0.1cm}\vec{r}_2 |) \psi_{v'}(\vec{r}_2) \psi_{c'}(\vec{r}_1) d\vec{r}_1 d\vec{r}_2,\nonumber\\\end{aligned}$$ $$\begin{aligned} \label{eq:ex} K_{vc,v'c'}\hspace{-0.15cm}=\hspace{-0.15cm}\int\hspace{-0.25cm}\int\hspace{-0.15cm}\psi_{v}^{*}(\vec{r}_1)\psi_{c}^{*}(\vec{r}_2)V(|\vec{r}_1\hspace{-0.1cm}-\hspace{-0.1cm}\vec{r}_2 |)\psi_{v}(\vec{r}_1)\psi_{c'}(\vec{r}_2) d\vec{r}_1 d\vec{r}_2.\nonumber\\\end{aligned}$$ Screening model --------------- The Coulomb interaction potential $V(|\vec{r}_1-\vec{r}_2 |)$ in two dimensions exhibits a nontrivial form as compared to tri-dimensional bulk materials due to non-local screening effects. We adopted the model of Rodin *et al.* for the Coulomb interaction between charges confined in a two-dimensional material sandwiched between a substrate with dielectric constant $\varepsilon_{sub}$ and vacuum [@rodin2014]. This is given by: $$\label{eq:dielectric} V(r) = \frac{q^2}{4\pi\varepsilon_0} \frac{\pi}{2 \kappa r_0} \left[ H_0\left( \frac{r}{r_0}\right)-Y_0\left( \frac{r}{r_0}\right)\right],$$ where $r_0 = 2\pi \alpha_{2D}/\kappa$, $\kappa = (1+ \varepsilon_{sub})/2$. $H_0$ and $Y_0$ are the Struve and Neumann functions. The parameter $\alpha_{2D} = 4.1$ nm represents the polarizability of a single BP layer in vacuum, and it was determined by Rodin *et al.*, using a density-functional calculations [@rodin2014]. Dipole matrix elements ---------------------- The first-order radiative recombination lifetime of the excitonic states $\Psi_\lambda$ is obtained by using the Fermi’s golden rule [@desousa2012; @Leung; @Ouisse] $$\label{eq:rectime} \frac{1}{\tau_\lambda} = \frac{4 n \alpha \omega_{\lambda}^3}{3 c^2} | D_\lambda |^2,$$ where $n = \sqrt{\epsilon_0}$ is the refractive index, $\alpha$ is the fine structure constant, $c$ is the speed of light in vacuum, $\omega_\lambda=E_\lambda / \hbar$, and $D_\lambda$ represents the dipole matrix elements: $$\label{eq:dipole} D_\lambda = \sum_{v,c} C_{v,c}^{\lambda} \langle \psi_c | \vec{E}_0 \cdot \vec{r} | \psi_v \rangle,$$ where $\vec{E}_0$ is the light polarisation direction. The excitonic absorption cross section can be calculated using Fermi’s golden rule as: $$\label{eq:abs} \sigma(\omega) \propto \sum_{\lambda} |D_{\lambda}|^2 \delta(\hbar \omega - E_{\lambda}).$$ The average exciton lifetime is obtained with: $$\label{eq:rectime2} \frac{1}{\tau} = \frac{ \sum_{\lambda} \tau_{\lambda}^{-1} e^{-(E_\lambda - E_0)/k_B T}} {\sum_{\lambda}e^{-(E_\lambda - E_0)/k_B T}},$$ where $E_0$ is the lowest exciton energy and $k_B$ is the Boltzmann constant. \[sec:results\]Results ====================== \[sec:singleparticle\]Single-particle spectra --------------------------------------------- Figure \[fig:BPQDs\](d) shows the size-dependent single-particle spectra of ML-QDs up to 10 nm of diameter. The horizontal lines represent the conduction $e_{cbm}$ and valence $e_{vbm}$ band edges of the bulk BP monolayer, where $e_{cbm}-e_{vbm} = E_g^{(bulk)} = 1.84$ eV. The blue lines indicate the size-dependent conduction $e_1(R)$ and valence $h_1(R)$ band edges. The size-dependent single-particle bandgap is defined as $E_g(R) = e_1(R) - h_1(R)$, where electron (hole) states are labeled as $e_n$ ($h_m$), and the index $n$ ($m$) grows using as reference $e_{cbm}$ ($e_{vbm}$). There are deep interface states within the bandgap of the QDs and the width of the band of interface states fluctuates with QD size because the QDs are not perfectly circular and exhibit mixed types of edges. Figure \[fig:wf\] shows the squared wavefunctions of a $10$ nm wide ML-BPQD. The six lowest (highest) confined states in the conduction (valence) band exhibit an increasing number of nodes compatible with two-dimensional quantum confinement with anisotropic effective masses in both conduction and valence bands, whereas the effective masses of electrons and holes in the zigzag ($y$) direction are larger than the ones in the armchair direction. The size-dependent single-particle bandgap of isolated ML-QDs is shown in Figure \[fig:gap\] (black symbols). This quantity can be fitted with the following power law: $$\label{eq:gap} E_g(R) = E_g^{(bulk)} + \frac{0.7641}{R^{1.41}},$$ where energies and sizes are in eV and nm units, respectively. Within the effective mass approximation (EMA) framework, it would be expected a size dependence of the type $E_{g} \propto R^{-2}$. The discrepancy between exponents reveals that EMA is not suitable to model the size-dependent bandgap of BPQDs because their actual confinement barrier is not infinite. From the single-particle gap, one can perturbatively estimate the excitonic gap as $E_X = E_g - E_B$, where $E_B=J_{e_1h_1,e_1h_1}$ is the exciton binding energy of the $(e_1,h_1)$ pair. The effect of the substrate is included in the dielectric screening model of Equation (\[eq:dielectric\]). We have adopted three different substrates: vacuum ($\varepsilon_{vac} = 1$), SiO$_2$ ($\varepsilon_{SiO_2} = 3.9$) and Si ($\varepsilon_{Si} = 11.7$). The size-dependent excitonic gaps (shown in top panel of Figure \[fig:gap\]) of ML-QDs deposited on Si (red symbols) and SiO$_2$ (magenta symbols) substrates are, respectively, well fitted by the following expressions: $$\label{eq:exsi_a} E_X^{(Si)}(R) = 1.69 + \frac{0.6713}{R^{1.41}},$$ $$\label{eq:exsio2_a} E_X^{(SiO_2)}(R) = 1.59 + \frac{0.4415}{R^{1.82}},$$ but the excitonic gap $E_X^{(vac)}(R)$ for isolated ML-QDs (in vacuum, blue symbols) seems to exhibit two size-dependent regimes and cannot be fitted by a single power law. The size dependence of $E_B$ (bottom panel of Figure \[fig:gap\]) evidences the strong effect of the substrate on the excitonic gap. In vacuum, $E_B$ varies from $1.1$ eV to $0.47$ eV, when the QD size reduces from $1$ nm to $10$ nm of diameter. For ML-QDs deposited on SiO$_2$ (Si), the binding energies reduces from $0.7$ eV ($0.39$ eV) to $0.25$ eV ($0.11$ eV) for the same size variation. The trend $E_B^{(vac)}>E_B^{(SiO_2)}>E_B^{(Si)}$ is explained by the fact that the electron-hole interaction is inversely proportional to the dielectric constant of the substrate (see Equation (\[eq:dielectric\])). Assuming that the calculations of $E_X(R) = E_g(R)- E_B(R)$ up to $R = 5$ nm are sufficient to capture the bulk behavior when fitting the datasets with $E_X(R) = E_X^{(bulk)} + A/R^n$, we can compare the calculated excitonic gaps with recent photoluminescence (PL) measurements in bulk monolayer BP with the fitted values of $E_X^{(bulk)}$ in Equations (\[eq:exsi\_a\]) and (\[eq:exsio2\_a\]). Zhang *et al.* reported a PL peak at $1.67$ eV for monolayer BP deposited on quartz (same dielectric constant of SiO$_2$) [@zhang2016], and Li *et al.* reported a PL peak at $1.73$ eV for BP deposited on sapphire (same dielectric constant of Si) [@li2017], as shown by the dashed lines in red and magenta in the top panel of Figure \[fig:gap\], respectively. If we compare the PL peak of Li (Zhang) at $1.73$ eV ($1.67$ eV) with our fitted $E_X^{(bulk)}$ value of $1.69$ eV ($1.59$ eV), we obtain a difference of $0.04$ eV ($0.08$ eV) that corresponds to errors of $\approx2.3\%$ ($\approx4.8\%$). This is a strong evidence of the robustness of our TB approach. Thus, the bulk estimates of the exciton binding energies in ML-BP are $E_B^{(Si)} = 0.15$ eV and $E_B^{(SiO_2)} = 0.25$ eV. When comparing the size-dependence of the exciton interaction and quantum confinement energies (through the ratio $\beta = E_B/E_{conf}$) for the different substrates (see the inset of bottom panel in Figure \[fig:gap\]), the transition from strong ($\beta<1$) to weak quantum ($\beta>1$) confinement regimes occurs at different sizes, depending on the type of substrate. For ML-BPQDs in vacuum, strong confinement regime only occur for very tiny QDs ($R\leq 1$ nm). As the size of isolated QDs increases, the exciton interaction becomes much stronger than the quantum confinement (eg. $\beta \approx 10$ for $R = 5$ nm). On the other hand, the transition from strong to weak confinement regime occur $R \approx 5$ nm for QDs deposited in Si. Even for large sizes ($R = 5$ nm), the quantum confinement energy is still moderately large compared to the exciton interaction ($\beta \approx 2$). This explain why the exponent of the size-dependence of $E_X^{(Si)}(R)$ is the same of the single-particle gap $E_g(R)$. In the case of SiO$_2$, the relatively low dielectric constant causes the strong-to-weak confinement transition to occur at $R \approx 2$ nm. For large $R$, the exciton interaction quickly becomes dominant ($\beta = 5$, for $R=5$ nm). In this case, the size-dependence exponent $E_X^{(SiO_2)}(R)$ becomes different from the exponent of the size-dependent single-particle gap. Excitonic spectra ----------------- Figure \[fig:exc\] shows the size-dependent excitonic spectra of BPQDs, where the bandgap interface states were disregarded. Those spectra were calculated using the CI formalism (six states from conduction and valence bands) considering the possibility of pure (blue lines) and mixed (red lines) exciton configurations. In the mixed configuration, the exciton states are formed by a linear combination of electron-hole $(e_n,h_m)$ pairs, whereas in the pure configuration, only degenerate exciton states are allowed to mix. Furthermore, in the pure configuration, all GS excitons are formed by the single $(e_1,h_1)$ pair, while for the mixed configuration, the composition of the GS exciton is size- and substrate-dependent (see Table \[tab:composition\]). In the Si substrate, the GS exciton in the QD with $1$ nm of diameter is $99.7\%$ formed by the $(e_1,h_1)$ pair. In the QD with diameter of $10$ nm, the GS exciton is formed by the pairs $(e_1,h_1)$ ($77.6\%$) and $(e_2,h_2)$ ($13.8\%$). In the SiO$_2$ substrate, the exciton composition is more complex due to the enhanced Coulomb interaction (compared to Si substrate) that favor the participation of deeper conduction and valence states even in the GS exciton. For example, in small QDs (up to $2$ nm of diameter), the GS exciton is nearly $100\%$ formed by the $(e_1,h_1)$ pair. The contribution of this pair reduces as the QD size increases, being as low as $50\%$ for QDs of $10$ nm of diameter. The mixed CI method lowers the exciton band gap, as shown in the inset panels of Figure \[fig:exc\]. This reduction in energy depends on the dielectric constant of the substrate, being of the order of $0.02$ eV for Si, and $0.05$ eV for SiO$_2$. If we use the largest QD size ($10$ nm of diameter) as ruler to compare our calculations with the available experiments in bulk BP, this energy reduction makes the excitonic gap of BP on Si to agree even better with the $1.73$ eV PL peak of Li *et al.* [@li2017] (sapphire substrate), as compared to the perturbative excitonic gap of Equation (\[eq:exsi\_a\]) (for $R\rightarrow \infty$). On the other hand, the many body interactions included in the CI method improves very little the agreement of Equation (\[eq:exsio2\_a\]) (for $R\rightarrow \infty$) with the $1.67$ eV PL peak of Zhang *et al.* [@zhang2016] (quartz substrate). Unfortunately, up to now there are no experimental reports of ML-BP deposited in substrates with dielectric constants lower than $\varepsilon_{SiO_2}$ to compare with our calculations of isolated BPQDs. BPQDs display a rich and complex size-dependent excitonic structure, exhibiting dark and bright exciton states, where only bright excitons contribute to light-emitting processes. The squared dipole matrix elements $|D_\lambda|^2$ are shown in Figure \[fig:osc\]. A strong anisotropy associated to the orientation of light polarization is observed. The light polarization pointing to $x$ direction (parallel to the armchair direction) results in matrix elements $2$ orders of magnitude larger than for the light polarization in $y$ direction (parallel to zigzag direction). This is compatible with recent PL experiments in bulk BP, which demonstrated that its PL emission has no optical signal in $y$ direction [@li2017]. Besides that, the polarization in $x$ direction (bottom panels) exhibits strong optical activity in the lower part of the excitonic spectra, while the polarization in $y$ direction (top panels) exhibits weak optical activity at higher energies. Experimental absorption peaks are proportional to $|D_\lambda|^2$. Thus it is expected that the absorption peaks increase either with the QD size or with a reduction of the substrate dielectric constant. In Figure \[fig:exc\_compare\] we compare the fine structure of excitons in different substrates. For the case of the BPQD with $5$ nm of diameter (top panel), the GS exciton is the brightest one in both substrates. The first excited exciton states is dark and located $56$ meV and $52$ meV above the GS for SiO$_2$ and Si substrates, respectively. The next bright excitons are $94$ meV ($48\%$ weaker than the GS) and $90$ meV ($59\%$ weaker than the GS) above GS for SiO$_2$ and Si substrates, respectively. The BPQD with $9$ nm of diameter on SiO$_2$ exhibits almost doubly degenerated GS exciton ($\Delta E\approx 3$ meV), where the GS is bright and the first excited state is dark ($|D_0|^2 << |D_1|^2 \approx 10^1$). The next bright state is $13$ meV above (88% weaker than) the GS. For the BPQD on Si with $9$ nm of diameter, the separation between the two lowest bright states is $22$ meV, with the second bright exciton being 58% weaker than the GS. It is instructive to investigate the temperature dependence of the average excitons lifetime for light polarization in $x$ direction, shown in Figure \[fig:lifetime\]. At low temperatures, the exciton lifetime is inversely proportional to the BPQD size, while at room temperature this relationship becomes more complicated, probably because of changes in the QD interface as the size of the BPQDs grows, affecting the energy distribution and wavefunctions of excited single-particle states, and consequently, excitonic states. The low temperature dependence is in general dominated by the lifetime of the ground state exciton, which also exhibits an inversely proportional relationship with QD size (shown in the inset for Si (solid lines) and SiO$_2$ (dashed lines) substrates). The lifetime of excitons in small BPQDs are insensitive to temperature (see black curves in Figure \[fig:lifetime\] for dots with $2$ nm of diameter, respectively), while for larger BPQDs, the exciton lifetimes exhibit a monotonic increase with temperature. The substrate has a dramatic effect: the average exciton lifetime is inversely proportional to $\varepsilon_{sub}$. The exciton lifetimes for light polarization in $y$ direction are not shown here because they are many orders of magnitude larger than the lifetimes for $x$ polarization. \[sec:discussion\]Discussion and conclusions ============================================ BPQDs exhibit a size-dependence single-particle bandgap $E_g(R) - E^{bulk}_{g} \propto R^{-1.41}$ in disagreement with simple models based on the EMA, where an exponent $n = 2$ is expected even for two-dimensional QDs. For example, Si nanocrystals have been intensively investigated in the nineties by different atomistic methods [@lwwang; @hill1995; @ogut1997; @rohlfing1998; @delerue2000; @vasiliev2001], and those studies also found exponents $n < 2$ for the size-dependent Si nanocrystal bandgaps. This discrepancy between EMA and atomistic theories to explain size-dependence of the bandgap of quantum dots is well known. The exponent $n = 2$ arises from infinite confinement barriers (vanishing wavefunctions) to simplify boundary conditions. Exponents with $n < 2$ can be obtained if one considers finite confinement barriers. However, the exponent $n = 1.41$ seems to be related to mixture of border geometries in our circular BPQDs. This conclusion is based on the recent theoretical study of de Sousa *et al.* [@desousa2016] showing that the band gap of zigzag and armchair BP nanoribbons scales with $1/D$ and $1/D^2$ ($D$ is the width of the nanoribbon), respectively. Our exponent $n = 1.41$ is in qualitative agreement with the fact that BPQDs with mixed borders should exhibit an intermediate exponent between 1 and 2. When excitonic effects are taken into account, the bandgap strongly depends on the substrate. For Si and SiO$_2$, the excitonic gap (calculated perturbatively using a single-particle approach) obeys a single power law $E_X(R) - E^{bulk}_{g} \propto R^{-n}$, where the exponent $n$ is substrate-dependent. For isolated QDs (vacuum as substrate), the size-dependent excitonic gap seems to obey a combination of power laws to describe two different regimes of strong (small QDs) and weak (large QDs) quantum confinement. One can generalize the size dependence of the excitonic band gap of QDs with the simple expression $$\label{eq:gapgeneral} E_X(R) = E_g^{(bulk)} + \frac{A}{R^m} - \frac{B}{R^n},$$ where the second and third terms represent the power laws describing the quantum confinement and exciton binding energies, respectively. The parameters $A$, $B$, $m$ and $n$ depend on several factors like dimensionality of the quantum confinement, surface passivation, effectives masses, and dielectric mismatch between QD and the external materials. In the case of our unpassivated BPQDs, $E_g^{(bulk)}$, $A$ and $m$ are known (see Equation (\[eq:gap\])). However, some phenomenological assumptions can be made. For example, it is known that: (i) $m \leq 2$ ($m=2$ for infinite confinement barriers within EMA); (ii) $n\leq 1$; and (iii) $m>n$. It also known that $B^{-1} \propto \Gamma(\varepsilon_{in},\varepsilon_{out})$, where $\Gamma(\varepsilon_{in},\varepsilon_{out})$ represents a relationship describing the dielectric mismatch. In the most general form of Equation (\[eq:gapgeneral\]), the two size regimes are separated by a minimum point (see the vacuum case in Figure \[fig:gap\], and the vacuum and SiO$_2$ cases in Figure \[fig:exc\]). The appearance of this minimum point is unexpected compared to the single power law observed both theoretical and experimentally in many types of quantum confined structures. This minimum point $R_{min}$ is located at $R_{min}^{m-n} = (m/n)(A/B)$, and the double power law behavior disappears when $R_{min}\rightarrow \infty$. In the case of BP, we have $B \propto \varepsilon_{sub}^{-1}$. Therefore, the position of the minimum point is directly proportional to $\varepsilon_{sub}$ in qualitative agreement with Figure \[fig:exc\]. In Figure \[fig:gapgeneral\], we fit the excitonic gaps calculated with the CI method with Equation (\[eq:gapgeneral\]). We obtain that the parameter $B$ ($n$) is inversely (directly) proportional to $\varepsilon_{sub}$. The perturbative approach adopted in Section \[sec:singleparticle\] allows to estimate the ground state exciton of 1.59 and 1.69 for ML-BP deposited on SiO$_2$ and Si, respectively. Those results exhibit a remarkable agreement when respectively, compared to the measurements of Zhang (1.67 eV, quartz substrate) and Li (1.73 eV, sapphire substrate) [@zhang2016; @li2017]. The errors between theory and experiments are 0.08 eV for SiO$_2$ and 0.04 eV for Si substrates. The estimated bulk exciton binding energies are $E_B^{(SiO_2)} = 0.25$ eV and $E_B^{(Si)} = 0.15$. Zhang *et al.* used a simple TB model to explain their measurements [@zhang2016] (their bulk bandgap was $2.12$ eV), resulting in an exciton binding energy of $0.45$ eV (quartz substrate), which is $80\%$ larger than our estimate using SiO$_2$ as substrate ($\varepsilon_{quartz}\approx \varepsilon_{SiO_2}$). Li *et al.* explained their measurements with a simple TB model [@li2017], but using a bulk bandgap of $1.8$ eV, leading to a binding energy of $0.07$ meV for monolayer BP on Si substrate. Here, our estimated binding energy is $50\%$ larger than the value of Li *et al.*. Despite of those discrepancies, our method is in very good quantitative and qualitative agreement with those state-of-the-art measurements. The actual values of single-particle gaps and exciton binding energies are still under debate. Several theoretical and experimental reports in the literature use bulk bandgaps varying between $1.52$ eV and $2.12$ eV [@rudenko2014; @rudenko2015; @zhang2016; @li2017], and accurate values are necessary in order to determine actual values of exciton binding energies. We believe that the ten-parameter TB scheme of Rudenko *et al.* is, so far, the most accurate band structure description of BP in the literature [@rudenko2015]. In addition, another critical issue is the understanding of the role of dielectric screening in two-dimensional materials [@castellanos2015; @rodin2014; @cartoixa2005; @cudazzo2011; @berkelbach2013; @latini2015; @olsen2016]. The inclusion of many-body effects within the CI framework allows us to calculate a number of features which cannot be predicted by simple single-particle methods. We have calculated the excitonic spectra for BPQDs on different substrates and their optical properties. Several experimental studies reported an extraordinary dependence of the optical properties of BP with respect to the direction of light polarization with a rich set of optical resonances appearing for light polarisation in the armchair direction [@zhang2016; @li2017; @zhang2014b]. For example Li *et al*. reported strong PL and absorption signals polarized in $x$ direction (armchair direction) and no signal at all with $y$ polarisation (zigzag direction). This is in good qualitative agreement with the ratio of $10^2$ between the calculated squared dipole matrix elements for polarisation in $x$ and $y$ directions. Finally, Zhang *et al*. reported strong temperature dependence of the Raman phonon modes in few layer BP, which is consistent with the strong temperature dependence of the excitonic lifetimes of BPQDs with diameter larger than $4$ nm [@zhang2014b]. The single-particle perturbative approach provided good estimates to the excitonic gaps determined by the CI method. The agreement is particularly good for Si substrate. For SiO$_2$ substrate exhibits some discrepancies for diameter ranging between $3$ nm and $7$ nm. The size-dependent excitonic gaps of the CI method clearly exhibit a shape that resembles a sum of power laws, as in the case of isolated QDs (See Figure \[fig:gap\]). Using Equation (\[eq:gapgeneral\]) to fit all excitonic gaps calculated with CI method and extrapolating the results for very large sizes (see Figure \[fig:gapgeneral\]), the fitted gaps seem to converge to values very close to the measurements of Zhang and Li [@zhang2016; @li2017]. It is remarkable that QDs as large as $R = 5$ nm are still far from monolayer bulk behavior when deposited in substrates with very low dielectric constants. We remark that subtle effects like the coupling of charges in BP and substrate polarons induces broadening of the single-particle gap and renormalization of effective masses (specially on zigzag direction). For example, Mogulkoc have shown the single-particle gap of ML-BP deposited on SiO$_2$ are enlarged by 30 meV [@mogulkoc2016]. If such effects were included in our model, the agreement of our calculations using SiO$_2$ as substrate with the experimental results of Zhang [@zhang2016] would be even better. Another possible ingredient to improve the quantitative agreement with experimental measurements is the increase of the CI basis size with more than six electron and hole states, because the enhanced Coulomb interaction in substrates with low dielectric constants may mix even deep electron-hole pairs. Despite of the good agreement of our calculations if the limit of large BPQDs with experimental measurements in ML-BP, one might argue that our model does not take into account complicated edges effects. Liang *et al.* studied edges reconstruction in ML-BP combining scanning tunelling spectroscopy (STS) and theoretical methods based on the Density Functional Theory (DFT) [@liang2014]. They reported that most dangling bonds self-passivate such that the coordination number of phophorus increase from 3 (in the middle of BP layer) to 4 or 5 at the edges, depending on the type of edge geometry. They calculated the electrostatic potential in zigzag BP nanoribbons to account for local fields near the edges due to reconstruction of dangling bonds. They show that the edge reconstruction creates a localized short-range ($\approx$ 0.15 nm) confining potential of 0.15 eV at the edges of a BP layer in vacuum. For supported BP layers, as shown in our calculations, this local edge files would be inversely proportional to the dielectric constant of the substrate. Even for dielectric constants as low as the one of SiO$_2$, the edges contribution would represent a small perturbation compared to the actual size-dependent single-particle band gap of small BPQDs. For larger BPQDs, their effects should be negligible. On the other hand, passivation of the dangling bonds with other atomic species like hydrogen and oxygen due to the exposition of BPQDs to air are expected to eliminate interface states and lower band gaps [@garoufalis2006]. Anyhow, a clear picture of the effects of edge reconstruction and/or passivation in the excitonic properties of BPQDs is an open question that must be further investigated. In conclusion, we studied the excitonic interactions in ML-BPQDs with a realistic TB scheme to calculate single-particle states and the CI method to account for many-body effects. 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[cl]{} diameter & GS exciton composition\ \ 1 nm & $(e_1,h_1)$: 99.3%\ 2 nm & $(e_1,h_1)$: 97.6%\ 3 nm & $(e_1,h_1)$: 92.2% , $(e_2,h_2)$: 5.0%\ 4 nm & $(e_1,h_1)$: 56.5% , $(e_1,h_2)$: 16.2% , $(e_2,h_1)$: 11.5%, $(e_2,h_2)$: 8.7%\ 5 nm & $(e_1,h_1)$: 81.9% , $(e_4,h_6)$: 6.6%\ 6 nm & $(e_1,h_1)$: 74.6% , $(e_3,h_1)$: 5.0% , $(e_4,h_6)$: 9.8%\ 7 nm & $(e_1,h_1)$: 60.7% , $(e_2,h_2)$: 22.6% , $(e_3,h_3)$: 8.2%\ 8 nm & $(e_1,h_2)$: 33.7% , $(e_2,h_1)$: 19.6% , $(e_2,h_3)$: 22.1% , $(e_3,h_2)$: 8.4% , $(e_3,h_4)$: 10.2%\ 9 nm & $(e_1,h_2)$: 26.1% , $(e_2,h_1)$: 12.0% , $(e_2,h_3)$: 30.2%, $(e_3,h_2)$: 11.5%, $(e_3,h_4)$: 12.9%\ 10 nm & $(e_1,h_2)$: 19.4% , $(e_2,h_1)$: 10.0% , $(e_2,h_3)$: 28.8%, $(e_3,h_2)$: 10.6%, $(e_3,h_4)$: 17.1% , $(e_5,h_3)$: 5.1%\ \ 1 nm & $(e_1,h_1)$: 99.5%\ 2 nm & $(e_1,h_1)$: 98.4%\ 3 nm & $(e_1,h_1)$: 94.1%\ 4 nm & $(e_1,h_1)$: 76.4% , $(e_2,h_2)$: 10.7%\ 5 nm & $(e_1,h_1)$: 86.9%\ 6 nm & $(e_1,h_1)$: 79.9% , $(e_4,h_6)$: 6.5%\ 7 nm & $(e_1,h_1)$: 65.4% , $(e_2,h_2)$: 21.3% , $(e_3,h_3)$: 6.2%\ 8 nm & $(e_1,h_1)$: 70.2% , $(e_2,h_2)$: 18.9% , $(e_3,h_3)$: 5.3%\ 9 nm & $(e_1,h_1)$: 64.5% , $(e_2,h_2)$: 23.4% , $(e_3,h_3)$: 7.2%\ 10 nm & $(e_1,h_1)$: 50.0% , $(e_2,h_2)$: 28.3% , $(e_3,h_3)$: 11.2%\ \ 1 nm & $(e_1,h_1)$: 99.7%\ 2 nm & $(e_1,h_1)$: 99.2%\ 3 nm & $(e_1,h_1)$: 96.8%\ 4 nm & $(e_1,h_1)$: 87.7%, $(e_2,h_2)$: 7.8%\ 5 nm & $(e_1,h_1)$: 92.8%\ 6 nm & $(e_1,h_1)$: 88.3%\ 7 nm & $(e_1,h_1)$: 77.9%, $(e_2,h_2)$: 17.5%\ 8 nm & $(e_1,h_1)$: 80.0%, $(e_2,h_2)$: 11.7%\ 9 nm & $(e_1,h_1)$: 82.6%, $(e_2,h_2)$: 10.1%\ 10 nm & $(e_1,h_1)$: 77.6%, $(e_2,h_2)$: 13.8%\ {width=".8\textwidth"} {width=".6\textwidth"} {width=".7\textwidth"} {width=".75\textwidth"} {width=".9\textwidth"} {width=".75\textwidth"} {width=".75\textwidth"} {width=".85\textwidth"}

--- abstract: 'This paper presents a new feedback shift register-based method for embedding deterministic test patterns on-chip suitable for complementing conventional BIST techniques for in-field testing. Our experimental results on 8 real designs show that the presented approach outperforms the bit-flipping approach by 24.7% on average. We also show that it is possible to exploit the uneven distribution of don’t care bits in test patterns in order to reduce the area required for storing deterministic test patterns more than 3 times with less than 2% fault coverage drop.' author: - | Nan Li       Elena Dubrova\ Royal Institute of Technology (KTH), Forum 120, 164 40 Kista, Sweden\ {nan3,dubrova}@kth.se bibliography: - 'bib.bib' title: 'Embedding of Deterministic Test Data for In-Field Testing' --- BIST, top-off test patterns, feedback shift register, NLFSR, in-filed testing. Introduction ============ Large test data volume is widely recognized as a major contributor to the testing cost of integrated circuits [@WaC05]. The test data volume in 2017 is expected to be 10 times larger than the one in 2012 [@ITRS]. On the contrary, the size of the Automatic Test Equipment (ATE) memory is expected to grow only twice [@ITRS]. A number of efficient on-chip test compression techniques have been proposed as a solution for reducing ATE memory requirements, including [@WaC05; @KoB01; @RaTKM04; @MiK04; @CzM11]. A test set for the circuit under test is compressed to a smaller set, which is stored in ATE memory. An on-chip decoder is used to generate the original test set from the compressed one during test application. Test compression has already established itself as a mainstream design-for-test methodology for manufacturing testing [@CzM11]. However, it cannot be used for in-field testing where ATE is not available [@MaA10]. For in-field testing, Built-In Self Test (BIST) including use of JTAG is applied, in which either pseudo-random test patterns are generated within the system or pre-computed deterministic test patterns are stored in system memory [@MaA10]. In terms of test application time and fault coverage, deterministic test patterns are obviously more effective than pseudo-random ones. The fault coverage achieved with pseudo-random test patterns can be as low as 65% [@DaT00]. Several methods for increasing BIST test coverage have been proposed, including modification of the circuit under test [@EiL83], insertion of control and observe points into the circuit [@RaTKM04], modification of the LFSR to generate a sequence with a different distribution of 0s and 1s [@ChM84], embedding of deterministic test patterns into LFSR’s patterns by LFSR re-seeding [@Jer08] or bit-flipping [@WuK96], or storing them in an on-chip memory [@SaDB84]. The idea of complementing pseudo-random patterns with deterministic patterns is particularly attractive because the deterministic patterns can also solve the problem with transition or delay faults which are not handled efficiently by the pseudo-random patterns. However, the area required to store deterministic test patterns within the system can be prohibitively high. For example, the memory required to store them may exceed 30% of the memory used in a conventional ATPG approach [@HeF99]. In this paper, we propose a new method for embedding deterministic test patterns on-chip suitable for complementing conventional techniques for in-field testing. We generate deterministic test patters using a structure known as [*binary machine*]{}. This name was introduced by S. Golomb in his seminal book [@Golomb_book]. Binary machines can be considered as a more general type of Non-Linear Feedback Shift Registers (NLFSRs) [@Ja89] in which every stage is updated by its own feedback function. Binary machines are typically smaller and faster than NLFSRs generating the same sequence. For example, consider the 4-stage NLFSR with the feedback function $$f(x_0,x_1,x_2,x_3) = x_0 \oplus x_3 \oplus x_1 \cdot x_2 \oplus x_2 \cdot x_3,$$ where “$\oplus$” is the XOR (addition modulo 2), “$\cdot$” is the AND, and $x_i$ is the variable representing the value of the stage $i$, $i \in \{0,1,2,3\}$. If this NLFSR is initialized to the state $(x_0 x_1 x_2 x_3) = (1000)$, it generates the output sequence $$\label{bs} (1,0,0,0,1,1,0,1,0,1,1,1,1,0,0)$$ with the period 15. The same sequence can be generated by the 4-stage binary machine with the feedback functions $$\begin{array}{lcl} f_3(x_0,x_3) & = & x_0 \oplus x_3 \\ f_2(x_1,x_2,x_3) & = & x_3 \oplus x_1 \cdot x_2 \\ f_1(x_2) & = & x_2 \\ f_0(x_1) & = & x_1. \end{array}$$ We can see that the binary machine uses 3 binary operations, while the NLFSR uses 5 binary operations. Furthermore, the depth of feedback functions of the binary machine is smaller than the depth of the feedback function of the NLFSR. Thus, the binary machine has a smaller propagation delay than the NLFSR. While binary machines can potentially be smaller and faster than NLFSRs, the search space for finding a best binary machine for a given sequence is much larger than the corresponding one for NLFSRs. Algorithms for constructing binary machines were presented in [@Du10aj; @Du11a]. Both algorithms result in binary machines with the minimum number of stages for a given binary sequence. However, they do not minimize the circuit complexity of feedback functions. For Finite State Machines (FSM), it is known that an FSM with a non-minimal number of stages, e.g. encoded using one-hot encoding, often has a smaller total size than an FSM with a minimal number of stages [@MiBS85]. In this paper, we present an algorithm with constructs binary machines with a non-minimal number of stages. Our experimental results show that binary machines constructed by the presented algorithm are 63.28% smaller on average compared to the one constructed by the algorithm [@Du11a]. The presented algorithm is particularly efficient for incompletely specified sequences, which are important for testing. The rest of the paper is organized as follows. Section \[bm\] gives an introduction to binary machines. Section \[sa1\], describes the new algorithm for constructing binary machines. Section \[exp\] presents the experimental results. Section \[con\] concludes the paper and discusses open problems. Binary Machines {#bm} =============== An $n$-stage binary machine consists of $n$ binary storage elements, called [*stages*]{} [@Golomb_book]. Each stage $i \in \{0,1, \ldots,n-1\}$ has an associated [*state variable*]{} $x_i \in \{0,1\}$ which represents the current value of the stage $i$ and a [*feedback function*]{} $f_i: \{0,1\}^n \rightarrow \{0,1\}$ which determines how the value of $x_i$ is updated (see Figure \[bin\_machine\]). A [*state*]{} of a binary machine is a vector of values of its state variables. At every clock cycle, the next state of a binary machine is determined from its current state by updating the values of all stages simultaneously to the values of the corresponding feedback functions. An $n$-stage binary machine has $2^n$ states corresponding to the set $\{0,1\}^n$ of all possible binary $n$-tuples. ![The general structure of an $n$-stage binary machine.[]{data-label="bin_machine"}](figure_bm-crop.pdf){width="3.5in"} The [*degree of parallelization*]{} of an $n$-stage binary machine, $k$, is the number of output bits generated at each clock cycle, $1 \leq k \leq n$. The [*dependence set*]{} of a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is defined by $$dep(f) = \{j \ | \ f(X)|_{x_j=0} \not = f(X)|_{x_j=1}\},$$ where $f(X)|_{x_j=k} = f(x_0, \ldots, x_{j-1}, k, x_{j+1}, \ldots, x_{n-1})$ for $k \in \{0,1\}$. The [*Algebraic Normal Form (ANF)*]{} [@CuS09] of a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ (also called [*Reed-Muller canonical form*]{} [@Gr91]) is an expression in the Galois Field or order 2, $GF(2)$, of type $$f(x_0, x_1,\ldots,x_{n-1}) = \sum_{i=0}^{2^n-1} c_i \cdot x_0^{i_0} \cdot x_1^{i_1} \cdot \ldots \cdot x_{n-1}^{i_{n-1}},$$ where $c_i \in \{0,1\}$ are constants and $(i_0 i_1 \ldots i_{n-1})$ is the binary expansion of $i$. Related Work ============  \[prev\] The first algorithm for constructing a binary machine with the minimum number of stages for a given binary sequence was presented in [@Du10aj]. This algorithm exploits the unique property of binary machines that [*any*]{} binary $n$-tuple can be the next state of a given current state. The algorithm assigns every 0 of a sequence a unique even integer and every 1 of a sequence a unique odd integer. Integers are assigned in an increasing order starting from 0. For example, if an 8-bit sequence 00101101 is given, the sequence of integers 0,2,1,4,3,5,6,7 can be used. This sequence of integers is interpreted as a sequence of states of a binary machine. The largest integer in the sequence of states determines the number of stages. In the example above, $\lceil \log_2 7 \rceil = 3$, thus the resulting binary machine has 3 stages. The feedback functions $f_0, f_1, f_2$ implementing the resulting current-to-next state mapping are derived using the traditional logic synthesis techniques [@espr]. Note that, in general, any permutation of integers can be used as a sequence of binary machine’s states, as long as the selected integer modulo 2 is equal to the corresponding bit of the output sequence. Different state assignments result in different feedback functions. The size of these functions may vary substantially. In [@Du11a], the algorithm [@Du10aj] was extended to binary machines generating $k$ bits of the output sequence per clock cycle. The main idea is to encode a binary sequence into an $m$-ary sequence which can be generated in a simpler way. As an example, suppose that we use the 4-ary encoding (00) = 0, (01) = 1, (10) = 2, (11) = 3 to encode the binary sequence 00101101 from the example above into the quaternary sequence 0231. Then, we can construct a parallel binary machine generating 00101101 2-bits per clock cycle with a sequence of states 0, 2, 3, 1. Note that $\lceil \log_2 3 \rceil = 2$, so the resulting parallel binary machine has one stage less than the binary machine constructed above. This is surprising taking into account that all existing techniques for the parallelization of LFSRs [@PeZ92; @MuS06] and NLFSRs [@canniere-trivium; @hell-grain] have area penalty. In was shown in [@Du11a] that, for random sequences, parallel binary machines can be an order of magnitude smaller than parallel LFSRs or NLFSRs generating the same sequence. Synthesis of binary machines {#sa1} ============================ The problem of finding a best binary machine for a given sequence can be divided into three sub-problems: 1. Selecting an optimal degree of parallelization for a given binary sequence. 2. Choosing an optimal state assignment for a given degree of parallelization. 3. Finding a best circuit for feedback functions for a given state assignment. Optimal degree of parallelization --------------------------------- The degree of parallelization determines how many output bits are generated per clock cycle. The size of binary machines may differ substantially for different parallelization degrees. The degree of parallelization is [*optimal*]{} if it minimizes the size of the resulting binary machine. In order to construct a binary machine with the degree of parallelization $p$, we map a binary sequence into an $2^p$-ary sequence by partitioning the binary sequence into vectors of length $p$. The resulting vectors are treated as binary expansions of elements of an $2^p$-ary sequence. The same approach was used in [@Du11a]. Let us denote by $N_i$ the number of occurrences of a digit $i$ in the $2^p$-ary sequence, $0 \leq i < 2^p$. Let $N_{max}$ be the largest of $N_i$. In [@Du11a], it was shown that the minimum number of stages in a binary machine generating a given binary sequence with the degree of parallelization $p$ is equal to $$\label{min_st} k = \lceil log_2 N_{max} \rceil + p.$$ From (\[min\_st\]) we can see that if $N_{max} = 1$, then $k = p$. Such a case is called [*full parallelization*]{}. On the base of our experimental results, we hypothesise that the optimal degree of parallelization belongs to the interval $$\label{opt_st} 1 \leq p_{opt} \leq \lceil \log_2 n \rceil$$ where $n$ is the sequence length. Note that for some applications, including testing, the degree of parallelization is specified by the user. For example, for testing it is equal to the number of scan chains. Optimal state assignment ------------------------ A state assignment determines a sequence of states which a binary machine follows. Different sequences of states give raise to different current-to-next state mappings and, thus, to different updating functions. The state assignment is [*optimal*]{} if it minimizes the size of the resulting binary machine. Since a binary machine is a deterministic finite state automaton, any current state has a unique next state. For a given $2^p$-ary encoding, the minimal number of bits which has to be added to $p$-tuples to make the current-to-next state mapping unique is $\lceil log_2 N_{max} \rceil$. The minimal number of stages in the resulting binary machine is given by (\[min\_st\]). The strategy for state assignment presented in this paper has two major differences from the one in [@Du11a]. First, we use a non-minimal number of stages, namely $$k \geq \lceil log_2 \frac{n}{p} \rceil + p.$$ Second, we assign states so that the feedback functions implementing the current-to-next state mapping depend on the minimum number of state variables. It is known that a Boolean function of $k$ variables needs $O(2^k/k)$ gates to be implemented (Shannon-Lupanov bound) [@We87]. Feedback functions of binary machines are random functions. For random functions, their actual size is very close to the upper bound. So, each extra variable nearly doubles the size of the function. In our method, the feedback functions of an $(m+p)$-stage binary machine depend on $m = \lceil log_2 \frac{n}{p} \rceil$ variables only. In [@Du11a], the feedback functions can potentially depend on all state variables. The pseudocode of the presented state assignment algorithm is shown as Algorithm \[alg1\]. The input of the algorithm in a binary sequence $A = (a_0, a_1, \ldots, a_n)$ and the desired degree of parallelization $p$. The output is a sequence $S = (s_0, s_1, \ldots, s_{r-1})$ of binary vectors $s_i = (s_{i,0}, s_{i,1}, \ldots, s_{i,p+m-1}) \in \{0,1\}^{p+m}$, where $r = \lceil n/p \rceil$ and $m = \lceil log_2 r \rceil$, corresponding to the states of an $(p+m)$-stage binary machine generating $A$ with the degree of parallelization $p$. The algorithm partitions $A$ into $p$-tuples and appends at the beginning of each $i$th $p$-tuple $m$ extra bits. These extra bits correspond to the binary expansion of the $i$th element of the permutation vector $\Pi$. Next, we define a mapping $s_i \mapsto s_{i+1}$, for all $i \in \{0,1,\ldots,r-2\}$. Since $\Pi$ is a permutation, each state in the resulting sequence of states has a unique next state, so the mapping is well-defined. The last state $s_{r-1}$ and each of the $2^{p+m}-r$ remaining states of the resulting binary $(p+m)$-stage machine are mapped to don’t cares values. This gives us the possibility to specify the functions $f_0, f_1, \ldots, f_{p+m}$ implementing the current-to-next state mapping in a way which minimizes their size. Since $r \leq 2^m$, we can treat them as functions depending on the first $m$ variables only. This is very important, because, as we mentioned above, for random functions, the size nearly doubles with each extra variable. Since, by construction, the first $p$ bits of each state $s_i$ in $S = (s_0, s_1, \ldots, s_{r-1})$ correspond to the $i$th $p$-tuple of $A$, the resulting binary machine generates $A$ with the degree of parallelization $p$. [$r := \lceil n/p \rceil$ $m := \lceil log_2 r \rceil$ $\Pi := (\pi_0, \pi_1, \ldots, \pi_{2^m-1})$ is a permutation of $(0,1,\ldots,2^m-1)$ Let $\pi_{i,j}$ be the $j$th element of the binary expansion of $\pi_i$, $j \in \{0,\ldots,m-1\}$ $s_{i,j} := a_{i*p+j}$ $s_{i,p+k} := \pi_{i,k}$ $s_i := (s_{i,0}, s_{i,1}, \ldots, s_{i,p+m-1})$ Return $S = (s_0, s_1, \ldots, s_{r-1})$ ]{} As an example, let us construct a binary machine which generates the following 20-bit binary sequence with the degree of parallelization 2: $$A = (0,0,1,1,0,1,1,1,0,0,1,0,1,1,1,0,1,1,0,0).$$ Since $n = 20$ and $p = 2$, we get $r = 10$ and $m = 4$. Suppose we use the following permutation of $(0,1,\ldots,15)$: $$\Pi = (1, 8, 4, 2, 9, 12, 6, 11, 5, 10, 13, 14, 15, 7, 3, 0)$$ Then, we get the following sequence of states: $$\begin{array}{l} S = (000100, 100011, 010001, 001011, 100100, 110010, \\ ~~~~~ 011011, 101110, 010111, 101000) \end{array}$$ The functions implementing the resulting current-to-next state mapping have the following defining table: $$\begin{tabular}{|c|c@{}c@{}c@{}c@{}c@{}c|} \hline $x_5 x_4 x_3 x_2$ & $f_5$ & $f_4$ & $f_3$ & $f_2$ & $f_1$ & $f_0$ \\ \hline 0~0~0~1 & 1&0&0&0&1&1 \\ 1~0~0~0 & 0&1&0&0&0&1 \\ 0~1~0~0 & 0&0&1&0&1&1 \\ 0~0~1~0 & 1&0&0&1&0&0 \\ 1~0~0~1 & 1&1&0&0&1&0 \\ 1~1~0~0 & 0&1&1&0&1&1 \\ 0~1~1~0 & 1&0&1&1&1&0 \\ 1~0~1~1 & 0&1&0&1&1&1 \\ 0~1~0~1 & 1&0&1&0&0&0 \\ 1~0~1~0 & -&-&-&-&-&- \\ \hline \end{tabular}$$ where “-” stands for a don’t care value. Recall that the functions depend of the four variables $x_5, x_4, x_3, x_2$ only. The remaining 6 input assignments are mapped to don’t cares. We can implement the above functions as: $$\begin{array}{l} f_5 = x_2 \oplus x_3\\ f_4 = x_5\\ f_3 = x_4\\ f_2 = x_3\\ f_1 = x_2 \oplus x_4\\ f_0 = (x_2 \oplus x_3)' \oplus x'_3 x'_3 x'_3\\ \end{array}$$ where “$'$” stands for a complement. It is important to use permutations $\Pi$ which have a low-cost implementation. Examples of such permutations are sequences of states generated by counters, LFSRs, or NLFSRs with simple feedback functions [@Du12]. In the example above, we used the sequence of states of an LFSR with the generator polynomial $1+x+x^4$. Best circuit for feedback functions ----------------------------------- The problem of finding a best circuit for a given Boolean function is known to be notoriously hard. The exact solutions are known only for up to five variable functions [@Kn98]. However, there are many powerful heuristic algorithms for multi-level circuit optimization which are capable of finding good circuits for larger functions [@espr]. We optimize feedback functions using UC Berkeley’s tool ABC [@abc]. Our experimental results show that, even for random functions, ABC is capable of reducing the size of the original, non-optimized circuit by 30% on average. Experimental Results {#exp} ==================== Comparison to previous BM synthesis algorithms ---------------------------------------------- In the first experiment, we compared the presented algorithm to the algorithm [@Du11a]. Using both algorithms, we constructed binary machines for random sequences of length $2^{20}$ with a different number of don’t care bits. The results are summarized in Table \[ta1\] and Figure \[ff\]. As we can see, the presented algorithm is significantly more efficient than the algorithm [@Du11a] for sequences with many don’t cares. For the case of 99% don’t cares, it outperforms the algorithm [@Du11a] by 93.4%. ------------ ---------------------- ------------------ ------- % of don’t care bits Alg. [@Du11a], $G_1$ Presented, $G_2$ 0 303307 243734 19.64 25 311528 203615 34.64 50 313591 150919 51.87 60 308210 127683 58.57 70 295038 101134 65.72 80 275313 72440 73.69 90 238762 39710 83.37 95 189995 21680 88.59 99 78323 5167 93.40 average 557118 107342 63.28 ------------ ---------------------- ------------------ ------- : Results for random sequences of length $2^{20}$.[]{data-label="ta1"} ![Results for random sequences of length $2^{20}$.[]{data-label="ff"}](bm_vs_fg_crop.pdf){width="3.5in"} --------- -------------- --------------- ----------- ---------------- ------------ ----------------- ----------------- ------- Bit-flipping Presented Name \# Gates \# Scan cells \# Faults \# Scan chains \# Top-off \# Gates, $G_1$ \# Gates, $G_2$ a 34113 2511 91834 128 207 5049 4565 9.59 b 22104 1726 66390 128 214 4969 4515 9.14 c 19621 1726 48920 128 40 973 472 51.52 d 21984 1727 66554 128 211 5003 4470 10.65 e 19275 1727 50076 128 66 1224 684 44.11 f 39277 3022 89108 128 133 3820 3352 12.25 g 31726 2690 83936 128 213 5125 4674 8.80 h 29418 2690 67654 128 40 1191 577 51.55 average 3420 2880 24.7 --------- -------------- --------------- ----------- ---------------- ------------ ----------------- ----------------- ------- --- ------- ----- ---------- ----------------- ------- ----- ---------- ----------------- ------ % Test Presented % Test Presented Coverage \# Gates, $G_2$ Coverage \# Gates, $G_3$ a 34113 207 99.98 4565 13.29 102 98.21 2004 5.87 b 22104 214 99.98 4515 20.28 63 98.30 1210 5.47 c 19621 40 99.98 472 2.24 40 99.98 427 2.17 d 21984 211 99.98 4470 20.18 67 99.13 1212 5.51 e 19275 66 99.98 684 3.38 66 99.98 647 3.36 f 39277 133 99.99 3352 8.45 100 99.46 2049 5.22 g 31726 213 99.98 4674 14.63 92 99.89 1771 5.58 h 29418 40 99.99 577 1.85 40 99.99 542 1.84 --- ------- ----- ---------- ----------------- ------- ----- ---------- ----------------- ------ Comparison to previous approaches for embedding deterministic test patterns --------------------------------------------------------------------------- In the second experiment, we compared the presented algorithms to the bit-flipping approach for embedding deterministic test patterns which, in our opinion, is one of the most efficient ones [@WuK96]. The results presented in this section were obtained using our implementation of the bit-flipping algorithm. We applied both algorithms to 8 real designs with the number of gates varying from 19K to 39K. The results are summarized in Table \[ta2\]. We first applied 9000 pseudo-random patterns to all designs. Then, we computed the top-off patterns required to reach maximum achievable stuck-at faults coverage using a commercial ATPG tool. We used bit-flipping and the presented algorithms to represent these top-off patterns. As we can see from Table \[ta2\], the presented approach outperforms the bit-flipping approach by 24.7% on average. The difference in the number of gates required in both approaches can be up to 51.5%. What is even more important, the area overhead of the presented approach goes down as the number of scan chains grows. On the contrary, the area overhead of the bit-flipping approach goes up (see Figure \[fs\]). However, in spite of the improvements, the percentage of the overall chip area required to store deterministic test patterns can be prohibitively high for some designs (see column 6 of Table \[ta3\]). It is known that the size of representation for a data is related to the entropy of data [@Sh48]. Entropy puts a theoretical limit on the size of the minimal representation that can be achieved. If a lower fault coverage is acceptable, then the area overhead can be reduced by exploiting the fact that don’t care bits are normally unevenly distributed among test patterns. As an example, consider the diagram in Figure \[dma\]. Each point on this diagram shows the number of don’t care bits in a test pattern of [*dma*]{} benchmark (in total 411 patterns of length 1720 bits each). These test patterns were generated by a commercial ATPG tool with dynamic compaction turned on and random fill turned off. They cover 100% of detectable stuck-at faults. The total percentage of specified bits is 6.45%. We can see that only the first few test patterns are highly specified. If we chop off the first 5% of test patterns, the entropy of the remaining patterns reduces twice. Therefore, they can be represented with a twice smaller representation than the one required for the whole set of test patterns. By using the last 95% of test patterns, we can achive 95.7% test coverage for stuck-at faults. In Table \[ta3\] we show that, by using a subset of the top-off patterns only, we can reduce the area required for their representation more than 3 times in some cases, while sacrificing the fault coverage by less than 2%. ![Area overhead as a function of the number of chains.[]{data-label="fs"}](chains_crop.pdf){width="3.5in"} ![Distribution of don’t care bits in the test patterns of [*dma*]{} benchmark.[]{data-label="dma"}](dma_crop.pdf){width="3.5in"} Conclusion {#con} ========== We presented a new method for embedding deterministic test patterns on-chip based on binary machines. The presented algorithm for synthesis of binary machines is significantly more efficient than previous work, especially for test data with many don’t cares. Our experimental results on 8 real designs show that the proposed approach outperforms the bit-flipping approach by 24.7% on average. We also show that it is possible to exploit uneven distribution of don’t care bits in test patterns to reduce the area required for generating top-off patterns more than 3 times with less than 2% decrease in fault coverage. We believe that the presented algorithm for synthesis of binary machines is quite close to an optimal. What can be improved in the proposed method is the strategy for selecting a subset of top-off patterns which maximizes the fault coverage and minimizes the area overhead. At present, we use a simple greedy algorithm which selects top-off patterns based on the number of don’t care bits and the number of covered faults. A more sophisticated approach is likely to bring better results. Binary machines can potentially be used for storing compressed test patterns for on-chip test compression techniques. This would eliminate the dependence of test compression on ATE memory. We are currently investigating the feasibility of such an approach on large industrial designs. Acknowledgement {#ack} =============== This work was supported in part by a research grant No 2011-03336 from the Swedish Governmental Agency for Innovation Systems VINNOVA.

--- abstract: 'We investigate the scalar Green function for spherically symmetric spacetimes expressed as a coordinate series expansion in the separation of the points. We calculate the series expansion of the function $V(x,x'')$ appearing in the Hadamard parametrix of the scalar Green function to very high order. This expansion is then used to investigate the convergence properties of the series and to estimate its radius of convergence. Using the method of Padé approximants, we show that the series can be extended beyond its radius of convergence to within a short distance of the normal neighborhood boundary.' author: - Marc Casals - Sam Dolan - 'Adrian C. Ottewill' - Barry Wardell title: Padé Approximants of the Green Function in Spherically Symmetric Spacetimes --- Introduction {#sec:intro} ============ Quasi-local series expansions – expansions in the separation of two points $x$ and $x'$ – are a frequently used tool for calculations of fields on curved spacetimes. Often, as a final step in the calculation, the coincidence limit, $x' \rightarrow x$, is taken. In these cases, the precise convergence properties of the series is of little interest. However, there are cases where we would like the points to remain separated [@Casals:Dolan:Ottewill:Wardell:2009; @Quinn:Wald:1997; @Quinn:2000; @Mino:Sasaki:Tanaka:1996; @DeWitt:1960]. In particular, we are motivated by the calculation of the *quasilocal* (QL) contribution to the scalar self-force [@Ottewill:Wardell:2008; @Ottewill:Wardell:2009] (for a review, see [@Poisson:2003; @Detweiler:2005]) on a scalar particle, $$\label{eq:QLSFInt} f^{a}_\mathrm{QL} (z(\tau))= \lim_{\epsilon \rightarrow 0} q^2 \int_{\tau - \Delta \tau}^{\tau-\epsilon} \nabla^{a} G_{ret} \left( z(\tau),z(\tau') \right) d\tau ',$$ where $z(\tau$) describes the worldline of the particle. This requires a quasi-local expansion of the retarded Green function, $G_{ret}(x,x')$, which is a solution of the scalar wave equation with point source, $$(\square - \xi R){G_{\text{ret}}}(x, x^{\prime}) = -4 \pi \frac{\delta^4(x^{\mu} - x^{\mu \prime})}{\sqrt{-g}}, \label{eq:gf-waveeq}$$ where $\xi$ is the curvature coupling constant and $R$ is the Ricci scalar. As expression requires the Green function for the points separated up to an amount $\Delta\tau$ along a world-line, it begs the question: how large can the separation of the points be before the series expansion is no longer a valid representation of the Green function? To the authors’ knowledge, this question has not yet been quantitatively answered. It is well known that the Hadamard parametrix for the Green function (upon which quasi-local calculations are based) is valid provided $x$ and $x'$ lie within a *normal neighborhood*[^1] [@Friedlander]. However, this does not necessarily guarantee that a series representation will be convergent everywhere within this normal neighborhood. In fact, we will show that the series is only convergent within a smaller region, the size of which is given by the *circle of convergence* of the series. However, this does not preclude the use of the quasi-local expansion to calculate the Green function *outside* the circle of convergence (but within the normal neighborhood). As we will show, Padé resummation techniques, which have been extremely successful in other areas [@Damour:Iyer:Sathyaprakash; @Porter:Sathyaprakash], are also effective in extending the series beyond its circle of convergence. In this paper, we will focus in particular on calculating the Green function for two spherically symmetric spacetimes: Schwarzschild and Nariai. The Nariai spacetime [@Nariai:1950; @Nariai:1951] arises naturally from efforts to consider a simplified version of Schwarzschild [@Casals:Dolan:Ottewill:Wardell:2009; @Cardoso:Lemos:2003; @Zerbini:Vanzo:2004]. It retains some of the key features of Schwarzschild (such as the presence of an unstable photon orbit and a similar effective radial potential which diminishes exponentially on one side), but frequently yields more straightforward calculations. This makes it an ideal testing ground for new methods which are later to be applied to the more complicated Schwarzschild case. In the present work, we will use the line element of the static region of the Nariai spacetime (with cosmological constant $\Lambda=1$ and Ricci scalar $R=4$) in the form $$ds^2 = -(1-{\rho}^2) dt^2 + (1-{\rho}^2)^{-1} d{\rho}^2 + d\Omega_2^2, \quad \quad \quad d\Omega_2^2=d\theta^2+\sin^2\theta d\phi^2 \label{eq:Nariai-le}.$$ where ${\rho}\in (-1,+1),t\in (-\infty,+\infty),\theta\in [0,\pi],\phi\in [0,2\pi)$. In this form, it yields a wave equation with potential which is seen to closely resemble that of the Schwarzschild metric, $$ds^2 = -\left(1-\frac{2M}{r}\right) dt^2 + \left(1-\frac{2M}{r}\right)^{-1} dr^2 + r^2 d\Omega^2_2, \label{eq:Schw-le}$$ In Sec. \[sec:WKB\] we use an adaptation of the Hadamard-WKB method developed by Anderson and Hu [@Anderson:2003] to efficiently calculate the coordinate series expansion of $V(x,x')$ to very high order for both Nariai and Schwarzschild spacetimes. In Sec. \[sec:convergence\] we use convergence tests to determine the radius of convergence of our series and show that, as expected, it lies within the convex normal neighborhood. We also estimate the local truncation error arising from truncating the series at a specific order. Using the method of Padé approximants, we show in Sec. \[sec:Pade\] how the domain of validity of the coordinate series can be extended beyond its radius of convergence to give an accurate representation of $V(x,x')$ to within a small distance of the edge of the normal neighborhood. Hadamard-WKB Calculation of the Green Function {#sec:WKB} ============================================== For the present quasilocal calculation, we need to consider the retarded Green function only for the points $x$ and $x'$ lying within a *normal neighborhood*. This allows us to express the retarded Green function in the Hadamard parametrix [@Hadamard; @Friedlander], $$\label{eq:Hadamard} G_{ret}\left( x,x' \right) = \theta_{-} \left( x,x' \right) \left\lbrace U \left( x,x' \right) \delta \left( \sigma \left( x,x' \right) \right) - V \left( x,x' \right) \theta \left( - \sigma \left( x,x' \right) \right) \right\rbrace ,$$ where $\theta_{-} \left( x,x' \right)$ is analogous to the Heaviside step-function (i.e. $1$ when $x'$ is in the causal past of $x$, $0$ otherwise), $\delta \left( \sigma\left( x,x' \right) \right)$ is the standard Dirac delta function, $U \left( x,x' \right)$ and $V \left( x,x' \right)$ are symmetric bi-scalars having the benefit that they are regular for $x' \rightarrow x$, and $\sigma \left( x,x' \right)$ is the Synge [@Synge; @Poisson:2003; @DeWitt:1965] world function (i.e., half the square of the geodesic distance). The term involving $U(x,x')$ is only non-zero for null connected points whereas the quasi-local self-force calculation which motivates us requires the Green function within the light-cone only. We will therefore only concern ourselves here with the calculation of the function $V(x,x')$. The fact that $x$ and $x'$ are close together suggests that an expansion of $V(x,x')$ in powers of the separation of the points, $$\label{eq:CoordGreen} V\left( x,x' \right) = \sum_{i,j,k=0}^{\infty} v_{ijk}(r) \left( t - t' \right)^{2i} \left( \cos \gamma - 1 \right)^j (r-r')^k,$$ where $\gamma$ is the angular separation of the points, may give a good representation of the function within the quasi-local region. Note that, as a result of the spherical symmetry of the spacetimes we will be considering, the expansion coefficients, $v_{ijk}(r)$, are only a function of the radial coordinate, $r$. Anderson and Hu [@Anderson:2003] have developed a Hadamard-WKB method for calculating these coefficients. They applied their method to the Schwarzschild case and subsequently found the coefficients to $14^{th}$ order using the *Mathematica* computer algebra system [@Anderson:Eftekharzadeh:Hu:2006]. In the present work, we adapt their method to allow for spacetimes of the Nariai form, (\[eq:Nariai-le\]). In particular, we consider a class of spacetimes of the general form $$\label{eq:diagonal-metric} ds^2 = -f(r) dt^2 + f^{-1}(r) dr^2 + g(r) \left( d\theta^2 + \sin^2\theta d\phi^2 \right),$$ where $f(r)$ and $g(r)$ are arbitrary functions of the radial coordinate, $r$, and previously Anderson and Hu had set $g(r) = r^2$, but allowed $g_{rr}$ and $g_{tt}$ to be independent functions of $r$. The form of the Nariai metric given in Eq. (\[eq:Nariai-le\]) falls into the class , with $f(r) = 1- r^2$ and $g(r) = 1$. For $f(r) = 1-\frac{2M}{r}$ and $g(r)=r^2$, this is the Schwarzschild metric of Eq. (\[eq:Schw-le\]). The method presented in this section differs from that of Ref. [@Anderson:2003] in the details of the WKB approach used, but otherwise remains very similar. Our alternative WKB approach, based on that of Refs. [@Howard:1985; @Winstanley:2007] proves extremely efficient when implemented in a computer algebra package. Following the prescription of Ref. [@Anderson:2003], the Hadamard parametrix for the real part of the Euclidean Green function (corresponding to the Euclidean metric arising from the change of coordinate $\tau = i t$) is [^2] $$\label{eq:HadamardEuclideanGF} \Re \left[ G_E (-i \tau,\vec{x};-i\tau',\vec{x}') \right]= \frac{1}{2\pi} \left( \frac{U(x,x')}{\sigma(x,x')} + V(x,x') \ln (|\sigma(x,x')|) + W(x,x') \right) ,$$ where $U(x,x')$, $V(x,x')$ and $W(x,x')$ are real-valued symmetric bi-scalars. Additionally, for the points $x$ and $x'$ separated farther apart in the time direction than in other directions, $$\sigma (x,x') = -\frac{1}{2} f(r) (t-t')^2 + O\left[(x-x')^3\right]$$ so the logarithmic part of Eq. (\[eq:HadamardEuclideanGF\]) is given by $$\label{eq:VlnTau} \frac{1}{\pi} V(x,x') \ln(\tau - \tau').$$ Therefore, in order to find $V(x,x')$, it is sufficient to find the coefficient of the logarithmic part of the Euclidean Green function. We do so by considering the fact that the Euclidean Green function also has the exact expression for the spacetimes of the form given in Eq. (\[eq:diagonal-metric\]): $$\label{eq:EuclideanGF} G_E (-i \tau,x;-i\tau',x') = \frac{1}{\pi} \int_{0}^{\infty} d\omega \cos \left[ \omega (\tau - \tau') \right] \sum_{l=0}^{\infty} (2l+1) P_l (\cos \gamma) C_{\omega l} p_{\omega l}(r_<) q_{\omega l}(r_>),$$ where $p_{\omega l}$ and $q_{\omega l}$ are solutions (normalised by $C_{\omega l}$) to the homogeneous radial equation for the scalar wave equation in the curved background (\[eq:diagonal-metric\]) (and where $r_< $, $r_>$ are the smaller/larger of $r$ and $r'$, respectively), along with the fact that $$\begin{aligned} \label{eq:WKB-int-log-relation} \int_\lambda^{\infty} d\omega \cos \left[ \omega (\tau - \tau') \right] \frac{1}{\omega^{2n+1}} &= \frac{(-1)^{n+1}}{(2n)!}(\tau-\tau')^{2n}\log\left(\tau-\tau'\right) + \cdots \nonumber\\ &= \frac{-1}{(2n)!}(t-t')^{2n}\log\left(\tau-\tau'\right)+\cdots ,\end{aligned}$$ where $\lambda$ is a low frequency cut-off justified by the fact that we will only need the $\log(\tau-\tau')$ term from the integral. We can therefore find $V(x,x')$ as an expansion in powers of the time separation of the points by expressing the sum, $$\label{eq:WKBsum} \sum_{l=0}^{\infty} (2l+1) P_l (\cos \gamma) C_{\omega l} p_{\omega l}(r_<) q_{\omega l}(r_>),$$ of Eq. (\[eq:EuclideanGF\]) as an expansion in inverse powers of $\omega$. This is achieved using a WKB-like method based on that of Refs. [@Howard:1985; @Winstanley:2007]. Given the form (\[eq:EuclideanGF\]) for the Euclidean Green function, the radial functions $S(r)=p_{\omega l}(r)$ and $S(r)=q_{\omega l}(r)$ must both satisfy the homogeneous wave equation, $$\label{eq:WKB-radial-eq} f \frac{d^2 S}{dr^2} + \frac{1}{g} \frac{d}{dr}(f g) \frac{dS}{dr} - \left[ \frac{\omega^2}{f} + \frac{l(l+1)}{g} + m_{\text{field}}^2 + \xi R\right] S = 0$$ where $m_{\text{field}}$ is the scalar field mass and $\xi$ is the coupling to the scalar curvature, $R$. Next, given the Wronskian $W(r) = C_{\omega l}(p_{\omega l}q_{\omega l}' - q_{\omega l}p_{\omega l}')$, its derivative is $$W' = C_{\omega l}(p_{\omega l}q_{\omega l}'' - q_{\omega l}p_{\omega l}'') = - \frac{1}{fg} (fg)' W$$ and the Wronskian condition is therefore $$\label{eq:WKBWronskian} C_{\omega l}(p_{\omega l}q'_{\omega l} - q_{\omega l}p'_{\omega l}) = -\frac{1}{fg}.$$ We now explicitly assume that $r>r'$ and define the function $$B(r,r') = C_{\omega l} p_{\omega l}(r') q_{\omega l}(r).$$ Since the sum of Eq. (\[eq:WKBsum\]) (and hence $B(r,r')$) is only needed as an expansion in powers of $(r-r')$, we expand $B(r,r')$ about $r'=r$ and (using Eq. (\[eq:WKB-radial-eq\]) to replace second order and higher derivatives of $B(r,r')$ with expressions involving $B(r,r')$ and $\partial_{r'}B(r,r')$) find that $$B(r,r') = \beta(r) + \alpha(r) (r'-r) + \left\{ \left[\frac{2 (\eta+\chi^2)}{(fg)^2}\right] \beta(r) - \left[\ln(fg) \right]' \alpha(r) \right\} \frac{(r'-r)^2}{2} + \cdots\qquad(\textrm{for\ }r > r')$$ where $$\begin{aligned} \beta(r) \equiv& \left[ B(r,r') \right]_{r' \to r^-} = C_{\omega l} p_{\omega l}(r) q_{\omega l}(r), & \alpha(r) \equiv \left[\partial_{r'}B(r,r')\right]_{r' \to r^-}=& C_{\omega l} p'_{\omega l}(r) q_{\omega l}(r)\end{aligned}$$ and $$\begin{aligned} \label{eq:eta_defn} \eta(r) &\equiv -\frac{1}{4}f g+\left(m_{\text{field}}^2+\xi R\right) f g^2\\ \label{eq:chi_defn} \chi^2(r) &\equiv \omega^2 g^2 + f g \left(l+\frac{1}{2}\right)^2 .\end{aligned}$$ It will therefore suffice to calculate $\beta(r)$ and $\alpha(r)$. Furthermore, using Eq.  we can relate $\alpha(r)$ to the derivative of $\beta(r)$, $$\alpha (r) = \frac{\beta'(r)}{2} + \frac{1}{2f(r)g(r)},$$ so it will, in fact, suffice to find $\beta(r)$ and its derivative, $\beta'(r)$. Using Eqs. (\[eq:WKB-radial-eq\]) and (\[eq:WKBWronskian\]), it is immediate to see that $\beta(r)$ must satisfy the nonlinear differential equation $$\label{eq:beta-ode} f g \frac{d}{dr}\left(f g \frac{d\sqrt{\beta}}{dr}\right)-\left(\eta +\chi ^2\right) \sqrt{\beta}+ \frac{1}{4 \beta^{3/2}}=0.$$ The short distance behaviour of the Green function is determined by the high-$\omega$ and/or high-$l$ behaviour of the integrand of Eq. , so we seek to express $\beta(r)$ as an expansion in inverse powers of $\chi$. To keep track of this expansion we may replace $\chi$ in Eq. (\[eq:beta-ode\]) by $\chi/\epsilon$ where $\epsilon$ is a formal expansion parameter which we eventually set to 1. Then, to balance at leading order we require $$\begin{aligned} (\chi/\epsilon)^2 \sqrt{\beta} \sim \frac{1}{4 \beta^{3/2}} \quad \implies \beta \sim \frac{\epsilon}{2 \chi} .\end{aligned}$$ We now write $$\begin{aligned} \label{eq:beta} \beta(r)&= \epsilon \beta_0(r) + \epsilon^2\beta_1(r) + \ldots\end{aligned}$$ where $\beta_0(r) \equiv 1/(2 \chi(r))$, insert this form for $\beta(r)$ in Eq. (\[eq:beta-ode\]), and solve formally order by order in $\epsilon$ to find a recursion relation for the $\beta_n(r)$. On doing so and using Eq. (\[eq:chi\_defn\]) to eliminate $(l+\frac12)^2$ in favour of $\omega^2$ and $\chi^2$ we find that we can write $$\begin{aligned} \label{eq:beta-n} \beta_n(r) = \sum\limits_{m=0}^{2n} \frac{A_{n,m}(r) \omega^{2m} }{\chi^{2n+2m+1}}\end{aligned}$$ so, for example, $$\begin{aligned} \beta_1(r) &=\frac{A_{1,0}(r)}{\chi^3}+\frac{A_{1,1}(r) \omega^2}{\chi^5}+\frac{A_{1,2}(r)\omega^4}{\chi^7}\\ \beta_2(r) &=\frac{A_{2,0}(r)}{\chi^5}+\frac{A_{2,1}(r) \omega^2}{\chi^7}+\frac{A_{2,2}(r)\omega^4}{\chi^9}+\frac{A_{2,3}(r) \omega^6}{\chi^{11}}+\frac{A_{2,4}(r)\omega^8}{\chi^{13}}. \end{aligned}$$ The recursion relations for $\beta_n(r)$ may then be re-expressed to allow us to recursively solve for the $A_{n,m}(r)$. Such a recursive calculation is ideally suited to implementation in a computer algebra system (CAS). Even on a computer, this recursive calculation becomes very long as $n$ becomes large and, in fact, dominates the time required to calculate the series expansion of $V(x,x')$ as a whole. For this reason, we have made available an example implementation in *Mathematica*, including precalculated results for several spacetimes of interest [@Hadamard-WKB-Code]. This code calculates analytic results for $A_{n,m}(r)$ on a moderate Linux workstation up to order $n\sim 25$, corresponding to a separation $|x-x'|^{50}$, in Schwarzschild and Nariai space-times in the order of 1 hour of CPU time. We also note at this point that knowledge of the $A_{n,m}(r)$ (and their $r$ derivative, which is straightforward to calculate) are all that is required to find the series expansion of $\beta'(r)$ and hence $\alpha(r)$. This can be seen by differentiating Eqs.  and with respect to $r$ to get $$\begin{aligned} \beta'(r) &= \epsilon \beta'_0 + \epsilon^2 \beta'_1 + \ldots,\end{aligned}$$ with $$\begin{aligned} \label{eq:betap-n} \beta'_n(r) &= \sum_{m=0}^{2n} \left[ \frac{A'_{n,m}(r) \omega^{2m} }{\chi^{2n+2m+1}} - (n+m+\frac12) \frac{2 \chi \chi' A_{n,m}(r)}{\chi^{2n+2m+3}} \right]\nonumber\\ &= \sum_{m=0}^{2n+1} \left\{ A'_{n,m}(r) - (n+m+\frac12) A_{n,m}(r)\frac{(fg)'}{fg} - (n+m-\frac12)A_{n,m-1}(r)\left[ (g^2)'-\frac{(fg)'}{fg}g^2\right]\right\} \frac{\omega^{2m} }{\chi^{2n+2m+1}}\end{aligned}$$ where we have used Eq.  to write $2 \chi \chi' = \chi^2 (fg)'/(fg) + \omega^2 ((g^2)' -g^2 (fg)'/(fg))$, and we use the convention $A_{n,-1}=A_{n,2m+1} = 0$. With the $A_{n,m}(r)$ and their first derivatives calculated, we are faced with the sum over $l$ in Eq. (\[eq:WKBsum\]), where Eqs.  and yield sums of the form $$\label{eq:WKB-simplified-sum} \sum_{l=0}^{\infty} 2 (l+{\textstyle\frac{1}{2}}) P_l (\cos \gamma) \frac{D_{n,m}(r) \omega^{2m}}{\chi^{2n+2m+1}}.$$ with $$D_{n,m} = \begin{cases} A_{n,m}(r)\\ A'_{n,m}(r) - (n+m+\frac12) A_{n,m}(r)\frac{(fg)'}{fg} - (n+m-\frac12)A_{n,m-1}(r)\left[ (g^2)'-\frac{(fg)'}{fg}g^2\right] \end{cases}$$ Since we are considering the points $x$ and $x'$ to be close together, we can treat $\gamma$ as a small quantity and expand the Legendre polynomial in a Taylor series about $\gamma=0$, or, more conveniently, in powers of $(\cos \gamma -1)$ about $(\cos \gamma -1)=0$. It is straightforward to express each term in this series as a polynomial in even powers of $(l+\frac{1}{2})$: $$\begin{aligned} P_l(\cos \gamma) = {}_2F_1\left(-l,l+1;1;(1-\cos\gamma)/2\right) =\sum_{p=0}^l \frac{\bigl((l+\frac12)^2-(1-\frac12)^2\bigr)\cdots \bigl((l+\frac12)^2-(p-\frac12)^2\bigr)}{2^p (p!)^2} (\cos\gamma-1)^p\end{aligned}$$ The calculation of the sum in Eq. (\[eq:EuclideanGF\]) therefore reduces to the calculation of sums of the form $$\label{eq:WKBsum2} 2 D_{n,m}(r) \sum_{l=0}^{\infty} \frac{(l+{\textstyle\frac{1}{2}})^{2p+1} \omega^{2m}}{\chi^{2n+2m+1}}.$$ For fixed $\omega$ and large $l$ the summand behaves as $l^{2(p-n-m)}$, and so only converges if $p<n+m$. If $p \geq n+m$, we first split the summand as $$\begin{aligned} \frac{(l+{\textstyle\frac{1}{2}})^{2p+1} \omega^{2m}}{\chi^{2m+2n+1}} =&\frac{(l+{\textstyle\frac{1}{2}})^{2(p-m-n)}\omega^{2m}}{(fg)^{m+n+1/2}}\left(1+\frac{\omega^2 g/f}{ \left(l+\frac{1}{2}\right)^2} \right)^{-m-n-1/2}\\ =&\frac{(l+{\textstyle\frac{1}{2}})^{2(p-m-n)}\omega^{2m}}{(fg)^{m+n+1/2}}\left\{ \sum_{k=0}^{p-m-n} \frac{(-1)^k(m+n+1/2)_k}{k!} \left(\frac{\omega^2 g/f}{ \left(l+\frac{1}{2}\right)^2} \right)^{k} +\right.\nonumber \\ &+ \left. \left[ \left(1+\frac{\omega^2 g/f}{ \left(l+\frac{1}{2}\right)^2} \right)^{-m-n-1/2}-\sum_{k=0}^{p-m-n} \frac{(-1)^k (m+n+1/2)_k}{k!} \left(\frac{\omega^2 g/f}{ \left(l+\frac{1}{2}\right)^2} \right)^{k} \right]\right\}\end{aligned}$$ where $(\alpha)_k=\Gamma(\alpha+k)/\Gamma(\alpha)$ is the Pochhammer symbol and the sum correspond to the first $(p-n-m)$ terms in the expansion of $ (1+x)^{-n-m-1/2}$ about $x\equiv(\omega^2 g/f)/(l+\frac12)^2=0$. The terms outside the square brackets correspond to positive powers of $\omega$ and so contribute to the light cone singularity, not the tail term $V(x,x')$ with which we are concerned in this paper. By contrast, the term in square brackets behaves as $(l+\frac12)^{-2}$ and so converges as $l\to\infty$ and will contribute to the tail term. We denote this term, with its prefactor as $$\begin{aligned} \label{eq:reg_integrand} \left[\frac{(l+{\textstyle\frac{1}{2}})^{2p+1} \omega^{2m}}{\chi^{2m+2n+1}}\right]_\mathrm{reg} &= \frac{\omega^{2(p-n)}}{f^{p+1/2}g^{2(n+m)-p+1/2}} x^{m+n-p} \left( \left(1+ x \right)^{-m-n-1/2} -\sum_{k=0}^{p-m-n} \frac{(-1)^k (m+n+1/2)_k}{k!} x^{k} \right)\end{aligned}$$ and adopt the understanding that the sum vanishes if $p<m+n$. To proceed further, we use the Sommerfeld-Watson formula [@Watson:1918], $$\sum_{l=0}^{\infty} F(l+{\textstyle\frac12}) = \Re \left[ \frac{1}{i} \int_\gamma dz \> F(z) \tan (\pi z) \right] = \int_{0}^{\infty} F\left(\lambda\right) d\lambda - \Re \left( i \int_{0}^{\infty} \frac{2}{1+e^{2\pi \lambda}} F\left(i \lambda \right) d\lambda \right)$$ which is valid provided we can rotate the contour of integration for $F(z) \tan(\pi z)$ from just above the real axis to the positive imaginary axis. Defining $z \equiv (f/g)^{1/2} \lambda/\omega = 1/\sqrt{x}$, where $\lambda\equiv (l+1/2)$, the sum can then be written as the contour integral, $$\label{eq:WKBintegrals} \frac{\omega^{2(p-n)+1}}{f^{p+1}g^{2(m+n)-p}}\left[ \int_{0}^{\infty} \mathrm{d}z\>\left[ \frac{ z^{2p+1} }{(1 +z^2)^{m+n+1/2}}\right]_\mathrm{reg} + (-1)^p \Re \left( \int_{0}^{\infty} \frac{2\mathrm{d}z}{1+e^{2\pi z \omega \sqrt{g/f}}} \frac{ z^{2p+1}}{(1 - z^2)^{m+n+1/2}} \right)\right] \ .$$ Note that there is no need to include the regularization terms in the second integral as their contribution is manifestly imaginary and so will not contribute to the final answer. For $p<m+n$ the first integral in Eq. (\[eq:WKBintegrals\]) may be performed immediately as $$\int_{0}^{\infty} \mathrm{d}z\> \frac{ z^{2p+1} }{(1 +z^2)^{m+n+1/2}} = \frac{p!}{2 (m+n-p-1/2)_{p+1}} \ .$$ For $p\geq m+n$, we use the regularised integrand arising from Eq. (\[eq:reg\_integrand\]) and temporarily introduce an ultraviolet cutoff $1/\epsilon^2$ to get $$\int_{\epsilon}^{\infty} \mathrm{d}x \> x^{m+n-p-3/2}\left[(1+x)^{-m-n-1/2} - \sum_{k=0}^{p-m-n} \frac{(-1)^k (m+n+1/2)_k}{k!} x^{k} \right]\ .$$ Integrating by parts $p-m-n+1$ times, the regularisation subtraction terms ensure that boundary term go to zero in the limit $\epsilon\to 0$ and we are left with $$(-1)^{p-m-n+1} \frac{(m+n+1/2)_{p-m-n+1}}{(1/2)_{p-m-n+1}}\int_{\epsilon}^{\infty} \mathrm{d}x \> x^{-1/2} (1+x)^{-p-3/2} = \frac{(-1)^{p-m-n+1} \pi p!}{\Gamma(m+n+1/2)\Gamma(p-m-n+3/2)}\>,$$ where the last equality reflects that after these integrations by parts the remaining integral is finite in the limit $\epsilon\to 0$. The second integral in Eq. (\[eq:WKBintegrals\]) is understood as a contour integral as illustrated in Fig. \[fig:wkbcontour\]. The integrand is understood to be defined on the complex plane cut from $z=-1$ to $1$ and additionally possesses singularities at $z=-1$ and $z=1$. To handle these in a fashion consistent with the Watson-Sommerfeld prescription we consider the contour in 3 parts: $\mathcal{C}_1$ running just below the cut from $0$ to $1-\epsilon$, $\mathcal{C}_2$ a semicircle of radius $\epsilon$ about $z=1$, and $\mathcal{C}_3$ running along the real axis from $z=1+\epsilon$ to $\infty$. ![The second integral in Eq. (\[eq:WKBintegrals\]) has a pole at $z=1$, so we split it into three parts: (1) An integral from $0$ to $1-\epsilon$, (2) An arc of radius $\epsilon$ about $z=1$ (3) An integral from $z=1+\epsilon$ to $\infty$.[]{data-label="fig:wkbcontour"}](wkbcontour.pdf){width="6cm"} The integrand along $\mathcal{C}_3$ is manifestly imaginary and so gives zero contribution. Writing the integrand as $$\frac{ G(z)}{(1 - z)^{m+n+1/2}}$$ where $$\label{eq:Gdef} G(z) = \frac{z^{2p+1}}{(1+e^{2\pi z \omega \sqrt{g/f}})(1+z)^{m+n+1/2}}$$ it is straightforward to see that $$\begin{aligned} \Re \int_{\mathcal{C}_2} \frac{ G(z)}{(1 - z)^{m+n+1/2}} \mathrm{d}z = - \sum\limits_{k=0}^\infty \frac{(-1)^{k}}{k!} G^{(k)}(1) \frac{\epsilon^{k-m-n+1/2}}{k-m-n+\frac{1}{2}}\ ,\end{aligned}$$ while integrating by parts $m+n$ times $$\begin{aligned} \int_{\mathcal{C}_1} \frac{ G(z)}{(1 - z)^{m+n+1/2}} \mathrm{d}z =& \sum\limits_{k=0}^{m+n-1} \frac{(-1)^{k}}{k!} G^{(k)}(1) \frac{\epsilon^{k-m-n+1/2}-1}{k-m-n+\frac{1}{2}}\nonumber \\ & + \int\limits_0^{1-\epsilon} \frac{ \mathrm{d} z}{(1-z)^{m+n+1/2}} \left[ G(z) - \sum\limits_{k=0}^{m+n-1} \frac{(-1)^{k}}{k!} G^{(k)}(1) (1-z)^k \right]\ .\end{aligned}$$ Adding the contributions from these components, it is clear that the $\epsilon \to 0$ divergences cancel and we are left with $$\begin{aligned} \label{eq:Gresult} - \sum\limits_{k=0}^{m+n-1} \frac{(-1)^{k}}{k!} \frac{G^{(k)}(1)}{k-m-n+\frac{1}{2}} + \int\limits_0^{1} \frac{ \mathrm{d} z}{(1-z)^{m+n+1/2}} \left[ G(z) - \sum\limits_{k=0}^{m+n-1} \frac{(-1)^{k}}{k!} G^{(k)}(1) (1-z)^k \right]\ ,\end{aligned}$$ where the subtraction terms in the integrand ensure the integral here is well-defined. While we can take this analysis further [@Ottewill:Winstanley:Young:2009], for our current purpose we note that we only need the expansion of Eq. (\[eq:Gresult\]) in terms of an inverse *powers* of $\omega$ as $\omega\to \infty$. From Eq. (\[eq:Gdef\]), it is immediate that the terms in the sum in (\[eq:Gresult\]) are exponentially small and so may be ignored for our purposes here. Indeed we may simultaneously increase the upper limit on the two sums in Eq. (\[eq:Gresult\]) without changing the result. Increasing it by 1 (or more), the integand increases from 0, peaks and then decreases to 0 at 1 with the peak approaching 0 as $\omega\to\infty$. Standard techniques from statistical mechanics then dictate that the $\omega\to\infty$ asymptotic form of the integral follows from the expanding the integrand, aside from the ‘Planck factor’, about $z=0$ and extending the upper limit to $\infty$. Again, in doing so the contribution from the summation within the integrand give exponentially small contribution so that the powers of $\omega$ are determined simply by $$\begin{aligned} \int\limits_0^{\infty} \frac{ \mathrm{d} z}{(1+e^{2\pi z \omega \sqrt{g/f}})} \mathop{\mathrm{Series}}\limits_{z=0} \left[\frac{z^{2p+1}}{(1-z^2)^{m+n+1/2}}\right]\end{aligned}$$ The coefficients in the series are known analytically, so to expand in inverse powers of $\omega$ we only need to compute integrals of the form $$\int_0^{\infty} \frac{z^{2N-1}}{1+e^{2\pi z \omega \sqrt{g/f}}} \, dz$$ which have the exact solutions [@GradRyz] $$\left(1-2^{1-2N}\right)\frac{f^N}{g^N \omega^{2N}}\frac{|B_{2N}|}{4N},$$ where $B_N$ is the $N$-th Bernoulli number. This expression allows us to calculate the integrals very quickly. Applying this method for summation over $l$, Eq. (\[eq:EuclideanGF\]) takes the form required by Eq. (\[eq:WKB-int-log-relation\]) so we now have the logarithmic part of the Euclidean Green function and therefore $V(x,x')$ as the required power series in $(t-t')$, $(\cos \gamma - 1)$ and $(r-r')$. Due to the length of the expressions involved, we have made available online [@Hadamard-WKB-Code] a *Mathematica* code implementing this algorithm, along with precalculated results for several spacetimes of interest including Schwarzschild, Nariai and Reissner-Nordström. Convergence of the Series {#sec:convergence} ========================= We have expressed $V(x,x')$ as a power series in the separation of the points. This series will, in general, not be convergent for all point separations – the maximum point separation for which the series remains convergent will be given by its *radius of convergence*. In this section, we explore the radius of convergence of the series in the Nariai and Schwarzschild spacetimes and use this as an estimate on the region of validity of our series. For simplicity, we will consider points separated only in the time direction so we will have a power series in $(t-t')$, $$\label{eq:CoordGreenT} V\left( x,x' \right) = \sum_{n=0}^{\infty} v_{n}(r) \left( t - t' \right)^{2n}.$$ where $v_{n}(r)$ is a real function of the radial coordinate, $r$ only. We will also consider cases where the points are separated by a fixed amount in the spatial directions, or where the separation in other directions can be re-expressed in terms of a time separation, resulting in a similar power series in $(t-t')$, but with the coefficients, $v_{n}(r)$, being different. This will give us sufficient insight without requiring overly complicated convergence tests. In the next section, we review some tests that will prove useful. In Secs. \[sec:Nariai-tests\] and \[sec:Schw-tests\] we present the results of applying those tests in Nariai and Schwarzschild spacetimes, respectively. Tests for Estimating the Radius of Convergence {#sec:Tests} ---------------------------------------------- ### Convergence Tests For the power series (\[eq:CoordGreenT\]), there are two convergence tests which will be useful for estimating the radius of convergence. The first of these, the *ratio test*, gives an estimate of the radius of convergence, $\Delta t_{RC}$, $$\Delta t_{RC} = \lim_{n\rightarrow\infty}\sqrt{\left|\frac{v_{n}}{v_{n+1}}\right|}.$$ Although strictly speaking, the large $n$ limit must be taken, in practice we have calculated enough terms in the series to get a good estimate of the limit by simply looking at the last two terms. The ratio test falls into difficulties, however, when one of the terms in the series is zero. Unfortunately, this occurs frequently for many cases of interest. It is possible to avoid this issue somewhat by considering non-adjacent terms in the series, i.e. by comparing terms of order $n$ and $n+m$, $$\label{eq:ratio-test} \Delta t_{RC} = \lim_{n\rightarrow\infty}\left|\frac{v_{n}}{v_{n+m}}\right|^{\frac{1}{2m}}.$$ Using the ratio test in this way gives better estimates of the radius of convergence, although the results are still somewhat lacking. To get around this difficulty, we also use a second test, the *root test*, $$\Delta t_{RC} = \limsup_{n\rightarrow\infty} \left|\frac{1}{v_n}\right|^{\frac{1}{2n}},$$ which is well suited to power series. This gives us another estimate of the radius of convergence. Again, in practice the last calculated term in the series will give a good estimate of the limit. It may appear that only the (better behaved) root test is necessary for estimating the radius of convergence of the series. However, extra insight can be gained from including both tests. This is because for the power series , the ratio test typically gives values *increasing* in $n$ while the root test gives values *decreasing* in $n$, effectively giving lower and upper bounds on the radius of convergence. ### Normal Neighborhood {#subsubsec:NN} The Hadamard parametrix for the retarded Green function, (\[eq:Hadamard\]), is only guaranteed to be valid provided $x$ and $x'$ are within a *normal neighborhood* (see footnote \[def:causal domain\]). This arises from the fact that the Hadamard parametrix involves Synge’s world function, $\sigma(x,x')$, which is only defined for the points $x$ and $x'$ separated by a *unique geodesic*. It is therefore plausible that the radius of convergence of our series could exactly correspond to the normal neighborhood size, $t_{\text{NN}}$. This turns out to not be the case, although it is still helpful to give consideration to $t_{\text{NN}}$ as it should place an upper bound on the radius of convergence of the series. The normal neighborhood size will be given by the minimum time separation of the spacetime points such that they are connected by two geodesics. For typical cases of interest for self-force calculations there will be a particle following a time-like geodesic, so $t_{\text{NN}}$ will be given by the minimum time taken by a null geodesic intersecting the particle’s world-line twice. In typical black hole spacetimes, this geodesic will orbit the black hole once before re-intersecting the particle’s world-line. Another case of interest is that of the points $x$ and $x'$ at constant spatial points, separated by a constant angle $\Delta\phi$. Initially, (i.e. when $t=t'$), the points will be separated only by spacelike geodesics. After sufficient time has passed for a null geodesic to travel between the points (going through and angle $\Delta\phi$, they will be connected first by a null geodesic and subsequently by a sequence of unique timelike geodesics. Since the geodesics are unique, $t_{\text{NN}}$ will not be given by this *first* null geodesic time. Rather, $t_\text{NN}$ will be given by the time taken by the *second* null geodesic (passing through an angle $2\pi - \Delta\phi$). This subtle, but important, distinction will be clearly evident when we study specific cases in the next sections. ### Relative Truncation Error Knowledge of the radius of convergence alone does not give information about the accuracy of the series representation of $V(x,x')$. The series is necessarily truncated after a finite number of terms, introducing a *truncation error*. As was done previously in Refs. [@Ottewill:Wardell:2008; @Anderson:Wiseman:2005] the local fractional truncation error can be estimated by the ratio between the highest order term in the expansion ( $O \left( \Delta \tau ^n \right)$, say) and the sum of all the terms up to that order, $$\label{eq:trunc-error} \epsilon \equiv \frac{f^a_{\rm QL}\left[ n \right]}{\sum_{i=0}^{n} f^a_{\rm QL}\left[ i \right]}.$$ Refs. [@Ottewill:Wardell:2008; @Anderson:Wiseman:2005] considered only the first two terms in the series when producing these estimates. Since we now have a vastly larger number of terms available, it is worthwhile considering these again to determine the accuracy of the high order series. Nariai Spacetime {#sec:Nariai-tests} ---------------- ### Normal Neighborhood {#subsec:Nariai-NN} Allowing only the time separation of the points to change, there are two cases of interest for the Nariai spacetime: 1. The static particle which has a normal neighborhood determined by the minimum coordinate time taken by a null geodesic circling the origin (${\rho}=0$) before returning. This is the time taken by a null geodesic, starting at ${\rho}={\rho}_1$ and returning to ${\rho}'={\rho}_1$, while passing through an angle $\Delta\phi=2\pi$. \[enum:static-nariai\] 2. Points at fixed radius, ${\rho}_1$, separated by an angle, say $\pi/2$. In this case, there is a null geodesic which goes through an angle $\Delta\phi=\pi/2$ when traveling between the points. However, there will not yet be any other geodesic connecting them, so this will not give $t_{\text{NN}}$. Instead, it is the the next null geodesic, which goes through $\Delta\phi=3\pi/2$, that gives the normal neighborhood boundary. \[enum:static-nariai-angle\] In both cases, the coordinate time taken by the null geodesic to travel between the points is given by [@Casals:Dolan:Ottewill:Wardell:2009] $$t_{NN} = 2 \tanh^{-1}({\rho}_1) + \ln \left( \frac{1-{\rho}_1 {\text{sech}}^2(\Delta\phi) + \tanh(\Delta\phi) \sqrt{1 - {\rho}_1^2 {\text{sech}}^2(\Delta\phi)}}{ 1+{\rho}_1 {\text{sech}}^2(\Delta\phi) - \tanh(\Delta\phi) \sqrt{1 - {\rho}_1^2 {\text{sech}}^2(\Delta\phi) }} \right) \label{eq:Nariai-geodesic-time}.$$ ### Results For the Nariai spacetime, the ratio test suffers from difficulties arising from zeros of the terms in the series. This is because the $i^{th}$ term of the series (of order $(t-t')^{2i}$) has $2i$ roots in $\rho$. In other words, the higher order the terms considered, the more likely one of the coefficients is to be near a zero, and not give a useful estimate of the radius of convergence. We have therefore compared non-adjacent terms to avoid this issue as much as possible, by choosing $m=n/2$ in Eq. . Fortunately, the root test is much less affected by such issues and can be used without any adjustments. In Fig. \[fig:nariai-roc2\], we fix the radial position of the points at $\rho=\rho'=1/2$ and $\rho=\rho'=1/99$ and plot the results of applying the root test (blue dots) and ratio test (brown dots) for cases (\[enum:static-nariai\]) Static particle (left) and (\[enum:static-nariai-angle\]) Points at fixed spatial points separated by an angle $\gamma=\pi/2$ (right) as a function of the maximum order, $n_\text{max}$, of the terms of the series considered, $v_\text{nmax} \left(t-t^\prime \right)^{2n_\text{max}}$. It seems in both cases that the plot is limiting towards a constant value for the radius of convergence. In the case (\[enum:static-nariai\]), we see that this gives a radius of convergence that is considerably smaller than the normal neighborhood size (purple dashed line). For the case (\[enum:static-nariai-angle\]), we again see that the root test is limiting to a value for the radius of convergence. However, as discussed in Sec. \[subsubsec:NN\], it is not the first null geodesic (lower dashed purple line), but the second null geodesic (upper dashed purple line) that determines the normal neighborhood and places an upper bound on the radius of convergence of the series. ![*Radius of convergence as a function of the number of terms in the series for the Nariai spacetime with curvature coupling $\xi=1/6$.* The limit as $n\rightarrow\infty$ will give the actual radius of convergence, but it appears that just using the terms up to $n=30$ is giving a good estimate of this limit. The radius of convergence is estimated by the root test (blue dots) and ratio test (brown dots) and is compared against the normal neighborhood size (purple dashed line) calculated from considerations on null geodesics (see Sec. \[subsubsec:NN\] and Sec. \[subsec:Nariai-NN\]). Note that plots for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:nariai-roc2"}](nariai-static-roc2.pdf "fig:"){width="6cm"} ![*Radius of convergence as a function of the number of terms in the series for the Nariai spacetime with curvature coupling $\xi=1/6$.* The limit as $n\rightarrow\infty$ will give the actual radius of convergence, but it appears that just using the terms up to $n=30$ is giving a good estimate of this limit. The radius of convergence is estimated by the root test (blue dots) and ratio test (brown dots) and is compared against the normal neighborhood size (purple dashed line) calculated from considerations on null geodesics (see Sec. \[subsubsec:NN\] and Sec. \[subsec:Nariai-NN\]). Note that plots for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:nariai-roc2"}](nariai-static-roc3.pdf "fig:"){width="6cm"} ![*Radius of convergence as a function of the number of terms in the series for the Nariai spacetime with curvature coupling $\xi=1/6$.* The limit as $n\rightarrow\infty$ will give the actual radius of convergence, but it appears that just using the terms up to $n=30$ is giving a good estimate of this limit. The radius of convergence is estimated by the root test (blue dots) and ratio test (brown dots) and is compared against the normal neighborhood size (purple dashed line) calculated from considerations on null geodesics (see Sec. \[subsubsec:NN\] and Sec. \[subsec:Nariai-NN\]). Note that plots for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:nariai-roc2"}](nariai-angle-roc2.pdf "fig:"){width="6cm"} ![*Radius of convergence as a function of the number of terms in the series for the Nariai spacetime with curvature coupling $\xi=1/6$.* The limit as $n\rightarrow\infty$ will give the actual radius of convergence, but it appears that just using the terms up to $n=30$ is giving a good estimate of this limit. The radius of convergence is estimated by the root test (blue dots) and ratio test (brown dots) and is compared against the normal neighborhood size (purple dashed line) calculated from considerations on null geodesics (see Sec. \[subsubsec:NN\] and Sec. \[subsec:Nariai-NN\]). Note that plots for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:nariai-roc2"}](nariai-angle-roc3.pdf "fig:"){width="6cm"} In Fig. \[fig:nariai-roc\] we use the root test (blue dots) and ratio test (brown line)[^3] to investigate how the the radius of convergence of the series varies as a function of the radial position of the points, ${\rho}_1$. Again, we look at both cases (\[enum:static-nariai\]) (left) and (\[enum:static-nariai-angle\]) (right). As a reference, we compare to the normal neighborhood size (purple dashed line). We find that, regardless of the radial position of the points, the radius of convergence of the series is well within the normal neighborhood, by an almost constant amount. As before, in the case (\[enum:static-nariai-angle\]) of the points separated by an angle, we find that it is the second, not the first null geodesic that gives the normal neighborhood size. ![*Estimates of the domain of validity (i.e. the radius of convergence) of the series expansion of $V(x,x')$ as a function of radial position in Nariai spacetime.* Root test on $O\left[(t-t')^{60}\right]$ series given as blue dots, ratio test given by brown line (see footnote \[fn:blips\]) and normal neighborhood estimate from null geodesics given by dashed purple lines. The left plot is for the case (\[enum:static-nariai\]), the static particle, right plot for the case (\[enum:static-nariai-angle\]), points separated by an angle of $\pi/2$. In both cases, the series is clearly divergent before the normal neighborhood boundary. This boundary is sometimes given by the second, rather than first null geodesic as can be seen in the plot on the right. In particular, this is the case for the points separated by an angle $\gamma=\pi/2$ since they are initially separated by only spacelike geodesics (see Sec. \[subsec:Nariai-NN\]).[]{data-label="fig:nariai-roc"}](nariai-static-roc.pdf "fig:"){width="6cm"} ![*Estimates of the domain of validity (i.e. the radius of convergence) of the series expansion of $V(x,x')$ as a function of radial position in Nariai spacetime.* Root test on $O\left[(t-t')^{60}\right]$ series given as blue dots, ratio test given by brown line (see footnote \[fn:blips\]) and normal neighborhood estimate from null geodesics given by dashed purple lines. The left plot is for the case (\[enum:static-nariai\]), the static particle, right plot for the case (\[enum:static-nariai-angle\]), points separated by an angle of $\pi/2$. In both cases, the series is clearly divergent before the normal neighborhood boundary. This boundary is sometimes given by the second, rather than first null geodesic as can be seen in the plot on the right. In particular, this is the case for the points separated by an angle $\gamma=\pi/2$ since they are initially separated by only spacelike geodesics (see Sec. \[subsec:Nariai-NN\]).[]{data-label="fig:nariai-roc"}](nariai-angle-roc.pdf "fig:"){width="6cm"} With knowledge of radius of convergence of the series established, it is also important to estimate the accuracy of the series within that radius. To that end, we plot in Fig. \[fig:nariai-trunc\] the relative truncation error, (\[eq:trunc-error\]), as a function of the time separation of the points at a fixed radius, $\rho=\rho'=1/2$. We find that the $60^{th}$ order series is extremely accurate to within a short distance of the radius of convergence of the series. ![*Relative truncation error in Nariai spacetime* arising from truncating the series expansion for $V(x,x')$ at order $|x-x'|^{60}$ (i.e. $n_\text{max}=30$) for cases (\[enum:static-nariai\]) the static particle (left) and (\[enum:static-nariai-angle\]) points separated by an angle $\pi/2$ (right). In both cases, the radial points are fixed at $\rho=\rho'=1/2$. The series is extremely accurate until we get close to the radius of convergence (see Fig.\[fig:nariai-roc2\]).[]{data-label="fig:nariai-trunc"}](nariai-static-trunc.pdf "fig:"){width="6cm"} ![*Relative truncation error in Nariai spacetime* arising from truncating the series expansion for $V(x,x')$ at order $|x-x'|^{60}$ (i.e. $n_\text{max}=30$) for cases (\[enum:static-nariai\]) the static particle (left) and (\[enum:static-nariai-angle\]) points separated by an angle $\pi/2$ (right). In both cases, the radial points are fixed at $\rho=\rho'=1/2$. The series is extremely accurate until we get close to the radius of convergence (see Fig.\[fig:nariai-roc2\]).[]{data-label="fig:nariai-trunc"}](nariai-angle-trunc.pdf "fig:"){width="6cm"} Schwarzschild Spacetime {#sec:Schw-tests} ----------------------- ### Normal Neighborhood {#normal-neighborhood} For a fixed spatial point at radius $r_1$ in the Schwarzschild spacetime, we would like to find the null geodesic that intersects it twice in the shortest time. This geodesic will orbit the black hole once before returning to $r_1$. Clearly, the coordinate time $t_{\text{NN}}$ for this orbit can only depend on $r_1$. The periapsis radius, $r_p$, will be reached half way through the orbit. For the radially inward half of this motion, the geodesic equations can be rearranged to give $$\begin{aligned} \pi = \int_0^\pi d\phi &= - \int_{r_1}^{r_p} \frac{dr}{r^2 \sqrt{\frac{1}{r_p^2}\left(1-\frac{2M}{r_p}\right)-\frac{1}{r^2}\left(1-\frac{2M}{r}\right)}}\\ \frac{t_{NN}}{2} = \int_0^{t_{NN}/2} dt &= - \int_{r_1}^{r_p} \frac{dr}{\left(1-\frac{2M}{r}\right)\sqrt{1-\frac{r_p^2}{r^2}\left(1-\frac{2M}{r}\right)\left(1-\frac{2M}{r_p}\right)^{-1}}}\end{aligned}$$ For a given point $r_1$, we numerically solve the first of these to find the periapsis radius, $r_p$, then solve the second to give the normal neighborhood size, $t_{\text{NN}}$. ### Results The ratio test proves more stable for Schwarzschild than it was for Nariai. The series coefficients still have a large number of roots in $r$, but most are within $r=6M$ and therefore don’t have an effect for the physically interesting radii, $r\ge6M$. Fig. \[fig:schw-static-roc-terms\] shows that the radius of convergence $ \Delta t_{RC}$ given by the root test is a decreasing function of the order of the term used, while that given by the ratio test is increasing. This effectively gives an upper and lower bound on the radius of convergence of the series. There is some ‘noise’ in the ratio test plot at lower radii (where that test is failing to give meaningful results), but we simply ignore this and omit the ratio test in this case. For the root test, the terms up to order $(t-t')^{52}$ was used, while for the ratio test, adjacent terms in the series up order $(t-t')^{52}$ were compared. ![*Radius of convergence as a function of the number of terms considered.* Root test (blue dots), ratio test (brown line – see footnote \[fn:blips\]), first null geodesic (purple dashed line). Left: Static particle at $r_1=100M$ Right: Static particle at $r_1=10M$. The radius of convergence is given by the limit as the number of terms $n_{\text{max}} \to \infty$. It is apparent that the plots are asymptoting to a value near, but slightly lower than the normal neighborhood size. Note that curves for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:schw-static-roc-terms"}](schw-static-roc2.pdf "fig:"){width="6cm"} ![*Radius of convergence as a function of the number of terms considered.* Root test (blue dots), ratio test (brown line – see footnote \[fn:blips\]), first null geodesic (purple dashed line). Left: Static particle at $r_1=100M$ Right: Static particle at $r_1=10M$. The radius of convergence is given by the limit as the number of terms $n_{\text{max}} \to \infty$. It is apparent that the plots are asymptoting to a value near, but slightly lower than the normal neighborhood size. Note that curves for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:schw-static-roc-terms"}](schw-static-roc4.pdf "fig:"){width="6cm"} In Fig. \[fig:schw-static-roc-radius\] we apply the root (blue dots) and ratio (brown line – see footnote \[fn:blips\]) tests for the case of a static particle at a range of radii in Schwarzschild spacetime. Using the root test as an upper bound and the ratio test as a lower bound, it is clear that the radius of convergence is near, but likely slightly lower than the normal neighborhood size (purple dashed line). In this case, for the root test, the term of order $(t-t')^{52}$ was used, while for the ratio test, terms of order $(t-t')^{52}$ and $(t-t')^{26}$ were compared. ![*Radius of convergence as a function of radial position for a static point in Schwarzschild.* Root test (blue dots), ratio test (brown line – see footnote \[fn:blips\]), first null geodesic (purple dashed line).[]{data-label="fig:schw-static-roc-radius"}](schw-static-roc.pdf){width="6cm"} In Fig. \[fig:schw-circ-roc\], we repeat for the case of the points separated on a circular timelike geodesic. The results are very similar to the static particle case and show the same features. ![*Radius of convergence as a function of radial position for a points separated along a circular geodesic in Schwarzschild.* Root test (blue dots), ratio test (brown line – see footnote \[fn:blips\]), first null geodesic (purple dashed line). Note that curves for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:schw-circ-roc"}](schw-circ-roc.pdf "fig:"){width="6cm"} ![*Radius of convergence as a function of radial position for a points separated along a circular geodesic in Schwarzschild.* Root test (blue dots), ratio test (brown line – see footnote \[fn:blips\]), first null geodesic (purple dashed line). Note that curves for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:schw-circ-roc"}](schw-circ-roc2.pdf "fig:"){width="6cm"} ![*Radius of convergence as a function of radial position for a points separated along a circular geodesic in Schwarzschild.* Root test (blue dots), ratio test (brown line – see footnote \[fn:blips\]), first null geodesic (purple dashed line). Note that curves for the ratio test were omitted in cases where it did not give meaningful results.[]{data-label="fig:schw-circ-roc"}](schw-circ-roc4.pdf "fig:"){width="6cm"} Extending the Domain of Series Using Padé Approximants {#sec:Pade} ====================================================== In the previous section it was shown that the circle of convergence of the series expansion of $V(x,x')$ is smaller than the size of the normal neighborhood. This is not totally unexpected. We would expect the normal neighborhood size to place an upper limit on the radius of convergence, but we can not necessarily expect the radius of convergence to be exactly the normal neighborhood size. However, since the Hadamard parametrix for the Green function is valid everywhere within the normal neighborhood, it is reasonable to hope that it would be possible to find an alternative series representation for $V(x,x')$ which is valid in the region outside the circle of convergence of the original series, while remaining within the normal neighborhood. The radius of convergence found in the previous section locates the distance (in the complex plane) to the closest singularity of $V(x,x')$. However, that singularity could lie anywhere on the (complex) circle of convergence and will not necessarily be on the real line. In fact, given that the Green function is clearly not singular at the radius of convergence on the real line, it is clear that the singularity of $V(x,x')$ does not lie on the real line. There are several techniques which can be employed to extend a series beyond its radius of convergence. Provided the circle of convergence does not constitute a *natural boundary* of the function, the method of *analytic continuation* can be used to find another series representation for $V(x,x')$ valid outside the circle of convergence of the original series . This could then be applied iteratively to find series representations covering the entire range of interest of $V(x,x')$. Although analytic continuation should be capable of extending the series expansion of $V(x,x')$, there is an alternative method, the method of *Padé approximants* which yields impressive results with little effort. The method of Padé approximants [@Bender:Orszag; @NumericalRecipes] is frequently used to extend the series representation of a function beyond the radius of convergence of the series. It has been employed in the context of General Relativity data analysis with considerable success [@Damour:Iyer:Sathyaprakash; @Porter:Sathyaprakash]. It is based on the idea of expressing the original series as a rational function (i.e. a ratio of two polynomials $V(x,x')=R(x,x')/S(x,x')$) and is closely related to the continued fraction representation of a function [@Bender:Orszag]. This captures the functional form of the singularities of the function on the circle of convergence of the original series. The Padé approximant, $P_M^N (t-t')$ is defined as $$P_M^N (t-t') \equiv \frac{\sum_{n=0}^N A_n (t-t')^n}{\sum_{n=0}^M B_n (t-t')^n}$$ where $B_0 = 1$ and the other $(M+N+1)$ terms are found by comparing to the first $(M+N+1)$ terms of the original power series. The choice of $M$ and $N$ is arbitrary provided $M+N\le n_{\text{max}}$ where $n_{\text{max}}$ is the highest order term that has been computed for the original series. There are, however, choices for $M$ and $N$ which give the best results. In particular, the diagonal, $P^N_N$, and sub-diagonal, $P^N_{N+1}$, Padé approximants yield optimal results. Nariai {#subsec:padeNariai} ------ The Green function in Nariai spacetime is known to be given exactly by a quasinormal mode sum [@Casals:Dolan:Ottewill:Wardell:2009; @Beyer:1999] at sufficiently late times. We can therefore use the Green function calculated from a quasinormal mode sum to determine the effectiveness of the Padé resummation. Figure \[fig:padeCompareSeriesNariai\] compares the quasinormal mode calculated Green function[^4] with both the original Taylor series representation and the Padé resummed series for $V(x,x')$ for a range of cases. We use the Padé approximant $P_{30}^{30}$, computed from the $60^{th}$ order Taylor series. In each case, the series representation (blue dashed line) diverges near its radius of convergence, long before the normal neighborhood boundary is reached. The Padé resummed series (red line), however, remains valid much further and closely matches the quasinormal mode Green function (black dots) up to the point where the normal neighborhood boundary is reached. ![*Comparing Padé approximant and Taylor series for $\theta(-\sigma(x,x')) V(x,x')$ to ‘exact’ Green function from quasinormal mode sum in Nariai spacetime with curvature coupling $\xi=1/8$ and $\xi=1/6$.* The Padé approximated $V(x,x')$ (red line) is in excellent agreement with the quasinormal mode Green function (black dots) up to the normal neighborhood boundary, $t_{NN}\approx6.56993$ (top) and $t_{NN}\approx4.9956$ (bottom).[]{data-label="fig:padeCompareSeriesNariai"}](padeCompareSeriesNariaiStaticXi8.pdf "fig:"){width="6cm"} ![*Comparing Padé approximant and Taylor series for $\theta(-\sigma(x,x')) V(x,x')$ to ‘exact’ Green function from quasinormal mode sum in Nariai spacetime with curvature coupling $\xi=1/8$ and $\xi=1/6$.* The Padé approximated $V(x,x')$ (red line) is in excellent agreement with the quasinormal mode Green function (black dots) up to the normal neighborhood boundary, $t_{NN}\approx6.56993$ (top) and $t_{NN}\approx4.9956$ (bottom).[]{data-label="fig:padeCompareSeriesNariai"}](padeCompareSeriesNariaiStaticXi6.pdf "fig:"){width="6cm"} ![*Comparing Padé approximant and Taylor series for $\theta(-\sigma(x,x')) V(x,x')$ to ‘exact’ Green function from quasinormal mode sum in Nariai spacetime with curvature coupling $\xi=1/8$ and $\xi=1/6$.* The Padé approximated $V(x,x')$ (red line) is in excellent agreement with the quasinormal mode Green function (black dots) up to the normal neighborhood boundary, $t_{NN}\approx6.56993$ (top) and $t_{NN}\approx4.9956$ (bottom).[]{data-label="fig:padeCompareSeriesNariai"}](padeCompareSeriesNariaiPi2Xi8.pdf "fig:"){width="6cm"} ![*Comparing Padé approximant and Taylor series for $\theta(-\sigma(x,x')) V(x,x')$ to ‘exact’ Green function from quasinormal mode sum in Nariai spacetime with curvature coupling $\xi=1/8$ and $\xi=1/6$.* The Padé approximated $V(x,x')$ (red line) is in excellent agreement with the quasinormal mode Green function (black dots) up to the normal neighborhood boundary, $t_{NN}\approx6.56993$ (top) and $t_{NN}\approx4.9956$ (bottom).[]{data-label="fig:padeCompareSeriesNariai"}](padeCompareSeriesNariaiPi2Xi6.pdf "fig:"){width="6cm"} As was shown in Refs. [@Kay:Radzikowski:Wald:1997; @Casals:Dolan:Ottewill:Wardell:2009], the Green function in Nariai spacetime is singular whenever the points are separated by a null geodesic. Furthermore, in Ref. [@Casals:Dolan:Ottewill:Wardell:2009] we have derived the functional form of these singularities and shown that they follow a four-fold pattern: $\delta(\sigma)$, $1/\pi \sigma$, $-\delta(\sigma)$, $-1/\pi \sigma$, depending on the number of caustics the null geodesic has passed through (this was also previously shown by Ori [@Ori1]). Within the normal neighborhood (where the Hadamard parametrix, , is valid), the $\delta(\sigma)$ singularities (i.e. at *exactly* the null geodesic times) will be given by the term involving $U(x,x')$. However, a times other than the *exact* null geodesic times, the Green function will be given fully by $V(x,x')$. For this reason, we expect $V(x,x')$ to reflect the singularities of the Green function near the normal neighborhood boundary The Padé approximant attempts to model the singularity of the function $V(x,x')$ (which occurs at the null geodesic time) by representing it as a rational function, i.e. a ratio of two power series. By its nature, this will only faithfully reproduce singularities of integer order. In the Nariai case, however, the asymptotic form of the the singularities is known exactly near the singularity times, $t_c$ [@Casals:Dolan:Ottewill:Wardell:2009]. In cases where the points are separated by an angle $\gamma\in (0,\pi)$ (i.e. away from a caustic), the singularities are expected to have a $1/(t-t'-t_c)$ behavior and it is reasonable to expect the Padé approximant to reproduce the singularity well. When the points are not separated in the angular direction (i.e. at a caustic), however, the singularities behave like $1/(t-t'-t_c)^{3/2}$ and we cannot reasonably expect the Padé approximant to accurately reflect this singularity without including a large number of terms in the denominator. Given knowledge of the functional form of the singularity, however, it is possible to improve the accuracy of the Padé approximant further. For a singularity of the form $1/S(t)$, we first multiply the Taylor series by $S(t)$. The result should then have either no singularity, or have a singularity which can be reasonably represented by a power series. The Padé approximant of this new series is then calculated and the result is divided by $S(t)$ to give an *improved Padé approximant*. This yields an approximant which includes the exact form of the singularity and more closely matches the exact Green function near the singularity. In Fig. \[fig:padeSqrtNariai\], we illustrate the improvement with an example case. We consider a static point in Nariai spacetime and compute the error in the Padé approximant relative to the quasinormal mode Green function (with $n\le8$). The regular Padé approximant is shown in green while the improved Padé approximant is show in orange. The relative error remains small closer to the singularity for the improved Padé approximant case than for the the standard Padé approximant. Note that the error for early times arises from the failure of the quasinormal mode sum to converge and does not reflect error in the series approximations. ![*Relative Error in Improved vs Regular Padé Approximant.* The relative error in the improved Padé approximant (orange) remains small closer to the singularity than the regular Padé approximant (green).[]{data-label="fig:padeSqrtNariai"}](padeErrorNariaiStaticXi8.pdf){width="6cm"} Schwarzschild {#subsec:padeSchw} ------------- For the Schwarzschild case, there is no quasinormal mode sum with which to compare the Padé approximated series[^5]. However, given the success in the Nariai case, we remain optimistic that Padé approximation will be successful for Schwarzschild. In an effort to estimate the effectiveness of the Padé approximant, we compare in Fig. \[fig:padeCompareSeriesSchw\] the series expression for $V(x,x')$ with two different Padé resummations, $P_{26}^{24}$ and $P_{26}^{26}$. The Padé approximant extends the validity beyond the radius of convergence of the series, but is less successful at reaching the normal neighborhood boundary ($t-t'=t_{NN}\approx 46.2471M$) than in the Nariai case. ![*Comparing Padé to Taylor series for Schwarzschild* in the case of a static particle at r=10M. The Padé approximants $P^{26}_{26}$ (red line) and $P^{24}_{26}$ (brown line) are likely to represent $V(x,x')$ more accurately near the normal neighborhood boundary (at $t-t'\approx46.2471M$) than the regular Taylor series (blue dashed line).[]{data-label="fig:padeCompareSeriesSchw"}](padeCompareSeriesSchwStaticR10.pdf){width="6cm"} The failure of the Padé approximant to reach the normal neighborhood boundary can be understood by the presence of extraneous singularities in the Padé approximant. The zeros of the denominator, $S(t-t')=0$, give rise to singularities which occur at times earlier that the null geodesic time. It is possible that this problem could be reduced to a certain extent using the knowledge of the functional form of the singularities to compute an *improved* Padé approximant (as was successful in the Nariai case). However, to the authors knowledge, the structure of the singularities in Nariai spacetime is not yet known. While it may be possible to adapt the work of Ref. [@Casals:Dolan:Ottewill:Wardell:2009] to find the asymptotic form of the singularities in Schwarzschild, without knowledge of the exact Green function we would not be able to dermine whether an improved Padé approximant would truly give an improvement. We therefore leave such considerations for later work. Convergence of the Padé Sequence {#subsec:Pade-convergence} -------------------------------- The use of Padé approximants has shown remarkable success in improving the accuracy and domain of the series representation of $V(x,x')$. However, this improvement has not been quantified. There is no general way to determine whether the Padé approximant is truly approximating the correct function, $V(x,x')$ or the domain in which it is valid [@NumericalRecipes]. In this subsection, we nonetheless attempt to gain some insight into the validity of the Padé approximants. The first issue to consider is the presence of extraneous poles in the Padé approximants. In Sec. \[subsec:padeNariai\], the Padé approximant was unable to exactly represent the $1/(t-t'-t_c)^{3/2}$ singularity at $t_c \approx 6.12$ and instead represented it by three (real-valued) simple poles (at $t-t' \approx 6.522, 6.854 \text{ and } 9.488$). This leads to the Padé approximant being a poor representation of the function near the poles. As was shown in Fig. \[fig:padeSqrtNariai\], having exact knowledge of the singularity allows the calculation of an improved Padé approximant without extraneous singularities[^6]. With extraneous poles dealt with, we consider the convergence of the Padé sequence of diagonal and sub-diagonal Padé approximants, $$P = \{P_0^0,~ P_1^0,~ P_1^1,~ P_2^1,~ P_2^2,~ P_3^2,~ P_3^3, \cdots \},$$ with $P_N$ being the $N$-th element of the sequence. The convergence of the Padé approximant sequence is determined by the behavior of the denominators, $S_N$ for large $N$ [@Bender:Orszag]. Provided $S(x,x')_N$ is not small, the Padé sequence will converge quickly toward the actual value of $V(x,x')$. When the first root of the denominator is at the null geodesic time (i.e. the normal neighborhood boundary), we can, therefore, be optimistic that the Padé sequence will remain convergent until this root is reached and the Padé approximants will accurately represent the function. To highlight the improvements made by using Padé approximants over regular Taylor series, we introduce the Taylor sequence (i.e. the sequence of partial sums of the series), $T=\{T_0, T_1, T_2, \cdots\}$, with $N$-th element, $$T_N = \sum_{n=0}^{N/2} v_{n} (t-t')^{2n}.$$ The two sequences $P_N$ and $T_N$ require approximately the same number of terms in the original Taylor series, so a direct comparison of their convergence will illustrate the improved convergence of the Padé approximants. In Fig. \[fig:padeConvergenceNariai\], we plot the Padé sequence (blue line) and Taylor sequence (purple line) for the case of static points at $\rho=1/2$ in the Nariai spacetime, with $\xi=1/8$. For early times (eg. $(t-t')=2$, it is clear that both Padé and Taylor sequences converge very quickly. At somewhat later times (eg. $(t-t')=3.3$), both sequences appear to remain convergent, but the Padé sequence is clearly converging much faster than the Taylor sequence. Outside the radius of convergence of the Taylor series (eg. $(t-t')=5, 6.3$), the Padé sequence is slower to converge, but appears to still do so. ![*Convergence of the Taylor and Padé Sequences* for the case of static points at $\rho=1/2$ in the Nariai spacetime, with $\xi=1/8$. Within the radius of convergence of the Taylor series, both Padé (blue line) and Taylor (purple line) sequences converge to the exact Green function (black dotted line) as calculated from a quasinormal mode sum with $n\le6$. The Padé sequence converges faster, particularly at larger times. Outside the radius of convergence of the Taylor series, only the Padé sequence is convergent. The top left plot is at a time $(t-t')=2$, top right plot is at $(t-t')=3.3$, bottom left at $(t-t')=5$, bottom right at $(t-t')=6.3$.[]{data-label="fig:padeConvergenceNariai"}](padeConvergenceNariaiStaticXi8T2.pdf "fig:"){width="6cm"} ![*Convergence of the Taylor and Padé Sequences* for the case of static points at $\rho=1/2$ in the Nariai spacetime, with $\xi=1/8$. Within the radius of convergence of the Taylor series, both Padé (blue line) and Taylor (purple line) sequences converge to the exact Green function (black dotted line) as calculated from a quasinormal mode sum with $n\le6$. The Padé sequence converges faster, particularly at larger times. Outside the radius of convergence of the Taylor series, only the Padé sequence is convergent. The top left plot is at a time $(t-t')=2$, top right plot is at $(t-t')=3.3$, bottom left at $(t-t')=5$, bottom right at $(t-t')=6.3$.[]{data-label="fig:padeConvergenceNariai"}](padeConvergenceNariaiStaticXi8T33.pdf "fig:"){width="6cm"} ![*Convergence of the Taylor and Padé Sequences* for the case of static points at $\rho=1/2$ in the Nariai spacetime, with $\xi=1/8$. Within the radius of convergence of the Taylor series, both Padé (blue line) and Taylor (purple line) sequences converge to the exact Green function (black dotted line) as calculated from a quasinormal mode sum with $n\le6$. The Padé sequence converges faster, particularly at larger times. Outside the radius of convergence of the Taylor series, only the Padé sequence is convergent. The top left plot is at a time $(t-t')=2$, top right plot is at $(t-t')=3.3$, bottom left at $(t-t')=5$, bottom right at $(t-t')=6.3$.[]{data-label="fig:padeConvergenceNariai"}](padeConvergenceNariaiStaticXi8T5.pdf "fig:"){width="6cm"} ![*Convergence of the Taylor and Padé Sequences* for the case of static points at $\rho=1/2$ in the Nariai spacetime, with $\xi=1/8$. Within the radius of convergence of the Taylor series, both Padé (blue line) and Taylor (purple line) sequences converge to the exact Green function (black dotted line) as calculated from a quasinormal mode sum with $n\le6$. The Padé sequence converges faster, particularly at larger times. Outside the radius of convergence of the Taylor series, only the Padé sequence is convergent. The top left plot is at a time $(t-t')=2$, top right plot is at $(t-t')=3.3$, bottom left at $(t-t')=5$, bottom right at $(t-t')=6.3$.[]{data-label="fig:padeConvergenceNariai"}](padeConvergenceNariaiStaticXi8T63.pdf "fig:"){width="6cm"} In Fig. \[fig:padeConvergenceSchw\], we again plot the Padé sequence, this time for the case of static points at $r=10M$ in the Schwarzschild spacetime. As in the Nariai case, for early times (eg. $(t-t')=10M$, it both Padé and Taylor sequences are converging very quickly. At slightly later times (eg. $(t-t')=20M, 27M$) the convergence of the Padé sequence is better than the Taylor sequence. Outside the radius of convergence of the Taylor series (eg. $(t-t')=32M$), the Padé sequence is slower to converge, but appears to still do so. The convergence of the series is slower in the Schwarzschild case than in the Nariai case. This is an indication that using more terms may yield a better result[^7]. ![*Convergence of the Taylor and Padé Sequences* for the case of static points at $r=10M$ in the Schwarzschild spacetime. The top left plot is at a time $(t-t')=10M$, top right plot is at $(t-t')=20M$, bottom left plot is at $(t-t')=27M$, bottom right plot is at $(t-t')=32M$.[]{data-label="fig:padeConvergenceSchw"}](padeConvergenceSchwStaticT10.pdf "fig:"){width="6cm"} ![*Convergence of the Taylor and Padé Sequences* for the case of static points at $r=10M$ in the Schwarzschild spacetime. The top left plot is at a time $(t-t')=10M$, top right plot is at $(t-t')=20M$, bottom left plot is at $(t-t')=27M$, bottom right plot is at $(t-t')=32M$.[]{data-label="fig:padeConvergenceSchw"}](padeConvergenceSchwStaticT20.pdf "fig:"){width="6cm"} ![*Convergence of the Taylor and Padé Sequences* for the case of static points at $r=10M$ in the Schwarzschild spacetime. The top left plot is at a time $(t-t')=10M$, top right plot is at $(t-t')=20M$, bottom left plot is at $(t-t')=27M$, bottom right plot is at $(t-t')=32M$.[]{data-label="fig:padeConvergenceSchw"}](padeConvergenceSchwStaticT27.pdf "fig:"){width="6cm"} ![*Convergence of the Taylor and Padé Sequences* for the case of static points at $r=10M$ in the Schwarzschild spacetime. The top left plot is at a time $(t-t')=10M$, top right plot is at $(t-t')=20M$, bottom left plot is at $(t-t')=27M$, bottom right plot is at $(t-t')=32M$.[]{data-label="fig:padeConvergenceSchw"}](padeConvergenceSchwStaticT32.pdf "fig:"){width="6cm"} Conclusions =========== In this paper we have presented an extension of the Hadamard-WKB method of Anderson and Hu [@Anderson:2003] to the Nariai spacetime. We have also demonstrated the use of an alternative WKB method [@Winstanley:2007; @Howard:1985], which allows very high order terms in the Taylor series to be calculated efficiently (on a computer) for Schwarzschild, Nariai and other spherically symmetric spacetimes. This allowed the series expansion of $V(x,x')$ (appearing in the Hadamard parametrix of the Green function) to be computed to significantly higher order than was done previously in Refs. [@Anderson:2003; @Anderson:Eftekharzadeh:Hu:2006]. These high order expansions facilitated an investigation of the convergence properties of the series. We also demonstrated the huge benefit of Padé approximants to improving the domain and convergence of the series. This paper serves a dual purpose 1. To discuss the calculation of the quasilocal Green function in Nariai spacetime, as required by Ref. [@Casals:Dolan:Ottewill:Wardell:2009]. 2. To investigate the potential for applying the same techniques in the Schwarzschild spacetime, with the goal of computing an accurate quasilocal Green function for use in a matched expansion calculation of the self-force. We have found that, using Padé approximants, it is possible compute the quasilocal Green function in Nariai spacetime to high accuracy to within a short distance of the normal neighborhood boundary. Even without the use of Padé approximants, the Taylor approximated series gives good accuracy within a large part of the quasilocal region. This gives confidence in their use for matched expansion calculations in Ref. [@Casals:Dolan:Ottewill:Wardell:2009]. With regard to the Schwarzschild case, we find that the quasilocal calculation of the Green function is in good standing and should be usable in matched expansion calculations once techniques for computing the ‘distant-past’ Green function have been fully developed. Both Padé and Taylor sequences remain convergent within a large part of the normal neighborhood. The use of Padé approximants is not quite as successful as for the Nariai case, but we remain optimistic that knowledge of the structure of the singularities in Schwarzschild may allow for the use of *improved* Padé approximants as discussed in Sec. \[subsec:padeSchw\]. The orders of the series calculated for Nariai ($60$-th) and Schwarzschild ($52$-nd) were the maximum possible within a reasonable time ($\sim 1$ day on a modern Linux desktop). Although these are considerably high order series, one may still wonder whether they are sufficiently high for matched expansion calculations. It is clear from Sec. \[subsec:Pade-convergence\] that the higher order terms only have a significant contribution near the radius of convergence (for the Taylor series) or normal neighborhood boundary (for the Padé approximant). As the Hadamard parametrix is only valid within the normal neighborhood, we consider the fact that the Padé approximant is accurate to within a short distance of the normal neighborhood boundary to be confirmation that the series has been calculated to sufficiently high order (in particular for the Nariai case). Additionally, within the matching region used in Ref. [@Casals:Dolan:Ottewill:Wardell:2009] for Nariai, we clearly have computed a sufficient number of coefficients to give the Green function to high accuracy. We can be optimistic that this is also the case for Schwarzschild: the quasilocal series is accurate long after the time when the quasinormal mode sum is expected to be convergent ($t-t'= 2 r_* \approx 12.77 M$, for the case of a static particle at $r=10M$ considered here). Our analysis has remained focused primarily on the case of one dimensional series. This was done for reasons of simplicity and clarity. For multi-dimensional series, one could re-express each of the coordinates in terms of a single parameter, as was done in Sec. \[sec:convergence\] for the case of a circular geodesic in Schwarzschild. Alternatively, one could make use of the extension of the Padé approximant to double and higher dimensional power series as developed by Chisholm [@Chisholm:1973; @Chisholm:McEwan:1974]. Acknowledgments =============== MC is grateful to the Department of Physics and Astronomy of the University of Mississippi for its hospitality during the preparation of this paper. MC was partially funded by Fundação para a Ciência e Tecnologia (FCT) - Portugal through project PTDC/FIS/64175/2006. MC, BW and SD are supported by the Irish Research Council for Science, Engineering and Technology, funded by the National Development Plan. [37]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , (), . , ****, (), . , ****, (), . , , , ****, (), . , ****, (). , ****, (), . , ****, (), . , ****, (), . , ****, (), . , ** (, , ), ISBN . , , , ****, (). , ****, (), . , ****, (). , ****, (). , ****, (), . , ****, (), . , ****, (), . , ** (, ), ISBN . , ** (, , ), ISBN . , ** (, , ). , , , ****, (), . , ****, (). , ****, (), . . , ****, (). , , (), . , ** (, ). , ****, (), . , ** (, ). , ** (, ), ed. , ** (, ). , , , , ** (, ), ed. , ****, (). , , , ****, (). (), , ****, (). , ****, (). [^1]: More precisely, the Hadamard parametrix requires that $x$ and $x'$ lie within a *causal domain* – a *convex normal neighbourhood* with causality condition attached. This effectively requires that $x$ and $x'$ be connected by a unique non-spacelike geodesic which stays within the causal domain. However, as we expect the term *normal neighbourhood* to be more familiar to the reader, we will use it throughout this paper, with implied assumptions of convexity and a causality condition.\[def:causal domain\] [^2]: Note that this definition of the Green function differs from that of Ref. [@Anderson:2003] by a factor of $4\pi$ and the definition of $V(x,x')$ differs by a further factor of 2. [^3]: \[fn:blips\]Note that the ‘blips’ in the ratio test are an artifact of the zeroes of the series coefficients used and are not to be taken to have any physical meaning. In fact, the ‘blips’ occur at different times when considering the series at different orders, so they should be ignored altogether. The ratio test plots should therefore only be fully trusted *away* from the ‘blips’. Near the blips, it is clear that one could interpolate an approximate value, however we have not done so here as the plot is to be taken only as an *indication* of the radius of convergence. [^4]: There are some caveats with how the quasinormal Green function was used. The *fundamental mode* ($n=0$) Green function was used and a *singularity time offset* applied as described in Ref. [@Casals:Dolan:Ottewill:Wardell:2009]. In the case where two singularities are present, two sets of fundamental mode Green functions were used, each shifted by an appropriate singularity time offset and matched at an intermediate point. [^5]: A quasinormal mode sum could be computed for the Schwarzschild case, but would be augmented by a branch cut integral [@Casals:Dolan:Ottewill:Wardell:2009]. This calculation is in progress but has yet to be completed [^6]: The improved Padé approximant has zeros in its denominator at $t-t' \approx 2.64, 6.51, 6.59 \text{and} 8.73$. The apparently extraneous singularity within the normal neighborhood (at $t-t'\approx 2.64$) does not cause any difficulty as the numerator also goes to zero at this point. [^7]: In the Nariai case (with $\xi=1/8$), the Taylor series for $V(x,x')$ starts at order $(t-t')^0$ and has been calculated to order $(t-t')^{60}$, while for Schwarzschild it starts at $(t-t')^4$ and has been calculated to order $(t-t')^{52}$. Since the Nariai series has several extra orders, it is reasonable to expect it to be a better approximation than the Schwarzschild series.

--- abstract: '[The marginal likelihood of a model is a key quantity for assessing the evidence provided by the data in support of a model. The marginal likelihood is the normalizing constant for the posterior density, obtained by integrating the product of the likelihood and the prior with respect to model parameters. Thus, the computational burden of computing the marginal likelihood scales with the dimension of the parameter space. In phylogenetics, where we work with tree topologies that are high-dimensional models, standard approaches to computing marginal likelihoods are very slow. Here we study methods to quickly compute the marginal likelihood of a single fixed tree topology. We benchmark the speed and accuracy of 19 different methods to compute the marginal likelihood of phylogenetic topologies on a suite of real datasets. These methods include several new ones that we develop explicitly to solve this problem, as well as existing algorithms that we apply to phylogenetic models for the first time. Altogether, our results show that the accuracy of these methods varies widely, and that accuracy does not necessarily correlate with computational burden. Our newly developed methods are orders of magnitude faster than standard approaches, and in some cases, their accuracy rivals the best established estimators.]{}' author: - | Mathieu Fourment$^1$, Andrew F. Magee$^{2}$, Chris Whidden$^{3}$, Arman Bilge$^3$,\ Frederick A. Matsen IV$^{3,\ast}$, Vladimir N. Minin$^{4,\ast}$ bibliography: - 'main.bib' title: 19 dubious ways to compute the marginal likelihood of a phylogenetic tree topology --- RH: THE MARGINAL LIKELIHOOD OF A TREE TOPOLOGY [ *$^1$University of Technology Sydney, ithree institute, Ultimo NSW 2007, Australia\ $^2$Department of Biology, University of Washington, Seattle, WA, 98195, USA\ $^3$Fred Hutchinson Cancer Research Center, Seattle, WA, 98109, USA\ $^4$Department of Statistics, University of California, Irvine, CA, 92697, USA*]{} [**Corresponding authors:**]{} Frederick A. Matsen IV, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave. N, Mail stop M1-B514, Seattle, WA, 98109, USA; E-mail: matsen@fredhutch.org & Vladimir Minin, Department of Statistics, University of California, Irvine, CA, 92697, USA; E-mail: vminin@uci.edu\ [**Keywords:**]{} Bayesian inference, model selection, evidence, importance sampling, variational Bayes Introduction {#introduction .unnumbered} ============ In phylogenetic inference, the tree topology forms a key object of inference. In Bayesian phylogenetics, this translates to approximating the posterior distribution of tree topologies. Typically, a joint posterior distribution of tree topologies and continuous parameters, including branch lengths and substitution model parameters, is approximated directly via Markov chain Monte Carlo (MCMC), as done in the popular Bayesian phylogenetics software MrBayes [@ronquist2012mrbayes]. However, MCMC over topologies is computationally expensive [@lakner2008efficiency; @hohna2008clock]. These MCMC algorithms spend a nontrivial amount of time marginalizing over branch lengths and substitution models parameters and discarding them so that the estimated posterior probability of a tree topology is the proportion of MCMC iterations in which it appears. Therefore, fast marginalization over continuous phylogenetic parameters may offer a boon to MCMC algorithm efficiency or even allow one to perform Bayesian phylogenetic inference without MCMC. In this paper, we review existing methods and develop new ones to compute the posterior probabilities of tree topologies by quickly marginalizing out branch lengths to compute the marginal likelihood of a given topology. We compare speed and accuracy of 19 methods and examine whether there is a speed-accuracy trade off. Given that the bulk of Bayesian inference is performed with methods that work because they allow the marginal likelihood to be avoided, why would one want to compute them at all? One potential application of these marginal likelihood computations is the development of fast, MCMC-free Bayesian phylogenetic inference. To make such an advance, first one would need to identify a large enough set of *a posteori* highly probable tree topologies, such as with a new optimization-based method called phylogenetic topographer (PT) [@pt]. Once a set of promising tree topologies is formed, we can compute their marginal likelihoods, then renormalize these marginal likelihoods (perhaps after multiplying by a prior) to obtain approximate posterior probabilities of tree topologies — the key output of Bayesian phylogenetic inference. Luckily, we can tap into a substantial body of research on computing the marginal likelihood of purely continuous statistical models in order to integrate out continuous parameters for any given tree topology [@hans2007shotgun; @lenkoski2011computational]. It is therefore high time we consider the possibility of constructing the posterior distribution on topologies without MCMC. To do so, we must know: how well, and how quickly, can we compute the marginal likelihood of a topology? In this paper, we address this question by benchmarking a wide range of methods for calculating the marginal likelihood of a topology with respect to branch lengths under the JC69 model, the simplest nucleotide substitution model. These approaches include very fast approximations including several based on the Laplace approximation [@tierney1986accurate; @kass1995bayes] and variational approaches [@ranganath2014black]. There are also approaches that require some sampling (though not of topologies), including those that make use of MCMC samples ( bridge sampling, [@overstall2010default; @gronau2017tutorial]) and approaches that employ importance sampling ( na[ï]{}ve Monte Carlo, [@hammersley1964general; @raftery1991stopping]). We also include approaches that make use of a set of so-called power posteriors, including the path sampling [@ogata1989monte; @gelman1998simulating; @lartillot2006computing; @baele2012improving] method frequently used in phylogenetics. Using a set of empirical datasets and a common inference framework, we benchmark 19 methods for computing the marginal likelihood of tree topologies. These 19 methods include some well-known in the phylogenetics literature, some we apply for the first time in phylogenetics, and others that we develop explicitly for this problem. We find that some of these new methods provide estimates that compare favorably to the precise (but slow) state-of-the-art approaches, while running orders of magnitudes more quickly. The title of our paper is adapted from the classic review of matrix exponentiation methods by @Moler1978-gi [@Moler2003-au]; it is not meant to cast doubt on the methods presented here, although we do find that some rather “dubious” methods making strong simplifying assumptions perform surprisingly well! Methods {#methods .unnumbered} ======= [lll]{} Abbreviation & Full name & $\#$ MCMC chains\ ELBO & Evidence Lower Bound & 0\ & Gamma Laplus Importance Sampling & 0$^*$\ & Varational Bayes Importance Sampling & 0$^*$\ BL &  Laplus & 0\ & Gamma Laplus & 0\ & Lognormal Laplus & 0\ MAP & Maximum un-normalized posterior probability & 0\ ML & Maximum likelihood & 0\ NMC & Naïve Monte Carlo & 0$^*$\ BS & Bridge Sampling & 1\ CPO & Conditional Predictive Ordinates & 1\ HM & Harmonic Mean & 1\ & Stabilized Harmonic Mean & 1\ NS & Nested Sampling & multiple short chains\ & Pointwise Predictive Density & 1\ & Path Sampling & 50\ & Modified Path Sampling & 50\ & Stepping Stone & 50\ & Generalized Stepping Stone & 50\ Marginal likelihoods {#marginal-likelihoods .unnumbered} -------------------- Consider a (fixed) unrooted topology $\tau$ for $S$ species with unconstrained branch length vector $\boldsymbol{\theta} = (\theta_1, \theta_2, \dots ,\theta_{2S-3})$ and the JC69 (Jukes-Cantor) model [@jukes1969evolution]. The JC69 model does not any have free parameters as it assumes equal base frequencies and equal substitution rate for all pairs of nucleotides. If branch lengths are measured in units of the expected number of substitutions per site and the JC69 substitution model is employed, the posterior distribution is given by: $$p(\boldsymbol{\theta} \mid \tau, D) = \frac{p(D \mid \boldsymbol{\theta}, \tau) p(\boldsymbol{\theta}\mid \tau)}{\int_{[0,\infty]^{2S-3}} p(D \mid \boldsymbol{\theta}, \tau) p(\boldsymbol{\theta} \mid \tau) \text{d}\boldsymbol{\theta}}.$$ The normalizing constant in the denominator of the right hand side is the marginal likelihood of the phylogenetic tree topology model $\tau$, $p(D \mid \tau)$. It is this marginal likelihood (of a sequence alignment given a topology) that is the quantity of interest in this manuscript. As is typical, we place independent exponential priors on branch lengths with a prior expectation of 0.1 substitutions, such that $p(\boldsymbol{\theta} \mid \tau) = p(\boldsymbol{\theta}) = \prod_{i=1}^{2S-3} p(\theta_i)$, where $p(x)$ is the exponential density. Calculating marginal likelihoods is an area of active statistical research, both inside and outside of phylogenetics. A complete review of all the methods that have been proposed for this purpose is outside the scope of this paper, and we refer readers to reviews by @gelman1998simulating and @gronau2017tutorial. We will first provide a basic sketch of the types of methods we employ (see Table \[tab:method\_names\] for abbreviations). Second, we describe some new methods for calculating the marginal likelihood designed specifically for topologies. Finally, a more detailed explanation of all the methods used in this paper can be found in the supplementary materials. Methods for calculating the marginal likelihood can be broken down into two main categories: sampling-free methods and sampling-based methods. The majority of sampling-free methods revolve around replacing the intractable posterior distribution with one whose normalizing constant can be more easily computed. These approaches include the Laplace approximation [@tierney1986accurate; @kass1995bayes], three new variations on this theme that we introduce here (the Laplus approximations), and a variational Bayes approximation [@ranganath2014black] from which we derive the evidence lower bound (ELBO). We additionally investigate the performance of the maximum likelihood and maximum a posteriori estimators to approximate the marginal likelihood. The sampling-based approaches can further be broken down into importance sampling and MCMC-based aproaches. In importance sampling, samples drawn from a tractable proposal distribution are used to calculate the marginal likelihood using simple identities. How well an importance sampling method works depends on how close the proposal distribution is to the true posterior. We examine three importance sampling approaches, na[ï]{}ve Monte Carlo (NMC) [@hammersley1964general; @raftery1991stopping], which uses the prior distribution as the proposal distribution, and two approaches using more sophisticated proposal distributions. Lastly, the MCMC-based methods can be broken down into those that can be used with a single chain, and those that require many chains. Among single-chain methods, we include the well-known harmonic mean (HM) estimator [@newton1994approximate], a variation thereof known as the stabilized harmonic mean () [@newton1994approximate], bridge sampling (BS) [@overstall2010default; @gronau2017tutorial], conditional predictive ordinates (CPO) [@lewis2014posterior], and the pointwise predictive density () [@vehtari2017practical]. Finally, the nested sampling (NS) method sits somewhere in between the single- and multiple-chain categories as it requires simulations from multiple short MCMC runs [@skilling2004nested; @skilling2006nested; @maturana2018nested]. The final set of methods all require multiple chains, which are “heated” with a heating parameter that interpolates between the posterior distribution and some other distribution. For the path sampling [@ogata1989monte; @gelman1998simulating; @lartillot2006computing; @friel2008marginal; @baele2012improving] and stepping stone (SS) methods [@xie2010improving], the power posterior path links the posterior to the prior distribution. @fan2010choosing proposed the generalized stepping stone (GSS) method in which the path is defined between the posterior and a reference distribution, hence avoiding issues associated with sampling from vague priors. A number of the above methods have been previously applied to phylogenetics, including all power posterior approaches, the harmonic mean, and conditional predictive ordinates. In phylogenetics, path sampling and stepping stone are currently the most widely used methods, and are included in popular inference programs like BEAST [@drummond2012bayesian] and MrBayes [@ronquist2012mrbayes]. ### Laplus {#laplus .unnumbered} The Laplace approximation [@tierney1986accurate; @kass1995bayes] replaces the true log-posterior distribution with a multivariate normal distribution. The mean is taken to be the joint posterior mode ($\boldsymbol{\tilde{\theta}} = (\tilde{\theta}_1,\tilde{\theta}_2, \dots ,\tilde{\theta}_{2S-3})$, and the covariance matrix is taken to be the inverse of the observed information matrix of $l(\boldsymbol{\theta}) = \log(p(D|\boldsymbol{\theta}, \tau)\,p(\boldsymbol{\theta} \mid \tau))$ evaluated at $\boldsymbol{\tilde{\theta}}$. Previous studies have approximated the likelihood surface of phylogenies using multivariate normal distributions [@thorne1998estimating; @guindon2010bayesian], including the use of parameter transformations to account for positivity and skew [@reis2011approximate]. However, the posterior distribution of branch lengths may have its mode at 0 in some dimensions, which is not a shape that can be attained by any transformation of a normal distribution. In related work, the conditional posterior distribution of single branch lengths has been approximated with a gamma distribution, which can accommodate the zero mode, enabling independence sampling [@aberer2015efficient]. We depart from the aforementioned approaches and introduce a novel framework to approximate the joint posterior distribution on branch lengths. For simplicity, in all cases we assume that *a posteriori* branch lengths are independent. This is obviously not true in practice, but we find that posterior correlations are often quite small, and that our independence assumption works well. This assumption also greatly reduces the computational burden by allowing us to sidestep computing every second partial derivative. Our “Laplus” approximation then takes the maximum *a posteriori* (MAP) vector of branch lengths $\boldsymbol{\tilde{\theta}}$ and the vector of second derivatives $\left(\frac{\partial^2l}{\partial \theta_1^2}, \frac{\partial^2l}{\partial \theta_2^2}, \dots, \frac{\partial^2l}{\partial \theta_{2S-3}^2}\right)$ and finds the parameters of our approximating distributions for each branch, $\boldsymbol{\phi}_i$, by matching modes and second derivatives of the approximating and posterior distributions of branch lengths. Unlike the method of moments and maximum likelihood estimation, our approach is fast as it does not require a set of samples to estimate the parameters of the distribution. We consider three distributions for approximating the marginal posteriors of branch lengths: lognormal, gamma, and beta$'$ ( beta prime). The general procedure for the Laplus approximations is similar regardless of what distribution ( the choice of $q$ in $q(x; \boldsymbol{\phi}_{i})$) is chosen to approximate the posterior, and is written here algorithmically: 1. Find the (joint) MAP branch lengths, $\boldsymbol{\tilde{\theta}} = (\tilde{\theta}_1,\tilde{\theta}_2, \dots ,\tilde{\theta}_{2S-3})$ 2. For $i = 1,\dots, 2S-3$ 1. Compute $\frac{\partial^2l}{\partial \theta_i^2}$, the second derivative of the log unnormalized posterior with respect to the $i^{\textnormal{th}}$ branch 2. Find parameters of $\boldsymbol{\phi}_{i}$ by solving $$\begin{aligned} \frac{d^2}{dx^2}\log(q(x; \boldsymbol{\phi}_{i})) &=\frac{\partial^2l}{\partial \theta_i^2}\Big|_{\theta_i = \tilde{\theta}_i}, \\ \text{mode}(q(x; \boldsymbol{\phi}_{i})) &= \tilde{\theta}_i \end{aligned}$$ 3. Catch exceptions 3. Compute the marginal likelihood as $\hat{p}_{\text{Laplus}}(D \mid \tau) = \frac{p(D \mid \boldsymbol{\tilde{\theta}},\tau)p(\boldsymbol{\tilde{\theta}} \mid \tau)}{\prod_i q(\tilde{\theta}_i ; \boldsymbol{\phi}_{i})}$. Exceptions occur when elements of $\boldsymbol{\phi}_{i}$ are outside of the domain of support, when the second derivative is nonnegative (so the posterior has a mode at 0), or when elements of $\boldsymbol{\phi}_{i}$ are otherwise suspect (such as producing particularly high-variance distributions with very short branches). Exceptions and their handling depend on the distributional kernel (choice of $q$), and we defer a full discussion of this to the supplementary material. ### Variational inference {#variational-inference .unnumbered} The main idea behind variational inference is to transform posterior approximation into an optimization problem using a family of approximate densities. The aim is to find the member of that family with the minimum Kullback-Leibler (KL) divergence to the posterior distribution of interest: $$\boldsymbol{\phi}^{*} = \argmin_{\boldsymbol{\phi} \in \boldsymbol{\Phi}} \mathrm{KL}(q(\boldsymbol{\theta}; \boldsymbol{\phi}) \parallel p(\boldsymbol{\theta} \mid D, \tau)),$$ where $q(\boldsymbol{\theta}; \boldsymbol{\phi})$ is the variational distribution parametrized by a vector $\boldsymbol{\phi} \in \boldsymbol{\Phi}$ and KL is defined as $$\mathrm{KL}(q \parallel p)=\int_{\boldsymbol{\theta}} q(\boldsymbol{\theta}; \boldsymbol{\phi}) \log \frac{q(\boldsymbol{\theta}; \boldsymbol{\phi})}{p(\boldsymbol{\theta} \mid D, \tau)}.$$ To minimize the KL divergence, we first rewrite the KL equation: $$\begin{aligned} \mathrm{KL}(q(\boldsymbol{\theta}; \boldsymbol{\phi}) \parallel p(\boldsymbol{\theta} \mid D, \tau)) &= \mathop{\mathbb{E}}[\log q(\boldsymbol{\theta}; \boldsymbol{\phi})] - \mathop{\mathbb{E}}[\log p(\boldsymbol{\theta} \mid D, \tau)] \\ & = \mathop{\mathbb{E}}[\log q(\boldsymbol{\theta}; \boldsymbol{\phi})] - \mathop{\mathbb{E}}[\log p(\boldsymbol{\theta}, D \mid \tau)] + \log p(D \mid \tau), \end{aligned}$$ where the expectations are taken with respect to the variational distribution $q$. The third term $\log p(D \mid \tau)$ on the right hand side of the last equality is a constant with respect to the variational distribution so it can be ignored for the purpose of the minimization. After switching the sign of the other two terms, the minimization problem can be framed as a maximization problem of the function $$\textrm{ELBO}(\boldsymbol{\phi}) = \mathop{\mathbb{E}}[\log p(\boldsymbol{\theta}, D \mid \tau)] - \mathop{\mathbb{E}}[\log q(\boldsymbol{\theta}; \boldsymbol{\phi})].$$ The ELBO is easier to calculate than the KL divergence as it does not involve computing the intractable posterior normalisation term $p(D \mid \tau)$. The ELBO gives a lower bound of the marginal likelihood, the very measure we are interested in estimating here. Here we use the ELBO estimate $\hat{p}_{\textrm{ELBO}}(D \mid \tau) := \max_{\boldsymbol{\phi} \in \boldsymbol{\Phi}}\textrm{ELBO}(\boldsymbol{\phi})$ to approximate the marginal likelihood of a topology. We used a Gaussian variational mean-field approximation applied to log-transformed branch lengths to ensure that the variational distribution stays within the support of the posterior. The mean-field approximation assumes complete factorisation of the distribution over each of the $2S-3$ branch length variables and each factor is governed by its own variational parameters $\boldsymbol{\phi}_i$: $$q(\theta_1, \dots, \theta_{2S-3}; \boldsymbol{\phi}) = \prod_{i=1}^{2S-3} q(\theta_i; \boldsymbol{\phi}_i),$$ where $q(\theta_i; \boldsymbol{\phi}_i)$ is a log-normal density and $\boldsymbol{\phi}_i = (\mu_i, \sigma_i)$. As in the Laplus approximation, this model also assumes that there is no correlation between branches. The variational parameters are estimated using stochastic gradient ascent using a black box approach [@ranganath2014black] similar to the algorithm implemented in Stan [@kucukelbir2015automatic]. ### Importance sampling {#importance-sampling .unnumbered} The Laplus and variational Bayes approximations of the marginal likelihood are fast, but in practice the approximate posterior does not always match the posterior of interest well. Since these methods rely on independent univariate probability distributions ( gamma, normal, etc), samples can be efficiently drawn from the approximate posterior distributions. We thus also used importance sampling to reduce the bias of the Laplus and variational Bayes methods using the approximate posterior distribution as the importance instrument distribution. The importance sampling estimate of $p(D \mid \tau)$ using an approximate normalized probability distribution (instrument distribution) $g$ is $$\hat{p}_{\textrm{IS}}(D \mid \tau) = \frac{1}{N} \sum_{i=1}^N \frac{p(D \mid \tilde{\boldsymbol{\theta}}_i, \tau) p(\tilde{\boldsymbol{\theta}}_i \mid \tau)}{g(\tilde{\boldsymbol{\theta}}_i)}\text{, where } \tilde{\boldsymbol{\theta}}_i \sim g(\boldsymbol{\theta}).$$ Benchmarks {#benchmarks .unnumbered} ---------- We benchmark the 19 methods for estimating fixed-tree marginal phylogenetic likelihood on 5 empirical datasets from a suite of standard test datasets [@lakner2008efficiency; @hohna2011guided; @larget2013estimation; @whidden2015quantifying], which we call DS1 through DS5. These datasets vary from 25 to 50 taxa, with alignment number of sites ranging from 378 to 2520. Instead of focusing primarily on the accuracy of the estimate of the single-tree marginal likelihoods, we focus on the approximate posterior of topologies we obtain by applying our marginal likelihood methods to each and normalizing the result as described below. We take measures of the goodness of these posteriors that directly address approximation error in quantities of interest, namely the posterior probabilities of topologies and the probabilities of tree splits. These are compelling choices because Bayesian phylogenetic inference is not performed to answer the question “what is the marginal likelihood of this topology" but rather to quantify support for evolutionary relationships/hypotheses. We note that the posterior of trees is also useful in other contexts, such as examining the information content of a dataset [@lewis2016estimating]. To compare marginal likelihood methods’ accuracy and precision, we need to establish a ground truth for $p(\tau_i \mid D)$ for each tree topology $\tau_i$. To do this, we use the extensive runs (called golden runs) of MrBayes from @whidden2015quantifying, which consist of 10 chains run for 1 billion generations each (subsampled every 1000 generations), with 25% discarded as burnin and all chains pooled when computing posterior summaries. This results in 7.5 million MCMC samples from 7.5 billion generations, with common diagnostics showing convergence of the chains. The credible sets contain between 5 and 1,141,881 topologies. For datasets DS1 to DS4, we run each of the 19 methods for calculating marginal likelihoods on every tree in the 95% posterior credible set. DS5 has a credible set that is too large (over one million topologies), so we consider only the 1000 most probable trees from this dataset. The only input for each of the 19 methods from the golden runs is the tree topology without branch lengths. In the Golden runs, `MrBayes` was set up to use a uniform prior for topologies and independent exponential priors with mean 0.1 for the branch lengths. After arriving at a set of trees for each benchmark dataset, we renormalize `MrBayes` posterior probabilities so that they sum to one over the selected trees: $\sum_i P(\tau_i \mid D) = 1$. We assume these probabilities form the true posterior mass function of tree topologies and measure accuracy with respect to this function. We use the Bayes rule to convert our approximations of the marginal likelihood to the posterior probability: $$\hat{p}(\tau_i \mid D) = \frac{\hat{p}(D \mid \tau_i) p(\tau_i)}{\sum_j \hat{p}(D \mid \tau_j) p(\tau_j)} = \frac{\hat{p}(D \mid \tau_i)}{\sum_j \hat{p}(D \mid \tau_j)},$$ where the last equality holds because we assumed the uniform prior over the tree topologies. The marginal likelihood estimations were replicated 10 times for each combination of method and dataset, allowing us to derive the standard deviation of the marginal likelihood estimates. We employ two different measures to determine closeness of an approximate posterior to the golden run posterior. Since many questions in phylogenetics concern the probabilities of individual splits, we consider the error in their estimated posterior probabilities. We calculate the root mean-squared deviation (RMSD) of the probabilities of splits, computed as $\text{RMSD} = \sqrt{\frac{1}{S} \sum_i (\hat{f}(s_i) - f(s_i))^2}$, where $s_i$ is a split (or bipartition) and $S$ the number of splits in the tree topology set. The probabilities of a split are given by $f(s_i) = \sum_j p(\tau_j \mid D)\,\mathbbm{1}_{s_i \in \tau_j}$ and $\hat{f}(s_i) = \sum_j \hat{p}(\tau_j \mid D)\,\mathbbm{1}_{s_i \in \tau_j}$, that is, they are the sums of posterior probabilities of the topologies that contain that split. To assess how well the posterior probabilities of topologies are estimated, we use the Kullback-Leibler (KL) divergence from $\hat{\mathbf{p}}= (\hat{p}(\tau_1 \mid D), \dots, \hat{p}(\tau_N \mid D))$ to $\mathbf{p}= (p(\tau_1 \mid D), \dots, p(\tau_N \mid D))$, where $N$ is the number of unique topologies in the 95% posterior credible set of the golden run. This is computed as as $\mathrm{KL}(\mathbf{p} \parallel \hat{\mathbf{p}})=\sum_i p(\tau_i \mid D) \log \frac{p(\tau_i \mid D)}{\hat{p}(\tau_i \mid D)}$. Given that these 19 marginal likelihood calculation methods vary widely in their computational efficiency, we also seek to benchmark the speed of the methods. As our measure of speed, we take the average time (per dataset) required to compute the marginal likelihood of a topology. The speed of these methods depends on a number of dataset-specific features (including on the size of the dataset and the number of phylogenies in the credible set), on run-time decisions (such as the number of MCMC iterations), and on the code that implements them. By incorporating multiple datasets (to average over dataset-specific effects) and implementing the methods in a single package (to control for run-time and implementation-specific effects), we are able to examine the general tradeoff between speed and accuracy, and highlight the use-cases we think the methods are suited for. Every method was implemented within the phylogenetic package `physher` [@fourment2014novel] (<https://github.com/4ment/physher>) and we used the same priors as in the golden runs of MrBayes. Datasets and scripts used in this study are available from <https://github.com/4ment/marginal-experiments/>. We note that this study used a single-threaded version of physher, leaving much room to improve the speed of these embarrassingly parallelizable algorithms. All analyses were run on Intel Xeon E5-26972.60GHz processors running CentOS release 6.1 with 244 GB of RAM. Results {#results .unnumbered} ======= Accuracy and precision {#accuracy-and-precision .unnumbered} ---------------------- ### RMSD {#rmsd .unnumbered} When comparing multiple replicate MCMC analyses (multiple runs), a standard metric in phylogenetics is the average standard deviation of split frequencies (ASDSF). Typically an ASDSF below 0.01 is taken to be evidence that two MCMC analyses are sampling the same distribution. We use the related (but stricter) RMSD as our measure of approximation error (Figure \[fig:rmsd\_by\_ds\]). By considering the plots of split probabilities organized by their RMSD, (Figure \[fig:split\_probs\], Supplementary Figures \[fig:split\_probs\_1\], \[fig:split\_probs\_2\], \[fig:split\_probs\_3\], and \[fig:split\_probs\_4\]), we developed two cutoffs for RMSD to classify method performance. We call methods with RMSD less than 0.01 to be in “good” agreement with ground truth, while we say that methods with RMSD between 0.01 and 0.05 are in “acceptable” agreement. RMSD above 0.05 indicates substantial disagreement between ground truth and estimates. Most of the 19 methods’ estimates fall within these categories consistently across the 5 datasets. MAP, ML,  and BL span the boundary between good and acceptable, while  spans all three categories. Recall that all methods abbreviations are in Table \[tab:method\_names\]. ![Average split posterior RMSD for 10 replicate runs of each dataset. , , BL, MAP, and ML are deterministic and therefore only one replicate is shown. The horizontal dashed and solid lines depict RMSDs of 0.05 and 0.01, respectively.[]{data-label="fig:rmsd_by_ds"}](RMSD_by_method_and_dataset_with_replicates.pdf){width="0.8\linewidth"} ### KL divergence {#kl-divergence .unnumbered} Broadly speaking, there is concordance between the performance of approximations whether measured by KL divergence or RMSD (Figures \[fig:split\_probs\], \[fig:approx\_vs\_srf\]). This is reassuring, as a good approximation should estimate the marginal likelihoods well, which should result in good approximations to the posterior, and thus good estimation of the split probabilities. We also find that the methods do a better job approximating the marginal likelihood of more probable trees than less probable trees (seen as triangular shapes of scatter points in Figure \[fig:approx\_vs\_srf\]). However, even methods that lead to notable scatter between truth and approximation, such as , can yield quite good estimates of the probabilities of splits. Additionally, if the only quantity of interest is the 50% majority-rule consensus tree, then even methods that estimate the marginal likelihood quite poorly can lead to reasonable trees (Figure \[fig:consensus\_trees\]). To get the same consensus tree, a method must merely place the same splits in the upper 50% range of posterior probability, so this measure can hide a substantial amount of variability in the estimated marginal likelihoods. ![The posterior probabilities of all the splits observed in DS5 for a single replicate. MrBayes posteriors are plotted on the x-axis versus the denoted approximation on the y-axis. Points are colored by the thresholds we discuss: RMSD $<$ 0.01 is a good approximation (green), 0.01 $\leq$ RMSD $<$ 0.05 is a potentially acceptable approximation (yellow), and RMSD $\geq$ 0.05 is poor (red). Panels are ordered by RMSD in increasing order.[]{data-label="fig:split_probs"}](DS5_split_probs.pdf){width="0.8\linewidth"} ![The approximate posterior probabilities of the topologies in DS5 versus the ground truth posterior probabilities from MrBayes, plotted on the log scale for clarity. The rank-ordering of the methods is closest to average for DS5. Results are for a single run of each method. Panels are ordered by RMSD in increasing order.[]{data-label="fig:approx_vs_srf"}](DS5_scatterplot.pdf){width="0.8\linewidth"} Speed {#speed .unnumbered} ----- Fast methods can give accurate results, while slow methods need not be accurate (Figure \[fig:time\_vs\_accuracy\]). Indeed,  is very fast to compute and gives good results,  is only slightly slower and gives excellent results, while NS is slow to compute and gives rather bad results for this problem. Method speed is primarily determined by the amount of sampling performed by the method: the more sampling required by a method, the slower it is. The fastest methods are deterministic and do not perform sampling at all, with MAP and ML being the fastest of the 19, requiring only optimization. There is a minor added computational cost of calculating additional derivatives of the phylogenetic likelihood function (here purely the derivatives with respect to branch lengths) in the case of the Laplus approximations. The calculation of the ELBO is slightly slower due to the cost of optimizing the variational parameters through stochastic gradient ascent. The next jump in speed is to methods that perform importance sampling. The single-chain methods are very consistent in time requirements since the computation time is largely dominated by the MCMC. They are notably slower than the importance sampling methods, because MCMC here used one million samples per tree, while we use 10,000 for importance sampling. The slowest methods require running multiple MCMC chains, and aside from  time requirements are essentially identical between these methods. We used 50 power posteriors in our analysis of stepping stone and path sampling methods, and as expected we find that they are very nearly 50 times slower than the single-chain methods. The consistency of the number of chains and the time requirement of the method clearly demonstrates that the largest computational effort is in the MCMC. It is worth noting, though, that after an MCMC analysis has run (power posteriors or single chains), any appropriate method can be used to post-process the chains and calculate the marginal likelihood, as MrBayes does with arithmetic and harmonic means. As an implementation detail of this study, every single-chain method uses the same MCMC samples to estimate the marginal likelihood and similarly, the power posterior-based methods use the same power posterior samples. ![ Average RMSD of splits in the approximate posterior against running time. Text denotes method used, while superscripts label applications to individual datasets. Four methods are omitted for visual clarity: MAP is essentially identical to ML, BL is nearly identical to , and  and  are both similar to . The horizontal dashed and solid lines depict RMSDs of 0.01 and 0.05 respectively. The RMSD is calculated using the average marginal likelihood of each tree from each of 10 replicate analyses. The running time is calculated using the average running time of each tree from each of 10 replicate analyses.[]{data-label="fig:time_vs_accuracy"}](RMSD_vs_time.pdf){width="0.8\linewidth"} Monte Carlo error {#monte-carlo-error .unnumbered} ----------------- No method to estimate the posterior probability of a tree is without sources of error. Monte Carlo error is a feature of all of sampling-based methods we benchmarked, including the methods using at least one MCMC chain and importance sampling methods (marked by asterisks in Table \[tab:method\_names\]). For these methods, and the variational approach (which uses stochastic optimization with noisy gradient estimates and thus also has inter-run variability) we ran 10 replicate analyses (Figure \[fig:estimator\_variability\]). Interestingly, we find that the inter-run variability of the methods is correlated with the goodness of the estimates (and hence the rank-orderings of the methods are similar in Figure \[fig:estimator\_variability\] and Figure \[fig:rmsd\_by\_ds\]). In discussing how well the methods approximate the posterior distribution of trees, to diminish the effects of Monte Carlo error, we use the average estimated marginal likelihood across the replicate analyses. Summary trees {#summary-trees .unnumbered} ------------- The accuracy of summary trees was correlated as expected with the accuracy of the posterior estimate on splits (Figure \[fig:consensus\_trees\]). We use majority-rule consensus trees [@margush1981consensusn], where a split appears in the consensus tree only if it appears in tree topologies whose posterior probabilities sum to at least 0.5. Thus for two approximate posteriors to produce the same summary tree, they must only agree on whether a split probability is above or below this threshold, meaning this is a less sensitive measure of how good an approximate posterior is than RMSD or KL. In Figure \[fig:consensus\_trees\], we show consensus trees for methods representing good approximations (RMSD $<$ 0.01), acceptable approximations (0.01 $\leq$ RMSD $<$ 0.05), and poor approximations (RMSD $\geq$ 0.05) for DS5 for a single run of each method. In this run, every good approximate posterior and most (0.59%) acceptable approximate posteriors produced a consensus tree identical to the golden run consensus tree. A small portion (0.25%) of poor approximate posteriors also produced identical consensus trees. ![Majority rule consensus trees DS5 based on four sources for posterior probabilities of trees. Each taxon is assigned a unique color and the branch leading to that taxon is colored the same in all 4 trees to show differences. The golden run and  trees are identical, while the tree for ML has a Robinson-Foulds distance of 4 to those trees and the tree for NMC a distance 14 (and 10 from the ML tree). Nodes with red circles denote parts of the tree different from the golden run tree.[]{data-label="fig:consensus_trees"}](DS5_consensus_trees.pdf){width="0.8\linewidth"} Discussion {#discussion .unnumbered} ========== In this paper, we present the most comprehensive benchmark to date of methods for computing marginal likelihoods of phylogenetic tree topologies. A number of estimators we benchmark are well known to the phylogenetics community, namely power posterior methods ( ) and the HM. We also include estimators that have been used less frequently in phylogenetics and are mainly more recent proposals: CPO, NS, and the . Three estimators, , , and NMC, to the best of our knowledge, have not previously been used in phylogenetics. Variational approaches have been proposed for models of heterogeneous stationary frequencies [@dang2018stochastic], otherwise intractable phylogenetic models [@jojic2004efficient; @wexler2007variational; @cohn2010mean], and to fit approximations to distributions on trees [@zhang2018sbn], but to our knowledge this is the first application of the ELBO to phylogenetic model comparison. One goal of this paper is to find methods that could work well with MCMC-free tree exploration approaches like PT, which requires evaluating the marginal likelihoods of hundreds or thousands of topologies. Aside from the ELBO, none of the above methods are fast enough to be suitable for this purpose. To this end, we develop the Laplus approximations and importance sampling methods based on Laplus and variational approximations. We also consider simply using the ML and the MAP. Choosing a method to use in practical scenarios {#choosing-a-method-to-use-in-practical-scenarios .unnumbered} ----------------------------------------------- As expected, methods differ drastically in runtime in proportion to the required Monte Carlo sampling effort. The fastest methods took less than one second per topology on all datasets analyzed, while the slowest took over 10,000. Perhaps surprisingly, there is no general tradeoff between speed and accuracy; while the slowest methods are among the most accurate, there are fast methods that are as good. We break the methods down into four categories: slow, moderately slow, fast, and ultrafast, and will now reviewing the methods by category—from slow and well-known to fast and novel—highlighting the best performers and their use cases. At the slow end of the spectrum, we find that the tried-and-true power posterior methods perform quite well, with  providing the best (and most precise) estimates of all 19 methods. The boost in performance compared from  relative to the other power posterior methods comes at the cost of a marginal increase in computation time due to the estimation and multiple evaluation of the reference distribution. The approximations produced by , , and  are all acceptable (i.e. RMSD $< 0.05$), with most of approximations falling into the good category (i.e. RMSD $<0.01$), and are similar in terms of speed, accuracy, and precision. The power posterior methods remain the best general-purpose tools for phylogenetic model comparisons, though they are certainly too slow to explore the tree space produced by PT. In the middle of the speed axis, we find that  is the most promising method, with performance that is on par with , , and . As  requires an order of magnitude less time than these power posterior-based methods, if it is extended to incorporate sampling trees (perhaps following @baele2015genealogical) it could become a valuable general-purpose model selection tool. The other estimators in this category span from poor to acceptable. The HM is a very bad estimator of the marginal likelihood, though the related  produces posteriors that are acceptable. Two other methods similar in spirit to the HM, CPO (a harmonic sitewise approach) and  (a sitewise arithmetic approach), both perform much better than the HM or the . NS would appear to be an unwise choice for estimating the marginal likelihoods of topologies, as it produces poor approximate posteriors. We note that this is a somewhat different application of NS than the recent work by @maturana2018nested, who report better results of using NS when averaging over (ultrametric) trees.  is the best fast method, and one of the best among the 19. With 10,000 samples, it produces estimates of the marginal likelihood on par with  while working three orders of magnitude more quickly.  produces marginal likelihoods almost as good but is somewhat slower. The ELBO, while faster than either  or  (which uses the variational approximation as the importance distribution) is notably worse. It is possible that this approach suffers from getting stuck in local minima, and that multiple starting points could improve its performance, and consequently the performance of . The worst method in this speed category with regards to accuracy, indeed of all 19 methods, is NMC. Among the ultrafast methods, the best candidate is . All the Laplus approximations are capable of yielding quite good estimates of the posterior distribution on trees, though they are quite variable in performance between methods, and  can produce poor approximate posteriors. MAP and ML are faster than any of the Laplus approximations, but are not as good. However, the success of all of these methods is truly remarkable. Empirical posterior distributions on branch lengths are clearly not point-masses, and yet simply normalizing the unnormalized posterior at the maximum outperforms 6 of the 19 tested methods. The success of the Laplus approximations suggests that our assumption of independence of branch lengths may not be too unreasonable, though their rather large inter-dataset variability and the improvement from importance sampling ( ) suggest that relaxing this assumption may improve performance. Future directions {#future-directions .unnumbered} ----------------- We restricted ourselves here to fixed-topology inference under the simplest substitution model. Future work should generalize beyond this simplest model to obtain a marginal likelihood across all continuous model parameters for more complex models. Another direction for future work is to investigate the effect of modelling correlation between model parameters, including within branch lengths. Although our preliminary results suggest that correlation between branch lengths is not strong, this assumption is not likely to hold for other parameters in more sophisticated models, such as the coalescent model in which the tree height/length is likely to be positively correlated with parameters governing population dynamics. Another future research avenue is to find some way to reduce the inter-dataset variability of the Laplus approximations. While this class of methods does very well on some datasets, in others there is a subset of topologies that present difficulties, possibly due to short branches with odd posterior distributions. The problems of identifying these branches and what to do with them remain open, but solving them may greatly improve the performance of the Laplus approximation. For fixed topology models, our results suggest bridge sampling is an accurate estimator that does not require as much compute time as the power posterior-based methods. To apply this method more broadly to the phylogenetic field we must develop novel bridge sampling proposal distributions, perhaps modeling correlation between parameters other than branch lengths, and more importantly proposals that sample a variety of tree topologies. However there has been some work on developing approximations of the posterior probability of trees [@hohna2011guided; @larget2013estimation; @zhang2018sbn], notably within the  framework [@baele2015genealogical]. Another avenue for research would be to develop a diagnostic to determine an appropriate number of power posteriors that is required to accurately estimate marginal likelihoods. Preliminary analyses have shown that the estimates calculated from 100 power posteriors were similar to estimates using 50 steps, it is however possible that fewer steps would be sufficient. Perhaps more enticing, though, is the prospect of integrating one of the fast or ultrafast methods with PT. PT currently uses ML—the fastest method of the 19—because speed is important, but  is comparable in speed while producing much better marginal likelihood estimates, so its inclusion in PT is worth investigating. For the added time cost of drawing samples and calculating additional likelihoods,  achieves an even more impressive estimate of the marginal likelihood than . However, given that PT explores far more trees than it eventually stores, this added time cost is almost certainly prohibitive, unless the number of importance samples can be drastically reduced. Nonetheless, once PT has found a set of high-likelihood trees, it seems prudent to use  on this set to produce the final approximate posterior. Acknowledgements {#acknowledgements .unnumbered} ================ This research funded by National Science Foundation grant CISE-1561334, CISE-1564137 and National Institutes of Health grant U54-GM111274. The research of FAM was supported in part by a Faculty Scholar grant from the Howard Hughes Medical Institute and the Simons Foundation. Computational facilities were provided to MF by the UTS eResearch High Performance Computer Cluster. Methods ======= In the Bayesian framework, the marginal likelihood or evidence of data $D$ conditioned on model $\tau$ with associated parameters $\boldsymbol{\theta} = (\theta_1, \theta_2, \dots, \theta_N)$ is $$p(D \mid \tau) = \int p(D \mid \boldsymbol{\theta}, \tau) p(\boldsymbol{\theta} \mid \tau) d\boldsymbol{\theta},$$ where $p(D \mid \boldsymbol{\theta}, \tau)$ is the probability of the data given parameters $\boldsymbol{\theta}$, $p(\boldsymbol{\theta} \mid \tau)$ is the prior on $\boldsymbol{\theta}$, and the integral is of dimension N. Dependence on model $\tau$ is suppressed in the rest of the document to simplify notation. Laplace method -------------- ### Classical Laplace The Laplace method [@tierney1986accurate] approximates the marginal likelihood by approximating the posterior distribution using a multivariate normal distribution with mean equal to the maximum a posteriori estimates $\boldsymbol{\tilde{\theta}}$, and covariance $\tilde{\Sigma}=(-H)^{-1}$ where $H$ is the Hessian matrix of second derivatives of $\log(p(D \mid \boldsymbol{\theta})p(\boldsymbol{\theta}))$. Specifically, let us define $l(\boldsymbol{\theta}) = \log(p(D \mid \boldsymbol{\theta})p(\boldsymbol{\theta}))$ and Taylor-expand $l(\boldsymbol{\theta})$ around $\boldsymbol{\tilde{\theta}}$. Exponentiating this quadratic approximation leads to a normal distribution with $\boldsymbol{\tilde{\mu}} = \boldsymbol{\tilde{\theta}}$ and $\tilde{\Sigma} = -H^{-1}$. Integrating the normal distribution yields the Laplace marginal likelihood estimator $$\hat{p}_{L}(D) \approx (2 \pi)^{d/2} \det(\tilde{\Sigma})^{1/2} p(D \mid \boldsymbol{\tilde{\theta}}) p(\boldsymbol{\tilde{\theta}}),$$ where $\det(\tilde{\Sigma})$ is the determinant of the covariance matrix. Unfortunately, the above normal approximation is not always accurate in practice. In our specific phylogenetic setting, the positivity of branch lengths creates problems for the normal approximation. It is however possible to improve the normal approximation of the posterior and the Laplace method if we transform each variable $\theta_i$ using a one-to-one twice differentiable function $g$ such as $\theta_i = g(z_i)$ and $z_i = g^{-1}(\theta_i)$. Applying the chain rule, the Hessian of the posterior for the transformed parameters is $$H_{i,j}^z = \frac{\partial^2 l}{\partial z_i \partial z_j} = \begin{cases} \frac{\partial l}{\partial \theta_i} \frac{\partial^2 \theta_i}{\partial z_i^2} + H_{ii} \left(\frac{\partial \theta_i}{\partial z_i}\right)^2 & \text{for } i = j, \\ H_{ij} \frac{\partial \theta_i}{\partial z_i} \frac{\partial \theta_j}{\partial z_j} & \text{otherwise}. \end{cases}$$ The transformation requires an adjustment to account for the distortion of the distribution hence insuring that the distribution integrates to 1. Therefore, given $\boldsymbol{z} \sim \mathcal{N}(\boldsymbol{\mu} ,\boldsymbol{\Sigma })$ the density of $\boldsymbol\theta$ is $$p(\boldsymbol{\theta}) = \mathcal{N}(g^{-1}(\boldsymbol\theta) \mid \boldsymbol{\mu} ,\boldsymbol{\Sigma })\, | \det J_{g^{-1}}(\boldsymbol\theta) |,$$ where $| \det J_{g^{-1}}(\boldsymbol\theta) |$ is the absolute value of the determinant of the Jacobian matrix evaluated at $\boldsymbol\theta$. However, we find in practice that some branch length posteriors are monotonically decreasing functions with modes at 0, and thus the transformation approach is not sufficient to make the normal approximation accurate. ### The Laplus approximations However, while some transformations may work well for a branch or subset of branches, we find in practice that there is no one transformation that works well for all branches on a tree. As an alternative we use a family of approximations inspired by the Laplace that we call the Laplus approximations (in recognition of the fact that they are like the Laplace but designed for parameters on $\mathbb{R}^+$). We share with the Laplace approximation the assumption that the posterior is concentrated around the mode, $\boldsymbol{\tilde{\theta}}$. Unlike the Laplace approximation, we assume that branch lengths are mutually independent, such that we can make the approximation $$p(\boldsymbol{\theta} \mid \tau, D) \approx \prod_i q(\theta_i;\boldsymbol{\phi}_i)$$ Here $q$ is a parametric distribution with known normalizing constant (such as the gamma distribution) that we will use to approximate the posterior distributions for each branch. For a given branch, $\boldsymbol{\phi}_i$ are the parameters of $q$ that approximate the marginal posterior of that branch. Let $C$ be a constant such that $$p(\boldsymbol{\theta} \mid \tau, D) = C \times p(D \mid \tau, \boldsymbol{\theta}) p(\boldsymbol{\theta})$$ That is, $C$ is the inverse of the marginal likelihood that we seek to estimate, and using our approximation above, $$C = \frac{p(\boldsymbol{\theta} \mid \tau, D)}{p(D \mid \tau, \boldsymbol{\theta}) p(\boldsymbol{\theta})} \approx \frac{\prod_i q(\theta_i;\boldsymbol{\phi}_i)}{p(D \mid \tau, \boldsymbol{\theta}) p(\boldsymbol{\theta})}$$ Finally, by applying this equation at the posterior mode, our resulting estimate of the marginal likelihood is $$\hat{p}_{\text{Laplus}}(D) = \hat{C}^{-1} = \frac{p(D|\boldsymbol{\tilde{\theta}})p(\boldsymbol{\tilde{\theta}})}{\prod_i q(\tilde{\theta}_i ; \boldsymbol{\phi}_{i})}$$ The general procedure for the Laplus approximations is similar regardless of parametric distributional family assumption $q$. Our goal is to take the joint MAP estimates of the branch lengths $\boldsymbol{\tilde{\theta}}$ and the vector of second derivatives of the log-posterior $(\frac{\partial^2l}{\partial \theta_1^2}, \frac{\partial^2l}{\partial \theta_2^2}, \dots, \frac{\partial^2l}{\partial \theta_n^2})$ and find the parameters of our approximating distributions for each branch, $\boldsymbol{\phi}_i$, by matching modes and second derivatives of the approximating and posterior distributions of branch lengths. The complete procedure is written here algorithmically. 1. Find the (joint) MAP branch lengths, $\boldsymbol{\tilde{\theta}} = (\tilde{\theta}_1,\tilde{\theta}_2, \dots ,\tilde{\theta}_n)$ 2. for $i$ in $1:n$ 1. Compute $\frac{\partial^2l}{\partial \theta_i^2}$, the second derivative of the log unnormalized posterior with respect to the $i^{\textnormal{th}}$ branch 2. Find parameters of $\boldsymbol{\phi}_{i} $ by solving $$\begin{aligned} \frac{d^2}{dx^2}\log(q(x; \boldsymbol{\phi}_{i})) &= \frac{\partial^2l}{\partial \theta_i^2}\Big|_{\theta_i = \tilde{\theta}_i} \\ \text{mode}(q(x; \boldsymbol{\phi}_{i})) &= \tilde{\theta}_i \end{aligned}$$ 3. Catch exceptions 3. Compute the marginal likelihood as $\hat{p}_{\text{Laplus}}(D) = \frac{p(D|\boldsymbol{\tilde{\theta}})p(\boldsymbol{\tilde{\theta}})}{\prod_i q(\tilde{\theta}_i ; \boldsymbol{\phi}_{i})}$ Exceptions occur when elements of $\boldsymbol{\phi}_{i}$ are outside of the domain of support, when $H_{ii}$ is nonnegative (so the posterior has a mode at 0), or when elements of $\boldsymbol{\phi}_{i}$ are otherwise suspect (such as producing particularly high-variance distributions with very short branches). Exceptions and their handling depend on the distributional assumption, and so we describe exception handling in the section for each distribution individually. We consider three choices for $q$, the gamma distribution, the  distribution, and the lognormal distribution. Since the Laplus method is not derived through a Taylor expansion of the unnormalised posterior, it is not subject to some of the assumptions required by Laplace’s method. Although both methods require the function to be twice differentiable, Laplace’s method assumes that the global maxima $\tilde{\boldsymbol{\theta}}$ is not at the boundary of the interval of integration so that the first derivatives vanishes at $\tilde{\boldsymbol{\theta}}$. Zero-length branches have typically non-zero (i.e. negative) first derivatives and positive second derivatives making the Laplus method attractive. And while it is obvious that there must be some dependence between branch lengths, we find in practice that the posterior correlations between branch lengths are often quite small. ### Gamma-Laplus Here we seek to approximate the marginal posteriors of all branch lengths with gamma distributions. The vector $\boldsymbol{\phi}_{i} = (\alpha_i, \beta_i)$ contains the shape and rate parameters of the gamma distribution; the log probability density function of the gamma is $$\log(\text{Gamma}(x;\alpha,\beta)) = \alpha \log(\beta) - \log(\Gamma(\alpha)) + (\alpha - 1) \log(x) - \beta x.$$ The first and second derivatives of the log gamma distribution with respect to $x$ are given by $$\begin{aligned} \frac{d}{d x}\log(\text{Gamma}(x;\alpha,\beta)) &= \frac{\alpha - 1}{x} - \beta,\\ \frac{d^2}{d x^2}\log(\text{Gamma}(x;\alpha,\beta)) &= -\frac{\alpha - 1}{x^2}.\\\end{aligned}$$ We make use of the second derivative of the log-posterior at the mode, $H_{ii} = \frac{\partial^2l}{\partial \theta_i^2}\Big|_{\theta_i = \tilde{\theta}_i}$ to estimate $\hat{\alpha}_i$ using the second derivative of the log of the gamma distribution. Then we solve for $\hat{\beta}_i$ using the analytic formula for gamma mode: $\tilde{\theta}_i = \frac{\alpha_i - 1}{\beta_i}$. $$\begin{aligned} H_{ii} &= -\frac{\hat{\alpha}_i - 1}{\tilde{\theta}_i^2}\\ \hat{\alpha}_i &= 1 - \tilde{\theta}_i^2 H_{ii}\\ \hat{\beta}_i &= \frac{\hat{\alpha}_i - 1}{\tilde{\theta}_i} = \frac{-\tilde{\theta}_i^2 H_{ii}}{\tilde{\theta}_i} = -\tilde{\theta}_i H_{ii}\end{aligned}$$ We note two exceptions to handle with the  approach. The first case are branches with a mode at 0, which have posteriors that are monotonically decreasing. The second case are branches that are short with oddly large variances. We detect branches of the first type by checking whether $\tilde{\theta}_i < \epsilon_1$ or $H_{ii} >= 0$. These branches are handled by fixing $\hat{\alpha_i} = 1$ (to ensure that the approximation is monotonically decreasing) and fitting $\hat{\beta_i}$ directly using the log-posterior calculated at $N$ points spaced evenly (on the log-scale) between $\tilde{\theta}_i$ and 0.5. We detect branches of the second type by checking whether $\tilde{\theta}_i < \epsilon_2$ and $\frac{\alpha_i}{\beta_i^2} > 0.1$. These branches are handled by fitting $\alpha_i, \beta_i$ to $N$ points spaced evenly (on the log-scale) between $\tilde{\theta}_i$ and 0.5, while constraining $\tilde{\theta}_i = \frac{\hat{\alpha}_i - 1}{\hat{\beta}_i}$ (such that the mode of the approximation to be the mode of the posterior). We use $N = 10$, $\epsilon_1 = 10^{-6}$, and $\epsilon_2 = 10^{-4}$. ### -Laplus Here we seek to approximate the marginal posteriors of all branch lengths as beta$'$ distributions. In this case, the vector $\boldsymbol{\phi}_{i} = (\alpha_i, \beta_i)$ concatenates the shape parameters of the beta$'$ distribution with log probability density function is $$\log(\text{Beta}'(x;\alpha,\beta)) = -\log(B(\alpha,\beta)) + (\alpha - 1) \log(x) - (\alpha+\beta)\log(x+1),$$ where $B$ is the beta function. The first and second derivatives of the log beta$'$ distribution with respect to $x$ are given by $$\begin{aligned} \frac{d}{d x}\log(\text{Beta}'(x;\alpha,\beta)) &= \frac{\alpha - 1}{x} - \frac{\alpha + \beta}{x + 1},\\ \frac{d^2}{d x^2}\log(\text{Beta}'(x;\alpha,\beta)) &= -\frac{\alpha - 1}{x^2} + \frac{\alpha + \beta}{(x + 1)^2}.\\\end{aligned}$$ When $\alpha \leq 1$, the beta$'$ distribution collapses to a monotonically decreasing distribution. When $\alpha = 1$, $$\begin{aligned} \log(\text{Beta}'(x;1,\beta_i)) &= -\log(B(1,\beta_i)) + (1 - 1) \log(x) - (1+\beta_i)\log(x+1),\\ \log(\text{Beta}'(x;1,\beta_i)) &= -\log(B(1,\beta_i)) - (1+\beta_i)\log(x+1),\\ \frac{d}{d x}\log(\text{Beta}'(x; 1,\beta_i)) &= -\frac{1 + \beta_i}{x + 1}.\end{aligned}$$ We make use of the second derivative at the mode, $H_{ii} = \frac{\partial^2l}{\partial \theta_i^2}\Big|_{\theta_i = \tilde{\theta}_i}$ to estimate $\hat{\beta}_i$. Then we solve for $\hat{\alpha}_i$ using the fact that $\tilde{\theta}_i = \frac{\hat{\alpha}_i - 1}{\hat{\beta}_i + 1}$. $$\begin{aligned} H_{ii} &= -\frac{\hat{\alpha}_i - 1}{\tilde{\theta}_i^2} + \frac{\hat{\alpha}_i + \hat{\beta}_i}{(\tilde{\theta}_i + 1)^2}\\ &= -\frac{1}{\tilde{\theta}_i} \frac{\hat{\alpha}_i - 1}{\frac{\hat{\alpha}_i - 1}{\hat{\beta}_i + 1}} + \frac{1}{\tilde{\theta}_i + 1} \frac{\hat{\alpha}_i + \hat{\beta}_i}{\frac{\hat{\alpha}_i + \hat{\beta}_i}{\hat{\beta}_i + 1}}\\ &= -\frac{1}{\tilde{\theta_i}} (\hat{\beta}_i + 1) + \frac{1}{\tilde{\theta}_i + 1} (\hat{\beta}_i + 1)\\ &= (\hat{\beta}_i + 1) \Big(\frac{1}{\tilde{\theta}_i + 1} - \frac{1}{\tilde{\theta}_i}\Big)\\ &= \frac{\hat{\beta}_i + 1}{\tilde{\theta}_i(\tilde{\theta}_i + 1)}\\ \hat{\beta}_i &= -H_{ii} (\tilde{\theta}_i + 1) \tilde{\theta}_i - 1\\ \hat{\alpha}_i &= \tilde{\theta}_i(\hat{\beta}_i + 1) + 1.\end{aligned}$$ We note two exceptions to handle with the BL approach. To start, we check if $\hat{\beta}_i < 0$, which implies $H_{ii} > 0$, meaning the posterior should be monotonically decreasing. In this case, we set $\hat{\alpha}_i = 1$ and use the equations outlined below to fit $\hat{\beta}_i$. We then check if $\hat{\beta}_i < 2$, in which case our approximate posterior has suspiciously high variance, in which case we fit $\alpha_i, \beta_i$ to $N$ points spaced evenly (on the log-scale) between $\tilde{\theta}_i$ and 0.5, while constraining $\tilde{\theta}_i = \frac{\hat{\alpha}_i - 1}{\hat{\beta}_i + 1}$ (such that the mode of the approximation to be the mode of the posterior). When we set $\hat{\alpha}_i = 1$ we can use the first derivative of the log-posterior, $\nabla_{i} = \frac{\partial l}{\partial \theta_i} \Big|_{\theta_i = \tilde{\theta}_i}$, to fit $\hat{\beta}_i$: $$\begin{aligned} \nabla_i &= -\frac{1 + \hat{\beta}_i}{\tilde{\theta}_i + 1},\\ \hat{\beta_i} &= -\nabla_i (\tilde{\theta}_i + 1) - 1.\end{aligned}$$ ### Lognormal-Laplus Here we seek to approximate the marginal posteriors of all branch lengths as lognormal distributions. The vector $\boldsymbol{\phi}_{i} = (\mu_i, \sigma_i)$ concatenates the mean and standard deviation parameters of the lognormal distribution with log probability density function $$\log(\text{Lognormal}(x;\mu_i,\sigma_i)) = -\frac{\log(2\pi)}{2} - \log(x) - \log(\sigma_i) - \frac{(\log(x) - \mu_i)^2}{2\sigma_i^2}.$$ The first and second derivatives of the log lognormal distribution with respect to $x$ are given by $$\begin{aligned} \frac{d}{d x}\log(\text{Lognormal}(x;\mu_i,\sigma_i)) &= -\frac{1}{x} - \frac{\log(x) - \mu_i}{x\sigma_i^2},\\ \frac{d^2}{d x^2}\log(\text{Lognormal}(x;\mu_i,\sigma_i)) &= \frac{1}{x^2} - \frac{-\log(x) + \mu_i + 1}{x^2\sigma_i^2}.\\\end{aligned}$$ We make use of the second derivative at the mode, $H_{ii} = \frac{\partial^2l}{\partial \theta_i^2}\Big|_{\theta_i = \tilde{\theta}_i}$, and the fact that $\tilde{\theta}_i = e^{\mu_i - \sigma_i^2}$ to estimate $\hat{\sigma}_i^2$. Then we solve for $\hat{\mu}_i$ using the fact that $\log(\tilde{\theta}_i) = \mu_i - \sigma_i^2$. $$\begin{aligned} H_{ii} &= \frac{1}{\tilde{\theta}_i^2} - \frac{-\log(\tilde{\theta}_i) + \hat{\mu}_i + 1}{\tilde{\theta}_i^2 \hat{\sigma}_i^2}\\ &= \frac{1}{\tilde{\theta}_i^2} - \frac{-(\hat{\mu}_i - \hat{\sigma}_i^2) + \hat{\mu}_i + 1}{\tilde{\theta}_i^2 \hat{\sigma}_i^2}\\ &= \frac{1}{\tilde{\theta}_i^2} - \frac{1}{\tilde{\theta}_i^2 \hat{\sigma}_i^2} - \frac{\hat{\sigma}_i^2}{\tilde{\theta}_i^2 \hat{\sigma}_i^2},\\ \hat{\sigma}_i^2 &= - \frac{1}{\tilde{\theta}_i^2 H_{ii}},\\ \hat{\mu}_i &= \log(\tilde{\theta}_i) + \hat{\sigma}_i^2.\\\end{aligned}$$ We note two exceptions to handle with the  approach. The first case are branches with a mode at 0, which have posteriors that are monotonically decreasing. The second case are branches that are short with oddly large variances. We nest the cases such that we first check for branches that fall in either category, checking $\tilde{\theta}_i < \epsilon_1$ or $H_{ii} >= 0$ or $\hat{\mu} > 5$ (which happens when $\hat{\sigma}$ is suspiciously large). As there is no parameter regime in which the lognormal is monotonically decreasing, and suspiciously high-variance branches are not fit any better by a lognormal distribution than a gamma distribution, at this point we switch to approximating branches as gamma distributions and proceed with exceptions as in the  approach. Importance sampling ------------------- Importance sampling uses a reference or importance distribution from which values are drawn, allowing summaries to be calculated for an unknown distribution by taking into account the importance weights (probabilities of drawing the sampled values). If $g$ is an importance distribution then $$\begin{aligned} p(D) &= \int p(D \mid \boldsymbol{\theta}) p(\boldsymbol{\theta}) \text{d}\boldsymbol{\theta}\\ &= \int \frac{p(D \mid \boldsymbol{\theta}) p(\boldsymbol{\theta})} {g(\boldsymbol{\theta})} g(\boldsymbol{\theta}) \text{d}\boldsymbol{\theta} \\ &= \mathbb{E}_g \left( \frac{p(D \mid \boldsymbol{\theta}) p(\boldsymbol{\theta})} {g(\boldsymbol{\theta})} \right). \end{aligned}$$ For a normalized density $g$, the estimate is given by, $$\hat{p}_{\text{IS}}(D) = \frac{1}{N} \sum_{i=1}^N \frac{p(D \mid \tilde{\boldsymbol{\theta}_i}) p(\tilde{\boldsymbol{\theta}_i})}{g(\tilde{\boldsymbol{\theta}_i})}, \tilde{\boldsymbol{\theta}}_i \sim g(\boldsymbol{\theta}).$$ For an unnormalized density $q$, the self normalized importance sampling estimate [@mcbook] is given by $$\hat{p}_{\text{IS}}(D) = \frac{\sum_{i=1}^N p(D \mid \tilde{\boldsymbol{\theta}}_i) w(\tilde{\boldsymbol{\theta}}_i)}{\sum_{i=1}^N w(\tilde{\boldsymbol{\theta}}_i)}, \tilde{\boldsymbol{\theta}}_i \sim q(\boldsymbol{\theta}),$$ where $w(\tilde{\boldsymbol{\theta}}_i)$ is the importance weight given by $w(\tilde{\boldsymbol{\theta}}_i)=\frac{p(\tilde{\boldsymbol{\theta}}_i)}{q(\tilde{\boldsymbol{\theta}}_i)}$. Naive Monte Carlo ----------------- The simplest Monte Carlo estimator of the marginal likelihood is defined as the expected value of the likelihood with respect to the prior distribution [@hammersley1964general; @raftery1991stopping]. The so called naive Monte Carlo () estimator can be approximated by drawing $N$ samples ${\boldsymbol{\theta}_1, \boldsymbol{\theta}_2, \dots, \boldsymbol{\theta}_N}$ from the prior distribution and calculating the arithmetic mean of the likelihood. $$\hat{p}_{\text{NMC}}(D) = \frac{1}{N} \sum_{i=1}^N p(D \mid \tilde{\boldsymbol{\theta}_i}), \tilde{\boldsymbol{\theta}}_i \sim p(\boldsymbol{\theta}).$$ Although this approach is fast and unbiased, the high-likelihood region can be distant from the high-prior region. Most $\tilde{\boldsymbol{\theta}}_i$s will therefore be sampled from a region of the likelihood with low probability yielding high variance [@newton1994approximate]. Harmonic mean ------------- The harmonic mean () estimator only requires samples from the posterior generated by a single MCMC or other samplers and is therefore appealing to the user [@newton1994approximate]. The harmonic mean estimator of marginal estimator is equivalent to an importance sampling estimator of $1/p(D)$ with importance distribution $p(\boldsymbol{\theta} \mid D)$: $$\hat{p}_{\text{HM}}(D) = \frac{1}{\frac{1}{N} \sum_{i=1}^N \frac{1}{p(D \mid \tilde{\boldsymbol{\theta}_i})}}, \tilde{\boldsymbol{\theta}}_i \sim p(\boldsymbol{\theta} \mid D).$$ This estimator is unstable due to the possible occurrence of small likelihood values the estimator and hence this estimator has infinite variance. Although the Law of Large Numbers guarantees that this estimator is consistent, the number of samples required to get an accurate estimate can be prohibitively high. Stabilized harmonic mean ------------------------ @newton1994approximate also proposed the stabilized harmonic mean () estimator to address the instability of the  estimator. The  estimator is based on importance sampling scheme where the importance sampling distribution is a mixture of the prior and the posterior: $p^\star(\boldsymbol{\theta}) = \delta p(\boldsymbol{\theta}) + (1 - \delta)p(\boldsymbol{\theta} \mid D)$ where $\delta$ is small, such that $$\hat{p}_{\text{SHM}^*}(D) = \frac{\sum_{i=1}^n \frac{p(D \mid \tilde{\boldsymbol{\theta}}_i)}{\delta \hat{p}_{\text{SHM}^*}(D) + (1 - \delta)p(D \mid \tilde{\boldsymbol{\theta}}_i)} }{\sum_{i=1}^n \{\delta \hat{p}_{\text{SHM}^*}(D) + (1 - \delta)p(D \mid \tilde{\boldsymbol{\theta}}_i)\}^{-1}}, \tilde{\boldsymbol{\theta}}_i \sim p^\star(\boldsymbol{\theta}).$$ Unfortunately this estimator requires simulating from both the posterior and prior. Newton and Raftery proposed to simulate from the posterior and assume that a further $\frac{\delta n}{(1-\delta)}$ observations are drawn from the prior, all of them with their likelihoods equal to their expected value $p(D)$. The likelihood of the imaginary samples drawn from the prior is $p(D \mid \theta_j) = \hat{p}_{SHM}$ for $j=1, \dots, \frac{\delta n}{1 - \delta}$. Then, the approximate marginal likelihood $\hat{p}_{\text{SHM}}(D)$ satisfies the following equation: $$\hat{p}_{\text{SHM}}(D) = \frac{\frac{\delta n}{1 - \delta} + \sum_{i=1}^n \frac{p(D \mid \tilde{\boldsymbol{\theta}}_i)}{\delta \hat{p}_{\text{SHM}}(D) + (1 - \delta)p(D \mid \tilde{\boldsymbol{\theta}}_i)}}{\frac{\delta n}{(1-\delta)\hat{p}_{\text{SHM}}(D)} + \sum_{i=1}^n \{\delta \hat{p}_{\text{SHM}}(D) + (1 - \delta)p(D \mid \tilde{\boldsymbol{\theta}}_i)\}^{-1}}, \tilde{\boldsymbol{\theta}}_i \sim p(\boldsymbol{\theta} \mid D),$$ which is solved by an iterative scheme that updates an initial guess of the marginal likelihood (e.g. harmonic mean estimate) until a stopping criterion is satisfied. In our implementation the recursion stops when the absolute change in $\log \hat{p}_{\text{SHM}}(D)$ is less than $10^{-7}$. @newton1994approximate advocate $\delta = 0.01$ while @lartillot2006computing use $\delta = 0.1$. In this study we used the $\hat{p}_{\text{SHM}}$ with $\delta = 0.01$. Bridge sampling --------------- Bridge sampling () was initially developed to estimate Bayes factors [@kass1995bayes] and was more recently adapted to approximate the marginal likelihood of a single model [@overstall2010default; @gronau2017tutorial]. Following a derivation by @gronau2017tutorial, the bridge sampling estimator is derived from the following identity: $$1 = \frac{\int p(D \mid \boldsymbol{\theta}) p(\boldsymbol{\theta}) h(\boldsymbol{\theta}) g(\boldsymbol{\theta}) d \boldsymbol{\theta}}{\int p(D \mid \boldsymbol{\theta}) p(\boldsymbol{\theta}) h(\boldsymbol{\theta}) g(\boldsymbol{\theta}) d \boldsymbol{\theta}},$$ where $g(\boldsymbol{\theta})$ is the proposal distribution and $h(\boldsymbol{\theta})$ is the bridge function. The bridge function ensures that the denominator in the identity is not zero. Multiplying both sides of the above identity by $p(D)$ the bridge sampling estimator of the marginal likelihood is $$p_{\text{BS}}(D) = \frac{\int p(D \mid \boldsymbol{\theta}) p(\boldsymbol{\theta}) h(\boldsymbol{\theta}) g(\boldsymbol{\theta}) d \boldsymbol{\theta}}{\int h(\boldsymbol{\theta}) g(\boldsymbol{\theta}) p(\boldsymbol{\theta} \mid D) d \boldsymbol{\theta}} = \frac{\mathbb{E}_{g(\boldsymbol{\theta})} (p(D \mid \boldsymbol{\theta}) p(\boldsymbol{\theta}) h(\boldsymbol{\theta}))}{\mathbb{E}_{p(\boldsymbol{\theta} \mid D)} (h(\boldsymbol{\theta}) g(\boldsymbol{\theta}))}.$$ The marginal likelihood is approximated using $n_1$ samples from the posterior distribution and $n_2$ samples from the proposal distribution $$\hat{p}_{\text{BS}}(D) = \frac{1/n_2 \sum_{i=1}^{n_2} (p(D \mid \tilde{\boldsymbol{\theta}_i}) p(\tilde{\boldsymbol{\theta}}_i) h(\tilde{\boldsymbol{\theta}}_i))}{1/n_1 \sum_{j=1}^{n_1} h(\boldsymbol{\theta}_j^*) g(\boldsymbol{\theta}_j^*)}, \tilde{\boldsymbol{\theta}}_i \sim g(\boldsymbol{\theta}), \boldsymbol{\theta}_j^* \sim p(\boldsymbol{\theta} \mid D).$$ Several bridge functions can be used including the so called *optimal bridge function* [@meng1996simulating]: $$h(\boldsymbol{\theta}) = \frac{C}{s_1 p(D \mid \boldsymbol{\theta}) p(\boldsymbol{\theta}) + s_2 p(D) g(\boldsymbol{\theta})},$$ where $s_1 = n_1/(n_1 + n_2)$ and $s_2 = n_2/(n_1 + n_2)$ and $C$ is a constant that cancels out. The definition of the optimal bridge function depends on the marginal likelihood itself, suggesting an iterative scheme to approximate $p(D)$ starting from an initial guess, such as the  estimate. @gronau2017tutorial provide a detailed description of an algorithm. Thermodynamic integration (aka path sampling, power posterior) -------------------------------------------------------------- The thermodynamic integration estimator was introduced by @lartillot2006computing in the phylogenetic context, borrowing ideas from path sampling [@gelman1998simulating] and the physics literature where a large body of research is dedicated to the estimation of normalisation constants. Lartillot and Philippe defined a path going from the prior to the unnormalised posterior $q$ using $$q_\beta = p(D \mid \boldsymbol{\theta})^\beta p(\boldsymbol{\theta})$$ for $\beta \in [0,1]$. The normalisation constant $Z_\beta$ of the tempered unnormalised posterior is therefore $$Z_\beta = \int_{\boldsymbol{\theta}} p(D \mid \boldsymbol{\theta})^\beta p(\boldsymbol{\theta}) d\boldsymbol{\theta}$$ and the log marginal likelihood of the model follows from the path sampling identity: $$\log p(D) = \log Z_1 - \log Z_0 = \int_0^1 \frac{\partial Z_\beta}{\partial \beta} d\beta = \int_0^1 E_{\boldsymbol{\theta} \mid D,\beta}(\log p(D \mid \boldsymbol{\theta})) d\beta .$$ @friel2008marginal worked on similar ideas but differ in the choice of temperature schedule and how the integral over \[0,1\] is approximated. @lartillot2006computing approximate the integral using the Simpson’s rule while @friel2008marginal applied the trapezoidal rule. The interval $\beta \in [0,1]$ is discretized such that $0=\beta_0 < \beta_1< \dots < \beta_K = 1$ and for each $\beta_i$ samples are drawn from $p(\boldsymbol{\theta} \mid D, \beta_i)$ to estimate $E_{\boldsymbol{\theta} \mid D, \beta_i}( \log p(D \mid \boldsymbol{\theta}))$. For example, using the trapezoidal rule the log marginal likelihood of a given model is $$\log \hat{p}_{\text{PS}}(D) \approx \sum_{i=1}^K (\beta_i - \beta_{i-1})\left(\frac{E_{i-1} + E_i}{2}\right),$$ where $E_i = E_{\boldsymbol{\theta} \mid \beta_i} \log p(D \mid \boldsymbol{\theta})$ is the expectation of the log deviance at $\beta_i$. @lartillot2006computing used equally spaced inverse temperatures between 0 and 1, while @friel2008marginal set $\beta_i = (i/K)^5$. It is clear that other temperature schedules can be exploited such as a schedule based on the quantiles of parametric distribution [@xie2010improving] (see stepping stone section) and the adaptive scheme proposed by @friel2014improving. @friel2014improving subsequently proposed a modified trapezoidal rule that uses the variance of the samples to improve the approximation: $$\log \hat{p}_{\text{MPS}}(D) \approx \sum_{i=1}^K (\beta_i - \beta_{i-1})\left(\frac{E_{i-1} + E_i}{2}\right) - \sum_{i=1}^K \frac{(\beta_i - \beta_{i-1})^2}{12} \left(V_{i} - V_{i+1}\right),$$ where $V_i = V_{\boldsymbol{\theta} \mid \beta_i}(\log p(D \mid \boldsymbol{\theta}))$ is the variance of the log deviance at $\beta_i$. Stepping stone -------------- @xie2010improving proposed the stepping stone () algorithm that is related to the path sampling approach described in the previous section. It uses a series of distributions defining a path between the prior and posterior and therefore inherits the computational burden of path sampling. Thermodynamic integration and stepping stone differ in the choice of $\beta$ values: @xie2010improving set $\beta_1, \dots, \beta_n$ equal to the quantiles of a density with fixed parameters (e.g. beta distribution). This approach allows for a more intensive sampling of power posteriors with small $\beta$ values, for which the posterior is changing rapidly. Let’s define the unnormalized power posterior distribution $q_\beta = p(D \mid \boldsymbol\theta)^\beta p(\boldsymbol\theta)$ and normalized power posterior distribution $p_\beta = \frac{q_\beta}{c_\beta}$, where $c_\beta$ is the power marginal likelihood of the data. The aim of the method is to estimate the ratio $r_{\text{SS}} = c_{1.0}/c_{0.0}$, which is equal to $c_{1.0}$ if the prior is proper. This ratio can be expanded into a series of telescopic product of ratios using intermediate power posteriors $$r_{\text{SS}} = \frac{c_{1.0}}{c_{0.0}} = \prod_{k=1}^K \frac{c_{\beta_k}}{c_{\beta_{k-1}}} = \prod_{k=1}^K r_{\text{SS},k},$$ where $r_{\text{SS},k} = c_{\beta_k}/c_{\beta_{k-1}}$ for $k = 1, \dots, K$. @xie2010improving estimate each ratio $c_{\beta_k}/c_{\beta_{k-1}}$ by importance sampling using $p_{\beta_{k-1}}$ as the importance distribution. Using the definition of importance sampling the $k^{th}$ ratio is $$\hat{r}_{\text{SS},k} = \frac{1}{n} \sum_{i=1}^n \frac{p(D \mid \boldsymbol{\theta}_{k-1,i})^{\beta_k}}{p(D \mid \boldsymbol{\theta}_{k-1,i})^{\beta_{k-1}}} = \frac{1}{n} \sum_{i=1}^n p(D \mid \boldsymbol{\theta}_{k-1,i})^{\beta_k-\beta_{k-1}},$$ where $p(D \mid \theta_{k-1,i})$ is the likelihood function evaluated at $\theta_{k-1,i}$, the $i^{th}$ MCMC sample sampled from $p_{\beta_{k-1}}$. The product of the $K$ ratios $\hat{r}_{\text{SS},k}$ yields the estimate of the marginal likelihood $$\hat{p}_{\text{SS}} = \prod_{k=1}^K \hat{r}_{\text{SS},k}.$$ Generalized stepping stone -------------------------- Although stepping stone proved to be more accurate than other approaches, such as path sampling [@xie2010improving], sampling distributions close to the prior (i.e., small $\beta$ values) can be difficult, particularly if the prior is diffuse. @fan2010choosing proposed to generalize the stepping stone method using a reference distribution that approximates the posterior distribution of interest using samples from the posterior distribution to parametrize the reference distribution. The reference distribution can be independent probability densities from the same family as the prior distribution or the product of densities with the same support. In our study the priors are exponential distributions, but we used gamma distributions that are parametrized using the method of moments. The shape and rate parameters are estimated by matching the first two moments of the gamma distribution to the marginal posterior sample mean and variance. In the same vein as the  method, the unnormalized and normalized power posterior distributions in the generalized stepping stone () approach are $$\begin{aligned} q_\beta &= \big(p(D \mid \boldsymbol\theta) p(\boldsymbol\theta)\big)^\beta \big(p_0(\boldsymbol\theta; \boldsymbol\phi)\big)^{1-\beta},\\ p_\beta &= \frac{q_\beta}{c_\beta},\end{aligned}$$ where $p(D \mid \boldsymbol\theta)$ is the likelihood function, $p(\boldsymbol\theta)$ is the prior distribution, $p_0$ is the reference distribution parametrized by $\boldsymbol\phi$, and $c_\beta$ is the (power) marginal likelihood of the data. The key difference with the  approach is that for $\beta=0$ the power posterior is equivalent to the reference distribution. As for the  method, the aim of this method is to estimate the ratio $r_{\text{GSS}} = c_{1.0}/c_{0.0}$ using importance sampling. The ratio $\hat{r}_{\text{GSS}, k}$ is estimated using $n$ samples from $p_{\beta_{k-1}}$: $$\hat{r}_{\text{GSS},k} = \frac{1}{n} \sum_{i=1}^n \left( \frac{p(D \mid \boldsymbol\theta_{k-1, i}) p(\boldsymbol\theta_{k-1,i})}{p_0(\boldsymbol\theta_{k-1,i}; \boldsymbol\phi)} \right)^{\beta_k - \beta_{k-1}}.$$ Combining $\hat{r}_{\text{GSS},k}$ for all $K$ ratios yields the marginal likelihood estimator: $$\hat{p}_{\text{GSS}} = \prod_{k=1}^K \hat{r}_{\text{GSS},k}.$$ Nested sampling --------------- Nested sampling is a Monte Carlo method that aims at calculating the marginal likelihood using a change of variable [@skilling2004nested; @skilling2006nested]. It transforms the multidimensional evidence integral over the parameter space into a more manageable one-dimensional integral over the likelihood space. Skilling defines the prior volume as $dX = p(\boldsymbol{\theta}) d \boldsymbol{\theta}$ so that $$X(\lambda) = \int _{\mathcal{L}(\boldsymbol{\theta}) > \lambda} p(\boldsymbol{\theta}) d \boldsymbol{\theta}, \label{eq:nsX}$$ where $\mathcal{L}(\boldsymbol{\theta})$ is the likelihood function and the integral is taken over the region bounded by the iso-likelihood contour $\mathcal{L}(\boldsymbol{\theta}) = \lambda$. The marginal likelihood becomes a one-dimensional integral over unit range $$p_{\text{NS}}(D) = \int_0^1 L(X) dX,$$ where $L(X)$ is the inverse function of $X(\lambda)$. Assuming that $L(X)$ can be computed for a sequence of decreasing values $0 < X_m < \dots < X_0 = 1$, the unit integral can be approximated using quadrature techniques as the weighted sum: $$\hat{p}_{\text{NS}}(D) \approx \sum_{i=1}^m L(X_i) w_i,$$ where $w_i = X_{i} - X_{i-1}$. The nested sampling algorithm uses a clever process of sampling from the prior (hence $dX$) and conditioning on the likelihood being above a given size (to achieve the likelihood condition of ) to approximate the input to such a quadrature technique [@skilling2006nested; @maturana2018nested]. The algorithm is initialized with $N$ samples $\{\boldsymbol{\theta}_1, \dots, \boldsymbol{\theta}_N\}$ drawn from the prior and their corresponding likelihoods are calculated $\{\mathcal{L}(\boldsymbol{\theta}_1), \dots, \mathcal{L}(\boldsymbol{\theta}_N)\}$. The sample with the lowest likelihood $L_{\min}$ is discarded from the set and replaced by a new sample $\boldsymbol{\theta}^*$ drawn from the prior subject to the constraint $L > L_{\min}$. When we use the discarded point as an $X_i$, the other points in the set of course satisfy the likelihood constraint. There are a variety of choices for terminating the algorithm [@maturana2018nested]. We choose to terminate when the absolute change in $\log(\hat{p}_{\text{NS}}(D))$ is less than $10^{-6}$. Posterior predictive model selection ------------------------------------ As an alternative to the marginal likelihood, the fit of a model can be assessed through the accuracy of its predictions [@gelman1996posterior]. The probability distribution of a new data set $\tilde{D}$ having observed data set $D$ is defined as $$p(\tilde{D} \mid D) = \int p(\tilde{D} \mid \boldsymbol{\theta}) p(\boldsymbol{\theta} \mid D) d \boldsymbol{\theta}.$$ ### Log pointwise predictive density A related quantity is the expected log pointwise predictive density [@vehtari2017practical] for a new data set, with $n$ data points, is defined as $$\text{elpd} = \sum_{i=1}^n \int p_t(\tilde{D}_i) \log p(\tilde{D}_i \mid D) d \tilde{D_i},$$ where $p_t(\tilde{D}_i)$ is the distribution representing the true data-generating process for $\tilde{D}_i$. In the phylogenetic framework, the observation $D_i$ corresponds to a single site in the alignment. Since the $p_t$ is not known, one can use cross-validation to approximate elpd (see next section). As in [@vehtari2017practical], we define the log pointwise predictive density $$\text{lpd} = \sum_{i=1}^n \log p(D_i \mid D) = \sum_{i=1}^n \log \int p(D_i \mid \boldsymbol{\theta}) p(\boldsymbol{\theta} \mid D) d \boldsymbol{\theta},$$ where $p(D_i \mid \boldsymbol{\theta})$ is the likelihood of the $i^{th}$ observation. The log pointwise predictive density can be estimated using $S$ draws $\boldsymbol{\theta}_1, \dots, \boldsymbol{\theta}_S$ from the posterior distribution $p(\boldsymbol{\theta} \mid D)$, by summing over the $n$ data points $$\widehat{\text{lpd}} = \sum_i^n \log \Big(\frac{1}{S}\sum_{s=1}^S p(D_i \mid \boldsymbol{\theta}_s)\Big), \boldsymbol{\theta}_s \sim p(\boldsymbol{\theta} \mid D).$$ We compared the fit of our topology models using the predictive accuracy approximation $\widehat{\text{lpd}}$ $$\log \hat{p}_{\text{PPD}}(D) = \widehat{\text{lpd}}$$ as an estimate of the log marginal likelihood. Although we are not aware of others using it in this way, we have found that it provides a reasonable approximation. However, the lpd of observed data $D$ is an overestimate of the elpd for future data [@vehtari2017practical]. ### Conditional predictive ordinates A related approach is the conditional predictive ordinates (CPO) method based on Bayesian leave-one-out (LOO). The leave-one-out estimate of the predictive density for a datapoint is $$\text{elpd}_{\text{loo}} = \sum_{i=1}^n \log p(D_i \mid D_{-i}) = \sum_{i=1}^n \log \ \int p(D_i \mid D_{-i}, \boldsymbol{\theta}) p(\boldsymbol{\theta} \mid D_{-i}) d \boldsymbol{\theta},$$ where $p(D_i \mid D_{-i})$ is the leave-one-out predictive density (aka conditional predictive ordinate) given the data without the $i^{th}$ data point.\ The CPO estimate of this is given by $$\hat{p}(D_i \mid D_{-i}) = \frac{1}{\frac{1}{S}\sum_{i=1}^S \frac{1}{p(D_i \mid \boldsymbol{\theta}_s)}}, \boldsymbol{\theta}_s \sim p(\boldsymbol{\theta} \mid D).$$ The resulting estimate of the log marginal likelihood (called the log pseudo-marginal likelihood by @lewis2014posterior) is given by $$\log \hat{p}_{\text{CPO}}(D) = \widehat{\text{lpd}_\text{loo}} = \sum_{i=1}^n \log \hat{p}(D_i \mid D_{-i})$$ Variational inference --------------------- Variational Bayes methods provide an analytical approximation to the posterior probability and a lower bound for the marginal likelihood. The main idea is to choose a family of distributions $q$ parametrised with parameters $\boldsymbol{\phi}$ and to minimize the Kullback Leibler (KL) divergence from variational distribution $q$ to the posterior distribution $p$ of interest $$\boldsymbol{\phi}^{*} = \argmin_{\boldsymbol{\phi} \in \boldsymbol{\Phi}} \mathrm{KL}(q(\boldsymbol{\theta}; \boldsymbol{\phi}) \parallel p(\boldsymbol{\theta} \mid D)).$$ It is difficult to minimise the KL divergence directly but much easier to minimize a function that is equal to it up to a constant. Expanding the KL divergence we get $$\begin{aligned} \mathrm{KL}(q(\boldsymbol{\theta}; \boldsymbol{\phi}) \parallel p(\boldsymbol{\theta} \mid D)) &= \mathop{\mathbb{E}}[\log q(\boldsymbol{\theta}; \boldsymbol{\phi})] - \mathop{\mathbb{E}}[\log p(\boldsymbol{\theta} \mid D)] \\ & = \mathop{\mathbb{E}}[\log q(\boldsymbol{\theta}; \boldsymbol{\phi})] - \mathop{\mathbb{E}}[\log p(\boldsymbol{\theta}, D)] + \log p(D)\\ & = -\textrm{ELBO}(\boldsymbol{\phi}) + \log p(D), \end{aligned}$$ where $\textrm{ELBO}(\boldsymbol{\phi}) = \mathop{\mathbb{E}}[\log p(\boldsymbol{\theta}, D)] - \mathop{\mathbb{E}}[\log q(\boldsymbol{\theta}; \boldsymbol{\phi})]$. This equation suggests that the $\textrm{ELBO}(\boldsymbol{\phi})$ is the lower bound of the evidence: $\log p(D) \geq \textrm{ELBO}(\boldsymbol{\phi})$. Instead of minimizing KL divergence, we maximize the evidence lower bound: $$\textrm{ELBO}(\boldsymbol{\phi}) = \mathop{\mathbb{E}}_{q(\boldsymbol{\theta}; \boldsymbol{\phi})}[\log p(D, \boldsymbol{\theta}) - \log q(\boldsymbol{\theta}; \boldsymbol{\phi})].$$ Several variational distributions can be used including the mean-field and fullrank Gaussian distributions. The fullrank model uses a multivariate Gaussian distribution to model the correlation between variables while the meanfield distribution assumes a diagonal covariance matrix. In this study we used the meanfield model hence taking the assumption that there is no correlation between the branch lengths of the phylogeny: $$q(\boldsymbol{\theta}; \boldsymbol{\phi}) = \mathcal{N}(\boldsymbol{\theta}; \boldsymbol{\mu}, \diag(\boldsymbol{\sigma}^2)) = \prod_{i=1}^n \mathcal{N}(\theta_i; \mu_i, \sigma_i^2).$$ It is common to use stochastic gradient ascent algorithm to maximise the ELBO as long as the model is differentiable [@ranganath2014black; @kucukelbir2015automatic]. In the phylogenetic context the derivative of posterior with respect to the branch lengths can be derived analytically without resorting to approximations such as finite differences. We used a log transform on the branch lengths to ensure that the variational distribution stays within the support of the posterior. Given an optimized variational model we used the ELBO as an approximation of the marginal likelihood $$\hat{p}_{\text{ELBO}}(D) = \max_{\boldsymbol{\phi} \in \boldsymbol{\Phi}}\textrm{ELBO}(\boldsymbol{\phi}).$$ The ELBO estimates can have high variance and might be of little use to discriminate between closely related models (in the KL sense). We used importance sampling to calculate the marginal likelihood of a model using the variational distribution $q$ as the importance distribution. This yields the $\hat{p}_{\text{VBIS}}(D)$ estimator: $$\hat{p}_{\text{VBIS}}(D) = \frac{1}{N} \sum_{i=1}^N \frac{p(D \mid \tilde{\boldsymbol{\theta}_i}) p(\tilde{\boldsymbol{\theta}_i})}{q_{\text{ELBO}}(\tilde{\boldsymbol{\theta}_i})}, \tilde{\boldsymbol{\theta}}_i \sim q_{\text{ELBO}}(\boldsymbol{\theta}).$$ Supplementary Figures ===================== For completion, we include here equivalents of Figure \[fig:approx\_vs\_srf\] and Figure \[fig:split\_probs\] for datasets DS1-4. We also include versions of Figure \[fig:time\_vs\_accuracy\] and Figure \[fig:rmsd\_by\_ds\] that use KL divergence instead of RMSD as the measure of accuracy. The KL and RMSD results are qualitatively similar. ![The posterior probabilities of all the splits observed in DS1 for a single replicate. MrBayes posteriors are plotted on the x-axis versus the denoted approximation on the y-axis. The line $y=x$ is provided for ease of interpretation, and points are colored by the thresholds we discuss: RMSD $<$ 0.01 is a good approximation (green), 0.01 $\leq$ RMSD $<$ 0.05 is a potentially acceptable approximation (yellow), and RMSD $\geq$ 0.05 is poor (red). Panels are ordered by RMSD in increasing order.[]{data-label="fig:split_probs_1"}](DS1_split_probs.pdf){width="0.8\linewidth"} ![The posterior probabilities of all the splits observed in DS2 for a single replicate. MrBayes posteriors are plotted on the x-axis versus the denoted approximation on the y-axis. The line $y=x$ is provided for ease of interpretation, and points are colored by the thresholds we discuss: RMSD $<$ 0.01 is a good approximation (green), 0.01 $\leq$ RMSD $<$ 0.05 is a potentially acceptable approximation (yellow), and RMSD $\geq$ 0.05 is poor (red). Panels are ordered by RMSD in increasing order.[]{data-label="fig:split_probs_2"}](DS2_split_probs.pdf){width="0.8\linewidth"} ![The posterior probabilities of all the splits observed in DS3 for a single replicate. MrBayes posteriors are plotted on the x-axis versus the denoted approximation on the y-axis. The line $y=x$ is provided for ease of interpretation, and points are colored by the thresholds we discuss: RMSD $<$ 0.01 is a good approximation (green), 0.01 $\leq$ RMSD $<$ 0.05 is a potentially acceptable approximation (yellow), and RMSD $\geq$ 0.05 is poor (red). Panels are ordered by RMSD in increasing order.[]{data-label="fig:split_probs_3"}](DS3_split_probs.pdf){width="0.8\linewidth"} ![The posterior probabilities of all the splits observed in DS4 for a single replicate. MrBayes posteriors are plotted on the x-axis versus the denoted approximation on the y-axis. The line $y=x$ is provided for ease of interpretation, and points are colored by the thresholds we discuss: RMSD $<$ 0.01 is a good approximation (green), 0.01 $\leq$ RMSD $<$ 0.05 is a potentially acceptable approximation (yellow), and RMSD $\geq$ 0.05 is poor (red). Panels are ordered by RMSD in increasing order.[]{data-label="fig:split_probs_4"}](DS4_split_probs.pdf){width="0.8\linewidth"} ![The approximate posterior probabilities of the topologies in DS1 versus the ground truth posterior probabilities from MrBayes, plotted on the log scale for clarity. Results are for a single run of each method. Panels are ordered by RMSD in increasing order.[]{data-label="fig:approx_vs_srf_1"}](DS1_scatterplot.pdf){width="0.8\linewidth"} ![The approximate posterior probabilities of the topologies in DS2 versus the ground truth posterior probabilities from MrBayes, plotted on the log scale for clarity. Results are for a single run of each method. Panels are ordered by RMSD in increasing order.[]{data-label="fig:approx_vs_srf_2"}](DS2_scatterplot.pdf){width="0.8\linewidth"} ![The approximate posterior probabilities of the topologies in DS3 versus the ground truth posterior probabilities from MrBayes, plotted on the log scale for clarity. Results are for a single run of each method. Panels are ordered by RMSD in increasing order.[]{data-label="fig:approx_vs_srf_3"}](DS3_scatterplot.pdf){width="0.8\linewidth"} ![The approximate posterior probabilities of the topologies in DS4 versus the ground truth posterior probabilities from MrBayes, plotted on the log scale for clarity. Results are for a single run of each method. Panels are ordered by RMSD in increasing order.[]{data-label="fig:approx_vs_srf_4"}](DS4_scatterplot.pdf){width="0.8\linewidth"} ![Average Kullback-Leibler (KL) divergence from MrBayes posteriors to approximate posteriors for each method on each dataset for 10 replicates. , , BL, MAP, and ML are deterministic and therefore only one replicate is shown.[]{data-label="fig:kl_divergences"}](kl_by_method_and_dataset_with_replicates.pdf){width="0.8\linewidth"} ![Average Kullback-Leibler (KL) divergence from MrBayes posteriors to approximate posteriors of splits in the approximate posterior against running time. Text denotes method used, while superscripts label applications to individual datasets. Four methods are omitted for visual clarity: MAP is essentially identical to ML, BL is nearly identical to , and  and  are both similar to . The horizontal dashed and solid lines depict RMSDs of 0.01 and 0.05 respectively. The KL divergence is calculated using the average marginal likelihood of each tree from each of 10 replicate analyses. The running time is calculated using the average running time of each tree from each of 10 replicate analyses.[]{data-label="fig:time_vs_accuracy_kl"}](KL_vs_time.pdf){width="0.8\linewidth"} ![Standard error of the Monte-Carlo-based estimators. Each point represents the standard error of an individual tree across the 10 replicate analyses for each estimator.[]{data-label="fig:estimator_variability"}](marginal-se.pdf){width="0.8\linewidth"}

--- abstract: 'We propose a numerical algorithm for finding optimal measurements for quantum-state discrimination. The theory of the semidefinite programming provides a simple check of the optimality of the numerically obtained results. With the help of our algorithm we calculate the minimum attainable error rate of a device discriminating among three particularly chosen non-symmetric qubit states.' address: 'Department of Optics, Palacky University, 17. listopadu 50, 77200 Olomouc, Czech Republic' author: - 'M. Ježek, J. Řeháček[^1], and J. Fiurášek' title: 'Finding optimal strategies for minimum-error quantum-state discrimination' --- Nonorthogonality of quantum states is one of the basic features of quantum mechanics. Its deep consequences are reflected in all quantum protocols. For instance, it is well known that perfect decisions between two nonorthogonal states cannot be made. This has important implications for the information processing at the microscopic level since it sets a limit on the amount of information that can be encoded into a quantum system. Although perfect decisions between nonorthogonal quantum states are impossible, it is of importance to study measurement schemes performing this task in the optimum, though imperfect, way. Two conceptually different models of decision tasks have been studied. The first one is based on the minimization of the Bayesian cost function, which is nothing but a generalized error rate [@helstrom]. In the special case of linearly independent pure states, the second model — unambiguous discrimination of quantum states — makes an interesting alternative. The latter scheme combines the error-less discrimination with a certain fraction of inconclusive results [@ivanovic; @dieks; @peres]. Ambiguous as well as unambiguous discrimination schemes have been intensively studied over the past few years. In consequence, the optimal measurements distinguishing between pair, trine, tetrad states, and linearly independent symmetric states are now well understood [@yuen; @Chefles_Barnett_1997; @Chefles_1998; @Chefles_Barnett_1998; @Phillips_Barnett_Pegg_1998; @sasaki; @Chefles_2000; @walgate; @Virmani_et_al_2001; @Zhang_et_al_2001; @Barnett_2001; @Chefles_2001; @barnett]. Many of the theoretically discovered optimal devices have already been realized experimentally, mainly with polarized light [@Huttner_et_al_1996; @Barnett_Riis_1997; @Clarke_Chefles_Barnett_Riis_2001; @Clarke_et_al_2001]. As an example of the practical importance of the optimal decision schemes let us mention their use for the eavesdropping on quantum cryptosystems [@Ekert_et_al_1994; @Dusek_et_al_2000]. The purpose of this paper is to develop universal method for optimizing ambiguous discrimination between generic quantum states. Assume that Alice sets up $M$ different sources of quantum systems living in $p$-dimensional Hilbert space. The complete quantum-mechanical description of each source is provided by its density matrix. Alice chooses one of the sources at random using a chance device and sends the generated quantum system to Bob. Bob is also given $M$ numbers $\{\xi_i\}$ specifying probabilities that $i$-th source is selected by the chance device. Bob is then required to tell which of the $M$ sources $\{\rho_i\}$ generated the quantum system he had obtained from Alice. In doing this he should make as few mistakes as possible. It is well known [@helstrom] that each Bob’s strategy can be described in terms of an $M$-component probability operator measure (POM) $\{\Pi_j\}$, $0<\Pi_i<1$, $\sum_j \Pi_j=1$. Each POM element corresponds to one output channel of Bob’s discriminating apparatus. The probability that Bob points his finger at the $k$-th source while the true source is $j$ is given by the trace rule: $P(k|j)={\rm Tr}\rho_j\Pi_k$. Taking the prior information into account, the average probability of Bob’s success in repeated experiments is $$\label{probab} P_s=\sum_{j=1}^M\xi_j{\rm Tr}\rho_j\Pi_j.$$ Since the objective is to keep Bob’s error rate as low as possible we should maximize this number over the set of all $M$-component POMs. In compact form the problem reads: $$\label{problem} \begin{array}{l} \rm{maximize}\; P_s\;\mbox{subject to constraints}\\[2pt] \Pi_j\ge 0, \quad j=1\ldots M,\\[3pt] \sum_j\Pi_j=1. \end{array}$$ Unfortunately, attacking this problem by analytical means has chance to succeed only in the simplest cases ($M=2$) [@helstrom], or cases with symmetric or linearly independent states [@yuen; @sasaki; @Barnett_2001; @barnett; @ban]. In most situations one must resort to numerical methods. In the following we will use the calculus of variations to derive a simple iterative algorithm that provides a convenient way of dealing with the problem (\[problem\]). This approach has already found its use in the optimization of teleportation protocols [@teleportation] and maximum-likelihood estimation of quantum measurements [@ml-measurements]. We are going to seek the global maximum of the success functional $P_s$ subject to the constraints given in Eq. (\[problem\]). To take care of the first constraint we will decompose the POM elements as follows $\Pi_j=A_j^{\dag}A_j,\; j=1\ldots M$. The other constraint (completeness) can be incorporated into our model using the method of uncertain Lagrange multipliers. Putting all things together, the functional to be maximized becomes $$\label{funct} {\cal L}=\sum_j \xi_j\rm{Tr}\{\rho_j A_j^{\dag}A_j\}- \rm{Tr}\{\lambda\sum_j A_j^{\dag}A_j\},$$ where $\lambda$ is a Hermitian Lagrange operator. This expression is now to be varied with respect to $M$ independent variables $A_j$ to yield a necessary condition for the extremal point in the form of a set of $M$ extremal equations for the unknown POM elements: $\xi_j\rho_j\Pi_j=\lambda\Pi_j,\; j=1\ldots M$. For our purposes it is advantageous to bring these equations to an explicitly positive semidefinite form, $$\label{extremal} \Pi_j=\xi_j^2\lambda^{-1}\rho_j\Pi_j\rho_j\lambda^{-1}, \quad j=1\ldots M.$$ Lagrange operator $\lambda$ is obtained by summing Eq. (\[extremal\]) over $j$, $$\label{lagrange} \lambda=\left(\sum_j\xi_j\rho_j\Pi_j\rho_j\right)^{1/2}.$$ The iterative algorithm comprised of the $M+1$ equations (\[extremal\]) and (\[lagrange\]) is the main formal result of this paper. One usually starts from some “unbiased” trial POM $\{\Pi^0_j\}$. After plugging it in Eq. (\[lagrange\]) the first guess of the Lagrange operator $\lambda$ is obtained. This operator is, in turn, used in Eq. (\[extremal\]) to get the first correction to the initial-guess strategy $\{\Pi^0_j\}$ [@pretty-good]. The procedure gets repeated, until, eventually, a stationary point is attained. Notice that both the positivity and completeness of the initial POM are preserved in the course of iterating. Since equations (\[extremal\]) and (\[lagrange\]) represent only a necessary condition for the extreme, one should always check the optimality of the stationary point. In the following we will make use of the theory of the semidefinite programming (SDP) [@semidef] to derive a criterion of the optimality of the iteratively obtained POM, which turns out to be the well-known Helstrom condition [@helstrom]. SDP theory also provides alternative means of solving the problem (\[problem\]) numerically. Recently, it has been pointed out [@semidef-appl] that many problems of the quantum-information processing can be formulated as SDP problems. For instance, let us compare our original problem (\[problem\]) to the SDP dual problem that is defined as follows: $$\label{semi-dual} \begin{array}{l} {\rm maximize}\; -{\rm Tr} F_0 Z,\\[2pt] Z\ge 0,\\[3pt] {\rm Tr} F_i Z=c_i,\quad i=1\ldots m, \end{array}$$ where data are $m+1$ Hermitian matrices $F_i$ and a complex vector $c\in {\mathbb C}^m$, and $Z$ is a Hermitian variable. As can be easily checked, our problem (\[problem\]) reduces to a dual SDP problem upon the following substitutions: $$\begin{aligned} \label{correspond} F_0&=&-\bigoplus_{j=1}^m \xi_j\rho_j,\quad Z=\bigoplus_{j=1}^m \Pi_j,\nonumber\\ F_i&=&\bigoplus_{j=1}^m\Gamma_i,\quad c_i={\rm Tr}\Gamma_i, \quad i=1\ldots p^2.\end{aligned}$$ Here operators $\{\Gamma_i, i=1\ldots p^2\}$ comprise an orthonormal operator basis in the $p^2$-dimensional space of Hermitian operators acting in the Hilbert space of our problem: ${\rm Tr}\Gamma_j\Gamma_k=\delta_{jk},\; j,k=1\ldots p^2$. For simplicity, let us take $\Gamma_1$ proportional to the unity operator, then all $c_i$ apart from $c_1$ vanish. An important point is that there exists a primal problem associated with the dual one, $$\label{semi-primal} \begin{array}{l} {\rm minimize} \; c^T x, \\ F(x)=F_0+\sum_i x_iF_i\ge 0. \end{array}$$ Here data $F_i$ and $c_i$ are the same as in Eq. (\[semi-dual\]), and vector $x$ is now the variable. The advantage of the SDP formulation of the quantum-state discrimination problem is that there are strong numerical tools designed for solving SDP problems. In particular, these methods are guaranteed to converge to the real solution. This might become important when the iterative algorithm derived above encounters convergence problems. A primal (dual) SDP problem is called “strictly feasible” if there exists $x$ ($Z$) satisfying the constraints in Eq. (\[semi-primal\]) \[Eq. (\[semi-dual\])\] with sharp inequalities. One can easily check that both the primal and dual problems associated with the quantum-state discrimination problem are strictly feasible. Hence we can use a powerful result of SDP theory saying that in this case, $x$ is optimal if and only if $x$ is primal feasible and there is a dual feasible $Z$ such that $$\label{compl-slackness} Z F(x)=0.$$ This condition is called the complementary slackness condition. Now, taking $x_i$ to be the coordinates of the Lagrange operator $\lambda$ in $\Gamma_i$ basis, $x_i={\rm Tr}\lambda \Gamma_i$, $i=1\ldots p^2$, the complementary slackness condition is seen to be equivalent to the extremal equation. Since the dual feasibility of the iteratively obtained POM elements is guaranteed by construction, our extremal equation becomes a necessary and sufficient condition on the maximum of $P_s$ once the positive semidefiniteness of $F(x)$ is verified. In terms of states and POM elements this latter condition reads: $$\label{criterion} \lambda-\xi_j\rho_j\ge 0,\quad j=1\ldots M$$ One perceives that complementary slackness condition (\[compl-slackness\]) together with the criterion of optimality (\[criterion\]) are nothing else than the well-known Helstrom equations [@helstrom] for POM maximizing the success probability (\[probab\]). Let us illustrate the utility of our algorithm on a simple, albeit nontrivial example of discriminating between three non-symmetric coplanar qubit states. The geometry of this problem is shown in Fig. \[fig-vectors\]. ![A cut through the Bloch sphere showing the states to be discriminated.[]{data-label="fig-vectors"}](fig1.eps){width="0.5\columnwidth"} $\Psi_1$ and $\Psi_2$ taken to be equal-prior states, $\xi_1=\xi_2=\xi/2$, symmetrically placed around the $z$ axis; the third state lies in the direction of $x$ or $y$. A similar configuration (with $\Psi_3$ lying along $z$) has recently been investigated by Andersson [*et al.*]{} [@barnett]. Exploiting the mirror symmetry of their problem the authors derived analytic expressions for POMs minimizing the average error rate. For a given angle $\varphi$ the optimum POM turned out to have two or three nonzero elements depending on the amount of the prior information $\xi$. Our problem is a bit more complicated one due to the lack of the mirror symmetry. Let us see whether the transition from the mirror-symmetric configuration to a non-symmetric one has some influence on the qualitative behavior of the optimal POMs. Minimal error rates calculated using the proposed iterative procedure \[Eqs. (\[extremal\]) and (\[lagrange\])\] for the fixed angle of $\varphi=\pi/16$ are summarized in Fig. \[fig-error\]. ![Average error rate ($1-P_s$) in dependence on Bob’s prior information $\xi$; $\varphi=\pi/16$. Regions I, II, and III are regions where the optimum discriminating device has two, three, and two output channels, respectively.[]{data-label="fig-error"}](fig2-g.eps){width="0.9\columnwidth"} The conclusions that can be drawn from the numerical results partly coincide with that of Ref. [@barnett]. For large $\xi$ (region III) the optimum strategy consist in the optimal discrimination between states $\Psi_1$ and $\Psi_2$. When $\xi$ becomes smaller than a certain $\varphi$-dependent threshold (region II), state $\Psi_3$ can no longer be ignored and the optimum POM has three nonzero elements. Simple calculation yields $$\label{boundary} \xi_{\rm II,III}=\frac{1}{1+\sin\varphi\cos\varphi}$$ for the threshold value of the prior. However, when $\xi$ becomes still smaller (region I), the optimum POM will eventually become a two-element POM again – the optimal strategy now being the optimal discrimination between states $\Psi_1$ and $\Psi_3$. This last regime is absent in the mirror-symmetric case. The transition between regions I and II is governed by a much more complicated expression than Eq. (\[boundary\]), and will not be given here. We will close the example noting that already a few iterations are enough to determine the optimum discriminating device to the precision the elements of the realistic experimental setup can be controlled with in the laboratory. In this paper we derived a simple iterative algorithm for finding optimal devices for quantum-state discrimination. Utility of our procedure was illustrated on a non-trivial example of discriminating between three non-symmetric states. From the mathematical point of view, the problem of quantum-state discrimination is a problem of the semidefinite programming. Such correspondence is a good news since there exist robust numerical tools designed to deal with SDP problems. 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--- abstract: 'We confirm the importance of standard medium effects (hadronic rescattering) in heavy-ion collisions by using a pQCD-based model to investigate the dilepton spectra from Pb+Au collisions at 158 $GeV$/$nucleon$. These same effects, namely prompt $\pi \rho \to \pi e^+e^-$, have been studied in several CERN SPS systems[@me1]. In addition, the contribution from $\eta$’s produced by intermediate-stage scattering of pions and previously unscattered projectile nucleons to the dilepton spectrum has been included. The results presented here are consistent with previous studies stating that this type of rescattering effect explains a portion of the “excess” lepton pairs seen by the CERES experiment, but not the entire effect.' address: - ' Department of Physics, Linfield College, McMinnville, OR 97128' - ' National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824' - ' Department of Physics and Astronomy, St. Cloud State University, St. Cloud, MN 56301' author: - 'J. Murray' - 'W. Bauer' - 'K. Haglin' title: Revisiting lepton pairs at the SPS --- In future heavy-ion programs, considerable effort will be spent on gaining information about the space-time history of the collision. This information becomes increasingly important if a phase transition to quark-gluon degrees of freedom occurs during the reaction. In order to have an understanding of this transition, as well as being able to determine its existence, it is necessary to find signals that will remain intact during all stages of the collision. Although hadrons are abundant in the final state, they are only truly sensitive to the system dynamics after hadronization occurs. In addition, the resulting hadronic environment will participate in multiple interactions before reaching the detector. Therefore, reconstructing any information that hadrons might contain about the initial stages of the collision would be a daunting task. This is the reason electromagnetic signals, such as dileptons and direct photons, have gained in popularity. These probes, due to a large mean free path in hadronic matter, appear in the detector after almost no interaction with the medium. This property makes electromagnetic signals ideal for studying the early stages of a heavy-ion collision where this type of phase transition might occur. This is why understanding these probes in current heavy-ion experiments is essential. In order to shed some light on the higher temperature and density experiments where “new” physics should occur, one must fully understand how electromagnetic signals behave in lower temperature and density systems. With this in mind, we revisit dilepton data taken by the CERES collaboration[@ceres]. As in the S+Au lepton pair measurements, the invariant mass spectra of dileptons from a Pb projectile, with an energy of 158 $GeV$/$nucleon$, incident on a Au target reported an excess of dileptons over the collaboration’s “cocktail” predictions[@ceres]. Since purely conventional explanations for the excess seem to be insufficient[@drees], the nature of the enhancement suggests several possibilities. Among the most prominent are studies into medium modifications resulting in a shifted rho mass[@rhoshift; @both] and consequences arising from modifications in the $\pi^+\pi^-\to e^+e^-$ reaction[@rapp; @both]. The study in this paper will be restricted to a more conventional approach[@first_suggestion; @haglin; @me1] using non-resonant scattering of pions and rhos to partially explain the enhancement of electron-positron pairs. To describe the initial stages of an ultra-relativistic heavy-ion collision, it is necessary to use parton degrees of freedom. With this in mind, there are efforts underway to construct so-called parton cascades[@geiger1; @geiger2]. These models are based on perturbative QCD and are therefore attractive candidates for a space-time transport theory in this energy regime. However, we have shown that there are severe problems with causality violations[@caus] and with the time-ordering of soft-gluon emission[@kort]. Therefore, under these circumstances, a much simpler approach might provide more reliable results: geometrical folding of the results of event generators for the elementary processes. This prescription is followed, for example, in HIJING[@hijing]. The simulation used in this study is similar to HIJING. It employs pQCD and parton distribution functions to characterize the individual nucleon-nucleon collisions and uses Glauber-type geometry[@ncoll] to determine the scaling. The kinematics of the nucleon-nucleon collisions are handled by PYTHIA and JETSET[@sjostrand], high energy event-generators using pQCD matrix elements as well as the Lund fragmentation scheme. We refer the reader to our previous work for a more detailed description of the model[@me1]. Dileptons from pseudoscalars ($\pi^0$, $\eta$, $\eta^{\prime}$) and vectors ($\omega$, $\rho^0$, $\phi$) produced in the primary scattering phase are not enough to account for the hadron-induced data measured by the CERES collaboration. Therefore, in addition to this type of lepton pairs, our model also incorporates secondary scattering of hadronic resonances. All pions and rhos formed during the primary collisions of nucleons will have a chance to scatter amongst themselves before decaying. The reactions we consider are of two types, one which produces a resonance that decays to dileptons and the other which goes to dileptons directly. Of the first type, $\pi^+ \pi^- \to \rho^0$ $\to e^+ e^-$ and $\pi^0 \rho^{\pm} \to {a_1}^{\pm}$ $\to \pi^{\pm} e^+ e^-$ have been included. Of the second type, $\pi^0 \rho^{\pm} \to \pi^{\pm} e^+e^-$ has been included and the other isospin channels ($\pi^{\pm} \rho^{0} \to \pi^{\pm} e^+e^-$, $\pi^{\mp} \rho^{\pm} \to \pi^{0} e^+e^-$) are assumed to be of the same magnitude. To accomplish these types of scattering, pions and rhos must of course appear in the final state of the model described in the previous section. As the default, JETSET automatically decays all hadronic resonances, but it also contains provisions to prohibit them. We thus allow pions and rhos to scatter when conditions are favorable. Technically, the steps involved in secondary scattering are similar to those for primary (nucleon-nucleon) scattering. Since the S+Au dilepton study was published[@me1], several additions have been made to the simulation. Previously, the only pions and rhos that were allowed to secondary scatter came from the string fragmentation stage of the simulation. Pions and rhos resulting from a hadronic decay chain were prevented from rescattering. These particles are now allowed to scatter and contribute to the dilepton spectra. Since our calculation includes dileptons from $\pi^+ \pi^-$ resonant production as well as $\pi \rho$ scattering, adjustments should be made to compensate for possible double counting. Charged pions that annihilate to form a lepton pair cannot also scatter with a neutral rho to form a lepton pair and a pion. This effect has been accounted for in our results. It should be noted that any effects arising from the lifetime of pions and rhos have not been accounted for, as it is possible for the pions and rhos to decay inside the reaction zone before having a chance to interact. In the final addition made to our simulation, we allow the original projectile nucleons that didn’t participate in the primary scattering stage to scatter with pions in the secondary (intermediate) stage. These unscattered projectile nucleons could contribute significantly to the overall multiplicity, provided that they exist in sufficient numbers. Based on the largest region of excess in the lepton pair spectra, we focused on the production of the $\eta$ meson by this mechanism. This type of rescattering is handled entirely by PYTHIA. The total dilepton yield from our model is the sum of lepton pairs from primary plus secondary scattering. The invariant mass distributions of the dileptons from all contributions will be discussed in the last section. A reasonable candidate for a successful model description of ultra-relativistic heavy-ion collisions must at minimum be able to reproduce the rapidity distributions and transverse spectra of pions produced in the collisions. We have performed these tests with our model and compared the results to available experimental data at CERN[@me1]. We will not repeat this analysis here, but only state the results: the total number of produced pions is reproduced to better than a factor of two; the shape of the rapidity distribution shows the correct degree of stopping; the slope of the transverse momenta is reproduced. It should be noted that our model was also tested against the proton-induced interactions (p+Be and p+Au) at CERN[@me1]. In both cases, our simulation reproduced the lepton pair data by only considering the primary decays of pseudoscalar and vector mesons. It is also reassuring that the dilepton spectra from these decays in our model were consistent with the cocktail predictions made by the CERES collaboration for all systems: p+Be, p+Au, S+Au, and Pb+Au. With the inclusion of the secondary scattering described in the previous section, the invariant mass distributions of dileptons are shown in Fig.1 for S+Au and Fig.2 for Pb+Au. Both systems have a marked increase in lepton-pair production between an invariant mass of 200 and 500 MeV as well as a noticeable increase in the higher mass region when compared with the spectra without secondary scattering included. The bulk of the increase is attributed to non-resonant $\pi$ $\rho$ scattering, not pseudoscalar and vector meson production by secondary scattering of projectile nucleons with pions. The latter effect is minimal, as on average only about 20 (5) projectile nucleons rescatter with pions in the Pb- (S-) induced collisions. The proton-induced reactions do not show any significant increase in dilepton production with secondary scattering included. It is not surprising that secondary scattering becomes important in the nucleus-nucleus systems, as a denser nuclear medium is created during these collisions as compared to the proton-induced collisions. When comparing the differences between our calculations and the experimental results for the S+Au dilepton spectrum to those found in Pb+Au, one finds that our calculations are much closer to experiment for the heavier projectile. Since the difference between conventional theory and experiment is presumed to be due to in-medium vector meson mass shifts [@rhoshift], this tendency is puzzling. 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--- author: - 'Vineet Gupta[^1] Tomer Koren Yoram Singer[^2]' title: 'Shampoo: Preconditioned Stochastic Tensor Optimization' --- [^1]: Google Brain. Email: `{vineet,tkoren}@google.com` [^2]: Princeton University and Google Brain. Email: `y.s@cs.princeton.edu`

--- abstract: | The subject of this work is the study of ${{\rm LS}}_+$-perfect graphs defined as those graphs $G$ for which the stable set polytope ${{\rm STAB}}(G)$ is achieved in one iteration of Lovász-Schrijver PSD-operator ${{\rm LS}}_+$, applied to its edge relaxation ${{\rm ESTAB}}(G)$. In particular, we look for a polyhedral relaxation of ${{\rm STAB}}(G)$ that coincides with ${{\rm LS}}_+({{\rm ESTAB}}(G))$ and ${{\rm STAB}}(G)$ if and only if $G$ is ${{\rm LS}}_+$-perfect. An according conjecture has been recently formulated (${{\rm LS}}_+$-Perfect Graph Conjecture); here we verify it for the well-studied class of claw-free graphs. stable set polytope, ${{\rm LS}}_+$-perfect graphs, claw-free graphs author: - Silvia Bianchi - Mariana Escalante - Graciela Nasini - 'Annegret Wagler\' title: 'Lovász-Schrijver PSD-operator on Claw-Free Graphs [^1]' --- Introduction ============ The context of this work is the study of the stable set polytope, some of its linear and semi-definite relaxations, and graph classes for which certain relaxations are tight. Our focus lies on those graphs where a single application of the Lovász-Schrijver positive semi-definite operator introduced in [@LovaszSchrijver1991] to the edge relaxation yields the stable set polytope. The *stable set polytope* ${{\rm STAB}}(G)$ of a graph $G=(V,E)$ is defined as the convex hull of the incidence vectors of all stable sets of $G$ (in a stable set all nodes are mutually nonadjacent). Two canonical relaxations of ${{\rm STAB}}(G)$ are the *edge constraint stable set polytope* $${{\rm ESTAB}}(G) = \{\mathbf{x} \in [0,1]^V: x_i + x_j \: \le \: 1, ij \in E \},$$ and the *clique constraint stable set polytope* $${{\rm QSTAB}}(G) = \{\mathbf{x} \in [0,1]^V: \sum_{i \in Q} \, x_i \: \le \: 1,\ Q \subseteq V\ \mbox{ maximal clique of $G$} \}$$ (in a clique all nodes are mutually adjacent, hence a clique and a stable set share at most one node). We have ${{\rm STAB}}(G) \subseteq {{\rm QSTAB}}(G) \subseteq {{\rm ESTAB}}(G)$ for any graph, where ${{\rm STAB}}(G)$ equals ${{\rm ESTAB}}(G)$ for bipartite graphs, and ${{\rm QSTAB}}(G)$ for perfect graphs only [@Chvatal1975]. According to a famous characterization achieved by Chudnovsky et al. [@ChudnovskyEtAl2006], perfect graphs are precisely the graphs without chordless cycles $C_{2k+1}$ with $k \geq 2$, termed *odd holes*, or their complements, the *odd antiholes* $\overline C_{2k+1}$ as node induced subgraphs (where the complement $\overline G$ has the same nodes as $G$, but two nodes are adjacent in $\overline G$ if and only if they are non-adjacent in $G$). Then, odd holes and odd antiholes are the only *minimally imperfect graphs*. Perfect graphs turned out to be an interesting and important class with a rich structure and a nice algorithmic behavior [@GrotschelLovaszSchrijver1988]. However, solving the stable set problem for a perfect graph $G$ by maximizing a linear objective function over ${{\rm QSTAB}}(G)$ does not work directly [@GrotschelLovaszSchrijver1981], but only via a detour involving a geometric representation of graphs [@Lovasz1979] and the resulting semi-definite relaxation ${{\rm TH}}(G)$ introduced in [@GrotschelLovaszSchrijver1988]. For some $N \in {\mathbf{Z}}_+$, an orthonormal representation of a graph $G=(V,E)$ is a sequence $(\mathbf{u_i} : i \in V)$ of $|V|$ unit-length vectors $\mathbf{u_i} \in {\mathbf{R}}^N$, such that $\mathbf{u_i}^T\mathbf{u_j} = 0$ for all $ij \not\in E$. For any orthonormal representation of $G$ and any additional unit-length vector $\mathbf{c} \in {\mathbf{R}}^N$, the orthonormal representation constraint is $\sum_{i \in V} (\mathbf{c}^T\mathbf{u_i})^2 x_i \leq 1$. ${{\rm TH}}(G)$ denotes the convex set of all vectors $\mathbf{x} \in {\mathbf{R}}_+^{|V|}$ satisfying all orthonormal representation constraints for $G$. For any graph $G$, $${{\rm STAB}}(G) \subseteq {{\rm TH}}(G) \subseteq {{\rm QSTAB}}(G)$$ holds and approximating a linear objective function over ${{\rm TH}}(G)$ can be done with arbitrary precision in polynomial time [@GrotschelLovaszSchrijver1988]. Moreover, if ${{\rm TH}}(G)$ is a rational polytope, an optimal solution can be obtained in polynomial time. This fact gives a great relevance to the beautiful characterization of perfect graphs obtained by the same authors: $$\label{equivalencias} G \textrm{ is perfect} \Leftrightarrow {{\rm TH}}(G)={{\rm STAB}}(G) \Leftrightarrow {{\rm TH}}(G)={{\rm QSTAB}}(G). $$ For all imperfect graphs, ${{\rm STAB}}(G)$ does not coincide with any of the above relaxations. It is, thus, natural to study further relaxations and to combinatorially characterize those graphs where ${{\rm STAB}}(G)$ equals one of them. #### Linear relaxations and related graphs. A natural generalization of the clique constraints are rank constraints associated with arbitrary induced subgraphs $G' \subseteq G$. By the choice of the right hand side $\alpha(G')$, denoting the size of a largest stable set in $G'$, rank constraints $$\mathbf{x}(G') = \sum_{i \in G'} \, x_i \, \le \, \alpha(G')$$ are valid for ${{\rm STAB}}(G)$. A graph $G$ is called *rank-perfect* by [@Wagler2000] if and only if ${{\rm STAB}}(G)$ is described by rank constraints only. By definition, rank-perfect graphs include all perfect graphs. By restricting the facet set to rank constraints associated with certain subgraphs, several well-known graph classes are defined, e.g., *near-perfect graphs* [@Shepherd1994] where only rank constraints associated with cliques and the whole graph are allowed, or *t-perfect* [@Chvatal1975] and *h-perfect graphs* [@GrotschelLovaszSchrijver1988] where rank constraints associated with edges, triangles and odd holes resp. cliques of arbitrary size and odd holes suffice. As common generalization of perfect, t-perfect, and h-perfect graphs, the class of *a-perfect graphs* was introduced in [@Wagler2004_4OR] as graphs $G$ where ${{\rm STAB}}(G)$ is given by rank constraints associated with antiwebs. An *antiweb* $A^k_n$ is a graph with $n$ nodes $0, \ldots, n-1$ and edges $ij$ if and only if $k \leq |i-j| \leq n-k$ and $i \neq j$. Antiwebs include all complete graphs $K_n = A^1_n$, odd holes $C_{2k+1} = A^k_{2k+1}$, and their complements $\overline C_{2k+1} = A^2_{2k+1}$. Antiwebs are $a$-perfect by [@Wagler2004_4OR], further examples of $a$-perfect graphs were found in [@Wagler2005_4OR]. A more general type of inequalities is obtained from complete joins of antiwebs, called *joined antiweb constraints* $$\sum_{i \leq k} \frac{1}{\alpha(A_i)} x(A_i) + x(Q) \leq 1,$$ associated with the join of some antiwebs $A_1, \ldots, A_k$ and a clique $Q$ (note that the inequality is scaled to have right hand side 1). This includes, e.g., all odd (anti)wheels (the join of a single node with an odd (anti)hole). We denote the linear relaxation of ${{\rm STAB}}(G)$ obtained by all joined antiweb constraints by ${{\rm ASTAB}}^*(G)$. By construction, we see that $${{\rm STAB}}(G) \subseteq {{\rm ASTAB}}^*(G) \subseteq {{\rm QSTAB}}(G) \subseteq {{\rm ESTAB}}(G).$$ In [@CPW_2009], a graph $G$ is called *joined a-perfect* if and only if ${{\rm STAB}}(G)$ coincides with ${{\rm ASTAB}}^*(G)$. Besides all a-perfect graphs, further examples of joined a-perfect graphs are *near-bipartite graphs* (where the non-neighbors of every node induce a bipartite graph) due to [@Shepherd1995]. #### A semi-definite relaxation and ${{\rm LS}}_+$-perfect graphs. In the early nineties, Lovász and Schrijver introduced the PSD-operator ${{\rm LS}}_+$ (called $N_+$ in [@LovaszSchrijver1991]) which, applied to ${{\rm ESTAB}}(G)$, generates a positive semi-definite relaxation of ${{\rm STAB}}(G)$ stronger than ${{\rm TH}}(G)$ (see Section \[Sec:N+perfect\] for details). In order to simplify the notation we write ${{\rm LS}}_+(G)={{\rm LS}}_+({{\rm ESTAB}}(G))$. As in the case of perfect graphs, the stable set problem can be solved in polynomial time for the class of graphs for which ${{\rm LS}}_+(G)={{\rm STAB}}(G)$ by [@LovaszSchrijver1991]. These graphs are called *${{\rm LS}}_+$-perfect*, and all other graphs *${{\rm LS}}_+$-imperfect* (note that they are also called $N_+$-(im)perfect, see e.g. [@BENT2011]). In addition, every subgraph of an ${{\rm LS}}_+$-perfect graph is also ${{\rm LS}}_+$-perfect. This motivates the definition of *minimally* ${{\rm LS}}_+$-*imperfect graphs* as the ${{\rm LS}}_+$-imperfect graphs whose proper induced subgraphs are all ${{\rm LS}}_+$-perfect. The two smallest of such graphs (regarding its number of nodes) were found by [@EMN2006] and [@LiptakTuncel2003] and are depicted in Figure \[grafos6\]. ![The graphs $G_{LT}$ (on the left) and $G_{EMN}$ (on the right).[]{data-label="grafos6"}](NIP_GltGenm.eps) In [@BENT2011], the authors look for a characterization of ${{\rm LS}}_+$-perfect graphs similar to the characterization (\[equivalencias\]) for perfect graphs: they intend to find an appropriate polyhedral relaxation $P(G)$ of ${{\rm STAB}}(G)$ such that $G$ is ${{\rm LS}}_+$-perfect if and only if ${{\rm STAB}}(G)=P(G)$. A conjecture has been recently proposed in [@BENT2013], which can be equivalently reformulated as follows [@ENW2014]: \[conjecture\] A graph $G$ is ${{\rm LS}}_+$-perfect if and only if ${{\rm STAB}}(G)={{\rm ASTAB}}^*(G)$. In [@LovaszSchrijver1991] it is shown that, for every graph $G$ $$\label{LS1} {{\rm LS}}_+(G) \subseteq {{\rm ASTAB}}^*(G).$$ Thus, the conjecture states that ${{\rm LS}}_+$-perfect graphs coincide with joined a-perfect graphs and ${{\rm ASTAB}}^*(G)$ is the polyhedral relaxation of ${{\rm STAB}}(G)$ playing the role of $P(G)$ in . Conjecture \[conjecture\] has been already verified for near-perfect graphs by [@BENT2011], for *fs-perfect graphs* (where the only facet-defining subgraphs are cliques and the graph itself) by [@BENT2013], for *webs* (the complements $W^k_n = \overline A^k_n$ of antiwebs) by [@EN2014] and for *line graphs* (obtained by turning adjacent edges of a root graph into adjacent nodes of the line graph) by [@ENW2014], see Section \[Sec:N+perfect\] for details. #### The ${{\rm LS}}_+$-Perfect Graph Conjecture for Claw-free graphs. The aim of this contribution is to verify Conjecture \[conjecture\] for a well-studied graph class containing all webs, all line graphs and the complements of near-bipartite graphs: the class of *claw-free graphs* (i.e., the graphs not containing as node induced subgraph the complete join of a single node and a stable set of size three). Claw-free graphs attracted much attention due to their seemingly asymmetric behavior w.r.t. the stable set problem. On the one hand, the first combinatorial algorithms to solve the problem in polynomial time for claw-free graphs [@Minty1980; @Sbihi1980] date back to 1980. Therefore, the polynomial equivalence of optimization and separation due to [@GrotschelLovaszSchrijver1988] implies that it is possible to optimize over the stable set polytope of a claw-free graph in polynomial time. On the other hand, the problem of characterizing the stable set polytope of claw-free graphs in terms of an explicit description by means of a facet-defining system, originally posed in [@GrotschelLovaszSchrijver1988], was open for several decades. This motivated the study of claw-free graphs and its different subclasses, that finally answered this long-standing problem only recently (see Section \[Sec:ClawFree\] for details). The paper is organized as follows: In Section 2, we present the State-of-the-Art on ${{\rm LS}}_+$-perfect graphs (including families of ${{\rm LS}}_+$-imperfect graphs needed for the subsequent proofs) and on claw-free graphs, their relevant subclasses and the results concerning the facet-description of their stable set polytopes from the literature. In Section 3, we verify, relying on the previously presented results, Conjecture \[conjecture\] for the studied subclasses of claw-free graphs. As a conclusion, we obtain as our main result: \[thm\_main\] The ${{\rm LS}}_+$-Perfect Graph Conjecture is true for all claw-free graphs. We close with some further remarks and an outlook to future lines of research. State-of-the-Art {#Sec:N+perfect} ================ About ${{\rm LS}}_+$-perfect graphs ----------------------------------- In order to introduce the ${{\rm LS}}_+$-operator we denote by $\e_0, \e_1, \dots, \e_n$ the vectors of the canonical basis of ${\mathbf{R}}^{n+1}$ (where the first coordinate is indexed zero), ${{\mathbf{1}}}$ the vector with all components equal to $1$ and $S_+^{n}$ the convex cone of symmetric and positive semi-definite $(n \times n)$-matrices with real entries. Let $K\subset [0,1]^n$ be a convex set and $$\operatorname{cone}(K)= \left\{\left(\begin{array}{c} x_0\\ \x \end{array} \right) \in {\mathbf{R}}^{n+1}: \x=x_0 \y; \;\; \y \in K \right\}.$$ Then, the convex set $M_+(K)$ is defined as: $$\begin{aligned} M_{+}(K) = \left\{Y \in S_+^{n+1}: \right. & & Y\e_0 = \operatorname{diag}(Y),\\ & & Y\e_i \in \operatorname{cone}(K), \\ & & \left. Y (\e_0 - \e_i) \in \operatorname{cone}(K), \; i=1,\dots,n \right\}, \end{aligned}$$ where $\operatorname{diag}(Y)$ denotes the vector whose $i$-th entry is $Y_{ii}$, for every $i=0,\dots,n$. Projecting this lifting back to the space ${\mathbf{R}}^n$ results in $${{\rm LS}}_{+}(K) = {\left\{ \x \in [0,1]^n : \left(\begin{array}{c} 1\\ \x \end{array} \right)= Y \e_0, \mbox{ for some } Y \in M_{+}(K)\right\}}.$$ In [@LovaszSchrijver1991], Lovász and Schrijver proved that ${{\rm LS}}_+(K)$ is a relaxation of the convex hull of integer solutions in $K$ and that $${{\rm LS}}_+^n(K)={\mathrm{conv}}(K\cap \{0,1\}^n),$$ where ${{\rm LS}}_+^0(K)=K$ and ${{\rm LS}}_+^k(K)={{\rm LS}}_+({{\rm LS}}_+^{k-1}(K))$ for every $k\geq 1$. In this work we focus on the behavior of a single application of the ${{\rm LS}}_+$-operator to the edge relaxation ${{\rm ESTAB}}(G)$ of the stable set polytope of a graph. Recall that we write ${{\rm LS}}_+(G)={{\rm LS}}_+({{\rm ESTAB}}(G))$ to simplify the notation and that graphs for which ${{\rm LS}}_+(G)={{\rm STAB}}(G)$ holds are ${{\rm LS}}_+$-perfect. Exhibiting one ${{\rm LS}}_+$-imperfect subgraph $G'$ in a graph $G$ certifies the ${{\rm LS}}_+$-imperfection of $G$. Hereby, characterizing ${{\rm LS}}_+$-imperfect graphs within a certain graph class turns out to be a way to attack the conjecture for this class. Recall that $G_{LT}$ and $G_{EMN}$ are the smallest ${{\rm LS}}_+$-imperfect graphs. In [@BENT2011] the authors showed that they are the two smallest members of an infinite family of ${{\rm LS}}_+$-imperfect graphs having stability number two that will play a central role in some subsequent proofs: \[[@BENT2011]\] \[mnpmascero\] Let $G$ be a graph with $\alpha(G)=2$ such that $G-v$ is an odd antihole for some node $v$. $G$ is ${{\rm LS}}_+$-perfect if and only if $v$ is completely joined to $G-v$. Further ${{\rm LS}}_+$-imperfect graphs can be obtained by applying operations preserving ${{\rm LS}}_+$-imperfection. In [@LiptakTuncel2003], the *stretching* of a node $v$ is introduced as follows: Partition its neighborhood $N(v)$ into two nonempty, disjoint sets $A_1$ and $A_2$ (so $A_1 \cup A_2 = N(v)$, and $A_1 \cap A_2 = \emptyset$). A stretching of $v$ is obtained by replacing $v$ by two adjacent nodes $v_1$ and $v_2$, joining $v_i$ with every node in $A_i$ for $i \in \{1, 2\}$, and subdividing the edge $v_1 v_2$ by one node $w$. In [@LiptakTuncel2003] it is shown: \[[@LiptakTuncel2003]\] \[stretching\] The stretching of a node preserves ${{\rm LS}}_+$-imperfection. Hence, all stretchings of $G_{LT}$ and $G_{EMN}$ are ${{\rm LS}}_+$-imperfect, see Figure \[fig\_stretchings\] for some examples. ![Some node stretchings ($v_1,w,v_2$ in black) of $G_{LT}$ and $G_{EMN}$.[]{data-label="fig_stretchings"}](NIP_oddStretchings.eps) Using stretchings of $G_{LT}$ and $G_{EMN}$ and exhibiting one more minimally ${{\rm LS}}_+$-imperfect graph, namely the web $W_{10}^2$, ${{\rm LS}}_+$-perfect webs are characterized in [@EN2014] as follows: \[[@EN2014]\] \[webs\] A web is ${{\rm LS}}_+$-perfect if and only if it is perfect or minimally imperfect. The proof shows that all imperfect not minimally imperfect webs with stability number 2 contain $G_{EMN}$ and all webs $W_{n}^2$ different from $W_{7}^2, W_{10}^2$, some stretching of $G_{LT}$. Furthermore, all other webs contain some ${{\rm LS}}_+$-imperfect $W_{n'}^2$ and are, thus, also ${{\rm LS}}_+$-imperfect. Another way to attack the conjecture is from the polyhedral point of view. A graph $G$ is said to be *facet-defining* if ${{\rm STAB}}(G)$ has a full-support facet. Observe that verifying Conjecture \[conjecture\] is equivalent to prove that the only facet defining ${{\rm LS}}_+$-perfect graphs are the complete joins of antiwebs. That is why we need to rely on structural results and complete facet-descriptions of stable set polytope of the graphs. Using this approach, in [@ENW2014], the authors characterized ${{\rm LS}}_+$-perfect line graphs by showing: \[[@ENW2014]\] \[hipomatch\] A facet-defining line graph $G$ is ${{\rm LS}}_+$-perfect if and only if $G$ is a clique or an odd hole. The proof relies on a result due to Edmonds & Pulleyblank [@EdmondsPulleyblank1974] who showed that a line graph $L(H)$ is facet-defining if and only if $H$ is a 2-connected hypomatchable graph (that is, for all nodes $v$ of $H$, $H-v$ admits a perfect matching). Such graphs $H$ have an ear decomposition $H_0, H_1, \dots, H_k = H$ where $H_0$ is an odd hole and $H_i$ is obtained from $H_{i-1}$ by adding an odd path (ear) between distinct nodes of $H_{i-1}$. In [@ENW2014], it is shown that the line graph $L(H_1)$ is a node stretching of $G_{LT}$ or $G_{EMN}$ and, thus, ${{\rm LS}}_+$-imperfect by [@LiptakTuncel2003]. Moreover, it is proved that the only minimally ${{\rm LS}}_+$-imperfect line graphs are stretchings of $G_{LT}$ and $G_{EMN}$. About claw-free graphs {#Sec:ClawFree} ---------------------- In several respects, claw-free graphs are generalizations of line graphs. An intermediate class between line graphs and claw-free graphs form *quasi-line graphs*, where the neighborhood of any node can be partitioned into two cliques (i.e., quasi-line graphs are the complements of near-bipartite graphs). Quasi-line graphs can be divided into two subclasses: fuzzy circular interval graphs and semi-line graphs. Let ${\cal C}$ be a circle, ${\cal I}$ a collection of intervals in ${\cal C}$ without proper containments and common endpoints, and $V$ a multiset of points in ${\cal C}$. A *fuzzy circular interval graph* $G(V,{\cal I})$ has node set $V$ and two nodes are adjacent if both belong to one interval $I \in {\cal I}$, where edges between different endpoints of the same interval may be omitted. [*Semi-line graphs*]{} are either line graphs or quasi-line graphs without a representation as a fuzzy circular interval graph. It turned out that so-called *clique family inequalities* suffice to describe the stable set polytope of quasi-line graphs. Given a graph $G$, a family $\mathcal F$ of cliques and an integer $p<n=|\mathcal F|$, the clique family inequality ($\mathcal F$, $p$) is the following valid inequality for ${{\rm STAB}}(G)$ $$(p-r)\sum_{i\in W} x_i+ (p-r-1) \sum_{i\in W_o} x_i \leq (p-r) \left\lfloor \frac{n}{p}\right\rfloor$$ where $r=n \, mod\, p$ and $W$ (resp. $W_o$) is the set of nodes contained in at least $p$ (resp. exactly $p-1$) cliques of $\mathcal F$. This generalizes the results of Edmods [@Edmonds1965] and Edmonds & Pulleyblank [@EdmondsPulleyblank1974] that ${{\rm STAB}}(L(H))$ is described by clique constraints and rank constraints $$\label{Eq_hypomatch-constraints} x(L(H')) \le \frac{1}{2}(|V(H')|-1)$$ associated with the line graphs of 2-connected hypomatchable induced subgraphs $H' \subseteq H$. Note that the rank constraints of type (\[Eq\_hypomatch-constraints\]) are special clique family inequalities. Chudnovsky and Seymour [@ChudnovskySeymour2004] extended this result to semi-line graphs, for which ${{\rm STAB}}(G)$ is given by clique constraints and rank constraints of type (\[Eq\_hypomatch-constraints\]). Then, semi-line graphs are rank-perfect with line graphs as only facet-defining subgraphs. Moreover, in [@GalluccioSassano97] Galluccio and Sassano prove that if a rank constraint is facet-defining for a claw-free graph $G$ then, either, $G$ is a clique or $G$ contains the line graph of a minimal 2-connected hypomatchable graph $H$ or $G$ contains $W_{\alpha k +1}^{k-1}$ with $k\geq 3$ and $\alpha =\alpha(G)$. Eisenbrand et al. [@EisenbrandEtAl2005] proved that clique family inequalities suffice to describe the stable set polytope of fuzzy circular interval graphs. Stauffer [@Stauffer] verified a conjecture of [@PecherWagler2006] that every facet-defining clique family inequality of a fuzzy circular interval graph $G$ is *associated with a web in* $G$. All these results together complete the picture for quasi-line graphs. However, there are claw-free graphs which are not quasi-line. In particular, every graph with stability number 2 is claw-free and the 5-wheel is the smallest claw-free not quasi-line graph. Due to Cook (see [@Shepherd1995]), all facets for graphs $G$ with $\alpha(G)=2$ are $1,2$-valued [*clique-neighborhood constraints*]{}. This is not the case for graphs $G$ with $\alpha(G) = 3$. In fact, all the known difficult facets of claw-free graphs occur in this class. Some non-rank facets with up to five different non-zero coefficients are presented in [@GilesTrotter1981; @LieblingEtAl2004]. All of these facets turned out to be so-called [*co-spanning 1-forest constraints*]{} due to [@PecherWagler2010], where it is also shown that it is possible to build a claw-free graph with stability number three inducing a co-spanning 1-forest facet with $b$ different left hand side coefficients, for every positive integer $b$. The problem of characterizing ${{\rm STAB}}(G)$ when $G$ is a connected claw-free but not quasi-line graph with $\alpha(G) \geq 4$ was studied by Galluccio et al.: In a series of results [@GGV2008; @GGV2014a; @GGV2014b], it is shown that if such a graph $G$ does not contain a clique cutset, then 1,2-valued constraints suffice to describe STAB($G$). Here, besides 5-wheels, different rank and non-rank facet-defining inequalities of the geared graph $G$ shown in Fig. \[Fig:5wheelStrips\] play a central role. In addition, graphs of this type can be decomposed into strips. A *strip* $(G,a,b)$ is a (not necessarily connected) graph with two designated simplicial nodes $a$ and $b$ (a node is *simplicial* if its neighborhood is a clique). A claw-free strip containing a 5-wheel as induced subgraph is a [*5-wheel strip*]{}. Given two node-disjoint strips $(G_1,a_1,b_1)$ and $(G_2,a_2,b_2)$, their *composition* is the union of $G_1\setminus\{a_1,b_1\}$ and $G_2\setminus\{a_2,b_2\}$ together with all edges between $N_{G_1}(a_1)$ and $N_{G_2}(a_2)$, and between $N_{G_1}(b_1)$ and $N_{G_2}(b_2)$ [@ChudnovskySeymour2004]. As shown in [@OPS2008], this composition operation can be generalized to more than two strips: Every claw-free but not quasi-line graph $G$ with $\alpha(G) \geq 4$ admits a decomposition into strips, where at most one strip is quasi-line and all the remaining ones are 5-wheel strips having stability number at most 3. There are only three “basic” types of 5-wheel strips (see Fig. \[Fig:5wheelStrips\]) which can be extended by adding nodes belonging to the neighborhood of the 5-wheels (see [@OPS2008] for details). Note that a claw-free but not quasi-line graph $G$ with $\alpha(G) \geq 4$ containing a clique cutset may have a facet-defining subgraph $G'$ with $\alpha(G') = 3$ (inside a 5-wheel strip of type 3), see [@PietropaoliWagler2008] for examples. Taking all these results together into account gives the complete list of facets needed to describe the stable set polytope of claw-free graphs. ![The three types of basic 5-wheel strips.[]{data-label="Fig:5wheelStrips"}](N+P_5wheelStrips.eps) ${{\rm LS}}_+$-Perfect Graph Conjecture for claw-free graphs {#results} ============================================================ In this section, we verify the ${{\rm LS}}_+$-Perfect Graph Conjecture for all relevant subclasses of claw-free graphs. Graphs with $\alpha(G) = 2$ --------------------------- The graphs with $\alpha(G) = 2$ play a crucial role in this context. Relying on the behavior of the stable set polytope under taking complete joins [@Chvatal1975] and the result on ${{\rm LS}}_+$-(im)perfect graphs $G$ with $\alpha(G) = 2$ (Theorem \[mnpmascero\]), we can prove: \[thm\_N+imperfect\_alpha2\] All facet-defining ${{\rm LS}}_+$-perfect graphs $G$ with $\alpha(G) = 2$ are odd antiholes or complete joins of odd antihole(s) and a (possibly empty) clique. Let $G$ be a facet-defining ${{\rm LS}}_+$-perfect graph with stability number 2. We first observe that $G$ is imperfect (because it has a full-support facet, but it is different from a clique). Thus, $G$ contains an odd antihole $\overline C$ by [@ChudnovskyEtAl2006]. If $G = \overline C$, we are done. If $G \neq \overline C$, every node $u$ outside $\overline C$ is completely joined to $\overline C$ due to Theorem \[mnpmascero\] (otherwise, $u$ and $\overline C$ induce an ${{\rm LS}}_+$-imperfect subgraph of $G$, a contradiction to $G$ ${{\rm LS}}_+$-perfect). Therefore, $G$ is the complete join of $\overline C$ and $G-\overline C$. Note that $G-\overline C$ is again ${{\rm LS}}_+$-perfect, facet-defining by Chvátal [@Chvatal1975], and $\alpha(G-\overline C) \leq 2$. If $\alpha(G-\overline C) = 2$, we apply the same argument as for $G$; if $\alpha(G-\overline C) = 1$, it is a clique. $\Box$ This shows that all facet-defining ${{\rm LS}}_+$-perfect graphs $G$ with $\alpha(G) = 2$ are joined $a$-perfect, and we conclude: \[Cor\_alpha2\_2\] The ${{\rm LS}}_+$-Perfect Graph Conjecture is true for graphs with stability number 2. Quasi-line graphs ----------------- Recall that quasi-line graphs divide into the two subclasses of semi-line graphs and fuzzy circular interval graphs. Chudnovsky and Seymour [@ChudnovskySeymour2004] proved that the stable set polytope of a semi-line graph is given by rank constraints associated with cliques and the line graphs of 2-connected hypomatchable graphs. Together with the result from [@ENW2014] (presented in Theorem \[hipomatch\]), we directly conclude that the ${{\rm LS}}_+$-Perfect Graph Conjecture holds for semi-line graphs. Based on the results of Eisenbrand et al. [@EisenbrandEtAl2005] and Stauffer [@Stauffer], combined with the characterization of ${{\rm LS}}_+$-imperfect webs from [@EN2014] (Theorem \[webs\]), we are able to show: \[thm\_N+imperfect\_fcig\] All facet-defining ${{\rm LS}}_+$-perfect fuzzy circular interval graphs are cliques, odd holes or odd antiholes. Let $G$ be a fuzzy circular interval graph such that it is a facet-defining ${{\rm LS}}_+$-perfect graph. If $G$ is a clique, the result is immediate. Otherwise, $G$ is the support graph of a clique family inequality with parameters $(\mathcal F, p)$ $$(p-r)\sum_{i\in W} x_i+ (p-r-1) \sum_{i\in W_o} x_i \leq (p-r) \left\lfloor \frac{n}{p}\right\rfloor \!,$$ associated with a web $W_n^{p-1}$ with $V(W_n^{p-1})\subset W$ ([@EisenbrandEtAl2005; @Stauffer]). More precisely, if for any node $v\in V$, ${\cal{I}}_v=\{I\in {\cal{I}}: v\in I\}$, there exist $I_l(v)$ and $I_r(v)$ in ${\cal{I}}_v$ such that $I_l(v)\cup I_r(v)=\bigcup_{I \in {\cal{I}}_v} I$. The clique family inequality associated with $W^{p-1}_n$ is the clique family inequality having parameters ${\cal{F}}=\{K(I_l(v)): v\in V(W_n^{p-1})\}$ and $p$ where $K(I_l(v))=\{u\in I_l(v): u \text{ is adjacent to } v\}$. By Theorem \[webs\], $W_n^{p-1}$ is ${{\rm LS}}_+$-perfect if and only if it is an odd hole or an odd antihole. That is, since $G$ is ${{\rm LS}}_+$-perfect then $n=2k+1$ and $p=2$ or $p=k\geq 3$. In both cases, $r=1$ follows. Consider first the case in which $p=2$. Then the clique family inequality $(\mathcal F, p)$ takes the form $$\sum_{i\in W} x_i\leq \left\lfloor \frac{n}{p}\right\rfloor.$$ Suppose there exists $v\in W\setminus V(W_{2k+1}^1)$. Then, $v$ belongs to $s\geq 2$ consecutive cliques in $\cal{F}$ implying that $v$ is connected to exactly $s+1$ consecutive nodes in $W_{2k+1}^1$. Observe that $s\leq 3$ since $G$ is a claw-free graph. Then, if $s=2$ (resp. $s=3$) $G$ contains an odd subdivision of $G_{LT}$ (resp. $G_{EMN}$). Since $G$ is $LS_+$-perfect then $W=V(W_{2k+1}^1)$ or, equivalently, $G=W_{2k+1}^1$. Now suppose that $p=k\geq 3$. Let us call $\{1,2,\ldots,2k+1\}$ the nodes in $V(W_{2k+1}^{k-1})$. Suppose there exists $v\in (W_o\cup W)\setminus V(W_{2k+1}^{k-1})$. Then $v$ belongs to at least $s\geq k-1$ consecutive cliques of the family $\mathcal F$. W.l.o.g we may assume that the $k-1$ of the $s$ consecutive cliques are the ones that contain the sets of nodes $\{1,\ldots,k\}$, $\{2,\ldots,k+1\}$, ... $\{k-1,\ldots,2k-2\}$. Then $v$ is connected to at least $2k-2$ consecutive nodes in $W_{2k+1}^{k-1}$. Moreover, since $G$ is quasi-line, $v$ is connected with at most $2k$ nodes. It follows that the subgraph of $G$ induced by $V(W_{2k+1}^{k-1})\cup \{v\}$ has stability number two, and from Theorem \[thm\_N+imperfect\_alpha2\] it is ${{\rm LS}}_+$-imperfect. But from our assumption that $G$ is ${{\rm LS}}_+$-perfect, we conclude $W_o\cup W=V(W_{2k+1}^{k-1})$ or, equivalently, $G=W_{2k+1}^{k-1}$. $\Box$ As a consequence, every ${{\rm LS}}_+$-perfect fuzzy circular interval graph is a-perfect. This verifies the ${{\rm LS}}_+$-Perfect Graph Conjecture for fuzzy circular interval graphs. Since the class of quasi-line graphs divides into semi-line graphs and fuzzy circular interval graphs, we obtain as direct consequence: \[Cor\_quasi-line\] The ${{\rm LS}}_+$-Perfect Graph Conjecture is true for quasi-line graphs. Claw-free graphs that are not quasi-line ---------------------------------------- It is left to treat the case of claw-free graphs that are not quasi-line. Here, we distinguish two cases according to their stability number. To treat the case of claw-free not quasi-line graphs $G$ with $\alpha(G) \geq 4$, we rely on the decomposition of such graphs into strips, where at most one strip is quasi-line and all the remaining ones are 5-wheel strips [@OPS2008]. By noting that 5-wheel strips of type 3 contain $G_{LT}$ and exhibiting ${{\rm LS}}_+$-imperfect line graphs in the other two cases, we are able to show: \[thm\_not\_quasi-line\_4\] Every facet-defining claw-free not quasi-line graph $G$ with $\alpha(G) \geq 4$ is ${{\rm LS}}_+$-imperfect. Let $G$ be a facet-defining claw-free not quasi-line graph with $\alpha(G) \geq 4$. According to [@OPS2008], $G$ has a decomposition into strips, where at most one strip is quasi-line and all the remaining ones have stability number at most 3 and contain a 5-wheel each. Recall that there are only three types of 5-wheel strips, Fig. \[Fig:5wheelStrips\] shows the “basic” types, which can be extended by adding nodes belonging to the neighborhood of the 5-wheels [@OPS2008]. Since $G$ is not quasi-line, it contains at least one 5-wheel strip $G'$. If $G'$ is of type 3, then $G'$ contains $G_{LT}$, induced by the squared nodes indicated in Fig. \[Fig:5wheelStrips\], and we are done. Hence, let $G'$ be of type 1 or 2. Note further that $G'$ is a proper subgraph of $G$ (by $\alpha(G') \leq 3$ but $\alpha(G) \geq 4$) and connected to $G-G'$ (since $G$ is facet-defining and, thus, cannot have a clique cutset by Chvátal [@Chvatal1975]). According to the strip composition, there are nodes in $G-G'$ playing the role of the two simplicial nodes of $G'$ (the two black nodes in Fig. \[Fig:5wheelStrips\]), and they are connected by a path $P$ with nodes exclusively in $G-G'$ (again, since $G$ cannot contain a clique cutset). If $G'$ is of type 1, then $G$ has, as induced subgraph, a node stretching of $G_{EMN}$ (resp. of $G_{LT}$) if $P$ is even (resp. odd), see the squared nodes in Fig. \[Fig:5wheelStrip1\]. ![$N_+$-imperfect subgraphs if $G'$ is of type 1.[]{data-label="Fig:5wheelStrip1"}](N+P_5wheelStrip1.eps) If $G'$ is of type 2, then $G$ has, as induced subgraph, a node stretching of $G_{LT}$ (resp. of $G_{EMN}$) if $P$ is even (resp. odd), see the squared nodes in Fig. \[Fig:5wheelStrip2\]. ![${{\rm LS}}_+$-imperfect subgraphs if $G'$ is of type 2.[]{data-label="Fig:5wheelStrip2"}](N+P_5wheelStrip2.eps) Hence, in all cases, $G$ contains an ${{\rm LS}}_+$-imperfect line graph and is itself ${{\rm LS}}_+$-imperfect. $\Box$ For graphs having stability number three, there is no decomposition known yet. Relying only on the behavior of the stable set polytope under clique identification [@Chvatal1975] and the result on ${{\rm LS}}_+$-(im)perfect graphs from Theorem \[mnpmascero\], we can prove: \[thm\_not\_quasi-line\_3\] Every facet-defining claw-free not quasi-line graph $G$ with $\alpha(G) = 3$ is ${{\rm LS}}_+$-imperfect. Let $G$ be a facet-defining claw-free graph with $\alpha(G) = 3$ that is not quasi-line. Then, there is a node $v$ in $G$ such that $G' = G[N(v)]$ cannot be partitioned into 2 cliques. Hence, in the complement $\overline G$ of $G$, the subgraph $\overline G'$ cannot be partitioned into two stable sets. Thus, $\overline G'$ is non-bipartite and contains an odd cycle. Let $C$ be the shortest odd cycle in $\overline G'$. Then $C$ is not a triangle (otherwise, $\overline C$ and $v$ induce a claw in $G$). Hence, $C$ is an odd hole (because it is an odd cycle of length $\geq 5$, but has no chords according to its choice). Therefore, $\overline C$ is an odd antihole in $G$; let $u_1, \ldots, u_{2k+1}$ denote it nodes and $u_iu_{i+k}$ be its non-edges. Furthermore, let $G_v = G[N(v) \cup \{v\}]$ and $W = N(v)-\overline C$. From now on, we will use $\overline C$ to denote both, the node set and the odd antihole when it is clear from the context. \[claim1\] $G_v$ has stability number 2 and\ - either contains an $LS_+$-imperfect subgraph - or is the complete join of $v$, $\overline C$ and $W$. Firstly, note that $N(v)$ does not contain a stable set of size 3 (otherwise, $G$ clearly contains a claw). Hence, $\alpha(G_v) = 2$ follows. If $W = \emptyset$, we are done. If there is a node $w \in W$, then for each such node, either $w$ is completely joined to $\overline C$ or else the subgraph of $G$ induced by $w$ and $\overline C$ is $LS_+$-imperfect due to Theorem \[mnpmascero\]. $\Diamond$\ We are done if $G_v$ is $LS_+$-imperfect. Hence, assume in the sequel of this proof that $G_v$ is the complete join of $v$, $\overline C$ and $W$. Since $\alpha(G) = 3$ holds, $G_v$ is a proper subgraph of $G$. We partition the nodes in $G-G_v$ into 3 different subsets:\ - $X$ containing all nodes from $G-G_v$ having a neighbor in $W$, - $Y$ having all nodes from $G-G_v$ having no neighbor in $W$, but a neighbor in $\overline C$, - $Z$ containing all nodes from $G-G_v$ having no neighbor in $W \cup \overline C$. \[claim2\] Every node $x \in X$\ - either induces together with $\overline C$ an $LS_+$-imperfect subgraph of $G$, - or is completely joined to $\overline C$. No node $x \in X$ can belong to a stable set $S = \{x,u_i,u_{i+k}\}$ (otherwise, any neighbor $w \in W$ of $x$ induces together with $S$ a claw in $G$). Hence, for every $x \in X$, the subgraph $G[\overline C \cup \{x\}]$ has stability number 2 and is either $LS_+$-imperfect or an odd antiwheel by Theorem \[mnpmascero\]. $\Diamond$\ We are done if some node $x \in X$ yields an $LS_+$-imperfect graph $G[\overline C \cup \{x\}]$. Hence, assume in the sequel of this proof that $\overline C$ and $X$ are completely joined. \[claim3\] $X$ is a clique. Otherwise, $G$ contains a claw containing some node $u_i \in \overline C$ as central node, $v$ and two non-adjacent nodes $x,x' \in X$. $\Diamond$\ Let $G_X$ denote the subgraph of $G$ induced by $v$, $N(v)$ and $X$. \[claim4\] We have $\alpha(G_X) = 2$. We know already that $\alpha(G_v) = 2$ by Claim 1. If $G_X$ contains a stable set $S$ of size 3, then $x \in S$ for some $x \in X$. This implies $S \cap \overline C = \emptyset$ (recall that we assume that $X$ and $\overline C$ are completely joined). In addition, $v \not\in S$ (since $v$ is adjacent to all nodes in $W$ (so we would have $S \cap W = \emptyset$ if $v \in S$), but $S$ cannot contain 2 nodes from $X$ (since $X$ is a clique by Claim 3)). Finally, $S$ cannot contain 2 non-adjacent nodes $w,w' \in W$ (otherwise, any node $u_i \in \overline C$ induces with $S$ a claw in $G$). Hence, there is no such stable set $S$ in $G_X$. $\Diamond$\ By $\alpha(G) = 3$ and $\alpha(G_X) = 2$, there is a node in $Y \cup Z$. We conclude that $Y \neq \emptyset$ (otherwise, $X$ would constitute a clique cutset of $G$, separating $Z$ from $G'$, a contradiction to $G$ facet-defining by Chvátal [@Chvatal1975]. \[claim5\] $W$ induces a clique. Otherwise, $G$ contains a claw induced by some node $y \in Y$, a neighbor $u_i \in \overline C$ of $y$, and two non-adjacent nodes $w,w' \in W$. $\Diamond$\ Hence, $G_v$ is in fact the complete join of a clique $Q = \{v\} \cup W$ and $\overline C$. In addition, $X$ is a clique and completely joined to $\overline C$, $Y$ is non-empty, and there is no edge between $Q$ and $Y$. We further obtain: \[claim6\] Every node $y \in Y$ is completely joined to $X$. Otherwise, $G$ contains a claw induced by some node $y \in Y$, a neighbor $u_i \in \overline C$ of $y$, $v$ and a non-neighbor $x \in X$ of $y$. $\Diamond$\ Note that, according to Theorem \[mnpmascero\], each node $y \in Y$ has three possibilities for its connections to $\overline C$:\ - either $y$ induces together with $\overline C$ an $LS_+$-imperfect subgraph of $G$, - or $y$ is completely joined to $\overline C$, - or $y$ belongs to a stable set $S_y = \{y,u_i,u_{i+k}\}$ (recall that, by Theorem \[mnpmascero\], whenever $\{y\} \cup \overline C$ has stability number 2, it is either $LS_+$-imperfect or an odd antiwheel). If a node $y \in Y$ gives rise to an $LS_+$-imperfect subgraph of $G$, we are done. Hence, assume in the sequel of this proof that $Y$ is partitioned into two subsets $Y_*$ and $Y_S$ containing all nodes $y$ that are completely joined to $\overline C$ resp. belong to a stable set $S_y = \{y,u_i,u_{i+k}\}$. We next show: \[claim7\] $Y_S \neq \emptyset$. Assume to the contrary that we have $Y = Y_*$. Then, $Y$ also induces a clique (otherwise, there is a claw in $G$ induced by $v$, some node $u_i \in \overline C$ and two non-adjacent nodes $y,y' \in Y$). This implies that $G[G_v \cup X \cup Y]$ has stability number 2 (by $\alpha(G_X) = 2$ due to Claim 4, $Y$ completely joined to $X$ due to Claim 6, and $Y = Y_*$ clique). Thus, $Z$ is non-empty (because $\alpha(G) = 3$). Hence, $G$ contains a clique cutset $X \cup Y$, separating $Z$ from $G_v$ (recall that every node in $Z$ has only neighbors in $X$ or $Y$, but not in $G_v$), a contradiction to $G$ facet-defining by Chvátal [@Chvatal1975]. Therefore, we conclude that $Y = Y_*$ cannot hold. $\Diamond$\ Having ensured the existence of a stable set $S_y = \{y,u_i,u_{i+k}\}$ in $G$, we next observe: \[claim8\] $X = \emptyset$. Otherwise, $G$ contains a claw induced by $S_y$ and any node $x \in X$ (recall: we assume that $X$ and $\overline C$ are completely joined (otherwise, $G$ is $LS_+$-imperfect by Claim 2), and have that $X$ and $Y$ are completely joined by Claim 6). $\Diamond$\ This implies particularly that no node outside $G_v$ has a neighbor in $W$. We next study the connections between $\overline C$ and $Y$ in more detail and obtain the following important fact: \[claim9\] $\overline C = C_5$ and each node $y \in Y_S$ has exactly two consecutive neighbors on $\overline C$. Consider some node $y \in Y_S$ and the stable set $S_y = \{y,u_i,u_{i+k}\}$. By construction of $Y$, $y$ has a neighbor $u_j \in \overline C$. This node $u_j$ (and any further neighbor of $y$ in $\overline C$) cannot be a common neighbor of $u_i$ and $u_{i+k}$ (otherwise, $u_j$ induces together with $S_y$ a claw in $G$). Hence, $u_j$ equals either $u_{i-1}$ (which is not adjacent to $u_{i+k}$) or else $u_{i+k+1}$ (which is not adjacent to $u_{i}$). W.l.o.g., say that $y$ has $u_{i-1}$ as neighbor in $\overline C$. Then $\overline C = C_5$ follows because for any $k \geq 3$, the graph induced by $y$ and $\overline C$ contains a claw with center $u_{i-1}$ and the nodes $u_{i+1},u_{i+k+2},y$ (or else $u_{i+1}$ or $u_{i+k+2}$ induce with $S_y$ a claw if $y$ is adjacent to $u_{i+1}$ or $u_{i+k+2}$). Moreover, we observe that $y$ is also adjacent to $u_{i-2}$ (otherwise, there is a claw with center $u_{i-1}$ and the nodes $u_{i},u_{i-2},y$). This shows the assertion that each node $y \in Y_S$ has exactly two consecutive neighbors on $\overline C = C_5$. $\Diamond$\ We next observe: \[claim10\] $Z$ induces a clique. Otherwise, $G$ contains a stable set of size 4, consisting of two non-adjacent nodes in $\overline C$ and two non-adjacent nodes in $Z$ (recall: by definition of $Z$, there is no edge between $\overline C$ and $Z$). $\Diamond$\ Hence, so far we have the following: $G_v$ is the complete join of a clique $Q = \{v\} \cup W$ and $\overline C = C_5$. $G-G_v$ is partitioned into two subsets $Y$ and $Z$ where\ - $Y$ is non-empty and partitions into two subsets $Y_*$ and $Y_S$ consisting of all nodes $y$ that are completely joined to $\overline C = C_5$ resp. belong to a stable set $S_y = \{y,u_i,u_{i+k}\}$ and have exactly two consecutive neighbors on the $C_5$; - $Z$ induces a clique and no node in $Z$ has a neighbor in $G_v$. We continue to explore the composition of $Y$ and its connections to $\overline C = C_5$: \[claim11\] If two nodes $y,y' \in Y$ share a same neighbor $u_j \in \overline C$, then $y$ and $y'$ are adjacent. Otherwise, $G$ contains a claw with center $u_j$ and the nodes $v,y,y'$. $\Diamond$ \[claim12\] There are at least two nodes in $Y_S$. By Claim 7, there is a node $y \in Y_S$. Then $Y \neq \{y\}$ follows (otherwise, the only two and consecutive neighbors of $y$ on $\overline C = C_5$ (by Claim 9) form a clique cutset in $G$, separating $y$ from $Q$, a contradiction to $G$ facet-defining by Chvátal [@Chvatal1975]). If all nodes from $Y - \{y\}$ belong to $Y_*$, then $Y$ induces a clique by Claim 11 (because all share a common neighbor in $\overline C$), and $Y - \{y\}$ together with the two and consecutive neighbors of $y$ on $\overline C$ form a clique cutset in $G$, separating $y$ from $Q$, again a contradiction to $G$ facet-defining. Hence, $Y_S$ contains at least two nodes. $\Diamond$\ Using similar arguments, we next show: \[claim13\] Not all nodes in $Y_S$ have the same two consecutive neighbors on $\overline C$. Otherwise, $G$ has a clique cutset (consisting of $Y_*$ and the two common, consecutive neighbors on $\overline C$ of all nodes in $Y_S$), separating $Y_S$ from $Q$, again a contradiction to $G$ facet-defining. $\Diamond$ \[claim14\] If all nodes in $Y_S$ share a common neighbor $u_i$ on $\overline C$, then $G$ contains $G_{EMN}$ as induced subgraph. By assumption, we have only three types of nodes in $Y$: nodes $y$ with $N_{\overline C}(y) = \{u_{i-1},u_i\}$, nodes $y'$ with $N_{\overline C}(y') = \{u_i,u_{i+1}\}$, and nodes $y_* \in Y_*$. $Y$ induces a clique (by Claim 11) and $Z = \emptyset$ follows (otherwise, $Y$ is a clique cutset separating $Z$ from $Q$, again a contradiction to $G$ facet-defining). Claim 12 combined with Claim 13 shows that there is at least one node $y$ adjacent to $u_{i-1},u_i$ and at least one node $y'$ adjacent to $u_i,u_{i+1}$. Moreover, there is also at least one node $y_* \in Y_*$ (otherwise, $G$ equals the gear (induced by $v$, $\overline C$, $y$ and $y'$, see Figure \[Fig\_N+P\_alpha3\_claim14\]) with possible replications of $v$, $y$, $y'$ and is not facet-defining, a contradiction). Hence, $G$ contains $G_{EMN}$ (induced by $y_*$ and the nodes $v$, $u_{i-1}$, $u_{i+1}$, $y$, and $y'$, see Figure \[Fig\_N+P\_alpha3\_claim14\]). $\Diamond$\ ![Subgraph induced by $\overline C$ and $v$, $y$, $y'$, $y_*$ (removing $y_*$ yields the gear, the bold edges indicate the $G_{EMN}$).[]{data-label="Fig_N+P_alpha3_claim14"}](N+P_alpha3_claim14.eps) This shows that $G$ is either $LS_+$-imperfect (and we are done) or $Y_S$ has two nodes with distinct neighbors on $\overline C$. Let us assume the latter. \[claim15\] If two nodes $y y' \in Y_S$ with distinct neighbors on $\overline C$ are adjacent, then $G$ contains $G_{LT}$ as induced subgraph. W.l.o.g., let $N_{\overline C}(y) = \{u_{1},u_2\}$ and $N_{\overline C}(y') = \{u_3,u_{4}\}$. If $y$ is adjacent to $y'$, then $u_1$, $y$, $y'$, $u_4$, $u_5$ induce together with $v$ a $G_{LT}$. $\Diamond$\ Hence, assume that no two nodes in $y y' \in Y_S$ with distinct neighbors on $\overline C$ are adjacent (otherwise, we are done). \[claim16\] If $Z \neq \emptyset$, then $G$ contains a node stretching of $G_{EMN}$ as induced subgraph. If there is a node $z \in Z$, then $z$ is adjacent to every node in $Y_S$ (otherwise, there is a node $y \in Y_S$ such that $z$ together with $S_y$ forms a stable set of size 4). Recall that we assume that $Y_S$ contains two non-adjacent nodes $y$, $y'$ with distinct neighbors on $\overline C$. Then $z$ together with $y$, $y'$ and $\overline C$ induce a node stretching of $G_{EMN}$. $\Diamond$\ Hence, we are done if $Z \neq \emptyset$ (because $G$ contains an $LS_+$-imperfect subgraph). So let us assume $Z = \emptyset$ from now on. Furthermore, assume w.l.o.g. that $y$ with $N_{\overline C}(y) = \{u_{1},u_2\}$ and $y'$ with $N_{\overline C}(y') = \{u_3,u_{4}\}$ is a pair of nodes in $Y_S$ having distinct neighbors on $\overline C$. Since $G$ is facet-defining (and, thus, without clique cutset), there is a path connecting $y$ and $y'$; let $P$ denote the shortest such path. Note that $P$ has length $\geq 2$ (recall: $y$ and $y'$ are supposed to be non-adjacent, otherwise $G$ contains a $G_{LT}$ by Claim 15). \[claim17\] If $P$ has length $2$, then $G$ is $LS_+$-imperfect. So, let $P$ have length $2$ and denote by $t$ its only internal node. Then $t \in Y$ follows (by $Z = \emptyset$ and because there is no common neighbor of $y$ and $y'$ in $\overline C$). We conclude that $t \not\in Y_*$ holds (otherwise, $G$ has a claw induced by $t$ and $S_y$) and that $t$ and $u_5$ are non-adjacent (otherwise, $G$ has a claw induced by $t$ and $y$, $y'$, $u_5$). In fact, $N_{\overline C}(t) = \{u_{2},u_3\}$ follows by our assumption that no two nodes in $Y_S$ with distinct neighbors on $\overline C$ are adjacent (otherwise, $G$ contains a $G_{LT}$ by Claim 15). Since the graph induced by $G_v$ together with $y,t,y'$, called a 3-gear (see Figure \[Fig\_alpha3\_claim17\]), is not facet-defining, there must be another node $y''$ in $Y$ (recall: we have $Z = \emptyset$). We are done if there is a node $y'' \in Y_*$ (because $G$ contains a $G_{EMN}$ induced by $u_1, y, t, u_3, v$ and $y''$ in this case). Hence, assume $y'' \in Y_S$. W.l.o.g., let $N_{\overline C}(y'') = \{u_{4},u_5\}$ (note: the case $N_{\overline C}(y'') = \{u_{1},u_5\}$ is symmetric, and in all other cases, $G$ is still a 3-gear with some replicated nodes, thus not facet-defining). Then $y''$ and $y'$ are adjacent by Claim 11. If $y''$ is also adjacent to $y$ or $t$, then we are done since then $G$ contains a $G_{LT}$ by Claim 15. If $y''$ is neither adjacent to $y$ nor to $t$, then $G$ contains an $LS_+$-imperfect line graph induced by $u_1, y, t, u_3, v$ and $y',y'',u_3$ (being a node stretching of $G_{EMN}$). Note that $G$ still contains one of the above $LS_+$-imperfect subgraphs if $G$ contains more nodes than considered so far. $\Diamond$\ ![Subgraph induced by $\overline C$ and $v$, $y$, $y'$, $t$, called a 3-gear.[]{data-label="Fig_alpha3_claim17"}](N+P_alpha3_claim17.eps) Thus, we are done if $P$ has length $2$. Let us finally assume that $P$ has length $\geq 3$ and $y$ and $y'$ have no common neighbor (recall: $P$ is a shortest path connecting them). Then $Y_* = \emptyset$ follows (because each node $y'' \in Y_*$ shares a common neighbor with $y$ and $y'$ on $\overline C$ and is, thus, adjacent to both $y$ and $y'$ by Claim 11, a contradiction to the choice of $P$ as shortest path connecting them). Moreover, there is a neighbor $\overline y$ of $y$ in $Y_S$ (otherwise, $\{u_{1},u_2\}$ forms a clique cutset separating $y$ from $Q$, a contradiction to $G$ facet-defining by Chvátal [@Chvatal1975]). In addition, $N_{\overline C}(\overline y)$ is different from $\{u_{3},u_4\}$ and from $\{u_{4},u_5\}$ (otherwise, $y$ and $y'$ would be a pair of nodes with distinct neighbors on $\overline C$ and connected by a path of length 2, hence $G$ is $LS_+$-imperfect by Claim 17 and we are done). We clearly have $N_{\overline C}(\overline y) \neq \{u_{2},u_3\}$ (otherwise, $\overline y$ adjacent to $y'$ follows by Claim 11 and we have again a pair of nodes with distinct neighbors on $\overline C$ and connected by a path of length 2, hence $G$ is $LS_+$-imperfect by Claim 17 and we are done). If there is no node in $Y_S$ having $\{u_{1},u_5\}$ as neighbors on $\overline C$, then $\{u_{1},u_2\}$ forms still a clique cutset separating $y$ from $Q$, again a contradiction. Hence, let $N_{\overline C}(\overline y) = \{u_{1},u_5\}$. Then, there is no node in $Y_S$ having $\{u_{4},u_5\}$ as neighbors on $\overline C$ (this node $y''$ would be adjacent to $\overline y$ by Claim 11, so that $\overline y$ would be a common neighbor of $y$ and $y''$, leading to an $LS_+$-imperfect subgraph of $G$ by Claim 17 and we are done). Similarly, there is no node in $Y_S$ having $\{u_{3},u_4\}$ as neighbors on $\overline C$. This finally implies that $\{u_{3},u_4\}$ is a clique cutset separating $y'$ from $Q$, a contradiction to $G$ facet-defining by Chvátal [@Chvatal1975]). That all further nodes of $G$ are either replicates of $y$, $y'$ or $\overline y$ (and $\{u_{3},u_4\}$ remains a clique cutset in all cases) finishes the proof. $\Box$ Hence, the only facet-defining subgraphs $G'$ of ${{\rm LS}}_+$-perfect claw-free not quasi-line graphs $G$ with $\alpha(G) = 3$ have $\alpha(G') = 2$ and are, by Theorem \[thm\_N+imperfect\_alpha2\], cliques, odd antiholes or their complete joins. We conclude that ${{\rm LS}}_+$-perfect facet-defining claw-free not quasi-line graphs $G$ with $\alpha(G) = 3$ are joined $a$-perfect and, thus, the ${{\rm LS}}_+$-Perfect Graph Conjecture is true for this class. This together with Theorem \[thm\_not\_quasi-line\_3\] shows that the only facet-defining subgraphs $G'$ of ${{\rm LS}}_+$-perfect claw-free not quasi-line graphs $G$ with $\alpha(G) \geq 4$ have $\alpha(G') = 2$ and are, by Theorem \[thm\_N+imperfect\_alpha2\], cliques, odd antiholes or their complete joins. Thus, every ${{\rm LS}}_+$-perfect claw-free not quasi-line graph $G$ with $\alpha(G) \geq 4$ is joined a-perfect and, thus, the ${{\rm LS}}_+$-Perfect Graph Conjecture holds true for this class. Combining Corollary \[Cor\_alpha2\_2\] with the above results shows that all ${{\rm LS}}_+$-perfect claw-free but not quasi-line graphs are joined $a$-perfect and we obtain: \[Cor\_not\_quasi-line\] The ${{\rm LS}}_+$-Perfect Graph Conjecture is true for all claw-free graphs that are not quasi-line. Finally, we obtain our main result (Theorem \[thm\_main\]) as direct consequence of Corollary \[Cor\_quasi-line\] and Corollary \[Cor\_not\_quasi-line\]: The ${{\rm LS}}_+$-Perfect Graph Conjecture is true for all claw-free graphs. Conclusion and future research ============================== The context of this work was the study of ${{\rm LS}}_+$-perfect graphs, i.e., graphs where a single application of the Lovász-Schrijver PSD-operator ${{\rm LS}}_+$ to the edge relaxation yields the stable set polytope. Hereby, we are particularly interested in finding an appropriate polyhedral relaxation $P(G)$ of ${{\rm STAB}}(G)$ that coincides with ${{\rm LS}}_+(G)$ and ${{\rm STAB}}(G)$ if and only if $G$ is ${{\rm LS}}_+$-perfect. An according conjecture has been recently formulated (${{\rm LS}}_+$-Perfect Graph Conjecture); here we verified it for the well-studied class of claw-free graphs (Theorem \[thm\_main\]). For that, it surprisingly turned out that it was not necessary to make use of the description of STAB$(G)$ for claw-free not quasi-line graphs $G$ - with $\alpha(G) = 2$ (by Cook, see [@Shepherd1994]), - with $\alpha(G) = 3$ (by Pêcher, Wagler [@PecherWagler2010]), - with $\alpha(G) \geq 4$ (by Galluccio, Gentile, Ventura [@GGV2008; @GGV2014a; @GGV2014b]). From the presented results and proofs, we can draw some further conclusions. First of all, we can determine the subclass of joined $a$-perfect graphs to which all ${{\rm LS}}_+$-perfect claw-free graphs belong to. In [@KosterWagler_IRII], it is suggested to call a graph $G$ *$m$-perfect* if the only facets of ${{\rm STAB}}(G)$ are associated with cliques and minimally imperfect graphs. According to [@CPW_2009], $G$ is *joined $m$-perfect* if ${{\rm STAB}}(G)$ is given only by facets associated with cliques, minimally imperfect graphs and their complete joins. Theorem \[hipomatch\] together with the results from Section \[results\] provide the complete list of all facet-defining ${{\rm LS}}_+$-perfect claw-free graphs: - cliques, - odd holes and odd antiholes, - complete joins of odd antihole(s) and a (possibly empty) clique. Hence, we conclude: All ${{\rm LS}}_+$-perfect claw-free graphs are joined $m$-perfect. Among these possible facets, only complete joins of odd antihole(s) and a non-empty clique are non-rank. This directly implies: A rank-perfect ${{\rm LS}}_+$-perfect claw-free graph has as only facet-defining subgraphs cliques, odd holes, odd antiholes, or complete joins of the latter. Note that Galluccio and Sassano provided in [@GalluccioSassano97] a complete characterization of the rank facet-defining claw-free graphs: they either belong to one of the following three families of rank-minimal graphs - cliques, - partitionable webs $W^{\omega - 1}_{\alpha \omega + 1}$ (where $\alpha$ and $\omega$ stand for stability and clique number, resp.), - line graphs of minimal 2-connected hypomatchable graphs $H$ (where $H-e$ is not hypomatchable anymore for any edge $e$), or can be obtained from them by means of two operations, sequential lifting and complete join. Our results show: an ${{\rm LS}}_+$-perfect claw-free graph $G$ has, besides cliques, only odd holes and odd antiholes as rank-minimal subgraphs; cliques are the only subgraphs in $G$ that can be sequentially lifted to larger rank facet-defining subgraphs, where complete joins can only be taken of odd antiholes. Note further that, besides verifying the ${{\rm LS}}_+$-Perfect Graph Conjecture for claw-free graphs, we obtained the complete list of all minimally ${{\rm LS}}_+$-imperfect claw-free graphs. In fact, the results in [@BENT2011; @EN2014; @ENW2014] imply that the following graphs are minimally ${{\rm LS}}_+$-imperfect: - graphs $G$ with $\alpha(G)=2$ such that $G-v$ is an odd antihole for some node $v$, not completely joined to $G-v$, - the web $W_{10}^2$, - ${{\rm LS}}_+$-imperfect line graphs (which are all node stretchings of $G_{LT}$ or $G_{EMN}$). Our results from Section \[results\] on facet-defining ${{\rm LS}}_+$-perfect claw-free graphs imply that they are the only minimally ${{\rm LS}}_+$-imperfect claw-free graphs. Finally, the subject of the present work has parallels to the well-developed research area of perfect graph theory also in terms of polynomial time computability. In fact, it has the potential of reaching even stronger results due the following reasons. Recall that calculating the value $$\eta_+(G) = \max {{\mathbf{1}}}^Tx, x \in {{\rm LS}}_+(G)$$ can be obtained with arbitrary precision in polynomial time for every graph $G$, even in the weighted case, by [@LovaszSchrijver1991]. Thus, the stable set problem can be solved in polynomial time for a strict superset of perfect graphs, the ${{\rm LS}}_+$-perfect graphs, by $\alpha(G) = \eta_+(G)$. Hence, our future lines of research include to find - new families of graphs where the conjecture holds (e.g., by characterizing the minimally ${{\rm LS}}_+$-imperfect graphs within the class), - new subclasses of ${{\rm LS}}_+$-perfect or joined a-perfect graphs, - classes of graphs $G$ where ${{\rm STAB}}(G)$ and ${{\rm LS}}_+(G)$ are “close enough” to have $\alpha(G) = \lfloor \eta_+(G) \rfloor$. In particular, the class of graphs $G$ with $\alpha(G) = \lfloor \eta_+(G) \rfloor$ can be expected to be large since ${{\rm LS}}_+(G)$ is a much stronger relaxation of ${{\rm STAB}}(G)$ than ${{\rm TH}}(G)$. In all cases, the stable set problem could be approximated with arbitrary precision in polynomial time in these graph classes by optimizing over ${{\rm LS}}_+(G)$. Finally, note that ${{\rm LS}}_+(P(G))$ with $${{\rm STAB}}(G) \subseteq P(G) \subseteq {{\rm ESTAB}}(G)$$ clearly gives an even stronger relaxation of ${{\rm STAB}}(G)$ than ${{\rm LS}}_+(G)$. However, already approximating with arbitrary precision over ${{\rm LS}}_+({{\rm QSTAB}}(G))$ cannot be done in polynomial time anymore for all graphs $G$ by [@LovaszSchrijver1991]. Hence, ${{\rm LS}}_+$-perfect graphs or their generalizations satisfying $\alpha(G) = \lfloor \eta_+(G) \rfloor$ are the most promising cases in this context. [10]{} S. Bianchi, M. Escalante, G. Nasini, and L. 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M. Grötschel, L. Lovász, and A. Schrijver, [*Geometric Algorithms and Combinatorial Optimization.*]{} Springer-Verlag (1988) A. Koster and A. Wagler, *Comparing imperfection ratio and imperfection index for graph classes,* RAIRO Operations Research, vol.  42, pp.  485-500, 2008. T.M. Liebling, G. Oriolo, B. Spille, and G. Stauffer. On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs. Mathematical Methods of Operations Research, 59(1):25–35, 2004. L. Lipták and L. Tunçel, “Stable set problem and the lift-and-project ranks of graphs,” *Math. Programming A*, vol. 98, pp. 319–353, 2003. L. Lovász, “On the Shannon capacity of a graph,” *IEEE Trans. Inf. Theory*, vol. 25, pp. 1–7, 1979. L. Lovász and A. Schrijver, “Cones of matrices and set-functions and 0-1 optimization,” *SIAM J. on Optimization*, vol. 1, pp. 166–190, 1991. G. Minty, “On maximal independent sets of vertices in claw-free graphs”, *J. of Combinatorial Theory*, vol.  28, pp. 284–304, 1980. G. Oriolo, U. Pietropaoli, and G. Stauffer, “A new algorithm for the maximum weighted stable set problem in claw-free graphs”, in A. Lodi, A. Panconesi and G. Rinaldi (eds), IPCO 2008, Vol. 5035 of Lecture Notes in Computer Science, Springer, Bertinoro, Italy, May 26-28, pp. 77–96, 2008. A. Pêcher, A. Wagler, “Almost all webs are not rank-perfect”, Math.  Programming B 105 (2006) 311–328 A. Pêcher, A. Wagler, “On facets of stable set polytopes of claw-free graphs with stability number three”, Discrete Mathematics 310 (2010) 493–498 U. Pietropaoli, A. Wagler, “Some results towards the description of the stable set polytope of claw-free graphs”, Proc. of ALIO/EURO Workshop on Applied Combinatorial Optimization, Buenos Aires, 2008 N. Sbihi, “Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile”, Discrete Mathematics 29, 53–76, 1980. F.B. Shepherd, “Near-Perfect Matrices,” *Math. Programming*, vol. 64, pp. 295–323, 1994. F.B. Shepherd, “Applying Lehman’s Theorem to Packing Problems,” *Math. Programming*, vol. 71, pp. 353–367, 1995. G. Stauffer, “On the Stable set polytope of claw-free graphs”. Phd thesis, 2005, Swiss Institute of Technology in Lausanne. A. Wagler, [*Critical Edges in Perfect Graphs.*]{} PhD thesis, TU Berlin and Cuvillier Verlag Göttingen (2000). A. Wagler, “Antiwebs are rank-perfect,” *4OR*, vol. 2, pp. 149-152, 2004. A. Wagler, “On rank-perfect subclasses of near-bipartite graphs,” *4OR*, vol. 3, pp. 329–336, 2005. [^1]: This work was supported by a MATH-AmSud cooperation (PACK-COVER), PID-CONICET 0277, and PICT-ANPCyT 0586.

--- abstract: 'Recent work (@corradi2015, @jones2016) has shown that the phenomenon of extreme abundance discrepancies, where recombination line abundances exceed collisionally excited line abundances by factors of 10 or more, seem to be strongly associated with planetary nebulae with close binary central stars. To further investigate, we have obtained spectra of a sample of nebulae with known close binary central stars, using FORS2 on the VLT, and we have discovered several new extreme abundance discrepancy objects. We did not find any non-extreme discrepancies, suggesting that a very high fraction of nebulae with close binary central stars also have an extreme abundance discrepancy.' author: - 'R. Wesson,$^{1,2}$[^1] D. Jones,$^{3,4}$ J. García-Rojas$^{3,4}$, R.L.M. Corradi,$^{3,4,5}$ and H.M.J. Boffin,$^{6}$' bibliography: - 'references.bib' --- Introduction ============ The abundances of heavy elements in photoionised nebulae may be determined from their strong, bright collisionally excited lines (CELs), whose emissivity depends exponentially on the temperature, or from the much weaker recombination lines (RLs), the emissivity of which has a weak power law dependence on the temperature. Abundances derived from RLs and CELs do not agree, with the values from RLs exceeding those from CELs by a factor most commonly around 2–3, but reaching extreme values of 10 or more in about 10 per cent of planetary nebulae (PNe). This abundance discrepancy problem has been known since the 1940s (@wyse1942), but a full understanding of its causes remains elusive. The abundance discrepancy factor ([*adf*]{}) for an ion is defined as the ratio of its RL abundance to its CEL abundance. O$^{2+}$ is much the easiest ion to measure an [*adf*]{} for, being usually the most abundant heavy element, and having RLs and CELs in the optical. In this article, unless otherwise specified, [*adf*]{} refers to [*adf(O$^{2+}$)*]{}. A number of plausible mechanisms have been postulated which could account for the abundance discrepancy. These include temperature fluctuations (@peimbert1967), density fluctuations (@viegas1994), hydrogen-deficient clumps (@liu2000), X-ray illuminated quasi-neutral clumps (@ercolano2009), and non-thermally distributed electrons (@nicholls2012). All except the hydrogen-deficient clump theory account for the discrepancy in the context of a chemically homogeneous nebula. Since 2006 there have been suggestions that whatever the mechanism, it may be related to central star binarity. Hf 2–2, one of the most extreme objects known with an abundance discrepancy of 80, has a binary central star with an orbital period so short that it must have undergone a common envelope (CE) phase (@liu2006). The difficulties in identifying such short period binaries meant that until recently, the numbers of planetary nebulae for which binarity was established and abundance discrepancy measured was small. However, recently the picture has become clearer, with @corradi2015 finding that three known post-CE binaries had nebulae with extreme abundance discrepancies, and then @jones2016 strengthening the link with a study of the post-CE nebula NGC 6778, finding an abundance discrepancy of a factor of nearly 20 on average, and peaking at around 40 in the centre of the nebula. We have obtained new spectra of a sample of PNe known to have close binary central stars. The sample has revealed a number of objects with extreme abundance discrepancies, and no low-discrepancy objects. There thus appears to be an almost perfect correlation between close binarity and extreme abundance discrepancies. Observations ============ We obtained spectra of about 25 planetary nebulae using FORS2 mounted on UT1 (Antu) at the Very Large Telescope in Chile, in ESO programmes 093.D-0038(A) and 096.D-0080(A). The spectra covered wavelengths from 3600-5000 and 5800-7200[Å]{} at a resolution of 1.5[Å]{}, sufficient to resolve recombination lines in the blended features at 4070[Å]{} and 4650[Å]{}. The target nebulae were southern objects with known close binary central stars, for which there were no recombination line abundances in the literature. The exception to that was Hf 2–2, known to have an extreme abundance discrepancy, which we included in our sample both as a control to verify our strategy and methodology, and with the hope that our spectra would be deep enough to permit a spatially resolved study. The programme was designed as a filler to be carried out in almost any weather conditions, and so a number of objects in the sample were observed in less than ideal conditions, but several excellent spectra were obtained. The data on NGC 6778, obtained early in the programme, revealed a new extreme object with an abundance discrepancy of $\sim$20, presented in @jones2016. Upon the completion of the programme, we have detected recombination lines in seven further objects, including Hf 2–2. Analysis ======== Spectra were reduced using standard <span style="font-variant:small-caps;">starlink</span> routines. Cosmic rays were removed from the 2D frames using a combination of <span style="font-variant:small-caps;">starlink</span>’s figaro routines and a python implementation of the LAcosmic algorithm [@vandokkum2001]. Emission line fluxes were measured using [alfa]{} [@wesson2016a], which we used in @jones2016 to perform a similar analysis for the high [*adf*]{} PN NGC 6778. [alfa]{} derives fluxes by optimising the parameters of Gaussian fits to line profiles using a genetic algorithm, after subtracting a globally-fitted continuum. [neat]{} [@wesson2012] was then used to derive final ionic and elemental abundances from these emission line fluxes. The code corrects for interstellar extinction using the ratios of H$\gamma$ and H$\delta$ to H$\beta$ (the H$\alpha$/H$\beta$ ratio was not used to calculate the extinction but rather as a sanity check to ensure that line fluxes measured from the non-overlapping red and blue spectra were consistent) and the Galactic extinction law of @howarth1983; temperatures and densities are then derived from the standard diagnostics [see @wesson2012 for full details]. Ionic abundances are then calculated from flux-weighted averages of the emission lines of each species using the previously derived temperatures and densities, and total abundances estimated using the ionisation correction scheme of @delgado-inglada2014 The atomic data and ionisation correction functions used were as in @jones2016. Abundance discrepancies and evidence for H-deficient material ============================================================= We detected recombination lines in eight of our sample objects. In all eight cases, the abundance discrepancy was extreme. The values we obtained (including the Jones et al. 2016 value for NGC 6778) are listed in Table \[adftable\]. Among the eight objects was Hf 2–2, for which we derive very similar results in the integrated spectrum to those of Liu et al. (2006). ---------------- ------------------------------------------ ---------------------- ----------- ----------------- ----------------- [**Object**]{} [**O$^{2+}_{RLs}$ / O$^{2+}_{CELs}$**]{} T$_e$(\[O [iii]{}\]) T$_e$(BJ) T$_e$(He [i]{}) T$_e$(O [ii]{}) Hf 2–2 80 8800 800 2000 2000 MPA 1759 80 11500 – 4000 – Pe 1-9 75 10000 – 4000 – NGC 6326 50 14500 8000 4000 – NGC 6337 30 12500 – 3000 $<$1000 NGC 6778 20 8800 4100 3000 1300 Fg 1 20 10450 – 6000 $<$1000 Hen 2–283 13 8800 – – 3000 ---------------- ------------------------------------------ ---------------------- ----------- ----------------- ----------------- : The abundance discrepancy for O$^{2+}$ and temperature diagnostics in the objects where recombination lines were detected[]{data-label="adftable"} We then estimated upper limits to the abundance discrepancy for the objects where recombination lines were not detected, using a spectral synthesis code to calculate the O$^{2+}$ recombination line spectrum, and varying the abundance until we found the value for which the RLs would have been detectable. We found that the limits from our non-detections were not strong, and that an extreme abundance discrepancy could not be ruled out in any case. Temperatures measured from various diagnostics support the picture of hydrogen-deficient material in these extreme objects. The standard \[O [iii]{}\] line ratios give temperatures typical of a photoionised gas of ‘normal’ composition - 8–12kK. Three diagnostics give generally much lower temperatures: the Balmer jump (BJ), which lies close to the edge of our spectral coverage, but is detected with sufficient signal to noise in several nebulae; He [i]{} emission line ratios, and O [ii]{} recombination lines, which should probe the coldest and most metal-rich regions of the gas (@mcnabb2013). The temperatures implied by the various diagnostics are listed in Table \[adftable\]. Discussion ========== The association between binarity and the abundance discrepancy suggests two hypotheses. Firstly, that all PNe with a close binary central star have an extreme abundance discrepancy. This work supports that hypothesis and almost doubles the number of objects known to have both a binary central star and an extreme abundance discrepancy. However, there is currently thought to be at least one object which contradicts that. NGC 5189 has a binary central star with a period of just over 4 days, but has an unusually low abundance discrepancy of 1.6 (@garcia-rojas2013), albeit measured from a very small region of the nebula. The Necklace is also a definite post-CE object, but no recombination lines were detected in deep spectra by @corradi2011. We have recently obtained observations of NGC 5189 covering most of the nebula, from which we will be able to see if the low value found by @garcia-rojas2013 is representative of the whole nebula or a chance observation of a low-[*adf*]{} region. If these two objects really have a low [*adf*]{}, then in total we now have 14 of 16 extreme-[*adf*]{} objects in the sample of close binaries with known chemistry. The complemenary hypothesis, that all nebulae with extreme abundance discrepancies have a close binary central star, is not yet well tested, but a number of extreme [*adf*]{} objects whose central star status is not yet confirmed would be fruitful grounds for investigation: Abell 30, Abell 58, NGC 1501, M 1-42 and M 3-32 are among the most extreme [*adf*]{} objects whose central star status is still unknown. Spatially resolved studies of many high-[*adf*]{} objects have found that the RL abundances are strongly centrally peaked, further associating the phenomenon with the central star. See e.g. @liu2000, @jones2016 and @garcia-rojas2016 [ and this volume]. Whether the link between them is universal or not, it is nevertheless clear that central star binarity and nebular chemistry are strongly linked. Any explanation for the most extreme abundance discrepancy must account for the fact that they are preferentially found in objects with close binary central stars. The strong similarities long noted between high [*adf*]{} PNe and old nova shells such as CP Pup and DQ Her, which exhibit extremely low plasma temperatures and strong recombination lines, suggests that some kind of nova-like outburst from the close binary central star could be responsible for ejecting H-deficient material into the nebulae. The $\sim$90% hit rate for close binaries having an extreme abundance discrepancy would suggest that this eruption must happen soon after the formation of the main nebula. [^1]: E-mail: rw@nebulousresearch.org

--- abstract: 'We introduce certain $C^*$-algebras and $k$-graphs associated to $k$ finite dimensional unitary representations $\rho_1,...,\rho_k$ of a compact group $G$. We define a higher rank Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,...,\rho_k}$, constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this $C^*$-algebra is isomorphic to a corner in the $C^*$-algebra of a row finite rank $k$ graph $\Lambda$ with no sources. For $G$ finite and $\rho_i$ faithful of dimension at least $2$, this graph is irreducible, it has vertices $\hat{G}$ and the edges are determined by $k$ commuting matrices obtained from the character table of the group. We illustrate with some examples when ${\mathcal{O}}_{\rho_1,...,\rho_k}$ is simple and purely infinite, and with some $K$-theory computations.' address: | Valentin Deaconu\ Department of Mathematics & Statistics\ University of Nevada\ Reno NV 89557-0084\ USA author: - Valentin Deaconu title: '$C^*$-algebras from $k$ group representations' --- introduction ============ The study of graph $C^*$-algebras was motivated among other reasons by the Doplicher-Roberts algebra ${\mathcal{O}}_\rho$ associated to a group representation $\rho$, see [@MRS; @KPRR]. It is natural to imagine that a rank $k$ graph is related to a fixed set of $k$ representations $\rho_1,...,\rho_k$ satisfying certain properties. Given a compact group $G$ and $k$ finite dimensional unitary representations $\rho_i$ on Hilbert spaces $\mathcal H_i$ of dimensions $d_i$ for $i=1,...,k$, we first construct a product system $\mathcal E$ indexed by the semigroup $({{\mathbb{N}}}^k,+)$ with fibers $\mathcal E_{n}=\mathcal H_1^{\otimes n_1}\otimes\cdots \otimes\mathcal H_k^{\otimes n_k}$ for $n=(n_1,...,n_k)\in {{\mathbb{N}}}^k$. Using the representations $\rho_i$, the group $G$ acts on each fiber of ${\mathcal{E}}$ in a compatible way, so we obtain an action of $G$ on the Cuntz-Pimsner algebra ${\mathcal{O}}({\mathcal{E}})$. This action determines the crossed product ${\mathcal{O}}({\mathcal{E}})\rtimes G$ and the fixed point algebra ${\mathcal{O}}({\mathcal{E}})^G$. Inspired from Section 7 of [@KPRR] and Section 3.3 of [@AM], we define a higher rank Doplicher-Roberts algebra $\mathcal O_{\rho_1,...,\rho_k}$ associated to the representations $\rho_1,...,\rho_k$. This algebra is constructed from intertwiners $Hom (\rho^n, \rho^m)$, where $\rho^n=\rho_1^{\otimes n_1}\otimes\cdots\otimes \rho_k^{\otimes n_k}$ acting on ${\mathcal{H}}^n=\mathcal H_1^{\otimes n_1}\otimes\cdots\otimes \mathcal H_k^{\otimes n_k}$ for $n=(n_1,...,n_k)\in \mathbb N^k$. We show that $\mathcal O_{\rho_1,...,\rho_k}$ is isomorphic to ${\mathcal{O}}({\mathcal{E}})^G$. If the representations $\rho_1,...,\rho_k$ satisfy some mild conditions, we construct a $k$-coloured graph $\Lambda$ with vertex space $\Lambda^0=\hat{G}$, and with edges $\Lambda^{{\varepsilon}_i}$ given by some matrices $M_i$ indexed by $\hat{G}$. Here ${\varepsilon}_i=(0,...,1,...,0)\in{{\mathbb{N}}}^k$ with $1$ in position $i$ are the canonical generators. The matrices $M_i$ have entries $$M_i(w,v)=|\{e\in \Lambda^{{\varepsilon}_i}: s(e)=v, r(e)=w\}|=\dim Hom(v,w\otimes \rho_i),$$ the multiplicity of $v$ in $w\otimes \rho_i$ for $i=1,...,k$. The matrices $M_i$ commute because $\rho_i\otimes\rho_j\cong \rho_j\otimes \rho_i$ for all $i,j=1,...,k$ and therefore $$\dim Hom(v,w\otimes \rho_i\otimes\rho_j)=\dim Hom(v,w\otimes \rho_j\otimes\rho_i).$$ By a particular choice of isometric intertwiners in $Hom(v,w\otimes \rho_i)$ for each $v,w\in \hat{G}$ and for each $i$, we can choose bijections $$\lambda_{ij}:\Lambda^{{\varepsilon}_i}\times_{\Lambda^0}\Lambda^{{\varepsilon}_j}\to \Lambda^{{\varepsilon}_j}\times_{\Lambda^0}\Lambda^{{\varepsilon}_i},$$ obtaining a set of commuting squares for $\Lambda$. For $k\ge 3$, we need to check the associativity of the commuting squares, i.e. $$(id_\ell\times \lambda_{ij})(\lambda_{i\ell}\times id_j)(id_i\times \lambda_{j\ell})=(\lambda_{j\ell}\times id_i)(id_j\times \lambda_{i\ell})(\lambda_{ij}\times id_\ell)$$ as bijections from $\Lambda^{{\varepsilon}_i}\times_{\Lambda^0}\Lambda^{{\varepsilon}_j}\times_{\Lambda^0}\Lambda^{{\varepsilon}_\ell}$ to $\Lambda^{{\varepsilon}_\ell}\times_{\Lambda^0}\Lambda^{{\varepsilon}_j}\times_{\Lambda^0}\Lambda^{{\varepsilon}_i}$ for all $i<j<\ell$, see [@FS]. If these conditions are satisfied, we obtain a rank $k$ graph $\Lambda$, which is row-finite with no sources, but in general not unique. In many situations, $\Lambda$ is cofinal and it satisfies the aperiodicity condition, so $C^*(\Lambda)$ is simple. For $k=2$, the $C^*$-algebra $C^*(\Lambda)$ is unique when it is simple and purely infinite, because its $K$-theory depends only on the matrices $M_1, M_2$. It is an open question what happens for $k\ge 3$. Assuming that the representations $\rho_1,...,\rho_k$ determine a rank $k$ graph $\Lambda$, we prove that the Doplicher-Roberts algebra $\mathcal O_{\rho_1,...,\rho_k}$ is isomorphic to a corner of $C^*(\Lambda)$, so if $C^*(\Lambda)$ is simple, then $\mathcal O_{\rho_1,...,\rho_k}$ is Morita equivalent to $C^*(\Lambda)$. In particular cases we can compute its $K$-theory using results from [@E]. The product system ================== Product systems over arbitrary semigroups were introduced by N. Fowler [@F], inspired by work of W. Arveson, and studied by several authors, see [@SY; @CLSV; @AM]. In this paper, we will mostly be interested in product systems ${\mathcal{E}}$ indexed by $( {{\mathbb{N}}}^k , +)$, associated to some representations $\rho_1,...,\rho_k$ of a compact group $G$. We remind some general definitions and constructions with product systems, but we will consider the Cuntz-Pimsner algebra ${\mathcal{O}}({\mathcal{E}})$ and we will mention some properties only in particular cases. Let $(P, \cdot)$ be a discrete semigroup with identity $e$ and let $A$ be a $C^*$-algebra. A [*product system*]{} of $C^*$-correspondences over $A$ indexed by $P$ is a semigroup ${\mathcal{E}}=\bigsqcup_{p\in P}{\mathcal{E}}_p$ and a map ${\mathcal{E}}\to P$ such that - for each $p\in P$, the fiber ${\mathcal{E}}_p\subset {\mathcal{E}}$ is a $C^*$-correspondence over $A$ with inner product $\langle\cdot,\cdot\rangle_p$; - the identity fiber ${\mathcal{E}}_e$ is $A$ viewed as a $C^*$-correspondence over itself; - for $p,q\in P\setminus\{e\}$ the multiplication map $${\mathcal{M}}_{p,q}:{\mathcal{E}}_p\times {\mathcal{E}}_q\to {\mathcal{E}}_{pq},\;\; {\mathcal{M}}_{p,q}(x,y)= xy$$ induces an isomorphism ${\mathcal{M}}_{p,q}:{\mathcal{E}}_p\otimes_A {\mathcal{E}}_q\to {\mathcal{E}}_{pq}$; - multiplication in ${\mathcal{E}}$ by elements of ${\mathcal{E}}_e=A$ implements the right and left actions of $A$ on each ${\mathcal{E}}_p$. In particular, ${\mathcal{M}}_{p,e}$ is an isomorphism. Let $\phi_p:A\to {\mathcal{L}}({\mathcal{E}}_p)$ be the homomorphism implementing the left action. The product system ${\mathcal{E}}$ is said to be [*essential*]{} if each ${\mathcal{E}}_p$ is an essential correspondence, i.e. the span of $\phi_p(A){\mathcal{E}}_p$ is dense in ${\mathcal{E}}_p$ for all $p\in P$. In this case, the map ${\mathcal{M}}_{e,p}$ is also an isomorphism. If the maps $\phi_p$ take values in ${\mathcal{K}}({\mathcal{E}}_p)$, then the product system is called [*row-finite*]{} or [*proper*]{}. If all maps $\phi_p$ are injective, then ${\mathcal{E}}$ is called [*faithful*]{}. Given a product system ${\mathcal{E}}\to P$ over $A$ and a $C^*$-algebra $B$, a map $\psi:{\mathcal{E}}\to B$ is called a [*Toeplitz representation*]{} of ${\mathcal{E}}$ if - denoting $\psi_p:=\psi|_{{\mathcal{E}}_p}$, then each $\psi_p:{\mathcal{E}}_p\to B$ is linear, $\psi_e:A\to B$ is a $*$-homomorphism, and $$\psi_e({\langle}x,y{\rangle}_p)=\psi_p(x)^*\psi_p(y)$$ for all $x,y\in {\mathcal{E}}_p$; - $\psi_p(x)\psi_q(y)=\psi_{pq}(xy)$ for all $p,q\in P, x\in {\mathcal{E}}_p, y\in {\mathcal{E}}_q$. For each $p\in P$ we write $\psi^{(p)}$ for the homomorphism ${\mathcal{K}}({\mathcal{E}}_p)\to B$ obtained by extending the map $\theta_{\xi, \eta}\mapsto \psi_p(\xi)\psi_p(\eta)^*$, where $$\theta_{\xi, \eta}(\zeta)=\xi{\langle}\eta, \zeta{\rangle}.$$ The Toeplitz representation $\psi:{\mathcal{E}}\to B$ is [*Cuntz-Pimsner covariant*]{} if $\psi^{(p)}(\phi_p(a))=\psi_e(a)$ for all $p\in P$ and all $a\in A$ such that $\phi_p(a)\in {\mathcal{K}}({\mathcal{E}}_p)$. There is a $C^*$-algebra ${\mathcal{T}}_A({\mathcal{E}})$ called the Toeplitz algebra of ${\mathcal{E}}$ and a representation $i_{\mathcal{E}}:{\mathcal{E}}\to {\mathcal{T}}_A({\mathcal{E}})$ which is universal in the following sense: ${\mathcal{T}}_A({\mathcal{E}})$ is generated by $i_{\mathcal{E}}({\mathcal{E}})$ and for any representation $\psi :{\mathcal{E}}\to B$ there is a homomorphism $\psi_*:{\mathcal{T}}_A({\mathcal{E}})\to B$ such that $\psi_*\circ i_{\mathcal{E}}=\psi$. There are various extra conditions on a product system ${\mathcal{E}}\to P$ and several other notions of covariance, which allow to define the Cuntz-Pimsner algebra ${\mathcal{O}}_A({\mathcal{E}})$ or the Cuntz-Nica-Pimsner algebra ${\mathcal{N}}{\mathcal{O}}_A({\mathcal{E}})$ satisfying certain properties, see [@F; @SY; @CLSV; @AM; @DK] among others. We mention that ${\mathcal{O}}_A({\mathcal{E}})$ (or ${\mathcal{N}}{\mathcal{O}}_A({\mathcal{E}})$) comes with a covariant representation $j_{\mathcal{E}}:{\mathcal{E}}\to {\mathcal{O}}_A({\mathcal{E}})$ and is universal in the following sense: ${\mathcal{O}}_A({\mathcal{E}})$ is generated by $j_{\mathcal{E}}({\mathcal{E}})$ and for any covariant representation $\psi :{\mathcal{E}}\to B$ there is a homomorphism $\psi_*:{\mathcal{O}}_A({\mathcal{E}})\to B$ such that $\psi_*\circ j_{\mathcal{E}}=\psi$. Under certain conditions, ${\mathcal{O}}_A({\mathcal{E}})$ satisfies a gauge invariant uniqueness theorem. For a product system ${\mathcal{E}}\to P$ with fibers ${\mathcal{E}}_p$ nonzero finitely dimensional Hilbert spaces, in particular $A={\mathcal{E}}_e={{\mathbb{C}}}$, let us fix an orthonormal basis ${\mathcal{B}}_p$ in ${\mathcal{E}}_p$. Then a Toeplitz representation $\psi:{\mathcal{E}}\to B$ gives rise to a family of isometries $\{\psi(\xi): \xi\in {\mathcal{B}}_p\}_{p\in P}$ with mutually orthogonal range projections. In this case ${\mathcal{T}}({\mathcal{E}})={\mathcal{T}}_{{\mathbb{C}}}({\mathcal{E}})$ is generated by a colection of Cuntz-Toeplitz algebras which interact according to the multiplication maps ${\mathcal{M}}_{p,q}$ in ${\mathcal{E}}$. A representation $\psi:{\mathcal{E}}\to B$ is Cuntz-Pimsner covariant if $$\sum_{\xi\in {\mathcal{B}}_p}\psi(\xi)\psi(\xi)^*=\psi(1)$$ for all $p\in P$. The Cuntz-Pimsner algebra ${\mathcal{O}}({\mathcal{E}})={\mathcal{O}}_{{\mathbb{C}}}({\mathcal{E}})$ is generated by a collection of Cuntz algebras. N. Fowler proved in [@F1] that if the function $p\mapsto \dim {\mathcal{E}}_p$ is injective, then the algebra ${\mathcal O}({\mathcal{E}})$ is simple and purely infinite. For other examples of multidimensional Cuntz algebras, see [@B]. A row-finite $k$-graph with no sources $\Lambda$ (see [@KP]) determines a product system ${\mathcal{E}}\to {{\mathbb{N}}}^k$ with ${\mathcal{E}}_0=A=C_0(\Lambda^0)$ and ${\mathcal{E}}_n=\overline{C_c(\Lambda^n)}$ for $n\neq 0$ such that we have a ${{\mathbb{T}}}^k$-equivariant isomorphism ${\mathcal{O}}_A({\mathcal{E}})\cong C^*(\Lambda)$. Recall that the universal property induces a gauge action on ${\mathcal{O}}_A({\mathcal{E}})$ defined by $\gamma_z(j_{\mathcal{E}}(\xi))=z^nj_{\mathcal{E}}(\xi)$ for $z\in {{\mathbb{T}}}^k$ and $\xi\in {\mathcal{E}}_n$. The following two definitions and two results are taken from [@DHS], see also [@Ka]. An action $ \beta $ of a locally compact group $ G $ on a product system $ {\mathcal{E}}\to P $ over $A$ is a family $ (\beta^{p})_{p \in P} $ such that $ \beta^{p} $ is an action of $ G $ on each fiber ${\mathcal{E}}_{p} $ compatible with the action $\alpha=\beta^e$ on $A$, and furthermore, the actions $(\beta^p)_{p\in P}$ are compatible with the multiplication maps ${\mathcal{M}}_{p,q}$ in the sense that $$\beta^{p q}_g({\mathcal{M}}_{p,q}(x \otimes y)) = {\mathcal{M}}_{p,q}(\beta^{p}_g(x) \otimes \beta^{q}_g(y))$$ for all $ g \in G $, $ x \in {\mathcal{E}}_{p} $ and $ y \in {\mathcal{E}}_{q} $. \[cp\] If $ \beta $ is an action of $ G $ on the product system ${\mathcal{E}}\to P $, we define the crossed product ${\mathcal{E}}\rtimes_{\beta} G $ as the product system indexed by $ P $ with fibers $ {\mathcal{E}}_{p} \rtimes_{\beta^{p}} G $, which are $ C^{\ast} $-correspondences over $ A \rtimes_{\alpha} G $. For $ \zeta \in C_c(G,{\mathcal{E}}_{p}) $ and $ \eta \in C_c(G,{\mathcal{E}}_{q}) $, the product $ \zeta \eta \in C_c(G,{\mathcal{E}}_{p q}) $ is defined by $$(\zeta \eta)(s) = \int_G{\mathcal{M}}_{p,q}(\zeta(t) \otimes \beta^{q}_t(\eta(t^{- 1} s)))dt.$$ \[p1\] The set $ {\displaystyle}{\mathcal{E}}\rtimes_{\beta} G = \bigsqcup_{p \in P} {\mathcal{E}}_{p} \rtimes_{\beta^{p}} G $ with the above multiplication satisfies all the properties of a product system of $ C^{\ast} $-correspondences over $ A \rtimes_{\alpha} G $. \[p2\] Suppose that a locally compact group $ G $ acts on a row-finite and faithful product system $ {\mathcal{E}}$ indexed by $ P = ({{\mathbb{N}}}^{k},+) $ via automorphisms $ \beta^{p}_{g} $. Then $ G $ acts on the Cuntz-Pimsner algebra ${\mathcal{O}}_{A}({\mathcal{E}}) $ via automorphisms denoted by $ \gamma_{g} $. Moreover, if $ G $ is amenable, then $ {\mathcal{E}}\rtimes_{\beta} G $ is row-finite and faithful, and $${\mathcal{O}}_{A}({\mathcal{E}}) \rtimes_{\gamma} G \cong {\mathcal{O}}_{A \rtimes_{\alpha} G}({\mathcal{E}}\rtimes_{\beta} G).$$ Now we define the product system associated to $k$ representations of a compact group $G$. We limit ourselves to finite dimensional unitary representations, even though the definition makes sense in greater generality. \[ps\] Given a compact group $G$ and $k$ finite dimensional unitary representations $\rho_i$ of $G$ on Hilbert spaces $\mathcal H_i$ for $i=1,...,k$, we construct the product system ${\mathcal{E}}={\mathcal{E}}(\rho_1,...,\rho_k)$ indexed by the commutative monoid $(\mathbb N^k,+)$, with fibers $$\mathcal E_n={\mathcal{H}}^n=\mathcal H_1^{\otimes n_1}\otimes\cdots\otimes \mathcal H_k^{\otimes n_k}$$ for $n=(n_1,...,n_k)\in {{\mathbb{N}}}^k$, in particular, $A=\mathcal E_0=\mathbb C$. The multiplication maps ${\mathcal{M}}_{n,m}:{\mathcal{E}}_n\times {\mathcal{E}}_m\to {\mathcal{E}}_{n+m}$ in ${\mathcal{E}}$ are defined using repeatedly the standard isomorphisms $\rho_i\otimes\rho_j\cong \rho_j\otimes \rho_i$ for all $i<j$. The associativity in ${\mathcal{E}}$ follows from the fact that $${\mathcal{M}}_{n+m,p}\circ ({\mathcal{M}}_{n,m}\times id)={\mathcal{M}}_{n,m+p}\circ ( id\times {\mathcal{M}}_{m,p})$$ as maps from ${\mathcal{E}}_n\times {\mathcal{E}}_m\times {\mathcal{E}}_p$ to ${\mathcal{E}}_{n+m+p}.$ Then ${\mathcal{E}}={\mathcal{E}}(\rho_1,...,\rho_k)$ is called the product system of the representations $\rho_1,...,\rho_k$. Similarly, a semigroup $P$ of unitary representations of a group $G$ would determine a product system ${\mathcal{E}}\to P$. With notation as in Definition \[ps\], assume $d_i=\dim {\mathcal{H}}_i\ge 2$. Then the Cuntz-Pimsner algebra ${\mathcal{O}}({\mathcal{E}})$ associated to the product system ${\mathcal{E}}\to {{\mathbb{N}}}^k$ described above is isomorphic with the $C^*$-algebra of a rank $k$ graph $\Gamma$ with a single vertex and with $|\Gamma^{{\varepsilon}_i}|=d_i$. This isomorphism is equivariant for the gauge action. Moreover, $${\mathcal{O}}({\mathcal{E}})\cong \mathcal O_{d_1}\otimes\cdots\otimes \mathcal O_{d_k},$$ where $\mathcal O_n$ is the Cuntz algebra. Indeed, by choosing a basis in each ${\mathcal{H}}_i$, we get the edges $\Gamma^{{\varepsilon}_i}$ in a $k$-coloured graph $\Gamma$ with a single vertex. The isomorphisms $\rho_i\otimes\rho_j\cong \rho_j\otimes \rho_i$ determine the factorization rules of the form $ef=fe$ for $e\in \Gamma^{{\varepsilon}_i}$ and $f\in \Gamma^{{\varepsilon}_j}$ which obviously satisfy the associativity condition. In particular, the corresponding isometries in $C^*(\Gamma)$ commute and ${\mathcal{O}}({\mathcal{E}})\cong C^*(\Gamma)\cong\mathcal O_{d_1}\otimes\cdots\otimes \mathcal O_{d_k}$, preserving the gauge action. For $d_i\ge 2$, the $C^*$-algebra ${\mathcal{O}}({\mathcal{E}})\cong C^*(\Gamma)$ is always simple and purely infinite since it is a tensor product of simple and purely infinite $C^*$-algebras. If $d_i=1$ for some $i$, then ${\mathcal{O}}({\mathcal{E}})$ will contain a copy of $C({{\mathbb{T}}})$, so it is not simple. Of course, if $d_i=1$ for all $i$, then ${\mathcal{O}}({\mathcal{E}})\cong C({{\mathbb{T}}}^k)$. For more on single vertex rank $k$ graphs, see [@DY; @DY1]. The compact group $G$ acts on each fiber ${\mathcal{E}}_n$ of the product system $\mathcal E$ via the representation $\rho^n=\rho_1^{\otimes n_1}\otimes\cdots\otimes \rho_k^{\otimes n_k}$. This action is compatible with the multiplication maps and commutes with the gauge action of ${{\mathbb{T}}}^k$. The crossed product $\mathcal E\rtimes G$ becomes a row-finite and faithful product system indexed by $\mathbb N^k$ over the group $C^*$-algebra $C^*(G)$. Moreover, $${\mathcal{O}}({\mathcal{E}}) \rtimes G \cong {\mathcal{O}}_{C^*(G)}({\mathcal{E}}\rtimes G).$$ Indeed, for $g\in G$ and $\xi\in {\mathcal{E}}_n={\mathcal{H}}^n$ we define $g\cdot\xi=\rho^n(\xi)$ and since $\rho_i\otimes \rho_j\cong \rho_j\otimes \rho_i$, we have $g\cdot(\xi\otimes \eta)=g\cdot\xi\otimes g\cdot \eta$ for $\xi\in {\mathcal{E}}_n, \eta\in{\mathcal{E}}_m$. Clearly, $$g\cdot\gamma_z(\xi)=g\cdot(z^n\xi)=z^n(g\cdot\xi)=\gamma_z(g\cdot\xi),$$ so the action of $G$ commutes with the gauge action. Using Proposition \[p1\], $\mathcal E\rtimes G$ becomes a product system indexed by $\mathbb N^k$ over $C^*(G)\cong {{\mathbb{C}}}\rtimes G$ with fibers ${\mathcal{E}}_n\rtimes G$. The isomorphism ${\mathcal{O}}({\mathcal{E}}) \rtimes G \cong {\mathcal{O}}_{C^*(G)}({\mathcal{E}}\rtimes G)$ follows from Proposition \[p2\]. Since the action of $G$ commutes with the gauge action, the group $G$ acts on the core algebra ${\mathcal{F}}={\mathcal{O}}({\mathcal{E}})^{{{\mathbb{T}}}^k}$. The Doplicher-Roberts algebra ============================= The Doplicher-Roberts algebras ${\mathcal{O}}_\rho$, denoted by ${\mathcal O}_G$ in [@DR1], were introduced to construct a new duality theory for compact Lie groups $G$ which strengthens the Tannaka-Krein duality. Here $\rho$ is the $n$-dimensional representation of $G$ defined by the inclusion $G\subseteq U(n)$ in some unitary group $U(n)$. Let ${\mathcal T}_G$ denote the representation category whose objects are tensor powers $\rho^p=\rho^{\otimes p}$ for $p\ge 0$, and whose arrows are the intertwiners $Hom(\rho^p, \rho^q)$. The group $G$ acts via $\rho$ on the Cuntz algebra ${\mathcal O}_n$ and ${\mathcal O}_G={\mathcal O}_\rho$ is identified in [@DR1] with the fixed point algebra ${\mathcal O}_n^G$. If $\sigma$ denotes the restriction to ${\mathcal O}_\rho$ of the canonical endomorphism of ${\mathcal{O}}_n$, then ${\mathcal T}_G$ can be reconstructed from the pair $({\mathcal O}_\rho,\sigma)$. Subsequently, Doplicher-Roberts algebras were associated to any object $\rho$ in a strict tensor $C^*$-category, see [@DR2]. Given finite dimensional unitary representations $\rho_1, ...,\rho_k$ of a compact group $G$ on Hilbert spaces $\mathcal H_1, ..., \mathcal H_k$ we will construct a Doplicher-Roberts algebra $\mathcal O_{\rho_1,...,\rho_k}$ from intertwiners $$Hom (\rho^n, \rho^m)=\{T\in{\mathcal{L}}({\mathcal H}^n, {\mathcal H}^m)\;\mid \; T\rho^n(g)=\rho^m(g)T\;\;\forall\;g\in G\},$$ where for $n=(n_1,...,n_k)\in \mathbb N^k$ the representation $\rho^n=\rho_1^{\otimes n_1}\otimes\cdots\otimes \rho_k^{\otimes n_k}$ acts on ${\mathcal{H}}^n=\mathcal H_1^{\otimes n_1}\otimes \cdots \otimes \mathcal H_k^{\otimes n_k}$. Note that $\rho^0=\iota$ is the trivial representation of $G$, acting on ${\mathcal{H}}^0={{\mathbb{C}}}$. This Doplicher-Roberts algebra will be a subalgebra of ${\mathcal{O}}({\mathcal{E}})$ for the product system ${\mathcal{E}}$ as in Definition \[ps\]. \[cat\] Consider $${\mathcal{A}}_0=\bigcup_{m,n\in {{\mathbb{N}}}^k}{\mathcal{L}}(\mathcal H^n,\mathcal H^m).$$ Then the linear span of ${\mathcal{A}}_0$ becomes a $*$-algebra ${\mathcal{A}}$ with appropriate multiplication and involution. This algebra has a natural ${{\mathbb{Z}}}^k$-grading coming from a gauge action of ${{\mathbb{T}}}^k$. Moreover, the Cuntz-Pimsner algebra ${\mathcal{O}}({\mathcal{E}})$ of the product system ${\mathcal{E}}={\mathcal{E}}(\rho_1,...,\rho_k)$ is equivariantly isomorphic to the $C^*$-closure of ${\mathcal{A}}$ in the unique $C^*$-norm for which the gauge action is isometric. Recall that the Cuntz algebra ${\mathcal{O}}_n$ contains a canonical Hilbert space ${\mathcal{H}}$ of dimension $n$ and it can be constructed as the closure of the linear span of ${\displaystyle}\bigcup_{p,q\in {{\mathbb{N}}}}{\mathcal{L}}(\mathcal H^p,\mathcal H^q)$ using embeddings $${\mathcal{L}}({\mathcal{H}}^p,{\mathcal{H}}^q)\subseteq {\mathcal{L}}({\mathcal{H}}^{ p+1},{\mathcal{H}}^{ q+1}),\;\;T\mapsto T\otimes I$$ where ${\mathcal{H}}^p=\mathcal H^{\otimes p}$ and $I:{\mathcal{H}}\to{\mathcal{H}}$ is the identity map. This linear span becomes a $*$-algebra with a multiplication given by composition and an involution (see [@DR1] and Proposition 2.5 in [@KPW]). Similarly, for all $r\in {{\mathbb{N}}}^k$, we consider embeddings ${\mathcal{L}}({\mathcal{H}}^n,{\mathcal{H}}^m)\subseteq {\mathcal{L}}({\mathcal{H}}^{n+r},{\mathcal{H}}^{m+r})$ given by $T\mapsto T\otimes I_r$, where $I_r:{\mathcal H}^r\to {\mathcal H}^r$ is the identity map, and endow ${\mathcal{A}}$ with a multiplication given by composition and an involution. More precisely, if $S\in {\mathcal{L}}({\mathcal{H}}^n,{\mathcal{H}}^m)$ and $T\in {\mathcal{L}}({\mathcal{H}}^q,{\mathcal{H}}^p)$, then the product $ST$ is $$(S\otimes I_{p\vee n-n})\circ (T\otimes I_{p\vee n-p})\in {\mathcal{L}}({\mathcal{H}}^{q+p\vee n-p},{\mathcal{H}}^{m+p\vee n-n}),$$ where we write $p\vee n$ for the coordinatewise maximum. This multiplication is well defined in ${\mathcal{A}}$ and is associative. The adjoint of $T\in {\mathcal{L}}({\mathcal{H}}^n,{\mathcal{H}}^m)$ is $T^*\in {\mathcal{L}}({\mathcal{H}}^m,{\mathcal{H}}^n)$. There is a natural $\mathbb Z^k$-grading on ${\mathcal{A}}$ given by the gauge action $\gamma$ of ${{\mathbb{T}}}^k$, where for $z=(z_1,...,z_k)\in {{\mathbb{T}}}^k$ and $T\in {\mathcal{L}}({\mathcal{H}}^n,{\mathcal{H}}^m)$ we define $$\gamma_z(T)(\xi)=z_1^{m_1-n_1}\cdots z_k^{m_k-n_k}T(\xi).$$ Adapting the argument in Theorem 4.2 in [@DR2] for ${{\mathbb{Z}}}^k$-graded $C^*$-algebras, the $C^*$-closure of ${\mathcal{A}}$ in the unique $C^*$-norm for which $\gamma_z$ is isometric is well defined. The map $$(T_1,...,T_k)\mapsto T_1\otimes\cdots\otimes T_k,$$ where $$T_1\otimes\cdots\otimes T_k: \mathcal H^n\to \mathcal H^m,\; (T_1\otimes\cdots\otimes T_k)(\xi_1\otimes\cdots\otimes \xi_k)=T_1(\xi_1)\otimes\cdots\otimes T_k(\xi_k)$$ for $T_i\in {\mathcal{L}}(\mathcal H_i^{n_i},\mathcal H_i^{m_i})$ for $i=1,...,k$ preserves the gauge action and it can be extended to an equivariant isomorphism from ${\mathcal{O}}({\mathcal{E}})\cong {\mathcal{O}}_{d_1}\otimes\cdots\otimes{\mathcal{O}}_{d_k}$ to the $C^*$-closure of ${\mathcal{A}}$. Note that the closure of ${\displaystyle}\bigcup_{n\in {{\mathbb{N}}}^k}{\mathcal{L}}(\mathcal H^n,\mathcal H^n)$ is isomorphic to the core ${\mathcal{F}}={\mathcal{O}}({\mathcal{E}})^{{{\mathbb{T}}}^k}$, the fixed point algebra under the gauge action, which is a UHF-algebra. To define the Doplicher-Roberts algebra $\mathcal O_{\rho_1,...,\rho_k}$, we will again identify $Hom(\rho^n,\rho^m)$ with a subset of $Hom(\rho^{n+r},\rho^{m+r})$ for each $r\in \mathbb N^k$, via $T\mapsto T\otimes I_r$. After this identification, it follows that the linear span ${}^0{\mathcal O}_{\rho_1, ..., \rho_k}$ of ${\displaystyle}\bigcup_{m,n\in{{\mathbb{N}}}^k}Hom(\rho^n, \rho^m)\subseteq {\mathcal{A}}_0$ has a natural multiplication and involution inherited from ${\mathcal{A}}$. Indeed, a computation shows that if $S\in Hom(\rho^n, \rho^m)$ and $T\in Hom(\rho^q,\rho^p)$, then $S^*\in Hom(\rho^m, \rho^n)$ and $$(S\otimes I_{p\vee n-n})\circ (T\otimes I_{p\vee n-p})\rho^{q+p\vee n-p}(g)=$$$$=\rho^{m+p\vee n-n}(g)(S\otimes I_{p\vee n-n})\circ (T\otimes I_{p\vee n-p}),$$ so $(S\otimes I_{p\vee n-n})\circ (T\otimes I_{p\vee n-p})\in Hom(\rho^{q+p\vee n-p}, \rho^{m+p\vee n-n})$ and ${}^0{\mathcal O}_{\rho_1, ..., \rho_k}$ is closed under these operations. Since the action of $G$ commutes with the gauge action, there is a natural $\mathbb Z^k$-grading of ${}^0{\mathcal O}_{\rho_1,...,\rho_k}$ given by the gauge action $\gamma$ of ${{\mathbb{T}}}^k$ on ${\mathcal{A}}$. It follows that the closure ${\mathcal O}_{\rho_1,..., \rho_k}$ of ${}^0{\mathcal O}_{\rho_1, ...,\rho_k}$ in ${\mathcal{O}}({\mathcal{E}})$ is well defined, obtaining the Doplicher-Roberts algebra associated to the representations $\rho_1, ...,\rho_k$. This $C^*$-algebra also has a $\mathbb Z^k$-grading and a gauge action of ${{\mathbb{T}}}^k$. By construction, ${\mathcal O}_{\rho_1,..., \rho_k}\subseteq {\mathcal{O}}({\mathcal{E}})$. For a compact Lie group $G$, our Doplicher-Roberts algebra ${\mathcal O}_{\rho_1,..., \rho_k}$ is Morita equivalent with the higher rank Doplicher-Roberts algebra ${\mathcal{D}}$ in [@AM]. It is also the section $C^*$-algebra of a Fell bundle over ${{\mathbb{Z}}}^k$. Let $\rho_i$ be finite dimensional unitary representations of a compact group $G$ on Hilbert spaces $\mathcal H_i$ of dimensions $d_i\ge 2$ for $i=1,...,k$. Then the Doplicher-Roberts algebra ${\mathcal O}_{\rho_1,...,\rho_k}$ is isomorphic to the fixed point algebra ${\mathcal O}({\mathcal{E}})^G\cong (\mathcal O_{d_1}\otimes\cdots\otimes \mathcal O_{d_k})^G$, where ${\mathcal{E}}={\mathcal{E}}(\rho_1,...,\rho_k)$ is the product system described in Definition \[ps\]. We known from Lemma \[cat\] that ${\mathcal O}({\mathcal{E}})$ is isomorphic to the $C^*$-algebra generated by the linear span of ${\displaystyle}{\mathcal{A}}_0= \bigcup_{m,n\in{{\mathbb{N}}}^k}{\mathcal{L}}({\mathcal H}^n, {\mathcal H}^m)$. The group $G$ acts on ${\mathcal{L}}({\mathcal H}^n, {\mathcal H}^m)$ by $$(g\cdot T)(\xi)=\rho^m(g)T(\rho^n(g^{-1})\xi)$$ and the fixed point set is $Hom(\rho^n, \rho^m)$. Indeed, we have $g\cdot T=T$ if and only if $T\rho^n(g)=\rho^m(g)T$. This action is compatible with the embeddings and the operations, so it extends to the $*$-algebra ${\mathcal{A}}$ and the fixed point algebra is the linear span of ${\displaystyle}\bigcup_{m,n\in{{\mathbb{N}}}^k}Hom(\rho^n, \rho^m)$. It follows that ${}^0{\mathcal O}_{\rho_1,...,\rho_k}\subseteq {\mathcal O}({\mathcal{E}})^G$ and therefore its closure ${\mathcal O}_{\rho_1,...,\rho_k}$ is isomorphic to a subalgebra of ${\mathcal O}({\mathcal{E}})^G$. For the other inclusion, any element in ${\mathcal O}({\mathcal{E}})^G$ can be approximated with an element from ${}^0{\mathcal O}_{\rho_1,...,\rho_k}$, hence ${\mathcal O}_{\rho_1,...,\rho_k}={\mathcal{O}}({\mathcal{E}})^G$. By left tensoring with $I_r$ for $r\in {{\mathbb{N}}}^k$, we obtain some canonical unital endomorphisms $\sigma_r$ of ${\mathcal O}_{\rho_1, ...,\rho_k}$. In the next section, we will show that in many cases, $\mathcal O_{\rho_1,...,\rho_k}$ is isomorphic to a corner of $C^*(\Lambda)$ for a rank $k$ graph $\Lambda$, so in some cases we can compute its $K$-theory. It would be nice to express the $K$-theory of ${\mathcal{O}}_{\rho_1,...,\rho_k}$ in terms of the endomorphisms $\pi\mapsto \pi\otimes \rho_i$ of the representation ring ${\mathcal{R}}(G)$. The rank $k$ graphs =================== For convenience, we first collect some facts about higher rank graphs, introduced in [@KP]. A rank $k$ graph or $k$-graph $(\Lambda, d)$ consists of a countable small category $\Lambda$ with range and source maps $r$ and $s$ together with a functor $d : \Lambda \rightarrow {{\mathbb{N}}}^k$ called the degree map, satisfying the factorization property: for every $\lambda \in \Lambda$ and all $m, n \in {{\mathbb{N}}}^k$ with $d( \lambda ) = m + n$, there are unique elements $\mu , \nu \in \Lambda$ such that $\lambda = \mu \nu $ and $d( \mu ) = m$, $d( \nu ) = n$. For $n \in {{\mathbb{N}}}^k$ we write $\Lambda^n := d^{-1} (n)$ and call it the set of paths of degree $n$. The elements in $\Lambda^{{\varepsilon}_i}$ are called edges and the elements in $\Lambda^0$ are called vertices. A $k$-graph $\Lambda$ can be constructed from $\Lambda^0$ and from its $k$-coloured skeleton $\Lambda^{{\varepsilon}_1}\cup\cdots \cup\Lambda^{{\varepsilon}_k}$ using a complete and associative collection of commuting squares or factorization rules, see [@S]. The $k$-graph $\Lambda$ is [*row-finite*]{} if for all $n\in {{\mathbb{N}}}^k$ and all $v\in \Lambda^0$ the set $v\Lambda^n := \{\lambda\in\Lambda^n : r(\lambda) = v\}$ is finite. It has no sources if $v\Lambda^n\neq \emptyset$ for all $v\in \Lambda^0$ and $n\in{{\mathbb{N}}}^k$. A $k$-graph $\Lambda$ is said to be [*irreducible*]{} (or [*strongly connected*]{}) if, for every $u,v\in \Lambda^0$, there is $\lambda\in \Lambda$ such that $u = r(\lambda)$ and $v = s(\lambda)$. Recall that $C^*(\Lambda)$ is the universal $C^*$-algebra generated by a family $\{S_\lambda: \lambda\in \Lambda \}$ of partial isometries satisfying: - $\{S_v:v\in \Lambda^0\}$ is a family of mutually orthogonal projections, - $S_{\lambda\mu}=S_\lambda S_\mu$ for all $\lambda, \mu\in \Lambda$ such that $s(\lambda) = r(\mu)$, - $S_\lambda^*S_\lambda = S_{s(\lambda)}$ for all $\lambda\in \Lambda$, - for all $v\in \Lambda^0$ and $n\in {{\mathbb{N}}}^k$ we have $$S_v=\sum_{\lambda\in v\Lambda^n}S_\lambda S_\lambda^*.$$ A $k$-graph $\Lambda$ is said to satisfy the [*aperiodicity condition*]{} if for every vertex $v\in \Lambda^0$ there is an infinite path $x\in v\Lambda^\infty$ such that $\sigma^mx\neq \sigma^nx$ for all $m\neq n$ in ${{\mathbb{N}}}^k$, where $\sigma^m:\Lambda^\infty\to \Lambda^\infty$ are the shift maps. We say that $\Lambda$ is [*cofinal*]{} if for every $x\in \Lambda^\infty$ and $v\in \Lambda^0$ there is $\lambda\in \Lambda$ and $n\in {{\mathbb{N}}}^k$ such that $s(\lambda)=x(n)$ and $r(\lambda)=v$. Assume that $\Lambda$ is row finite with no sources and that it satisfies the aperiodicity condition. Then $C^*(\Lambda)$ is simple if and only if $\Lambda$ is cofinal (see Proposition 4.8 in [@KP] and Theorem 3.4 in [@RS]). We say that a path $\mu\in \Lambda$ is a loop with an entrance if $s(\mu)=r(\mu)$ and there exists $\alpha\in s(\mu)\Lambda$ such that $d(\mu)\ge d(\alpha)$ and there is no $\beta\in \Lambda$ with $\mu= \alpha\beta$. We say that every vertex [*connects to a loop with an entrance*]{} if for every $v\in \Lambda^0$ there are a loop with an entrance $\mu\in \Lambda$ and a path $\lambda\in \Lambda$ with $r(\lambda)=v$ and $s(\lambda)=r(\mu)=s(\mu)$. If $\Lambda$ satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, then $C^*(\Lambda)$ is purely infinite (see Proposition 4.9 in [@KP] and Proposition 8.8 in [@S06]). Given finitely dimensional unitary representations $\rho_i$ of a compact group $G$ on Hilbert spaces $\mathcal H_i$ for $i=1,...,k$, we want to construct a rank $k$ graph $\Lambda=\Lambda(\rho_1,...,\rho_k)$. Let $R$ be the set of equivalence classes of irreducible summands $\pi:G\to U({\mathcal{H}}_\pi)$ which appear in the tensor powers $\rho^n=\rho_1^{\otimes n_1}\otimes\cdots \otimes \rho_k^{\otimes n_k}$ for $n\in {{\mathbb{N}}}^k$ as in [@MRS]. Take $\Lambda^0=R$ and for each $i=1,...,k$ consider the set of edges $\Lambda^{{\varepsilon}_i}$ which are uniquely determined by the matrices $M_i$ with entries $$M_i(w,v)=|\{e\in \Lambda^{{\varepsilon}_i}: s(e)=v, r(e)=w\}|=\dim Hom(v,w\otimes \rho_i),$$ where $v,w\in R$. The matrices $M_i$ commute since $\rho_i\otimes\rho_j\cong \rho_j\otimes \rho_i$ and therefore $$\dim Hom(v,w\otimes \rho_i\otimes\rho_j)=\dim Hom(v,w\otimes \rho_j\otimes\rho_i)$$ for all $i<j$. This will allow us to fix some bijections $$\lambda_{ij}:\Lambda^{{\varepsilon}_i}\times_{\Lambda^0}\Lambda^{{\varepsilon}_j}\to \Lambda^{{\varepsilon}_j}\times_{\Lambda^0}\Lambda^{{\varepsilon}_i}$$ for all $1\le i<j\le k$, which will determine the commuting squares of $\Lambda$. As usual, $$\Lambda^{{\varepsilon}_i}\times_{\Lambda^0}\Lambda^{{\varepsilon}_j}=\{(e,f)\in \Lambda^{{\varepsilon}_i}\times \Lambda^{{\varepsilon}_j}: s(e)=r(f)\}.$$ For $k\ge 3$ we also need to verify that $\lambda_{ij}$ can be chosen to satisfy the associativity condition, i.e. $$(id_\ell\times \lambda_{ij})(\lambda_{i\ell}\times id_j)(id_i\times \lambda_{j\ell})=(\lambda_{j\ell}\times id_i)(id_j\times \lambda_{i\ell})(\lambda_{ij}\times id_\ell)$$ as bijections from $\Lambda^{{\varepsilon}_i}\times_{\Lambda^0}\Lambda^{{\varepsilon}_j}\times_{\Lambda^0}\Lambda^{{\varepsilon}_\ell}$ to $\Lambda^{{\varepsilon}_\ell}\times_{\Lambda^0}\Lambda^{{\varepsilon}_j}\times_{\Lambda^0}\Lambda^{{\varepsilon}_i}$ for all $i<j<\ell$. Many times $R=\hat{G}$, so $\Lambda^0=\hat{G}$, for example if $\rho_i$ are faithful and $\rho_i(G)\subseteq SU({\mathcal{H}}_i)$ or if $G$ is finite, $\rho_i$ are faithful and $\dim \rho_i\ge2$ for all $i=1,...,k$, see Lemma 7.2 and Remark 7.4 in [@KPRR]. Given representations $\rho_1,...,\rho_k$ as above, assume that $\rho_i$ are faithful and that $R=\hat{G}$. Then each choice of bijections $\lambda_{ij}$ satisfying the associativity condition determines a rank $k$ graph $\Lambda$ which is cofinal and locally finite with no sources. Indeed, the sets $\Lambda^{{\varepsilon}_i}$ are uniquely determined and the choice of bijections $\lambda_{ij}$ satisfying the associativity condition will be enough to determine $\Lambda$. Since the entries of the matrices $M_i$ are finite and there are no zero rows, the graph is locally finite with no sources. To prove that $\Lambda$ is cofinal, fix a vertex $v\in \Lambda^0$ and an infinite path $x\in \Lambda^\infty$. Arguing as in Lemma 7.2 in [@KPRR], any $w\in \Lambda^0$, in particular $w=x(n)$ for a fixed $n$ can be joined by a path to $v$, so there is $\lambda\in \Lambda$ with $s(\lambda)=x(n)$ and $r(\lambda)=v$. See also Lemma 3.1 in [@MRS]. Note that the entry $M_i(w,v)$ is just the multiplicity of the irreducible representation $v$ in $w\otimes \rho_i$ for $i=1,...,k$. If $\rho_i^*=\rho_i$, the matrices $M_i$ are symmetric since $$\dim Hom(v, w\otimes \rho_i)=\dim Hom(\rho^*_i\otimes v,w).$$ Here $\rho^*_i$ denotes the dual representation, defined by $\rho_i^*(g)=\rho_i(g^{-1})^t$, and equal in our case to the conjugate representation $\bar{\rho_i}$. For $G$ finite, these matrices are finite, and the entries $M_i(w,v)$ can be computed using the character table of $G$. For $G$ infinite, the Clebsch-Gordan relations can be used to determine the numbers $M_i(w,v)$. Since the bijections $\lambda_{ij}$ in general are not unique, the rank $k$ graph $\Lambda$ is not unique, as illustrated in some examples. It is an open question how the $C^*$-algebra $C^*(\Lambda)$ depends in general on the factorization rules. To relate the Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,...,\rho_k}$ to a rank $k$ graph $\Lambda$, we mimic the construction in [@MRS]. For each edge $e\in \Lambda^{{\varepsilon}_i}$, choose an isometric intertwiner $$T_e: {\mathcal{H}}_{s(e)}\to {\mathcal{H}}_{r(e)}\otimes {\mathcal{H}}_i$$ in such a way that $${\mathcal{H}}_\pi\otimes {\mathcal{H}}_i=\bigoplus_{e\in \pi\Lambda^{{\varepsilon}_i}}T_eT_e^*({\mathcal{H}}_\pi\otimes {\mathcal{H}}_i)$$ for all $\pi\in \Lambda^0$, i.e. the edges in $\Lambda^{{\varepsilon}_i}$ ending at $\pi$ give a specific decomposition of ${\mathcal{H}}_\pi\otimes {\mathcal{H}}_i$ into irreducibles. When $\dim Hom(s(e), r(e)\otimes \rho_i)\ge 2$ we must choose a basis of isometric intertwiners with orthogonal ranges, so in general $T_e$ is not unique. In fact, specific choices for the isometric intertwiners $T_e$ will determine the factorization rules in $\Lambda$ and whether they satisfy the associativity condition or not. Given $e\in \Lambda^{{\varepsilon}_i}$ and $f\in \Lambda^{{\varepsilon}_j}$ with $r(f)=s(e)$, we know how to multiply $T_e\in Hom(s(e),r(e)\otimes \rho_i)$ with $T_f\in Hom(s(f),r(f)\otimes \rho_j)$ in the algebra ${\mathcal{O}}_{\rho_1,...,\rho_k}$, by viewing $Hom(s(e),r(e)\otimes \rho_i)$ as a subspace of $Hom(\rho^n,\rho^m)$ for some $m,n$ and similarly for $Hom(s(f),r(f)\otimes \rho_j)$. We choose edges $e'\in \Lambda^{{\varepsilon}_i}, f'\in \Lambda^{{\varepsilon}_j}$ with $s(f)=s(e'), r(e)=r(f'), r(e')=s(f')$ such that $T_eT_f=T_{f'}T_{e'}$, where $T_{f'}\in Hom(s(f'),r(f')\otimes \rho_j)$ and $T_{e'}\in Hom(s(e'),r(e')\otimes \rho_i)$. This is possible since $$T_eT_f=(T_e\otimes I_j)\circ T_f\in Hom(s(f),r(e)\otimes \rho_i\otimes \rho_j),$$ $$T_{f'}T_{e'}=(T_{f'}\otimes I_i)\circ T_{e'}\in Hom(s(e'),r(f')\otimes \rho_j\otimes \rho_i),$$ and $\rho_i\otimes \rho_j\cong \rho_j\otimes \rho_i$. In this case we declare that $ef=f'e'$. Repeating this process, we obtain bijections $\lambda_{ij}:\Lambda^{{\varepsilon}_i}\times_{\Lambda^0}\Lambda^{{\varepsilon}_j}\to \Lambda^{{\varepsilon}_j}\times_{\Lambda^0}\Lambda^{{\varepsilon}_i}$. Assuming that the associativity conditions are satisfied, we obtain a $k$-graph $\Lambda$. We write $T_{ef}=T_eT_f=T_{f'}T_{e'}=T_{f'e'}$. A finite path $\lambda\in \Lambda^n$ is a concatenation of edges and determines by composition a unique intertwiner $$T_\lambda:{\mathcal{H}}_{s(\lambda)}\to {\mathcal{H}}_{r(\lambda)}\otimes{\mathcal{H}}^n.$$ Moreover, the paths $\lambda\in \Lambda^n$ with $r(\lambda)=\iota$, the trivial representation, provide an explicit decomposition of ${\mathcal{H}}^n={\mathcal{H}}_1^{\otimes n_1}\otimes\cdots\otimes{\mathcal{H}}_k^{\otimes n_k}$ into irreducibles, hence $${\mathcal{H}}^n=\bigoplus_{\lambda\in\iota \Lambda^n}T_\lambda T_\lambda^*({\mathcal{H}}^n).$$ Assuming that the choices of isometric intertwiners $T_e$ as above determine a $k$-graph $\Lambda$, then the family $$\{T_\lambda T^*_\mu: \lambda\in\Lambda^m, \mu\in\Lambda^n, r(\lambda)=r(\mu)=\iota, s(\lambda)=s(\mu)\}$$ is a basis for $Hom(\rho^n, \rho^m)$ and each $T_\lambda T^*_\mu$ is a partial isometry. Each pair of paths $\lambda, \mu$ with $d(\lambda)=m, d(\mu)=n$ and $r(\lambda)=r(\mu)=\iota$ determines a pair of irreducible summands $T_\lambda({\mathcal{H}}_{s(\lambda)}), T_\mu({\mathcal{H}}_{s(\mu)})$ of ${\mathcal{H}}^m$ and $ {\mathcal{H}}^n$ respectively. By Schur’s lemma, the space of intertwiners of these representations is trivial unless $s(\lambda)=s(\mu)$ in which case it is the one dimensional space spanned by $T_\lambda T_\mu^*$. It follows that any element of $Hom(\rho^n, \rho^m)$ can be uniquely represented as a linear combination of elements $T_\lambda T_\mu^*$ where $s(\lambda)=s(\mu)$. Since $T_\mu$ is isometric, $T_\mu^*$ is a partial isometry with range ${\mathcal{H}}_{s(\mu)}$ and hence $T_\lambda T_\mu^*$ is also a partial isometry whenever $s(\lambda)=s(\mu)$. \[t1\] Consider $\rho_1,..., \rho_k$ finite dimensional unitary representations of a compact group $G$ and let $\Lambda$ be the $k$-coloured graph with $\Lambda^0=R\subseteq \hat{G}$ and edges $\Lambda^{{\varepsilon}_i}$ determined by the incidence matrices $M_i$ defined above. Assume that the factorization rules determined by the choices of $T_e\in Hom(s(e),r(e)\otimes \rho_i)$ for all edges $e\in \Lambda^{{\varepsilon}_i}$ satisfy the associativity condition, so $\Lambda$ becomes a rank $k$ graph. If we consider $P\in C^*(\Lambda)$, $$P=\sum_{\lambda\in\iota \Lambda^{(1,...,1)}}S_\lambda S_\lambda^*,$$ where $\iota$ is the trivial representation, then there is a $*$-isomorphism of the Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,...,\rho_k}$ onto the corner $PC^*(\Lambda)P$. Since $C^*(\Lambda)$ is generated by linear combinations of $S_\lambda S_\mu^*$ with $s(\lambda)=s(\mu)$ (see Lemma 3.1 in [@KP]), we first define the maps $$\phi_{n,m}:Hom(\rho^n, \rho^m)\to C^*(\Lambda),\;\; \phi_{n,m}(T_\lambda T_\mu^*)=S_\lambda S_\mu^*$$ where $s(\lambda)=s(\mu)$ and $r(\lambda)=r(\mu)=\iota$. Since $S_\lambda S_\mu^*=PS_\lambda S_\mu^*P$, the maps $\phi_{n,m}$ take values in $PC^*(\Lambda)P$. We claim that for any $r\in {{\mathbb{N}}}^k$ we have $$\phi_{n+r,m+r}(T_\lambda T_\mu^*\otimes I_r)=\phi_{n,m}(T_\lambda T_\mu^*).$$ This is because $${\mathcal{H}}_{s(\lambda)}\otimes {\mathcal{H}}^r=\bigoplus_{\nu\in s(\lambda)\Lambda^r}T_\nu T_\nu^*({\mathcal{H}}_{s(\lambda)}\otimes {\mathcal{H}}^r),$$ so that $$T_\lambda T_\mu^*\otimes I_r=\sum_{\nu\in s(\lambda)\Lambda^r}(T_\lambda\otimes I_r)(T_\nu T_\nu^*)(T_\mu^*\otimes I_r)=\sum_ {\nu\in s(\lambda)\Lambda^r} T_{\lambda\nu}T_{\mu\nu}^*$$ and$$S_\lambda S_\mu^*=\sum_{\nu\in s(\lambda)\Lambda^r}S_\lambda(S_\nu S_\nu^*)S_\mu^*=\sum_{\nu\in s(\lambda)\Lambda^r}S_{\lambda\nu}S_{\mu\nu}^*.$$ The maps $\phi_{n,m}$ determine a map $\phi: {}^0{\mathcal{O}}_{\rho_1,...,\rho_k}\to PC^*(\Lambda)P$ which is linear, $*$-preserving and multiplicative. Indeed, $$\phi_{n,m}(T_\lambda T_\mu^*)^*=(S_\lambda S_\mu^*)^*=S_\mu S_\lambda^*=\phi_{m,n}(T_\mu T_\lambda^*).$$ Consider now $T_\lambda T_\mu^*\in Hom(\rho^n, \rho^m),\;\; T_\nu T_\omega^*\in Hom (\rho^q,\rho^p)$ with $s(\lambda)=s(\mu), s(\nu)=s(\omega), r(\lambda)=r(\mu)=r(\nu)=r(\omega)=\iota$. Since for all $n\in {{\mathbb{N}}}^k$ $$\sum_{\lambda\in\iota\Lambda^n}T_\lambda T_\lambda^*=I_n,$$ we get $$T_\mu^*T_\nu= \begin{cases} T_\beta^*\;\; \text{if}\;\; \mu=\nu\beta\\ T_\alpha \;\;\text{if}\;\; \nu=\mu\alpha\\0 \;\;\text{otherwise,}\end{cases}$$ hence $$\phi((T_\lambda T_\mu^*)(T_\nu T_\omega^*))=\begin{cases} \phi(T_\lambda T_{\omega\beta}^*)=S_\lambda S_{\omega\beta}^*\;\;\text{if} \;\; \mu=\nu\beta\\ \phi(T_{\lambda\alpha}T_\omega^*)=S_{\lambda\alpha}S_\omega^*\;\;\text{if}\;\; \nu=\mu\alpha\\0 \;\;\text{otherwise.}\end{cases}$$ On the other hand, from Lemma 3.1 in [@KP], $$S_\lambda S_\mu^* S_\nu S_{\omega}^*=\begin{cases} S_\lambda S_{\omega\beta}^*\;\;\text{if} \;\; \mu=\nu\beta\\ S_{\lambda\alpha}S_\omega^*\;\;\text{if}\;\; \nu=\mu\alpha\\0 \;\;\text{otherwise,}\end{cases}$$ hence $$\phi((T_\lambda T_\mu^*)(T_\nu T_\omega^*))=\phi(T_\lambda T_\mu^*)\phi(T_\nu T_\omega^*).$$ Since $PS_\lambda S_\mu^*P=\phi_{n,m}(T_\lambda T_\mu^*)$ if $r(\lambda)=r(\mu)=\iota$ and $s(\lambda)=s(\mu)$, it follows that $\phi$ is surjective. Injectivity follows from the fact that $\phi$ is equivariant for the gauge action. If the $k$-graph $\Lambda$ associated to $\rho_1, ...,\rho_k$ is cofinal, it satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, then the Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,...,\rho_k}$ is simple and purely infinite, and is Morita equivalent with $C^*(\Lambda)$. This follows from the fact that $C^*(\Lambda)$ is simple and purely infinite and because $PC^*(\Lambda)P$ is a full corner. There is a groupoid ${\mathcal{G}}_\Lambda$ associated to a row-finite rank $k$ graph $\Lambda$ with no sources, see [@KP]. By taking the pointed groupoid ${\mathcal{G}}_\Lambda(\iota)$, the reduction to the set of infinite paths with range $\iota$, under the same conditions as in Theorem \[t1\], we get an isomorphism of the Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,...,\rho_k}$ onto $C^*({\mathcal{G}}_\Lambda(\iota))$. Examples ======== Let $G=S_3$ be the symmetric group with $\hat{G}=\{\iota, \epsilon, \sigma\}$ and character table $(1)$ $(12)$ $(123)$ ------------ ------- -------- --------- $\iota$ $1$ $1$ $1$ $\epsilon$ $1$ $-1$ $1$ $\sigma$ $2$ $0$ $-1$ Here $\iota$ denotes the trivial representation, $\epsilon$ is the sign representation and $\sigma$ is an irreducible $2$-dimensional representation, for example $$\sigma((12))=\left[\begin{array}{rr}-1&-1\\0&1\end{array}\right],\;\;\sigma((123))=\left[\begin{array}{rr}-1&-1\\1&0\end{array}\right].$$ By choosing $\rho_1=\sigma$ on ${\mathcal{H}}_1={{\mathbb{C}}}^2$ and $\rho_2=\iota+\sigma$ on ${\mathcal{H}}_2={{\mathbb{C}}}^3$, we get a product system ${\mathcal{E}}\to {{\mathbb{N}}}^2$ and an action of $S_3$ on ${\mathcal{O}}({\mathcal{E}})\cong \mathcal O_2\otimes \mathcal O_3$ with fixed point algebra ${\mathcal{O}}({\mathcal{E}})^{S_3}\cong {\mathcal{O}}_{\rho_1,\rho_2}$ isomorphic to a corner of the $C^*$-algebra of a rank $2$ graph $\Lambda$. The set of vertices is $\Lambda^0=\{\iota,\epsilon, \sigma\}$ and the edges are given by the incidence matrices $$M_1=\left[\begin{array}{ccc}0&0&1\\0&0&1\\1&1&1\end{array}\right], \;\; M_2=\left[\begin{array}{ccc}1&0&1\\0&1&1\\1&1&2\end{array}\right].$$ This is because $$\iota\otimes\rho_1=\sigma,\; \epsilon\otimes \rho_1=\sigma,\; \sigma\otimes \rho_1=\iota+\epsilon+\sigma,$$ $$\iota\otimes\rho_2=\iota+\sigma,\; \epsilon\otimes\rho_2=\epsilon+\sigma,\; \sigma\otimes\rho_2=\iota+\epsilon+2\sigma.$$ We label the blue edges by $e_1, ..., e_5$ and the red edges by $f_1,...,f_8$ as in the figure $$\begin{tikzpicture} \renewcommand{\ss}{\scriptstyle} \node[inner sep=1.0pt, circle, fill=black] (u) at (-6,0) {}; \node[below] at (u.south) {$\ss \iota$}; \node[inner sep=1.0pt, circle, fill=black] (v) at (-4,0) {}; \node[below] at (v.south) {$\ss \epsilon$}; \node[inner sep=1.0pt, circle, fill=black] (w) at (-2,0) {}; \node[below] at (w.south) {$\ss \sigma$}; \draw[-stealth, semithick, blue] (u) to [out=45, in=135] node[above,black] {$\ss e_1$} (w); \draw[-stealth, semithick, blue] (w) to [out=-135, in=-45] node[below,black] {$\ss e_2$} (u); \draw[-stealth, semithick, blue] (v) to [out=20, in=160] node[above,black] {$\ss e_3$} (w); \draw[-stealth, semithick, blue] (w) to [out=-160, in=-20] node[below,black] {$\ss e_4$} (v); \draw[-stealth, semithick, blue] (w) .. controls (-.5,-1) and (-.5, 1) .. node[left,black] {$\ss e_5$} (w); \node[inner sep=1.0pt, circle, fill=black] (x) at (1,0) {}; \node[inner sep=1.0pt, circle, fill=black] (y) at (3,0) {}; \node[inner sep=1.0pt, circle, fill=black] (z) at (5,0) {}; \draw[-stealth, semithick, red] (x) to [out=45, in=135] node[above,black] {$\ss f_4$} (z); \draw[-stealth, semithick, red] (z) to [out=-135, in=-45] node[below,black] {$\ss f_3$} (x); \draw[-stealth, semithick, red] (y) to [out=20, in=160] node[above,black] {$\ss f_6$} (z); \draw[-stealth, semithick, red] (z) to [out=-160, in=-20] node[below,black] {$\ss f_5$} (y); \draw[-stealth, semithick, red] (x) .. controls (-0.5,1) and (-0.5, -1) .. node[right,black] {$\ss f_1$} (x); \draw[-stealth, semithick, red] (z) .. controls (4,1.5) and (6, 1.5) .. node[above,black] {$\ss f_7$} (z); \draw[-stealth, semithick, red] (y) .. controls (1.5,1) and (1.5,-1) .. node[right,black] {$\ss f_2$} (y); \draw[-stealth, semithick, red] (z) .. controls (4,-1.5) and (6, -1.5) .. node[below,black] {$\ss f_8$} (z); \node[below] at (x.south) {$\ss \iota$}; \node[below] at (y.south) {$\ss \epsilon$}; \node[right] at (z.east) {$\ss \sigma$}; \end{tikzpicture}$$ The isometric intertwiners are $$T_{e_1}:{\mathcal{H}}_\iota\to {\mathcal{H}}_\sigma\otimes {\mathcal{H}}_1, \; T_{e_2}:{\mathcal{H}}_\sigma\to {\mathcal{H}}_\iota\otimes {\mathcal{H}}_1, \;T_{e_3}:{\mathcal{H}}_\epsilon\to {\mathcal{H}}_\sigma\otimes {\mathcal{H}}_1,$$ $$T_{e_4}:{\mathcal{H}}_\sigma\to {\mathcal{H}}_\epsilon\otimes {\mathcal{H}}_1,\; T_{e_5}:{\mathcal{H}}_\sigma\to{\mathcal{H}}_\sigma\otimes{\mathcal{H}}_1,$$ $$T_{f_1}:{\mathcal{H}}_\iota\to {\mathcal{H}}_\iota\otimes{\mathcal{H}}_2,\; T_{f_2}:{\mathcal{H}}_\epsilon\to {\mathcal{H}}_\epsilon\otimes{\mathcal{H}}_2,\; T_{f_3}:{\mathcal{H}}_\sigma\to {\mathcal{H}}_\iota\otimes{\mathcal{H}}_2,$$ $$T_{f_4}:{\mathcal{H}}_\iota\to {\mathcal{H}}_\sigma\otimes{\mathcal{H}}_2,\; T_{f_5}:{\mathcal{H}}_\sigma\to{\mathcal{H}}_\epsilon\otimes {\mathcal{H}}_2,\; T_{f_6}:{\mathcal{H}}_\epsilon\to{\mathcal{H}}_\sigma\otimes {\mathcal{H}}_2,$$ $$T_{f_7}, T_{f_8}:{\mathcal{H}}_\sigma\to {\mathcal{H}}_\sigma\otimes{\mathcal{H}}_2$$ such that $$T_{e_1}T_{e_1}^*+T_{e_3}T_{e_3}^*+T_{e_5}T_{e_5}^*=I_\sigma\otimes I_1, \; T_{e_2}T_{e_2}^*=I_\iota\otimes I_1,\; T_{e_4}T_{e_4}^*=I_\epsilon\otimes I_1,$$ $$T_{f_1}T_{f_1}^*+T_{f_3}T_{f_3}^*=I_\iota\otimes I_2,\; T_{f_2}T_{f_2}^*+T_{f_5}T_{f_5}^*=I_\epsilon\otimes I_2,$$$$T_{f_4}T_{f_4}^*+T_{f_6}T_{f_6}^*+T_{f_7}T_{f_7}^*+T_{f_8}T_{f_8}^*=I_\sigma\otimes I_2.$$ Here $I_\pi$ is the identity of ${\mathcal{H}}_\pi$ for $\pi\in\hat{G}$ and $I_i$ the identity of ${\mathcal{H}}_i$ for $ i=1,2$. Since $$M_1M_2=\left[\begin{array}{ccc}1&1&2\\1&1&2\\2&2&4\end{array}\right]$$ and $$T_{e_2}T_{f_4}, T_{f_3}T_{e_1}\in Hom(\iota, \iota\otimes\rho_1\otimes\rho_2),$$ $$T_{e_2}T_{f_6}, T_{f_3}T_{e_3}\in Hom(\epsilon, \iota\otimes\rho_1\otimes\rho_2),$$ $$T_{e_2}T_{f_7}, T_{e_2}T_{f_8}, T_{f_1}T_{e_2}, T_{f_3}T_{e_5}\in Hom(\sigma, \iota\otimes\rho_1\otimes\rho_2),$$ $$T_{e_4}T_{f_4}, T_{f_5}T_{e_1}\in Hom(\iota, \epsilon\otimes\rho_1\otimes\rho_2),$$ $$T_{e_4}T_{f_6}, T_{f_5}T_{e_3}\in Hom(\epsilon, \epsilon\otimes\rho_1\otimes\rho_2),$$ $$T_{e_4}T_{f_7}, T_{e_4}T_{f_8}, T_{f_2}T_{e_4}, T_{f_5}T_{e_5}\in Hom(\sigma, \epsilon\otimes\rho_1\otimes\rho_2),$$ $$T_{e_1}T_{f_1}, T_{e_5}T_{f_4}, T_{f_7}T_{e_1}, T_{f_8}T_{e_1}\in Hom(\iota, \sigma\otimes\rho_1\otimes\rho_2),$$ $$T_{e_3}T_{f_2}, T_{e_5}T_{f_6}, T_{f_7}T_{e_3}, T_{f_8}T_{e_3}\in Hom(\epsilon,\sigma\otimes\rho_1\otimes\rho_2),$$ $$T_{e_5}T_{f_7}, T_{e_5}T_{f_8}, T_{e_3}T_{f_5}, T_{e_1}T_{f_3}, T_{f_6}T_{e_4}, T_{f_4}T_{e_2}, T_{f_7}T_{e_5}, T_{f_8}T_{e_5}\in Hom(\sigma, \sigma\otimes\rho_1\otimes\rho_2),$$ a possible choice of commuting squares is $$e_2f_4=f_3e_1,\; e_2f_6=f_3e_3,\; e_2f_7=f_1e_2,\; e_2f_8=f_3e_5,\; e_4f_4=f_5e_1,\; e_4f_6=f_5e_3$$ $$e_4f_7= f_2e_4,\; e_4f_8=f_5e_5,\; e_1f_1=f_7e_1,\; e_5f_4=f_8e_1,\; e_3f_2=f_7e_3,\; e_5f_6=f_8e_3,$$ $$e_5f_7=f_6e_4,\; e_5f_8=f_4e_2,\; e_3f_5=f_7e_5,\; e_1f_3=f_8e_5.$$ This data is enough to determine a rank $2$ graph $\Lambda$ associated to $\rho_1, \rho_2$. But this is not the only choice, since for example we could have taken $$e_2f_4=f_3e_1,\; e_2f_6=f_3e_3,\; e_2f_8=f_1e_2,\; e_2f_7=f_3e_5,\; e_4f_4=f_5e_1,\; e_4f_6=f_5e_3$$ $$e_4f_8= f_2e_4,\; e_4f_7=f_5e_5,\; e_1f_1=f_7e_1,\; e_5f_4=f_8e_1,\; e_3f_2=f_8e_3,\; e_5f_6=f_7e_3,$$ $$e_5f_7=f_6e_4,\; e_5f_8=f_4e_2,\; e_3f_5=f_7e_5,\; e_1f_3=f_8e_5,$$ which will determine a different $2$-graph. A direct analysis using the definitions shows that in each case, the $2$-graph $\Lambda$ is cofinal, it satisfies the aperiodicity condition and every vertex connects to a loop with an entrance. It follows that $C^*(\Lambda)$ is simple and purely infinite and the Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,\rho_2}$ is Morita equivalent with $C^*(\Lambda)$. The $K$-theory of $C^*(\Lambda)$ can be computed using Proposition 3.16 in [@E] and it does not depend on the choice of factorization rules. We have $$K_0(C^*(\Lambda))\cong\text{coker}[I-M_1^t\;\; I-M_2^t]\oplus\ker\left[\begin{array}{c}M_2^t-I\\I-M_1^t\end{array}\right]\cong \mathbb Z/2\mathbb Z,$$ $$K_1(C^*(\Lambda))\cong\ker[I-M_1^t\;\;I-M_2^t]/\text{im}\left[\begin{array}{c}M_2^t-I\\I-M_1^t\end{array}\right]\cong 0.$$ In particular, ${\mathcal{O}}_{\rho_1,\rho_2}\cong {\mathcal{O}}_3$. On the other hand, since $\rho_1, \rho_2$ are faithful, both ${\mathcal{O}}_{\rho_1}, {\mathcal{O}}_{\rho_2}$ are simple and purely infinite with $$K_0({\mathcal{O}}_{\rho_1})\cong {{\mathbb{Z}}}/2{{\mathbb{Z}}},\; K_1({\mathcal{O}}_{\rho_1})\cong 0,\; K_0({\mathcal{O}}_{\rho_2})\cong {{\mathbb{Z}}},\; K_1({\mathcal{O}}_{\rho_2})\cong {{\mathbb{Z}}},$$ so ${\mathcal{O}}_{\rho_1,\rho_2}\ncong {\mathcal{O}}_{\rho_1}\otimes {\mathcal{O}}_{\rho_2}$. With $G=S_3$ and $\rho_1=2\iota, \rho_2=\iota+\epsilon$, then $R=\{\iota, \epsilon\}$ so $\Lambda$ will have two vertices and incidence matrices $$M_1=\left[\begin{array}{cc}2&0\\0&2\end{array}\right],\;\; M_2=\left[\begin{array}{cc}1&1\\1&1\end{array}\right],$$ which give $$\begin{tikzpicture} \renewcommand{\ss}{\scriptstyle} \node[inner sep=1.0pt, circle, fill=black] (u) at (-3,0) {}; \node[below] at (u.south) {$\ss \iota$}; \node[inner sep=1.0pt, circle, fill=black] (v) at (-1,0) {}; \node[below] at (v.south) {$\ss \epsilon$}; \node[inner sep=1.0pt, circle, fill=black] (w) at (1,0) {}; \node[below] at (w.south) {$\ss \iota$}; \node[inner sep=1.0pt, circle, fill=black] (x) at (3,0) {}; \node[below] at (x.south) {$\ss \epsilon$}; \draw[-stealth, semithick, blue] (u) .. controls (-4,1.5) and (-2, 1.5) .. node[below,black] {$\ss e_1$} (u); \draw[-stealth, semithick, blue] (u) .. controls (-2,-1.5) and (-4, -1.5) .. node[above,black] {$\ss e_2$} (u); \draw[-stealth, semithick, blue] (v) .. controls (-2,1.5) and (0, 1.5) .. node[below,black] {$\ss e_3$} (v); \draw[-stealth, semithick, blue] (v) .. controls (0,-1.5) and (-2, -1.5) .. node[above,black] {$\ss e_4$} (v); \draw[-stealth, semithick, red] (w) .. controls (-0.5,1) and (-0.5, -1) .. node[right,black] {$\ss f_1$} (w); \draw[-stealth, semithick, red] (w) to [out=30, in=150] node[above,black] {$\ss f_2$} (x); \draw[-stealth, semithick, red] (x) to [out=-150, in=-30] node[below,black] {$\ss f_3$} (w); \draw[-stealth, semithick, red] (x) .. controls (4.5,-1) and (4.5,1) .. node[left,black] {$\ss f_4$} (x); \end{tikzpicture}$$ Again, a corresponding choice of isometric intertwiners will determine some factorization rules, for example $$e_1f_1=f_1e_2,\; e_2f_1=f_1e_1,\; e_1f_3=f_3e_3,\; e_2f_3=f_3e_4,$$ $$e_3f_2=f_2e_1,\; e_4f_2=f_2e_2,\; e_3f_4=f_4e_4,\; e_4f_4=f_4e_3.$$ Even though $\rho_1, \rho_2$ are not faithful, the obtained $2$-graph is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, so ${\mathcal{O}}_{\rho_1,\rho_2}$ is simple and purely infinite with trivial $K$-theory. In particular, ${\mathcal{O}}_{\rho_1,\rho_2}\cong {\mathcal{O}}_2$. Note that since $\rho_1, \rho_2$ have kernel $N={\langle}(123){\rangle}\cong {{\mathbb{Z}}}/3{{\mathbb{Z}}}$, we could replace $G$ by $G/N\cong {{\mathbb{Z}}}/2{{\mathbb{Z}}}$ and consider $\rho_1,\rho_2$ as representations of ${{\mathbb{Z}}}/2{{\mathbb{Z}}}$. Consider $G={{\mathbb{Z}}}/2{{\mathbb{Z}}}=\{0,1\}$ with $\hat{G}=\{\iota,\chi\}$ and character table $0$ $1$ --------- ----- ------ $\iota$ $1$ $1$ $\chi$ $1$ $-1$ Choose the $2$-dimensional representations $$\rho_1=\iota+\chi,\; \rho_2=2\iota,\; \rho_3=2\chi,$$ which determine a product system ${\mathcal{E}}$ such that ${\mathcal{O}}({\mathcal{E}})\cong {\mathcal{O}}_2\otimes{\mathcal{O}}_2\otimes {\mathcal{O}}_2$ and a Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,\rho_2,\rho_3}\cong {\mathcal{O}}({\mathcal{E}})^{{{\mathbb{Z}}}/2{{\mathbb{Z}}}}$. An easy computation shows that the incidence matrices of the blue, red and green graphs are $$M_1=\left[\begin{array}{cc}1&1\\1&1\end{array}\right],\; M_2=\left[\begin{array}{cc}2&0\\0&2\end{array}\right],\; M_3=\left[\begin{array}{cc}0&2\\2&0\end{array}\right].$$ $$\begin{tikzpicture} \renewcommand{\ss}{\scriptstyle} \node[inner sep=1.0pt, circle, fill=black] (u) at (-5,0) {}; \node[below] at (u.south) {$\ss \iota$}; \node[inner sep=1.0pt, circle, fill=black] (v) at (-3,0) {}; \node[below] at (v.south) {$\ss \chi$}; \node[inner sep=1.0pt, circle, fill=black] (w) at (-1,0) {}; \node[below] at (w.south) {$\ss \iota$}; \draw[-stealth, semithick, blue] (u) .. controls (-6.5,1) and (-6.5, -1) .. node[right,black] {$\ss e_1$} (u); \draw[-stealth, semithick, blue] (u) to [out=30, in=150] node[above,black] {$\ss e_2$} (v); \draw[-stealth, semithick, blue] (v) to [out=-150, in=-30] node[below,black] {$\ss e_3$} (u); \draw[-stealth, semithick, blue] (v) .. controls (-1.5,-1) and (-1.5,1) .. node[left,black] {$\ss e_4$} (v); \node[inner sep=1.0pt, circle, fill=black] (x) at (1,0) {}; \node[inner sep=1.0pt, circle, fill=black] (y) at (3,0) {}; \node[inner sep=1.0pt, circle, fill=black] (z) at (5,0) {}; \draw[-stealth, semithick, red] (w) .. controls (-2,1.5) and (0, 1.5) .. node[below,black] {$\ss f_1$} (w); \draw[-stealth, semithick, red] (w) .. controls (0,-1.5) and (-2, -1.5) .. node[above,black] {$\ss f_2$} (w); \draw[-stealth, semithick, red] (x) .. controls (0,1.5) and (2, 1.5) .. node[below,black] {$\ss f_3$} (x); \draw[-stealth, semithick, red] (x) .. controls (2,-1.5) and (0, -1.5) .. node[above,black] {$\ss f_4$} (x); \draw[-stealth, semithick, green] (z) to [out=100, in=80] node[above,black] {$\ss g_1$} (y); \draw[-stealth, semithick, green] (z) to [out=160, in=20] node[above,black] {$\ss g_2$} (y); \draw[-stealth, semithick, green] (y) to [out=-80, in=-100] node[below,black] {$\ss g_4$} (z); \draw[-stealth, semithick, green] (y) to [out=-20, in=-160] node[below,black] {$\ss g_3$} (z); \node[below] at (x.south) {$\ss \chi$}; \node[below] at (y.south) {$\ss \iota$}; \node[below] at (z.east) {$\ss \chi$}; \end{tikzpicture}$$ With labels as in the figure, we choose the following factorization rules $$e_1f_1=f_2e_1,\; e_1f_2=f_1e_1,\; e_2f_1=f_4e_2,\; e_2f_2=f_3e_2,$$ $$e_3f_3=f_2e_3,\; e_3f_4=f_1e_3,\; e_4f_4=f_3e_4,\; e_4f_3=f_4e_4,$$ $$f_1g_1=g_2f_3,\; f_1g_2=g_1f_3,\; f_2g_1=g_2f_4,\; f_2g_2=g_1f_4,$$ $$f_3g_3=g_4f_1,\; f_3g_4=g_3f_1,\; f_4g_3=g_4f_2,\; f_4g_4=g_3f_2,$$ $$e_1g_1=g_2e_4,\; e_1g_2=g_1e_4,\; e_2g_1=g_3e_3,\; e_2g_2=g_4e_3,$$ $$e_3g_3=g_1e_2,\; e_3g_4=g_2e_2,\; e_4g_3=g_4e_1,\; e_4g_4=g_3e_1.$$ A tedious verification shows that all the following paths are well defined $$e_1f_1g_1,\; e_1f_1g_2,\; e_1f_2g_1, \; e_1f_2g_2,\; e_2f_1g_1,\; e_2f_1g_2,\; e_2f_2g_1,\; e_2f_2g_2,$$ $$e_3f_3g_3,\; e_3f_3g_4,\; e_3f_4g_3,\; e_3f_4g_4,\; e_4f_3g_3,\; e_4f_3g_4,\; e_4f_4g_3,\; e_4f_4g_4,$$ so the associativity property is satisfied and we get a rank $3$ graph $\Lambda$ with $2$ vertices. It is not difficult to check that $\Lambda$ is cofinal, it satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, so $C^*(\Lambda)$ is simple and purely infinite. Since $\partial_1=[I-M_1^t\; I-M_2^t\; I-M_3^t]:{{\mathbb{Z}}}^6\to {{\mathbb{Z}}}^2$ is surjective, using Corollary 3.18 in [@E], we obtain $$K_0(C^*(\Lambda))\cong \ker\partial_2/\text{im}\; \partial_3\cong 0,\;K_1(C^*(\Lambda))\cong \ker\partial_1/\text{im}\; \partial_2\oplus \ker\partial_3\cong 0,$$ where $$\partial_2=\left[\begin{array}{ccc} M_2^t-I&M_3^t-I&0\\I-M_1^t&0&M_3^t-I\\0&I-M_1^t&I-M_2^t\end{array}\right],\;\;\partial_3=\left[\begin{array}{c}I-M_3^t\\M_2^t-I\\I-M_1^t\end{array}\right],$$ in particular ${\mathcal{O}}_{\rho_1,\rho_2,\rho_3}\cong{\mathcal{O}}_2$. Let $G={{\mathbb{T}}}$. We have $\hat{G}=\{\chi_k:k\in {{\mathbb{Z}}}\}$, where $\chi_k(z)=z^k$ and $\chi_k\otimes\chi_\ell=\chi_{k+\ell}$. The faithful representations $$\rho_1=\chi_{-1}+\chi_0,\; \rho_2=\chi_0+\chi_1$$ of ${{\mathbb{T}}}$ will determine a product system ${\mathcal{E}}$ with ${\mathcal{O}}({\mathcal{E}})\cong {\mathcal{O}}_2\otimes {\mathcal{O}}_2$ and a Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,\rho_2}\cong {\mathcal{O}}({\mathcal{E}})^{{\mathbb{T}}}$ isomorphic to a corner in the $C^*$-algebra of a rank $2$ graph $\Lambda$ with $\Lambda^0=\hat{G}$ and infinite incidence matrices, where $$M_1(\chi_k,\chi_\ell)=\begin{cases}1 &\text{if}\; \ell=k \;\text{or}\; \ell=k-1\\0& \text{otherwise,}\end{cases}$$ $$M_2(\chi_k,\chi_\ell)=\begin{cases} 1 &\text{if}\; \ell=k \;\text{or}\; \ell=k+1\\0& \text{otherwise.}\end{cases}$$ The skeleton of $\Lambda$ looks like $$\begin{tikzpicture} \renewcommand{\ss}{\scriptstyle} \node[inner sep=1.0pt, circle, fill=black] (u) at (-3,0) {}; \node[right] at (u.east) {$\ss \chi_{-1}$}; \node[left] at (u.west) {$ \cdots$}; \node[inner sep=1.0pt, circle, fill=black] (v) at (-1,0) {}; \node[right] at (v.east) {$\ss \chi_0$}; \node[inner sep=1.0pt, circle, fill=black] (w) at (1,0) {}; \node[right] at (w.east) {$\ss \chi_1$}; \node[inner sep=1.0pt, circle, fill=black] (x) at (3,0) {}; \node[right] at (x.east) {$\ss \chi_2$}; \node[right] at (x.east) {$\hspace{5mm}\cdots$}; \draw[-stealth, semithick, blue] (u) .. controls (-4,1.5) and (-2, 1.5) .. (u); \draw[-stealth, semithick, red] (u) .. controls (-4,-1.5) and (-2, -1.5) .. (u); \draw[-stealth, semithick, blue] (v) .. controls (-2,1.5) and (0, 1.5) .. (v); \draw[-stealth, semithick, red] (v) .. controls (-2,-1.5) and (0, -1.5) .. (v); \draw[-stealth, semithick, blue] (w) .. controls (0,1.5) and (2, 1.5) .. (w); \draw[-stealth, semithick, red] (w) .. controls (0,-1.5) and (2, -1.5) .. (w); \draw[-stealth, semithick, blue] (x) .. controls (2,1.5) and (4, 1.5) .. (x); \draw[-stealth, semithick, red] (x) .. controls (2,-1.5) and (4, -1.5) .. (x); \draw[-stealth, semithick, blue] (u) to [out=30, in=150] (v); \draw[-stealth, semithick, red] (v) to [out=-150, in=-30] (u); \draw[-stealth, semithick, blue] (v) to [out=30, in=150] (w); \draw[-stealth, semithick, red] (w) to [out=-150, in=-30] (v); \draw[-stealth, semithick, blue] (w) to [out=30, in=150] (x); \draw[-stealth, semithick, red] (x) to [out=-150, in=-30] (w); \end{tikzpicture}$$ and this $2$-graph is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, so $C^*(\Lambda)$ is simple and purely infinite. Let $G=SU(2)$. It is known (see p.84 in [@BD]) that the elements in $\hat{G}$ are labeled by $V_n$ for $n\ge 0$, where $V_0=\iota$ is the trivial representation on ${{\mathbb{C}}}$, $V_1$ is the standard representation of $SU(2)$ on ${{\mathbb{C}}}^2$, and for $n\ge 2$, $V_n=S^nV_1$, the $n$-th symmetric power. In fact, $\dim V_n=n+1$ and $V_n$ can be taken as the representation of $SU(2)$ on the space of homogeneous polynomials $p$ of degree $n$ in variables $z_1,z_2$, where for ${\displaystyle}g=\left[\begin{array}{cc} a&b\\c&d\end{array}\right]\in SU(2)$ we have $$(g\cdot p)(z)=p(az_1+cz_2, bz_1+dz_2).$$ The irreducible representations $V_n$ satisfy the Clebsch-Gordan formula $$V_k\otimes V_\ell=\bigoplus_{j=0}^qV_{k+\ell-2j},\; q=\min\{k,l\}.$$ If we choose $\rho_1=V_1, \rho_2=V_2$, then we get a product system ${\mathcal{E}}$ with ${\mathcal{O}}({\mathcal{E}})\cong {\mathcal{O}}_2\otimes {\mathcal{O}}_3$ and a Doplicher-Roberts algebra ${\mathcal{O}}_{\rho_1,\rho_2}\cong {\mathcal{O}}({\mathcal{E}})^{SU(2)}$ isomorphic to a corner in the $C^*$-algebra of a rank $2$ graph with $\Lambda^0=\hat{G}$ and edges given by the matrices $$M_1(V_k,V_\ell)=\begin{cases}1&\text{if}\; k=0\;\text{and}\; \ell=1\\ 1& \text{if}\; k\ge 1\;\text{and}\; \ell \in\{k-1,k+1\}\\0&\text{otherwise,}\end{cases}$$ $$M_2(V_k, V_\ell)=\begin{cases} 1 &\text{if}\; k=0\;\text{and}\; \ell=2\\1&\text{if}\; k=1\;\text{and}\; \ell\in \{1,3\}\\ 1&\text{if}\; k\ge 2\;\text{and}\; \ell\in\{k-2,k,k+2\}\\0&\text{otherwise.}\end{cases}$$ The skeleton looks like $$\begin{tikzpicture} \renewcommand{\ss}{\scriptstyle} \node[inner sep=1.0pt, circle, fill=black] (u) at (-5,0) {}; \node[right] at (u.east) {$\ss V_0$}; \node[inner sep=1.0pt, circle, fill=black] (v) at (-3,0) {}; \node[right] at (v.east) {$\ss V_1$}; \node[inner sep=1.0pt, circle, fill=black] (w) at (-1,0) {}; \node[right] at (w.east) {$\ss V_2$}; \node[inner sep=1.0pt, circle, fill=black] (x) at (1,0) {}; \node[right] at (x.east) {$\ss V_3$}; \node[inner sep=1.0pt, circle, fill=black] (y) at (3,0) {}; \node[right] at (y.east) {$\ss V_4$}; \node[inner sep=1.0pt, circle, fill=black] (z) at (5,0) {}; \node[right] at (z.east) {$\ss V_5$}; \node[right] at (z.east) {$\hspace{5mm}\cdots$}; \draw[-stealth, semithick, red] (v) .. controls (-2.5,-1) and (-3.5, -1) .. (v); \draw[-stealth, semithick, red] (w) .. controls (-0.5,-1) and (-1.5, -1) .. (w); \draw[-stealth, semithick, red] (x) .. controls (1.5,-1) and (0.5, -1) .. (x); \draw[-stealth, semithick, red] (y) .. controls (3.5,-1) and (2.5, -1) .. (y); \draw[-stealth, semithick, red] (z) .. controls (5.5,-1) and (4.5, -1) .. (z); \draw[-stealth, semithick, blue] (u) to [out=30, in=150] (v); \draw[-stealth, semithick, red] (u) to [out=45, in=135] (w); \draw[-stealth, semithick, red] (w) to [out=-135, in=-45] (u); \draw[-stealth, semithick, red] (v) to [out=45, in=135] (x); \draw[-stealth, semithick, red] (x) to [out=-135, in=-45] (v); \draw[-stealth, semithick, red] (w) to [out=45, in=135] (y); \draw[-stealth, semithick, red] (y) to [out=-135, in=-45] (w); \draw[-stealth, semithick, red] (x) to [out=45, in=135] (z); \draw[-stealth, semithick, red] (z) to [out=-135, in=-45] (x); \draw[-stealth, semithick, blue] (v) to [out=-150, in=-30] (u); \draw[-stealth, semithick, blue] (v) to [out=30, in=150] (w); \draw[-stealth, semithick, blue] (w) to [out=-150, in=-30] (v); \draw[-stealth, semithick, blue] (w) to [out=30, in=150] (x); \draw[-stealth, semithick, blue] (x) to [out=-150, in=-30] (w); \draw[-stealth, semithick, blue] (x) to [out=30, in=150] (y); \draw[-stealth, semithick, blue] (y) to [out=-150, in=-30] (x); \draw[-stealth, semithick, blue] (y) to [out=30, in=150] (z); \draw[-stealth, semithick, blue] (z) to [out=-150, in=-30] (y); \end{tikzpicture}$$ and this $2$-graph is cofinal, satisfies the aperiodicity condition and every vertex connects to a loop with an entrance, in particular ${\mathcal{O}}_{\rho_1,\rho_2}$ is simple and purely infinite. 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--- abstract: 'The Gaussian Process (GP) framework flexibility has enabled its use in several data modeling scenarios. The setting where we have unavailable or uncertain inputs that generate possibly noisy observations is usually tackled by the well known Gaussian Process Latent Variable Model (GPLVM). However, the standard variational approach to perform inference with the GPLVM presents some expressions that are tractable for only a few kernel functions, which may hinder its general application. While other quadrature or sampling approaches could be used in that case, they usually are very slow and/or non-deterministic. In the present paper, we propose the use of the unscented transformation to enable the use of any kernel function within the Bayesian GPLVM. Our approach maintains the fully deterministic feature of tractable kernels and presents a simple implementation with only moderate computational cost. Experiments on dimensionality reduction and multistep-ahead prediction with uncertainty propagation indicate the feasibility of our proposal.' author: - | Daniel Augusto R. M. A. de Souza, César Lincoln C. Mattos and João Paulo P. Gomes\ Department of Computer Science, Universidade Federal do Ceará\ Fortaleza, Brazil\ `danielramos@lia.ufc.br, cesarlincoln@dc.ufc.br, jpaulo@lia.ufc.br` bibliography: - 'references.bib' title: 'Unscented Gaussian Process Latent Variable Model: learning from uncertain inputs with intractable kernels' --- Introduction ============ Gaussian Processes (GP) models have been widely used in the machine learning community as a Bayesian approach to nonparametric kernel-based learning, enabling full probabilistic predictions [@rasmussen2006]. Due to its flexibility, the GP framework has been applied in several contexts, such as semi-supervised learning [@Damianou:semidescribed15], dynamical modeling [@frigola2014; @mattos2016_iclr; @eleftheriadis2017identification], variational autoencoders [@eleftheriadis2016variational; @casale2018gaussian] and hierarchical modeling [@salimbeni2017doubly; @havasi2018inference]. The works above have in common an important building block: the GP Latent Variable Model (GPLVM), proposed by [@lawrence2004] to handle learning scenarios where we have uncertain inputs. GPLVM was extended by [@titsias2010] with a Bayesian training approach (Bayesian GPLVM) and later by [@damianou2013] in a multilayer setting (Deep GPs). The variational approach by [@titsias2010] for the Bayesian GPLVM presents calculations that are tractable for few choices for the kernel function, such as the radial basis function (RBF) kernel. However, it has been pointed out that the RBF kernel presents limited extrapolation capability [@mackay1998introduction]. Some authors have been tackling that issue. [@duvenaud2013structure; @lloyd2014automatic] pursue a compositional approach to build more expressive kernels from simpler ones. [@wilson2013gaussian] propose the spectral mixture kernel to automatically discover patterns and extrapolate beyond the training data. [@wilson2016deep; @wilson2016stochastic; @al2017learning] propose the use of deep neural networks to learn kernel functions directly from data. Although those proposals achieve more flexible kernels, they turn some GPLVM expressions intractable. [@eleftheriadis2017identification; @salimbeni2017doubly] handle non-RBF kernels with uncertain inputs by using the so-called “reparametrization trick” [@KingmaWelling2014; @RezendeEtAl2014] in the doubly stochastic variational inference framework, introduced by [@titsias2014doubly]. Such approach results in a flexible inference methodology, but, since it resorts to stochastic sampling, it strays from the deterministic advantage of standard variational methods. In the present paper, we aim to handle the issue of propagating the uncertainty in the GPLVM while maintaining the deterministic framework presented by [@titsias2010]. We tackle the intractabilities of uncertain inputs and/or non-RBF kernels by exploiting the unscented transformation (UT), a deterministic technique to approximate nonlinear mappings of a probability distribution [@julier2004unscented; @menegaz2015systematization]. The UT projects a finite number of *sigma points* through a nonlinear function and use their computed statistics to estimate the transformed mean and covariance, resulting in a method more scalable than, for instance, the Gauss-Hermite quadrature. A few authors have already considered the UT in the context of GP models. [@ko2007gp; @ko2009gp] use the Unscented Kalman Filter (UKF) with GP-based transition and observation functions. The resulting GP-UKF has been successfully applied by other authors [@anger2012unscented; @wang2014human; @safarinejadian2014fault]. [@ko2011learning] extends the previous works by considering the original GPLVM, where the latent variables are optimized instead of integrated. [@steinberg2014extended] tackle other kinds of intractabilities and present the Unscented GP (UGP) model to handle supervised tasks with non-Gaussian likelihoods, such as binary classification. However, the above works do not consider the Bayesian GPLVM, where the latent variables which represent uncertain inputs are approximately integrated. Thus, we aim to use the UT to handle the intractabilities of the Bayesian GPLVM. More specifically, we propose to approximate the integrals that arise from the convolutions of the kernel function with a Gaussian density in the variational framework by [@titsias2010]. Our methodology enables the use of any kernel, including the ones obtained via auxiliary parametric models in a kernel learning setup. We evaluate our approach in the GPLVM original task of dimensionality reduction and in the task of uncertainty propagation during a free simulation (multistep-ahead prediction) of dynamical models. The results indicate the feasibility of our proposal, both in terms of quantitative metrics and computational effort. In summary, our main contributions are: (*i*) an extension to the Bayesian GPLVM using the UT to deterministically handle intractable integrals and enable the use of any kernel; (*ii*) a set of experiments comparing the proposed approach and alternative approximations using Gauss-Hermite quadrature and Monte Carlo sampling in tasks involving dimensionality reduction and dynamical free simulation. The remainder of the paper is organized as follows. In Section \[SEC\_THEORY\] we present the theoretical background by summarizing the GPLVM framework and the UT approximation. In Section \[SEC\_PROPOSE\] we detail our proposal to apply the UT within the Bayesian GPLVM setting. In Section \[SEC\_EXP\] we present and discuss the obtained empirical results. We conclude the paper in Section \[SEC\_CONC\] and present ideas for further work. Theoretical Background {#SEC_THEORY} ====================== In this section, we summarize the GP and the Bayesian GPLVM models, as well as the UT. The Gaussian Process Framework ------------------------------ Let $N$ inputs $\bm{x}_i \in \mathbb{R}^{D_x}$, organized in a matrix $\bm{X} \in \mathbb{R}^{N \times D_x}$ be mapped via $f : \mathbb{R}^{D_x} \rightarrow \mathbb{R}^{D_y}$ to a $N$ correspondent outputs $\bm{f}_i \in \mathbb{R}^{D_y}$, organized in the matrix $\bm{F} \in \mathbb{R}^{N \times D_y}$. We observe $\bm{Y} \in \mathbb{R}^{N \times D_y}$, a noisy version of $\bm{F}$. Considering an observation noise $\bm{\epsilon} \sim \mathcal{N}(\bm{0}, \sigma^2\bm{I})$, we have $\bm{f}_{:d} = f(\bm{x}_i)$ and $\bm{y}_{:d} = \bm{f}_{:d} + \bm{\epsilon}$, where $\bm{y}_{:d} \in \mathbb{R}^N$ denotes the vector comprised of the $d$-th component of each observed sample, i.e., the $d$-th column of the matrix $\bm{Y}$. If we choose a multivariate zero mean Gaussian prior for each dimension of $\bm{F}$, we get [@rasmussen2006]: $$\begin{aligned} %\nonumber %p(\bm{F}|\bm{X}) & = \prod_{d=1}^{D_y} \mathcal{N}( \bm{f}_{:d} | \bm{0}, \bm{K}_f ), \\ %\nonumber %p(\bm{Y}|\bm{F},\bm{X}) & = \prod_{d=1}^{D_y} \mathcal{N}( \bm{y}_{:d} | \bm{f}_{:d}, \sigma^2 \bm{I}) \mathcal{N}( \bm{f}_{:d} | \bm{0}, \bm{K}_f ), \\ \label{EQ:GPLVM} p(\bm{Y}|\bm{X}) & = \prod_{d=1}^{D_y} \mathcal{N}( \bm{y}_{:d} | \bm{0}, \bm{K}_f + \sigma^2 \bm{I}), \end{aligned}$$ where we were able to analytically integrate out the non-observed (*latent*) variables $\bm{f}_{:d}|_{d=1}^{D_y}$. The elements of the covariance matrix $\bm{K}_f \in \mathbb{R}^{N \times N}$ are calculated by $[\bm{K}_f]_{ij} = k(\bm{x}_i, \bm{x}_j), \forall i,j \in \{1,\cdots,N\}$, where $k(\cdot, \cdot)$ is the so-called covariance (or *kernel*) function. The Bayesian Gaussian Process Latent Variable Model {#section_BGPLVM} --------------------------------------------------- The Gaussian Process Latent Variable Model (GPLVM), proposed by @lawrence2004, extends the GP framework for scenarios where we do not have the inputs $\bm{X}$, which generated the observations $\bm{Y}$ via the modeled function. The GPLVM was originally proposed in the context of nonlinear dimensionality reduction[^1], which can be done choosing $D_x < D_y$. However, the approach has proved to be flexible enough to be used in several other scenarios. For instance, in supervised tasks, the matrix $\bm{X}$ can be seen as a set of observed but uncertain inputs [@damianou2016variational]. The Bayesian GPLVM, proposed by @titsias2010, considers a variational approach [@jordan1999; @blei2016] to approximately integrate the latent variables $\bm{X}$. Inspired by Titsias’ variational sparse GP framework [@titsias2009], the Bayesian GPLVM avoids overfitting by considering the uncertainty of the latent space and enables the determination of $D_x$ by using a kernel function with ARD (*automatic relevance determination*) hyperparameters. Following [@titsias2010], we start by including $M$ inducing points $\bm{z}_{:d} \in \mathbb{R}^M$ associated to each output dimension and evaluated in $M$ pseudo-inputs $\bm{\zeta}_j|_{j=1}^M \in \mathbb{R}^{D_x}$, where $p(\bm{z}_{:d}) = \mathcal{N}(\bm{z}_{:d} | \bm{0}, \bm{K}_z)$ and $\bm{K}_z \in \mathbb{R}^{M \times M}$ is the kernel matrix computed from the pseudo-inputs. The joint distribution of all the variables in the GPLVM is now given by (with omitted dependence on $\bm{\zeta}_j$) $$\label{EQ:GPLVM_JOINT} p(\bm{Y}, \bm{X}, \bm{F}, \bm{Z}) = \left( \prod_{d=1}^{D_y} p(\bm{y}_{:d} | \bm{f}_{:d}) p(\bm{f}_{:d} | \bm{z}_{:d}, \bm{X}) p(\bm{z}_{:d}) \right) p(\bm{X}).$$ Applying Jensen’s inequality to Eq. gives a lower bound to the marginal log-likelihood $\log p(\bm{Y})$: $$p(\bm{Y}) = \int p(\bm{Y}, \bm{X}, \bm{F}, \bm{Z}) \mathrm{d}\bm{X} \mathrm{d}\bm{F} \mathrm{d}\bm{Z} \geq Q \log \left[ \frac{p(\bm{Y}, \bm{X}, \bm{F}, \bm{Z})}{Q} \right] \mathrm{d}\bm{X} \mathrm{d}\bm{F} \mathrm{d}\bm{Z},$$ where $Q$ is the variational distribution, chosen to be given by the form $Q = q(\bm{X}) q(\bm{Z}) p(\bm{F} | \bm{Z}, \bm{X})$, where $p(\bm{F} | \bm{Z}, \bm{X})$ is an analytical conditional distribution of Gaussians and the variational distributions $q(\bm{X}) = \prod_i^N q(\bm{x}_i)$ and $q(\bm{Z}) = \prod_d^{D_y} q(\bm{z}_{:d})$ respectively approximate the posteriors of the variables $\bm{X}$ and $\bm{Z}$ by products of multivariate Gaussians. The final analytical bound derived by [@titsias2010], which may be directly used to perform model selection, depends on the three terms below, named $\Psi$-statistics: $$\begin{aligned} \label{EQ:PSI_0} \Psi_0 &= \sum_{i=1}^N \Psi_0^{(i)} \in \mathbb{R}, \quad \text{ where } \Psi_0^{(i)} = \int k(\bm{x}_i,\bm{x}_i) q(\bm{x}_i) \mathrm{d}\bm{x}_i, \\ \label{EQ:PSI_1} \bm{\Psi}_1 & \in \mathbb{R}^{N \times M}, \quad\text{ where } [\bm{\Psi}_1]_{ij} = \int k(\bm{x}_i,\bm{\zeta}_j) q(\bm{x}_i) \mathrm{d}\bm{x}_i \\ \label{EQ:PSI_2} \bm{\Psi}_2 &= \sum_{i=1}^N \bm{\Psi}_2^{(i)} \in \mathbb{R}^{M \times M}, \quad \text{ where }[\bm{\Psi}_2^{(i)}]_{jm} = \int k(\bm{x}_i,\bm{\zeta}_j) k(\bm{x}_i,\bm{\zeta}_m) q(\bm{x}_i) \mathrm{d}\bm{x}_i.\end{aligned}$$ The above expressions represent convolutions of the kernel function with the variational distribution $q(\bm{X})$ and are tractable only for a few kernel functions, such as the RBF and the linear kernels. The Unscented Transformation {#sec_ut} ---------------------------- The unscented transformation (UT) is a method for estimating the first two moments of the mapping of a random variable under an arbitrary function. Proposed by @uhlmann1995dynamic in the context of Kalman filters (KF), the transformation itself is decoupled from the so-called Unscented KF. In the UT, the mean and covariance of the transformed random variable are approximated with a weighted average of transformed *sigma points* $\bm{s}_n$, derived from the first two moments of the original input. Let $\bm{x} \sim \mathcal{N}(\bm{\mu},\bm{\Sigma})$ be a $D$-dimensional variable which will pass through an arbitrary transformation $f\colon \mathbb{R}^D \rightarrow \mathbb{R}^Q$. Given uniform weights for the sigma points, the output moments are computed by: $$\mathbb{E}(f(\bm{x})) \approx \frac{1}{2D}\sum_{n=1}^{2D} f(\bm{s}_n) = \bm{m}, \quad \mathrm{cov}(f(\bm{x})) \approx \frac{1}{2D}\sum_{n=1}^{2D} (f(\bm{s}_n)-\bm{m})(f(\bm{s}_n)-\bm{m})^\top.$$ There are several strategies to select sigma points (see e.g. @menegaz2015systematization). We follow the original scheme by @uhlmann1995dynamic, with uniform weights and sigma points chosen from the columns of the squared root of $D\bm{\Sigma}$, an efficient way to generate a symmetric distribution of sigma points. Let $\mathrm{chol}(\bm{\Sigma})$ be the Cholesky decomposition of the matrix $\bm{\Sigma}$. Then, the sigma points $\bm{s}_n$ are defined as: $$\bm{s}_n = \bm{\mu} + [\mathrm{chol}(D\bm{\Sigma})]_{:n}, \quad \bm{s}_{n+D} = \bm{\mu} - [\mathrm{chol}(D\bm{\Sigma})]_{:n}, \quad \forall n \in [1,D],$$ where $[\mathrm{chol}(D\bm{\Sigma})]_{:n}$ denotes the $n$-th column of the lower triangular matrix $\mathrm{chol}(D\bm{\Sigma})$. Since only a relatively small number of sigma points is used ($2D$, where $D$ is the dimensionality of the input) and their computation is completely deterministic, the UT presents a viable alternative to other quadrature methods, such as Gauss-Hermite and Monte Carlo sampling. Proposed Methodology {#SEC_PROPOSE} ==================== From Section \[section\_BGPLVM\], we can see that the computation of the $\Psi$-statistics in Eqs. - is the only part that prevents the application of the Bayesian GPLVM with arbitrary kernels. Since the $\Psi$-statistics are actually Gaussian expectations of nonlinear functions, we propose to approximate their computation in intractable cases using the UT. We emphasize that the use of the UT to solve the $\Psi$-statistics is convenient since we are often able to limit the dimension of the integrand when learning latent spaces. Furthermore, as noted by @honkela2004approximating, the UT is most suited for Gaussian integrals with lower dimensionality, which is usually the case with the Bayesian GPLVM. A similar result was pointed out by @zhang2009accuracy when comparing UT with other sampling strategies. Besides enabling the use of non-analytical kernels in the Bayesian GPLVM, our choice of using UT-based approximations in place of, for instance, the Gauss-Hermite (GH) quadrature, brings great computational benefits, due to the number of points that are evaluated to compute the Gaussian integral. Given a $D$-dimensional random variable, the UT requires just a linear number of $2D$ sampled points for evaluation, while the GH quadrature requires $H^D$ points, where $H$ is a user chosen order parameter. Even for $H=2$ and moderate dimensionality values, e.g. $D=20$, the GH approach would require at least $2^{20}$ evaluations per approximation, which is infeasible. We also note that the UT and the GH quadrature have similar forms in the single dimension case. In the Bayesian GPLVM, the amount of sampled points is relevant, since the approximations are computed at each step of the variational lower bound optimization. Thus, the number of times we evaluate the $\Psi$-statistics gives a raw estimate of the chosen approximation computational budget. To verify how accuracy evolves with dimension when using UT in the context of the Bayesian GPLVM, we computed $\bm{\Psi}_1$ (see Eq. ) considering a RBF kernel on random data of varying dimension. We use both the UT and the GH quadrature. Since we actually compute only a column of the matrix $\bm{\Psi}_1$, we can measure the approximations accuracies by comparing the error norm between the approximation and the analytical solution, which in this case is feasible. [0.49]{} ![Comparison between the UT and the GH quadrature for computing $\Psi$-statistics.[]{data-label="fig:utvsgh"}](figuras/utghtest/accuracy3.pdf "fig:"){width="\linewidth"} [0.49]{} ![Comparison between the UT and the GH quadrature for computing $\Psi$-statistics.[]{data-label="fig:utvsgh"}](figuras/utghtest/points_used3.pdf "fig:"){width="\linewidth"} Fig. \[fig:utvsgh\] indicates that the error norm ratio between the UT and the GH ($E_{\text{UT}}/E_{\text{GH}}$) presents the tendency of smaller errors for the GH quadrature as the input dimension increases. However, we can see that the UT has slightly better accuracy than the GH on low dimensions ($\leq 8$). Importantly, the UT requires exponentially fewer sample points, as illustrated in the bottom plot of Fig. \[fig:utvsgh\]. Experiments {#SEC_EXP} =========== We consider two standard tasks for the GPLVM: dimensionality reduction and free simulation of dynamical models with uncertainty propagation. We compare our UT approach with the Gauss-Hermite (GH) quadrature and the reparametrization trick based Monte Carlo (MC) sampling for computing the $\Psi$-statistics of the Bayesian GPLVM. In the tractable cases, we also consider the analytical expressions. All experiments were implemented in Python using the GPflow framework [@matthews2017gpflow]. The implementation can be found at <https://github.com/danisson/UnscentedGPLVM>. For the GH experiments, to maintain a reasonable computational cost, we use $2^D$ points, where $D$ is the input dimension. For the MC approximations, we use two different numbers of samples: the same number used by UT and the same number used by GH. Each MC experiment was run ten times, with averages and standard deviations reported. The MC approximation is similar to the one in the doubly stochastic variational framework [@titsias2014doubly], but without mini-batch updates. Dimensionality Reduction ------------------------ The dimensionality reduction task is especially suitable for our UT-based approach, since the dimension of the integrand in the $\Psi$-statistics is usually small for the purposes of data visualization. We used two datasets which were referred by @lawrence2004 and @titsias2010, the Oil flow dataset and the USPS digit dataset. In both cases, we compare the analytic Bayesian GPLVM model with the RBF kernel against a kernel with non-analytic $\Psi$-statistics. We have considered the following kernels: Matérn 3/2, the periodic kernel[^2] and a Multilayer Perceptron (MLP) composed on a RBF kernel, similar to the manifold learning approach by [@calandra2016manifold]. The means of the variational distribution were initialized based on standard Principal Component Analysis (PCA) and the latent variances were initialized to $0.1$. Also, $30$ points from the initial latent space were selected as inducing pseudo-inputs and appropriately optimized during training. Each scenario was evaluated following two approaches: a qualitative analysis of the learned two-dimensional latent space; a quantitative metric in which we take the known labels from each dataset and find the predictive accuracy of which class a point in the learned latent space should belong to. The method used to that end is a five-fold cross validated 1-nearest neighbor (1-NN). For the quantitative results, we also show the accuracy on the PCA projection as a sanity check. ### Oil flow dataset The multiphase Oil flow dataset consists of 1000 observations with 12 attributes, where each one belongs to one of three classes [@bishop1993]. We apply GPLVM with five latent dimensions and select the two dimensions with the greatest inverse lengthscales. For the approximations using GH quadrature, we have used $2^5=32$ samples in total. This contrasts with the UT, which only uses $2 \times 5=10$ samples. We note that we have attempted to follow @titsias2010 and use ten latent dimensions, but that would require the GH to evaluate $2^{10}=1024$ samples at each optimization step, which made the method too slow on the used hardware. [.24]{} ![Two-dimensional projections of the Oil flow and USPS 0-4 digits datasets for GPLVM with different kernels and approximations. The projections shown are the best ones obtained in the crossvalidation steps. 1-NN mislabels are marked in red.[]{data-label="fig:oilRBF"}](figuras/oil/analytic_SquaredExponential_with_errors.pdf "fig:"){width="\linewidth"} [.24]{} ![Two-dimensional projections of the Oil flow and USPS 0-4 digits datasets for GPLVM with different kernels and approximations. The projections shown are the best ones obtained in the crossvalidation steps. 1-NN mislabels are marked in red.[]{data-label="fig:oilRBF"}](figuras/oil/gaussHermite2_Matern32_with_errors.pdf "fig:"){width="\linewidth"} [.24]{} ![Two-dimensional projections of the Oil flow and USPS 0-4 digits datasets for GPLVM with different kernels and approximations. The projections shown are the best ones obtained in the crossvalidation steps. 1-NN mislabels are marked in red.[]{data-label="fig:oilRBF"}](figuras/oil/unscented_Matern32_with_errors.pdf "fig:"){width="\linewidth"} [.24]{} ![Two-dimensional projections of the Oil flow and USPS 0-4 digits datasets for GPLVM with different kernels and approximations. The projections shown are the best ones obtained in the crossvalidation steps. 1-NN mislabels are marked in red.[]{data-label="fig:oilRBF"}](figuras/oil/montecarlo32_Matern32_5_with_errors.pdf "fig:"){width="\linewidth"} ------------------------------------------------------------------------ [.24]{} ![Two-dimensional projections of the Oil flow and USPS 0-4 digits datasets for GPLVM with different kernels and approximations. The projections shown are the best ones obtained in the crossvalidation steps. 1-NN mislabels are marked in red.[]{data-label="fig:oilRBF"}](figuras/usps1/analytic_SquaredExponential.pdf "fig:"){width="\linewidth"} [.24]{} ![Two-dimensional projections of the Oil flow and USPS 0-4 digits datasets for GPLVM with different kernels and approximations. The projections shown are the best ones obtained in the crossvalidation steps. 1-NN mislabels are marked in red.[]{data-label="fig:oilRBF"}](figuras/usps1/gaussHermite2_MLP.pdf "fig:"){width="\linewidth"} [.24]{} ![Two-dimensional projections of the Oil flow and USPS 0-4 digits datasets for GPLVM with different kernels and approximations. The projections shown are the best ones obtained in the crossvalidation steps. 1-NN mislabels are marked in red.[]{data-label="fig:oilRBF"}](figuras/usps1/unscented_MLP.pdf "fig:"){width="\linewidth"} [.24]{} ![Two-dimensional projections of the Oil flow and USPS 0-4 digits datasets for GPLVM with different kernels and approximations. The projections shown are the best ones obtained in the crossvalidation steps. 1-NN mislabels are marked in red.[]{data-label="fig:oilRBF"}](figuras/usps1/montecarlo32_MLP_3.pdf "fig:"){width="\linewidth"} [ ll S\[table-format=3.1(3),table-align-uncertainty=true\] S\[table-format=3.1(3),table-align-uncertainty=true\] S\[table-format=3.1(3),table-align-uncertainty=true\] ]{} & &\ & & [**Oil flow**]{} & [**USPS 0-4 digits**]{} & [**USPS All digits**]{}\ PCA & - & 78.9(65) & 65.0(13) & 36.8(45)\ \[1ex\] Analytic & RBF & 98.0(27) & 72.4(15) & 40.8(61)\ \[1ex\] & Matérn $3/2$ & 95.0(61) & [-]{} & [-]{}\ & MLP \[2,30,60\] & [-]{} & 85.0(28) & 51.4(4)\ & RBF & 98.0(27) & 71.2(22) & 42.4(65)\ \[1ex\] & Matérn $3/2$ & 100.0(0) & [-]{} & [-]{}\ & MLP \[2,30,60\] & [-]{} & 86.5(15) & 52.6(35)\ & RBF & 98.0(27) & 71.6(26) & 40.1(59)\ \[1ex\] & Matérn $3/2$ & 94.3(4) & [-]{} & [-]{}\ & MLP \[2,30,60\] & [-]{} & 53.0(45) & 24.7(31)\ & RBF & 93.4(73) & 48.4(24) & 29.5(31)\ \[1ex\] & Matérn $3/2$ & 94.7(4) & [-]{} & [-]{}\ & MLP \[2,30,60\] & [-]{} & 58.3(5) & 24.9(09)\ & RBF & 95.7(4) & 49.3(4) & 32.5(48)\ In Fig. \[fig:oilRBF\] we can see that independent of the chosen method to solve the $\Psi$-statistics, either the analytic expressions or any approximation yields similar overall qualitative results. Tab. \[tab:dimred\] contains the 1-NN predicted accuracy results for all kernels and approximation methods. As expected, all the nonlinear approaches performed better than regular PCA. We can see that the RBF results for the deterministic approaches are all similar, while the Matérn 3/2 kernel with the UT approximation obtained slightly better results overall. However, when using MC sampling with the same amount of points that UT and GH used, the results for all kernels were worse than both UT and GH. ### USPS digit dataset The USPS digit dataset contains $7000$ $16 \times 16$ gray-scale images of handwritten numerals from $0$ to $9$. We considered two scenarios. First, we replicated the same setup by @lawrence2004, using only $3000$ samples from the digits $0$ to $4$. The second experiment uses all classes and a subset of $5000$ samples. We used a GPLVM with five latent dimensions on all kernels but the MLP kernel, where we use two latent dimensions. We follow the same evaluation methodology previously described. We expected the MLP kernel to fare better than the RBF kernel. This is due to the well known capabilities of neural networks to find lower dimensional representations of higher dimensional structured data [@wilson2016deep]. As seen in Tab. \[tab:dimred\], this was indeed the case. Bayesian GPLVM with a neural network in the kernel function obtained the best results. Fig. \[fig:oilRBF\] shows a comparison between the analytic solution with RBF versus the approximate solutions using a MLP kernel with a single hidden layer and \[2, 30, 60\] neurons (input, hidden and output, respectively). For the experiment with all digits, we can see in Tab. \[tab:dimred\] that the difference between kernels was even higher. Tab. \[tab:dimred\] also presents that all the MC experiments were worse performing than PCA for this specific task. We conjecture that such weakness was due to the dimensionality of the optimization surface, as well as noisy approximations of the objective function and its gradients, which can misdirect the optimization. This problem shines a light on the potential weakness of non-deterministic approximations when used to optimize the model kernel hyperparameters and variational parameters. Dynamical Free Simulation ------------------------- Free simulation, or multistep-ahead prediction, is a task that consists in forecasting the values of a dynamical system arbitrarily far into the future based on past predicted values. In most simple models, such as the GP-NARX model [@kocijan2005dynamic], each prediction does not depend on the uncertainty of past predictions, but only past mean predicted values. To propagate the uncertainty of each prediction to the next implies to perform predictions with uncertain inputs. This task has been approached before, for instance by [@girard2003], but for GP models using the RBF kernel. In this section, we first train a GP-NARX without considering uncertain inputs, following the regular NARX approach [@kocijan2005dynamic]. Then, we apply the same optimized kernel hyperparameters in a GPLVM, selecting all the training inputs as pseudo-inputs. Finally, the GPLVM is used to perform a free simulation with uncertain inputs formed by the past predictive distributions. Since we apply approximations for computing the $\Psi$-statistics in the predictions, we can choose any valid kernel. ### Airline passenger dataset [ lll S\[table-format=2.2(2),table-align-uncertainty=true\] S\[table-format=2.2(2),table-align-uncertainty=true\] ]{} **Model** & **Method** & **Kernel** & [**NLPD**]{} & [**RMSE**]{}\ GP-NARX & & RBF+Linear & 21.82 & 69.39\ & & Periodic+RBF+Linear & 14.00 & 44.99\ GPLVM & Analytic & RBF+Linear & 13.23 & 68.92\ & & RBF+Linear & 13.22 & 68.88\ & & Periodic+RBF+Linear & 09.49 & 45.03\ & & RBF+Linear & 13.23 & 68.88\ & & Periodic+RBF+Linear & 09.59 & 45.28\ & & RBF+Linear & 13.30(24) & 68.70(27)\ & & Periodic+RBF+Linear & 9.50(17) & 45.50(27)\ & & RBF+Linear & [-]{} & [-]{}\ & & Periodic+RBF+Linear & [-]{} & [-]{}\ [0.49]{} ![Illustration of the results obtained in the dynamical free simulation experiments. Best obtained runs are shown. Note that the MC run with 24 samples presented numerical issues and could not be completed.[]{data-label="fig:air"}](figuras/passengers/NARX_analytic_SquaredExponential_Linear.pdf "fig:"){width="\linewidth"} [0.49]{} ![Illustration of the results obtained in the dynamical free simulation experiments. Best obtained runs are shown. Note that the MC run with 24 samples presented numerical issues and could not be completed.[]{data-label="fig:air"}](figuras/passengers/GPLVM_analytic_SquaredExponential_Linear.pdf "fig:"){width="\linewidth"} [0.49]{} ![Illustration of the results obtained in the dynamical free simulation experiments. Best obtained runs are shown. Note that the MC run with 24 samples presented numerical issues and could not be completed.[]{data-label="fig:air"}](figuras/passengers/GPLVM_gaussHermite2_Periodic_SquaredExponential_Linear.pdf "fig:"){width="\linewidth"} [0.49]{} ![Illustration of the results obtained in the dynamical free simulation experiments. Best obtained runs are shown. Note that the MC run with 24 samples presented numerical issues and could not be completed.[]{data-label="fig:air"}](figuras/passengers/GPLVM_unscented_Periodic_SquaredExponential_Linear.pdf "fig:"){width="\linewidth"} [0.49]{} ![Illustration of the results obtained in the dynamical free simulation experiments. Best obtained runs are shown. Note that the MC run with 24 samples presented numerical issues and could not be completed.[]{data-label="fig:air"}](figuras/passengers/GPLVM_montecarlo4096_Periodic_SquaredExponential_Linear_1.pdf "fig:"){width="\linewidth"} [0.49]{} ![Illustration of the results obtained in the dynamical free simulation experiments. Best obtained runs are shown. Note that the MC run with 24 samples presented numerical issues and could not be completed.[]{data-label="fig:air"}](figuras/passengers/GPLVM_montecarlo24_Periodic_SquaredExponential_Linear_8.pdf "fig:"){width="\linewidth"} We consider the Airline passenger numbers dataset, which was recorded monthly from 1949 to 1961 [@timeseriesdata]. The first four years were used for training and the rest was left for testing. We chose an autoregressive lag of 12 past observations as input. After the GP-NARX kernel hyperparameters are optimized, as previously mentioned, we choose the variance of the variational distribution in the GPLVM to be equal to the optimized noise variance. We perform a free simulation from the beginning of the training set until the end of the test set, using past predicted variances as variational variances of the uncertain inputs, which enables approximate uncertainty propagation during the simulation. We used the following kernels: a mixture of a RBF kernel with a linear kernel; a mixture of periodic, RBF and linear kernels. The latter combination of kernels was chosen due to our prior knowledge that airplane ticket sales follow a periodic trend and have an overall upward tendency because of the popularity increase and decrease in the tickets prices. We emphasize that the choice of such a flexible combination of kernels would not be possible without the use of approximate methods when considering the uncertain inputs scenario and the GPLVM framework. Quantitative evaluation is done by computing the root mean squared error (RMSE), given by $\mathrm{RMSE} = \sqrt{\frac{1}{N_*} \sum_{i=1}^{N_*} (y_i - \tilde{\mu}_i)^2}$, where $N_*$ is the number of test samples, $y_i$ is the true output and $\tilde{\mu}_i$ is the predicted mean output. We also compute the average negative log-predictive density (NLPD), given by $\mathrm{NLPD} = \frac{1}{2}\log2\pi + \frac{1}{2N_*} \sum_{i=1}^{N_*} \left[ \log\tilde{\sigma}_i^2 + \frac{(y_i - \tilde{\mu}_i)^2}{\tilde{\sigma}_i^2} \right],$ where $\tilde{\sigma}_i^2$ is the $i$-th predicted variance. We note that both metrics are “the lower, the better” and are computed only for the test set. Tab. \[tab:airAcc\] presents the obtained results. Although with similar RMSE, all GPLVM variants presented better NLPD values when compared to their standard GP-NARX counterparts. That is expected, since the uncertainty of each prediction is being approximately propagated to the next predictions. Since this experiment deals with 12-dimensional inputs, following the discussion in Section \[SEC\_PROPOSE\], the GH approximation might have better accuracy than the UT approximation. However, even using only $2 \times 12=24$ points against GH’s $2^{12}=4096$ points, the difference between UT’s and GH’s accuracies is negligible, given that predictions using UT runs much faster. As shown in Fig. \[fig:air\], visual difference between the two methods is subtle. The results obtained by the MC approximation with the same amount of samples as the GH quadrature (4096) are similar, but when using the same quantity as the UT (24 samples), the model presents numerical issues and do not obtain meaningful outputs. Conclusion {#SEC_CONC} ========== In this work, we have considered the problem of learning GP models from unavailable or uncertain inputs within the Bayesian GPLVM framework. We have tackled the intractabilities that arise in the original variational methodology by @titsias2010 when non-RBF and nonlinear kernels are used by proposing the use of the unscented transformation. We have performed computational experiments on two tasks: dimensionality reduction and free simulation of dynamical models with uncertainty propagation. In both cases, our UT-based approach scaled much better than the compared Gauss-Hermite quadrature, while obtaining a similar overall approximation. The UT results were also more stable and consistent than the ones obtained by Monte Carlo sampling, which may also require a larger number of samples. Importantly, our method is simple to implement and does not impose any stochasticity, maintaining the deterministic feature of the standard Bayesian GPLVM variational framework. For future work we aim to evaluate the UT in more scenarios where inference with GP models falls into intractable integrals. For instance, we intend to tackle intractable expressions that arise with hierarchical GP models and GP-based Bayesian optimization with uncertain inputs. [^1]: The GPLVM is a nonlinear extension of the probabilistic Principal Component Analysis [@lawrence2004]. [^2]: As defined by @mackay1998introduction, in Eq. (47).

--- abstract: 'In this paper we generalize Artin-Verdier, Esnault and Wunram construction of McKay correspondence to arbitrary Gorenstein surface singularities. The key idea is the definition and a systematic use of a degeneracy module, which is an enhancement of the first Chern class construction via a degeneracy locus. We study also deformation and moduli questions. Among our main result we quote: a full classification of special reflexive MCM modules on normal Gorenstein surface singularities in terms of divisorial valuations centered at the singularity, a first Chern class determination at an adequate resolution of singularities, construction of moduli spaces of special reflexive modules, a complete classification of Gorenstein normal surface singularities in representation types, and an study on the deformation theory of MCM modules and its interaction with their pullbacks at resolutions. For the proof of these theorems it is crucial to establish several isomorphisms between different deformation functors, that we expect that will be useful in further work as well.' address: - 'Javier Fernández de Bobadilla: (1) IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain (2) BCAM Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Basque Country, Spain (3) Academic Colaborator at UPV/EHU' - 'Agustín Romano-Velázquez: (1) CIMAT, Jalisco S/N, Mineral de Valenciana, 36023, Guanajuato, México' author: - Javier Fernández de Bobadilla - 'Agustín Romano-Velázquez' date: '13-3-2019' title: 'Reflexive modules on normal Gorenstein Stein surfaces, their deformations and moduli' --- [^1] Introduction ============ McKay correspondence [@McK] is a bijection between the irreducible representations of a finite subgroup of $\mathrm{SL}(2,{{\mathbb{C}}})$ and the irreducible components of the exceptional divisor of the minimal resolution of the associated quotient surface singularity, such a bijection extends to an isomorphism of the McKay quiver (associated to the structure of representations with respect to direct sums and tensor products), and the dual graph of the exceptional divisor of the minimal resolution of singularities. McKay noticed the correspondence by a case by case study using the classification of finite subgroups of $\mathrm{SL}(2,{{\mathbb{C}}})$. After its discovery by McKay, a conceptual geometric understanding of the correspondence was achieved by a series of papers by Gonzalez-Springberg and Verdier [@GoVe], by Artin and Verdier [@AV] for the correspondence at the level of vertices and by Esnault and Knörrer [@EsKn] at the level of edges. At the level of vertices the correspondence can be summarized as follows: let $\pi:{\tilde{X}}\to X$ be the minimal resolution of the quotient singularity. To an irreducible representation $\rho$ one associates an indecomposable reflexive ${\mathcal{O}_{X}}$-module $M$. The module $\pi^*M/\mathrm{Torsion}$ was proved to be locally free, and its first Chern class $c_1(\pi^*M/\mathrm{Torsion})$ is the Poincaré dual of a curvette hitting transversely an unique irreducible component of the exceptional divisor. Such a component is the image of the representation $\rho$ under McKay correspondence. Artin and Verdier proved that the first Chern class determines the module $M$ along with the representation $\rho$. Conversely, for any irreducible component of the exceptional divisor there is a representation and a module realizing it. We do not spell the correspondence at the level of edges since the main concern of this paper is a wide generalization of the results of Artin and Verdier at the level of vertices, and the subsequent contributions of Esnault, Wunram and Kahn which we describe below. Esnault [@Es] improved Artin and Verdier correspondence by working on arbitrary rational surface singularity. She discovered that already for quotient singularities by finite subgroups of $\mathrm{GL}(2,{{\mathbb{C}}})$ not included in $\mathrm{SL}(2,{{\mathbb{C}}})$ the pair given by the first Chern class $c_1(\pi^*M/\mathrm{Torsion})$ and the rank of the module is not enough to determine the reflexive module. A satisfactory McKay correspondence for arbitrary rational surface singularities was provided by Wunram [@Wu]. For this he defined a reflexive module to be [*special*]{} if we have the vanishing $R^1\pi_*(\pi^*M)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}=0$ (where $(\quad)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ denotes the dual with respect to the structure sheaf). He proved that the first Chern class construction of Artin and Verdier defines a bijection between: a) the set of special indecomposable reflexive ${\mathcal{O}_{X}}$-modules and b) the set of irreducible components of the exceptional divisor of the minimal resolution. Moreover the first Chern class $c_1(\pi^*M/\mathrm{Torsion})$ determines special reflexive ${\mathcal{O}_{X}}$-modules. The reader may consult the survey of Riemenschneider [@Rie], for a more complete account of results described up to now, a summary of other approaches to McKay correspondence and a nice characterization of special reflexive modules. Beyond the case of rational singularities the only complete study of reflexive modules was provided by Kahn [@Ka] for the case of minimally elliptic singularities. He studied reflexive ${\mathcal{O}_{X}}$-modules $M$ via the associated locally free ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}:=(\pi^*M)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ restricted to a cycle supported at the exceptional divisor. In the simply elliptic case he provides a full classification of reflexive modules building on Atiyah’s classification of vector bundles on elliptic curves. On normal surface singularities, reflexive modules coincide with Maximal Cohen Macaulay (MCM) ones. A singularity has finite, tame or wild Cohen-Macaulay representation type if the maximal dimension of the families of indecomposable MCM modules is 0, 1 or unbounded respectively. A consequence of Kahn’s results is that simply elliptic singularities are of tame representation type. His results were completed by Drodz, Greuel and Kashuba [@DrGrKa], who proved that cusp singularities, and more generally any log-canonical surface singularity are of tame Cohen-Macaulay representation type, and posed the conjecture that non-canonical surface singularities are of wild representation type (we prove this conjecture in this paper for normal Gorenstein surface singularities). Due to a result of Eisenbud [@Ei], in the hypersurface case MCM modules correspond to matrix factorizations. Therefore any result proven for MCM modules have a translation into matrix factorizations. Using this equivalence, Knörrer [@Kn] and Buchweitz, Greuel, Schreyer [@BuGrSch] proved in arbitrary dimension that the isolated hypersurface singularities of finite Cohen-Macaulay representation type are exactly the simple (ADE type) ones. Besides the results described above the knowledge on the classification of MCM modules on other singularities available in the literature is quite limited. McKay correspondence admits generalizations and extensions in many directions (some of them hinted in Riemenschneider survey [@Rie]). An important one is the study of Auslander and Reiten categories of MCM modules over a singularity. The reader may consult the book of Yoshino [@Yos] for a rather complete account of known results. In this paper we focus in the classification of the objects of this categories for arbitrary normal Gorenstein singularities, leaving the structure of the category ready for later work. Following the line of the first Chern class correspondence described above it is natural to ask whether there is a similar description of indecomposable MCM modules on other singularities, and on the other on the characterization of the irreducible components of the exceptional divisor in terms of reflexive modules. Once the singularities are not rational, reflexive modules come in families, and deformation and moduli problems are important. We obtain quite complete answers to these questions for the case of Gorenstein normal singularities, including: - a full classification of special reflexive MCM modules in terms of divisorial valuations centered at the singularity, which can be seen as a generalization of McKay correspondence, - a first Chern class determination at an adequate resolution of singularities, - construction of moduli spaces of special reflexive modules, - a complete classification of Gorenstein normal surface singularities in Cohen-Macaulay representation types, confirming Drodz, Greuel and Kashuba conjecture for this class, - a study on the deformation theory of MCM modules and its interaction with their pullbacks at resolutions, A detailed, non technical, description of the results of this paper is provided in the next section. Although the papers quoted above on McKay correspondence and classification of MCM modules work over singularities or complete local rings, we prove many of our results in the more general setting of reflexive modules over Stein normal surfaces (usually with Gorenstein singularities). We feel that this generalization will have interesting applications. For example it allows to apply our results to affine patches of reflexive modules on projective surfaces, opening the way to obtain applications in the global projective case. It should be expected that our results have applications in other areas. MCM modules on Gorenstein singularities have become recently even more important due to the equivalence with matrix factorizations explained above. Following an idea of Kontsevich, the work of Kasputin and Li (see  [@KaLi]), and Orlov (see [@Or1], [@Or2]) showed that matrix factorizations have applications to the study of Landau-Ginzburg models appearing in string theory, and to the study of Kontsevich’s homological mirror symmetry. By Khovanov and Rozansky [@KoRo1], [@KoRo2], MCM modules have interesting applications to link invariants. Matrix factorizations also have applications to cohomological field theories [@PV2]. Besides these applications, matrix factorizations have connections with representation theory, Hodge theory and other topics; for more information see for example the references of the paper by Eisenbud and Peeva [@EiPe], where matrix factorizations are generalized to complete intersections. Description of results ====================== In this section we offer a detailed summary of the main results and techniques contained in this paper. The reader should be able to get a picture of the paper by only reading this section. Along this section we cross-reference each of the results so the reader may jump to the appropriate part of the paper for more details. We describe each of the sections of the paper, but first we need to set the terminology. Setting and notation -------------------- Throughout this article, unless otherwise stated, we denote by $X$ a Stein normal surface. By $(X,x)$ we denote either a complex analytic normal surface germ, or the spectrum of a normal complete ${{\mathbb{C}}}$-algebra of dimension $2$. As usual ${\mathcal{O}_{X}}$ denotes the structure sheaf and ${\mathcal{O}_{X,y}}$ is the local ring at a (non-necessarily closed) point $y\in X$. In this situation $X$ has a dualizing sheaf $\omega_X$, and by abuse of notation we denote also by $\omega_X$ its stalk at $x$, which is the dualizing module of the ring ${\mathcal{O}_{X,x}}$. If $X$ has Gorenstein singularities then the dualizing sheaf is an invertible sheaf. This means that if $(X,x)$ is Gorenstein then the dualizing module coincides with ${\mathcal{O}_{X,x}}$. Unless otherwise stated, we denote by $\pi \colon {\tilde{X}}\to X$ a resolution of singularities (in a few instances it will denote a proper modification from a normal origin). The exceptional set is denoted by $E:=\pi^{-1}(x)$, and its decomposition into irreducible components is $E=\cup_{i}E_i$. By a curvette we will mean a (multi)-germ of a curve centered at the exceptional divisor. If $(X,x)$ is Gorenstein, there is a Gorenstein $2$-form $\Omega_{{\tilde{X}}}$ which is meromorphic in ${\tilde{X}}$, and has neither zeros nor poles in ${\tilde{X}}\setminus E$; it is called the [*Gorenstein form*]{}. Let $div(\Omega_{{\tilde{X}}})=\sum q_iE_i$ be the divisor associated with the Gorenstein form. If $X$ is a Stein surface with Gorenstein singularities then there exist holomorphic $2$-forms $\Omega_{{\tilde{X}}}$ in ${\tilde{X}}\setminus E$ so that $div(\Omega_{{\tilde{X}}})=A+\sum q_iE_i$, where $A$ is disjoint from $E$. The coefficients $q_i$ are independent of the form $\Omega_{{\tilde{X}}}$ having these properties. Next, we extend to Stein normal surfaces some usual notions for singularities and define a new one: \[def:smallresgor\] Let $\pi \colon {\tilde{X}}\to X$ be a resolution either of a Stein normal surface with Gorenstein singularities, or of a Gorenstein surface singularity. The [*canoninal cycle*]{} is defined by $Z_k:=\sum_i -q_iE_i$, where the $q_i$ are the coefficients defined above. We say that ${\tilde{X}}$ is [*small with respect to the Gorenstein form*]{} if $Z_k$ is greater than or equal to $0$. The [*geometric genus*]{} of $X$ is defined to be the dimension as a $\mathbb{C}$-vector space of $R^1\pi_*{\mathcal{O}_{{\tilde{X}}}}$ for any resolution. \[rem:smallnotmodifiesreg\] If a resolution $\pi \colon {\tilde{X}}\to X$ of a Stein normal surface with Gorenstein singularities is small with respect to the Gorenstein form, then it is an isomorphism over the regular locus of $X$. We will only consider resolutions which are isomorphisms over the regular locus of $X$. Given $X$ a normal Stein surface, or $(X,x)$ a normal surface singularity germ, we denote by $i:U\to X$ the inclusion of $U:=X\setminus Sing(X)$ in $X$. The degeneracy module correspondences {#sec:introdegmodcorr} ------------------------------------- We study reflexive modules via resolutions. Let $X$ be a Stein normal surface and $\pi:{\tilde{X}}\to X$ be a resolution. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module. Its associated full ${\mathcal{O}_{{\tilde{X}}}}$-module is ${\mathcal{M}}:=(\pi^*M)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. A basic proposition of Kahn (Proposition \[fullcondiciones\]) characterizes full ${\mathcal{O}_{{\tilde{X}}}}$-modules and establishes a bijection between reflexive ${\mathcal{O}_{X}}$-modules and full ${\mathcal{O}_{{\tilde{X}}}}$-modules. In Section \[chapter:artin-verdier-esnault\] we improve and systematize Artin-Verdier’s first Chern class construction by proving that there is a bijective correspondence between reflexive ${\mathcal{O}_{X}}$-modules (respectively full ${\mathcal{O}_{{\tilde{X}}}}$-modules) of rank $r$ equipped with a system of $r$ sufficiently generic sections, and Cohen Macaulay ${\mathcal{O}_{X}}$-modules of dimension $1$ together with a system of $r$ generators (respectively dimension $1$ Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-modules with $r$ global sections generating as a ${\mathcal{O}_{{\tilde{X}}}}$-module). These correspondences are the main tool in establishing the main results of this paper; we describe them in detail right below. Let $\pi:Y\to X$ denote a proper birational map (which could be the identity or a resolution). Let ${\mathcal{M}}$ be a ${\mathcal{O}_{Y}}$-module which is locally free of rank $r$ and generated by global sections except at a finite subset of $Y$. The degeneracy ${\mathcal{O}_{Y}}$-module ${\mathcal{A}}$ (see Section \[sec:degeneracymodule\]) is defined to be the quotient of ${\mathcal{M}}$ by the submodule generated by the sections. When ${\mathcal{M}}$ is the sheaf of sections of a vector bundle the support of ${\mathcal{A}}$ represents the first Chern class. We prove that ${\mathcal{A}}$ is Cohen-Macaulay of dimension $1$, with support $A$ meeting the exceptional divisor at finitely many points, and such that ${\mathcal{A}}_y\cong{\mathcal{O}_{A,y}}$ for generic $y$. We call such a module a [*generically reduced Cohen-Macaulay modules of dimension $1$*]{}. A set of [*nearly generic sections*]{} is a set of $r$ sections so that its associated degeneracy module is generically reduced Cohen-Macaulay of dimension $1$. We prove in Section \[sec:CM1\] that such modules admit a double inclusion $${\mathcal{O}_{A}}\subset {\mathcal{A}}\subset {\mathcal{O}_{\tilde{A}}}$$ where $\tilde{A}$ is the normalization of $A$. Based on this inclusion we associate to ${\mathcal{A}}$ a numerical invariant $\mathfrak{A}$ called the [*set of orders*]{} to ${\mathcal{A}}$ which consists of a subset of ${{\mathbb{N}}}^l$ and is similar to the semigroup of a curve with $l$ branches. As in the case of curve semigroups the set $\mathfrak{A}$ has a [*minimal conductor*]{} element $cond(\mathfrak{A})$. When $Y={\tilde{X}}$ is a resolution of singularities, for any pair $({\mathcal{A}},(\psi_1,...,\psi_r))$ we define the [*Containment Condition*]{} (see Definition \[def:containment\]) which measures the interaction of $({\mathcal{A}},(\psi_1,...,\psi_r))$ with the canonical sheaf $\omega_{{\tilde{X}}}$. The following theorem is a precise statement of the correspondences announced above (see Theorems \[th:corrsing\] and \[th:corres\] in the main body of the paper) \[th:introcorr\] Let $\pi:{\tilde{X}}\to X$ be a resolution of a normal Stein surface. - If $X$ has Gorenstein singularities there is a bijection between a) the set of pairs $(M,(\phi_1,...,\phi_r))$ of reflexive ${\mathcal{O}_{X}}$-modules with a set of nearly generic sections and b) the set of pairs $({\mathcal{C}},(\psi_1,...,\psi_r))$ of generically reduced Cohen-Macaulay modules of dimension $1$ with $r$ generators. If the sections are generic the module $M$ has free factors if and only if the system of generators $(\psi_1,...,\psi_r)$ is not minimal. - There is a bijection between a) the set of pairs $({\mathcal{M}},(\phi_1,...,\phi_r))$ of full ${\mathcal{O}_{{\tilde{X}}}}$-modules with a set of nearly generic sections and b) the set of pairs $({\mathcal{A}},(\psi_1,...,\psi_r))$ of generically reduced Cohen-Macaulay modules of dimension $1$ with $r$ generators as a ${\mathcal{O}_{{\tilde{X}}}}$-module satisfying the Containment Condition. The key to proof of the theorem consists mainly of a cohomological analysis for which the previous study on the structure of the degeneracy modules plays a central role. As we will see later, the degeneracy modules are the crucial new ingredient that allows us to extend McKay correspondence techniques to general surface singularities. In Propositions \[prop:dirressing\] and \[prop:invressing\] we investigate the relation between the correspondences at $X$ and at a resolution, and the relation of the correspondences at different resolutions. It is not easy to handle the Containment Condition directly. In Section \[sec:valuative\] we introduce a numerical condition for $({\mathcal{A}},(\psi_1,...,\psi_r))$, called the [*Valuative Condition*]{} which requires the set of orders $\mathfrak{A}$ to be contained in another subset of ${{\mathbb{N}}}^l$, called the [*Canonical Set of Orders*]{}, which codifies the interaction of $({\mathcal{A}},(\psi_1,...,\psi_r))$ with the canonical sheaf $\omega_{{\tilde{X}}}$ (see Definition \[def:canoset2\]). In Proposition \[prop:contval\] we show that the Containment Condition implies the Valuative Condition. In Proposition \[prop:consecuenciaspracticas\] we prove that the converse is true in sufficiently many cases, that cover all the needs of this paper. The structure of reflexive modules via the degeneracy module correspondences {#sec:introstruct} ---------------------------------------------------------------------------- Each reflexive module over a singular surface has a favorite resolution of singularities (for rational singularities it coincides with the minimal resolution). Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$ over a normal Stein surface $X$. The [*minimal adapted resolution*]{} of $M$ is the minimal resolution of singularities $\pi:{\tilde{X}}\to X$ such that the full ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}$ associated with $M$ is generated by global sections. In Proposition \[prop:minadap\] we prove its existence and uniqueness. In Proposition \[prop:minadapnumchar\] we characterize numerically the minimal adapted resolution of a reflexive module in terms of the minimal conductor of the Canonical Set of Orders of the pair $({\mathcal{A}},(\psi_1,...,\psi_r))$, obtained by applying the correspondence of Theorem \[th:introcorr\] to the pair $({\mathcal{M}},(\phi_1,...,\phi_r))$ formed by the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated with $M$ and a system of generic sections. Similar to Wunram’s generalization of McKay correspondence we obtain the most detailed results for special reflexive modules. The right generalization of special module is provided in Definitions \[def:especial\] and \[def:espmodule\]. Let $\pi:{\tilde{X}}\to X$ be a resolution of a normal Stein surface, $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$ and ${\mathcal{M}}$ its associated full ${\mathcal{O}_{{\tilde{X}}}}$-module. We say that ${\mathcal{M}}$ is a [*special full sheaf*]{} if we have the equality ${\dim_{\mathbb{C}}(R^1\pi_*\left({\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right))}=rp_g$. The [*specialty defect*]{} of ${\mathcal{M}}$ is the number $d({\mathcal{M}}):={\dim_{\mathbb{C}}(R^1\pi_*\left({\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right))}-rp_g$. We prove that the specialty defect is non-negative. We say $M$ to be a [*special reflexive*]{} module if for any resolution of singularities of $X$ its associated full sheaf is special. We study the behavior of the specialty defect under dominant maps $\sigma:{\tilde{X}}_2\to{\tilde{X}}_1$ between resolutions, and various properties of special modules in Sections \[sec:minimaladapted\] and \[sec:indecomposables\]. The main results are: 1. the specialty defect of the full ${\mathcal{O}_{{\tilde{X}}_2}}$-module ${\mathcal{M}}_2$ associated to $M$ is greater than or equal to the specialty defect of the full ${\mathcal{O}_{{\tilde{X}}_1}}$-module ${\mathcal{M}}_1$ associated to $M$. We have the equality if $\rho$ is an isomorphism over the locus where ${\mathcal{M}}_1$ is not generated by global sections (see Proposition \[prop:specialtybehaviour\]). 2. Given a reflexive ${\mathcal{O}_{X}}$-module, if at the minimal adapted resolution the specialty defect of the full sheaf associated to $M$ vanishes, then it vanishes for any resolution (see Theorem \[th:characspecial\]). 3. Assume that the Stein normal surface $X$ has Gorenstein singularities. Let $({\mathcal{C}},(\psi_1,...,\psi_r))$ be the pair associated with $(M,(\phi_1,...,\phi_r))$ by the correspondence of Theorem \[th:introcorr\]. If $M$ is a special ${\mathcal{O}_{X}}$-reflexive module over $X$ and $(\phi_1,...,\phi_r)$ are generic sections, then ${\mathcal{C}}$ is isomorphic to $n_*{\mathcal{O}_{\tilde{C}}}$, where $n:\tilde{C}\to C$ is the normalization of the support of ${\mathcal{C}}$. The converse is true (see Proposition \[prop:Aeslanormalizacion\]). 4. Assume now that $(X,x)$ is a normal Gorenstein surface singularity and that $M$ is special without free factors. Let $C$ be like in the previous property. There is a bijection between the indecomposable direct summands of $M$ and the irreducible components of $C$. Property (3) gives a geometric understanding of specialty under the degeneracy module correspondence, and also allows to produce full sheaves with prescribed Chern classes. If $(X,x)$ is a germ, the normalization map $n:\tilde{C}\to C$ is an arc at the singularity if $C$ is irreducible. Hence Property (3) also establishes a link between arc spaces and reflexive modules. Property (4) is important because it shows how the degeneracy module correspondence recovers the decomposition into indecomposables in a very geometric way. Properties (3) and (4) are false for non-special modules. A crucial tool in the study of reflexive modules is the computation of the first cohomology of full sheaves (see Theorem \[formuladimensionM\]): \[th:introformuladimensionM\] Let $X$ be a Stein normal surface with Gorenstein singularities. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $\pi:{\tilde{X}}\to X$ be a small resolution with respect to the Gorenstein form, let $Z_k$ be the canonical cycle at ${\tilde{X}}$. Let ${\mathcal{M}}$ be the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M$. Let $d({\mathcal{M}})$ be the specialty defect of ${\mathcal{M}}$. Then we have the equality $${\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{M}})} = rp_g - [c_1({\mathcal{M}})] \cdot [Z_k] + d({\mathcal{M}}).$$ The proof occupies the whole Section \[sec:cohomology\]. An interesting Corollary is the fact that the minimal adapted resolution $\pi:{\tilde{X}}\to X$ of a special reflexive ${\mathcal{O}_{X}}$-module $M$ over a Stein normal surface with Gorenstein singularities is characterized by the fact that ${\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{M}})} = rp_g$, where ${\mathcal{M}}$ is the associated full ${\mathcal{O}_{{\tilde{X}}}}$-module (see Corollary \[cor:dimMadap\]). This has the interesting consequence that the cycle representing first Chern class of ${\mathcal{M}}$ does not meet the support of the canonical cycle $Z_K$. It also shows that ${\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{M}})} = rp_g$ can be used as an invariant controlling the minimal adapted resolution process. The above tools allow us to prove some of the main results of the paper. The first is a determination of special reflexive modules in terms of a first Chern class (Theorem \[th:Chernresolucionadapted\]): \[th:introChernresolucionadapted\] Let $X$ be a Stein normal surface with Gorenstein singularities. Let $M$ be a special ${\mathcal{O}_{X}}$-module without free factors. Let $\pi:{\tilde{X}}\to X$ be the minimal resolution adapted to $M$, and ${\mathcal{M}}$ the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M$. The module ${\mathcal{M}}$ (and equivalently $M$) is determined by its first Chern class in $\text{Pic}({\tilde{X}})$. The classification of reflexive modules {#sec:introclass} --------------------------------------- In Section \[sec:combclass\] we provide a combinatorial classification of reflexive modules. Given a normal surface singularity $(X,x)$ and a reflexive ${\mathcal{O}_{X}}$-module $M$, we define its associated graph ${\mathcal{G}}_M$ to be the dual graph of the minimal good resolution dominating the minimal adapted resolution to $M$, decorated adding as many arrows to each of its vertices $v$ as the number $c_1({\mathcal{M}})(E_v)$, where $E_v$ is the component of the exceptional divisor corresponding to $v$ and $c_1({\mathcal{M}})$ is the first Chern class of the associated full ${\mathcal{M}}$-module. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be the pair associated with $({\mathcal{M}},(\phi_1,...,\phi_r))$ under the correspondence of Theorem \[th:introcorr\], where $(\phi_1,...,\phi_r)$ are generic sections. Proposition \[prop:minadapspproperty\] shows that the support of ${\mathcal{A}}$ is a disjoint union of as many smooth curvettes as arrows has ${\mathcal{G}}_M$, each of them meeting transversely the irreducible component of the exceptional divisor corresponding to the vertex where the arrow is attached. In Theorem \[th:charresgraphsp\] we characterize combinatorially the graphs of special modules over Gorenstein surface singularities. We prove that these are precisely the graphs such that 1. the graph is numerically Gorenstein. 2. if a vertex has genus $0$, self intersection $-1$ and has at most two neighboring vertices, then it supports at least $1$ arrow. 3. if a vertex supports arrows then its coefficient in the canonical cycle equals $0$. However, a much stronger classification result is the following one (Corollary \[Cor:finalprincipal\] in the body of the paper). Given a normal surface singularity a [*irreducible divisor over*]{} $x$ is the same that a divisorial valuation of the function field of $X$ centered at $x$. An irreducible divisor over $x$ [*appears at*]{} a model $\pi:{\tilde{X}}\to X$ if its center at ${\tilde{X}}$ is a divisor. \[th:introfinalprincipal\] Let $(X,x)$ be a Gorenstein surface singularity. Then there exists a bijection between the following sets: 1. The set of special indecomposable reflexive ${\mathcal{O}_{X}}$-modules up to isomorphism. 2. The set of irreducible divisors $E$ over $x$, such at any resolution of $X$ where $E$ appears, the Gorenstein form has neither zeros nor poles along $E$. This theorem specializes to classical Mckay correspondence in the case of rational double points. By taking direct sums one obtains a full classification of special reflexive modules over Gorenstein surface singularities. Deformations and families {#def:introdefs_fam} ------------------------- Having studied reflexive modules as individual objects we turn to deformations and moduli questions for the rest of the paper. Here we work as generally as possible: we allow deformations of the underlying space when we deform reflexive modules; when we deform full sheaves we allow simultaneous deformation of the space and of the resolution. In Section \[sec:deffunctors\] we define the relevant deformation functors, morphisms between them and establish the existence of versal deformations. Let $X$ be a Stein normal surface and $M$ be a reflexive ${\mathcal{O}_{X}}$-module. A deformation of $(X,M)$ over a base $(S,s)$ consists of a flat deformation ${\mathcal{X}}$ of the $X$ over $(S,s)$ together with a ${\mathcal{O}_{{\mathcal{X}}}}$-module which is flat over $S$ and whose fibre over $s$ is isomorphic to $M$ (see Definition \[def:deforfam\]). Since reflexivity is an open property in flat families this is an adequate notion of deformations of reflexive sheaves. This definition leads to a deformation functor $\mathbf{Def_{X,M}}$. The functor of deformations fixing the base space $X$ is a sub-functor. On the other hand fullness is not an open property in flat families. Therefore the right definition of the deformation functor of full sheaves needs a cohomological condition: let $X$ and $M$ as before. Let $\pi:{\tilde{X}}\to X$ be a resolution and ${\mathcal{M}}$ the full ${\mathcal{O}_{{\tilde{X}}}}$ module associated with $M$. A deformation of $({\tilde{X}},X,{\mathcal{M}})$ is formed by a very weak simultaneous resolution $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ of a flat deformation ${\mathcal{X}}$ of $X$ over $S$, and a ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-module $\overline{{\mathcal{M}}}$ which is flat over $S$, whose specialization over $s$ is isomorphic to ${\mathcal{M}}$ and such that $R^1\Pi_*\overline{{\mathcal{M}}}$ is flat over $S$ (see Definition \[def:deformationfull\]). This leads to a deformation functor $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$. One has subfunctors fixing the underlying space and/or the resolution. In Proposition \[prop:fullopen\] it is shown that fullness is an open property in families defined as above; the proof uses the flatness of $R^1\Pi_*\overline{{\mathcal{M}}}$ in a crucial way. This shows that our definition is the correct notion of deformation within the category of full sheaves. In Proposition \[prop:naturaltrans\] we show that the push-forward functor $\Pi_*$ defines a natural transformation from $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$ to $\mathbf{Def_{X,M}}$. Since Kahn’s result (Proposition \[fullcondiciones\]) establishes a bijection between reflexive ${\mathcal{O}_{X}}$-modules and full ${\mathcal{O}_{{\tilde{X}}}}$-modules, and this bijection is via the push-forward functor, one could naively expect that this $\Pi_*$ is an isomorphism of functors. This is not the case as we will see below. Analyzing the functors $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$ and $\mathbf{Def_{X,M}}$, and the transformation $\Pi_*$ directly seems a difficult task. The extension to deformation functor isomorphisms of the correspondences of Theorem \[th:introcorr\] is the crucial tool in our subsequent analysis. The correspondences as isomorphisms of deformation functors {#def:isodef} ----------------------------------------------------------- First we enrich the functors $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$ and $\mathbf{Def_{X,M}}$ and define deformation functors $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$. Given $X$ and $M$ as above, and $(\phi_1,...,\phi_r)$ a set of $r=rank(M)$ nearly generic sections, the deformation functor $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$ associates to $(S,s)$ a deformation $({\mathcal{X}},\overline{M})$ in $\mathbf{Def_{X,M}}(S,s)$ along with a set of sections $(\overline{\phi}_1,...,\overline{\phi}_r)$ extending $(\phi_1,...,\phi_r)$ (see Definition \[def:enhanceddef1\]). The definition of $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ is similar (see Definition \[def:enhanceddef2\]). There are obvious forgetful functors from $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$ to $\mathbf{Def_{X,M}}$, and from $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ to $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$. In order to be able to prove an analog of Theorem \[th:introcorr\] for deformations we need deformation functors of generically reduced $1$-dimensional Cohen-Macaulay modules together with sets of generators. The relevant definitions are the following (see Definitions \[def:enhanceddef1\] and \[def:enhanceddef2\]): let $X$ be as above, let $({\mathcal{C}},(\psi_1,...,\psi_r))$ be a generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{X}}$-module, together with a system of generators as a ${\mathcal{O}_{X}}$-module. A deformation of $(X,{\mathcal{C}},(\psi_1,...,\psi_r))$ over a germ $(S,s)$ consists of a flat deformation ${\mathcal{X}}$ of the space $X$ over $(S,s)$, a ${\mathcal{O}_{{\mathcal{X}}}}$-module $\overline{{\mathcal{C}}}$ which is flat over $S$ and specializes to ${\mathcal{C}}$ over $s$, and a set of sections $(\overline{\psi}_1,...,\overline{\psi}_r)$ of $\overline{{\mathcal{C}}}$ which specialize to $(\psi_1,...,\psi_r)$ over $s$. The resulting deformation functor is denoted by $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$. Like in the case of deformations of full sheaves the straightforward generalization of this functor to the case of resolutions does not work; we need to add a further condition to the families, that in a certain sense is the analog of the flatness condition of $R^1\Pi_*\overline{{\mathcal{M}}}$ in the case of deformations of full sheaves. Let $\pi:{\tilde{X}}\to X$ be a resolution of singularities and ${\mathcal{A}}$ be a $1$-dimensional generically reduced Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module whose support meets the exceptional divisor at finitely many points. Let $(\psi_1,...,\psi_r)$ be a set of global sections of ${\mathcal{A}}$ generating it as a ${\mathcal{O}_{{\tilde{X}}}}$-module. A [*specialty defect constant deformation of*]{} $({\mathcal{A}},(\psi_1,...,\psi_r))$ over a germ $(S,s)$ consists of a very weak simultaneous resolution $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ of a flat deformation of $X$ over $(S,s)$, a ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-module $\overline{{\mathcal{A}}}$ which is flat over $S$ and specializes to ${\mathcal{A}}$ over $s$, and a set of sections $(\overline{\psi}_1,...,\overline{\psi}_r)$ which specialize to $(\psi_1,...,\psi_r)$ over $s$, and which are so that the cokernel $\overline{{\mathcal{D}}}$ of the natural mapping $\Pi_*{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to \Pi_*\overline{{\mathcal{A}}}$ induced by the sections is flat over $S$. The resulting deformation functor is denoted by $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$. The functor $\Pi_*$ defines a natural transformation from $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$ to $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$, where ${\mathcal{C}}$ is the submodule of $\pi_*{\mathcal{A}}$ generated by $(\psi_1,...,\psi_r)$. The following is our main tool in studying deformation and moduli functors (see Theorems \[th:dirXdef\] and \[th:dirresdef\] for a more precise version). \[th:introdefcorr\] Let $\pi:{\tilde{X}}\to X$ be a resolution of a Stein normal surface with Gorenstein singularities. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$ and ${\mathcal{M}}$ be the associated full ${\mathcal{O}_{{\tilde{X}}}}$-module. 1. Let $(\phi_1,...,\phi_r)$ be nearly generic sections of $M$ , let $({\mathcal{C}},(\psi_1,...,\psi_r))$ be the pair associated with $(M,(\phi_1,...,\phi_r))$ under the correspondence of Theorem \[th:introcorr\]. There is an isomorphism between the functors $\mathbf{Def_{X,M}}^{(\phi_1,...,\phi_r)}$ and $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$. 2. Let $(\phi_1,...,\phi_r)$ be nearly generic sections of ${\mathcal{M}}$, let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be the pair associated with $({\mathcal{M}},(\phi_1,...,\phi_r))$ under the correspondence of Theorem \[th:introcorr\]. There is an isomorphism between the functors $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}^{(\phi_1,...,\phi_r)}$ and $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$. The proof of this Theorem gets quite technical and occupies several pages of the paper, but its subsequent applications makes the effort worthwhile. In Propositions \[prop:defdirressing\] and \[prop:comparecorrdef\] we explain the behavior of the correspondences of Theorem \[th:introdefcorr\] under the functor $\Pi_*$. An important corollary of this theorem is that the specialty defect remains constant in a deformation of $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$ (see Corollary \[cor:specialtydefectconstant\]). In Example \[ex:liftsnotlifts\] we give an example of deformation of a special reflexive module such that the generic member of the family is not special. As a consequence we produce a deformation which does not lift to the minimal resolution. This shows that $\Pi_*$ does not induce an isomorphism of functors from $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$ to $\mathbf{Def_{X,M}}$. In the previous example, the reason for which the natural transformation of functors $\Pi_*:\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}\to\mathbf{Def_{X,M}}$ is not an isomorphism is that, in general, deformations in $\mathbf{Def_{X,M}}(S,s)$ do not lift to ${\tilde{{\mathcal{X}}}}$. This motivates Section \[sec:liftingdefs\], in which we study systematically the liftability problem for families using the correspondences of Theorem \[th:introdefcorr\]. Let $({\mathcal{X}},\overline{M})$ be an element in $\mathbf{Def_{X,M}}(S,s)$. Let $(\overline{\phi}_1,...,\overline{\phi}_r)$ be $r=rank(M)$ sections of $\overline{M}$ which specialize to nearly generic sections over $s$. Let $({\mathcal{X}},\overline{{\mathcal{C}}},(\psi_1,...,\psi_r))$ be the result of applying the correspondence of Theorem \[th:introdefcorr\] to $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r))$. Let $\overline{C}$ be the support of $\overline{{\mathcal{C}}}$. We say that $C$ [*lifts to*]{} ${\tilde{{\mathcal{X}}}}$ if the fibre over $s$ of the strict transform of $C$ by $\Pi$ coincides with the strict transform by $\pi$ of the fibre of $C$ over $s$. This notion is introduced at Definition \[def:liftlocus\], where also the notion of liftability for $({\mathcal{X}},\overline{{\mathcal{C}}},(\psi_1,...,\psi_r))$ is defined. Assume that $(S,s)$ is a reduced base. In Proposition \[prop:necessarylifting\] we prove that there is a deformation in $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}(S,s)$ which transforms under $\Pi_*$ to $({\mathcal{X}},\overline{M})$ if and only if $({\mathcal{X}},\overline{{\mathcal{C}}},(\psi_1,...,\psi_r))$ lifts to ${\tilde{{\mathcal{X}}}}$ according with Definition \[def:liftlocus\]. Moreover this implies that the support $\overline{C}$ lifts to ${\tilde{{\mathcal{X}}}}$. In Example \[ex:nonlifting\] we exhibit a deformation in $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}(S,s)$ that does not lift to ${\tilde{{\mathcal{X}}}}$ because the support $\overline{C}$ does not lift. In Example \[ex:liftsnotlifts\] we find a deformation in $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}(S,s)$ that does not lift to ${\tilde{{\mathcal{X}}}}$ because $({\mathcal{X}},\overline{{\mathcal{C}}},(\psi_1,...,\psi_r))$ does not lift to ${\tilde{{\mathcal{X}}}}$, even if the support $\overline{C}$ does lift. In Proposition \[prop:genericlifting\] we prove that for any deformation over a reduced base $(S,s)$ there exists a Zariski dense open subset on $(S,s)$ over which the deformation lifts to a full family. For our later applications we need a sufficient condition for lifting of deformations in $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}(S,s)$ that is easier to handle than the liftability of $({\mathcal{X}},\overline{{\mathcal{C}}},(\psi_1,...,\psi_r))$ predicted in Proposition \[prop:necessarylifting\]. This is worked out in Section \[sec:suffcondlift\], where it is proved that under certain conditions the liftability of the support $\overline{C}$ is enough. In Definition \[def:deltaconstant\] we introduce the notion of simultaneously normalizable deformations of reflexive modules: let $X$ be a normal Stein surface, let ${\mathcal{X}}$ be a deformation of $X$ over a reduced base $(S,s)$, let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. A deformation $({\mathcal{X}},\overline{M})$ of $(X,M)$ over a reduced base $(S,s)$ is said to be [*simultaneously normalizable*]{} if the degeneracy locus $\overline{C}$ of $\overline{M}$ for a generic system of $r$ sections admits a simultaneous normalization over $S$. Using this definition we prove (see Theorem \[th:sufficientlifting\]): \[th:introsufficientlifting\] Let $X$ be a normal Gorenstein surface singularity. Let ${\mathcal{X}}$ be a deformation of $X$ over a normal base $(S,s)$. Let $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ be a very weak simultaneous resolution. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module and $({\mathcal{X}},\overline{M})$ be a simultaneously normalizable deformation of $(X,M)$ over the base $(S,s)$, so that for each $s'\in S$ the module $\overline{M}|_{s'}$ is special. If the support of the degeneracy module of $\overline{M}$ for a generic system of sections is liftable to ${\tilde{{\mathcal{X}}}}$, then the family $({\mathcal{X}},\overline{M},\iota)$ lifts to a full family on ${\tilde{{\mathcal{X}}}}$. Moduli spaces of reflexive modules, Cohen-Macaulay representation types {#sec:introapplications} ----------------------------------------------------------------------- Now we describe two applications of the machinery developed up to now. The first application appears in Section \[sec:fintamewild\] and confirms a conjecture of Drodz, Greuel and Kashuba [@DrGrKa] and, together with previous work in [@AV] and [@DrGrKa] completes the classification of Gorenstein normal surface singularities in Cohen-Macaulay representation types. Let us recall that a surface singularity $(X,x)$ is of finite, tame or wild Cohen-Macaulay representation type if there are at most finite, $1$-dimensional or unbounded dimensional families of indecomposable Maximal Cohen-Macaulay ${\mathcal{O}_{X,x}}$-modules respectively. Here we prove (see Theorem \[th:reptype\]). A Gorenstein surface singularity is of finite Cohen-Macaulay representation type if and only if it is a rational double point. Gorenstein surface singularities of tame Cohen-Macaulay representation type are precisely the log-canonical ones. The remaining Gorenstein surface singularities are of wild Cohen-Macaulay representation type. The second application is the construction of fine moduli spaces of special modules without free factors of prescribed graph and rank on Gorenstein normal surface singularities. This enhances the classification Theorem \[th:introfinalprincipal\]. It is provided in Section \[sec:moduli\]. Let ${\mathcal{G}}$ be the graph of a special reflexive ${\mathcal{O}_{X}}$-module on a Gorenstein normal surface singularity $X$. Ler $r$ be a positive integer. In Definition \[def:modulifunctor\] a moduli functor $\mathbf{Mod_{{\mathcal{G}}}^r}$ is defined in a similar way as the deformation functors above. It parametrizes flat families of special reflexive modules without free factors of rank $r$ and graph ${\mathcal{G}}$, over normal base spaces. The main result is Theorem \[theo:moduli\]: The functor $\mathbf{Mod_{{\mathcal{G}}}^r}$ is represented by an algebraic variety. Moreover in its proof we see that the variety representing the functor has a very nice geometric description: it parametrizes sequences of infinitely near points to $x$ in the singularity $X$ with a given combinatorial type. Reflexive modules and full sheaves ================================== See [@BrHe], [@Har1] and  [@Ne] as basic references on dualizing sheaves, modules and normal surface singularities. Cohen-Macaulay modules and reflexive modules -------------------------------------------- Let $X$ be a normal surface along this section. Let $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{X}}}(\bullet,\bullet)$ and by $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^i_{{\mathcal{O}_{X}}}(\bullet,\bullet)$ the sheaf theoretic $Hom$ and $Ext$ functors. The dual of a ${\mathcal{O}_{X}}$-module $M$ is $M^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}:=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{X}}}(M,{\mathcal{O}_{X}})$. The $\omega_X$-dual is $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{X}}}(M,\omega_X)$. A module ${\mathcal{O}_{X}}$-module $M$ is called *reflexive* if the natural homomorphism from $M$ to $M^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ is an isomorphism. It is called $\omega_X$-reflexive if the natural map $M\to\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{X}}}(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{X}}}(M,\omega_X),\omega_X)$ is an isomorphism. A ${\mathcal{O}_{X}}$-module $M$ is called *Cohen-Macaulay* if the depth of each of its stalks $M_y$ is equal to the dimension of the module. If the depth of $M_y$ is equal to the dimension of ${\mathcal{O}_{X,y}}$, then the module $M_y$ is called *maximal Cohen-Macaulay*; this definitions extend to sheaves if we ask that they hold for every stalk. If $M_x$ is Cohen-Macaulay then $M$ is Cohen-Macaulay at a neighborhood of $x$. A module is [*indecomposable*]{} if it can not be written as a direct sum of two non trivial submodules. By [@Yos Proposition 1.5] and [@Har3 Section 1] some basic properties of maximal Cohen-Macaulay modules are: 1. If ${\mathcal{O}_{X,y}}$ is a regular local ring, then any maximal Cohen-Macaulay module over it is free. 2. If ${\mathcal{O}_{X,y}}$ is a reduced local ring of dimension one, then an ${\mathcal{O}_{X,y}}$-module $M$ is maximal Cohen-Macaulay if and only if it is torsion free, that is, when the natural homomorphism $M \to M^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ is a monomorphism. 3. If ${\mathcal{O}_{X,y}}$ is normal of dimension two, then an ${\mathcal{O}_{X,y}}$-module $M$ is maximal Cohen-Macaulay if and only if it is reflexive. 4. A consequence of the previous properties is that reflexive sheaves over regular rings ${\mathcal{O}_{X,y}}$ of dimension at most $2$ are free. 5. Let $M$ be a ${\mathcal{O}_{X}}$-module. Then $M_x$ is a reflexive ${\mathcal{O}_{X,x}}$-module if and only if the adjunction morphism $M\to i_*i^*M$ is an isomorphism. 6. If a ${\mathcal{O}_{X}}$-module is reflexive at $x$, then it is reflexive at an open neighbourhood of $x$ in $X$. The canonical module and Cohen-Macaulay modules have the following properties, which are a special case of [@BrHe Theorem 3.3.10]. \[Th:Herzog\] Let $X$ be a normal surface. For $t=0,1,2$ and all Cohen-Macaulay ${\mathcal{O}_{X}}$-modules $M$ of dimension $t$ one has 1. $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^{2-t}(M,\omega_X)$ is Cohen-Macaulay of dimension t, 2. $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^{i}(M,\omega_X)=0$ for all $i \neq 2-t$, 3. there exists an isomorphism $M \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^{2-t}\left(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^{2-t}(M,\omega_X),\omega_X\right)$ which in the case $t=2$ is just the natural homomorphism from $M$ into the $\omega_X$-bidual of $M$. A consequence of the previous Theorem and Property (3) above is that $\omega_X$-reflexivity is equivalent to reflexivity. The following proposition will be useful: \[prop:extCM1\] Let $X$ be a normal surface. Let ${\mathcal{A}}$ be a $1$-dimensional ${\mathcal{O}_{X}}$-module. Then the ${\mathcal{O}_{X}}$-module $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{X}}}({\mathcal{A}},\omega_X)$ is Cohen-Macaulay of dimension $1$. Consider the exact sequence $0\to{\mathcal{B}}\to{\mathcal{A}}\to{\mathcal{A}}'\to 0$, where ${\mathcal{B}}$ is the submodule of elements with support at $x$. No element of ${\mathcal{A'}}$ has support at $x$, and hence ${\mathcal{A}}'$ is Cohen-Macaulay of dimension $1$. Applying $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}(\bullet,\omega_X)$ and considering the associated exact sequence we obtain the isomorphism $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{X}}}({\mathcal{A}},\omega_X)\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{X}}}({\mathcal{A}}',\omega_X)$, which implies the result by the first assertion of Theorem \[Th:Herzog\]. Full sheaves ------------ We are interested in studying the reflexive modules on Stein normal surfaces, or in normal surface singularities, via a resolution. For this we will use the notion of full sheaves, introduced by Esnault [@Es] for rational surface singularities and generalized by Kahn [@Ka]. Along this section let $X$ be a Stein normal surface and $\pi:{\tilde{X}}\to X$ be a resolution. All results of this section are valid replacing $x$ by $(X,x)$, which is either a germ of normal surface singularity or the spectrum of a normal complete ${{\mathbb{C}}}$-algebra of dimension $2$. A ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}$ is called *full* if there is a reflexive ${\mathcal{O}_{X}}$-module $M$ such that ${\mathcal{M}} \cong \left(\pi^* M\right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. A ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}$ is [*generically generated by global sections*]{} if it is generated by global sections except in a finite set. \[fullcondiciones\] A locally free sheaf ${\mathcal{M}}$ on ${\tilde{X}}$ is full if and only if 1. ${\mathcal{M}}$ is generically generated by global sections. 2. The natural map $H^1_E({\tilde{X}},{\mathcal{M}}) \to H^1({\tilde{X}},{\mathcal{M}})$ is injective. If ${\mathcal{M}}$ is the full sheaf associated to $M$, then $\pi_* {\mathcal{M}}=M$. Kahn’s proof is for singularities. The proof for Stein normal surfaces is the same if one uses that a ${\mathcal{O}_{X}}$-module on a Stein space is generated by global sections. The last assertion is implicit in Kahn’s proof, and it gives us a natural bijection between reflexive ${\mathcal{O}_{X}}$-modules and full ${\mathcal{O}_{{\tilde{X}}}}$-modules. The following two lemmas that will be used later. \[lema:ceroggsg\] If ${\mathcal{M}}$ is a full ${\mathcal{O}_{{\tilde{X}}}}$-module, then $R^1 \pi_* ({\mathcal{M}} \otimes {\omega_{{\tilde{X}}}})=0$. If ${\mathcal{M}}$ is generated by global sections, Grauert-Riemenschneider Vanishing Theorem implies that $R^1 \pi_* \left( {\mathcal{M}} \otimes {\omega_{{\tilde{X}}}} \right)$ is equal to zero. If ${\mathcal{M}}$ is almost generated by global sections, consider ${\mathcal{M}}'$ the subsheaf of ${\mathcal{M}}$ generated by global sections, therefore we get the exact sequence $0 \to {\mathcal{M}}' \to {\mathcal{M}} \to {\mathcal{G}} \to 0$, with $\text{Supp}({\mathcal{G}})$ zero dimensional. Applying the functor $-\otimes {\omega_{{\tilde{X}}}}$ to the previous exact sequence, we get the desired vanishing via the long exact sequence of the the functor $\pi_* -$. \[lema:dualM\] If ${\mathcal{M}}$ is a full sheaf, then $\pi_* \left({\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right) = \left( \pi_* {\mathcal{M}} \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. Consider the following cohomology exact sequence $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { 0 & H_{E}^{0}\left ( {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) & H^{0}\left ( {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) & H^{0}\left (U, {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) & \\ & H_{E}^{1}\left ( {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) & H^{1}\left ( {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) & H^{1}\left (U, {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) & \dots \\ }; \foreach \y [remember=\y as \lasty (initially 1)] in {1, 2} { \foreach \x [remember=\x as \lastx (initially 2)] in {3,...,4} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); } } \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-2-4) -- (m-2-5); \draw[densely dotted,-stealth] (m-1-4) to [out=355, in=175] (m-2-2); \end{tikzpicture}$$ Since ${\mathcal{M}}$ is locally free we have that $H_{E}^{0}\left ( {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) = 0$, $H_{E}^{1}\left ( {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) \cong H^{1}\left ( {\mathcal{M}} \otimes {\omega_{{\tilde{X}}}} \right)$, by Serre duality. By Lemma \[lema:ceroggsg\] we get $H^{1}\left ( {\mathcal{M}} \otimes {\omega_{{\tilde{X}}}} \right) = 0$. Hence $\pi_*({\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}})=H^{0}\left ( {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right) \cong H^{0}\left (U, {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right)$. Now denote by $M:= \pi_* {\mathcal{M}}$. Since $M^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ is reflexive we get the equalities $M^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} = i_{*} i^{*} \left(M^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right) = i_{*}\left({\mathcal{M}}_{|_U}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right) = H^{0}\left (U, {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right)$. Therefore we have the isomorphism $M^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \cong \pi_* \left({\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right)$. Another notion that will be important in this work is the concept of specialty. Previously Wunram [@Wu] and Riemenschneider [@Rie] defined a special full sheaf as a full sheaf which its dual has the first cohomology group is equal to zero. Using this definition Wunram generalized McKay correspondence in the following sense: he proved that in the case of a rational surface singularity and taking the minimal resolution, there is a bijection between isomorphism classes of special full sheaves and irreducible components of the exceptional divisor. Their specialty notion is adapted to the case of rational singularities. For us the definition of special is as follows. \[def:especial\] A full ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}$ on ${\tilde{X}}$ of rank $r$ is called *special* if ${\dim_{\mathbb{C}}(R^1 \pi_* \left({\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right))} = rp_g$. *The defect of specialty of ${\mathcal{M}}$* is the number ${\dim_{\mathbb{C}}(R^1 \pi_* \left( {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right))}-rp_g$. Notice that this definition is a generalization of the concept given by Wunram and Riemenschneider and both definitions coincide in the case of a rational singularity. Since the definition of being special depends on the resolution, we have a another related notion. \[def:espmodule\] Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module. We say that $M$ is *a special module* if for any resolution the full sheaf associated to $M$ is special. Later it will become clear the importance of these concepts. Enhancing the Chern class and the degeneracy modules correspondences {#chapter:artin-verdier-esnault} ==================================================================== In [@AV] Artin and Verdier interpret geometrically McKay correspondence as follows: given an indecomposable reflexive ${\mathcal{O}_{X}}$-module, with $X$ a rational double point, they assign to it the first Chern class of the bi-dual of its pull-back to the minimal resolution. It turns out that the Chern class determines the module and that it hits precisely the exceptional divisor that McKay correspondence associates to the module. If $X$ is not a rational double point the fisrt Chern class does not determine the module [@Es]. In this section we refine Artin-Verdier construction in the following sense. The first Chern class may be constructed as the degeneracy locus of a set of as many generic sections as the rank of the module. Here we, to the same set of sections, we associate a [*degeneracy module*]{}, which is a Cohen-Macaulay module of dimension $1$ that whose support is the degeneracy locus. This refined correspondence is one of the main tools in our study of reflexive modules. Degeneracy modules of vector bundles {#sec:degeneracymodule} ------------------------------------ In this section we refine the construction of the first Chern class of a vector bundle as explained above. In order to avoid introducing more notation we only work in the generality needed in this paper, but the construction apply to more situations. Let $X$ be a normal Stein surface and $\pi:{\tilde{X}}\to X$ a proper birational map from a normal space ${\tilde{X}}$ ([*not necessarily a resolution, for example $\pi$ may be the identity map*]{}). Let $E:=\pi^{-1}(x)$. Let ${\mathcal{M}}$ be a reflexive ${\mathcal{O}_{{\tilde{X}}}}$-module that is generically generated by global sections. Let $S\subset E$ be the finite subset which is the union of the singular locus $Sing({\tilde{X}})$ and the locus where ${\mathcal{M}}$ is not generated by global sections. Denote by $M:=\pi_*{\mathcal{M}}$ the ${\mathcal{O}_{X}}$-module of global sections of ${\mathcal{M}}$. Suppose that $\text{rank}({\mathcal{M}})=r$ and take $\phi_1, \dots, \phi_r$ generic sections. Consider the exact sequence given by the sections $$\label{exctseq:directaM} 0 \to {\mathcal{O}_{{\tilde{X}}}}^r \stackrel{(\phi_1,...,\phi_r)}{\longrightarrow} {\mathcal{M}} \to {\mathcal{A}}' \to 0.$$ \[def:degeneracymodule\] Given ${\mathcal{M}}$ a reflexive ${\mathcal{O}_{{\tilde{X}}}}$-module of rank $r$ as above and $(\phi_1, \dots, \phi_r)$ a set of $r$ sections, the ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{A}}'$ defined by the previous exact sequence is called the [*degeneracy module*]{} of ${\mathcal{M}}$ associated with the given sections. The sections are called [*weakly generic*]{} if the support of the degeneracy module is a proper closed subset. \[prop:generalizationAV0\] Given ${\mathcal{M}}$ a ${\mathcal{O}_{{\tilde{X}}}}$-module of rank $r$ as above and a weakly generic set of $r$ sections, the associated degeneracy module is Cohen-Macaulay. Dualize the exact sequence (\[exctseq:directaM\]) with respect to a dualizing sheaf $\omega_{{\tilde{X}}}$ (which exists since the surface ${\tilde{X}}$ is normal, and hence Cohen-Macaulay) to obtain $$0 \to \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\tilde{X}}}({\mathcal{M}},\omega_{{\tilde{X}}}) \to \omega_{{\tilde{X}}}^r \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}', \omega_{{\tilde{X}}} \right) \to 0,$$ and dualizing again with respect to $\omega_{{\tilde{X}}}$ we get $$\label{exact:AesCM} 0 \to \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\tilde{X}}}(\omega_{{\tilde{X}}},\omega_{{\tilde{X}}})^r \to \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}} \left(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}} \left({\mathcal{M}}, \omega_{{\tilde{X}}} \right), \omega_{{\tilde{X}}} \right) \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}', \omega_{{\tilde{X}}} \right), \omega_{{\tilde{X}}} \right) \to 0.$$ Theorem \[Th:Herzog\] applied to ${\mathcal{O}_{{\tilde{X}}}}$ yields that the first term is isomorphic to ${\mathcal{O}_{{\tilde{X}}}}^r$. Since ${\mathcal{M}}$ is reflexive we have that the middle term is isomorphic to ${\mathcal{M}}$. Functoriality of the double $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}$ implies then that the first morphism of the last exact sequence coincides with the first morphism of the exact sequence (\[exctseq:directaM\]). Therefore we get that ${\mathcal{A}}'$ is isomorphic to $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}', \omega_{{\tilde{X}}} \right), \omega_{{\tilde{X}}} \right)$. Finally since ${\mathcal{A}}'$ has dimension one we conclude that ${\mathcal{A}}'$ is a Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module of dimension one by Proposition \[prop:extCM1\]. The following Bertini-Type Proposition is a generalization of Lemma 1.2 of [@AV]. \[prop:generalizationAV\] Let ${\mathcal{M}}$ a reflexive ${\mathcal{O}_{{\tilde{X}}}}$-module of rank $r$ that is generically generated by global sections as above and $(\phi_1, \dots, \phi_r)$ a set of $r$ generic sections. Let $C$ be the support of the degeneracy module ${\mathcal{A}}'$ with reduced structure. 1. For any point $x\in{\tilde{X}}$ not contained in $S$ we have the isomorphism ${\mathcal{A}}'_x\cong{\mathcal{O}_{C,x}}$. 2. The support of $C$ meets $E$ in finitely many points. Let $Z\subset{\tilde{X}}$ be any finite set disjoint with $S$, a sufficiently generic choice of the sections $(\phi_1, \dots, \phi_r)$ ensures that $C$ is smooth outside $S$, that $C$ does not meet $Z$, and that $E$ and $C$ meet in a transversal way at those meeting points not contained in $S$. However $C$ contains $S$ and at these points it may meet $E$ in a non-transversal way. The restriction ${\mathcal{M}}|_{{\tilde{X}}\setminus S}$ of the sheaf ${\mathcal{M}}$ to ${\tilde{X}}\setminus S$ is locally free and generated by global sections of ${\mathcal{M}}$ over ${\tilde{X}}$. Let $\psi_1, \dots, \psi_k$ be global sections of ${\mathcal{M}}$ such that they generate it over ${\tilde{X}}\setminus S$. Denote by $E$ the vector bundle over ${\tilde{X}}\setminus S$ whose sheaf of sections is ${\mathcal{M}}|_{{\tilde{X}}\setminus S}$ and by $({\tilde{X}}\setminus S) \times \mathbb{C}^k$ the trivial vector bundle of rank $k$ over ${\tilde{X}}\setminus S$. The generating global sections $(\psi_1,...,\psi_k)$ induce a surjective morphism of vector bundles $$\Psi:({\tilde{X}}\setminus S) \times \mathbb{C}^k\to E,$$ defined by $\Psi(x,(c_1, \dots, c_k))=\sum_{j=1}^k \psi_j(x)c_j$. Since holomorphic vector bundles over Stein spaces are trivial, and ${\tilde{X}}\setminus S$ admits a finite Stein cover, there exist a finite trivializing covering for $E$. Let $U$ be an trivializing open set. Consider the local trivialization $E|_{U}\to U \times \mathbb{C}^r$. In the open set $U$ the global sections $\psi_1, \dots, \psi_k$ can be written as follows $$A= \begin{tikzpicture}[baseline=(m-2-1.base)] \matrix (m)[matrix of math nodes, left delimiter=(,right delimiter=)]{ a_{11} & a_{12} & \dots & a_{1k} \\ \vdots & \vdots & \vdots & \vdots \\ a_{r1} & a_{12} & \dots & a_{rk} \\}; \end{tikzpicture},$$ where the entries of the $i$-th column are the coordinates of $\psi_i$. Notice that the matrix $A$ has entries in ${\mathcal{O}_{{\tilde{X}}}}(U)$. In the trivialization over $U$ the restriction of the map $\Psi$ is $\Psi_U:U \times \mathbb{C}^k \to U \times \mathbb{C}^r$, where $\Psi_U(x,(c_1, \dots, c_k))=(x, A(x)(c_1, \dots, c_k)^{\intercal})$. Now for each matrix $B$ in $\text{Mat}\left( k \times r, \mathbb{C} \right)$, we get sections $\phi_1, \dots, \phi_r$ of ${\mathcal{M}}$ by the formula $$(\phi_1, \dots, \phi_r) = (\psi_1, \dots, \psi_k)B,$$ and a choice of generic sections amounts to the choice of a generic matrix $B$. In the trivializing frame over the open set $U$ the coordinates of the sections $\phi_i$ is the $i$-th column of the matrix product $AB$: $$(\phi_1, \dots, \phi_r) = AB.$$ So, in the open set $U$ the exact sequence is $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=3.5em,row sep=2em] { 0 & {\mathcal{O}_{U}}^r & {\mathcal{O}_{U}}^r & {\mathcal{A}}'|_U & 0. \\ & {\mathcal{O}_{U}}^k & & &\\}; \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-1-2) edge node[auto]{$(\phi_1, \dots, \phi_r)$} (m-1-3); \draw[-stealth] (m-1-3) -- (m-1-4); \draw[-stealth] (m-1-4) -- (m-1-5); \draw[-stealth] (m-1-2) edge node[auto]{$B$} (m-2-2); \draw[-stealth] (m-2-2) edge node[below]{$A$} (m-1-3); \end{tikzpicture}$$ Then we have that $$\label{eq:A'explicito} \text{Supp}({\mathcal{A}}')\cap U = \{x \in U \, | \, \det (AB) = 0\}.$$ Consider the stratification by rank in the set $\text{Mat}\left( r \times r, \mathbb{C} \right)$ and denote by $$\text{Mat}\left( r \times r, \mathbb{C} \right)^i := \{ c \in \text{Mat}\left( r \times r, \mathbb{C} \right) \, | \, \operatorname{corank}(c) \geq i \}.$$ We have that $\operatorname{codim}(\text{Mat}\left( r \times r, \mathbb{C} \right)^i) = i^2$ and $\dim({\tilde{X}}\setminus S) =2.$ Now consider the map $\Theta \colon \left( U \setminus S \right) \times \text{Mat}(k \times r, \mathbb{C}) \to \text{Mat}(r \times r, \mathbb{C})$, given by $(x,B) \mapsto A(x)B$. Since the sections $\{\psi_1 ,\dots, \psi_k\}$ generate ${\mathcal{M}}$ over the set $U \setminus S$, we get that the map $\Theta$ is a submersion and therefore it is transverse to the rank stratification. By the Parametric Transversality Theorem, for almost every $B$ in $\text{Mat}(k \times r, \mathbb{C})$, the map $$\Theta_B: U \setminus S \to Mat(r \times r, \mathbb{C})$$ defined by $x \mapsto A(x)B$ is transverse to the rank stratification and to the sets $Z$ and $E$. By the finiteness of the trivializing cover we can choose a matrix $B$ generic such that in each trivialization the map $\Theta_B$ is transverse to the rank stratification and to $Z$. By Equation (\[eq:A’explicito\]) and transversality we have that $\text{Supp}({\mathcal{A}}')\cap U$ is smooth of dimension $1$, disjoint to $Z$ and transversal to $E$. The tranversality of $\Theta_B$ to the rank stratification also implies that for any $x\in U$ we have the isomorphism ${\mathcal{A}}'_x\cong {\mathcal{O}_{C,x}}$, where $C$ is the support of ${\mathcal{A}}'$. Since the trivializing open sets cover ${\tilde{X}}\setminus S$ the proposition is proved. The previous Proposition motivates the following definition. \[def:nearlygeneric\] Let ${\mathcal{M}}$ be a reflexive ${\mathcal{O}_{{\tilde{X}}}}$-module of rank $r$. A collection $(\phi_1, \dots, \phi_r)$ of $r$ sections is called [*nearly generic*]{} if the following conditions are satisfied. Let $C$ be the support of the degeneracy module ${\mathcal{A}}'$. 1. For any point $x\in{\tilde{X}}$ except in a finite collection we have the isomorphism ${\mathcal{A}}'_x\cong({\mathcal{O}_{C}})_x$. 2. The support of $C$ meets $E$ in finitely many points (notice that this condition is void in the case $X={\tilde{X}}$). Proposition \[prop:generalizationAV\] states that a generic set of sections is in particular nearly generic. Cohen-Macaulay modules of dimension 1 {#sec:CM1} ------------------------------------- In this section we study the structure of Cohen-Macaulay modules of dimension $1$ which are of rank 1 and generically reduced. The concrete description that we are about to obtain will be very important in our study of reflexive modules. \[def:genredCM\] Let $Y$ be an analytic space and ${\mathcal{C}}$ a ${\mathcal{O}_{Y}}$-module dimension $1$. Let $C$ be the support of ${\mathcal{C}}$, with reduced structure. The module ${\mathcal{C}}$ is [*rank 1 generically reduced* ]{} if ${\mathcal{C}}$ is isomorphic to ${\mathcal{O}_{C}}$ except in finitely many points of the support. \[rem:obsss\] Definition \[def:nearlygeneric\] and the first assertion of Proposition \[prop:generalizationAV\] states that degeneracy modules (see Definition \[def:degeneracymodule\]) for nearly generic sections are rank 1 generically reduced Cohen-Macaulay modules of dimension $1$. \[prop:genredCM\] Let $Y$ be an analytic space, and ${\mathcal{C}}$ be a rank 1 generically reduced Cohen-Macaulay ${\mathcal{O}_{Y}}$-module of dimension $1$. Let $C$ be the support of ${\mathcal{C}}$ with reduced structure. Denote by $n \colon \tilde{C} \to C$ the normalization. 1. The sheaf ${\mathcal{C}}$ is a ${\mathcal{O}_{C}}$-module, that is, the ideal of $C$ is contained in the annihilator of ${\mathcal{C}}$. 2. Let $n:\tilde{C}\to C$ be the normalization. If $C$ is Stein then there exists an inclusion of ${\mathcal{C}}$ as ${\mathcal{O}_{C}}$-submodule of $n_*{\mathcal{O}_{\tilde{C}}}$ which contains ${\mathcal{O}_{C}}$. In other words, we have the chain of inclusions $${\mathcal{O}_{C}} \subset {\mathcal{C}} \subset n_*{\mathcal{O}_{\tilde{C}}}.$$ For the first assertion let $f$ be an element of the ideal of $C$. Assume that there exists a section $c$ of ${\mathcal{C}}$ such that $f\cdot c$ is different from zero. Since ${\mathcal{C}}$ is rank 1 generically reduced the support of $f\cdot c$ is a finite set of points, but this forces $f\cdot c$ to vanish, since otherwise ${\mathcal{C}}$ would not have depth $1$. Now we prove the second assertion. Consider the following map $$\begin{aligned} h\colon {\mathcal{C}} &\to {\mathcal{C}} \otimes_{{\mathcal{O}_{C}}} {\mathcal{O}_{\tilde{C}}}/(Torsion),\\ c &\mapsto c\otimes 1.\end{aligned}$$ By hypothesis we have that ${\mathcal{C}}$ is isomorphic to ${\mathcal{O}_{C}}$ except in finitely many points. Therefore the support of the kernel of $h$ is finite and since ${\mathcal{C}}$ does not have any finitely supported section we get that the map $h$ is injective. Now notice that $n^* {\mathcal{C}}/(Torsion) = {\mathcal{C}} \otimes_{{\mathcal{O}_{C}}} {\mathcal{O}_{\tilde{C}}}/(Torsion)$ is a torsion-free ${\mathcal{O}_{\tilde{C}}}$-module of rank one, and hence isomorphic to ${\mathcal{O}_{\tilde{C}}}$ (since $\tilde{C}$ is smooth and Stein). So we have an injection $h\colon {\mathcal{C}} \to {\mathcal{O}_{\tilde{C}}}$. Consider the (multi)-germ of $\tilde{C}$ at the support of ${\mathcal{O}_{\tilde{C}}}/h({\mathcal{C}})$. Enumerate the branches of the multi-germ as $(\tilde{C}_j,p_j)$ for $j=1,...,l$. Each ${\mathcal{O}_{\tilde{C}_j,p_j}}$ is a discrete valuation ring for any $j$. Let $t_j$ be a uniformizing parameter of ${\mathcal{O}_{\tilde{C}_j,p_j}}$. Let us denote by $h(c)_j$ the germ of $h(c)$ in ${\mathcal{O}_{\tilde{C}_j,p_j}}$. For any section $c$ of ${\mathcal{C}}$ we define $\text{ord}(h(c)) := \left(\dots,\text{ord}_{t_j}\left(h(c)_j \right),\dots \right),$ where $\text{ord}_{t_j}$ denotes the valuation of the ring ${\mathcal{O}_{\tilde{C}_j}}$. Notice that $\text{ord}(h(c))$ belongs to the set $\mathbb{N}^l$. Now for any generic $\lambda$ and $\mu$ in $\mathbb{C}$ and $c$ and $c'$ in ${\mathcal{C}}$ we have that $$\text{ord}(h(\lambda c+ \mu c' ))= \text{min}\{\text{ord}(h(c)), \text{ord}(h(c')) \},$$ where the minimum is taken componentwise. As a consequence, for a generic section $c_0$ of ${\mathcal{C}}$ we have that $\text{ord}(h(c_0))$ is the absolute minimum of the image of $ord$. Denote by $(n_1, \dots, n_l) =\text{ord}(h(c_0))$. By genericity and the fact that $h({\mathcal{C}})$ spans ${\mathcal{O}_{\tilde{C}}}$ outside $\{p_1,...,p_l\}$ we also know that $h(c_0)$ does not vanish outside $\{p_1,...,p_l\}$. Consider the commutative diagram $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { {\mathcal{C}} & {\mathcal{O}_{\tilde{C}}} \\ & {\mathcal{O}_{\tilde{C}}}[t_1^{-1},...,t_l^{-1}] \\}; \draw[-stealth] (m-1-1) edge node[auto]{$h$} (m-1-2); \draw[-stealth] (m-1-1) edge node[below]{$g$} (m-2-2.north west); \draw[-stealth] (m-1-2) edge node[auto]{$\cdot \frac{1}{h(c_0)}$} (m-2-2); \end{tikzpicture}$$ where $\cdot \frac{1}{h(c_0)}$ is multiplication by $\frac{1}{h(c_0)}$. By construction the map $g$ is an ${\mathcal{O}_{C}}$-monomorphism and the image of $g$ is contained in ${\mathcal{O}_{\tilde{C}}}$ because $\text{min}\{\text{ord}(g(c)) \, | \, \text{$c$ is a section of ${\mathcal{C}}$}\} = \text{ord}(g(c_0))=(0,\dots,0).$ Moreover we have the equality $g(c_0)=1$. This implies that $g$ provides the needed chain of embeddings. The previous structure result allows to define some invariants of rank 1 generically reduced Cohen-Macaulay modules of dimension $1$ which will be important for us. Consider the notations of the previous proposition and its proof. We have defined an embedding of ${\mathcal{O}_{C}}$-modules $$\iota:{\mathcal{C}}\to{\mathcal{O}_{\tilde{C}}},$$ such that ${\mathcal{O}_{C}}$ is included in $\iota({\mathcal{C}})$. Therefore the support of ${\mathcal{O}_{\tilde{C}}}/\iota({\mathcal{C}})$ is contained in the pre-image by $n$ of the singular set of $C$. Let now be $\{p_1,...,p_l\}$ the pre-image by $n$ of the singular set, and denote by $(\tilde{C}_j,p_j)$ the germ of $\tilde{C}$ at $p_j$. As in the previous proof we have an order function $$ord:{\mathcal{O}_{\tilde{C}}}\to{{\mathbb{N}}}^l.$$ \[lem:indeporderset\] The image of the composition $ord{{\circ}}\iota$ is independent of the embedding $\iota:{\mathcal{C}}\to{\mathcal{O}_{\tilde{C}}}$ as long as ${\mathcal{O}_{C}}$ is included in $\iota({\mathcal{C}})$. Any two embeddings differ by multiplication by a unit in ${\mathcal{O}_{\tilde{C}}}$. \[def:orderset\] The [*set of orders*]{} $\mathfrak{C}$ of ${\mathcal{C}}$ is the image in ${{\mathbb{N}}}^l$ of the composition $ord{{\circ}}\iota$ for an embedding $\iota:{\mathcal{C}}\to{\mathcal{O}_{\tilde{C}}}$ of ${\mathcal{O}_{C}}$-modules such that ${\mathcal{O}_{C}}$ is included in $\iota({\mathcal{C}})$. Given a subset $\mathfrak{B}\subset\mathbb{Z}^l$, and a vector $(d_1,...,d_l)$, we denote by $(d_1,...,d_l)+\mathfrak{B}\subset\mathbb{Z}^l$ the translation of $\mathfrak{B}$ in the direction of the vector $(d_1,...,d_l)$. \[rem:tanslationinv\] Since ${\mathcal{C}}$ is a ${\mathcal{O}_{Y}}$-module, the set of orders $\mathfrak{C}$ is stable by translation in the direction given by any vector $ord(f|_C)$ for $f\in{\mathcal{O}_{Y}}$ (where $f|_C$ is the restriction of $f$ to $C$). It is well known that ${\mathcal{O}_{C}}$ has a conductor ideal (see for example [@Ab (19.21)]). In our case we define the conductor of ${\mathcal{C}}$ as follows (compare with [@Ab (19.21)]). \[def:conductorsett\] Let ${\mathcal{C}}$ be an ${\mathcal{O}_{C}}$-module such that $${\mathcal{O}_{C}} \subset {\mathcal{C}} \subset n_* {\mathcal{O}_{\tilde{C}}}.$$ The ${\mathcal{O}_{C}}$-submodule $$\left \{s \in {\mathcal{C}} \, | \, s\cdot n_* {\mathcal{O}_{\tilde{C}}} \subset {\mathcal{C}} \right \},$$ is called *the conductor of ${\mathcal{C}}$*. The image of the conductor under the order function is called the [*conductor set of*]{} ${\mathcal{C}}$. It is a subset of $\mathfrak{C}$. Any element of the conductor set of ${\mathcal{C}}$ is called [*a conductor of*]{} $\mathfrak{C}$. \[rem:propertiesconductor\] Since ${\mathcal{O}_{C}}$ is contained in ${\mathcal{C}}$, we get that the conductor of ${\mathcal{C}}$ is non-empty. Since the conductor of ${\mathcal{C}}$ is closed by taking ${\mathcal{O}_{X}}$-linear combinations, the conductor set of $\mathfrak{C}$ has an absolute minimum, which is denoted by $cond(\mathfrak{C})$. The conductor set of $\mathfrak{C}$ satisfies the following property: if $c=(c_1,\dots,c_l)$ is a conductor of $\mathfrak{C}$ then for any vector $w$ in $\mathbb{N}^l$ we have that $c+w$ belongs to the set $\mathfrak{C}$. This motivates the following \[def:conductorset\] Let $\mathfrak{C}$ be a subset of ${{\mathbb{Z}}}^l$, the [*conductor set*]{} of $\mathfrak{C}$ is the (perhaps empty) set $$\{v\in \mathfrak{C} \,| \, v+{{\mathbb{N}}}^l\subset\mathfrak{C} \}.$$ The correspondence at the Stein surface --------------------------------------- In this section we let $X$ be a Stein surface with Gorenstein singularities; in many of the cases $X$ will be a Milnor representative of a normal Gorenstein surface singularity. The Gorenstein assumption and Theorem \[Th:Herzog\] allow us to understand the relation between reflexive ${\mathcal{O}_{X}}$-modules with nearly generic sections and rank 1 generically reduced Cohen-Macaulay ${\mathcal{O}_{X}}$-modules with a system of generators. \[sec:corrsing\] \[th:corrsing\] Let $X$ be a Stein surface with Gorenstein singularities. There is a bijective correspondence between the set of pairs $(M,(\phi_1,...,\phi_r))$ of rank $r$ reflexive ${\mathcal{O}_{X}}$-modules with $r$ nearly generic sections and the set of pairs $({\mathcal{C}},(\psi_1,...,\psi_r))$ of rank 1 generically reduced Cohen-Macaulay ${\mathcal{O}_{X}}$-modules with a system of generators of ${\mathcal{C}}$ as ${\mathcal{O}_{X}}$-module. Under this correspondence, if the system of generators $(\psi_1,...,\psi_r)$ is not minimal then the module $M$ contains free factors. As a partial converse: if $M$ contains free factors and $(\phi_1,...,\phi_r)$ are generic, then the system of generators $(\psi_1,...,\psi_r)$ is not minimal. The correspondence from the first set to the second is called the [*direct correspondence at $X$*]{}, its inverse is called the [*inverse correspondence at $X$*]{}. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$ and $(\phi_1,...,\phi_r)$ be $r$ nearly generic sections. We obtain the exact sequence defining the degeneracy module given by the sections $$\label{exsq1} 0 \to {\mathcal{O}_{X}}^r \to M \to {\mathcal{C}}' \to 0.$$ Since the sections are nearly generic the module ${\mathcal{C}}'$ is of rank and 1 generically reduced by Remark \[rem:obsss\]. Dualizing the exact sequence with respect to ${\mathcal{O}_{X}}$ we get $$\label{exsq11} 0 \to N \to {\mathcal{O}_{X}}^r \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}', {\mathcal{O}_{X}} \right) \to 0.$$ where $N$ is the dual of $M$. By Proposition \[prop:extCM1\] the module ${\mathcal{C}}:=\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}', {\mathcal{O}_{X}} \right)$ is Cohen-Macaulay of dimension one. Let $C$ be the support of ${\mathcal{C}}'$, which coincides with the support of ${\mathcal{C}}$. A direct computation of ${\mathcal{C}}$ shows that, at a smooth point $y\in X$, if ${\mathcal{C}}'_y$ is isomorphic to ${\mathcal{O}_{C,y}}$ then ${\mathcal{C}}_y$ is isomorphic to ${\mathcal{O}_{C,y}}$ too. This shows that ${\mathcal{C}}$ is rank 1 generically reduced . Therefore we associate to the reflexive module $M$ with the given sections, the module ${\mathcal{C}}$ with the generators induced by the previous exact sequence. Conversely, let $({\mathcal{C}},(\psi_1,...,\psi_r))$ be a rank 1 generically reduced $1$-dimensional Cohen-Macaulay module with a system of generators. Define $N$ to be the kernel of the morphism ${\mathcal{O}_{X}}^r \to {\mathcal{C}}$ induced by the generators. We have the exact sequence $$\label{exsq2} 0 \to N \to {\mathcal{O}_{X}}^r \to {\mathcal{C}} \to 0.$$ Dualizing the exact sequence we get $$0 \to {\mathcal{O}_{X}}^r \to M \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1\left({\mathcal{C}}, {\mathcal{O}_{X}} \right) \to 0,$$ where $M$ is the dual of $N$, and hence it is reflexive. To the pair $({\mathcal{C}},(\psi_1,...,\psi_r))$ we associate the pair $(M,(\phi_1,...,\phi_r))$, where the sections are induced by the second morphism of the previous exact sequence. The sections are nearly generic since the module $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1\left({\mathcal{C}}, {\mathcal{O}_{X}} \right)$ is generically reduced, for being ${\mathcal{C}}$ rank 1 generically reduced . The correspondences are mutually inverse due to part (iii) of Theorem \[Th:Herzog\]. The assertion relating free factors with minimality of the system of generators follows [@Es]. Suppose that the system of generators $(\psi_1,...,\psi_r)$ of ${\mathcal{C}}$ is not minimal. Then an obvious computation shows that the module of relations $N$ has free factors. Since $M$ is the dual of $N$ then $M$ has free factors. Conversely, suppose that $M$ has a free factors, that is $M\cong M_1\oplus{\mathcal{O}_{X}}^a$. Then sequence (\[exsq1\]) becomes $$0 \to {\mathcal{O}_{X}}^{r-a}\oplus{\mathcal{O}_{X}}^a \to M_1\oplus{\mathcal{O}_{X}}^a \to {\mathcal{C}}' \to 0.$$ The genericity of the choice of the system of generators $(\phi_1,...,\phi_r)$ imply that the morphism ${\mathcal{O}_{X}}^a \to {\mathcal{O}_{X}}^a$ obtained by the triple composition of the natural inclusion of ${\mathcal{O}_{X}}^a$ into ${\mathcal{O}_{X}}^{r-a}\oplus{\mathcal{O}_{X}}^a$, the first morphism of the sequence, and the canonical projection of $M_1\oplus{\mathcal{O}_{X}}^a$ to ${\mathcal{O}_{X}}^a$ is an isomorphism. Dualizing we obtain $$0 \to N_1\oplus{\mathcal{O}_{X}}^a \to {\mathcal{O}_{X}}^{r-a}\oplus{\mathcal{O}_{X}}^a \to {\mathcal{C}} \to 0.$$ Since the corresponding morphism ${\mathcal{O}_{X}}^a \to {\mathcal{O}_{X}}^a$ is an isomorphism we conclude that the system of generators of ${\mathcal{C}}$ is not minimal. We may also understand the module of relations of the generators of the Cohen-Macaulay module ${\mathcal{C}}$. \[inversaabajo\] Let ${\mathcal{C}}$ be an Cohen-Macaulay ${\mathcal{O}_{X}}$-module of dimension one, $\{\phi_1, \dots, \phi_n\}$ a set of generators of ${\mathcal{C}}$ as ${\mathcal{O}_{X}}$-module and consider the exact sequence obtained by the generators $$\label{exctseq:directaNabajo} 0 \to N \to {\mathcal{O}_{X}}^r \to {\mathcal{C}} \to 0.$$ Then the module, $N$ is reflexive. Dualizing the exact sequence and denoting by $M:= N^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$, we obtain the exact sequence $$\label{exctseq:directaMabajo} 0 \to {\mathcal{O}_{X}}^r \to M \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}}\right) \to 0.$$ Since ${\mathcal{C}}$ is Cohen-Macaulay of dimension one then by Theorem \[Th:Herzog\] the module $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}},{\mathcal{O}_{X}}\right)$ is Cohen-Macaulay of dimension one and ${\mathcal{C}} \cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left( \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}},{\mathcal{O}_{X}}\right),{\mathcal{O}_{X}}\right)$. Now dualizing and using the previous identification we obtain the exact sequence $$0 \to N^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \to {\mathcal{O}_{X}}^r \to {\mathcal{C}} \to 0,$$ hence we have the identification $N=N^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ and $N$ is reflexive. The correspondence at the resolution {#sec:corrres} ------------------------------------ In this section we generalize the correspondence given by Artin-Verdier [@AV], Esnault [@Es] and Wunram [@Wu] at the resolution to the general case, that is $(X,x)$ is any normal surface singularity; in fact we go further and allow $X$ to be a Stein normal surface with possibly several singularities. We will obtain a bijection as in previous section. One of the sets is formed by pairs of modules together with systems of nearly generic sections. The other set is formed by rank 1 generically reduced $1$-dimensional Cohen-Macaulay modules together with sets of sections that satisfy a certain property. Before we state the main result of the section we need to explain what the property means precisely. ### The Containment Condition {#sec:containment} Let $\pi:{\tilde{X}}\to X$ be a resolution of singularities of a Stein normal surface $X$ which is an isomorphism at the regular locus of $X$. Let $E$ denote the exceptional divisor and $U={\tilde{X}}\setminus E$. Let ${\mathcal{A}}$ be a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module. Let $(\psi_1,...,\psi_r)$ be $r$ global sections spanning ${\mathcal{A}}$ as ${\mathcal{O}_{{\tilde{X}}}}$-module. The set of sections $(\psi_1,...,\psi_r)$ define a morphism ${\mathcal{O}_{\tilde{X}}}^r\to{\mathcal{A}}$. Tensoring with the dualizing sheaf $\omega_{\tilde{X}}=\Lambda^2\Omega^1_{{\tilde{X}}}$ and taking sections in $U$ we obtain a morphism $$\label{eq:deltaprimeravez} \delta:H^0(U,\omega_{\tilde{X}}^r)\to H^0(U,{\mathcal{A}}\otimes\omega_{\tilde{X}}).$$ We also have a restriction morphism $$\gamma_1:H^0({\tilde{X}},{\mathcal{A}}\otimes\omega_{\tilde{X}})\to H^0(U,{\mathcal{A}}\otimes\omega_{\tilde{X}}).$$ \[def:containment\] The pair $({\mathcal{A}},(\psi_1,...,\psi_r))$ satisfies the [*Containment Condition*]{} if we have the inclusion $\mathrm{Im}\gamma_1\subset \mathrm{Im}\delta$. ### The correspondence at the resolution {#the-correspondence-at-the-resolution} \[th:corres\] Let $\pi:{\tilde{X}}\to X$ be a resolution of singularities of a Stein normal surface which is an isomorphism at the regular locus of $X$. There is a bijective correspondence between the set of pairs $({\mathcal{M}},(\phi_1,...,\phi_r))$ formed by a locally free ${\mathcal{O}_{{\tilde{X}}}}$-module which is almost generated by global sections, and a set of $r$ nearly generic sections, and the set of pairs $({\mathcal{A}},(\psi_1,...,\psi_r))$ formed by a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module, whose support meets $E$ in finitely many points, and a set of $r$ global sections spanning ${\mathcal{A}}$ as ${\mathcal{O}_{{\tilde{X}}}}$-module. Moreover ${\mathcal{M}}$ is full if and only if $({\mathcal{A}},(\psi_1,...,\psi_r))$ satisfies the Containment Condition (see Definition \[def:containment\]). We start defining the first bijection. Given $({\mathcal{M}},(\phi_1,...,\phi_r))$ we consider the exact sequence induced by the sections: $$\label{exctseq:da1} 0 \to {\mathcal{O}_{{\tilde{X}}}}^r \to {\mathcal{M}} \to {\mathcal{A}}' \to 0.$$ The degeneracy module ${\mathcal{A}}'$ is $1$-dimensional Cohen-Macaulay by Proposition \[prop:generalizationAV0\], and is rank 1 generically reduced with support intersecting the exceptional divisor $E$ in a finite set, by definition of nearly-generic sections (Definition \[def:nearlygeneric\]). Dualizing the sequence we obtain $$\label{exact:da2} 0 \to {\mathcal{N}} \to {\mathcal{O}_{{\tilde{X}}}}^r \to {\mathcal{A}} \to 0,$$ where ${\mathcal{A}}=\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\tilde{X}}}}}({\mathcal{A}}',{\mathcal{O}_{{\tilde{X}}}})$ is a rank 1 generically reduced $1$-dimensional Cohen-Macaulay module (same arguments than in the proof of Theorem \[th:corrsing\]), whose support meets $E$ in finitely many points (since it coincides with the support of ${\mathcal{A'}})$. Let $(\psi_1,...,\psi_r)$ be the generators of ${\mathcal{A}}$ as ${\mathcal{O}_{{\tilde{X}}}}$-module given by the previous exact sequence. We define the direct correspondence as the correspondence sending the pair $({\mathcal{M}},(\phi_1,...,\phi_r))$ to the pair $({\mathcal{A}},(\psi_1,...,\psi_r))$. Conversely, given $({\mathcal{A}},(\psi_1,...,\psi_r))$ we consider the exact sequence (\[exact:da2\]) given by the sections. Dualizing it we obtain the exact sequence (\[exctseq:da1\]), and we define the inverse correspondence sending $({\mathcal{A}},(\psi_1,...,\psi_r))$ to $({\mathcal{M}},(\phi_1,...,\phi_r))$, where $(\phi_1,...,\phi_r)$ are the sections induce by the sequence (\[exctseq:da1\]). The direct and inverse correspondences are inverse to each other, for the same reasons appearing in the proof of Theorem \[th:corrsing\]. In order to prove the Theorem we have to show that ${\mathcal{M}}$ is full if and only if $({\mathcal{A}},(\psi_1,...,\psi_r))$ satisfies the Containment Condition. For this we use the characterization of Proposition \[fullcondiciones\]. We start showing that the inverse correspondence always gives a ${\mathcal{O}_{{\tilde{X}}}}$-module that is generically generated by global sections. This is stated in a separate lemma. \[lem:gengen\] If $({\mathcal{A}},(\psi_1,...,\psi_r))$ is formed by a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module, whose support meets $E$ in finitely many points, and a set of $r$ global sections spanning ${\mathcal{A}}$ as ${\mathcal{O}_{{\tilde{X}}}}$-module, then the ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}$ obtained by applying inverse correspondence is generically generated by global sections. Applying the functor $\pi_* -$ to the exact sequence we get $$0 \to {\mathcal{O}_{X}}^r \to \pi_*{\mathcal{M}} \to \pi_*{\mathcal{A}}' \to R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}^r \to R^1 \pi_*{\mathcal{M}} \to 0.$$ Denote by ${\mathcal{G}}$ the image of $\pi_* {\mathcal{M}}$ in $\pi_*{\mathcal{A}}'$, so we obtain the following two exact sequences of ${\mathcal{O}_{X}}$-modules $$0 \to {\mathcal{O}_{X}}^r \to \pi_*{\mathcal{M}} \to {\mathcal{G}} \to 0,$$ $$\label{exctseq:ggsg} 0 \to {\mathcal{G}} \to \pi_* {\mathcal{A}}' \to R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}^r \to R^1 \pi_*{\mathcal{M}} \to 0.$$ Since the support of ${\mathcal{A}}'$ intersects the exceptional divisor in a finite collection of points, then we can identify $\pi_* {\mathcal{A}}'$ with ${\mathcal{A}}'$, viewed as a ${\mathcal{O}_{X}}$-module. Then ${\mathcal{G}}$ is a sub ${\mathcal{O}_{X}}$-module of ${\mathcal{A'}}$. Denote by ${\mathcal{M}}'$ the subsheaf of ${\mathcal{M}}$ generated by its global sections, and by ${\mathcal{G}}'$ the sub-${\mathcal{O}_{{\tilde{X}}}}$-module of ${\mathcal{A}}'$ spanned by ${\mathcal{G}}$. We have the following diagram of coherent ${\mathcal{O}_{{\tilde{X}}}}$-modules $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=1.5em]{ & 0 & 0 & 0 & \\ 0 & {\mathcal{O}_{{\tilde{X}}}}^r & {\mathcal{M}}' & {\mathcal{G'}} & 0 \\ 0 & {\mathcal{O}_{{\tilde{X}}}}^r & {\mathcal{M}} & {\mathcal{A}}' & 0 \\ & 0 & {\mathcal{F}} & {\mathcal{F}}' & 0 \\ & & 0 & 0 & \\ }; \foreach \y [remember=\y as \lasty (initially 1)] in {2,3,4} { \foreach \x [remember=\x as \lastx (initially 2)] in {3,4} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); \draw[-stealth] (m-\lasty-\x) -- (m-\y-\x); } } \draw[-stealth] (m-1-2) -- (m-2-2); \draw[-stealth] (m-2-2) -- (m-3-2); \draw[-stealth] (m-2-1) -- (m-2-2); \draw[-stealth] (m-3-1) -- (m-3-2); \draw[-stealth] (m-2-4) -- (m-2-5); \draw[-stealth] (m-3-4) -- (m-3-5); \draw[-stealth] (m-3-2) -- (m-4-2); \draw[-stealth] (m-4-4) -- (m-4-5); \draw[-stealth] (m-4-3) -- (m-5-3); \draw[-stealth] (m-4-4) -- (m-5-4); \end{tikzpicture}$$ It is enough to prove that the support of ${\mathcal{F}}$, which coincides with the support of ${\mathcal{F}}'$, is a finite set. For this it is enough to show that ${\dim_{\mathbb{C}}({\mathcal{F}}')}={\dim_{\mathbb{C}}(coker({\mathcal{G}}'\to{\mathcal{A}}'))}<\infty$. But we have the inequalities $${\dim_{\mathbb{C}}(coker({\mathcal{G}}'\to{\mathcal{A}}'))}\leq {\dim_{\mathbb{C}}(coker({\mathcal{G}}\to\pi_*{\mathcal{A}}'))}\leq rp_g$$ by the Exact Sequence (\[exctseq:ggsg\]). In order to finish the proof of Theorem \[th:corres\] we have to prove that the map $H^1_E({\mathcal{M}})\to H^1({\mathcal{M}})$ is injective if and only if $({\mathcal{A}},(\psi_1,...,\psi_r))$ satisfies the Containment Condition. By Serre duality the Containment Condition is equivalent to the surjection of the natural map $H^1_E\left({\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \otimes {\omega_{{\tilde{X}}}} \right) \to H^1\left({\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \otimes {\omega_{{\tilde{X}}}} \right)$, hence we will study this map. As before we denote ${\mathcal{N}}:={\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. Apply the functor $- \otimes {\omega_{{\tilde{X}}}}$ to the exact sequence (\[exact:da2\]), $$\label{exctseq:directaWN} 0 \to {\mathcal{N}}\otimes {\omega_{{\tilde{X}}}} \to {\omega_{{\tilde{X}}}}^r \to {\mathcal{A}}\otimes {\omega_{{\tilde{X}}}} \to 0.$$ Taking the long exact sequence of cohomology and local cohomology for the previous exact sequence we obtain a diagram of exact sequences: (m)\[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=1.7em,row sep=2em\][ H\^[0]{}\_E( [\_]{}) & H\^[0]{}\_E([\_]{}\^r) & H\^0\_E([\_]{}) & H\^[1]{}\_E([\_]{})& H\^[1]{}\_E([\_]{}\^r)\ H\^[0]{}([\_]{}) & H\^[0]{}([\_]{}\^r) & H\^0([\_]{}) & H\^[1]{}([\_]{})& H\^[1]{}([\_]{}\^r)\ H\^[0]{}(U;[\_]{}) & H\^[0]{}(U;[\_]{}\^r) & H\^0(U;[\_]{}) & H\^[1]{}(U;[\_]{})& H\^[1]{}(U;[\_]{}\^r)\ ]{}; in [3,...,5]{} [ (m-1-) – (m-1-); (m-2-) – (m-2-); (m-3-) – (m-3-); (m-2-) – (m-3-); ]{} (m-2-1) – (m-3-1); (m-2-5) – (m-3-5); (m-1-4) – (m-2-4); (m-1-5) – (m-2-5); (m-1-1) – (m-2-1); (m-1-2) – (m-2-2); (m-1-3) – (m-2-3); (m-1-1) – (m-1-2); (m-2-1) – (m-2-2); (m-3-1) – (m-3-2); (m-3-1.south) to \[out = -90, in = 90, looseness = .7\] (m-1-4.north); (m-3-2.south) to \[out = -90, in = 90, looseness = .7\] (m-1-5.north); We have $H^1({\omega_{{\tilde{X}}}})=0$ by Grauert-Riemenschneider Vanishing Theorem and $H^0_E({\mathcal{A}}\otimes {\omega_{{\tilde{X}}}})=0$ because ${\mathcal{A}}\otimes {\omega_{{\tilde{X}}}}$ has depth one and its support intersects the exceptional divisor in a finite set. Therefore we have the following diagram of exact sequences $$\label{subdiagram} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { \quad & 0 & H_{E}^1\left({\mathcal{N}} \otimes {\omega_{{\tilde{X}}}}\right) & \quad \\ H^{0}\left({\omega_{{\tilde{X}}}}^r\right) & H^0\left({\mathcal{A}}\otimes {\omega_{{\tilde{X}}}}\right) & H^{1}\left({\mathcal{N}} \otimes {\omega_{{\tilde{X}}}}\right) & 0 \\ H^{0}\left(U;{\omega_{{\tilde{X}}}}^r\right) & H^0\left(U;{\mathcal{A}}\otimes {\omega_{{\tilde{X}}}}\right) & H^{1}\left(U;{\mathcal{N}} \otimes {\omega_{{\tilde{X}}}}\right) & \quad \\}; \draw[-stealth] (m-1-2) -- (m-2-2); \draw[-stealth] (m-2-1) -- node[auto]{$\beta$} (m-2-2); \draw[-stealth] (m-2-2) -- node[auto]{$\alpha$} (m-2-3); \draw[-stealth] (m-2-3) -- (m-2-4); \draw[-stealth] (m-3-1) -- node[auto]{$\delta$} (m-3-2); \draw[-stealth] (m-3-2) -- node[auto]{$\gamma_2$} (m-3-3); \draw[-stealth] (m-2-3) -- node[auto]{$\varphi$} (m-3-3); \draw[-stealth] (m-2-1) -- (m-3-1); \draw[-stealth] (m-2-2) -- node[auto]{$\gamma_1$} (m-3-2); \draw[-stealth] (m-1-3) -- node[auto]{$\theta$} (m-2-3); \end{tikzpicture}$$ A diagram chase shows that $H_{E}^1({\mathcal{N}} \otimes {\omega_{{\tilde{X}}}}) \stackrel{\theta}{\longrightarrow} H^1({\mathcal{N}} \otimes {\omega_{{\tilde{X}}}})$ is an epimorphism if and only if $$\operatorname{Im}\gamma_1 \subset \operatorname{Im}\delta,$$ which is precisely the Containment Condition. Working with the Containment Condition is quite difficult. Because of this, in the next section, we introduce a numerical condition which is implied by the Containment Condition, and that, in sufficiently many cases for our applications, is equivalent to it. ### The Valuative Condition {#sec:valuative} Let $\pi:{\tilde{X}}\to X$ be a resolution of singularities of a Stein normal surface $X$ which is an isomorphism over the regular locus of $X$. Let $E$ denote the exceptional divisor and $U:={\tilde{X}}\setminus E$. Let $C\subset X$ be a curve and $\overline{C}$ be its strict transform to ${\tilde{X}}$. Let $n:\tilde{C}\to \overline{C}$ be the normalization. Let $\{p_1,...,p_l\}$ be the preimage by $n$ of $E$. Let $(\tilde{C}_j,p_j)$ be the germ of $\tilde{C}$ at $p_j$. The ring ${\mathcal{O}_{\tilde{C}_j,p_j}}$ is a discrete valuation ring, and its valuation is denoted by $ord_{\tilde{C_j}}$. Let $\beta$ be a meromorphic differential $2$-form in $H^0(U,\omega_{{\tilde{X}}})$. We define a $l$-uple $ord(\beta)$ in $\left({{\mathbb{Z}}}\cup\{+\infty\}\right)^l$ as follows. For any $j$ we let $q_j:=n(p_j)$ and choose a non-vanishing holomorphic differential $2$-form germ $\omega_{q_j}$ at $q_j$. Then $\beta=h_j\omega_{q_j}$ where $h_j$ is a meromorphic function. Define $$\label{eq:orddiffform} ord(\beta):=(ord_{\tilde{C_1}}(h_1),...,ord_{\tilde{C_l}}(h_l)).$$ It is clear that the definition does not depend on the choice of the forms $\omega_{q_j}$. This defines an order function $$\label{eq:ordercanonical} ord:H^0(U,\omega_{{\tilde{X}}})\to({{\mathbb{Z}}}\cup\{\infty\})^l.$$ \[def:canoset\] The [*canonical set of orders of the curve*]{} $C$ [*at the resolution*]{} $\pi$ is the set $$\mathfrak{K}_\pi:=ord(H^0(U,\omega_{{\tilde{X}}}))\subset{{\mathbb{Z}}}^l.$$ Given two subsets $\mathfrak{A},\mathfrak{B}\subset{{\mathbb{Z}}}^l$ we denote by $\mathfrak{A}+\mathfrak{B}$ the subset of sums $a+b$ where $a\in\mathfrak{A}$ and $b\in\mathfrak{B}$. Let $\mathfrak{S}$ be the semigroup of orders of ${\mathcal{O}_{C}}$; since $H^0(U,\omega_{{\tilde{X}}})$ is a ${\mathcal{O}_{X}}$-module, we have the equality $\mathfrak{K}_\pi+\mathfrak{S}=\mathfrak{K}_\pi$. Given $\alpha,\beta\in H^0(U,\omega_{{\tilde{X}}})$, for generic $\lambda,\mu\in{{\mathbb{C}}}$ we have the equality $$\label{eq:ordenminimoforma} ord(\lambda\alpha+\mu\beta)=min(ord(\alpha),ord(\beta));$$ hence the canonical set of orders of the curve $C$ has an absolute minimum. \[def:tranlation\] We define the [*canonical vector*]{} $(d_1(\overline{C}),...,d_l(\overline{C}))$ of the curve $\overline{C}\in{\tilde{X}}$ to be the absolute minimum of the canonical set of orders. \[rem:canonicalcondset\] Since we have the equality $\mathfrak{K}_\pi+\mathfrak{S}=\mathfrak{K}_\pi$, the set $(d_1(\overline{C}),...,d_l(\overline{C}))+\mathfrak{S}$ is included in $\mathfrak{K}_\pi$. Then $\mathfrak{K}_\pi$ has a non-empty conductor set, which contains the conductor set of $\mathfrak{S}$ translated by the vector $(d_1(\overline{C}),...,d_l(\overline{C}))$. Since $\mathfrak{K}_\pi$ is closed by taking minima, the conductor set of $\mathfrak{K}_\pi$ has a unique absolute minimun, which we denote by $cond(\mathfrak{K}_\pi)$. Now assume that $X$ has Gorenstein singularities. Then there exist a holomorphic 2-form $\Omega\in H^0(U,\omega_{{\tilde{X}}})$ whose associated divisor is $$\text{div}(\Omega) =A+ \sum q_iE_i,$$ where each $q_i$ is a integer, the $E_i$’s are the irreducible components of the exceptional divisor and $A$ is a divisor disjoint with $E$. The integers $q_i$ are independent on the choice of $\Omega$. We call $\Omega$ a [*Gorenstein form*]{}. \[rem:translationgor\] If $X$ has Gorenstein singularities, and adopting the previous notation, the canonical vector $(d_1(\overline{C}),...,d_l(\overline{C}))$ of the curve $\overline{C}$ is given by the formulae $$d_i(\overline{C}):=\sum_{j}q_j\overline{C}_i\cdot E_j,$$ where $\overline{C}_i\cdot E_j$ denotes intersection multiplicity. \[rem:gorordcan\] If $X$ has Gorenstein singularities then the canonical set of orders of the curve $C$ at the resolution $\pi$ is equal to $(d_1(\overline{C}),...,d_l(\overline{C}))+\mathfrak{S}$, and its conductor set is the conductor set of $\mathfrak{S}$ translated by the vector $(d_1(\overline{C}),...,d_l(\overline{C}))$. Let ${\mathcal{A}}$ be a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module whose support equals $\overline{C}$. Let $\mathfrak{A}$ be its set of orders (see Definition \[def:orderset\]). Let $(\psi_1,...,\psi_r)$ be $r$ global sections spanning ${\mathcal{A}}$ as ${\mathcal{O}_{{\tilde{X}}}}$-module. The ${\mathcal{O}_{X}}$-module ${\mathcal{C}}$ spanned by $(\psi_1,...,\psi_r)$ is rank 1 generically reduced $1$-dimensional Cohen-Macaulay. Let $\mathfrak{C}$ be the set of orders of ${\mathcal{C}}$, which is the subset of $\mathfrak{A}$ obtained following Definition \[def:orderset\]. The set of sections $(\psi_1,...,\psi_r)$ define a epimorphism ${\mathcal{O}_{\tilde{X}}}^r\to{\mathcal{A}}$. Tensoring with $\omega_{\tilde{X}}$ we obtain an epimorphism $\omega_{\tilde{X}}^r\to{\mathcal{A}}\otimes\omega_{\tilde{X}}$. Taking global sections in $U$ we obtain a morphism $$\label{eq:deltaprimeravez2} \delta:H^0(U,\omega_{\tilde{X}}^r)\to H^0(U,{\mathcal{A}}\otimes\omega_{\tilde{X}}).$$ Since $\omega_{\tilde{X}}$ is locally free of rank $1$ and ${\mathcal{A}}$ has Stein support, the restriction of $\omega_{\tilde{X}}$ to the support of ${\mathcal{A}}$ is isomorphic to the structure sheaf of the support. Hence there is an isomorphism ${\mathcal{A}}\cong {\mathcal{A}}\otimes\omega_{\tilde{X}}$. We have fixed an embedding of ${\mathcal{A}}$ into ${\mathcal{O}_{\tilde{C}}}$, which naturally embeds $H^0(U,{\mathcal{A}})$ into the total fraction ring $K(\tilde{C})$. Such total fractions ring maps into the direct sum ring $\oplus_{i=1}^l K(\tilde{C}_i,p_i)$ (here $K(\tilde{C}_i,p_i)$ is the quotient field of ${\mathcal{O}_{\tilde{C}_i,p_i}}$). \[def:canoset2\] We define the canonical set of orders of $({\mathcal{A}},(\psi_1,...,\psi_r))$ to be the image $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$ of the composition $$H^0(U,\omega_{\tilde{X}}^r)\stackrel{\delta}{\longrightarrow}H^0(U,{\mathcal{A}}\otimes\omega_{\tilde{X}})\cong H^0(U,{\mathcal{A}})\subset \oplus_{i=1}^l K(\tilde{C}_i,p_i)\stackrel{ord}{\longrightarrow} ({{\mathbb{Z}}}\cup\{+\infty\})^l.$$ It is easy to verify that the previous definition does not depend on the choice of the isomorphism ${\mathcal{A}}\cong{\mathcal{A}}\otimes\omega_{\tilde{X}}$. \[def:valuative\] With the notations introduced above, the pair $({\mathcal{A}},(\psi_1,...,\psi_r))$ satisfy the [*Valuative Condition*]{} if the inclusion $$\mathfrak{A}\subset\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$$ is satisfied. \[rem:valuative\] We have the inclusion $$\mathfrak{C}+\mathfrak{K}_\pi\subset\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}.$$ If $X$ has Gorenstein singularities and $(d_1(\overline{C}),...,d_l(\overline{C}))$ is the Gorenstein vector at the resolution $\pi$, then we have the equalities $$\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}=\mathfrak{C}+\mathfrak{K}_\pi=(d_1(\overline{C}),...,d_l(\overline{C}))+ \mathfrak{C}.$$ Therefore, in the Gorenstein case the valuative condition holds for $({\mathcal{A}},\psi_1,...,\psi_r))$ if and only if the inclusion $$\mathfrak{A}\subset (d_1(\overline{C}),...,d_l(\overline{C}))+ \mathfrak{C}$$ is satisfied. The first inclusion follows easily from the definition of $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$. The equalities follow because in the Gorenstein case any element of $H^0(U,\omega_{\tilde{X}}^r)$ is equal to a Gorenstein form multiplied by a global regular function. \[rem:conductorsuma\] Since both $\mathfrak{C}$ and $\mathfrak{K}_\pi$ have non empty conductor sets and $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$ is closed by taking minima, the conductor set of $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$ is non-empty and has an absolute minimum denoted by $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})$. Moreover we have the inequalities $$\label{eq:ineqcond} cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq cond(\mathfrak{C})+cond(\mathfrak{K}_\pi)\leq cond(\mathfrak{C})+(d_1(\overline{C}),...,d_l(\overline{C})).$$ In the Gorenstein case this inequality becomes the equality $$\label{eq:eqcondgor} cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})=cond(\mathfrak{C})+(d_1(\overline{C}),...,d_l(\overline{C})).$$ Likewise, the conductor set of $\mathfrak{A}$ has an absolute minimum denoted by $cond(\mathfrak{A})$. \[rem:valuativeparticular\] The following set of easy observations will have very useful consequences later. 1. If $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq (0,...,0)$, then the Valuative Condition is satisfied. 2. If $\mathfrak{A}$ equals ${{\mathbb{N}}}^l$, then the valuative condition is equivalent to the inequality $$cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq (0,...,0).$$ 3. If $X$ has Gorenstein singularities, then the condition $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq (0,...,0)$ holds if and only if $cond(\mathfrak{C})\leq -(d_1(\overline{C}),...,d_l(\overline{C}))$. \[prop:contval\] The Containment Condition implies the Valuative Condition. We have to translate the Containment Condition $\mathrm{Im}\gamma_1\subset \mathrm{Im}\delta$ into the inclusion of sets $$\mathfrak{A}\subset\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}.$$ We need the concrete description of the sheaf ${\mathcal{A}}$ obtained in Proposition \[prop:genredCM\]: let $\overline{C}$ be the support of ${\mathcal{A}}$, $n:\tilde{C}\to\overline{C}$ be its normalization. We have a chain of inclusions $$\label{eq:chainlocal18} {\mathcal{O}_{\overline{C}}}\subset{\mathcal{A}}\subset n_*{\mathcal{O}_{\tilde{C}}},$$ which is not necessarily unique. We fix one of such chains of inclusions. Since ${\omega_{{\tilde{X}}}}$ is an invertible sheaf and ${\mathcal{A}}$ has support contained in a Stein open subset of $\tilde{X}$ we have the isomorphism ${\mathcal{A}} \otimes {\omega_{{\tilde{X}}}}\cong {\mathcal{A}}$. Hence $$H^0({\tilde{X}},{\mathcal{A}} \otimes {\omega_{{\tilde{X}}}}) \cong H^0({\tilde{X}},{\mathcal{A}}),$$ and $$H^0(U, {\mathcal{A}} \otimes {\omega_{{\tilde{X}}}}) \cong H^0(U, {\mathcal{A}})\subset K(\tilde{C}),$$ where $K(\tilde{C})$ denotes the total fraction ring of the ring $H^0(\tilde{C},{\mathcal{O}_{\tilde{C}}})$. The image $\operatorname{Im}\gamma_1$ is then identified with the inclusion $$H^0({\tilde{X}},{\mathcal{A}})\subset H^0(\tilde{C},{\mathcal{O}_{\tilde{C}}})\subset K(\tilde{C}).$$ As a consequence, if we consider the order function $$ord:K(\tilde{C})\to{{\mathbb{Z}}}^l,$$ we obtain the equality of sets $$\label{eq:1} ord(\operatorname{Im}\gamma_1)=\mathfrak{A}.$$ The morphism $\delta$ is induced by the sections $\{\psi_1, \dots, \psi_r\}$ of ${\mathcal{A}}$. The definition of $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$ gives the equality $$\label{eq:2} ord(\operatorname{Im}\delta)=\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}.$$ Now the result follows, because of equalities (\[eq:1\]) and (\[eq:2\]). Before we extract some useful consequences of Remark \[rem:valuativeparticular\] and Theorem \[th:corres\] we need a further lemma: \[lema:formagorenstein\] Let $(X,x)$ be a complex analytic germ of a normal two-dimensional Gorenstein singularity and $\pi \colon {\tilde{X}}\to X$ be a resolution with exceptional divisor $E=\bigcup_{i=1}^n E_i$. Then for any component $E_j$ where the Gorenstein form $\Omega$ has a pole and for any component $E_k$ where the Gorenstein form has a zero, we have that $E_j \cap E_k = \emptyset$. If the singularity is rational the Gorenstein form does not have poles at any resolution. The result follows for that case. If the singularity is non-rational the Gorenstein form have strict poles at any component of the exceptional divisor of the minimal resolution. If $p$ is a point at a resolution $\tilde{X}$, $E=\cup_{i=1}^r E_i$ is the decomposition in irreducible components of the exceptional divisor at $\tilde{X}$, and $E_p$ is the exceptional divisor at the blow up at $p$, then the order of $\Omega$ at $E_p$ equals $$ord_{E_p}(\Omega):=1+\sum_{i=1}^r ord_{E_i}(\Omega) mult_{p}(E_i).$$ Using induction on the number of blows ups that are necessary in order to obtain the resolution $\pi \colon {\tilde{X}}\to X$ from the minimal resolution the proposition is proved easily in the non-rational case. The consequences announced above are: \[prop:consecuenciaspracticas\] Let $\pi:{\tilde{X}}\to X$ be a resolution. Let ${\mathcal{A}}$ be a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module. Let $\mathfrak{A}$ be its set of orders (see Definition \[def:orderset\]). Let $\overline{C}$ be the support of ${\mathcal{A}}$ and $n:\tilde{C}\to \overline{C}$ be the normalization. Let $(d_1(\overline{C}),...,d_l(\overline{C}))$ be the canonical vector of $\overline{C}$ and $\mathfrak{K}_\pi$ be the canonical set of orders of $\overline{C}$. Let $(\psi_1,...,\psi_r)$ be $r$ global sections spanning ${\mathcal{A}}$ as ${\mathcal{O}_{{\tilde{X}}}}$-module. The ${\mathcal{O}_{X}}$-module ${\mathcal{C}}$ spanned by $(\psi_1,...,\psi_r)$ is rank 1 generically reduced $1$-dimensional Cohen-Macaulay. Let $\mathfrak{C}$ be the set of orders of ${\mathcal{C}}$. Let $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$ be the canonical set of orders of $({\mathcal{A}},(\psi_1,...,\psi_r))$. Let $({\mathcal{M}},(\phi_1,...,\phi_r))$ be the pair associated with $({\mathcal{A}},(\psi_1,...,\psi_r))$ in the proof of Theorem \[th:corres\]. 1. If $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq 0$, then ${\mathcal{M}}$ is full. 2. For the previous condition it is enough to have the inequality $cond(\mathfrak{C})\leq -(d_1(\overline{C}),...,d_l(\overline{C}))$. 3. Suppose that the curve $\overline{C}$ is smooth and meets the exceptional divisor transversely at smooth points. Then ${\mathcal{M}}$ is full if and only if we have the inequality $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq 0$. In the Gorenstein case the inequality becomes $cond(\mathfrak{C})\leq -(d_1(\overline{C}),...,d_l(\overline{C}))$. 4. If there is an index $i$ such that the strict inequality $d_i(\overline{C})>0$ holds, then ${\mathcal{M}}$ is not full. In the Gorenstein case, if $\overline{C}$ meets a component of the exceptional divisor where the Gorenstein form has a zero, then ${\mathcal{M}}$ is not full. 5. If $X$ has Gorenstein singularities, $\pi$ is small with respect to the Gorenstein form and ${\mathcal{C}}={\mathcal{A}}$ (that is, if the module ${\mathcal{M}}$ is special), then ${\mathcal{M}}$ is full. 6. Suppose that $(d_1(\overline{C}),...,d_l(\overline{C}))\leq 0$ and that ${\mathcal{C}}=\pi_*n_*{\mathcal{O}_{\tilde{C}}}$. Then ${\mathcal{M}}$ is a special full ${\mathcal{O}_{\tilde{X}}}$-module. In the Gorenstein case the first inequality holds when $\overline{C}$ does not meet an exceptional divisor where the Gorenstein form has a $0$. If $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq 0$ then $n_*{\mathcal{O}_{\tilde{C}}}$ is contained in $\mathrm{Im}\delta$. Since $\mathrm{Im} \gamma_1$ is contained in $n_*{\mathcal{O}_{\tilde{C}}}$ the Containment Condition holds and Assertion (1) follows by Theorem \[th:corres\]. Assertion (2) is a consequence of Assertion (1) and Inequality (\[eq:ineqcond\]). In order to prove Assertion (3) notice that if $\overline{C}$ is smooth and meets the exceptional divisor transversely at smooth points then ${\mathcal{A}}$ is equal to ${\mathcal{O}_{\overline{D}}}=n_*{\mathcal{O}_{\tilde{D}}}$ and then we have $\mathfrak{A}={{\mathbb{N}}}^l$. In this situation the Valuative Condition is equivalent to the inequality $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq 0$. In the Gorenstein case Equality (\[eq:eqcondgor\]) transforms the previous inequality into $cond(\mathfrak{C})\leq -(d_1(\overline{C}),...,d_l(\overline{C}))$. This proves Assertion (3). In order to prove Assertion (4) notice that the vector $(0,...,0)$ is always contained in $\mathfrak{A}$, since ${\mathcal{O}_{\overline{C}}}$ is contained in ${\mathcal{A}}$. On the other hand, if there is an index $i$ such that the strict inequality $d_i(\overline{C})>0$ then $(0,...,0)$ is not contained in $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$, and the Valuative Condition does not hold. By Lemma \[lema:formagorenstein\], in the Gorenstein case there is an index $i$ such that the strict inequality $d_i(\overline{C})>0$ holds if and only if $\overline{C}$ meets a component of the exceptional divisor where the Gorenstein form has a zero. This proves Assertion (4). For Assertion (6) notice that if ${\mathcal{C}}=\pi_*n_*{\mathcal{O}_{\tilde{C}}}$ then automatically we have the equality ${\mathcal{C}}={\mathcal{A}}$, and ${\mathcal{M}}$ is special. If $(d_1(\overline{C}),...,d_l(\overline{C}))\leq 0$, it is easy to show using the equality ${\mathcal{C}}=\pi_*n_*{\mathcal{O}_{\tilde{C}}}$, that we have the inequality $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq 0$. So by Assertion (1), if $(d_1(\overline{C}),...,d_l(\overline{C}))\leq 0$ then ${\mathcal{M}}$ is full. This proves the assertion, except the addendum about Gorenstein singularities, which follows from Lemma \[lema:formagorenstein\]. Assertion (5) is a bit harder. We will proof the Containment Condition directly by a cohomological argument: we have to prove that image of $\gamma_1$ is contained in the image of $\delta$ in Diagram (\[subdiagram\]). Since we have a small resolution with respect to the Gorenstein form (see Definition \[def:smallresgor\]), we can consider the exact sequence $$\label{eq:exactseqCs} 0 \to {\omega_{{\tilde{X}}}} \to {\mathcal{O}_{{\tilde{X}}}} \to {\mathcal{O}_{Z_K}} \to 0.$$ Now apply the functor $-\otimes-$ to the sequences and , $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em]{ & 0 & 0 & & \\ 0 & {\mathcal{N}} \otimes {\omega_{{\tilde{X}}}} & {\omega_{{\tilde{X}}}}^r & {\mathcal{A}}\otimes {\omega_{{\tilde{X}}}} & 0 \\ 0 & {\mathcal{N}} & {\mathcal{O}_{{\tilde{X}}}}^r & {\mathcal{A}} & 0 \\ & {\mathcal{N}}\otimes {\mathcal{O}_{Z_k}} & {\mathcal{O}_{Z_k}}^r & {\mathcal{A}} \otimes {\mathcal{O}_{Z_k}} & 0 \\ & 0 & 0 & 0 & \\}; \foreach \y [remember=\y as \lasty (initially 2)] in {3, 4} { \foreach \x [remember=\x as \lastx (initially 2)] in {3,4} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); \draw[-stealth] (m-\lasty-\x) -- (m-\y-\x); } } \draw[-stealth] (m-1-2) -- (m-2-2); \draw[-stealth] (m-1-3) -- (m-2-3); \draw[-stealth] (m-2-2) -- (m-2-3); \draw[-stealth] (m-2-3) -- (m-2-4); \draw[-stealth] (m-2-2) -- (m-3-2); \draw[-stealth] (m-2-1) -- (m-2-2); \draw[-stealth] (m-3-1) -- (m-3-2); \draw[-stealth] (m-2-4) -- (m-2-5); \draw[-stealth] (m-3-4) -- (m-3-5); \draw[-stealth] (m-3-2) -- (m-4-2); \draw[-stealth] (m-4-4) -- (m-4-5); \draw[-stealth] (m-4-2) -- (m-5-2); \draw[-stealth] (m-4-3) -- (m-5-3); \draw[-stealth] (m-4-4) -- (m-5-4); \end{tikzpicture}$$ By the last diagram we get the following commutative diagram $$\begin{tikzpicture} \matrix (m) [matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em]{ & H^0\left({\tilde{X}},{\omega_{{\tilde{X}}}}^r\right)& & \vphantom{H^0}H^0\left({\tilde{X}},{\mathcal{A}} \otimes {\omega_{{\tilde{X}}}}\right) & \\ H^0\left({\tilde{X}},{\mathcal{O}_{{\tilde{X}}}}^r\right) & & \vphantom{H^0}H^0\left({\tilde{X}},{\mathcal{A}}\right) & \\ & H^0\left(U,{\omega_{{\tilde{X}}}}^r\right) & & H^0\left(U,{\mathcal{A}} \otimes {\omega_{{\tilde{X}}}}\right) \\ H^0\left(U,{\mathcal{O}_{{\tilde{X}}}}^r\right) & & H^0\left(U,{\mathcal{A}}\right) & \\}; \path[-stealth] (m-1-2) edge (m-1-4) edge (m-2-1) edge [densely dotted] (m-3-2) (m-1-4) edge node[auto]{$\gamma_1$} (m-3-4) edge node[auto]{$\beta$} (m-2-3) (m-2-1) edge [-,line width=6pt,draw=white] (m-2-3) edge node[left=17pt, below=1pt]{$\rho$} (m-2-3) edge node[auto]{$\nu$}(m-4-1) (m-3-2) edge [densely dotted] node[left=30pt, below=1pt]{$\delta$} (m-3-4) edge [densely dotted] node[auto]{$\theta$} (m-4-1) (m-4-1) edge node[auto]{$\delta '$} (m-4-3) (m-3-4) edge node[auto]{$\alpha$} (m-4-3) (m-2-3) edge [-,line width=6pt,draw=white] (m-4-3) edge node[left=5pt,below=-25pt]{$\gamma_1'$} (m-4-3); \end{tikzpicture}$$ We need to prove that $$\label{conditionauxiliar} \text{im}(\gamma_1) \subset \text{im}(\delta).$$ Notice that the maps $\alpha$ and $\theta$ are isomorphisms because the support of ${\mathcal{O}_{Z_K}}$ does not intersect $U$. Since $\alpha$ is injective, the condition is equivalent to $$\operatorname{Im}(\alpha \gamma_1) \subset \operatorname{Im}(\alpha \delta).$$ Since the diagram is commutative and $\theta$ is onto we get $$\text{im}(\alpha \delta) = \text{im}(\delta ' \theta) = \text{im}(\delta ').$$ Hence it is enough to prove that the image of $(\alpha \gamma_1)$ is contained in the image of $\delta '$. Using again that the diagram is commutative and $\rho$ is onto because ${\mathcal{M}}$ is special, we get $$\text{im}(\alpha \gamma_1) = \text{im}(\gamma_1 ' \beta) \subset \text{im}(\gamma_1') = \text{im}(\gamma_1 ' \rho) = \text{im}(\delta ' \nu) \subset \text{im}(\delta '),$$ as we wish. A comparison of correspondences {#sec:comparacion} ------------------------------- In order to compare the two correspondences we need to impose that $X$ is a normal Stein surface with Gorenstein singularities. The following propositions compare the correspondence at the Stein surface (Theorem \[th:corrsing\]) and the correspondences at various resolutions (Theorem \[th:corres\]). \[prop:dirressing\] Let $X$ be a normal Stein surface with Gorenstein singularities. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $\pi_1:{\tilde{X}}_1\to X$ be a resolution and $\rho:{\tilde{X}}_2\to{\tilde{X}}_1$ be the blow up at a point $p$. Denote by $\pi_2:{\tilde{X}}_2\to X$ the composition $\pi_2=\pi_1{{\circ}}\rho$. Denote by ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ the full sheaves associated with $M$ at each of the resolutions. Let $(\phi_1,...,\phi_r)$ be $r$ generic sections of $M$. Let $({\mathcal{A}}_1,(\psi^1_1,...,\psi^1_r))$ and $({\mathcal{A}}_2,(\psi^2_1,...,\psi^2_r))$ be the pairs associated with $({\mathcal{M}}_1,(\phi_1,...,\phi_r))$ and $({\mathcal{M}}_2,(\phi_1,...,\phi_r))$ under Theorem \[th:corres\]. Let $({\mathcal{C}},(\psi^0_1,...,\psi^0_r))$ be the pair associated with $(M,(\phi_1,...,\phi_r))$ under Theorem \[th:corrsing\]. 1. There are inclusions ${\mathcal{C}}\subset (\pi_1)_*{\mathcal{A}}_1\subset (\pi_2)_*{\mathcal{A}}_2$. Under this inclusion the sections $(\psi^i_1,...,\psi^i_r)$ are identified for $i=0,1,2$. The dimension of the quotient $(\pi_i)_*{\mathcal{A}}_i/{\mathcal{C}}$ as a ${{\mathbb{C}}}$-vector space equals $\dim_{{{\mathbb{C}}}}(R^1(\pi_i)_*{\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}_i)-rp_g$ for $i=1,2$. 2. We have the inclusion ${\mathcal{A}}_1\subset \rho_*{\mathcal{A}}_2$ and the dimension of the quotient $\rho_*{\mathcal{A}}_2/{\mathcal{A}}_1$ as a ${{\mathbb{C}}}$-vector space equals $\dim_{{{\mathbb{C}}}}(R^1\rho_*{\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}_2)$. For $i=1,2$ consider the exact sequence $$0\to{\mathcal{O}_{{\tilde{X}}_i}}^r\to {\mathcal{M}}_i\to {{\mathcal{A}}}'_i\to 0,$$ whose first morphism is induced by the sections $(\phi_1,...,\phi_r)$. Dualizing we obtain $$\label{eq:otravez1} 0\to{\mathcal{N}}_i\to{\mathcal{O}_{{\tilde{X}}_i}}^r\to {{\mathcal{A}}}_i\to 0,$$ where the second morphism is induced by the sections $(\psi^i_1,...,\psi^i_r)$. The sections $(\phi_1,...,\phi_r)$ also induce the exact sequence $$0\to {\mathcal{O}_{X}}^r\to M\to{\mathcal{C}}'\to 0,$$ and dualizing it we obtain $$\label{eq:otravez2} 0\to N\to{\mathcal{O}_{X}}^r\to{\mathcal{C}}\to 0,$$ where the second morphism is induced by $(\psi^0_1,...,\psi^0_r)$. Applying $R(\pi_i)_*$ to (\[eq:otravez1\]), we obtain the exact sequence $$\label{exseq:918} 0\to (\pi_i)_*{\mathcal{N}}_i\to{\mathcal{O}_{X}}^r\to (\pi_i)_*{\mathcal{A}}_i\to R^1(\pi_i)_*{\mathcal{N}}_i\to R^1(\pi_i)_*{\mathcal{O}_{X}}^r\to 0.$$ By Lemma \[lema:dualM\] and its proof the first morphism of the previous sequence (for $i=1,2$) coincides with the first morphism of sequence (\[eq:otravez2\]). This implies that the image of the second morphism of the previous sequence (for $i=1,2$) coincides with ${\mathcal{C}}$ and that, under this identification the systems of sections $(\psi^i_i,...,\psi^i_r)$ coincide for $i=0,1,2$. This proves the inclusions ${\mathcal{C}}\subset (\pi_i)_*{\mathcal{A}}_i$ for $i=1,2$ and the identification of the sections. Since ${\mathcal{A}}_i$ is generated by $(\psi^i_1,...,\psi^i_r)$ as a ${\mathcal{O}_{{\tilde{X}}_i}}$-module and ${\mathcal{O}_{{\tilde{X}}_2}}$ contains ${\mathcal{O}_{{\tilde{X}}_1}}$, the inclusion $(\pi_1)_*{\mathcal{A}}_1\subset (\pi_2)_*{\mathcal{A}}_2$ also holds. The equality $${\dim_{\mathbb{C}}(\pi_i)_*{\mathcal{A}}_i/{\mathcal{C}})}=\dim_{{{\mathbb{C}}}}(R^1(\pi_i)_*{\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}_i)-rp_g$$ follows from Exact Sequence (\[exseq:918\]). This shows the first assertion. The proof of the second assertion follows similarly, by applying $R\rho_*$ to Sequence (\[eq:otravez1\]), and taking into account the identification of the sections $(\psi^i_i,...,\psi^i_r)$ for $i=1,2$. For later use we need to compare the sets $\mathfrak{K}_{({\mathcal{A}}_1,(\psi^1_1,...,\psi^1_r))}$ and $\mathfrak{K}_{({\mathcal{A}}_2,(\psi^2_1,...,\psi^2_r))}$ (see Section \[sec:valuative\] for the corresponding definition). \[prop:compcanord\] Consider the same situation than in the previous proposition, but allow non-Gorenstein normal singularities. There exist a non-negative integer vector $(d_1,...,d_l)$ such that we have the equality $$\label{eq:transorder} \mathfrak{K}_{({\mathcal{A}}_2,(\psi^2_1,...,\psi^2_r))}=(d_1,...,d_{l})+\mathfrak{K}_{({\mathcal{A}}_1,(\psi^1_1,...,\psi^1_r))}.$$ As a consequence we have also the equality $$\label{eq:transcond} cond(\mathfrak{K}_{({\mathcal{A}}_2,(\psi^2_1,...,\psi^2_r))})=(d_1,...,d_{l})+cond(\mathfrak{K}_{({\mathcal{A}}_1,(\psi^1_1,...,\psi^1_r))}).$$ The vector is strictly positive if and only if the blowing up center of $\rho$ meets the support of ${\mathcal{A}}_1$. Denote by $E^i$ the exceptional divisor of $\pi_i$. Let $C$ be the support of ${\mathcal{C}}$, let $\overline{C}^i$ be the support of ${\mathcal{A}}_i$. We have a birational morphisms $\rho|_{\overline{C}^2}:\overline{C}^2\to\overline{C}^1$ and $\pi_1|_{\overline{C}^1}: \overline{C}^1\to C$. Consider the normalization $n:\tilde{C}\to \overline{C}^2$. Let $p_j$ for $j=1,...,l$ be the points of $\tilde{C}$ which map via $n$ to a point of the exceptional divisor. Let $(\tilde{C}_j,p_j)$ be the germ at $p_j$. Denote $p^2_j:=n(p_j)$ and $p^1_j:=\rho(p^2_j)$ for $j=1,...,l$. The proposition is trivial if the center of the blowing up $\rho$ does not coincide with $p^1_j$ for any $j$. By notational convenience we assume that $p^1_1$ is the blowing-up center. We denote by $E^2_1$ the exceptional divisor of $\rho$. Choose local coordinates $(x^i_j,y^i_j)$ of ${\tilde{X}}^i$ around each point $p^i_j$. The choice is made so that if two points $p^i_j$ coincide for different $j$, then the corresponding coordinates are also the same, and so that, if $\rho(p^2_j)=p^1_1$, then $\rho$ expresses in local coordinates as $\rho(x^2_j,y^2_j)=(x^2_j,x^2_jy^2_j)$. Let $\beta_1,...,\beta_n$ be a system of generators of $H^0(U,\omega_{{\tilde{X}}})$ as a ${\mathcal{O}_{X}}$-module. The differential form $\beta_k$ expresses in each of the local chart around $p^i_j$ as $\beta=h_{k,j}^idx^i_j\wedge dy^i_j$, where $h^i_{k,j}$ is a germ of meromorphic function at $p^i_j$. If $\rho(p^2_j)=p^1_1$ then $h^2_{k,j}=x^2_j\rho^*h^1_{k,1}$. If $\rho(p^2_j)=p^1_m$ for $m\neq 1$ then $h^2_{k,j}=\rho^*h^1_{k,m}$. In order to compare $\mathfrak{K}_{({\mathcal{A}}_1,(\psi^1_1,...,\psi^1_r))}$ and $\mathfrak{K}_{({\mathcal{A}}_2,(\psi^2_1,...,\psi^2_r))}$ we compare the images of $H^0(U,\omega_{{\tilde{X}}})$ in the ring $\bigoplus_{i=1}^lK(\tilde{C}_i,p_i)$ according with Definition \[def:canoset2\]; we denote each of the images by $Im^i$ for $i=1,2$. Since, by Proposition \[prop:dirressing\] the sections $(\psi^1_1,...,\psi^1_r)$ and $(\psi^2_1,...,\psi^2_r)$ are identified, each of them define the same $l$-uple in $\bigoplus_{i=1}^lK(\tilde{C}_i,p_i)$, which we denote by $(\psi_v|_{\tilde{C}_1},...,\psi_v|_{\tilde{C}_1})$ for $v\in\{1,...,r\}$. Then $Im^i$ is the ${\mathcal{O}_{X}}$-module spanned by $$\left \{(h^i_{1}|_{\tilde{C}_1}\psi_v|_{\tilde{C}_1},...,h^i_{l}|_{\tilde{C}_l}\psi_v|_{\tilde{C}_l}):v\in\{1,...,r\} \right \}.$$ Enumerate the points $p^2_1,...,p^2_l$ so that those whose image by $\rho$ equals $p^1_1$ are $p^2_1,...,p^2_{l_1}$ for $l_1\leq l$. We have the equality $$Im^2=(x^2_1|_{\overline{C}^2_1},...,x^2_{l_1}|_{\overline{C}^2_{l_1}},1,...,1)Im^1.$$ As a consequence, if for any $w\leq l_1$ we define the intersection multiplicity $d_w:=I_{p^2_w}(E^2_1,\overline{C}^2_w)$ we have the equality $$\mathfrak{K}_{({\mathcal{A}}_2,(\psi^2_1,...,\psi^2_r))}=(d_1,...,d_{l_1},0,...,0)+\mathfrak{K}_{({\mathcal{A}}_1,(\psi^1_1,...,\psi^1_r))}.$$ \[prop:invressing\] Let $\pi:{\tilde{X}}\to X$ be a resolution of a normal Stein surface with Gorenstein singularities, which is an isomorphism over the regular locus of $X$. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be a pair formed by a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module, whose support meets $E$ in finitely many points, and a set of $r$ global sections spanning ${\mathcal{A}}$ as ${\mathcal{O}_{{\tilde{X}}}}$-module and satisfying the Containment Condition. Let ${\mathcal{C}}$ be the ${\mathcal{O}_{X}}$-module spanned by $\psi_1,...,\psi_r$. Then ${\mathcal{C}}$ is a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{X}}$-module. Let $({\mathcal{M}},(\phi_1,...,\phi_r))$ and $(M,(\phi'_1,...,\phi'_r))$ be the results of applying the correspondences of Theorems \[th:corres\] and \[th:corrsing\] at ${\tilde{X}}$ and at the $X$ to $({\mathcal{A}},(\psi_1,...,\psi_r))$ and $({\mathcal{C}},(\psi_1,...,\psi_r))$ respectively. Then we have the equalities $\pi_*{\mathcal{M}}=M$ and $(\phi_1,...,\phi_r)=(\phi'_1,...,\phi'_r)$. According with the proof of Theorem \[th:corres\] and its proof the module ${\mathcal{N}}$ in the sequence $0\to{\mathcal{N}}\to{\mathcal{O}_{{\tilde{X}}}}^r\to{\mathcal{A}}\to 0$ is the dual of ${\mathcal{M}}$. Pushing down by $\pi_*$ we obtain $$0\to\pi_*{\mathcal{N}}\to{\mathcal{O}_{X}}^r\to\pi_*{\mathcal{A}}\to R^1\pi_*{\mathcal{N}}\to R^1\pi_*{\mathcal{O}_{{\tilde{X}}}}^r\to 0,$$ and the image of the map ${\mathcal{O}_{X}}^r\to\pi_*{\mathcal{A}}$ is the ${\mathcal{O}_{X}}$-module spanned by $\psi_1,...,\psi_r$, that is, the module ${\mathcal{C}}$. So we obtain the sequence $$0\to\pi_*{\mathcal{N}}\to{\mathcal{O}_{X}}^r\to{\mathcal{C}}\to 0.$$ According with Theorem \[th:corrsing\] and its proof the module $\pi_*{\mathcal{N}}$ is isomorphic to the dual of $M$. By Lemma \[lema:dualM\] the module $\pi_*{\mathcal{N}}$ is isomorphic to the dual of $\pi_*{\mathcal{M}}$. This concludes the proof of the equality $\pi_*{\mathcal{M}}=M$. Under the equality, the coincidence of the sections is straightforward. The minimal adapted resolution ============================== In this section we show that, given a Stein normal surface $X$ and a reflexive ${\mathcal{O}_{X}}$-module, there is a minimal resolution for which the associated full sheaf is generated by global sections. This resolution will be crucial later. \[prop:minadap\] Let $X$ be a Stein normal surface. If $M$ is a reflexive ${\mathcal{O}_{X}}$-module, then there exists a unique minimal resolution $\rho \colon {\tilde{X}}' \to X$ such that the associated full ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}:= \left ( \rho^* M \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ is generated by global sections. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module, $\pi \colon {\tilde{X}}\to X$ be the minimal resolution with exceptional divisor $E$ and denote by ${\mathcal{M}}= \left ( \pi^* M \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. If ${\mathcal{M}}$ is generated by global sections, then we are done. If ${\mathcal{M}}$ is not generated by global sections, then there exists a finite set of points $S=\{p_1, \dots, p_n\} \subset E$ where ${\mathcal{M}}$ fails to be generated by global sections. Assume that the rank of ${\mathcal{M}}$ is $r$. Take $r$ generic sections of ${\mathcal{M}}$ and consider the exact sequence given by the sections $$0 \to {\mathcal{O}_{{\tilde{X}}}}^r \to {\mathcal{M}} \to {\mathcal{A}}' \to 0.$$ By the degeneracy module definition (Definition \[def:degeneracymodule\]) we have the inclusion $S \subset \text{Supp}({\mathcal{A}}')$. Let $\overline{C}$ be the support of ${\mathcal{A}}'$ and $(d_1(\overline{C}),...,d_r(\overline{C}))$ be the associated canonical vector (see Definition \[def:tranlation\]). By Proposition \[prop:consecuenciaspracticas\], (4) we have the inequality $$\label{eq:recordatorio} (d_1(\overline{C}),...,d_r(\overline{C})) \leq (0,...,0).$$ Denote by $\sigma_{S} \colon {\tilde{X}}' \to X$ the blow up at the set of centers $S$. Therefore we have the following commutative diagram $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em]{ {\tilde{X}}' & {\tilde{X}}\\ & X \\}; \draw[-stealth] (m-1-1) edge node [above] {$\sigma_{S}$} (m-1-2); \draw[-stealth] (m-1-1) edge node [right] {$\pi'$} (m-2-2); \draw[-stealth] (m-1-2) edge node [right] {$\pi$} (m-2-2); \end{tikzpicture}$$ Denote by ${\mathcal{M}}'$ the full ${\mathcal{O}_{\tilde{X}'}}$-module associated to $M$. If ${\mathcal{M}}'$ is generated by global sections, then we are done, otherwise we repeat the procedure. In order to prove that this process eventually ends we use $(d_1(\overline{C}),...,d_r(\overline{C}))$ as a resolution invariant. We take the same generic global sections for ${\mathcal{M}}'$ and ${\mathcal{M}}$ (in both cases the set of global sections is $M$). The support $\overline{C}'$ of the degeneracy module of ${\mathcal{M}}'$ for this sections is the strict transform of $\overline{C}$ by $\sigma_S$. The normalization $\tilde{C}$ of $\overline{C}$ and $\overline{C}'$ is the same. Let $\{(\tilde{C}_i,p_i)\}_{i=1}^l$ be the branches of $\tilde{C}$ considered at the beginning of Section \[sec:valuative\]. Let $\beta$ be meromorphic differential form in $\tilde{X}$. By the behavior of the poles of the pullback of a meromorphic differential form by a blow up at a point in a smooth surface, we obtain that the order of $\sigma_S^*\beta$ at the different branches $(\tilde{C}_i,p_i)$ for $i=1,...,l$ is greater or equal that the order of $\beta$. Moreover the order is strictly greater if the blowing up center meets the component that we are dealing with. This implies the strict inequality $$(d_1(\overline{C}),...,d_r(\overline{C}))< (d_1(\overline{C}'),...,d_r(\overline{C}')).$$ This together with Inequality (\[eq:recordatorio\]) shows that the process terminates after finitely many steps. \[def:minadap\] Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module. The minimal resolution $\rho:{\tilde{X}}\to X$ where the associated full ${\mathcal{O}_{{\tilde{X}}}}$-module is generated by global sections is called the [*minimal adapted resolution*]{} to $M$. \[rem:minadapgor\] Let $X$ be a Stein surface with Gorenstein singularities. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module. Then the minimal adapted resolution to $M$ is small with respect to the canonical form. This is a consequence of Inequality (\[eq:recordatorio\]) and Proposition \[prop:consecuenciaspracticas\], (4). At the minimal adapted resolution the degeneracy module of the full sheaf for a generic set of sections has special properties, and Lemma 1.2 of [@AV] holds as stated there. \[lem:ArtinVerdierminadap\] Let $X$ be a Stein normal surface. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $\rho:{\tilde{X}}\to X$ be a resolution which dominates the minimal adapted resolution to $M$. The degeneracy module ${\mathcal{A'}}$ of a set of $r$ generic global sections is isomorphic to ${\mathcal{O}_{D}}$, where $D\subset{\tilde{X}}$ is a smooth curve meeting the exceptional divisor transversely at its smooth locus. Moreover, by changing the sections the meeting points of $D$ with the exceptional divisor also change. It is a simplification of the proof of Proposition \[prop:generalizationAV\]. Our aim now is to characterize numerically the minimal adapted resolution. \[prop:minadapnumchar\] Let $X$ be a Stein normal surface. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $(\phi_1,...,\phi_r)$ be $r$ generic sections. Let $\pi:{\tilde{X}}\to X$ be a resolution, denote by ${\mathcal{M}}$ the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated with $M$. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be the pair associated with $({\mathcal{M}},(\phi_1,...,\phi_r))$ under Theorem \[th:corres\]. Genericity of the sections imply that the associated canonical set of orders $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$ (see Definition \[def:canoset2\]), and its minimal conductor $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})$ are independent of the chosen sections. Then the following properties hold: 1. At the minimal adapted resolution we have the equality $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})=(0,...,0)$. 2. At any resolution we have the inequality $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})\leq (0,...,0)$. 3. A resolution dominates the minimal adapted resolution if and only if we have the equality $$cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})=(0,...,0).$$ Let $\pi:{\tilde{X}}_1\to X$ be the minimal adapted resolution to $M$ and ${\mathcal{M}}_1$ the full ${\mathcal{O}_{{\tilde{X}}_1}}$-module associated with $M$. Let $E_1$ be the exceptional divisor. Consider the decomposition in irreducible components $E_1=\cup_{j=1}^{m}E_{1,j}$. Let $({\mathcal{A}}_1,(\psi_1,...,\psi_r))$ be the pair associated with $({\mathcal{M}}_1,(\phi_1,...,\phi_r))$ by Theorem \[th:corres\]. Let $\overline{C}^1$ be the support of ${\mathcal{A}}_ 1$. By Lemma \[lem:ArtinVerdierminadap\], $\overline{C}^1$ is a smooth curve meeting the exceptional divisor $E_1$ transversely at smooth points and we have the isomorphism ${\mathcal{A}}_1\cong{\mathcal{O}_{\overline{C}^1}}$. Then, by Proposition \[prop:consecuenciaspracticas\] (3), at the minimal adapted resolution $\pi:{\tilde{X}}_1\to X$ we have the inequality $cond(\mathfrak{K}_{({\mathcal{A}}_1,(\psi^1_1,...,\psi^1_r))})\leq (0,...,0)$. Suppose that the inequality is strict. We will derive a contradiction, which will prove Assertion (1). Up to reindexing we may assume that the first coordinate of $cond(\mathfrak{K}_{({\mathcal{A}}_1,(\psi_1,...,\psi_r))})$ is negative. Since $\overline{C}^1$ is smooth it coincides with its normalization. Let $p_1$ be the point of $\overline{C}^1\cap E_1$ so that the order at the branch $(\overline{C}^1,p_1)$ gives the first coordinate of the order function (see Section \[sec:valuative\] to recall the corresponding definitions). Let $\rho:{\tilde{X}}_2\to{\tilde{X}}_1$ be the blow up at $p_1$; define $\pi_2:=\pi_1{{\circ}}\rho$. Let $\overline{C}^2$ be the strict transform of $\overline{C}^1$ by $\rho$. Define ${\mathcal{A}}_2:={\mathcal{O}_{\overline{C}^2}}$. Since $\rho$ induces an isomorphism between ${\mathcal{A}}_2={\mathcal{O}_{\overline{C}^2}}$ and ${\mathcal{A}}_1={\mathcal{O}_{\overline{C}^1}}$ the sections $(\psi_1,...,\psi_r)$ of ${\mathcal{A}}_1$ may also be regarded as sections of ${\mathcal{A}}_2$. A computation like in the proof of Proposition \[prop:compcanord\] shows the equality $cond(\mathfrak{K}_{({\mathcal{A}}_2,(\psi_1,...,\psi_r))})=(1,0,...,0)+cond(\mathfrak{K}_{({\mathcal{A}}_1,(\psi_1,...,\psi_r))})$. Therefore we have the inequality $cond(\mathfrak{K}_{({\mathcal{A}}_2,(\psi_1,...,\psi_r))})\leq (0,...,0)$, and by Proposition \[prop:consecuenciaspracticas\] (1), the correspondence of Theorem \[th:corres\] assigns to $({\mathcal{A}}_2,(\psi_1,...,\psi_r))$ a pair $({\mathcal{M}}_2,(\phi^2_1,...,\phi^2_r))$, where ${\mathcal{M}}_2$ is a full ${\mathcal{O}_{{\tilde{X}}_2}}$-module and $(\phi^2_1,...,\phi^2_r)$ is a system of nearly generic global sections. An application of Proposition \[prop:invressing\] shows the equalities $(\pi_2)_*{\mathcal{M}}_2=(\pi_1)_*{\mathcal{M}}_1$ and $(\phi^2_1,...,\phi^2_r)=(\phi_1,...,\phi_r)$. By Lemma \[lem:ArtinVerdierminadap\], there is a slight perturbation $(\phi'_1,...,\phi'_r)$ of the sections $(\phi_1,...,\phi_r)$ such that if we denote by ${\mathcal{A}}_1'$ the degeneracy module of $({\mathcal{M}}_1,(\phi'_1,...,\phi'_r))$, then its support $(\overline{C}^1)'$ satisfies - it does not meet the blowing up center $\overline{C}_1\cap E$; - we have the equality of intersection numbers $(\overline{C}^1)'\cdot E_{1,j}=\overline{C}^1\cdot E_{1,j}$ for any irreducible component $E_{1,j}$. The support of the degeneracy module of $({\mathcal{M}}_2,(\phi_1,...,\phi_r))$ is equal to $\overline{C}^2$. On the other hand, since $(\overline{C}^1)'$ does not meet the blowing up center $\overline{C}_1\cap E$, the support of the degeneracy module of $({\mathcal{M}}_2,(\phi'_1,...,\phi'_r))$ is the strict transform of $(\overline{C}^1)'$ to ${\tilde{X}}_2$. Let $F$ be the exceptional divisor of $\rho$. Observe that $\overline{C}^2\cdot F=1$, and by property (2) above $(\overline{C}^1)'\cdot F=0$. Since the Poincare dual of the support of the degeneracy locus is the first Chern class of the module ${\mathcal{M}}_2$, we have two different Chern class representations intersecting differently the cycle $F$. This is a contradiction which proves Assertion (1). Assertions (2) and (3) are simple Corollaries of Assertion (1) and Proposition \[prop:compcanord\]. It is convenient to specialize the previous Proposition to the Gorenstein case: \[cor:minadapnumchargor\] With the notation of the previous Proposition, and of Proposition \[prop:consecuenciaspracticas\], if $X$ has Gorenstein singularities we have: 1. At the minimal adapted resolution we have the equality $cond(\mathfrak{C})=(-d_1(\overline{C}),...,-d_l(\overline{C}))$. 2. At any resolution we have the inequality $cond(\mathfrak{C})\leq (-d_1(\overline{C}),...,-d_l(\overline{C}))$. 3. A resolution dominates the minimal adapted resolution if and only if we have the equality $$cond(\mathfrak{C})=(-d_1(\overline{C}),...,-d_l(\overline{C})).$$ The behaviour of the speciality defect and minimal adapted resolutions of special modules {#sec:minimaladapted} ----------------------------------------------------------------------------------------- We study the behaviour of the specialty defect under blow up, and show that if the specialty defect vanishes at a given resolution, it vanishes at any resolution. As a corollary we establish the existence of special reflexive modules and show an interesting property of their generic degeneracy modules, which links them with arcs in the singularity. The following two propositions control the behavior of the specialty defect under blow up. \[prop:specialtybehaviour\] Let $X$ be a Stein normal surface. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module. Let $\pi \colon {\tilde{X}}\to X$ be a resolution and $p$ be a point in ${\tilde{X}}$. Denote by $\sigma \colon {\tilde{X}}' \to {\tilde{X}}$ the blow up of the point $p$. We have the following diagram $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em]{ {\tilde{X}}' & {\tilde{X}}\\ & X \\}; \draw[-stealth] (m-1-1) edge node [above] {$\sigma$} (m-1-2); \draw[-stealth] (m-1-1) edge node [right] {$\rho$} (m-2-2); \draw[-stealth] (m-1-2) edge node [right] {$\pi$} (m-2-2); \end{tikzpicture}$$ where $\rho:= \pi \circ{} \sigma$. Denote by ${\mathcal{M}}= \left( \pi^* M \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ and ${\mathcal{M}}'= \left( \rho^* M \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. Then the specialty defect of ${\mathcal{M}}$ is less or equal to the specialty defect of ${\mathcal{M}}'$. Moreover if ${\mathcal{M}}$ is generated by global sections at $p$, then the specialty defect of ${\mathcal{M}}$ equals the specialty defect of ${\mathcal{M}}'$. Denote by $$\begin{aligned} {\mathcal{N}}' &= {\mathcal{M}}'^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}},\\ {\mathcal{N}} &= {\mathcal{M}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}.\end{aligned}$$ Since $\rho= \pi \circ{} \sigma$, in order to compute $R^1 \rho_*{\mathcal{N}}'$ we use the Leray spectral sequence. In this case the page $E_2^{(p,q)}$ of the spectral sequence is given by $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells, nodes={minimum width=5ex,minimum height=3ex,outer sep=-5pt}, column sep=1ex, row sep=1ex]{ & & & & \\ 2 & 0 & 0 & 0 & \\ 1 & \pi_*\left(R^1 \sigma_* {\mathcal{N}}'\right ) & R^1 \pi_*\left(R^1 \sigma_* {\mathcal{N}}'\right ) & 0 & \\ 0 & \pi_*\left(\sigma_* {\mathcal{N}}' \right ) & R^1 \pi_*\left(\sigma_* {\mathcal{N}}' \right ) & 0 & \\ \quad\strut & 0 & 1 & 2 & \quad\strut \\}; \draw[thick] (m-1-1.east) -- (m-5-1.east) ; \draw[thick] (m-5-1.north) -- (m-5-5.north) ; \end{tikzpicture}$$ The spectral sequence degenerates, therefore we obtain the following exact sequence $$\label{exact:cambioN} 0 \to R^1 \pi_*\left(\sigma_* {\mathcal{N}}' \right ) \to R^1 \rho_* {\mathcal{N}}' \to \pi_*\left(R^1 \sigma_* {\mathcal{N}}'\right ) \to 0.$$ Now by adjuction we have the following identification $$\begin{aligned} \begin{split}\label{eq:cambioN} R^1 \pi_*\left(\sigma_* {\mathcal{N}}' \right ) &= R^1 \pi_* \left(\sigma_* \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}'}}}\left(\sigma^* \pi^* M, {\mathcal{O}_{{\tilde{X}}'}}\right )\right )\\ &=R^1 \pi_* \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left(\pi^* M, \sigma_* {\mathcal{O}_{{\tilde{X}}'}}\right )\\ &=R^1 \pi_* \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left(\pi^* M, {\mathcal{O}_{{\tilde{X}}}}\right )\\ &=R^1 \pi_* {\mathcal{N}}. \end{split}\end{aligned}$$ By and we get $$\label{exact:cambioN2} 0 \to R^1 \pi_* {\mathcal{N}} \to R^1 \rho_* {\mathcal{N}}' \to \pi_*\left(R^1 \sigma_* {\mathcal{N}}'\right ) \to 0.$$ Therefore the specialty defect of ${\mathcal{M}}$ is less or equal to the specialty defect of ${\mathcal{M}}'$. Assume that ${\mathcal{M}}$ is generated by global sections at $p$. By the previous exact sequence we only need to prove that $\pi_{*}\left(R^1 \sigma_* {\mathcal{N}}'\right )=0$. Now consider the exact sequence given by the natural map from $\pi^* M$ to its double dual $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0 & T & \pi^* M & {\mathcal{M}} & {\mathcal{S}} & 0, \\}; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,6} { \draw[-stealth] (m-1-\lastx) -- (m-1-\x); } \end{tikzpicture}$$ where $T$ is the kernel and ${\mathcal{S}}$ is the cokernel. Notice that the support of ${\mathcal{S}}$ is the set $S$ (the points where ${\mathcal{M}}$ fails to be generated by global sections, see Section \[chapter:artin-verdier-esnault\]). The last exact sequence can be split as follows $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0 & T & \pi^* M & \pi^* M / T & 0, \\ 0 & \pi^* M / T & {\mathcal{M}} & {\mathcal{S}} & 0. \\}; \foreach \y in {1,2} { \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,5} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); } } \end{tikzpicture}$$ Applying the functor $\sigma^* -$ to the last two exact sequences we obtain $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0 & K_1 & \sigma^*T & \rho^* M & \sigma^* \pi^* M / T & 0, \\ 0 & K_2 & \sigma^* \pi^* M / T & \sigma^*{\mathcal{M}} & \sigma^*{\mathcal{S}} & 0, \\}; \foreach \y in {1,2} { \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,6} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); } } \end{tikzpicture}$$ where $K_1$ and $K_2$ are the modules that make the last sequences exact (recall that $\sigma^* -$ is just a right exact functor). Hence we split the previous exact sequences as follows $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em]{ 0 & K_1 & \sigma^*T & H_1 & 0, \\ 0 & H_1 & \rho^* M & \sigma^*\pi^* M / T & 0, \\ 0 & K_2 & \sigma^* \pi^* M / T & H_2 & 0, \\ 0 & H_2 & \sigma^*{\mathcal{M}} & \sigma^*{\mathcal{S}} & 0.\\}; \foreach \y in {1,2,3,4} { \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,5} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); } } \end{tikzpicture}$$ Dualizing the first, second and third exact sequences we get $$\begin{aligned} H_1^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} &\cong 0, \quad \text{because $\sigma^* T$ is supported in the exceptional divisor},\\ \left(\sigma^*\pi^* M / T\right )^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} &\cong (\rho^* M)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}, \quad \text{by the previous identification}, \\ H_2^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} &\cong \left(\sigma^*\pi^* M / T\right )^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}, \quad \text{because $K_2$ is supported in the exceptional divisor}.\end{aligned}$$ Hence, as we have ${\mathcal{N}}' = {\mathcal{M}}'^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \cong (\rho^* M)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ we get ${\mathcal{N}}' \cong \left(\sigma^*\pi^* M / T\right )^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\cong H_2^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. Finally dualizing the fourth exact sequence and using the previous identifications we get the exact sequence $$0 \to \left(\sigma^*{\mathcal{M}}\right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \to {\mathcal{N}}' \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}'}}}^1\left(\sigma^* {\mathcal{S}}, {\mathcal{O}_{{\tilde{X}}'}}\right ).$$ Since the point $p$ does not belong to the support of ${\mathcal{S}}$, we get that the support of $\sigma^* {\mathcal{S}}$ is zero dimensional, therefore $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}'}}}^1\left(\sigma^* {\mathcal{S}}, {\mathcal{O}_{{\tilde{X}}'}}\right )$ is equal to zero. Hence we get $$R^1 \sigma_* \left(\sigma^*{\mathcal{M}}\right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \cong R^1 \sigma_* {\mathcal{N}}'.$$ Since ${\mathcal{M}}$ is locally free and we obtain ${\tilde{X}}'$ taking the blow up in the point $p$ we get $$R^1 \sigma_* \left ( \left (\sigma^*{\mathcal{M}}\right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right)= R^1 \sigma_* \left ( \left(\sigma^*{\mathcal{O}_{{\tilde{X}}}}^r\right )^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}} \right)= R^1 \sigma_* {\mathcal{O}_{{\tilde{X}}'}}^r = 0.$$ Hence $R^1 \sigma_* {\mathcal{N}}'$ is equal to zero. \[cor:especialminimal\] Let $X$ be a Stein normal surface. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module. If the full sheaf associated to $M$ at the minimal adapted resolution is special, the the full sheaf associated to $M$ at the minimal resolution of $X$ is also special. \[th:characspecial\] Let $X$ be a Stein normal surface. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module. The module $M$ is special if and only if the full sheaf associated with $M$ at its minimal adapted resolution is special. Denote by $\pi:{\tilde{X}}\to X$ the minimal resolution adapted to $M$ and by $\pi_{\text{min}} \colon {\tilde{X}}_{\text{min}} \to X$ be the minimal resolution of $X$. We need to prove that for any resolution $\rho \colon \hat{X} \to X$, the full sheaf $\hat{{\mathcal{M}}}= \left(\rho^* M\right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ is special (Definition \[def:espmodule\]). If the minimal resolution coincides with the minimal resolution adapted to $M$, then by Proposition \[prop:specialtybehaviour\] we are done. Suppose that the minimal resolution and the minimal resolution adapted to $M$ do not coincide. By taking a finite succession of blowing ups in different points we obtain a resolution $\breve{\rho} \colon \breve{X} \to X$ such that it satisfies the following diagram $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em]{ \breve{X} & {\tilde{X}}\\ \hat{X} & {\tilde{X}}_{min} \\}; \draw[-stealth] (m-1-1) edge node[auto]{$o$} (m-1-2); \draw[-stealth] (m-1-1) edge node[auto]{$\nu$} (m-2-1); \draw[-stealth] (m-1-2) edge node[auto]{$\pi$} (m-2-2); \draw[-stealth] (m-2-1) edge node[auto]{$\rho$} (m-2-2); \end{tikzpicture}$$ where $\nu$ and $o$ are a composition of blowings up in points and $\breve{\rho}= \rho \circ{}\nu$. By Proposition \[prop:specialtybehaviour\] the full sheaf associated to $M$ in the resolution $\breve{X}$ is special. Again by Proposition \[prop:specialtybehaviour\] the specialty defect of the full sheaf associated to $M$ in the resolution $\hat{X}$ is less or equal to zero, and hence equal to $0$. Therefore $M$ is special. \[def:mincanord\] Let $(X,x)$ be a normal surface singularity, $\pi:{\tilde{X}}\to X$ a resolution, and $F$ an irreducible component of the exceptional divisor. The [*minimal canonical order at*]{} $F$ is defined to be $$ord_F(K_X):=\min\left(\{ord_F(div(\beta):\beta\in H^0(U,\omega_{{\tilde{X}}})\}\right).$$ \[rem:obsmco\] The following easy observations hold: 1. The minimal canonical order at $F$ does not depend on the resolution where $F$ appears. 2. In the Gorenstein case the minimal canonical order at $F$ is the order at $F$ of the Gorenstein form. 3. In the Gorenstein case the minimal canonical order of a divisor appearing at the minimal resolution is non-positive. 4. If $F$ is a divisor and $F'$ is another divisor obtained by blowing up $F$ at a generic point then we have $ord_{F'}(K_X)=ord_F(K_X)+1$. \[cor:existenmodespeciales\] Let $(X,x)$ be a normal Gorenstein surface singularity. Then there exist non-free special reflexives modules. A consequence of the third and fourth assertions of Remark \[rem:obsmco\] is the existence of a resolution $\pi:{\tilde{X}}\to (X,x)$ such that a component $F$ of the exceptional divisor has minimal canonical order equal to $0$. Let $D$ be a irreducible smooth curve transverse to the exceptional divisor $F$ at a generic point. We choose ${\mathcal{A}}={\mathcal{O}_{D}}$ and consider $(\psi_1,...,\psi_r)$ generators of ${\mathcal{A}}$ as a ${\mathcal{O}_{X}}$-module. In that case $\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))}$ equals ${{\mathbb{N}}}$, and hence we have the equality $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})=0$. By Proposition \[prop:consecuenciaspracticas\] (1), if $({\mathcal{M}},(\phi_1,...,\phi_r))$ is the pair associated with $({\mathcal{A}},(\psi_1,...,\psi_r))$ by Theorem \[th:corres\], the sheaf ${\mathcal{M}}$ is full. The sheaf ${\mathcal{M}}$ is special. Indeed, since the sections $(\psi_1,...,\psi_r)$ generate ${\mathcal{A}}$ as a ${\mathcal{O}_{X}}$-module, the third map in the exact sequence $$\label{seq:largaotravez} 0\to\pi_*{\mathcal{N}}\to{\mathcal{O}_{X}}^r\stackrel{(\psi_1,...,\psi_r)}{\longrightarrow}\pi_*{\mathcal{A}}\to R^1\pi_*{\mathcal{N}}\to R^1\pi_*{\mathcal{O}_{{\tilde{X}}}}^r\to 0$$ is surjective. By Proposition \[prop:minadapnumchar\] the resolution $\pi:{\tilde{X}}\to (X,x)$ is the minimal adapted resolution to the module $\pi_*{\mathcal{M}}$. Finally, by Theorem \[th:characspecial\] the reflexive ${\mathcal{O}_{X}}$-module $\pi_*{\mathcal{M}}$ is special. Non-freeness holds because by construction ${\mathcal{M}}$ has non-trivial first Chern class. The next proposition explains the structure of degeneracy modules of special reflexive modules for sets of generic sections. . \[prop:Aeslanormalizacion\] Let $X$ be a Stein normal surface with Gorenstein singularities. Let $M$ be a special reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $(\phi_1,...,\phi_r)$ be $r$ generic sections. Let ${\mathcal{C}}$ be the degeneracy module of $(M,(\phi_1,...,\phi_r))$, and let $C$ be its support. Let $n:\tilde{C}\to C$ be the normalization. Then we have the isomorphism ${\mathcal{C}}\cong n_*{\mathcal{O}_{\tilde{C}}}$. Conversely, let $C\subset X$ be a reduced curve and $n:\tilde{C}\to C$ be its normalization. Let $(\psi_1,...,\psi_r)$ be a set of generators of $n_*{\mathcal{O}_{\tilde{C}}}$ as a ${\mathcal{O}_{X}}$-module. Let $(M,(\phi_1,...,\phi_r))$ be the pair associated with $(n_*{\mathcal{O}_{\tilde{C}}},(\psi_1,...,\psi_r))$ under the correspondence of Theorem \[th:corrsing\]. Then $M$ is a special reflexive module. Let $\pi:{\tilde{X}}\to X$ be the minimal adapted resolution to $M$ and ${\mathcal{M}}$ be the associated full sheaf. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be the pair associated with $({\mathcal{M}},(\phi_1,...,\phi_r))$ by Theorem \[th:corres\]. By Lemma \[lem:ArtinVerdierminadap\] we have the isomorphism ${\mathcal{A}}\cong{\mathcal{O}_{D}}$ for a smooth curve meeting the exceptional divisor transversely at smooth points. By specialty and the exact sequence (\[seq:largaotravez\]) we have the equality ${\mathcal{C}}=\pi_*{\mathcal{A}}$. For the converse let $(M,(\phi_1,...,\phi_r))$ be the pair associated with $(n_*{\mathcal{O}_{\tilde{C}}},(\psi_1,...,\psi_r))$ under the correspondence of Theorem \[th:corrsing\]. Consider the minimal resolution $\pi:{\tilde{X}}\to X$ adapted to $M$. Let $\overline{C}$ be the strict transform of $C$ and let ${\mathcal{A}}:=n_* {\mathcal{O}_{\tilde{C}}}$. By Proposition \[prop:consecuenciaspracticas\] $(6)$, if $({\mathcal{M}},(\phi_1,...,\phi_r))$ is the pair associated with $({\mathcal{A}},(\psi_1,...,\psi_r))$ under Theorem \[th:corres\], then the ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}$ is full and special. By Proposition \[prop:invressing\] we have that $\pi_*{\mathcal{M}}$ is isomorphic to $M$. Since ${\mathcal{M}}$ is special, the module $M$ is special by Theorem \[th:characspecial\]. Let us record an interesting property of minimal adapted resolutions of special reflexive modules: \[prop:minadapspproperty\] Let $X$ be a Stein normal surface with Gorenstein singularities. Let $M$ be a special reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $(\phi_1,...,\phi_r)$ be $r$ generic sections. Let $\pi:{\tilde{X}}\to X$ be the minimal adapted resolution to $M$ and ${\mathcal{M}}$ be the associated full sheaf. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be the pair associated with $({\mathcal{M}},(\phi_1,...,\phi_r))$ by Theorem \[th:corres\]. The support of ${\mathcal{A}}$ is a smooth curve meeting the exceptional divisor transversely at smooth points which are located at divisors where the minimal canonical order vanishes. We only need to prove that $D$ only meets at components where the minimal canonical order vanishes. Since $\pi:{\tilde{X}}\to X$ is the minimal adapted resolution we have the equality $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})=(0,...,0)$ by Proposition \[prop:minadapnumchar\]. This, together with the equality ${\mathcal{C}}=n_*{\mathcal{O}_{D}}$ (which follows from the previous proposition), implies that $D$ only meets at components where the minimal canonical order vanishes. The decomposition in indecomposables and the irreducible components of the degeneracy locus {#sec:indecomposables} ------------------------------------------------------------------------------------------- Here we determine the relation between the decomposition of a special reflexive module into indecomposables, and the decomposition of its generic degeneracy locus into irreducible components. Here we work in the local case instead of the more general Stein surface case: we consider $(X,x)$ to be a normal Gorenstein surface singularity. \[prop:decompesp\] Let $(X,x)$ be a normal Gorenstein surface singularity. Let $M$ be a special ${\mathcal{O}_{X}}$ module of rank $r$ and $(\phi_1,...,\phi_r)$ be a set of $r$ generic sections. Let $\pi:{\tilde{X}}\to X$ be any resolution. Let ${\mathcal{M}}$ be the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M$. Let $({\mathcal{C}},(\psi^0_1,...,\psi^0_r))$ and $({\mathcal{A}},(\psi^1_1,...,\psi^1_r))$ be the results of applying Theorems \[th:corrsing\] and  \[th:corres\] to $(M,(\phi_1,...,\phi_r))$ and $({\mathcal{M}},(\phi_1,...,\phi_r))$ respectively. Suppose that $M$ has no free direct factors. Then are natural bijections between the following sets 1. The indecomposable direct summands of $M$. 2. The indecomposable direct summands of ${\mathcal{M}}$. 3. The irreducible components of the support of ${\mathcal{C}}$. 4. The irreducible components of the support of ${\mathcal{A}}$. The first and second sets are in a bijection via $\pi_*$. The third and fourth sets are in bijection via $\pi$. Now we show a bijection between the first and third set. For this is convenient to choose $\pi:{\tilde{X}}\to X$ to be the minimal resolution adapted to $M$ (or at least dominating it). Let $C$ be the support of ${\mathcal{C}}$. By Lemma \[lem:ArtinVerdierminadap\] the support $\overline{C}$ of ${\mathcal{A}}$ is the strict transform of $C$ by $\pi$, and decomposes as a disjoint union $\overline{C}=\coprod_{j=1}^k\overline{C}_j$ of $k$ smooth curves meeting the exceptional divisor transversely at smooth points. By Proposition \[prop:Aeslanormalizacion\], we have the isomorphism ${\mathcal{C}}\cong \pi_*{\mathcal{O}_{\tilde{C}}}$. We have the isomorphism ${\mathcal{O}_{\overline{C}}}=\oplus_{j=1}^k{\mathcal{O}_{\overline{C}_j}}$. For each $j$ let $(\psi_{j,1},...,\psi_{j,r_j})$ be a minimal system of generators of $n_*{\mathcal{O}_{\tilde{C_j}}}$ as a ${\mathcal{O}_{X}}$-module. Applying Theorem \[th:corrsing\] to the pair $(n_*{\mathcal{O}_{\overline{C}}},(\psi_{1,1},...,\psi_{k,r_k}))$ we obtain a reflexive ${\mathcal{O}_{X}}$-module $M'$, which has no free factors by the minimality of the set of generators of $n_*{\mathcal{O}_{\overline{C}}}$. If we denote by $M'_j$ the reflexive ${\mathcal{O}_{X}}$-module obtained by applying Theorem \[th:corrsing\] to the pair $(n_*{\mathcal{O}_{\overline{C_j}}},(\psi_{j,1},...,\psi_{j,r_j}))$ then we have the direct sum decomposition $M'=\oplus_{j=1}^kM'_j$. By the proof of Theorem \[th:corrsing\] we have that each of $M$ and $M'$ are isomorphic to the dual of the module of relations of a minimal set of generators of $n_*{\mathcal{O}_{\overline{C}}}$ as ${\mathcal{O}_{X}}$-module. Hence $M$ and $M'$ are isomorphic. In order to finish the proof we have to show that each $M'_j$ is indecomposable. Let $M'_j=\oplus_{v=1}^{m_j} M'_{j,v}$ be the decomposition in indecomposable reflexive modules. Since $M$ does not have free factors no one of the factors is free. For each $v=1,...,m_j$ choose a generic system of sections $(\phi_{j,v,1},...,\phi_{j,v,n_{j,v}})$. Since the minimal resolution adapted to a reflexive module dominates the minimal resolution adapted to each of its direct factors we conclude that $\pi:{\tilde{X}}\to X$ dominates the minimal resolution adapted to $M_j$. Denote by ${\mathcal{M}}'_{j,v}$ be the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M'_{j,v}$. Let $({\mathcal{A}}_{j,v},(\psi_{j,v,1},...,\psi_{j,v,n_{j,v}}))$ by the pair associated with $({\mathcal{M}}'_{j,v},(\phi_{j,v,1},...,\phi_{j,v,n_{j,v}}))$ by Theorem \[th:corres\]. By Lemma \[lem:ArtinVerdierminadap\], the support of ${\mathcal{A}}_{j,v}$ is a disjoint union of smooth curves meeting the exceptional divisor transversely at smooth points. Such collection of curves is non-empty since the ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}'_{j,v}$ is non-free. The full ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}$ associated to $M$ is the direct sum of the modules ${\mathcal{M}}'_{j,v}$ when $j$ and $v$ vary. Taking the union of the system of sections $(\phi_{j,v,1},...,\phi_{j,v,n_{j,v}})$ letting $j$ and $v$ vary we find a system of sections of ${\mathcal{M}}$, whose degeneracy locus is a disjoint union of smooth curves in ${\tilde{X}}$ meeting the exceptional divisor transversely at smooth points. Moreover there is at least a meting point for each pair of indexes $(j,v)$, since the full sheaf ${\mathcal{M}}'_{j,v}$ is not free. Consequently, if at least one $M'_j$ is not indecomposable the number of meeting points is strictly larger than $k$. On the other hand we know that the degeneracy locus of ${\mathcal{M}}$ for a system of generic sections is a disjoint union of smooth curves in ${\tilde{X}}$ meeting the exceptional divisor transversely at $k$ smooth points. The system of sections obtained as the union of $(\phi_{j,v,1},...,\phi_{j,v,m_v})$ letting $j$ and $v$ vary, may be deformed continuously into a system of generic sections, and the corresponding degeneracy loci deform flatly. Since it is impossible to deform a curve meeting transversely the exceptional divisor in strictly more than $k$ points into a curve meeting transversely the exceptional divisor in precisely $k$ points, we deduce that each $M'_j$ is indecomposable. The cohomology of full sheaves on normal Stein surfaces with Gorenstein singularities {#sec:cohomology} ===================================================================================== The main objective of this section is to prove to following theorem. \[formuladimensionM\] Let $X$ be a Stein normal surface with Gorenstein singularities. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $\pi:{\tilde{X}}\to X$ be a small resolution with respect to the Gorenstein form, let $Z_k$ be the canocical cycle at ${\tilde{X}}$, (see Defintion \[def:smallresgor\]). Let ${\mathcal{M}}$ be the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M$. Let $d$ be the specialty defect of ${\mathcal{M}}$. Then we have the equality $${\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{M}})} = rp_g - [c_1({\mathcal{M}})] \cdot [Z_k] + d.$$ This theorem will be very important in the following section. It will allow us to prove that a full special sheaf on a Gorenstein surface is determined by its first Chern class in the minimal adapted resolution. An inmediate Corollary of the theorem and Proposition \[prop:minadapspproperty\] shows how to use the cohomology of full sheaves as a resolution invariant for reflexive modules. \[cor:dimMadap\] Let $X$ be a Stein normal surface with Gorenstein singularities. Let $M$ be a special reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $\pi:{\tilde{X}}\to X$ be a small resolution with respect to the Gorenstein form. Let ${\mathcal{M}}$ be the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M$. Then the resolution is the minimal adapted resolution if and only if we have the equality ${\dim_{\mathbb{C}}(R^1\pi_*{\mathcal{M}})}=rp_g$. The Theorem is local in the base $X$. Therefore we may assume that $(X,x)$ is a normal Gorenstein germ. Then we have dualizing modules $\omega_X={\mathcal{O}_{X}}$ and $\omega_{{\tilde{X}}}=\Lambda^2\Omega^1_{{\tilde{X}}}=\pi^{!}{\mathcal{O}_{X}}$. The proof occupies the rest of the section, including some intermediate results, which we will single out as separate Lemmata and Propositions. Let us start with some preliminary work. Let $M$, $\pi:{\tilde{X}}\to X$, $Z_k$, ${\mathcal{M}}$, $r$ and $d$ be as in the statement of the Theorem. Take $r$ generic sections and consider the exact sequence obtained by the sections $$\label{exctseq:directaM9} 0 \to {\mathcal{O}_{{\tilde{X}}}}^r \to {\mathcal{M}} \to {\mathcal{A}}' \to 0,$$ and its dual $$\label{exctseq:directaN9} 0 \to {\mathcal{N}} \to {\mathcal{O}_{{\tilde{X}}}}^r \to {\mathcal{A}} \to 0,$$ where ${\mathcal{A}}= \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}', {\mathcal{O}_{{\tilde{X}}}} \right)$. Since dualizing the first morphism of (\[exctseq:directaN9\]) we recover the first morphism of (\[exctseq:directaM9\]) back, we deduce that $$\label{Adobleext} {\mathcal{A}}' \cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right).$$ Applying the functor $\pi_* -$ to the exact sequence we get $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0 & N & {\mathcal{O}_{X}}^r & \pi_* {\mathcal{A}} & R^1 \pi_* {\mathcal{N}} & R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}^r & 0, \\}; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,7} { \draw[-stealth] (m-1-\lastx) -- (m-1-\x); } \end{tikzpicture}$$ where $N$ is $\pi_* {\mathcal{N}}$ and, by Lemma \[lema:dualM\], the module $N$ is equal to the module $M^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. The last exact sequence can be split as follows $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0 & N & {\mathcal{O}_{X}}^r & {\mathcal{C}} & 0, \\ 0 & {\mathcal{C}} & \pi_* {\mathcal{A}} & {\mathcal{D}} & 0, \\ 0 & {\mathcal{D}} & R^1 \pi_* {\mathcal{N}} & R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}^r & 0, \\}; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,5} { \draw[-stealth] (m-1-\lastx) -- (m-1-\x); \draw[-stealth] (m-2-\lastx) -- (m-2-\x); \draw[-stealth] (m-3-\lastx) -- (m-3-\x); } \end{tikzpicture}$$ where $\text{lenght}({\mathcal{D}})=d$. Dualizing the first and second exact sequence we obtain $$\label{exact:cortaM} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0& {\mathcal{O}_{X}}^r & M & \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right) & 0, \\}; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,5} { \draw[-stealth] (m-1-\lastx) -- (m-1-\x); } \draw[-stealth] (m-1-3) edge node [auto] {$h$} (m-1-4); \end{tikzpicture}$$ $$\label{exact:CyD} \begin{tikzpicture} \matrix (n)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0&\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left(\pi_* {\mathcal{A}}, {\mathcal{O}_{X}} \right) & \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right) & \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right) & 0. \\}; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,5} { \draw[-stealth] (n-1-\lastx) -- (n-1-\x); } \draw[-stealth] (n-1-2) edge node [auto] {$i$} (n-1-3); \end{tikzpicture}$$ Applying the functor $\pi_* -$ to the exact sequence , using the identification and comparing with the exact sequence we obtain the diagram $$\label{diagram:comparacionM} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=2em] { 0 & {\mathcal{O}_{X}}^r & M & \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right) & 0\\ 0 & {\mathcal{O}_{X}}^r & M & \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) & R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}^r & R^1 \pi_* {\mathcal{M}} & 0 \\}; \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-1-2) -- (m-1-3); \draw[-stealth] (m-1-3) edge node [auto] {$h$} (m-1-4); \draw[-stealth] (m-1-4) -- (m-1-5); \draw[-stealth] (m-2-1) -- (m-2-2); \draw[-stealth] (m-2-2) -- (m-2-3); \draw[-stealth] (m-2-3) edge node [auto] {$h$} (m-2-4); \draw[-stealth] (m-2-4) -- (m-2-5); \draw[-stealth] (m-2-5) -- (m-2-6); \draw[-stealth] (m-2-6) -- (m-2-7); \draw[-stealth] (m-1-2) edge node [auto] {$Id$} (m-2-2); \draw[-stealth] (m-1-3) edge node [auto] {$Id$} (m-2-3); \draw[-stealth] (m-1-4) edge node [auto] {$\theta$} (m-2-4); \end{tikzpicture}$$ where $Id$ is the identity and $\theta$ is the map that makes the diagram commute. Since $(X,x)$ is Gorenstein and $\pi:{\tilde{X}}\to X$ is small with respect to the Gorenstein form we have the exact sequence $$\label{eq:exactseqCs9} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0& {\omega_{{\tilde{X}}}} & {\mathcal{O}_{{\tilde{X}}}} & {\mathcal{O}_{Z_K}} & 0. \\}; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,5} { \draw[-stealth] (m-1-\lastx) -- (m-1-\x); } \draw[-stealth] (m-1-2) edge node [auto] {$c$} (m-1-3); \end{tikzpicture}$$ Applying the functor $\pi \left( \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}} \left( {\mathcal{A}}, - \right) \right)$ to the map $c$ we obtain the map $$\label{mapeoC} \pi \left( \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}} \left( {\mathcal{A}}, - \right) \right)(c) \colon \pi \left( \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}} \left( {\mathcal{A}}, {\omega_{{\tilde{X}}}} \right) \right) \to \pi \left( \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}} \left( {\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) \right).$$ Abusing notation, let us denote the previous map by $c$. Since the singularity $(X,x)$ is Gorenstein, the ring ${\mathcal{O}_{X}}$ is the dualizing module for the singularity [@BrHe Theorem 3.3.7]. In this case the Grothendieck duality for the map $\pi$ [@Har1 Ch. VII] establish the isomorphism $$R \pi_* R \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}\left(-, {\omega_{{\tilde{X}}}} \right) \cong R \operatorname{\text{Hom}\,}_{{\mathcal{O}_{X}}} \left(R \pi_* -, {\mathcal{O}_{X}} \right).$$ Applying this for ${\mathcal{A}}$, and using Grothendieck spectral sequence for the composition of two functors we obtain an isomorphism $$\label{eq:isofrakg9} \mathfrak{g}:\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right) \cong\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left(\pi_* {\mathcal{A}}, {\mathcal{O}_{X}} \right).$$ Now using , , and , we have the diagram $$\label{diagramacc} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=2em] { \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right) & \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left(\pi_* {\mathcal{A}}, {\mathcal{O}_{X}} \right) & \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right) & \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) \\ \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right) & & & \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right)\\}; \draw[-stealth] (m-1-1) edge node [auto] {$\mathfrak{g}$} (m-1-2); \draw[-stealth] (m-1-2) edge node [auto] {$i$} (m-1-3); \draw[-stealth] (m-1-3) edge node [auto] {$\theta$} (m-1-4); \draw[-stealth] (m-1-1) edge node [auto] {$Id$} (m-2-1); \draw[-stealth] (m-1-4) edge node [auto] {$Id$} (m-2-4); \draw[-stealth] (m-2-1) edge node [auto] {$c$} (m-2-4); \end{tikzpicture}$$ \[lema:tecnico\] The diagram commutes. Denote by $\mathfrak{c}$ the map given by the composition $\theta \circ{} i \circ{} \mathfrak{g}$. Consider the map $f:= (c - \mathfrak{c})$. Since the map $\pi \colon {\tilde{X}}\to X$ is an isomorphism outside the exceptional divisor, we have that for any section $s$ of $\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right)$, the section $f(s)$ is supported in the exceptional divisor, hence $f(s) \in H^0_E\left(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) \right)$ but this cohomology group is zero. Therefore for any section $s$ of $\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right)$ we have that $f(s)=0$ which it is equivalent to say that the maps $c$ and $\mathfrak{c}$ coincide. \[proposition:dimM\] We have the equality $${\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{M}})} = rp_g - {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{Z_k}} \right))} + {\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))}.$$ Applying the functor $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}(-,-)$ to the exact sequences and we get $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { & 0 & 0 & 0 & \\ 0 & {\omega_{{\tilde{X}}}}^r & {\mathcal{M}} \otimes {\omega_{{\tilde{X}}}} & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right) & 0 \\ 0 & {\mathcal{O}_{{\tilde{X}}}}^r & {\mathcal{M}} & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) & 0 \\ 0 & {\mathcal{O}_{Z_k}}^r & {\mathcal{M}} \otimes {\mathcal{O}_{Z_k}} & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{Z_k}} \right) & 0 \\ & 0 & 0 & 0 & \\}; \foreach \y [remember=\y as \lasty (initially 2)] in {3,4} { \foreach \x [remember=\x as \lastx (initially 2)] in {3,4} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); \draw[-stealth] (m-\lasty-\lastx) -- (m-\y-\lastx); } } \draw[-stealth] (m-2-4) edge node [auto] {$c$} (m-3-4); \draw[-stealth] (m-1-2) -- (m-2-2); \draw[-stealth] (m-1-4) -- (m-2-4); \draw[-stealth] (m-4-1) -- (m-4-2); \draw[-stealth] (m-1-3) -- (m-2-3); \draw[-stealth] (m-2-1) -- (m-2-2); \draw[-stealth] (m-2-2) -- (m-2-3); \draw[-stealth] (m-2-3) -- (m-2-4); \draw[-stealth] (m-2-4) -- (m-2-5); \draw[-stealth] (m-3-1) -- (m-3-2); \draw[-stealth] (m-3-4) -- (m-3-5); \draw[-stealth] (m-4-4) -- (m-4-5); \draw[-stealth] (m-4-2) -- (m-5-2); \draw[-stealth] (m-4-3) -- (m-5-3); \draw[-stealth] (m-4-4) -- (m-5-4); \draw[-stealth] (m-2-4) -- (m-3-4); \draw[-stealth] (m-3-4) -- (m-4-4); \end{tikzpicture}$$ Applying the functor $\pi_* -$ to the last commutative diagram we get $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=2em] { & 0 & 0 & 0 & \\ 0 & \pi_* {\omega_{{\tilde{X}}}}^r & \pi_* \left({\mathcal{M}} \otimes {\omega_{{\tilde{X}}}} \right) & \pi_*\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right) & 0 & 0 & 0\\ 0 & {\mathcal{O}_{X}}^r & M & \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) & R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}^r & R^1 \pi_* {\mathcal{M}} & 0 \\ 0 & \pi_* {\mathcal{O}_{Z_k}}^r & \pi_* \left( {\mathcal{M}} \otimes {\mathcal{O}_{Z_k}} \right) & \pi_*\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{Z_k}} \right) & 0 & 0 &\\}; \foreach \y [remember=\y as \lasty (initially 2)] in {3} { \foreach \x [remember=\x as \lastx (initially 2)] in {3,...,7} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); \draw[-stealth] (m-\lasty-\lastx) -- (m-\y-\lastx); } } \draw[-stealth] (m-3-3) edge node [auto] {$h$} (m-3-4); \draw[-stealth] (m-3-4) edge node [auto] {$\alpha$} (m-3-5); \draw[-stealth] (m-2-4) edge node [auto] {$c$} (m-3-4); \draw[-stealth] (m-1-4) -- (m-2-4); \draw[-stealth] (m-4-1) -- (m-4-2); \draw[-stealth] (m-2-7) -- (m-3-7); \draw[-stealth] (m-3-5) -- (m-4-5); \draw[-stealth] (m-3-6) -- (m-4-6); \draw[-stealth] (m-1-2) -- (m-2-2); \draw[-stealth] (m-1-3) -- (m-2-3); \draw[-stealth] (m-2-1) -- (m-2-2); \draw[-stealth] (m-2-2) -- (m-2-3); \draw[-stealth] (m-2-3) -- (m-2-4); \draw[-stealth] (m-2-4) -- (m-2-5); \draw[-stealth] (m-2-5) -- (m-2-6); \draw[-stealth] (m-2-6) -- (m-2-7); \draw[-stealth] (m-3-1) -- (m-3-2); \draw[-stealth] (m-4-2) -- (m-4-3); \draw[-stealth] (m-4-3) -- (m-4-4); \draw[-stealth] (m-3-2) -- (m-4-2); \draw[-stealth] (m-3-3) -- (m-4-3); \draw[-stealth] (m-3-4) -- (m-4-4); \draw[-stealth] (m-4-4) -- (m-4-5); \draw[densely dotted,-stealth] (m-4-2) to [out = -90, in = 90, looseness = .7] (m-2-5); \draw[densely dotted,-stealth] (m-4-3) to [out = -90, in = 90, looseness = .7] (m-2-6); \draw[densely dotted,-stealth] (m-4-4) to [out = -90, in = 90, looseness = .7] (m-2-7); \end{tikzpicture}$$ By this diagram we get $$\begin{aligned} \begin{split}\label{align:cuentas1} {\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{M}})} &= rp_g - {\dim_{\mathbb{C}}(\text{Im}(\alpha))},\\ {\dim_{\mathbb{C}}(\pi_*\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{Z_k}} \right))} &= {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right))}, \end{split}\end{aligned}$$ and $$\label{alphaauxiliar} {\dim_{\mathbb{C}}(\operatorname{Im}\alpha)} ={\dim_{\mathbb{C}}( \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \ker \alpha)} = {\dim_{\mathbb{C}}( \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \operatorname{Im}h)}.$$ Now by we have $\operatorname{Im}h = \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right)$. Hence by the previous equality, , and we get $$\begin{aligned} \begin{split}\label{imagenalfa1} {\dim_{\mathbb{C}}(\operatorname{Im}\alpha)} &={\dim_{\mathbb{C}}( \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \operatorname{Im}h)}\\ &={\dim_{\mathbb{C}}( \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right))} \\ &= {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left(\pi_* {\mathcal{A}}, {\mathcal{O}_{X}} \right))} -{\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))} \\ &= {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right))} -{\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))} . \end{split}\end{aligned}$$ Let $c$ and $\mathfrak{c}$ be the morphisms given in the diagram . Now by and by Lemma \[lema:tecnico\] we have $$\begin{aligned} \begin{split}\label{align:cuentas2} {\dim_{\mathbb{C}}(\operatorname{Im}\alpha)} &= {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right))} -{\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))}\\ &= {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \operatorname{Im}\mathfrak{c})} -{\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))}\\ &= {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \operatorname{Im}c)} -{\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))}\\ &= {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{{\tilde{X}}}} \right) / \pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\omega_{{\tilde{X}}}} \right))} - {\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))}. \end{split}\end{aligned}$$ Therefore by and we get the desired equality. By the Proposition \[proposition:dimM\] to prove Theorem \[formuladimensionM\] it is enough to prove the equalities $${\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{Z_k}} \right))} = [c_1({\mathcal{M}})] \cdot [Z_k].$$ $${\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))}=d.$$ The following lemma gives us the first equality. \[lemma:A1A2\] Let ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ be two Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-modules of dimension one such that ${\mathcal{A}}_1$ is contained in ${\mathcal{A}}_2$, the quotient ${\mathcal{A}}_2 / {\mathcal{A}}_1$ is finitely supported and the support of each sheaf intersects the exceptional divisor in finitely many points. Then we have the equality $${\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right))} = {\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2, {\mathcal{O}_{Z_K}} \right))}.$$ Let ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ as in the statement. We have the exact sequence $$0 \to {\mathcal{A}}_1 \to {\mathcal{A}}_2 \to {\mathcal{A}}_2/{\mathcal{A}}_1 \to 0.$$ Applying the functor $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}(-, {\mathcal{O}_{Z_K}})$ to the last exact sequence we get $$\label{diagram:A1A2} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { 0 & \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_2/{\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) & \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_2, {\mathcal{O}_{Z_K}} \right) & \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) & \\ & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2/{\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2, {\mathcal{O}_{Z_K}} \right) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) & \\ & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2/{\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2, {\mathcal{O}_{Z_K}} \right) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) & 0\\}; \foreach \y [remember=\y as \lasty (initially 1)] in {1, 2,3} { \foreach \x [remember=\x as \lastx (initially 2)] in {3,4} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); } } \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-3-4) -- (m-3-5); \draw[densely dotted,-stealth] (m-1-4) to [out=355, in=175] (m-2-2); \draw[densely dotted,-stealth] (m-2-4) to [out=355, in=175] (m-3-2); \end{tikzpicture}$$ Since ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ are Cohen-Macaulay sheaves of dimension one and the support of each sheaf intersects the exceptional divisor finitely we have $$\begin{aligned} \begin{split}\label{cerosA1A2} \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_2, {\mathcal{O}_{Z_K}} \right) &= \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) = 0, \\ \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2, {\mathcal{O}_{Z_K}} \right) &= \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) = 0. \end{split}\end{aligned}$$ The second equality uses Auslander-Buchbaum formula and the fact that each ${\mathcal{A}}_i$ has depth $1$. Since all the sheaves in are supported in a finite set we can work locally, therefore we assume that ${\mathcal{O}_{{\tilde{X}}}}$ is $\mathbb{C}[x,y]$ and $Z_K$ is ${\mathcal{O}_{{\tilde{X}}}}/(f)$ for some function $f$. Now by and we just need to prove the following equality $$\label{A1A2reduccion} {\dim_{\mathbb{C}}(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2/{\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right))} = {\dim_{\mathbb{C}}(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2/{\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right))}.$$ Consider the following resolution of ${\mathcal{O}_{Z_K}}$ $$\label{resolucionC} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=0.5em] { 0 & {\mathcal{O}_{{\tilde{X}}}} & {\mathcal{O}_{{\tilde{X}}}} & {\mathcal{O}_{Z_K}} & 0. \\}; \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-1-2) edge node [auto] {$\cdot f$} (m-1-3); \draw[-stealth] (m-1-3) -- (m-1-4); \draw[-stealth] (m-1-4) -- (m-1-5); \end{tikzpicture}$$ Applying the functor $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_2 / {\mathcal{A}}_1, - \right)$ to the last exact sequence we get $$\label{diagramaA1A2} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2em,row sep=2em] { 0 & \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) & \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) & \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) &\\ & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) &\\ & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) & 0\\ }; \foreach \y [remember=\y as \lasty (initially 1)] in {1, 2,3} { \foreach \x [remember=\x as \lastx (initially 2)] in {3,...,4} { \draw[-stealth] (m-\y-\lastx) -- (m-\y-\x); } } \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-3-4) -- (m-3-5); \draw[densely dotted,-stealth] (m-1-4) to [out=355, in=175] (m-2-2); \draw[densely dotted,-stealth] (m-2-4) to [out=355, in=175] (m-3-2); \end{tikzpicture}$$ Now since the support of ${\mathcal{A}}_2 / {\mathcal{A}}_1$ is zero dimensional, we have by Theorem \[Th:Herzog\] $$\begin{aligned} \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) &=0.\end{aligned}$$ By the previous equality and the exact sequence we get $$0 \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{{\tilde{X}}}} \right) \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2 / {\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right) \to 0.$$ Taking ${{\mathbb{C}}}$-dimensions we immediately obtain: $${\dim_{\mathbb{C}}(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1\left({\mathcal{A}}_2/{\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right))} = {\dim_{\mathbb{C}}(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^2\left({\mathcal{A}}_2/{\mathcal{A}}_1, {\mathcal{O}_{Z_K}} \right))}.$$ The equality ${\dim_{\mathbb{C}}(\pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{A}}, {\mathcal{O}_{Z_k}} \right))} = [c_1({\mathcal{M}})] \cdot [Z_k]$ holds. By the previous Lemma it is enough to assume that ${\mathcal{A}}$ is isomorphic to ${\mathcal{O}_{\overline{C}}}$, where $\overline{C}$ is the support of ${\mathcal{A}}$. A direct computation of $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}^1 \left({\mathcal{O}_{\overline{C}}}, {\mathcal{O}_{Z_k}} \right)$ gives the result. The equality ${\dim_{\mathbb{C}}(\operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^2 \left({\mathcal{D}}, {\mathcal{O}_{X}} \right))}={\dim_{\mathbb{C}}({\mathcal{D}})}=d$ holds. Using Theorem \[Th:Herzog\] one easily reduces by induction on ${\dim_{\mathbb{C}}({\mathcal{D}})}$ to the case ${\mathcal{D}}={{\mathbb{C}}}_p$, where ${{\mathbb{C}}}_p$ is the skyscraper sheaf at a point $p$ with stalk ${{\mathbb{C}}}$. A direct computation shows that case. The proof of Theorem \[formuladimensionM\] is complete now. The classification and structure of special reflexive modules {#sec:classsprefl} ============================================================= The combinatorial classification {#sec:combclass} -------------------------------- Let $(X,x)$ be a normal surface singularity. Lemma \[lem:ArtinVerdierminadap\] allows to define the resolution graph of a reflexive module. Denote by $M$ a reflexive ${\mathcal{O}_{X}}$-module, $\pi \colon (\tilde{X},E) \to (X,x)$ the minimal adapted resolution to $M$, ${\mathcal{M}}$ the full sheaf associated to $M$ and $r$ the rank of ${\mathcal{M}}$. Take $r$ generic sections of ${\mathcal{M}}$ and consider the exact sequence given by the sections $$0 \to {\mathcal{O}_{{\tilde{X}}}}^r \to {\mathcal{M}} \to {\mathcal{A}}' \to 0.$$ By Lemma \[lem:ArtinVerdierminadap\] the sheaf ${\mathcal{A}}'$ is isomorphic to ${\mathcal{O}_{D}}$, where $D$ is a smooth curve meeting the exceptional divisor transversely at smooth points. We construct a graph as follows: 1. Let ${\mathcal{G}}^o_M$ be the dual graph of ${\tilde{X}}$ of the minimal good resolution that dominates $\pi$, weighted with the self-intersection and the genus of each component (see [@Ne]). 2. In each vertex $v_i$, add as many arrows as the first Chern class of ${\mathcal{M}}$ intersects the exceptional divisor $E_i$. Call the resulting decorated graph ${\mathcal{G}}_M$. \[def:resgraphM\] The resolution graph ${\mathcal{G}}_M$ of the module $M$ is the graph described in the previous construction. In the next theorem we characterize combinatorially resolution graphs of special modules over Gorenstein surface singularities. By a negative plumbing graph we mean the dual graph of a good resolution of a surface singularity. The property of being numerically Gorenstein only depends on the plumbing graph. In this case there is a canonical cycle with integral coefficients (see [@Ne]). \[th:charresgraphsp\] Let ${\mathcal{G}}$ be a negative definite plumbing graph, such that to some of its vertices there are a finite number of arrows attached. There is a Gorenstein surface singularity $(X,x)$ and a special reflexive module whose resolution graph is isomorphic to ${\mathcal{G}}$ if and only if each of the following properties is satisfied: 1. the graph is numerically Gorenstein. 2. if a vertex has genus $0$, self intersection $-1$ and has at most two neighboring vertices, then it supports at least $1$ arrow. 3. if a vertex supports arrows then its coefficient in the canonical cycle equals $0$. Property $(1)$ is necessary because Gorenstein implies numerically Gorenstein. Property $(2)$ holds by the minimality of the good resolution dominating the minimal adapted resolution. Property $(3)$ is a direct consequence of Proposition \[prop:minadapspproperty\]. Conversely, let ${\mathcal{G}}$ be a graph satisfying all the properties. By [@PPP] there is a Gorenstein normal surface singularity $(X,x)$ who has a resolution with plumbing graph equal to the result of deleting the arrows of ${\mathcal{G}}$. Let $\pi:{\tilde{X}}\to X$ be such a resolution. Let $D\subset{\tilde{X}}$ be a smooth curvette meeting the exceptional divisor $E$ transversely at smooth points, and so that for each vertex $v$ of ${\mathcal{V}}$ the number of components of $D$ meeting the irreducible component $E_v$ of $E$ corresponding to $v$, is exactly the number of arrows attached to $v$. Define ${\mathcal{A}}:={\mathcal{O}_{D}}$ and let $\psi_1,...,\psi_r$ be a set of generators of $\pi_*{\mathcal{A}}$ as a ${\mathcal{O}_{X}}$-module. Since ${\mathcal{A}}$ is equal to ${\mathcal{O}_{D}}$ and the canonical order at the components $E_v$ met by $D$ is $0$, choosing $D$ generic enough we conclude that $cond(\mathfrak{K}_{({\mathcal{A}},(\psi_1,...,\psi_r))})=(0,...,0)$. Hence, by Proposition \[prop:consecuenciaspracticas\] (1), we deduce that if $({\mathcal{M}},(\phi_1,...,\phi_r))$ is the result of applying the bijection of Theorem \[th:corres\] to $({\mathcal{A}},(\psi_1,...,\psi_r))$ then ${\mathcal{M}}$ is full. It is also special since $\psi_1,...,\psi_r$ generate ${\mathcal{A}}$ as a ${\mathcal{O}_{X}}$-module (same argument than at the proof of Corollary \[cor:existenmodespeciales\]). By Proposition \[prop:minadapnumchar\] the resolution $\pi:{\tilde{X}}\to X$ is the minimal resolution adapted to $\pi_*{\mathcal{M}}$, and by Theorem \[th:characspecial\] the module $\pi_*{\mathcal{M}}$ is special. It is clear by construction that the resolution graph of $\pi_*{\mathcal{M}}$ equals ${\mathcal{G}}$. A consequence of the previous Theorem and Proposition \[prop:decompesp\] is the following corollary \[cor:charresgraphspindec\] Let ${\mathcal{G}}$ be a negative definite plumbing graph, such that to some of its vertices there are a finite number of arrows attached. There is a Gorenstein surface singularity $(X,x)$ and an indecomposable special reflexive module whose resolution graph is isomorphic to ${\mathcal{G}}$ if and only if the conditions of the previous Theorem hold and in addition ${\mathcal{G}}$ has only one arrow. The first Chern class of a module at its minimal adapted resolution {#sec:1stchernspecial} ------------------------------------------------------------------- Let $X$ be a Stein normal surface with Gorenstein singularities. Here we study the relation of a reflexive ${\mathcal{O}_{X}}$-module and its first Chern class in the Picard group of its minimal adapted resolution. We show that if the module is special then the first Chern class determines the module, providing a vast generalization of the corresponding result of Artin and Verdier for rational double points [@AV]. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $\pi:{\tilde{X}}\to X$ be the minimal resolution adapted to $M$, denote by $E$ the exceptional divisor. Let ${\mathcal{M}}$ be the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M$. The first Chern class of ${\mathcal{M}}$ in $Pic({\tilde{X}})$ is the class determined by the determinant bundle ${\mathcal{L}}:=det({\mathcal{M}})$. \[prop:extension1st\] The full ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{M}}$ is an extension of the determinant line bundle ${\mathcal{L}}$ by ${\mathcal{O}_{{\tilde{X}}}}^{r-1}$. Take $r$ generic sections $(\phi_1,...,\phi_r)$ of ${\mathcal{M}}$ and consider the exact sequence given by the sections $$\label{exctseq:directaM55} 0 \to {\mathcal{O}_{{\tilde{X}}}}^r \to {\mathcal{M}} \to {\mathcal{A}}' \to 0.$$ Since the resolution is the minimal adapted resolution we have that ${\mathcal{A}}'$ is isomorphic to ${\mathcal{O}_{D}}$, where $D$ is a smooth curve meeting the exceptional divisor $E$ transversely at smooth points by Lemma \[lem:ArtinVerdierminadap\]. Locally in a trivializing open subset $U$ of the locally free sheaf ${\mathcal{M}}$ we have that the sections can be written as follows $$Q= \begin{tikzpicture}[baseline=(m-2-1.base)] \matrix (m)[matrix of math nodes, left delimiter=(,right delimiter=)]{ q_{11} & q_{12} & \dots & q_{1r} \\ \vdots & \vdots & \vdots & \vdots \\ q_{r1} & q_{12} & \dots & q_{rr} \\}; \end{tikzpicture}$$ where each $q_{ij}$ is an element of ${\mathcal{O}_{{\tilde{X}}}}(U)$. Therefore $p$ belongs to $D$ if and only if the determinant of $Q(p)$ is equal to zero. Since $D$ is smooth, the matrix $Q$ must have at least $r-1$ columns linearly independent. Therefore we can choose $r-1$ sections linear independent everywhere. By genericity of the system of sections $(\phi_1,...,\phi_r)$ we may assume that these sections are $(\phi_1,...,\phi_{r-1})$. These sections give us the exact sequence $$\label{exact:picard} 0 \to {\mathcal{O}_{{\tilde{X}}}}^{r-1} \to {\mathcal{M}} \to {\mathcal{L}} \to 0,$$ where ${\mathcal{L}}$ is the line bundle $\text{det}({\mathcal{M}})$. Now we assume that $M$ is special and $(X,x)$ is Gorenstein and prove stronger properties. \[lemma:dimL\] The dimension of $R^1 \pi_* {\mathcal{L}}$ is $p_g$. By Proposition \[prop:minadapspproperty\] the resolution $\pi:{\tilde{X}}\to X$ is small with respect to the Gorenstein form (hence the canonical cycle $Z_K$ is non-negative), and moreover $D$ does not meet the support of $Z_K$. Therefore we have $$\begin{aligned} \text{Tor}_1^{{\mathcal{O}_{{\tilde{X}}}}}({\mathcal{O}_{D}},{\mathcal{O}_{Z_K}}) &= 0, \\ {\mathcal{O}_{D}}\otimes{\mathcal{O}_{Z_K}} &= 0.\end{aligned}$$ By these equalities, applying $- \otimes {\mathcal{O}_{Z_K}}$ the exact sequence $$\label{exctseq:detM1} 0 \to {\mathcal{O}_{{\tilde{X}}}} \to {\mathcal{L}} \to {\mathcal{O}_{D}} \to 0,$$ we get $${\mathcal{O}_{Z_K}} \cong {\mathcal{L}} \otimes {\mathcal{O}_{Z_K}}.$$ Now applying the functor $\pi_* -$ to the exact sequence and using the last isomorphism we obtain $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { 0 & \pi_* {\mathcal{O}_{{\tilde{X}}}} & \pi_* {\mathcal{L}} & \pi_* {\mathcal{O}_{D}}& R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}} & R^1 \pi_* {\mathcal{L}} & 0 \\ & & & 0 & R^1 \pi_* {\mathcal{O}_{Z_K}} & R^1 \pi_* {\mathcal{L}} \otimes {\mathcal{O}_{Z_K}} & 0 \\ }; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,7} { \draw[-stealth] (m-1-\lastx) -- (m-1-\x); } \foreach \x [remember=\x as \lastx (initially 4)] in {5,...,7} { \draw[-stealth] (m-2-\lastx) -- (m-2-\x); } \draw[-stealth] (m-1-5) -- (m-2-5); \draw[-stealth] (m-1-6) -- (m-2-6); \end{tikzpicture}$$ Since the diagram commutes and $R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}$ and $R^1 \pi_* {\mathcal{O}_{Z_K}}$ are isomorphic we conclude that $R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}$ and $R^1 \pi_* {\mathcal{L}} $ have the same dimension. \[th:Chernresolucionadapted\] Let $X$ be a Stein normal surface with Gorenstein singularities. Let $M$ be a special ${\mathcal{O}_{X}}$-module without free factors. Let $\pi:{\tilde{X}}\to X$ be the minimal resolution adapted to $M$, and ${\mathcal{M}}$ the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M$. The module ${\mathcal{M}}$ (and equivalently $M$) is determined by its first Chern class in $\text{Pic}({\tilde{X}})$. Applying the functor $\pi_* -$ to the exact sequence we get $$\label{exac:detMtemp} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { 0 & \pi_* {\mathcal{O}_{{\tilde{X}}}}^{r-1} & \pi_* {\mathcal{M}} & \pi_* {\mathcal{L}}& R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}}^{r-1} & R^1 \pi_* {\mathcal{M}} & R^1 \pi_* {\mathcal{L}} & 0. \\ }; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,8} { \draw[-stealth] (m-1-\lastx) -- (m-1-\x); } \end{tikzpicture}$$ Since $$\begin{aligned} {\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{M}})} &= rp_g \quad \text{by Corollary~\ref{cor:dimMadap} and}\\ {\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{L}})} &= {\dim_{\mathbb{C}}(R^1 \pi_* {\mathcal{O}_{{\tilde{X}}}})}=p_g \quad \text{by Lemma \ref{lemma:dimL}},\end{aligned}$$ we get that the exact sequence split as follows $$\label{eq:paraelrango} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { 0 & \pi_* {\mathcal{O}_{{\tilde{X}}}}^{r-1} & \pi_* {\mathcal{M}} & \pi_* {\mathcal{L}} & 0. \\ }; \foreach \x [remember=\x as \lastx (initially 1)] in {2,...,5} { \draw[-stealth] (m-1-\lastx) -- (m-1-\x); } \end{tikzpicture}$$ Therefore $\pi_* {\mathcal{M}} \in \operatorname{\text{Ext}\,}_{{\mathcal{O}_{X}}}^1 \left( \pi_* {\mathcal{L}}, {\mathcal{O}_{X}}^{r-1} \right)$. Since the module $\pi_* {\mathcal{M}}$ is reflexive and without free factors we conclude the proof by Lemma 1.9.ii in [@AV] (this Lemma globalizes to the Stein surface situation that we are considering here). The classification of special reflexive modules on Gorenstein surface singularities {#sec:clasdef} ----------------------------------------------------------------------------------- Before we state and prove the classification theorem we need the following lemma. \[lem:varioD\] Let $(X,x)$ be a normal Gorenstein surface singularity and $\pi:{\tilde{X}}\to X$ be a resolution which is small with respect to the canonical form such that for some irreducible component $E_i$ of the exceptional divisor $E$ we have $E_i \not \subseteq \text{Supp}(Z_K)$. If $D_1$ and $D_2$ are two irreducible curvettes, each one transverse to $E_i$ at regular points of $E$, then we have an isomorphism of line bundles ${\mathcal{O}_{{\tilde{X}}}}(-D_1) \cong {\mathcal{O}_{{\tilde{X}}}}(-D_2)$. We want to prove that ${\mathcal{O}_{{\tilde{X}}}}(-D_1+D_2)$ is isomorphic to ${\mathcal{O}_{{\tilde{X}}}}$. Consider the exponential exact sequence $$\begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { 0 & \mathbb{Z} & {\mathcal{O}_{{\tilde{X}}}} & {\mathcal{O}_{{\tilde{X}}}}^* & 0. \\ }; \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-1-2) -- (m-1-3); \draw[-stealth] (m-1-3) edge node[auto]{$\text{exp}$} (m-1-4); \draw[-stealth] (m-1-4) -- (m-1-5); \end{tikzpicture}$$ Applying the functor $\pi_* $ to the previous exact sequence we get $$\label{exact:largaexp} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=2.5em,row sep=2em] { \dots & H^1({\tilde{X}}, \mathbb{Z}) & H^1({\tilde{X}}, {\mathcal{O}_{{\tilde{X}}}}) & H^1({\tilde{X}}, {\mathcal{O}_{{\tilde{X}}}}^*) & H^2({\tilde{X}}, \mathbb{Z}) & 0. \\ }; \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-1-2) -- (m-1-3); \draw[-stealth] (m-1-3) edge node[auto]{$\text{exp}$} (m-1-4); \draw[-stealth] (m-1-4) edge node[auto]{$\delta$} (m-1-5); \draw[-stealth] (m-1-5) -- (m-1-6); \end{tikzpicture}$$ We know that the Picard group of ${\tilde{X}}$ is $H^1({\tilde{X}}, {\mathcal{O}_{{\tilde{X}}}}^*)$ and the morphism $\delta$ is given by taking the first Chern class in cohomology. By hypothesis we know that $\delta({\mathcal{O}_{{\tilde{X}}}}(-D_1+D_2)) = 0$. By the exact sequence we get that there exist an element $f$ in $H^1({\tilde{X}}, {\mathcal{O}_{{\tilde{X}}}})$ such that the line bundle given by $\text{exp}(f)$ is isomorphic to ${\mathcal{O}_{{\tilde{X}}}}(-D_1+D_2)$. Denote by $E$ the exceptional divisor of $\pi$. By the location of the curvettes $D_1$ and $D_2$, an easy Čech cohomology computation shows that there exists a finite Stein cover ${\mathcal{U}}=\{U_i\}_{i\in I}$ so that $f\in H^1({\tilde{X}}, {\mathcal{O}_{{\tilde{X}}}})=\check{H}^1({\mathcal{U}}, {\mathcal{O}_{{\tilde{X}}}})$ is represented by a $1$-cocycle $[f_{i,j}]$ with $f_{i,j}\in {\mathcal{O}_{{\tilde{X}}}}(U_i\cap U_j)$ so that $f_{i,j}=0$ unless $U_i\cap U_j\cap E$ is included in $E_i\setminus Sing(E)$. Since the resolution is small with respect to the Gorenstein form, we have the exact sequence $$\tag{\ref{eq:exactseqCs}} 0 \to {\omega_{{\tilde{X}}}} \to {\mathcal{O}_{{\tilde{X}}}} \to {\mathcal{O}_{Z_K}} \to 0.$$ Applying the functor $\pi_* -$ and by Grauert-Riemenschneider Vanishing Theorem we get that the homomorphism $$H^1({\tilde{X}}, {\mathcal{O}_{{\tilde{X}}}})\to H^1({\tilde{X}}, {\mathcal{O}_{Z_K}})$$ is an isomorphism. Since the image of the Čech cocycle $[f_{i,j}]$ under this isomorphism is obviously zero for having empty support ($Z_k$ does not have support in $E_i$), we deduce that $f=0$ in $H^1({\tilde{X}}, {\mathcal{O}_{{\tilde{X}}}})$. This implies that ${\mathcal{O}_{{\tilde{X}}}}(-D_1+D_2)$ is the trivial line bundle. Let $(X,x)$ be a normal surface singularity and $\pi \colon {\tilde{X}}\to X$ be a resolution. Any irreducible component $E_i$ of the exceptional divisor is called *a divisor over $X$*. \[rem:resoluciondivisores\] Let $E_1, \dots, E_n$ be a collection of divisors over $X$. Then there exists a unique minimal resolution $\pi \colon {\tilde{X}}\to X$ such that $E_1, \dots, E_n$ are irreducible components of the exceptional divisor. Now we present the classification theorem: \[Teo:final\] Let $(X,x)$ be a Gorenstein surface singularity. Then there exists a bijection between the following sets: 1. The set of special ${\mathcal{O}_{X}}$-modules without free factors up to isomorphism. 2. The set of finite pairs $(E_1, n_1), \dots, (E_l, n_l)$ where each $E_i$ is a divisor over $X$ and $n_i$ is a positive integer, such the minimal resolution given by Remark \[rem:resoluciondivisores\] is small with respect to the Gorenstein form and the Gorenstein form does not have any pole in the components $E_1, \dots, E_l$. Let $M$ be a special ${\mathcal{O}_{X}}$-module and $\pi \colon {\tilde{X}}\to X$ be the minimal resolution adapted to $M$ with exceptional divisor $E=\bigcup_{i=1}^l E_i$. Denote by ${\mathcal{M}}$ the full sheaf associated to $M$ and by $n_j=c_1({\mathcal{M}}) \cdot E_j$ for $j=1, \dots, l$. We associate to the module $M$ the pairs $(E_1,n_1), \dots, (E_k,n_k)$ such that $n_j$ is different form zero. In order to prove the surjectivity of the previous assignment consider $(E_1, n_1), \dots, (E_l, n_l)$ where each $E_i$ is a divisor over $X$ and $n_i$ is a positive integer and denote by $\pi \colon {\tilde{X}}\to X$ be the resolution given by Remark \[rem:resoluciondivisores\]. The divisors are so that $\pi \colon {\tilde{X}}\to X$ is small with respect to the Gorenstein form, and the coefficient of the canonical cycle at each of them vanishes. For each positive integer $n_j$ take a smooth curvette $D_j$ with $n_j$ irreducible components such that $D_j$ intersects only the irreducible component $E_j$ and the intersection is transverse. Denote by $D= D_1 \coprod \dots \coprod D_l$. Let $(\psi_1,...,\psi_r)$ be a minimal set of generators of $\pi_*{\mathcal{O}_{D}}$ as a ${\mathcal{O}_{X}}$-module. Since the $E_i$’s are not at the support of the canonical cycle we have that the minimal conductor $cond(\mathfrak{K}_{({\mathcal{O}_{D}},(\psi_1,...,\psi_r))})$ equals $0$. Let $({\mathcal{M}},(\phi_1,...,\phi_r))$ be pair associated to $({\mathcal{O}_{D}},(\psi_1,...,\psi_r))$ by the correspondence of Theorem \[th:corres\]. By Proposition \[prop:consecuenciaspracticas\], (1) the module ${\mathcal{M}}$ is full. Since $(\psi_1,...,\psi_r)$ generate of $\pi_*{\mathcal{O}_{D}}$ as a ${\mathcal{O}_{X}}$-module, we have that ${\mathcal{M}}$ is special. By Theorem \[th:characspecial\] the module $M:=\pi_*{\mathcal{M}}$ is special. The equality $cond(\mathfrak{K}_{({\mathcal{O}_{D}},(\psi_1,...,\psi_r))})=0$ implies that $\pi:{\tilde{X}}\to X$ is the minimal resolution adapted to $M$ by Proposition \[prop:minadapnumchar\]. By construction, the previous assignment applied to $M$ gives $(E_1, n_1), \dots, (E_l, n_l)$. In order to prove surjectivity we need that $M$ does not have free factors. If $M$ has free factors we write $M=M_0\oplus{\mathcal{O}_{X}}^a$, where $M_0$ is without free factors. Then the previous assignment applied to $M_0$ also gives $(E_1, n_1), \dots, (E_l, n_l)$ and surjectivity is proven. The injectivity follows from Theorem \[th:Chernresolucionadapted\] and Lemma \[lem:varioD\]. \[rem:Dgenericanofactorlibre\] If the union of curvettes $D$ in the previous proof is chosen generic then the obtained module $M$ does not have free factors. By the previous Theorem there is a unique reflexive ${\mathcal{O}_{X}}$-module $M_0$ without free factors associated with the set of pairs $(E_1, n_1), \dots, (E_l, n_l)$. Let $r$ be its rank and $(\phi_1,...,\phi_r)$ be a set of generic sections. Let $({\mathcal{C}},(\psi_1,...,\psi_r))$ be the pair associated with $(M_0,(\phi_1,...,\phi_r))$ by Theorem \[th:corrsing\]. Since $M_0$ is without free factors and the sections are generic $(\psi_1,...,\psi_r)$ is a minimal set of generators of ${\mathcal{C}}$. Taking $\pi:{\tilde{X}}\to X$ the minimal adapted resolution to $M_0$ and using specialty and Proposition \[prop:invressing\], we have that ${\mathcal{C}}=\pi_*{\mathcal{O}_{D}}$ for a curve $D$ as in the previous proof. Let $D'$ be a union of curvettes as in the proof of the previous Theorem. Let $(\psi_1,...,\psi_r)$ be a minimal set of generators of $\pi_*{\mathcal{O}_{D'}}$ as a ${\mathcal{O}_{X}}$-module. Let $(M',(\phi_1,...,\phi_r))$ be the pair associated with $(\pi_*{\mathcal{O}_{D'}},(\psi_1,...,\psi_r))$ by Theorem \[th:corrsing\]. According with Proposition \[prop:invressing\], if $M$ is the module associated with $D'$ by the previous proof, then we have the equality $M=M'$. The minimal number of generators of $\pi_*{\mathcal{O}_{D'}}$ as a ${\mathcal{O}_{X}}$-module is upper semi-continuous under deformation of $D'$. Then the minimal number of generators among all choices of $D'$ as in the previous proof is $r$: if it were smaller the module $M_0$ would have rank smaller than $r$. Now, if $D'$ is chosen generic the module $M$ associated with $D'$ by the previous proof has rank $r$ and contains $M_0$ as a direct factor, hence it is equal to $M_0$. \[Cor:finalprincipal\] Let $(X,x)$ be a Gorenstein surface singularity. Then there exists a bijection between the following sets: 1. The set of special, indecomposable ${\mathcal{O}_{X}}$-modules up to isomorphism. 2. The set of irreducible divisors $E$ over $x$, such at any resolution of $X$ where $E$ appears, the Gorenstein form has not either zeros or poles along $E$. It follows immediately from Theorem \[Teo:final\] and Proposition \[prop:decompesp\]. Notice that if $(X,x)$ is a rational double point, then the previous Corollary is the McKay correspondence given by Artin and Verdier [@AV]. Let $(X,x)$ be a rational double point and denote by $\pi\colon {\tilde{X}}\to X$ the minimal resolution with exceptional divisor $E=\bigcup_{i=1}^l E_i$. Then there exists a bijection between the following sets: 1. The set of reflexive, indecomposable ${\mathcal{O}_{X}}$-modules up to isomorphism. 2. The set of irreducible divisors $E_i$ where $E_i$ is an irreducible component of the exceptional divisor $E$. Since the singularity is a rational double point the following two sets are the same: 1. The set of irreducible divisors $E_i$ where $E_i$ is an irreducible component of the exceptional divisor $E$. 2. The set of irreducible divisors $E'_i$ where $E'_i$ is a divisor over $X$, such that the minimal resolution given by Lemma \[rem:resoluciondivisores\] is small with respect to the Gorenstein form and the Gorenstein form does not have any pole in the components $E'_i$. Now the Corollary follows immediately from Corollary \[Cor:finalprincipal\] and from the fact that any reflexive module on a rational double point singularity is special. Deformations of reflexive modules and full sheaves {#sec:defs} ================================================== In the next sections we study deformations of reflexive modules. We treat simultaneously deformations over complex spaces and over complex algebroid germs (spectra of noetherian complete ${{\mathbb{C}}}$-algebras). The deformation functors {#sec:deffunctors} ------------------------ We assume basic knowledge on Deformation Theory. We follow [@Har4] as a basic reference. In order to fix terminology we recall some known definitions. \[not:restrictionstalk\] Let ${\mathcal{Y}}\to S$ be flat morphism of two complex spaces, and $y,s$ be points in each of them. Let $\overline{M}$ be a ${\mathcal{O}_{{\mathcal{Y}}}}$-module. We will use the notation $\overline{M}|_s:=\overline{M}\otimes_{{\mathcal{O}_{S}}}({\mathcal{O}_{S}}/\mathfrak{m}_s)$, where $\mathfrak{m}_s$ denotes the maximal ideal at $s$. Clearly $\overline{M}|_s$ is a ${\mathcal{O}_{{\mathcal{Y}}_s}}$ module, where ${\mathcal{Y}}_s$ is the fibre of ${\mathcal{Y}}$ over $s$. Furthermore $(\overline{M}|_s)_y$ will denote the stalk of $\overline{M}|_s$ at $y$. \[def:deforfam\] Let $Y$ be a either complex space or an algebroid germ and $M$ be a ${\mathcal{O}_{Y}}$-module. 1. A [*deformation*]{} of $(Y,M)$ over a germ of complex space (or an algebroid germ) $(S,s)$ is a triple $({\mathcal{Y}},\overline{M},\iota)$ where ${\mathcal{Y}}$ is a flat deformation of $Y$ over $S$, $\overline{M}$ is a ${\mathcal{O}_{{\mathcal{Y}}}}$-module which is flat over $S$, and $\iota$ is an isomorphism from $M$ to the fibre $\overline{M}|_s$. 2. A [*deformation fixing*]{} $Y$ of $M$ over a germ of complex space $(S,s)$ is a deformation of $(Y,M)$ over $(S,s)$ such that ${\mathcal{Y}}$ is the trivial deformation $Y\times S$. 3. Given a flat morphism ${\mathcal{Y}}\to S$, a [*flat family of modules on*]{} ${\mathcal{Y}}$ is a ${\mathcal{O}_{{\mathcal{Y}}}}$-module $\overline{M}$ which is flat over $S$. 4. A [*flat family of ${\mathcal{O}_{Y}}$-modules fixing*]{} $Y$ is a flat family of modules on $Y\times S$. Deformations of a pair $(Y,M)$ form a contravariant functor $\mathbf{Def_{Y,M}}$ from the category of germs of complex spaces to the category of sets in the usual way: morphisms of germs are transformed into mappings of set via the pull-back of deformation. Likewise deformations of $M$ fixing the base form a contravariant functor $\mathbf{Def_{M}}$. If we restrict to the category of spectra of Artinian $\mathbb{C}$-algebras, we can view the functor as a covariant functor from Artinian $\mathbb{C}$-algebras to sets. It is easy to check that Schlessinger conditions $(H0)-(H2)$ of Theorem 16.2 of [@Har4] are satisfied for these two functors. Let us remark that reflexiveness is an open property (see the next Lemma), and hence the deformation notion of Definition \[def:deforfam\] is adequate as a deformation notion of reflexive sheaves. \[lema:reflexiveopen\] Let $\sigma:{\mathcal{Y}}\to S$ be a flat family of normal surfaces. Let $\overline{M}$ be a family of modules on ${\mathcal{Y}}$. Let $y\in{\mathcal{Y}}$ be a point. Suppose that the ${\mathcal{O}_{{\mathcal{Y}}_{\sigma(y)},y}}$-module $(\overline{M}|_{\sigma(y)})_y$ is reflexive. There exists an open neighborhood $U$ of $y$ in ${\mathcal{Y}}$ such that for any $y'\in U$ the ${\mathcal{O}_{{\mathcal{Y}}_{\sigma(y')},y'}}$-module $(M|_{\sigma(y')})_{y'}$ is reflexive. Reflexiveness is equivalent to being Cohen-Macaulay of dimension $2$. An straightforward adaptation of EGA IV [@Gr § 6.11] shows that the locus where $(M|_{s'})_{x'}$ is Cohen-Macaulay of dimension $2$ is open. Since being Gorenstein is equivalent to ask that the dualizing sheaf is an invertible sheaf, it is easy to prove that being Gorenstein is also an open property. In our work we do not use that property, so we omit the proof. On the other hand, if we work at the resolution, fullness is not an open property, so one needs to restrict the deformations of Definition \[def:deforfam\] in order to get a good notion of deformations and flat families of full sheaves. \[def:veryweak\] Let ${\mathcal{X}}\to S$ be a flat family of normal Stein surfaces. A [*very weak simultaneous resolution*]{} of ${\mathcal{X}}\to S$ is a proper birational morphism $\Pi:\tilde{{\mathcal{X}}}\to{\mathcal{X}}$ satisfying 1. $\tilde{{\mathcal{X}}}$ is flat over $S$, 2. For any closed point $s\in S$ the morphism $\Pi|_s:\tilde{{\mathcal{X}}}_s\to{\mathcal{X}}_s$ is a resolution of singularities. We will use the following notation: for any $s\in S$ denote the restriction $\Pi|_{{\tilde{{\mathcal{X}}}}_s}$ by $\Pi_s:{\tilde{{\mathcal{X}}}}_s\to {\mathcal{X}}_s$. \[lem:h0h1\] Let ${\mathcal{X}}\to S$ be a flat family of normal Stein surfaces and $\Pi:\tilde{{\mathcal{X}}}\to{\mathcal{X}}$ a very weak simultaneous resolution. Let $\overline{{\mathcal{M}}}$ be a ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-module which is flat over ${\mathcal{O}_{S}}$. Then the first 2 of the following 3 conditions are equivalent and imply the third: 1. $R^1\Pi_*\overline{{\mathcal{M}}}$ is flat as ${\mathcal{O}_{S}}$-module in a neighborhood of a point $s\in S$; 2. the natural map $(\Pi_*\overline{{\mathcal{M}}})|_s\to (\Pi_s)_*(\overline{{\mathcal{M}}}|_s)$ is an isomorphism; 3. $\Pi_*\overline{{\mathcal{M}}}$ is flat as ${\mathcal{O}_{S}}$-module in a neighborhood of a point $s\in S$. If $S$ is the spectrum of an artinian algebra then the third condition is equivalent to the first two conditions. For any morphism $\phi:S'\to S$ let $\Pi': {\tilde{{\mathcal{X}}}}\times_S S'\to {\mathcal{X}}\times_S S'$, $\tilde{\psi}:{\tilde{{\mathcal{X}}}}\times_S S'\to {\tilde{{\mathcal{X}}}}$ and $\psi:{\mathcal{X}}\times_S S'\to {\mathcal{X}}$ be the natural maps. If the $3$ previous conditions are satisfied for any $s\in S$ then the natural map $$\label{eq:basechange} \psi^*(R^1\Pi_*\overline{{\mathcal{M}}})\to R^1(\Pi')_*\left(\tilde{\psi}^*\overline{{\mathcal{M}}}\right),$$ is an isomorphism. The proof is an adaptation of the methods of Section III. 12 of [@Har2]. The main difference is that in our case the morphism $\Pi$ is not projective, and that $\Pi_*\overline{{\mathcal{M}}}$ is not coherent over $S$. Now we explain the changes needed in each of the results from Hartshorne book that we will use; we numerate the results as Hartshorne does. Our base $S$ in Hartshorne’s setting is the spectrum of a ring $A$. Easy adaptations of the proofs allow to modify Hartshorne statements as follows: Proposition 12.1 is true without modification. The complex $L^\bullet$ of Proposition 12.2 is bounded above, $L^i$ is finitely generated and free over $A$ if $i>0$, and only flat over $A$ if $i\leq 0$. Proposition 12.4 is true asking $W^i$ to be flat over $A$ instead of projective, if $i=0$, and not asking $Q$ to be finitely generated if $i=0$. Proposition 12.5 is true as stated. Corollary 12.6 is true if one asks $T^0(A)$ to be flat instead of projective. Proposition 12.10 works as stated; the only point of the proof of Proposition 12.10 that needs some care is the following: the Theorem on Formal Functions ([@Har2], Chapter III, Theorem 11.1) is used only for the $0$-th cohomology. This theorem assumes projectivity for the morphism $\Pi$, but for $0$-th cohomology the theorem works without this hypothesis. Now let us proceed to the proof using Hartshorne language. There are only two functors, $T^0$ and $T^1$. Thus $T^0$ is left exact and $T^1$ right exact. Condition $(1)$ translates in the flatness of $T^1(A)$, which by the adapted Corollary 12.6 of Hartshorne is equivalent to the exactness of $T^1$. This is equivalent to the exactness of $T^0$, and by the adapted Corollary 12.6 this implies the flatness of $T^0(A)$, which is exactly Condition $(3)$. Condition $(2)$ is a particular case of the isomorphism (\[eq:basechange\]). If the first condition hold then $T^1$ is exact and hence $T^0$ is right exact. Proposition 12.5 of Hartshorne implies the isomorphism (\[eq:basechange\]). Then a direct application of Proposition 12.10 of [@Har2]) gives that Condition $(2)$ implies the right exactness of $T^0$. This implies the exactness of $T^1$ and, by the adapted Corollary 12.6, Condition $(1)$ holds. Suppose that $S$ is the spectrum of an artinian algebra. The Artinian Principle of Exchange of [@Wa] gives the equivalence between the first and third conditions. \[def:deformationfull\] Let $X$ be a normal Stein surface, $\pi:{\tilde{X}}\to X$ be a resolution and ${\mathcal{M}}$ be a full ${\mathcal{O}_{{\tilde{X}}}}$-module. 1. A [*deformation*]{} of $({\tilde{X}},X,{\mathcal{M}})$ over a germ of complex space $(S,s)$ is a cuadruple $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},\iota)$, where ${\mathcal{X}}$ is a flat deformation of $X$ over $(S,s)$, there is a proper birational morphism $\Pi:{\tilde{{\mathcal{X}}}}\to{\mathcal{X}}$ which is a very weak simultaneous resolution, $\overline{{\mathcal{M}}}$ is ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-module which is flat over $S$, and $\iota$ is an isomorphism from ${\mathcal{M}}$ to $\overline{{\mathcal{M}}}|_s$. A [*deformation fixing $X$*]{} of $({\tilde{X}},X,{\mathcal{M}})$ is a deformation $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},\iota)$ where ${\mathcal{X}}$ is the trivial deformation. A [*deformation fixing $({\tilde{X}},X)$*]{} of $({\tilde{X}},X,{\mathcal{M}})$ is a deformation where ${\mathcal{X}}$ and ${\tilde{{\mathcal{X}}}}$ are trivial deformations. 2. A [*full deformation*]{} of $({\tilde{X}},X,{\mathcal{M}})$ is a deformation $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},\iota)$ such that $R^1\Pi_*\overline{{\mathcal{M}}}$ is flat as ${\mathcal{O}_{S}}$-module. 3. Given a morphism $\Pi:{\tilde{{\mathcal{X}}}}\to{\mathcal{X}}$ as above, a family of full modules on ${\tilde{{\mathcal{X}}}}$ is a triple $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}})$ where $\overline{{\mathcal{M}}}$ is a ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-module which is flat over $S$, the ${\mathcal{O}_{S}}$-module $R^1\Pi_*\overline{{\mathcal{M}}}$ is flat, and the ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}_s}}$-module $\overline{{\mathcal{M}}}|_{s}$ is full for all $s\in S$. A flat family fixing $X$ and/or $({\tilde{X}},X)$ is defined in the obvious way. The reader may have noticed that while in the definition of full family we ask that the ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}_s}}$-module $\overline{{\mathcal{M}}}|_{s}$ is asked to be full for all $s\in S$, we do not ask the same property for full deformations over a germ $(S,s)$. The reason is the following proposition, which shows that fullness is an open property in full deformations. \[prop:fullopen\] Let $X$ be a normal Stein surface, $\pi:{\tilde{X}}\to X$ be a resolution and ${\mathcal{M}}$ be a full ${\mathcal{O}_{{\tilde{X}}}}$-module. Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},\iota)$ be a full deformation of $({\tilde{X}},X,{\mathcal{M}})$ over $(S,s)$. There exists an open neighborhood $W$ of $s\in S$ such that for any $s'\in U$ the ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}_s}}$-module $\overline{{\mathcal{M}}}|_{s}$ is full. We will use the characterization of Proposition \[fullcondiciones\]. Since ${\mathcal{M}}$ is locally free, by flatness we have that $\overline{{\mathcal{M}}}$ is also locally free. Then $\overline{{\mathcal{M}}}|_{s'}$ is locally free for any $s'\in S$. Since $R^1\Pi_*\overline{{\mathcal{M}}}$ is flat over $S$, by Lemma \[lem:h0h1\] we have that $\Pi_*\overline{{\mathcal{M}}}$ is flat over $S$ and also the equality $$\label{eq:igualdadcommutada} (\Pi_*\overline{{\mathcal{M}}})|_{s'}=(\Pi|_{{\tilde{{\mathcal{X}}}}_{s'}})_*\overline{{\mathcal{M}}}|_{s'}.$$ This implies that $\Pi_*\overline{{\mathcal{M}}}$ is a flat deformation of the reflexive module $(\Pi|_{{\tilde{{\mathcal{X}}}}_{s}})_*{\mathcal{M}}$ (here we use that ${\mathcal{M}}$ is full). By openness of reflexivity (Lemma \[lema:reflexiveopen\]), for any $s'$ is a neighborhood of $s$ in $S$ we have that $(\Pi|_{{\tilde{{\mathcal{X}}}}_{s'}})_*\overline{{\mathcal{M}}}|_{s'}$ is a reflexive ${\mathcal{O}_{{\mathcal{X}}_{s'}}}$-module. Let $E_{s'}$ be the exceptional divisor at ${\tilde{{\mathcal{X}}}}_{s'}$. We have the local cohomology exact sequence $$0\to H^0_{E_{s'}}(\overline{{\mathcal{M}}}|_{s'})\to H^0({\tilde{{\mathcal{X}}}}_{s'},\overline{{\mathcal{M}}}|_{s'})\to H^0({\tilde{{\mathcal{X}}}}_{s'}\setminus E_{s'},\overline{{\mathcal{M}}}|_{s'})\to H^1_{E_{s'}}(\overline{{\mathcal{M}}}|_{s'})\to H^1({\tilde{{\mathcal{X}}}}_{s'},\overline{{\mathcal{M}}}|_{s'}).$$ The morphism $H^0({\tilde{{\mathcal{X}}}}_{s'},\overline{{\mathcal{M}}}|_{s'})\to H^0({\tilde{{\mathcal{X}}}}_{s'}\setminus E_{s'},\overline{{\mathcal{M}}}|_{s'})$ is surjective since $H^0({\tilde{{\mathcal{X}}}}_{s'},\overline{{\mathcal{M}}}|_{s'})$ is a reflexive ${\mathcal{O}_{{\mathcal{X}}_{s'}}}$-module and ${\tilde{{\mathcal{X}}}}_{s'}\setminus E_{s'}$ is identified with ${\mathcal{X}}_{s'}$ minus a finite set of points (where the modification takes place). Hence $H^1_{E_{s'}}(\overline{{\mathcal{M}}}|_{s'})\to H^1({\tilde{{\mathcal{X}}}}_{s'},\overline{{\mathcal{M}}}|_{s'})$ is injective as needed. Let $(\phi_1,...,\phi_m)$ be a collection of global sections of ${\mathcal{M}}$, which almost generate it except at a finite set $Z\subset{\tilde{X}}$. By Equality (\[eq:igualdadcommutada\]) there exist $(\overline{\phi}_1,...,\overline{\phi}_m)$, global sections of $\overline{{\mathcal{M}}}$ which specialize to $(\phi_1,...,\phi_m)$ at the fibre ${\tilde{X}}$ over $s$. Let $\overline{Z}\subset{\tilde{{\mathcal{X}}}}$ denote the locus where the sections $(\overline{\phi}_1,...,\overline{\phi}_m)$ do not generate $\overline{{\mathcal{M}}}$. Then we have the equality $\overline{Z}\cap {\tilde{X}}=Z$, and as a consequence there is an open neighborhood $U$ of $s$ in $S$ such that $\overline{Z}\cap {\tilde{{\mathcal{X}}}}_{s'}$ is finite for any $s'\in S$. Over $U$ we have that $\overline{{\mathcal{M}}}|_{s'}$ is generically generated by global sections. At the following proposition we introduce the relevant deformation functors. \[propdef:defuntors\] Deformations of $({\tilde{X}},X,{\mathcal{M}})$ form a contravariant functor from the category of germs of complex spaces to the category of sets. We denote it by $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}}$. The functors of deformations fixing $X$ and $({\tilde{X}},X)$ are denoted respectively by $\mathbf{Def_{{\tilde{X}},{\mathcal{M}}}}$ and $\mathbf{Def_{{\mathcal{M}}}}$. Full deformations of $({\tilde{X}},X,{\mathcal{M}})$ form a contravariant functor denoted by $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$. The functors of full deformations fixing $X$ and $({\tilde{X}},X)$ are denoted respectively by $\mathbf{FullDef_{{\tilde{X}},{\mathcal{M}}}}$ and $\mathbf{FullDef_{{\mathcal{M}}}}$. These functors, restricted to the artinian basis, may be seen as a covariant functor from the category of artinian algebras to sets. The only non-trivial point is to prove that pullback of full deformations are full deformations, but this follows from the isomorphism (\[eq:basechange\]). It is again easy to check that Schlessinger conditions $(H0)-(H2)$ are satisfied for the functors defined at the previous proposition. \[prop:naturaltrans\] Let $\pi:{\tilde{X}}\to X$ be a resolution of a normal Stein surface, let ${\mathcal{M}}$ be a full ${\mathcal{O}_{{\tilde{X}}}}$-module and $M=\pi_*{\mathcal{M}}$ be its associated reflexive module. The push forward operation $\Pi_*$ along the resolution map defines a natural transformation from $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$ to $\mathbf{Def_{X,M}}$. Analogous statements holds for the deformation functors fixing $X$ and/or ${\tilde{X}}$, and for families of full modules. For the assertion about deformations let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},\iota)$ be a full deformation of $({\tilde{X}},X,{\mathcal{M}})$. Flatness of the push down $\Pi_*\overline{{\mathcal{M}}}$ holds by Lemma \[lem:h0h1\]. The isomorphism from $(\Pi_*\overline{{\mathcal{M}}})|_s$ to $M$ is obtained composing the natural isomorphism (Lemma \[lem:h0h1\]) with the isomorphism $\Pi_*\iota$. The remaining assertions are proved similarly. Although there is a bijection between full sheaves and reflexive sheaves [@Ka], as we will see below the transformation $\mathbf{FullDef_{{\mathcal{M}}}}\to\mathbf{Def_M}$ is not an isomorphism of functors. \[prop:miniversalexist\] Let $\pi \colon {\tilde{X}}\to X$ be a resolution of a normal Stein surface. Let ${\mathcal{M}}$ be a full ${\mathcal{O}_{{\tilde{X}}}}$-module and $M=\pi_*{\mathcal{M}}$ be its associated reflexive ${\mathcal{O}_{X}}$-module. Then the deformation functors $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}}$, $\mathbf{Def_{X,{\mathcal{M}}}}$, $\mathbf{Def_{{\mathcal{M}}}}$, $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$, $\mathbf{FullDef_{X,{\mathcal{M}}}}$, $\mathbf{FullDef_{{\mathcal{M}}}}$, $\mathbf{Def_{X,M}}$ and $\mathbf{Def_M}$ have miniversal deformations. Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},\iota)$ be the miniversal deformation of $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}}$, then the miniversal deformation of $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$ is the stratum of the flattening stratification of $R^1\Pi_*\overline{{\mathcal{M}}}$ containing the origin. Analogous statements hold for $\mathbf{Def_{X,{\mathcal{M}}}}$ / $\mathbf{FullDef_{X,{\mathcal{M}}}}$ and $\mathbf{Def_{{\mathcal{M}}}}$ / $\mathbf{FullDef_{{\mathcal{M}}}}$. By Theorem 16.2 of [@Har4] in order to prove the existence of miniversal deformation we only have to prove Schlessinger $(H3)$ condition. For $\mathbf{Def_M}$ this amounts to prove that $\operatorname{\text{Ext}\,}^1_{{\mathcal{O}_{X}}}(M,M)$ is finite dimensional (see for example [@Tr]) Since $M$ is reflexive, it is locally free at $X\setminus Sing(X)$. Therefore $\operatorname{\text{Ext}\,}^1_{{\mathcal{O}_{X}}}(M,M)$ is finite dimensional as needed. The functor $\mathbf{Def_{X,M}}$ fibres over the functor of deformations of $X$, with fibre the functor $\mathbf{Def_M}$. Since both the base and fibre functors satisfy $(H3)$, so $\mathbf{Def_{X,M}}$ does it. Since ${\mathcal{M}}$ is locally free the local to global spectral sequence shows that $\operatorname{\text{Ext}\,}^1_{{\mathcal{O}_{{\tilde{X}}}}}({\mathcal{M}},{\mathcal{M}})$ is isomorphic to $R^1\pi_*\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{X}}}}}({\mathcal{M}},{\mathcal{M}})$, which is finite dimensional. Hence the Schlessinger condition $(H3)$ is satisfied for the functor $\mathbf{Def_{{\mathcal{M}}}}$. Since the functor $\mathbf{FullDef_{{\mathcal{M}}}}$ is a sub-functor of $\mathbf{Def_{{\mathcal{M}}}}$, the condition $(H3)$ also holds for it. For the functors $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}}$, $\mathbf{Def_{X,{\mathcal{M}}}}$, $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}}$, $\mathbf{FullDef_{X,{\mathcal{M}}}}$ we use fibration of functors arguments as before. The flattening stratification statement is by versality and definition of the functors. Let us give a basic proposition that we will use later. \[prop:dualMdeformado\] Let $\pi\colon {\tilde{X}}\to X$ be a resolution of a normal Stein surface, let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},\iota)$ be a full deformation of $({\tilde{X}},X,{\mathcal{M}})$ over $(S,s)$. Then $\Pi_* \left(\overline{{\mathcal{M}}}^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}\right) = \left( \Pi_* \overline{{\mathcal{M}}} \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$. The proof is an adaptation of Lemma \[lema:ceroggsg\] and Lemma \[lema:dualM\]. Let ${\omega_{{\tilde{{\mathcal{X}}}}|S}}$ be relative the canonical sheaf over ${\tilde{{\mathcal{X}}}}$. The sheaf $\overline{{\mathcal{M}}} \otimes {\omega_{{\tilde{{\mathcal{X}}}}|S}}$ is locally free, hence it is flat over $S$ and $(\overline{{\mathcal{M}}} \otimes {\omega_{{\tilde{X}}|S}})|_s \cong {\mathcal{M}}\otimes {\omega_{{\tilde{X}}}}$, then by Lemma \[lema:ceroggsg\] and the Cohomology and Base Change Theorem ([@Har2], Chapter III, Theorem 12.11) we have $R^1 \Pi_* (\overline{{\mathcal{M}}} \otimes {\omega_{{\tilde{X}}}})=0$. Now the proof is parallel to the proof of Lemma \[lema:dualM\]. The correspondence for deformations at families of normal Stein surfaces with Gorenstein singularities {#sec:singfam} ------------------------------------------------------------------------------------------------------ \[def:enhanceddef1\] Let $X$ be a normal Stein surface, let ${\mathcal{C}}$ be a rank $1$ generically reduced Cohen-Macaulay ${\mathcal{O}_{X}}$-module of dimension $1$, and a system of generators $(\psi_1,...,\psi_r)$ of ${\mathcal{C}}$ as ${\mathcal{O}_{X}}$ module. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$ and $(\phi_1,...,\phi_r)$ be $r$ sections. 1. A [*deformation of*]{} $(X,{\mathcal{C}},(\psi_1,...,\psi_r))$ over a germ $(S,s)$ is a cuadruple $({\mathcal{X}},\overline{{\mathcal{C}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$, where $({\mathcal{X}},\overline{{\mathcal{C}}},\rho)$ is a deformation of $(X,{\mathcal{C}})$ and $(\overline{\psi}_1,...,\overline{\psi}_r)$ are sections of $\overline{{\mathcal{C}}}$, which via the isomorphism $\rho$ restrict to $(\psi_1,...,\psi_r)$ over $s$. A deformation [*fixing*]{} $X$ is a deformation such that ${\mathcal{X}}$ is the trivial deformation of $X$. 2. A [*deformation of*]{} $(X,M,(\phi_1,...,\phi_r))$ over a germ $(S,s)$ is a cuadruple $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$, where $({\mathcal{X}},\overline{M},\iota)$ is a deformation of $M$ and $(\overline{\phi}_1,...,\overline{\phi}_r)$ are sections of $\overline{M}$, which via the isomorphism $\iota$ restrict to $(\phi_1,...,\phi_r)$ over $s$. A deformation [*fixing*]{} $X$ is a deformation such that ${\mathcal{X}}$ is the trivial deformation of $X$. The deformations defined above, together with the pullback operation, form two contravariant functors $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$ and $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$ from the category of germs of complex spaces to the category of sets. Deformations fixing the base form two contravariant functors called $\mathbf{Def_{{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$ and $\mathbf{Def_{M}^{(\phi_1,...,\phi_r)}}$ . Now we extend the correspondences at the Stein space (Theorem \[th:corrsing\]) to get an isomorphism of functors. First we need the following lemma, which can be found in Proposition 0.1 of [@dJvS], and also in [@SchE], Proposition 2.2. \[lem:extbasechange\] Let $Y$ and $(S,s)$ be a complex space and a germ of complex space. Let ${\mathcal{Y}}$ be a flat deformation of $Y$ over $(S,s)$. Let $\overline{{\mathcal{F}}}$ be a ${\mathcal{O}_{{\mathcal{Y}}}}$-module. Let $\mathfrak{m}_s$ be the maximal ideal at $s$. For any ${\mathcal{O}_{S}}$-module $L$ and any index $i$ there is a natural morphism $$\label{eq:basechangeext} \phi^i(L):\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{Y}}}}}^i(\overline{{\mathcal{F}}},{\mathcal{O}_{{\mathcal{Y}}}})\otimes_{{\mathcal{O}_{S}}} L\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{Y}}}}}^i(\overline{{\mathcal{F}}},{\mathcal{O}_{{\mathcal{Y}}}}\otimes_{{\mathcal{O}_{S}}} L).$$ The following assertions hold: 1. If $\phi^i({\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ is surjective, then $\phi^i(L)$ is an isomorphism for any $L$. 2. Assume that $\phi^i({\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ is surjective. Then $\phi^{i-1}({\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ is surjective if and only if $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{Y}}}}}^i(\overline{{\mathcal{F}}},{\mathcal{O}_{{\mathcal{Y}}}})$ is flat over $S$. 3. If $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{Y}}}}}^i({\mathcal{F}},{\mathcal{O}_{{\mathcal{Y}}}}\otimes {\mathcal{O}_{S,s}}/\mathfrak{m}_s)=0$ then $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{Y}}}}}^i({\mathcal{F}},{\mathcal{O}_{{\mathcal{Y}}}}\otimes_{{\mathcal{O}_{S}}} L)=0$ for all $L$. \[lem:isoext\] In the situation of the preceeding Lemma, if $\overline{{\mathcal{F}}}$ is flat over $S$ then we have the isomorphism $$\label{eq:isoext} \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{Y}}}}}^i(\overline{{\mathcal{F}}},{\mathcal{O}_{{\mathcal{Y}}}}\otimes {\mathcal{O}_{S,s}}/\mathfrak{m}_s)\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{Y}}}^i(\overline{{\mathcal{F}}}\otimes {\mathcal{O}_{S,s}}/\mathfrak{m}_s,{\mathcal{O}_{{\mathcal{Y}}}}\otimes {\mathcal{O}_{S,s}}/\mathfrak{m}_s).$$ It is straightforward if one compute $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}$ using a free resolution of ${\mathcal{F}}$. Now we are able to extend the correspondence given by Theorem \[th:corrsing\]. From now and for this and the following section we will always assume that $X$ is a normal Stein surface with Gorenstein singularities. \[th:dirXdef\] Let $X$ be a normal Stein surface with Gorenstein singularities. Let $(M,(\phi_1,..,\phi_r))$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$ and a collection $r$ nearly generic sections. Let $({\mathcal{C}},(\psi_1,...,\psi_r))$ be the rank 1 generically reduced Cohen-Macaulay ${\mathcal{O}_{X}}$-module of dimension $1$ with the collection of generators obtained from $(M,(\phi_1,..,\phi_r))$ by the direct correspondence at the $X$ (see Theorem \[th:corrsing\]). The correspondence defined at Theorem \[th:corrsing\] extends to define an isomorphism between the functors $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$. The isomorphism restricts to an isomorphism between $\mathbf{Def_{M}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{Def_{{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$. Let $(S,s)$ be a germ of complex space. Let $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ be a deformation of $(X,M,(\phi_1,...,\phi_r))$ over $(S,s)$. Let ${\mathcal{O}_{{\mathcal{X}}}}^r\to\overline{M}$ be the morphism induced by the sections, denote its cokernel by $(\overline{{\mathcal{C}}})'$. Then we have the exact sequence: $$\label{eq:dirsingfam1} 0\to {\mathcal{O}_{{\mathcal{X}}}}^r\to\overline{M}\to (\overline{{\mathcal{C}}})'\to 0.$$ The flatness of $\overline{M}$ over $S$, and the fact that the first mapping specializes over $s$ to an injection, implies the flatness of $(\overline{{\mathcal{C}}})'$ over $S$, by using the Local Criterion of Flatness. The specialization of the sequence to the fibre over $s$ is the exact sequence of ${\mathcal{O}_{X}}$-modules $$\label{eq:dirsingfam2} 0\to{\mathcal{O}_{X}}^r\to\overline{M}|_s\to (\overline{{\mathcal{C}}})'|_s\to 0,$$ and $\iota$ induces an isomorphism between this sequence and the exact sequence $$\label{eq:dirsingfam3} 0\to {\mathcal{O}_{X}}^r\to M\to {\mathcal{C}}'\to 0,$$ induced by the sections $(\phi_1,...,\phi_r)$. The dual of the last sequence is the sequence $$\label{eq:dirsingfam4} 0\to N\to {\mathcal{O}_{X}}^r\to {\mathcal{C}}\to 0,$$ where the last morphism of the sequence gives rise to the generators $(\psi_1,...,\psi_r)$ of ${\mathcal{C}}$ (see the proof of Theorem \[th:corrsing\]). Dualize the sequence (\[eq:dirsingfam1\]) with respect to ${\mathcal{O}_{{\mathcal{X}}}}$ and obtain the exact sequence $$0\to \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{N},{\mathcal{O}_{{\mathcal{X}}}})\to {\mathcal{O}_{{\mathcal{X}}}}^r\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}((\overline{{\mathcal{C}}})',{\mathcal{O}_{{\mathcal{X}}}})\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{M},{\mathcal{O}_{{\mathcal{X}}}}).$$ Define $\overline{N}:=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{M},{\mathcal{O}_{{\mathcal{X}}}})$. We claim that $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{M},{\mathcal{O}_{{\mathcal{X}}}})$ vanishes. Indeed, since $M$ is Cohen-Macaulay of dimension $2$ we have the vanishing $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{X}}}(M,{\mathcal{O}_{X}})=0$ by Theorem \[Th:Herzog\]. By Lemma \[lem:isoext\] we have the isomorphism $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{M},{\mathcal{O}_{X}})\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{X}}}(M,{\mathcal{O}_{X}})=0$. This vanishing, together with Lemma \[lem:extbasechange\] proves the claim. As a consequence of the claim we have the exact sequence $$\label{eq:dirsingfam5} 0\to \overline{N}\to {\mathcal{O}_{{\mathcal{X}}}}^r\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}((\overline{{\mathcal{C}}})',{\mathcal{O}_{{\mathcal{X}}}})\to 0.$$ We define $\overline{{\mathcal{C}}}:=\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}((\overline{{\mathcal{C}}})',{\mathcal{O}_{{\mathcal{X}}}})$. We claim that the following assertions hold: 1. the ${\mathcal{O}_{{\mathcal{X}}}}$-module $\overline{{\mathcal{C}}}$ is flat over $S$. 2. The specialization $\overline{N}|_s\to {\mathcal{O}_{X}}^r$ of the first morphism of the sequence coincides with $N\to{\mathcal{O}_{X}}^r$. Assume the claim. The second assertion induces an identification of ${\mathcal{O}_{X}}^r\to{\mathcal{C}}$ with ${\mathcal{O}_{X}}^r\to \overline{{\mathcal{C}}}|_s$. Let $\rho$ denote the isomorphism ${\mathcal{C}}\to\overline{{\mathcal{C}}}|_s$. The second morphism of Sequence (\[eq:dirsingfam5\]) induces a collection of sections $(\bar\psi_1,...,\bar\psi_r)$. The first assertion shows that $({\mathcal{X}},\overline{{\mathcal{C}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ is a deformation of $(X,{\mathcal{C}},(\psi_1,...,\psi_r))$ over $(S,s)$. Let us prove the claim. Since ${\mathcal{C}}'$ is Cohen-Macaulay of dimension $1$ and $X$ has Gorenstein singularities, Theorem \[Th:Herzog\] implies the vanishing $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^i_{{\mathcal{O}_{X}}}({\mathcal{C}}',{\mathcal{O}_{X}})=0$ if $i\geq 2$. Then, since $(\overline{{\mathcal{C}}})'$ is flat over $S$, using Lemma \[lem:isoext\] we obtain the vanishing $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^i_{{\mathcal{O}_{{\mathcal{X}}}}}((\overline{{\mathcal{C}}})',{\mathcal{O}_{X}})=0$ if $i\geq 2$. Now we will apply Lemma \[lem:extbasechange\] repeatedly for ${\mathcal{F}}=(\overline{{\mathcal{C}}})'$: by the vanishing $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^2_{{\mathcal{O}_{{\mathcal{X}}}}}((\overline{{\mathcal{C}}})',{\mathcal{O}_{X}})=0$ we deduce that $\phi^2({\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ is surjective; the Lemma \[lem:extbasechange\] (1) shows that $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^2((\overline{{\mathcal{C}}})',{\mathcal{O}_{{\mathcal{X}}}})$ vanishes (hence it is flat over $S$). By Lemma \[lem:extbasechange\], (2) we have that $\phi^1({\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ is surjective. Lemma \[lem:extbasechange\], (2) shows now that $\overline{{\mathcal{C}}}=\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}((\overline{{\mathcal{C}}})',{\mathcal{O}_{{\mathcal{X}}}})$ is flat over $S$ if and only if $\phi^0({\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ is surjective. This surjectivity holds since the target of this map vanishes because $(\overline{{\mathcal{C}}})'$ has proper support. This shows Assertion $(1)$ of the claim. For Assertion $(2)$ we apply Lemma \[lem:extbasechange\] for ${\mathcal{F}}=\overline{M}$. We need to show the isomorphism $\overline{N}|_s\cong N$, but this follows from Lemma \[lem:extbasechange\], (1), if we show that $\phi^0({\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ is surjective. By Lemma \[lem:extbasechange\], (2), this is reduced to prove the vanishing of $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\overline{M},{\mathcal{O}_{{\mathcal{X}}}}\otimes_{{\mathcal{O}_{S}}} {\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ and the flatness over $S$ of $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\overline{M},{\mathcal{O}_{{\mathcal{X}}}})$. The vanishing holds because $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\overline{M},{\mathcal{O}_{{\mathcal{X}}}}\otimes_{{\mathcal{O}_{S}}} {\mathcal{O}_{S,s}}/\mathfrak{m}_s)$ is isomorphic to $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1(M,{\mathcal{O}_{X}})$ by flatness of $\overline{M}$ and Lemma \[lem:isoext\], and the second module vanishes by Theorem \[Th:Herzog\]. The flatness of $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\overline{M},{\mathcal{O}_{{\mathcal{X}}}})$ holds because this module also vanishes (apply Lemma \[lem:isoext\] and Lemma \[lem:extbasechange\], (3) and (1) as before, starting from the vanishing of $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{X}}}(M,{\mathcal{O}_{X}})$). In order to have a natural transformation from $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$ to $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$ we have to show that the construction commutes with pullbacks. This follows from Lemma \[lem:extbasechange\], (1), if we show the isomorphism $\overline{{\mathcal{C}}}|_s\cong\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1((\overline{{\mathcal{C}}})',{\mathcal{O}_{{\mathcal{X}}}}\otimes_{{\mathcal{O}_{S}}} {\mathcal{O}_{S,s}}/\mathfrak{m}_s)$. By the flatness of $(\overline{{\mathcal{C}}})'$ over $S$ and Lemma \[lem:isoext\] the second module is isomorphic to ${\mathcal{C}}$, and then the desired isomorphism becomes the already proven identity $\overline{{\mathcal{C}}}|_s\cong {\mathcal{C}}$. Now we define the inverse natural transformation from $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$ to $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$. Let $({\mathcal{X}},\overline{{\mathcal{C}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be a deformation of $(X,{\mathcal{C}},(\psi_1,...,\psi_r))$ over $(S,s)$. Consider the exact sequence induced by the sections: $$\label{eq:invsingfam1} 0\to \overline{N}\to{\mathcal{O}_{{\mathcal{X}}}}^r\to \overline{{\mathcal{C}}}\to 0.$$ The flatness of $\overline{{\mathcal{C}}}$ over $S$ implies the flatness of $\overline{N}$ over $S$. The specialization of the sequence to the fibre over $s$ is the exact sequence of ${\mathcal{O}_{X}}$-modules $$\label{eq:invsingfam2} 0\to \overline{N}|_s\to{\mathcal{O}_{X}}^r\to \overline{{\mathcal{C}}}|_s\to 0,$$ and $\rho$ induces an isomorphism between this sequence and the exact sequence $$\label{eq:invsingfam3} 0\to N\to{\mathcal{O}_{X}}^r\to {\mathcal{C}}\to 0,$$ induced by the generators $(\psi_1,...,\psi_r)$. The dual of the last sequence is the sequence $$\label{eq:invsingfam4} 0\to{\mathcal{O}_{X}}^r\to M\to\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{X}}}({\mathcal{C}},{\mathcal{O}_{X}})\to 0,$$ where the sections $(\phi_1,...,\phi_r)$ are induced by the first map of the sequence (see the proof of Theorem \[th:corrsing\]). Dualize the sequence (\[eq:invsingfam1\]) with respect to ${\mathcal{O}_{{\mathcal{X}}}}$ and obtain the exact sequence $$\label{eq:invsingfam5} 0\to {\mathcal{O}_{{\mathcal{X}}}}^r\to \overline{M}\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{{\mathcal{C}}},{\mathcal{O}_{{\mathcal{X}}}})\to 0,$$ where we define $\overline{M}:=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{N},{\mathcal{O}_{{\mathcal{X}}}})$. We claim that the following assertions hold: 1. the ${\mathcal{O}_{{\mathcal{X}}}}$-module $\overline{M}$ is flat over $S$. 2. The specialization $ {\mathcal{O}_{X}}^r\to \overline{M}|_s$ of the first morphism of the sequence is isomorphic to ${\mathcal{O}_{X}}^r\to M$. Assume the claim. The second assertion induces an isomorphism from ${\mathcal{O}_{X}}^r\to M$ to ${\mathcal{O}_{X}}^r\to \overline{M}|_s$. Let $\iota$ denote the isomorphism $M\to\overline{M}|_s$. The first morphism of Sequence (\[eq:invsingfam5\]) induces a collection of sections $(\bar\phi_1,...,\bar\phi_r)$. The first assertion shows that $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ is a deformation of $(X,{\mathcal{C}},(\phi_1,...,\phi_r))$ over $(S,s)$. The proof of the claim is an application of Lemmata \[lem:isoext\] and \[lem:extbasechange\] competely analogous to the proof of the previous two claim in this proof. We omit it here. In order to have a natural transformation from $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$ to $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$ we have to show that the construction commutes with pullbacks, but this is an application of Lemmata \[lem:isoext\] and \[lem:extbasechange\], similar to the proof that the transformation from $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$ to $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$ commutes with pullbacks. Finally we need to check that the correspondences that we have just defined are inverse to each other, but this is clear by construction. It is obvious that the isomorphisms that we have defined restrict to an isomorphism between $\mathbf{Def_{M}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{Def_{{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$. The correspondence for deformations at simultaneous resolutions of families of normal Stein Surfaces with Gorenstein singularities {#sec:resfam} ---------------------------------------------------------------------------------------------------------------------------------- \[def:enhanceddef2\] Let $X$ be a normal Stein surface and $\pi:{\tilde{X}}\to X$ be a resolution. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be a rank $1$ generically reduced Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module of dimension $1$, and a system of generators as ${\mathcal{O}_{{\tilde{X}}}}$-module satisfying the Containment Condition. Let ${\mathcal{M}}$ be a full ${\mathcal{O}_{{\tilde{X}}}}$-module of rank $r$ and $(\phi_1,...,\phi_r)$ be $r$ nearly generic sections. 1. A [*deformation of*]{} $({\tilde{X}},X,{\mathcal{A}},(\psi_1,...,\psi_r))$ over a germ $(S,s)$ is a quintuple $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$, where $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},\rho)$ is a deformation of $({\tilde{X}},X,{\mathcal{A}})$, and $(\overline{\psi}_1,...,\overline{\psi}_r)$ are sections of $\overline{{\mathcal{A}}}$, which via the isomorphism $\rho$ restrict to $(\psi_1,...,\psi_r)$ over $s$. 2. A [*specialty defect constant deformation of*]{} $({\mathcal{A}},(\psi_1,...,\psi_r))$ over a germ $(S,s)$ is a deformation $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ such that the cokernel $\overline{{\mathcal{D}}}$ of the natural mapping $$\Pi_*{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to \Pi_*\overline{{\mathcal{A}}}$$ induced by the sections $(\overline{\psi}_1,...,\overline{\psi}_r)$ is flat over $S$. 3. A [*(full) deformation of*]{} $({\tilde{X}},X,{\mathcal{M}},(\phi_1,...,\phi_r))$ over a germ $(S,s)$ is a quintuple $$({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota),$$ where $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},\iota)$ is a (full) deformation of $({\tilde{X}},X,{\mathcal{M}})$ and $(\overline{\phi}_1,...,\overline{\phi}_r)$ are sections of $\overline{{\mathcal{M}}}$, which via the isomorphism $\iota$ restrict to $(\phi_1,...,\phi_r)$ over $s$. The deformations defined above give rise to functors $\mathbf{Def_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$, $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$, $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$. The obvious restricted deformation functors fixing ${\tilde{X}}$ or ${\tilde{X}}$ and $X$ are denoted by $\mathbf{Def_{X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$, $\mathbf{SDCDef_{X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$, $\mathbf{Def_{X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$, $\mathbf{FullDef_{X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$, $\mathbf{Def_{{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$, $\mathbf{SDCDef_{{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$, $\mathbf{Def_{{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{FullDef_{{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$. The assignements defined in the previous definition are in fact contravariant functors. The only non-trivial assertion is for $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$: we need to prove the preservation of the flatness of the cokernel ${\mathcal{D}}$ under pullback. This holds because the formation of cokernels commutes with pullbacks and flatness is preserved by pullbacks. \[lem:defsarcos\] In the setting of the previous definition, suppose that $(\psi_1,...,\psi_r)$ generate ${\mathcal{A}}$ as a ${\mathcal{O}_{X}}$-module (that is, they generate $\pi_*{\mathcal{A}}$). Then the functors $\mathbf{Def_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$ and $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$ coincide. The specialization of the cokernel of $\Pi_*:{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to \pi_*\overline{{\mathcal{X}}}$ to the fibre over $s$ vanishes, and hence the cokernel vanishes as well by Nakayama Lemma. \[th:dirresdef\] Let $X$ be a normal Stein surface with Gorenstein singularities. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be a Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module of dimension $1$, and a system of generators as ${\mathcal{O}_{{\tilde{X}}}}$-module satisfying the Containment Condition, let $({\mathcal{M}},(\phi_1,...,\phi_r))$ be a full ${\mathcal{O}_{{\tilde{X}}}}$-module of rank $r$ with $r$ nearly generic sections. Suppose that the pairs $({\mathcal{A}},(\psi_1,...,\psi_r))$ and $({\mathcal{M}},(\phi_1,...,\phi_r))$ are related by the correspondence of Theorem \[th:corres\]. There is an isomorphism of functors between $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$. The isomorphism restricts to isomorphisms between $\mathbf{FullDef_{X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{SDCDef_{X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$, and between $\mathbf{FullDef_{{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{SDCDef_{{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$. The restriction statements are obvious after finding an isomorphism between $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$. The proof of this isomorphism has two parts. In the first we find an isomorphism between $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{Def_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$. In the second part we prove that, under the defined isomorphism full deformations correspond to specialty defect constant deformations. <span style="font-variant:small-caps;">Part 1</span>. The first part runs parallel to the proof of Theorem \[th:dirXdef\], so we skip many details and include only what is needed to define the isomorphism and to set up the notation for the proof of Part 2. Let $(S,s)$ be a germ of complex space. Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ be an element of $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}(S,s)$. Consider the exact sequence induced by the sections: $$\label{eq:dirresfam1} 0\to {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to\overline{{\mathcal{M}}}\to (\overline{{\mathcal{A}}})'\to 0.$$ The flatness of $\overline{{\mathcal{M}}}$ over $S$, and the fact that the first mapping specializes over $s$ to an injection, implies the flatness of $(\overline{{\mathcal{A}}})'$ over $S$, by using the local criterion of flatness. The specialization of the sequence to the fibre over $s$ is the exact sequence of ${\mathcal{O}_{{\tilde{X}}}}$-modules $$\label{eq:dirresfam2} 0\to{\mathcal{O}_{{\tilde{X}}}}^r\to\overline{{\mathcal{M}}}|_s\to (\overline{{\mathcal{A}}})'|_s\to 0,$$ and $\iota$ induces an isomorphism between this sequence and the exact sequence $$\label{eq:dirresfam3} 0\to {\mathcal{O}_{{\tilde{X}}}}^r\to {\mathcal{M}}\to ({\mathcal{A}})'\to 0,$$ induced by the sections $(\phi_1,...,\phi_r)$. The dual of this last sequence is the sequence $$\label{eq:dirresfam4} 0\to {\mathcal{N}}\to {\mathcal{O}_{{\tilde{X}}}}^r\to {\mathcal{A}}\to 0,$$ where the last morphism of the sequence gives rise to the generators $(\psi_1,...,\psi_r)$ of ${\mathcal{A}}$ as a ${\mathcal{O}_{{\tilde{X}}}}$-module (see the proof of Theorem \[th:corres\]). Dualize the sequence (\[eq:dirresfam1\]) with respect to ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$ and obtain the exact sequence $$\label{eq:dirresfam5} 0\to \overline{{\mathcal{N}}}\to {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to \overline{{\mathcal{A}}}\to 0,$$ where $\overline{{\mathcal{N}}}:=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}(\overline{{\mathcal{M}}},{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}})$ and $\overline{{\mathcal{A}}}:=\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}((\overline{{\mathcal{A}}})',{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}})$. The following two assertions are proved like the corresponding ones in the proof of Theorem \[th:dirXdef\]. 1. the ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-module $\overline{{\mathcal{A}}}$ is flat over $S$. 2. The specialization $\overline{{\mathcal{N}}}|_s\to {\mathcal{O}_{X}}^r$ of the first morphism of the sequence is isomorphic to ${\mathcal{N}}\to{\mathcal{O}_{X}}^r$. The second assertion induces an isomorphism from ${\mathcal{O}_{{\tilde{X}}}}^r\to{\mathcal{A}}$ to ${\mathcal{O}_{{\tilde{X}}}}^r\to \overline{{\mathcal{A}}}|_s$. Let $\rho$ denote the isomorphism ${\mathcal{A}}\to\overline{{\mathcal{A}}}|_s$. The second morphism of Sequence (\[eq:dirresfam5\]) induces a collection of sections $(\bar\psi_1,...,\bar\psi_r)$. We have that $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ is a deformation of $({\tilde{X}},X,{\mathcal{A}},(\psi_1,...,\psi_r))$ over $(S,s)$ by the first assertion. So, we have defined map $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}(S,s)\to\mathbf{Def_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}(S,s)$. In order to have a natural transformation of functors from $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ to $\mathbf{Def_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$ we need to show that the transformation which we have defined commutes with pullbacks. This is analogous to the corresponding statement in the proof of Theorem \[th:dirXdef\]. Now we define the inverse natural transformation from $\mathbf{Def_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$ to $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$. Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be a deformation of $({\tilde{X}},X,{\mathcal{A}},(\psi_1,...,\psi_r))$ over $(S,s)$. Consider the exact sequence induced by the sections: $$\label{eq:invresfam1} 0\to \overline{{\mathcal{N}}}\to{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to \overline{{\mathcal{A}}}\to 0.$$ The flatness of $\overline{{\mathcal{A}}}$ over $S$ implies the flatness of $\overline{{\mathcal{N}}}$ over $S$. The specialization of the sequence to the fibre over $s$ is the exact sequence of ${\mathcal{O}_{{\tilde{X}}}}$-modules $$\label{eq:invresfam2} 0\to \overline{{\mathcal{N}}}|_s\to{\mathcal{O}_{{\tilde{X}}}}^r\to \overline{{\mathcal{A}}}|_s\to 0,$$ and $\rho$ induces an isomorphism between this sequence and the exact sequence $$\label{eq:invresfam3} 0\to {\mathcal{N}}\to{\mathcal{O}_{X}}^r\to {\mathcal{A}}\to 0.$$ induced by the generators $(\psi_1,...,\psi_r)$. The dual of this last sequence is the sequence $$\label{eq:invresfam4} 0\to{\mathcal{O}_{{\tilde{X}}}}^r\to {\mathcal{M}}\to\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\tilde{X}}}}}({\mathcal{A}},{\mathcal{O}_{{\tilde{X}}}})\to 0,$$ where the sections $(\phi_1,...,\phi_r)$ are induced by the first map of the sequence (see the proof of Theorem \[th:corres\]). Dualize the sequence (\[eq:invresfam1\]) with respect to ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$ and obtain the exact sequence $$\label{eq:invresfam5} 0\to {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to \overline{{\mathcal{M}}}\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}(\overline{{\mathcal{A}}},{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}})\to 0,$$ where $\overline{{\mathcal{M}}}:=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}(\overline{{\mathcal{N}}},{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}})$. The following two assertions are proved like the corresponding ones in the proof of Theorem \[th:dirXdef\]: 1. the ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-module $\overline{{\mathcal{M}}}$ is flat over $S$. 2. The specialization $ {\mathcal{O}_{{\tilde{X}}}}^r\to \overline{{\mathcal{M}}}|_s$ of the first morphism of the sequence is isomorphic to ${\mathcal{O}_{{\tilde{X}}}}^r\to{\mathcal{M}}$. The second assertion induces an isomorphism from ${\mathcal{O}_{{\tilde{X}}}}^r\to {\mathcal{M}}$ to ${\mathcal{O}_{{\tilde{X}}}}^r\to \overline{{\mathcal{M}}}|_s$. Let $\iota$ denote the isomorphism ${\mathcal{M}}\to\overline{{\mathcal{M}}}|_s$. The first morphism of Sequence (\[eq:invresfam5\]) induces a collection of sections $(\bar\phi_1,...,\bar\phi_r)$. The first assertion shows that $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ is a deformation of $({\tilde{X}},X,{\mathcal{M}},(\phi_1,...,\phi_r))$ over $(S,s)$. <span style="font-variant:small-caps;">Part 2</span>. Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ be an element of $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}(S,s)$. We have to show that the quintuple $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ associated to it in Part 1 is a specialty constant deformation. In this part the Gorenstein condition plays an important role. For this we have to show that the cokernel $\overline{{\mathcal{D}}}$ of the map $\Pi_*{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to \Pi_*\overline{{\mathcal{A}}}$ is flat over $S$. Denote by $\overline{{\mathcal{C}}}$ the image of the same map. We have the exact sequences $$\label{eq:Cbarra} 0\to\Pi_*\overline{{\mathcal{N}}}\to {\mathcal{O}_{{\mathcal{X}}}}^r\to \overline{{\mathcal{C}}}\to 0,$$ $$\label{eq:Dbarra} 0\to \overline{{\mathcal{C}}}\to\Pi_*\overline{{\mathcal{A}}}\to\overline{{\mathcal{D}}}\to 0.$$ Now we specialize Sequence (\[eq:Dbarra\]) at $s$. If we denote the maximal ideal of $s$ by $\mathfrak{m}_s$, and use the flatness of $\overline{{\mathcal{A}}}$ over $S$, we have the exact sequence $$0\to Tor_1^{{\mathcal{O}_{S}}}(\overline{{\mathcal{D}}},{\mathcal{O}_{S}}/\mathfrak{m}_s)\to (\overline{{\mathcal{C}}})|_s\to(\Pi_*\overline{{\mathcal{A}}})|_s\to (\overline{{\mathcal{D}}})|_s\to 0.$$ We claim that the second morphism of the sequence can be identified with the injective morphism ${\mathcal{C}}\to\pi_*{\mathcal{A}}$, where ${\mathcal{C}}$ is the ${\mathcal{O}_{X}}$-module spanned by $(\psi_1,...,\psi_r)$. If the claim is true we deduce the vanishing of $Tor_1^{{\mathcal{O}_{S}}}(\overline{{\mathcal{D}}},{\mathcal{O}_{S}}/\mathfrak{m}_s)$. This implies the flatness of $\overline{{\mathcal{D}}}$ using the local criterion of flatness. In order to prove the claim we produce natural identifications $(\Pi_*\overline{{\mathcal{A}}})|_s\cong \pi_*{\mathcal{A}}$ and $(\overline{{\mathcal{C}}})|_s\cong {\mathcal{C}}$, which show that the morphism $(\overline{{\mathcal{C}}})|_s\to(\Pi_*\overline{{\mathcal{A}}})|_s$ is injective, yielding the desired vanishing. For the first identification notice that $\Pi_*\overline{{\mathcal{A}}}$ coincides with $\overline{{\mathcal{A}}}$ with the ${\mathcal{O}_{{\mathcal{X}}}}$-module structure obtained by restriction of scalars; similarly $\pi_*{\mathcal{A}}$ coincides with ${\mathcal{A}}$ with the ${\mathcal{O}_{X}}$-module structure obtained by restriction of scalars. Then the needed identification is the isomorphism $\rho$ defined above. The second identification is a bit more involved (at the moment we do not even know the flatness of $\overline{{\mathcal{C}}}$ over $S$). We proceed like in Section \[sec:cohomology\] to obtain a diagram similar to (\[diagram:comparacionM\]). Dualizing sequence (\[eq:Cbarra\]) we obtain the sequence $$0 \to {\mathcal{O}_{{\mathcal{X}}}}^r \to \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{N}}},{\mathcal{O}_{{\mathcal{X}}}}) \to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right) \to 0.$$ Applying $\Pi_*$ to Sequence (\[eq:dirresfam1\]) and using the isomorphism $(\overline{{\mathcal{A}}})'\cong\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}(\overline{{\mathcal{A}}},{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}})$, which comes from the fact that the natural transformations are inverse to each other, we obtain the exact sequence $$0 \to {\mathcal{O}_{{\mathcal{X}}}}^r \to \Pi_*\overline{{\mathcal{M}}} \to \Pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}^1 \left(\overline{{\mathcal{A}}}, {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}} \right) \to R^1 \Pi_* {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r \to R^1 \Pi_* \overline{{\mathcal{M}}} \to 0.$$ We have the chain of equalities $$\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{N}}},{\mathcal{O}_{{\mathcal{X}}}})=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}(\overline{{\mathcal{M}}},{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}),{\mathcal{O}_{{\mathcal{X}}}})=$$ $$=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{M}}},{\mathcal{O}_{{\mathcal{X}}}}),{\mathcal{O}_{{\mathcal{X}}}})=\Pi_*\overline{{\mathcal{M}}}.$$ The first is because $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}(\overline{{\mathcal{M}}},{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}) \cong \overline{{\mathcal{N}}}$, the second is a consequence of Proposition \[prop:dualMdeformado\], and the third is a consequence of Proposition \[prop:naturaltrans\] and the fact that flat deformations of reflexive modules are reflexive. The last two exact sequences, together with the previous identifications yields the following commutative diagram: $$\label{diagram:comparacionMbarra} \begin{tikzpicture} \matrix (m)[matrix of math nodes, nodes in empty cells,text height=1.5ex, text depth=0.25ex, column sep=1.5em,row sep=2em] { 0 & {\mathcal{O}_{{\mathcal{X}}}}^r & \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{N}}},{\mathcal{O}_{{\mathcal{X}}}}) & \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right) & 0\\ 0 & {\mathcal{O}_{{\mathcal{X}}}}^r & \Pi_*\overline{{\mathcal{M}}} & \Pi_* \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}^1 \left(\overline{{\mathcal{A}}}, {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}} \right) & R^1 \Pi_* {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r & R^1 \Pi_* \overline{{\mathcal{M}}} & 0\\}; \draw[-stealth] (m-1-1) -- (m-1-2); \draw[-stealth] (m-1-2) -- (m-1-3); \draw[-stealth] (m-1-3) edge node [auto] {$h$} (m-1-4); \draw[-stealth] (m-1-4) -- (m-1-5); \draw[-stealth] (m-2-1) -- (m-2-2); \draw[-stealth] (m-2-2) -- (m-2-3); \draw[-stealth] (m-2-3) edge node [auto] {$h$} (m-2-4); \draw[-stealth] (m-2-4) -- (m-2-5); \draw[-stealth] (m-2-5) -- (m-2-6); \draw[-stealth] (m-2-6) -- (m-2-7); \draw[-stealth] (m-1-2) edge node [auto] {$Id$} (m-2-2); \draw[-stealth] (m-1-3) edge node [auto] {$Id$} (m-2-3); \draw[-stealth] (m-1-4) edge node [auto] {$\theta$} (m-2-4); \end{tikzpicture}$$ where $\theta$ is the only map making the diagram commutative. Since we have the isomorphism $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}(\overline{{\mathcal{N}}},{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}) \cong \overline{{\mathcal{M}}}$, dualizing the sequence (\[eq:dirresfam5\]), and using the exact sequence (\[eq:dirresfam1\]) we obtain that $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}^1 \left(\overline{{\mathcal{A}}}, {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}} \right)$ is isomorphic to $(\overline{{\mathcal{A}}}')$, and, as ${\mathcal{O}_{S}}$-module, $\Pi_*(\overline{{\mathcal{A}}}')$ is equal to $(\overline{{\mathcal{A}}}')$, which is flat over $S$. Then, the exactness of the lower row of the diagram and the flatness of $R^1 \Pi_* \overline{{\mathcal{M}}}$ imply that $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)$ is flat over $S$. As a consequence, specializing the first row of diagram (\[diagram:comparacionMbarra\]) over $s$ we obtain the exact sequence $$0 \to {\mathcal{O}_{X}}^r \to M \to \left(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)\right)|_s \to 0.$$ Comparing with the first row of diagram (\[diagram:comparacionM\]) we obtain the isomorphism $$\label{eq:espext} \left(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)\right)|_s\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1({\mathcal{C}},{\mathcal{O}_{X}}).$$ We have the chain of isomorphisms: $$\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^2(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right), {\mathcal{O}_{X}})\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^2(\left(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)\right)|_{s=0}, {\mathcal{O}_{X}})\cong$$ $$\cong\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^2(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right),{\mathcal{O}_{X}})=0.$$ The first isomorphism is due to the flatness of $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)$ and Lemma \[lem:isoext\], the second isomorphism is because of Equation (\[eq:espext\]), and the vanishing is due to the fact that ${\mathcal{C}}$ is Cohen-Macaulay of dimension $1$, Theorem \[Th:Herzog\] and the Gorenstein condition. The last vanishing shows, applying Lemma \[lem:extbasechange\] (3), (2), and (1) for ${\mathcal{F}}=\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)$, the isomorphism $$\label{eq:auxi1} \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right), {\mathcal{O}_{{\mathcal{X}}}})|_s\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right), {\mathcal{O}_{X}}).$$ Dualizing the first row of diagram (\[diagram:comparacionMbarra\]) we obtain the exact sequence $$0\to \operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{N}}},{\mathcal{O}_{{\mathcal{X}}}}),{\mathcal{O}_{{\mathcal{X}}}})\to{\mathcal{O}_{{\mathcal{X}}}}^r\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right), {\mathcal{O}_{{\mathcal{X}}}})\to$$ $$\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{N}}},{\mathcal{O}_{{\mathcal{X}}}}),{\mathcal{O}_{{\mathcal{X}}}}).$$ We have the chain of equalities $$\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{N}}},{\mathcal{O}_{{\mathcal{X}}}}),{\mathcal{O}_{{\mathcal{X}}}})=$$ $$=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{M}}},{\mathcal{O}_{{\mathcal{X}}}}),{\mathcal{O}_{{\mathcal{X}}}}),{\mathcal{O}_{{\mathcal{X}}}})=$$ $$=\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{M}}},{\mathcal{O}_{{\mathcal{X}}}})=\Pi_*\overline{{\mathcal{N}}}.$$ The first and third equalities are applications of Proposition \[prop:dualMdeformado\] and the second is because a triple dual coincides with a single dual. We also have $$\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}(\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{N}}},{\mathcal{O}_{{\mathcal{X}}}}),{\mathcal{O}_{{\mathcal{X}}}})= \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^1_{{\mathcal{O}_{{\mathcal{X}}}}}(\Pi_*\overline{{\mathcal{M}}},{\mathcal{O}_{{\mathcal{X}}}})=0.$$ The first equality has been shown in the chain of equalities prior to diagram , and the vanishing follows by an application of Lemmata \[lem:isoext\] and \[lem:extbasechange\], the fact that $\Pi_*\overline{{\mathcal{M}}}$ is a flat deformation of a reflexive ${\mathcal{O}_{X}}$-module and the Gorenstein condition. After these identifications the last exact sequence becomes: $$0\to \Pi_*\overline{{\mathcal{N}}}\to{\mathcal{O}_{{\mathcal{X}}}}^r\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right), {\mathcal{O}_{{\mathcal{X}}}})\to 0,$$ and this gives, by comparison with Exact Sequence (\[eq:Cbarra\]) the isomorphism $$\label{eq:nuevoC} \overline{{\mathcal{C}}}\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right), {\mathcal{O}_{{\mathcal{X}}}}).$$ The following chain of isomorphisms gives the needed identification: $$(\overline{{\mathcal{C}}})|_{s}\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right), {\mathcal{O}_{{\mathcal{X}}}})|_s\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right), {\mathcal{O}_{X}})\cong$$ $$\cong\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1((\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right))|_s, {\mathcal{O}_{X}})\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1(\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right), {\mathcal{O}_{X}})\cong {\mathcal{C}}.$$ The first isomorphism is by (\[eq:nuevoC\]); the second by (\[eq:auxi1\]); the third by flatness of $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)$ and Lemma \[lem:isoext\]; the fourth by (\[eq:espext\]); the fifth is by Theorem \[Th:Herzog\], using that ${\mathcal{C}}$ is Cohen-Macaulay of dimension $1$. We have proven that the pair $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ is a specialty defect constant deformation of the pair $({\tilde{X}},X,{\mathcal{A}},(\psi_1,...,\psi_r))$ over $(S,s)$. In order to finish Part 2 of the proof we let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be a specialty constant deformation of $({\tilde{X}},X,{\mathcal{A}},(\psi_1,...,\psi_r))$ over $(S,s)$, and consider $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$, the deformation assigned by the isomorphism of functors from $\mathbf{Def_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$ to $\mathbf{Def_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$. We have to prove that $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ is an element of $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}(S,s)$. For this we need to show that $R^1\Pi_*\overline{{\mathcal{M}}}$ is flat over $S$. As before denote by $\overline{{\mathcal{C}}}$ the image of $\Pi_*{\mathcal{O}_{{\mathcal{X}}}}\to \Pi_*\overline{{\mathcal{A}}}$ and consider the exact sequences (\[eq:Cbarra\]) and (\[eq:Dbarra\]). Denote by ${\mathcal{D}}$ the image of $\pi_*{\mathcal{O}_{X}}\to \pi_*{\mathcal{A}}$, and consider the exact sequences $$\label{eq:C} 0\to\pi_*{\mathcal{N}}\to {\mathcal{O}_{X}}^r\to {\mathcal{C}}\to 0,$$ $$\label{eq:D} 0\to {\mathcal{C}}\to\pi_*{\mathcal{A}}\to{\mathcal{D}}\to 0.$$ Sequence (\[eq:Dbarra\]), and the flatness of $\Pi_*\overline{{\mathcal{A}}}$ and $\overline{{\mathcal{D}}}$ implies the flatness of $\overline{{\mathcal{C}}}$ over $(S,s)$. Observe that $\Pi_*\overline{{\mathcal{A}}}$ and $\overline{{\mathcal{D}}}$ specialize over $s$ to $\pi_*{\mathcal{A}}$ and ${\mathcal{D}}$. Consequently, specializing Sequence (\[eq:Dbarra\]) over $s$ and comparing with Sequence (\[eq:D\]) we conclude the isomorphism $$\label{eq:Cspbien} \overline{{\mathcal{C}}}|_s\cong{\mathcal{C}}.$$ We have the chain of isomorphisms $$\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^2_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{{\mathcal{C}}},{\mathcal{O}_{X}})\cong\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^2_{{\mathcal{O}_{X}}}(\overline{{\mathcal{C}}}|_s,{\mathcal{O}_{X}})\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}^2_{{\mathcal{O}_{X}}}({\mathcal{C}},{\mathcal{O}_{X}})=0.$$ The first isomorphism is by flatness of $\overline{{\mathcal{C}}}$ and Lemma \[lem:isoext\], the second follows from (\[eq:Cspbien\]), and the vanishing by Theorem \[Th:Herzog\], the fact that ${\mathcal{C}}$ is Cohen-Macaulay of dimension $1$ and the Gorenstein condition. Then, by Lemma \[lem:extbasechange\] we have the isomorphism $$\label{eq:identcruc} \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)|_s\cong \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right).$$ Observe also the vanishing $\operatorname{\mathscr{H}\text{\kern -3pt {\calligra\large om}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}(\overline{{\mathcal{C}}},{\mathcal{O}_{X}})=0.$ Using the last isomorphism and the vanishing, Lemma \[lem:extbasechange\] (2) implies that $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)$ is flat over $S$. Specializing the first row of Diagram (\[diagram:comparacionMbarra\]) over $s$, we obtain the exact sequence $$0\to {\mathcal{O}_{X}}^r\to \Pi_*\overline{{\mathcal{M}}}|_s\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\mathcal{X}}}}}^1 \left(\overline{{\mathcal{C}}}, {\mathcal{O}_{{\mathcal{X}}}} \right)|_s\to 0.$$ Using the identification (\[eq:identcruc\]) and comparing with the sequence $$0\to{\mathcal{O}_{X}}^r\to\pi_*{\mathcal{M}}\to \operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{X}}}^1 \left({\mathcal{C}}, {\mathcal{O}_{X}} \right) \to 0,$$ obtained by dualizing Sequence (\[eq:C\]), we deduce the isomorphism $\Pi_*\overline{{\mathcal{M}}}|_s\cong \pi_*{\mathcal{M}}$. Hence Condition 2 of Lemma \[lem:h0h1\] holds and this concludes the proof. The previous theorem, together with Theorem \[th:corres\] gives the following set of corollaries: Assume that $({\mathcal{A}},(\psi_1,...,\psi_r))$ is a Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module of dimension $1$ with a collection of generators as a ${\mathcal{O}_{{\tilde{X}}}}$-module satisfying the Containment Condition. Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be a specialty defect constant deformation of $({\mathcal{A}},(\psi_1,...,\psi_r))$ over a germ $(S,s)$. Then for any $s'\in S$ the Cohen-Macaulay ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}_{s'}}}$-module with generators $(\overline{{\mathcal{A}}}|_{s'},(\overline{\psi}_1|_{s'},...,\overline{\psi}_r|_{s'}))$ satisfies the Containment Condition. Assume that $({\mathcal{A}},(\psi_1,...,\psi_r))$ is a Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module of dimension $1$ with a collection of generators as a ${\mathcal{O}_{{\tilde{X}}}}$-module satisfying the Containment Condition. Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be a specialty defect constant deformation of $({\mathcal{A}},(\psi_1,...,\psi_r))$ over a germ $(S,s)$. Applying the correspondence defined in Theorem \[th:dirresdef\] we obtain a full deformation of full ${\mathcal{O}_{{\tilde{X}}}}$-modules whose specialty defect is constant. \[cor:specialtydefectconstant\] The specialty defect is constant in a full deformation of a full sheaf. Given a full sheaf ${\mathcal{M}}$, let $(\phi_1,...,\phi_r)$ be generic sections. For any deformation $(\overline{{\mathcal{M}}},\iota)$ of ${\mathcal{M}}$ over $(S,s)$ let $(\overline{\phi}_1,...,\overline{\phi}_r)$ be an extension of the sections over $(S,s)$. Applying the correspondence of Theorem \[th:dirresdef\] to $(\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ we obtain a speciality defect constant deformation. Apply the inverse correspondence to get back $(\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ and use the previous Remark. Finally we need to compare the isomorphism between the functors $\mathbf{Def_{X,M}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{Def_{X,{\mathcal{C}}}^{(\psi_1,...,\psi_r)}}$ with the isomorphism between the functors $\mathbf{FullDef_{{\tilde{X}},X,{\mathcal{M}}}^{(\phi_1,...,\phi_r)}}$ and $\mathbf{SDCDef_{{\tilde{X}},X,{\mathcal{A}}}^{(\psi_1,...,\psi_r)}}$. The results we need are the following: \[prop:defdirressing\] Let $X$ be a normal Stein surface with Gorenstein singularities, let $(M,(\phi_1,...,\phi_r))$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$ together with $r$ generic sections, $\pi:{\tilde{X}}\to X$ be a resolution and ${\mathcal{M}}$ be the associated full ${\mathcal{O}_{{\tilde{X}}}}$-module. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be the result of applying the correspondence of Theorem \[th:corres\] to $({\mathcal{M}},(\phi_1,...,\phi_r))$, and $({\mathcal{C}},(\psi'_1,...,\psi'_r))$ the result of applying the correspondence of Theorem \[th:corrsing\] to $(M,(\phi_1,...,\phi_r))$. Consider a full deformation $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ of $({\tilde{X}},X,{\mathcal{M}},(\phi_1,...,\phi_r))$ over a base $S$. Let $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ be the deformation of $(X,M,(\phi_1,...,\phi_r))$ obtained applying $\Pi_*$ (see Proposition \[prop:naturaltrans\]). Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be the result of applying the correspondence of Theorem \[th:dirresdef\] to $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$, and let $({\mathcal{X}},\overline{{\mathcal{C}}},(\overline{\psi}'_1,...,\overline{\psi}'_r),\rho')$ be the result of applying the correspondence of Theorem \[th:dirXdef\] to $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$. Then there is a natural inclusion $\overline{{\mathcal{C}}}\subset \Pi_*\overline{{\mathcal{A}}}$ which extends the inclusion of ${\mathcal{C}}$ into $\pi_*{\mathcal{A}}$ predicted in Proposition \[prop:dirressing\]. Under this inclusion the sections $(\overline{\psi}_1,...,\overline{\psi}_r)$ are identified with $(\overline{\psi}'_1,...,\overline{\psi}'_r)$. The proof is a straightforward adaptation of the proof of Proposition \[prop:dirressing\] in which one should quote Proposition \[prop:dualMdeformado\] instead of Lemma \[lema:dualM\]. \[prop:comparecorrdef\] Let $\pi:{\tilde{X}}\to X$ be a resolution of a normal Stein surface with Gorenstein singularities. Let $({\mathcal{A}},(\psi_1,...,\psi_r))$ be a pair formed by a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module, whose support meets the exceptional divisor $E$ in finitely many points, and a set of $r$ global sections spanning ${\mathcal{A}}$ as ${\mathcal{O}_{{\tilde{X}}}}$-module and satisfying the Containment Condition. Let ${\mathcal{C}}$ be the ${\mathcal{O}_{X}}$-module spanned by $\psi_1,...,\psi_r$ (then ${\mathcal{C}}$ is a rank 1 generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{X}}$-module). Let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be a specialty constant deformation of $({\tilde{X}},X,{\mathcal{A}},(\psi_1,...,\psi_r))$, with $\Pi:{\tilde{{\mathcal{X}}}}\to{\mathcal{X}}$ the simultaneous resolution. Denote by $\overline{{\mathcal{C}}}$ the ${\mathcal{O}_{{\mathcal{X}}}}$-submodule of $\Pi_*\overline{{\mathcal{A}}}$ spanned by $(\overline{\psi}_1,...,\overline{\psi}_r)$. Then $({\mathcal{X}},\overline{{\mathcal{C}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ is a deformation of $(X,{\mathcal{C}},(\psi_1,...,\psi_r))$. Moreover, let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\rho)$ be the result of applying the correspondence of Theorem \[th:dirresdef\] to $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ and $({\mathcal{X}},\overline{M},(\overline{\phi}'_1,...,\overline{\phi}'_r),\rho')$ be the result of applying the correspondence of Theorem \[th:dirXdef\] to $({\mathcal{X}},\overline{{\mathcal{C}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$. Then we have the equality of deformations $$({\mathcal{X}},\Pi_*\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\rho)=({\mathcal{X}},\overline{M},(\overline{\phi}'_1,...,\overline{\phi}'_r),\rho').$$ In order to prove that $({\mathcal{X}},\overline{{\mathcal{C}}},(\overline{\psi}_1,...,\overline{\psi}_r),\iota)$ is a deformation of $(X,{\mathcal{C}},(\psi_1,...,\psi_r))$ we only need to prove the flatness of $\overline{{\mathcal{C}}}$ over the base $S$ of the deformation. This follows because we have the exact sequence $$0\to\overline{{\mathcal{C}}}\to\Pi_*\overline{{\mathcal{A}}}\to\overline{{\mathcal{D}}}\to 0,$$ the module $\overline{{\mathcal{D}}}$ is flat over $S$ because $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ is a specialty constant deformation and the module $\Pi_*\overline{{\mathcal{A}}}$ is also flat over $S$ because it coincides with $\overline{{\mathcal{A}}}$ as a ${\mathcal{O}_{S}}$-module. The remaining assertion runs parallel to the proof of Proposition \[prop:invressing\]: according with the proof of Theorem \[th:dirresdef\] and its proof the module $\overline{{\mathcal{N}}}$ in the sequence $0\to\overline{{\mathcal{N}}}\to{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to\overline{{\mathcal{A}}}\to 0$ is the dual of $\overline{{\mathcal{M}}}$. Pushing down by $\Pi_*$ we obtain $$0\to\Pi_*\overline{{\mathcal{N}}}\to{\mathcal{O}_{{\mathcal{X}}}}^r\to\Pi_*\overline{{\mathcal{A}}}\to R^1\Pi_*\overline{{\mathcal{N}}}\to R^1\Pi_*{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}^r\to 0,$$ and the image of the map ${\mathcal{O}_{{\mathcal{X}}}}^r\to\Pi_*\overline{{\mathcal{A}}}$ is the ${\mathcal{O}_{{\mathcal{X}}}}$-module spanned by $\overline{\psi}_1,...,\overline{\psi}_r$, that is, the module $\overline{{\mathcal{C}}}$. So we obtain the sequence $$0\to\Pi_*\overline{{\mathcal{N}}}\to{\mathcal{O}_{{\mathcal{X}}}}^r\to\overline{{\mathcal{C}}}\to 0.$$ According with Theorem \[th:dirXdef\] and its proof the module $\Pi_*\overline{{\mathcal{N}}}$ is isomorphic to the dual of $\overline{M}$. By Proposition \[prop:dualMdeformado\] the module $\Pi_*\overline{{\mathcal{N}}}$ is isomorphic to the dual of $\Pi_*\overline{{\mathcal{M}}}$. This concludes the proof of the equality $\Pi_*\overline{{\mathcal{M}}}=\overline{M}$. Classification of Gorenstein normal surface singularities in Cohen-Macaulay representation types {#sec:fintamewild} ================================================================================================ In this section we prove that non log-canonical Gorenstein surface singularities are of wild Cohen-Macaulay representation type. This confirms a conjecture by Drodz, Greuel and Kashuba [@DrGrKa], and completes the classification in Cohen-Macaulay representation types of Gorenstein normal surface singularities. The reader may consult [@DrGrKa] and the references quoted there for a full definition of finite, tame and wild Cohen-Macaulay representation types. For our purposes it is enough to know that if a singularity admits essential families of indecomposable reflexive modules then it is of wild Cohen-Macaulay representation type. \[def:reptype\] Let $(X,x)$ be a normal surface singularity. A family $\overline{M}$ of ${\mathcal{O}_{X}}$-modules over a variety $S$ is called [*essential*]{} for any reflexive module $M$, if the set of points $s\in S$ such that $\overline{M}|_s$ is isomorphic to $M$ is a $0$-dimensional analytic subvariety. \[prop:unboundedfam\] Let $(X,x)$ be a normal Gorenstein surface singularity. If the minimal canonical order of prime divisors over $x$ (see Definition \[def:mincanord\]) is unbounded from below, then there are essential families of indecomposable special reflexive ${\mathcal{O}_{X}}$-modules with arbitrarily high dimensional base. Thus $(X,x)$ is of wild Cohen-Macaulay representation type. Let $\pi:{\tilde{X}}\to X$ be a resolution which contains an irreducible component $F$ of the exceptional divisor $E$ of $\pi$ such that the minimal canonical order at $F$ equals $-d$. We are going to construct a $d$-dimensional essential family of indecomposable special reflexive ${\mathcal{O}_{X}}$-modules. Let $V$ be the set of sequences $(x_1,...,x_d)$ of infinitely near points to $x$ having the following inductive properties: the point $x_1$ is a smooth point of the exceptional divisor $E$ belonging to $F$. For $2\leq i\leq d-1$ let $\pi_{i-1}^{x_1,...,x_{i-1}}:{\tilde{X}}^{i-1}\to X$ be the composition of the blow-ups of ${\tilde{X}}$ at $x_1,...,x_{i-i}$, and the map $\pi$, let $E^{i-1}$ be the exceptional divisor of $\pi_{i-1}^{x_1,...,x_{i-1}}$ and $F^{i-1}$ be the exceptional divisor of the blow up at $x_{i-1}$; the point $x_i$ is a smooth point of the exceptional divisor $E^{i-1}$ belonging to $F^{i-1}$. According with the methods of [@Bo], Chapter 3, there is a variety $V$ parametrizing such sequences of infinitely near points, and a universal sequence of proper birational maps $$Z^{d-1}\stackrel{\Pi^{d-1}}{\longrightarrow} Z^{d-1}\to...\to Z^{1}\stackrel{\Pi^{1}}{\longrightarrow} {\tilde{X}}\times V\stackrel{\Pi^{0}}{\longrightarrow} X\times V,$$ so that $\Pi^0$ is the map $(\pi,Id_V)$ and for any $(x_1,...,x_d)\in V$ and any $k\leq d-1$, the fibre morphism $$\Pi^0_{(x_1,...,x_d)}{{\circ}}...{{\circ}}\Pi^{k}_{(x_1,...,x_d)}:Z^k_{(x_1,...,x_d)}\to X\times\{(x_1,...,x_d)\},$$ is equal to $\pi{{\circ}}\pi^{x_1}....{{\circ}}\pi_{k}^{x_1,...,x_{k}}$. We denote the composition $\Pi^0{{\circ}}...{{\circ}}\Pi^{d-1}$ by $\rho$. Let $\overline{D}\subset Z^{d-1}$ be a divisor such that its fibre over $(x_1,...,x_d)\in V$ is a smooth curvette meeting the exceptional divisor $E^{d-1}$ transversely through $x_d$. It is clear that $\overline{D}$ exists at least over a Zariski open subset of $V$. Let $r(x_1,...,x_d)$ be the minimal number of generators of $(\rho_*{\mathcal{O}_{\overline{D}}})_{(x_1,...,x_d)}$ as a ${\mathcal{O}_{X}}$-module. It is easy to prove that the function $r(x_1,...,x_d)$ is upper semi-continuous in $V$. By shrinking $V$ to a Zariski open subset we may assume the existence of sections $(\overline{\psi}_1,....,\overline{\psi}_r)$ of $\rho_*{\mathcal{O}_{\overline{D}}}$ that specialized over each point $(x_1,...,x_d)\in V$ gives a minimal set of generators of $(\rho_*{\mathcal{O}_{\overline{D}}})_{(x_1,...,x_d)}$. Applying the correspondence of Theorem \[th:dirXdef\] to $((\rho_*{\mathcal{O}_{\overline{D}}})_{(x_1,...,x_d)},(\overline{\psi}_1,....,\overline{\psi}_r))$ we obtain pair $(\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r))$, where $\overline{M}$ is a family of reflexive ${\mathcal{O}_{X}}$-modules. Since $R^1\rho_*{\mathcal{O}_{\overline{D}}}$ vanishes, Lemma \[lem:h0h1\] implies the equality $(\rho_*{\mathcal{O}_{\overline{D}}})_{(x_1,...,x_d)}=(\rho_{(x_1,...,x_d)})_*{\mathcal{O}_{\overline{D}|_{(x_1,...,x_d)}}}$ for any $(x_1,...,x_d)\in V$. Consequently the module $M|_{(x_1,...,x_d)}$ is the result of applying the correspodence of Theorem \[th:corrsing\] to the pair $((\rho_{(x_1,...,x_d)})_*{\mathcal{O}_{\overline{D}|_{(x_1,...,x_d)}}},(\overline{\psi}_1|_{(x_1,...,x_d)},....,\overline{\psi}_r|_{(x_1,...,x_d)}))$. Then, having chosen $\overline{D}$ generic, by Proposition \[prop:Aeslanormalizacion\], Remark \[rem:canonicalcondset\] and Proposition \[prop:decompesp\] we conclude that $M|_{(x_1,...,x_d)}$ is a special indecomposable module. The dimension of $V$ equals $d$, since each of the infinitely near points is free. Therefore, in order to finish the proof we only have to show that the family is essential. We claim that $(\rho_{(x_1,...,x_d)})_*$ is the minimal resolution adapted to $M|_{(x_1,...,x_d)}$. If the claim is true the modules $M|_{(x_1,...,x_d)}$ are pairwise non-isomorphic by Theorem \[Teo:final\] and we are done. The claim follows from Proposition \[prop:minadapnumchar\], noticing the facts that $$cond\left( \mathfrak{K}_{((\rho_{(x_1,...,x_d)})_*{\mathcal{O}_{\overline{D}|_{(x_1,...,x_d)}}},(\overline{\psi}_1|_{(x_1,...,x_d)},....,\overline{\psi}_r|_{(x_1,...,x_d)}))} \right)=0,$$ and that the minimal canonical order of $F^{i}$ vanishes if and only if $i=d$. \[th:reptype\] A Gorenstein surface singularity is of finite Cohen-Macaulay representation type if and only if it is a rational double point. Gorenstein surface singularities of tame Cohen-Macaulay representation type are precisely the log-canonical ones. The remaining Gorenstein surface singularities are of wild Cohen-Macaulay representation type. In [@Es] it is proved that a normal surface singularity has finitely many indecomposable reflexive modules if and only if it is a quotient singularity. This implies that the Gorenstein surface singularities of finite Cohen-Macaulay representation type are exactly the rational double points. In [@DrGrKa] it is proved that log-canonical surface singularities are of tame Cohen-Macaulay representation type. By the classification of Example 3.27 of [@Ko], if $X$ is a Gorenstein singularity which is not log-canonical, and $\sum q_i E_i$ is the divisor associated with its Gorenstein form at the minimal resolution, then either there is a $q_i<-1$, or the exceptional divisor have singularities of Milnor number at least $3$. In this case it is possible to obtain resolutions which are small with respect to the canonical cicle with arbitrarily negative coefficients for the divisor of the Gorenstein form. Proposition \[prop:unboundedfam\] shows now that non log-canonical singularities are of wild Cohen-Macaulay representation type. Lifting deformations {#sec:liftingdefs} ==================== Let $X$ be a normal Stein surface. Let $\pi:{\tilde{X}}\to X$ a resolution with exceptional divisor $E$. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$ and ${\mathcal{M}}$ be the associated full ${\mathcal{O}_{{\tilde{X}}}}$-module. We study when deformations of $M$ lift to full deformations of ${\mathcal{M}}$. \[def:liftlocus\] Let $X$ be a normal Stein surface. Let ${\mathcal{X}}$ be a deformation of $X$ over a base $(S,s)$. Let $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ be a very weak simultaneous resolution with exceptional divisor ${\mathcal{E}}$. The [*strict transform*]{} $\hat{A}$ of a subscheme $A\subset {\mathcal{X}}$ is the scheme theoretic closure of $(\Pi|_{{\tilde{{\mathcal{X}}}}\setminus {\mathcal{E}}})^{-1}(A\setminus\Pi({\mathcal{E}})$ in ${\tilde{{\mathcal{X}}}}$. A subscheme $A\subset {\mathcal{X}}$ [*lifs to*]{} ${\tilde{{\mathcal{X}}}}$ if the strict transform $\hat{A}$ to ${\tilde{{\mathcal{X}}}}$ is finite over $A$. Let ${\mathcal{C}}$ be a rank $1$ generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{X}}$-module, $(\psi_1,...,\psi_r)$ be a system of generators. A deformation $({\mathcal{X}},\overline{{\mathcal{C}}},(\psi_1,...,\psi_r),\iota)$ of $({\mathcal{C}},(\psi_1,...,\psi_r))$ [*lifs in a specialty defect constant way to*]{} ${\tilde{{\mathcal{X}}}}$ if - there is a rank $1$ generically reduced $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-module ${\mathcal{A}}$ meeting the exceptional divisor in finitely many points, and a set of generators $(\psi'_1,...,\psi'_r)$ as a ${\mathcal{O}_{{\tilde{X}}}}$ module such that the ${\mathcal{O}_{X}}$-submodule of $\pi_*{\mathcal{A}}$ generated by the sections $(\psi'_1,...,\psi'_r)$ is isomorphic to ${\mathcal{C}}$. Then, by abuse of notation we denote the sections $(\psi'_1,...,\psi'_r)$ by $(\psi_1,...,\psi_r)$. - There is a specialty defect constant deformation $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{A}}},(\overline{\psi}_1,...,\overline{\psi}_r),\iota')$ of $({\mathcal{A}},(\psi_1,...,\psi_r))$ whose image by $\Pi_*$ equals $({\mathcal{X}},\overline{{\mathcal{C}}},(\psi_1,...,\psi_r),\iota)$. \[rem:liftstrictcommuta\] In the setting of the previous definition, it is clear that $A$ lifts to ${\tilde{{\mathcal{X}}}}$ if and only if the fibre over $s$ of the strict transform of $A$ coincides with the strict transform of the fibre of $A$ over $s$. \[lem:curvecriterionliftability\] Let $X$ be a normal Stein surface. Let ${\mathcal{X}}$ be a deformation of $X$ over a reduced base $(S,s)$. Let $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ be a very weak simultaneous resolution with exceptional divisor ${\mathcal{E}}$. Let $A\subset {\mathcal{X}}$ be a closed subscheme such that the fibre $A_s$ of $A$ over $s$ is of dimension $1$, and such that the Zariski open subset $A\cap ({\mathcal{X}}\setminus\Pi({\mathcal{E}}))$ is flat over $S$. Then $A$ is liftable if and only if for any for any arc $\gamma:Spec(\mathbb{C}[[t]])\to (S,s)$ the subscheme $A\times_S Spec(\mathbb{C}[[t]])\subset {\mathcal{X}}\times_{S} Spec(\mathbb{C}[[t]])$ is liftable for the very weak simultaneous resolution obtained by pullback. If $A$ is liftable, then, by the previous Remark it is obvious that for any arc $\gamma:Spec(\mathbb{C}[[t]])\to (S,s)$ the subscheme $A\times_S Spec(\mathbb{C}[[t]])\subset {\mathcal{X}}\times_{S} Spec(\mathbb{C}[[t]])$ is liftable. It is clear that a subscheme $A$ is liftable if and only if each of the irreducible components of the corresponding reduced subscheme $A^{red}$ are liftable. Hence we may assume $A$ to be reduced and irreducible. Conversely, assume that $A$ is not liftable. Then, by the previous Remark the strict transform of the fibre $A_s$ to ${\tilde{{\mathcal{X}}}}_s$ is strictly contained in the fibre $\hat{A}|_s$ over $s$ of the strict transform of $A$ to ${\tilde{{\mathcal{X}}}}$. Let $a\in\hat{A}|_s$ a point not contained in the strict transform of $A_s$. On the other hand, there is a Zariski open subset $W$ of $S$ such that for any $s'\in W$ the fibre $\hat{A}|_{s'}$ coincides with the strict transform of the fibre over $s'$. Since $A$ is irreducible, so it is its strict transform $\hat{A}$. Then the strict transform $\hat{A}|_W$ is Zariski-dense in $\hat{A}$, and consequently $a$ is at the closure of $\hat{A}|_W$. By Curve Selection Lemma there exists an arc $\hat{\gamma}:Spec(\mathbb{C}[[t]])\to\hat{A}$ such that $\hat{\gamma}(0)=a$ and such that the generic point of $Spec(\mathbb{C}[[t]])$ is mapped to $\hat{A}|_W$. Let $\gamma:Spec(\mathbb{C}[[t]])\to S$ be the composition of $\hat{\gamma}$ with $\Pi$ and the projection to $S$. Then $\gamma$ is an arc so that $A\times_S Spec(\mathbb{C}[[t]])\subset {\mathcal{X}}\times_{S} Spec(\mathbb{C}[[t]])$ is not liftable for the very weak simultaneous resolution obtained by pullback. \[prop:necessarylifting\] Let $X$ be a normal Stein surface with Gorenstein singularities. Let ${\mathcal{X}}$ be a deformation of $X$ over a reduced base $(S,s)$. Let $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ be a very weak simultaneous resolution with exceptional divisor ${\mathcal{E}}$. Denote by $\pi:{\tilde{X}}\to X$ the fibre of $\Pi$ over $s$. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. Let $({\mathcal{X}},\overline{M},\iota)$ be a deformation of $M$ over $(S,s)$. Let ${\mathcal{M}}$ be the full ${\mathcal{O}_{{\tilde{X}}}}$-module associated to $M$. The first 3 of the following conditions are equivalent and imply the fourth and fifth. 1. There is a deformation $({\tilde{{\mathcal{X}}}},{\mathcal{X}},(\overline{{\mathcal{M}}}),\iota')$ of ${\mathcal{M}}$ which transforms under $\Pi_*$ to $({\mathcal{X}},\overline{M},\iota)$. 2. For any collection $(\phi_1,....,\phi_r)$ of nearly generic global sections of ${\mathcal{M}}$ and any extension $(\overline{\phi}_1,...,\overline{\phi}_r)$ as sections of $\overline{M}$, the deformation $({\mathcal{X}},{\mathcal{C}},(\psi_1,...,\psi_r),\rho)$ obtained applying the correspondence of Theorem \[th:dirXdef\] to $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ lifts in a specialty defect constant way to ${\tilde{{\mathcal{X}}}}$. 3. There exists a collection $(\phi_1,....,\phi_r)$ of nearly generic global sections of ${\mathcal{M}}$ and an extension $(\overline{\phi}_1,...,\overline{\phi}_r)$ as sections of $\overline{M}$, such that the deformation $({\mathcal{X}},{\mathcal{C}},(\psi_1,...,\psi_r),\rho)$ obtained applying the correspondence of Theorem \[th:dirXdef\] to $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$ lifts in a specialty defect constant way to ${\tilde{{\mathcal{X}}}}$. 4. For any collection $(\phi_1,....,\phi_r)$ of nearly generic global sections of ${\mathcal{M}}$ and any extension $(\overline{\phi}_1,...,\overline{\phi}_r)$ as sections of $\overline{M}$, the support of the degeneracy module of $(\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r))$ is liftable. 5. There exists a collection $(\phi_1,....,\phi_r)$ of nearly generic global sections of ${\mathcal{M}}$ and an extension $(\overline{\phi}_1,...,\overline{\phi}_r)$ as sections of $\overline{M}$, such that the support of the degeneracy module of $(\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r))$ is liftable. Suppose Condition $(1)$ holds. Consider a collection $(\phi_1,....,\phi_r)$ of nearly generic global sections of ${\mathcal{M}}$ and an extension $(\overline{\phi}_1,...,\overline{\phi}_r)$ as sections of $\overline{M}$. Applying the correspondence of Theorem \[th:dirresdef\] to $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota')$ we obtain the desired lifting. This proves Condition $(2)$. Condition $(2)$ implies Condition $(3)$ trivially. If Condition $(3)$ holds let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},{\mathcal{A}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be the specialty defect constant lifting. Applying the correspondence of Theorem \[th:dirresdef\] to it we obtain a deformation $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\overline{\phi}_1,...,\overline{\phi}_r),\iota')$. The cuadruple $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}}),\iota')$ obtained by forgetting the sections is the deformation that we need to obtain to show that Condition $(1)$ holds. If Condition $(2)$ holds let $({\tilde{{\mathcal{X}}}},{\mathcal{X}},{\mathcal{A}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be the specialty defect constant lifting. The support of ${\mathcal{A}}$ is the strict transform of the support of the degeneracy module of $(\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r))$. Since for any $s'\in S$ the support of ${\mathcal{A}}|_{s'}$ meets the exceptional divisor at finitely many points, Condition $(4)$ holds. Condition $(4)$ implies condition $(5)$ obviously. In the next example we use the non-liftability of the support of the degeneracy module to prove that a deformation of a reflexive module does not lift to a full deformation. \[ex:nonlifting\] Let $X=V(xz-y^2)\subset\mathbb{C}^3$. Let $S=Spec(\mathbb{C}[[s]])$. Define $\alpha:Spec(\mathbb{C}[[t,s]])\to X\times S$ by $(t,s)\to (t^2,t^3+st,(t^2+s)^2)$. Define ${\mathcal{C}}:=\alpha_*\mathbb{C}[[t,s]]$, and let $(\overline{\psi}_1,...,\overline{\psi}_r)$ be a system of generators of ${\mathcal{C}}$ as ${\mathcal{O}_{X\times S}}$-module. The correspondence of Theorem \[th:dirresdef\] defines a deformation $\overline{M}$ of reflexive modules which does not lift to a full deformation of full sheaves, since the support of the degeneracy module is not liftable. Here we show an example where the support of the degeneracy module is liftable, but there is no full deformation. \[ex:liftsnotlifts\] Let $X=V(x^3+y^3+z^3)$. This is a minimally elliptic singularity as studied in [@Ka]. The blowing up at the origin $\pi:{\tilde{X}}\to X$ produces its minimal resolution. Its exceptional divisor is a smooth elliptic curve $E$. Let $\tilde{C}$ be a smooth curvette embedded in ${\tilde{X}}$, which meets $E$ at a single point $p$ with intersection multiplicity equal to $2$. Let $t$ be a uniformizing parameter for the germ $(\tilde{C},p)$. The curve $C:=\pi(\tilde{C})$ has an ordinary cusp singularity at the origin of ${{\mathbb{C}}}^3$ (that is $C=Spec{{\mathbb{C}}}[[t^2,t^3]]$). Let $S:=Spec{{\mathbb{C}}}[[s]]$. We have the isomorphism ${\mathcal{O}_{\tilde{C}\times S}}\cong{{\mathbb{C}}}[[t,s]]$; this endows ${{\mathbb{C}}}[[t,s]]$ with structures of ${\mathcal{O}_{X\times S}}$-module and of ${\mathcal{O}_{{\tilde{X}}\times S}}$-module. Denote by ${\mathcal{C}}$ the ${\mathcal{O}_{X\times S}}$-submodule of ${{\mathbb{C}}}[[t,s]]$ spanned by $(s+t,t^2)$. We let ${\mathcal{D}}$ be its cokernel as ${\mathcal{O}_{X\times S}}$ module. It is easy to check that ${\mathcal{D}}$ is ${{\mathbb{C}}}[[s]]$-flat, and hence ${\mathcal{O}_{C}}$ is ${{\mathbb{C}}}[[s]]$-flat as well. Apply the correspondence of Theorem \[th:dirXdef\] to $({\mathcal{C}},(s+t,t^2))$ and obtain $(\overline{M},(\overline{\phi}_1,\overline{\phi}_2))$, where $\overline{M}$ is a family of reflexive ${\mathcal{O}_{X}}$-modules. The support of ${\mathcal{C}}$ equals $C\times S$, and its strict transform equals $\tilde{C}\times S$. So $C\times S$ lifts. On the other hand, if there is a family $\overline{{\mathcal{M}}}$ of full ${\tilde{X}}$-modules lifting $\overline{M}$, then applying the correspondence of Theorem \[th:dirresdef\] to $(\overline{{\mathcal{M}}},(\overline{\phi}_1,\overline{\phi}_2))$ we would obtain $(\overline{{\mathcal{A}}},(s+t,t^2))$, where $\overline{{\mathcal{A}}}$ is the ${\mathcal{O}_{\tilde{C}\times S}}$ module spanned by $(s+t,t^2)$. Since this module is not Cohen-Macaulay of dimension $2$ (it is not a free module over ${{\mathbb{C}}}[[t,s]]$), it can not be a flat family over ${{\mathbb{C}}}[[s]]$ of $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}}}$-modules. This is a contradiction which implies that the there is no family of full ${\tilde{X}}$-modules lifting $\overline{M}$. Another way of proving the non-existence of lifting is observing that the specialty defect of the full ${\mathcal{O}_{{\tilde{X}}}}$-module lifting $\overline{M}|_0$ is zero, that the specialty defect of the full ${\mathcal{O}_{{\tilde{X}}}}$-module lifting $\overline{M}|_{s}$ is not zero if $s\neq 0$, and using Corollary \[cor:specialtydefectconstant\]. \[prop:genericlifting\] Let $X$ be a normal Stein surface with Gorenstein singularities. Let ${\mathcal{X}}$ be a deformation of $X$ over a reduced base $(S,s)$. Let $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ be a very weak simultaneous resolution. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module and $({\mathcal{X}},\overline{M},\iota)$ be a deformation of $(X,M)$ over a reduced base $(S,s)$, so that for each $s'\in S$ the module $\overline{M}|_{s'}$ is special. There exists a dense Zariski open subset of $S$ where the deformation lifts as a full family of ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-modules. Let $(\overline{\phi}_1,...,\overline{\phi}_r)$ be generic sections of $\overline{M}$. Let $({\mathcal{X}},{\mathcal{C}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be the result of applying the correspondence of Theorem \[th:dirresdef\] to $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$. Let $A$ be the support of ${\mathcal{C}}$. There exists a dense Zariski open subset $U\subset S$ such that the following properties hold: 1. The subscheme $A|_U$ lifts to ${\tilde{{\mathcal{X}}}}|_U$. 2. The subscheme $A|_U$ is a flat family of reduced curves over $U$, which admits a simultaneous normalization $n:\tilde{A}|_U\to A|_U$. 3. The module $\overline{C}|_U$ is isomorphic to $n_*{\mathcal{O}_{\tilde{A}|_U}}$. Let $\hat{A}|_U$ be the strict transform of $A|_U$. Since $A|_U$ lifts to ${\tilde{{\mathcal{X}}}}|_U$ the normalization $\tilde{A}|_U$ dominates $\hat{A}|_U$. Hence ${\mathcal{O}_{\tilde{A}|_U}}$ is a ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}|_U}}$-module, flat over $U$. Applying the correspondence of Theorem \[th:dirresdef\] to $({\tilde{{\mathcal{X}}}}|_U,{\mathcal{X}}|_U,{\mathcal{O}_{\tilde{A}|_U}},\overline{\psi}_1|_U,...,\overline{\psi}_r|_U))$ we obtain the desired full family. \[prop:genericlifting\] Let $X$ be a normal Stein surface with Gorenstein singularities. Let ${\mathcal{X}}$ be a deformation of $X$ over a reduced base $(S,s)$. Let $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ be a very weak simultaneous resolution. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module and $({\mathcal{X}},\overline{M},\iota)$ be a deformation of $(X,M)$ over $(S,s)$. Then there exists a dense Zariski open subset of $S$ where the deformation lifts as a full family of ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-modules. Denote by $$\begin{aligned} \overline{{\mathcal{N}}}&:= \left( \Pi^* \overline{M} \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}},\\ \overline{{\mathcal{M}}}&:= \left( \Pi^* \overline{M} \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}},\end{aligned}$$ and also denote by ${\mathcal{M}}_{s'}:=\left( \Pi^*\overline{M}|_{s'}\right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ the full sheaf associated to $\overline{M}|_{s'}$ and ${\mathcal{N}}_{s'}:=\left( \Pi^*\overline{M}|_{s'}\right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}= \left( {\mathcal{M}}_{'s} \right)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}$ for any $s'\in S$. By genericity of flatness over a reduced base, there exists a Zariski dense open subset $U\subset S$ such that the following two properties hold: 1. The sheaves $\Pi^* \overline{M}$, $\overline{{\mathcal{N}}}$, $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}^1\left(\Pi^* \overline{M}, {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}} \right)$ and $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}}^2\left(\Pi^* \overline{M}, {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}} \right)$ are flat over $U$. 2. The sheaf $R^1 \Pi_* \overline{{\mathcal{M}}}$ is flat over $U$. For any $s'\in S$ we know that ${\tilde{{\mathcal{X}}}}|_{s'}$ is a smooth surface of dimension two, hence the Auslander-Buchsbaum Formula [@BrHe Theorem 1.3.3] implies that $\operatorname{\mathscr{E}\text{\kern -3pt {\calligra\large xt}}\,}_{{\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}|_s}^j\left(\Pi^* \overline{M}|_{s'}, {\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}|_{s'}\right)=0$ for any $j\geq 3$. The last vanishing, Assertion (1) of the previous list and a repeated application of Lemma \[lem:extbasechange\] and Lemma \[lem:isoext\] to the sheaf $\Pi^* \overline{M}$ implies the isomorphism $\overline{{\mathcal{N}}}|_{s'} \cong {\mathcal{N}}_{s'}$ for any $s' \in U$. So we have that $\overline{{\mathcal{N}}}|_U$ is locally free, and dualizing we obtain that $\overline{{\mathcal{M}}}|_U$ is flat over $U$ (locally free on ${\tilde{{\mathcal{X}}}}$) and the isomorphism $\overline{{\mathcal{M}}}|_{s'} \cong {\mathcal{M}}_{s'}$ for any $s' \in U$. Finally Assertion (2) of the previous list implies that $({\tilde{{\mathcal{X}}}}|_U, {\mathcal{X}}|_U, \overline{{\mathcal{M}}}|_U)$ is a family of full modules over $U$. In order to finish the proof we need to verify that the natural morphism of coherent ${\mathcal{O}_{{\mathcal{X}}}}$-modules $$\overline{M}|_U\to (\Pi|_U)_* \overline{{\mathcal{M}}}|_U$$ obtained as the composition $$\overline{M}|_U\to (\Pi|_U)_*(\Pi|_U)^*\overline{M}|_U\to (\Pi|_U)_*((\Pi|_U)^*\overline{M}|_U)^{{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}{\raise0.9ex\hbox{$\scriptscriptstyle\vee$}}}=(\Pi|_U)_* \overline{{\mathcal{M}}}|_U$$ is an isomorphism. By Lemma \[lem:h0h1\] the sheaf $(\Pi|_U)_* \overline{{\mathcal{M}}}|_U$ is flat over $U$. Using this and Nakayama’s Lemma we are reduced to prove that and for any $s'\in U$ we have the specialization $$\overline{M}|_{s'}\to ((\Pi|_U)_* \overline{{\mathcal{M}}}|_U)|_{s'}$$ is an isomorphism. Lemma \[lem:h0h1\] implies that this morphism coincides with $$\overline{M}|_{s'}\to (\Pi|_{s'})_* (\overline{{\mathcal{M}}}|_{s'}),$$ but the second sheaf in the module has been proved to be isomorphic to $(\Pi|_{s'})_* {\mathcal{M}}_{s'}$, where ${\mathcal{M}}_{s'}$ is the full ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}_{s'}}}$-module associated with $\overline{M}|_{s'}$. Then the morphism is an isomorphism as needed. Sufficient conditions for liftability to full deformations {#sec:suffcondlift} ---------------------------------------------------------- In this section we show sufficient conditions ensuring the liftability of a family of reflexive sheaves to a full family on a very weak simultaneous resolution. The result we prove probably is not the best that one can hope for, but is more than sufficient for the applications we have in mind. In particular, at some point we simplify things by working on a normal surface singularity rather than on a normal Stein surface. \[def:deltaconstant\] Let $X$ be a normal Stein surface. Let ${\mathcal{X}}$ be a deformation of $X$ over a reduced base $(S,s)$. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module of rank $r$. A deformation $({\mathcal{X}},\overline{M},\iota)$ of $(X,M)$ over a reduced base $(S,s)$ is said to be [*simultaneously normalizable*]{} if the degeneracy locus $\overline{C}$ of $\overline{M}$ for a generic system of $r$ sections admits a simultaneous normalization over $S$. That is, there exists a smooth family of curves $\overline{D}$ over $S$, and a morphism $n:\overline{D}\to\overline{C}$ such that for any $s'\in S$ the restriction $n|_{\overline{D}_{s'}}\colon \overline{D}_{s'}\to\overline{C}_{s'}^{red}$ is the normalization. \[rem:simnor\] 1. The existence of simultaneous normalization of $\overline{C}$ in the previous definition does not depend on the choice of the system of generic sections. 2. The reader may consult [@GreNor] and [@Ko0] for recent accounts on simultaneous normalization. 3. There is a way to define a functor of simultaneously normalizable deformations of reflexive modules, considering also non-reduced bases $(S,s)$. The definition has some subtlety since the family of supports $\overline{C}$ need not be a flat family of reduced curves. Since the applications presented in this paper do not need such a definition we avoid it. 4. If at the previous definition we assume $X$ to be a normal surface singularity (that is, a germ), we may assume that the normalization of $\overline{C}_s$ is a disjoint union of discs $\coprod_i D_i$. As a consequence if the family is simultaneously normalizable, then the predicted simultaneous normalization $\overline{D}$ is equal to $\coprod_i D_i\times S$. \[th:sufficientlifting\] Let $X$ be a normal Gorenstein surface singularity. Let ${\mathcal{X}}$ be a deformation of $X$ over a normal base $(S,s)$. Let $\Pi:{\tilde{{\mathcal{X}}}}\to {\mathcal{X}}$ be a very weak simultaneous resolution. Let $M$ be a reflexive ${\mathcal{O}_{X}}$-module and $({\mathcal{X}},\overline{M},\iota)$ be a simultaneously normalizable deformation of $(X,M)$ over the base $(S,s)$, so that for each $s'\in S$ the module $\overline{M}|_{s'}$ is special. If the support of the degeneracy module of $\overline{M}$ for a generic system of sections is liftable to ${\tilde{{\mathcal{X}}}}$, then the family $({\mathcal{X}},\overline{M},\iota)$ lifts to a full family on ${\tilde{{\mathcal{X}}}}$. Let $(\overline{\phi}_1,...,\overline{\phi}_r)$ be a system of generic global sections of $\overline{M}$. Let $({\mathcal{X}},\overline{{\mathcal{C}}},(\overline{\psi}_1,...,\overline{\psi}_r),\rho)$ be the result of applying the correspondence of Theorem \[th:dirXdef\] to $({\mathcal{X}},\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r),\iota)$. Let $\overline{C}$ be the support of $\overline{{\mathcal{C}}}$. Let $$n:\overline{D}\to\overline{C}$$ be the simultaneous normalization that exists because $({\mathcal{X}},\overline{M},\iota)$ is simultaneously normalizable. By Remark \[rem:simnor\] we may assume that $\overline{D}$ is the product of $S$ with a disjoint union of discs. Let $K(\overline{C})$ be the total fraction ring of $\overline{C}$. If $\overline{C}=\cup_{i=1}^m\overline{C}_i$ is the decomposition in irreducible components then the total fraction ring equals the direct product $K(\overline{C})=\prod_{i=1}^mK(\overline{C}_i)$ of the function fields of the components. Denote by $T$ the set of non-zero divisors of ${\mathcal{O}_{\overline{C}}}$. The localization $T^{-1}\overline{{\mathcal{C}}}$ is a $K(\overline{C})$-module, which expresses as $T^{-1}\overline{{\mathcal{C}}}=\prod_{i=1}^m\overline{{\mathcal{C}}}_i$, where $\overline{{\mathcal{C}}}_i$ is a $K(\overline{C}_i)$-vector space. Since $\overline{{\mathcal{C}}}$ is a flat family of $1$-dimensional rank 1 generically reduced Cohen-Macaulay modules over the normal base $S$, the natural map to the localization $\overline{{\mathcal{C}}}\to T^{-1}\overline{{\mathcal{C}}}$ is injective, and each $\overline{{\mathcal{C}}}_i$ is a $1$-dimensional vector space. Hence the localization $T^{-1}\overline{{\mathcal{C}}}$ is isomorphic to $\prod_{i=1}^mK(\overline{C}_i)=K(\overline{C})$. We have found a ${\mathcal{O}_{\overline{C}}}$-module monomorphism $\iota:\overline{{\mathcal{C}}}\hookrightarrow K(\overline{C}).$ Noticing that ${\mathcal{O}_{\overline{D}}}$ is a sub-ring of $K(\overline{C})$, it makes sense to define $\overline{{\mathcal{B}}}$ to be the ${\mathcal{O}_{\overline{D}}}$-submodule of $K(\overline{C})$ spanned by $\overline{{\mathcal{C}}}$. Since $\overline{{\mathcal{C}}}$ is generated as ${\mathcal{O}_{\overline{C}}}$ module by $(\overline{\psi}_1,...,\overline{\psi}_r)$, multiplying by the common denominator of $\iota(\overline{\psi}_1),...,\iota(\overline{\psi}_r)$ we may assume that the image of $\iota$ lies in ${\mathcal{O}_{\overline{D}}}$. We have got the chain of inclusions of ${\mathcal{O}_{\overline{C}}}$-modules $$\label{eq:inc1111} \overline{{\mathcal{C}}}\hookrightarrow\overline{{\mathcal{B}}}\hookrightarrow{\mathcal{O}_{\overline{D}}}.$$ The second inclusion is an inclusion of ${\mathcal{O}_{\overline{D}}}$-modules; thus $\overline{{\mathcal{B}}}$ is an ideal in ${\mathcal{O}_{\overline{D}}}$. Suppose that the zero set $V(\overline{{\mathcal{B}}})$ contains an irreducible component $Z$ that is dominant over $S$ by the natural projection map. Such a $Z$ is contained in a unique connected component of $\overline{D}$, and each of these connected components is isomorphic to the product of $S$ times a disc. Let $t$ be a coordinate of the disc. By the normality of $(S,s_0)$ there is a Weierstrass polynomial $P$ in ${\mathcal{O}_{S}}[[t]]$ whose zero locus defines $Z$. Then, dividing by the appropriate Weierstrass polynomials we may assume that the embedding $\iota$ is so that the zero set of the ideal $\overline{{\mathcal{B}}}$ does not dominate $S$. Hence there exists a Zariski dense open subset $U$ in $S$ where, under the embedding $\iota$ we have the equality $$\label{eq:inc1122} \overline{{\mathcal{B}}}|_U={\mathcal{O}_{\overline{D}|_U}}.$$ For any $s'\in S$, by specialty of $\overline{M}|_{s'}$ and Proposition \[prop:Aeslanormalizacion\] we have the isomorphism $\overline{{\mathcal{C}}}|_{s'}\cong{\mathcal{O}_{\overline{D}_{s'}}}$. This implies that $\overline{{\mathcal{B}}}|_{s'}$ is a monic ideal in ${\mathcal{O}_{\overline{D}_{s'}}}$ and the equality $\overline{{\mathcal{C}}}|_{s'}=\overline{{\mathcal{B}}}|_{s'}$. Then Equation (\[eq:inc1122\]) implies the equality $$\label{eq:2222} \overline{{\mathcal{C}}}|_{U}={\mathcal{O}_{\overline{D}|_{U}}}.$$ Define $\overline{{\mathcal{D}}}:={\mathcal{O}_{\overline{D}}}/\overline{{\mathcal{C}}}$. Denote by $\mathfrak{m}_S$ the maximal ideal of ${\mathcal{O}_{S,s}}$. Applying $\centerdot\otimes_{{\mathcal{O}_{S}}}{\mathcal{O}_{S}}/\mathfrak{m}_S$ to the exact sequence of ${\mathcal{O}_{\overline{C}}}$-modules $$0\to \overline{{\mathcal{C}}}\to{\mathcal{O}_{\overline{D}}} \to \overline{{\mathcal{D}}}\to 0,$$ we obtain the exact sequence $$0\to Tor_1^{{\mathcal{O}_{S}}}(\overline{{\mathcal{D}}},{\mathcal{O}_{S}}/\mathfrak{m}_S)\to \overline{{\mathcal{C}}}\otimes_{{\mathcal{O}_{S}}}{\mathcal{O}_{S}}/\mathfrak{m}_S\to{\mathcal{O}_{\overline{D}}}\otimes_{{\mathcal{O}_{S}}}{\mathcal{O}_{S}}/\mathfrak{m}_S\to \overline{{\mathcal{D}}}\otimes_{{\mathcal{O}_{S}}}{\mathcal{O}_{S}}/\mathfrak{m}_S\to 0.$$ The second group in the sequence is isomorphic to ${\mathcal{C}}|_{s_0}$, the third group is ${\mathcal{O}_{\overline{D}|_{s_0}}}$ and the morphism connecting them is injective at the generic points of the support of ${\mathcal{C}}|_{s_0}$. Then, since ${\mathcal{C}}|_{s_0}$ is a rank $1$ generically reduced Cohen-Macaulay ${\mathcal{O}_{X}}$-module of dimension $1$ the morphism $$\overline{{\mathcal{C}}}\otimes_{{\mathcal{O}_{S}}}{\mathcal{O}_{S}}/\mathfrak{m}_S\to{\mathcal{O}_{\overline{D}}}\otimes_{{\mathcal{O}_{S}}}{\mathcal{O}_{S}}/\mathfrak{m}_S$$ is injective and then $Tor_1^{{\mathcal{O}_{S}}}(\overline{{\mathcal{D}}},{\mathcal{O}_{S}}/\mathfrak{m}_S)$ vanishes. Then $\overline{{\mathcal{D}}}$ is flat over $S$ by the Local Criterion of Flatness. Since by Equality (\[eq:2222\]) the ${\mathcal{O}_{S}}$-module $\overline{{\mathcal{D}}}$ has proper support we conclude that $\overline{{\mathcal{D}}}$ vanishes. This proves that the inclusion (\[eq:inc1111\]) becomes the equality $$\label{eq:eq3333} \overline{{\mathcal{C}}}={\mathcal{O}_{\overline{D}}}.$$ Since $\overline{C}$ has been assumed to liftable to ${\tilde{{\mathcal{X}}}}$ the restriction of $\Pi$ to the strict transform $\hat{\overline{C}}$ of $\overline{C}$ to ${\tilde{{\mathcal{X}}}}$ is finite and birational over $\overline{C}$. This implies that $\overline{D}$ dominates $\hat{\overline{C}}$, and then we have a ring monomorphism ${\mathcal{O}_{\hat{\overline{C}}}}\hookrightarrow{\mathcal{O}_{\overline{D}}}$. As a consequence ${\mathcal{O}_{\overline{D}}}$ inherits a structure of ${\mathcal{O}_{{\tilde{{\mathcal{X}}}}}}$-module. Summing up we have shown that $({\tilde{{\mathcal{X}}}},{\mathcal{X}},{\mathcal{O}_{\overline{D}}},(\psi_1,...,\psi_r),Id|_{{\mathcal{O}_{\overline{D}_s}}})$ is a specialty defect constant deformation of $({\tilde{X}},X,{\mathcal{O}_{\overline{D}_s}},(\psi_1|_s,...,\psi_r|_s))$. Applying the correspondence of Theorem \[th:dirresdef\] to $({\tilde{{\mathcal{X}}}},{\mathcal{X}},{\mathcal{O}_{\overline{D}}},(\psi_1,...,\psi_r),Id|_{{\mathcal{O}_{\overline{D}_s}}})$ we obtain a full deformation $({\tilde{{\mathcal{X}}}},{\mathcal{X}},\overline{{\mathcal{M}}},(\phi_1,...,\phi_r),\iota)$. An application of Proposition \[prop:comparecorrdef\] concludes the proof. Moduli spaces of special reflexive sheaves on Gorenstein surface singularities {#sec:moduli} ============================================================================== Let $(X,x)$ be a Gorenstein normal surface singularity. In this section we consider deformations and families of ${\mathcal{O}_{X}}$-modules fixing the space $X$. Our aim is to construct moduli spaces of special reflexive modules with prescribed combinatorial type. \[def:modulifunctor\] Let $(X.x)$ be a Gorenstein surface singularity. Let ${\mathcal{G}}$ be the graph of a special reflexive ${\mathcal{O}_{X}}$-module and $r$ a positive integer. A [*family of special modules with graph*]{} ${\mathcal{G}}$ [*and rank*]{} $r$ over a complex space $S$ is a ${\mathcal{O}_{X\times S}}$-module $\overline{M}$ which is flat over $S$ and such that for any $s\in S$, the module $\overline{M}|_s$ is a special reflexive module of rank $r$ and graph ${\mathcal{G}}$. Define a moduli functor $\mathbf{Mod^r_{{\mathcal{G}}}}$ from the category of Normal Complex Spaces to the category of Sets, assigning to a normal complex space $S$ the set of families of special modules without free factors with graph ${\mathcal{G}}$ and rank $r$, and to a morphisms of complex spaces the corresponding pullback of families. In this section we prove that the previous functors are representable by a complex algebraic variety. Our moduli functor is somewhat restricted: we only consider families over normal complex spaces. It is an open problem to show that our moduli spaces represent the usual moduli functors. One should notice that even the definition of the moduli functor has some subtleties: restricting Example \[ex:liftsnotlifts\] to the base $Spec({{\mathbb{C}}}[[s]]/(s^2))$ one sees a flat deformation of a special reflexive ${\mathcal{O}_{X}}$-module, such that over each point of the base the corresponding reflexive module is special, but that should not be considered as a family of special modules. Let ${\mathcal{G}}$ be the graph of a special reflexive module (the possible graphs are classified in Theorem \[th:charresgraphsp\]). Deleting the arrows of ${\mathcal{G}}$ we obtain a resolution graph ${\mathcal{G}}^{o}$ of the singularity $X$. Two different resolutions with the same resolution graph only differ in the positions of the infinitely near points which are the centers of the blow ups occuring after the minimal resolution. Let $\mathfrak{M}'_{{\mathcal{G}}^{o}}$ be the set of pairs $(\pi,\varphi)$, where $\pi:{\tilde{X}}\to X$ is a resolution of singularities and $\varphi$ is a bijection from the vertices of ${\mathcal{G}}^{o}$ to the irreducible components of the exceptional divisor $E$ of $\pi$ inducing an isomorphism from ${\mathcal{G}}^{o}$ to the dual graph of the resolution. The group $Aut({\mathcal{G}}^{o})$ of automorphims of the graph ${\mathcal{G}}^{o}$ acts on $\mathfrak{M}'_{{\mathcal{G}}^{o}}$ by composition on the left at the second coordinate. The group $Aut({\mathcal{G}})$ is a subgroup of $Aut({\mathcal{G}}^{o})$. Define $\mathfrak{M}_{{\mathcal{G}}^{o}}$ and $\mathfrak{M}_{{\mathcal{G}}}$ to be the quotient of $\mathfrak{M}'_{\mathfrak{G}^{o}}$ by $Aut({\mathcal{G}}^{o})$ and $Aut({\mathcal{G}})$ respectively. The points of $\mathfrak{M}_{{\mathcal{G}}^{o}}$ and $\mathfrak{M}_{{\mathcal{G}}}$ are equivalence classes which will be denoted by $(\pi,\varphi)$ for simplicity. \[lem:varietyresolutions\] The sets $\mathfrak{M}_{{\mathcal{G}}}$ and $\mathfrak{M}_{{\mathcal{G}}^o}$ have a natural structure of algebraic variety, and the natural map from the first to the second is an etale covering. The variety $\mathfrak{M}_{{\mathcal{G}}}$ is a moduli space of resolutions of $X$, and has a universal family in the following sense: there is a birational morphism $$\Pi:\tilde{{\mathcal{X}}}\to X\times \mathfrak{M}_{{\mathcal{G}}},$$ such that for any $(\pi,\varphi)\in \mathfrak{M}_{{\mathcal{G}}}$ the pullback morphism $$\Pi|_{[\tilde{{\mathcal{X}}}_{(\pi,\varphi)}}:\tilde{{\mathcal{X}}}_{(\pi,\varphi)}\to X\times\{(\pi,\varphi)\}$$ coincides with $\pi$, where $\tilde{{\mathcal{X}}}_{(\pi,\varphi)}$ denotes the fibre of $\tilde{{\mathcal{X}}}$ over $(\pi,\varphi)$ by the composition of $\Pi$ with the projection to the second factor. See [@Bo], Chapter 3. \[not:restrU\] Let $U$ be an open subset of $\mathfrak{M}_{{\mathcal{G}}}$, the restriction of the universal family over $U$ is denoted by $$\Pi|_U:\tilde{{\mathcal{X}}}|_U\to X\times U.$$ Now we associate a special reflexive module to each point of $\mathfrak{M}_{{\mathcal{G}}}$. Let $(\pi,\varphi)\in\mathfrak{M}_{\mathfrak{G}}$. A curve $D\subset\tilde{{\mathcal{X}}}|_{(\pi,\varphi)}$ is $(\pi,\varphi)$-[*appropriate*]{} if 1. it is a disjoint union of smooth curvettes which meet the exceptional divisor of $\pi$ transversely, and for any vertex of $\mathfrak{M}_{{\mathcal{G}}}$ the number of curvettes meeting the divisor that $\varphi$ assigns to this vertex is exactly the number of arrows attached to this vertex, 2. the minimal number of generators of $\pi_*{\mathcal{O}_{D}}$ as a ${\mathcal{O}_{X}}$ module is minimal among the curves with the previous property. In Remark \[rem:Dgenericanofactorlibre\] and its proof it is shown that a generic curve having the first property also satisfies the second. Let $D$ be a $(\pi,\varphi)$-appropriate curve. Let $(\psi_1,...,\psi_r)$ be a minimal set of generators of the ${\mathcal{O}_{X}}$-module $\pi_*{\mathcal{O}_{D}}$. Applying the correspondence of Theorem \[th:corrsing\] to $(\pi_*{\mathcal{O}_{D}},(\psi_1,...,\psi_r))$ we obtain $(M_{(\pi,\varphi)},(\phi_1,...,\phi_r))$, where $M_{(\pi,\varphi)}$ is special reflexive ${\mathcal{O}_{X}}$-module of rank $r$ (specialty is by Proposition \[prop:Aeslanormalizacion\]). In Sections \[sec:1stchernspecial\] and \[sec:clasdef\] it is proved that $M_{(\pi,\phi)}$ does not depend on the $(\pi,\varphi)$-appropriate curve. However, there is no reason for which the rank $r$ should be independent on the combinatorial type. In fact the rank of $M_{(\pi,\varphi)}$ induces a stratification in $\mathfrak{M}_{{\mathcal{G}}}$ because of the following lemma. \[lem:rankuppersem\] The function $rank(M_{(\pi,\varphi)})$ is upper-semicontinuous in $\mathfrak{M}_{{\mathcal{G}}}$ for the Zariski topology. The assertion is local. Consider $(\pi,\varphi)\in\mathfrak{M}_{{\mathcal{G}}}$. Given a small neighborhood $U$ of $(\pi,\varphi)$ it is easy to construct a subscheme $\overline{D}\subset \tilde{{\mathcal{X}}}|_U$ such that for any $(\pi',\varphi')\in U$ we have that the fibre $\overline{D}|_{(\pi',\varphi')}$ is a $(\pi',\varphi')$-appropriate curve and the ${\mathcal{O}_{\tilde{{\mathcal{X}}}|_U}}$-module ${\mathcal{O}_{\overline{D}}}$ is flat over $U$ (this is easy by the genericity statement given in Remark \[rem:Dgenericanofactorlibre\]). Since $R^1\Pi_*{\mathcal{O}_{\overline{D}}}$ vanishes, Lemma \[lem:h0h1\] implies that $\Pi_*{\mathcal{O}_{\overline{D}}}$ is flat over $U$ and that $(\Pi_*{\mathcal{O}_{\overline{D}}})|_{(\pi',\varphi')}$ equals $(\pi')_*({\mathcal{O}_{\overline{D}}}|_{(\pi',\varphi')})$. Then, the minimal number of generators of $(\Pi_*{\mathcal{O}_{\overline{D}}})|_{(\pi',\varphi')}$ as ${\mathcal{O}_{X}}$-module is upper-semicontinuous in the Zariski topology, and coincides with the rank of $M_{(\pi',\varphi')}$. Denote by $\mathfrak{M}^r_{{\mathcal{G}}}$ to be the locally closed subset corresponding to modules of rank $r$. By Theorem \[Teo:final\], the closed points of $\mathfrak{M}^r_{{\mathcal{G}}}$ are in a bijection with the set of reflexive modules without free factors of graph ${\mathcal{G}}$ and rank $r$. Our next step is to construct a universal family over each $\mathfrak{M}^r_{{\mathcal{G}}}$. \[lem:uniqueunivfamily\] Let $U$ be an open subset of $\mathfrak{M}^r_{{\mathcal{G}}}$. For $i=1,2$ let $\overline{D}^i\subset \tilde{{\mathcal{X}}}|_U$ such that for any $(\pi',\varphi')\in U$ we have that the fibre $\overline{D}^i|_{(\pi',\varphi')}$ is a $(\pi',\varphi')$-appropriate curve and the ${\mathcal{O}_{\tilde{{\mathcal{X}}}|_U}}$-module ${\mathcal{O}_{\overline{D}^i}}$ is flat over $U$. Let $(\overline{\psi}^i_1,...,\overline{\psi}^i_r)$ be a system of generators of ${\mathcal{O}_{\overline{D}^i}}$ as a ${\mathcal{O}_{X\times S}}$-module. Let $(\overline{M}^i,(\overline{\phi}^i_1,...,\overline{\phi}^i_r))$ be the result of applying the correspondence of Theorem \[th:dirXdef\] to $(\Pi_*{\mathcal{O}_{\overline{D}^i}},(\overline{\psi}^i_1,...,\overline{\psi}^i_r))$. Then we have the equality $\overline{M}^1=\overline{M}^2$. Let $(\overline{{\mathcal{M}}}^i,(\overline{\phi}^i_1,...,\overline{\phi}^i_r))$ be the result of applying the correspondence of Theorem \[th:dirresdef\] to $({\mathcal{O}_{\overline{D}^i}},(\overline{\psi}^i_1,...,\overline{\psi}^i_r))$. By Proposition \[prop:comparecorrdef\] we have the equality $\Pi_*\overline{{\mathcal{M}}}^i=\overline{M}^i$. Repeating in family the arguments of Section \[sec:1stchernspecial\] we obtain that $\overline{M}^i$ is determined by the line bundle ${\mathcal{O}_{\tilde{{\mathcal{X}}}|_U}}(-\overline{D}^i)$. An argument like in Lemma \[lem:varioD\] show the isomorphism ${\mathcal{O}_{\tilde{{\mathcal{X}}}|_U}}(-\overline{D}^1)\cong {\mathcal{O}_{\tilde{{\mathcal{X}}}|_U}}(-\overline{D}^2)$. \[lem:univfamily\] There exists a unique family of special modules $\overline{M}^r_{{\mathcal{G}}}$ with graph ${\mathcal{G}}$ and rank $r$ over $\mathfrak{M}^r_{{\mathcal{G}}}$ such that for any $(\pi,\varphi)\in\mathfrak{M}^r_{{\mathcal{G}}}$ we have the isomorphism $\overline{M}^r_{{\mathcal{G}}}|_{(\pi,\varphi)}=M_{(\pi,\varphi)}$. The previous Lemma shows how to construct the family locally, and it shows, using Theorem \[th:dirresdef\] for suitable generic sections, that the result is unique up to the choices made. So the procedure glues well to a global universal family. \[theo:moduli\] The variety $\mathfrak{M}^{r}_{{\mathcal{G}}}$ represents the functor $\mathbf{Mod^r_{{\mathcal{G}}}}$. Let $\overline{M}$ be a family of special modules without free factor with graph ${\mathcal{G}}$ and rank $r$ over a normal base $S$. Consider the mapping $$\theta:S\to \mathfrak{M}^r_{{\mathcal{G}}}$$ sending $s$ to the unique point of $(\pi,\varphi)\in\mathfrak{M}^r_{{\mathcal{G}}}$ such that we have the isomorphism $\overline{M}|_s\cong \overline{M}^r_{{\mathcal{G}}}|_{(\pi,\varphi)}$. We have to prove that the map is a complex analytic morphism, and that the pullback of the the universal family over $\mathfrak{M}^{r}_{{\mathcal{G}}}$ gives back the family $\overline{M}$. The infinitely near points that one need to blow up to get $\pi$ from the minimal resolution in $X$ are partially ordered as follows: the first generation points is the set of points appearing in the minimal resolution, the second generation set of points are those appearing in the resolution obtained by blow up the first generation points, and succesively. Let $k$ be the maximal generation order of the infinitely near points needed to obtain $\pi$. Let ${\mathcal{G}}^{o}_i$ be the dual graph of the result of blowing up the $i$-th generation points. We have a natural sequence of morphisms $$\label{eq:sequencegeneration} \mathfrak{M}^r_{{\mathcal{G}}}\hookrightarrow\mathfrak{M}_{{\mathcal{G}}}\to \mathfrak{M}_{{\mathcal{G}}^{o}}=\mathfrak{M}_{{\mathcal{G}}^{o}_k}\to \mathfrak{M}_{{\mathcal{G}}^{o}_{k-1}}\to ...\to \mathfrak{M}_{{\mathcal{G}}^{o}_{1}},$$ where the first morphism is a locally closed inclusion and the second morphism is an etale covering. We will prove by induction that the composition $$\theta_i:S\to \mathfrak{M}_{{\mathcal{G}}^{o}_i}$$ is a complex analytic morphism for any $i$. The initial step of the induction runs as follows. Let $\pi:{\tilde{X}}_{min}\to X$ be the minimal resolution. Let and $\Pi:{\tilde{X}}_{min}\times S\to X\times S$ be the product of the map $\pi$ with the identity at $S$. Denote by $E_{\Pi}$ the exceptional divisor of $\Pi$. For any $s\in S$ denote by ${\mathcal{M}}_s$ the full ${\mathcal{O}_{{\tilde{X}}_{min}}}$-module associated with $\overline{M}|_s$. By Proposition \[prop:genericlifting\] there is a Zariski open subset $V$ in $S$ and a flat ${\mathcal{O}_{{\tilde{X}}_{min}\times S}}$-module $\overline{{\mathcal{M}}}_V$ such that $R^1\Pi_*\overline{{\mathcal{M}}}_V$ is flat, for any $s\in V$ the restriction $\overline{{\mathcal{M}}}_V|_s$ is isomorphic to ${\mathcal{M}}_s$, and we have $\Pi_*\overline{{\mathcal{M}}}_V=\overline{M}|_V$. Fix $s_0\in S$. Choose a neighborhood $U$ of $s_0$ in $S$ and sections $(\overline{\phi}_1,...,\overline{\phi}_r)$ of $\overline{M}|_U$ so that the sections $(\overline{\phi}_1|_{s},...,\overline{\phi}_r|_{s})$ are nearly generic for - the ${\mathcal{O}_{{\tilde{X}}_{min}}}$-module ${\mathcal{M}}_{s}$ for any $s$ in $U\cap V$ or $s=s_0$, - the ${\mathcal{O}_{X}}$-module $\overline{M}_{s}$ for any $s\in U$. Choosing $(\overline{\phi}_1,...,\overline{\phi}_r)$ generic and perhaps shrinking $U$ suffices to guarantee the properties above. Since the assertion we want to prove is local in $S$ we may assume $U=S$ and $V\subset U$ to save some notation. Applying the correspondence of Theorem \[th:dirXdef\] to $(\overline{M},(\overline{\phi}_1,...,\overline{\phi}_r))$ we obtain $(\overline{{\mathcal{C}}},(\overline{\psi}_1,...,\overline{\psi}_r))$. Since Theorem \[th:dirXdef\] establishes an isomorphism of deformation functors, applying the correspondence of Theorem \[th:corrsing\] to $(\overline{M}|_{s},(\overline{\phi}_1|_{s},...,\overline{\phi}_r|_{s}))$, we obtain the pair $(\overline{{\mathcal{C}}}|_{s},(\overline{\psi}_1|_s,...,\overline{\psi}_r)|_s))$. The correspondence of Theorem \[th:corres\] applied to $({\mathcal{M}}_{s},(\overline{\phi}_1|_{s},...,\overline{\phi}_r|_{s}))$ gives a pair $({\mathcal{A}}_{s},(\psi_{1,s},...,\psi_{r,s}))$, where ${\mathcal{A}}_{s}$ is $1$-dimensional Cohen-Macaulay ${\mathcal{O}_{{\tilde{X}}_{min}}}$-module generated by the sections $(\psi_{1,s},...,\psi_{r,s})$. By Proposition \[prop:dirressing\] and specialty we have that $\pi_*{\mathcal{A}}_{s}$ equals $\overline{{\mathcal{C}}}|_{s}$, and we have the identification of sections $(\psi_{1,s},...,\psi_{r,s})=(\overline{\psi}_{1}|_s,...,\overline{\psi}_{r}|_s)$. The correspondence of Theorem \[th:dirresdef\] applied to $(\overline{{\mathcal{M}}}_V,(\overline{\phi}_1,...,\overline{\phi}_r))$ gives a pair $(\overline{{\mathcal{A}}}_V,(\overline{\psi}_{1,V},...,\overline{\psi}_{r,V}))$. By Proposition \[prop:defdirressing\] and specialty we have that $\Pi_*(\overline{{\mathcal{A}}}_V)$ equals $\overline{{\mathcal{C}}}|_V$, and the sections $(\overline{\psi}_{1,V},...,\overline{\psi}_{r,V})$ are identified with the restriction $(\overline{\psi}_1|_V,...,\overline{\psi}_r|_V)$ of the sections $(\overline{\psi}_1,...,\overline{\psi}_r)$ over $V$. Since Theorem \[th:dirresdef\] establishes an isomorphism of deformation functors, for any $s\in V$ we have the equality of pairs $$({\mathcal{A}}_{s},(\psi_{1,s},...,\psi_{r,s}))=(\overline{{\mathcal{A}}}_V|_s,(\overline{\psi}_1|_s,...,\overline{\psi}_r)|_s).$$ For any $s\in V$, let $A_s$ be the support of ${\mathcal{A}}_{s}$. By the construction of the minimal adapted resolution (see the proof of Proposition \[prop:minadap\]), the intersection of $A_s\cap E_{\Pi}$ is a finite set containing the set of infinitely near points of first generation of $\Pi|_s$. If $A_V$ is the support of $\overline{{\mathcal{A}}}_V$ we have that the fibre $A_V|_s$ of $A_V$ over $s$ equals $A_s$. This implies that the graph of the restriction $$\theta_1|_V:V\to\mathfrak{M}_{{\mathcal{G}}^{o}_1}$$ is a complex analytic subvariety of $V\times \mathfrak{G}^{o}_1$. Using Zariski’s Main Theorem this implies that $\theta_1|_V$ is a complex analytic morphism since $V$ is normal. <span style="font-variant:small-caps;">Claim I</span>. Let $\overline{C}\subset X\times S$ be the support of $\overline{{\mathcal{C}}}$. Then $\overline{C}$ lifts to ${\tilde{X}}_{min}\times S$. Assume that the claim is true. Denote by $\tilde{C}$ the strict transform $\tilde{C}$ of $C$. Then, for any $s\in S$ the intersection of $\tilde{C}|_s\cap E_{\Pi}$ is a finite set and moreover the fibre $\tilde{C}|_s$ is the support of ${\mathcal{A}}_s$ for $s\in S$. Consequently $\tilde{C}|_s\cap E_{\Pi}$ contains the set of infinitely near points of first generation of $\Pi|_s$. This implies that the graph of $\theta_1$ is a complex analytic subvariety of $U\times \mathfrak{G}^{o}_1$, and using Zariski’s Main Theorem this implies that $\theta_1|_U$ is a complex analytic morphism since $U$ is normal. Let us prove the claim. By Lemma \[lem:curvecriterionliftability\] we may assume that $(S,s_0)$ is a germ of smooth curve. The fibre of $\hat{C}_s$ over a generic point of $S$ equals $A_s$. The fibre $\hat{C}|_{s_0}$ equals $A_{s_0}+F$ where $F$ is a divisor in ${\tilde{X}}_{min}$ with support in $E$, which is non-negative. Now, since the modules $\overline{M}|_s$ have the same combinatorial type for any $s\in S$, we have the equality of intersection numbers $E_i\centerdot A_s=E_i\centerdot A_{s_0}$ for any component $E_i$ of $E$. This implies the vanishing $E_i\centerdot F=0$, and hence $F=0$ by the non-degeneracy of the intersection form. This shows that $F$ vanishes, and shows the liftability. This sets up the initial step of the induction. The inductive step is completely similar. So, we have shown that $\theta_k$ is an analytic morphism. Now, since the first morphism of the sequence (\[eq:sequencegeneration\]) is a locally closed inclusion, and the second is an etale map, we conclude that $\theta$ is a complex analytic morphism. Observe that our induction procedure shows that the support $\overline{C}$ of $\overline{{\mathcal{C}}}$ lifts to the universal family of minimal adapted resolutions ${\tilde{{\mathcal{X}}}}$. Then an application of Theorem \[th:sufficientlifting\] shows that $\overline{M}$ lifts to a full family $\overline{{\mathcal{M}}}$ on ${\tilde{{\mathcal{X}}}}$. In order to prove that $\overline{M}$ coincides with the pullback by $\theta$ of the universal family it is enough to show it locally, but this is a consequence of Lemma \[lem:uniqueunivfamily\] and the procedure in which $\overline{{\mathcal{M}}}$ is constructed in the proof of Theorem \[th:sufficientlifting\]. [99]{} S. Abhyankar. 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[^1]: The first author is partially supported by IAS and by ERCEA 615655 NMST Consolidator Grant, MINECO by the project reference MTM2016-76868-C2-1-P (UCM), by the Basque Government through the BERC 2018-2021 program and Gobierno Vasco Grant IT1094-16, by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718 and by Bolsa Pesquisador Visitante Especial (PVE) - Ciencias sem Fronteiras/CNPq Project number: 401947/2013-0. The second author is partially supported by IAS and by ERCEA 615655 NMST Consolidator Grant, Bolsa Pesquisador Visitante Especial (PVE) - Ciencias sem Fronteiras/CNPq Project number: 401947/2013-0, CONACYT 253506 and 286447.

--- abstract: 'Explicit construction of local observable algebras in quasi-Hermitian quantum theories is derived in both the tensor product model of locality and in models of free fermions. The latter construction is applied to several cases of a $\mathcal{PT}$-symmetric toy model of particle-conserving free fermions on a 1-dimensional lattice, with nearest neighbour interactions and open boundary conditions. Despite the locality of the Hamiltonian, local observables do not exist in generic collections of sites in the lattice. The collections of sites which do contain nontrivial observables depends strongly depends on the complex potential.' author: - Jacob Barnett bibliography: - 'mybib4locality.bib' title: '**Nonlocality of Observables in Quasi-Hermitian Quantum Theory**\' --- Introduction ============ While sufficient, Hermiticity in a fixed Hilbert space is not necessary to ensure observables of a quantum theory have real eigenvalues, nor is it necessary for unitarity. In finite Hilbert spaces, the necessary and sufficient condition for real spectra [@QuasiHerm92; @bender2010pt; @williams1969operators; @mostafazadeh2002pseudo2; @mostafazadeh2002pseudo3] as well as unitarity [@QuasiHerm92; @mannheim2013pt; @bender2002complex] is *quasi-Hermiticity*, quantum theories with observables satisfying $$\eta O = O^\dag \eta \label{operator},$$ for some hermitian, positive definite operator $\eta$, referred to as the *metric*. The postulate of time evolution remains unchanged, dictated by Schrödinger evolution with a quasi-Hermitian Hamiltonian. Unitarity and expectation values are defined with respect to the physical inner product, $$\begin{aligned} \braket{\psi|\phi}_\eta &= \braket{\psi|\eta|\phi}, \label{PhysInnProd} \\ \braket{O}_\eta &= \frac{\braket{\psi|\eta O|\psi}}{\braket{\psi|\eta|\psi}}.\end{aligned}$$ Quasi-Hermitian quantum theory is often claimed to be a genuine extension of quantum theory. However, the theory mentioned above is simply quantum theory where the physical Hilbert space inner product and adjoint are defined through $\braket{\cdot|\cdot}_\eta$. In addition, motivated by the well known theorem that any two separable Hilbert spaces of the same dimension are isomorphic, there is an equivalent theory expressed in terms of the inner product $\braket{\cdot|\cdot}$ which can be constructed through a similarity transform [@mosta2003equivalence; @kretschmer2001interpretation; @kretschmer2004quasi]. Despite the mathematical equivalence of these two pictures of quantum theory, we emphasize that *local* quantum theory is generalized by using a quasi-Hermitian representation. More generally, the usage of quasi-Hermitian representations plays a role when there exists an additional physical significance to an inner product structure aside from a role in computing expectation values. To elaborate, consider the tensor product model of locality, which is defined on a Hilbert space with a tensor product factorization, $H \simeq H_A \otimes H_B$. A common choice for the physical inner product is one which factorizes with $H_A \otimes H_B$. This assumption is unnecessarily restrictive, since a general inner product is deduced from a metric operator which may not factorize in the form $\eta = \eta_A \otimes \eta_B$. Operators local to subsystem $A$ in this model are defined as those which decompose as a tensor product with the identity operator on subsystem $B$, $$\begin{aligned} O = O_A \otimes \mathbbm{1}_B, \label{locality}\end{aligned}$$ and vice versa for local operators in subsystem $B$. Motivating this definition is the observation that expectation values of local observables can be computed from the local state, a partial trace of the density matrix [@densityOperator], without referencing the entire Hilbert space, $$\begin{aligned} \rho_A &= \text{Tr}_B\, \frac{\ket{\psi} \bra{\psi} \eta}{\braket{\psi|\eta|\psi}}, \\ \braket{O}_\eta &= \text{Tr}_A\, \rho_A O_A.\end{aligned}$$ In addition, observables in disjoint subsystems can be simultaneously measured without affecting each other, a result known as no-signalling [@noSignalling; @normieNonLocality]. Note the definition of a local operator is independent of the inner product structure. Thus, quasi-Hermitian theories with differing metric operators can in principle contain distinct local observables algebras, even with distinct, possibly vanishing dimensions. Generically, while the aforementioned similarity transform maps a quasi-Hermitian theory to a Hermitian theory, it does not preserve the notion of local observable algebras. This is a consequence of the *nonlocality* of the similarity transform, demonstrated for instance in [@freeFermionMetric; @jin2009solutions; @mostafazadeh2005anharmonic]. In fact, the space of local quasi-Hermitian models similar to local Hermitian models was found in [@MetricTensorStructure], and is smaller than the total space of local quasi-Hermitian models. The use of quasi-Hermitian models allows for the discovery and exploration of a broader set of local quantum theories, despite their mathematical equivalence to nonlocal Hermitian theories. This idea could prove useful for finding new quantum field theories, where the space of theories is heavily constrained by principles such as gauge symmetry and renormalizability. In addition, it has promise for problems in quantum gravity, where the evolution is expected to be local, but, due to diffeomorphism invariance, observables are nonlocal. The goal of this work is to explicitly construct local observable algebras in quasi-Hermitian theories in general, to discuss their properties, and derive when they existence in several toy models. As the observables in a subsystem $A$ could be further localized to a subsystem contained in $A$, we introduce the notion of an local observable: An observable is *extensively* local over subsystem $A$ if this observable is not local to any subsystem of $A$. Note that if two subsystems $A, B$ have extensively local observables, then their union does as well, proven by taking a suitable linear combination of observables local to $A$ and $B$. In \[theorems\], a theorem relating the existence of observables local to a subsystem to the Schmidt decomposition of the metric is proven. Simple, illustrative examples and corollaries are presented. In addition, theorems addressing extensively local observables specific to free fermions are introduced in \[free fermion locality\], having the classic advantage generic with free fermion models of computational simplicity. For systems of fermions, there is a notion of locality which is more naturally related to the anti-commutation relations. A review of this definition is presented in \[fermion locality\], and the theorems of \[free fermion locality\] are adapted to this notion of locality as well. Our toy model is a local[^1] , one-dimensional, many-body Hamiltonian of free fermions, a tight-binding model with a complex on-site potential and anisotropic tunneling amplitudes, $$\begin{aligned} H_{\mathcal{PT}} &= \left(\gamma a^{\dag}_m a^{}_m + \gamma^{*} a^{\dag}_{\overline{m}} a^{}_{\overline{m}} \right) + \sum^n_{i = 1} \left(V_i a^\dag_i a^{}_i\right) \nonumber\\&+ \sum^{n-1}_{i = 1} \left( t^{*}_{n-i} a^{\dag}_i a^{}_{i+1} + t_i a^{\dag}_{i+1} a^{}_i \right),\end{aligned}$$ where $V_i = V_{n-i} \in \mathbbm{R}$, $\overline{m} = n-m+1$, and $\text{arg} \, t_{n-i} = \text{arg} \, t_i$. This toy model is symmetric under combined parity and time-reversal symmetries, $\mathcal{PT}$, and is known to be quasi-Hermitian in some special cases [@freeFermionMetric; @InfiniteLattice; @Babbey; @jin2009solutions; @MyFirstPaper; @ZnojilModel; @ZnojilGeneralized; @JoglekarSaxena; @PTModels]. Surprisingly, even though $H_{\mathcal{PT}}$ is local, there generically exist subsystems with no local observables. The locality profile appears to depend strongly on the potential. While for $\gamma \in \mathbbm{R}$, there is a metric such that local observables exist in *every* subsystem, the metrics for $n = 2m, \gamma \notin \mathbbm{R}$ studied in this paper are only compatible with observables local to $\mathcal{P}$-symmetric subsystems. For other choices of $m$, there exist observables local to subsystems which are not $\mathcal{P}$-symmetric. For instance, for $m=1, t_i = t$, there are local observables in every connected subsystem with more than two sites. As discussed in \[Symmetric Ham Locality\], the $\mathcal{PT}$-symmetry expresses itself in a weaker form: If the metric is $\mathcal{PT}$-symmetric, and subsystem $A$ contains local observables, then the $\mathcal{P}$ dual of $A$ also contains local observables. $\mathcal{PT}$-symmetric free fermions {#review} ====================================== The structure of quasi-Hermitian theory {#fundamentals} --------------------------------------- A brief review of some central foundational results in quasi-Hermitian theory in finite Hilbert spaces is provided in this section. The interested reader is referred to two review articles for additional depth, [@MakingSenseNonHerm; @MostaReview]. The aim of this section is to prove a spectral theorem for quasi-Hermitian operators, emphasized in [@mosta2003equivalence; @mostafazadeh2008metric], and to discuss some properties of quasi-Hermitian theories through its proof. \[diagonalizable\] An operator, $O$, in a finite-dimensional Hilbert space $\mathcal{H}$ is diagonalizable with a real spectrum if and only if $O$ is quasi-Hermitian with respect to some Hermitian positive-definite metric, $\eta$. Note this theorem is more powerful than the textbook claim that Hermiticity in a fixed Hilbert space is sufficient for a real spectrum, as quasi-Hermiticity is the sufficient and necessary condition for a real spectrum. This theorem readily generalizes to the case of infinite dimensional operators with discrete spectrum [@kretschmer2001interpretation]. This proof begins by constructing a metric operator, $\eta$, for every diagonalizable operator with real spectrum, $O$. Consider an orthonormal basis under the inner product given through $\mathcal{H}$. $O$ is diagonalizable in this basis, and can be expressed in the form $O = \mathcal{U} D \mathcal{U}^{-1}$, where the diagonal matrix $D$ is Hermitian. The most general metric operator associated to $O$ is thus $$\begin{aligned} \eta^{-1} = \mathcal{U} d \mathcal{U}^\dag, \label{generalMetric}\end{aligned}$$ where $d$ is a Hermitian, positive-definite matrix which commutes with $D$. Note the choice of metric is not unique. A single operator of physical significance, such as the Hamiltonian, is compatible many realizations of inner products, which give different resulting quasi-Hermitian theories. Choosing a particular metric can be done by requiring additional operators to have physical significance [@QuasiHerm92; @kretschmer2001interpretation]. Other, less physical choices, are to pick a set of eigenvectors $\mathcal{U}$ and set $d = \mathbbm{1}$, or perhaps to choose one metric operator which has an explicit analytic realization. The remaining direction of the proof can be performed in multiple ways. The author chooses to follow one method performed initially in [@williams1969operators]. Assume the existence of a metric operator, $\eta$. Since $\eta$ is positive definite, its square root exists, is Hermitian, and is invertible $$\eta = \Omega^2, \,\,\, \Omega = \Omega^{\dag}. \label{Omega}$$ The square root, $\Omega$, constructs a similarity transformation from $O$ to a Hermitian operator, $$h := \Omega O \Omega^{-1} = h^\dag. \label{Similar}$$ Since $O$ is similar to a Hermitian operator, $O$ is diagonalizable with a real spectrum. The map from observables to Hermitian operators constructed in the proof of \[diagonalizable\], $O \rightarrow \Omega O \Omega^{-1}$, maps a quasi-Hermitian theory to a Hermitian theory with Hermitian inner product [@mosta2003equivalence]. However, these two theories can have different notions of locality, since the operator $\Omega$ may contain nonlocal properties. To briefly comment on foundational issues in the theory with infinite Hilbert spaces, note that the theorem of this section freely uses finite dimensional concepts, such as diagonalizability and the assumption that positive operators have inverses defined on the entirety of $\mathcal{H}$. Critically, the similarity transformation defined through $\Omega$ may not exist if either the metric or its inverse is an unbounded operator. Typically, the metric is assumed to be bounded to avoid issues with operator domain equalities in the quasi-Hermiticity condition. To guarantee the mathematical validity of the similarity transform, one may be tempted to assume the metric’s inverse is bounded as well. However, for certain non-Hermitian Hamiltonians with a real spectrum and an associated metric operator, there exists no metric with a bounded inverse [@siegl2012metric]. Without the assumption of a bounded metric inverse, a spectral theorem of the sort mentioned above can’t exist. The quasi- Hermitian operator described in [@dieudonne] with complex spectrum, is one such counter-example. However, the conclusion that the eigenvalues are real still holds (note the spectrum of an operator is in general larger than the space of eigenvalues). ### Relation to $\mathcal{PT}$-symmetry Some treatments of quasi-Hermitian theory start from symmetries of the Hamiltonian, as opposed to the metric. The more relaxed condition that the energy eigenvalues come in complex conjugate pairs ($E_n$, $E^{*}_n$) is equivalent to the existence of a discrete, anti-linear symmetry, $\Theta$ [@mostafazadeh2002pseudo2] $$[H, \Theta] = 0,$$ If the $\Theta$-symmetry is *unbroken*, so that at least one set of eigenstates of $H$ are also eigenstates of $\Theta$, then the energies are strictly real [@bender1999pt]. Generic classes of $\Theta$-symmetric Hamiltonians experience a boundary in parameter space, referred to as a *phase transition*, beyond which $\Theta$ is *broken*, and inside which $\Theta$ is unbroken [@bender1998real; @InfiniteLattice; @Babbey; @freeFermionMetric; @jin2009solutions; @MyFirstPaper; @PTRing]. For any quasi-Hermitian operator, there exists a decomposition, $$\Theta = \mathcal{PT}$$ into a linear operator $\mathcal{P}$ and the *time reversal*, or complex conjugation, operator $\mathcal{T}$, which commute and square to $\mathbbm{1}$ [@bender2010pt]. Following the historic work of [@bender1998real], the special case of $\Theta$ which is a product of *parity* and time-reversal symmetries is often used to construct quasi-Hermitian models, and the field of quasi-Hermitian quantum theory is generally synonymous with $\mathcal{PT}-\textit{Symmetric}$ quantum theory[^2]. Interestingly, the metric of \[generalMetric\] is also $\mathcal{PT}-$symmetric in the case where $H$ is non-degenerate. In the case of a degeneracy, only the choice of $d = \mathbbm{1}$ in \[generalMetric\] is guaranteed to be $\mathcal{PT}-$symmetric, where $\mathcal{U}$ must be chosen to contain a set of $\mathcal{PT}-$unbroken eigenstates. Local Hamiltonians {#local Ham} ------------------ To allow for nontrivial interactions between subsystems, the locality condition for a Hamiltonian is weaker than the locality definition used for observables. Qualitatively, a Hamiltonian is local if after a brief period of Schrödinger evolution on a generic state, only the qualities of nearby pairs of subsystems influence each other. Given a Hilbert space, $\mathcal{H}$, with a factorization into *sites*, $i$, in a *lattice*, $S$, $\mathcal{H} \simeq \bigotimes_{i \in S} \mathcal{H}_i$, and a graph, $G = (S, E), E\subseteq S \times S$, a Hamiltonian is said to be local if it is a sum over operators local to pairs of vertices in the graph, $$H = \sum_{(i,j) \in E} H_{ij} \sum_{\alpha, \beta} O^\alpha_i O^\beta_j,$$ where $O^{\alpha}_i$ forms an orthonormal basis for the operators local to the site $i$. Lattice Fermions ---------------- A natural setting for studying aspects of locality is the space of quantum many-body problems, since their Hilbert spaces have a natural tensor product decomposition. Free fermions are a cherished example of many-body problems; due to their relationship with a *first quantized* quantum theory on a Hilbert space which scales linearly with the number of sites, many features of free fermionic models can be computed with only polynomial computational resources. Examples of such features include the ground state energy [@nielsen2005fermionic] as well as entanglement entropies [@FreeFermionEntanglement]. This is in constrast to solving a generic many-body problem, which has an exponential complexity. Examples of $\mathcal{PT}$-symmetric free fermions have been well-studied [@InfiniteLattice; @jin2009solutions; @Babbey; @freeFermionMetric; @MyFirstPaper; @PTRing]. Free fermion models are constructed with a realization of the *canonical anti-commutation relations*, a relationship amongst a set of $n$ creation $a^{\dag}_i$ and annihilation $a_j$ operators [@nielsen2005fermionic] $$\{a^{\dag}_i, a^{}_j\} = \delta_{ij} \mathbbm{1}, \,\,\,\,\, \{a^{\dag}_i, a^\dag_j\} = 0, \label{CCR}$$ where $\mathbbm{1}$ is the identity operator and $\delta$ is the Kronecker delta. Lowercase latin indices, such as $i, j$ above, are elements of the set $[n]$, which is the set of integers ranging from $1$ to $n$. A *vacuum* is defined as a state annihilated by all $a_i$, $a_i \ket{0} = 0$. For the rest of this report, it is assumed that the vacuum is *unique*. As a consequence, every state in $\mathcal{H}$ can be constructed through linear combinations of repeated application of creation operators on the vacuum [@nielsen2005fermionic], and the Hilbert space of this representation is $N = 2^n$ dimensional. For notational simplicity, a Hilbert space will be denoted with a subscript, which is a set whose elements refer to a labelling of a basis of the Hilbert space. Let $\mathbbm{P}([n])$ denote the power set of $[n]$, so the Hilbert space of free fermion models will be denoted $\mathcal{H}_{\mathbbm{P}([n])}$. A useful generalization of the creation and annihilation operators is to construct a representation, $a, a^\dag: \mathbbm{C}^n \rightarrow \text{End}(\mathcal{H}_{\mathbbm{P}([n])})$[^3], of the CAR algebra over $\mathbbm{C}^n$, $$\begin{aligned} a(f) &= \sum_{i \in [n]} f^*_i a^{}_i, \\ a^\dag(g) &= \sum_{i \in [n]} g^{}_i a^\dag_i. \label{generalCAR}\end{aligned}$$ The space of operators on $\mathcal{H}_{\mathbbm{P}([n])}$ can be expressed through linear combinations of products of creation and annihilation operators, $$\begin{aligned} \text{End}(\mathcal{H}_{\mathbbm{P}([n])}) = \text{span} \left \lbrace \left(\prod_{i \in S_1} a^\dag_i \right) \left(\prod_{j \in S_2} a^{}_j\right): S_1, S_2 \subseteq [n]\right \rbrace. \label{OperatorSpanBasic}\end{aligned}$$ Equivalently, the space of operators on $\mathcal{H}_{\mathbbm{P}([n])}$ can be generated from linear combinations of creation and annihilation operators, so long as the linear combinations arise from vectors which form a linearly independent basis of $\mathbbm{C}^n = \text{span}\{v^\mu|\mu \in [n]\} = \text{span} \{{w^\nu|\nu \in [n]}\}$, $$\begin{aligned} \text{End}(\mathcal{H}_{\mathbbm{P}([n])}) = \text{span} \left \lbrace \arraycolsep=1.4pt\def\arraystretch{1.8} \begin{array}{l} \left(\prod\limits_{\mu \in S_1} a^\dag(v^\mu)\right) \left( \prod\limits_{\nu \in S_2} a(w^\nu)\right):\\ S_1, S_2 \subseteq [n] \end{array}\right \rbrace. \label{OperatorSpan}\end{aligned}$$ In particular, using the specific basis of $\mathbbm{C^n} = \text{span} \{e_i:i\in [n]\}$, where $$\begin{aligned} (e_{i})_j = \delta_{ij}, \label{CnBasis}\end{aligned}$$ the decomposition of \[OperatorSpan\] reduces to that of \[OperatorSpanBasic\] ### Fermionic Locality {#fermion locality} Given an abstract Hilbert space, $\mathcal{H}$, with no a priori tensor product structure, locality is defined through a unitary transformation, $\iota$, to a theory on a Hilbert space with a tensor product factorization, $\mathcal{H}_A \otimes \mathcal{H}_B$. For a quasi-Hermitian theory defined on $\mathcal{H}$, the metric $\eta$ transforms to the metric $\eta_{\text{TPS}} \in \text{End}(\mathcal{H}_A\otimes \mathcal{H}_B)$ via $$\begin{aligned} \iota: \mathcal{H} \rightarrow \mathcal{H}_A \otimes \mathcal{H}_B, \,\,\,\,\, \iota \eta \iota^\dag = \eta_{\text{TPS}}.\end{aligned}$$ This map is referred to as a *tensor product structure* [@TPS]. A tensor product structure for $\mathcal{H}_{\mathbbm{P}([n])}$ is given by a unitary map, the Jordan-Wigner transform [@JordanWigner; @nielsen2005fermionic], into the tensor product of $n$ two-dimensional Hilbert spaces (each equipped with a Pauli matrix, $Z_i$, and a lowering operator, $\sigma_i = \ket{0}_i \bra{1}_i$), $$\begin{aligned} \iota_p: \mathcal{H}_{\mathbbm{P}([n])} &\rightarrow \bigotimes_{i\in [n]} \mathcal{H}^i_{[2]}, \\ \iota^{}_p a^{}_{p(i)} \iota^\dag_p &= \bigotimes_{j < i} Z_j \otimes \sigma_i, \label{a to sigma} \\ \iota_p \ket{0} &= \bigotimes_{i \in [n]} \ket{0}_i,\end{aligned}$$ where $p$ is a permutation of sites, a one-to-one map $p:[n]\rightarrow [n]$. A common choice for $p$ is the identity map. A Jordan-Wigner transform can be inverted, which produces the important identity $$Z_j = \iota_p \left( a_{p(j)} a^\dag_{p(j)} - a^\dag_{p(j)} a_{p(j)} \right) \iota^\dag_p. \label{a to Z}$$ In addition, when there are observables local to $\mathcal{H}_{\mathbbm{P}(A)} \simeq \otimes_{i \in A} \mathcal{H}^i_{[2]}$, we will let statements of the form “$\mathbbm{P}(A)$ contains observables” be synonymous to “$A$ contains observables”. Notice the lack of a choice of Jordan-Wigner transform which localizes all $a_i$. This is a consequence $a_i$ satisfying anti-commutation relations, while operators local to disjoint subsystems necessarily commute. To find a notion of locality directly from the canonical anti-commutation relations, the space of operators must be restricted to a set of commuting operators. Even products of pairs of creation and annihilation operators satisfy this criteria, motivating the following alternative definition of the space of operators local to a subsystem [@BravyiKitaev]. An operator, $O_S$, is said to be *Bravyi-Kitaev local* to a collection of sites, $S \subseteq [n]$, if and only if it’s a linear combination of an even product of creation and annhilation operators with indices in this collection, $$\begin{aligned} O_{S} \in \text{span} \left\{ \arraycolsep=1.4pt\def\arraystretch{1.8} \begin{array}{l} \left(\prod\limits_{\tiny { i \in A \subseteq \mathbbm{P}(S)}} a^\dag_i \right)\left( \prod\limits_{\tiny { j \in B \subseteq \mathbbm{P}(S)}} a^{}_j \right): \\ |A|+|B| \equiv 0 {\ (\mathrm{mod}\ 2)} \end{array} \right\},\end{aligned}$$ where $|A|$ is the cardinality of $A$. This observable is *extensively* local over $S$ if and only if $$\begin{aligned} O_{S} \notin \text{span} \left\{ \arraycolsep=1.4pt\def\arraystretch{1.8} \begin{array}{l} \left(\prod\limits_{\tiny { i \in A \subseteq \mathbbm{P}(S_1)}} a^\dag_i \right)\left( \prod\limits_{\tiny { j \in B \subseteq \mathbbm{P}(S_1)}} a^{}_j \right): \\ |A|+|B| \equiv 0 {\ (\mathrm{mod}\ 2)} \end{array} \right\}\end{aligned}$$ for all proper subsets $S_1 \subset S$. Operator-based definitions of locality were related to tensor product structures in [@ObservableLocality]. Explicitly, a set of algebras, $\mathcal{A}_i$, associated to a set of disjoint subsystems, $i \in \Lambda$, needs to satisfy three axioms to derive an equivalent tensor product structure: 1. The algebras are independent, $\mathcal{A}_i \cap \mathcal{A}_j = \mathbb{1} \, \forall i \neq j$ 2. Two distinct local subalgebras commute, $[ \mathcal{A}_i, \mathcal{A}_j] = 0 \, \forall i \neq j$ \[subalgebra commutation\] 3. The algebras generate the entire space of operators on $\mathcal{H}$, $\vee_{i} \mathcal{A}_i = \text{End}(\mathcal{H})$. \[generators\] The second axiom is critical to ensure a lack of signalling between subsystems. Bravyi-Kitaev locality doesn’t satisfy the generation axiom of locality, \[generators\], since odd products of creation and annihilation operators are assumed to be unphysical. Thus, Bravyi-Kitaev Locality is weaker than a tensor product structure. ### Free fermions Free fermion Hamiltonians are constructed from products of pairs of creation and annihilation operators. This paper restricts itself to the particle-conserving case, where $$H = \sum_{i,j} \Gamma_{ij} a^{\dag}_i a_j. \label{general_H}$$ $H$ is Hermitian if and only if the $n\times n$ matrix $\Gamma$, the *first quantized Hamiltonian*, is Hermitian A physical inner product requires construction of a metric associated to $H$. One choice of metric follows from a metric associated to the first quantized Hamiltonian, $$M \Gamma = \Gamma^\dag M \label{first quantized metric},$$ where the adjoint for matrices is taken to be complex conjugate transposition, and $M$ is Hermitian and positive-definite. The unique solution, $\eta \in \text{End}(\mathcal{H}_{\mathbbm{P}([n])})$, to the operator equations $$\begin{aligned} \eta a^{\dag}_i &= \sum_j M_{j i} a^{\dag}_j \eta, \\ \label{reduced metric to metric} \eta \ket{0} &= \ket{0}\end{aligned}$$ is a valid metric for $H$ [@PTModels]. To avoid confusion, the metric $\eta$ will be referred to as the *total metric*, and $M$ will be referred to as the *reduced metric*. Note that the *number operator*, $\hat{n} = \sum_i a^{\dag}_i a_i$, is an observable with this choice of metric. Since the number operator is an observable, two Hamiltonians related by a chemical potential, $H' = H + \mu \hat{n}$, have the same choices of reduced metrics. Toy model {#toy model} --------- A simple testing ground for the locality theorems proven in \[theorems\] is a generalization of the models studied in [@freeFermionMetric; @Babbey; @JoglekarSaxena; @MyFirstPaper; @ZnojilGeneralized; @InfiniteLattice] $$\begin{aligned} H_{\mathcal{PT}} &= \left(\gamma a^{\dag}_m a_m + \gamma^{*} a^{\dag}_{\overline{m}} a_{\overline{m}} \right) + \sum_{i \in [n]} \left(V_i a^\dag_i a_i\right)\nonumber\\&+\sum_{i \in [n-1]} \left(t^*_{n-i} a^{\dag}_i a_{i+1} + t_i a^{\dag}_{i+1} a_i \right) , \label{PT}\end{aligned}$$ with $V_i = V_{n-i} \in \mathbb{R}$ and $n>1$. The special sites $m, \overline{m} = n-m+1$ are referred to as *impurities*, and the complex parameters $t_i$ are referred to as *hopping amplitudes*. The toy model only includes one-dimensional nearest neighbour interactions, so $\Gamma_{ij} \neq 0 \Leftrightarrow |i-j| \leq 1$. All free fermions Hamiltonians with nearest neighbour interactions are local with respect to the one-dimensional graph $G_{[n]} = ([n], E_{[n]})$, $E_{[n]} = \{(i, i+1) , i \in [n-1]\}$ and the tensor product structure associated to the standard Jordan-Wigner transform, $\iota_{1}$. Despite being non-Hermitian, this Hamiltonian has an anti-linear symmetry, $\mathcal{PT}$, the product of combined (linear) *Parity* and (anti-linear) *Time reversal* $$\begin{aligned} \mathcal{P} a^\dag_i &= a^\dag_{\, \bar{i}} \mathcal{P}, \,\,\,\,\, \mathcal{P} \ket{0} = \ket{0}, \label{Parity} \\ \mathcal{T} a^\dag_i &= a^\dag_i \mathcal{T}, \,\,\,\,\, \mathcal{T} \ket{0} = \ket{0}, \label{timeReverse}\end{aligned}$$ where $\bar{i} = n-i+1$. Equations  and imply $\mathcal{P}^2 = \mathcal{T}^2 = 1$, $\mathcal{P} = \mathcal{P}^\dag, \mathcal{T} = \mathcal{T}^\dag$, and $[\mathcal{P},\mathcal{T}] = 0$. This paper doesn’t construct local observable algebras for metrics compatible with $H_{\mathcal{PT}}$ in full generality. Rather, we make two simplifications on the space of parameters: Firstly, we will assume that the phases of the hopping amplitudes, $t_j = |t_j| e^{i \theta_j}$, are parity symmetric, so $\theta_j = \theta_{n-j}$. The phase symmetry serves two purposes: it simplifies the metric, and it helps ensure the reality of the spectrum. Phase symmetry isn’t a necessary criteria for the reality of the spectrum, exemplified by [@PTModels]. Interestingly, if the phases are symmetric, they do not affect the spectrum of the Hamiltonian, as evidenced by writing $H$ in terms of an alternative representation of the canonical anti-commutation relations, $$\begin{aligned} b_i &:= e^{-i \chi_i} a_i, \label{alt CAR rep}\\ \{b_i,b_j^\dag\} &= \delta_{ij} \mathbbm{1}, \\ \chi_{i+1} &:= \sum_{j \in [i]} \theta_j, \,\,\,\,\, \chi_1 = 0, \\ H_{\mathcal{PT}} &= \left(\gamma b^{\dag}_m b_m + \gamma^{*} b^{\dag}_{\overline{m}} b_{\overline{m}} \right) \nonumber \\&+ \sum_{i \in [n-1]} \left( |t_{n-i}| e^{i (\theta_i-\theta_{n-i})} b^{\dag}_i b_{i+1} + |t_i| b^{\dag}_{i+1} b_i \right).\end{aligned}$$ However, changing the phases changes the space of metrics, and thus, changes the associated observables. Yet, the question of the existence of extensively local observables with locality defined in either the $a_i,a_j^\dag$ or $b_i, b_j^\dag$ representation yields the same answer. This is a consequence of the transformation defined in \[alt CAR rep\] not relating creation or annihilation operators at distinct sites. Secondly, we study two special cases, summarized in the subsections below: ### Farthest Impurities Properties of $H_{\mathcal{PT}}$ investigated in this section assume $t_i = t$. Secondly, we assume $m=1$. Thirdly, we choose $t \in \mathbbm{R}$, due to the spectral equivalence and equivalence of Bravyi-Kitaev locality between $t_i = t$ and $t_i = |t|$. A convenient choice of units adopted in this paper sets $t=1$[^4] . Explicitly, $$H_{\text{XX}} = \left(\gamma a^{\dag}_1 a_1 + \gamma^{*} a^{\dag}_{n} a_{n} \right) + \sum_{i \in [n-1]} \left(a^{\dag}_i a_{i+1} + a^{\dag}_{i+1} a_i \right). \label{XX}$$ This model is quasi-Hermitian in a region of parameters known numerically [@PTModels], and known to contain the unit disk $|\gamma| = 1$ [@freeFermionMetric]. One analytic solution to the reduced metric in this case is [@farImpurityMetric; @SSHMetric], $$M_{ij} = \begin{cases} \,\,\,\,\,1 & i = j \\ -i \,\text{Im} \gamma\, \left(\gamma^{*} \right)^{j-i-1} & i < j \\ \,\,\,\,\,i \,\text{Im} \gamma\, \left(\gamma \right)^{i-j-1} & i > j \end{cases}. \label{not positive}$$ While satisfying Hermiticity and \[first quantized metric\], $M$ fails to be a valid reduced metric for all complex-valued $\gamma$ in the $\mathcal{PT}$-unbroken region, due to a lack of positive definiteness [@PTModels]. ### Nearest neigbour impurities The second special case fixes $n = 2m$, but leaves the amplitudes $t_i \neq 0$ arbitrary. A 1-parameter family of reduced metrics is given in \[badass metric\] [@PTModels]. Importantly, the metric decomposes into parity blocks, $$M_{ij} \neq 0 \Leftrightarrow i = \bar{j}.$$ Its matrix elements are given by the following recurrence relations $$\begin{aligned} \begin{pmatrix} M_{mm} & M_{m \, m+1} \\ M_{m+1 \, m} & M_{m+1 \, m+1} \end{pmatrix} &= \begin{pmatrix} 1 & \dfrac{\beta - i \text{Im} \gamma}{t_m}\\ \dfrac{\beta + i \text{Im} \gamma}{t^{*}_m} & 1 \end{pmatrix} \\ M_{ii} &= \dfrac{t_{n-i}}{t_i} M_{i+1\, i+1}\\ M_{i \bar{i}} = M^*_{\bar{i} i} &= \dfrac{t_{n-i}}{t^*_i} M_{i+1\, n-i} \\ M_{\bar{i} \bar{i}} &= \dfrac{t_{n-i}}{t_i} M_{n-i\, n-i} \label{badass metric},\end{aligned}$$ where the index $i < m$ and where $\beta \in \mathbbm{R}$ satisfies $\beta^2 + (\text{Im}\gamma )^2/|t_m|^2 < 1$ [@PTModels]. In addition, it’s known that the $\mathcal{PT}$-symmetry breaking boundary is $\text{Im} \gamma = |t_m|$. The nearest neighbour impurity case experiences an unusual phase transition: the entire spectrum is purely imaginary for $\text{Im}\gamma > |t_m|$ [@MyFirstPaper], and $H$ has only $n/2$ eigenvalues at $\text{Im} \gamma = |t_m|$. Local Observable Algebras in Quasi-Hermitian Theories {#theorems} ===================================================== Local Observables in the Tensor Product Picture {#GeneralTheorems} ----------------------------------------------- A natural clue for the studies of locality would be the tensor product structure of the metric, since tensor products are used to define locality, and the metric is used to define observables. Importance of the metric’s tensor product structure is exemplified for instance by [@MetricTensorStructure], the central claim of which is that *a quasi-Hermitian algebra contains local operator algebras generated by Hermitian observables if and only if a tensor product,* $$\eta = \eta_A \otimes \eta_B. \label{eta_tensor}$$ A general operator in $\mathcal{H}$ can’t be written as a tensor product of the form , however, a general operator has an *operator Schmidt decomposition* [@operatorSchmidt] to a subsystems $A$ and $B$$$\eta = \sum_{i} \sqrt{\chi_i} \eta^i_A \otimes \eta^i_B,$$ where $\chi_i \geq 0$, and in the case where $\eta$ is a Hilbert-Schmidt operator, $\eta^i_{A,B}$ are orthonormal under the Hilbert-Schmidt inner product $$\text{Tr}\, {\eta^i_A}^{\dag} \eta^{\,j}_A = \delta_{ij}, \,\,\,\,\,\,\, \text{Tr}\, {\eta^i_B}^{\dag} \eta^{\,j}_B = \delta_{ij}. \label{SubMetricOrth}$$ In the more general case where $\eta$ is not a Hilbert-Schmidt operator, it’s still the case that $\eta^{i}_{A,B}$ are linearly independent. Let’s refer to such a set of operators, $\eta^i_{A,B}$ as a set of Schmidt operators associated to $\eta$. The *Schmidt Number* of this decomposition, $n_{AB}$, is the number of nonzero $\chi_i$. While a generic operator may admit different Schmidt decompositions, the Schmidt number for all decompositions is the same. Note the existence of a Schmidt decomposition such that $\eta^i_{A,B} = {\eta^i}^\dag_{A,B}$ follows from the existence of a basis for the space of Hermitian operators in $\text{End}(\mathcal{H})$ such that the basis elements are tensor product operators. However, since the space of positive definite operators is not a vector space, in general, the operators $\eta^i_{A,B}$ aren’t positive definite. With the above in mind, the following theorem relating to the existence of local observables can now be proven: \[quasilocal\_theorem\] The following are equivalent: 1. There exists an operator local to subsystem $B$, $O = \mathbbm{1}_A \otimes O_B$, which is a quasi-Hermitian observable with respect to a metric, $\eta$. In addition, the operator $O_B$ is not a multiple of the identity operator.\ 2. There exists a simultaneous solution, $O_B$, to the operator equations $$\eta^{\,j}_B O_B = O^{\dag}_B \eta^{\,j}_{B} \,\,\,\,\,\,\,\,\,\,\, \forall j : \chi_j > 0, \label{block metrics}$$ where $O_B$ is not a multiple of the identity operator.\ 3. There exists a partitioning of $\mathcal{H}_R \oplus \mathcal{H}_{B-R} = \mathcal{H}_B$ such that any set of Schmidt operators, $\{\eta^i_B\}$, associated to the metric are simultaneously reducible under a invertible transformation, $S$, of the form $$S^{\dag} \eta^i_B S = \left(\begin{matrix} \eta^i_R & 0 \\ 0 & \eta^i_{B-R} \end{matrix} \right). \label{reduce metric}$$ Keeping $O_B$ distinct from a multiple of the identity is a sort of triviality condition. Such an operator is trivially a local observable, however, this observable contains no physical information, as a measurement of such an observable always yields the degenerate eigenvalue. In addition, we only provide a proof of the third item for the case of compact observables, where one can safely diagonalize an operator in the normal sense. The validity of this item outside of this case is outside the scope of this report. Proof of $\textit{1}\Leftrightarrow \textit{2}$: Assume an observable has a local decomposition, $O = \mathbbm{1}_A \otimes O_B$. Let $\phi^i_{A,B}: \text{End}(\mathcal{H}_{A,B}) \rightarrow \mathbb{C}$ denote a linear functional satisfying $$\begin{aligned} \phi^i_{A,B} (\eta^{j}_{A,B}) = \delta_{ij}.\end{aligned}$$ Applying the map $\phi^i_{A} \otimes \mathbbm{1}_B$ to both sides of the quasi-Hermiticity condition, \[operator\], for $O$ imposes the constraints \[block metrics\] on $O_B$. In the case where $\eta$ is a Hilbert-Schmidt operator, this map is realized as taking a partial trace over $A$ after multiplication by ${\eta^{\,j}_{A}}^{\dag} \otimes \mathbbm{1}$. Conversely, if $O_B$ satisfies \[block metrics\], $O = \mathbb{1}_A \otimes O_B$ is quasi-Hermitian with respect to $\eta$. Proof of $\textit{1} \Rightarrow \textit{3}$: Using \[diagonalizable\], and noting that $O$ is diagonalizable if and only if $O_B$ is diagonalizable, $O_B$ has a diagonalization $O_B = S D S^{-1}$, with $D = D^{\dag}$. Substituting this into \[block metrics\]$$S^{\dag} \eta^{i}_B S D = D S^{\dag} \eta^i_B S.$$ For $O$ to be nontrivial, there must be at least two distinct elements $d_1 \neq d_2$ of $D$. For any $\ket{d_1}$ and $\ket{d_2}$ from the eigenspaces of $d_1$ and $d_2$ respectively, the matrix elements of $S^{\dag} \eta^i_B S$ vanish $$\begin{aligned} \braket{d_1|S^{\dag} \eta^{i}_B S D|d_2} = \braket{d_1|D S^{\dag} \eta^i_B S|d_2} = 0.\end{aligned}$$ Thus, $S^{\dag} \eta^i_B S$ is reducible to the eigenspaces of $D$, completing this direction of the proof. Proof of $\textit{3} \Rightarrow \textit{1}$: If $S$ satisfying \[reduce metric\] exists, then \[block metrics\] can be rewritten as $$\left(\begin{matrix} \eta^i_R & 0 \\ 0 & \eta^i_{B-R} \end{matrix} \right) (S^{-1} O_B S) = (S^{-1} O_B S)^{\dag} \left(\begin{matrix} \eta^i_R & 0 \\ 0 & \eta^i_{B-R} \end{matrix} \right).$$ A nontrivial solution for $O_B$ can be constructed with distinct eigenvalues $d_1, d_2$, $$O_B = S \left(\begin{matrix} d_1 \mathbbm{1}_R & 0 \\ 0 & d_2 \mathbbm{1}_{B-R} \end{matrix} \right) S^{-1}.$$ The existence of observables local to a subsystem $A$ doesn’t imply that they are extensively local. There could exist a partitioning of $A$ into smaller subsystems $A_1, A_2$ such that every observable in $A$ is of the form $O_A = \mathbbm{1}_{A_1} \otimes O_{A_2}$. Theorem  offers limited support in this matter: it’s simple to demonstrate that if every observable in $A$ is of the form $O_A = \mathbbm{1}_{A_1} \otimes O_{A_2}$, then $\{ \eta^i_{A_1} \}$, is irreducible and $\{\eta^i_{A_2} \}$ is reducible. The converse of this statement is not true: a simple counterexample entails picking $\mathcal{H}_{A_2}$ to be a tensor product of $\mathcal{H}_{A_3}\otimes \mathcal{H}_{A_4}$, where $\{\eta^i_{A_4} \}$ is reducible, $\{\eta^i_{A_3} \}$ is irreducible, and observables exist in $A_1 \cup A_3$. \[Schmidt\_lower\_bound\] If $n_{AB} = 1$, so $\eta = \eta_A \otimes \eta_B$, extensively local observables exist in both $A$ and $B$. This can be proven through explicit construction of local observables $$O = O_{A} \eta_{A} \otimes O_{B} \eta_{B}, \,\,\, O_{A} = {O_A}^{\dag}, \, O_{B} = {O_B}^{\dag},$$ though we’d like to point out that this is a corollary of \[quasilocal\_theorem\]. Since the metric $\eta$ is both Hermitian, $\eta_{A,B}$ are also Hermitian, so they’re diagonalizable, and thus reducible to the spaces of their eigenvectors. In the case of a finite Hilbert space, if $n_{AB} > (\min \{|A|^2, |B|^2\}-1)^2 + 1$, no local observables exist in the smaller of $A$ and $B$. For simplicity assume without loss of generality that $|B| \leq |A|$. The proof proceeds by contradiction. Assume such a local observable exists in $B$. Construct a set of Hermitian Schmidt operators, $\eta^i_{A,B} = {\eta^i}^\dag_{A,B}$. By \[quasilocal\_theorem\], the Schmidt operators, $\eta^i_B$, must be simultaneously reducible. The decomposition fixes $2(\mathcal{M}-|R|)|R|$ matrix elements of each $\eta^i_B$ in a suitable basis, which is minimized by a block of size $|R| = 1$. This leaves $\mathcal{M}^2-2 \mathcal{M} + 2$ unfixed parameters in $\eta^i_B$. If the dimension of $\text{span} \{ \eta^i_B: i \in [n_{AB}] \}$ exceeds this bound, the Schmidt operators must be linearly dependent, a contradiction. Tighter bounds on the Schmidt number than those presented above can’t exist. The tightness of the lower bound is demonstrated by a metric with a Schmidt number of two and no local observables. Such a metric can be constructed on a Hilbert space over two qubits, $$\eta_{\text{min}} = (\mathbbm{1} + \beta \sigma_x) \otimes (\mathbbm{1} + \beta \sigma_x) + \beta^2 \sigma_Y \otimes \sigma_Y,$$ where $\beta \in \mathbbm{R}$ is chosen sufficiently small so $\eta_{\text{min}}$ is positive definite, and $\sigma_{x,y}$ are Pauli matrices. A brief calculation shows \[block metrics\] has no solutions, thus, this metric has no observables of the form $O = \mathbbm{1} \otimes O_B$ or $O = O_A \otimes \mathbbm{1}$. The tightness of the upper bound is demonstrated by a construction of a metric with Schmidt number $n_{AB} = (\min\{|A|^2,|B|^2\}-1)^2+1$ and local observables. Defining the orthonormal set of matrices $A^{ij}$ such that $A^{ij}_{kl} = \delta_{ik} \delta_{jl}$, one such metric is $$\eta_{\text{max}} = \alpha \mathbbm{1} + \sum_{ij} \left(\begin{matrix} 0 & 0 \\ 0 & A^{ij} \end{matrix} \right)_A \otimes \left(\begin{matrix} 0 & 0 \\ 0 & A^{ij} \end{matrix} \right)_B,$$ where $\alpha > 0$ is chosen to be sufficiently large so that $\eta_{\text{max}}$ is positive definite. Note $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \otimes \mathbbm{1}$ is an observable with respect to this metric. Local Observables in Free Fermions {#free fermion locality} ---------------------------------- Applying \[quasilocal\_theorem\] to a generic many-body problem requires simultaneously reducing matrices whose dimensions scale exponentially in the size of the corresponding subsystems, themselves derived from an operator which scales exponentially with the lattice size. However, certain aspects of locality for free fermions can be found with only polynomial computations. As with other polynomial-time calculations for free fermions, the technique is to reduce the problem into a first quantized setting. To quantify locality through observables in a first quantized setting, a correspondance between observables associated to the total metric and observables associated to the reduced metric is desired. Explicitly, $o$ is an observable in a quasi-Hermitian theory with metric $M$, $$M o = o^\dag M, \label{reduced observables}$$ if and only if $$O = \sum_{ij} o_{i j} a^\dag_i a_j \label{proj_obs}$$ is a quasi-Hermitian observable with respect to a metric $\eta$ which reduces to $M$ via \[reduced metric to metric\]. The subclass of operators of the form \[proj\_obs\] which are Bravyi-Kitaev local to a subsystem $A \subset [n]$ are simply those satisfying $o_{ij} = 0$ if $i \in A'$ or $j \in A'$, where $A' = [n]-A$ denotes the complement of $A$. Bravyi-Kitaev extensively local observables satisfy the additional constraint that for all $i \in A$, there exists $j \in A$ such that either $o_{ij} \neq 0$ or $o_{ji} \neq 0$. Let’s refer to such matrices $o$ as extensively local reduced observables. The existence of extensively local reduced observables turns out to be necessary for the existence of extensively local observables extent in the theory of free fermions, as will be shown shortly. Before proving this result, some elaborations on the extensively local reduced observables will be presented. Since extensively local reduced observables are block matrices, let’s define some notation relating to block decompositions of matrices. Let $M^{AB}$ denote the block of matrix elements $M_{ij}$ with $i \in A, j \in B$. In particular (rearranging columns and rows in $M$ as necessary), $$M = \begin{pmatrix} M^{AA} & M^{A A'} \\ M^{A' A} & M^{A' A'} \end{pmatrix},$$ and $M^{\{i\} [n]}$ denotes the $i^\text{th}$ row of $M$. When necessary, the matrix elements $M^{AB}$ will be considered as an operator on $\text{span} \{e_i:i\in B\}$. In addition, let $K(A) = \dim \ker M^{A' A}$, and let $\{w^\mu | \mu \in [K(A)]\}$ denote a basis of $\ker M^{A' A}$. A brief examination of \[reduced observables\] shows that the most general local reduced observable is a matrix of the form $$o_{ij} = \begin{cases} \sum\limits_{\mu,\nu \in [K(A)]} \alpha_{\mu \nu} \left( w^\mu {w^\nu}^\dag M^{AA} \right)_{ij} & \text{if}\, i, j \in A \,\\ 0 & \text{otherwise} \end{cases},$$ where $\alpha_{\mu \nu} = \alpha^*_{\nu \mu} \in \mathbbm{C}$. Note $K(A)$ is a measure of how many local observables are in subsystem $A$. The above statements relating to $\ker M^{A' A}$ are readily generalized to the case of observables in the full theory of free fermions, as demonstrated in the following theorem: \[polynomial theorem\] Bravi-Kitaev extensively local observables in subsystem $A$ which are quasi-Hermitian with respect to the metric of \[reduced metric to metric\] exist if and only if $$K(A) := \dim \ker M^{A' A} > \dim \ker M^{S' S} \label{Kernel equation}$$ for all proper subsets $S \subset A$. In addition, an operator, $O$, is a Bravyi-Kitaev local observable if and only if it can be expressed of the form (using the notation of \[generalCAR\]) $$O = \sum_{{\tiny S_1, S_2}} O_{S_1 S_2} \left(\prod_{\mu \in S_1} a^\dag(w^\mu)\right) \left( \prod_{\nu \in S_2} a(M^{AA} w^\nu) \right)\label{GeneralBKQH},$$ where $S_1, S_2 \in \mathbbm{P}([K(A)])$, $O_{S_1 S_2} = O^*_{S_2 S_1}$, $O_{S_1 S_2} = 0$ when $|S_1|+|S_2| \equiv 1 \mod 2$, and $\{w^\mu|\mu \in [K(A)] \}$ is a basis of $\ker M^{A' A}$. Let $O_A$ denote a nonzero operator in $\text{End}(\mathcal{H}_{\mathbbm{P}([n])})$ which is Bravyi-Kitaev local to subsystem $A$. Using \[OperatorSpan\], for every $O_A$, there exists linearly independent sets of vectors $\{f^\mu| \mu \in [\mathcal{F}]\}, \{g^\nu|\nu \in [\mathcal{[G]}]\}\subset \mathbbm{C}^n$ satisfying $f^\mu,g^\nu \in \text{span}\{e_i:i\in A\}$ such that $$\begin{aligned} O_A &= \sum_{S_1 \in \mathbbm{P}([\mathcal{F}])}\sum_{S_2 \in \mathbbm{P}(\mathcal{G})} O_{S_1 S_2} \times \nonumber \\ &\left(\prod_{\mu \in S_1} a^\dag(f^\mu)\right) \left( \prod_{\nu \in S_2} a(g^\nu) \right) , \label{sum}\end{aligned}$$ where $O_{S_1 S_2} \in \mathbbm{C}$. Without loss of generality, $f^\mu$ and $g^\nu$ can be chosen such that every such vector appears in the sum in \[sum\] at least once. The quasi-Hermiticity condition, \[operator\], applied to $O_A$ implies $$\begin{aligned} O^\dag &= \eta O \eta^{-1} \\ &= \sum_{S_1 \in \mathbbm{P}([\mathcal{F}])}\sum_{S_2 \in \mathbbm{P}(\mathcal{[G]})} O_{S_1 S_2} \times \nonumber \\ &\left( \prod_{\mu \in S_1} a^\dag(M f^\mu) \right) \left( \prod_{\nu \in S_2} a(M^{-1} g^\nu) \right).\end{aligned}$$ Observing that $O^\dag$ is also Bravyi-Kitaev local to $A$ results in the following vector identities $$\begin{aligned} f^\mu &\in \ker M^{A' A},\\ g^\mu &\in \ker {M^{-1}}^{A' A}.\end{aligned}$$ Note $2\times 2$ matrix inversion results in the following kernel identity, $$\begin{aligned} \ker {M^{-1}}^{A' A} = M^{A A}( \ker M^{A' A}).\end{aligned}$$ Thus, expressing $f,g$ in terms of a basis $\{w^\mu|\mu\in [K(A)]\}$ of $\ker M^{A' A}$, the operator $O_A$ can be re-expressed in the form of \[GeneralBKQH\]. The constraint $O_{S_1 S_2} = O^*_{S_2 S_1}$ follows from demanding quasi-Hermiticity. Importantly, note that extensively local reduced observables exist if and only if $K(A) > K(S)$ for all subsets $S \subset A$. Intuitively, the vectors in $\ker M^{S' S}$ correspond to observables local to $S$, so vectors in $\ker M^{A' A}$ which do not belong to any subset $\ker M^{S' S}$ can not correspond to an observable local to any subset $S$, thus, their corresponding observables are extensively local to $A$. Due to the potentially nonlocal string of $Z$ factors in the Jordan Wigner transform, an observable of the form \[proj\_obs\] is not necessarily local in the sense given by a tensor product structure, so \[polynomial theorem\] is not simple to generalize to the case where locality in free fermion theories is defined via a Jordan-Wigner transform. However, in the case when the subsystem is *connected*, that is, for every $i, j \in A$ and every integer $k$ satisfying $p(i)<p(k)<p(j)$, $k \in A$, the inserted $Z$ factors in \[proj\_obs\] are local, so such a Bravyi-Kitaev local observable is additionally local under the Jordan-Wigner transform defined by the map $p$. Note for every subsystem $A$ containing Bravyi-Kitaev local observables, there exists a $p$ such that every observable in $A$ is also local with respect to a Jordan-Wigner transform. The validity of the \[polynomial theorem\] in non-connected subsystems and the context of a Jordan-Wigner transform will not be commented on in this report. Notice that altering the diagonal entries of the metric bears no impact on the existence of observables local to a subsystem, since the diagonal entries never contribute to $M^{A' A}$. Simple examples of reduced metrics with an analytical understanding of locality are presented below. Suppose the reduced metric block reduces to a set $S$, so $$M_{ij} = 0 \,\,\,\,\, \forall i \in S, \, \forall j \notin S. \label{M block}$$ Then there exists an observable which is both extensively local with respect to any Jordan-Wigner transform, $\iota_p$, and Bravyi-Kitaev local extensively, over subsystem $S$, $$\hat{n}_S = \sum_{i \in S} a^\dag_i a_i. \label{how many particles}$$ This observable is not quasi-hermitian if \[M block\] doesn’t hold in $S$. In particular, this implies that diagonal reduced metrics have observables in every subsystem. A special case of reduced metrics are those which reduce to $1\times 1$ and $2\times 2$ blocks. Equivalently, there exists an *associated involution*, $f:[n]\rightarrow[n], f \circ f = 1$, such that $M_{i j} \neq 0 \, \Leftrightarrow \, i = f(j) \, \text{or} \, i = j$. These reduced metrics are special since the existence of nontrivial local observables can be read off directly from $f$: \[Involution theorem\] Given a reduced metric which decomposes into $1 \times 1$ and $2 \times 2$ blocks, extensively local observables exist in subsystems, $A \subset [n]$, if and only if the subsystem’s image under the reduced metric’s associated involution, $f$, is itself $f(A) = A$. In addition, the most general Bravyi-Kitaev local observable in this case is $$\begin{aligned} O = \sum_{{\tiny S_1,S_2 \in \mathbbm{P}([A])}} O_{S_1 S_2} \left(\prod_{i \in S_1} a^\dag(e_i)\right) \left( \prod_{j \in S_2} a(M^{AA} e_j) \right)\label{GeneralBKQH},\end{aligned}$$ where $e_i \in \mathbbm{C}^n$ is defined in \[CnBasis\] and $O_{S_1 S_2} = O^*_{S_2 S_1}$. The construction of extensively local observables in subsystems closed under $f$ follows from $M^{A' A} = 0$, so that $\ker M^{A' A} = \text{span}\{e_i :i \in A\}$. Given a site $i \in A$ whose dual satisfies $f(i) \notin A$, and a vector $v \in \ker M^{A'A}$, the equation $M^{\{f(i)\} A} v = 0$ immediately implies $v_i = 0$, and the involution symmetry of $M$ implies $(M^{AA} v)_i = 0$. As a consequence, any observable of the form \[GeneralBKQH\] is local to the subsystem $S = A-\{i\}$, and therefore is not extensive. The special case of observables localized at a single site is quite simple to analyse. \[single site theorem\] Quasi-Hermitian observables, with respect to the metric of \[reduced metric to metric\], which are extensively local to a single site, $i$, exist if and only if the reduced metric is a block matrix, $M_{ij} = M_i \delta_{ij}$. If $M$ block reduces, $\hat{n}_{\{i\}}$ is an observable. If $M$ doesn’t block reduce, $\ker M^{A' A} = \emptyset$, and there are no observables. ### Application to toy models {#ApplicationOfTheorems} The theorems of the last section are readily applied to the toy models introduced in \[toy model\]. #### Nearest neighbor impurities ($n=2m$) Corollary  is quite strong in the case of nearest neighbour impurities, $n = 2m$, where the metric of \[badass metric\] block decomposes into Parity sectors. Thus, for the metric of \[badass metric\] with either $\text{Im} \gamma \neq 0$ or $\beta \neq 0$, *extensively local observables exist in and only in Parity symmetric subsystems.* In the case of $\text{Im} \gamma = \beta = 0$, there are extensively local observables in *every* subsystem, since the metric is diagonal in this case. It appears the off-diagonal elements of the Hamiltonian are irrelevant in determing which subsystems contain local observables. #### Farthest Impurities $(m = 1, \text{Im} \gamma \neq 0)$ This section will assume the choice \[not positive\] for the metric. This metric is only positive definite on a portion of $\mathcal{PT}$-unbroken region, demonstrated in [@PTModels], so it’s unclear whether there exists a metric with the same properties concerning the existence of local observables in the case where $\gamma$ is not in this region. Theorem  mandates calculating kernels of blocks of the reduced metric, $\ker M^{A' A}$. For this model, the existence of observables local to a subsystem is related to whether the subsystem is connected. Some related notation is defined in the following paragraph: Consider the graph $G_A = (A, E_A)$ with vertices $A$ and edges $E_A = \{(i, i+1): i, i+1 \in A\}$. Let $\mathfrak{C}_A$ denote the set of connected components of $G_A$. A distance between components, $d_A:\mathfrak{C}_A \times \mathfrak{C}_A: \rightarrow \mathbbm{R}$, is defined as $$\begin{aligned} d_A(C_1, C_2) = \min \left\{d(i,j):i \in C_1, j \in C_2 \right\},\end{aligned}$$ where $d$ is the geodesic distance in $G_{[n]}$. Intuitively, $d_A$ measures the number of sites between $C_1$ and $C_2$. Finally, denote the *leftmost*, $C_L$, and *rightmost*, $C_R$, or collectively *edge* components of $G_A$ to be the connected components containing $\min A$ and $\max A$ respectively. Lastly, define $$\begin{aligned} A_{<i} &= \{k<i:i \in A\}, \\ A_{>i} &= \{k>i:i\in A\}, \label{A<} \\ v_{<i} &= \sum_{j<i} v_j e_j, \\ v_{>i} &= \sum_{j>i} v_j e_j, \label{v<}\end{aligned}$$ where the sums are set to zero if they sum over an empty set. When $\gamma^* \gamma = 1$, the extensively local observables are comparatively simple to construct. \[UnitDiskLocality\] For quasi-Hermitian theories with respect to the metric of \[not positive\] and $\gamma^* \gamma = 1$, extensively local observables in subsystem $A$ exist if and only if either $G_A$ contains no connected components with exactly one site or $A = \{1,n\} \cup B$, where $B$ contains no connected components with exactly one site. Suppose $A$ contains a connected component with exactly one site, $i$. Suppose there exists $v \in K(A)$. Assuming $i \notin \{1,n\}$, the kernel equation, $\sum_j M^{A' A}_{ij} v_j = 0$, for indices $i-1,i+1$ is $$\begin{aligned} \begin{pmatrix} \gamma^{-2} & -1 & {\gamma^*}^{2} \\ 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} M^{\{i+1\} A_{<i}}\, v_{<i} \\ i \text{Im} \gamma\, v_i \\ M^{\{i+1\} A_{>i}}\, v_{>i} \end{pmatrix} = 0.\end{aligned}$$ For the second case of the proposition, suppose $i = 1 \in A$, but $2, n \notin A$, $n>3$ (the cases $n = 2,3$ are trivial and follow from \[single site theorem\]). The other case, $i = n \in A, 1, n-1 \notin A$, follows from $\mathcal{PT}-$symmetry. Then the kernel equations at sites $2, n$ are $$\begin{pmatrix} 1 & 1 \\ \gamma^{n-2} & -(\gamma^{*})^{4-n} \end{pmatrix} \begin{pmatrix} i \text{Im} \gamma\, v_1 \\ M^{\{2\} A_{>1}}\, v_{>1} \end{pmatrix} = 0.$$ In all cases mentioned above, since $\gamma^* \gamma = 1$, these equations imply $v_i = 0$. In addition, note $M^{\{i\} A-\{i\}} v= 0$ since $$\begin{aligned} M^{\{i\}\, A-\{i\}} v = \gamma^{-1} M^{\{i+1\} A_{<i}}\, v_{<i} + \gamma^{*} M^{\{i+1\} A_{>i}}\, v_{>i} = 0.\end{aligned}$$ Thus, $M^{S' S} v = 0$ for $S = A - \{i\}$. Thus, by \[polynomial theorem\], if either $A$ has a single-site connected component between the endpoints of the lattice, or exactly one of $1,n$ is in $A$, no extensively local observables exist in $A$. The converse follows from explicit construction of extensively local observables. If $C$ is a connected subset of $A$, then $\ker M^{C' C} = ((1, \gamma^*, \dots {\gamma^*}^{|C|})^\dag)^\perp$, so $K(C) = |C|-1$. If $C = \{1, n\}$, then $\left(1, \gamma^{n-3} \right)^\intercal \in \ker M^{C' C}$. As a consequence, extensively local observables exist in every connected subset of $[n]$, as well as the subset $\{1,n\}$. Taking suitable linear combinations of the above vectors demonstrates the existence of extensively local observables in the subsystem $A = \{1,n\} \cup B$, where $B$ is a union of connected components with at least two sites. The remainder of the section is devoted to the case $\gamma^* \gamma \neq 1$. The final result is summarized in proposition . Some examples of subsystems containing local observables are shown in \[ExampleRegions\]. To simplify the analysis, we start with the special case where the subsystem is connected. \[ConnectedRegions\] For a quasi-Hermitian model with the metric of \[not positive\] for $\gamma^* \gamma \neq 1$, every subset $C \subset [n]$ such that $G_C$ is connected with at least three sites contains Bravyi-Kitaev extensively local observables. In addition, $\{1,2\}$ and $\{n-1,n\}$ contain Bravyi-Kitaev extensively local observables. No other connected subgraph contains Bravyi-Kitaev local observables. By \[single site theorem\], if $|C| = 1$, there are no nontrivial local observables. In the case of connected subsystems, $\dim \ker M^{C' C}$ is easy to find, since the rows labelled by indices to the left of $C$ are all multiples of each other, and similarly for indices to the right of $C$. Note the set of rows of $M^{C' C}$ to the left or right of $C$ doesn’t exist if either $1 \in C$ or $n \in C$, so in these cases, $K(C)$ increases by one. Thus, $$\begin{aligned} \ker M^{C' C} &= \text{span} \left\{\begin{array}{l} 1_{C'}(\{1\})(1, \gamma^*, \dots, {\gamma^*}^{|C|})^\dag, \\1_{C'}(\{n\})({\gamma}^{|C|}, \dots, \gamma, 1)^\dag \end{array} \right\}^\perp,\\ K(C) &= |C|-2+1_A(\{1\})+1_A(\{n\}),\end{aligned}$$ where $1_S:\mathbbm{P([n])}\rightarrow \mathbbm{P([n])}$ is the indicator function, $$\begin{aligned} 1_S(T) = 1-\delta_{T\cap S \,\emptyset}.\end{aligned}$$ $K(C)$ is nonzero if and only if $|C| \geq 3$, $C = \{1,2\}$, or $C = \{n-1,n\}$, proving that these subsystems are the only connected subsystems with local observables. Note that removing any number of sites from $C$ necessarily reduces $K(C)$, so the subsystem $C$ also contains extensively local observables. If a subsystem is a union of disjoint connected subsystems of the form above, there are observables which are extensively local to said subsystem. It only remains to check subsystems which have an isolated site or pair of sites. \[conds\] Bravyi-Kitaev observables which are extensively local to a subsystem, $A \subseteq [n]$, and quasi-Hermitian under the metric of \[not positive\] exist only if the following conditions on its connected components, $A = \cup \mathfrak{C}_A$, are met: 1. If $A$ contains a connected component, $C \in \mathfrak{C}_A$ with $|C|\leq2$, and $C \notin \{\{1,2\},\, \{n-1,n\}\}$, then $A$ must contain at least one more connected component, so $A \neq C$.\ 2. For every connected component with a single site, $C = \{i\}$, then $i-2, i+2 \in A$ when $i-2, i+2 \in [n]$. Intuitively, this connected component is separated from the rest of the substem by at most one site from both the left and the right.\ 3. The connected components with two sites, $|C| = 2$, satisfy $\min d_A(C, \mathfrak{C}_A) \leq 2,$ unless $C = \{1,2\}$ or $C = \{n-1,n\}$. Intuitively, this connected component is separated from the rest of the subsystem by at most one sites from either the left or the right.\ 4. The edge components may have a single site only if that component is $C_L = \{1\}$ or $C_R = \{n\}$. Otherwise, $|C_{L,R}| > 1$. The set of all subsets $A \subseteq [n]$ satisfying the above criteria will be referred to as $\mathcal{R}$. 1. Trivial consequence of lemma  2. Assume $\{i\}$ is a connected component of $A$. Consider the kernel conditions $\left(M^{A' A} v\right)_j = 0$ for the following choices of $j \in \{i-2,i-1,i+1,i+2\}\cap [n]$, using the notation of \[A&lt;,v&lt;\], $$\begin{aligned} V^\intercal &:= \left(M^{\{i-1\} A_{<i}}\, v_{<i}, i \text{Im} \gamma\, v_i, M^{\{i-1\} A_{>i}}\, v_{>i} \right)\\ V &\in X := \text{span}\left\{\begin{array}{l}\arraycolsep=1.4pt\def\arraystretch{1.8} 1_{A'}(\{i-2\})\left(\gamma^{-1},-{\gamma}^*,\gamma^*\right)^\dag\\1_{A'}(\{i-1\})\left(1, -1, 1\right)^\dag,\\ 1_{A'}(\{i+1\})\left(\gamma^2, 1, (\gamma^{*})^{-2} \right)^\dag\\ 1_{A'}(\{i+2\})\left(\gamma^3,\gamma,(\gamma^*)^{-3} \right)^\dag\end{array}\right\}^\perp\end{aligned}$$ For a nontrivial solution $V$ to exist, $\dim X \leq 2$, which only happens if either $\gamma^* \gamma = 1$ or $i-2,i+2\in A$ when $i-2,i+2\in [n]$. Thus, when $\gamma^* \gamma \neq 1$, since $V = 0$, the vector $v_i$ is also in the kernel of $M^{S' S}$, where $S = A - \{i\}$. Therefore, there is no observable extensively local to the site $i$ in this case.\ 3. Suppose $i, i+1 \in A$, and assume $n>4$, since $n=4$ reduces to \[ConnectedRegions\]. Consider the kernel conditions $(M^{A'A}v)_j = 0$ for the choices of $j\in \{i-2, i-1, i+2, i+3\}\cap [n]$: $$\begin{aligned} V^\intercal &:= \left(M^{\{i-2\} A_{<i}} v_{A_{<i}}, i \text{Im} \gamma v_i, i \text{Im} \gamma v_{i+1}, M^{\{i-2\} A_{>i+1}} v_{A_{>i+1}}\right), \\ V \in X &:=\text{span} \left\{ \begin{array}{l} 1_{A'}(\{i-2\})\left(1, -\gamma^*, -{\gamma^*}^2, 1\right)^\dag \\ 1_{A'}(\{i-1\})\left(\gamma, -1, -\gamma^{*}, {\gamma^{*}}^{-1}\right)^\dag \\ 1_{A'}(\{i+2\})\left(\gamma^4, \gamma, 1, {\gamma^{*}}^{-4}\right)^\dag \\ 1_{A'}(\{i+3\})\left(\gamma^5, \gamma^2, \gamma, {\gamma^{*}}^{-5}\right)^\dag \end{array} \right\}^\perp. \label{|C|=2}\end{aligned}$$ For $V$ to be nontrivial, $\dim X \leq 3$. This only happens when $|\gamma| = 1$, or $i-2\in A$ when $i-2 \in [n]$, or $i+3 \in A$ when $i+3 \in [n]$. 4. Suppose $i$ is the leftmost site, $i-1 \in [n]$, and $i+1 \in A'$. Then $$\begin{aligned} \begin{pmatrix} -1 & 1 \\ 1 & {\gamma^{*}}^{-2} \end{pmatrix} \begin{pmatrix} i \text{Im} \gamma v_i\\ M^{\{i-1\} A_{>i}} v_{>i} \end{pmatrix} = 0.\end{aligned}$$ Thus, $v_i = M^{\{i-1\} A_{>i}} v_{>i} = 0$, so $M^{\{i\} A} v = 0$. Thus, for every $v \in \ker A$, $v_{>i} \in \ker{M^{S' S}}$ with $S = A - \{i\}$, so by \[polynomial theorem\], there are no extensively local observables in $A$. The case where $i$ is the rightmost site follows from $\mathcal{PT}$ symmetry. The remainder of this section is dedicated to showing that subsystems $A$ satisfying the enumerated criteria of lemma , $A \in \mathcal{R}$, do contain extensively local observables. \[m=1Final\] Bravyi-Kitaev extensively local observables to a subsystem $A=\cup \mathfrak{C}_A \subseteq [n]$, which are quasi-Hermitian under the metric of \[not positive\], exist if and only if the conditions of lemma  are met. Define $$\begin{aligned} \mathfrak{C}_k &= \{C \in \mathfrak{C}_A: |C| \leq k\}\\ \mathcal{R}_k &= \{A \in \mathcal{R}: \mathfrak{C}_A = \mathfrak{C}_k\}.\end{aligned}$$ Intuitively, $\mathcal{R}_k$ is the set of all subsystems $A$ whose connected components contain at most $k$ sites. Note $\mathcal{R}_{n} = \mathcal{R}$. We’ll use an inductive argument to prove that for each $\mathcal{R}_k$, every element contains extensively local observables. Consider first the base case where $A \subset \mathcal{R}_2$, where all connected components have cardinality at most 2. Note $|\mathfrak{C}_2|-|\mathfrak{C}_1|$ denotes the number of connected components in $\mathfrak{C}_A$ with cardinality exactly two. A simple argument by counting the number of linearly independent rows in $M^{A' A}$ results in the identity $$\begin{aligned} K(A) \geq |\mathfrak{C}_2|-|\mathfrak{C}_1| -1 + 1_A(\{1\}) + 1_A(\{n\}).\end{aligned}$$ Thus, local observables exist in $A$. The following proves by contradiction that the observables constructed above are extensively local. Suppose such an observable is not extensive, but is local to $S \subset A$ with cardinality at most $k$. This subset doesn’t satisfy the criteria of lemma , so the observable must be local to a subset $S$ with cardinality at most $k-1$. Repeating this argument inductively until $k = 1$ would imply the existence of a observable local to a single site, which contradicts corollary . To prove the inductive hypothesis, we’ll demonstrate that for every $A \in \mathcal{R}_k$, with $k>2$, there exists a decomposition $A = \cup_i A_i$ such that either $A_i \in \mathcal{R}_{k-1}$, or $A_i$ is a union of connected components, $A_i = \cup C_{k,2} \subseteq [n]$ with $|C_{k,2}| \geq 3$. Since $A_i$ either is assumed to have extensively local observables in the first case, or known to have extensively local observables by lemma  in the latter case, $A$ must have extensively local observables. Suppose $A$ has $l$ connected components $C \in \mathfrak{C}_A$ with cardinality $|C| = k$. We’ll express $A$ as a union of the form $A = B_1 \cup B_2 \cup B_3$ such that $B_1, B_2 \in \mathcal{R}_k$, $B_3$ is connected with cardinality $|A_3|>2$, and $B_1$ and $B_2$ combined have $l-1$ connected components with cardinality $|C| = k$. An inductive argument on $l$ thus constructs the decomposition $A_i$ from the previous paragraph. Pick one connected component $C \in \mathfrak{C}_A$ with cardinality $|C| = k$. The construction of $B_1, B_2, B_3$ splits into four cases: If $A = C$, the construction $B_1 = B_2 = \emptyset, B_3 = C$ is trivial. If $\min d(C, \mathfrak{C}_A)\geq 3$, then the sets $B_1 = A-C, B_2 = \emptyset$ must satisfy the axoims of lemma , and $B_3 = C$ is of the desired form. If there is a unique set $C_1$ such that $d(C, C_1) = 2$, set $B_3 = C$, $B_2 = \emptyset$. In the case where $\max C_1 < \min C$, set $B_1 = A_{<\min C + 2}$, else, set $B_1 = A_{>\max C - 2}$. In the final case, there are two sets $C_1, C_2$ such that $d(C,C_1) = d(C,C_2) = 2$. Without loss of generality, assume $\max C_1 < \min C < \max C < \min C_2$. Set $B_3 = C$, $B_1 = A_{<\min C + 2}$, $B_2 = A_{>\max C - 2}$. Symmetry Properties of Local Observables {#Symmetric Ham Locality} ---------------------------------------- Observe that in both toy models, if a subsystem has local observables, its parity dual also has local observables. This statement is also true for any theory with a $\mathcal{PT}-$symmetric metric, such as the common choice $\eta = \mathcal{PC}$ [@bender2002complex]. This is proven by explicit construction of an observable in the parity dual of a subsystem which is known to contain local observables. Explicitly, given a Hilbert space which factorizes $\mathcal{H} = \otimes_i \mathcal{H}_i$, a quasi-Hermitian observable, $O_A$, local to a collection of sites $A$, and supposing $\mathcal{PT}$ decomposes with the factorization of the Hilbert space, $\mathfrak{PT}:\text{End}(\mathcal{H}_i)\rightarrow \text{End}(\mathcal{H}_{\bar{i}}), \mathfrak{PT}(O) = \mathcal{PT} O \mathcal{PT}$, the following observable is local to $A_{\mathcal{PT}}= \{\bar{i}: i \in A\}$: $$\begin{aligned} O_{\bar{i}} &= \mathcal{PT} O_i \mathcal{PT} \\ \eta O_{\bar{i}} &= O^\dag_{\bar{i}} \eta.\end{aligned}$$ Results ======= This paper presents several theorems which construct local operator algebras in quasi-Hermitian theories. In addition, it applies them to several $\mathcal{PT}$-symmetric toy models of free fermions with nearest neighbor interactions. The first set of theorems, presented in \[GeneralTheorems\] apply when locality is defined by a tensor product structure, and relates the existence of local observables to the Schmidt decomposition of the metric. Numerical application of this theorem to a generic many body problem in practice requires some care, as the number of matrix elements of the metric scales exponentially with lattice size. The second set of theorems, presented in \[free fermion locality\], applies to models of free fermions. Locality is defined directly from the canonical anti-commutation relations rather than through a tensor product structure. It seems that the structure of local observable algebras depends more strongly on the non-Hermiticity of the potential than the non-Hermiticity of the hopping amplitudes, and it’s more sensitive to the location of non-Hermitian impurities than their relative strength. When the non-Hermitian impurities are closest to each other, nontrivial observables exist if and only if the subsystem containing them is Parity symmetric. When the impurities are farthest from each other, the existence of local observables depends on certain connectivity properties of the subsystem, outlined and proven in the various propositions of \[ApplicationOfTheorems\]. For a precise statement of these connectivity properties see lemma  and proposition . Interestingly, in the case where the impurities are farthest from each other, the space of subsystems with local observables is broader in the special case $|\gamma| = |t|$, where $\gamma$ is the non-Hermitian potential and $t$ is the non-Hermitian hopping amplitude. We note this may be connected to a special spectral property of the Hamiltonian: when $\gamma = e^{i \theta}$, only one eigenvalue of $H$ depends on $\theta$ [@PTModels]. Lastly, the subsystems containing nontrivial local observables with respect to a $\mathcal{PT}-$symmetric inner product have a weak relation to $\mathcal{PT}$-symmetry, assuming $\mathcal{PT}$ admits a suitable tensor product factorization: If a subsystem contains observables, then its $\mathcal{PT}$ dual also contains observables. In particular, this result holds for theories constructed with a $\mathcal{CPT}$ inner product, and for non-degenerate Hamiltonians. Outlook ======= Due to the equivalence between a quasi-Hermitian theory in the Hilbert space with inner product $\braket{\cdot|\cdot}$ and a Hermitian theory in the Hilbert space with inner product $\braket{\cdot|\cdot}_\eta$, a natural question to ask is whether the generalized notion of locality discussed in this paper can be obtained without the use of a quasi-Hermitian description of a model. The discrepancy between the algebra of local operators and the algebra of physical observables stems from the distinction between the physical inner product, $\braket{\cdot|\cdot}_\eta$, and the inner product given by a tensor product structure. The author suggests that assuming these two inner products are the same is an unnecessarily restrictive assumption of quantum theory, and exploration of interesting nonlocal phenomena will follow from breaking this assumption. Quasi-Hermitian descriptions assist this process in cases where an understanding of the local degrees of freedom precedes an understanding of the dynamics. Reversing the roles of the Hamiltonian and locality suggests a procedure for starting with a physical inner product, and from there defining the tensor product structure and local degrees of freedom. In this sense, spacetime emerges from the fundamental degrees of freedom associated with $\braket{\cdot|\cdot}_\eta$. When the additional constraint that the Hamiltonian is local in the emergent degrees of freedom is applied, the choice of a tensor product structure is generically unique [@LocalFromSpec]. In addition to the aforementioned application of our interpretation of quasi-Hermitian theory to the emergence of spacetime and generalizing local quantum theory, we mention several natural extrapolations of the results and strategy of this work below: 1. A generalization of \[quasilocal\_theorem\] relating to the existence criteria of extensively local observables.\ 2. A discussion of local observables in non-Hermitian QFTs. Is it possible to find a theory with observables nonlocal in time with quasi-Hermiticity? This would bring this formalism one step closer to a bridge with quantum gravity.\ 3. A strategy for proving whether there exists a metric associated to a Hamiltonian which is compatible with local observables. Such a strategy may not apparent from the tensor product structure of $H$ alone, as $H_{\mathcal{PT}}$ is an example of a Hamiltonian with Schmidt numbers not identical with its metric.\ 4. Since quasi-Hermitian theories often emerge through renormalization schemes [@LeeModel; @LeeModelPT; @quantumRG; @blochFeschbach], it would be interesting to see how nonlocality emerges through an appropriate renormalization procedure.\ 5. What can be said about the locality properties of observables for a Hamiltonian which is not local in the sense of \[local Ham\], but is rather $k$-local for some $k>2$ [@LocalFromSpec]?\ 6. A discussion of the local observable algebras in the case where the metric is time-dependent. In this case, for unitarity, the generator of time-evolution is no longer an observable, but satisfies $$i \hbar \frac{d}{dt} \eta = H^\dag \eta - \eta H$$ instead [@timeDependentInnerProd].\ 7. The complete set of metrics associated to the first quantized $m=1$ Hamiltonian with uniform tunnelling amplitudes is known [@SSHMetric], how do more general choices of metrics change the existence properties of local observables?\ 8. An understanding of how *entanglment* of the metric operator affects properties of local observable algebras, where an entangled metric operator is defined in the same fashion as an entangled mixed state [@werner1989quantum].\ 9. A generalization \[polynomial theorem\] to metric operators compatible with models of fermions with pair creation and annihilation. Acknowledgements {#acknowledgements .unnumbered} ================ The author would like to thank Richard Cleve, Yogesh Joglekar, Yasha Neiman, Nic Shannon, Lee Smolin, and Neil Turok for insightful discussions. This research was supported in part by the Perimeter Institute. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada, and by the province of Ontario through the Ministry of Research and Innovation. This work was also funded and by a visiting research position at the Okinawa Institute for Science and Technology. [^1]: The definition of a local Hamiltonian is different from that of a local observable, and is reviewed in \[local Ham\]. [^2]: $\mathcal{PT}$-symmetry and quasi-Hermiticity are distinct for infinite dimensional Hilbert spaces [@PTnotQuasi] [^3]: $\text{End}(\mathcal{H})$ denotes the set of linear operators over $\mathcal{H}$ [^4]: The case $t = 0$ is trivial. In this case, if $\gamma \not\in \mathbbm{R}$, the model is unphysical due to a complex spectra, and for $\gamma \in \mathbbm{R}$, the Hamiltonian is Hermitian.

--- abstract: 'We examine statistical properties of a laser beam propagating in a turbulent medium. We prove that the intensity fluctuations at large propagation distances possess Gaussian probability density function and establish quantitative criteria for realizing the Gaussian statistics depending on the laser propagation distance, the laser beam waist, the laser frequency and the turbulence strength. We calculate explicitly the laser envelope pair correlation function and corrections to its higher order correlation functions breaking Gaussianity. We discuss also statistical properties of the brightest spots in the speckle pattern.' author: - 'Igor Kolokolov$^{1,2}$, Vladimir Lebedev$^{1,2}$, and Pavel M. Lushnikov$^{1,3}$' title: Statistical properties of the laser beam propagating in a turbulent medium --- Introduction ============ Propagation of either the acoustic beam or the electromagnetic beam in a turbulent medium is the classical problem which has been a subject of numerous investigations which especially intensified after the advent of lasers in 1960-ies. The shape of the beam wavefront is progressively disturbed with the increase of the propagation distance due to the turbulent fluctuations. These distortions are random due to the chaotic nature of turbulence. Therefore they should be described statistically. Below we refer specifically to the laser beam while assuming that the same theory can be applied for the acoustic beam. The early advances in beam propagation through turbulent medium outlined in Refs. [@TatarskiiBook1961; @TatarskiiBook1971; @Goodman; @StrohbehnBook1978] were focused on lower order statistical moments of the laser intensity. Among such moments often a scintillation index $\sigma_I^2\equiv \langle I^2\rangle/\langle I\rangle^2-1$ is used. The index $\sigma_I^2$ is the measure of the strength of the fluctuations of the irradiance $I$ (the laser beam intensity) at the target plane. Here and below we denote by $\langle \ldots \rangle$ an average over the ensemble of the atmospheric turbulence realizations or, equivalently, over time. At relatively small propagation distances, where irradiance fluctuation are small ($\sigma_I^2\ll1$), the classic perturbative approach well describes modification of the laser beam propagation due to turbulence [@TatarskiiBook1961; @TatarskiiBook1971]. Statistically averaged beam characteristics at larger distances in strong scintillation regimes ($\sigma_I^2\gg 1 $) were addressed through semi-heuristic theory [@AndrewsPhillipsBook1998]. At such distances, the laser beam disintegrates into speckles with lower order statistical moments providing only very limited information about structure of the intensity fluctuations. It was found in Ref. [@LachinovaVorontsovJOpt2016] that a significant fraction of deviation between the theoretical value of $\sigma_I^2$ [@AndrewsPhillipsBook1998] and simulations is due to rare large fluctuations of laser beam intensity. Such giant fluctuations were also observed in numerical experiments [@VorontsovEtAlAMOSConf2010; @VorontsovEtAlAMOSConf2011]. The work [@LushnikovVladimirovaJETP2018] studied the structure of large fluctuations and proposed to use them for the efficient delivery of laser energy over long distances by triggering the pulse laser operations only during the times of such rare fluctuations. It was demonstrated in Ref. [@LushnikovVladimirovaJETP2018] that after $7$ km propagation of the laser beam with the initial beam waist $1.5$ cm in typical atmospheric conditions, about $0.1\%$ atmospheric realizations carry $\gtrsim 28\%$ of initial power in a single giant fluctuation (a single speckle) of the laser intensity. In this paper we address the statistical properties of the laser intensity fluctuations and identify analytically the spatial profiles of the intense laser speckles. We consider a propagation of the initially Gaussian beam at large distances corresponding to $\sigma_I^2\gg 1 $. We assume that the typical transverse size of speckles is much smaller than the Gaussian beam width which is also ensured at large enough distances of propagation through turbulent medium. We use the ladder sequence of diagrams of the stochastic perturbation theory to show that the probability density function (PDF) of laser intensity fluctuations at such distances is well approximated by the Gaussian stochastic process. The properties of such stochastic process at each propagation distance $z$ is fully determined by the pair correlation function of the laser intensity fluctuation which has the explicit expression in the integral form. The spatial structure of large fluctuations (the bright spots of the laser intensity) of such stochastic process has the form of the transverse correlation function of the Gaussian process with the transverse width scaling as $\propto z^{-3/5}$ for the Kolmogorov-Obukhov spectrum of the atmospheric turbulence [@TatarskiiBook1961; @TatarskiiBook1971]. We also found non-Gaussian corrections to the higher order correlation functions and established that these corrections decrease with the propagation distance. The plan of the paper is the following. In Section \[sec:basiceq\] we introduce basic equations in the physical variables, define the fluctuations of the refractive index (Section \[sec:Fluctuationsn1\]) and discuss the characteristic propagation lengths which controls the properties of laser beam fluctuations (Section \[sec:Spatialscales\]). Section \[sec:dimensionless\] introduces scaled dimensionless variables. Section \[sec:Gaussianfluctuations\] provides a reduction of the refractive index fluctuations to the Gaussian stochastic process with the power law pair correlation function. Section \[sec:Parametersregions\] discusses the parameters and different spatial regions in the dimensionless variables. Section \[sec:corr\] analyzes the pair, fourth and higher order correlation functions as well as address the statistic of the intensity fluctuations. Section \[sec:Effective action\] develops the ladder approximation as well as discusses the corrections beyond the ladder approximation identifying the region of the applicability of the Poisson statistics. Section \[sec:Conclusion\] summarize the results and discuss their applicability conditions. Appendices provide the details of the ladder approximation, corrections beyond it and the calculations of the statistics of the large fluctuations (bright spots) of the laser intensity. Basic equations {#sec:basiceq} =============== A propagation of a monochromatic beam with a single polarization through turbulent media is described by the linear Schrödinger equation (LSE) (also called as Leontovich equation) [@TatarskiiBook1961; @TatarskiiBook1971; @VlasovPetrishchevTalanovRdiofiz1971] for the spatiotemporal envelope of the electric field $$\begin{aligned} \label{nlsdimensionall} {{i}}\frac{\partial }{\partial Z}\Psi+\frac{1}{2k_{0}}\nabla_\perp^2\Psi +k _{0}n_1\Psi=0, \end{aligned}$$ where all quantities are functions of ${\bm R}_\perp,Z,t$. Here the beam is aligned along $Z$-axis, ${\bm R}_\perp\equiv (X,Y)$ are the transverse coordinates, $t$ is the time, $\nabla_\perp\equiv(\partial _X, {\partial _Y}),$ $k_0=2\pi n_0/\lambda_0$ is the wavevector in the medium, $\lambda_0$ is the wavelength in the vacuum, $n=n_0+n_1$ is the linear index of refraction with the average value $n_0$ and the fluctuation contribution $n_1$, with zero average $\langle n_1 \rangle=0$. The beam intensity is expressed as $I=|\Psi|^2$. We assume that the inverse time of the laser beam propagation is much larger than characteristic rates of the turbulent motions. Then corrections related to the dependence of $n_1$ on $t$ are negligible and we can operate in terms of flash realizations of the refractive index $n_1$. Statistics of such flash realizations can be characterized by simultaneous correlation functions of $n_1$. Thus $t$ does not explicitly enter Eq. [nlsdimensionall]{}, but serves as the parameter distinguishing different atmospheric realizations, so below we omit $t$ in arguments of all functions. The neglect of the temporal derivative (that derivative results from the chromatic dispersion) is justified for laser beams with typical durations longer than a few nanoseconds. Linear absorbtion (results in exponential decay of the laser intensity with propagation distance) is straightforward to include into Eq. [nlsdimensionall]{}. Kerr nonlinearity can be also added to Eq. [nlsdimensionall]{} resulting in nonlinear Schrödinger Eq. which describes the catastrophic self-focusing (collapse) of laser beam for laser powers $P$ above critical power $P_c$ ($P_c\sim3$GW for $\lambda_0= 1064\text{nm)}$ [@VlasovPetrishchevTalanovRdiofiz1971; @ZakharovJETP1972; @LushnikovDyachenkoVladimirovaNLSloglogPRA2013] and multiple filamentation for $P\gg P_c$ [@LushnikovVladimirovaOptLett2010]. At distances well below the nonlinear length, one can consider Kerr nonlinearity as perturbation (see, e.g., Ref. [@VasevaFedorukRubenchikTuritsyn]) combining it with the effect of atmospheric turbulence. Such additions are beyond the scope of this paper. We consider propagation of the beam produced by a laser located at $Z=0$. We assume that at the laser output the beam has a Gaussian shape $$\label{Gaussianbeam} \Psi_{in}=A\exp\left(- R_\perp^2/w_0^2\right),$$ where $w_0$ is the initial Gaussian beam waist, $A$ is the initial beam amplitude. The expression (\[Gaussianbeam\]) should be treated as the initial condition to the equation (\[nlsdimensionall\]) posed at $Z=0$. At small $Z$ one can neglect the term with $n_1$ in Eq. (\[nlsdimensionall\]). Then we find the explicit solution of Eq. (\[nlsdimensionall\]) with the initial condition (\[Gaussianbeam\]) given by the diffraction-limited Gaussian beam $$\Psi=\frac{w_0^2}{w_0^2+2iZ/k_0} \exp\left(-\frac{R_\perp^2}{w_0^2+2iZ/k_0}\right). \label{freesolution}$$ Note that at distances $Z\gg Z_R$, where $Z_R$ is the Rayleigh length $$Z_R=k _0w_0^2/2, \label{Rayleighlength}$$ the beam width can be estimated as $Z/(k_0w_0)\gg w_0$, whereas the phase varies on a much smaller transverse spatial scale $\sim \sqrt{Z/k_0}$. Fluctuations of the refractive index {#sec:Fluctuationsn1} ------------------------------------ The refractive index variation $n_{1}$ is proportional to the density fluctuation of the turbulent medium. Description of principal properties of the atmospheric turbulence can be found in monograph [@Monin]. Typically, $l_0$ is in the range of few mm or even smaller, while $L_0$ is ranging from many meters to kilometers. The main contribution to $n_{1}$ stems from the turbulent fluctuations with scales of the order of the integral length (the outer scale) of the turbulence $L_0$. We are however interested in atmospheric fluctuations on scales of the order of the laser beam width, that is assumed to be much smaller than $L_{0}$. In terms of Eq. [nlsdimensionall]{} such fluctuations on the scale $L_0$ result in the change of the phase of $\Psi$ which are nearly homogenous and do not affect the laser intensity. Besides, the beam width is assumed to be much larger than Kolmogorov scale (the inner scale of turbulence) $l_0$. Then $l_0$ does not significantly affect PDF of laser beam fluctuations. In this situation the turbulent fluctuations relevant for the problem belong to the inertial scale of turbulence where they possess definite scaling properties [@Frisch] and are both homogeneous and isotropic. Atmospheric fluctuations can be characterized by the structure function of the refractive index fluctuations that is the simultaneous average $\langle [n_1(\bm R_\perp,Z)-n_1({\bm 0},0)]^2\rangle$. Remind that the angular brackets $\langle \ldots \rangle$ mean averaging over realizations of $n_1$ or, equivalently, averaging over time. In the inertial range of turbulence the Kolmogorov-Obukhov law is valid $$\label{KolmogorovObukhov} \langle [n_1(\bm R_\perp,Z)-n_1({\bm 0},0)]^2\rangle =C_n^2 \rho^{2/3},$$ where $\rho=\sqrt{R_\perp^2+Z^2}$ and the factor $C_n^2$ characterizes strength of the turbulence. The expression (\[KolmogorovObukhov\]) is correct provided $l_0\ll \rho\ll L_0$. We now consider a little bit more general case not restricting to the Kolmogorov-Obukhov law [KolmogorovObukhov]{}. It is still natural to assume homogeneity and isotropy of the turbulence on scales smaller than its integral scale $L_0$. Then the structure function $\langle [n_1(\bm R_\perp,Z)-n_1({\bm 0},0)]^2\rangle$ depends solely on $\rho$, which is the absolute value of the separation between points. We assume the following general power law coordinate dependence on scales from the inertial interval: $$\label{KolmogorovObukhovgeneral} \langle [n_1(\bm R_\perp,Z)-n_1({\bm 0},0)]^2\rangle =C_n^2 \rho^{\mu},$$ where $\mu<1$ is the scaling exponent, characterizing the refractive index fluctuations. The particular case of the Kolmogorov-Obukhov law [KolmogorovObukhov]{} corresponds to $\mu=2/3$. Using Eq. [KolmogorovObukhovgeneral]{}, one can represent the pair correlation function of $n_1$ by the expansion in three dimensional Fourier harmonics as follows (see e.g. Refs. [@TatarskiiBook1961; @TatarskiiBook1971]) $$\begin{aligned} & \langle n_1(\bm R_\perp,Z)n_1({\bm 0},0)\rangle =\frac{{ C_n^2 \Gamma(\mu+2)}}{4\pi^2}\sin{\left (\frac{\pi\mu}{2} \right )} \nonumber\\ & \times\int {d^2 q\,dq_z} \exp(i\bm q \cdot\bm R_\perp+i q_z Z) (q^2+q_z^2)^{-3/2-\mu/2}, \label{paircorr2dimensional} \end{aligned}$$ where $q^2=q_x^2+q_y^2$. Here we assumed that the divergence of the integral at small $q,q_z$ is removed by a cutoff at the large scale $\rho\sim L_0$ which can be also taken care by considering the structure function [KolmogorovObukhovgeneral]{} instead of the pair correlation function. The propagation length $Z$ is assumed to be much larger than the beam width, i.e. one can use the paraxial approximation with the characteristic wavevector $q$ much larger than the characteristic component $q_z$. Therefore $q_z$ can be neglected in Eq. (\[paircorr2dimensional\]) in comparison with $q$. Then one obtains by replacing $q^2+q_z^2\to q^2$ that $$\begin{aligned} & \langle n_1(\bm R_\perp,Z)n_1({\bm 0},0)\rangle =\frac{{ C_n^2 \Gamma(\mu+2)}}{2\pi}\sin{\left (\frac{\pi\mu}{2} \right )} \nonumber\\ & \times\delta(Z) \int {d^2 q} \exp(i\bm q \cdot\bm R_\perp) q^{-3-\mu}. \label{paircorr2dimensional2} \end{aligned}$$ The main contribution to the above integral stems from small wavevectors $q$ (of the order of the inverse integral scale of turbulence $L_0$) giving an $\bm R_\perp$-independent constant. Extracting also an $\bm R_\perp$-dependent contribution one obtains that $$\begin{aligned} & \langle n_1(\bm R_\perp,Z)n_1({\bm 0},0)\rangle =\delta(Z) C_n^2 \LARGE [\mathrm{Const+} \nonumber\\ & \left .+\frac{\Gamma(\mu+2)\Gamma(-1/2-\mu/2)}{2^{\mu+2}\Gamma(3/2+\mu/2)}\sin{\left (\frac{\pi\mu}{2} \right )} R_\perp^{\mu+1} \right]. \label{paircorr2delta} \end{aligned}$$ Note that Eq. [paircorr2delta]{} can be obtained from Eq. [paircorr2dimensional2]{} using the regularized version of the integral through the replacement of the power law [KolmogorovObukhovgeneral]{} with the corresponding Von Kármán spectrum [@TatarskiiBook1961] obtained by the substitution $ q^{-3-\mu}\to (q^2+q_0^2)^{-3/2-\mu/2},$ where $q_0\sim 1/L_0.$ Spatial scales {#sec:Spatialscales} -------------- The basic equation (\[nlsdimensionall\]) is a linear equation with multiplicative noise. There are some characteristic distances for the laser beam propagation in the random medium which can be extracted by comparison of different terms in Eq. (\[nlsdimensionall\]) and taking into account the expression (\[paircorr2delta\]). The first scale is determined by the propagation distance at which the scintillation index $\sigma^2_I$ of the initially plane wave, $\Psi|_{Z=0}\equiv const$, becomes $\sim 1$. Using the perturbation technique of Refs. [@TatarskiiBook1971; @RytovKravtsovTatarskiiBook1989] we obtain that $$\begin{aligned} \label{sigmaR} \sigma_I^2\equiv \sigma_R^2=p_1C_n^2k_0^{3/2-\mu/2}Z^{3/2+\mu/2}.\end{aligned}$$ Here $$\label{p1def} p_1\equiv \sin \left(\frac{1}{2} \pi\mu\right) \cos \left(\frac{\pi (\mu+1)}{4}\right) \Gamma \left(-\frac{\mu}{2}-\frac{3}{2}\right) \Gamma (\mu+2),$$ which is the generalization of Eq. (47.31) of Ref. [@TatarskiiBook1971] beyond the particular case $\mu=2/3$. The quantity $\sigma_R^2$ is called the Rytov variance and for $\mu=2/3$ it recovers the standard expression $$\begin{aligned} \label{sigmaR5p3} \sigma_R^2=\frac{\sqrt{3}}{4} \sqrt{2- \sqrt{3}} \Gamma \left(-\frac{11}{6}\right) \Gamma \left(\frac{8}{3}\right)C_n^2k_0^{7/6}Z^{11/6} \nonumber\\ =1.22871\ldots C_n^2k_0^{7/6}Z^{11/6} \end{aligned}$$ (see e.g. Ref. [@AndrewsPhillipsBook1998], where $1.22871\ldots $ was replaced by $1.23$ following the approximate numerical value provided in Refs. [@TatarskiiBook1971; @RytovKravtsovTatarskiiBook1989]). Fluctuations of the intensity $I$ are strong at $\sigma^2_R\gtrsim 1$, so we define the characteristic distance $Z_{rytov}$ from the condition that $\sigma_R^2=1$, which gives together with Eq. [sigmaR]{} $$Z_{rytov}=\left (p_1C_n^2k_0^{3/2-\mu/2}\right)^{-2/(3+\mu)}. \label{rytov}$$ We call the distance (\[rytov\]) as the [* Rytov length*]{}. In our work the Rytov length is assumed to be larger than the Rayleigh length (\[Rayleighlength\]), $$Z_{rytov}\gtrsim Z_R, \label{narrow}$$ which implies a smallness of the initial beam waist $w_0$ and a relative weakness of the refractive index fluctuations to make sure that the condition [narrow]{} is satisfied. We investigate the propagation distances $Z>Z_{rytov}$ where the Gaussian beam is already disintegrated into speckles. It follows from Eq. (\[nlsdimensionall\]) that the envelop $\Psi$ changes with $Z$ due to $n_1$ as $\Psi\propto \exp[ i k_0\int _0^ZdZ' n_1(\bm R_\perp,Z')]$. Then we obtain from Eq. (\[paircorr2delta\]) that the average square of the phase difference between two points at the same $Z$ but different $\bm R_\perp$ caused by the refractive index fluctuations is $$\begin{aligned} k_0^2 \left\langle\left\{\int \limits_0^ZdZ\, '[n_1(\bm R_\perp,Z')-n_1(\bm 0,Z')]\right\}^2 \right\rangle \nonumber \\ =-\frac{k_0^2\Gamma(\mu+2)\Gamma(-1/2-\mu/2)}{2^{\mu+1}\Gamma(3/2+\mu/2)}\sin{\left (\frac{\pi\mu}{2} \right )}C_n^2 R_\perp^{\mu+1} Z. \nonumber \end{aligned}$$ Equating this phase difference to unity, we find the phase correlation length $$R_{ph}\sim (k_0^2 C_n^2 Z)^{-1/(\mu+1)}. \label{phasecorrelation}$$ Thus qualitatively the beam front at a given $Z$ can be considered as multiple cells of the transverse size $R_{ph}$ with independent phases between different cells. In accordance with the Huygens-Fresnel principle, the wave amplitude can be considered as a superposition of waves emitted by the secondary sources at a wavefront. The source of a transverse size $R_0$ produces the beam of the transverse length $\sim \theta Z $ at the propagation distance $Z$, where $\theta \equiv 1 /(k_0 R_{0})$ is the corresponding beam divergence. Respectively, the sources of the transverse size $\sim R_{ph}$ located at $Z=Z_1$ produce the beams of the transverse length $\sim Z _1/(k_0 R_{ph})$ at the distance $Z=2Z_1$. This length becomes larger than the transverse size $2Z_1/(k_0 w_0)$ at $Z=2Z_1$ of the initial beam if $R_{ph}\gtrsim w_0/2$. This condition is satisfied at the distance $Z> Z_\star, $ where $Z_\star$ is estimated from the condition that $w_0\sim R_{ph}|_{Z=Z_\star}$ and Eq. [phasecorrelation]{} as $$Z_\star \sim k_0^{-2} C_n^{-2} w_0^{-\mu-1}. \label{Andrews}$$ At $Z>Z_\star$ the total transverse beam width $R_{width}$ looses memory of the initial beam waist $w_0$ and is estimated as $$R_{width}\sim Z /(k_0 R_{ph}) \sim (C_n^2 Z^{\mu+2}k_0^{1-\mu})^{1/(\mu+1)}. \label{beamwidth}$$ Note that the inequality (\[narrow\]) results in $Z_\star\gtrsim Z_{rytov}$. Dimensionless variables {#sec:dimensionless} ======================= Here we introduce dimensionless parameters, that we use below. Namely, the dimensionless coordinates $\bm r=(x,y)$ and $z$ are defined as follows $$\label{dimensionless} x=X/w_0, \, y= Y/w_0, \, \bm r= \bm R_\perp/ w_0, \, z=Z/(4Z_R).$$ where $w_0$ is the initial Gaussian beam waist, see Eq. (\[Gaussianbeam\]), and $Z_R$ is Rayleigh length (\[Rayleighlength\]). Then we obtain from Eq. [nlsdimensionall]{} the following dimensionless stochastic equation $$\label{laser1} {{i}}\frac{\partial }{\partial z}\Psi+\nabla^2\Psi +\xi ({\bm r},z)\Psi=0,$$ where the random factor $$\label{nxirelation} \xi=2k_0 ^2w_0^2n_1,$$ determines stochastic properties of the envelope $\Psi$. The initial condition (\[Gaussianbeam\]) in the dimensionless units takes the following form $$\Psi_\mathrm{in}(\bm r)= \exp(-r^2), \label{initial}$$ where the initial beam amplitude is set to one without loss of the generality because we consider the linear equation for the wave amplitude. In the dimensionless units the relation (\[paircorr2delta\]) is rewritten as $$\begin{aligned} \langle \xi(\bm r_1,z_1)\xi(\bm r_2,\bm z_2)\rangle =\left({\rm const}- D r_{12}^{\mu+1}\right) \delta(z_1-z_2), \label{noise} \end{aligned}$$ where $r_{12}=|\bm r_1-\bm r_2|$ and the factor $D$ is $$\begin{aligned} D=- \frac{ c_n^2 }{2^{\mu+1}} \frac{\Gamma(\mu+2)\Gamma(-1/2-\mu/2)}{\Gamma(3/2+\mu/2)}{}\sin{\left (\frac{\pi\mu}{2} \right )}. \label{Ddef} \end{aligned}$$ Here we used the dimensionless turbulence strength $c^2_n$ first introduced in Ref. [@LushnikovVladimirovaJETP2018] as $$\label{cndef} \begin{split} c_n^2\equiv k_0^3w_0^{\mu+3}C_n^2. \end{split}$$ For $\mu=2/3,$ Eq. [Ddef]{} implies that $D/c_n^2=2.91438\ldots$ Reduction of the refractive index fluctuations to Gaussian stochastic process with power law pair correlation function {#sec:Gaussianfluctuations} ---------------------------------------------------------------------------------------------------------------------- We introduce the Fourier transform $$\tilde\xi(\bm k, z)=\int d^2r\ \exp(-i\bm k \bm r) \xi(\bm r,z).$$ Then Eq. (\[noise\]) implies that $$\begin{aligned} \langle \tilde\xi(\bm k,z_1) \tilde \xi(\bm q,z_2) \rangle =\frac{D(2\pi)^2}{p_0 k^{3+\mu}}\delta(z_1-z_2)\delta(\bm k+\bm q), \nonumber \\ p_0\equiv\frac{1}{2^{2+\mu}\pi (\mu+1)} \frac{\Gamma(1/2-\mu/2)}{\Gamma(3/2+\mu/2)}. \label{spectrum} \end{aligned}$$ Of course, the expression (\[spectrum\]) corresponds to Eq. (\[noise\]). Since $\xi$ is short-correlated in $z$, it can be regarded to possess Gaussian statistics by the central limit theorem (see e.g. [@FellerBook1957]). The probability distribution describing fluctuations of $\xi$ corresponding to Eq. (\[spectrum\]) can be written as $$\begin{aligned} {\cal P}\propto \exp\left[ -\frac{p_0}{2D} \int dz\ \int \frac {d^2 q}{(2\pi)^2} q^{3+\mu} |\tilde\xi|^2\right], \label{probab} \end{aligned}$$ where the integration over $z$ is taken over the propagation length of the laser beam. The consistency of Eqs. (\[spectrum\]) and [probab]{} can be immediately verified by replacing the integrals in Eq. [probab]{} by discrete sums with Gaussian integrals explicitly evaluated which allow to take the continuous limit back from sum to integrals. The probability density [probab]{} can be a starting point for calculating complicated averages over the $\xi$-statistics. Parameters and regions {#sec:Parametersregions} ---------------------- In the dimensionless variables [dimensionless]{}, using Eqs. [p1def]{}, [cndef]{} and [Ddef]{}, the Rytov length [rytov]{} takes the following form $$\begin{aligned} z_{rytov}\equiv \frac{Z_{rytov}}{4Z_R} =\frac{1}{2} (p_1{c_n^2})^{-\frac{2}{\mu+3}}=p_2D^{-\frac{2}{\mu+3}}, \label{rytovdimensionless} \end{aligned}$$ where $$\begin{aligned} \label{p2def} p_2\equiv\frac{1}{2} \Gamma \left(\frac{ \text{} 2^{{\mu}+3} \pi\cos \left[\frac{1}{4} \pi (\mu+1)\right] }{(\mu+3)^2 \Gamma \left(-\frac{\mu}{2}-\frac{3}{2}\right)\cos \left(\frac{\pi \mu}{2}\right)}\right)^{-\frac{2}{\mu+3}}. $$ For $\mu=2/3$, Eq. [p2def]{} reduces to $p_2=0.80088\ldots.$ The condition [narrow]{} together with [rytov]{} can be rewritten as $D \lesssim 1$. In the dimensionless variables [dimensionless]{}, the variations of the phase in the transverse direction $\bm r$ due to the noise become of order unity at the scale $r\sim r_{ph}$ where $$r_{ph}= R_{ph}/w_0\sim (Dz)^{-1/(\mu+1)}, \label{phasel}$$ and $R_{ph}$ is defined by Eq. (\[beamwidth\]). This quantity can be treated as the phase correlation length of the envelope $\Psi$ in the transverse direction. Eq. (\[Andrews\]), with Eqs. [Ddef]{} and [cndef]{} taken into account, transforms in the dimensionless variables [dimensionless]{} into $$z_\star=Z_\star/(4Z_R)\sim D^{-1}. \label{star}$$ Note that $z_\star\gtrsim z_{rytov}$ because of the inequality (\[narrow\]). As discussed in section \[sec:Spatialscales\], the total transverse beam width $r_{width}$ at the distance $z\gtrsim z_\star$ is determined by the random diffraction, where $r_{width}$ is determined from Eq. [beamwidth]{} as $$r_{width}=R_{width}/w_0\sim(Dz)^{1/(\mu+1)}z. \label{tord8}$$ The width (\[tord8\]) grows as $z$ increases faster than the pure diffraction case since $\mu<1$. The random diffraction leads to a random phase of the field $\Psi$ at $z\gg z_\star$. Therefore $\Psi$ possesses Gaussian statistics at the scales. Correspondingly, the intensity $I=|\Psi|^2$ has the Poissonian statistics. We examine below the accuracy of this statement by calculating corrections to the Poissonian statistics. Correlation functions {#sec:corr} ===================== Statistical properties of the electric field envelope $\Psi$ can be examined in terms of its correlation functions. The second-order correlation function is the average $$F_2(\bm r_1,\bm r_2,z) =\left\langle \Psi(\bm r_1, z) \Psi^\star(\bm r_2, z)\right\rangle, \label{per2}$$ taken at a given $z$. We examine also higher order correlation functions $$\begin{aligned} F_{2n}(\bm r_1, \dots , \bm r_{2n},z) = \qquad \nonumber \\ \langle \Psi(\bm r_1,z) \dots \Psi(\bm r_n,z) \Psi^\star(\bm r_{n+1},z) \dots \Psi^\star(\bm r_{2n},z) \rangle, \label{cof1} \end{aligned}$$ The correlation functions (\[per2\],\[cof1\]) are obviously invariant under a homogeneous phase shift. Therefore they are insensitive to the refraction index fluctuations at the integral scale of turbulence. The angular brackets in Eqs. (\[per2\]) and (\[cof1\]) designate averaging over the statistics of $\xi$. Principally, one should solve Eq. (\[laser1\]) for any realization $\xi$ at a given initial condition, then calculate the product in the angular brackets in Eq. (\[cof1\]) and then average over the realizations with the weight dictated by Eq. (\[noise\]). Of course, this procedure cannot be performed explicitly. However, Eq. (\[laser1\]) and the expression (\[noise\]) admit derivation of closed equations for correlation functions $F_{2n}$. First such procedure for the pair correlation function was proposed by Kraichnan [@Kraichnan] and Kazantsev [@Kazantsev] in the contexts of the passive scalar turbulence and turbulent dynamo, correspondingly. It was independently obtained in the optical context in Ref. [@TatarskiiJETP1969] for both the pair and higher order correlation functions. To obtain the equations for $F_{2n}$, one may start with the relation $$\Psi(\bm r, z_2)=\mathrm T\exp\left[i\int_{z_1}^{z_2}dz\ (\nabla^2+\xi)\right]\Psi(\bm r,z_1), \label{tord}$$ where $\mathrm T\exp$ means an $z$-ordered exponent. The relation (\[tord\]) is a direct consequence of the equation (\[laser1\]). The equation (\[tord\]) enables one to relate a product of $\Psi(z_2)$, $\Psi^\star(z_2)$ to the corresponding product of $\Psi(z_1)$, $\Psi^\star(z_1)$. Due to short in $z$ correlations of $\xi$ one can independently average the average of $\Psi(z_1)$, $\Psi^\star(z_1)$ and the exponents. Say, $$\begin{aligned} F_2(\bm r_1,\bm r_2,z_2) =\left\langle \Psi(\bm r_1, z_2) \Psi^\star(\bm r_2, z_2)\right\rangle \nonumber \\ =\left\langle \mathrm T\exp\left\{i\int_{z_1}^{z_2}dz\ [\nabla_1^2+\xi(\bm r_1)]\right\} \right. \nonumber \\ \left. \mathrm T\exp\left\{-i\int_{z_1}^{z_2}dz\ [\nabla_2^2+\xi(\bm r_2)]\right\} \right\rangle \nonumber \\ F_2(\bm r_1,\bm r_2,z_1). \label{tord1} \end{aligned}$$ Analogous relations can be obtained for other products. Analyzing close $z_1$ and $z_2$, one can expand the exponents. Since $\langle \xi \rangle=0$, we should expand terms with $\xi$ up to the second order as follows $$\begin{aligned} \mathrm T\exp\left[i\int_{z_1}^{z_2}dz\ (\nabla^2+\xi)\right]\approx 1+i\int_{z_1}^{z_2}dz\ (\nabla^2+\xi) \nonumber \\ -\int_{z_1}^{z_2}dz\ \xi(\bm r, z) \int_{z_1}^z d\zeta\ \xi(\bm r,\zeta). \label{tord2} \end{aligned}$$ Substituting the expansion (\[tord2\]) and the analogous expressions for the other exponents, keeping terms up to the second order in $\xi$ and averaging, we relate $F_2(z_2)$ to $F_2(z_1)$. Since $\langle\xi(\bm r,z_1)\xi(\bm r,z_2)\rangle$ is a narrow symmetric function of $z_1-z_2$, one should take $$\int^{z_2} dz_1\, \langle\xi(\bm r,z_1)\xi(\bm r,z_2)\rangle=\mathrm{const}/2,$$ see Eq. (\[noise\]). As a result, we obtain an increment of $F_2$, proportional to $z_2-z_1$. Passing from the (small) increment to the differential equation, one finds the equation for the pair correlation function (\[per2\]) $$\begin{aligned} \partial_z F_2=i(\nabla_1^2-\nabla_2^2)F_2 -D r^{\mu+1} F_2, \label{per1} \end{aligned}$$ where $\bm r=\bm r_1-\bm r_2$. The constant, which appears in Eq. (\[noise\]), drops from the equation, as it should be. The equation (\[per1\]) can be rewritten as $$\partial_z F_2=2i\frac{\partial^2}{\partial\bm R \partial\bm r} F_2 -D r^{\mu+1} F_2, \label{per3}$$ where $\bm R=(\bm r_1+\bm r_2)/2$. Pair correlation function ------------------------- A formal solution of the equation (\[per1\]) can be written in terms of the two-point Green function ${\cal G}$: $$\begin{aligned} F_2(\bm r_1,\bm r_2,z)= \int d^2x_1\, d^2x_2\, {\cal G}(\bm r_1,\bm r_2,\bm x_1,\bm x_2,z) \nonumber \\ \Psi_{in}(\bm x_1) \Psi_{in}(\bm x_2), \label{tord3} \end{aligned}$$ where $\Psi_{in}(\bm x_1) \Psi_{in}(\bm x_2)$ is the initial value of the pair correlation function, see Eq. (\[initial\]). The Green function ${\cal G}$ is equal to zero at $z<0$ and satisfies the equation $$\begin{aligned} \partial_z {\cal G}=i(\nabla_1^2-\nabla_2^2){\cal G} -D r^{\mu+1} {\cal G} \nonumber \\ +\delta(z) \delta(\bm r_1-\bm x_1) \delta(\bm r_2-\bm x_2). \label{tord4} \end{aligned}$$ Note that the Green function by itself does not know about initial conditions for the envelope $\Psi$. To find the Green function, one can pass to the Fourier transform $\tilde {\cal G}$ as follows $${\cal G}(\bm R,\bm r,\bm X,\bm x,z)=\int \frac{d^2 k}{(2\pi)^2} \exp(i\bm k \bm R) \tilde {\cal G}(\bm k,\bm r, \bm X, \bm x,z), \nonumber$$ where $\bm r=\bm r_1-\bm r_2$, $\bm R=(\bm r_1+\bm r_2)/2$, $\bm x=\bm x_1-\bm x_2$, $\bm X=(\bm x_1+\bm x_2)/2$. Then the equation (\[tord4\]) is rewritten as $$\begin{aligned} \partial_z \tilde {\cal G}=-2\bm k \nabla \tilde {\cal G} -D r^{\mu+1} \tilde {\cal G} \nonumber \\ =\delta(z) \delta(\bm r-\bm x) \exp(-i\bm X \bm k), \nonumber \end{aligned}$$ where $\nabla=\partial/\partial\bm r$. Solving that equation by characteristics, one finds that $$\begin{aligned} \tilde {\cal G}=\theta(z) \delta(\bm r-2\bm kz-\bm x) \exp(-i\bm X \bm k) \nonumber \\ \exp\left[-D\int_0^z d\zeta\ |\bm r-2 \bm k \zeta|^{\mu+1}\right]. \label{per4} \end{aligned}$$ Returning to the real space, one finds $$\begin{aligned} {\cal G}= \frac{\theta(z)}{16 \pi^2 z^2} \exp\left[\frac{i}{2z}(\bm r-\bm x)(\bm R-\bm X) \right. \nonumber \\ \left. -{Dz}\int_0^1 d\chi\, |\chi \bm x+(1-\chi)\bm r|^{\mu+1} \right]. \label{tord5} \end{aligned}$$ This quantity is symmetric in permutation of the initial and terminal points, as it should be. Integrating the expression (\[tord5\]) over $\bm R$ at a given $\bm r$, one obtains that $$\begin{aligned} \int d^2R\ {\cal G}= {\theta(z)}\delta(\bm r-\bm x) \exp(-Dzr^{\mu+1}). \label{tord9} \end{aligned}$$ Substituting the expression into Eq. (\[tord3\]), we find that $$\int d^2R\ F_2=\frac{\pi}{2} \exp\left(-\frac{r^2}{2}-Dzr^{\mu+1}\right), \label{tord10}$$ where the expression (\[initial\]) is substituted. The first term in the exponent dominates at $z\ll z_\star$ and the second term dominates in the opposite limit, $z\gg z_\star$. The dominance of the term $-Dzr^{\mu+1}$ in the second regime $z\gg z_\star$ implies that the transverse correlation length $r_{ph}\sim (Dz)^{-1/(\mu+1)}$ in full agrement with the qualitative analysis of sections \[sec:Spatialscales\] and \[sec:Parametersregions\] including Eq. [phasel]{}. Further we are interested in distances $z\gg z_\star$, where effects of random diffraction are relevant. Then the characteristic values of $\bm x$ and $\bm r$ are determined by the integral in the exponent in Eq. (\[tord5\]). Equating the integral to unity, we find the estimate $|\bm x|\sim |\bm r|\sim r_{ph}$, where $r_{ph}$ is the phase correlation length (\[phasel\]). The correlation length is related to the random diffraction on fluctuations of the refraction index destroying phase correlations. Equating then the first term in the exponent in Eq. (\[tord5\]), we find the characteristic value $R\sim z/r_{ph}=r_{width}$ where $r_{width}$ is determined by Eq. (\[tord8\]). The quantity has the meaning of the beam width, caused by the random diffraction. At distances $z\gg z_\star$ the width is much larger than the pure diffraction width $z$. In analyzing the pair correlation function in accordance with Eq. (\[tord3\]), one should take into account the initial width of the beam. Just the initial width determines the characteristic value of $\bm X$, it can be estimated as unity. Thus in the case $z\gg z_\star$ one obtains from Eq. (\[tord3\]) $$\begin{aligned} \langle \Psi(\bm R+\bm r/2) \Psi^\star(\bm R-\bm r/2) \rangle =\int \frac{d^2x\, d^2X}{16 \pi^2 z^2} \nonumber \\ \exp\left[\frac{i}{2z}\bm r\bm R -{Dz}\int_0^1 d\chi\, |\chi \bm r+(1-\chi)\bm x|^{\mu+1} \right] \nonumber \\ \Psi_{in}(\bm X+\bm x/2)\Psi_{in}(\bm X-\bm x/2), \label{pairc3} \end{aligned}$$ where we neglected $\bm X$, $\bm x$ in the first term in the exponent. If $r\ll r_{ph}$ then the value of $x$ is determined by the second term in the exponent, then $x$ can be estimated as $r_{ph}$. If $r_{ph}\ll r \ll 1$ then the value of $x$ is determined by the second term in the exponent as well, however, $x$ can be estimated as $r$. After integration over $\bm x$ in the expression (\[pairc3\]), there remains a dependence on $\bm r$ with the characteristic value $r\sim r_{ph}$. Since at $r\ll 1$, $x\ll 1$ as well, we can neglect $\bm x$ in the product $\Psi_{in}(\bm X+\bm x/2)\Psi_{in}(\bm X-\bm x/2)$ in Eq. (\[pairc3\]). Then one can integrate over $\bm X$ to obtain $$\begin{aligned} \langle \Psi(\bm R+\bm r/2) \Psi^\star(\bm R-\bm r/2) \rangle = \int \frac{d^2x}{32 \pi z^2} \nonumber \\ \exp\left[\frac{i}{2z}\bm r\bm R -{Dz}\int_0^1 d\chi\, |\chi \bm r+(1-\chi)\bm x|^{\mu+1} \right]. \label{pairc4} \end{aligned}$$ Analyzing the expression (\[pairc4\]) we conclude that the characteristic value of $r$ is determined by the expression (\[phasel\]). Thus, the quantity $r_{ph}$ plays the role of the beam correlation length in the transverse direction as well. We neglected the factor $\exp [-{i}\bm x\bm R/(2z)]$ in the expression (\[pairc4\]). For $R$ larger than $z/R_{ph}=D^{1/{\mu+1}}z^{1/{\mu+1}+1}$, the exponent is fast oscillating. That leads to diminishing the expression of the pair correlation function in comparison with the expression (\[phasel\]). Thus, the quantity (\[tord8\]) is the beam width for $z\gg z_\star$, indeed. The quantity is determined solely by fluctuations. Fourth-order correlation function --------------------------------- The equation for the fourth-order correlation function $$F_4=\langle \Psi(\bm r_1,z) \Psi(\bm r_2,z) \Psi^\star(\bm r_3,z) \Psi^\star(\bm r_4,z) \rangle \label{per9}$$ can be derived, similar the equation (\[per1\]), from the representation (\[tord\]). The corresponding equation is give by $$\begin{aligned} \partial_z F_4= i(\nabla_1^2+\nabla_2^2-\nabla_3^2-\nabla_4^2)F_4 \qquad \label{per10} \\ -D\left[-r_{12}^{\mu+1}+r_{13}^{\mu+1}+r_{14}^{\mu+1}-r_{34}^{\mu+1} +r_{23}^{\mu+1}+r_{24}^{\mu+1}\right]F_4, \nonumber \end{aligned}$$ where $r_{12}=|\bm r_1-\bm r_2|$ and so on. Generally, the separations between the points in the different spots are of the order of $r_{width}$ (\[tord8\]). Then the real factor in the right-hand side of the equation (\[per10\]) is $\sim Dr_{width}^{\mu+1}$. At $z\gg z_\star$ one finds $Dr_{width}^{\mu+1}z\gg1$. That leads to a strong suppression of the fourth-order correlation function. However, in the geometry where separations between the points $\bm r_1$ and $\bm r_3$, $\bm r_2$ and $\bm r_4$ (or between the points $\bm r_1$ and $\bm r_4$, $\bm r_2$ and $\bm r_3$) are much smaller than $r_{width}$, the factor appears to be much smaller. Therefore the fourth-order correlation function has sharp maxima in the geometries. Further we examine just this case. Having in mind the geometry where separations $r_{13}$ and $r_{24}$ are much smaller than $r_{width}$, we rewrite the equation (\[per10\]) as $$\begin{aligned} \frac{\partial F_4}{\partial z}=2i\left(\frac{\partial^2F_4}{\partial\bm R_1 \partial \bm \rho_1} +\frac{\partial^2F_4}{\partial\bm R_2 \partial \bm \rho_2}\right) \qquad \nonumber \\ - [D(\rho_1^{\mu+1}+\rho_2^{\mu+1})+U] F_4, \qquad \label{perr} \\ U/D =|\bm R+\bm\rho_1/2+\bm\rho_2/2|^{\mu+1} -|\bm R+\bm\rho_1/2-\bm\rho_2/2|^{\mu+1} \nonumber \\ -|\bm R-\bm\rho_1/2+\bm\rho_2/2|^{\mu+1} +|\bm R-\bm\rho_1/2-\bm\rho_2/2|^{\mu+1}, \quad \label{peru} \end{aligned}$$ where we introduced $$\begin{aligned} \bm r_1=\bm R_1+\bm\rho_1/2, \ \bm r_3=\bm R_1-\bm\rho_1/2, \nonumber \\ \bm r_2=\bm R_2+\bm\rho_2/2, \ \bm r_4=\bm R_2-\bm\rho_2/2, \nonumber \end{aligned}$$ and $\bm R=\bm R_1-\bm R_2$. If $R\sim R_{width}\gg \rho_1,\rho_2$. Then the terms in the quantity $U$ (\[peru\]) cancel each other and we can neglect $U$ in the equation (\[perr\]). Then the operator in the equation becomes a sum of two operators in the equation for the pair correlation function (\[tord4\]). Therefore in the geometry, $F_4$ is a product of two pair correlation functions, $F_2(\bm r_1,\bm r_3)F_2(\bm r_2,\bm r_4)$. Let us now estimate accuracy of the approximation. For the purpose we should evaluate corrections to the fourth-order correlation function $F_4$. In accordance with the equation (\[perr\]), the first correction is written as $$\begin{aligned} \delta F_4= -\int_0^z d\zeta \int d^2x_1 d^2x_2d^2x_3d^2x_4 \nonumber \\ U(\bm x_1, \bm x_2,\bm x_3,\bm x_4) F_2(\bm x_1,\bm x_3,\zeta) F_2(\bm x_2,\bm x_4,\zeta) \nonumber \\ {\cal G}(\bm r_1,\bm r_3,\bm x_1,\bm x_3,z-\zeta) {\cal G}(\bm r_2,\bm r_4,\bm x_2,\bm x_4,z-\zeta). \label{peru2} \end{aligned}$$ In this integral the separations $x_{13}\sim x_{24}\sim r_{ph}$, $x_{14}\approx x_{23}\sim r_{width}$, and the quantity $$U\sim D x_{14}^{c-2}x_{13}^2\sim D r_{ph}^2 r_{width}^{c-2}.$$ Evaluating the Green functions ${\cal G}$ in Eq. (\[peru2\]) as $z^{-2}$ in accordance with the expression (\[tord5\]), we find that the correction (\[peru2\]) is evaluated as $\alpha F_2(\bm r_1,\bm r_3)F_2(\bm r_2,\bm r_4)$, where $\alpha$ is the parameter $$\alpha=(z_{rytov}/z)^{(1-\mu)(\mu+3)/(\mu+1)}. \label{smallpar}$$ The parameter is small provided $z\gg z_{rytov}$. A more accurate calculation is presented in Appendix \[sec:crossbar\]. To analyze the fourth-order correlation function $F_4$ in the geometry, where all separations between the points are much smaller than $r_{width}$, one should use the Green function $${\cal G}_4(\bm r_1,\bm r_2,\bm r_3,\bm r_4,\bm x_1,\bm x_2,\bm x_3,\bm x_4,z),$$ of the equation (\[per10\]), satisfying $$\begin{aligned} \partial_z {\cal G}_4- i(\nabla_1^2+\nabla_2^2-\nabla_3^2-\nabla_4^2){\cal G}_4+ \nonumber \\ D\left[-r_{12}^{\mu+1}+r_{13}^{\mu+1}+r_{14}^{\mu+1}- r_{34}^{\mu+1}+r_{23}^{\mu+1}+r_{24}^{\mu+1}\right]{\cal G}_4 \nonumber \\ =\delta(z) \delta(\bm r_1-\bm x_1)\delta(\bm r_2-\bm x_2) \delta(\bm r_3-\bm x_3)\delta(\bm r_4-\bm x_4). \label{peru3} \end{aligned}$$ The fourth-order correlation function is expressed as $$\begin{aligned} F_4(\bm r_1,\bm r_2,\bm r_3,\bm r_4,z) =\int d^2 x_1 d^2 x_2d^2x_3d^2x_4 \nonumber \\ {\cal G}_4(\bm r_1,\bm r_2,\bm r_3,\bm r_4,\bm x_1,\bm x_2,\bm x_3,\bm x_4,z) \nonumber \\ \Psi_{in}(\bm x_1) \Psi_{in}(\bm x_2) \Psi_{in}(\bm x_3)\Psi_{in}(\bm x_4). \label{peru4} \end{aligned}$$ One can apply the same arguments as for the fourth order correlation function, to the Green function ${\cal G}$. Thus, ${\cal G}$ as a function of $\bm r_1,\bm r_2,\bm r_3,\bm r_4$ has the sharp maxima in the geometry where separations between the points $\bm r_1$ and $\bm r_3$, $\bm r_2$ and $\bm r_4$ (or between the points $\bm r_1$ and $\bm r_4$, $\bm r_2$ and $\bm r_3$) are much smaller than $r_{width}$. Therefore in the main approximation $$\begin{aligned} {\cal G}_4(\bm r_1,\bm r_2,\bm r_3,\bm r_4,\bm x_1,\bm x_2,\bm x_3,\bm x_4,z) \approx \nonumber \\ {\cal G}(\bm r_1,\bm r_3,\bm x_1,\bm x_3,z) {\cal G}(\bm r_2,\bm r_4,\bm x_2,\bm x_4,z) \nonumber \\ {\cal G}(\bm r_1,\bm r_4,\bm x_1,\bm x_4,z) {\cal G}(\bm r_2,\bm r_3,\bm x_2,\bm x_3,z) . \label{peru5} \end{aligned}$$ Since the equation for the correlation function ${\cal G}_4$ in terms of $\bm x_i$ is the same as in terms of $\bm r_i$ (\[peru3\]), the approximation (\[peru5\]) is correct in terms of $\bm x_i$ as well. As for any Green function, one may write $$\begin{aligned} {\cal G}_4(\bm r_1,\bm r_2,\bm r_3,\bm r_4,\bm x_1,\bm x_2,\bm x_3,\bm x_4,z) = \nonumber \\ \int d^2y_1 d^2y_2 d^2y_3d^2y_4 \nonumber \\ {\cal G}_4(\bm r_1,\bm r_2,\bm r_3,\bm r_4,\bm y_1,\bm y_2,\bm y_3,\bm y_4,\zeta) \nonumber \\ {\cal G}_4(\bm y_1,\bm y_2,\bm y_3,\bm y_4,\bm x_1,\bm x_2,\bm x_3,\bm x_4,z-\zeta). \qquad \label{peru6} \end{aligned}$$ In integration over $\bm y_i$, the main contribution to the integral is produced just by the regions where separations between the points $\bm y_1$ and $\bm y_3$, $\bm y_2$ and $\bm y_4$ (or between the points $\bm y_1$ and $\bm y_4$, $\bm y_2$ and $\bm y_3$) are much smaller than $r_{width}$. Such integration reproduces the approximation (\[peru5\]). We conclude that one can use the approximation (\[peru5\]) for any geometry of the points $\bm r_i,\bm x_i$. Higher order correlation functions ---------------------------------- One can easily generalize the above procedure for correlation functions of arbitrary order. The equation for $2n$-th order correlation function is $$\begin{aligned} \partial_z F_{2n}= i(\nabla_1^2+\dots +\nabla_n^2- \nabla_{n+1}^2 - \dots -\nabla_{2n}^2) F_{2n} \quad \label{cof2} \\ +D \left[ {\sum_{i=1}^n}\sum_{j=i+1}^n r_{ij}^{\mu+1} +\sum_{i=n+1}^{2n} \sum_{j=i+1}^{2n} r_{ij}^{\mu+1} \right. \nonumber \\ \left. -\sum_{i=1}^n \sum_{j=n+1}^{2n}r_{ij}^{\mu+1} \right]F_{2n}, \nonumber \end{aligned}$$ where, as above, $r_{ij}=|\bm r_i-\bm r_j|$. Analogously to Eqs. (\[tord3\],\[peru4\]), a formal solution of the equation (\[cof2\]) can be written as $$\begin{aligned} F_{2n}=\int d^2x_1 \dots d^2x_{2n} \nonumber \\ {\cal G}_{2n}(\bm r_1,\dots, \bm r_{2n},\bm x_1,\dots, \bm x_{2n},z) \nonumber \\ \Psi_{in}(\bm x_1) \dots \Psi_{in}( \bm x_{2n}), \label{tord6} \end{aligned}$$ where the Green function ${\cal G}_{2n}$ is equal to zero at $z<0$ and satisfies the equation $$\begin{aligned} \partial_z {\cal G}_{2n}= i(\nabla_1^2+\dots +\nabla_n^2- \nabla_{n+1}^2 - \dots -\nabla_{2n}^2) {\cal G}_{2n} \quad \nonumber \\ +D \left[ {\sum_{i=1}^n}\sum_{j=i+1}^n r_{ij}^{\mu+1} +\sum_{i=n+1}^{2n} \sum_{j=i+1}^{2n} r_{ij}^{\mu+1} \right. \nonumber \\ \left. -\sum_{i=1}^n \sum_{j=n+1}^{2n}r_{ij}^{\mu+1} \right]{\cal G}_{2n} \nonumber \\ +\delta(z)\delta(\bm r_1-\bm x_1)\dots \delta(\bm r_{2n}-\bm x_{2n}). \qquad \label{tord7} \end{aligned}$$ Under the condition $z\gg z_\star,$ the Green function ${\cal G}_{2n}$ have sharp maxima in the geometry where the points are split into $n$ close pairs $\bm r_i,\bm r_j$ ($i=1,\dots, n$, $j=n+1,\dots,2n$), where the separations are smaller or of the order of $r_{ph}$ (\[phasel\]). In the geometry, the “large” differences (of the order of $r_{width}$) in Eq. (\[tord7\]) cancel each other. Then we pass to the differential operator that is a sum of the differential operators of the type appearing in the equation for the pair correlation function (\[per1\]). Therefore the $2n$-th order correlation function in the geometry is a product of $n$ pair correlation functions. There are $n!$ of such geometries, and in the main approximation the $2n$-th order correlation function can be presented as a sum of $n!$ terms that are products of the pair correlation functions, like the expression (\[peru6\]). This is just the case subjected by Wick theorem where $\Psi$ possesses Gaussian statistics. The property is a consequence of random diffraction that makes the phase of $\Psi$ a random variable. One can estimate corrections to the Gaussian statistics. For this one should evaluate the terms that were discarded in the geometry of close pairs. An analysis, analogous to one produced for the fourth order correlation function, shows that the approximation is justified by the same small parameter $\alpha$ (\[smallpar\]). Statistics of intensity ----------------------- Let us analyze statistical properties of the intensity $I=|\Psi|^2$, taken at the distance $z\gg z_\star$ inside the diffraction spot $r\ll r_{width}$. One expects that the quantity has the Poisson statistics because of the randomness of the phase of $\Psi$ caused by the refractive index fluctuations [@Halperin]. We prove the conjecture and give the quantitative criterion determining the applicability region of the statistics. The average $\langle I^n \rangle$ can be written as $$\begin{aligned} \langle I^n \rangle= \int d^2x_1 \dots d^2x_{2n} {\cal G}_{2n}(\bm r,\dots, \bm r,\bm x_1,\dots, \bm x_{2n},z) \nonumber \\ \Psi_{in}(\bm x_1) \dots \Psi_{in}( \bm x_{2n}), \qquad \label{tord26} \end{aligned}$$ in accordance with Eq. (\[tord6\]). Thus we should establish properties of the Green function ${\cal G}_{2n}$ in the situation where the final points coincide. We use the following property of any Green function $$\begin{aligned} {\cal G}_{2n}(\bm r_1,\dots, \bm r_{2n},\bm x_1,\dots, \bm x_{2n},z) =\int d^2y_1 \dots d^2y_{2n} \nonumber \\ {\cal G}_{2n}(\bm r_1,\dots, \bm r_{2n},\bm y_1,\dots, \bm y_{2n},\zeta) \qquad \nonumber \\ {\cal G}_{2n}(\bm y_1,\dots, \bm y_{2n},\bm x_1,\dots, \bm x_{2n},z-\zeta) \qquad \label{tord27} \end{aligned}$$ If we choose $\zeta\sim z,$ then the characteristic value of $y$ in the integral can be estimated as $R_{width}$ (\[tord8\]). However, both Green functions under the integral have have sharp maxima in the geometry where the points are split into $n$ close pairs $\bm y_i,\bm y_j$ ($i=1,\dots, n$, $j=n+1,\dots,2n$), provided the separations are smaller or of the order of $R_{ph}$ (\[phasel\]). For the second Green function, we established the property above. For the first Green function it follows from the fact that in terms of the variables $y,$ it satisfies the same equation as in terms of $r$. Thus both Green functions are represented as sums of the products of the pair Green functions. Therefore we arrive at the Gaussian statistics for $\Psi(\bm r)$ and, consequently, at the Poisson statistics for $I$. In other words, the probability distribution function of $I$ is exponential. Corrections to the Gaussian approximation are controlled by the same small parameter $\alpha$ (\[smallpar\]). Now we can estimate the region of applicability of the Poisson approximation. If we analyze $\langle I^n\rangle$, then the relative correction, associated with the neglected terms in the equation for the Green functions, is estimated as $n^2 \alpha$, for large $n$. Thus the Poisson expression is valid if $n \ll 1/\sqrt{\alpha}$. In other words, the exponential probability distribution is correct one if $I\ll \langle I \rangle /\sqrt{\alpha}$. Effective action {#sec:Effective action} ================ Here we propose an alternative language for describing effects associated with fluctuations of the refractive index. Correlation functions of the field $\Psi$ can be examined in the framework of an effective quantum field theory [@MSR73; @Janssen; @Domin]. The theory produces a diagrammatic expansion of the type first developed by Wyld in the context of hydrodynamic turbulence [@Wyld]. Applications of the technique to the optical problems can be found in the works [@Kolokol; @Churkin; @Vergeles]. In the framework of the effective quantum theory, the correlation functions of the field $\Psi$ can be found as functional integrals over $\Psi,\Psi^\star,P,P^\star$ (where $P,P^\star$ are auxiliary fields) with the weight $$\exp\left\{-{\cal S}+\int d^2r\ \left[P \Psi_{in}^\star +P^\star \Psi_{in}\right]\right\}, \label{weight}$$ where the effective action ${\cal S}$ is constructed in accordance with the equation (\[laser1\]). Here $\Psi_{in}$ is the initial condition for the field $\Psi$ posed at $z=0$. In our setup, the initial condition is determined by Eq. (\[initial\]). To find ${\cal S}$, we start from the weight $$\exp\left\{-{\cal I}+\int d^2r\ \left[P \Psi_{in}^\star +P^\star \Psi_{in}\right]\right\}, \label{weight1}$$ where ${\cal I}$ forces the equation (\[laser1\]): $$\begin{aligned} {\cal I}=\int d^2r\ dz\ P^\star\left(i\partial_z\Psi + \nabla^2 \Psi +\xi\Psi\right) \nonumber \\ -\int d^2r\ dz\ P\left(i\partial_z\Psi^\star - \nabla^2 \Psi^\star -\xi\Psi\right). \label{weight2} \end{aligned}$$ The weight (\[weight\]) is obtained by averaging over the statistics of the refractive index fluctuations in accordance with Eq. (\[noise\]). The constant term corresponding to homogeneous phase fluctuations cannot contribute to the effective action. Thus we arrive at the additional condition $$\int d^2r\, (P^\star \Psi+ P\Psi^\star ) =0, \label{gauge}$$ to be imposed on the field $P$. Then the constant drops from the consideration. As a result, we find that the effective action ${\cal S}$ is the sum ${\cal S}= {\cal S}_{(2)} +{\cal S}_{int}$, where $$\begin{aligned} {\cal S}_{(2)}=\int d^2r\ dz\ P^\star\left(i\partial_z\Psi + \nabla^2 \Psi \right) \nonumber \\ -\int d^2r\ dz\ P\left(i\partial_z\Psi^\star - \nabla^2 \Psi^\star \right), \label{lader4} \\ {\cal S}_{int}=\frac{1}{2}\int d^2r_1\, d^2r_2\, dz\ (P_1^\star \Psi_1+ P_1\Psi^\star_1 ) \nonumber \\ (D |\bm r_1-\bm r_2|^{\mu+1}-{\rm const}) (P_2^\star \Psi_2+ P_2\Psi_2^\star ). \label{lasem4} \end{aligned}$$ Here the first terms describes the free laser beam diffraction, whereas the second term describes the influence of the fluctuations of the refractive index, that is the random diffraction. Correlation functions of the fields $\Psi,\Psi^\star,P,P^\star$ can be found in the framework of the perturbation theory. For the purpose one expands $\exp(-{\cal S}_{int})$ in ${\cal S}_{int}$ and calculate explicitly the resulting expressions that are Gaussian integrals with the weight $$\exp\left\{-{\cal S}_{(2)}+\int d^2r\ \left[P \Psi_{in} +P^\star \Psi_{in}\right]\right\},$$ that is the exponent of the quadratic in the fields quantity. The terms of the perturbation series can be represented as Feynman diagrams. The diagrammatic expansion of this type was first used by Wyld in the context of hydrodynamic turbulence [@Wyld]. Analytical expressions caused by the diagrams are constructed from the propagators, that are the correlation functions $$\begin{aligned} G=\langle \Psi(\bm r,z) P^\star(0,0)\rangle =- \frac{\theta(z)}{4\pi z} \exp\left(i\frac{r^2}{4z}\right), \label{prop1} \\ G^\star=\langle \Psi^\star(\bm r,z) P(0,0)\rangle =- \frac{\theta(z)}{4\pi z} \exp\left(-i\frac{r^2}{4z}\right). \label{prop2} \end{aligned}$$ Here angular brackets mean averaging with the weight $\exp[-{\cal S}_{(2)}]$ and $\theta(z)$ is Heaviside step function. Let us stress that there are no corrections, caused by the interaction term (\[lasem4\]), to the expressions (\[prop1\]) and (\[prop2\]) because of causality. Ladder approximation {#sec:Ladderapproximation} -------------------- The pair correlation function of the envelope at a given $z$ is written as the integral (\[tord3\]). The Green function in the relation can be expressed in terms of the correlation function of the introduced fields $${\cal G}=\langle \Psi(\bm r_1,z) \Psi^\star(\bm r_2,z) P(\bm x_1,0)P^\star(\bm x_2,0)\rangle. \label{dia1}$$ The expression (\[dia1\]) can be derived if to expand the weight (\[weight\]) up to the second order in $\Psi_{in}$. (200,110) (-20,0) (-3.5,3.5) – (-1,3.5); (-1,2.5) – (-3.5,2.5); (0.5,3.5) – (2.5,3.5); (2.5,2.5) – (0.5,2.5); (2.5,3.5) – (4.5,3.5); (4.5,2.5) – (2.5,2.5); (-3.5,1) – (-1,1); (-1,1) – (1.5,1); (1.5,1) – (4,1); (4,0) – (1.5,0); (1.5,0) – (-1,0); (-1,0) – (-3.5,0); (2.5,2.5) – (2.5,3.5); (-1,0) – (-1,1); (1.5,0) – (1.5,1); In zero approximation ${\cal G}=G(\bm r_1-\bm r_4,z)G^\star(\bm r_2-\bm r_3,z)$. Contributions to ${\cal G}$, related to the interaction term (\[lasem4\]), can be presented by ladder diagrams, see Fig. \[fig:ladder\]. Here a line directed to the right represents the average $G$ (\[prop1\]) and a line directed to the left represents the average $G^\star$ (\[prop2\]). The dotted line represents the factor $-D r^{\mu+1}$, where $r$ is the separation between the points. Summation of the ladder diagrams depicted in Fig. \[fig:ladder\] leads to an integral equation for ${\cal G}$, analysed in Appendix \[sec:ladder\]. Solving the integral equation, one obtains the expression (\[tord5\]) obtained above by other method. Let us stress that the expression is the exact one in our setup due to the shortness of the refractive index correlations in $z$. Let us analyze momenta of the intensity $I$. In the framework of our scheme, the moment $\langle I^n \rangle$ can be written as $$\begin{aligned} \langle I^n \rangle= \int d^2x_1 \dots d^2x_{2n} \left\langle \left[\Psi(0,z) \Psi^\star(0,z)\right]^n \right. \nonumber \\ \left. P(\bm x_1,0) \dots P(\bm x_n,0) P^\star(\bm x_{n+1},0) \dots P^\star(\bm x_{2n},0) \right\rangle \nonumber \\ \Psi_{in}(\bm x_1) \dots \Psi_{in}(\bm x_{2n}) . \label{dial1} \end{aligned}$$ The problem is how to calculate the average in the expression (\[dial1\]). Below we use the ladder approximation, where the average in the expression (\[dial1\]) is reduced to a product of factors corresponding to the ladder diagrams. Then one finds $$\begin{aligned} \langle I^n \rangle= \int d^2x_1 \dots d^2x_{2n} \bigl[ {\cal G}(0,0,\bm x_1,\bm x_{n+1},z) \dots \nonumber \\ {\cal G}(0,0,\bm x_n,\bm x_{2n},z) +\dots \bigr] \Psi_{in}(\bm x_1) \dots \Psi_{in}(\bm x_{2n}), \label{powerl} \end{aligned}$$ where the number of summands is $n!$. Each summand produces $\langle I\rangle^n$ and we find $\langle I^n \rangle= n!\langle I\rangle^n$. In other words, we arrive at the Poisson statistics for $I$ with the probability density ${\cal P}(I)=\langle I \rangle^{-1} \exp(-I/\langle I \rangle)$. The validity of the ladder approximation should be checked separately. Correlation functions of $I$ can be analyzed in the same ladder approximation. If we separate the points in the correlation function $\langle I(\bm r_1,z) I(\bm r_2,z) \dots \rangle$, a part of the ladders are switched off when the separation becomes larger than $R_{ph}$. The corresponding analysis is analogous to one made for the pair correlation function $\langle\Psi \Psi^\star\rangle$. Thus cumulants (irreducible parts) of the averages like $\langle I^n(\bm r_1,z) I^n(\bm r_2,z)\rangle$ becomes parametrically smaller where $\bm r=\bm r_1-\bm r_2$ exceeds $r_{ph}$. (200,110) (-20,0) at (4.5,3.3)[$\bm y_1$]{}; at (4.5,2.3)[$\bm y_2$]{}; at (4.5,1.2)[$\bm y_3$]{}; at (4.5,0.2)[$\bm y_4$]{}; at (-3.5,3.3)[$\bm x_1$]{}; at (-3.5,2.3)[$\bm x_2$]{}; at (-3.5,1.2)[$\bm x_3$]{}; at (-3.5,0.2)[$\bm x_4$]{}; at (0.5,3.3)[$\bm r_1$]{}; at (0.2,2.3)[$\bm r_2$]{}; at (0.8,1.2)[$\bm r_3$]{}; at (0.5,0.2)[$\bm r_4$]{}; (-3.5,1) – (-2.5,1); (-2.5,1) – (-1,1); (-1,1) – (0.5,1); (0.5,1) – (2,1); (2,1) – (3.5,1); (3.5,1) – (4.5,1); (4.5,0) – (3.5,0); (3.5,0) – (2,0); (2,0) – (0.5,0); (0.5,0) – (-1,0); (-1,0) – (-2.5,0); (-2.5,0) – (-3.5,0); (-2.5,0) – (-2.5,1); (-1,0) – (-1,1); (2,0) – (2,1); (3.5,0) – (3.5,1); (0.5,1) – (0.5,2.5); (-3.5,3.5) – (-2.5,3.5); (-2.5,3.5) – (-1,3.5); (-1,3.5) – (0.5,3.5); (0.5,3.5) – (2,3.5); (2,3.5) – (3.5,3.5); (3.5,3.5) – (4.5,3.5); (4.5,2.5) – (3.5,2.5); (3.5,2.5) – (2,2.5); (2,2.5) – (0.5,2.5); (0.5,2.5) – (-1,2.5); (-1,2.5) – (-2.5,2.5); (-2.5,2.5) – (-3.5,2.5); (-2.5,2.5) – (-2.5,3.5); (-1,2.5) – (-1,3.5); (2,2.5) – (2,3.5); (3.5,2.5) – (3.5,3.5); Let us now analyze corrections to the ladder approximation. The first correction is determined by the diagrams including a crossbar connecting two ladders, see Fig. \[fig:crossbar\]. Corrections to the averages like $\langle I^n \rangle$ are analyzed in Appendix \[sec:crossbar\]. It is shown there that the small parameter justifying the applicability condition of the ladder approximation is $\alpha$ (\[smallpar\]). Now we can estimate the region of applicability of the Poisson approximation. If we analyze $\langle I^n\rangle$, then the relative correction, associated with the crossbar is estimated as $n^2 \alpha$, for large $n$. Thus the Poisson expression is valid if $n \ll 1/\sqrt{\alpha}$. In other words, the exponential probability distribution is correct if $I\ll \langle I \rangle /\sqrt{\alpha}$. Due to the refraction index fluctuations, the beam cross-section is divided into a large number of speckles $N$. The number of the speckles in the pattern can be estimated as the ratio of the beam cross-section area to the square of the correlation length (\[phasel\]). Thus we obtain $$\begin{aligned} N\sim D^{4/(\mu+1)}z^{2+4/(\mu+1)}, \qquad z \gg z_\star. \label{speckle2} \end{aligned}$$ One can think about statistics of the beam intensity in the brightest speckle. For this purpose, we assume that the intensity statistics in the speckles are independent and are determined by the Poisson statistics. The corresponding analysis is made in Appendix \[sec:maximum\]. The main result of the analysis is that the average intensity in the brightest speckle can be estimated as $\langle I \rangle \ln N$. Due to slowness of the logarithmic function this value remains inside the applicability region of Poisson statistics determined by the condition $I\ll \langle I \rangle /\sqrt{\alpha}$. That justifies our conclusions. Conclusion {#sec:Conclusion} ========== We analyzed statistical properties of the speckle pattern of light intensity in the cross-section of the laser beam propagating in the turbulent fluid (atmosphere). The pattern is formed due to refraction index fluctuations caused by the turbulent fluctuations. We demonstrated that there are two characteristic dimensionless propagation length, $z_{rytov}$ and $z_\star$, determined by the relations (\[rytovdimensionless\]) and (\[star\]). Our theory is valid under the condition $z_{rytov}\lesssim z_\star$. We are interested in the region $z\gg z_\star$ where effects of the random diffraction dominate. If the propagation length $z$ satisfies the condition $z \gg z_\star,$ then the beam contains many speckles and its width $r_{width}$ becomes much wider than in the case of the free diffraction propagation. This is due to the refractive index fluctuations. The beam width is determined by the expression (\[beamwidth\]), that is $$r_{width}\propto z^{1/(\mu+1)+1}.$$ The correlation length of the signal $r_{ph}$, related to fluctuations of its phase, appears to be much smaller than the beam width and can be estimated in accordance with Eq. (\[phasel\]) by $$r_{ph}\propto z^{-1/(\mu+1)}.$$ We found the analytical expression for the pair correlation function (\[pairc4\]) in the region $z \gg z_\star$ and demonstrated that higher order correlation functions of the envelope $\Psi$ are split into products of the pair correlation functions. This result is in accordance with the expectation that strong phase fluctuations lead to an effective Gaussianity of the envelope statistics. We analyzed also non-Gaussian corrections to the higher order correlation functions and established that they are controlled by the $z$-dependent parameter $\alpha\propto z^{\mu+1-4/(\mu+1)}$, see Eq. (\[smallpar\]). As one expects, the parameter diminishes as $z$ grows due to the increasing role of the random diffraction. We developed also the diagrammatic technique for calculating corrections to the correlations functions of the envelope related to the random diffraction. In the diagrammatic language, the effective Gaussianity of the envelope statistics is explained as the approximation where the so-called ladder diagrams are taken into account. Let us stress that the approximation implies a deep resummation of the diagrams. The diagrammatic technique gives a powerful tool to go beyond the scope of the Gaussian approximation and enables one to calculate analytically non-Gaussian corrections. The Gaussianity of the envelope statistics results in the Poisson statistics of the beam intensity $I$. We established the applicability region of the statistics is given by $I\ll \langle I \rangle /\sqrt{\alpha}$, where $\langle I \rangle$ is the average intensity inside the pattern and $\alpha$ is the parameter (\[smallpar\]). We examined the statistics of the brightest spot among the large number $N\sim (r_{width}/r_{ph})^2$ in the speckle pattern. The average value of the intensity inside the brightest spot can be estimated as $\langle I \rangle \ln N$. The quantity lies inside the applicability region of the Poisson approximation. In the opposite limit $z_{rytov}\gg z_\star$, compare to considered in this paper, there appears an intermediate region $z_{rytov}\gg z\gg z_\star$, of the dimensionless propagation lengths $z$, where fluctuation effects are extremely strong. The approach developed in this paper is not applicable to that region, being, however, applicable to the region $z\gg z_{rytov}$. The region $z_{rytov}\gg z\gg z_\star$ needs a special analysis based on the diagrammatic technique and the related quantum field theory. Such analysis is outside of the scope of this paper. We thank support of the Russian Ministry of Science and High Education, program 0033-2019-0003. The work of P.M.L. was supported by the state assignment “Dynamics of the complex systems". The work of P.M.L. was supported by the National Science Foundation, grant DMS-1814619. Simulations were performed at the Texas Advanced Computing Center using the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF Grant ACI-1053575. Ladder representation {#sec:ladder} ===================== Here, we demonstrate how to obtain the expression (\[tord5\]) for the Green function (\[dia1\]) using the ladder representation for the object. The quantity ${\cal G}$ is depicted by the sum of the ladder sequence of the diagrams depicted in Fig. \[fig:ladder\]. The ladder representation leads to the following integral equation $$\begin{aligned} {\cal G}(\bm r_1, \bm r_2, \bm r_3, \bm r_4,z) =G(\bm r_1-\bm r_4,z) G^\star(\bm r_2-\bm r_3,z) \nonumber \\ -\int d\zeta \, d^2 r_5 d^2 r_6 D |\bm r_{56}|^{\mu+1} G(\bm r_{15},z-\zeta) G^\star(\bm r_{26},z-\zeta) \nonumber \\ {\cal G}(\bm r_5, \bm r_6, \bm r_3, \bm r_4,\zeta). \qquad \label{ladf1} \end{aligned}$$ Using the relation $$\begin{aligned} (\partial_z -i\nabla_1^2+i\nabla_2^2) [G(\bm r_1,z)G^\star(\bm r_2,z)] \nonumber \\ =\delta(z)\delta(\bm r_1)\delta(\bm r_2). \nonumber \end{aligned}$$ following from Eqs. (\[prop1\],\[prop2\]), one obtains from Eq. (\[ladf1\]) $$\begin{aligned} (\partial_z -i\nabla_1^2+i\nabla_2^2+D r_{12}^{\mu+1}){\cal G} \nonumber \\ =\delta(z)\delta(\bm r_1-\bm r_4)\delta(\bm r_2-\bm r_3). \label{ladf2} \end{aligned}$$ Passing to the variables $\bm R=(\bm r_1+\bm r_2)/2$, $\bm r=\bm r_1-\bm r_2$, we rewrite the equation (\[ladf2\]) as $$\begin{aligned} \left(\partial_z -2i\frac{\partial^2}{\partial \bm r \partial \bm R}+D r^{\mu+1}\right){\cal G} \nonumber \\ =\delta(z)\delta(\bm r-\bm x)\delta(\bm R-\bm X), \label{ladf3} \end{aligned}$$ where $\bm X=(\bm r_3+\bm r_4)/2$, $\bm x=\bm r_4-\bm r_3$. The equation (\[ladf3\]) is equivalent to Eq. (\[tord4\]). Corrections to the ladder approximation {#sec:crossbar} ======================================= Here we demonstrate how to find corrections to the ladder approximation. For this purpose we consider the first correction to the product of two ladders giving the main contribution to the fourth-order correlation function of $\Psi$. The correction is determined by the diagrams of the type depicted in Fig. \[fig:crossbar\], containing the only crossbar connecting two ladders. The sum of the ladder diagrams of the type presented in Fig. \[fig:crossbar\] gives the first correction to the product ${\cal G}(\bm y_1,\bm y_2,\bm x_2,\bm x_1,z){\cal G}(\bm y_3,\bm y_4,\bm x_4,\bm x_3,z)$. After summation of the ladder sequences, the diagrams depicted in Fig. \[fig:crossbar\] give the following analytical expression $$\begin{aligned} -D\int d\zeta\ d^2r_1 d^2r_2 d^2 r_3 d^2r_4 |\bm r_2-\bm r_3|^{\mu+1} \nonumber \\ {\cal G}(\bm y_1,\bm y_2,\bm r_2,\bm r_1,z-\zeta) {\cal G}(\bm y_3,\bm y_4,\bm r_4,\bm r_3,z-\zeta) \nonumber \\ {\cal G}(\bm r_1,\bm r_2,\bm x_2,\bm x_1,\zeta) {\cal G}(\bm r_3,\bm r_4,\bm x_4,\bm x_3,\zeta), \nonumber \end{aligned}$$ where the integral over $\zeta$ goes from $0$ to $z$. In derivation of the expression we used the relations $$\begin{aligned} \int d^2x\, G(\zeta,\bm x) G(z-\zeta, \bm r-\bm x) =-i G(z,\bm r), \nonumber \\ \int d^2x\, G^\star(\zeta,\bm x) G^\star(z-\zeta, \bm r-\bm x) =i G^\star(z,\bm r), \label{prop10} \end{aligned}$$ for the functions (\[prop1\],\[prop2\]), that can be checked directly. Taking into account the expression (\[lasem4\]) for the interaction and the relations (\[prop10\]), one finds ultimately the following correction $\Theta$ to the product ${\cal G}(\bm y_1,\bm y_2,\bm x_2,\bm x_1,z){\cal G}(\bm y_3,\bm y_4,\bm x_4,\bm x_3,z)$ $$\begin{aligned} \Theta=-D\int d\zeta\ d^2r_1 d^2r_2 d^2 r_3 d^2r_4 \left(|\bm r_2-\bm r_3|^{\mu+1} \right. \nonumber \\ \left. +|\bm r_1-\bm r_4|^{\mu+1} -|\bm r_1-\bm r_3|^{\mu+1}-|\bm r_2-\bm r_4|^{\mu+1}\right) \nonumber \\ {\cal G}(\bm y_1,\bm y_2,\bm r_2,\bm r_1,z-\zeta) {\cal G}(\bm y_3,\bm y_4,\bm r_4,\bm r_3,z-\zeta) \nonumber \\ {\cal G}(\bm r_1,\bm r_2,\bm x_2,\bm x_1,\zeta) {\cal G}(\bm r_3,\bm r_4,\bm x_4,\bm x_3,\zeta), \label{prop11} \end{aligned}$$ Let us consider the case $\bm x_i=\bm y_i=0$. We introduce the quantities $$\begin{aligned} \bm r_1=\bm X +\bm R/2 +\bm y/2, \ \bm r_2=\bm X +\bm R/2 -\bm y/2, \nonumber \\ \bm r_3=\bm X -\bm R/2 +\bm s/2, \ \bm r_4=\bm X -\bm R/2 -\bm s/2, \label{varia} \end{aligned}$$ where $\bm s$ and $\bm y$ measure the ladder thickness whereas $\bm R$ is the separation between the ladders. Taking the expression (\[tord5\]) into account, we conclude that integration over $\bm X$ in Eq. (\[prop11\]) produces a $\delta$-function fixing $\bm y=-\bm s$. After integrating over $\bm s$, one arrives at the expression $$\begin{aligned} \Theta=\frac{D}{2^{14}\pi^6}\int \frac{d\zeta\ d^2R\, d^2y}{z^2 \zeta^2 (z-\zeta)^2} \nonumber \\ \left(|\bm R+\bm y|^{\mu+1} +|\bm R-\bm y|^{\mu+1}-2R^{\mu+1}\right) \nonumber \\ \exp\left[ \frac{iz}{2\zeta (z-\zeta)}\bm y \bm R -\frac{2Dz}{2+\mu}y^{\mu+1}\right]. \label{prop12} \end{aligned}$$ One can take the integral over $\bm R$ in the expression (\[prop12\]) explicitly to obtain that $$\begin{aligned} \Theta=\frac{2^{2c} c D}{2^{8} \pi^4 z^{3-c}} \frac{\Gamma(2+\mu/2)}{\Gamma(1-c/2)} \int_0^1d\chi\, [\chi(1-\chi)]^{\mu+1} \nonumber \\ \int_0^\infty \frac{d y}{y^{2+\mu}} \exp\left(-\frac{2Dzy^{\mu+1}}{2+\mu}\right) \sin^2\left[\frac{y^2}{4z\chi(1-\chi)}\right]. \label{prop14} \end{aligned}$$ One can worry about a singular contribution related to small $\chi(1-\chi)$. At $y^2\lesssim z\chi(1-\chi)$ the integral (\[prop14\]) converges. Thus, at small $\chi(1-\chi)$ the integral over $y$ produces a singular contribution $\propto [\chi(1-\chi)]^{-(\mu+1)/2}$, suppressed by the factor $[\chi(1-\chi)]^{\mu+1}$ in Eq. (\[prop14\]). Thus we arrive at the natural estimates $\zeta\sim z$, $y\sim (Dz)^{-1/{\mu+1}}$, $R\sim D^{1/(\mu+1)}z^{1+1/(\mu+1)}\gg y$. Therefore the integral (\[prop14\]) is estimated as $$\Theta\sim \frac{D}{z^{5-c}} y^{3-\mu} \sim \frac{1}{z^4}\alpha. \label{prop15}$$ The factor $\alpha$ (\[smallpar\]) characterizes smallness of the corrections to the ladder diagrams. Let us examine the pair correlation function $\langle I I \rangle$ at distances much larger than the correlation length $R_{ph}$ (\[phasel\]). The main contribution to the irreducible part of the correlation function is determined by the same diagram depicted in Fig. \[fig:crossbar\]. If $z\ll z_\star$, then $\bm x_i$ are of order of $a$ and can be neglected. Then $\bm y=-\bm s\sim (Dz)^{-1/(\mu+1)}$ and we arrive at the same smallness (\[smallpar\]). If $z\gg z_\star$ then $x_{12}\sim x_{34}\sim (Dz)^{-1/(\mu+1)}$ and $y\sim s\sim (Dz)^{-1/(\mu+1)}$ as well. And we return to the same smallness (\[smallpar\]). The correlation length of the correction to the pair correlation function $\langle I I \rangle$ is $r_{width}$ (\[tord8\]). An analogous analysis can be done for the other correlation functions of $I$. Statistics of a largest value {#sec:maximum} ============================= We examine here the case where independent variables $x_1,\dots, x_{N+1}$ have identical Poisson probability density functions $$\label{pu1} {\cal P}(x_j)=\frac{1}{q}\exp\left(-\frac{x_j}{q}\right), \quad \langle x_j \rangle=q.$$ We are interested in the statistics of the largest value in the sequence $x_1,\dots, x_{N+1}$. Let us designate the largest value as $y$. Then the probability density function of $y$ can be written as $$\begin{aligned} {\cal P}(y)=\frac{N+1}{q}\exp\left(-\frac{y}{q}\right) \left[\frac{1}{q}\int\limits_{0}^{y}dx\,e^{-x/q}\right]^N \label{su1} \\ =\frac{N+1}{q}\exp\left(-\frac{y}{q}\right) \left(1-e^{-y/q}\right)^N. \nonumber \end{aligned}$$ Further we assume $N\gg 1$ and, consequently, the average value $\langle y\rangle \gg q$. In this case it is possible to rewrite the expression (\[su1\]) as $$\begin{aligned} {\cal P}(y)\approx \frac{N}{q} \exp\left(-\frac{y}{q}-Ne^{-y/q}\right). \label{su2}\end{aligned}$$ Then it is possible to calculate easily the average value $$\begin{aligned} \langle y\rangle =\int dy\ y{\cal P}(y) \approx q\ln N. \label{su3}\end{aligned}$$ We see that the average value $\langle y\rangle$ (\[su3\]) is much larger than $q$, indeed, since $N\gg 1$. That justifies our conclusions. Bright speckles =============== Here we consider bright speckles, that are characterized by the inequality $I\gg \langle |\Psi(\bm y)|^2 \rangle$. Here $\bm y$ is the center of the speckle and $I$ is the beam intensity at the point. We are interested in the shape of such bright speckle. To solve the problem, one can use the saddle-point approach. However, we assume that we are still inside the Gaussian approximation. Let us analyze the probability density $P$, that at some point $\bm y$ the beam intensity is $I$. In the Gaussian approximation the probability density can be written as $$\begin{aligned} P(I,\bm y)=\int D\Psi\ D\Psi^\star\ {\cal N} e^{-{\cal S}} \delta\left[\Psi^\star(\bm y) \Psi(\bm y)-I\right], \label{bright1} \end{aligned}$$ where ${\cal N}$ is the normalization factor and ${\cal S}$ is the effective action. It is written as $${\cal S}=\int d^2 r\ \Psi^\star \hat{K} \Psi, \label{bright6}$$ where $\hat{K}$ is the operator, related to the pair correlation function: $$\hat K F_2(\bm r_1,\bm r_2)=\delta(\bm r_1-\bm r_2). \label{bright2}$$ Obviously, $\int dI\ P(I,\bm y)=1$. We rewrite the expression (\[bright1\]) as $$\begin{aligned} P(I,\bm y)=\int D\Psi\ D\Psi^\star\ {\cal N} \int \frac{d\lambda}{2\pi i} \nonumber \\ \exp\left[-\int d^2 r\ \Psi^\star \hat{K} \Psi +\lambda \Psi^\star(\bm y) \Psi (\bm y) -\lambda I \right], \label{bright3} \end{aligned}$$ where integration over $\lambda$ goes along the imaginary axis. If $I$ is high, then the integration in the expression (\[bright3\]) can be performed in the saddle-point approximation. The saddle-point equation for $\Psi$ is $$\hat K \Psi=\lambda \Psi(\bm y) \delta(\bm r-\bm y). \label{bright4}$$ Taking into account the equation (\[bright2\]), one finds from Eq. (\[bright4\]) $$\Psi(\bm r)=\lambda \Psi(\bm y) F_2(\bm r,\bm y). \label{bright5}$$ Therefore $$\lambda F_2(\bm y,\bm y)=1. \label{bright7}$$ Multiplying the relation (\[bright5\]) by $\Psi^\star(\bm y)$, one obtains $$\Psi(\bm r)\Psi^\star(\bm y)=\lambda I F_2(\bm r,\bm y). \label{bright8}$$ where the self-consistency condition $\Psi^\star(\bm y) \Psi(\bm y)=I$ is used. As we see from Eq. (\[bright8\]), the saddle-point profile is determined by the pair correlation function. As it follows from Eqs. (\[bright3\]) and (\[bright4\]), the saddle-point value of the probability density $P(I,\bm y)$ is $\exp(-\lambda I)$. Using the relation (\[bright7\]), one obtains $$P(I,\bm y)\sim \exp\left[-\frac{I}{F_2(\bm y,\bm y)}\right] =\exp\left[-\frac{I}{\langle |\Psi(\bm y)|^2 \rangle}\right]. \label{bright9}$$ Thus we return to the Poisson statistics. In addition, we see that the applicability condition of the saddle-point approximation is $I\gg \langle |\Psi(\bm y)|^2 \rangle$, indeed. 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--- abstract: 'In the present work, we investigate existence of deformations and algebraic approximability for certain uniruled Kähler threefolds. In the first part, we establish existence of infinitesimal deformations for all conic bundles with relative Picard number one over a non-algebraic compact Kähler surface $S$ and existence of positive-dimensional families of deformations in all but some special cases. In the second part, we study the question of algebraic approximability for projective bundles over $S$ and threefolds bimeromorphic to $\mathbb{P}_1\times S$.' address: | Mathematisches Institut\ Universität Bayreuth\ 95440 Bayreuth\ Germany author: - Florian Schrack bibliography: - 'literatur.bib' title: Algebraic approximation of Kähler threefolds --- Introduction ============ An interesting class of complex manifolds to study is the class of compact manifolds which arise as limits of projective manifolds in the following sense: Let $X$ be a compact complex manifold. A proper holomorphic submersion $\pi\colon\mathcal{X}\to T$ between complex manifolds $\mathcal{X}$ and $T\ni0$ is called an *algebraic approximation of $X$* if (i) the central fiber $\mathcal{X}_0:=\pi^{-1}(0)$ is isomorphic to $X$ and (ii) there exists a sequence $(t_k)_{k\in\mathbb{N}}\subset T$ converging to $0$ such that for each $k$, the fiber $\mathcal{X}_{t_k}:=\pi^{-1}(t_k)$ is a projective manifold. We call $X$ *algebraically approximable* if there exists an algebraic approximation of $X$. It is a corollary of Kodaira’s classification of complex surfaces that every compact Kähler surface is algebraically approximable. A new, more conceptual proof of this result not relying on classification results has recently been given by Buchdahl in [@Buc06] and [@Buc08]. It was conjectured for a long time that indeed every compact Kähler manifold is algebraically approximable. This was shown to be false by C. Voisin in [@Voi04], where she proved that in every dimension $\ge 4$ there exist compact Kähler manifolds not even having the homotopy type of a projective manifold. It is still open whether every compact Kähler threefold is algebraically approximable. Since the classification theory of Kähler threefolds is still a little bit vague at the time being, the most one can possibly hope for is settling a reasonably large class of threefolds carrying some special structure. In the present article, we undertake a first attempt towards this goal. Our strategy is to use Kodaira’s approximation results for surfaces in order to construct algebraic approximations for threefolds carrying some type of $\mathbb{P}_1$-fibration over a Kähler surface. An intermediate problem is to study the existence of deformations of these threefolds. For conic bundles, we establish the existence of infinitesimal deformations by a careful study of the geometry of discriminant loci in section \[sec:conic-bundl-discr\]: \[thm:infpos\] Let $S$ be a compact non-algebraic Kähler surface and $f\colon X\to S$ a conic bundle with $\rho(X)=\rho(S)+1$. Then $H^1(T_X)\ne0$. In most cases, we can even show the existence of a positive-dimensional family of deformations: \[thm:infconi\] Let $S$ be a compact non-algebraic Kähler surface and $f\colon X\to S$ a conic bundle with $\rho(X)=\rho(S)+1$. Then $$h^1(T_X)>h^2(T_X),$$ except possibly in the following cases: (i) $S$ is a torus and $E:=f_*(K_{X/S}^*)$ is projectively flat, i.e. $\mathbb{P}(E)$ is given by a representation $\pi_1(S)\to\mathrm{PGL}(3,\mathbb{C})$. Furthermore, $X$ is not isomorphic to $\mathbb{P}(V)$ for any rank-two bundle $V$ on $S$. (ii) $S$ is a minimal properly elliptic surface (i.e. $\kappa(S)=1$) and all singular fibers of the elliptic fibration are multiples of smooth elliptic curves. By studying vector bundles over K3 surfaces and tori in section \[sec:vector-bundles-k3\], we lay the basis for understanding projectivized rank-two bundles over surfaces of Kodaira dimension $0$. We obtain the following result: \[thm:projr2b\] Let $S$ be a compact Kähler surface with $\kappa(S)=0$ and $V$ a holomorphic rank-two vector bundle on $S$. Then the projective bundle $\mathbb{P}(V)$ is algebraically approximable. By combining techniques from the previous sections, we extend our results in various directions in section \[sec:algebr-appr-cert\]. First, we prove algebraic approximability of threefolds bimeromorphic to a product of $\mathbb{P}_1$ with a compact Kähler surface: \[thm:main\] Let $X$ be a compact Kähler threefold for which there exists a compact Kähler surface $S$ and a bimeromorphic map $\phi\colon \mathbb{P}_1\times S\dashrightarrow X$. Then $X$ is algebraically approximable. Finally, we prove algebraic approximability for certain types of conic bundles: \[thm:ellcon\] Let $S$ be a compact Kähler surface with an elliptic fibration $r\colon S\to C$ over a smooth curve $C$. If $f\colon X\to S$ is a conic bundle such that the rank-three bundle $f_*(K_{X/S}^*)$ is trivial when restricted to the general fiber of $r$, then $X$ is algebraically approximable. \[thm:splitcon\] Let $S$ be a K3 surface and $f\colon X\to S$ a conic bundle such that $f_*(K_{X/S}^*)$ splits as a direct sum of line bundles. Then $X$ is algebraically approximable. [**Acknowledgements.**]{} In large parts, this paper is a condensed version of my Ph.D. thesis prepared at the University of Bayreuth [@Schrack10]. I would like to thank my advisor Thomas Peternell for many fruitful discussions and suggestions. The work was supported by DFG Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds.” The concept of algebraic approximability ======================================== In this section, we first review some approximability results for elliptic surfaces we will use in the following. After that we investigate the behavior of algebraic approximability with respect to certain types of maps between complex manifolds. We start with general remark concerning the algebraic approximation of *Kähler* manifolds: Let $\mathcal{S}\to T\ni 0$ be a deformation of a compact Kähler manifold $\mathcal{S}_0$. Then for any $t\in T$ in a sufficiently small neighborghood of $0$, the fiber $\mathcal{S}_t$ is also Kähler. Since a compact Kähler manifold is projective if and only if it is Moishezon, finding an approximation of $\mathcal{S}_0$ by projective manifolds is equivalent to finding an approximation by Moishezon manifolds. The elliptic surface case ------------------------- \[prop:elsurf\] Let $S$ be a compact Kähler surface with an elliptic fibration $r\colon S\to C$ over a smooth curve $C$. Then there exist a complex manifold $T\ni0$ and a deformation $\pi\colon\mathcal{S}\to T$ of $S=\mathcal{S}_0$ with the following properties: (i) There is an elliptic fibration $R\colon\mathcal{S}\to C\times T$ with $\operatorname{pr}_2\circ R=\pi$ and $R|_{\mathcal{S}_0}=r$. (ii) For every point $c\in C$ there exists an open neighborhood $U\subset C$ of $c$ such that $R^{-1}(U\times T)\cong r^{-1}(U)\times T$ over $C\times T$. (iii) The set of points $t\in T$ where $\mathcal{S}_t$ is projective is dense in $T$. This proposition is the essence of an intricate study of elliptic fibrations first carried out by Kodaira (cf. [@FM94] for details). A much more elegant proof has been given by Buchdahl in [@Buc08]. Some general criteria --------------------- The following criterion follows easily from Kodaira’s result about stability of subspaces: Let $X\subset Y$ be an inclusion of compact complex manifolds. If $Y$ is algebraically approximable and $H^1(N_{X|Y})=0$, then also $X$ is algebraically approximable. As étale covers are topological objects, an algebraic approximation can obviously be lifted to finite étale covers: Let $X$ be a compact complex manifold which is algebraically approximable. Then any finite étale cover of $X$ is also algebraically approximable. The following is slightly more complicated: Let $f\colon X\to Y$ be a holomorphic surjection from a compact Kähler manifold $X$ to some manifold $Y$. Assume that $f$ has connected fibers and that for general $y\in Y$, $X_y$ is projective with $H^1(\mathcal{O}_{X_y})=0$. Suppose furthermore that $\operatorname{Ext}^2(\Omega^1_{X/Y},\mathcal{O}_X)=0$. Then if $Y$ is algebraically approximable, $X$ is also algebraically approximable. Let $\psi\colon\mathcal{Y}\to T\ni0$ be an algebraic approximation of $Y=\mathcal{Y}_0$. By [@Hor74 Theorem 6.1], the assumption $\operatorname{Ext}^2(\Omega^1_{X/Y},\mathcal{O}_X)=0$ implies (after shrinking $T$ sufficiently) that there exist a deformation $\pi\colon\mathcal{X}\to T\ni 0$ of $X=\mathcal{X}_0$ and a holomorphic map $F\colon\mathcal{X}\to\mathcal{Y}$ over $T$ such that $F|_{\mathcal{X}_0}=f$. It is easy to see that $F$ has connected fibers and $H^1(\mathcal{O}_{\mathcal{X}_y})=0$ for general $y\in\mathcal{Y}$. Since $X$ is Kähler, also $\mathcal{X}_t$ is Kähler for sufficiently small $t$. Now if $t\in T$ is such that $\mathcal{Y}_t$ is projective, $\mathcal{X}_t$ is also projective by [@Fuj83a Proposition 7]. We also state the following property: Let $X$ be a compact Kähler manifold, $\hat{X}\to X$ the blow-up of $X$ along a compact submanifold $Y\subset X$ of codimension $\ge2$. Then $\hat{X}$ is algebraically approximable if and only if there exists a $Y$-stable algebraic approximation of $X$. One can generalize the concept of algebraic approximability to arbitrary complex spaces. By using generalizations of Horikawa’s deformation theory from [@Ran91], one can for instance prove the following result: Let $X$ be a three-dimensional normal complex variety having at most terminal singularities and $f\colon X\dashrightarrow X'$ a composition of Mori contractions and flips. Then, if $X$ is algebraically approximable, so is $X'$. Conic bundles: discriminant loci and deformations {#sec:conic-bundl-discr} ================================================= In this section, we want to investigate the geometry of discriminant loci of conic bundles over non-algebraic surfaces: Let $f\colon X\to S$ be a conic bundle over a compact surface $S$ (i.e. $X$ is a complex manifold and every fiber of $f$ is isomorphic to a curve of degree $2$ in $\mathbb{P}_2$). If we let $E:=f_*{K_{X/S}^*}$, then $X$ is a divisor on $\mathbb{P}(E)$ inside the linear system $\lvert\mathcal{O}_{\mathbb{P}(E)}(2)\otimes\pi^*\det E^*\rvert$, where $\pi\colon\mathbb{P}(E)\to S$ is the natural projection. Thus $X$ is given by a section $\sigma\in H^0(S^2E\otimes\det E^*)$. Via the canonical embedding $$S^2E\otimes\det E^*\subset E\otimes E\otimes\det E^*\cong\mathcal{H}\mathit{om}_{\mathcal{O}_S}(E^*,E\otimes\det E^*),$$ the section $\sigma$ induces a non-zero section $$\det\sigma\in\mathrm{Hom}(\det E^*,\det (E\otimes\det E^*))\cong H^0(\det E^*),$$ which defines a divisor $\Delta_f$ on $S$, the so-called *discriminant locus* of $f$. Obviously, $\Delta_f$ consists of those points $s\in S$ for which the fiber $X_s$ is a singular conic. The discriminant locus has an interesting property: \[prop:pdisc\] For any conic bundle $f\colon X\to S$ over a compact surface $S$ with $\rho(X)=\rho(S)+1$, the discriminant locus $\Delta_f$ is a reduced divisor with the property that any smooth rational component of $\Delta_f$ meets other components of $\Delta_f$ in at least two points. cf. [[@Miy83 Remark 4.2 and Lemma 4.5]]{} This motivates the following definition: A divisor $D$ on a compact complex surface is said to have *property (P)* if it is reduced and for any smooth rational component $C$ of $D$ we have $$C\cdot(D-C)\ge2.$$ Using the terminology of this definition, Proposition \[prop:pdisc\] states that $\Delta_f$ has property (P). Property (P) is preserved under blow-downs: \[prop:pblow\] Let $S$ be a compact surface and $p\colon S\to S'$ the blow-down of a $(-1)$-curve $C_0$ on $S$. Then for any divisor $D$ on $S$ which has property (P), the divisor $D':=p_*D$ on $S'$ also has property (P). Since $D$ is reduced, there are distinct irreducible curves $C_1$, $\dots$, $C_k$ different from $C_0$ such that $$\text{either}\quad D=\sum_{i=1}^kC_i\qquad\text{or}\quad D=C_0+\sum_{i=1}^kC_i.$$ If we let $C_1':=p_*C_1$, $\dots$, $C_k':=p_*C_k$, then $C_1'$, $\dots$, $C_k'$ are distinct irreducible curves on $S'$ and we have $$D'=p_*D=\sum_{i=1}^kC_i'.$$ If now $C_i'$ is smooth rational for some $i$, then also $C_i$ is smooth rational, so by hypothesis we have $$C_i\cdot(D-C_i)\ge2.$$ From this we obtain: $$\begin{split} C_i'\cdot(D'-C_i')&=p^*C_i'\cdot p^*(p_*D-C_i')\\ &=p^*C_i'\cdot(D-C_i)\\ &=C_i\cdot(D-C_i)+(C_0\cdot C_i)C_0\cdot(D-C_i). \end{split}$$ The claim now follows easily because $C_0\cdot C_i\in\{0,1\}$ and either $C_0$ is not a component of $D$, whence $C_0\cdot(D-C_i)\ge0$, or $C_0$ is a smooth rational component of $D$ and thus $$C_0\cdot(D-C_i)=C_0\cdot(D-C_0)+C_0^2-C_0\cdot C_i\ge0$$ by hypothesis. We shall use Propositions \[prop:pdisc\] and \[prop:pblow\] to study discriminant loci of conic bundles over surfaces by induction over blow-ups. For surfaces with algebraic dimension $0$, we obtain: \[lem:disca0\] Let $S$ be a compact Kähler surface with algebraic dimension $a(S)=0$. Then any conic bundle $f\colon X\to S$ with $\rho(X)=\rho(S)+1$ is a $\mathbb{P}_1$-bundle. By Proposition \[prop:pdisc\] the discriminant locus $\Delta_f$ has property (P), so it is sufficient to show that if $D$ is a divisor on $S$ having property (P), then necessarily $D=0$. Since $a(S)=0$, however, any connected curve on $S$ is a tree of rational curves which never satisfies property (P). Using more intricate arguments, we can also handle the case where $S$ has algebraic dimension $1$: \[lem:disca1\] Let $S$ be a compact complex surface with algebraic dimension $a(S)=1$ and $f\colon X\to S$ a conic bundle with $\rho(X)=\rho(S)+1$. Then we have $$\Delta_f^2\ge3\Delta_f\cdot K_S+4K_S^2.$$ By Proposition \[prop:pdisc\], it is sufficient to show that for any divisor $D$ on $S$ satisfying property (P), we have $$\label{eq:discineq} D\cdot(D-3K_S)-4K_S^2\ge0.$$ In order to show inequality , we first assume $S$ to be minimal. Since $a(S)=1$, the surface $S$ is elliptic. Let $r\colon S\to B$ be the (unique) elliptic fibration over a smooth curve $B$. Then any irreducible curve on $S$ is contained in a fiber of $r$. So, by Kodaira’s classification of elliptic fibers, any irreducible curve on $S$ is either a $(-2)$-curve or numerically proportional to a fiber of $r$. If $F$ is a fiber of $r$ and $L$ is an arbitrary line bundle on $S$, then $F\cdot L=0$, so for numerical purposes we can assume that $D$ is a sum of distinct $(-2)$-curves. By an easy calculation, property (P) then implies that $D^2=0$. This in turn implies that $D\cdot L=0$ for any line bundle $L$ on $S$, so inequality  follows from $K_S^2\le0$. If $S$ is not minimal, we let $p\colon S\to S'$ be the blow-down of a $(-1)$-curve $C_0\subset S$. By Proposition \[prop:pblow\], the divisor $D':=p_*D$ has property (P), so we may inductively assume that $$\label{eq:indineq} D'\cdot(D'-3K_{S'})-4K_{S'}^2\ge0.$$ Now we have $K_S=p^*K_{S'}+C_0$. Furthermore, if we write $D=p_*^{-1}D'+\varepsilon C_0$ (note that $\varepsilon\in\{0,1\}$ because D is reduced) and let $\mu$ be the multiplicity of the divisor $D'$ at the point $p(C_0)$, we obtain $$\label{eq:pulb} D=p^*D'+(\varepsilon-\mu)C_0.$$ So we can calculate $$D\cdot(D-3K_S)-4K_S^2=D'\cdot(D'-3K_S)-4K_S^2+4+(\mu-\varepsilon)(\varepsilon-\mu-3).$$ By inequality , in order to show inequality , it thus remains to show that $$\label{eq:mueps} (\mu-\varepsilon)(\mu-\varepsilon+3)\le 4.$$ Since $a(S')=a(S)=1$, the surface $S'$ is also elliptic with all irreducible curves contained in fibers of the elliptic fibration. By Kodaira’s classification of fibers, every reduced curve on $S'$ has multiplicity at most $3$ in any of its points, so in particular we have $\mu\le 3$. Because $C_0$ is smooth rational and $C_0\cdot(D-C_0)=\mu-\varepsilon+1$, property (P) yields the implication $$\label{eq:pimp} \varepsilon=1\Rightarrow\mu\ge2.$$ Hence, in order to prove inequality , it remains to exclude the following two cases: (i) $\mu=3$, (ii) $\mu=2$ and $\varepsilon=0$. We first assume $\mu=3$. Using Kodaira’s classification of singular elliptic fibers, we find that the only possible way for the reduced divisor $D'$ to contain a point of multiplicity $3$ is that $D'$ contains three smooth rational components $C_1'$, $C_2'$ and $C_3'$ intersecting in an ordinary triple point. Now, using implication  for further blow-downs of $S'$, we see that $D'$ cannot contain any $(-1)$-curve lying in the same elliptic fiber as the $C_i'$ but distinct from the $C_i'$. Thus we find that the divisor $C_1'+C_2'+C_3'$ is a connected component of $D'$. If we let $C_i:=p^{-1}_*C_i'$, then $C_i$ is a smooth rational component of $D$ such that, by equation , $$C_i\cdot(D-C_i)=(p^*C_i'-C_0)\cdot(p^*(D'-C_i')+(\varepsilon-2)C_0)=\varepsilon,$$ contradicting property (P). It remains to consider the case $\mu=2$. We must show that in this case, $\varepsilon=1$, i.e. $C_0$ is a component of $D$. To prove this, we can assume that $D'$ consists of exactly one connected component, namely the one containing the point $p(C_0)$. By Kodaira’s classification and implication , we thus obtain the following possibilities for $D'$: (i) $D'$ is an irreducible curve having an ordinary double point or a cusp at $p(C_0)$. In this case the strict transform $p^{-1}_*D'$ is a smooth rational curve, so property (P) implies that $C_0$ must be a component of $D$. (ii) $D'=C_1'+C_2'$ where $C_1'$ and $C_2'$ are smooth rational curves intersecting only at $p(C_0)$ (with multiplicity $2$). Then $p^{-1}_*D'$ consists of two smooth rational curves intersecting transversally at exactly one point, so again by property (P), $C_0$ must be a component of $D$. (iii) $D'$ is a cycle of smooth rational curves. Then $p^{-1}_*D'$ is a chain of rational curves. In this case, we can apply property (P) to the curves at the end of this chain to conclude that $C_0$ must be a component of $D$. We can use the preceding lemmas to estimate the dimension of the deformation space of a conic bundle via the Riemann-Roch theorem. To do so, we need the following basic result: \[prop:conbun\] Let $f\colon X\to S$ be a conic bundle over a compact surface $S$. Then we have $\operatorname{Ext}^3(\Omega^1_{X/S},\mathcal{O}_X)=0$ and $H^3(T_X)=0$. We first prove that the relative cotangent sheaf $\Omega^1_{X/S}$ is torsion-free: If we let $E:=f_*K_{X/S}^*$, then $X$ is a submanifold of $\mathbb{P}(E)$ and we get a short exact sequence $$0\longrightarrow N^*_{X|\mathbb{P}(E)}\longrightarrow\Omega^1_{\mathbb{P}(E)/S}|_X\longrightarrow\Omega^1_{X/S}\longrightarrow0.$$ Dividing out the torsion of $\Omega^1_{X/S}$ yields the sequence $$0\longrightarrow K\longrightarrow\Omega^1_{\mathbb{P}(E)/S}|_X\longrightarrow\Omega^1_{X/S}/\mathrm{tor}\longrightarrow0,$$ where $K$ is a reflexive rank-$1$ subsheaf (and thus a sub-line bundle) of the rank-$2$ bundle $\Omega^1_{\mathbb{P}(E)/S}|_X$. We obviously have $$\label{eq:torincl} N^*_{X|\mathbb{P}(E)}\subset K.$$ Smoothness of $X$ readily implies that $\Omega^1_{X/S}$ is locally free (and thus torsion-free) outside codimension $2$. This means that is an isomorphism outside codimension $2$. Since the left and right sides of  are both line bundles, this implies that is actually an equality, so $\Omega^1_{X/S}$ is torsion-free. Now, for a general point $s\in S$, the fiber $X_s$ is isomorphic to $\mathbb{P}_1$, and thus $K_X|_{X_s}\cong\Omega^1_{X/S}|_{X_s}\cong\Omega^1_{X_s}\cong\mathcal{O}_{\mathbb{P}_1}(-2)$. So we have $H^0((\Omega^1_{X/S}\otimes K_X)|_{X_s})=0$ for general $s$ and thus $H^0(\Omega^1_{X/S}\otimes K_X)=0$ by torsion-freeness of $\Omega^1_{X/S}$. Serre duality yields $\operatorname{Ext}^3(\Omega^1_{X/S},\mathcal{O}_X)\cong H^0(\Omega^1_{X/S}\otimes K_X)^*$, and the first part of the claim follows. Finally, again by Serre duality, $H^3(T_X)=0$ is equivalent to $H^0(\Omega^1_X\otimes K_X)=0$, which is clearly true since the bundle $\Omega^1_X\otimes K_X$ is negative on any smooth fiber of $f$. We have now gathered all the necessary tools to estimate the dimension of the deformation space of a non-algebraic conic bundle: \[prop:exdefo\] Let $S$ be a compact non-algebraic Kähler surface and $f\colon X\to S$ a conic bundle with $\rho(X)=\rho(S)+1$. Then $$h^1(T_X)\ge h^2(T_X).$$ If $h^1(T_X)=h^2(T_X)$, then $H^0(T_X)=0$ and $$c_1^2(S)=c_2(S)=c_1^2(f_*(K_{X/S}^*))=c_2(f_*(K_{X/S}^*))=0.$$ The pushforward $$E:=f_*(K_{X/S}^*)$$ is a rank-$3$ vector bundle on $S$ such that there exists a canonical embedding $X\subset\mathbb{P}(E)$. We have $H^3(T_X)=0$ by Proposition \[prop:conbun\], so by a standard Riemann-Roch calculation (see [@Schrack10 Proposition 4.2] for details), we obtain the formula $$\label{eq:riero} \begin{split} h^1(T_X)-h^2(T_X)&=h^0(T_X)+\bigl(c_2(E)-c_1(E)c_1(S)-\tfrac{4}{3}c_1^2(S)\bigr)\\ &\quad-\tfrac{2}{3}c_1^2(S)+7\chi(\mathcal{O}_S). \end{split}$$ $S$ being a non-algebraic Kähler surface, we have $c_1^2(S)\le0$ and $\chi(\mathcal{O}_S)\ge0$. Furthermore, by [@BL87 Théorème 3.1], we have $$\label{eq:banlep} c_2(E)-\tfrac{1}{3}c_1^2(E)\ge0,$$ so, by , in order to prove $h^1(T_X)\ge h^2(T_X)$, it remains to show that $$\label{eq:chern} c_1^2(E)\ge 3c_1(E) c_1(S)+4c_1^2(S).$$ If $a(S)=1$, this follows directly from Lemma \[lem:disca1\] by observing that $\Delta_f\in\lvert\det E^*\rvert$; if $a(S)=0$, then $c_1(E)=0$ by Lemma \[lem:disca0\] and the inequality follows because $c_1^2(S)\le0$. Now if $h^1(T_X)=h^2(T_X)$, all summands on the right side of  must be zero, so we directly obtain $h^0(T_X)=0$ and $c_1^2(S)=c_2(S)=0$. Since $S$ is non-algebraic, $c_1^2(S)=0$ implies that $c_1(S)\cdot L=0$ for any line bundle $L$ on $S$. Furthermore, we must have equality in and , so we also obtain $c_1^2(E)=c_2(E)=0$. We can now prove two of the theorems announced in the introduction: By Proposition \[prop:exdefo\], we only need to exclude the case $h^1(T_X)=h^2(T_X)=0$: Suppose $H^1(T_X)=H^2(T_X)=0$. Since we have $\operatorname{Ext}^3(\Omega^1_{X/S},\mathcal{O}_X)=0$ by Proposition \[prop:conbun\], applying $\operatorname{Hom}(\cdot,\mathcal{O}_X)$ to the relative cotangent sequence $$0\longrightarrow f^*\Omega^1_S \longrightarrow \Omega^1_X \longrightarrow \Omega^1_{X/S} \longrightarrow 0$$ yields $$0=H^2(T_X)\longrightarrow H^2(f^* T_S) \longrightarrow \operatorname{Ext}^3(\Omega^1_{X/S},\mathcal{O}_X)=0$$ and thus (using Serre duality) $$H^0(\Omega^1_S\otimes K_S)^*\cong H^2(T_S) \cong H^2(f^*T_S) =0.$$ We have $c_1^2(S)=0$ by Proposition \[prop:exdefo\], so $S$ is minimal. Together with $c_2(S)=0$ and $H^2(T_S)=0$, this implies that $S$ is properly elliptic by surface classification theory. Let $r\colon S\to C$ be the elliptic fibration of $S$ over a smooth curve $C$. Tensoring the relative cotangent sequence of the elliptic fibration $r$ by $K_S$ and taking global sections, we obtain the long exact sequence $$0\longrightarrow H^0(r^*K_C\otimes K_S) \longrightarrow H^0 (\Omega^1_S\otimes K_S) \longrightarrow H^0(\Omega^1_{S/C}\otimes K_S) \longrightarrow\dots,$$ so we obtain $$\label{eq:canvan} H^0(r^*K_C\otimes K_S)=0.$$ Now, by relative duality, $$\label{eq:reldual} r_*K_S=K_C\otimes (R^1r_*\mathcal{O}_S)^*,$$ where by Riemann-Roch on $C$, $$\begin{split} \deg(R^1r_*\mathcal{O}_S)^*&=-\deg(R^1r_*\mathcal{O}_S)=-\chi(R^1r_*\mathcal{O}_S)+\chi(\mathcal{O}_C)\\ &=h^0(\mathcal{O}_C)-h^1(\mathcal{O}_C)-h^0(R^1r_*\mathcal{O}_S)+h^1(R^1r_*\mathcal{O}_S). \end{split}$$ Now the Leray spectral sequence for $r$ yields $$\begin{split} h^0(\mathcal{O}_S)&=h^0(\mathcal{O}_C),\\ h^1(\mathcal{O}_S)&=h^1(\mathcal{O}_C)+h^0(R^1r_*\mathcal{O}_S),\\ h^2(\mathcal{O}_S)&=h^1(R^1r_*\mathcal{O}_S), \end{split}$$ hence $\deg(R^1r_*\mathcal{O}_S)^*=\chi(\mathcal{O}_S)=0$ (cf. [@BPV04 p. 213f.]). We obtain $$\deg(r_*(r^*K_C\otimes K_S))=4g(C)-4,$$ thus by equation , we conclude that $g(C)\le1$. On the other hand, as $S$ is a non-algebraic Kähler surface, we must have $H^0(K_S)=H^{2,0}(S)\ne0$, so implies $g(C)\ge1$. So $C$ must be an elliptic curve, in particular we have $K_C\cong\mathcal{O}_C$. But then, equation  contradicts $H^0(K_S)\ne0$. In order to prove Theorem \[thm:infconi\], we need to describe the exceptional cases of Proposition \[prop:exdefo\] in more detail: Suppose that $h^1(T_X)=h^2(T_X)$. Then by Proposition \[prop:exdefo\], we have $c_1^2(S)=0$, which yields that $S$ is a minimal surface. From $c_2(S)=0$, we obtain that $S$ must either be a torus or a minimal properly elliptic surface such that all singular fibers of the elliptic fibration on $S$ are multiples of smooth elliptic curves (cf. [@Kod63b p. 14]). Now suppose that $S$ is a torus. By Proposition \[prop:exdefo\], we have $c_1^2(E)=c_2(E)=0$. By [@Yan89 Theorem 5.12], this implies that $E$ is projectively flat. Now, suppose that there exists a rank-2 bundle $V$ on $S$ such that $X\cong\mathbb{P}(V)$. An easy calculation shows that we have $H^q(X, T_{X/S})\cong H^q(S, S^2V\otimes\det V^*)$ for all $q$, so by $H^0(T_X)=0$, it follows that $H^0(S^2V\otimes\det V^*)=0$ and therefore, by Serre duality, also $H^2(X,T_{X/S})\cong H^2(S,S^2V\otimes\det V^*)=0$. The relative tangent sequence then yields $h^1(T_X)\ge h^1(T_S)=4$ and $h^2(T_X)\le h^2(T_S)=2$, so in particular $h^1(T_X)>h^2(T_X)$. Vector bundles on K3 surfaces and tori {#sec:vector-bundles-k3} ====================================== The aim of this section is to prove algebraic approximability of projectivized rank-two bundles over Kähler surfaces of Kodaira dimension $0$. The most important step is to examine vector bundles over K3 surfaces and tori: Let $S$ be a K3 surface or a two-dimensional torus and $V$ a vector bundle on $S$. The objective is to study whether one can construct an algebraic approximation $\pi\colon\mathcal{S}\to T\ni0$ of $S=\mathcal{S}_0$ such that there exists a vector bundle $\mathcal{V}$ on $\mathcal{S}$ with $\mathcal{V}|_{\mathcal{S}_0}\cong V$. We will show that this is always possible (up to a twist of $V$ by a line bundle) if $V$ has rank $2$. We begin by examining the case of line bundles. In order to deal with higher-rank bundles later on, it is important to have the following statement about “simultaneous extendability” of line bundles: \[prop:k3torlb\] Let $S$ be a K3 surface or a torus. Let $\mathcal{K}\ni0$ be the Kuranishi space and $\pi\colon\mathcal{S}\to\mathcal{K}$ a versal deformation of $S=\mathcal{S}_0$. Then there exists an $(h^{1,1}(S)-\rho(S))$-dimensional smooth analytic subset $T$ of $\mathcal{K}$ containing $0$ such that for any line bundle $L$ on $S$, there is a line bundle $\mathcal{L}$ on $\mathcal{S}_T:=\pi^{-1}(T)$ with $\mathcal{L}|_{\mathcal{S}_0}\cong L$. We let $h:=h^{1,1}(S)$. Then $(\mathcal{K},0)$ is an $h$-dimensional smooth germ of a complex manifold and we have a $C^\infty$-isomorphism $$\mathcal{S}\cong_{C^\infty}S\times \mathcal{K}.$$ This isomorphism allows us to naturally identify $H^2(\mathcal{S}_t,\mathbb{C})$ with $H^2(S,\mathbb{C})$ for all $t\in\mathcal{K}$. If we now denote by $\langle\cdot,\cdot\rangle$ the $\mathbb{C}$-bilinear extension to $H^2(S,\mathbb{C})$ of the standard intersection form on $H^2(S,\mathbb{Z})$, then by Torelli’s theorem, the period map $$\begin{aligned} \mathcal{P}\colon\mathcal{K}&\to\mathbb{P}(H^2(S,\mathbb{C})),\\ t&\mapsto H^{2,0}(\mathcal{S}_t),\end{aligned}$$ gives a local isomorphism of $\mathcal{K}$ onto the period domain $$\Omega:=\{\,[\omega]\in\mathbb{P}(H^2(S,\mathbb{C}))\mid \langle\omega,\omega\rangle=0,\langle\omega,\overline{\omega}\rangle>0\,\}.$$ We now let $$T:=\mathcal{P}^{-1}(NS(S)^\perp\cap\Omega),$$ where $NS(S)^\perp\subset\mathbb{P}(H^2(S,\mathbb{C}))$ is the orthogonal space with respect to the bilinear form $\langle\cdot,\cdot\rangle$. Since $\langle\cdot,\cdot\rangle$ is non-degenerate, the projective space $NS(S)^\perp$ has dimension $h+1-\rho(S)$. Since $NS(S)^\perp$ is defined over $\mathbb{R}$, the inequality $\langle\omega,\overline{\omega}\rangle>0$ in the definition of $\Omega$ implies that $NS(S)^\perp\cap\Omega$ is smooth at every point. This implies that $T$ is smooth of dimension $h-\rho(S)$. Now for any line bundle $L$ on $S$ and for any $t\in T$, we have that $$\langle c_1(L),\mathcal{P}(t)\rangle=0,$$ i.e. $c_1(L)\in H^{1,1}(\mathcal{S}_t)$ by orthogonality of the Hodge decomposition. We consider the following excerpt from the push-forward of the exponential sequence on $\mathcal{S}_T$ via the restriction $\pi_T:=\pi|_{\mathcal{S}_T}$: $$R^1\pi_{T*}\mathcal{O}\stackrel{\psi}{\longrightarrow} R^1\pi_{T*}\mathcal{O}^*\stackrel{c_1}{\longrightarrow} R^2\pi_{T*}\mathbb{Z}\stackrel{\varphi}\longrightarrow R^2\pi_{T*}\mathcal{O}.$$ The constant extension of $c_1(L)$ defines a section $\zeta\in H^0(R^2\pi_{T*}\mathbb{Z})$ which satisfies $\varphi(\zeta)=0$ (because $c_1(L)\in H^{1,1}(\mathcal{S}_t)$ for all $t\in T$). By the above sequence, we obtain a section in $H^0(R^1\pi_{T*}\mathcal{O}^*)$ which, by the Leray spectral sequence, gives a line bundle $\tilde{\mathcal{L}}$ on $\mathcal{S}_T$ with $c_1(\tilde{\mathcal{L}}|_{\mathcal{S}_0})=c_1(L)$. Thus the line bundle $L\otimes\tilde{\mathcal{L}}^*|_{\mathcal{S}_0}$ is numerically trivial on $\mathcal{S}_0$, so is given by an element $\nu\in H^1(\mathcal{O}_{\mathcal{S}_0})$. The sheaf $R^1\pi_{T*}\mathcal{O}$ is locally free, so $\nu$ can be extended to give a section $\tilde{\nu}\in H^0(R^1\pi_{T*}\mathcal{O})$. If we let $\mathcal{N}$ be the line bundle on $\mathcal{S}$ given by $\psi(\tilde{\nu})$, then $\mathcal{L}:=\tilde{L}\otimes \mathcal{N}$ is a line bundle on $\mathcal{S}$ with $\mathcal{L}|_{\mathcal{S}_0}\cong L$. It is well-known that any non-trivial deformation of a K3 surface or a torus is an algebraic approximation. So, since $\rho(S)<h^{1,1}(S)$ for any non-algebraic surface $S$, the deformation $\pi$ given by Proposition \[prop:k3torlb\] is indeed an algebraic approximation. In order to deal with rank-two bundles arising as extensions of line bundles, it is important to achieve constancy of cohomology dimensions for the extended line bundles. This is automatic for K3 surfaces: \[prop:k3coh\] Let $S$ be a non-algebraic K3 surface, $T\ni0$ a complex manifold and $\pi\colon\mathcal{S}\to T$ a deformation of $S=\mathcal{S}_0$ with the property that for any effective divisor $D$ on $S$ there exists a line bundle $\mathcal{L}_D$ on $\mathcal{S}$ with $\mathcal{L}_D|_{\mathcal{S}_0}\cong\mathcal{O}_S(D)$. Then for any line bundle $\mathcal{L}$ on $\mathcal{S}$, there exists an open neighborhood $U\subset T$ of $0$ such that for every $q$, the map $U\ni t\mapsto h^q(\mathcal{L}|_{\mathcal{S}_t})$ is constant. For each $t\in T$, we let $\mathcal{L}_t:=\mathcal{L}|_{\mathcal{S}_t}$. If both $h^0(\mathcal{L}_0)\ne0$ and $h^2(\mathcal{L}_0)\ne0$, then $\mathcal{L}_0\cong\mathcal{O}_{\mathcal{S}_0}$. But then all $\mathcal{L}_t$ must be numerically trivial and hence trivial because $h^1(\mathcal{O}_{\mathcal{S}_t})=0$ for all $t$. So we can assume that either $h^0(\mathcal{L}_0)=0$ or $h^2(\mathcal{L}_0)=0$. By using Serre duality, we can further reduce to the case $h^2(\mathcal{L}_0)=0$. Then, by semi-continuity, we have $h^2(\mathcal{L}_t)=0$ for all sufficiently small $t$. Thus, by constancy of the holomorphic Euler characteristic $\chi(\mathcal{L}_t)$, it remains to show that $h^0(\mathcal{L}_t)$ is constant, i.e., that any section $s\in H^0(\mathcal{L}_0)$ can be extended to give a section $\tilde{s}\in H^0(\mathcal{L}_U)$ with $\tilde{s}|_{\mathcal{S}_0}=s$. So, let $s\in H^0(\mathcal{L}_0)$. Then $s$ defines an effective divisor $D=\sum_i D_i$, where the $D_i$ are (not necessarily distinct) prime divisors. By hypothesis, there exist line bundles $\mathcal{L}_i$ with $\mathcal{L}_i|_{\mathcal{S}_0}\cong\mathcal{O}_{\mathcal{S}_0}(D_i)$. Now, since $\mathcal{S}_0$ is a non-algebraic K3 surface, every $D_i$ is either a $(-2)$-curve or the reduction of a fiber of an elliptic fibration of $S$. In either case, an easy calculation shows that $h^1(\mathcal{O}_{\mathcal{S}_0}(D_i))=h^2(\mathcal{O}_{\mathcal{S}_0}(D_i))=0$. By semi-continuity and constancy of holomorphic Euler characteristic, this implies that $h^0(\mathcal{L}_i|_{\mathcal{S}_t})$ is independent of $t$. So, if we write $s=\prod_is_i$ with $s_i\in H^0(\mathcal{O}_{\mathcal{S}_0}(D_i))$, the $s_i$ can be extended to give sections $\tilde{s}_i\in H^0(\mathcal{L}_i)$ with $\tilde{s}_i|_{\mathcal{S}_0}=s_i$. But now $(\bigotimes_i\mathcal{L}_i)\otimes \mathcal{L}^*$ is numerically trivial on each $\mathcal{S}_t$ and thus trivial, so $\bigotimes_i\mathcal{L}_i\cong\mathcal{L}$ and the product $\tilde{s}:=\prod_i\tilde{s}_i$ defines a section in $H^0(\mathcal{L})$ with $\tilde{s}|_{\mathcal{S}_0}=s$. For tori, we need to choose the extended bundles appropriately: \[prop:torcoh\] Let $S$ be a non-algebraic torus and $\pi\colon\mathcal{S}\to T$ the deformation of $S=\mathcal{S}_0$ from Proposition \[prop:k3torlb\]. Then for any line bundle $L$ on $S$, there exists a line bundle $\mathcal{L}$ on $\mathcal{S}$ and an open neighborhood $U\subset T$ of $0$ such that $\mathcal{L}|_{\mathcal{S}_0}\cong L$ and $U\ni t\mapsto h^q(\mathcal{L}|_{\mathcal{S}_t})$ is a constant map for every $q$. As in the proof of Proposition \[prop:k3coh\], we can assume $h^2(L)=0$ and $h^0(L)\ne 0$. We then have $L\cong\mathcal{O}_S(D)$ for some divisor $D>0$ on $S$. Since there are no curves on a torus of algebraic dimension $0$, we have $a(S)=1$ and thus obtain an elliptic fibration $r\colon S\to C$ over some elliptic curve $C$. Now all irreducible curves on $S$ are fibers of $r$, so there exists an effective divisor $\delta$ on $C$ such that $D=r^*\delta$. Since $L$ extends to $\mathcal{S}$ by Proposition \[prop:k3torlb\], we have $a(\mathcal{S}_t)\ge1$ for all $t\in T$, thus, there exists a relative algebraic reduction $R\colon\mathcal{S}\to\mathcal{C}$ over $T$, where $\mathcal{C}\to T$ is a deformation of $C=\mathcal{C}_0$ and $R|_{\mathcal{S}_0}=r$. We can now choose an effective divisor $\tilde{\delta}$ on $\mathcal{C}$, which is flat over $T$, such that $\tilde{\delta}|_{\mathcal{S}_0}=\delta$. We then obtain the desired extension of the line bundle $L$ by letting $\mathcal{L}:=R^*\mathcal{O}_{\mathcal{C}}(\tilde{\delta})$. The preceding propositions enable us to extend rank-two bundles having a section: \[lem:k3torr2\] Let $S$ be a K3 surface or a torus and $V$ a rank-two vector bundle on $S$ with $H^0(V)\ne0$. Then there exists an algebraic approximation $\pi\colon\mathcal{S}\to T\ni0$ of $S=\mathcal{S}_0$ and a rank-two bundle $\mathcal{V}$ on $\mathcal{S}$ such that $\mathcal{V}|_{\mathcal{S}_0}\cong V$. Let $0\ne s\in H^0(V)$. If we denote by $G$ the divisorial part of the zero locus of $s$, we obtain a short exact sequence $$\label{eq:r2seq} 0\longrightarrow\mathcal{O}_S\longrightarrow V\otimes\mathcal{O}_S(-G)\longrightarrow L\otimes\mathcal{I}_Y\longrightarrow0,$$ where $L$ is some line bundle on $S$, and $\mathcal{I}_Y$ is the ideal sheaf of some locally complete intersection subvariety $Y\subset S$ of codimension at least $2$. We observe that sequence  is given by an extension class in $\operatorname{Ext}^1(L\otimes\mathcal{I}_Y,\mathcal{O}_S)$, which by Serre duality is isomorphic to $H^1(L\otimes\mathcal{I}_Y)$. Suppose we have managed to construct an algebraic approximation $\pi\colon\mathcal{S}\to T\ni0$ of $S=\mathcal{S}_0$, a line bundle $\mathcal{L}$ on $\mathcal{S}$ with $\mathcal{L}|_{\mathcal{S}_0}\cong L$ and a locally complete intersection subvariety $\mathcal{Y}\subset\mathcal{S}$, flat over $T$, with $\mathcal{Y}\cap\mathcal{S}_0=Y$. We define $\mathcal{Y}_t:=\mathcal{Y}\cap\mathcal{S}_t$ and suppose further that $h^1(\mathcal{L}_t\otimes\mathcal{I}_{\mathcal{Y}_t})$ is independent of $t$ (we remark that flatness of $\mathcal{Y}$ implies $\mathcal{I}_{\mathcal{Y}}|_{\mathcal{S}_t}\cong\mathcal{I}_{\mathcal{Y}_t}$). Then the extension class in $H^1(L\otimes\mathcal{I}_Y)=H^1(\mathcal{L}_0\otimes\mathcal{I}_{\mathcal{Y}_0})$ can be extended to give a class in $H^1(\mathcal{L}\otimes\mathcal{I}_{\mathcal{Y}})$. We thus obtain a short exact sequence $$0\longrightarrow\mathcal{O}_{\mathcal{S}}\longrightarrow\mathcal{V}'\longrightarrow\mathcal{L}\otimes\mathcal{I}_{\mathcal{Y}}\longrightarrow0$$ defining a rank-$2$ vector bundle $\mathcal{V}'$ on $\mathcal{S}$ with $\mathcal{V}'|_{\mathcal{S}_0}\cong V\otimes\mathcal{O}_S(-G)$. If we have chosen the algebraic approximation such that every effective divisor can be extended as a line bundle, there is a line bundle $\mathcal{G}$ on $\mathcal{S}$ with $\mathcal{G}|_{\mathcal{S}_0}\cong\mathcal{O}_S(D)$, and we can just set $\mathcal{V}:=\mathcal{V}'\otimes\mathcal{G}$. We now describe how to satisfy the assumptions we made above: We first remark that for any deformation $\pi\colon\mathcal{S}\to T$ of $S=\mathcal{S}_0$, any line bundle $\mathcal{L}$ on $\mathcal{S}$ and any codimension $\ge2$ subvariety $\mathcal{Y}\subset\mathcal{S}$ flat over $T$, we have for every $t\in T$ a short exact sequence $$0\longrightarrow(\mathcal{L}\otimes\mathcal{I}_{\mathcal{Y}})|_{\mathcal{S}_t}\longrightarrow\mathcal{L}_t\longrightarrow\mathcal{L}|_{\mathcal{Y}_t}\longrightarrow0,$$ whose long exact cohomology sequence shows that $h^1(\mathcal{L}_t\otimes\mathcal{I}_{\mathcal{Y}_t})$ is constant provided that $h^0(\mathcal{L}_t\otimes\mathcal{I}_{\mathcal{Y}_t})$ and all the $h^q(\mathcal{L}_t)$ are constant. Now, if $h^0(L\otimes\mathcal{I}_Y)=0$, we can just choose $\pi$ according to Proposition \[prop:k3torlb\]. We can then choose $\mathcal{L}$ with the desired properties according to Propositions \[prop:k3coh\] and \[prop:torcoh\]. The constancy of $h^0(\mathcal{L}_t\otimes\mathcal{I}_{\mathcal{Y}_t})$ then follows by semi-continuity for any flat extension $\mathcal{Y}\subset\mathcal{S}$ of $Y$. So, we can now assume that $Y\ne\emptyset$ and that $h^0(L\otimes\mathcal{I}_Y)>0$. Then also $h^0(L)>0$ and we have $L\cong\mathcal{O}_S(D)$ for some divisor $D>0$ on $S$. We first consider the case $a(S)=0$: Then $S$ is a K3 surface and $D$ is composed of $(-2)$-curves intersecting (at most) transversally. Furthermore, we have $h^0(L)=1$ and thus $Y\subset D$ as an inclusion of complex spaces. If we choose the algebraic approximation $\pi\colon\mathcal{S}\to T$ according to Proposition \[prop:k3torlb\], we can find by Propositions \[prop:k3torlb\] and \[prop:k3coh\] a unique effective divisor $\mathcal{D}$ on $\mathcal{S}$ with $\mathcal{D}\cap\mathcal{S}_0=D$. Letting $\mathcal{D}_t:=\mathcal{D}\cap\mathcal{S}_t$ for $t\in T$, each $\mathcal{D}_t$ is again composed of $(-2)$-curves sharing the same intersection pattern as $D$. Since the $Y$ consists of isolated points only (together with infinitesimal neighborhoods) we can obviously construct a codimension $\ge2$ subvariety $\mathcal{Y}\subset\mathcal{S}$, flat over $T$ with $\mathcal{Y}_0=Y$, such that $\mathcal{Y}\subset\mathcal{D}$ as an inclusion of complex spaces. Setting $\mathcal{L}:=\mathcal{O}_{\mathcal{S}}(\mathcal{D})$, this means that the defining section of $\mathcal{D}_t$ inside $H^0(\mathcal{L}_t)$ actually yields a section inside $H^0(\mathcal{L}_t\otimes\mathcal{I}_{\mathcal{Y}_t})$. We have thus proved $h^0(\mathcal{L}_t\otimes\mathcal{I}_{\mathcal{Y}_t})=1$ for any $t$. It remains to consider the case that $a(S)=1$: We first assume $S$ to be a K3 suface. Then we choose the algebraic approximation $\pi\colon\mathcal{S}\to T$ according to Proposition \[prop:elsurf\]. As before, we can deform any divisor on $\mathcal{S}$ to a divisor on the neighboring $\mathcal{S}_t$ such that locally, its structure remains unchanged. Thus, by suitably choosing $\mathcal{Y}\subset\mathcal{S}$, we can ensure extendability of all sections in $H^0(\mathcal{S}_t\otimes\mathcal{I}_{\mathcal{Y}_t})$ as before. If now $S$ is a torus with $a(S)=1$, we investigate the linear system of divisors on $S$ defined by the subspace $H^0(L\otimes\mathcal{I}_Y)\subset H^0(L)$. We can decompose this linear system as $$\label{eq:divdec} \lvert H^0(L\otimes\mathcal{I}_Y)\rvert=D_0+\lvert L\otimes\mathcal{O}_S(-D_0)\rvert,$$ where $D_0$ is an effective divisor on $S$ containing $Y$ as a complex subspace. We choose the algebraic approximation $\pi\colon\mathcal{S}\to T$ according to Proposition \[prop:k3torlb\] and line bundles $\mathcal{M}$ and $\mathcal{N}$ on $\mathcal{S}$ with $\mathcal{M}|_{\mathcal{S}_0}\cong\mathcal{O}_S(D_0)$ and $\mathcal{N}|_{\mathcal{S}_0}\cong L\otimes\mathcal{O}_S(-D_0)$ according to Proposition \[prop:torcoh\]. We obtain a section $\tilde{s}_0\in H^0(\mathcal{M})$ such that $\tilde{s}_0|_{\mathcal{S}_0}$ is just the defining section of the divisor $D_0$. By choosing $\tilde{s}_0$ suitably, we can achieve that there exists a codimension $\ge2$ subvariety $\mathcal{Y}\subset\mathcal{S}$ which is flat over $T$, such that $\mathcal{Y}_0=Y$ and $\tilde{s}_0$ vanishes along $\mathcal{Y}$. If we now let $\mathcal{L}:=\mathcal{M}\otimes\mathcal{N}$, we can extend any section $\zeta\in H^0(L\otimes\mathcal{I}_Y)$ to a section in $H^0(\mathcal{L}\otimes\mathcal{I}_{\mathcal{Y}})$ by writing $\zeta=s_0\cdot \zeta_0$ according to , extending $\zeta_0$ to a section $\tilde{\zeta}_0\in H^0(\mathcal{N})$ according to Proposition \[prop:torcoh\] and choosing $\tilde{s}_0\cdot\tilde{\zeta}_0$ as extension of $\zeta$. To conclude the proof, we remark that each of the algebraic approximations of $S$ chosen above has the property that any effective divisor on $S$ extends as a line bundle on $\mathcal{S}$ (cf. Propositions \[prop:elsurf\] and \[prop:k3torlb\]), so the line bundle $\mathcal{G}$ mentioned at the beginning indeed exists. Using different arguments, we can also treat the case of simple rank-two bundles: \[lem:simpbun\] Let $S$ be a K3 surface or a torus. Then there exists an algebraic approximation $\pi\colon\mathcal{S}\to T\ni 0$ of $S=\mathcal{S}_0$ such that for any simple rank-two vector bundle $V$ on $S$, there exists a rank-two bundle $\mathcal{V}$ on $\mathcal{S}$ such that $\mathcal{V}|_{\mathcal{S}_0}\cong V$. We let $\pi\colon\mathcal{S}\to T$ be the algebraic approximation of $S=\mathcal{S}_0$ according to Proposition \[prop:k3torlb\]. We consider the projective bundle $\mathbb{P}(V)$ over $S$. An easy calculation shows that $h^2(T_{\mathbb{P}(V)/S})=h^2(S^2V\otimes\det V^*)=0$. This implies by [@Hor74 Theorem 6.1] that there exists a deformation $\psi\colon\mathcal{X}\to T$ of $\mathbb{P}(V)=\mathcal{X}_0$ and a holomorphic map $F\colon\mathcal{X}\to\mathcal{S}$ extending the natural projection $f\colon\mathbb{P}(V)\to S$. We observe that the bundle $V$ is given by $V=f_*\mathcal{O}_{\mathbb{P}(V)}(1)$ and that $\mathcal{O}_{\mathbb{P}(V)}(2)\cong K_{\mathbb{P}(V)/S}^*\otimes f^*\det V$. By Proposition \[prop:k3torlb\], there exists a line bundle $\mathcal{G}$ on $\mathcal{S}$ with $\mathcal{G}|_{\mathcal{S}_0}\cong\det V$. We define a line bundle $\mathcal{L}$ on $\mathcal{X}$ as $\mathcal{L}:=K_{\mathcal{X}/\mathcal{S}}\otimes F^*\mathcal{G}$. Now obviously the bundle $\mathcal{L}_0\cong\mathcal{O}_{\mathbb{P}(V)}(2)$ has a square root. The obstruction to having a square root is purely topological, so by the topological triviality of $\psi$, also $\mathcal{L}$ has a square root, i.e. a line bundle $\mathcal{K}$ with $\mathcal{K}^{\otimes2}\cong\mathcal{L}$. We can then define $\mathcal{V}:=F_*\mathcal{L}$ to obtain a rank-$2$ bundle on $\mathcal{S}$ with $\mathcal{V}|_{\mathcal{S}_0}\cong V$. The following lemma allows to carry over the above statements to blown-up K3 surfaces and tori: \[lem:blowind\] Let $S$ be a compact surface, $p\colon\hat{S}\to S$ the blow-up of a point $s_0\in S$ and $\hat{V}$ a vector bundle on $\hat{S}$. Suppose that there exists an algebraic approximation $\pi\colon\mathcal{S}\to T\ni0$ of $S=\mathcal{S}_0$ and a vector bundle $\mathcal{V}$ on $\mathcal{S}$ such that $$\mathcal{V}|_{\mathcal{S}_0}\cong (p_*\hat{V})^{**}.$$ Then, after shrinking $T$ sufficiently, there exists an algebraic approximation $\hat{\mathcal{S}}\to T$ of $\hat{S}=\hat{\mathcal{S}}_0$, a proper modification $P\colon\hat{\mathcal{S}}\to\mathcal{S}$ over $T$ with $P|_{\hat{\mathcal{S}}_0}=p$ and a vector bundle $\hat{\mathcal{V}}$ on $\hat{\mathcal{S}}$ such that $$\hat{\mathcal{V}}|_{\hat{\mathcal{S}}_0}\cong\hat{V}.$$ We choose a submanifold $\mathcal{C}\subset\mathcal{S}$ which is mapped isomorphically onto $T$ by $\pi$ such that $\mathcal{C}\cap\mathcal{S}_0=\{s_0\}$. We now let $P\colon\hat{\mathcal{S}}\to\mathcal{S}$ be the blow-up of $\mathcal{C}\subset\mathcal{S}$. The deformation $\hat{\pi}:=\pi\circ P\colon\hat{\mathcal{S}}\to T$ is then an algebraic approximation of $\hat{S}=\hat{\mathcal{S}}_0$. It remains to construct the desired bundle $\hat{\mathcal{V}}$ on $\hat{\mathcal{S}}$: We let $V:=(p_*\hat{V})^{**}$. Since $V$ is a reflexive sheaf on the two-dimensional manifold $S$, it is locally free. The bidual of the natural map $p^*p_*\hat{V}\to\hat{V}$ induces a map $$\iota\colon p^*V\to\hat{V}^{**}=\hat{V}.$$ If we let $\mathcal{E}\subset\hat{\mathcal{S}}$ be the exceptional divisor of $P$, then $E:=\mathcal{E}\cap\mathcal{S}_0\subset\mathcal{S}_0$ is the exceptional divisor of $p$ and $\iota$ is an isomorphism over $\hat{\mathcal{S}}_0\setminus E$. We thus obtain a short exact sequence $$\label{eq:blowindseq} 0\longrightarrow p^*V\longrightarrow\hat{V}\longrightarrow Q\longrightarrow 0,$$ given by a cohomology class in $\operatorname{Ext}^1(Q,p^*V)$, where $Q$ is a coherent sheaf on $\hat{\mathcal{S}}_0$ with support contained in $E$. We want to construct an extension of sequence  to all of $\hat{\mathcal{S}}$. By hypothesis, there is a vector bundle $\mathcal{V}$ on $\mathcal{S}$ with $\mathcal{V}|_{\mathcal{S}_0}\cong V$. We can choose an open neighborhood of $\mathcal{E}$ inside $\hat{\mathcal{S}}$ such that $P^*\mathcal{V}|_{\mathcal{U}}$ is trivial and such that $\hat{\pi}$ induces an isomorphism $\mathcal{U}\cong U\times T$ for some open neighborhood $U$ of $E$ in $\hat{\mathcal{S}}_0$. Via this isomorphism, we can extend the sheaf $Q$ constantly to give a coherent sheaf on $\mathcal{U}$. The support of this sheaf being contained in $\mathcal{E}$, we can extend by zero to obtain a coherent sheaf $\mathcal{Q}$ on $\hat{\mathcal{S}}$. By construction, we have for every point $t\in T$: $$\operatorname{Ext}^1(\mathcal{Q}|_{\hat{\mathcal{S}}_t},(P^*\mathcal{V})|_{\hat{\mathcal{S}}_t})\cong\operatorname{Ext}^1(Q|_U,\mathcal{O}_U),$$ so in particular, the dimension $$\dim\operatorname{Ext}^1(\mathcal{Q}|_{\hat{\mathcal{S}}_t},(P^*\mathcal{V})|_{\hat{\mathcal{S}}_t})$$ is independent of $t$. By [@BPS80 Satz 3], this implies that the base change morphism $$\operatorname{\mathcal{E}\mathit{xt}}^1_{\hat{\pi}}(\mathcal{Q},P^*\mathcal{V})\otimes\mathbb{C}(t)\to\operatorname{Ext}^1(\mathcal{Q}|_{\hat{\mathcal{S}}_t},(P^*\mathcal{V})|_{\hat{\mathcal{S}}_t})$$ is an isomorphism for any $t\in T$. This means that (after shrinking $T$ sufficiently) we can extend the class of sequence  to a class in $\operatorname{Ext}^1(\mathcal{Q},P^*\mathcal{V})$ giving a short exact sequence $$0\longrightarrow P^*\mathcal{V}\longrightarrow\hat{\mathcal{V}}\longrightarrow\mathcal{Q}\longrightarrow0,$$ which defines a bundle $\hat{\mathcal{V}}$ as desired. We can now prove the central result of this section: It is sufficient to construct an algebraic approximation $\pi\colon\mathcal{S}\to T$ of $S=\mathcal{S}_0$ such that there exists a vector bundle $\mathcal{V}$ on $\mathcal{S}$ with $\mathcal{V}|_{\mathcal{S}_0}\cong V\otimes L$ for some line bundle $L$ on $S$ (we can then take $\mathbb{P}(\mathcal{V})$ as an algebraic approximation of $\mathbb{P}(V)$). By Lemma \[lem:blowind\], we can assume $S$ to be minimal, so $S$ is either a K3 surface or a torus. Now, if $H^0(V \otimes L)\ne0$ for some $L$, we are done by Lemma \[lem:k3torr2\]. If $H^0(V\otimes L)=0$ for every line bundle $L$ on $S$, then in particular $V$ is simple, so we can apply Lemma \[lem:simpbun\]. Further results {#sec:algebr-appr-cert} =============== Threefolds bimeromorphic to a product ------------------------------------- \[lem:surfapp\] Let $S$ be a compact Kähler surface. Then there exists an algebraic approximation $\pi\colon\mathcal{S}\to T\ni0$ of $S$ with the following property: For any compact curve $C\subset S$ there exists an open subset $U\subset S$ with $C\subset U$ and an open subset $\mathcal{U}\subset\mathcal{S}$ such that (i) $\mathcal{U}\cap\mathcal{S}_0=U$ and (ii) $\mathcal{U}$ is isomorphic to the product $U\times T$ over $T$. If $a(S)=1$ then $S$ is elliptic and any connected compact curve on $S$ is contained in a fiber a the elliptic fibration. The claim then follows directly from Proposition \[prop:elsurf\]. If $a(S)=0$, we first observe that we can assume $S$ to be minimal. There are no compact curves on a torus of algebraic dimension $0$, so we are left with the case that $S$ is a K3 surface. Then $C$ is a configuration of $(-2)$-curves. We take $\pi$ to be the deformation from Proposition \[prop:k3torlb\]. Then, by Proposition \[prop:k3coh\], $C$ can be extended to a curve $\mathcal{C}\subset\mathcal{S}$ which is proper and flat over $T$, such that for each $t\in T$, the intersection $\mathcal{C}_t:=\mathcal{C}\cap\mathcal{S}_t$ is a configuration of $(-2)$-curves isomorphic to $C$. The claim now follows because isomorphic configurations of $(-2)$-curves have isomorphic neighborhoods. Lemma \[lem:surfapp\] is the essential ingredient to the By Hironaka’s Chow Lemma (cf. [@Hir75]) and resolution of singularities, there exists a compact complex manifold $\hat X$ together with a holomorphic map $p\colon \hat{X}\to X$ obtained as a finite composition of blow-ups along smooth centers such that there exists a proper modification $f\colon\hat{X}\to \mathbb{P}_1\times S$ making the diagram $$\xymatrix{ &\hat{X}\ar[d]^p\ar[dl]_f\\ \mathbb{P}_1\times S\ar@{-->}[r]_\phi&X }$$ commute. The center of $f$ has codimension at least $2$ in $\mathbb{P}_1\times S$, so there exists a (not necessarily connected) compact curve $C\subset S$ such that $f$ is an isomorphism over $\mathbb{P}_1\times(S\setminus C)$. We now take an algebraic approximation $\pi\colon\mathcal{S}\to T$ of $S$ according to Lemma \[lem:surfapp\]. Then there is an open neighborhood $U\subset S$ of $C$ and an open subset $\mathcal{U}\subset\mathcal{S}$ with $\mathcal{U}\cap\mathcal{S}_0=U$ and $\mathcal{U}\cong U\times T$ over $T$. We observe that $\tilde{\pi}:=\operatorname{pr}_2\circ(\operatorname{id}\times\pi)\colon\mathbb{P}_1\times\mathcal{S}\to T$ is an algebraic approximation of $\mathbb{P}_1\times S$ such that the restriction $\tilde{\pi}|_{\mathbb{P}_1\times\mathcal{U}}$ is isomorphic to the natural projection $\mathbb{P}_1\times U\times T\to T$. The modification $f$ thus trivially induces a modification $\tilde{f}\colon f^{-1}(\mathbb{P}_1\times U)\times T\to\mathbb{P}_1\times\mathcal{U}$ which can be glued together with the identity on $\mathbb{P}_1\times(\mathcal{S}\setminus\mathcal{U})$ to give a proper modification $F\colon\hat{\mathcal{X}}\to\mathbb{P}_1\times\mathcal{S}$ such that the composition $\tilde{\pi}\circ F\colon\hat{\mathcal{X}}\to T$ is an algebraic approximation of $\hat{X}$. Since $p$ is a composition of blow-ups along smooth centers, we have $p_*\mathcal{O}_{\hat{X}}=\mathcal{O}_X$ and $R^1p_*\mathcal{O}_{\hat{X}}=0$. Using [@Hor76 Theorem 8.2] we conclude (after shrinking $T$ sufficiently) the existence of a proper holomorphic submersion $\psi\colon\mathcal{X}\to T$ and a holomorphic map $P\colon\hat{\mathcal{X}}\to\mathcal{X}$ such that $P\circ\psi=\tilde{\pi}\circ F$, $\psi^{-1}(0)\cong X$ and $P|_{(\tilde{\pi}\circ F)^{-1}(0)}=p$. It is now easy to see that $\psi$ is an algebraic approximation of $X$. By Fujiki’s classification results ([@Fuj83 Proposition 14.1]), every non-algebraic uniruled Kähler threefold carries a meromorphic $\mathbb{P}_1$-fibration. Conic bundles ------------- We let $E:=f_*K_{X/S}^*$. By hypothesis, $r_*E$ is a rank-3 bundle on $C$ and we get an exact sequence $$\label{eq:ellseq} 0\longrightarrow r^*r_*E\stackrel{\alpha}{\longrightarrow}E\longrightarrow Q\longrightarrow 0,$$ where $Q$ is a torsion sheaf on $S$. There exists a finite set $Z\subset C$ such that the support of $Q$ is contained in $r^{-1}(Z)$. This means that $\alpha|_{r^{-1}(C\setminus Z)}$ is an isomorphism. We choose an algebraic approximation $\pi\colon\mathcal{S}\to T\ni0$ of $S=\mathcal{S}_0$ and an extension $R\colon\mathcal{S}\to C\times T$ of $r$ according to Proposition \[prop:elsurf\]. By the local triviality of $R$ over $T$, there exists a coherent sheaf $\mathcal{Q}$ on $\mathcal{S}$, flat over $T$, with $\mathrm{supp}(\mathcal{Q})\subset R^{-1}(Z\times T)$, such that $\mathcal{Q}|_{\mathcal{S}_0}\cong Q$ and $$\operatorname{Ext}^1(\mathcal{Q}|_{\mathcal{S}_t},\mathcal{O}_{\mathcal{S}_t})\cong\operatorname{Ext}^1(Q,\mathcal{O}_S)$$ for all $t\in T$. This implies the existence of a rank-$3$ bundle $\mathcal{E}$ on $\mathcal{S}$ sitting in a short exact sequence $$\label{eq:ellseqext} 0\longrightarrow R^*\operatorname{pr}_1^*r_*E\stackrel{\tilde{\alpha}}{\longrightarrow}\mathcal{E}\longrightarrow\mathcal{Q}\longrightarrow0$$ whose restriction to $\mathcal{S}_0$ yields sequence . In particular, $\tilde{\alpha}|_{R^{-1}((C\setminus Z)\times T)}$ is an isomorphism. By construction, the section $$\sigma\in H^0(S^2E\otimes\det E^*)$$ yields a section $$\tilde{\sigma}\in H^0\left((S^2\mathcal{E}\otimes\det\mathcal{E}^*)|_{R^{-1}((C\setminus Z)\times T)}\right)$$ with $\tilde{\sigma}|_{R^{-1}((C\setminus Z)\times\{0\})}=\sigma|_{r^{-1}(C\setminus Z)}$. By the local triviality of the situation, however, any point $p\in Z$ posesses an open neighborhood $U\subset C$ such that there exists a section $$\tilde{\tilde{\sigma}}\in H^0((S^2\mathcal{E}\otimes\det\mathcal{E}^*)|_{R^{-1}(U\times T)})$$ with $\tilde{\tilde{\sigma}}|_{R^{-1}(U\times\{0\})}=\sigma|_{r^{-1}(U)}$ and $\tilde{\tilde{\sigma}}|_{R^{-1}((U\setminus\{p\})\times T)}=\tilde{\sigma}|_{R^{-1}((U\setminus\{p\})\times T)}$. The sections $\tilde{\sigma}$ and $\tilde{\tilde{\sigma}}$ glue together to give a section $$s\in H^0(S^2\mathcal{E}\otimes\det\mathcal{E}^*)$$ with $s|_{\mathcal{S}_0}=\sigma$. The section $s$ defines a conic bundle over $\mathcal{S}$ yielding an algebraic approximation of $X$. Let $E:=f_*K_{X/S}^*$. Then $X\subset\mathbb{P}(E)$ is given by a section $s\in H^0(S^2E\otimes\det E^*)$. Let $\pi\colon\mathcal{S}\to T$ be the algebraic approximation of $S=\mathcal{S}_0$ from Proposition \[prop:k3torlb\]. We write $E\cong L_1\oplus L_2\oplus L_3$ and choose line bundles $\mathcal{L}_i$ on $\mathcal{S}$ with $\mathcal{L}_i|_{\mathcal{S}_0}\cong L_i$ for $i=1$, $2$, $3$. Letting $\mathcal{E}:=\mathcal{L}_1\oplus\mathcal{L}_2\oplus\mathcal{L}_3$, we have $\mathcal{E}|_{\mathcal{S}_0}\cong E$, and by Proposition \[prop:k3coh\], the section $s$ extends to a section $\tilde{s}\in H^0(S^2\mathcal{E}\otimes\det\mathcal{E}^*)$ with $\tilde{s}|_{\mathcal{S}_0}= s$, defining a conic bundle $\mathcal{X}\subset\mathbb{P}(\mathcal{E})$ over $\mathcal{S}$ yielding an algebraic approximation of $\mathcal{X}_0\cong X$.

--- abstract: 'The potential complementarity between chondrules and matrix of chondrites, the Solar System’s building blocks, is still a highly debated subject. Complementary superchrondritic compositions of chondrite matrices and subchondritic chondrules may point to formation of these components within the same reservoir or, alternatively, to mobilization of elements during secondary alteration on chondrite parent bodies. Zinc isotope fractionation through evaporation during chondrule formation may play an important role in identifying complementary relationships between chondrules and matrix and is additionally a mobile element during hydrothermal processes. In an effort to distinguish between primary Zn isotope fractionation during chondrule formation and secondary alteration, we here report the Zn isotope data of five chondrule cores, five corresponding igneous rims and two matrices of the relatively unaltered Leoville CV3.1 chondrite. The detail required for these analyses necessitated the development of an adjusted Zn isotope analyses protocol outlined in this study. This method allows for the measurement of 5 ng Zn fractions, for which we have analyzed the isotope composition with an external reproducibility of 120 ppm. We demonstrate that we measure primary Zn isotope signatures within the sampled fractions of Leoville, which show negative $\updelta$$^{66}$Zn values for the chondrule cores ($\updelta$$^{66}$Zn = –0.43$\pm$0.14 $\permil$), more positive values for the igneous rims ($\updelta$$^{66}$Zn = –0.01$\pm$0.30 $\permil$) and chondritic values for the matrix ($\updelta$$^{66}$Zn = 0.19$\pm$0.14 $\permil$). In combination with elemental compositions and petrology of these chondrite fractions, we argue that chondrule cores, igneous rims and matrix could have formed within the same reservoir in the protoplanetary disk. The required formation mechanism involves Zn isotope fractionation through sulfide loss during chondrule core formation and concurrent thermal processing of matrix material. Depleted olivine-bearing grains representing this processed matrix would have accreted to the depleted chondrule cores and subsequently reabsorbed material (including $^{66}$Zn-rich) from a complementary volatile-rich gas, thereby forming the igneous rims. This would have allowed the rims to move towards an isotopically chondritic composition, similar to the non-processed matrix in Leoville. We note that Zn isotope analyses of components in other chondrites (f.e., CM, CO, EC) are necessary to identify if this complementarity relationship is generic or unique for each chondrite group. The development of a Zn isotope protocol for singularly small samples is a step forward in that direction.' address: - 'Université de Paris, Institut de Physique du Globe de Paris, CNRS UMR 7154, 1 rue Jussieu, 75238 Paris, France' - 'Université de Paris, Institut de Physique du Globe de Paris, CNRS UMR 7154, 1 rue Jussieu, 75238 Paris, France; Institut Universitaire de France, Paris, France' author: - Elishevah van Kooten - Frédéric Moynier bibliography: - 'Zntechnical2.bib' title: 'Zinc isotope analyses of singularly small samples ($<$5 ng Zn): investigating chondrule-matrix complementarity in Leoville' --- Zn isotopes ,CV chondrules ,igneous rims Introduction ============ Chondrites represent the remnant building blocks of our Solar System and have been relatively unaltered since their formation approximately 4.56 billion years ago. Most chondrites mainly consist of chondrules, once molten silicate droplets formed by transient heating events in the protolanetary disk. They are cemented together by submicron-sized dust grains called matrix and further contain less abundant refractory inclusions (the first condensed solids, i.e., CAIs and AOAs, [@Connelly2012]) and FeNi metal [@Krot2009; @Scott2014]. Even though chondrites have never been molten and, hence, maintain their initial accretionary texture, most chondrites have been modified to some degree on their parent bodies by thermal metamorphism, aqueous alteration or both. These changes affect the chemical and isotope composition of bulk chondrites as well as of individual components by processes such as diffusion, fluid interaction and evaporation. In addition, chondrules may have experienced volatile-loss (or perhaps gain) and concurrent stable isotope fractionation during their formation and before their accretion into chondrite parent bodies [@Ebel2018]. To identify these different processes is paramount to decipher the pre-accretionary history of chondritic components from post-accretionary secondary alteration of asteroids.\ Zinc is a moderately volatile element with a relatively low condensation temperature (T$_{c}$ = 726 K, [@Lodders2003]) and is well-suited element to study volatility-related processes during the formation of the Solar System. Zinc isotope fractionation reported in bulk chondrites and achondrites has shed light on processes related to volatile-loss. Lunar basalts and un-brecciated eucrites show correlations between heavy isotope enrichment and volatile-depletion, consistent with kinetic fractionation during evaporation [@Paniello2012; @Paniello2012a; @Kato2015; @Moynier2017]. Bulk carbonaceous chondrites on the other hand (with the exception of CH and CB chondrites), show light isotope enrichment with volatile depletion, a trend opposite to what is expected from these materials if they experienced evaporation [@Luck2005; @Albarede2009; @Pringle2017; @Sossi2018]. Similarly, chondrules of the CV chondrite Allende are also enriched in light Zn isotopes, an observation attributed to sulfide loss during chondrule formation when heavy Zn isotopes preferentially partitioned into the sulfides [@Pringle2017].\ With the Zn isotope analyses of Allende chondrules, a first step has been taken into tracing chondrule formation and the link between Zn isotope variations of bulk carbonaceous chondrites and the relative abundance of their components. However, many new questions arise from this study. It is unclear, for example, how the high degree of hydrothermal alteration in Allende ($>$CV3.6; [@Bonal2016]) relates to the Zn isotope variability in individual chondrules, since Zn is highly mobile in aqueous fluids. Moreover, chondrules from other chondrite groups might not show the same variability and magnitude of Zn isotope fractionation and it is unknown how this would relate to the bulk isotope signature of their parent bodies and the supposed complementarity between chondrules and matrix. Furthermore, multiple melting events of chondrules could have chemically and isotopically affected any initial relationships between chondritic components. These numerous issues require a detailed investigation of chondrule and matrix fractions, which is presently highly limited by the size of these constituents. Zinc isotope analyses usually rely on hundreds of ng Zn to reach an external reproducibility (2SD) of 40 ppm for $^{66}$Zn/$^{64}$Zn and 50 ppm for $^{68}$Zn/$^{64}$Zn [@Chen2013]. Recent analyses of Zn isotope ratios in seawater have reduced the amount of Zn to 6-7 ng with an external reproducibility of 200 ppm [@Conway2013]. However, the seawater matrix is completely different from solid rock materials and the use of double-spike in this method prevents the determination of mass-dependent relationships between $\updelta$$^{66}$Zn and $\updelta$$^{68}$Zn or the possible determination of non-mass dependent isotopic variations. @Pringle2017 recently reported chondrule Zn isotope data with total Zn abundances between 30 and 500 ng, but without measurements of correspondingly low standards to test the accuracy and precision of these data. Here, we report a fine-tuned method to measure Zn isotopes in unprecedentedly small samples, with Zn abundances of 2-5 ng and with an external reproducibility of 120 ppm (5 ng Zn) for $^{66}$Zn/$^{64}$Zn for a range of terrestrial standards. We use this method in a proof of concept to demonstrate Zn isotope variations between individual chondrules of the relatively unaltered CV3.1 [@Bonal2016] chondrite Leoville, as well as relationships between core, igneous rim and surrounding matrix of these chondrules. Materials and methods ===================== Materials --------- We have sampled multiple terrestrial standards with varying bulk compositions and with known Zn isotope signatures. Per standard, five samples were weighed (50-200 $\upmu$g per sample) containing approximately 5-10 ng Zn each, from bulk powders of BCR-2 (USGS basalt), AGV2 (USGS andesite) and CP1 (metasediment from the Queyras Schiste lustrés Complex, [@Inglis2017]). In addition, we have selected larger aliquots of BHVO-2 (USGS basalt) and CV chondrite NWA 12523 (5 mg), with totals of $\sim$600 ng Zn each. We have, furthermore, sampled five chondrule cores, five igneous rims and two surrounding matrices of the Leoville CV3.1 chondrite. This CV chondrite was selected based on its low degree of alteration relative to other CV chondrites [@Bonal2016]. Although Kaba has experienced a similarly low degree of thermal metamorphism, it has undergone pervasive aqueous alteration [@Keller1990]. The Leoville fractions were extracted by New Wave microdrill with tungsten carbide drill bits at the Institute de Physique du Globe de Paris (IPGP) and thereafter transferred to clean Savillex beakers following methods used in @vankooten2017a. The chondrule sizes ranged between 0.5 and 2 mm and care was taken not to drill too deep ($<$200 $\upmu$m) to avoid contamination from surrounding materials. All drillspots were carefully examined under a plain-light microscope and suspicious spots were discarded. We have drilled approximately three cylindrical holes (100 $\upmu$m diameter, 200 $\upmu$m depth) per sample, yielding 2-5 ng of Zn.\ Elemental composition {#methods_composition} --------------------- To determine the elemental composition of the Leoville chondrule and matrix samples, we have analyzed major and minor element compositions by Agilent 7900 ICP-Q-MS at IPGP. Elemental compositions were analyzed from sample solutions taken from the 1.5M HBr cut acquired from anion column chromatography (see section \[methods\_column\]). Since sample digestion with HF acid leads to complete loss of Si, it was not possible to determine the total weight and corresponding weight percentages of elements in the samples. Hence, we present elemental compositions as ratios normalized to Mg. The terrestrial standards AGV2 and BCR-2 were measured alongside Leoville chondrule fractions to determine potential elemental fractionation on the columns. The percentual uncertainty was calculated by comparing these standard analyses to literature values. Besides bulk compositions of the sampled chondrule and matrix fractions, we have also acquired elemental maps using the Zeiss EVO MA10 scanning electron microscope (SEM) at IPGP. Column chromatography {#methods_column} --------------------- We have developed a protocol for Zn isotope analyses, adapted from the method used by @Moynier2006 to accommodate for the small sample sizes used in this work ($<$10 ng of Zn). The major limitations of measuring such small samples stem from 1) Zn contamination from acids used during purification, 2) from resin organics and elemental impurities that cause matrix effects on the mass spectrometer and 3) from low signal/noise ratios during isotope analyses. Hence, our adjusted purification protocol is aimed at reducing these factors to insignificant levels. The Zn blank produced in previous digestion and column chromatography methods for $>$1 $\upmu$g Zn aliquots is typically 5-20 ng [@Arnold2010; @Chen2013; @Moynier2015], comprising 100-400 % of a 5 ng sample. It is unclear what the major contributing factors to the blank are in previous studies, thus, we have carefully analyzed the Zn abundance in all acids used during our procedure (Table \[tab:blanks\]). First, we have compared the Zn blank during sample digestion in 1 ml 3:1 concentrated HNO$_{3}$/HF mixtures, with and without the use of Parr bombs. The digestion with four different Parr bombs included 24 hours at 150$^{\circ}$C and 48 hours at 210$^{\circ}$C inside a furnace. The use of Parr bombs contributes $<$0.6 ng to the Zn blank, whereas digestion of samples excluding the Parr bombs adds insignificant amounts of Zn to the samples. Leoville samples have been digested in Parr bombs to ensure full assimilation of Zn-containing refractory spinels. Subsequently, the samples were dried down and left for 24 hours on a hotplate at 140$^{\circ}$C in 1 ml aqua regia, which contained negligible Zn blanks. After drying down the samples, they were taken up in 100 $\upmu$l 1.5M HBr and were left to flux on a hotplate for 1 hour before loading the samples onto the columns.\ Our Zn purification protocol employs an anion AG-1 X8 (200-400 mesh) exchange resin in teflon 4:1 shrinkage tubes, of which the resin bed is 2 mm $\times$ 2.5 mm, with a resin volume of 100 $\upmu$l. The columns are cleaned by alternating 7M HNO$_{3}$ with MQ-water three times, after which they are conditioned with 400 $\upmu$l 1.5M HBr. @Moynier2015 use similar columns during a second and third purification step with a total acid volume of 11 ml per column (6 ml 1.5M HBr and 5 ml 0.5M HNO$_{3}$). We have carried out repeated elution tests using samples with varying compositions (e.g., BHVO-2 and CV chondrite NWA 12523) to reduce the amount of acids and, thus, the Zn contribution from these acids (Fig. \[fig:FigZnelutionprofile1\] and \[fig:FigZnelutionprofileMgFe\]). We show consistently that 800 $\upmu$l of 1.5M HBr is needed to elute $>$99.5 % of the sample matrix (Fig. \[fig:FigZnelutionprofile1\] and \[fig:FigZnelutionprofileMgFe\]). Figure \[fig:FigZncutcontamination\] shows that the size of the sample load over the column affects the matrix contribution to the Zn-cut. Samples smaller than $\sim$100 $\upmu$g have an increasingly contaminated Zn-cut by siderophile elements (e.g., Fe, Ni, Cr). Zinc is eluted in 800 $\upmu$l 0.5M HNO$_{3}$, with Zn yields $>$99.9 %. This step is repeated once more to remove trace amounts of sample matrix. Over a period of four months we have analyzed five procedural blanks reflecting the Zn contamination of the column chromatography procedure. We find that the total column blank is non-reproducible, but in all cases $<$0.35 ng (on average $<$0.2 ng), which represents $<$7 % of a 5 ng Zn sample. Our HBr and HNO$_{3}$ acid blanks are typically lower than the procedural blank ($<$0.01 ng), suggesting that the contamination is derived from random Zn impurities within the anion resin related to the initial manufacturing of the resin [@Shiel2009], or perhaps from laboratory wear such as gloves [@Garcon2017]. Hence, the Zn procedural blank would only be lowered by use of a different resin, an even smaller column size or not at all. Nevertheless, for our purpose, the total Zn blank is surmountable for sample sizes of 5-10 ng Zn, since for observed natural Zn isotope fractionation ranges (i.e., $\sim$2 $\permil$, [@Moynier2017]), the added error from blank contamination falls within the external reproducibility of our measurements (see section \[limitations\]). We note that this purification procedure is suited for total sample sizes $<$5 mg. Loading sample solutions onto 100 $\upmu$l columns above this level resulted in a stagnation of acid elution. Neptune Plus MC-ICPMS --------------------- Zinc isotope ratios were measured using a Thermo Scientific Neptune Plus Multi-Collector Inductively-Coupled-Plasma Mass-Spectrometer (MC-ICP-MS) at IPGP. The analytical setup is identical to @Moynier2015, barring the use of an APEX IF introduction system instead of a spray chamber. This allowed for a 3-4 times higher Zn signal. With one block of 30 measurements and an integration time of 8.389 seconds, 5 ng Zn aliquots in 500 $\upmu$l 0.1M HNO$_{3}$ (10 ppb) solutions could typically be measured at $\sim$0.7-1.0 V of $^{64}$Zn. Blank solutions were analyzed before and after measurements and generally contained $\sim$ 3-5 mV of $^{64}$Zn. At the beginning of each session (three sessions in total), the BHVO-2 standard (with a total of 600 ng Zn) was measured three times in aliquots of 5 ng Zn and bracketed by a JMC-Lyon standard. This was done to provide an indication of the potential accuracy and reproducibility of such small samples. All data are represented in the delta notation as permil deviations from the JMC-Lyon standard: $$\updelta^\textrm{x}\textrm{Zn} =\left [ \frac{(^\textrm{x}\textrm{Zn}/^{64}\textrm{Zn})_\textrm{\scriptsize sample}}{(^\textrm{x}\textrm{Zn}/^{64}\textrm{Zn})_\textrm{\scriptsize JMC-Lyon}} -1 \right] \times 10^{3}$$ Results ======= Terrestrial standards --------------------- Nine repeat analyses of the BHVO-2 standard with a total Zn abundance of 600 ng were carried out using fractions of 5 ng Zn per measurement. The resulting average $\updelta$$^{66}$Zn and $\updelta$$^{68}$Zn values were 0.34$\pm$0.08 $\permil$ (2SD) and 0.69$\pm$0.16 $\permil$ (2SD), respectively (Table \[tab:standardsZn\]). The $\updelta$$^{66}$Zn value is in agreement with accepted literature values [@Moynier2017] of 0.29$\pm$0.09 $\permil$ (2SD) and $\updelta$$^{66}$Zn and $\updelta$$^{68}$Zn values define a mass-dependent relationship (Fig. \[fig:FigZnstandards\]). The 2SD error of our measurements (80 ppm for $\updelta$$^{66}$Zn) is similar to that of repeat cross-laboratory analyses of BHVO-2 ($>$1 $\upmu$g Zn per measurement) and give us the expected lower limit external reproducibility of our analyses. Five Zn isotope analyses of BCR-2 from separate digestions of 5-10 ng Zn yield average $\updelta$$^{66}$Zn and $\updelta$$^{68}$Zn values of 0.16$\pm$0.12 $\permil$ (2SD) and 0.34$\pm$0.32 $\permil$ (2SD), respectively. A similar average $\updelta$$^{66}$Zn value is achieved for repeat measurements of AGV2 ($\updelta$$^{66}$Zn = 0.21$\pm$0.16 $\permil$, $\updelta$$^{68}$Zn = 0.58$\pm$0.60 $\permil$), although with a larger error on $\updelta$$^{66}$Zn and $\updelta$$^{68}$Zn. Both geological standards are within error of reported average literature values of $\updelta$$^{66}$Zn (0.25$\pm$0.08 $\permil$ for BCR-2, 0.29$\pm$0.06 $\permil$ for AGV2; [@Moynier2017]). Our analyses of CP1 metasediments show that the obtained Zn isotope data ($\updelta$$^{66}$Zn = –0.07$\pm$0.10 $\permil$, $\updelta$$^{68}$Zn = –0.17$\pm$0.11 $\permil$) are also in agreement with previously reported data ($\updelta$$^{66}$Zn = 0.00$\pm$0.08 $\permil$, [@Inglis2017]). All data lie on the mass-dependent correlation line (Fig. \[fig:FigZnstandards\]). Even though all Zn isotope standards are in good agreement with reported values, we note that a small negative offset ($<$0.08 $\permil$ on $\updelta$$^{66}$Zn) may exist between our dataset and reported values. However, this potential offset is unresolvable, within error of the external reproducibility of our dataset, which we estimate to be 120 ppm on $\updelta$$^{66}$Zn. Although the uncertainty of AGV2 is higher than this estimation, this is likely the result of matrix element contamination to the Zn-cut (see section \[limitations\]). Leoville chondrule cores, igneous rims and matrix ------------------------------------------------- ### Petrology A total of five Leoville type-I (FeO-poor) chondrules were selected for Zn isotope analyses (Fig. \[fig:FigBSEchondrules1\], \[fig:FigBSEchondrules2\] and \[fig:MapsRims\]). Four out of five chondrule cores consist of porphyritic subhedral/euhedral forsterite inside a glassy mesostasis. Subhedral/euhedral enstatite has grown between olivine crystals, zoned by Ca-rich pyroxene. One chondrule core contains barred olivine with interspersed enstatite needles (Ch3). Most chondrule cores contain minor FeNi metal or troilite inclusions ($<$5 vol.%), except for Ch5, which contains a large proportion of metal ($\sim$30 vol.%). The cores are surrounded by sometimes thin and sharp (Ch3 and Ch5), but mostly irregular rings of FeNi metal and/or troilite aggregates (Ch6). In Ch1 and Ch2 the distinction between metal/sulfide rim and outer igneous rim cannot be made as the metal/sulfide is interspersed between the silicates. The igneous rims of the Leoville chondrules are more fine-grained than their cores. The rims consist of small 10-100 $\upmu$m anhedral to euhedral forsterite phenocrysts ($\sim$ 300 $\upmu$m in Ch6), which are to variable extent overgrown by low-FeO pyroxene (Ch1 $>$ Ch3 $>$ Ch2 $\approx$ Ch6 $>$ Ch5). The forsterite grains that are overgrown are typically anhedral and are indistinguishable in composition to the core olivines (i.e., low FeO content). The rim mineralogy is complemented by $<$100 $\upmu$m rounded metal and troilite grains, which appear identical in composition to the metal/sulfide rims between the cores and igneous rims. The rims typically contain more metal and especially troilite relative to the cores. These rim minerals are set in a glassy mesostasis that is similar in composition to that of the core, with the exception of Ch5, where the rim mesostasis is relatively Al-poor and Na-rich. The average thickness of the igneous rims is fairly constant between chondrules (200-300 $\upmu$m), even though the chondrule core size can vary significantly (0.2 - 2 mm). Within a rim, however, the thickness can vary considerably by a factor 5. Some chondrules also have surrounding fine-grained dust rims that show similar variations in thickness and these rims appear to complement the thickness of the igneous rims (Ch2 and Ch3). Hence, where the igneous rims are relatively thin, the dust rims are thicker. The boundaries between igneous and dust rims are generally irregular. Fractions of igneous rim can be found as inclusions in the dust rim (Ch3). The matrix locations from Ch1 and Ch2 are sampled close to the chondrule boundaries, although we could not distinguish between intra-chondrule matrix and dust rims. In most cases the dust rims of Leoville chondrules probably overlap with each other. In regions where clear dust rims are distinguished, a very narrow strip of intra-chondrule matrix seperates the two rims. Hence, in regions where chondrules lie closer together, the dust rims are directly connected to each other. The dust rims are generally more FeO-rich and fine-grained compared to the intra-chondrule matrix (Fig. \[fig:FigBSEchondrules2\], Ch3). The latter typically contains larger (mostly elongated) Fe sulfide and calcite grains. ### Composition Major and minor element compositions of Leoville chondrule fractions can be divided according to their 50% condensation temperatures, where Ca, Al, and Ti are the most refractory elements and Co$>$Fe$>$Ni$>$Cr$>$Mn$>$Na are increasingly volatile [@Lodders2003]. These divisions show similar elemental patterns between cores, rims and matrices (Fig. \[fig:Majorelements\], Table \[tab:majorel\]). For example, Ca/Mg, Al/Mg and Ti/Mg ratios are on average higher in the chondrules cores relative to the igneous rims, whereas they show a smaller variability and are near-chondritic (chondritic being the CI ratio in Fig. \[fig:Majorelements\]) between rims and matrix. The cores have typically sub-chondritic Cr/Mg and Ni/Mg ratios, whereas the igneous rims are chondritic or super-chondritic. The Mn/Mg ratios of the cores are also sub-chondritic, whereas the igneous rims are close in composition to bulk CV chondrites and the matrix super-chondritic. The cores and rims have a larger spread in Na/Mg than in Mn/Mg ratios and both are close in composition to bulk CV chondrites. We note that for these volatile elements the bulk CV composition is sub-chondritic. The matrix Na/Mg ratios are sub-chondritic but higher than the cores and rims. Finally, the Fe/Mg ratios are sub-chondritic for the cores, chondritic for the rims and super-chondritic for the matrix. Overall, the igneous rims are very close in composition to bulk CV chondrites for a range of elements with different condensation temperatures, whereas the cores are super-chondritic for refractory elements and sub-chondritic for more volatile elements (with the exception of Na/Mg). The compositions of the rims are similar to those of the cores, except for Cr/Mg, Ni/Mg and Fe/Mg ratios, which can be linked to a higher abundance of metal and sulfide in the igneous rims. The matrix is generally chondritic or super-chondritic for a range of elemental compositions. ### Zinc isotope analyses {#Results_Znisotopes} We report on the Zn isotope analyses of five chondrule cores, five igneous rims surrounding these cores and two matrix samples in the near vicinity of Ch1 and Ch2 (Fig. \[fig:ChondruleZnisotopes\], Table \[tab:LeovilleZn\]). During the collection of the samples by microdrill it was difficult to estimate the total amount of Zn in each sample, since the Zn concentration in chondrules is variable and we could not accurately measure the total weight of each sample. Consequently, only four out of twelve samples contained approximately 5 ng of Zn, whereas the other chondrule fractions contained $\sim$2 ng. These less concentrated aliquots were analyzed after repeat measurements of 5 ppb BHVO-2 (n=9) and bulk CV chondrite (n=12) solutions, to redetermine the analytical uncertainty of these smaller samples. The $\updelta$$^{66}$Zn values of the reference standards are reproduced accurately, albeit with a relatively larger uncertainty for the CV chondrite ($\updelta$$^{66}$Zn$_{BHVO-2}$ = 0.27$\pm$0.05 $\permil$ \[2SD\], $\updelta$$^{66}$Zn$_{CV}$ = 0.27$\pm$0.19 $\permil$ \[2SD\]), whereas the $\updelta$$^{68}$Zn values are less accurate and precise ($\updelta$$^{68}$Zn$_{BHVO-2}$ = 0.34$\pm$0.13 $\permil$ \[2SD\], $\updelta$$^{68}$Zn$_{CV}$ = 0.49$\pm$1.17 $\permil$ \[2SD\]). The 2SD error of the CV chondrite for a 5 ppb solution is more realistic than the small error found for similarly sized BHVO-2 aliquots, since we measure with half the signal intensity. The good reproducibility of BHVO-2 is likely related to optimal circumstances for instrumental drift. Most of the individual measurements are not mass-dependent due to the lesser precision of the $\updelta$$^{68}$Zn values, most likely because of the lower signal/noise ratio of the analyses. Nevertheless, our $\updelta$$^{66}$Zn standard data are reproducible within an uncertainty of 190 ppm and in excellent agreement with literature values of BHVO-2 and bulk CV chondrites. For this reason, we report only the $\updelta$$^{66}$Zn values of the chondrule fractions, although $\updelta$$^{68}$Zn values can be found in Table \[tab:LeovilleZn\]. The chondrule cores have $\updelta$$^{66}$Zn values ranging between –0.36$\pm$0.12 $\permil$ and –0.54$\pm$0.19 $\permil$, and are tightly clustered with an average of –0.43$\pm$0.14 $\permil$. These values coincide with the most negative whole chondrule data (e.g. core and rim) obtained for Allende [@Pringle2017]. The igneous rims have more positive $\updelta$$^{66}$Zn signatures, ranging between –0.14$\pm$0.19 $\permil$ and 0.23$\pm$0.19 $\permil$, with an average of –0.01$\pm$0.30$\permil$. We note that the sample with the outlying value of 0.23$\permil$ (Ch5) has a distinct rim mineralogy compared to the other chondrules, olivine being absent (Fig. \[fig:MapsRims\]). The two matrix analyses from locations surrounding Ch1 and Ch2 yield similar Zn isotope signatures with $\updelta$$^{66}$Zn values of 0.24$\pm$0.12 $\permil$ and 0.14$\pm$0.12 $\permil$, respectively. These values coincide with the average bulk $\updelta$$^{66}$Zn value of CV chondrites (0.24$\pm$0.12 $\permil$, [@Pringle2017]), but are somewhat lower than the matrix-rich aliquot taken from Allende (0.35 $\permil$, [@Pringle2017]). Discussion ========== Size limitations on Zn isotope analyses {#limitations} --------------------------------------- Repeat analyses of 5 ng Zn aliquots from the BHVO-2 standard yield accurate and reproducible $\updelta$$^{66}$Zn values, in agreement with literature data. The external reproducibility of 80 ppm (2SD) on $\updelta$$^{66}$Zn is identical to the spread found in inter-laboratory analyses of BHVO-2. This error is lower than the reproducibility of separate digestions of geological standards, each containing 5 ng of Zn (120 ppm on $\updelta$$^{66}$Zn). The higher error can be explained in two ways. First, we find a positive correlation between the $^{62}$Ni/$^{64}$Zn ratio and the deviation from the mass-dependent fractionation line for terrestrial standards (Fig. \[fig:NiZnthreshold\]). Since we correct for the interference of $^{64}$Ni on $^{64}$Zn by monitoring the abundance of $^{62}$Ni [@Moynier2015], the $^{62}$Ni/$^{64}$Zn ratio likely reflects the abundance of residual matrix elements in the Zn-cut after purification, which will introduce matrix effects during Zn isotope analyses. This ratio appears to be independent from the initial Ni/Zn ratio before purification. For example, the Ni/Zn ratio of AGV2 is a factor 2 lower (0.22) than that of CP1 (0.42), but has higher residual Ni/Zn ratios. Moreover, AGV2 has an intermediate Zn concentration (86 ppm) relative to CP1 (39 ppm) and BCR-2 (127 ppm). Hence, it is unclear how the matrix composition of AGV2 would affect the deviation from the mass-dependent fractionation line. It is possible that during or after column purification the AGV2 samples were contaminated somehow, since it would only take $\sim$10 pg of added Ni to contaminate the sample. Nevertheless, future Zn isotope analyses on terrestrial samples with similarly small sample sizes would benefit from the dependency of the $^{62}$Ni/$^{64}$Zn ratio on a successful outcome of a measurement. A threshold ratio ($^{62}$Ni/$^{64}$Zn $<$ 1.5 $\times$ 10$^{-3}$) could be assigned to ensure measurements fall within the external reproducibility found in this study. This way, before all sample is consumed during analyses, another column pass might be performed to remove the residual matrix. However, this dependency is likely not relevant for Leoville fractions analyzed in this study. The lack of correlation between $^{62}$Ni/$^{64}$Zn and the mass-dependency in these samples (Fig. \[fig:NiZnthresholdchondrules\]) suggests that the large variations between $\Delta$$^{66}$Zn (= $\updelta$$^{68}$Zn/2-$\updelta$$^{66}$Zn, the deviation from the mass-dependent correlation line) with similar $^{62}$Ni/$^{64}$Zn ratios are probably the result of low signal/noise ratios on $^{68}$Zn for samples $<$5 ng Zn (see below).\ Another caveat in analyzing very small Zn aliquots is matrix effects from resin-derived organics during mass spectrometry measurements. Anion exchange resins have been shown to contaminate samples with organics stripped from the resin, even after extensive cleaning [@Pietruszka2008]. These organics produce matrix effects during MC-ICPMS measurements. Although our purification protocol significantly reduces the amounts of acids used, the relative column size is not significantly adjusted to the smaller samples. The samples in this study are 2-3 orders of magnitude smaller than previous studies, whereas the reduction of resin volume is only by a factor 3.5. Hence, the resulting Zn-cut contains a relatively large proportion of organics that can affect our data. Zinc isotope analyses of BCR-2, AGV2 and CP1 show that a small offset may exist between our dataset and reported values from literature ($<$0.08 $\permil$) even though our data are within error of these reported values and the offset is within the external reproducibility of our measurements. This potential negative offset is likely caused by matrix effects from resin-derived organics. Both the Zn procedural blank and the organics can be limited by decreasing the total column volume of the Zn purification protocol. Here, we are restricted by the amount of sample that can be passed over the column without stagnating the resin. The processing of even smaller samples may result in a relative increase of the residual matrix/Zn ratio (Fig. \[fig:FigZncutcontamination\]), requiring another column pass, which in turn would result in a larger organics/Zn ratio. Hence, a sample size of 5 ng Zn is probably at the limit of what is possible at the moment for reasonable Zn isotope data, reasonable meaning that the external reproducibility of our measurements is relatively small compared to the expected range of $\updelta$$^{66}$Zn values ($\sim$2 $\permil$, [@Moynier2017]).\ Our aim in this study was to proof the concept of measuring very small Zn aliquots by sampling fractions of Leoville chondrules. As reported in section \[Results\_Znisotopes\], most chondrule fractions contained $<$5 ng of Zn. For most of the individual measurements of these fractions, we cannot test the mass-dependency due to the lesser precision of the $\updelta$$^{68}$Zn values, most likely because of the lower signal/noise ratio of the analyses. Although our standard data using such small Zn concentrations reflect accurate values for $\updelta$$^{66}$Zn, the Leoville chondrule fractions may suffer from a relatively large blank contribution, which is based on our repeated blank tests conservatively estimated to be $<$0.935 ng, but on average $<$0.350 ng. This could contribute up to 50% of the total sample size. Hence, we may expect a maximal positive shift in $\updelta$$^{66}$Zn values of $<$0.36 $\permil$ for the chondrule cores, $<$0.15 $\permil$ for the igneous rims and $<$0.05 $\permil$ for the matrix. This correction uses $\updelta$$^{66}$Zn = 0 $\permil$ for the blank, which is based on measurements of the isotope composition of the procedural blank in this study. To this end, we have put four aliquots of 1 ng JMC Lyon standard through our Zn purification procedure and have measured these separately for their Zn isotope composition against a pure and unprocessed JMC Lyon standard (Table \[tab:standardsZn\]). The $\updelta$$^{66}$Zn values of the processed standards are all indistinguishable from the unprocessed standard, which suggests that the composition of the blank is identical. This is in agreement with similar Zn isotope measurements of bulk column blanks from Zn purification protocols using the same anion resin and reagents [@Shiel2009]. In detail, these authors have processed bulk column blanks of Zn elutes from anion resins (AG MP1, 200-400 mesh, macroporous version of AG1) and added that to a pure Zn standard solution (in-house standard PCIGR1, identical in Zn isotope composition to the JMC-Lyon standard used in this study; [@Shiel2009]) at different ratios between blank and standard (e.g., 0.2, 10 and 50 %). @Shiel2009 show, using the same HNO$_{3}$ flux sample treatment after purification as in this study, that the blank contaminated Zn standards are typically positive and within 2SD error ($\updelta$$^{66}$Zn = 0.1 $\permil$) of the pure Zn standard. Hence, the negative $\updelta$$^{66}$Zn values observed for the chondrule cores are not likely to be the result of a blank contribution with a very anomalous Zn isotope values. It is probable that the chondrule cores were somewhat more negative than we observe here. Furthermore, we note that the chondrule cores, although measured for different sample sizes (e.g., 1-3 ng Zn), exhibit identical $\updelta$$^{66}$Zn values (–0.43$\pm$0.14 $\permil$, 2SD), suggesting that the upper limit blank contribution of 0.9 ng Zn is an overestimation and that it is more realistic to use the average blank of 0.3 ng Zn or lower. Thus, such a blank contribution would yield corrected $\updelta$$^{66}$Zn values of –0.56 $\permil$ for the chondrule cores, which is within error of our measurements. Hence, we suggest that the Zn isotope data from the chondrule cores is accurate enough to allow for cosmochemical interpretations, even though being less robust than 5 ng Zn samples due to a relatively higher blank contribution and a lower signal/noise ratio during MC-ICPMS measurements.\ The applied blank correction may be counteracted (for negative $\updelta$$^{66}$Zn values) or enlarged (for positive $\updelta$$^{66}$Zn values) due to organic interferences. However, since the Zn isotope data for different chondrule fractions (cores, igneous rims and matrices) are tightly clustered within each group (using different Zn concentrations during isotope analyses) and the procedural blanks are variable, we suggest that this offset is likely minor.\ In summary, we show that Zn isotope analyses are size limited by matrix elements contaminating the Zn-cut, by resin derived organics, low signal/noise ratios and blank contributions. Samples $>$5 ng Zn result in mass-dependent, accurate and reproducible data when purified sufficiently, whereas samples $<$5 ng are mostly susceptible to low signal/noise ratios of $^{68}$Zn and relatively high blank contributions. Petrological and compositional relationships between Leoville fractions {#elements_complementarity} ----------------------------------------------------------------------- To understand the magnitude and direction of Zn isotope fractionation between cores, igneous rims and surrounding matrix within the Leoville CV3.1 chondrite, it is first important to discuss the petrological and compositional context of the chondritic fractions and how they complement each other. ### Compositional heteorogeneity of CV chondrite matrix @Rubin1987 were the first to measure the composition of multiple chondrule cores and their igneous rims within a CV chondrite (e.g., Allende). We show in Figure \[fig:Majorelements\] that the average core and rim compositions in this study closely match the average data of @Rubin1987. In the same study, bulk matrix compositions of Allende were measured using various analytical techniques, including defocused beam, neutron activation and wet chemical analyses. The Allende matrix composition differs significantly from our Leoville matrix analyses in that the latter is relatively enriched in volatiles. We observe a tentative correlation between the 50% condensation temperature and the fractionation factor between our dataset and that of @Rubin1987 and @Clarke1971 using neutron activation and wet chemical analyses, respectively (Fig. \[fig:FractionationCondensation\]). Note that we do not compare our data to defocused beam analyses, since these type of measurements may cause artificial variability in Mg amongst others [@Barkman2013; @Zanda2018]. The Fe/Mg ratio in the Leoville matrix is a factor $\sim$2 higher (T$_{c}$$<$1328$^{\circ}$C), whereas the Mn/Mg ratio is a factor $\sim$3 higher (T$_{c}$$<$1158$^{\circ}$C) and the Na/Mg ratio a factor $\sim$4 higher (T$_{c}$$<$958$^{\circ}$C) compared to the Allende matrix (Fig. \[fig:Majorelements\]). Defocused beam analyses of CV chondrite matrix including Leoville show that average matrix compositions are typically super-chondritic and fairly uniform within the same chondrite (Table \[tab:Matrix\], [@Clarke1971; @McSween1977; @Kracher1985; @Zolensky1993; @Klerner2001; @Huss2005; @Hezel2010; @Palme2015]). Even considering a $\sim$40% offset between microprobe and bulk analyses of Orgueil, the latter is more Fe/Mg rich [@Zanda2018], suggesting that Fe/Mg ratios measured by microprobe are underestimated. On a sub-mm scale, the variations in Fe/Mg become larger and matrix compositions as measured in this study are not uncommon [@Palme2015]. Individual point analyses of matrix in Efremovka and Mokoia have Fe/Mg and Mn/Mg ratios $<$4.4 and $<$0.045, respectively. Since our matrix samples have been sampled on a sub-mm scale, it is not suprising to find such highly super-chondritic values. The small scale matrix variations could be the result of nebular heterogeneity [@Grossman1996] or redistribution through parent body alteration, since it has been suggested that the most unaltered chondrites have CI-like matrix compositions [@Zanda2018]. During aqueous alteration, formation of secondary calcite, fayalitic olivine and phyllosilicate/sulfide compounds will redistribute elements throughout the matrix [@Wasson2009]. For example, the analyses of a matrix region enriched in secondary calcite, would be defined by higher Ca/Mg ratios, whereas regions where sulfide precipitated are characterized by higher Fe/Mg and S/Mg ratios. Contrary, our matrix analyses show a fairly continuous trend between elemental condensation temperature and fractionation (Fig. \[fig:FractionationCondensation\]), which cannot easily be explained by redistribution through aqueous alteration. Moreover, since the matrix regions sampled from our Leoville section show very limited evidence for alteration (e.g., no observations of larger secondary carbonate or sulfide agglomerates), we suggest that our matrix data mainly reflect nebular heterogeneity, rather than variability through aqueous alteration. This is likely true at least for refractory and moderately volatile elements. More volatile elements such as Na, K and S have been shown to transport into chondrules during very early stages of thermal metamorphism, during which the matrix is being depleted and the chondrules enriched [@Grossman2005]. This is in agreement with the Leoville matrix being lower in Na/Mg relative to the CI ratio, whereas the chondrule cores have higher values (Fig. \[fig:Majorelements\]). For other moderately volatile elements, the matrix composition is close to the CI ratio. Moreover, the expected fractionation factor for the Na/Mg ratio between Allende and Leoville matrix is offset from the general trend (Fig. \[fig:FractionationCondensation\]). For moderately volatile elements with 50% condensation temperatures $>$1000 $^{\circ}$C, we suggest that the Leoville matrix analyses reflect primary nebular compositions. Thus, we can use our data to infer genetic relationships between chondrule cores, igneous rims and the surrounding matrix. ### Complexity of chondrule petrology Chondrules are typically divided into type I (FeO-poor) and type II (FeO-rich) compositions, after which they are further classified based on porphyritic, barred or cryptocrystalline textures and mineralogies (olivine and/or low-Ca pyroxene phenocrysts). Most chondrules have experienced more than one (strong) melting event, as evidenced by primary and secondary cores (separated by a metal/sulfide rim, generally with the same mineralogy and oxidation degree) and surrounding igneous rims [@Rubin2010; @Scott2014]. These rims are usually finer-grained ($<$100 $\upmu$m) than their corresponding cores, even though they are often described as ’coarse-grained’ to avoid confusion with fine-grained matrix rims. Igneous rims have so far been observed enveloping chondrule cores in CV ($\sim$50 %, [@Rubin1984; @Rubin1987]), CO ($<$1 %, [@Rubin1984]), CR (unknown, [@Krot2004]) and ordinary chondrites ($\sim$10 %, [@Rubin1984; @Krot1995]). @Rubin1987 described the petrology and composition of CV chondrite cores and corresponding igneous rims and found that, similar to our chondrule data, individual core-rim pairs were not compositionally matched and probably did not form during the same heating event. It is generally acknowledged that igneous rims formed after the melting and solidification of the host chondrule [@Krot1995], unlike observed mineralogical zonation of chondrule cores that have displayed open system behaviour (i.e., low-Ca pyroxene rims, [@Friend2016; @Barosch2019]). The rims had to have been heated to lower temperatures (T$_{sol}$$<$1000-1200 $^{\circ}$C, [@Hewins2005; @Jones2018]) than the cores (T$_{liq}$$<$1700$^{\circ}$C), to avoid complete melting and homogenization of the chondrule [@Rubin1987]. Thus, the igneous rims may have experienced (liquid-solid) sintering rather than complete melting, considering the liquidus temperature. This begs the question whether igneous rims formed from similar precursor materials as their host chondrules and if this precursor dust is similar in composition to the surrounding matrix. With this in mind, we will discuss below scenarios of complementarity and non-complementarity (for a detailed overview of both models see [@Hezel2018; @Zanda2018]) during the formation of the Leoville chondrule cores and rims. ### Genetic relationships of Leoville fractions In recent complementarity studies that assess the genetic relationship between chondrules and matrix [@Bland2005; @Palme2015; @Hezel2018], chondrules are considered as single entities that have experienced one melting event, whereas in reality most chondrules likely experienced multiple heating events [@Bollard2017]. As such, the average composition of a chondrule core and its igneous rim cannot be complementary to the surrounding matrix. It is, however, possible to evaluate potential complementarity between the core and rim as well as the rim and the surrounding matrix. Here, we discuss the scenarios of complementarity and non-complementarity for our petrological observations as well as major and minor element data. In Figures \[fig:Scenarios\] and \[fig:SchematicChFor\] we present these scenarios with Cr/Mg versus Fe/Mg ratios as an example. Complementarity between chondrule and matrix requires the precursor material to have a CI-like composition (e.g., falling along the CI ratio line, Fig. \[fig:Scenarios\]) and the chondrules and matrix to have formed in the same reservoir [@Hezel2018]. If we take the average chondrule core composition as a starting point, the resulting primary dust rim around the core (from which the igneous rim formed) must be equal to Mx1 if the ratio between chondrule and dust rim was approximately 1:1. A calculation (using Adobe Illustrator PathArea plugin) of the surface areas of chondrule cores and igneous rims from the five chondrules analyzed in this study shows that this ratio is realistic. If the primary dust rim subsequently experienced another heating event and evaporative loss, then the final composition of the igneous rim could be near chondritic (again with a dust:chondrule ratio of 1:1), whereas the final matrix (Mx2) would be super-chondritic. While our data appear to be perfectly in agreement with the complementarity scenario, applying the same model to other major element plots from Figure \[fig:Majorelements\] yields very different chondrule:dust ratios. For example, using refractory elements Ti and Al would result in a near 100 % abundance of Mx1, whereas the Na/Mg ratios requires a near 100% abundance of chondrules. In the case of Ni/Mg and Mn/Mg it is not even possible to produce a Mx1 that lies between the igneous rim and Mx2 composition, when the precursor dust has to have a CI-like composition. Similar discrepancies in chondrule/matrix ratios have been previously observed for CR and CO chondrite assemblages [@Zanda2018].\ In a non-complementarity scenario, such as envisioned by @Zanda2018, the final matrix composition of a chondrite is CI-like and chondrules are not produced in the same chemical reservoir. Any deviations from a CI-like matrix composition are considered to be the result of aqueous alteration processes. If we consider a chondritic starting composition for the primary dust rim accreted on the chondrule cores (Figs. \[fig:Scenarios\] and \[fig:SchematicChFor\]), then the heating event that produced the igneous rims required thermobaric conditions that evaded significant loss of non-refractory elements such as Fe, Cr, Ni and Mn, since the final composition of the igneous rims is also chondritic. The matrix complementary to the chondrule cores (Mx1) must have been removed from the CV chondrule reservoir, or the chondrules were transported to a different disk region. While this scenario works demonstrated solely by the elemental ratios of cores, rims and matrix, it is difficult to explain the petrological features of the igneous rims (i.e., relict forsterite).\ Alternatively, in a scenario adopted from @Marrocchi2013 for the formation of pyroxene-porphyritic (PP) chondrules in Vigarano, the matrix material that was present during chondrule core formation was thermally processed alongside the cores. Depleted forsterite-rich dust was subsequently accreted to the chondrule cores, which could have interacted with a complementary volatile-rich gas, bringing the composition of the chondrule rims to (near-)chondritic [@Marrocchi2013]. In this scenario, the S, FeO and SiO-rich gas would have been reabsorbed into the silicate melt of the rims and would have co-crystallized sulfides and low-Ca pyroxene at sulfur saturation level. The resulting petrological features of this model are in agreement with our observations, namely the presence of relict forsterite grains overgrown by low-Ca pyroxene co-existing with troilite/metal assemblages. Moreover, this model also fits with the near-chondritic compositions of the igneous rims. We note that this model implies non-complementarity between chondrule cores and igneous rims in the classical sense [@Hezel2018], but necessitates formation of both components in the same chemical reservoir. In section \[chondrule\_formation\], we discuss these scenarios using Zn isotope analyses. The significance of Zn isotope fractionation in Leoville {#significance} -------------------------------------------------------- The Zn isotope compositions of the Leoville chondrule cores and rims correspond to the most negative and most positive $\updelta$$^{66}$Zn values measured for Allende chondrules, respectively (Fig. \[fig:ChondruleZnisotopes\], [@Pringle2017]). Although the petrology of the analyzed Allende chondrules is unknown, they likely reflect total Zn isotope signatures of combined cores and igneous rims. Hence, we suggest that the variable signatures of these chondrules ($\updelta$$^{66}$Zn = –0.45 - 0.18 $\permil$) reflect a mixing line between negative chondrule cores and more positive igneous rims of CV chondrules. Consequently, the $\updelta$$^{66}$Zn range observed for Allende (an oxidized $>$CV3.6 chondrite) chondrules is equivalent to the range observed for Leoville (reduced CV3.1) chondrules, suggesting that the degree of thermal metamorphism experienced by Allende is inconsequential for the Zn isotope variability in its chondrules. This suggests that the exchange of Zn between chondrules and matrix during thermal alteration must have been limited. However, Zn is considered to be highly mobile in aqueous fluids [@Pringle2017; @Zanda2018] and the effect of aqueous alteration on the Zn isotope composition of the Leoville chondrules must be discussed first.\ Zinc isotope variations in Leoville and Allende chondrules are potentially the result of aqueous alteration. @Alexander2019 has shown that the Zn concentration is correlated to the matrix abundance in carbonaceous chondrites, suggesting that chondrules were initially devoid of Zn. Thus, the limited alteration experienced by Leoville could have been sufficient to redistribute Zn from matrix to chondrules. It has been previously shown that other mobile elements such as Na and S can been extensively redistributed between chondrules and matrix in the most primitive ordinary chondrites [@Grossman2005]. At the earliest stages of fluid-assisted metamorphism (petrological type 3.00-3.1), Na diffuses into chondrule mesostasis during vitrification and albite formation. At petrological types $>$3.1, Na transfers back to the matrix where feldspar starts to crystallize. Hence, the direction of Na mobilization is dependent on the sink of Na at different stages of thermal metamorphism. This raises the question what the sink of Zn is during these metamorphic stages, since Zn can act as a lithophile as well as a chalcophile element. Zinc is chalcophile and stored in the sphalerite component of sulfides in the least altered chondrites Semarkona (LL3.00), ALH 77307 (CO3.0) and Kainsaz (CO3.1) [@Johnson1991]. Any petrological grade higher than Semarkona in ordinary chondrites has detectable and increasing amounts of Zn incorporated in chromites that have crystallized or re-equilibrated during thermal metamorphism [@Johnson1991; @Chikami1999]. Hence, mild reheating in chondrites could redistribute Zn, similarly to Na and S. Spinel analyses from Allende chondrules reflect considerable ZnO concentrations ($<$0.55 wt%; average = 0.28 wt%; [@Riebe]). The abundance of spinel in Allende chondrules is estimated to be 0.6 wt%, resulting in an average Zn contribution of $\sim$13 ppm in Allende chondrules from spinels. Total Zn concentrations of chondrules analyzed by @Pringle2017 range between 31 and 94 ppm, suggesting that the contribution from potentially secondary Zn in spinel does not account for the total Zn in chondrules. Hence, some Zn could be primary. However, without a detailed knowledge of the petrology of the Allende chondrules analyzed for Zn isotopes, we cannot say with any certainty to which extent the Zn in the chondrules is primary. Another potential source for secondary Zn could be derived from Fe sulfides redistributed to Allende chondrule interiors (Fig. \[fig:Allende\], [@Zanda1995; @Hewins1997; @Grossman2005]). Unlike Leoville, the relatively high degree of thermal metamorphism in Allende has resulted in FeS coarsening and redistribution towards the chondrules cores in some of the more altered chondrules (Fig. \[fig:Allende\]), suggesting that Zn redistribution in Allende chondrules has been more pervasive. Thus, in theory, Allende chondrules may have only contained secondary Zn. However, it is possible that the redistribution of sulfides and Zn in Allende chondrules would have taken place only within the chondrules, which would have acted as a closed system. In this case, FeS from the igneous rims would have been transferred to the chondrule interiors, but Zn would not have easily transferred from the matrix to the chondrules. This is a realistic possibility, since secondary minerals formed within the matrix (e.g., carbonates, sulfides, sulfates and phosphates) could have represented a considerable sink for Zn by itself, preventing redistribution towards the chondrules. This model would explain the similar range of $\updelta$$^{66}$Zn values observed in Leoville and Allende chondrules, in spite of their different alteration degrees. This model assumes that the sulfides in the igneous rims are primary unaltered components, not mobilized from the matrix. Indeed, similar to the CV3.1-3.4 chondrite Vigarano, the absence of magnetite-carbide-sulfide associations [@Krot2004] and the nature of metal-sulfide structures (i.e., the absence of sulfide rims around metal, [@Marrocchi2013]), suggest that the troilite grains in Leoville igneous rims are primary high-temperature components [@Marrocchi2013]. Furthermore, even though very pristine ordinary chondrites show redistribution of Zn to spinels, the carbonaceous chondrite Kainsaz reports an absence of secondary Zn in its chondrules. This implies that the Zn mobility was different for carbonaceous chondrites and that Leoville, being of similar petrological degree to Kainsaz, may have avoided significant redistribution of Zn. This is certainly in agreement with the tightly clustered $\updelta$$^{66}$Zn values of the Leoville chondrule cores. We would predict more variable $\updelta$$^{66}$Zn signatures depending on the chondrule core sizes and mineralogy if these values were dependent on the redistribution of Zn. However, sampled chondrule cores vary in size by a factor $\sim$10 and the texture and mineralogy of the chondrules varies considerably as well. Moreover, sphalerite dissolution of Zn would yield unfractionated or heavy Zn signatures of the hydrothermal fluid (depending on pH and pCO$_{2}$ conditions, [@Fujii2011]), whereas subsequent incorporation into spinels would result in even heavier Zn isotope values relative to sphalerite. In contrast, we observe negative $\updelta$$^{66}$Zn values for Leoville chondrule cores. In summary, we suggest that the Zn isotope data from the Leoville chondrules reflect primary rather than secondary signatures. Zn isotope behaviour during chondrule formation {#chondrule_formation} ----------------------------------------------- The negative $\updelta$$^{66}$Zn values for chondrules relative to the CV bulk have been attributed to quantitative removal of sulfides with heavy Zn isotope signatures during chondrule formation [@Pringle2017]. This process results in a correlation between the Zn concentration and $\updelta$$^{66}$Zn values of the chondrules [@Pringle2017]. We observe a similar correlation between Mg/Zn ratios and $\updelta$$^{66}$Zn values of Leoville chondrule cores, igneous rims and matrix (Fig. \[fig:ZnVsZnisotope\]). We note that the resulting regression may be scattered due to variable Mg concentrations between chondrule core and igneous rims [@Rubin1987], which are generally higher for the cores than the rims. Nevertheless, the igneous rims have higher Zn concentrations and $\updelta$$^{66}$Zn values than their corresponding cores. Our data are in agreement with the heavy Zn isotope fraction being hosted by the Fe sulfides, since the igneous rims have a higher abundance of these components. As discussed in section \[significance\], these sulfides are likely primary features of the Leoville chondrules, developed during chondrule formation. Sulfide abundances in Vigarano CV chondrules have been linked to the co-crystallization of low-Ca pyroxene [@Marrocchi2013]. Hence, the higher the abundance of low-Ca pyroxene, the higher the sulfide concentration and predictably the higher the $\updelta$$^{66}$Zn value of the chondrule. Indeed, the igneous rim in Ch5, with a significantly higher abundance of low-Ca pyroxene relative to forsterite also accomodates the highest $\updelta$$^{66}$Zn value (Table \[tab:LeovilleZn\]). The Ch1 igneous rim with the highest abundance of forsterite/enstatite has the lowest $\updelta$$^{66}$Zn value. Hence, our data is fully in agreement with the separation of silicate and sulfide melt during chondrule formation being at the core of Zn isotope fractionation. In this model, the chondrule cores must have lost their sulfide component to a considerable degree. This implies that the igneous rims maintained/retained a significant fraction of their sulfide inventory for them to show only slight depletions in $^{66}$Zn relative to the matrix. @Marrocchi2013 have suggested that the formation of low-Ca pyroxene (PP) chondrules in Vigarano must have been the result of a reaction between partially depleted olivine-bearing precursors with a volatile (sulfur)-rich gas. Consequently, the PP chondrule composition becomes (near-)chondritic. This agrees with the observation of relict forsterite grains overgrown by low-Ca pyroxene as well as with the near-chondritic compositions of the Leoville igneous rims. These olivine-bearing precursors may represent initially CI-like dust that was thermally processed during the chondrule formation episode that formed the Leoville cores. If the volatile-rich gas reflects the complementary component of the simultaneously formed depleted olivine-bearing dust and the depleted type-I chondrule cores, then the cores and igneous rims are in fact genetically related and formed from the same chemical reservoir. In this scenario, the depleted olivine-bearing dust accreted onto the chondrule cores and subsequently reacted with a volatile-rich gas. The heavy Zn isotope fraction that was initially lost to the chondrule cores and the forsterite-rich thermally processed dust by sulfide vaporization, would have reabsorbed back onto the igneous rims together with S, FeO and SiO from the volatile-rich gas. The amount of interaction with this gas would have affected the Zn isotope composition of the rims (e.g., a higher reabsorbtion of Zn would yield a higher $\updelta$$^{66}$Zn value). The completion of the reaction from forsterite to low-Ca pyroxene yields $\updelta$$^{66}$Zn values similar to the matrix, suggesting that the initial composition of the dust that formed the chondrule cores and igneous rims was similar to the thermally unprocessed surrounding matrix. Hence, the Zn isotope data suggests that all Leoville fractions sampled here formed within an isotopically similar reservoir in the protoplanetary disk.\ We have several side notes to this conclusion: 1) Although we have shown that chondrules and matrix could have formed within the same reservoir, the chondrules are not complementary to the matrix, since the matrix is isotopically chondritic, similar to bulk CV rather than super-chondritic; 2) Without analyzing the Zn isotope compositions of chondrules and matrices in other chondrite classes, it is unknown whether the isotopic relationships shown here are unique to the CV chondrite reservoir or represent more generic relationships [@Connelly2018]. It is, therefore, necessary to conduct a systematic study into the Zn isotope compositions of other chondrites. Our new Zn isotope method can now accommodate the analyses of singularly small samples such as CM, CO and EC chondrules ($<$300 $\upmu$m diameter); 3) Elemental complementarity scenarios do not account for more complex chondrule forming scenarios that include multiple melting events, thermal processing, melt-gas interaction and recondensation (Fig. \[fig:Scenarios\]). Conclusions =========== We report on an adopted Zn isotope analytical protocol to accommodate the measurement of small Zn aliquots (e.g., 2-5 ng of Zn). The newly developed method is successfully tested on various terrestrial standards and samples, which show that accurate and reproducible Zn isotope analyses are dependent on the following size-limiting factors: 1) A low level of contamination of matrix elements in the Zn-cut, 2) low level contamination from resin-derived organics and 3) high signal/noise levels. We further present as a proof of concept, Zn isotope analyses on small fractions (e.g., chondrule cores, igneous rims and matrix) from the relatively unaltered Leoville CV3.1 chondrite. First, we demonstrate that Zn isotope variations from Leoville samples reflect primary signatures related to nebular heterogeneity and/or chondrule forming processes, rather than aqueous alteration. These variations include tightly clustered negative $\updelta$$^{66}$Zn values for the chondrule cores ($\updelta$$^{66}$Zn = –0.43$\pm$0.14 $\permil$), more variable and positive values for the igneous rims ($\updelta$$^{66}$Zn = –0.01$\pm$0.30 $\permil$) and chondritic values for the matrix ($\updelta$$^{66}$Zn = 0.19$\pm$0.14 $\permil$). We further show that combined with the elemental composition and petrology, the Zn isotope analyses of the Leoville fractions point towards the following chondrule forming model as a mechanism to fractionate Zn isotopes: The sulfides containing isotopically heavy Zn must be vaporized during chondrule core formation, thereby lowering the $\updelta$$^{66}$Zn values of the chondrules. Concurrently, chondritic dust is thermally processed alongside the chondrules and subsequently accretes to the chondrule cores as volatile depleted forsterite-bearing grains, which represent the precursors to the igneous rims. These grains react with the complementary volatile-rich gas, which progressively enriches the igneous rims in heavy Zn isotopes as well as moderately volatile elements. This results in a near-chondritic elemental composition of the igenous rims, as well as forsterite relict grains overgrown by low-Ca pyroxene co-existing with troilite/metal assemblages, in agreement with our data. We suggest that all CV components are not complementary to each other in the classical sense proposed by [@Hezel2018], but could have formed from the same chemical and isotopic reservoir. We note that further Zn isotope investigations of other chondrite classes are necessary to distinguish between a generic scenario of isotope fractionation between chondrules and matrix or if these signatures are specific to CV chondrites. Our newly developed Zn isotope analytical protocol will be able to forward this research for small samples such as CM, CO and EC chondrules, and be applied to a large set of small rock samples (f.e., mineral separates and specimens from sample return missions). Acknowledgements {#acknowledgements .unnumbered} ================ This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No 786081. F.M. acknowledges funding from the European Research Council under the H2020 framework program/ERC grant agreement no. 637503 (Pristine) and financial support of the UnivEarthS Labex program at Sorbonne Paris Cité (ANR-10-LABX- 0023 and ANR-11-IDEX-0005-02). Parts of this work were supported by IPGP multidisciplinary PARI program, and by Region Île-de-France SESAME Grant no. 12015908. We thank Jian Huang and two anonymous reviewers for their constructive comments on this manuscript. We thank the Natural History Museum of Denmark for the generous loan of a thick section from Leoville. References {#references .unnumbered} ========== ------------------------------- -- -- -------- -------------------------------------- ------------------------ **Blank contribution** **Zn (ng)** 1.5M HBr (1) 1.6 ml in two-step ***$<$0.010*** 1.5M HBr (2) 7.80 anion column 6M HCl (1) 78.24 0.5 ml in sample digestion ***$<$0.005*** 6M HCl (2) 0.5M HNO$_{3}$ (1) 1.6 ml in two-step ***n.d.*** 0.5M HNO$_{3}$ (2) anion column 7M HNO$_{3}$ (1) 8.98 1 ml during digestion ***$<$0.08*** 7M HNO$_{3}$ (2) 19.35 27M HF (1) 7.48 100 $\upmu$l during digestion ***$<$0.004*** 27M HF (2) 38.54 7M HNO$_{3}$ + 27M HF (1) 6.19 Digestion of sample in ***$<$0.574*** 7M HNO$_{3}$ + 27M HF (2) 6.94 1 ml HNO$_{3}$/HF in 7M HNO$_{3}$ + 27M HF (3) 11.35 Parr bombs at 210$^{\circ}$C 7M HNO$_{3}$ + 27M HF (4) 5.48 1.5M HBr + 0.5M HNO$_{3}$ (1) 26.96 Total procedural blank ***$<$0.350*** 1.5M HBr + 0.5M HNO$_{3}$ (2) 112.83 of column chemistry 1.5M HBr + 0.5M HNO$_{3}$ (3) 16.22 1.5M HBr + 0.5M HNO$_{3}$ (4) 1.5M HBr + 0.5M HNO$_{3}$ (5) **Total Zn blank (with Parr bombs)** ***$<$0.924*** **Total Zn blank (no Parr bombs)** ***$<$0.350*** ------------------------------- -- -- -------- -------------------------------------- ------------------------ \[tab:blanks\] [rcccc]{} \ & Total amount & Amount of Zn & &\ & of Zn (ng) & individual analyses (ng) & &\ [  – *Continued from previous page*]{}\ & Total amount & Amount of Zn & &\ & of Zn (ng) & individual analyses (ng) & &\ \ BHVO-2 & 600 & 5 & 0.39 & 0.69\ & & 5 & 0.35 & 0.59\ & & 5 & 0.34 & 0.58\ & & 5 & 0.31 & 0.67\ & & 5 & 0.27 & 0.71\ & & 5 & 0.32 & 0.65\ & &5& 0.35 & 0.83\ &&5&0.41&0.75\ &&5&0.34&0.70\ ***Average*** & & & ***0.34*** & ***0.69***\ *2SD* & & & *0.08* & *0.16*\ & & 2.5 & 0.31 & 0.32\ & & 2.5 & 0.27 & 0.30\ & & 2.5 & 0.26 & 0.28\ & & 2.5 & 0.24 & 0.27\ & & 2.5 & 0.28 & 0.31\ & & 2.5 & 0.31 & 0.37\ & & 2.5 & 0.26 & 0.46\ & & 2.5 & 0.27 & 0.41\ & & 2.5 & 0.26 & 0.33\ ***Average*** & & & ***0.27*** & ***0.34***\ *2SD* & & & *0.05* & *0.13*\ ***Literature*** & & & ***0.28*** &\ *2SD* & & & *0.09* &\ & & & &\ BCR-2-1 & 5 & 5 & 0.06 & 0.18\ BCR-2-2 & 5 & 5 & 0.15 & 0.48\ BCR-2-3 & 5 & 5 & 0.14 & 0.34\ BCR-2-4 & 10 & 5 & 0.22 & 0.48\ & & 5 & 0.21 & 0.46\ BCR-2-5 & 5 & 5 & 0.20 & 0.12\ ***Average*** & & & ***0.16*** & ***0.34***\ *2SD* & & & *0.12* & *0.32*\ ***Literature*** & & & ***0.25*** &\ *2SD* & & & *0.08* &\ & & & &\ AGV2-1 & 5 & 5 & 0.32 & 0.59\ AGV2-2 & 5 & 5 & 0.25 & 0.21\ AGV2-3 & 10 & 5 & 0.15 & 0.55\ & 10 & 5 & 0.11 & 0.50\ AGV2-4 & 5 & 5 & 0.21 & 1.05\ **Average** & & & ***0.21*** & ***0.58***\ *2SD* & & & *0.16* & *0.60*\ ***Literature*** & & & ***0.29*** &\ *2SD* & & & *0.06* &\ & & & &\ CP1-1 & 5 & 5 & –0.07 & –0.19\ CP1-2 & 10 & 5 & –0.06 & –0.20\ & & 5 & –0.08 & –0.06\ CP1-3 & 5 & 5 & –0.05 & –0.16\ CP1-4 & 5 & 5 & 0.01 & –0.18\ CP1-5 & 10 & 5 & –0.10 & –0.24\ & & 5 & -0.15 & -0.14\ ***Average*** & & & ***–0.07*** & ***–0.17***\ *2SD* & & & *0.10* & *0.11*\ ***Literature*** & & & ***0.00*** &\ *2SD* & & & *0.08* &\ & & & &\ CV chondrite & 600 & 2.5 & 0.33 & 1.36\ & & 2.5 & 0.36 & 1.59\ & & 2.5 & 0.17 & 0.17\ & & 2.5 & 0.14 & 0.06\ & & 2.5 & 0.39 & 0.55\ & & 2.5 & 0.25 & 0.21\ & & 2.5 & 0.21 & 0.02\ & & 2.5 & 0.13 & 0.06\ & & 2.5 & 0.27 & 0.42\ & & 2.5 & 0.48 & 0.65\ & & 2.5 & 0.21 & 0.31\ & & 2.5 & 0.26 & 0.44\ ***Average*** & & & ***0.27*** & ***0.49***\ *2SD* & & & *0.19* & *1.17*\ ***Literature*** & & & ***0.24*** & ***0.44***\ *2SD* & & & *0.12* & *0.21*\ JMC Lyon (processed) & 1 & 1 & 0.06 &\ & & 1 & –0.08 &\ & & 1 & –0.10 &\ & & 1 & 0.00 &\ ***Average*** & & & ***–0.03*** &\ *2SD* & & & *0.15* &\ \[tab:standardsZn\] ------- -------- -------- --------- --------- -------- -------- -------- -------- -------- -------- -------- 1 2 3 5 6 Na/Mg 2.97 2.88 1.05 1.08 2.79 0.038 0.008 0.030 0.024 0.021 0.024 Al/Mg 8.66 8.29 3.28 3.31 0.75 0.35 0.10 0.22 0.10 0.10 0.18 Ca/Mg 3.76 3.44 2.42 2.36 2.61 0.34 0.08 0.22 0.12 0.10 0.17 Cr/Mg 0.0023 0.0016 0.00085 0.00083 1.46 0.020 0.016 0.012 0.025 0.017 0.018 Ti/Mg 0.47 0.58 0.644 0.625 2.89 0.0139 0.0039 0.0115 0.0041 0.0056 0.0078 Mn/Mg 0.07 0.07 0.067 0.070 5.48 0.0050 0.0036 0.0021 0.0035 0.0022 0.0033 Fe/Mg 4.10 4.33 4.16 4.47 6.37 0.41 0.61 0.11 1.44 0.28 0.57 Ni/Mg 0.0021 0.0018 0.0006 0.0005 27.42 0.010 0.028 0.002 0.095 0.027 0.032 Co/Mg 0.0016 0.0015 0.0019 0.0017 8.29 0.0063 0.0027 0.0320 0.0139 0.0072 0.0124 1 2 3 5 6 1 2 Na/Mg 0.022 0.010 0.030 0.022 0.022 0.021 0.034 0.044 0.039 Al/Mg 0.14 0.10 0.10 0.08 0.12 0.108 0.19 0.10 0.15 Ca/Mg 0.15 0.09 0.12 0.09 0.11 0.113 0.21 0.10 0.15 Cr/Mg 0.039 0.025 0.036 0.033 0.035 0.033 0.042 0.030 0.036 Ti/Mg 0.0077 0.0055 0.0061 0.0047 0.0053 0.0059 0.0073 0.0037 0.0055 Mn/Mg 0.0085 0.0192 0.0037 0.0156 0.0048 0.0125 0.0429 0.0192 0.0311 Fe/Mg 2.31 1.47 1.17 2.16 1.14 1.65 3.19 3.35 3.27 Ni/Mg 0.144 0.054 0.090 0.115 0.126 0.106 0.101 0.131 0.116 Co/Mg 0.0124 0.0036 0.0297 0.0074 0.0124 0.0131 0.0064 0.0069 0.0067 ------- -------- -------- --------- --------- -------- -------- -------- -------- -------- -------- -------- \[tab:majorel\] Zn (ng) Cycles **$\updelta$$^{66}$Zn ($\permil$)** **$\updelta$$^{68}$Zn ($\permil$)** -- ------------ --------- -------- ------------------------------------- ------------------------------------- ch1 5 30 0.24$\pm$0.12 0.51$\pm$0.32 ch2 5 30 0.14$\pm$0.12 0.59$\pm$0.32 ***Mean*** ***0.19***$\pm$***0.14*** ***0.55***$\pm$***0.12*** ch1 5 30 –0.14$\pm$0.12 0.54$\pm$1.17 ch2 2 30 –0.11$\pm$0.19 1.89$\pm$0.32 ch3 2 30 –0.04$\pm$0.19 1.08$\pm$1.17 ch5 5 30 0.23$\pm$0.12 0.45$\pm$1.17 ch6 3 30 0.01$\pm$0.19 0.76$\pm$1.17 ***Mean*** ***–0.01***$\pm$***0.30*** ***0.94***$\pm$***1.16*** ch1 1 15 –0.41$\pm$0.19 0.14$\pm$1.17 ch2 1 15 –0.41$\pm$0.19 0.10$\pm$1.17 ch3 2 30 –0.54$\pm$0.19 0.40$\pm$1.17 ch5 1 15 –0.36$\pm$0.19 –0.15$\pm$1.17 ch6 3 30 –0.41$\pm$0.12 0.04$\pm$0.32 ***Mean*** ***–0.43*** $\pm$ ***0.14*** ***0.10***$\pm$***0.40*** : Zn isotope data for Leoville chondrule cores, igneous rims and surrounding matrices, with the abundance of Zn given for each sample. Note that three chondrule cores only had 1 ng of Zn and were analyzed in a block of 15 cycles on the MC-ICPMS, rather than the 30 cycles used for the other samples. Standards measured with 15 cycles yield identical $\updelta$$^{66}$Zn values within 0.02 $\permil$. \[tab:LeovilleZn\] **Sample** **Authors** **Technique** ------------ ------ ------- --------------- --------------- Leoville 2.11 0.015 @Kracher1985 DBA Leoville 1.96 0.016 @McSween1977 DBA Leoville 3.27 0.031 This study WA Vigarano 2.46 0.011 @McSween1977 DBA Vigarano 3.23 0.019 @Zolensky1993 DBA Vigarano 2.71 0.022 @Klerner2001 DBA Efremovka 2.63 0.015 @McSween1977 DBA Efremovka 2.53 0.022 @Hezel2010 DBA Allende 1.79 0.013 @Clarke1971 WA Allende 2.03 0.013 @McSween1977 DBA Allende 2.36 0.014 @Rubin1984 DBA Allende 1.57 0.011 @Rubin1987 INAA : Matrix Fe/Mg and Mn/Mg ratios of various CV chondrites analyzed by different authors and using different analytical techniques. DBA = Defocused or broad beam analyses, WA = Wet chemical analyses, INAA = Instrumental neutron activation analysis. \[tab:Matrix\] ![Elution profiles of major elements from a BHVO-2 standard. One column volume reflects 100 $\upmu$l of acid. After eight column volumes all matrix elements are eluted with 1.5M HBr. We note that Zn peaks between 12 and 19 column volumes are analytical artifacts due to a low signal/noise ratio. We have repeated these elution tests using BHVO-2 and CV chondrite aliquots spiked with 100 ng Zn. These tests resulted in reproducible yields $>$99.9 %. \[fig:FigZnelutionprofile1\]](FigZnelutionprofile.pdf){width="90.00000%"} ![Elution profiles of Mg and Fe in 1.5M HBr for BHVO-2 (grey curves) and CV chondrite NWA 12523 (red curves). Repeat experiments show consistent elution profiles and purification of matrix elements after eight column volumes, independent of matrix composition. \[fig:FigZnelutionprofileMgFe\]](FigZnelutionprofileMgFe.pdf){width="80.00000%"} ![Analyses of several major and minor elements in the eluted Zn-cut from CV chondrite NWA 12523 after purification for various sample sizes, showing purification for each element $>$99.5 %. Aliquots $>$100 $\upmu$g generally result in a cleaner Zn-cut. $>$99.9 %. Sample sizes aimed at in this study ($\sim$50 $\upmu$g) have a relatively large abundance of matrix elements in the Zn-cut after a single column pass (Fe $\approx$30 ng and Ni $\approx$0.8 ng on 5 ng of Zn) and require a second column pass for acceptable Zn purification (Fe $\approx$0.08 ng and Ni $\approx$0.002 ng on 5 ng of Zn). \[fig:FigZncutcontamination\]](FigZncutcontamination.pdf){width="90.00000%"} ![Zn isotope data showing $\updelta$$^{66}$Zn and $\updelta$$^{68}$Zn values for terrestrial standards and samples. Errors are 2SD from average mean (Table \[tab:standardsZn\]). The shaded areas are (recommended) literature values for CP1 (purple, [@Inglis2017]) and BHVO-2, BCR-2 and AGV2 (orange, [@Moynier2017]). We note that literature only reports the $\updelta$$^{66}$Zn values. The solid and dashed lines represent kinetic and equilibrium fractionation laws, respectively. \[fig:FigZnstandards\]](FigZnstandards.pdf){width="90.00000%"} ![*Left:* BSE images for Leoville chondrules in which their cores, igneous rims (IgR) and matrices (Mx) are highlighted. *Met* = metal, *Sf* = sulfide, *Ol* = olivine, *Ms* = mesostasis. *Right:* BSE images of the chondrules after microdrilling. The drill locations for cores, rims and matrices are highlighted. Discarded drill holes are reflected by red coded areas. \[fig:FigBSEchondrules1\]](FigBSEchondrules1.jpg){width="60.00000%"} ![*Left:* BSE images for Leoville chondrules in which their cores, igneous rims (IgR) and matrices (Mx) are highlighted. *Met* = metal, *Sf* = sulfide, *Ol* = olivine, *Ms* = mesostasis. *Right:* BSE images of the chondrules after microdrilling. The drill locations for cores, rims and matrices are highlighted. Discarded drill holes are reflected by red coded areas. \[fig:FigBSEchondrules2\]](FigBSEchondrules2.jpg){width="90.00000%"} {width="110.00000%"} {width="110.00000%"} ![$\updelta$$^{66}$Zn values for Leoville cores, igneous rims and matrix, as well as their average compositions (black-rimmed symbols). Data for chondrules (blue) and matrix (yellow) from @Pringle2017 (P17) is also shown. \[fig:ChondruleZnisotopes\]](ChondruleZnisotopes.pdf){width="100.00000%"} ![The deviation from the Zn mass-dependent fractionation line ($\Delta$$^{66}$Zn $\permil$) versus the $^{62}$Ni/$^{64}$Zn ratio of analyzed terrestrial standards by MC-ICPMS after correction of the $^{64}$Ni interference on $^{64}$Zn. The positive correlation through the data is interpreted as matrix effects from contaminants in the Zn-cut that affect the mass-dependency of the measurements. We show that we can apply a threshold Ni/Zn ratio to the data, above which the quality of the data is poorer than the external reproducibility. \[fig:NiZnthreshold\]](NiZnthreshold.pdf){width="100.00000%"} ![The same Figure as Figure \[fig:NiZnthreshold\], but including Leoville chondrule data from this study. \[fig:NiZnthresholdchondrules\]](NiZnthresholdchondrules.pdf){width="100.00000%"} ![Matrix fractionation factors calculated from elemental ratios normalized to Mg from this work, @Rubin1987 (neutron activation analyses) and @Clarke1971 (wet chemical analyses) against 50% condensation temperatures [@Lodders2003]. The dashed regression line is taken from all data. RW87 = @Rubin1987 and C71 = @Clarke1971. \[fig:FractionationCondensation\]](FractionationCondensation.png){width="100.00000%"} {width="100.00000%"} {width="110.00000%"} ![Mg (red), Ca (green), Al (blue) and S (yellow) elemental maps of Leoville (A-B) and Allende (C-D) sections (chondrules in these images lie close to the ones sampled in this study). Note how the matrix of Leoville is mostly opaque and fine-grained, whereas the Allende matrix is coarse-grained with large sulfides and carbonates. The sulfides in Leoville are mainly present in the igneous rims or as sulfide rims to the chondrules, whereas in Allende, the sulfides have mobilized towards the interior cores in the most altered chondrules. \[fig:Allende\]](Allende_Leoville_compressed.pdf){width="100.00000%"} ![Zn isotope data of Leoville chondrule cores (closed green circles), igneous rims (open red circles) and matrix (yellow diamonds) against their Mg/Zn ratios. Mg concentrations have been obtained from ICPMS data and Zn concentrations are from MC-ICPMS analyses. Hence, the Mg/Zn ratios are relative and not absolute. The composition of Ch5 lies close to the matrix data and has a different mineralogy compared to the other igneous rims (see text for further explanation). \[fig:ZnVsZnisotope\]](ZnVsZnisotope.png){width="100.00000%"}

--- abstract: 'In a linear magnetoelectric the lattice is coupled to electric and magnetic fields: both affect the longitudinal-transverse splitting of zone-center optical phonons on equal footing. A response matrix relates the macroscopic fields $(D,B)$ to $(E,H)$ at infrared frequencies. It is shown that the response matrices at frequencies $0$ and $\infty$ fulfill a generalized Lyddane-Sachs-Teller relationship. The rhs member of such relationship is expressed in terms of weighted averages over the longitudinal and transverse excitations of the medium, and assumes a simple form for an harmonic crystal.' author: - Raffaele Resta title: 'Lyddane-Sachs-Teller relationship in linear magnetoelectrics' --- =5000 =10000 The original Lyddane-Sachs-Teller (LST) relationship [@Lyddane41] applies to the simple case of a cubic binary crystal in the harmonic regime. It relates four macroscopically measurable constants as $$\frac{\varepsilon(0)}{\varepsilon(\infty)} = \frac{\omega^2_{\rm L}}{\omega^2_{\rm T}} . \label{lst}$$ Here $\varepsilon(0)$ is the static dielectric constant, which includes the lattice contribution, $\varepsilon(\infty)$ is the so called “static high frequency” (a.k.a “clamped-ion”) dielectric constant, which accounts for the electronic response only, and $\omega_{\rm L}$ ($\omega_{\rm T}$) is the zone-center longitudinal (transverse) optical frequency [@textbook]. It is remarkable that all [*microscopic*]{} parameters (force constants, masses, Born effective charges, cell volume) disappear from [Eq. (\[lst\])]{}. Magnetoelectrics (MEs) are insulators where electric fields control magnetization, and conversely magnetic fields control polarization; they attracted considerable theoretical and technological interest in recent times [@Fiebig05; @Eerenstein06; @Iniguez08; @Hehl08; @Hehl09; @Essin09; @Wojdel09; @Essin10; @Wojdel10]. The simplest and most studied single-crystal linear ME is antiferromagnetic Cr$_2$O$_3$ [@Fiebig05; @Iniguez08; @Hehl08; @Hehl09]. In any linear ME the role of the dielectric function $\varepsilon(\omega)$ is played by the $2\times2$ response matrix—called ${{\cal R}}(\omega)$ here—which yields the macroscopic fields $(D,B)$ in terms of $(E,H)$ at frequency $\omega$: $$\left(\begin{array}{c} D \\ B \end{array} \right) = {{\cal R}}(\omega) \left(\begin{array}{c} E \\ H \end{array} \right) \equiv \left(\begin{array}{cc} \varepsilon(\omega) & \alpha(\omega) \\ \alpha(\omega) & \mu(\omega) \end{array} \right) \left(\begin{array}{c} E \\ H \end{array} \right) , \label{resp}$$ where $\varepsilon$, $\mu$, and $\alpha$ are permittivity, magnetic permeability, and ME coupling, respectively. In this Letter we are going to show that a generalized LST relationship holds in the form $$\frac{\mbox{tr } \{ {{\cal R}}^{-1}(\infty) {{\cal R}}(0) \} - 2}{2 - \mbox{tr } \{ {{\cal R}}^{-1}(0) {{\cal R}}(\infty) \} } = \frac{\omega^2_{\rm L}}{\omega^2_{\rm T}} . \label{rap}$$ In the simple case of a magnetically inert material (i.e. $\mu \equiv \mbox{constant}$, $\alpha \equiv 0$) the lhs of [Eq. (\[rap\])]{} equals indeed $\varepsilon(0)/\varepsilon(\infty)$. While in ordinary dielectrics the LO-TO splitting is due to the coupling of the ionic displacements to macroscopic electric fields, in linear MEs it is due to the coupling of both (electric and magnetic) fields on the same footing: this is perspicuous in the lhs of [Eq. (\[rap\])]{}. The simple form of [Eqs. (\[lst\]) and (\[rap\])]{} requires a crystalline system with only a single IR-active mode at the zone center. The additional requirement of cubic symmetry can be relaxed; it is nonetheless convenient to consider only crystals whose symmetry is orthorombic or higher; then all crystalline tensors can be simultaneously diagonalized (as e.g. in Cr$_2$O$_3$). This allows us to adopt simple scalar-like notations, as in [Eqs. (\[lst\]) and (\[rap\])]{} with the proviso that we deal separately with each principal direction. Over the years the LST relationship has been extended in several ways, to cover cases where more than one IR-active mode per direction exists [@Kurosawa61; @Cochran62], the crystal is in a low-symmetry class [@Lax71; @Gonze97], or even the material is noncrystalline and/or anharmonic [@Barker75; @Barker75b; @Noh89; @Sievers90]. The general case can be written as $$\frac{\varepsilon(0)}{\varepsilon(\infty)} = \frac{{\langle \omega^2 \rangle}_{\rm L}}{{\langle \omega^2 \rangle}_{\rm T}} . \label{lst2}$$ The quantities in the rhs are weighted averages, obtained from moments of the appropriate spectral functions. The derivation is based on general principles of statistical mechanics and does not require an Hamiltonian, even less an harmonic one [@Barker75; @Noh89]; in the special case of a single IR-active harmonic mode per direction [Eq. (\[lst2\])]{} is equivalent to [Eq. (\[lst\])]{}. In this Letter we generalize the viewpoint of Ref. [@Noh89] to the ME case, showing that $$\frac{\mbox{tr } \{ {{\cal R}}^{-1}(\infty) {{\cal R}}(0) \} - 2}{2 - \mbox{tr } \{ {{\cal R}}^{-1}(0) {{\cal R}}(\infty) \} } = \frac{{\langle \omega^2 \rangle}_{\rm L}}{{\langle \omega^2 \rangle}_{\rm T}} ,\label{lst3}$$ where the rhs is defined below, [Eqs. (\[m1\]) and (\[m2\])]{}. The presentation proceeds as follows. We start at a very general level without any [*microscopic*]{} assumption about the ME medium, and using only very general principles in order to arrive at [Eq. (\[lst3\])]{}. We will then apply the general results to a crystalline system in the harmonic regime, and finally we will show that [Eq. (\[lst3\])]{} reduces to [Eq. (\[rap\])]{} in the single-mode case. We write explicitly the linear response matrix of the ME medium as the sum of its real and imaginary part: ${{\cal R}}(\omega) = {{\cal R}}'(\omega) +i {{\cal R}}''(\omega)$, and analogously for its inverse; both ${{\cal R}}(\omega)$ and ${{\cal R}}^{-1}(\omega)$ obey the Kramers-Kronig relationships in the form [[R]{}]{}’() - [[R]{}]{}() &=& \_0\^d ’ , \[k1\]\ [[[R]{}]{}\^[-1]{}]{}’() - [[R]{}]{}\^[-1]{}() &=& \_0\^d ’ . \[k2\] From these, it follows immediately that the numerator and denominator in the lhs of [Eqs. (\[rap\]) and (\[lst3\])]{} are $$\mbox{tr} \{ {{\cal R}}^{-1}(\infty) {{\cal R}}(0) \} - 2 = \frac{2}{\pi} \int_0^\infty \frac{d\omega}{\omega} \mbox{tr} \{ {{\cal R}}^{-1}(\infty) {{\cal R}}''(\omega) \} , \label{kk1}$$ $$2 - \mbox{tr} \{ {{\cal R}}^{-1}(0) {{\cal R}}(\infty) \} = - \frac{2}{\pi} \int_0^\infty \frac{d\omega }{\omega} \mbox{tr} \{ {{{\cal R}}^{-1}}''(\omega) {{\cal R}}(\infty) \} . \label{kk2}$$ The integrands in the rhs of [Eqs. (\[kk1\]) and (\[kk2\])]{} are interpreted here as the transverse and longitudinal spectral weights, respectively, by means of which we [*define*]{} the second moments [\^2 ]{}\_[T]{} &=& \[m1\]\ [\^2 ]{}\_[L]{} &=& \[m2\] . The reason for the semantics ([*transverse*]{} and [*longitudinal*]{}) may appear obscure at this point; it will become clear when specializing [Eqs. (\[m1\]) and (\[m2\])]{} to an harmonic crystal—see also [Eq. (\[mean\])]{} below. In order to arrive at our main result, [Eq. (\[lst3\])]{}, we exploit the “superconvergence” theorem [@Altarelli72]. For large $\omega$ (i.e. for $\omega$ much larger than all the resonances of the medium) ${{\cal R}}''(\omega)$ vanishes; [Eqs. (\[k1\]) and (\[k2\])]{} yield, to leading order in $1/\omega^2$, $${{\cal R}}(\infty)^{-1} {{\cal R}}(\omega) \simeq {\cal I} - \frac{2}{\pi \omega^2} \int_0^\infty d \omega' \; \omega' {{\cal R}}(\infty)^{-1} {{\cal R}}''(\omega') , \label{kkk1}$$ $${{\cal R}}(\omega)^{-1} {{\cal R}}(\infty) \simeq {\cal I} - \frac{2}{\pi \omega^2} \int_0^\infty d \omega' \; \omega' {{{\cal R}}^{-1}}''(\omega') {{\cal R}}(\infty), \label{kkk2}$$ where ${\cal I}$ is the $2\times 2$ identity. Inversion of [Eq. (\[kkk1\])]{} to the same order gives the alternative expression $${{\cal R}}(\omega)^{-1} {{\cal R}}(\infty) \simeq {\cal I} + \frac{2}{\pi \omega^2} \int_0^\infty d \omega' \; \omega' {{\cal R}}(\infty)^{-1} {{\cal R}}''(\omega') . \label{inv}$$ Next we take the trace of [Eqs. (\[kkk2\]) and (\[inv\])]{}; permuting the matrices in the product we get the identity & & \_0\^d { [[R]{}]{}()\^[-1]{} [[R]{}]{}”()} &=& - \_0\^d { [[[R]{}]{}()\^[-1]{}]{}” [[R]{}]{}()} . \[iden\] In order to arrive at [Eq. (\[lst3\])]{} it is enough to put together Eqs. (\[kk1\]), (\[kk2\]), (\[m1\]), (\[m2\]), and (\[iden\]). We stress that at the root of the superconvergence identity, [Eq. (\[iden\])]{}, is the assumption that $\omega = \infty$ actually means $\omega$ much higher than all the frequencies of ionic motions, yet lower than the frequencies of electronic excitations [@textbook]. Therefore the clamped-ion response ${{\cal R}}(\infty)$ is a real matrix. Next we address an harmonic crystal. A zone-center optical mode is lattice-periodical; it is then expedient to consider the energy per cell of a (macroscopically homogeneous) solid with a frozen-in phonon distortion. This energy is well defined only when a prescription for taking the thermodynamic limit is given. We cut a sample in the shape of a slab parallel to the principal axes; we remind our assumption that all crystal tensors are diagonal on them. If the slab is free-standing in vacuo, all fields vanish outside ($E$=$D$=$H$=$B$=$0$), while the value of the fields inside depend on the polarization of the mode. Simple electrostatics and magnetostatics imply that if the phonon polarization is parallel to the slab (“transverse”), both $E$ and $H$ vanish, while if it is perpendicular (“longitudinal”) $D$ and $B$ vanish [@rap_a30]. The order of the limits (first a slab, then its thickness going to infinity) is essential, and the two energies (longitudinal and transverse) are indeed different in the thermodynamic limit. Similar reasonings apply if the lattice-periodical mode is regarded as the ${\bf k} \rightarrow 0$ limit of a finite-${\bf k}$ optical phonon [@textbook]. If the crystal has $N$ IR-active modes in each principal direction, we denote with $\omega_n$ the zone-center TO frequencies (i.e those with $E$=0 and $H$=0); equivalently, $\omega_n^2$ are the eigenvalues of [*the analytical part*]{} of the dynamical matrix at ${\bf k} = 0$. The free energy per cell in function of the normal coordinates $u_n$, $E$, and $H$ (taken as independent variables) is expanded to second order as [@Fiebig05; @Iniguez08] && F(E,H,{ u\_n }) = F\_0 + \_n \_n\^2 u\_n\^2 &-& \[ () E\^2 + 2 () E H + () H\^2 \] &-& \_n ( u\_n Z\_n\^\* E + u\_n \^\*\_n H ) , \[free\] where we adopt atomic Gaussian units [@units], and $\Omega$ is the cell volume. $Z_n^*$ are the mode-effective charges and $\zeta_n^*$ their magnetic analogues; notice that in ordinary units the normal-mode coordinates would include a factor with the dimensions of (mass)$^{1/2}$, while $Z_n^*$ and $\zeta_n^*$ would include a factor with the dimensions of (mass)$^{-1/2}$. The derivatives of $F$ provide the equations of motion in the form D &=& - = () E + () H + \_n Z\_n\^\* u\_n B &=& - = () E + () H + \_n \_n\^\* u\_n f\_n &=& - = - \_n\^2 u\_n + Z\_n\^\* E + \_n\^\* H. \[motion\] We then consider forced oscillations at frequency $\omega$, i.e. $f_n = -\omega^2 u_n$. Elimination of the $u_n$’s from [Eq. (\[motion\])]{} provides the linear ME response, including the lattice contribution; we cast it in compact form as [[R]{}]{}’() &=& [[R]{}]{}() + \_n \[ini\]\ [[R]{}]{}”() &=& \_n [[Z]{}]{}\_n [[Z]{}]{}\_n\^ (\_n\^2 - \^2) , where the ME lattice coupling vectors are $${{\cal Z}}_n = \left(\begin{array}{c} Z^*_n \\ \zeta^*_n \end{array} \right) , \qquad {{\cal Z}}_n^\dagger = \left(\begin{array}{cc} Z^*_n & \zeta^*_n \end{array} \right) .$$ [Eq. (\[ini\])]{} is the elegant result obtained in 2008 by J. Ìñiguez [@Iniguez08]; the TO frequencies $\omega_n$ are clearly the $N$ poles of ${{\cal R}}(\omega)$. Actual computations performed for the paradigmatic material Cr$_2$O$_3$ and based on [Eq. (\[ini\])]{} show that the magnetoelectric coupling $\alpha(0)$ is significantly enhanced by the lattice contribution [@Iniguez08]. Clearly, a large coupling is the key property to be exploited in device applications [@Fiebig05]. It is now expedient to express [Eq. (\[motion\])]{} in terms of $D$ and $B$. ( [c]{} E\ H ) &=& [[R]{}]{}\^[-1]{}() \[inverse\]\ - \^2 u\_n &=& - \_n\^2 u\_n - [[Z]{}]{}\_n\^[[R]{}]{}\^[-1]{}() \_[n’]{} [[Z]{}]{}\_[n’]{} u\_[n’]{} &+& [[Z]{}]{}\_n\^[[R]{}]{}\^[-1]{}() ( [c]{} D\ B ). \[motion2\] As explained above, in the LO modes $D$=0 and $B$=0 by definition [@rap_a30]. The first line of [Eq. (\[motion2\])]{} clearly shows that the LO eigenmodes are in general different from the TO ones; an explicit $N \times N$ diagonalization is needed in order to find the LO frequencies. We indicate with $\tilde{\omega}_n$ these frequencies, and we also transform the lattice coupling vectors to the LO eigenmodes as ${{\cal R}}^{-1}(\infty) {{\cal Z}}_n\rightarrow \tilde{{{\cal Z}}}_n$, in order to write [Eq. (\[motion2\])]{} as $$- \omega^2 u_n = - \tilde{\omega}_n^2 u_n + \tilde{{{\cal Z}}}_n^\dagger \left(\begin{array}{c} D \\ B \end{array} \right) . \label{motion3}$$ Elimination of the $u_n$’s from [Eqs. (\[inverse\]) and (\[motion3\])]{} provides ${{\cal R}}^{-1}(\omega)$ in the form [[R]{}]{}’\^[-1]{}() &=& [[R]{}]{}\^[-1]{}() - \_n \[ini2\]\ [[R]{}]{}”\^[-1]{}() &=& - \_n \_n \_n\^ (\_n\^2 - \^2) . Using the above results, we arrive at a very transparent expression for the rhs member of [Eq. (\[lst3\])]{}: $$\frac{{\langle \omega^2 \rangle}_{\rm L}}{{\langle \omega^2 \rangle}_{\rm T}} = \frac{\sum_n \frac{1}{\omega_n^2} {{\cal Z}}_n^\dagger {{\cal R}}^{-1}(\infty) {{\cal Z}}_n}{ \sum_n \frac{1}{\tilde{\omega}_n^2} \tilde{{{\cal Z}}}_n^\dagger {{\cal R}}(\infty) \tilde{{{\cal Z}}}_n } , \label{mean}$$ i.e. ${\langle \omega^2 \rangle}_{\rm T}$ is the weighted harmonic mean of the $\omega^2_n$’s, with weights ${{\cal Z}}_n^\dagger {{\cal R}}^{-1}(\infty) {{\cal Z}}_n$, and ${\langle \omega^2 \rangle}_{\rm L}$ is the weighted harmonic mean of the $\tilde{\omega}^2_n$’s, with weights $\tilde{{{\cal Z}}}_n^\dagger {{\cal R}}(\infty) \tilde{{{\cal Z}}}_n$. The two sets of weights are in general different, except when the transverse and longitudinal eigenmodes happen to be the same. However, “superconvergence”, [Eq. (\[iden\])]{}, implies the same normalization in any case: $$\sum_n {{\cal Z}}_n^\dagger {{\cal R}}^{-1}(\infty) {{\cal Z}}_n = \sum_n \tilde{{{\cal Z}}}_n^\dagger {{\cal R}}(\infty) \tilde{{{\cal Z}}}_n .$$ In the special case where a single IR-active mode exists ${\langle \omega^2 \rangle}_{\rm T} = \omega_{\rm T}^2$ and ${\langle \omega^2 \rangle}_{\rm L} = \omega_{\rm L}^2$: this concludes the proof of [Eq. (\[rap\])]{}. A well known bound requires the matrix ${{\cal R}}$ to be positive definite [@Fiebig05]. Since this is based on stability arguments, it only concerns ${{\cal R}}(0)$ and ${{\cal R}}(\infty)$: the former is a genuine static property of the real system, while the latter can be regarded as a static property in the infinite nuclear mass limit. At any other frequency the matrix ${{\cal R}}(\omega)$ accounts for the forced oscillations of the system, which is clearly out of equilbrium. Therefore ${{\cal R}}(\omega)$ is not required, in general, to be positive definite; because of the same reason, $\varepsilon(\omega)$ is not a positive real function in ordinary dielectrics [@textbook]. All of the above results reduce to previously known ones for a magnetically inert material, where the $2 \times 2$ matrix ${{\cal R}}(\omega)$ has the unique nontrivial entry $\varepsilon(\omega)$. It is worth examining [Eq. (\[mean\])]{} for $\zeta^*_n = 0$: $$\frac{{\langle \omega^2 \rangle}_{\rm L}}{{\langle \omega^2 \rangle}_{\rm T}} = \frac{1}{\varepsilon^2(\infty)}\frac{\sum_n \frac{1}{\omega_n^2} (Z^*_n)^2 }{ \sum_n \frac{1}{\tilde{\omega}_n^2} (\tilde{Z}^*_n)^2 } . \label{mean2}$$ The weights in the harmonic means $(Z^*_n)^2$ and $(\tilde{Z}^*_n)^2$ are the (squared) transverse and longitudinal effective charges, respectively. If (and only if) the transverse and longitudinal eigenmodes are the same, then $\tilde{Z}^*_n = {Z}^*_n/\varepsilon(\infty)$. Actually, this is the well known relationship between the transverse (a.k.a. Born) end longitudinal (a.k.a. Callen) effective charges. For an ordinary dielectric the $N$ poles of $\varepsilon(\omega)$ are the TO frequencies $\omega_n$, while the $N$ poles of $1/\varepsilon(\omega)$ are the LO ones $\tilde{\omega}_n$: see [Eqs. (\[ini\]) and (\[ini2\])]{}. Since the response is a single-component function, the poles of $1/\varepsilon(\omega)$ coincide with the zeros of $\varepsilon(\omega)$. Simple considerations about zeros and poles of $\varepsilon(\omega)$ eventually lead to the simple expression $$\frac{{\langle \omega^2 \rangle}_{\rm L}}{{\langle \omega^2 \rangle}_{\rm T}} = \prod_{n=1}^N \frac{\tilde{\omega}_n}{\omega_n} ,$$ first found in 1961 by Kurosawa [@Kurosawa61; @Sievers90]. This result [*does not*]{} generalize to the ME case, for the good reason that the response is a $2 \times 2$ matrix: the present formulation is based on [*traces*]{} throughout, and the inverse of the trace bears no simple relationship to the trace of the inverse, at variance with the purely electrical case. This Letter addresses the linear relationship between the pairs $(E,H)$ and $(D,B)$ throughout, starting with [Eq. (\[resp\])]{} onwards. Other pairings are possible. In particular the choice $(E,B)$ and $(D,H)$ looks like a more natural one for at least two reasons: (i) the microscopic forces are determined by the $(E,B)$ pair [@rap_a30], and (ii) a relativistic formulation can be elegantly cast in terms of two four-dimensional field tensors whose entries are $(E,B)$ and $(D,H)$, respectively [@Hehl08; @Hehl09]. Nonetheless, the same choice in the present context would not be a convenient one. In fact, as explained above (see also Ref. [@rap_a30]), in a transverse mode $E=0$ and $B \neq 0$, and in a longitudinal one $D=0$ and $H \neq 0$. Throughout this Letter we have stressed the formal equivalence of electric and magnetic fields in their coupling to the lattice in MEs. However, the orders of magnitude of electric and magnetic phenomena in condensed matter are not the same. ME effects are notoriously small [@Fiebig05], and the corrections to the LST relationship in most cases are expected to be small as well. In oxides the dielectric constants—either $\varepsilon(\infty)$ or $\varepsilon(0)$—are typically in the range 2 to 10, while $|\mu -1|$ is of the order $10^{-4}$ [@nota]; in the most studied linear ME, i.e. Cr$_2$O$_3$, even $\alpha$ is of the order $10^{-4}$ [@Iniguez08; @Hehl08]. An accurate evaluation of the lhs of [Eqs. (\[rap\]) and (\[lst3\])]{} in “conventional” ME materials would require a measurement of all the entries of the response matrices ${{\cal R}}$ to the same absolute error, which could be problematic. More perspicuous effects are expected in nonconventional materials [@Fiebig05], such as those where the ME effect can be tuned [@Lee10]. In conclusion, this Letter shows that the ratio $\varepsilon(0)/\varepsilon(\infty)$ entering the LST relationship must be replaced, for a linear ME, by the lhs of [Eqs. (\[rap\]) and (\[lst3\])]{}, whose ingredients are the full ME responses at frequency $0$ and $\infty$, i.e. static and clamped-ion. In the most general case the rhs member of our generalized LST relationship is the ratio between spectral moments of the longitudinal and transverse excitations of the medium. It assumes a simple form for a crystalline ME in the harmonic approximation, and finally is identical to the original LST one when only one mode is IR active. The relationship shows that the LO-TO splitting in a ME originates from the coupling of ionic displacements to both electric and magnetic macroscopic fields. I thank J. Ìñiguez and D. Vanderbilt for illuminating discussions about magnetoelectrics. Work supported by the ONR grant N00014-07-1-1095. [10]{} . . . . . . . . . . . . . . . . . . . . . . In the neighborhood of ferromagnetic transitions $|\mu -1|$ may become much larger, while in the presence of soft modes $\varepsilon(0)$ may increase by some orders of magnitude. J. H. Lee [*et al.*]{}., Nature [**466**]{}, 954 (2010).

--- author: - 'Takashi Oka, Hideo Aoki' title: Nonequilibrium Quantum Breakdown in a Strongly Correlated Electron System --- Introduction ============ During the past decades, there has been an increasing fascination and surprises with diverse [*quantum many-body effects*]{}. With the magical touch of interaction a simple electron system may assume insulating, metallic, magnetic or superconducting states according as the control parameters are changed. Strongly correlated electron systems, as exemplified by the high-Tc superconductors and their host materials realized in transition-metal oxides, as well as by organic metals, have provided us with an ideal playground, where various crystal structures with band-filling control and band-width control etc provide the richness in the phase diagram[@Imada1998]. On the other hand, there is a long history of the interests in [*non-equilibrium phase transitions*]{}. Statistical mechanically, there is an intriguing problem of how we can generally define the notion of a “phase" in non-equilibrium systems, but we can still discuss individual systems in specified non-equilibrium conditions to extract more general viewpoints. Now, if we combine the above two ingredients, namely, if we consider [*strongly-correlated electron systems in non-equilibrium*]{}, we plunge into an even more fascinating physics. In fact, recent years have witnessed an upsurge of interests in non-equilibrium states in many-body systems with drastic changes in the electronic states in strong dc electric fields, in intense laser fields, etc. ![ Non-linear transport and optical response []{data-label="title"}](TOfig0.eps){width="6.cm"} Developments in fabrication techniques such as realization of clean thin films with electrodes attached have triggered several groundbreaking experiments, e.g., non-linear transport measurements in thin films [@Asamitsu1997; @Ponnambalam1999; @Oshima1999; @Liu2000; @Baikalov2003; @Sawa2004], in layered systems [@Inagaki2004] and observations of clean metallic states in heterostructures [@OhtomoNature]. Non-linear phenomena in correlated electron systems now begin to attract interests in a wide range of researchers: One obvious area of application is future-generation electronic devices, where a high sensitivity of a system near a phase boundary to external conditions may lead to drastic functionalities[@Asamitsu1997]. However, even more attractive is its relevance to fundamental physics, especially, to non-equilibrium statistical physics, where we can observe the behavior of various phase transitions taking place under non-equilibrium conditions. The purpose of the present article is to discuss the nonequilibrium metal-insulator transition in strongly correlated electron systems [@Oka2005a; @Oka2003; @Oka2004a; @Oka2004b], which is known, for equilibrium systems, as Mott’s transition. Before going into detail, we first give a brief introduction of the transition, and discuss how quantum breakdown through non-adiabatic transitions in nonequilibrium becomes relevant in non-linear transports. The model we study is the single-band Hubbard model, which is the simplest possible one that captures many essential properties of correlated electron physics. The Hamiltonian reads $$\begin{aligned} H_0=-t_{{\rm hop}}\sum_{{\langle}i,j{\rangle}\sigma}\left(c^\dagger_{i\sigma}c_{j\sigma}+\mbox{h.c.}\right) +U\sum_{i}n_{i{\uparrow}}n_{i{\downarrow}},\end{aligned}$$ where $c_{i\sigma}$ annihilates an electron on site $i$ with spin $\sigma$, $n_{i\sigma}= c^\dagger_{i\sigma}c_{i\sigma}$ the number operator, $U$ the strength of the on-site Coulomb repulsion, and $t_{{\rm hop}}$ the hopping integral. The filling $\displaystyle{n=\frac{1}{L}\sum_{i=1}^L{\langle}n_{i{\uparrow}}+n_{i{\downarrow}} {\rangle}}$, with $L$ the number of sites, is an important parameter, which changes the groundstate property drastically. ![Metal-insulator transition in equilibrium due to doping: (a) A Mott insulator realized at half filling. (b) A hole-doped metallic state. (c) An electron-doped metallic state. []{data-label="TOfig1"}](TOfig1.eps){width="10.cm"} When the band is half-filled with one electron per site on average($n=1$), each electron tends to be localized on a separate lattice site and the spin tends to be antiferromagnetically correlated. When $U/t$ is large enough, the groundstate is insulating, which is called Mott’s insulator (Fig.\[TOfig1\](a)), and the groundstate is separated from the charge excited states with a many-body energy gap — Mott gap. When we inject carriers (usually with a chemical doping by adding or replacing to other elements) to increase (electron doping) or decrease (hole doping) the filling from unity, the Mott gap collapses for large enough doping, and the system becomes metallic. This is the metal-insulator transition or the Mott transition, which is widely observed in strongly correlated materials. In these materials, a state occupied simultaneously by up an down-spin electrons — which we call a doublon — and holes carry the current. After the discovery of the high-temperature superconductivity in cuprates, carrier-doped Mott insulators have been subject of a huge number of experimental and theoretical studies. ![(a) Schematic experimental configuration. (b) Carriers (doublons and holes) created by an external electric field. []{data-label="TOfig2"}](TOfig2.eps){width="11.cm"} Now, let us consider what will happen if we attach a set of electrodes to a strongly-correlated sample, and apply a large bias voltage across the electrodes (Fig. \[TOfig2\] (a)). Although the setup may seem simple enough, there is a profound physics involved. In regions near the electrodes (or near the interface in the case of heterojunctions between a strongly-correlated and ordinary materials), a “band bending” similar to doped semiconductors can take place and lead to an interface Mott transition when the filling becomes one [@OkaNagaosa]. The width of the insulating layer changes as the applied bias is changed, which dominates the behavior of the non-linear transport (i.e., the $I-V$ characteristics). The result (with DMRG + Hartree potential) for the band-bending effects in this case can be understood if we assume a [*local equilibrium*]{} for the relation between the density of electrons and the potential. The local properties are determined by the Hartree potential governed by Poisson’s equation, which in turn determines the local chemical potential and controls the metal-insulator transition. Even more interesting, however, is the case where we no longer have local equilibrium. Specifically, a [*quantum many-body breakdown of a Mott’s insulator*]{} takes place when the applied electric field is large enough and creates doublons and holes in the Mott insulating groundstate (Fig. \[TOfig2\] (b)) [@Oka2003; @Oka2004b; @Oka2005a]. ![Dielectric breakdown of a Mott insulator in a strong electric field due to many-body Landau-Zener transition: The groundstate and excited states with charge excitations are separated by an energy barrier, and quantum tunneling among many-body states takes place when the electric field is strong enough. []{data-label="fig:energydiagram"}](TOfig3.eps){width="11.5cm"} The creation mechanism is a many-body analog of the “Zener breakdown", well-known in semiconductor physics [@Zener]. Namely, while we cannot use the notion of the electronic band structure for correlated electron systems, we can envisage the carrier-creation process as a tunneling across a kind of barrier. As displayed in Fig. \[fig:energydiagram\], production of carriers occurs through tunneling between the Mott insulating groundstate and excited states with doublons and holes. If we denote the distance between a doublon and a hole by $l_{{\rm dh}}$, the energy profile as a function of $l_{{\rm dh}}$ roughly reads $$\Delta E\sim U-l_{{\rm dh}}F,$$ where $F$ is the strength of the electric field. The profile curve reaches the energy before the creation of the doublon-hole pair for the separation at which $$\Delta E\sim U-\bar{l}_{\rm dh}F=0, \label{eq:delEzero}$$ to which the tunneling becomes possible. There is a threshold field strength for this process to occur. This is because larger quantum fluctuations are required to have a larger separation between doublon and hole in the Mott insulator. In other words, the overlaps of many-body wave function of the groundstate and excited states decrease rapidly for large $l_{\rm dh}$. We can formulate this with the Landau-Zener picture in the time-dependent gauge for the external electric field, for which, as we shall show below[@Oka2005a](eq.(\[eq:lzHub\])), the threshold field strength is given by $$F_{{\rm th}}=\frac{\Delta_{{\rm c}}(U)^2}{8t_{{\rm hop}}},$$ where $\Delta_{{\rm c}}(U)$ is the charge gap (i.e., the Mott gap), and the tunneling rate per length is given by $$\Gamma(F)/L=-\frac{2F}{h}a\ln\left[1-p(F)\right],$$ where $p(F)=e^{-\pi\frac{F_{{\rm th}}}{F}}$ is the tunneling probability and $a$ a non-universal constant depending on the detail of the system. The tunneling rate $\Gamma(F)/L$, being related to the production rate of carriers, is directly related to physical properties in the bulk if the interface effect is neglected. Indeed, such a non-linearity in the $I-V$ characteristics has been observed in real materials, most prominently in a one-dimensional copper oxide [@tag]. Another interesting consequence of eq.(\[eq:delEzero\]) is that it gives the “critical separation" of the doublon-hole excitation $\bar{l}_{{\rm dh}}=U/F_{{\rm th}}$. This has to do with the convex shape of the energy profile against $l_{\rm dh}$ (Fig.\[fig:energydiagram\]), which is reminiscent of the energy profile for the standard nucleation theory that treats the critical size of a stable-phase droplet to grow without being crushed, although the physics involved is quite different. In the present case, when the field is greater than the threshold, the [*electric-field induced metallic state*]{}, where doublon-hole pairs continue to be created, becomes the stable phase. The first goal of this article is to derive the relations presented above and study the creation mechanism of carriers (this part is the extended argument of our papers [@Oka2003; @Oka2005a]). We need to treat the process quantum mechanically and in a many-body formulation. In doing so, we present a renewed and unified interpretation of the Zener transition of insulators. The key quantity is the effective Lagrangian of quantum dynamics (see §\[sec:Heisenberg-Euler\] for a detailed introduction) which is define by [@Oka2005a] $$\mathcal{L}(F)=-\frac{i}{L^d}\lim_{t\to \infty}\frac{1}{t}\ln\Xi(t),$$ where $\Xi(t)$ is the groundstate-to-groundstate transition amplitude and $L^d$ is the volume of the $d$ dimensional system and $L$ the linear size. There is a deep relation between the theories of dielectric breakdown in condensed matter and a branch in quantum field theory known as non-linear quantum electrodynamics (QED) (table: \[fig:theories\]). The effective Lagrangian defined above coincides with the Heisenberg-Euler effective Lagrangian for non-adiabatic evolution [@Heisenberg1936; @Oka2005a]. The effective Lagrangian have been used to study the Schwinger mechanism of electron-positron pair production from the QED vacuum in strong electric fields [@Schwinger1951]. In fact, we show that the Schwinger mechanism and the Zener tunneling are equivalent, where the effective action coincides if we consider the breakdown of simple Dirac type band insulators. Furthermore, the effective action gives the non-adiabatic extension of the Berry phase theory of polarization. dielectric breakdown in cond. matter non-linear QED ------------------------- ---------------------------------------------- -------------------------------------- mechanism Zener breakdown[@Oka2003] Schwinger mechanism [@Schwinger1951] excitation electron (doublon)-hole pair electron-positron pair effect of interaction many-body Landau-Zener back reaction non-linear polarization cross correlation (ME effect) photon-photon interaction non-adiabatic Berry phase theory [@Oka2005a] - : Relation between the theory of dielectric breakdown in condensed matter and non-linear QED from the point of view of the effective Lagrangian.[]{data-label="fig:theories"} In the latter part of the article we shall discuss the effect of annihilation of doublon-hole pairs (Fig. \[fig:annihilation\]). In a one-dimensional system, a doublon and a hole cannot pass each other without being pair-annihilated even as virtual processes. Since the groundstate is locally stable, the many-body state tends to remain in the ground state, but there should be a finite probability for the state to “branch into" excited states through many paths in the many-body energy space. Thus, the long-time behavior of the wave function involves numerous scattering processes in the energy space, where, as we shall see, the phase interference plays a key role. We can indeed regard the phase before the dielectric breakdown takes place as a dynamical localization in the many-body energy space, which reduces the tunneling rate and makes the groundstate survive[@Oka2004b; @Oka2004a]. A statistical mechanical treatment helps in understanding this, and we briefly discuss it in terms of the quantum walk. ![Annihilation processes for carriers in a correlated electron system. []{data-label="fig:annihilation"}](TOfig4.eps){width="11.5cm"} A brief comment on the numerical methods used in the article: In order to understand the non-equilibrium processes, we need to integrate the time-dependent, many-body Schrödinger equation to look at the evolution of the many-body wave function in, say, the Hubbard model in strong electric fields. This is a formidable task, for which no analytically exact treatment is known, so that we rely on several numerical methods, which include the exact diagonalization and the time-dependent density matrix renormalization group method (td-DMRG; The version of td-DMRG we adopt is the one proposed by White and Feiguin[@White2004]). Non-adiabatic evolution and pair creation of carriers ===================================================== Electric fields and gauge transformation ---------------------------------------- When we describe a system in finite electric fields, we can choose from two gauges. One is the case where we have a slanted electrostatic potential, with the gauge field $A^\mu=(Fx,0)$ for a one-dimensional system with $F=eE$ being the electric field, while the other represents the electric field via a time-dependent vector potential, $A^\mu=(0,-Ft)$. In the first gauge, the tilted potential enters in the Hamiltonian as $$H(F)=H_0+F\hat{X},\quad \hat{X}=\sum_jjn_j,$$ where $\hat{X}$ is the position operator[@Resta1998] and $H_0$ the original Hamiltonian. This gauge is [*in*]{}compatible with systems with periodic boundary conditions. The Hamiltonian $H(F)$ in fact becomes an unbounded operator in an infinite system, since one can lower the energy indefinitely by moving an electron to $j\to -\infty$. In the other gauge, which we call the time-dependent gauge, the hopping term of the Hamiltonian becomes time-dependent as $$H(\phi(t))=-t_{{\rm hop}}\sum_{i\sigma} \left(e^{i\frac{2\pi}{L}\phi(t)}c^\dagger_{i+1\sigma}c_{i\sigma}+\mbox{h.c}\right) +\hat{V}, \label{2HamiltoniangeneralTD}$$ where $\Phi(t)$ represents a time-dependent Aharonov-Bohm(AB) flux, $$\phi(t)\equiv \Phi(t)/\Phi_0=FLt/h. \label{2timeperiod}$$ Physically, this gauge amounts to considering a periodic system (a ring) and a magnetic flux piercing the ring, where the time-dependent flux induces electric fields by Faraday’s law (Fig. \[fig:twogauges\]). For a higher-dimensional system, the ring becomes a (generalized) torus. The time-dependent gauge is suited for periodic systems since it is compatible with the lattice translation symmetry. The electric current operator is obtained by differentiating the Hamiltonian by $A^1$ as $$J(\phi)=-\frac{dH(\phi)}{dA^1}=-it_{{\rm hop}}\sum_{i\sigma} \left(e^{i\frac{2\pi}{L}\phi}c^\dagger_{i+1\sigma}c_{i\sigma} -e^{-i\frac{2\pi}{L}\phi}c^\dagger_{-i\sigma}c_{i+1\sigma}\right).$$ There exists an important operator relation among $H$, $J$ and $\hat{X}$, $$J(\phi)=\frac{i}{\hbar}\left[H(\phi),\hat{X}\right],$$ which comes from Heisenberg’s equation of motion for the current operator, $J(\phi)=\frac{d}{dt}\hat{X}$. We can relate the two gauges with a twist operator[@Avron1992] defined by $$g(\phi)=e^{-i\frac{2\pi}{L}\phi\hat{X}},$$ and the two Hamiltonians are related by a gauge transformation generated by the twist operator, i.e., $$H(F)=g^\dagger(\phi(t))H(\phi(t))g(\phi(t))-i g^\dagger (\phi(t)){\partial}_t g(\phi(t)). \label{2gaugetransformation}$$ Heisenberg-Euler effective Lagrangian {#sec:Heisenberg-Euler} ------------------------------------- We first discuss the non-adiabatic evolution of electron wave functions in insulators (either one-body or many-body) in strong electric fields. Let us consider an insulator at $T=0$ and $F=0$, which is described by the groundstate wave function $|\Psi_0{\rangle}$. We then switch on the electric fields at $t=0$ to study the quantum mechanical evolution of the system. We limit our discussions to coherent dynamics and ignore the effect of dissipation due to heat bath degrees of freedom as well as boundary effects near the electrodes. A key quantity to study the non-adiabatic evolution and quantum tunneling in strong electric fields is the (condensed-matter counterpart to the) effective Lagrangian introduced for QED by Heisenberg and Euler[@Heisenberg1936]. In the time-independent gauge, the electrons are described by the solution of the Schrödinger equation, $$|\Psi(t){\rangle}=e^{-itH(F)}|\Psi_0{\rangle},$$ where we have put $\hbar=1$. The overlap of the solution with the groundstate for $F=0$ — groundstate-to-groundstate transition amplitude — should contain the information on the tunneling processes, so we define $$\Xi(t)={\langle}\Psi_0|e^{-itH(F)}|\Psi_0{\rangle}e^{itE_0},$$ where we have factored out the trivial dynamical phase of the groundstate, $E_0={\langle}\Psi|H(F=0)|\Psi{\rangle}$. In the case of the time-dependent gauge, we need to be careful, since the groundstate is $\phi$ dependent. If we denote $|0;\phi{\rangle}$ as the instantaneous groundstate of $H(\phi)$, the groundstate-to-groundstate transition amplitude becomes $$\begin{aligned} \Xi (\tau) ={\langle}0;\phi(\tau)|\hat{T}e^{-\frac{i}{\hbar} \int_0^\tau H(\phi(s))ds}|0;\phi(0){\rangle}e^{\frac{i}{\hbar}\int_0^\tau E_0(\phi(s))ds}, \label{ggamplitude}\end{aligned}$$ where $\hat{T}$ stands for the time ordering, and $ E_0(\phi)={\langle}0;\phi(\tau)|H(\phi)|0;\phi{\rangle}$ the dynamical phase of the groundstate. ![The original problem studied by Callan and Coleman in which quantum tunneling from an unstable vacuum is considered [@CallanColeman1977]. []{data-label="fig:breakdown"}](TOfig6.eps){width="3.5cm"} We define[@Oka2005a] the effective Lagrangian by $$\mathcal{L}(F)=-\frac{i}{L^d}\lim_{t\to \infty}\frac{1}{t}\ln\Xi(t),$$ where $L^d$ is the volume of the $d$ dimensional system with a linear dimension of $L$. We can also regard the Lagrangian as the exponent of the asymptotic behavior of the amplitude, $\Xi (\tau)\sim e^{i\tau L^d\mathcal{L}(F)}$. When the electric field is large enough, the groundstate becomes unstable with the quantum tunneling to excited states activated. The tunneling rate is described by the imaginary part of the effective Lagrangian, $$\Gamma(F)/L^d \equiv 2\mbox{Im}\;\mathcal{L}(F),$$ which gives the rate of the exponential decay of the vacuum (groundstate). In the quantum field theory, the decay rate of an unstable vacuum has been discussed by Callan and Coleman, where the tunneling takes place when the potential is suddenly changed by an external field[@CallanColeman1977] (Fig. \[fig:breakdown\]). As we shall see later in several models, in the theory of dielectric breakdown, the tunneling corresponds to creation of charge carriers. In band insulators the carriers are electrons and holes, while in Mott insulators they are doublons and holes. If we neglect boundary effects and assume that all the carriers are absorbed by electrodes, we can conclude that the tunneling rate is proportional to the leakage current, i.e., $$J_{{\rm leak}}\propto \Gamma(F)/L^d.$$ Indeed, this is the original picture of Zener when he calculated the leakage current in a simple band insulator \[fig:zener\]. Zener has studied the dielectric breakdown in a simple one-dimensional insulator using the time-independent gauge [@Zener1934] as well as the time-dependent gauge [@Zener]. In the former, he has calculated the tunneling probability of Bloch functions in constant electric fields to obtain the tunneling rate. In the time-dependent gauge, he has described the problem as a system with a time-dependent Hamiltonian represented by a two-by-two matrix to study the tunneling near the level anti-crossing, which is now known as the Landau-Zener transition [@Landau; @Zener]. The reason why we have called $\mathcal{L}(F)$ the effective Lagrangian is that it coincides with the Heisenberg-Euler effective Lagrangian in QED [@Heisenberg1936]. They have studied the dynamics and nonlinear responses of the QED vacuum in strong electric fields by calculating the effective Lagrangian (for a review, see e.g. [@Dittrich]). By integrating out high-energy degrees of freedom (polarization processes due to electron-positron creation/annihilation) they arrived at an effective description of the low-energy degrees of freedom, namely the quantum correction, originating from the fluctuation of the QED vacuum, to the Maxwell theory of electromagnetism. Indeed, if we apply our formalism to band insulators with Dirac-type (mass-gapped) dispersions, the effective Lagrangian coincides with the Heisenberg-Euler Lagrangian with some modifications coming from the Brillouin zone structure of the Bloch waves as will be shown in the next section. The correspondence between the two phenomena is straightforward: The ground-state of the insulator translates to the QED vacuum, charge excitations to the electron-positron pairs. The tunneling rate also has its QED counterpart, namely the vacuum decay rate due to the Schwinger mechanism — creation of electron-positron pairs in strong electric fields [@Schwinger1951]. ### Related theories We conclude this section with comments on the relation of the effective Lagrangian approach to earlier theoretical frameworks. #### Berry’s phase theory of polarization: In the Berry’s phase theory of polarization[@Resta1992; @KingSmith1993; @Resta1998; @Resta1999; @Nakamura2002], the ground-state expectation value of the twist operator $e^{-i\frac{2\pi}{L}\hat{X}}$, which shifts the phase of electron wave functions on site $j$ by $-\frac{2\pi}{L}j$ [@Nakamura2002], plays a crucial role. It was revealed that the real part of a quantity $$w=\frac{-i}{2\pi}\ln{\langle}0|e^{-i\frac{2\pi}{L}\hat{X}}|0{\rangle}$$ gives the linear-response electric polarization, $P_{\rm el}=-\mbox{Re}w$ [@Resta1998], while its imaginary part gives a criterion for metal-insulator transition, i.e., $D=4\pi\mbox{Im}w$ is finite in insulators and divergent in metals [@Resta1999]. The present effective action is regarded as a non-adiabatic (finite electric field) extension of $w$. To give a more accurate argument, recall that the effective Lagrangian can be expressed as $$\mathcal{L}(F)\sim \frac{-i\hbar}{\tau L}\ln\left( {\langle}0|e^{-\frac{i}{\hbar}\tau(H+F\hat{X})} |0{\rangle}e^{\frac{i}{\hbar}\tau E_0}\right)$$ for $d=1$. Let us set $\tau=h/LF$ and consider the small $F$ limit. For insulators we can replace $H$ with the groundstate energy $E_0$ to have $\mathcal{L}(F)\sim wF$ in the linear-response regime. Thus the real part of Heisenberg-Euler’s expression[@Heisenberg1936] for the non-linear polarization $P_{\rm HE}(F)=-{\partial}\mathcal{L}(F)/{\partial}F$ naturally reduces to the Berry’s phase formula $P_{\rm el}$ in the $F\to 0$ limit (cf. eq.(\[2Schwinger2\]) below). Its imaginary part, which is related to the decay rate as $\mbox{Im}P_{\rm HE}(F)=-\frac{\hbar}{2} \frac{{\partial}\Gamma(F)/L}{{\partial}F}$, reduces to $ -D/4\pi$ and gives the criterion for the transition, originally proposed for the zero field case. #### Non-Hermitian quantum mechanics The dielectric breakdown of Mott insulators was also studied in the framework of non-Hermitian quantum mechanics[@Fukui; @NakamuraHatano06]. Fukui and Kawakami studied a non-Hermitian Hubbard model in which the leftward and rightward hopping integral are assumed to be unequal [@Fukui]. The non-Hermiticity is assumed to represent the coupling of the system with a “dissipative environment". With the Bethe ansatz solution they have observed the gap between the groundstate and the first excited state to close when the hopping asymmetry is large enough. It seems that the remaining question is to relate this result with measurable quantities. Zener breakdown of band insulators revisited — Non-adiabatic geometric phase and the Schwinger mechanism -------------------------------------------------------------------------------------------------------- ![(a) Energy levels of a Dirac band in 2D. (b) A one dimensional slice of the higher dimensional Dirac band in which carriers (doublons and holes) are created by an external electric field in the $k_\parallel$ direction. []{data-label="TOfig8"}](TOfig8.eps){width="12.cm"} Before examining the dielectric breakdown of correlated electron systems, let us first discuss the dielectric breakdown of band insulators in an electric field ${\mathbf}{F}$ within the effective-mass picture. This will turn out to be heuristic, since we can obtain an analytic expression for the effective Lagrangian which can be readily applied to general band insulators. For simplicity we take a pair of hyperbolic bands ${\varepsilon}_\pm({\mathbf}{k})=\pm \sqrt{V^2+v^2k^2}$ (considered here in $d$ spatial dimensions), where $2V$ is the band gap, $-(+)$ denote the valence (conduction) band, and $v$ the asymptotic slope of the dispersion. We first obtain the groundstate-to-groundstate transition amplitude with the time-dependent gauge in the periodic boundary condition. There, a time-dependent AB-flux in units of the flux quantum, $\phi(\tau)=FL\tau/h$ (with the electronic charge $e=1$ and $L$ being the system size), is introduced to induce an electric field $F$, which makes the Hamiltonian time dependent as $$H(\phi(\tau))= \sum_{\bf{k},\alpha=\pm}{\varepsilon}_\alpha\left({\mathbf}{k}+\frac{2\pi}{L}\phi(\tau){\mathbf}{e}_{\parallel}\right) c_{\alpha}^\dagger({\mathbf}{k})c_{\alpha}({\mathbf}{k}).$$ Here ${\mathbf}{e}_{\parallel}$ is the unit vector parallel to ${\mathbf}{F}$, and $c_{\alpha}^\dagger({\mathbf}{k})$ the creation operator with spin indices dropped. If we denote the ground state of $H(\phi)$ as $|0;\phi{\rangle}$ and its energy as $E_0(\phi)$, the groundstate-to-groundstate transition amplitude reads $$\begin{aligned} \Xi (\tau) ={\langle}0;\phi(\tau)|\hat{T}e^{-\frac{i}{\hbar} \int_0^\tau H(\phi(s))ds}|0;\phi(0){\rangle}e^{\frac{i}{\hbar}\int_0^\tau E_0(\phi(s))ds}, \label{ggamplitude}\end{aligned}$$ where $\hat{T}$ stands for the time ordering. The effective Lagrangian $\mathcal{L}(F)$ for the quantum dynamics is defined from the asymptotic behavior, $\Xi (\tau)\sim e^{\frac{i}{\hbar} \tau L^d\mathcal{L}(F)}$. The dynamics of the one-body model can be solved analytically (Fig.\[TOfig8\] (b)), since we can cut the dispersion in $d$ spatial dimensions into slices, each of which reduces to Landau-Zener’s two band model in 1D[@Landau; @Zener]. Namely, if we decompose the $k$ vector as $({\mathbf}{k}_\perp,k_\parallel)$, where ${\mathbf}{k}_\perp$ ($k_\parallel$) is the component perpendicular (parallel) to ${\mathbf}{F}$, where each slice for a given ${\mathbf}{k}_\perp$ is a copy of Landau-Zener’s model with a gap $\Delta_{\rm band}({\mathbf}{k})\equiv 2\sqrt{V^2+v^2k_\perp^2}$. The Landau-Zener transition takes place around the level anti-crossing for which $k_\parallel+\frac{2\pi}{L}\phi(\tau )$ moves across the Brillouin zone(BZ) in a time interval $\delta \tau=h/F$. The process can be expressed as a scattering and the Bogolubov coefficients between the “in" and “out" states (see Fig.\[TOfig8\]) is given by the solution to the two band problem, i.e., $$\begin{aligned} c_+^\dagger({\mathbf}{k})\to \sqrt{1-p({\mathbf}{k})}e^{-i\chi({\mathbf}{k})}c_ +^\dagger({\mathbf}{k})+\sqrt{p({\mathbf}{k})}c_-^\dagger({\mathbf}{k}),\nonumber\\ c_-^\dagger({\mathbf}{k})\to -\sqrt{p({\mathbf}{k})}c_+^\dagger({\mathbf}{k})+ \sqrt{1-p({\mathbf}{k})}e^{i\chi({\mathbf}{k})}c_-^\dagger({\mathbf}{k}). \label{2Bogoliubov}\end{aligned}$$ Here the tunneling probability for each ${\mathbf}{k}$ is given by the Landau-Zener(LZ) formula[@Landau; @Zener], $$p({\mathbf}{k})=\exp\left[-\pi \frac{(\Delta_{\rm band}({\mathbf}{k})/2)^2}{vF} \right]. \label{LZS1}$$ On the other hand, the phase $\chi({\mathbf}{k})=-\theta({\mathbf}{k})+\gamma({\mathbf}{k})$ appearing in the Bogolubov coefficients consists of the trivial dynamical phase, $ \hbar \theta({\mathbf}{k})= \int_0^{\delta \tau}{\varepsilon}_{+}({\mathbf}{k}+ \frac{2\pi}{L}\phi(s){\mathbf}{e}_\parallel) ds, $ and the Stokes phase[@Zener; @Kayanuma1993], $$\begin{aligned} \gamma({\mathbf}{k})=\frac{1}{2}\mbox{Im}\;\int_0^\infty ds\frac{e^{-i(\Delta_{\rm band}({\mathbf}{k})/2)^2 s}}{s} \left[\cot (vFs)-\frac{1}{vFs}\right]. \label{2StokesphaseDDimBand}\end{aligned}$$ The Stokes phase, a non-adiabatic extension of Berry’s geometric phase [@Berry1984], depends not only on the topology of the path but, unlike the adiabatic counterpart, also on the field strength $F$ [@Kayanuma1997]. In terms of the fermion operators the groundstate is obtained by filling the lower band $|0;\phi{\rangle}=\prod_{{\mathbf}{k}} c^\dagger_-({\mathbf}{k}-\frac{2\pi}{L}\phi{\mathbf}{e}_{\parallel})|{\rm vac}{\rangle}$, where $|{\rm vac}{\rangle}$ is the fermion vacuum with $c_{\pm}({\mathbf}{k})|{\rm vac}{\rangle}=0$. If we assume that excited charges are absorbed by electrodes we obtain from eqs.(\[ggamplitude\]), (\[2Bogoliubov\]) $$\begin{aligned} \mbox{Re}\;\mathcal{L}(F) &=&-F \int_{\rm BZ} \frac{d{\mathbf}{k}}{(2\pi)^{d}}\frac{\gamma ({\mathbf}{k})}{2\pi},\nonumber\\ \mbox{Im}\;\mathcal{L}(F) &=&-F\int_{\rm BZ} \frac{d{\mathbf}{k}}{(2\pi)^{d}}\frac{1}{4\pi}\ln \left[ 1-p({\mathbf}{k})\right], \label{2Schwinger2}\end{aligned}$$ where the dynamical phase $\theta$ cancels the factor $e^{\frac{i}{\hbar}\int_0^\tau E_0(\phi(s))ds}$ in eq.(\[ggamplitude\]). ![The dependence of the conductivity on the electric field in the non-linear regime for band insulators with spatial dimension $d=1,2,3$. The inset zooms in the threshold region. []{data-label="fig:conductivity"}](TOfig9.eps){width="10.cm"} Integration over ${\mathbf}{k}$ in eq.(\[2Schwinger2\]) leads to the groundstate decay rate per volume for a $d$-dimensional hyperbolic band, $$\begin{aligned} \Gamma(F)/L^d&=& \frac{F}{(2\pi)^{d-1}h}\left(\frac{F}{v}\right)^{(d-1)/2}\nonumber\\ &&\times \sum_{n=1}^\infty \frac{1}{n^{(d+1)/2}} e^{-\pi n\frac{V^2}{vF}} \left[ \mbox{erf}\left(\sqrt{\frac{nv\pi^3}{F}}\right)\right]^{d-1}. \label{BandSchwingerFormula}\end{aligned}$$ The threshold for the tunneling is governed by the most nonlinear (actually essentially singular) factor in the above formula, namely $e^{-\pi n\frac{V^2}{vF}}$, so that the threshold electric field is given by $$F_{{\rm th}}=\frac{V^2}{v}.$$ Although an analytic integration eq.(\[2Schwinger2\]) is possible for a Dirac band (= hyperbolic valence and conduction bands), the expression is valid for general band dispersions. In fig. \[fig:conductivity\], the leakage current divided by the field strength, which is proportional to $\Gamma(F)/FL^d$, is plotted for the spatial dimension $d=1,2,3$. The $F$-dependence is essentially $$J_{{\rm leakage}}\propto F^{(d+1)/2} e^{-\pi \frac{F_{{\rm th}}}{F}},$$ which has a threshold behavior as shown in the inset of the figure. Above the threshold, two regimes exist. In the medium field regime, the current scales as $j_{{\rm leakage}}\sim F^{(d+1)/2}$ where the power depends on $d$. However, when the field strength is even stronger, the error function appearing in eq.(\[BandSchwingerFormula\]), which is due to the lattice structure (with the ${\mathbf}{k}$ integral restricted to the BZ), starts to take an asymptote (erf$(x) \sim (2/\sqrt{\pi})x$). Then various factors (including a power of $F$) cancel with each other, and the leakage current in the $F\to \infty$ limit approaches a [ *universal*]{} function, $$J_{{\rm leakage}}\propto \Gamma(F)/L^d\to -\frac{F}{h}\ln \left[1-\exp\left(-\pi \frac{F_{\rm th}}{F} \right)\right], \label{eq:leakagecurrent}$$ where the $d$ dependence disappears up to a trivial $d$-dependent numerical factor. This prediction on the non-linear transport can be checked experimentally including low-dimensional systems such as carbon nanotubes ($d=1$). Graphene ($d=2$) is also interesting, but this system has a massless Dirac dispersion, so that a special treatment should be required. #### Comparison to Heisenberg-Euler-Schwinger’s results in non-linear QED Let us have a closer look at the decay of the QHE vacuum. In 1936, Heisenberg and Euler studied Dirac particles in strong electric fields, and discussed non-linear optical responses of the QED vacuum — vacuum polarization — in terms of an effective Lagrangian [@Heisenberg1936]. Later, Schwinger refined their approach and calculated the vacuum decay rate [@Schwinger1951][^1]. Up to the one-loop level, Schwinger calculated the vacuum-to-vacuum transition amplitude using the proper time regularization method to obtain $$\Delta\mathcal{L}^{{\rm QED}}(F)=\frac{1}{8\pi^2}\int_0^\infty\frac{ds}{s^2} \left[F\cot(Fs)-\frac{1}{s}\right]e^{-ism_e^2} \label{eq:HES}$$ for (3+1)-dimensional QED, where $m_e$ is the electron mass. The integrand has a pole in the complex domain and has an imaginary part, which gives $$\Gamma(F)^{{\rm QED}}/L^d=\frac{\alpha F^2}{2\pi^2}\sum_{n=1}^\infty \frac{1}{n^2}\exp\left(-\frac{n\pi m_e^2}{|F|}\right), \label{eq:qedschwinger}$$ the famous Schwinger’s formula for the electron-positron pair creation rate [@Schwinger1951], where $\alpha=1/137$ is the fine-structure constant. Thus the expression for the QED effective Lagrangian, eq.(\[eq:HES\]), coincides with the Stokes phase for the non-adiabatic Landau-Zener tunneling, except for a difference in the momentum integral. As we have mentioned above, an important difference in lattice systems is that the momentum integral is limited to the Brilliouin zone, and the decay-rate acquires an extra factor (compare eq.(\[BandSchwingerFormula\]) with erf with eq.(\[eq:qedschwinger\])). This modification changes the strong field limit of the leakage current which leads to the universal expression (eq.(\[eq:leakagecurrent\])). Another important difference, which is quantitative, appears in the threshold voltage: The threshold for band insulators $E_{\rm th}^{\rm band}=F_{\rm th}^{\rm band}/e=V^2/vae$ ($a$: lattice constant) is many orders smaller than the threshold for the QED instability $E^{\rm QED}=\frac{m_e^2c^3}{\hbar}\sim 10^{16}\;\mbox{V/cm}$. For example, if we have an insulator with parameters $a=10^{-7}\mbox{cm},\;v=2t_{{\rm hop}}=1\mbox{eV},\; V=1\mbox{eV}$, then the threshold becomes as small as $E_{\rm th}^{\rm band}=10^7 \mbox{V/cm}$. Heisenberg and Euler’s original aim was to discuss non-linear optical properties of the vacuum in terms of $\Delta\mathcal{L}$. In fact, they calculated the effective Lagrangian in the presence of both electric and magnetic fields [@Heisenberg1936], and obtained $$\begin{aligned} \Delta\mathcal{L}^{{\rm QED}}(F)=C\frac{{\mathbf}{E}^2-{\mathbf}{B}^2}{2}+\frac{2\alpha^2}{45m_e^4} \left[({\mathbf}{E}^2-{\mathbf}{B}^2)^2+7({\mathbf}{E}\cdot{\mathbf}{B})^2\right]+\ldots, \label{eq:qedeffectivelagrangianperturbation}\end{aligned}$$ where $C$ is a diverging constant that we drop after renormalization. The electric polarization can be obtained from the real part of the effective action via $$\Delta P(F)=\frac{{\partial}}{{\partial}F}\Delta \mathcal{L}(F).$$ If we plug in eq(\[eq:qedeffectivelagrangianperturbation\]), the non-linear polarization of Dirac particles becomes $$\begin{aligned} \Delta P&=&\frac{2\alpha^2}{45m_e^4}\left( -4B^2E+14B_\parallel^2E+4E^3 \right)+\ldots,\\ &=&\sum_{n=1}^\infty P^{(n)}({\mathbf}{B})E^n,\end{aligned}$$ where $B_\parallel$ is the component of ${\mathbf}{B}$ parallel to ${\mathbf}{E}$, and $ P^{(n)}({\mathbf}{B})$ the $n-$th order non-linear polarization. Thus we can examine nonlinear polarizations and cross correlations (a combined effect of ${\mathbf}{E}, {\mathbf}{B}$) with the effective Lagrangian, as touched upon in Table \[fig:theories\]. Dielectric breakdown in a Mott insulator — many-body Landau-Zener transition and a nonequilibrium phase diagram --------------------------------------------------------------------------------------------------------------- Before applying the effective Lagrangian approach to the dielectric breakdown of Mott insulators, we need to examine the excitation spectra, which is displayed in fig.\[fig:adiabaticLevels\]. There we plot, for the half-filling, the many-body energy levels of the Hamiltonian, $$\begin{aligned} H(\phi(t))=-t_{{\rm hop}}\sum_{i\sigma} \left(e^{i\frac{2\pi}{L}\phi(t)}c^\dagger_{i+1\sigma} c_{i\sigma}+\mbox{h.c.}\right)+U\sum_in_{i{\uparrow}}n_{i{\downarrow}}+V\sum_i(-1)^in_i. \label{eq:phiHubbard}\end{aligned}$$ Here $U$ is the Hubbard repulsion, $V$ a staggered potential to introduce valence and conduction bands, so that $U=V=0$ corresponds to a noninteracting system in a free space, $U=0,\;V\ne 0$ a band insulator, and a large $U$ and $V=0$ a Mott insulator. In the figure, we have only plotted charge excitations (where the charge rapidities are excited in the language of Bethe-ansatz solution). As can been seen, levels cross in the free model while in the band and the Mott insulators an energy gap separates the ground state from excited states. The gap is $2V$ for the band insulator. The Hubbard Hamiltonian eq.(\[eq:phiHubbard\]) with $U\ne 0,\;V=0$ is also exactly solvable in 1D. Woynarovich used the Bethe ansatz method [@Woynarovich1982; @Woynarovich19822] to study the ground state as well as the excited states (see also [@Sut; @Kabe; @Ari]). The two solid lines in Mott insulator’s spectrum correspond to the ground state and the charge-excited state with one pair of complex charge rapidities, quantum numbers appearing in the Bethe-ansatz solution. The energy gap $\Delta E(U)$ between these states are known to converge to the Mott gap $\Delta_{\rm Mott}(U)$ in the limit of infinite system, $$\Delta E(U)\to\Delta_{\rm c}(U).$$ An important feature in the spectrum of the Mott insulator is that level repulsion occurs at many places over the excited states. The repulsion comes from Umklapp electron-electron scattering, i.e., a scattering process in which the momentum sum changes by reciprocal lattice vectors. In band insulators level repulsions obviously come from one-body scattering as we have seen above. Why have we first looked at the adiabatic spectrum? There is an important relation between the adiabatic energy and the current expectation value. From the Hellmann-Feynman theorem, i.e., $\displaystyle{\frac{dE}{d\lambda}= \frac{{\langle}\Psi|{\partial}H/{\partial}\lambda|\Psi{\rangle}}{{\langle}\Psi|\Psi{\rangle}}}$ for $H(\lambda)|\Psi(\lambda){\rangle}=E(\lambda)|\Psi(\lambda){\rangle}$, we obtain $$\begin{aligned} J_n(\phi)&=&{\langle}n;\phi|J(\phi)|n;\phi{\rangle}\nonumber\\ &=&\left(\frac{L}{2\pi}\right)\frac{{\partial}E_n(\phi)}{{\partial}\phi}\,\end{aligned}$$ which is valid for all $\phi$. If we expand it around $\phi=0$, we get $$J_n(\phi)=J_n(0)+\left(\frac{L}{2\pi}\right)\frac{{\partial}^2 E_n(0)}{{\partial}\phi^2}\phi +O(\phi^2).$$ Using $\phi=FLt/h$ and defining the transport coefficients $\mathcal{D}_n$ by $J_n(\phi)=J_n(0)+\mathcal{D}_n Ft+O(F^2)$, we obtain $$\mathcal{D}_n(L) =\left(\frac{L}{2\pi}\right)^2\frac{{\partial}^2 E_n(0)}{{\partial}\phi^2}.$$ When we focus on a dissipationless adiabatic transport at $T=0$, the total current thus reads $${\langle}J(t){\rangle}=\mathcal{D}_0(L)Ft,$$ which is determined by the Drude weight (charge stiffness) $\mathcal{D}_0(L)$. As we can see in Fig.\[fig:adiabaticLevels\], even for insulators ((b) and (c)), the Drude weight $\mathcal{D}_0(L)$ of a finite system is not necessarily zero. If we remember [*Kohn’s criterion*]{}[@Kohn1963] for metal-insulator transitions, stated as $$\begin{aligned} \lim_{L\to \infty}\mathcal{D}_0(L)=\left\{ \begin{array}{cl} 0&\quad\mbox{insulator},\\ \mbox{finite}&\quad\mbox{perfect metal}, \end{array} \right. \label{2KohnCriterion}\end{aligned}$$ we can see that we must go to the limit of infinite systems to distinguish metals from insulators. Indeed, the problem of taking the infinite-size limit is also occurs in the study of dielectric breakdown in Mott insulators as we shall see later. ### Short-time behavior — an exact diagonalization result Since the time evolution of many-body systems cannot be treated analytically, we employ numerical methods to time-integrate in two steps — for short-time behavior and long-time behavior. For the short-time evolution in dielectric breakdown of Mott insulators we exactly diagonalize the time-dependent Schrödinger equation as follows: First we start from the ground state of $H(\phi=0)$ at time $t=0$. The wave function evolves with the phase that increases as $$\phi(t)= 0 \rightarrow FLt/h.$$ Here $F=eaE$ is the field strength, $L$ the length of the chain. We numerically solve the time-dependent Schrödinger equation, $$i\frac{d}{dt}|\Psi (t){\rangle}=H(\phi(t))|\Psi (t){\rangle}.$$ We choose the initial state to be the ground-state $|0{\rangle}$ of $H(0)$, which is obtained here by the Lanczos method. The time integration of the state vector, which, being a many-body state, has a huge dimension, requires a reliable algorithm. So we adopt here the Cranck-Nicholson method that guarantees the unitary time evolution, where the time evolution is put into a form, $$|\Psi (t+\Delta t){\rangle}=e^{ -i\int_t^{t+\Delta t} H(t)}\; dt\;|\Psi (t){\rangle}\simeq \frac{1-i\Delta t/2H(t+\Delta t/2)}{1+i\Delta t/2H(t+\Delta t/2)}\;|\Psi (t){\rangle},$$ which is unitary by definition. Here the time step is taken to be small enough ($dt=1.0\times 10^{-2}$ with the time in units of $\hbar/t$ hereafter) to ensure convergence for $L\leq 10$, for which the dimension of the Hamiltonian is $\sim 10^4$. We have concentrated on the total $S^z=0$ subspace with $N_\uparrow=N_\downarrow=L/2$. ![ (a) The sample geometry, where an AB flux, $\phi(t) = LFt$, increasing linearly with time induces an electric force through Faraday’s law. (b) Time evolution of the current, $J(t)$, for a half-filled, 10-site Hubbard model for various strengths of the Hubbard repulsion, $0\leq U/W\leq 5$ ($W=4t_{\rm hop}$ is the non-interacting band width) for a fixed electric field $F=1/10L$. Time is measured in units of $\tau_t\equiv \hbar/t_{\rm hop}$, $LF$ in $t_{\rm hop}$, and $J(t)$ in $1/\tau_t$. The range of the time in this panel is restricted to a range of the AB-flux $0\le\phi\le 1$. (c) A wider plot of the current for various values of $F$ with a fixed $U/W=0.25$, again for the half-filled case. Here the horizontal axis is $\phi$. (d) A plot similar to (c) for a non-half-filled case ($N_\uparrow=N_\downarrow=3<L/2=5$). []{data-label="currents"}](TOfig11.eps){width="10.5cm"} #### Evolution of the total current We first plot in Fig.\[currents\](b) the result for the expectation value of the current density averaged over the sites, $ J=-\frac{it}{L} \sum_{i,\sigma}\left( e^{i\frac{2\pi}{L}\phi(t)}\;c_{i+1\sigma}^\dagger c_{i\sigma} - {\rm h.c.} \right). $ The behavior of $J(t)$ for a fixed value of the electric field $F$ is seen to fall upon three regimes when $U$ is varied: A perfect metallic behavior ($J(t)\propto t$) when the electrons are free ($U/W=0$), an insulating behavior ($J(t)=0$) when the interaction is strong enough ($U/W\gg 1$), and an intermediate regime of $U/W$ where $J$ is finite with some oscillations for finite systems. In contrast, a non-half-filled system in nonequilibrium ($F\neq 0$) has a time evolution that is distinct from the ground-state behavior (Fig.\[currents\](d)). The difference has its root in the spectral property as will be discussed later. If we look at the behavior over several periods ($0<\phi<10$) for a fixed value of $U/W$ for the half-filled (Fig.\[currents\](c)) and for a non-half-filled case (Fig.\[currents\](d)), the result may be summarized as follows: (i) : Small $F$ regime (Mott insulator preserved at half filling)\ A drastic difference between the half-filled and doped systems appears for small $F$. When half-filled, $J(t)$ in the limit of $F\rightarrow 0$ smoothly approaches a periodic saw-tooth behavior with periodicity $\phi=1$, which is the AB-oscillation of the ground-state current. (ii) : Moderate $F$ regime (metal)\ In this regime, the current in the half-filled case is non-zero and shows oscillatory behaviors (seen typically in data for $LF=0.008$ in Fig.\[currents\](c)). (iii) : Large $F$ regime (perfect metal)\ When the electric field $F$ becomes large enough, the system behaves as a kind of metal. The current $J(t)$ exhibits a long-period ($\Delta\Phi=\Phi_0 L$) oscillation, which is the Bloch oscillation, a hallmark of a metal. The averaged current, $$\langle J\rangle= \frac{1}{T}\int_0^{T}{\langle}J(t){\rangle}dt,$$ integrated over a quarter of the Bloch period (with $\phi(T)=\frac{L}{4}$) is plotted against $F$ in Fig.\[IV\] for various values of $U$. We can see that $\langle J \rangle$ becomes nonzero rather abruptly at the metallization as $F$ is increased, where the threshold electric field increases and the $F$- dependence becomes weaker when we increase $U/t$. Just after the metallization some oscillation (in the $F$- dependence this time) is seen for finite systems. ![Dependence of the averaged current $\langle J(t)\rangle$ on $F$ for various values of $U/t$ for the half-filled Hubbard model with $L=6$. []{data-label="IV"}](TOfig12.eps){width="9cm"} #### Evolution of the survival probability ![ (a) Spectrum of the half-filled Hubbard model $H(\phi)$ for $0\le\phi\le 1$. Bold lines represent the ground state and the first state among the tunneling-allowed excited states, respectively. (b) $U/W$-dependence of the Mott gap $\Delta E$, encircled in (a). $W=4t_{\rm hop}$ is the non-interacting band width, and system size is $L=10$ and $U/t_{\rm hop}=1.5$. []{data-label="lzHub1"}](TOfig13.eps){width="11.5cm"} In order to calculate the decay rate introduced above, we compute the temporal evolution of the ground-state survival probability, $$P_0(t)= |{\langle}0;\phi(s)|\hat{T}e^{-\frac{i}{\hbar}\int_0^tH(\phi(s))ds}|0;0{\rangle}|^2,$$ where $|0;\phi{\rangle}$ denotes the ground-state of $H(\phi)$. The survival probability is related to the decay rate of the ground-state by $P_0(t)=e^{-\Gamma t}$. The short-time feature in the survival probability is expected to be described by the single Landau-Zener transition between the ground-state and the lowest excited state, displayed by the two bold lines in the figure[^2], that takes place around $\phi=\frac{1}{2}$ (Fig.\[lzHub1\](a)). If we concentrate on the two levels, the time evolution operator at $t=\Delta t$ ($\Delta t=\frac{h}{FL}$ is defined as the time when $\phi(\Delta t)=1$ is reached) is approximated by a $2\times 2$ matrix, $$\begin{aligned} U_{\rm{2level}}(t=\Delta t)= \left( \begin{array}{cc} \sqrt{1-p}e^{-i\chi}&-\sqrt{p}\\ \sqrt{p}&\sqrt{1-p}e^{i\chi} \end{array} \right),\end{aligned}$$ where the tunneling probability $p$ is given by the Landau-Zener formula [@Landau; @Zener; @St], $$\begin{aligned} &&\hspace{-2cm} p=\exp\left(-\pi \frac{F_{\rm th}^{\rm LZ}}{F}\right), \quad F_{\rm th}^{\rm LZ}=\frac{\left[\Delta_c(U)/2\right]^2}{v}. \label{eq:lzHub}\end{aligned}$$ Here, $\Delta_c(U)$ is the excitation gap (Fig.\[lzHub1\](b)), $v=2t_{{\rm hop}}$, and $\chi$ the sum of dynamical and Stokes phases. In order to verify eq.(\[eq:lzHub\]), we can numerically calculate the survival probability $P_0(t)$ from $t=0$ to $t=\Delta t$ for various $U$ and $F$ (Fig.\[lzHub2\]). After determining the tunneling probability from $p=1-P_0(\Delta t)$, we plot it against the diabaticity parameter $\frac{(\Delta_c(U)/2)^2}{vF}$. The data points(Fig.\[lzHub2\](b)) for various values of $U$ fall around a common line, which is just the prediction of the Landau-Zener formula. The agreement is better for smaller values of $U$ where we can treat the Umklapp term as a perturbation. ![ (a) Short-time temporal evolution of the survival probability $P_0(t)$ in the half-filled Hubbard model ($L=10, N_\uparrow=N_\downarrow=5$) for various values of $F$ with $U/t=0.1$. The inset shows the solutions of the LZS equation with its asymptotic values indicated as dashed horizontal lines. (b)The transition probability $p$, with $P(t=\Delta t)=1-p$, plotted against the diabaticity parameter. The dashed line is the prediction of the Landau-Zener formula. []{data-label="lzHub2"}](TOfig14.eps){width="10.cm"} ### Long-time behavior — a time-dependent DMRG result The conclusion obtained in the previous section with the exact diagonalization is that the short-time behavior after the electric field is switched on is dominated by the single Landau-Zener transition between the ground state and the first excited state. However, several important questions remain, e.g., Will the first transition remain finite in the infinite-size limit? : Indeed, Kohn’s criterion (\[2KohnCriterion\]) asserts that the $\phi$ dependence of the ground-state energy of a Mott insulator should vanish for $L\rightarrow \infty$. This implies that the adiabatic flow (Fig. \[fig:adiabaticLevels\](c)) should become flat in this limit, which may seem to indicate that the transition will be washed out. However, this contradicts with the expression for the threshold $F_{\rm Zener}=\frac{[\Delta_c(U)/2]^2}{v}$ (eq.(\[eq:lzHub\])), which remains finite in the $L\to \infty$ limit. Since this expression is obtained in a small system and in the small $U$ limit, there is a possibility that this breaks down. Surprisingly, we shall show that this expression survives in large systems even when $U$ is not small (Fig. \[Fth\]). The effect of pair annihilation : After the first transition, we expect the system to undergo further transitions to higher-energy levels. This process, however, should be couterbalanced by another processes, the [*pair annihilation*]{} of doublons and holes. These processes, which do not conserve the total momentum in general, are caused by the Umklapp scattering. Thus the pair creation (= Landau-Zener transitions to high-energy states) tends to be offset by pair annihilation, which implies that the decay rate of the ground state may become smaller compared to the single Landau-Zener transition case[^3]. These questions have motivated us to study the dielectric breakdown in the half-filled Hubbard model for longer time periods, which is accomplished by the time-dependent density matrix renormalization group method. A version of the real-time DMRG was first intruduced by Cazalilla and Marston with a truncated DMRG Hilbert space and a renormalized Hamiltonian [@tddmrg]. Precision of their method degrades rapidly in the long-time limit, since an update of the Hilbert space is lacking. Recently, Vidal proposed an improved method for simulating time-dependent phenomena in one-dimensional lattice systems employing the Trotter-Suzuki decomposition [@Vidal2003; @Vidal2004]. White and Feiguin [@White2004] as well as other groups [@Daley2004] modified this idea and combined it with the finite-size DMRG algorithm. If we denote the DMRG wave function as $$|\Psi{\rangle}=\sum_{l\alpha_j\alpha_{j+1} r}\psi_{l\alpha_j\alpha_{j+1} r} |l{\rangle}|\alpha_j{\rangle}|\alpha_{j+1}{\rangle}|r{\rangle},$$ where $|l{\rangle},\;|r{\rangle}$ is the basis of the truncated Hilbert space with dimension $m$ and $|\alpha_j{\rangle},\;|\alpha_{j+1}{\rangle}$ are the two sites that bridge the left and right blocks in the DMRG procedure. By employing the Trotter-Suzuki decomposition, $$e^{-idt H}\simeq e^{-idtH_1/2}e^{-idtH_2/2}\ldots e^{-idtH_2/2}e^{-idtH_1/2},$$ we can apply the time-evolution operator $e^{-idtH_j/2}$ to the $j$-th wave function as $$\left(e^{-idtH_j/2}\psi\right)_{l\alpha_j\alpha_{j+1} r}= \sum_{\alpha_j'\alpha_{j+1} '} \left(e^{-idtH_j/2}\right)_{\alpha_j\alpha_{j+1};\alpha_j'\alpha_{j+1}'} \psi_{l\alpha_j'\alpha_{j+1}' r} .$$ After applying $e^{-idtH_j/2}$, we diagonalize the density matrix and move to the next link just as in the usual finite-size algorithm. One cycle of this procedure results in an evolution of time by $dt$, and we can repeat it as many times as we wish. Compared with the version by Cazalilla-Marston [@tddmrg], this algorism has higher precision and we can simulate non-equilibrium excited states efficiently[@Daley2004], although one drawback of the t-dependent DMRG is that we can only treat systems with open boundary conditions. Here we study transient behaviors of the one-dimensional Hubbard model with open boundary condition. We use the time-independent gauge, for which the Hamiltonian is $$H(F)=-t_{\rm hop}\sum_{j,\sigma}\left(c_{j+1\sigma}^\dagger c_{j\sigma}+\mbox{h.c.}\right)+U\sum_j n_{j{\uparrow}}n_{j{\downarrow}}+F\hat{X},$$ where $\hat{X}=\sum_jjn_j$ is the position operator representing the tilted potential. As in the previous section, we start the time evolution from the $F=0$ ground-state $|0{\rangle}$ obtained by the usual finite-size DMRG. The wave function in this gauge is simply $$|\Psi(t){\rangle}=e^{-itH(F)}|0{\rangle},$$ which is obtained with the t-dependent DMRG. #### Evolution of the charge density ![ Temporal evolution of the charge density $n_j(t)$ in the half-filled Hubbard model with $m=150, L=50, dt=0.02$ for $F=0.1$(a) and $F=1.0$(b). (c) depicts a cross section of (a) for $t=4$. []{data-label="densityU4"}](TOfig15.eps){width="12cm"} We first discuss the temporal evolution of the charge density, $n_j(t)={\langle}\Psi(t)|n_j|\Psi(t){\rangle}$, after the electric field is switched on at $t=0$. At half-filling the initial distribution is $n_j(t)=1$. After the application of the electric field, a charge density wave (CDW) pattern is formed when the electric field is not too strong (Fig. \[densityU4\](a)). This state is stationary and the density profile do not change any further. The pattern is formed because the boundary condition breaks the translational symmetry, where the amplitude of the pattern corresponds to the polarization $\Delta P(F)$ induced by the field. When the electric field becomes stronger, charge transfers start to occur, with charge accumulation and charge depletion being formed around the edges in an open-boundary chain (Fig.\[densityU4\]). This is a sign that the ground state collapses due to quantum tunneling. #### The decay rate of the ground state The groundstate-to-groundstate transition amplitude is, in the time-independent gauge, $$\Xi(t)={\langle}0|e^{-\frac{i}{\hbar} \tau\left(H+F\hat{X}\right)}|0{\rangle}e^{\frac{i}{\hbar} t E_0}, \label{eq:transitionamplitude2}$$ where we denote the ground state of $H$ as $|0{\rangle}$ and its energy as $E_0$. Figure \[ImSU35\](a) shows the temporal evolution of the ground-state survival probability $|\Xi(t)|^2$ for a system with $U/t_{\rm hop}=3.5$. As time evolves, the slope of $-\ln |\Xi(t)|^2$ ($\propto$ the decay rate) decreases after an initial stage, which implies a suppression of the tunneling from the short-time behavior. This should indicate that charge excitations are initially produced due to the Landau-Zener tunneling from the ground state to the first excited states, but that scattering among the excited states become important as the population of the excitations grows. In other words, pair annihilation of carriers becomes important and acts to suppress the tunneling rate. We have determined $\Gamma(F)$ from the long-time behavior with a fitting $-\ln |\Xi(t)|^2=\Gamma(F)t+{\rm const}$. The decay rate per length $\Gamma(F)/L$ is plotted in Fig.\[ImSU35\](b), where we have varied the system size ($L=30, 50$) to check the convergence. $\Gamma(F)/L$ is seen to remain vanishingly small until the field strength exceeds a threshold. To characterize the threshold $F_{\rm th}(U)$ for the breakdown we can evoke the form obtained above for the one-body system. The formula (eq.(\[BandSchwingerFormula\]) for $d=1$ with the error function ignored and the factor of $2$ recovered for the spin degeneracy), $$\Gamma(F)/L=-\frac{2F}{h}a(U)\ln \left[1-\exp\left(-\pi \frac{F_{\rm th}^{\rm Mott}(U)}{F}\right)\right], \label{fittingfunction}$$ is originally derived for one-body problem, and an obvious interest here is whether the formula can be applicable if we replace the one-body $F_{\rm th}^{\rm band}$ with the many-body $F_{\rm th}^{\rm Mott}(U)$. In the above we have added a factor $a(U)$, a parameter representing the suppression of the quantum tunneling. The dashed line in Fig.\[ImSU35\](b) is the fitting to the formula for $U/t_{\rm hop}=3.5$, where we can see that the fitting, including the essentially singular form in $F$, is surprisingly good, given a small number of fitting parameters. The value of $a(U)$ turns out to be close to but smaller than unity (taking between $0.77$ to $0.55$ as $U/t$ is increased from $2.5$ to $5.0$). If we perform this for various values of $U$ we can construct a “nonequilibrium (dielectric-breakdown) phase diagram", as displayed in Fig.\[Fth\], which plots the $U$ dependence of $F_{\rm th}^{\rm Mott}$. The dashed line is the prediction of the Landau-Zener formula [@Oka2003], $$F_{\rm th}^{\rm LZ}(U)=\frac{[\Delta_{\rm c}(U)/2]^2}{v}. \label{LZMott}$$ For the size of the Mott (charge) gap we use the Bethe-ansatz result,[@Lieb:1968AM] $$\Delta_{\rm c}(U)=\frac{8t_{\rm hop}}{U}\int_1^\infty\frac{\sqrt{y^2-1}} {\sinh(2\pi yt_{\rm hop}/U)}dy,$$ with $v/t_{\rm hop}=2$. As can be seen, the DMRG result and the Landau-Zener result agrees surprisingly well. Long-time behavior and a mapping to a quantum random walk --------------------------------------------------------- Since many levels should be involved in the above pair creation/annihilation processes, next thing we want to have is a statistical mechanical setup for the time evolution of the Mott insulator. The problem at hand is a closed quantum system in external driving forces (e.g., electric fields), which are represented by a time varying parameter $\phi(t)$ of the Hamiltonian. We want to discuss the asymptotic solution of the time-dependent Schrödinger equation $$i\hbar\frac{d}{dt}|\Psi(t){\rangle}=H(\phi(t))|\Psi(t){\rangle}.$$ We introduce $|n;\phi{\rangle}$ as the set of eigenstates of the time-dependent Hamiltonian $H(\phi)$, and denote the energy eigenvalue as $E_n(\phi)$, i.e., $H(\phi)|n;\phi{\rangle}=E_n(\phi)|n;\phi{\rangle}$ (Fig.\[LZ2QW\]). Since $|n;\phi{\rangle}$ forms a complete orthonormal basis, the wave function $|\Psi(t){\rangle}$ can be expanded as $$|\Psi(t){\rangle}=\sum_n\psi(n,t) e^{-\frac{i}{\hbar}\int_0^tE_n(s)ds} |n;\phi(t){\rangle}\label{5WaveVector}$$ with coefficients $\psi(n,t) ={\langle}n;\phi(t)|\hat{T}e^{-i\int_0^tH(\phi(s))ds}|0;0{\rangle}/e^{-\frac{i}{\hbar}\int_0^tE_n(s)ds}$. Note that we have removed the contribution from the dynamical phase $\int_0^tE_n(s)ds/\hbar$ in the definition of $\psi(n,t)$. Although the evolution depends on the detail of the system ($H(\phi)$), we can deduce some [*universal*]{} features that depend only on the feature of the energy levels, i.e., distribution of level repulsion in the spectrum. ![ (a) The spectrum of the half-filled Hubbard model. The circle corresponds to Landau-Zener transition between the two energy levels which can be expressed by a $2\times 2$ unitary matrix (eq.(\[eq:lzunitarywalk\])). (b) Idealized energy levels where level anti-crossings are expressed by circles. Quantum interference takes place when contributions from different paths are considered. []{data-label="LZ2QW"}](TOfig18.eps){width="11cm"} Each energy level is subject to the Landau-Zener tunneling to neighboring levels in a time period $\Delta t/2$, and is most conveniently expressed in terms of the transfer matrix representation[@Oka2004a; @Nakamura1987]. To this end, we denote the pairs as $$\Psi(n,\tau)= \left( \begin{array}{c} \psi_L(n,\tau)\\ \psi_R(n,\tau)) \end{array} \right),$$ and the time evolution “rule" can be expressed as $$\Psi(n,\tau+1)=P_{n+1}\Psi(n+1,\tau)+Q_{n-1}\Psi(n-1,\tau), \label{BQW1}$$ where $P_n$ ($Q_n$) is the upper (lower) half of a $2\times 2$ unitary matrix, $$U_n=\left( \begin{array}{cc} a_n&b_n\\ c_n&d_n \end{array} \right),\quad P_n=\left( \begin{array}{cc} a_n&b_n\\ 0&0 \end{array} \right),\quad Q_n=\left( \begin{array}{cc} 0&0\\ c_n&d_n \end{array} \right).$$ ![ Application of $Q_n$ and $P_n$ in the quantum random walk. []{data-label="5PandQ"}](TOfig19.eps){width="3.5cm"} The diagonal elements of $U_n$ represent the Landau-Zener transition from the $n$-th level to $(n-1)$ or $(n+1)$-th level, where the explicit form is $$U_n= \left( \begin{array}{cc} \sqrt{p_n}e^{i\beta_n}&\sqrt{1-p_n}e^{i\gamma_n}\\ -\sqrt{1-p_n}e^{-i\gamma_n}&\sqrt{p_n}e^{-i\beta_n} \end{array} \right). \label{eq:lzunitarywalk}$$ Here the Landau-Zener tunneling probability $p_n$ depends on the ratio of the Zener threshold field $F_{\rm Zener}^n$ and the electric field $F$ as $$p_n=\exp\left(-\pi\frac{F_{\rm Zener}^n}{F}\right),$$ where $F_{\rm Zener}^n$ generically depends on $n$. If we regard $\Psi(n,\tau)=\left( \begin{array}{c} \psi_L(n,\tau)\\ \psi_R(n,\tau) \end{array} \right)$ as a “qubit" on “site" $n$, eq.(\[BQW1\]) defines an evolution of a [*one-dimensional quantum walk*]{} with a reflecting boundary at $n=0$ corresponding to the ground-state (Fig.\[LZ2QW\](c)). A quantum walk is a quantum counterpart of the classical random walk. Models with essentially equivalent ideas have appeared in various fields: to name a few, quantum transport and dissipation [@Blatter1988; @Oka2003], quantum Hall effect [@Chalker1988], optics [@Bouwmeeste2000; @Wojcik2004] and recently in quantum information [@Nayak2000; @Konno2003; @Konno2002b; @KonnoReviewPQRS; @Mackay2002; @KonnoNamikiSoshi; @Konno2002a; @Bach2002; @Yamasaki2003; @Tregenna2003; @Kempe2003; @Tregenna2003; @Ambainis2003]. In the field of quantum information (see e.g. [@NielsenBook]), introduced by Aharonov, Ambainis, Kempe and Vazirani in 2001 [@Aharonov2001], the quantum walk is arousing interest in hope of revealing new features in the quantum algorithms (for reviews see [@Kempe2003; @Tregenna2003; @Ambainis2003]). Researches stem into many directions, e.g., the effect of absorbing boundary conditions [@Bach2002; @Yamasaki2003; @Konno2003], higher-dimensional systems [@Mackay2002; @Tregenna2003; @Inui2004twodim], localization in systems with internal degrees of freedom [@Inui2004multi], and many powerful analytical techniques are being developed. ![ The time evolution of the distributions of wave function amplitude $\rho(n,t)=|\psi(n,t)|^2$ in energy space. The vertical axis $n$ is the index of the energy levels. (a)For small tunneling, the distribution is localized at the ground state. (b) For intermediate tunneling, a localized state remains, while the amplitude starts to bifircate into excited states of the wave function is excited. (c) When the tunneling is larger than the threshold, the localized state disappears. []{data-label="1ThreeAsymptotics"}](TOfig20.eps){width="11.5cm"} An important feature of the quantum walk, as opposed to the classical walk, is that different transition paths interfere with each other quantum mechanically. We in fact find that the quantum interference leads to a [*dynamical localization*]{}, an analog of Anderson’s localization taking place in the energy space rather than in the position space. In our previous work [@Oka2004a] we have employed the PRQS method, a technique to treat quantum walks, to perform the path integral, and obtained the exact asymptotic distributions of the wave function for a simplified model. The resultant states can be categorized in three types depending on the strength of the electric field, as schematically plotted in Fig. \[1ThreeAsymptotics\]. (A) is an adiabatic evolution that takes place in weak driving forces (electric fields). The dominant part of the distribution $\rho(n,t)$ against energy is a delta function localized around the ground-state. When the driving force become stronger, quantum tunneling broadens the delta function, as plotted in (B). The shape of the peak is maintained by a balance between tunneling and dynamical localization. (C) is the case where the driving force overwhelms the effect of dynamical localization, and the system is driven rapidly into the excited states. This, in our view, corresponds to the dielectric breakdown. Experimental implications {#3Experimentalresults} ------------------------- Now we discuss experimental implication of the many-body Landau-Zener transition mechanism. In fact, there are several mechanisms which may lead to breakdown of insulators. For example, Fröhlich’s electric avalanche mechanism may take place, in which a small number of excited electrons act as a seed and become accelerated by the electric field until they cooperatively destroy the insulator. We can distinguish Landau-Zener transition from the avalanche mechanism through the temperature dependence and from interface effects by changing the size of the sample. Another important effect is the band bending near an interface of the Mott insulator and electrodes. For a thin sample, this may lead to injection of carriers, and results in the interface Mott transition[@OkaNagaosa]. Dielectric breakdown of one-dimensional Mott insulators was experimentally studied by Taguchi [*et al.*]{}, who obtained the $J-E$ characteristics (Fig.\[4IVExp\]) of Sr$_2$CuO$_3$ and SrCuO$_2$ samples[@tag], which are both quasi-1D, strongly correlated electron systems. Experiments were done by placing small single crystals in circuits as shown in Fig.\[4IVExp\](a), and the voltage drop $V$ was measured while the current density $J$ was fixed. Depending on the strength of the electric field $F$, transport properties change drastically, as summarized in the following. In weak electric fields, the $J-E$ characteristics shows an Ohmic behavior at finite temperatures. When the electric field exceeds a threshold value, the current shows a dramatic increase. Such drastic changes cannot be explained by perturbation in $F$, and we must consider non-perturbative effects, i.e., a behavior essentially singular in $F$ like $J\sim\mbox{function of} \exp\left(-F_{\rm th}/F\right),$ which is a typical tunneling effect with threshold $F_{\rm th}$. The temperature dependence (Fig.\[4FTemperature\]) of the threshold can be fit well by $F_{\rm th}(T)/F_{\rm th}(0)\sim \exp\left(-T/T_0\right)$. This excludes the avalanche mechanism, for which an activation type temperature dependence ($F_{\rm th}^{\rm avalanche}(T)/F_{\rm th}(0)\sim \exp\left(T_0/T\right)$) is expected. One indication that the breakdown is indeed quantum in nature is that the threshold extrapolates to a finite value for $T \rightarrow 0$. From the extrapolation (Fig.\[4FTemperature\](a)), we obtain a threshold, $$F_{\rm th}^{\rm exp}\sim 10^6-10^7\;(\mbox{eV/cm}),$$ for Sr$_2$CuO$_3$ and SrCuO$_2$. The Landau-Zener result (intended for $T=0$) of the threshold (eq.(\[eq:lzHub\])) is $$F_{\rm th}^{\rm LZ}(U)=\frac{[\Delta_{\rm c}(U)/2]^2}v \sim (1\mbox{eV})^2/(10^{-7}\mbox{eV/cm}) \sim 10^6\;(\mbox{eV/cm})$$ is comparable with the experimental result. Interestingly, the decay rate $\Gamma(U)$ we have introduced theoretically can be measured experimentally[@tag]. This is done by studying the transient behavior of the current after the electric field is switched on at $t=0$. At first the current density is zero, and then becomes non-zero after a certain delay time $t=\tau (F)$(Fig.\[4FTemperature\](b), lower inset). The authors in [@tag] have introduced a phenomenological percolation model to relate the delay time with the production rate $P(F)$ of the conductive domains (see [@tag]). In this model, conductive domains are envisaged to grow in the sample, and the current density is assumed to become finite when the left and right electrodes are connected by these domains (Fig.\[4FTemperature\](c)). This leads to a relation, $$P(F)=-\left(F\frac{d\tau}{dF}\right)^{-1}.$$ The experimental result for the production rate in Fig.\[4FTemperature\](b) is obtained in this way. The nature and the microscopic origin of the “conductive domain" are not clear, but if we interpret them to be domains with a high density of charge excitations produced by the Landau-Zener transition, the vacuum decay rate per volume $\Gamma(F)/L^d$ characterizes quantum tunneling from the ground-state to excited states. With the identification we expect that the decay rate and the production rate are identical, i.e., $$P(F)\sim \Gamma(F).$$ This identification is encouraged by the field dependence of $P(F)$ (lower panel of Fig.\[4FTemperature\](b)), which is close to the expected form ($\Gamma/L\sim -\frac{2F}{h}a(U) \ln\left[1-\exp\left( -\pi\frac{F_{\rm th}^{\rm Mott}}{F}\right) \right]$) of the decay rate. In this experiment a scaling study — a systematic change of the size of the sample — was also performed to confirm that the nonlinear effect occurs in the bulk. From these observations, we conclude that the experiment by Taguchi [*et al.*]{}[@tag] can be explained by the many-body Landau-Zener tunneling mechanism. However, to be more confident, we need to know the temperature dependence of the threshold theoretically, which is still a challenging task in the present many-body system. Conclusion ---------- In this article, we have explained how dielectric breakdown of Mott insulators can be explained from the nonequilibrium behaviors of charge carriers, especially from their creation and annihilation processes. Both processes are the result of many-body Landau-Zener nonadiabatic tunneling transition between many-body energy levels, where charge creation processes are counterbalanced by annihilation processes. From numerical result we have obtained a nonequilibrium (dielectric-breakdown) phase diagram. If the coherence of the dynamics is preserved at sufficiently low temperatures, a quantum interference, as modeled by a quantum walk in energy space, may lead to dynamical localization, which saturates the creation process and leads to a non-equilibrium stable state. The decay rate $\Gamma(F)$ that we have discussed is a measurable quantity: it is the production rate observed by Taguchi [*et al.*]{} [@tag] in copper oxides. The experimental result is consistent with our prediction $\Gamma(F) \sim \frac{F}{2\pi}\ln (1-p)$ when an extrapolation to zero temperature is made. It is an interesting future topic to understand the properties of the non-equilibrium stable state in more detail. An important open question is how the energy dissipation processes take place in nonequilibrium situations. Here we have stressed that the many-body processes act effectively as a source of dissipation through scattering, but an explicit incorporation of heat-bath effects, electrode effects, etc, is left to a future problem. [*Acknowledgements:*]{} We wish to thank Rytaro Arita and Norio Konno for the collaboration in the workes described here and for illuminating discussions. We also indebted to Yshai Avishai and Paul Wiegmann for illuminating discussions. TO acknowledges Masaaki Nakamura, Kazuma Nakamura, Shuichi Murakami, and Naoto Nagaosa for helpful comments. 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--- abstract: 'In this talk I address two high impact physics programs that require the use of polarized and unpolarized positron beams in addition to using electron beams of the same energy. First, I address what will be gained from using positron beams in addition to electron beams in the extraction of the Compton Form Factors (CFFs) and generalized parton distributions (GPDs) from Deeply Virtual Compton Scattering (DVCS) on a proton target. As a second high impact science program I discuss an experimental scenario using unpolarized positrons to measure elastic scattering on protons in an effort to determine definitively the 2-photon exchange contributions in order to resolve a longstanding discrepancy in the determination of the proton’s electric and magnetic form factors.' author: - 'Volker D. Burkert' title: | Positrons at JLab\ Advancing Nuclear Science in Hall B --- INTRODUCTION {#intro} ============ The challenge of understanding nucleon electromagnetic structure still continues after six decades of experimental scrutiny. From the initial measurements of elastic form factors to the accurate determination of parton distributions through deep inelastic scattering, the experiments have increased in statistical and systematic accuracy. During the past two decades it was realized that the parton distribution functions represent special cases of a more general, much more powerful, way to characterize the structure of the nucleon, the generalized parton distributions (GPDs) (see [@Diehl:2003ny; @Belitsky:2005qn] for reviews). ![The [CLAS12]{} detector in Hall B. The detector was designed for inclusive, semi-inclusive, as well as exclusive processes such as DVCS. The construction and commissioning of the detector system was completed recently. [CLAS12]{} is part of the DOE funded energy upgrade of the Jefferson Lab CEBAF accelerator from 6 GeV to 12 GeV, and may play an important role in programs that make use of positron beams at Jefferson Lab..[]{data-label="fig:clas12"}](CLAS12.png){height="0.25\textheight"} The GPDs are the Wigner quantum phase space distribution of quarks in the nucleon describing the simultaneous distribution of particles with respect to both position and momentum in a quantum-mechanical system. In addition to the information about the spatial density and momentum density, these functions reveal the correlation of the spatial and momentum distributions, [*i.e.*]{} how the spatial shape of the nucleon changes when probing quarks of different momentum fraction of he nucleon. The concept of GPDs has led to completely new methods of “spatial imaging” of the nucleon in the form of (2+1)-dimensional tomographic images, with 2 spatial dimensions and 1 dimension in momentum [@Belitsky:2003nz; @Ji:2003ak; @Burkardt:2002hr]. The second moments of GPDs are related to form factors that allow us to quantify how the orbital motion of quarks in the nucleon contributes to the nucleon spin, and how the quark masses and the forces on quarks are distributed in transverse space, a question of crucial importance for our understanding of the dynamics underlying nucleon structure and the forces leading to color confinement. The four leading twist GPDs $H$, $\tilde{H}$, $E$, and $\tilde{E}$, depend on the 3 variable $x$, $\xi$, and $t$, where $x$ is the longitudinal momentum fraction of the struck quark, $\xi$ is the longitudinal momentum transfer to the quark ($\xi \approx x_B/(2-x_B)$), and $t$ is the invariant 4-momentum transfer to the proton. The mapping of the nucleon GPDs, and a detailed understanding of the spatial quark and gluon structure of the nucleon, have been widely recognized as key objectives of nuclear physics of the next decades. This requires a comprehensive program, combining results of measurements of a variety of processes in electron–nucleon scattering with structural information obtained from theoretical studies, as well as with expected results from future lattice QCD simulations. The [CLAS12]{} detector, shown in Fig. \[fig:clas12\], has recently been completed and has begun the experimental science program in the 12 GeV era Jefferson Lab. ![Leading order contributions to the production of high energy single photons from protons. The DVCS handbag diagram contains the information on the unknown GPDs.[]{data-label="fig:handbag"}](figure1){height="0.2\textheight"} ACCESSING GPD IN DVCS {#gpds} ===================== The most direct way of accessing GPDs at lower energies is through the measurement of Deeply Virtual Compton Scattering (DVCS) in a kinematical domain where the so-called handbag diagram shown in Fig.  \[fig:handbag\] makes the dominant contributions. However, in DVCS as in other deeply virtual reactions, the GPDs do not appear directly in the cross section, but in convolution integrals, e.g. $$\int_{-1}^{+1}{{H^q(x,\xi,t)dx}\over {x - \xi + i\epsilon}} = \int_{-1}^{+1}{{H^q(x,\xi,t)dx}\over {x - \xi }} + i\pi H^q(\xi,\xi,t)~,$$ where the first term on the r.h.s. corresponds to the real part and the second term to the imaginary part of the scattering amplitude. The superscript $q$ indicates that GPDs depend on the quark flavor. From the above expression it is obvious that GPDs, in general, can not be accessed directly in measurements. However, in some kinematical regions the Bethe-Heitler (BH) process where high energy photons are emitted from the incoming and scattered electrons, can be important. Since the BH amplitude is purely real, the interference with the DVCS amplitude isolates the imaginary part of the DVCS amplitude. The interference of the two processes offers the unique possibility to determine GPDs directly at the singular kinematics $x=\xi$. At other kinematical regions a deconvolution of the cross section is required to determine the kinematic dependencies of the GPDs. It is therefore important to obtain all possible independent information that will aid in extracting information on GPDs. The interference terms for polarized beam $I_{LU}$, longitudinally polarized target $I_{UL}$, transversely (in scattering plane) polarized target $I_{UT}$, and perpendicularly (to scattering plane) polarized target $I_{UP}$ are given by the expressions: $$I_{LU} \sim \sqrt{\tau^\prime} [F_1 H + \xi (F_1+F_2) \tilde{H} + \tau F_2 E]$$ $$I_{UL} \sim \sqrt{\tau^\prime} [F_1 \tilde{H} + \xi (F_1 + F_2) H + (\tau F_2 - \xi F_1)\xi \tilde{E}]$$ $$I_{UP} \sim {\tau}[F_2 H - F_1 E + \xi (F_1 + F_2)\xi \tilde{E}$$ $$I_{UT} \sim {\tau}[F_2 \tilde{H} + \xi (F_1 + F_2) E - (F_1+ \xi F_2) \xi \tilde{E}]$$ where $\tau = -t/4M^2$, $\tau^\prime = (t_0 - t)/4M^2$. By measuring all 4 combinations of interference terms one can separate all 4 leading twist GPDs at the specific kinematics $x=\xi$. Experiments at JLab using 4 to 6 GeV electron beams have been carried out with polarized beams  [@Jo:2015ema; @Gavalian:2008aa; @Girod:2007aa; @Camacho:2006qlk; @Stepanyan:2001sm] and with longitudinal target [@Pisano:2015iqa; @Seder:2014cdc; @Chen:2006na], showing the feasibility of such measurements at relatively low beam energies, and their sensitivity to the GPDs. Techniques of how to extract GPDs from existing DVCS data and what has been learned about GPDs can be found in  [@Guidal:2013rya; @Kumericki:2012yz]. In the following sections we discuss what information may be gained by employing both electron and positron beams in deeply virtual photon production. Differential cross section for polarized electrons and positrons (leptons) {#crs-leptons} -------------------------------------------------------------------------- The structure of the differential cross section for polarized beam and unpolarized target is given by: $$\sigma_{\vec{e}p\rightarrow e\gamma p} = \sigma_{BH} + e_\ell \sigma_{INT} + P_\ell e_\ell \tilde\sigma_{INT} + \sigma_{VCS} + P_\ell \tilde\sigma_{VCS} \label{diffcrs}$$ where $\sigma$ is even in azimuthal angle $\phi$, and $\tilde\sigma$ is odd in $\phi$. The interference terms $\sigma_{INT} \sim \rm{Re} {\it A}_{\gamma^*N\rightarrow \gamma N} $ and $\tilde\sigma_{INT} \sim \rm{Im} {\it A}_{\gamma^*N\rightarrow \gamma N} $ are the real and imaginary parts, respectively of the Compton amplitude. Using polarized electrons the combination $-\tilde\sigma_{INT} + \tilde\sigma_{VCS}$ can be determined by taking the difference of the beam helicities. The electron-positron charge difference for unpolarized beams determines $\sigma_{INT}$. For fixed beam polarization and taking the electron-positron difference one can extract the combination $P_\ell\tilde\sigma_{INT} + \sigma_{INT}$. If only a polarized electron beam is available one can separate $\tilde\sigma_{INT}$ from $\tilde\sigma_{VCS}$ using the Rosenbluth technique [@Rosenbluth:1950yq]. This requires measurements at two significantly different beam energies which reduces the kinematical coverage that can be achieved with this method. With polarized electrons and polarized positrons both $\sigma_{INT}$ can be determined and $\tilde\sigma_{INT}$ can be separated from $\tilde\sigma_{VCS}$ in the full kinematic range available at the maximum beam energy. ![The beam spin asymmetry showing the DVCS-BH interference for 11 GeV beam energy [@e12-06-119]. Left panel: $x=0.2$, $Q^2=3.3$GeV$^2$, $-t=0.45$GeV$^2$. Middle and right panels: $\phi=90^{\circ}$, other parameters same as in left panel. Many other bins will be measured simultaneously. The curves represent various parameterizations within the VGG model [@Vanderhaeghen:1999xj]. Projected uncertainties are statistical.[]{data-label="fig:dvcs_alu_12gev"}](dvcs){height=".24\textheight"} Differential cross section for polarized proton target {#crs-protons} ------------------------------------------------------ The structure of the differential cross section for polarized beam and polarized target contains the polarized beam term of the previous section and an additional term related to the target polarization [@Belitsky:2001ns; @Diehl:2005pc]: $$\sigma_{\vec{e}\vec{p}\rightarrow e\gamma p} = \sigma_{\vec{e}p\rightarrow e\gamma p} + T[P_\ell\Delta\sigma_{BH} + e_\ell \Delta\tilde\sigma_{INT} + P_\ell e_\ell\Delta\sigma_{INT} + \Delta\tilde\sigma_{VCS} + P_\ell\Delta\sigma_{VCS}]$$ where the target polarization $T$ can be longitudinal or transverse. If only unpolarized electrons are available, the combination $-\Delta\tilde\sigma_{INT} + \Delta\tilde\sigma_{VCS}$ can be measured from the differences in the target polarizations. If unpolarized electrons and unpolarized positrons are available the combination $T\Delta\tilde\sigma_{INT} + \sigma_{INT}$ can be determined at fixed target polarization. With both polarized electron and polarized positron beams, the combination $T\Delta\tilde\sigma_{INT} + TP_\ell\Delta\sigma_{INT} + P_\ell\tilde\sigma_{INT} + \sigma_{INT}$ can be measured at fixed target polarization. Availability of both polarized electron and polarized positron beams thus allows the separation of all contributing terms. ![Electron-positron DVCS charge asymmetries: Top-left: Azimuthal dependence of the charge asymmetry for positron and electron beam at 11 GeV beam. Top-right: Moment in $\cos(\phi)$ of the charge asymmetry versus momentum transfer $t$ to the proton. Bottom-left: Charge asymmetries for polarized electron and positron beams at fixed polarization (LU). Bottom right: Charge asymmetry for longitudinally polarized protons at fixed polarization (UL). The error bars are estimated for a 1000 hrs run with positron beam and luminosity $L = 2\times 10^{34}$ cm$^{-2}$sec$^{-1}$ at a beam polarization $P=0.6$. Electron luminosity $L = 10 \times10^{34}$ cm$^{-2}$sec$^{-1}$, and electron beam polarization $P = 0.8$. The error bars are statistical for a single bin in $Q^2$, $x$, and $t$ as shown in the top-left panel. Other bins are measured simultaneously.[]{data-label="fig:cross_section"}](dvcs_asymmetries.png){height=".40\textheight"} If only polarized electron beams are available a Rosenbluth separation with different beam energies can separate the term $\Delta\tilde\sigma_{INT}$ from $\Delta\tilde\sigma_{VCS}$, again in a much more limited kinematical range and with likely larger systematic uncertainties. The important interference term $\Delta\sigma_{INT}$ can only be determined using the combination of polarized electron and polarized positron beams. Estimates of charge asymmetries for different lepton charges ------------------------------------------------------------ For quantitative estimates of the charge differences in the cross sections we use the acceptance and luminosity achievable with [CLAS12]{} as basis for measuring the process $ep \rightarrow e\gamma p$ at different beam and target conditions. A 10 cm long liquid hydrogen is assumed with an electron current of 40nA, corresponding to an operating luminosity of $10^{35}$cm$^{-2}$sec$^{-1}$. For the positron beam a 5 times lower beam current of 8nA is assumed. In either case 1000 hours of beam time is used for the rate projections. For quantitative estimates of the cross sections the dual model [@Guzey:2008ys; @Guzey:2006xi] is used. It incorporates parameterizations of the GPDs $H$ and $E$. As shown in Fig. \[fig:cross\_section\], effects coming from the charge asymmetry can be large. In case of unpolarized beam and unpolarized target the cross section for electron scattering has only a small dependence on azimuthal angle $\phi$, while the corresponding positron cross section has a large $\phi$ modulation. The difference is directly related to the term $\sigma_{INT}$ in equation (\[diffcrs\]). Experimental Setup for DVCS Experiments {#dvcs} --------------------------------------- Figure \[dvcs\_experiment\] shows generically how the electron-proton and the positron-proton DVCS experiments could be configured. Electrons and positrons would be detected in the forward detection system of CLAS12. However, for the positron run the Torus magnet would have the reversed polarity so that positron trajectories would look identical to the electron trajectories in the electron-proton experiment, and limit systematic effects in acceptances. The recoil proton in both cases would be detected in the Central Detector at the same solenoid magnet polarity, also eliminating most systematic effects in the acceptances. However, there is a remaining systematic difference in the two configuration, as the forward scattered electron/positron would experience different transverse field components in the solenoid, which will cause the opposite azimuthal motion in $\phi$ in the forward detector. A good understanding of the acceptances in both cases is therefore important. The high-energy photon is, of course, not affected by the magnetic field configuration. ![CLAS12 configuration for the two electron and positron experiments (generic). The central detector will detect the protons, and the bending in teh magnetic solenoid field will be identical for the same kinematics. The electron and the positron, as well as the high-energy DVCS photon will be detected in the forward detector part. The electron and positron will be deflected in the Torus magnetic field in the same way as the Torus field direction will be opposite in the two experiments. The deflection in $\phi$ due to the solenoid fringe field will be of same magnitude $\Delta\phi$ but opposite in direction. The systematic of this shift can be controlled by doing the same experiment with opposite solenoid field directions that would result in the sign change of the $\Delta\phi$. []{data-label="dvcs_experiment"}](dvcs_exp.pdf){height="180pt" width="350pt"} In the next section we discuss a possible solution to the, so-far, not conclusive experimental studies of two-photon effects in elastic electron-proton scattering and their effect on the ratio of electric to magnetic form factors $G_E/G_M$ versus $Q^2$. 2-PHOTON EFFECTS IN ELASTIC SCATTERING OFF PROTONS {#2-photon-effects} ================================================== In the electromagnetic physics community it is well known that two experimental approaches, the Rosenbluth separation and the beam polarization transfer approach results in conflicting values for the $G_E/G_M$ ratio when plotted as a function of $Q^2$. The results of the different experimental methods are compiled in Fig. \[GeGm\]. The trends of the two data sets are inconsistent with each other, although there is a large spread in the Rosenbluth data samples. The latter seem to be more consistent with near $Q^2$-independent behavior, while the polarization data have a strong downward behavior with $Q^2$. Furthermore, the uncertainties in the former are much larger and within the individual data sets there seem to be discrepancies as well. The difference of the two methods may been attributed to 2-photon exchange effects, which are expected to be much more important in the cross section subtraction method than in the polarization transfer method. Recent efforts to quantify 2-photon exchange contributions ---------------------------------------------------------- It is obviously important to resolve the discrepancy with experiments that have sensitivity to 2-photon contributions. The most straightforward process to evaluate 2-photon contribution is the measurement of the ratio of elastic $e^+p/e^-p$ scattering, which in leading order is given by the expression: $R_{2\gamma} = 1 - 2\delta_{\gamma\gamma}$. Several experiments have recently been carried out to measure the 2-photon exchange contribution in elastic scattering: the VEPP-3 experiment at Novosibirsk [@Rachek:2014fam], the CLAS experiment at Jefferson Lab [@Rimal:2016toz; @Adikaram:2014ykv], and the Olympus experiment at DESY [@Henderson:2016dea]. The kinematic reach of each experiment is shown in the right panel of Fig. \[GeGm\]. The kinematic coverage is much smaller in these experiments $Q^2 < 2$ GeV$^2$, and $\epsilon > 0.5$, where the 2-photon effects are expected to be small, and systematics of the measurements must be extremely well controlled. The combined evaluation of all three experiments led the authors of the review article Ref. [@Afanasev:2017gsk] to the conclusion that the results of the experiments are inconsistent with the $\delta_{\gamma\gamma} = 0$ hypothesis at 99.5% confidence. At the same time, they state that [*“the results of these experiments are by no means definitive”*]{}, and [*“there is a clear need for similar experiments at larger $Q^2$ and at $\epsilon < 0.5$.”*]{} ![Left panel: The ratio of the proton electric and magnetic form factors $G_E/G_M$. The cyan markers are results of experiments based on the Rosenbluth method. The red markers are from JLab Hall A experiments. Right panel: Kinematics covered by the three recent experiments to measure the 2-photon exchange contribution to the elastic ep cross section. Both figures are taken from a recent review article [@Afanasev:2017gsk].[]{data-label="GeGm"}](GeGm-Q2eps.png){height="160pt" width="320pt"} Conclusions From Previous Experiments ------------------------------------- In the following I discuss a possible experiment with CLAS12 at Jefferson Lab that may be able to remedy the shortcomings of the previous measurements. What are these shortcomings? - Kinematics coverage in $Q^2$ and in $\epsilon$ are mostly where 2-photon effects are expected to be small - Systematic uncertainties are marginal in some cases - Higher $Q^2$ and small $\epsilon$ corresponding to high energy and large electron scattering angles were out of reach Can we do better with a setup using the modified CLAS12 detector? To address this question we begin with the close to ideal kinematic coverage that this setup provides. Figure \[angle\_reach\] shows the angle coverage for both the electron (left) and for the proton (right). There is a one-to-one correlation between the electron scattering angle and the proton recoil angle. For the kinematics of interest, say $\epsilon < 0.6$ and $Q^2 > 2$ GeV$^2$ for the chosen beam energies from 2.2 to 6.6 GeV, nearly all of the electron scattering angles fall into a polar angle range from $40^\circ$ to $125^\circ$, and corresponding to the proton polar angle range from $8^\circ$ to $35^\circ$. While these kinematics are most suitable for accessing the 2-photon exchange contributions, the setup will be able to also measure the reversed kinematics with the electrons at forward angle and the protons at large polar angles. This is in fact the standard CLAS12 configuration of DVCS and most other experiments, however it will not cover the kinematics with highest sensitivity to the 2-photon exchange contributions. Figure \[Q2eps\] shows the expected elastic scattering rates covering the ranges of highest interest, with $\epsilon < 0.6$ and $Q^2 = 2 - 10$ GeV$^2$. Sufficiently high statistics of ${\sigma_N / N} < 1\%$ can be achieved within 10 hrs for the lowest energy and within 1000 hrs for the highest energy, to cover the full range in kinematics. Note that all kinematic bins will be measured simultaneously at a given energy, and the shown rates are for the individual bins in $Q^2$ - $\epsilon$ space. A New Experimental Setup - Kinematic Coverage and Rate Estimates ---------------------------------------------------------------- ![Polar angle and $\epsilon$ coverage for electron detection (left) and for proton detection (right). []{data-label="angle_reach"}](angle_reach.png){height="180pt" width="320pt"} ![Estimated elastic event rates per hour for selected standard CEBAF beam energies of 2.2, 4.4, 6.6 GeV in the $\epsilon$ - $Q^2$ plane. Rates are given only for the lowest and highest $Q^2$ bin.[]{data-label="Q2eps"}](eps_Q2.png){height="180pt" width="330pt"} ![CLAS12 configuration for the elastic $e^-p/e^+p$ scattering experiment (generic). The central detector will detect the electron/positrons, and bending in the solenoid magnetic field will be identical for the same kinematics. The proton will be detected in the forward detector part. The Torus field direction will be the same in both cases. The deflection in $\phi$ due to the solenoid fringe field will be of same in magnitude of $\Delta\phi$ but opposite in direction. The systematic of this shift can be controlled by doing the same experiment with opposite solenoid field directions that would result in the sign change of the $\Delta\phi$.[]{data-label="2gamma-exp"}](2gamma-exp.pdf){height="180pt" width="350pt"} In order to achieve the desired reach in $Q^2$ and $\epsilon$ the [CLAS12]{} detection system has to be used with reversed detection capabilities for electrons. The main modification will involve replacing the current Central Neutron Detector (CND) with a central electromagnetic calorimeter (CEC) . The CEC will not need very good energy or angle resolution (both are provided by the tracking detectors) but will be used for trigger purposes and to aid in electron/pion separation. The over constrained kinematics of measured scattered electrons and recoil protons should be sufficient to select the elastic kinematics and eliminated any background (this will have to be demonstrated by detailed simulations). For the rate estimates and the kinematical coverage we have made a number of assumptions that are not overly stringent: - Positron beam currents (unpolarized): $I_{e^+} \approx 60$ nA. - Beam profile: $\sigma_x,~\sigma_y < 0.4$ mm. - Polarization: not required, so phase space at the source maybe chosen for optimized yield and beam parameters. - Obtain the electron beam from the same source as the positrons to keep systematic under control. - Switching from $e^+$ to $e^-$ operation should be doable in reasonable time frame ( $< 1$ day) to keep machine stable, and systematics under control. - Operate experiment with 5cm liquid H$_2$ target and luminosity of $0.8 \times 10^{35}$ cm$^2$sec$^{-1}$ - Use the CLAS12 Central Detector for lepton ($e^+/e^-$) detection at $\Theta_l = 40 - 125^\circ$. - Use CLAS12 Forward Detector for proton detection at $\Theta_p = 7^\circ - 35^\circ$ The CLAS12 configuration suitable for this experiment is shown in Fig. \[2gamma-exp\]. SUMMARY ======= Availability of a 11 GeV positron beam at JLab can significantly enhance the experimental program using the [CLAS12]{} detector in Hall B [@Burkert:2012rh]. I discussed two high profile programs that would very significantly benefit from a high performance polarized positron source and accelerated beam. The first program fits well into the already developed 3D-imaging program with electron beams, where the imaginary part of the DVCS amplitude can be extracted. The program with polarized positrons enables access to the azimuthally even BH-DVCS interference terms that are directly related to the real part of the scattering amplitude. Moreover, by avoiding use of the Rosenbluth separation technique, the leading contributions to the cross sections may be separated in the full kinematical range available at the JLab 12 GeV upgrade. Even at modest polarized positron beam currents of 8nA good statistical accuracy can be achieved for charge differences and charge asymmetries. For efficient use of polarized targets higher beam currents of up to 40nA are needed to compensate for the dilution factor of $\sim 0.18$ inherent in the use of currently available polarized proton targets based on ammonia as target material, and to allow for a more complete DVCS and GPD program at 12 GeV. The second program requiring positron beams is the measurement of the 2-photon exchange contributions in the elastic electron-proton scattering. The measurements we outlined, if properly executed with excellent control of systematic uncertainties, should close the book on the discrepancies in the ratio of electric to magnetic form factors when measured with two different methods. In this talk we have focussed on experiments with a large acceptance detector, which may be the only option given the low current expected for polarized positron beams of high polarization and good beam parameters. 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--- abstract: 'Waiter–Client and Client–Waiter games are two–player, perfect information games, with no chance moves, played on a finite set (board) with special subsets known as the winning sets. Each round of the biased $(1:q)$ game begins with Waiter offering $q+1$ previously unclaimed elements of the board to Client, who claims one. The $q$ elements remaining are then claimed by Waiter. If Client fully claims a winning set by the time all board elements have been offered, he wins in the Client–Waiter game and loses in the Waiter–Client game. We give an estimate for the threshold bias of the $(1:q)$ Waiter–Client and Client–Waiter versions of two different games: the non–2–colourability game, played on the complete $k$–uniform hypergraph, and the $k$–SAT game. In particular, we show that the unique value of $q$ at which the winner of the Client–Waiter version of the non–2–colourability game changes is $\frac{1}{n}\binom{n}{k}2^{-k(1+o_k(1))}$ and, for the Waiter–Client version, the corresponding value of $q$ is $\frac{1}{n}\binom{n}{k}2^{\Theta_k(k)}$. Additionally, we show that the threshold bias for the Waiter–Client and Client–Waiter versions of the $k$–SAT game is $\frac{1}{n}\binom{n}{k}$ up to a factor that is exponential and polynomial in $k$ respectively. This shows that these games exhibit the “probabilistic intuition”.' author: - 'Wei En Tan [^1]' bibliography: - 'WCCW2col\_kgraph.bib' title: 'Waiter–Client and Client–Waiter colourability games on a $k$–uniform hypergraph and the $k$–SAT game' --- Introduction {#intro} ============ In this paper, we estimate the value of the threshold bias for two biased $(1:q)$ Waiter–Client and Client–Waiter games: the non–2–colourability game, played on the complete $n$–vertex $k$–uniform hypergraph $K_n^{(k)}$, and the $k$–SAT game. These belong to a wider class of games known as “positional games”. A positional game is a two–player perfect information game where each player takes turns to claim previously unclaimed (free) elements of a set (board) $X$ until all members of $X$ have been claimed. At this point, the game ends. The winner is determined by the winning criteria of the specific type of positional game in play. Such criteria are defined by a set $\mathcal{F}\subseteq 2^X$ of so–called “winning sets” which are known to both players before the game begins. A game with board $X$ and set $\mathcal{F}$ of winning sets is often denoted by the pair $(X,\mathcal{F})$. Popular examples of positional games include Tic–Tac–Toe and Hex [@Gale]. Since the influential papers of Hales and Jewett [@HJ], Lehman [@Lehman] and Erdős and Selfridge [@ES], positional games have become a widely researched area of combinatorics, and developments in this field have made a significant impact in spheres such as computer science, with regards to the de-randomisation of randomised algorithms (see *e.g.* [@ES]). In the literature, the board for these games is most commonly the edge set of a graph or hypergraph. For an extensive survey on positional games, the interested reader may refer to the monographs of Beck [@TTT] and Hefetz, Krivelevich, Stojaković and Szabó [@HKSS]. In biased $(1:q)$ Waiter–Client and Client–Waiter games, where $q$ is a positive integer, the two players, Waiter and Client, play in the following way. At the beginning of each round of the $(1:q)$ Waiter–Client game $(X,\mathcal{F})$, Waiter offers exactly $q+1$ free elements of $X$ to Client. Client claims one of these, and the remaining $q$ elements are then claimed by Waiter. If, in the last round, only $1\leqslant r<q+1$ free elements remain, Waiter claims all of them. Waiter wins the game if he can force Client to fully claim a winning set in $\mathcal{F}$. Otherwise, Client wins. In the $(1:q)$ Client–Waiter game $(X,\mathcal{F})$, each round begins with Waiter offering $1\leqslant t\leqslant q+1$ free elements of $X$ to Client. Client then claims one of these elements, and the remainder of the offering (if any) is claimed by Waiter. In this game, Client wins if he can fully claim a winning set in $\mathcal{F}$, otherwise Waiter wins. Since these games are finite, perfect information, two–player games with no chance moves and no possibility of a draw, a classical result from Game Theory guarantees a winning strategy (*i.e.* a strategy that, if followed, ensures a win regardless of how the opponent plays). Also, both games are bias monotone in Waiter’s bias $q$. This means that, if Client has a winning strategy for a $(1:q)$ Waiter–Client game then Client also has a winning strategy for the same game with bias $(1:q+1)$. The analogous implication is true when Waiter has a strategy to win a $(1:q)$ Client–Waiter game. Thus, for each $(1:q)$ Waiter–Client or Client–Waiter game $(X,\mathcal{F})$, there exists a unique value of $q$ at which the winner of the game changes. This is known as the *threshold bias* of the game. We give bounds on the threshold bias of two specific types of Waiter–Client and Client–Waiter games in this paper. The first game of interest is the non–2–colourability game $(E(K_n^{(k)}),\mathcal{NC}_2)$ played on the edge set $E(K_n^{(k)})$ of the complete $n$–vertex $k$–uniform hypergraph $K_n^{(k)}$, for some positive integer $k$. In this, the set $\mathcal{NC}_2$ of winning sets is defined to be $$\mathcal{NC}_2=\{F\subseteq E(K_n^{(k)}):\chi(F)>2\},$$ where $\chi$ denotes the weak chromatic number. The second game we consider is the $k$–SAT game $(\mathcal{C}^{(k)}_n,\mathcal{F}_{SAT})$, where $k$ is a positive integer. This is played on the set $\mathcal{C}^{(k)}_n$ of all $2^k\binom{n}{k}$ possible *$k$–clauses*, where each $k$–clause is the disjunction of exactly $k$ non–complementary literals taken from $n$ fixed boolean variables $x_1,\ldots,x_n$. By *literal*, we mean a boolean variable $x_i$ or its negation $\neg x_i$. The set $\mathcal{F}_{SAT}$ of winning sets is defined to be $$\mathcal{F}_{SAT}=\{\phi:\exists \mathcal{S}\subseteq\mathcal{C}^{(k)}_n\text{ s.t. }\phi=\bigwedge\mathcal{S}\text{ and }\phi\text{ is not satisfiable}\},$$ where $\bigwedge\mathcal{S}$ denotes the conjunction of all $k$–clauses in $\mathcal{S}$. These games are interesting from a computational point of view, since the complexity of deciding the 2–colourability or the satisfiability of a random instance is not known. Additionally, since extensive work (see *e.g.* [@Coja-Oghlan2012; @Ding2015]) has gone into understanding 2–colourable hypergraphs and satisfiable $k$–CNF boolean formulae in the random setting, they are also perfect candidates for developing our understanding of an interesting heuristic known as the *probabilistic intuition*. This suggests that the threshold bias of a game, in which both players play optimally, is roughly the same as it would be, with high probability (*i.e.* with probability tending to 1 as the size of the board tends to infinity), if both players play the game randomly instead. Investigating this notion serves as the main motivation for the research in this paper and we discuss it in greater detail in Section \[probint\]. The results that follow show that this phenomenon occurs for our chosen games. The Results {#results} ----------- ### The non–2–colourability game We prove that the threshold bias for the $(1:q)$ Waiter–Client and Client–Waiter versions of $(E(K_n^{(k)}),\mathcal{NC}_2)$ is $\frac{1}{n}\binom{n}{k}2^{\Theta_k(k)}$ and $\frac{1}{n}\binom{n}{k}2^{-k(1+o_k(1))}$ respectively. In Section \[probint\], we will see that these match the probabilistic intuition. \[th::WC2col\_kgraph\] Let $k$, $q$ and $n$ be positive integers, with $n$ sufficiently large and $k\geqslant 2$ fixed, and consider the $(1:q)$ Waiter–Client non–2–colourability game played on the edge set of the complete $k$–uniform hypergraph $K_n^{(k)}$ on $n$ vertices. If $q\leqslant\binom{\lceil n/2\rceil}{k}\frac{\ln 2}{2((1+\ln 2)n+\ln 2)}$, then Waiter can force Client to build a non–2–colourable hypergraph. Also, if $q\geqslant2^{k/2}e^{k/2+1}k\binom{n}{k}/n$, then Client can keep his hypergraph 2–colourable throughout the game. \[th::CW2col\_kgraph\] Let $k$, $q$ and $n$ be positive integers, with $n$ sufficiently large and $k\geqslant 2$ fixed, and consider the $(1:q)$ Client–Waiter non–2–colourability game played on the edge set of the complete $k$–uniform hypergraph $K_n^{(k)}$ on $n$ vertices. If $q\leqslant\binom{\lceil n/2\rceil}{k}\frac{\ln 2}{(1+\ln 2)n}$, then Client can build a non–2–colourable hypergraph. However, when $q\geqslant k^32^{-k+5}\binom{n}{k}/n$, Waiter can ensure that Client has a 2–colourable hypergraph at the end of the game. Thus, for the Waiter–Client version, we have a gap of $(1+o(1))(1+1/\ln 2)k\cdot 2^{3k/2+1}e^{k/2+1}$ between the upper and lower bounds of $q$ and, for the Client–Waiter version, we have a gap of $(1+o(1))(1+1/\ln 2) 2^5k^3$. \[generalr\_col\] Our proofs of Theorems \[th::WC2col\_kgraph\] and \[th::CW2col\_kgraph\] generalise easily to the Waiter–Client and Client–Waiter non–$r$–colourability game $(E(K_n^{(k)}),\mathcal{NC}_r)$, for any fixed $r,k\geqslant 2$, where $$\mathcal{NC}_r=\{F\subseteq E(K_n^{(k)}):\chi(F)>r\}.$$ In particular, their generalisation shows that the threshold bias for the $(1:q)$ Waiter–Client and Client–Waiter versions of $(E(K_n^{(k)}),\mathcal{NC}_r)$ is $\frac{1}{n}\binom{n}{k}r^{\Theta_k(k)}$ and $\frac{1}{n}\binom{n}{k}r^{-k(1+o_k(1))}$ respectively. In Section \[probint\], we will see that these threshold biases match the probabilistic intuition when $r$ is sufficiently large. For the sake of clarity and simplicity of presentation, and since the generalisation to $r$ colours is straightforward, we only include our proofs for the case $r=2$ in this paper. ### The $k$–SAT game We prove that the threshold bias for the $(1:q)$ Waiter–Client and Client–Waiter versions of $(\mathcal{C}^{(k)}_n,\mathcal{F}_{SAT})$ is $\frac{1}{n}\binom{n}{k}$ up to a factor that is exponential and polynomial in $k$ respectively. This approximately matches the probabilistic intuition, as we shall see in Section \[probint\]. \[th::WCkSAT\] Let $k$, $q$ and $n$ be positive integers, with $n$ sufficiently large and $k\geqslant 2$ fixed, and consider the $(1:q)$ Waiter–Client $k$–SAT game played on $\mathcal{C}^{(k)}_n$. When $q\leqslant\binom{n}{k}/(2n)$, Waiter can ensure that the conjunction of all $k$–clauses claimed by Client by the end of the game is not satisfiable. However, when $q\geqslant 2^{3k/2}e^{k/2+1}k\binom{n}{k}/n$, Client can ensure that the conjunction of all $k$–clauses he claims remains satisfiable throughout the game. \[th::CWkSAT\] Let $k$, $q$ and $n$ be positive integers, with $n$ sufficiently large and $k\geqslant 2$ fixed, and consider the $(1:q)$ Client–Waiter $k$–SAT game played on $\mathcal{C}^{(k)}_n$. When $q<\binom{n}{k}/n$, Client can ensure that the conjunction of all $k$–clauses he claims by the end of the game is not satisfiable. However, when $q\geqslant 16k^3\binom{n}{k}/n$, Waiter can ensure that the conjunction of all $k$–clauses claimed by Client is satisfiable throughout the game. Thus, for the Waiter–Client version, we have a gap of $k\cdot 2^{3k/2+1}e^{k/2+1}$ between the upper and lower bounds of $q$ and, for the Client–Waiter version, we have a gap of $16k^3$. The Probabilistic Intuition {#probint} --------------------------- As mentioned previously, the main motivation behind our results is to investigate a heuristic known as the *probabilistic intuition*. This was first employed by Chvátal and Erdős in [@CE] and we illustrate it as follows. Consider a $(1:q)$ Waiter–Client game $(X,\mathcal{F})$ and suppose both players play randomly. Then, at the end of the game, Client has a set of $\lfloor |X|/(q+1)\rfloor$ random elements of the board. If, with high probability (*whp*) this set contains a winning set, then the probabilistic intuition predicts that Waiter has a winning strategy for the $(1:q)$ game. On the other hand, it predicts that Client has a winning strategy for the $(1:q)$ game if *whp* his set of elements at the end of the game does *not* contain a winning set. In particular, suppose that the function $m_{\mathcal{F}}=m_{\mathcal{F}}(|X|)$ is such that, for every $m\gg m_{\mathcal{F}}$ (*i.e.* $\lim_{|X|\rightarrow\infty}m_{\mathcal{F}}/m=0$) any set of $m$ random elements of $X$ contains some $\mathcal{A}\in\mathcal{F}$ *whp* and, for every $m\ll m_{\mathcal{F}}$ any set of $m$ random elements of $X$ does *not* contain any $A\in\mathcal{F}$ *whp*. We refer to $m_{\mathcal{F}}$ as the *threshold* for the property of containing a winning set in $\mathcal{F}$. If the threshold bias $q_{\mathcal{F}}$ of the game $(X,\mathcal{F})$ has the same order of magnitude as $|X|/m_{\mathcal{F}}$, we say that the game exhibits the probabilistic intuition. Surprisingly, research into Waiter–Client and Client–Waiter games played on graphs has found the probabilistic intuition exhibited in many such games whose winning sets are defined by various graph properties (see *e.g.* [@HKTpcm; @BHKL; @BHL; @ctcs]). Our results give more examples of Waiter–Client and Client–Waiter games for which this is true. Random play by both Waiter and Client in the $(1:q)$ non–2–colourability game on $K_n^{(k)}$ leaves Client with a random $n$–vertex $k$–uniform hypergraph at the end of the game. Alon and Spencer [@AS] were the first to bound the values $c$ for which the random $n$–vertex $k$–uniform hypergraph with $m=cn$ edges, denoted by $\mathcal{H}_k(n,m)$, is 2–colourable. They showed that the threshold $c_{2,k}$ for $c$ (although only conjectured to exist) satisfies $\tilde{c}\cdot2^k/k^2<c_{2,k}<2^{k-1}\ln2-\ln2/2$ for some small constant $\tilde{c}>0$. The gap of order $k^2$ was reduced by Achlioptas, Kim, Krivelevich and Tetali to order $k$ in [@AKKT]. Subsequently, Achlioptas and Moore [@AM2002a] improved this further by showing that $c_{2,k}\geqslant2^{k-1}\ln2-\ln2/2-(1+\varepsilon)/2$ for any $\varepsilon>0$ and $k$ sufficiently large. Further still, Coja–Oghlan and Zdeborová [@Coja-Oghlan2012] improved this lower bound by an additive $(1-\ln 2)/2$. Hence, the edge threshold for the 2–colourability of a random $n$–vertex $k$–uniform hypergraph is $c_{2,k}n=2^{k(1+o_k(1))}n$. Therefore, the probabilistic intuition predicts that the threshold bias for the $(1:q)$ non–2–colourability game $(E(K_n^{(k)}),\mathcal{NC}_2)$ is $\frac{1}{n}\binom{n}{k}2^{-k(1+o_k(1))}$ which matches the threshold bias (up to the error term in the exponent) given by Theorems \[th::WC2col\_kgraph\] and \[th::CW2col\_kgraph\]. Thus, both the $(1:q)$ Waiter–Client and the $(1:q)$ Client–Waiter versions of the non–2–colourability game exhibit the probabilistic intuition. Such a conclusion also holds in the more general setting, concerning the $r$–colourability of $\mathcal{H}_k(n,cn)$ where $r\geqslant 2$. This is demonstrated by the following bounds on the corresponding threshold $c_{r,k}$ for $c$ described in the literature. By generalising a result of Achlioptas and Naor [@AN2005] on $r$–colouring a random graph (2–uniform hypergraph), Dyer, Frieze and Greenhill [@DFG2015] proved that $(r-1)^{k-1}\ln(r-1)\leqslant c_{r,k} \leqslant (r^{k-1}-1/2)\ln r.$ The lower bound was subsequently improved by Ayre, Coja–Oghlan and Greenhill [@ACG2015] to $(r^{k-1}-1/2)\ln r-\ln 2-1.01\ln r/r$ for sufficiently large $r$. Thus, for such $r$, the edge threshold for the $r$–colourability of a random $n$–vertex $k$–uniform hypergraph is $c_{r,k}n=r^{k(1+o_k(1))}n$. Therefore, the probabilistic intuition predicts that the threshold bias for the $(1:q)$ non–$r$–colourability game $(E(K_n^{(k)}),\mathcal{NC}_r)$ is $\frac{1}{n}\binom{n}{k}r^{-k(1+o_k(1))}$ when $r$ is large, which matches the threshold biases mentioned in Remark \[generalr\_col\] (up to the error term in the exponent). The $(1:q)$ Waiter–Client and Client–Waiter $k$–SAT games also exhibit the probabilistic intuition. Indeed, Coja–Oghlan and Panagiotou [@CojaOghlan2016] found that the threshold for the satisfiability of the conjunction of random $k$–clauses in $\mathcal{C}^{(k)}_n$ is $(2^{k}\ln 2-(1+\ln 2)/2+o_k(1))n$ (see [@FP; @CR; @FS; @AM2002b; @FW; @AM2006; @AP] for earlier work). Hence, the probabilistic intuition predicts that the threshold bias for the $(1:q)$ $k$–SAT game $(\mathcal{C}^{(k)}_n,\mathcal{F}_{SAT})$ is $\frac{1}{n}\binom{n}{k}(\ln 2-o_k(1))^{-1}$. This is matched, up to a constant factor, by the lower bound on the threshold bias for the Waiter–Client and Client–Waiter $k$–SAT games given in Theorem \[th::WCkSAT\] and \[th::CWkSAT\] respectively. Since the gap between the upper and lower bounds depends only on $k$ (exponentially in the Waiter–Client game and polynomially in the Client–Waiter game), the threshold bias for both versions of the $k$–SAT game has the same order of magnitude as that predicted by the probabilistic intuition. Related Work ------------ Waiter–Client and Client–Waiter games were first introduced by Beck (see *e.g.* [@becksec]) under the names Picker–Chooser and Chooser–Picker. They are interesting for a number of different reasons. Firstly, when Waiter plays randomly in a Waiter–Client (Client–Waiter) game, the game becomes the *avoiding* (*embracing*) Achlioptas process. Secondly, as we have seen, these games often exhibit the probabilistic intuition. Finally, recent findings indicate that these games may be useful in the study of the highly popular Maker–Breaker games (see *e.g.* [@becksec; @cdm; @Bednarska; @Knox]). Since their introduction, much work has gone into finding the threshold bias of Waiter–Client and Client–Waiter games, with winning sets defined by a variety of different graph properties including connectivity, non–planarity and containing a $K_t$–minor (see *e.g.* [@HKTpcm; @BHKL]). In [@HKTpcm], the $(1:q)$ Waiter–Client and Client–Waiter non–2–colourability games were considered within the more general non–$k$–colourability game, but instead of playing on $E(K_n^{(k)})$, Hefetz, Krivelevich and Tan considered play on the edge set $E(K_n)$ of the complete graph on $n$ vertices. They found that, for both the Waiter–Client and the Client–Waiter versions, the threshold bias has order $\Theta_{n,k}\left(n/(k\log k)\right)$, thereby showing that these games also exhibit the probabilistic intuition. It was by generalising the techniques used in their paper, that we obtained our results stated in Section \[results\]. To our knowledge, the $k$–SAT game has not yet been considered in the literature on positional games. However, the Achlioptas process for $k$–SAT has been studied (see *e.g.* [@SV; @Perkins; @Dani2013]). Preliminaries ============= Whenever necessary, we assume that the number $n$ of vertices or boolean variables is sufficiently large. We also omit floor and ceiling signs whenever these are not crucial. Hypergraph notation ------------------- Let $\mathcal{H}$ be any $k$–uniform hypergraph with vertex set $V(\mathcal{H})$ and edge set $E(\mathcal{H})\subseteq 2^{V(\mathcal{H})}$, where each edge consists of exactly $k$ vertices. For a vertex $v \in V(\mathcal{H})$, let $E_{\mathcal{H}}(v)$ denote the set of edges of $\mathcal{H}$ that contain $v$ and let $d_{\mathcal{H}}(v) = |E_{\mathcal{H}}(v)|$. For a set $S \subseteq V(\mathcal{H})$, let $\mathcal{H}[S]$ denote the hypergraph with vertex set $S$ and edge set $\{e\in E(\mathcal{H}):e\subseteq S\}$. The *maximum degree* of $\mathcal{H}$ is denoted by $\Delta(\mathcal{H}) = \max \{d_{\mathcal{H}}(v) : v \in V(\mathcal{H})\}$. We say that a set $A \subseteq V(\mathcal{H})$ is *independent* in $\mathcal{H}$ if $\{e\in E(\mathcal{H}):e\subseteq A\} = \emptyset$. The *independence number* of $\mathcal{H}$, denoted by $\alpha(\mathcal{H})$, is the maximum size of an independent set of vertices in $\mathcal{H}$. A subhypergraph $\mathcal{H}'\subseteq\mathcal{H}$ (*i.e.* a hypergraph $\mathcal{H}'$ with $V(\mathcal{H}')\subseteq V(\mathcal{H})$ and $E(\mathcal{H}')\subseteq E(\mathcal{H})$) is a *clique* in $\mathcal{H}$ if every set of $k$ vertices in $V(\mathcal{H}')$ is an edge of $\mathcal{H}'$. We sometimes refer to a clique with $\ell$ vertices as an $\ell$–clique. The *clique number* of $\mathcal{H}$, denoted by $\omega(\mathcal{H})$, is the largest $\ell$ such that $\mathcal{H}$ contains an $\ell$–clique. The *weak chromatic number* of $\mathcal{H}$, denoted by $\chi(\mathcal{H})$, is the smallest integer $k$ for which $V(\mathcal{H})$ can be partitioned into $k$ independent sets. For a set $F\subseteq E(\mathcal{H})$, we abuse notation slightly by using $\chi(F)$ to denote the chromatic number of the hypergraph with vertex set $V(\mathcal{H})$ and edge set $F$. Let us denote the complete $n$–vertex $k$–uniform hypergraph by $K_n^{(k)}$ (*i.e.* $K_n^{(k)}$ is an $n$–clique). At any given moment in a Waiter–Client or Client–Waiter game, played on $E(K_n^{(k)})$, let $E_W$ and $E_C$ denote the set of edges currently owned by Waiter and Client respectively. We denote the hypergraph with vertex set $V(K_n^{(k)})$ and edge set $E_W$ by $\mathcal{H}_W$, and the hypergraph with vertex set $V(K_n^{(k)})$ and edge set $E_C$ by $\mathcal{H}_C$. Moreover, let $\mathcal{H}_F$ be the hypergraph consisting of all edges of $K_n^{(k)}$ that are free at a given moment. $k$–SAT notation ---------------- Let $V=\{x_1,\ldots,x_n\}$ be a set of $n$ boolean variables. We define $B_i:=\{x_i,\neg x_i\}$ for each $i\in[n]$ and $\mathcal{B}:=\{B_i:i\in [n]\}$. The conjunction of any number of $k$–clauses is called a *$k$–CNF boolean formula*. For a set $\mathcal{A}$ of $k$–clauses, let $S(\mathcal{A})$ denote the set of literals that appear in the $k$–clauses of $\mathcal{A}$. In the case where $\mathcal{A}$ consists of a single $k$–clause $c$, we will abuse notation slightly and write $S(c)$ instead of $S(\{c\})$. For a set $L$ of literals, let $\mathcal{A}[L]$ denote the set of $k$–clauses of $\mathcal{A}$ whose literals lie in $L$. Let $N_{\mathcal{A}}(L)$ denote the set of $k$–clauses in $\mathcal{A}$ that contain at least one literal from $L$ and let $d_{\mathcal{A}}(L)=|N_{\mathcal{A}}(L)|$. Additionally, we use $\bigwedge\mathcal{A}$ to denote the conjunction of all $k$–clauses in $\mathcal{A}$ and, for any set $B\subseteq\cup\mathcal{B}$, we use $\bigvee B$ to denote the disjunction of all literals in $B$. Let us denote the set of all possible $2^k\binom{n}{k}$ $k$–clauses, each consisting of non–complementary literals taken from $n$ boolean variables $x_1,\ldots,x_n$, by $\mathcal{C}^{(k)}_n$. At any given moment in a Waiter–Client or Client–Waiter game, played on $\mathcal{C}^{(k)}_n$, let $\mathcal{C}_C$ denote the set of all $k$–clauses currently owned by Client and let $\mathcal{C}_F$ denote the set of $k$–clauses that are currently free. Useful Tools ------------ We will use the following two lemmas which result from a standard application of the Lovász Local Lemma (see *e.g.* Chapter 5 in [@ProbMethod], [@Gebauer2009]). \[degcondition\] Let $\mathcal{H}$ be a $k$–uniform hypergraph with maximum degree $\Delta(\mathcal{H})\leqslant 2^k/(8k)$. Then $\mathcal{H}$ is 2–colourable. \[satcondition\] Let $k\geqslant 2$ be an integer. Any $k$–CNF boolean formula in which no variable appears in more than $2^{k-2}/k$ $k$–clauses is satisfiable. We will also use the following game theoretic tools. The first two apply to the *transversal* game $(X,\mathcal{F}^*)$. For a finite set $X$ and ${\mathcal F}\subseteq 2^X$, the *transversal* family of ${\mathcal F}$ is ${\mathcal F}^* := \{A \subseteq X : A \cap B \neq \emptyset \textrm{ for every } B \in {\mathcal F}\}$. \[th::CwinsCWtrans\] Let $q$ be a positive integer, let $X$ be a finite set and let $\mathcal{F}$ be a family of subsets of $X$. If $$\sum_{A \in \mathcal{F}} \left(\frac{q}{q+1}\right)^{|A|} < 1,$$ then Client has a winning strategy for the $(1:q)$ Client–Waiter game $(X, \mathcal{F}^*)$. \[th::WaiterBES\] Let $q$ be a positive integer, let $X$ be a finite set and let ${\mathcal F}$ be a family of subsets of $X$. If $$\sum_{A \in {\mathcal F}} 2^{-|A|/(2q-1)} < 1/2 \,,$$ then Waiter has a winning strategy for the $(1 : q)$ Waiter–Client game $(X, {\mathcal F}^*)$. \[th::ClientBES\] Let $q$ be a positive integer, let $X$ be a finite set, let ${\mathcal F}$ be a family of (not necessarily distinct) subsets of $X$ and let $\Phi(\mathcal{F}) = \sum_{A \in \mathcal{F}} (q+1)^{-|A|}$. Then, when playing the $(1 : q)$ Waiter–Client game $(X, {\mathcal F})$, Client has a strategy to avoid fully claiming more than $\Phi(\mathcal{F})$ sets in ${\mathcal F}$. The rest of this paper is organised as follows: in Section \[sec::non2col\] we prove Theorems \[th::WC2col\_kgraph\] and \[th::CW2col\_kgraph\]. In Section \[sec::ksat\] we prove Theorems \[th::WCkSAT\] and \[th::CWkSAT\]. Finally, in Section \[sec::openprob\] we present some open problems. The non–2–colourability game {#sec::non2col} ============================ The Waiter–Client non–2–colourability game ------------------------------------------ **Proof of Theorem \[th::WC2col\_kgraph\]**. Fix $k\geqslant 2$. #### Waiter’s strategy: {#waiters-strategy .unnumbered} Suppose that $q\leqslant\binom{\lceil n/2\rceil}{k}\frac{\ln 2}{2((1+\ln 2)n+\ln 2)}$. Since $\omega(\mathcal{H}_W)=\alpha(\mathcal{H}_C)$ at the end of the game and, for any hypergraph $\mathcal{H}$ on $n$ vertices, $\chi(\mathcal{H})\alpha(\mathcal{H})\geqslant n$, it suffices to show that Waiter has a strategy to force Client to claim an edge in every $\lceil n/2\rceil$–clique of $K_n^{(k)}$. Let $\mathcal{F}$ denote the set of all $\lceil n/2\rceil$–cliques in $K_n^{(k)}$. Observe that $$\begin{aligned} \sum_{A\in\mathcal{F}}2^{-|A|/(2q-1)}\leqslant\binom{n}{\lceil n/2\rceil}2^{-\binom{\lceil n/2\rceil}{k}/(2q)}\leqslant\left(\frac{en}{\lceil n/2\rceil}\right)^{\lceil n/2\rceil}2^{-((1+1/\ln 2)n+1)}<\frac{1}{2}\left(\frac{e}{2^{1/\ln 2}}\right)^n=\frac{1}{2},\end{aligned}$$ where the second inequality holds by our choice of $q$. Hence, by Theorem \[th::WaiterBES\], Waiter can ensure that $\alpha(\mathcal{H}_C)<\lceil n/2\rceil$ to give $\chi(\mathcal{H}_C)>2$ by the end of the game. #### Client’s strategy: {#clients-strategy .unnumbered} Suppose $q\geqslant 2^{k/2}e^{k/2+1}k\binom{n}{k}/n$. Also, let $\mathcal{F}=\{F:\exists S\subseteq V(K_n^{(k)})\text{ s.t. }S\neq\emptyset, F\subseteq E(K_n^{(k)}[S])\text{ and }|F|=\frac{2}{k}|S|\}$. Observe that, $$\begin{aligned} \Phi(\mathcal{F})&=\sum_{A\in\mathcal{F}}(q+1)^{-|A|}\leqslant\sum_{t=k}^n\binom{n}{t}\binom{\binom{t}{k}}{2t/k}q^{-2t/k}\leqslant\sum_{t=k}^n\left[\frac{en}{t}\left(\frac{ek\binom{t}{k}}{2tq}\right)^{2/k}\right]^t\\ &\leqslant\sum_{t=1}^n\left[\frac{en}{t}\left(\frac{et^{k-1}}{2q(k-1)!}\right)^{2/k}\right]^t\leqslant\sum_{t=1}^n\left[\frac{en}{t}\left(\frac{1}{(2e)^{k/2}}\left(\frac{t}{n}\right)^{k-1}\right)^{2/k}\right]^t\\ &=\sum_{t=1}^n\left[\frac{1}{2}\left(\frac{t}{n}\right)^{\frac{2}{k}(k-1)-1}\right]^t<\sum_{t=1}^{\infty}\left[\frac{1}{2}\right]^t=1,\end{aligned}$$ where the fourth inequality follows from our choice of $q$ and since $n$ is assumed to be sufficiently large. Thus, by Theorem \[th::ClientBES\], Client can ensure that, for every $S\subseteq V(\mathcal{H}_C)$, there exists a vertex $v\in S$ with $d_{\mathcal{H}_C[S]}(v)\leqslant 1$ at the end of the game. Thus, in $V(\mathcal{H}_C)$, there exists a vertex $v_1$ contained in at most one edge of $\mathcal{H}_C$. In $V(\mathcal{H}_C)\setminus\{v_1\}$, there exists a vertex $v_2$ contained in at most one edge of $\mathcal{H}_C[V(\mathcal{H}_C)\setminus\{v_1\}]$. Continuing in this way, we obtain an ordering $v_1,\ldots,v_n$ of the vertices of $\mathcal{H}_C$ where, for each $v_i$, there is at most one edge of $\mathcal{H}_C$, consisting of vertices in $\{v_i,\ldots,v_n\}$, that contains $v_i$. Therefore, by colouring the vertices of $\mathcal{H}_C$ greedily from $v_n$ to $v_1$, we may obtain a 2–colouring of $\mathcal{H}_C$. [$\Box$\ ]{} The Client–Waiter non–2–colourability game ------------------------------------------ **Proof of Theorem \[th::CW2col\_kgraph\]**. Fix $k\geqslant 2$. #### Client’s strategy: {#clients-strategy-1 .unnumbered} Suppose that $q\leqslant\binom{\lceil n/2\rceil}{k}\frac{\ln 2}{(1+\ln 2)n}$. Since $\omega(\mathcal{H}_W)=\alpha(\mathcal{H}_C)$ at the end of the game and, for any hypergraph $\mathcal{H}$ on $n$ vertices, $\chi(\mathcal{H})\alpha(\mathcal{H})\geqslant n$, it suffices to show that Client has a strategy to claim an edge in every $\lceil n/2\rceil$–clique of $K_n^{(k)}$. Let $\mathcal{F}$ denote the set of all $\lceil n/2\rceil$–cliques in $K_n^{(k)}$. Observe that, $$\begin{aligned} \sum_{A \in \mathcal{F}} \left(\frac{q}{q+1}\right)^{|A|}&\leqslant\sum_{A\in\mathcal{F}}2^{-|A|/q}\leqslant\binom{n}{\lceil n/2\rceil}2^{-\binom{\lceil n/2\rceil}{k}/q}\\ &\leqslant\left(\frac{en}{\lceil n/2\rceil}\right)^{\lceil n/2\rceil}2^{-(1+1/\ln 2)n}<\left(\frac{e}{2^{1/\ln 2}}\right)^n=1,\end{aligned}$$ where our third inequality follows from our choice of $q$. Hence, by Theorem \[th::CwinsCWtrans\], Client can ensure that $\omega(\mathcal{H}_W)<\lceil n/2\rceil$, which means that $\chi(\mathcal{H}_C)>2$. #### Waiter’s strategy: {#waiters-strategy-1 .unnumbered} Suppose that $q\geqslant k^32^{-k+5}\binom{n}{k}/n$ and let us first consider when $k\geqslant 7$. By Lemma \[degcondition\], it suffices for Waiter to ensure that $\Delta(\mathcal{H}_C)\leqslant2^k/(8k)$ at the end of the game. He achieves this as follows. In the first round, Waiter offers $q+1$ arbitrary free edges. After this, whenever Client claims an edge, say $e$ consisting of vertices $v_1,\ldots,v_k$ ordered arbitrarily, Waiter responds in the following way. Let $F_1\subseteq\{e\in E(\mathcal{H}_F):v_1\in e\}$ with size $|F_1|=\min\{d_{\mathcal{H}_F}(v_1),\lfloor (q+1)/k\rfloor\}$ and, for each $2\leqslant j\leqslant k$, let $F_j\subseteq\{e\in E(\mathcal{H}_F):v_j\in e\}\setminus\cup\{F_\ell:1\leqslant\ell<j\}$ with size $|F_j|=\min\{|\{e\in E(\mathcal{H}_F):v_j\in e\}\setminus\cup\{F_\ell:1\leqslant\ell<j\}|,\lfloor (q+1)/k\rfloor\}$. Immediately after Client has claimed $e$, Waiter offers all edges in $\cup\{F_i:i\in [k]\}$. (Recall that, in any round of a Client–Waiter game, Waiter may offer less than $q+1$ edges if he desires.) If no free edge touches $e$, Waiter performs his response on an edge that Client claimed earlier on in the game. If no free edges touch any of Client’s previously claimed edges, Waiter simply offers $\min\{q+1,|E(\mathcal{H}_F)|\}$ arbitrary free edges. It is clear that, by responding to each of Client’s moves in this way, Waiter offers every edge of $K_n^{(k)}$ in the game. We claim that Waiter’s strategy ensures $\Delta(\mathcal{H}_C)\leqslant2^k/(8k)$ at the end of the game. Indeed, let $u\in V(K_n^{(k)})$ be an arbitrary vertex. Every time Client claims an edge containing $u$, Waiter offers at least $\lfloor (q+1)/k\rfloor$ free edges containing $u$, except for perhaps the last time he offers edges at $u$ when there may be less than $\lfloor (q+1)/k\rfloor$ such edges available. Every time Waiter offers edges containing $u$, Client may claim at most one of these. Hence, at the end of the game, $$d_{\mathcal{H}_C}(u)\leqslant \frac{\binom{n-1}{k-1}}{\lfloor (q+1)/k\rfloor}+1\leqslant\frac{2k\binom{n-1}{k-1}}{q}+1\leqslant\frac{2^k}{8k},$$ where the final inequality follows from our choice of $k$ and $q$. In the case where $2\leqslant k\leqslant 6$, Waiter still performs the above strategy, ensuring that $$d_{\mathcal{H}_C}(u)\leqslant \frac{\binom{n-1}{k-1}}{\lfloor (q+1)/k\rfloor}+1<\frac{\binom{n-1}{k-1}k}{q+1-k}+1\leqslant 2,$$ at the end of the game, where the final inequality follows from $k\leqslant 6$, our choice of $q$ and for sufficiently large $n$. Thus, every vertex in Client’s hypergraph is contained in at most one edge of $\mathcal{H}_C$. It is clear therefore, that $\mathcal{H}_C$ is 2–colourable. [$\Box$\ ]{} The $k$–SAT game {#sec::ksat} ================ The Waiter–Client $k$–SAT game ------------------------------ **Proof of Theorem \[th::WCkSAT\]**. Fix $k\geqslant 2$. #### Waiter’s strategy: {#waiters-strategy-2 .unnumbered} Let $q\leqslant\binom{n}{k}/(2n)$ and let $$\mathcal{F}=\{\mathcal{A}\subseteq \mathcal{C}^{(k)}_n: |S(\mathcal{A})|=n\text{ and }\bigvee B\in\mathcal{A},\forall B\subseteq S(\mathcal{A})\text{ with }|B|=k\}.$$ Note that, since no $k$–clause of $\mathcal{C}^{(k)}_n$ contains a pair of complementary literals, $S(\mathcal{A})$ cannot contain a pair of complementary literals, for each $\mathcal{A}\in\mathcal{F}$. Hence, observe that $$\begin{aligned} \sum_{\mathcal{A}\in\mathcal{F}}2^{-|\mathcal{A}|/(2q-1)}<2^{n-\binom{n}{k}/(2q)}=1,\end{aligned}$$ where the final equality follows from our choice of $q$. So, by Theorem \[th::WaiterBES\], Waiter can force Client to claim a $k$–clause in every $\mathcal{A}\in\mathcal{F}$ by the end of the game. Thus, for every partition $(V_1, V_2)$ of the literals in $S(\mathcal{C}^{(k)}_n)$, where no pair of complementary literals lie in the same part, Client owns two $k$–clauses, $c_1$ and $c_2$, satisfying $S(c_i)\subseteq V_i$ for every $i\in[2]$. This means that, regardless of how one assigns the values 0 and 1 to the boolean variables, Client will always have a $k$–clause consisting entirely of 0–valued literals, since every $\{0,1\}$–assignment defines such a partition of $S(\mathcal{C}^{(k)}_n)$. Thus, at the end of the game, $\bigwedge\mathcal{C}_C$ is not satisfiable. #### Client’s strategy: {#clients-strategy-2 .unnumbered} Let $q\geqslant 2^{3k/2}e^{k/2+1}k\binom{n}{k}/n$. Observe that, with $$\mathcal{F}=\left\{\mathcal{A}:\exists\mathcal{D}\subseteq\mathcal{B}\text{ s.t. }\mathcal{D}\neq\emptyset,\mathcal{A}\subseteq \mathcal{C}^{(k)}_n\left[\bigcup\mathcal{D}\right],\text{ and }|\mathcal{A}|=\frac{2}{k}|\mathcal{D}|\right\},$$ we have $$\begin{aligned} \Phi(\mathcal{F})&=\sum_{\mathcal{A}\in\mathcal{F}}(q+1)^{-|\mathcal{A}|}<\sum_{t=k}^n\binom{n}{t}\binom{\binom{t}{k}2^k}{2t/k}q^{-2t/k}\leqslant\sum_{t=k}^n\left[\frac{en}{t}\left(\frac{ek2^k\binom{t}{k}}{2tq}\right)^{2/k}\right]^t\\ &\leqslant\sum_{t=1}^n\left[\frac{en}{t}\left(\frac{et^{k-1}2^{k-1}}{q(k-1)!}\right)^{2/k}\right]^t\leqslant\sum_{t=1}^n\left[\frac{en}{t}\left(\frac{1}{(2e)^{k/2}}\left(\frac{t}{n}\right)^{k-1}\right)^{2/k}\right]^t\\ &=\sum_{t=1}^n\left[\frac{1}{2}\left(\frac{t}{n}\right)^{\frac{2}{k}(k-1)-1}\right]^t<\sum_{t=1}^{\infty}\left[\frac{1}{2}\right]^t=1,\end{aligned}$$ where the fourth inequality follows from our choice of $q$ and since $n$ is assumed to be sufficiently large. Thus, by Theorem \[th::ClientBES\], Client can ensure that, for every $\mathcal{D}\subseteq\mathcal{B}$, there exists some $B\in\mathcal{D}$ such that $|B\cap S(\mathcal{C}_C\left[\bigcup\mathcal{D}\right])|\leqslant 1$ at the end of the game. Consequently, there exists an ordering $B_{i_1},\ldots,B_{i_n}$ of the elements of $\mathcal{B}$ satisfying the following. For every $1\leqslant j< n$, there is a literal $v_{i_j}\in B_{i_j}$ such that every $k$–clause $c\in\mathcal{C}_C\left[\cup\{B_{i_k}:k\geqslant j\}\right]$ satisfies $S(c)\cap B_{i_j}=\{v_{i_j}\}$ or $S(c)\cap B_{i_j}=\emptyset$. Assigning the value, 0 or 1, to the variable $x_{i_j}$ such that $v_{i_j}=1$, for every $j\in [n]$, provides a satisfying truth assignment for $\bigwedge\mathcal{C}_C$. [$\Box$\ ]{} The Client–Waiter $k$–SAT game ------------------------------ **Proof of Theorem \[th::CWkSAT\]**. Fix $k\geqslant 2$. #### Client’s strategy: {#clients-strategy-3 .unnumbered} Let $q<\binom{n}{k}/n$. With $$\mathcal{F}=\{\mathcal{A}\subseteq \mathcal{C}^{(k)}_n: |S(\mathcal{A})|=n\text{ and }\bigvee B\in\mathcal{A},\forall B\subseteq S(\mathcal{A})\text{ with }|B|=k\},$$ observe that $$\begin{aligned} \sum_{\mathcal{A}\in\mathcal{F}}\left(\frac{q}{q+1}\right)^{|\mathcal{A}|}\leqslant\sum_{\mathcal{A}\in\mathcal{F}}2^{-|\mathcal{A}|/q}\leqslant 2^{n-\binom{n}{k}/q}<1,\end{aligned}$$ where the final inequality follows from our choice of $q$. Hence, by Theorem \[th::CwinsCWtrans\], Client can claim a $k$–clause in every $\mathcal{A}\in\mathcal{F}$ by the end of the game. Thus, for every partition $(V_1, V_2)$ of the literals in $S(\mathcal{C}^{(k)}_n)$, where no pair of complementary literals lie in the same part, Client owns two $k$–clauses, $c_1$ and $c_2$, satisfying $S(c_i)\subseteq V_i$ for every $i\in[2]$. This means that, regardless of how one assigns the values 0 and 1 to the boolean variables, Client will always have a $k$–clause consisting entirely of 0–valued literals, since every $\{0,1\}$–assignment defines such a partition of $S(\mathcal{C}^{(k)}_n)$. Hence, $\bigwedge\mathcal{C}_C$ is not satisfiable at the end of the game. #### Waiter’s strategy: {#waiters-strategy-3 .unnumbered} Let $q\geqslant 16k^3\binom{n}{k}/n$ and first consider when $k\geqslant 6$. By Lemma \[satcondition\], it suffices to show that Waiter can ensure no variable appears in more than $2^{k-2}/k$ $k$–clauses of $\mathcal{C}_C$ at the end of the game. Equivalently, this means that at most $2^{k-2}/k$ $k$–clauses of $\mathcal{C}_C$ are allowed to contain a literal in $B_i$, for every $i\in[n]$. Waiter plays as follows. In the first round, Waiter offers $q+1$ arbitrary free $k$–clauses. After this, Waiter responds to every $k$–clause claimed by Client in the following way. Suppose Client has just claimed the $k$–clause $c$ with $S(c)=\{v_1,\ldots,v_k\}$. Then, for each $i\in[k]$, there exists $j_i\in [n]$ such that $v_i\in B_{j_i}$. Let $\mathcal{D}_1\subseteq N_{\mathcal{C}_F}(B_{j_1})$ with size $|\mathcal{D}_1|=\min\{d_{\mathcal{C}_F}(B_{j_1}),\lfloor(q+1)/k\rfloor\}$, and for each $2\leqslant i\leqslant k$, let $\mathcal{D}_i \subseteq N_{\mathcal{C}_F}(B_{j_i})\setminus\cup\{\mathcal{D}_{\ell}:1\leqslant\ell<i\}$ with size $|\mathcal{D}_i|=\min\{|N_{\mathcal{C}_F}(B_{j_i})\setminus\cup\{\mathcal{D}_{\ell}:1\leqslant\ell<i\}|,\lfloor(q+1)/k\rfloor\}$. Waiter immediately offers all $k$–clauses in $\cup\{\mathcal{D}_i:i\in[k]\}$ (recall that Waiter may offer less than $q+1$ $k$–clauses in a Client–Waiter game). If $\cup\{\mathcal{D}_i:i\in[k]\}=\emptyset$, then Waiter performs the described response on a $k$–clause claimed by Client earlier in the game. If no free $k$–clause contains a literal of any $k$–clause previously claimed by Client, then Waiter simply offers $\min\{q+1,|\mathcal{C}_F|\}$ free $k$–clauses. It is clear that Waiter offers every $k$–clause of $\mathcal{C}^{(k)}_n$ at some point in the game if he plays in this way. We claim that the described strategy ensures that at most $2^{k-2}/k$ $k$–clauses of $\mathcal{C}_C$ contain a literal in $B_i$, for every $i\in[n]$, at the end of the game. Indeed, fix an arbitrary $i\in [n]$ and consider $B_i$. Each time Client claims a $k$–clause that contains a literal in $B_i$, Waiter offers at least $\lfloor(q+1)/k\rfloor$ free $k$–clauses that also contain a literal in $B_i$, except for perhaps the last time when there may be less than $\lfloor (q+1)/k\rfloor$ such free $k$–clauses. Each time Waiter offers these $k$–clauses, Client claims at most one of them. So at the end of the game, the number of $k$–clauses of $\mathcal{C}_C$ containing a literal of $B_i$ is at most $$\frac{\binom{n-1}{k-1}2^k}{\lfloor (q+1)/k\rfloor}+1\leqslant \frac{2^{k+1}k\binom{n-1}{k-1}}{q}+1\leqslant \frac{2^{k-2}}{k},$$ where the final inequality follows from our choice of $k$ and $q$. In the case where $2\leqslant k\leqslant 5$, Waiter still performs the above strategy, ensuring that the number of $k$–clauses of $\mathcal{C}_C$ containing a literal of $B_i$ is at most $$\frac{\binom{n-1}{k-1}2^k}{\lfloor (q+1)/k\rfloor}+1<\frac{\binom{n-1}{k-1}2^kk}{q+1-k}+1\leqslant 2,$$ at the end of the game, where the final inequality follows from $k\leqslant 5$, our choice of $q$ and for $n$ sufficiently large. Thus, each boolean variable appears in at most one $k$–clause of $\mathcal{C}_C$. It is clear therefore, that $\bigwedge\mathcal{C}_C$ is satisfiable at the end of the game. [$\Box$\ ]{} Concluding remarks and open problems {#sec::openprob} ==================================== In this paper, we proved that the threshold bias of the $(1:q)$ non–2–colourability game $(E(K_n^{(k)}),\mathcal{NC}_2)$ is $\frac{1}{n}\binom{n}{k}2^{\Theta_k(k)}$ and $\frac{1}{n}\binom{n}{k}2^{-k(1+o_k(1))}$ for the Waiter–Client and Client–Waiter versions respectively. In addition, we showed that the threshold bias for both the $(1:q)$ Waiter–Client and Client–Waiter versions of the $k$–SAT game $(C_n^{(k)},\mathcal{F}_{SAT})$ is $\frac{1}{n}\binom{n}{k}$ up to an exponential and polynomial factor in $k$ respectively. This shows that these games exhibit the probabilistic intuition. However, there is room to improve the bounds on the threshold bias for all four games, especially in the Waiter–Client versions where the gap is exponential in $k$. In particular, we believe that the threshold bias of these games is asymptotically equivalent to that predicted by the probabilistic intuition in the following sense. Let the threshold bias for the $(1:q)$ Waiter–Client and Client–Waiter non–2–colourability games $(E(K_n^{(k)}),\mathcal{NC}_2)$ be denoted by $b_{\mathcal{NC}_2}^{WC}$ and $b_{\mathcal{NC}_2}^{CW}$ respectively. Then $$\lim_{k\rightarrow\infty}\left\{\lim_{n\rightarrow\infty}\frac{1}{n}\binom{n}{k}\frac{(b_{\mathcal{NC}_2}^{WC})^{-1}}{2^{k-1}\ln 2-\ln 2/2}\right\}=\lim_{k\rightarrow\infty}\left\{\lim_{n\rightarrow\infty}\frac{1}{n}\binom{n}{k}\frac{(b_{\mathcal{NC}_2}^{CW})^{-1}}{2^{k-1}\ln 2-\ln 2/2}\right\}=1.$$ Let the threshold bias for the $(1:q)$ Waiter–Client and Client–Waiter $k$–SAT games $(\mathcal{C}_n^{(k)},\mathcal{F}_{SAT})$ be denoted by $b_{\mathcal{F}_{SAT}}^{WC}$ and $b_{\mathcal{F}_{SAT}}^{CW}$ respectively. Then $$\lim_{k\rightarrow\infty}\left\{\lim_{n\rightarrow\infty}\frac{1}{n}\binom{n}{k}\frac{(b_{\mathcal{F}_{SAT}}^{WC})^{-1}}{\ln 2}\right\}=\lim_{k\rightarrow\infty}\left\{\lim_{n\rightarrow\infty}\frac{1}{n}\binom{n}{k}\frac{(b_{\mathcal{F}_{SAT}}^{CW})^{-1}}{\ln 2}\right\}=1.$$ Such similarity between the threshold bias and the probabilistic intuition has been observed before in other Waiter–Client and Client–Waiter games. For example, the threshold bias for the Waiter–Client $K_t$–minor game, played on the edge set $E(K_n)$ of the complete graph $K_n$, matches the probabilistic intuition to this degree (see [@HKTpcm]). Thus, it would be interesting to see if the same is true for the games studied here. [^1]: School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom. Email: WET916@bham.ac.uk.

--- abstract: 'In this paper, we construct an infinite family of $\frac{q-1}{2}$-ovoids of the generalized quadrangle $Q(4,q)$, for $q\equiv 1 (\text{mod}\ 4)$ and $q>5$. Together with [@feng2016family] and [@bamberg2009tight], this establishes the existence of $\frac{q-1}{2}$-ovoids in $Q(4,q)$ for each odd prime power $q$.' address: 'School of Mathematical Sciences, Zhejiang University, 38 Zheda Road, Hangzhou 310027, Zhejiang P.R China' author: - Tao Feng - Ran Tao title: 'An infinite family of $m$-ovoids of $Q(4,q)$' --- $m$-ovoid ,Generalized quadrangle ,Ovoid ,Parabolic quadric 51E12 ,05B25 ,51E20 Introduction ============ A *generalized quadrangle* (GQ) of order $(s,t)$ is an incidence structure of points and lines with the properties that every two points are incident with at most one line, every point is incident with $t+1$ lines, every line is incident with $s+1$ points, and for any point $P$ and line $\ell$ that are not incident there is a unique point on $\ell$ collinear with $P$. The point-line dual of a GQ of order $(s,t)$ is a GQ of order $(t,s)$. In the case $s=t$, we say that the GQ has order $s$. We are only concerned with thick generalized quadrangles, i.e. those GQs of order $(s,t)$ with $s>1$ and $t>1$. The classical GQs are the point-line incidence structures arising from the finite classical polar spaces of rank 2. For more background on GQ, please refer to the monograph [@MR2508121]. In this paper, we are concerned with $m$-ovoids of the classical GQ $Q(4,q)$. The points and lines of $Q(4,q)$ are respectively the totally singular points and totally singular lines contained in a parabolic quadric of $\text{PG}(4,q)$. An ovoid of $Q(4,q)$ is a point set of $Q(4,q)$ that intersects each totally singular line (i.e. generator) in exactly one point. Ovoids of $Q(4,q)$ are of great importance in finite geometry. For instance, its point-line dual gives rise to spreads of $W(3,q)$, which further give rise to translation planes of order $q^2$ by the Bruck-Bose/André construction. The list of known ovoids in $Q(4,q)$ is very short. Please refer to [@penttila2000ovoids] for a summary of known ovoids of $Q(4,q)$. The concept of $m$-ovoid is first introduced by Thas [@thas1989interesting], as a generalization of ovoids. The notion of $m$-ovoids is now known as a special case of intriguing sets, the latter first introduced in [@bamberg2009tight] by Bamberg et al. for generalized quadrangles and later generalized to finite polar spaces in [@bamberg2007tight]. In short, an $m$-ovoid of $Q(4,q)$ is a point set of $Q(4,q)$ that intersects each totally singular line in exactly $m$ points. It is known that such point sets can give rise to strongly regular graphs and projective two-weight codes, cf. [@bamberg2009tight; @calderbank1986geometry]. We now make a summary of known constructions of $m$-ovoids in $Q(4,q)$. For $q$ even, the GQ $W(3,q)$ is isomorphic to $Q(4,q)$. Cossidente et al. [@cossidente2008m] have shown that $W(3,q)$ has $m$-ovoids for all integers $m$, $1\leq m\leq q$. Therefore, $Q(4,q)$ also has $m$-ovoids for all integers $m$, $1\leq m\leq q$. For $q$ odd, Cossidente and Penttila [@cossidente2005hemisystems] constructed a hemisystem of $H(3,q^2)$ for all odd $q$ and subsequently there have been more constructions of hemisystems in $H(3,q^2)$ cf. [@bamberg2018new; @Korchm2017hemisystem; @bamberg2010hemisystems]. By duality, this gives rise to a $\frac{q+1}{2}$-ovoid of $Q^-(5,q)$, whose intersection with a non-tangent hyperplane yields a $\frac{q+1}{2}$-ovoid in $Q(4,q)$. In [@feng2016family], the first author and collaborators constructed the first infinite family of $m$-ovoid of $Q(4,q)$ with $m=\frac{q-1}{2}$ for $q\equiv 3 (\text{mod}\ 4)$, which generalized some sporadic examples listed in [@bamberg2009tight]. The following table taken from [@bamberg2009tight] lists the known $m$-ovoids for small $q$. In the end of [@feng2016family], it is commented that “As for future research, it would be interesting to generalize the examples of $\frac{q-1}{2}$-ovoids of $Q(4,q)$ (with $q=5$ or 9) in Example 5 of [@bamberg2009tight] into an infinite family. We could not see any general pattern for the prescribed automorphism groups in those examples.” In this paper, we are able to construct an infinite family of $\frac{q-1}{2}$-ovoids of $Q(4,q)$ for $q\equiv 1 (\text{mod}\ 4)$ and $q>5$. The construction and proof technique in this paper is very similar to that in [@feng2016family]. The main difficulty is to choose the right prescribed automorphism group out of the rich subgroup structures of the group $\text{PGO}(5,q)$. Our family does not seem to generalize the sporadic examples listed in [@bamberg2009tight] by examining their automorphism groups. **$q$** **Known $m$** **Unknown $m$** --------- --------------- ----------------- 3 1,2,3 - 5 1,2,3,4,5 - 7 1,3,4,5,7 2,6 9 1,3,4,5,6,7,9 2,8 11 1,5,6,7,11 2,3,4,8,9,10 : m-ovoids of $Q(4,q)$[]{data-label="tab1"} The model and the prescribed group ================================== A model for $Q(4,q)$ -------------------- Let $q\equiv 1 (\text{mod}\ 4)$ be a prime power, and set $V=\mathbb{F}_{q}\times\mathbb{F}_{q}\times\mathbb{F}_{q^2}\times\mathbb{F}_{q}$. We regard $V$ as a 5-dimensional vector space over $\mathbb{F}_{q}$, and write its element in the form $(x,y,\alpha,z)$, where $x,y,z\in\mathbb{F}_{q}$ and $\alpha\in\mathbb{F}_{q^2}$. We define the quadratic form $Q$ on $V$ as follows $$Q((x,y,\alpha,z))=xy+\alpha^{q+1}+z^2.$$ The polar form $B$ of $Q$ is given by $$B((x,y,\alpha,z),(x',y',\alpha',z'))=xy'+x'y+\alpha\alpha'^{q}+\alpha^{q}\alpha'+2zz'.$$ It is easy to check that the quadratic form $Q$ defined as above is non-degenerate and the associated quadric is a parabolic quadric $Q(4,q)$. Therefore, we can define $Q(4,q)$ as $$Q(4,q)=\{\langle(x,y,\alpha,z)\rangle|0\neq(x,y,\alpha,z)\in V,\,Q((x,y,\alpha,z))=0\}.$$ In the remaining part of this paper, we will use $(x,y,\alpha,z)$ instead of $\langle(x,y,\alpha,z)\rangle$ to denote a projective point of $\text{PG}(4,q)$ for simplicity. We choose this model such that the prescribed automorphism group $G$ we introduce now has a good presentation. For each $\lambda\in\mathbb{F}_{q}^{*}$ and $\mu\in\mathbb{F}_{q^2}^{*}$ with $\mu^{\frac{q+1}{2}}=1$, we define $$T_{\lambda,\mu}:(x,y,\alpha,z)\mapsto(x\lambda,y\lambda^{-1},\alpha\lambda^{\frac{q-1}{2}}\mu,z),$$ which is an isometry of $Q(4,q)$. Let $H$ be the group generated by all such $T_{\lambda,\mu}$’s, i.e., $H=\langle T_{\lambda,\mu}:\lambda\in\mathbb{F}^{*}_{q},\mu\in\mathbb{F}_{q^2}^{*},\mu^{\frac{q+1}{2}}=1\rangle$. The group $H$ defined above is a cyclic subgroup of order $\frac{q^2-1}{2}$ of $\text{PGO}(5,q)$. \[lem1\] It is clear that $H$ is the direct product of the two cyclic subgroups $\langle T_{\lambda,1}:\lambda\in\mathbb{F}_{q}^{*}\rangle$ and $\langle T_{1,\mu}: \mu\in\mathbb{F}_{q^2}^{*},\mu^{\frac{q+1}{2}}=1\rangle$, which has order $q-1$ and $\frac{q+1}{2}$ respectively. Since $\text{gcd}(q-1,\frac{q+1}{2})=1$ for $q\equiv 1 (\text{mod}\ 4)$, the claim follows. Furthermore, we define an involution $\tau$ in $\text{PGO}(5,q)$ as follows: $$\tau:(x,y,\alpha,z)\mapsto(y,x,\alpha^{q},z),$$ which is also an isometry of $Q(4,q)$. Set $G:=\langle H,\tau\rangle$, which is isomorphic to $C_{\frac{q^2-1}{2}}\rtimes C_{2}$, where $C_{k}$ denotes a cyclic group of order $k$. This is the prescribed automorphism group for our putative $m$-ovoids in $Q(4,q)$. The $G$-orbits ============== We now describe the $G$-orbits of $Q(4,q)$ with $G$ as defined above. For a point $P\in \text{PG}(4,q)$, let $O(P)$ denote the $G$-orbit containing $P$. Let $\gamma$ be a fixed element of $\mathbb{F}_{q^2}$ with $\gamma^{q+1}=-1$, and let $\square_q$ (resp. $\square_{q^2}$) and $\blacksquare_q$ (resp. $\blacksquare_{q^2}$) be the set of nonzero squares of $\mathbb{F}_q$ (resp. $\mathbb{F}_{q^2}$) and nonsquares of $\mathbb{F}_q$ (resp. $\mathbb{F}_{q^2}$) respectively. All the $G$-orbits have length $\frac{q^2-1}{2}$ or $q^2-1$ with the following exceptions: 1. a unique orbit $O(1,0,0,0)=\{(1,0,0,0),(0,1,0,0)\}$ of length 2; 2. a unique orbit $O(1,-1,0,1)=C\setminus O(1,0,0,0)$ of length $q-1$ ,where $C=\{(x,y,0,z)\in V|xy+z^2=0, (x,y,0,z)\neq (0,0,0,0)\}$ is a conic; 3. a unique orbit $O(0,0,\gamma,1)$ of length $q+1$. We call the orbits of length $\frac{q^2-1}{2}$ *short* orbits, the orbits of length $q^2-1$ *long* orbits and the remaining orbits *exceptional*. Let $x,y\in\mathbb{F}^{*}_{q}$, $\alpha\in\mathbb{F}^{*}_{q^2}$, and $(x,y,\alpha,1)\in Q(4,q)$. Then the $G$-orbit $O(x,y,\alpha,1)$ is a short orbit if and only if $xy(1+xy)\in\square_q$, or equivalently $\alpha^{q+1}(1+\alpha^{q+1})\in\square_{q}$. \[lem2\] $O(x,y,\alpha,1)$ is a short orbit if and only if its stabilizer in $G$ has order 2 since the order of $G$ is $q^2-1$. If $T_{\lambda,\mu}$ stabilizes $(x,y,\alpha,1)$, then there exists a constant $c\in\mathbb{F}^{*}_{q}$ such that $(x\lambda,y\lambda^{-1},\alpha\lambda^{\frac{q-1}{2}}\mu,1)=c\cdot(x,y,\alpha,1)$. It follows that $c=1=\lambda=\lambda^{-1}=\lambda^{\frac{q-1}{2}}\mu$, which gives $\lambda=\mu=1$. If $T_{\lambda,\mu}\cdot\tau$ stabilizes $(x,y,\alpha,1)$, then there exists a constant $c'\in\mathbb{F}_{q}^{*}$ such that $(y\lambda,x\lambda^{-1},\alpha^{q}\lambda^{\frac{q-1}{2}}\mu,1)=c'\cdot(x,y,\alpha,1)$. We get $c'=1$, $\lambda=xy^{-1}$, $\mu=(yx^{-1})^{\frac{q-1}{2}}\alpha^{1-q}$, which indicates that the value of $\lambda,\mu$ is uniquely determined by $x,y,\alpha$ respectively. In order for $T_{\lambda,\mu}\cdot\tau$ to be in $G$, we need $\mu^{\frac{q+1}{2}}=1$, i.e. $(yx^{-1})^{\frac{q-1}{2}}\alpha^{\frac{1-q^2}{2}}=1$, which is equivalent to $xy\alpha^{q+1}$ is a square of $\mathbb{F}_{q}^{*}$. The claim now follows since $xy+\alpha^{q+1}+1=0$ and $-1\in\square_{q}$ for $q\equiv 1 (\text{mod}\ 4)$. Let $\omega$ be a fixed primitive element of $\mathbb{F}_{q^2}$ and let $\gamma$ be a fixed element of $\mathbb{F}_{q^2}$ with $\gamma^{q+1}=-1$ as introduced above. According to Lemma \[lem2\], we are now ready to give an explicit description of short and long orbits below. For the size of $S\cap (S'-1)$, where $S,S'\in\{\square_{q},\,\blacksquare_{q}\}$, please refer to Remark \[rem1\] in the next section. Here $S'-1=\{x-1|\,x\in S'\}$. There are a total of $q-1$ short orbits of length $\frac{q^2-1}{2}$, which we list below. 1. The point set $\{(x,y,\alpha,0)\in Q(4,q)|\,\alpha\neq 0\}$ splits into two orbits, $O(1,-\omega^{2(q+1)},\omega^2,0)$ and $O(1,-\omega^{q+1},\omega,0)$. Both orbits have size $\frac{q^2-1}{2}$. 2. For each $y\in\square_{q}\cap(\square_{q}-1)$, there are two orbits $O(1,y,\alpha,1)$ and $O(1,y,\alpha^{q},1)$ with $y+\alpha^{q+1}+1=0$. In total, there are $2\cdot|\square_{q}\cap(\square_{q}-1)|=\frac{q-5}{2}$ such orbits of length $\frac{q^2-1}{2}$. 3. For each $y\in\blacksquare_{q}\cap(\blacksquare_{q}-1)$, there are two orbits $O(1,y,\alpha,1)$ and $O(1,y,\alpha^{q},1)$ with $y+\alpha^{q+1}+1=0$. In total, there are $2\cdot|\blacksquare_{q}\cap(\blacksquare_{q}-1)|=\frac{q-1}{2}$ such orbits of length $\frac{q^2-1}{2}$. There are a total of $\frac{q+3}{2}$ long orbits of length $q^2-1$, which we list below. 1. There are two orbits of length $q^2-1$ with points whose second coordinate zero, namely, $O(1,0,\gamma,1)$ and $O(1,0,\gamma\omega^{q-1},1)$. 2. For each $y\in\square_{q}\cap(\blacksquare_{q}-1)$, there is exactly one orbit $O(1,y,\alpha,1)$ with $y+\alpha^{q+1}+1=0$. In total, there are $|\square_{q}\cap(\blacksquare_{q}-1)|=\frac{q-1}{4}$ such orbits of length $q^2-1$. 3. For each $y\in\blacksquare_q\cap(\square_q-1)$, there is exactly one orbit $O(1,y,\alpha,1)$ with $y+\alpha^{q+1}+1=0$. In total, there are $|\blacksquare_{q}\cap(\square_{q}-1)|=\frac{q-1}{4}$ such orbits of length $q^2-1$. In particular, there are 7 orbits in which a representative has a coordinate being zero: $O(1,0,0,0)$, $O(1,-1,0,1)$, $O(0,0,\omega^{\frac{q-1}{2}},1)$, $O(1,-\omega^{2(q+1)},\omega^2,0)$, $O(1,-\omega^{q+1},\omega,0)$, $O(1,0,\omega^{\frac{q-1}{2}},1)$ and $O(1,0,\omega^{\frac{3(q-1)}{2}},1)$. The construction of $\frac{q-1}{2}$-ovoids in $Q(4,q)$ ====================================================== We are now ready to describe the construction of $\frac{q-1}{2}$-ovoids in $Q(4,q)$. Let $\omega$ be a primitive element of $\mathbb{F}_{q^2}^{*}$ as introduced above. Fix a pair $(a,b)$ in $\mathbb{F}_{q}^{*}$ such that $$1+a^2=b^2. \label{1paseqba}$$ Now we define $$\begin{array}{rl} \mathcal{M}= &O(1,-1,0,1)\cup O(1,-\omega^{2(q+1)},\omega^2,0)\cup\\ &O(1,0,\omega^{\frac{q-1}{2}},1) \cup \mathcal{T}\cup O(1,-b^2,a,1). \label{eq:1} \end{array}$$ where $$\mathcal{T}=\{(x,y,\alpha,1)\in Q(4,q)\ |\ 1+b^{-2}xy\in\square_{q},\ xy\alpha\neq0\ \}. \label{mathcalT}$$ The set $\mathcal{T}$ is $G$-invariant with $\frac{(q^2-1)(q-5)}{2}$ points. \[lemt\] It is straightforward to check that $\mathcal{T}$ is $G$-invariant. It remains to compute the size of $\mathcal{T}$. We have $$\begin{aligned} |\mathcal{T}| &=(q-1)|\{z\in\mathbb{F}_{q}^{*},\,\alpha\in\mathbb{F}_{q^2}^{*}|\,z+1+\alpha^{q+1}=0,\,1+b^{-2}z\in\square_{q}\}|\\ &=(q^2-1)|\{z\in\mathbb{F}_{q}|\,1+b^{-2}z\in\square_{q},\,z\neq0,-1\}|\\ &=(q^2-1)(|\{z\in\mathbb{F}_{q}|\, 1+b^{-2}z\in \square_{q}\}|-2)\\ &=(q^2-1)\left(\frac{q-1}{2}-2\right)=\frac{(q^2-1)(q-5)}{2}.\end{aligned}$$ Here, we have made use of the fact that $1-b^{-2}=a^2b^{-2}$ is a square in the third equality and $|b^2(\square_{q}-1)|=\frac{q-1}{2}$ in the fourth equality. This proof is complete. By Lemma \[lemt\], we have $$|\mathcal{M}|=(q-1)+2(q^2-1)+\frac{(q^2-1)(q-5)}{2}=\frac{q-1}{2}(q^2+1),$$ which is exactly the size of a $\frac{q-1}{2}$-ovoid in $Q(4,q)$. Let $\eta$ be the quadratic (multiplicative) character of $\mathbb{F}_{q}$, i.e., $$\eta(x)=\left\{\begin{array}{lll}1 & \text{if\ } x\in\square_{q},\\-1 & \text{if\ } x\in\blacksquare_{q}, \\0 & \text{if\ } x=0. \end{array}\right. \label{defeta}$$ Furthermore, we define the Kronecker delta function $[[\mathcal{P}]]$ as follows $$[[\mathcal{P}]]=\left\{\begin{array}{ll}1, & \text{if the property\ }\mathcal{P} \text{\ holds},\\0, & \text{otherwise. } \end{array}\right. \label{Kronecker}$$ [@lidl1997finite Theorem 5.48] Let $g(x)=a_{2}x^{2}+a_{1}x+a_{0}\in\mathbb{F}_{q}[x]$ with $q$ odd and $a_{2}\neq0$. Set $d=a_{1}^{2}-4a_{0}a_{2}$ and let $\eta$ be the quadratic character of $\mathbb{F}_{q}$. Then $$\sum_{c\in\mathbb{F}_{q}}\eta(g(c))=\left\{\begin{array}{ll}-\eta(a_{2}) & \text{if\ } d\neq 0,\\(q-1)\eta(a_{2}) & \text{if\ } d=0. \end{array}\right.$$ \[lem6\] \[rem1\] Consider the special case $g(x)=x(x+1)$, i.e. $a_1=a_2=1$, $a_0=0$. We have $d=a_{1}^{2}-4a_{0}a_{2}=1$, so $\sum_{c\in\mathbb{F}_{q}}\eta(g(c))=-1$. Let $n_{1},\,n_{2},\,n_{3},\,n_{4}$ be the number of $x$ such that $(\eta(x),\eta(x+1))=$ $(1,1)$, $(1,-1)$, $(-1,1)$, $(-1,-1)$ respectively. Then $$\begin{aligned} &n_{1}+n_{2}=\frac{q-1}{2}-1,\ n_{3}+n_{4}=\frac{q-1}{2},\\ &n_{1}+n_{3}=\frac{q-1}{2}-1,\ n_{2}+n_{4}=\frac{q-1}{2}. \end{aligned}$$ From $\sum_{c\in\mathbb{F}_{q}}\eta(g(c))=-1$, we get $n_{1}-n_{2}-n_{3}+n_{4}=-1$. We solve from these equations that $$n_{1}=\frac{q-5}{4},\ n_{2}=\frac{q-1}{4},\ n_{3}=\frac{q-1}{4},\ n_{4}=\frac{q-1}{4}.$$ In particular, $n_{1}$ is the size of $\square_{q}\cap(\square_{q}-1)$, $n_{2}$ is the size of $\square_{q}\cap(\blacksquare_{q}-1)$, $n_{3}$ is the size of $\blacksquare_{q}\cap(\square_{q}-1)$, and $n_{4}$ is the size of $\blacksquare_{q}\cap(\blacksquare_{q}-1)$ by definition. The point set $\mathcal{M}$ in (\[eq:1\]) is a $\frac{q-1}{2}$-ovoid of $Q(4,q)$ for $q\equiv 1 (\text{mod}\ 4)$ and $q>5$. We take the same technique as in [@feng2016family], i.e., we show that each line of $Q(4,q)$ meets $\mathcal{M}$ in $\frac{q-1}{2}$ points. Each line of $Q(4,q)$ intersects the hyperplane $\{(x,y,\alpha,z)\in V|\;y=0\}$ in at least one point. There are four $G$-orbits of $Q(4,q)$ that have a representative with second coordinate zero, namely, $O(1,0,0,0)$, $O(0,0,\omega^{\frac{q-1}{2}},1)$, $O(1,0,\omega^{\frac{q-1}{2}},1)$ and $O(1,0,\omega^{\frac{3(q-1)}{2}},1)$. Since $\mathcal{M}$ is $G$-invariant, we only need to consider the lines through $(1,0,0,0)$, $(0,0,\omega^{\frac{q-1}{2}},1)$, $(1,0,\omega^{\frac{q-1}{2}},1)$ or $(1,0,\omega^{\frac{3(q-1)}{2}},1)$. By the assumption of $q>5$ and Lemma \[lemt\], the set $\mathcal{T}$ is non-empty.\ **Case 1.** The line $\ell$ of $Q(4,q)$ passes through $P=(1,0,0,0)$. The line $\ell$ intersects the hyperplane $\{(x,y,\alpha,z)\in V:\,x=0\}$ in exactly one point $Q$, perpendicular to $P$, with $Q=(0,y_{1},\alpha_{1},z_{1})$. Since $B(P,Q)=0$, we have $y_{1}=0$ and $\alpha_{1}^{q+1}+z_{1}^2=0$, $z_{1}\neq0$ by the definition of $B$ and $Q(4,q)$. So we can set $z_{1}=1$. Therefore, $\ell=\langle P,Q\rangle$ with $Q=(0,0,\alpha_{1},1)$ for some $\alpha_{1}\in\mathbb{F}_{q^2}^{*}$ with $\alpha_{1}^{q+1}+1=0$. The line $\ell$ can be denoted as $\ell=\{(t,0,\alpha_{1},1)|t\in\mathbb{F}_{q}\}\cup\{(1,0,0,0)\}$. It is clear that $P=(1,0,0,0)\notin\mathcal{M}$ in this case. For any $(x,y,\alpha,z)\in$ $O(1,-1,0,1)$, $O(1,-\omega^{2(q+1)},\omega^2,0)$, $\mathcal{T}$ or $O(1,-b^2,a,1)$, we have $y\neq 0$, so it can not lie on the line $\ell$. Therefore, $|\ell\cap\mathcal{M}|=|\ell\cap O(1,0,\omega^{\frac{q-1}{2}},1)|$. We now show that this size is $\frac{q-1}{2}$. The orbit $O(1,0,\omega^{\frac{q-1}{2}},1)$ is a long orbit with $q^2-1$ elements. It is the union of $$U_{1}=\{(\lambda,0,\lambda^{\frac{q-1}{2}}\mu\omega^{\frac{q-1}{2}},1)|\lambda\in\mathbb{F}_{q}^{*},\mu\in\mathbb{F}_{q^2}^{*},\mu^{\frac{q+1}{2}}=1\}, \label{eq:3}$$ and $$U_{2}=\{(0,\lambda,-\lambda^{\frac{q-1}{2}}\mu^{-1}\omega^{-\frac{q-1}{2}},1)|\lambda\in\mathbb{F}_{q}^{*},\mu\in\mathbb{F}_{q^2}^{*},\mu^{\frac{q+1}{2}}=1\}. \label{eq:4}$$ It is clear that $U_{2}\cap\ell=\emptyset$ by examining the second coordinate. Suppose that the point $(t,0,\alpha_{1},1)$ of $\ell$ lies in $U_{1}$, where $t\in\mathbb{F}_{q}$. Then there exists $\lambda\in\mathbb{F}_{q}^{*}$, $\mu\in\mathbb{F}_{q^2}^{*}$ with $\mu^{\frac{q+1}{2}}=1$ such that $$(t,0,\alpha_{1},1)=c\cdot(\lambda,0,\lambda^{\frac{q-1}{2}}\mu\omega^{\frac{q-1}{2}},1)$$ for some $c\in\mathbb{F}_{q}^{*}$. The last coordinate gives that $c=1$ and then the first coordinate gives that $\lambda=t$, which means $t\neq0$. By comparing third coordinate, we get $$\mu=\alpha_{1}t^{-\frac{q-1}{2}}\omega^{-\frac{q-1}{2}}$$ and $$\mu^{\frac{q+1}{2}}=\alpha_{1}^{\frac{q+1}{2}}t^{-\frac{q-1}{2}}\omega^{-\frac{q^2-1}{4}}=1.$$ Therefore, $$\begin{aligned} |\ell\cap U_{1}| &=\#\{t\in\mathbb{F}_{q}^{*}|\,t^{\frac{q-1}{2}}=\alpha_{1}^{\frac{q+1}{2}}\omega^{-\frac{q^2-1}{4}}\}\\ &=\frac{q-1}{2}.\end{aligned}$$ The last equality holds since $$\left(\alpha_{1}^{\frac{q+1}{2}}\omega^{-\frac{q^2-1}{4}}\right)^{2}=(\alpha_{1}^{q+1})(\omega^{-\frac{q^2-1}{2}})=1$$ by $\alpha_{1}^{q+1}+1=0$ and then $\alpha_{1}^{\frac{q+1}{2}}\omega^{-\frac{q^2-1}{4}}\in\{1,\,-1\}$.\ **Case 2.** The line $\ell$ of $Q(4,q)$ passes through $P=(0,0,\omega^{\frac{q-1}{2}},1)$. The line $\ell$ intersects the hyperplane $\{(x,y,\alpha,z)\in V|\,z=0\}$ in exactly one point $Q=(x_{1},y_{1},\alpha_{1},0)$, where $x_{1}y_{1}+\alpha_{1}^{q+1}=0$. If $\alpha_{1}=0$, then $x_{1}y_{1}=0$. It means $Q=(1,0,0,0)$ or $(0,1,0,0)$, which lies in the same $G$-orbit, and we are reduced to Case 1. We assume that $\alpha_{1}\neq0$ below. From $P\perp Q$, we get $\text{Tr}_{\mathbb{F}_{q^2}/\mathbb{F}_{q}}(\alpha_{1}\omega^{\frac{q^2-q}{2}})=0$, i.e. $\alpha_{1}=\omega^{-1}a_{1}$ for some $a_{1}\in\mathbb{F}_{q}^{*}$. We can assume that the point $Q$ equals $(1,y_{1},\alpha_{1},0)$ with $y_{1}+\alpha_{1}^{q+1}=0$ since $x_{1}y_{1}\neq0$ and the line $$\ell=\{(1,y_{1},\alpha_{1}+t\omega^{\frac{q-1}{2}},t)|t\in\mathbb{F}_{q}\}\cup\{(0,0,\omega^{\frac{q-1}{2}},1)\}.$$ Obviously, the point $P=(0,0,\omega^{\frac{q-1}{2}},1)\notin\mathcal{M}$ in this case. It is clear that $\alpha_{1}=\omega^{-1}a_{1}$ is a nonsquare of $\mathbb{F}_{q^2}^{*}$. Below we calculate the intersection number of $\ell$ with each part of $\mathcal{M}$ respectively. 1. $|\ell\cap O(1,-1,0,1)|=0$.\ Suppose that the point $(1,y_{1},\alpha_{1}+t\omega^{\frac{q-1}{2}},t)$ of $\ell$ lies in $O(1,-1,0,1)$, where $t\in\mathbb{F}_{q}$. Then its third coordinate must be 0, i.e., $\alpha_{1}+t\omega^{\frac{q-1}{2}}=0$. We have shown that $\alpha_{1}$ is a nonsquare of $\mathbb{F}_{q^2}^{*}$ above. On the other hand, $-\omega^{\frac{q-1}{2}}t$ is a square or 0 of $\mathbb{F}_{q}$ since $q\equiv 1(\text{mod}\ 4)$. This contradiction completes the proof. 2. $|\ell\cap O(1,-\omega^{2(q+1)},\omega^2,0)|=0$.\ Each element in this intersection has a zero last coordinate and the only element in $\ell$ with this property is $Q$. On the other hand, $Q$ is not in $O(1,-\omega^{2(q+1)},\omega^2,0)=\{(\lambda,-\omega^{2(q+1)}\lambda^{-1},\omega^2\lambda^{\frac{q-1}{2}}\mu,0)|\lambda\in\mathbb{F}_{q}^{*},\mu\in\mathbb{F}_{q^2}^{*},\mu^{\frac{q+1}{2}}=1\}$ since otherwise there exists $\lambda\in\mathbb{F}_{q}^{*}$ such that $y_{1}=-\omega^{2(q+1)}\lambda^{-2}$ by comparing the second coordinate, and we get a contradiction since $y_{1}=-\alpha_{1}^{q+1}$ is a nonsquare of $\mathbb{F}_{q}$. 3. $|\ell\cap O(1,0,\omega^{\frac{q-1}{2}},1)|=0$.\ Each element in this intersection has a zero second coordinate and the only element in $\ell$ with this property is $P$ since $y_{1}\neq0$. However, $P=(0,0,\omega^{\frac{q-1}{2}},1)\notin O(1,0,\omega^{\frac{q-1}{2}},1)$ because $O(0,0,\omega^{\frac{q-1}{2}},1)$ and $O(1,0,\omega^{\frac{q-1}{2}},1)$ are different $G$-orbits. 4. $|\ell\cap\mathcal{T}|=\frac{q-1}{2}$.\ Suppose that the point $(1,y_{1},\alpha_{1}+t\omega^{\frac{q-1}{2}},t)$ of $\ell$ lies in $\mathcal{T}$, where $t\in\mathbb{F}_{q}$. Then there exists $(x,y,\alpha,1)\in\mathcal{T}$ such that $$(1,y_1,\alpha_{1}+t\omega^{\frac{q-1}{2}},t)=c\cdot(x,y,\alpha,1)$$ for some $c\in\mathbb{F}_{q}^{*}$. The last coordinate gives that $c=t$. In particular, $t\neq0$. By comparing the other coordinates, we get $$1=tx,\,y_{1}=ty,\,\alpha_{1}+t\omega^{\frac{q-1}{2}}=t\alpha.$$ It follows that $x=t^{-1}$, $y=t^{-1}y_{1}$ and $\alpha=t^{-1}\alpha_{1}+\omega^{\frac{q-1}{2}}$. Therefore, $$\begin{aligned} |\ell\cap\mathcal{T}|=\#\{t\in\mathbb{F}_{q}^{*}|1+(tb)^{-2}y_{1}\in\square_{q}\} \end{aligned}$$ by the definition of $\mathcal{T}$ in Eqn. (\[mathcalT\]). As we mentioned above, $y_{1}$ is a nonsquare in $\mathbb{F}_{q}^{*}$. Hence $$\begin{aligned} |\ell\cap\mathcal{T}|&=2\cdot|b^{-2}y_{1}\square_{q}\cap (\square_{q}-1)|\\ &=2\cdot|\blacksquare_{q}\cap(\square_{q}-1)|=\frac{q-1}{2}. \end{aligned}$$ 5. $|\ell\cap O(1,-b^{2},a,1)|=0$.\ The orbit $O(1,-b^{2},a,1)$ is a short orbit with $$O(1,-b^{2},a,1)=\{(\lambda,-b^2\lambda^{-1},a\lambda^{\frac{q-1}{2}}\mu,1):\lambda\in\mathbb{F}_{q}^{*},\mu\in\mathbb{F}_{q^2}^{*},\mu^{\frac{q+1}{2}}=1\}. \label{express:Oba}$$ Suppose that the point $(1,y_1,\alpha_{1}+t\omega^{\frac{q-1}{2}},t)$ of $\ell$ lies in $O(1,-b^{2},a,1)$, where $t\in\mathbb{F}_{q}$. Then by comparing the coordinates, we get $y_{1}=-t^2b^2$, a contradiction to the fact that $y_{1}$ is a nonsquare of $\mathbb{F}_{q}$. To sum up, we get $|\ell\cap\mathcal{M}|=\frac{q-1}{2}$. This completes the proof for Case 2.\ **Case 3.** The line $\ell$ of $Q(4,q)$ passes through $P=(1,0,\omega^{\frac{q-1}{2}},1)$. Similar to Case 2, we only need to consider the case that $\ell$ passes through a point $Q=(1,-\alpha_{1}^{q+1},\alpha_{1},0)$ for some $\alpha_{1}\in\mathbb{F}_{q^2}^{*}$. From $P\perp Q$, we deduce that $-\alpha_{1}^{q+1}+\text{Tr}_{\mathbb{F}_{q^2}/\mathbb{F}_{q}}(\alpha_{1}\omega^{\frac{q^2-q}{2}})=0$. Set $$y_{1}=-\alpha_{1}^{q+1},\quad \beta=\alpha_{1}\omega^{\frac{-(q-1)}{2}}.$$ Then $$\beta^{q+1}=(\alpha_{1}\omega^{\frac{-(q-1)}{2}})^{q+1}=-\alpha_{1}^{q+1}=y_{1}, \label{y1eqbetaqp1}$$ and $$\text{Tr}_{\mathbb{F}_{q^2}/\mathbb{F}_{q}}(\beta)=-\text{Tr}_{\mathbb{F}_{q^2}/\mathbb{F}_{q}}(\alpha_{1}\omega^{\frac{q^2-q}{2}})=-\alpha_{1}^{q+1}=y_{1}. \label{y1eqTrbeta}$$ In particular, $\beta^{q+1}=\beta+\beta^{q}$ with $\beta\in\mathbb{F}_{q^2}^{*}$, which is equivalent to $$\beta^{q}=\beta^{q-1}+1 \label{betaqeqbetaqm1p1}$$ and $$\beta-1=\beta^{-(q-1)}. \label{betam1eqbetaqm1}$$ We claim that $\beta\in\mathbb{F}_{q}$ if and only if $\beta=2$ : if $\beta\in\mathbb{F}_{q}$, then $\beta=\beta^{q}=\beta^{q-1}+1=2$ by Eqn. (\[betaqeqbetaqm1p1\]); the converse is obvious. The proof for the case $\beta=2$ and $\beta\neq 2$ are basically the same and the former is easier, so we will only prove the case $\beta\neq2$ below. In this case, we know that the minimal polynomial of $\beta$ over $\mathbb{F}_{q}$ is $X^{2}-y_{1}X+y_{1}$ by Eqn. (\[y1eqbetaqp1\]) and Eqn. (\[y1eqTrbeta\]), and its discriminant $d'=y_{1}(y_{1}-4)$ is a nonsquare of $\mathbb{F}_{q}$ since $\beta\notin\mathbb{F}_{q}$. With all the preparations, we are now ready to compute the intersection size of $\ell$ with each part of $\mathcal{M}$. We have $$\ell=\{(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)|t\in\mathbb{F}_{q}\}\cup\{(1,0,\omega^{\frac{q-1}{2}},1)\}$$ with $y_1\neq 4$ (i.e. $\beta\neq 2$) and $\beta=\alpha_{1}\omega^{\frac{-(q-1)}{2}}$. It is clear that $P=(1,0,\omega^{\frac{q-1}{2}},1)\in O(1,0,\omega^{\frac{q-1}{2}},1)\subset\mathcal{M}$ in this case. 1. $|\ell\cap O(1,-1,0,1)|=0$.\ Suppose that the point $(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)$ of $\ell$ lies in $O(1,-1,0,1)$, where $t\in\mathbb{F}_{q}$. Comparing the third coordinate gives that $\beta=-t\in\mathbb{F}_{q}$, i.e., $y_1=4$. This is a contradiction. 2. $|\ell\cap O(1,-\omega^{2(q+1)},\omega^2,0)|=[[y_{1}\in\square_{q}]]$.\ Suppose that the point $(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)$ of $\ell$ lies in the G-orbit $O(1,-\omega^{2(q+1)},\omega^2,0)$. The last coordinate must be zero, then $t=0$. We have the point $(1,y_{1},\beta\omega^{\frac{q-1}{2}},0)\in\ell\cap O(1,-\omega^{2(q+1)},\omega^2,0)$ if and only if $y_{1}$ is a nonzero square of $\mathbb{F}_{q}$ by comparing the coordinates. Therefore, this claim follows by the definition of the Kronecker delta function in Eqn. (\[Kronecker\]). 3. $|\ell\cap O(1,0,\omega^{\frac{q-1}{2}},1)|=1$.\ On one hand, $P$ is a common point of $\ell$ and $O(1,0,\omega^{\frac{q-1}{2}},1)$. On the other hand, suppose that there is a point $(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)$ of $\ell$ that lies in $O(1,0,\omega^{\frac{q-1}{2}},1)$ with $y_{1}\neq 0$. Since $y_{1}\neq 0$, this point must be in $U_{2}$ by Eqn. (\[eq:3\]) and Eqn. (\[eq:4\]). So we obtain that $t=-1$. There exist $\lambda\in\mathbb{F}_{q}^{*}$, $\mu\in\mathbb{F}_{q^2}^{*}$ with $\mu^{\frac{q+1}{2}}=1$ such that $$(0,y_{1},(\beta-1)\omega^{\frac{q-1}{2}},-1)=c\cdot(0,\lambda,-\lambda^{\frac{q-1}{2}}\mu^{-1}\omega^{-\frac{q-1}{2}},1)$$ for some $c\in\mathbb{F}_{q}^{*}$. It follows that $c=-1$, $\lambda=-y_{1}$, $$\mu=\lambda^{\frac{q-1}{2}}(\beta-1)^{-1}\omega^{-(q-1)},$$ and $$\mu^{\frac{q+1}{2}}=-\lambda^{\frac{q-1}{2}}(\beta-1)^{-\frac{q+1}{2}}=1.$$ Then we get $$y_{1}^{\frac{q-1}{2}}=(-\lambda)^{\frac{q-1}{2}}=-(\beta-1)^{\frac{q+1}{2}}=-(\beta^q-1)^{\frac{q^2+q}{2}}. \label{y1betaanre}$$ We deduce that $y_{1}^{\frac{q-1}{2}}=-\beta^{\frac{q^2-1}{2}}$ by Eqn. (\[betaqeqbetaqm1p1\]) and Eqn. (\[y1betaanre\]). On the other hand, $y_{1}=\beta^{q+1}$ by Eqn. (\[y1eqbetaqp1\]), which gives $y_{1}^{\frac{q-1}{2}}=\beta^{\frac{q^2-1}{2}}$. This is a contradiction. 4. $|\ell\cap\mathcal{T}|=\frac{q-3}{2}-\frac{1}{2}(\eta(y_{1}(4b^2-y_{1}))+1)-[[y_{1}\in\square_{q}]]$.\ Suppose that the point $(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)$ of $\ell$ lies in $\mathcal{T}$, where $t\in\mathbb{F}_{q}$. Then there exists $(x,y,\alpha,1)\in\mathcal{T}$ such that $$(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)=c\cdot(x,y,\alpha,1)$$ for some $c\in\mathbb{F}_{q}^{*}$. The last coordinate gives that $c=t$. In particular, $t\neq0$ and $t\neq-1$ since $t+1=tx$, $tx\neq0$ and $t=c$. By comparing the other coordinates, we get $$t+1=tx,\,y_{1}=ty,\,(\beta+t)\omega^{\frac{q-1}{2}}=t\alpha.$$ It follows that $x=1+t^{-1}$, $y=t^{-1}y_{1}$ and $\alpha=t^{-1}(\beta+t)\omega^{\frac{q-1}{2}}$. Therefore, $$\begin{aligned} |\ell\cap\mathcal{T}|&=\#\{t\in\mathbb{F}_{q}|\,1+b^{-2}(1+t^{-1})t^{-1}y_{1}\in\square_{q},\,t\neq 0,-1\}\\ &=\#\{t\in\mathbb{F}_{q}|\,b^{2}t^2+y_{1}t+y_{1}\in\square_{q},\,t\neq 0,-1\}. \end{aligned}$$ Set $g(X)=b^2X^2+y_{1}X+y_{1}$. We have $g(0)=y_{1}$ and $g(-1)=b^2\in\square_{q}$. Let $\eta$ be the quadratic character of $\mathbb{F}_{q}$ as introduced in Eqn. (\[defeta\]). The discriminant of $g(X)$ is nonzero, i.e., $y_{1}^{2}-4b^2y_{1}\neq0$. Otherwise, $y_{1}=4b^2$ and $d'=y_{1}(y_{1}-4)=4b^2(4(b^2-1))=16b^2a^2\in\square_{q}$, a contradiction. Hence $g(X)=0$ has 2 or 0 solutions in $\mathbb{F}_{q}$ according as the discriminant $y_{1}^{2}-4b^2y_{1}$ is a square or not. In other words, the number of solutions to $g(X)=0$ in $\mathbb{F}_{q}$ equals $\eta(y_{1}(4b^2-y_{1}))+1$. Then we obtain that $$\begin{aligned} &|\ell\cap\mathcal{T}|=\frac{1}{2}\sum_{t\in\mathbb{F}_{q}, b^2t^2+y_{1}t+y_{1}\neq0}\left(\eta(b^2t^2+y_{1}t+y_{1})+1\right)-[[y_{1}\in\square_{q}]]-1\\ &=\frac{1}{2}\sum_{t\in\mathbb{F}_{q}}\left(\eta(b^2t^2+y_{1}t+y_{1})+1\right)-\frac{1}{2}|\{t\in\mathbb{F}_{q}|\,b^2t^2+y_{1}t+y_{1}=0\}|-[[y_{1}\in\square_{q}]]-1\\ &=\frac{q-2}{2}+\frac{1}{2}\sum_{t\in\mathbb{F}_{q}} \eta(b^2t^2+y_{1}t+y_{1})-\frac{1}{2}|\{t\in\mathbb{F}_{q}|\,b^2t^2+y_{1}t+y_{1}=0\}|-[[y_{1}\in\square_{q}]]\\ &=\frac{q-2}{2}+\frac{1}{2}(-\eta(b^2))-\frac{1}{2}|\{t\in\mathbb{F}_{q}|\,b^2t^2+y_{1}t+y_{1}=0\}|-[[y_{1}\in\square_{q}]]\\ &=\frac{q-3}{2}-\frac{1}{2}\left(\eta(y_{1}(4b^2-y_{1}))+1\right)-[[y_{1}\in\square_{q}]]. \end{aligned}$$ We get the fourth equality by $y_{1}^2-4b^2y_{1}\neq0$ and Lemma \[lem6\]. 5. $|\ell\cap O(1,-b^{2},a,1)|=\frac{1}{2}\left(\eta(y_{1}(4b^2-y_{1}))+1\right)$.\ In the case of $\eta(y_{1}(4b^2-y_{1}))=-1$, i.e., $y_{1}(4b^2-y_{1})\in\blacksquare_{q}$, we show that $|\ell\cap O(1,-b^2,a,1)|=0$. Recall that the elements of $O(1,-b^{2},a,1)$ are listed in Eqn. (\[express:Oba\]). Suppose that the point $(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)$ of $\ell$ lies in $O(1,-b^2,a,1)$, then there exists $\lambda\in\mathbb{F}_{q}^{*}$, $\mu\in\mathbb{F}_{q^2}^{*}$ with $\mu^{\frac{q+1}{2}}=1$ such that $$(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)=c\cdot(\lambda,-b^2\lambda^{-1},a\lambda^{\frac{q-1}{2}}\mu,1)$$ for some $c\in\mathbb{F}_{q}^{*}$. By comparing coordinates, we get $$c=t,\,t+1=c\lambda,\,y_{1}=-cb^2\lambda^{-1},\,(\beta+t)\omega^{\frac{q-1}{2}}=ca\lambda^{\frac{q-1}{2}}\mu.$$ In particular, $t\neq0,-1$ since $c\neq0$ and $\lambda\neq0$. It follows that $\lambda=1+t^{-1}$. Plugging into the third equation above, we get $$y_{1}\lambda+cb^2=y_{1}(1+t^{-1})+tb^2=0,\ \textup{i.e.},\ b^2t^2+y_{1}t+y_{1}=0.$$ It means that $b^2X^2+y_{1}X+y_{1}=0$ has a solution in $\mathbb{F}_{q}$. However, the equation $b^2X^2+y_{1}X+y_{1}=0$ has discriminant $y_{1}(4b^2-y_{1})\in\blacksquare_{q}$ by assumption, so has no solution in $\mathbb{F}_{q}$, a contradiction. Next, we consider the case $\eta(y_{1}(4b^2-y_{1}))=1$, i.e., $y_{1}(4b^2-y_{1})\in\square_{q}$, and we show that $|\ell\cap O(1,-b^2,a,1)|=1$. Suppose that the point $(t+1,y_1,(\beta+t)\omega^{\frac{q-1}{2}},t)$ of $\ell$ lies in $O(1,-b^2,a,1)$. By the same argument, we have $t\neq0,-1$ and there exists $\lambda\in\mathbb{F}_{q}^{*}$, $\mu\in\mathbb{F}_{q^2}^{*}$ with $\mu^{\frac{q+1}{2}}=1$ such that $$\lambda=t^{-1}+1,\,b^2t^2+y_{1}t+y_{1}=0,$$ and $$\mu=(\beta+t)\omega^{\frac{q-1}{2}}(at(t^{-1}+1)^{\frac{q-1}{2}})^{-1}.$$ In this case, $b^2X^2+y_{1}X+y_{1}=0$ has two distinct solutions $t_{1}$, $t_{2}$ in $\mathbb{F}_{q}$ such that $$\left\{\begin{array}{l} t_{1}+t_{2}=-y_{1}b^{-2},\\ t_{1}t_{2}=y_{1}b^{-2}. \end{array}\right. \label{eqt1t2}$$ We deduce that $$t_1^{-1}+t_{2}^{-1}=\frac{t_{1}+t_{2}}{t_{1}t_{2}}=-1$$ from (\[eqt1t2\]) and $$(t_{1}^{-1}+1)(t_{2}^{-1}+1)=(t_{1}t_{2})^{-1}+(t_{1}^{-1}+t_{2}^{-1})+1=b^2y_{1}^{-1}. \label{t1t2inv}$$ Let $\lambda_{i},\,\mu_{i}$ be the corresponding value of $\lambda,\mu$ when $t=t_{i}$, $i=1,2$. We now show that exactly one of the $\mu_{i}$’s satisfies that $\mu_{i}^{\frac{q+1}{2}}=1$ and the claim follows. We compute that $$\begin{aligned} \mu_{i}^{q+1}&=\omega^{\frac{q^2-1}{2}}(\beta^q+t_{i})(\beta+t_{i})(a^2t_{i}^2)^{-1}(t_{i}^{-1}+1)^{-(q-1)}\\ &=-(\beta^{q+1}+(\beta^q+\beta)t_{i}+t_{i}^2)(a^2t_{i}^2)^{-1}\\ &=-(y_{1}+t_{i}y_{1}+t_{i}^2)(a^2t_{i}^2)^{-1}\\ &=-(-(b^2-1)t_{i}^2)(a^2t_{i}^2)^{-1}=1.\end{aligned}$$ for $i=1,\,2$. Here we have made use of Eqn. (\[y1eqbetaqp1\]), Eqn. (\[y1eqTrbeta\]), Eqn. (\[1paseqba\]) and the fact that $a,\,t_{i}\in\mathbb{F}_{q}$, $b^2t_{i}^2+y_{1}t_{i}+y_{1}=0$ for $i=1,\,2$. Hence, we have $\mu_{i}^{\frac{q+1}{2}}\in\{1,\,-1\}$ for $i\in\{1,\,2\}$. We next compute that $$\begin{aligned} \mu_{1}^{\frac{q+1}{2}}\mu_{2}^{\frac{q+1}{2}}&=\left(\frac{(\beta+t_{1})\omega^{\frac{q-1}{2}}}{at_{1}(t_{1}^{-1}+1)^{\frac{q-1}{2}}}\cdot\frac{(\beta+t_{2})\omega^{\frac{q-1}{2}}}{at_{2}(t_{2}^{-1}+1)^{\frac{q-1}{2}}}\right)^{\frac{q+1}{2}}\\ &=\omega^{\frac{q^2-1}{2}}((t_{1}^{-1}+1)(t_{2}^{-1}+1))^{-\frac{q-1}{2}}\left(\frac{\beta^2+(t_{1}+t_{2})\beta+t_{1}t_{2}}{a^2t_{1}t_{2}}\right)^{\frac{q+1}{2}}\\ &=-y_{1}^{\frac{q-1}{2}}\left[(\beta^2-y_{1}b^{-2}(\beta-1))\left({a^2y_{1}b^{-2}}\right)^{-1}\right]^{\frac{q+1}{2}}\\ &=-y_{1}^{\frac{q-1}{2}}\left[(\beta^2-y_{1}b^{-2}\beta^{-(q-1)})(a^2y_{1}b^{-2})^{-1}\right]^{\frac{q+1}{2}}\\ &=-y_{1}^{\frac{q-1}{2}}\left((b^2-1)\beta^2(a^{2}y_{1})^{-1}\right)^{\frac{q+1}{2}}\\ &=-y_{1}^{\frac{q-1}{2}}\cdot y_{1}\cdot y_{1}^{-\frac{q+1}{2}}=-1.\end{aligned}$$ Here, we have made use of Eqn. (\[t1t2inv\]), Eqn. (\[eqt1t2\]), Eqn. (\[betam1eqbetaqm1\]) and Eqn. (\[1paseqba\]). Therefore, there is exactly one $\mu_{i}$ such that $\mu_{i}^{\frac{q+1}{2}}=1$ for $i\in\{1,2\}$. To sum up, we deduce that $|\ell\cap\mathcal{M}|=\frac{q-1}{2}$, and the proof for Case 3 is complete.\ **Case 4.** The line $\ell$ of $Q(4,q)$ passes through $P=(1,0,\omega^{\frac{3(q-1)}{2}},1)$. This case is almost the same as Case 3 and we omit the details.\ To sum up, each line of $Q(4,q)$ intersects $\mathcal{M}$ in $\frac{q-1}{2}$ points. Thus, $\mathcal{M}$ is a $\frac{q-1}{2}$-ovoid of $Q(4,q)$. This completes the proof. We define an isometry of order 2 of $Q(4,q)$ as follows: $$\sigma: (x,y,\alpha,z)\rightarrow (y,x,\alpha,z)$$ Then $\sigma$ stabilizes our $\frac{q-1}{2}$-ovoid $\mathcal{M}$ by direct check, and $\langle\sigma\rangle$ normalizes $G$. For $q=9,\,13,\,17$, we have checked with Magma [@cannon2006handbook] that $\langle G,\sigma\rangle$, which is isomorphic to $C_{\frac{q^2-1}{2}}\rtimes(C_{2}\times C_{2})$, is the full stabilizer of $\mathcal{M}$ in $\text{PGO}(5,q)$. Concluding remarks ================== In this paper, we have constructed $\frac{q-1}{2}$-ovoids in $Q(4,q)$ for $q\equiv 1 (\text{mod}\ 4)$, $q>5$. Together with the results in [@feng2016family] and [@bamberg2009tight], this shows that $\frac{q-1}{2}$-ovoids exist in $Q(4,q)$ for all odd prime power $q$. Our technique is similar to that in [@feng2016family] and our main contribution is that we find the correct prescribed automorphism group to make the technique in [@feng2016family] applicable in our case. This was a challenge due to the rich subgroup structure of $\text{PGO}(5,q)$. The determination of the spectrum of $m$-ovoids in $Q(4,q)$, i.e., to determine for which $m$ there is an $m$-ovoid in $Q(4,q)$, seems out of reach for the moment. Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported by National Natural Science Foundation of China under Grant No. 11771392. [10]{} J. Bamberg, M. Giudici, G.F. Royle, Every flock generalized quadrangle has a hemisystem, , 42(2010),795–810. J. Bamberg, S. Kelly, M. Law, T. Penttila, Tight sets and $m$-ovoids of finite polar spaces, , 114(2007)1293–1314. J. Bamberg, M. Law, T. Penttila, Tight sets and $m$-ovoids of generalised quadrangles, , 29(2009)1–17. J. Bamberg, M. Lee, K. Momihara, Q. Xiang, A new infinite family of hemisystems of the Hermitian surface, , 38(2018)43–66. W. Bosma, J. Cannon, , Sydney, 1995. R. Calderbank, W.M. Kantor, The geometry of two-weight codes, , 18(1986)97–122. A. Cossidente, C. Culbert, G.L. Ebert, G. Marino, On m-ovoids of $W_{3}(q)$, , 14(2008)76–84. A. Cossidente, T. Penttila, Hemisystems on the Hermitian surface, , 72(2005)731–741. T. Feng, K. Momihara, Q. Xiang, A family of $m$-ovoids of parabolic quadrics, , 140 (2016)97-111. G. Korchmáros, G.P. Nagy, P. Speziali, , , (2017). R. Lidl, H. Niederreiter, Finite fields, , 1997. S.E. Payne, J.A. Thas, Finite generalized quadrangles, , 2009. T. Penttila, B. Williams, Ovoids of parabolic spaces, , 82(2000)1–19. J.A. Thas, Interesting pointsets in generalized quadrangles and partial geometries, , 114(1989)103–131.

--- abstract: 'Pairing between spinless fermions can generate Majorana fermion excitations that exhibit intriguing properties arising from non-local correlations. But simple models indicate that non-local correlation between Majorana fermions becomes unstable at non-zero temperatures. We address this issue by showing that anisotropic interactions between dipolar fermions in optical lattices can be used to significantly enhance thermal stability. We construct a model of oriented dipolar fermions in a square optical lattice. We find that domains established by strong interactions exhibit enhanced correlation between Majorana fermions over large distances and long times even at finite temperatures, suitable for stable redundancy encoding of quantum information. Our approach can be generalized to a variety of configurations and other systems, such as quantum wire arrays.' author: - 'Fei Lin and V.W. Scarola' title: Enhancing the Thermal Stability of Majorana Fermions with Redundancy Using Dipoles in Optical Lattices --- [*Introduction:*]{} The wide variety of optical lattice geometries offer unprecedented tunability in manipulating quantum degenerate gases into complex quantum states [@greiner:2002]. Recent developments in the cooling of molecules (e.g., $^{40}\text{K} ^{87}\text{Rb}$) [@ospelkaus:2008] and magnetic atoms (e.g., $^{161}$Dy) [@lu:2012] imply that anisotropy in dipolar interactions will soon provide further opportunity to explore some of the most elusive yet compelling quantum states, entangled Majorana fermions (MFs). Seminal lattice models demonstrate particle-like excitations that behave as MFs thanks to non-local symmetries [@kitaev:2001; @kitaev:2003]. They entangle with each other over large distances through string operator (SO) correlations. In simple models SOs have straightforward definitions, e.g., fermion parity [@kitaev:2001], with non-trivial consequences. They signal underlying topological order with fascinating properties that have motivated proposals for topologically protected qubits [@kitaev:2003; @nayak:2008]. The crossing of SOs is responsible for unusual anyonic braid statistics [@kitaev:2003; @nussinov:2009]. And SOs connecting these excitations also underlie theories of quantum state teleportation [@semenoff:2007; @tewari:2008]. The zero-temperature properties of models hosting topological order set the stage for work connected to experiments. Kitaev’s two-dimensional (2D) Toric Code Hamiltonian [@kitaev:2003] motivated early proposals in optical lattices [@duan:2003; @micheli:2006; @weimer:2010]. But the 1D Kitaev chain model [@kitaev:2001] is one of the simplest models supporting MF excitations. Anticipation of non-local MF properties in 1D led to experimental proposals and experiments in both optical lattices [@jiang:2011; @diehl:2011; @kraus:2012] and solids [@kitaev:2001; @lutchyn:2010; @mourik:2012]. But prospects for observing non-local correlation of MF pairs over long times and distances hinge on the stability of SOs [@nussinov:2009; @chesi:2010]. SOs in important lattice models are unstable at non-zero temperatures. For example, SOs in the 2D Toric Code model vanish at long times and distances because of thermal excitations [@nussinov:2009; @chesi:2010; @castelnovo:2007; @hastings:2011]. Recent work also argues that MFs in lattice models of topological $p$-wave superconductors are sensitive to thermal fluctuations [@cheng:2010; @bauer:2012]. A general theorem [@hastings:2011] sets strict criteria for non-local correlations to remain resilient against thermal fluctuations. Fortunately, recent calculations indicate that topological phases can be enhanced through: disorder [@wootton:2011], and proximity coupling [@fidkowski:2011; @fu:2008] to a reservoir in topological superconducting wires [@lutchyn:2010]. There are also proposals to go beyond 1D wires to multi-channel or 2D MF arrays [@potter:2010]. ![(Color online) Schematic of dipolar fermions (spheres) in a 2D optical lattice. Dipolar moments $\vec{p}$ (arrows on each sphere) align along an applied field, at an angle $\theta$ with the $x$-axis.[]{data-label="dipole_lattice"}](lattice.pdf){width="80mm"} We propose that dipolar interactions in optical lattices [@baranov:2012] offer a powerful tool to stabilize the SOs in MF models. We show that anisotropy in both the lattice and dipolar interactions electrostatically copy SOs to force excitations to form arrays of strings which we call [*domains*]{} in this work. We thus propose a robust mechanism, the formation of domains with redundant MF edges, as a route to stabilize MFs, akin to quantum error correction schemes using redundant qubits [@shor:1995]. We pair two methods \[quantum Monte Carlo (QMC) and mean field theory\] to solve a model of dipolar fermions to demonstrate that domain formation in electrostatically coupled Kitaev chains significantly enhances the stability of SOs. QMC here is unbiased and shows the thermal stability of domains while our mean field theory (which agrees with QMC within regimes of applicability) explicitly reveals MFs. [*Model:*]{} We first consider a Hubbard model of dipolar fermions in an $L\times L$ optical lattice and then discuss a specific parameter regime. In Fig. \[dipole\_lattice\] fermions with dipolar moment $\vec{p}$ can hop between nearest neighbor (NN) sites. A large optical lattice depth along the $y$ direction strongly suppresses hopping in the $y$ direction. $V_x(\theta)=D^2(1-3\cos^2\theta)/r_0^3$ ($V_y=D^2/r_0^3$) is the $x$ ($y$) component of the NN dipolar fermion interaction. Here $D^2\sim \vec{p}^2$ and $r_0$ is lattice constant. We can tune $\theta$ so that the NN dipolar interaction is attractive along the $x$-direction. We construct a Hubbard model capturing the above features: $$\begin{aligned} & &H_D=-\sum_{i,j}\left(t_{x}a_{i,j}^{\dagger}a_{i+1,j}^{\phantom{\dagger}}+t_{y}a_{i,j}^{\dagger}a_{i,j+1}^{\phantom{\dagger}}+h.c.\right)\nonumber\\ &+&\sum_{i,j}\left[V_x(\theta)n_{i,j}n_{i+1,j}+V_y n_{i,j}n_{i,j+1}-\mu_{0}n_{i,j}\right], \label{dipolarH}\end{aligned}$$ where we have open(periodic) boundary condition in the $x(y$) directions. $a_{i,j}^{\dagger}$ creates a spinless fermion at the site $(i,j)$ and $n_{i,j}=a_{i,j}^{\dagger}a_{i,j}^{\phantom{\dagger}}$. $t_{x}(t_{y})$ is the hopping energy between NN sites in the $x(y)$ direction. $\mu_{0}$ is the chemical potential. For a range of $\theta$ yielding $V_{x}<0$ the ground state of Eq. (\[dipolarH\]) is stable and exhibits $p$-wave pairing. For $t_{x}=t_{y}$ functional renormalization group [@bhongale:2012] and mean field theory [@liu:2012] calculations show a BCS paired state for long-range dipolar interactions consistent with short-range interactions in Eq. (\[dipolarH\]) [@cheng:2010]. $p$-wave pairing between neighbors along $x$-rows can be modeled by real-space attraction: $\exp(\bold{i}\Phi_{i,j})\vert \Delta \vert a_{i+1,j}^{\dagger}a_{i,j}^{\dagger} + h.c.$, where $\Phi_{i,j}$ and $\vert \Delta \vert$ are the phase and magnitude of the pairing field within an $x$-row. But for $t_{y} \ll t_{x}$ the system can be analyzed with Luttinger liquid theory to show that weakly coupled 1D dipolar systems also posses $p$-wave pairing order with algebraically decaying pairing correlations [@huang:2009]. For $t_y\ll \vert \Delta \vert$, Josephson tunneling between paired states contributes an energy: $\sim -t_{y} ^{2} \cos( \Phi_{i,j}-\Phi_{i,j+1})$, which aligns the phase of the pairing field between each $x$-row, $\Phi_{i,j}-\Phi_{i,j+1}\rightarrow 0$. Hereafter, we assume a uniform pairing field to motivate a thermally stable MF model. Increasing $t_y$ should adiabatically connect the coupled-1D [@huang:2009] and 2D square lattice limits [@bhongale:2012; @liu:2012]. [*Effective Model:*]{} We perform a mean field decoupling of the attractive dipolar interaction term in Eq. (\[dipolarH\]) to establish the centerpiece of our study [@supplementarymaterial]: $$H_F=\sum_j H_K^j+V_y\sum_{i,j}\left (n_{i,j}-\frac{1}{2} \right ) \left (n_{i,j+1}-\frac{1}{2}\right ), \label{fermionH}$$ where the Hamiltonian for the $j$th Kitaev chain is $H_K^j=-t\sum_{i}\left (a_{i,j}^{\dagger}-a_{i,j}^{\phantom{\dagger}}\right) \left (a_{i+1,j}^{\dagger}+a_{i+1,j}^{\phantom{\dagger}}\right )-\mu n_{i,j}$. At the Hartree-Fock level the chemical potential renormalizes to $\mu=\mu_{0}+2\langle n_{i,j}\rangle |V_x(\theta)|-V_y/2$ and the hopping becomes $t=t_{x}- |V_x(\theta)| \langle a_{i+1,j}^{\dagger}a_{i,j}^{\vphantom{\dagger}} \rangle$, which is our energy unit. In Eq. (\[fermionH\]), we tuned $V_{x}$ to match the pairing term with the renormalized hopping by setting $t_{x}=|V_x(\theta)|\langle a_{i+1,j}^{\dagger}a_{i,j}^{\dagger}+ a_{i+1,j}^{\dagger}a_{i,j}^{\vphantom{\dagger}}\rangle$. MFs can arise away from this particular point, which is guaranteed by the presence of a gap in the energy spectrum of $H_{F}$ [@dorier:2005]. $t_y$ is energetically negligible but is included as a second order effect by setting $\Phi_{i,j}=0$. We work near half filling $\langle n \rangle=1/2$, i.e., $\mu=0$. Eq. (\[fermionH\]) describes an array of strongly interacting Kitaev chains, whose ground state is $2^L$-fold degenerate [@supplementarymaterial], which is not explicit in Eq. (\[dipolarH\]). Our direct QMC simulations on Eq. (\[dipolarH\]) show the emergence of precisely the same set of degeneracies expected from Eq. (\[fermionH\]) for the parameters given by the Hartree-Fock decoupling [@supplementarymaterial; @lin:2013]. ![(Color online) The thermal expectation value of SOs from QMC as a function of an applied global field for several system sizes for $V_y=4.8t$ and $\mu=0$. The top (bottom) panel shows data for a characteristic low (high) temperature. The insets show schematic examples of a MF domain that breaks up into two MF domains at high temperatures. “+” in the figures is fermion parity for the entire chain, and each chain has the same parity for the one configuration drawn. Empty dashed circles denote empty MF edge states; hatched circles denote MF edge states occupied by one particle per row.[]{data-label="PvsZ2H"}](P_vs_z2h_scaling.pdf){width="4in"} ![(Color online) The thermal expectation value of SOs from QMC as a function of an applied global field for several system sizes for $V_y=4.8t$ and $\mu=0$. The top (bottom) panel shows data for a characteristic low (high) temperature. The insets show schematic examples of a MF domain that breaks up into two MF domains at high temperatures. “+” in the figures is fermion parity for the entire chain, and each chain has the same parity for the one configuration drawn. Empty dashed circles denote empty MF edge states; hatched circles denote MF edge states occupied by one particle per row.[]{data-label="PvsZ2H"}](one_domain.png){width="1.5in"} ![(Color online) The thermal expectation value of SOs from QMC as a function of an applied global field for several system sizes for $V_y=4.8t$ and $\mu=0$. The top (bottom) panel shows data for a characteristic low (high) temperature. The insets show schematic examples of a MF domain that breaks up into two MF domains at high temperatures. “+” in the figures is fermion parity for the entire chain, and each chain has the same parity for the one configuration drawn. Empty dashed circles denote empty MF edge states; hatched circles denote MF edge states occupied by one particle per row.[]{data-label="PvsZ2H"}](two_domains.png){width="1.4in"} [*Mechanism for Stabilizing MFs:*]{} Eq. (\[fermionH\]) is a highly non-trivial many-body model. It maps onto an intractable quantum spin compass model [@supplementarymaterial; @dorier:2005]. Below we argue that the inter-chain interactions stabilize correlation between edge $y$-columns of MFs. We use mean field theory to show that Eq. (\[fermionH\]) reduces to a MF model [@supplementarymaterial]. Consider a pair of MF operators, $c_{2i,j}$ and $c_{2i-1,j}$, for each site of the lattice, $(i,j)$, where $a_{i,j}^{\dagger}=(c_{2i-1,j}-\bold{i}c_{2i,j})/2$ [@kitaev:2001]. We impose a mean field decoupling of the $V_y$ term, using a 2-site unit cell along the $y$ direction. Each site of the unit cell corresponds to sublattice A or B. We thus have $H_M^{\alpha}=\bold{i}t\sum_{i}c_{2i,\alpha}c_{2i+1,\alpha} +(\bold{i}\tilde{\mu}_{\alpha}/2)\sum_{i}c_{2i-1,\alpha}c_{2i,\alpha}$, where $\alpha \in \{ A,B\}$ denotes sublattice and the renormalized chemical potential, $\tilde{\mu}_{\alpha}=\mu+\bold{i}V_y\langle c_{2i-1,\alpha}c_{2i,\alpha} \rangle$. Furthermore, we can show [@supplementarymaterial] that the ground state avoids strong $V_{y}$ by setting $\langle c_{2i-1,\alpha}c_{2i,\alpha} \rangle=0$ for $V_y>4t$. This leads to two columns of localized MF states, one at each edge. Solutions of $H_M$ exhibit domains with MF edge states along $y$-columns (Fig. \[PvsZ2H\]) [@supplementarymaterial]. Note that the $V_{y}$ term in Eq. (\[fermionH\]) leads to a chemical potential staggered along $y$ columns, which binds MFs along $y$ but leaves them to propagate along $x$. An energy penalty, $\sim V_{y}$, will result if only one row changes its parity. The inter-row interaction therefore increases the dimension of the MF edge state (from a point particle to a $y$-column) to establish the mechanism for enhancing the stability of the non-local MF state against thermal fluctuations. The entire ground state can thus be regarded as a redundantly encoded qubit of several MFs. Along these lines, mean field theory suggests the following Gutzwiller projected wave function: $ \prod_{i,j=1}^{L}\left(1-n_{i,j}n_{i,j+1}\right)\phi^j_{\rm BCS}, $ where $\phi^j_{\rm BCS}$ is the BCS wave function hosting MFs in the $j$th $x$-row. Thermally stable non-local correlation implies that $y$-columns of MF pairs at $i=1$ and $i=L$ host real dipoles in a superposition that remains robust against thermal excitations. To establish robustness we note that the Hilbert space of Eq. (\[fermionH\]) possesses a spectral gap, $\Delta E$, above a degenerate manifold of states for the parameters we consider here [@dorier:2005]. But the entropy gain, $S$, in the free energy cost to create excitations, $\Delta E-TS$, can overwhelm the energy gap depending on the effective dimensionality of excitations. Strong interactions, $V_{y}>4t$, require the creation of entire domains (with a perimeter $\sim L$, $\Delta E\sim L$, and $S\sim L$) to destroy non-local correlations as opposed to $\Delta E\sim \mathcal{O}(1)$ and $S\sim \log L$ for $V_{y}<4t$. Favorable entropy scaling implies that non-local correlation between MF $y$-columns in 2D is much more thermodynamically stable than between pairs of individual MFs in 1D. [*QMC Test of Thermal Stability:*]{} We test the robustness of SOs of MFs with QMC simulations [@sandvik:2002] on Eq. (\[fermionH\]) [@supplementarymaterial]. The non-local correlation between edge states at $i=1$ and $i=L$ is captured by a set of $L$ SOs that stretch across each $x$-row: $P_j\equiv\prod_{i=1}^{L}(1-2n_{i,j})=(-1)^{\sum_in_{i,j}},$ where $j=1,2,\cdots,L$ along $y$. $P_j$ is equivalent to the fermion parity for the $j$th row. The expectation value of the SOs, $P_j$, act as order parameters. Unique values, $\langle P_{j} \rangle=\pm 1$, can be used to define each sector and therefore indicate stability in the non-local correlations between MFs. But $\langle P \rangle=0$ indicates that thermal excitations destroy any distinction between sectors. We compute $\langle P_{j} \rangle$ to show spontaneous breaking of these discrete symmetries for $V_{y}>4t$ even at non-zero temperatures. To detect such a symmetry breaking we perturb the above spinless fermion model with a weak global field: $H=H_F-\tilde{h}\sum_{j=1}^{L}P_j.$ The global field, $P=L^{-1}\sum_{j=1}^{L}P_j$, imposes a splitting between the otherwise degenerate states. We define $\tilde{h}=hL$ to ensure that the perturbing term imposes a non-zero energy splitting per particle, $h$, between degenerate sectors even in the limit $L\rightarrow\infty$. $h> 0$ favors $\langle P \rangle=1$. ![(Color online) Top: The susceptibility of the string-string correlation function $O$ from QMC simulations for different $L$’s at $V_y=4.8t$ and $\mu=0$. The SOs tend to order along the $y$ direction for $T<T_{c}$. The inset shows a schematic of an ordered domain with MFs forming columns at the ends (dashed lines). The domains shrink for $T>T_{c}$. Bottom: $T_{c}$ extrapolated to $L\rightarrow\infty$. The solid line is a linear chi-squared fit. []{data-label="Tc_vs_L"}](Tc_vs_L.pdf){width="4in"} ![(Color online) Top: The susceptibility of the string-string correlation function $O$ from QMC simulations for different $L$’s at $V_y=4.8t$ and $\mu=0$. The SOs tend to order along the $y$ direction for $T<T_{c}$. The inset shows a schematic of an ordered domain with MFs forming columns at the ends (dashed lines). The domains shrink for $T>T_{c}$. Bottom: $T_{c}$ extrapolated to $L\rightarrow\infty$. The solid line is a linear chi-squared fit. []{data-label="Tc_vs_L"}](domains.pdf){width="1.5in"} We first compute $\langle P \rangle$ in the limit $V_{y}<4t$ using QMC. For $V_{y}=3.2t$ we find $\langle P \rangle \rightarrow 0$ with increasing $L$. This indicates that the SOs in 1D $x$-rows alone are extremely sensitive to thermal fluctuations, as expected from the entropy argument above, even with $\Phi_{i,j}$ held constant. Our calculations are time independent. One may find $|\langle P \rangle|>0$ at short times. We now calculate $\langle P \rangle$ in the strongly interacting case, $V_{y}=4.8t$, where we expect arrays of strings to form stable domains. Fig. \[PvsZ2H\] shows $\langle P \rangle$ at low and high temperatures. At high $T$ the bottom panel shows that a large value of $h$ is needed to stabilize the SOs. But at low $T$ (top panel) we find that very small fields tend to force all $x$-rows to spontaneously occupy the lowest energy state in the limit $h\rightarrow 0$, which indicates that $y$-columns of MFs located at $i=1$ and $i=L$ can be prepared in a long-lasting entangled state stretching over large distances even at finite temperatures. [*Thermal Stability of Domains:*]{} The arrays of SOs defining domains are stable at low temperatures but eventually break up at large $T$. To find the critical temperature for domain formation, we define a string-string order parameter that captures the ordering strength along the $y$ direction: $\langle O \rangle\equiv L^{-2}\sum_{j,j'=1}^{L} \langle P_j P_{j'} \rangle.$ The operator $O$ is similar to the static structure factor, $S_{k_{y}}\propto\sum_{j,j'=1}^{L} \exp{[-\bold{i} k_{y}(j-j')]} \langle n_j n_{j'} \rangle$, but with the replacement $n_j n_{j'}\rightarrow P_j P_{j'}$ and with wavevector $k_y=0$. We look for long-range order in the susceptibility of $O$, $\chi_{O}=L^2(\langle O^2\rangle-\langle O\rangle^2)/T$. A peak in $\chi_{O}$ versus $T$ indicates the critical temperature $T_c$ at which the large domain breaks up along the $y$ direction. For $V_{y}<4t$ we find no peaks in our simulations and therefore no domain formation for weakly interacting chains, i.e., $T_{c}=0$. We observe domain formation in $\chi_{O}$ for $V_{y}>4t$. The top panel of Fig. \[Tc\_vs\_L\] shows $\chi_{O}$ as a function of temperature for $V_{y}=4.8t$. Above $T_{c}$ the $y$-columns of MFs are no longer ordered. The bottom panel extracts $T_{c}$ in the thermodynamic limit, yielding $T_c=0.275(4)t$. Our results agree with studies on the quantum compass model showing a thermal phase transition in the universality class of the 2D Ising model [@wenzel:2008]. ![(Color online) The main panel plots the energy splitting between two sectors defined by $P_{j}=\pm1$ for all $x$-rows as a function of chemical potential for Eq. (\[fermionH\]) at $T=0.16t$ and $V_y=4.8t$. Inset (a) shows a weak linear increase in density with increasing $\mu$ inside the topological phase ($\mu\lesssim 1.5t$). Inset (b) shows a schematic phase diagram established by the lifting of the degeneracy, horizontal arrow. The vertical arrow indicates the thermal phase transition explored in Fig. \[Tc\_vs\_L\]. MFT denotes the mean field theory result.[]{data-label="ESplitting"}](DE_vs_hz_combined_n_vs_mu.pdf){width="3.6in"} ![(Color online) The main panel plots the energy splitting between two sectors defined by $P_{j}=\pm1$ for all $x$-rows as a function of chemical potential for Eq. (\[fermionH\]) at $T=0.16t$ and $V_y=4.8t$. Inset (a) shows a weak linear increase in density with increasing $\mu$ inside the topological phase ($\mu\lesssim 1.5t$). Inset (b) shows a schematic phase diagram established by the lifting of the degeneracy, horizontal arrow. The vertical arrow indicates the thermal phase transition explored in Fig. \[Tc\_vs\_L\]. MFT denotes the mean field theory result.[]{data-label="ESplitting"}](phase_diagram.png){width="2.03in"} The robustness of the ground state degeneracy also reveals the stability of the SOs. We denote each ground state energy sector by $E(P_1,P_2,\cdots)$. We found that this degeneracy was not lifted with a weak staggered chemical potential, inter-chain hopping, or a uniform chemical potential shift [@lin:2013]. We present representative results for the uniform chemical potential shift. Fig. \[ESplitting\] shows the energy splitting per particle of two different sectors of the $P_j$ operator: $\delta E\equiv E(-1,-1,\cdots)-E(1,1,\cdots)$, as a function of $\mu$. The flat portion for $\mu/t\ll1$ indicates a robust degeneracy. Above $\mu\approx 1.5t$ the energy splitting acquires a size dependence, as expected for $\mu>\Delta E$. Inset (a) shows that the particle density has weak linear dependence for $\mu/t\ll1$ which is also captured by the mean field theory. Our results are consistent with the formation of a thermally robust topological phase, shown in inset (b) of Fig. \[ESplitting\]. [*Detection in Optical Lattices:*]{} Domain formation can be observed directly in time-of-flight measurements. Noise correlations between shots of individual time-of-flight images relate to $S_{k}$ [@altman:2004]. In the topological phase we anticipate the formation of lines, rather than peaks, in noise correlations because the $V_{y}$ term correlates the density along just the $y$ direction for $T<T_{c}$. Observations of these lines should therefore allow identification of $T_{c}$. Correlation between MFs could be demonstrated through non-local measures similar to those proposed in quantum wires [@tewari:2008]. Local spectroscopic probes [@jiang:2011; @kraus:2012] applied at each domain edge could be adapted to detect the response of one domain edge when dipoles are added to alternating Kitaev chains on the opposite edge. The particle number parity in the opposite edge should respond with signatures of non-local correlations in dynamics [@tewari:2008]. Recent experiments using high resolution spectroscopy to measure particle number parity [@simon:2011] and SOs [@endres:2011] could be used to explicitly measure response. [*Fluctuations in Pairing:*]{} We connected a model of oriented fermionic dipoles, Eq. (\[dipolarH\]), to a pairing model, Eq. (\[fermionH\]). The pairing model itself demonstrates significantly enhanced stability of MF state via domain formation at $T>0$. But our specific implementation still allows fluctuations of the pairing field between $x$-rows. Fortunately, the long-range dipolar interaction has been found to enhance the stability of $p$-wave superfluidity [@liu:2012]. Coherent reservoirs can further suppress pairing field fluctuations via the proximity effect [@diehl:2011; @kraus:2012; @fu:2008]. We can show that an optical lattice geometry allowing proximity coupling is possible [@supplementarymaterial]. We note, however, that excitations in the system may couple to those in the reservoir [@fidkowski:2011]. [*Conclusion:*]{} We considered an effective model of oriented dipolar fermions in a 2D lattice that allows hopping along directions where the dipoles attract but suppresses hopping along directions where dipoles repel. In the $p$-wave superfluid regime we model the system with repulsive Kitaev chains. Each chain experiences a self-consistently renormalized chemical potential due to its neighbor to impose an energy penalty for excitations. This energy penalty is the mechanism behind MF domain formation and therefore enhances correlation between columns of MFs along each domain edge. Unbiased QMC confirms that string operators defining non-local MF states remain robust to thermal fluctuations. 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Supplementary Material for “Enhancing the thermal stability of Majorana fermions with redundancy using dipoles in optical lattices” =================================================================================================================================== Derivation of Effective Model ----------------------------- In this section we derive the effective model $H_F$ \[Eq. (2) in the main text\] from the dipolar model $H_D$ \[Eq. (1) in the main text\] at the Hartree-Fock level. This shows that, deep in the superfluid phase, $H_{F}$ captures the essential physics of $H_{D}$. All of our numerical calculations in the paper are performed on $H_F$. The attractive interaction term along $x$-rows, $V_x(\theta)n_{i,j}n_{i+1,j}$, in $H_D$ decouples in the Hartree-Fock approximation: $$\begin{aligned} n_{i,j}n_{i+1,j} &\approx&\langle n_{i,j}\rangle n_{i+1,j}+\langle n_{i+1,j}\rangle n_{i,j} \nonumber \\ &-&\langle a_{i,j}^{\dagger}a_{i+1,j}\rangle a_{i+1,j}^{\dagger}a_{i,j}+\langle a_{i,j}^{\dagger}a_{i+1,j}^{\dagger}\rangle a_{i+1,j}a_{i,j}\nonumber\\ &-&C + h.c.,\end{aligned}$$ where $C\equiv\langle n_{i,j}\rangle\langle n_{i+1,j}\rangle-\langle a_{i,j}^{\dagger}a_{i+1,j}\rangle\langle a_{i+1,j}^{\dagger}a_{i,j}\rangle+\langle a_{i,j}^{\dagger}a_{i+1,j}^{\dagger} \rangle\langle a_{i,j}a_{i+1,j}\rangle$. We define the renormalized chemical potential $\mu=\mu_0+2\langle n_{i,j}\rangle|V_x(\theta)|-V_y/2$ and the renormalized hopping $t=t_x-|V_x(\theta)|\langle a_{i+1,j}^{\dagger}a_{i,j}\rangle$. We further assume that by tuning $V_x(\theta)$ the renormalized hopping $t$ matches the pairing amplitude $t=|V_x(\theta)|\langle a_{i+1,j}^{\dagger}a_{i,j}^{\dagger}\rangle$. As argued in the main text, we also take the $t_y=0$ limit to arrive at the effective model $H_F$ in Eq. (2) of the main text. Ground State Degeneracy ----------------------- In this section we show that the ground state of $H_F$ in the main text is $2^L$ fold degenerate for our cylindrical geometry [@doucotsupp:2005; @doriersupp:2005] for $t_{y}=0$. We then discuss the $t_{y}\rightarrow 0$ limit. In the main text we defined a set of SOs $P_j$ along the $x$ direction, which commute with $H_F$. Similarly, we define a set of SOs $Q_i$ along the $y$ axis, which also commute with $H_F$, $$Q_i=\prod_j(2\tilde{a}_{i,j}),$$ where $i=1,2,\cdots,L$ and $$\tilde{a}_{i,j}\equiv F_{i,j}(a_{i,j}^{\dagger}+a_{i,j}^{\phantom{\dagger}})/2, \label{Sx_definition}$$ where the transformation coefficients are given by: $$F_{i,j}=\prod_{j'<j}\prod_k(1-2n_{k,j'})\prod_{i'<i}(1-2n_{i',j}).$$ Note that the operator $\tilde{a}_{i,j}$ corresponds to a spin $\frac{1}{2}$ operator along the $x$ direction in spin space, $S_{i,j}^x$, based on the Jordan-Wigner transformation [@chensupp:2007]. One can check that $\{P_j, Q_i\}=0$. To see the degeneracy explicitly, suppose that we have a common eigenstate $\phi_0$ of $H_F$ and $Q_i$. If we act $P_j$ on the state $\phi_0$, we get $\phi_1=P_j\phi_0$. Since $P_j$ does not commute with $Q_i$, $\phi_1$ must be different from $\phi_0$. However, $\phi_1$ is still an eigenstate of $H_F$ with the same eigenvalue as $\phi_0$, because $P_j$ commutes with $H_F$. Each eigenstate is, therefore, at least 2-fold degenerate. Furthermore, since $[P_kP_j,Q_i]=0$, $\phi_1$ is also an eigenstate of the operator product $P_kP_j$. We then have $\phi_1=P_j\phi_0 \propto (P_kP_j)P_j\phi_0=P_k\phi_0$, which means that acting $P_k (k\neq j)$ on $\phi_0$ will not generate a different state than $\phi_1=P_j\phi_0$. Every eigenstate, including the ground state, is therefore, 2-fold degenerate. Exact diagonalization studies in combination with $L^{\text{th}}$ order perturbation theory show that in the $L\rightarrow\infty$ limit the low-lying $2^L-2$ excited states will collapse with the exact 2-fold degenerate ground state, thus forming a $2^L$-fold degenerate ground state in the equivalent spin-quantum compass model [@doriersupp:2005] (For a mapping to the quantum compass model see the section “QMC Simulations”). The gap between the ground state and the low-lying $2^L-2$ excited states was found to collapse as $\sim (2t_{x}/V_{y})^{L}$ for $V_{y}>4t_{x}$ [@doriersupp:2005]. Note that the $2^L$-fold degeneracy arises even in the large $V_y$ limit. We now consider the $t_y\rightarrow 0$ limit, i.e., non-zero hopping along the $y$ direction. In our model, with $t_y=0$, edge MFs are unable to hybridize with those in neighboring rows. In the $t_y\rightarrow 0$ limit we also observe a $2^L$ degeneracy in spite of edge MF coupling (hybridization) effects discussed in the literature [@pottersupp:2010]. Our model is different from these works because it is very strongly interacting. Even with a small $t_y$ hopping, we believe that hybridization is still strongly suppressed because of the strong $V_y$ term, which will give a large energy penalty if a single fermion hops between chains. We have performed direct numerical simulations of Eq.(1) in the main text for various lattice sizes, $L=4,6,$ and 8, to confirm, within numerical accuracy, the emergence of such a set of degeneracies in the ground state. For example, we find degeneracies for $t_x=1$, $V_y=1.2$, and $V_x=-0.053$, that are immune to small $t_y$ perturbations. Validating a Mean Field Picture ------------------------------- To show the existence of MFs and domains we perform a mean field decoupling of Eq. (2) in the main text along the $y$ direction. The mean field theory presented in this section is in terms of real fermions but is equivalent to the MF mean field theory presented in the next section, Eq. (\[MFmajoranaH\]), and in the main text. We then verify the mean field theory by direct comparison with an unbiased QMC analysis. Finally we will discuss the parameter regimes of validity. To construct the mean field equations we divide the lattice into 2 sublattices, $A$ and $B$, along the $y$ direction, and decouple the interaction terms (staggered density assumption). We obtain the following 4 coupled mean field equations: $$\begin{aligned} H_1^{\alpha}&=&-8t\langle \tilde{a}_{i+1,\alpha}\rangle\tilde{a}_{i,\alpha}-\tilde{\mu}_{\alpha}\left (n_{i,\alpha}-\frac{1}{2} \right ),\nonumber\\ H_2^{\alpha}&=&-t\sum_i \left(a_{i,\alpha}^{\dagger}-a_{i,\alpha}^{\phantom{\dagger}}\right ) \left (a_{i+1,\alpha}^{\dagger}+a_{i+1,\alpha}^{\phantom{\dagger}} \right)\nonumber\\ & & -\tilde{\mu}_{\alpha}\sum_in_{i,\alpha}, \label{mftH}\end{aligned}$$ where $\tilde{\mu}_{\alpha}=\mu_{\alpha}-2V_y\langle n_{i,\alpha}-1/2\rangle$. $\mu_A$ and $\mu_B$ are applied staggered chemical potentials for $A$ and $B$ sublattices. In the spin language, the first equation defines a single spin in a magnetic field while the second is a quantum Ising model. We use the solutions of both of these models [@lieb:1961; @pfeuty:1970] to solve both models exactly and then the coupled equations, Eqs. (\[mftH\]), through iteration. ![ (Color online.) QMC (L=4, 6, 8) and mean field theory comparison of the staggered density (top), intra-$x$-row hopping and pairing correlation function (middle), and the inter-$x$-row density-density correlation function (bottom) at $V_y=4.8t$. We apply staggered chemical potentials $\mu_A$ and $\mu_B$ to the $A$ and $B$ sublattices, respectively. []{data-label="QMC_PMFT_cmp"}](qmc_pmft_cmp.pdf){width="\columnwidth"} Eqs. (\[mftH\]) assume a spatially uniform chemical potential (for each sublattice). If this assumption is correct, it implies that excitations for any given $x$-row are copied to all other $x$ rows to yield a domain. The existence of domains of string operators is therefore implicit in the mean field theory but we must validate Eq. (\[mftH\]) as a good approximation to Eq. (2) in the main text to justify this picture. We validate Eqs. (\[mftH\]) by direct comparison with QMC solutions to Eq. (2) in the main text. To compare we compute correlation functions using both mean field theory and QMC. The following local correlation functions define quantum bond order along the $x$ direction and density bond order along the $y$ direction. $$\begin{aligned} r_x&\equiv&\frac{1}{4}\langle (a_{i,j}^{\dagger}-a_{i,j}^{\phantom{\dagger}})(a_{i+1,j}^{\dagger}+a_{i+1,j}^{\phantom{\dagger}})\rangle,\nonumber\\ r_y&\equiv&\langle (\frac{1}{2}-n_{i,j})(n_{i,j+1}-\frac{1}{2})\rangle. \label{rdefinitions}\end{aligned}$$ Under the spin mapping these correlation functions have been studied in a corresponding spin model, the quantum compass model [@nussinov:2005; @scarola:2009]. Fig. \[QMC\_PMFT\_cmp\] shows that the mean field theory offers an excellent approximation to the QMC results. The large value of $V_{y}$ leads to bond ordering along $y$ (large $r_{y}$). But the non-zero values of $r_{x}$ show quantum correlations along the $x$ direction. Therefore both QMC and mean field theory show that the $y$-columns superpose throughout the lattice to yield a quantum entangled ground state at non-zero temperatures. The good agreement between QMC and mean field theory therefore supports the domain picture implicit in Eqs. (\[mftH\]). There are, however, small differences between QMC and mean field calculations for $T/t<4$ in Fig. \[QMC\_PMFT\_cmp\]. This is due to the fact that mean field calculations ignore quantum fluctuations (and therefore underestimate $r_x$) at low temperatures and exaggerate the effects of classical $V_y$ interactions (and therefore overestimate $r_y$). Despite this drawback, mean field calculations for $V_y>4t$ still capture the essential physics of the original model. To be specific, at low temperatures both QMC and mean field calculations give $r_y=1/4$, which means that $(\langle n_{i,j}\rangle+\langle n_{i,j+1}\rangle)/2-\langle n_{i,j}n_{i,j+1}\rangle=1/2$. At half filling for a uniform system, i.e., $\langle n_{i,j}\rangle+\langle n_{i,j+1}\rangle=1$, we have $\langle n_{i,j}n_{i,j+1}\rangle=0$, which shows that the system avoids large $V_y$ interactions. This explains why mean field calculations are accurate in this regime. The validity of our mean field theory crucially depends on the order parameter assumption (staggered density in a given column to avoid $V_y$ interactions). Mean field theory breaks down when different ordering appears. This is shown in inset (a) of Fig. 4 in the main text for the large $\mu/t$ limit. Here the topological phase disappears. In this limit a new order parameter is required to capture the effects of adding extra particles to the system. Mapping to Majorana Fermions ---------------------------- Here we prove that we can transform Eq. (2) in the main text into an interacting MF model by introducing two MF operators, $c_{2i,j}$ and $c_{2i-1,j}$, for each site of the lattice, $(i,j)$ [@kitaevsupp:2001] with $c^{\vphantom{\dagger}}_{a,b}c^{\vphantom{\dagger}}_{a',b'}=-c^{\vphantom{\dagger}}_{a',b'}c^{\vphantom{\dagger}}_{a,b}$ (for $\{a,b \}\neq\{a',b' \}$), $c^{\vphantom{\dagger}}_{a,b}=c_{a,b}^{\dagger}$ and $(c^{\vphantom{\dagger}}_{a,b})^2=1$. The absence of kinetics along the $y$ direction implies that each particle can be labeled with a specific $x$-row index, $j$. The MF operators then relate to the physical fermion operators by a complex superposition: $ a_{i,j}^{\dagger}=(c_{2i-1,j}-\bold{i}c_{2i,j})/2. $ We can now demonstrate the existence of edge states by mapping Eq. (2) in the main text to MF space: $$\begin{aligned} H_M=& &\bold{i}t\sum_{i,j}c_{2i,j}c_{2i+1,j} +\frac{\bold{i}\mu}{2}\sum_{i,j}c_{2i-1,j}c_{2i,j}\nonumber\\ &-&\frac{V_y}{4}\sum_{i,j}c_{2i-1,j}c_{2i,j}c_{2i-1,j+1}c_{2i,j+1}. \label{majoranaH}\end{aligned}$$ Here we see that the first two terms equate to the Kitaev chains \[the first term $\sum_j H_K^j$ in Eq. (2) in the main text\] and define a bilinear MF theory. States defined by the dangling operators, $c_{1,j}$ and $c_{2L,j}$, at the ends of each $x$-row establish two-fold degenerate MF states that can be entangled at $T=0$. Next we want to understand the effect of interactions, $V_y>0$, on the degenerate MF states in a mean field approximation (validated in the main text and in the previous section). We note that the MF correlation function is directly related to the real fermion number operator: $C^M_{i,j}\equiv (\bold{i}/2)c_{2i-1,j}c_{2i,j}=n_{i,j}-1/2$. From the mean field and QMC comparison result and discussions in the previous Supplementary Material section \[see $r_y$ in Eq. (\[rdefinitions\]) and Fig. \[QMC\_PMFT\_cmp\]\], we can see that at low temperatures for fixed index $i$ the MF correlation function $C^M_{i,j}$ has alternating values of $\frac{1}{2}$ and $-\frac{1}{2}$ along the $y$ direction. This minimizes the interaction energy. Therefore, we can do a mean field decoupling of the $V_y$ interaction term in the MF Hamiltonian, Eq. (\[majoranaH\]), to obtain the following Hamiltonian: $$H_M^{\alpha}=\bold{i}t\sum_{i=1}^{L-1}c_{2i,\alpha}c_{2i+1,\alpha} +\frac{\bold{i}\tilde{\mu}_{\alpha}}{2}\sum_{i=1}^{L}c_{2i-1,\alpha}c_{2i,\alpha}, \label{MFmajoranaH}$$ where $\alpha \in \{ A,B\}$ indexes sublattices and $\tilde{\mu}_{\alpha}=\mu+V_y\langle C^M_{i,\alpha} \rangle$. Eq. (\[MFmajoranaH\]) yields edge MFs only for certain parameter regimes. To see where, we solve the eigenequation $H_M^{\alpha}u_{\alpha}=0$ for the zero-energy eigenfunction $u_{\alpha}$ of the $\alpha$’th Kitaev chain. One real-space solution is [@kitaevsupp:2001]: $$u_{\alpha}\propto \bigg (1,0,\frac{\tilde{\mu}_{\alpha}}{2t},0,\left (\frac{\tilde{\mu}_{\alpha}}{2t}\right )^2,0,\cdots \bigg).$$ Here we see that the edge MF survives for $\tilde{\mu}_{\alpha}/2t\ll 1$. At half filling ($\mu=0$) this gives highly localized edge MFs, $u_{\alpha}\propto (1,0,0...)$. For $V_y>4t$ $C^M_{i,j}$ oscillates in sign for a single classical configuration but gives $\langle C^M_{i,j}\rangle=0$ in the quantum ground state. This shows that $\tilde{\mu}_{\alpha}=\mu$, i.e., the chemical potential for each Kitaev chain is not renormalized for $V_y>4t$. But the situation is different for $V_y<4t$. Here we have $\tilde{\mu}_{\alpha}\sim \mu+V_y$. In this regime, the large chemical potential prevents the formation of edge MFs. QMC Simulations --------------- In this section we describe our QMC simulations in more detail. We first show that, after mapping Eq. (2) in the main text to a spin model, we can compute correlation functions using the Stochastic Series Expansion (SSE) [@sandviksupp:2002] combined with the quantum Wang-Landau (QWL) algorithm [@troyersupp:2003]. QMC parameters are given. We then discuss the nature of the sign problem that arises when we add inter-chain tunneling to simulate Eq. (1) in the main text. We first show how to map Eq. (2) in the main text to a spin model. We use a Jordan-Wigner transformation that zig-zags through the lattice [@chensupp:2007]: $$\begin{aligned} a_{i,j}&=&\bigg( \prod_{i'<i,j'}\sigma_{i',j'}^{z}\prod_{j''=1}^{j-1}\sigma_{i,j''}^{z} \bigg)\sigma_{i,j}^{+}, \nonumber\\ \sigma_{i,j}^{z}&=&(-1)^{a_{i,j}^{\dagger}a_{i,j}^{\vphantom{\dagger}}},\end{aligned}$$ where $\sigma^x$, $\sigma^y$, and $\sigma^z$ are the Pauli matrices and $\sigma^{\pm}=(\sigma^x\pm\bold{i}\sigma^y)/2$, to map the model onto the quantum compass model [@chensupp:2007]: $$\begin{aligned} H_F=\sum_{i,j}\left [-t\sigma_{i,j}^{x}\sigma_{i+1,j}^{x}+\frac{V_y}{4}\sigma_{i,j}^z\sigma_{i,j+1}^z -\mu_{0}\frac{1-\sigma_{i,j}^{z}}{2}\right ]\nonumber \end{aligned}$$ To solve this model we perform QMC simulations with SSE [@sandviksupp:2002] combined with the QWL algorithm [@troyersupp:2003]. In the QWL approach the partition function is expanded as a series in powers of $\beta\equiv (k_{B}T)^{-1}$: $$\textrm{Tr} e^{-\beta H_F}=\sum_{n=0}^{N_{\textrm{max}}}S \vert g(n)\vert \beta^n,$$ where $N_{\textrm{max}}$ is the maximum expansion order. $N_{\textrm{max}}$ determines the lowest temperature that can be reached in the simulation and $g(n)$ corresponds to the classical density of states. $S$ is the overall sign. In the absence of a sign problem we have $S=1$ and $g(n)=\vert g(n)\vert$. In the presence of a sign problem we have $\langle S\rangle <1$. Severe sign problems, $\langle S\rangle \rightarrow 0$, prevent control of error in QMC sampling. The quantum compass model does not have a sign problem, implying that Eq. (2) in the main text does not have a sign problem. The distribution of $g(n)$ is obtained from a random sampling protocol [@troyersupp:2003]. It can be used to estimate the free energy, internal energy, entropy, heat capacity, and other properties of the system. We note that to measure other physical quantities, e.g., the density, density-density correlation, and the fermion parity operator, we need to accumulate their distributions at every order of the series expansion. In simulating $H_{F}$ we find that the energy barrier between different fermion parity operator sectors is very large. The large energy barrier dramatically increases the autocorrelation time in conventional QMC simulations with non-local updating. Without the QWL algorithm, the energy autocorrelation time for $V_y>4t$ is typically $\sim 10^3 - 10^4$ MC sweeps, which is prohibitively large for obtaining accurate QMC results. (We define 1 MC sweep as 1 diagonal update followed by $ N_{\rm max}/L_{\rm loop}$ loop updates with average loop length $L_{\rm loop}$.) We find that the QWL algorithm is necessary to reduce the autocorrelation time in QMC by enabling tunneling between different fermion parity sectors. We check the convergence of various physical quantities in the simulation with respect to $N_{\rm{max}}$. We find that local quantities such as internal energy, average density, density-density correlation function, etc., converge much faster than the non-local fermion parity operator, $P$, at low temperatures, which usually requires a much larger $N_{\rm{max}}$. In practice we find the following values for $N_{\rm{max}}$ to be enough for $P$ to converge in our simulations in the desired low temperature range: $N_{\rm{max}}=5000, 8000,$ and $10000$ for $L=4, 6$, and $8$, respectively. A typical QMC run on a single 2.53 GHz Intel Xeon CPU with the above $N_{\rm max}$ takes 1, 2, and 12 days, respectively, for the flat histogram to converge within $10^{-6}$. We usually do 10 such runs to estimate the error bars of various physical quantities for each set of parameters. We now discuss simulation of Eq. (1) in the main text. We map into a quantum spin model using the same Jordan-Wigner transformation [@chensupp:2007]: $$\begin{aligned} H_{QS}&=&\sum_{i,j}\bigg\{-t_x\sigma_{i,j}^{-}\sigma_{i+1,j}^{+}-t_y(-1)^{n_{d}(i,j;i,j+1)}\sigma_{i,j}^{-}\sigma_{i,j+1}^{+}\nonumber\\ & +&h.c.+\frac{V_x(\theta)}{4}\sigma_{i,j}^z\sigma_{i+1,j}^z+\frac{V_y}{4}\sigma_{i,j}^z\sigma_{i,j+1}^z\nonumber\\ &-&\mu_{0}\frac{1-\sigma_{i,j}^{z}}{2}\bigg\},\end{aligned}$$ where: $$n_{d}(i,j;i,j+1)\equiv\sum_{i'=i+1}^L(-1)^{\tilde{n}_{i',j}}+\sum_{i'=1}^{i-1}(-1)^{\tilde{n}_{i',j+1}},$$ counts the number of down spins between sites $(i,j)$ and $(i,j+1)$, exclusively. Here $\tilde{n}_{i',j}=1 (0) $ if there is a down (up) spin at site $(i',j)$. For $t_{y}=0$, $H_{QS}$ reduces to the quantum compass model discussed above (and therefore Eq. (2) in the main text). But the $t_y$ term introduces a sign problem in QMC simulations. Despite the sign problem, the above quantum spin model can also be simulated with SSE combined with the QWL algorithm. We find that, for small $t_y$, the sign problem is not severe. For example, for an $L=4$ system and $t_y=t_x/10$, we find $\langle S \rangle >0.2$ for $T>t_{y}$. For smaller $t_y$ values, we can approach lower temperatures. We have performed QMC simulations on the quantum spin model for $L=4, 6$, and $8$ to detect the emergence of the ground state degeneracy. We discuss an example result in the section, “Ground State Degeneracy”. System-Reservoir Optical Lattice Geometry ----------------------------------------- We show that an optical superlattice can be used to host a 2D “system” lattice parallel to a 2D “reservoir” lattice. The system lattice is an array of chains in the $x-y$ plane that allow strong tunneling along the $x$-direction and weak tunneling along the $y$-direction. The reservoir lattice is a square lattice with nearly equal tunneling along both the $x$ and $y$ direction. The increased dimensionality of the reservoir strengthens the pair superfluid in the reservoir. A tunable potential barrier controls the tunneling between the system and the reservoir. ![Plot of the potential defining a double well optical lattice along the $z$ direction for $v_{z}=-15E_{R}$, $\phi_{1}=0$, and $\phi_{2}=3\pi/2$. []{data-label="double_well"}](double_well_lattice.pdf){width="\columnwidth"} The optical lattice is formed from three laser beam pairs: 1) a double well optical lattice potential, $V_{zz}$, formed from the interference of counter propagating beams along the $z$ direction, 2) a pair of beams with the same polarization counter-propagating in the $x$-$z$ plane, to form $V_{xz}$, and 3) a similar pair of beams but in the $y$-$z$ plane, to form $V_{yz}$. If each beam pair does not interfere then the total potential experienced by the particles is: $V_{\text{tot}}(x,y,z)=V_{zz}(z)+V_{xz}(x,z)+V_{yz}(y,z)$. ![(Color online.) Plot of the total potential for the system-reservoir optical lattice, $V_{\text{tot}}$. Points are plotted for $v_{\text{tot}}< -10 E_{R}$. The parameters are chosen to be: $v_{z}=-15E_{R}$, $v_{x}= -0.5 E_{R} $, $v_{y}= -1 E_{R}$, $\phi_{1}=-(k\pi+2\pi/1.9)$, and $\phi_{2}=-(k\pi/2+2\pi/1.9)$. []{data-label="bath_lattice"}](bath_lattice.pdf){width="\columnwidth"} The system and reservoir are formed from the double well lattice along the $z$ direction. The potential $V_{zz}$ can be formed from the interference of two counter propagating lasers with differing wavelengths. The distance between the system and the reservoir can be changed by using different laser wavelengths to define the double well. We choose the wavelengths to differ by a factor of 2 to yield: $$\begin{aligned} V_{zz}(z)&=&\frac{v_{z}}{2}\left[ \cos\left(kz-\phi_{1}\right)-\cos\left(kz/2-\phi_{2}\right) \right] \nonumber \\\end{aligned}$$ Here the wavevector of the primary lattice is $k=2\pi/\lambda$. This potential is plotted in Fig. \[double\_well\]. We consider an arrangement where the potential established by the remaining beam pairs is given by: $$\begin{aligned} V_{xz}(x,z)&=& v_{x} \left[ \cos\left( k x \right)+ \cos\left( k z \right) \right]^{2} \nonumber \\ V_{yz}(y,z)&=& v_{y} \left[ \cos\left( k y \right)+ \cos\left( k z \right) \right]^{2}\end{aligned}$$ Because the beam pairs forming $V_{xz}$ and $V_{yz}$ each have the same polarization, they interfere to form a node in the $z$ direction at the location of the reservoir. The reservoir then experiences a nearly isotropic square lattice even with $v_{x}\neq v_{y}$. Fig. \[bath\_lattice\] plots an equipotential surface defined by $V_{\text{tot}}$. The potentials are defined in units of the lattice recoil, $E_{R}\equiv h^{2}/2m\lambda^{2}$. Here $m$ is the mass of the particles. Fig. \[bath\_lattice\] shows a configuration where the particles in the system lattice, near $z=0$, have little tunneling along $y$ whereas the reservoir lattice, near $z=-\lambda$, is essentially a 2D square lattice. This geometry allows a 2D dipolar superfluid in the reservoir to be placed in close proximity to the system lattice. [99]{} B. Doucot, M. V. Feigel’man, L. B. Ioffe, and A. S. Ioselevich, Phys. Rev. B [**71**]{}, 024505 (2005). J. Dorier, F. Becca, F. Mila, Phys. Rev. B **72**, 024448 (2005). A. C. Potter and P. A. Lee, Phys. Rev. Lett. [**105**]{}, 227003 (2010); G. Kells, D. Meidan, and P. W. Brouwer, Phys. Rev. B [**85**]{}, 060507(R) (2012); V. Lahtinen, A. W. W. Ludwig, J. K. Pachos, and S. Trebst, Phys. Rev. B [**86**]{}, 075115 (2012). P. Pfeuty, Ann. Phys. [**57**]{}, 79 (1970). E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. [**16**]{}, 407 (1961). Z. Nussinov and E. Fradkin, Phys. Rev. B **71**, 195120 (2005). V. W. Scarola, K. B. Whaley, and M. Troyer, Phys. Rev. B **79**, 085113 (2009). A. Y. Kitaev, Phys.-Usp. **44**, 131 (2001). H. Chen, C. Fang, J. Hu, and H. Yao, Phys. Rev. B **75**, 144401 (2007). O. F. Syljuasen and A. W. Sandvik, Phys. Rev. E **66**, 046701 (2002). M. Troyer, S. Wessel, and F. Alet, Phys. Rev. Lett. **90**, 120201 (2003).

--- abstract: 'We analyse the orbital kinematics of the Milky Way (MW) satellite system utilizing the latest systemic proper motions for 38 satellites based on data from [*Gaia*]{} Data Release 2. Combining these data with distance and line-of-sight velocity measurements from the literature, we use a likelihood method to model the velocity anisotropy, $\beta$, as a function of Galactocentric distance and compare the MW satellite system with those of simulated MW-mass haloes from the APOSTLE and Auriga simulation suites. The anisotropy profile for the MW satellite system increases from $\beta$$\sim$$-2$ at $r$$\sim$20 kpc to $\beta$$\sim$0.5 at $r$$\sim$200 kpc, indicating that satellites closer to the Galactic centre have tangentially-biased motions while those farther out have radially-biased motions. The motions of satellites around APOSTLE host galaxies are nearly isotropic at all radii, while the $\beta(r)$ profiles for satellite systems in the Auriga suite, whose host galaxies are substantially more massive in baryons than those in APOSTLE, are more consistent with that of the MW satellite system. This shape of the $\beta(r)$ profile may be attributed to the central stellar disc preferentially destroying satellites on radial orbits, or intrinsic processes from the formation of the Milky Way system.' author: - | Alexander H. Riley,$^{1}$[^1][](https://orcid.org/0000-0001-5805-5766) Azadeh Fattahi,$^{2}$ Andrew B. Pace,$^{1}$[^2] Louis E. Strigari,$^{1}$ Carlos S. Frenk,$^{2}$ Facundo A. G[ó]{}mez,$^{3,4}$ Robert J. J. Grand,$^{5,6,7}$ Federico Marinacci,$^{8}$ Julio F. Navarro,$^{9}$ R[ü]{}diger Pakmor,$^{7}$ Christine M. Simpson,$^{10,11}$ and Simon D. M. White$^{7}$\ $^{1}$Department of Physics and Astronomy, Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA\ $^{2}$Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\ $^{3}$Instituto de Investigaci[ó]{}n Multidisciplinar en Ciencia y Tecnolog[í]{}a, Universidad de La Serena, Ra[ú]{}l Bitr[á]{}n 1305, La Serena, Chile\ $^{4}$Departamento de F[í]{}sica y Astronom[í]{}a, Universidad de La Serena, Av. Juan Cisternas 1200 N, La Serena, Chile\ $^{5}$Heidelberger Institut f[ü]{}r Theoretische Studien, Schlo[ß]{}-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany\ $^{6}$Zentrum f[ü]{}r Astronomie der Universit[ä]{}t Heidelberg, Astronomisches Recheninstitut, M[ö]{}nchhofstr. 12-14, 69120 Heidelberg, Germany\ $^{7}$Max-Planck-Institut f[ü]{}r Astrophysik, Karl-Schwarzschild-Str. 1, D-85748, Garching, Germany\ $^{8}$Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA\ $^{9}$Department of Physics and Astronomy, University of Victoria, Victoria, BC, V8P 1A1, Canada\ $^{10}$Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637, USA\ $^{11}$Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA bibliography: - 'references.bib' - 'software.bib' date: 'Accepted [2019 April 4]{}. Received [2019 March 24]{}; in original form [2018 October 24]{}' title: The velocity anisotropy of the Milky Way satellite system --- \[firstpage\] galaxies: dwarf – Galaxy: kinematics and dynamics – dark matter Introduction {#sec:intro} ============ PM study $N_\text{sats}$ Satellites considered Methodology $N_\text{stars}$ --------------- ----------------- -------------------------------------- ------------------------------------- ------------------ @helmi 12 classical satellites, Bo[ö]{}tes I Iterative, using only DR2 data 115/339.5/23109 @simon 17 $M_V > -8, d_\odot < 100$ kpc, w/ RV Match RV members in DR2 2/8/68 @fritz () 39 $r<420$ kpc w/ RV Match RV members in DR2 2/18/2527 @kallivayalil 13 possible satellites of LMC Iterative, initial from RV 3/11/110 @massari 7 $M_V<-2.5$, $d_\odot < 70$ kpc Iterative, initial from RV or HB 29/53/189 @pace 14 satellites in DES footprint Probabilistic, incorporated DES DR1 5/15.5/67 Our presence within the Local Group offers it a special importance in astronomy. It is the only part of the Universe where we can detect small ($M_\ast \lesssim 10^5$ M$_\odot$) dwarf galaxies, resolve their stellar populations, and study their internal properties and kinematics. As the most dark-matter dominated galaxies in the Universe [@2012AJ....144....4M], these dwarfs provide crucial tests of the current structure formation paradigm – cold dark matter with a cosmological constant ([$\Lambda$CDM]{}). While several predictions of [$\Lambda$CDM]{} (e.g. large scale structure, temperature anisotropies of the cosmic microwave background) agree with observations extraordinarily well [@2005Natur.435..629S; @2012AnP...524..507F], the model faces a number of challenges on the scales of dwarf galaxy satellites (see for a recent review). [Many of these challenges, including the so-called missing satellites [@1999ApJ...522...82K; @1999ApJ...524L..19M] and too-big-to-fail [@2011MNRAS.415L..40B; @2012MNRAS.422.1203B] problems, have potential solutions through the inclusion of galaxy formation physics [@2000ApJ...539..517B; @2002MNRAS.333..156B; @2002ApJ...572L..23S] that have been reinforced by cosmological hydrodynamic simulations of galaxy formation [@2009MNRAS.399L.174O; @2014ApJ...786...87B; @2016MNRAS.457.1931S; @2016ApJ...827L..23W; @2018MNRAS.478..548S; @2018arXiv180604143G].]{} [A growing challenge to [$\Lambda$CDM]{} is that a large fraction of satellites seem to be located on a co-rotating plane around their host galaxy (the plane-of-satellites problem; see @2018MPLA...3330004P for a recent review). Such planes have been observed around the Milky Way , Andromeda [@2013Natur.493...62I], and galaxies outside of the Local Group [@2014Natur.511..563I; @2015ApJ...805...67I; @2018Sci...359..534M]. The degree to which planes of satellites pose a challenge to [$\Lambda$CDM]{} is contested; some analyses have concluded that a thin planar configuration of satellites is extremely unusual in [$\Lambda$CDM]{} [@2012MNRAS.423.1109P; @2015ApJ...800...34G], but a detailed statistical analysis (taking account of the “look elsewhere effect”) suggests that thin satellite planes like that of the MW and M31 occur in about 10% of galactic systems [@2015MNRAS.452.3838C].]{} Studies of the planes of satellites generally focus on two aspects of satellite kinematics: the clustering of orbital poles and the reconstruction of satellite orbits. The orbital poles of the MW satellites are more clustered than an isotropic distribution, with strong clustering measured for 8 of the 11 classical satellites [@2013MNRAS.435.2116P]. Orbit reconstruction is more challenging since the outcome is sensitive to the total and radial distribution of mass in the Milky Way, which are uncertain (@2015ApJS..216...29B [@2017MNRAS.465...76M]; see Figure 7 of @2019MNRAS.484.5453C for a comparison of recent measurements of the total mass). This translates into large uncertainties in the reconstructed orbits, [making direct comparisons to theoretical predictions more complicated.]{} To study the orbit structure of the satellite population in a potential-independent way, @cautun used the velocity anisotropy, $\beta$, to characterize the orbital properties of the satellites. Introduced by @1980MNRAS.190..873B to quantify the orbital structure of a spherical system, $\beta$ is most commonly used in spherical Jeans equation modeling to recover the mass of a system. In a Galactocentric spherical coordinate system where $r$ corresponds to radial distance, $\theta$ the polar angle, and $\phi$ the azimuthal angle, $\beta$ is defined as: $$\label{eqn:beta} \beta(r) = 1 - \frac{\sigma_\theta(r)^2 + \sigma_\phi(r)^2}{2\sigma_r(r)^2}$$ where $\sigma_r, \sigma_\theta, \sigma_\phi$ are the velocity dispersions along each coordinate direction. The $\beta$ parameter can take values in the range $-\infty$ to 1, where $\beta = 1$ corresponds to radial orbits plunging in and out of the Galactic centre, $\beta \rightarrow -\infty$ to circular orbits, and $\beta = 0$ to velocities being isotropically distributed at each point. [Studies of $\beta$ for the MW have predominantly used either halo stars [@2012MNRAS.424L..44D; @2013ApJ...766...24D; @2016ApJ...820...18C; @2018arXiv181012201C] or globular clusters [@2018ApJ...862...52S; @2019ApJ...873..118W; @2019MNRAS.484.2832V] as tracers. While these studies have largely focused on obtaining a single value of $\beta$ as an input for spherical Jeans equation modeling, the radial anisotropy profile $\beta(r)$ also contains interesting information on the accretion history of the MW. For example, @2018ApJ...853..196L used high-resolution cosmological hydrodynamic simulations to find that dips in the $\beta(r)$ profile of halo stars may be associated with localized perturbations from, or remnants of, destroyed satellites. Such perturbations have been observed in the $\beta(r)$ profile for halo stars in the Milky Way [@2018arXiv181012201C].]{} [To date, the only measurement of $\beta$ using MW satellite galaxies has been from @cautun. Using proper motions derived using the Hubble Space Telescope (HST) for the 10 brightest satellite galaxies, they obtained $\beta = -2.2 \pm 0.4$. The low number of MW satellites with measured proper motions prohibited further studies of the $\beta(r)$ profile until the second data release from [*Gaia*]{} . Since [*Gaia*]{} DR2, proper motions for nearly every MW satellite galaxy have been measured, which now motivates further studies of $\beta$ using the MW satellites as tracers.]{} In this paper, we compare the kinematics of the Milky Way satellites to expectations from [$\Lambda$CDM]{} using state-of-the-art cosmological hydrodynamic zoom [@1993ApJ...412..455K; @1996ApJ...472..460F; @2014MNRAS.437.1894O] simulations: APOSTLE [A Project Of Simulating The Local Environment; @2016MNRAS.457..844F; @2016MNRAS.457.1931S] and Auriga [@2017MNRAS.467..179G]. By focusing on $\beta$, our results only depend on the present-day kinematics of the MW satellites and not the total or radial distribution of MW mass. We use the latest satellite proper motion measurements as deduced from [[*Gaia*]{} DR2]{}, increasing the number of satellites used in an anisotropy analysis from the 10 in @cautun to 38. Furthermore, we utilize a likelihood method to determine the intrinsic $\sigma_i$’s of the MW satellite system. This more robust method, combined with the increased number of satellites spread over a wide range of Galactocentric distances ($\sim15 - 250$ kpc), allows us to perform the first measurement of $\beta(r)$ for the satellites of the Milky Way. This paper is organized as follows. In Section \[sec:data\] we review the new [*Gaia*]{} DR2 proper motions for MW satellites included in our analysis. In Section \[sec:sims\] we describe the cosmological hydrodynamic zoom simulations that provide our predictions within [$\Lambda$CDM]{}. In Section \[sec:analysis\] we detail our methodology for computing $\beta$. In Section \[sec:results\] we present the main results of our analysis and in Section \[sec:discussion\] we provide a possible interpretation of these results. In Section \[sec:conclusions\] we present our conclusions. Proper motions {#sec:data} ============== The public release of [*Gaia*]{} DR2 has profoundly impacted near-field cosmology in a very short period of time. The data release contains an all-sky catalog of the five-parameter astrometric solution (position on the sky, parallax, and proper motion) for more than 1.3 billion sources . These data have already been used in multiple studies of the kinematics of the Milky Way’s stellar halo [e.g. @2018ApJ...862L...1D], satellites [e.g. @2019MNRAS.484.5453C] and globular clusters [e.g. @2019MNRAS.484.2832V]. We use results from six studies (see Table \[tab:studies\] for a summary) which derive [*Gaia*]{} DR2 proper motions for MW satellites with comparable precision to those derived using the Hubble Space Telescope [for a review of proper motions with HST, see @2014ASPC..480...43V]. @helmi demonstrated [*Gaia*]{} DR2’s ability to constrain proper motions for the Magellanic Clouds, the classical (pre-SDSS) satellites, and ultra-faint dwarf Bo[ö]{}tes I. @simon presented the first proper motions for many nearby ($<$100 kpc) ultra-faint dwarf galaxies, while @fritz [hereafter ] extended the limit out to 420 kpc with the largest sample of 39 satellites. @kallivayalil derived proper motions for satellites located near the Magellanic Clouds, motivated by the possibility that some of them may be satellites of the LMC itself. @massari computed proper motions for seven dwarfs, three of which do not have spectroscopic information. @pace presented a probabilistic method of determining systemic proper motions that utilized the superb photometry from the first public data release of the Dark Energy Survey [@2018ApJS..239...18A]. The full list of satellites that we consider in this analysis is presented in Table \[tab:properties\], along with a summary of their properties. For this analysis we only consider satellites out to 300 kpc from the Galactic centre. We omit globular clusters and satellites whose nature is still under debate [e.g. Crater I; @2015ApJ...810...56K; @2016MNRAS.460.3384V]. We also do not consider overdensities that are thought to be tidally disrupting dwarf galaxies: Bo[ö]{}tes III [@2009ApJ...702L...9C; @2018ApJ...865....7C], Canis Major [@2004MNRAS.348...12M], and Hydra I [@2016ApJ...818...39H]. Furthermore, we restrict our analysis to satellites which have published line-of-sight velocities in order to have full 6-D kinematic information. Galactocentric coordinates {#subsec:coords} -------------------------- ![Tangential velocity excess of Milky Way satellites using proper motions from [but LMC and SMC proper motions from @helmi]. A ratio of radial to total kinetic energy $V_\text{rad}^2/V_\text{tot}^2 \lesssim 1/3$ indicates a tangentially biased motion.[]{data-label="fig:excess"}](vtan_excess.pdf){width="\linewidth"} In order to convert the line-of-sight velocity and proper motion measurements into Galactocentric coordinates, we use the distance measurements from Table \[tab:properties\]. The Galactocentric Cartesian coordinates are then computed assuming a distance from the Sun to the Galactic centre of $8.2\pm0.1$ kpc, a height of the Sun relative to the Galactic plane of $25\pm5$ pc, and a solar motion relative to the Galactic centre of ($10\pm1$, $248\pm3$, $7\pm0.5$) km s$^{-1}$ , in a frame where the x-axis points from the position of the Sun projected on to the Galactic plane to the Galactic centre, the y-axis points towards Galactic longitude $l=90^\circ$ (i.e. in the direction of Galactic rotation), and the z-axis points towards the North Galactic Pole. This right-handed Cartesian system is then converted into spherical coordinates, with $r$ the distance from the Galactic centre, polar angle $\theta$ defined from the z-axis, and azimuthal angle $\phi$ defined from the x-axis such that the Galactic rotation is in the $-\phi$ direction. We perform 2,000 Monte Carlo simulations drawing satellite proper motions, line-of-sight velocities, and heliocentric distances randomly from Gaussian distributions centred on their measured values with dispersions given by their respective errors. When drawing the proper motions we account for the correlation between $\mu_{\alpha*} \equiv \mu_\alpha\cos\delta$ and $\mu_\delta$ if provided in the proper motion study. The randomly drawn kinematic properties are then converted into Galactocentric spherical coordinates as described in the previous paragraph. The resulting Galactocentric positions and velocities (and corresponding uncertainties), obtained directly from the observed distance, line-of-sight velocity, and proper motion measurements, are summarized in Table \[tab:galactoprops\]. [To illustrate the tangential nature of the motions of the MW satellites, we show the ratio of of radial to total kinetic energy $V_\text{rad}^2/V_\text{tot}^2$ for each satellite in Figure \[fig:excess\]. A ratio $\lesssim$ 1/3 indicates a tangentially-biased motion. We find that $\sim$80% of MW satellites show a tangential velocity excess, comparable to @cautun who found that 9 of the 10 brightest MW satellites had tangentially-biased motions.]{} Sample selections {#subsec:PMstudy} ----------------- It is important to note that the proper motions derived by both @simon and were based only on matching spectroscopically confirmed member stars with [*Gaia*]{} DR2 data, in some cases depending on very few ($N\sim2-5$) stars to derive a systemic proper motion. The small number statistics could lead to a biased result; @massari found that the subsample of spectroscopic members in Segue 2 used by @simon and is systematically shifted in proper motion space relative to the full sample recovered using their iterative method. To avoid problems from this potential bias, as well as to confirm that our results do not depend strongly on the systematics associated with a particular study, we consider three different samples of proper motion data in our analysis: 1. [38 satellites, comprised of 36 satellites from plus the LMC and SMC. This is the full list of satellites in Table \[tab:properties\].]{} 2. a “gold” sample constructed by prioritizing studies which included steps in their analysis to increase the sample of member stars beyond the spectroscopic sample. For example, @pace used a probabilistic method incorporating photometry from the first public data release of the Dark Energy Survey [@2018ApJS..239...18A]. This “gold” sample consists of 32 satellites,[^3] with proper motions taken from the five other previous studies. 3. the same 32 satellites from the “gold” sample, but using the proper motions. Since [*Gaia*]{} DR2 proper motions for the Magellanic Clouds have only been reported by @helmi, we use these proper motion measurements in all samples. The exact study used for each satellite in the “gold” sample is shown in Table \[tab:properties\]. As detailed in Section \[sec:results\], we find that our results do not depend on which sample is used. We focus on the results for the full 38 satellite sample using proper motions in the following sections. Cosmological simulations {#sec:sims} ======================== [cccccccc]{} Run & $M_{200}$ & $R_{200}$ & $M_\ast$ & $R_{1/2,\ast}$ & $V_\text{circ}(8.2\text{ kpc})$ & $N_\text{subs}$ & $N_\text{subs}$\ & \[$10^{12}$ M$_\odot$\] & \[kpc\] & \[$10^{10}$ M$_\odot$\] & \[kpc\] & \[km s$^{-1}$\] & ($V_\text{max} > 5$ km s$^{-1}$) & ($M_\ast > 0$)\ \ & 1.57 & 238.8 & 2.3 & 8.2 & 173.6 & 1187 & 52\ & 1.02 & 206.8 & 1.1 & 6.2 & 124.3 & 1071 & 41\ & 1.16 & 216.2 & 1.1 & 5.0 & 155.3 & 1006 & 63\ & 1.12 & 213.7 & 1.6 & 4.7 & 148.8 & 1232 & 62\ & 2.04 & 260.6 & 2.2 & 5.5 & 172.6 & 1517 & 76\ & 1.07 & 210.3 & 1.2 & 7.4 & 136.7 & 999 & 27\ & 1.43 & 231.5 & 2.2 & 6.7 & 163.6 & 1105 & 35\ & 0.47 & 160.1 & 1.0 & 6.2 & 121.0 & 669 & 26\ & 0.90 & 198.5 & 1.0 & 3.3 & 150.4 & 810 & 36\ & 0.78 & 189.3 & 0.9 & 4.2 & 136.5 & 784 & 33\ \ Au6 & 1.01 & 211.8 & 6.3 & 4.7 & 224.2 & 517 & 74\ Au16 & 1.50 & 241.5 & 8.8 & 9.6 & 217.7 & 594 & 95\ Au21 & 1.42 & 236.7 & 8.6 & 7.7 & 231.7 & 621 & 97\ Au23 & 1.50 & 241.5 & 8.8 & 8.1 & 240.6 & 582 & 83\ Au24 & 1.47 & 239.6 & 8.5 & 8.4 & 219.0 & 629 & 87\ Au27 & 1.70 & 251.4 & 9.7 & 6.6 & 254.5 & 564 & 104\ MW & $1.1\pm0.3$ & 220.7$^a$ & $5\pm1$ & – & $238\pm15$ & – & $124^{+40}_{-27}$\ To compare our results with the expectations from the standard [$\Lambda$CDM]{} cosmology, we utilize suites of self-consistent cosmological hydrodynamic zoom simulations of Local Group analogs, APOSTLE [@2016MNRAS.457..844F; @2016MNRAS.457.1931S], and of Milky Way analogs, Auriga [@2017MNRAS.467..179G]. These two simulation suites have similar resolution and include similar baryonic processes (e.g. star formation, stellar, supernova and black hole feedback, uniform background UV field for reionization), though the numerical methods and prescriptions for subgrid physics are different (see references in following subsections for details). We also analyse dark-matter-only runs from these suites for comparison. APOSTLE ------- The APOSTLE project is a suite of cosmological hydrodynamic zoom simulations of twelve volumes using the code developed for the EAGLE project [@2015MNRAS.446..521S; @2015MNRAS.450.1937C]. The galaxy formation model includes metallicity-dependent star formation and cooling, metal enrichment, stellar and supernova feedback, homogeneous X-ray/UV background radiation (hydrogen reionization assumed at $z_{reion} = 11.5$), supermassive black-hole formation and growth, and AGN activity [@2009MNRAS.398...53B; @2015MNRAS.454.1038R]. The full details of the subgrid physics can be found in @2015MNRAS.446..521S. The APOSTLE volumes were selected to have similar dynamical properties as the Local Group; the full selection procedure is described in @2016MNRAS.457..844F and a discussion of the main simulation characteristics is given in @2016MNRAS.457.1931S. In summary, each volume consists of a MW/M31-like pair of haloes with [halo]{} mass[^4] ranging from $0.5-2.5\times10^{12}$ M$_\odot$, separated by $800\pm200$ kpc, approaching with radial velocity $<250$ km s$^{-1}$ and tangential velocity $< 100$ km s$^{-1}$. The haloes are isolated, with no additional halo larger than the smaller of the pair within 2.5 Mpc of the midpoint between the pair, and in environments where the Hubble flow is relatively unperturbed out to 4 Mpc. [The simulations adopt the WMAP-7 cosmological parameters [@2011ApJS..192...18K]: $\Omega_M = 0.272$, $\Omega_\Lambda = 0.728$, $h = 0.704$, $\sigma_8 = 0.81$, and $n_s = 0.967$.]{} The volumes were simulated at three resolution levels, the highest of which (and the only level considered here) has primordial gas (DM) particle masses approximately $1.0(5.0)\times10^4$ M$_\odot$, with a maximum force softening length of 134 pc. Five volumes have been simulated so far at this resolution, corresponding to AP-01, AP-04, AP-06, AP-10, and AP-11 in Table 2 of @2016MNRAS.457..844F. Each halo in a pair is treated separately in this analysis, resulting in ten high-resolution APOSTLE haloes being considered in this work. Auriga ------ The Auriga simulations [@2017MNRAS.467..179G] are a suite of cosmological magnetohydrodynamic zoom simulations of single MW-like haloes with [halo]{} masses in the range $1-2\times10^{12}$ M$_\odot$. They were performed with the moving mesh code AREPO [@2010MNRAS.401..791S] and a galaxy formation model that includes primordial and metal line cooling, a prescription for a uniform background UV field for reionization (completed at $z = 6$), a subgrid model for star formation and stellar feedback [@2003MNRAS.339..289S], magnetic fields [@2014ApJ...783L..20P; @2017MNRAS.469.3185P], and black hole seeding, accretion, and feedback. The Auriga haloes were identified as isolated in the $z=0$ snapshot of the parent dark-matter-only simulation with a 100 Mpc box-side length of the EAGLE project introduced in @2015MNRAS.446..521S. [To be considered isolated, the centre of any target halo must be located outside 9 times the [halo]{} radius of any other halo that has a mass greater than 3% that of the target.]{} The simulations assumed the cosmological parameters: $\Omega_M = 0.307$, $\Omega_\Lambda = 0.693$, $h = 0.6777$, $\sigma_8 = 0.8288$, and $n_s = 0.9611$. The volumes were simulated at three resolution levels, the highest of which (and the only level considered here) has baryonic element (DM particle) masses approximately $0.5(4.0)\times10^4$M$_\odot$, with a maximum force softening length of 185 pc, comparable to the highest resolution for APOSTLE. Thus far, six haloes have been resimulated at this high resolution, corresponding to Au6, Au16, Au21, Au23, Au24, and Au27 in Table 1 of @2017MNRAS.467..179G. Stellar discs ------------- ![Circular velocity profiles based on total mass for APOSTLE (blue) and Auriga (orange) haloes. The formation of stellar discs in Auriga is reflected in a much deeper potential near the centre of the halo. For comparison, we also show the circular velocity profile for an NFW [@1996ApJ...462..563N; @1997ApJ...490..493N] halo with $M_{200} = 10^{12}$ M$_\odot$ and $c=10$ (black, dashed), where $c$ is the ratio between the virial radius and the NFW scale radius.[]{data-label="fig:vcirc"}](vcirc.pdf){width="\linewidth"} Even though the APOSTLE and Auriga haloes have similar [halo]{} masses, the difference in their baryon content at $z=0$ affects the shape and depth of their potentials (and hence the dynamics of their satellite systems). A main difference between the two simulation suites is the mass and morphology of the stellar discs of their main galaxies.[^5] The Auriga simulations are able to produce radially extended and thin discs, with sizes comparable to that of the MW [@2017MNRAS.467..179G; @2018MNRAS.481.1726G], while their total stellar masses are slightly higher, close to $10^{11}$ M$_\odot$, than that of the MW. By contrast, the APOSTLE host galaxies have morphologies that are less disky with relatively low stellar mass, $\sim 10^{10}$ M$_\odot$.[^6] A summary of properties for each simulation run is shown in Table \[tab:simprops\]. The total circular velocity profiles, $V_\text{circ}=\sqrt{G M(<r)/r}$, for the two simulations are shown in Figure \[fig:vcirc\]. The different behaviour of the APOSTLE [and Auriga]{} haloes (blue [and yellow]{} curves) compared to the NFW circular velocity profile (black dashed curve) is due to the contraction of haloes in response to the presence of baryons. The much larger difference in the circular velocity profiles between the two simulation suites is due to the more massive stellar discs in Auriga (orange curves) combined with the enhanced dark matter contraction. These differences are useful in quantifying the effect of a stellar disc on $\beta$. The deepening of the potential due to the large baryonic disc, combined with the non-spherical potential of the disc, can affect the tidal stripping of subhaloes. Hydrodynamic simulations suggest that tidal effects from [a]{} baryonic disc near the centre of a host halo can reduce the number of dark substructures by up to a factor of two [@2014ApJ...786...87B; @2016MNRAS.458.1559Z; @2017MNRAS.467.4383S; @2018ApJ...859..129N], an effect that is reproduced in DMO simulations with an embedded disc potential [@2010ApJ...709.1138D; @2015MNRAS.452.2367Y; @2017MNRAS.471.1709G; @2018arXiv181112413K]. This tidal disruption preferentially affects radial orbits that come close to the disc [@2017MNRAS.471.1709G], implying that surviving subhaloes in the inner regions should be on [circular]{} orbits, resulting in a lower $\beta$ near the centre. Matching the [radial distributions]{} {#subsec:matchdata} ------------------------------------- ![The radial distribution of subhaloes with $V_\text{max} > 5$ km s$^{-1}$ in APOSTLE (blue) and Auriga (orange) compared to that of the MW satellites (black). The deepening of the potential in Auriga haloes results in a less centrally concentrated radial distribution compared to APOSTLE, due to more subhaloes being destroyed. The [magenta]{} contours correspond to matching the radial distribution but not number of satellites[, while the green contours correspond to additionally matching the number of satellites]{} (see Section \[subsec:matchdata\] for details) for both suites. Solid curves indicate the median and shaded regions the total spread.[]{data-label="fig:radialdist"}](radial_dist.pdf){width="\linewidth"} [In addition to comparing the results of simulated systems to that of the MW satellites, we would like to select samples of subhaloes that are more representative of the observed MW satellites’ radial distribution. By comparing the results of these subhalo samples to those of the full simulated systems, we will be able to understand the impact of the tracers’ radial distribution on our results. We will also provide a fairer comparison between the simulated systems and the MW satellites.]{} We begin by considering all subhaloes which, at $z$$=$0, have maximum circular velocity $V_\text{max} > 5$ km s$^{-1}$. [This is a conservative resolution limit for both APOSTLE and Auriga, roughly corresponding to subhalo masses of $\sim5\times10^6$ M$_\odot$ or containing $\sim100$ DM particles.]{} These subhaloes are a mix of dark and luminous [(i.e. contain stars)]{}; typically $\sim4$% contain stars in APOSTLE and $\sim15$% contain stars in Auriga (see Table \[tab:simprops\]). [We then create two subhalo samples resulting from]{} (1) matching the radial distribution [of subhaloes to that of the MW satellites]{} and (2) additionally matching the abundance of [subhaloes to that of the MW satellites]{}. When matching both [radial distribution and abundance (case 2)]{}, we simply [compare the Galactocentric distance of each MW satellite to the host-centric distance of each subhalo and select the closest match]{} (without replacement). When only matching the radial distribution and not the abundance [(case 1)]{}, we select subhaloes based on the following inverse transform sampling method: 1. Compute the cumulative distribution function (CDF) of [MW satellite Galactocentric distances]{}. 2. Generate a random number uniformly between 0 and 1 and map that number to a [distance]{} using the CDF from step 1. 3. Select the subhalo [that has a host-centric distance that is closest to the randomly generated distance]{} and add it to the sample if it is within 5 kpc of the randomly generated value. This subhalo is removed from possible selection in the future, i.e. without replacement. 4. Repeat steps (ii)-(iii) until a [distance]{} is generated that does not have a subhalo match within 5 kpc. The 5 kpc cutoff is meant to strike a balance between increasing the number of subhaloes in the sample (higher cutoff) and providing a close match [between the radial distribution of subhaloes]{} to [that of the]{} MW satellites (lower cutoff). Our results are not sensitive to the exact value of this cutoff. Using this method, we typically find [subhalo]{} radial distributions that are much closer to the MW satellite distribution than that of the original subhalo populations (see Figure \[fig:radialdist\]). Likelihood Analysis {#sec:analysis} =================== We seek to model the orbital kinematics of Milky Way satellites and compare the results with those of cosmological simulations using the velocity anisotropy parameter $\beta$. Two models are considered: (1) a constant value of $\beta$ at all radii and (2) one in which $\beta$ varies as a function of Galactocentric distance. To determine the posterior probability densities for each model, we use `emcee` [@emcee], an implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC). Framework and [constant $\beta$ model]{} {#subsec:uniform} ---------------------------------------- We assume that the velocity distribution of the MW satellite system in Galactocentric spherical coordinates ($r, \theta, \phi$) is a multivariate Gaussian with different means and dispersions in each direction. The resulting likelihood $F_i$ for a given satellite $i$ with velocity ${\boldsymbol{v}}_i = (v_{r,i}, v_{\theta, i}, v_{\phi,i})$ is then $$F_i = \frac{1}{\sqrt{(2\pi)^3 \left|{\boldsymbol{\Sigma}}_i\right|}}\exp\left[-\frac{({\boldsymbol{v}}_i-{\boldsymbol{v}}_\text{sys})^\text{T} {\boldsymbol{\Sigma}}_i^{-1}({\boldsymbol{v}}_i-{\boldsymbol{v}}_\text{sys})}{2} \right], \label{eqn:gaussian}$$ where ${\boldsymbol{v}}_\text{sys} = (v_r, v_\theta, v_\phi)$ are the intrinsic mean velocities of the system [(i.e. the entire population of MW satellites)]{} and the covariance matrix is $${\boldsymbol{\Sigma}}_i \equiv \begin{bmatrix} \sigma_r^2 + \delta_r^2 & C_{r\theta}\delta_r\delta_\theta & C_{r\phi}\delta_r\delta_\phi \\ C_{r\theta}\delta_r\delta_\theta & \sigma_\theta^2 + \delta_\theta^2 & C_{\theta\phi}\delta_\theta\delta_\phi \\ C_{r\phi}\delta_r\delta_\phi & C_{\theta\phi}\delta_\theta\delta_\phi & \sigma_\phi^2 + \delta_\phi^2 \end{bmatrix}.$$ Here, $(\sigma_r, \sigma_\theta, \sigma_\phi)$ are the intrinsic dispersions of the system [(i.e. the entire population of MW satellites)]{} and $(\delta_r, \delta_\theta, \delta_\phi, C_{r\theta}, C_{r\phi}, C_{\theta\phi})$ are the observed measurement errors and correlation coefficients for the velocities of the given satellite, which are obtained from the 2,000 Monte Carlo samples described in Section \[subsec:coords\]. Due to the conversion from heliocentric to Galactocentric spherical coordinates, the resulting satellite velocity errors are not necessarily Gaussian in each component. We find that approximating the errors as Gaussian is reasonable in most cases, though Draco II, Tucana III, and Willman 1 show significant skewness and kurtosis in both $v_\theta$ and $v_\phi$. The combined log-likelihood for the full satellite sample is then $$\begin{aligned} \ln \mathcal{L} &= \sum_{i=1}^{N_\text{sats}} \ln F_i = -\frac{1}{2}\sum_{i=1}^{N_\text{sats}} 3\ln{2\pi} + \ln{\left|{{\boldsymbol{\Sigma}}_i}\right|} + {\boldsymbol{u}}_i^\text{T} {\boldsymbol{\Sigma}}_i^{-1}{\boldsymbol{u}}_i \\ &\propto -\sum_{i=1}^{N_\text{sats}} {\boldsymbol{u}}_i^\text{T} {\boldsymbol{\Sigma}}_i^{-1}{\boldsymbol{u}}_i + \ln \left|{\boldsymbol{\Sigma}}_i\right|, \label{eqn:likelihood}\end{aligned}$$ where ${\boldsymbol{u}}_i \equiv {\boldsymbol{v}}_i - {\boldsymbol{v}}_\text{sys}$. Equation \[eqn:likelihood\] is the likelihood function used to probe the model parameter space with `emcee`. The first model we consider assumes constant velocity dispersions at all radii, resulting in a constant value for $\beta$. We impose spherical symmetry by requiring $v_r = v_\theta = 0$ and $\sigma_\theta^2 = \sigma_\phi^2$[, as is commonly assumed in Jeans equation modeling of the dynamics of a system]{}. In total this model then has 3 free parameters: a mean rotational motion $v_\phi$ and dispersions $\sigma_r$ and $\sigma_\theta$. We assume uniform priors for the mean motion $-500 < v_\phi < 500$ km s$^{-1}$ as well as for the dispersions $0 < \sigma_i < 300$ km s$^{-1}$. We repeat the analysis with Jeffreys prior $-3 < \log_{10}\sigma_i < 3$ and find that this does not meaningfully change our results. Variable dispersions with radius {#subsec:variable} -------------------------------- To take advantage of the increased number of satellites with proper motions over a wide range of Galactocentric distances, we include a separate likelihood analysis in which we adopt a simple model for the velocity dispersion to vary as a function of radius in each coordinate $j$: $$\label{eqn:variable} \sigma_j(r) = {\sigma_{j,0}}{\left(1 + \frac{r}{r_{j,0}}\right)^{-\alpha_j}}$$ where $\sigma_{j,0}$ and $r_{j,0}$ are the characteristic dispersion and length scales and $\alpha_j$ is the slope of the fall off at large radii. We then use the same likelihood function as in Section \[subsec:uniform\] (specifically Equation \[eqn:likelihood\]) with the additional parameters introduced in Equation \[eqn:variable\]. The $\beta(r)$ profile then follows from Equation \[eqn:beta\]. ![Posterior distributions for [all 38]{} Milky Way satellites assuming the [constant $\beta$ model, using proper motions]{}. From left to right the parameters are: $v_\phi$ (systemic rotational motion in km s$^{-1}$), $\sigma_r$, and $\sigma_\theta=\sigma_\phi$ (intrinsic velocity dispersions in km s$^{-1}$). The contours enclose 39.4, 85.5, 98.9% of the posterior distribution corresponding to $1, 2, 3 - \sigma$ confidence intervals. The dotted lines on the 1-D histograms are the 16[th]{}, 50[th]{}, and 84[th]{} percentiles and the numerical values are quoted above.[]{data-label="fig:uniformcorner"}](corner_uniform.pdf){width="\linewidth"} As in the [constant $\beta$ model]{}, we impose spherical symmetry by requiring $v_r = v_\theta = 0$ and $\sigma_\theta^2(r) = \sigma_\phi^2(r)$. In total this model then has 7 parameters: a mean rotational motion $v_\phi$ (which is held constant with $r$), the characteristic dispersion and length scales $\sigma_{i,0}$ and $r_{i,0}$, and the slope $\alpha_i$, for both Galactocentric spherical coordinates $r$ and $\theta$. We assume the same uniform prior for the mean motion $-500 < v_\phi < 500$ km s$^{-1}$ as in the [constant $\beta$ analysis]{}. For the $\sigma_i(r)$ parameters we assume uniform priors $50 < \sigma_{i,0} < 1000$ km s$^{-1}$, $10 < r_{i,0} < 1000$ kpc, and $0 < \alpha_i < 10$. We repeat the analysis with Jeffreys priors $-3 < \log_{10}\sigma_{i,0} < 3$ and $-3 < \log_{10} r_{i,0} < 3$ and again find that this change of priors does not meaningfully change our results. Results {#sec:results} ======= We now present the resulting posterior probability densities for $\beta(r)$ for the MW satellite system using the models described above. We show that satellites within $r \lesssim 100$ kpc have more tangentially-biased motions (lower $\beta$) than those farther away. This result is also seen in simulated MW analogs, but it is difficult to disentangle effects due to the central stellar disc from those imprinted at formation. From here onwards, we refer to dark-matter-only simulations from the APOSTLE and Auriga suites collectively as “DMO" and the haloes simulated with baryonic physics by their suite name. [Constant $\beta$ model]{} -------------------------- The posterior distribution of parameters for the [constant $\beta$ model]{} for the MW satellites is shown in Figure \[fig:uniformcorner\] and the resulting posterior for $\beta$ is shown in Figure \[fig:uniform\] (blue curve). We find that the satellites are overall on near tangential orbits, with $\beta = -1.02_{-0.45}^{+0.37}$. We do not find significant evidence for [the MW satellite population exhibiting]{} rotation parallel to the plane of the Milky Way [disc]{} ($v_\phi = -14_{-20}^{+20}$ km s$^{-1}$[; note that a star located in the disc would have $v_\phi$ on the order of $\sim$100 km s$^{-1}$]{}). [We also find that the constant $\beta$ results are similar when using the different samples described in Section \[subsec:PMstudy\] (“gold” sample and “gold” satellites with proper motions).]{} ![Posterior distributions for $\beta$ assuming the [constant $\beta$ model with proper motions.]{} The results are shown using [all 38]{} satellites (blue), satellites within 100 kpc (orange, [23 satellites]{}), and satellites outside of 100 kpc (green, [15 satellites]{}). The vertical black dashed line corresponds to the isotropic case $\beta=0$.[]{data-label="fig:uniform"}](uniform.pdf){width="\linewidth"} To better understand the results from the variable $\beta(r)$ model, we examine two radial bins. We split the satellites into two populations, one with $r < 100$ kpc (23 satellites) and the other with $r > 100$ kpc (15 satellites), and perform the same [constant]{} $\beta$ analysis on each. The inner and outer regions clearly have different posterior distributions (Figure \[fig:uniform\], orange and green curves respectively), with the inner region having a more negative (i.e., more tangentially biased) $\beta$ posterior than the outer region. These results do not change when considering each of the different proper motion samples described in Section \[subsec:PMstudy\]. This supports the finding in the $\beta(r)$ model (discussed below) that satellites in the inner region ($r \lesssim 100$ kpc) have more tangentially-biased motions than those farther away. ### [Comparison to Cautun & Frenk [(2017)]{}]{} [Our [constant]{} $\beta$ result agrees within 2$\sigma$ with the result of @cautun. These authors found $\beta = -2.2 \pm 0.4$ using HST proper motions of only 10 of the brightest satellites and simply computing $\beta$ from Monte Carlo realizations of the MW satellite system using observational errors. Using updated [*Gaia*]{} DR2 proper motions and our likelihood method, our result for that same subsample of 10 satellites is $\beta = -1.52_{-1.23}^{+0.86}$, which is consistent with @cautun. The small offset is likely due to different input data and analysis techniques.]{} Variable $\beta$ model ---------------------- ![Posterior distributions for the $\beta(r)$ model for [the 38]{} MW satellites [using proper motions]{}. Top: posterior for $\beta(r)$. The horizontal black dashed line corresponds to the isotropic case, $\beta=0$. Bottom: posterior for the systemic dispersion in $\sigma_r$ ([orange]{}) and $\sigma_\theta = \sigma_\phi$ ([green]{}). The [brown]{} ticks along the middle axis mark the radial positions of the MW satellites. For both panels, solid curves correspond to the median values and the shaded region the $16-84$% confidence interval.[]{data-label="fig:variable"}](variable.pdf){width="\linewidth"} The posterior distribution for the parametrized $\beta(r)$ model [for the MW satellites]{} is shown in Figure \[fig:variable\] [(top panel), along with the dispersions $\sigma_r$ and $\sigma_\theta = \sigma_\phi$ (bottom panel)]{}. We find that the radial profile dips in the inner ($<100$ kpc) region to $\beta$$\sim$$-2$ at $r\sim20$ kpc and flattens out to $\beta$$\sim$0.5 in the outer region. This again indicates that satellites near the centre of the Milky Way have tangentially-biased motions, while satellites in the outer region have more radially-biased motions. ![[Posterior distributions for the $\beta(r)$ model using different satellite samples. The “gold” sample and “gold” satellites using proper motions are as described in Section \[subsec:PMstudy\]. “No LMC satellites” excludes the candidate satellites of the LMC as identified by @kallivayalil. We define ultra-faint dwarf galaxies as those fainter than $10^5$ L$_\odot$ and refer to galaxies brighter than this limit as “classical.” The results from these different input samples are all consistent (within 68% confidence) with the original full sample of 38 satellites.]{}[]{data-label="fig:othersamples"}](variable_othersamples.pdf){width="\linewidth"} {width="\linewidth"} Using the [other]{} proper motion samples described in Section \[subsec:PMstudy\] does not impact the results; the “gold” sample and “gold” satellites with proper motions have nearly the same $\beta(r)$ profile. Furthermore, this dip in $\beta$ does not appear to be dependent on a particular satellite or population of satellites. We repeated the analysis removing Sagittarius (which has a well-constrained proper motion at $r\sim18$ kpc), removing satellites with luminosities above or below the median luminosity, and removing candidate satellites of the LMC identified by @kallivayalil: Horologium I, Carina II, Carina III, and Hydrus I. [The results from these different input samples are all consistent (within 68% confidence) with the original full sample of 38 satellites (see Figure \[fig:othersamples\]).]{} Taken together, these results indicate that satellites closer to the Galactic centre have more tangentially-biased (near-circular) motions than those farther away. This dip in $\beta(r)$ could be a reflection of the destruction of substructure by the central stellar disc, as discussed by @2017MNRAS.467.4383S and @2017MNRAS.471.1709G. To interpret this result for the MW satellite system, we move on to analyse the simulated systems in the APOSTLE and Auriga suites using the same methods. {width="\linewidth"} Simulations ----------- The posterior distributions for the $\beta(r)$ profiles of simulated MW analogs are shown in [Figure \[fig:apostle\] for APOSTLE and Figure \[fig:auriga\] for Auriga]{}. When considering all subhaloes with $V_\text{max} > 5$ km s$^{-1}$ (top row), it is clear that the presence of a massive stellar disc affects the radial $\beta$ profile. The $\beta(r)$ profiles for DMO hosts are nearly flat at $\beta\sim0$, indicating isotropic motions at all radii. The inclusion of baryons in APOSTLE does not have a noticeable effect on the $\beta(r)$ profiles, which are very similar to DMO. Only the Auriga haloes exhibit a dip in the $\beta$ profile near the centre, resulting from the massive central disc preferentially destroying radial orbits that pass near the galaxy. However, the $\beta(r)$ profile estimates of our simulated systems are sensitive to the radial distribution of the subhaloes. Matching the radial distribution of subhaloes with that of the MW satellites, following the procedure described in Section \[subsec:matchdata\], results in similar $\beta(r)$ estimates in some systems that do not contain stellar discs; the estimates for some of the APOSTLE and even DMO systems become consistent with the results for the MW satellites (second row). This similarity is even more pronounced when the subhaloes are also selected to match the total number of MW satellites in addition to the radial distribution (third row). While these corrections bring some DMO and APOSTLE estimates of $\beta(r)$ in line with that of the MW, the Auriga systems still provide the best agreement. There are many corrected DMO and APOSTLE systems that still have $\beta\sim0$ near the centre, but only a few corrected Auriga profiles that do not have a dip in $\beta$. These results suggest that the dip in the $\beta$ profile for the MW satellite system is likely best explained by effects due to the stellar disc, but also is sensitive to the radial distribution of tracers considered. Finally, we also consider the sample of subhaloes in APOSTLE and Auriga that contain stars at $z=0$ ($M_\ast > 0$, bottom row). Nearly every $\beta(r)$ profile matches that of the full subhalo population, albeit with increased scatter due to a smaller sample size (see Table \[tab:simprops\] for the number in each population). The only exception is the less massive halo of AP-01, whose $\beta$ profile is shifted to lower values at all radii but maintains the same shape. The agreement between the $\beta(r)$ profiles when considering all subhaloes [*vs.*]{} the subsample containing stars suggests that the $\beta$ profile (corrected for observational distance biases) traced by the MW satellites is likely indicative of the intrinsic profile for the MW, unaffected by the complex physics that dictate which subhaloes are populated by satellites [@2015MNRAS.448.2941S; @2017MNRAS.464.3108G; @2019ApJ...873...34N]. Discussion {#sec:discussion} ========== It is clear from our results that dwarf galaxy satellites closer to the centre of the Milky Way have tangentially-biased motions while those farther from the centre have radially-biased motions. In this Section we explore some interpretations of the $\beta(r)$ profile and place our results in the context of those from other tracers. Stellar disc ------------ The inclusion of baryonic processes in cosmological simulations has helped resolve a number of small-scale challenges to [$\Lambda$CDM]{}. A notable effect is the destruction of substructure due to the potential of a massive stellar disc. @2014ApJ...786...87B found that 6 of the 8 subhaloes in a DMO simulation that did not have a baryonic simulation counterpart had pericentric passages that took them within 30 kpc of the galaxy centre. @2017MNRAS.467.4383S found that the presence of baryons near the centre of APOSTLE haloes reduces the number of subhaloes by factors of $\sim1/4 - 1/2$ independently of subhalo mass but increasingly towards the host halo centre. @2017MNRAS.471.1709G found similar destruction of subhaloes in the Latte simulation suite and showed that simply embedding a central disc potential in DMO simulations reproduced these radial subhalo depletion trends, arguing that the additional tidal field from the central galaxy is the primary cause of subhalo depletion [see also @2010ApJ...709.1138D; @2015MNRAS.452.2367Y; @2017MNRAS.465L..59E]. We also note that @2016MNRAS.458.1559Z found similar results in simulations using the AREPO code, the same code with which Auriga was performed. A central stellar disc, whether artificially embedded in DMO simulations or formed through the inclusion of baryonic physics, preferentially destroys subhaloes on radial orbits that pass close to the disc. The surviving population then has tangentially-biased motions compared to DMO [@2017MNRAS.467.4383S; @2017MNRAS.471.1709G], which is expected to be reflected in a lower value of $\beta$. However, there is also a radial dependence of $\beta$ which has not yet been explored; with increasing distance from the central galaxy the destructive effects of the disc potential weaken, causing the $\beta(r)$ profile to rise to $\beta \sim 0.5$ in the outer region ($100\text{ kpc} < r < R_\text{vir}$) as subhaloes are more likely to be on their first infall [@2004MNRAS.352..535D; @2010MNRAS.402...21N]. Additionally, as surviving massive satellites pass near the stellar disc, both experience a torque and exchange angular momentum, likely inducing further circularization of the surviving satellite orbits [@2017MNRAS.465.3446G; @2017MNRAS.472.3722G]. Our simulation results are consistent with this interpretation. When considering all subhaloes with $V_\text{max} > 5$ km s$^{-1}$ in the APOSTLE and Auriga suites (top rows of [Figures \[fig:apostle\] and \[fig:auriga\]]{}), the $\beta(r)$ profiles for DMO and APOSTLE haloes, which have less massive central galaxies, are relatively constant with $\beta \gtrsim 0$. In stark contrast, the $\beta(r)$ profiles for Auriga haloes have $\beta \lesssim -0.5$ near the centre of the halo and increase to $\beta \gtrsim 0$ by $\sim 200$ kpc. These results are similar when considering, instead, subhaloes that contain stars at $z=0$ (bottom rows of [Figures \[fig:apostle\] and \[fig:auriga\]]{}). The Auriga simulations produce stellar discs that are massive, thin, and radially extended, like that of the MW, while APOSTLE forms less massive host galaxies with weaker discs. This distinction impacts the orbital distribution of subhaloes and results in the Auriga subhaloes showing a variation of $\beta$ similar to that of the MW satellite system. [However, it is worth noting the possibility that not all of the subhalo disruption is due to the physical effects of the stellar disc. @2018MNRAS.474.3043V and @2018MNRAS.475.4066V raise concerns about artificial subhalo disruption due to numerical effects, suggesting that tracking subhalo disruption requires many more particles than required for typical simulation convergence tests. For example, @2018MNRAS.474.3043V find that orbits passing within 10-20% of the virial radius of a host may require $N > 10^6$ particles for an accurate treatment. More work may be required to understand the differences between these results and those from typical convergence tests. This will, in turn, inform our understanding of how much subhalo disruption is due to physical effects of the stellar disc vs. numerical effects and how this impacts the inferred $\beta(r)$ profile.]{} The radial distribution ----------------------- This clean interpretation of a $\beta(r)$ profile caused by the tidal field of the central galaxy becomes muddier when accounting for the observed radial distribution of the Milky Way satellites. We know the current census of satellites is incomplete both radially, due to surface brightness and luminosity selection effects, and in area on the sky, as less than half of the sky has been covered by surveys capable of finding ultra-faint satellite galaxies [@2018PhRvL.121u1302K; @2018MNRAS.479.2853N]. This results in a satellite sample that is more centrally concentrated than those found in M31 and in cosmological simulations (@2014MNRAS.439...73Y [@2018arXiv180803654G; @2018arXiv181112413K], however, see @2019MNRAS.483.2000L), giving greater weight to satellites located closer to the centre. We attempt to account for this by matching the abundance and/or radial distribution of simulated subhaloes with $V_\text{max} > 5$ km s$^{-1}$ to that of the MW satellites. Applying these corrections to simulated MW-mass systems tends to lower $\beta$ estimates relative to when the full population is used (see [Figures \[fig:apostle\] and \[fig:auriga\], middle two rows]{}). As a result, the inferred $\beta(r)$ profiles for some DMO and APOSTLE haloes, which do not contain massive central galaxies, are consistent with that of the MW satellites. This is not to say that the impact of the central disc is not crucial to explaining the anisotropy of the MW satellite system. As shown in [Figures \[fig:apostle\] and \[fig:auriga\]]{}, for any given selection criterion applied to the subhaloes the dip in the $\beta(r)$ profiles is most prominent for the Auriga host haloes, which have massive central discs. However, a more complete analysis of the MW disc’s impact on the $\beta(r)$ profile would require understanding the true selection function for the MW satellites. Knowing this selection function, combined with a more [detailed]{} modeling procedure [(e.g. using a distribution function in action/angle coordinates, as in @2019MNRAS.484.2832V, would lend greater insight to orbital properties at the expense of assuming a MW potential)]{}, would enhance future studies of the $\beta(r)$ profile for the MW satellite galaxies. Comparison with other tracers {#subsec:othertracers} ----------------------------- ![Comparison of MW $\beta(r)$ results between different tracers. The [grey]{} points and contours correspond to studies using globular clusters, while the black points correspond to studies using halo stars. The blue contours are the results from this work. The horizontal black dashed line corresponds to the isotropic case, $\beta=0$.[]{data-label="fig:tracers"}](tracer_comparison.pdf){width="\linewidth"} Finally, it is interesting to compare our $\beta(r)$ results with those using other tracers of the MW potential (see Figure \[fig:tracers\]). @2018arXiv181012201C used HST proper motions of $N\sim200$ halo stars in four different fields, spherically averaging to find $\beta\sim0.5-0.7$ over $19 < r < 29$ kpc. This is higher than the values found in several other studies using line-of-sight velocities alone to constrain the anisotropy of the stellar halo, which tend to prefer isotropic or tangentially-biased $\beta$ values [for a summary, see Figure 6 in @2016ApJ...820...18C]. Using MW globular clusters (GCs), @2018ApJ...862...52S estimated $\beta = 0.609_{-0.229}^{+0.130}$ over $10.6 < r < 39.5$ kpc with proper motions from HST while @2019ApJ...873..118W found $\beta = 0.48_{-0.20}^{+0.15}$ over $2.0 < r < 21.1$ kpc with proper motions from [*Gaia*]{} DR2. These two values suggest a trend for the GC orbits to become more radially-biased with increasing distance. Indeed, @2019MNRAS.484.2832V modelled $\beta(r)$ for the MW GCs using a distribution function-based method in action/angle space and found a steady increase from $\beta\sim0.0$ at 0.5 kpc to $\beta\sim0.6$ at 200 kpc [see Figure 7 of @2019MNRAS.484.2832V], consistent with these other results. The dip in the $\beta(r)$ profile for the MW globular cluster system detected by @2019MNRAS.484.2832V is qualitatively similar in shape to what we find for the MW dwarf satellites, but is very different both in characteristic radial scale and in overall amplitude. At $r>100$ kpc the inferred values of $\beta$ are similar. [The dip in the globular cluster profile may possibly be attributed to the accretion history vs. in situ formation [@2001ApJ...561..751F; @2008ApJ...689..919P].]{} It is [also]{} possible that both the globular clusters and stellar halo are remnants of stars previously attached to subhaloes on radial orbits, which are preferentially destroyed by the stellar disc, and maintain the anisotropy of their progenitors (see @2001ApJ...548...33B [@2005ApJ...635..931B; @2008ApJ...680..295B] for halo stars; @1984ApJ...277..470P [@2006MNRAS.368..563M; @2017MNRAS.472.3120B] for globular clusters). This potential connection between different tracers of the MW $\beta(r)$ profile merits further modeling, possibly with a joint analysis of Milky Way halo stars, globular clusters, and dwarf galaxies. Summary {#sec:conclusions} ======= In this work we have analysed the kinematics of 38 Milky Way satellites focusing on an estimate of the velocity anisotropy parameter, $\beta$, and its dynamical interpretation. Utilizing the latest satellite proper motions inferred from [*Gaia*]{} DR2 data, we modelled $\beta$ using a likelihood method and, for the first time, estimated $\beta(r)$ for the MW satellite system. We then compared these results with expectations from [$\Lambda$CDM]{} using the APOSTLE and Auriga simulation suites. A summary of our main results is as follows: - The MW satellites have overall tangentially-biased motions, with best-fitting [constant]{} $\beta = -1.02_{-0.45}^{+0.37}$. By parametrizing $\beta(r)$, we find that the anisotropy profile for the MW satellite system increases from $\beta\sim-2$ at $r\sim20$ kpc to $\beta\sim0.5$ at $r\sim200$ kpc, indicating that satellites closer to the Galactic centre have tangentially-biased motions while those farther out have radially-biased motions. - Comparing these results with the APOSTLE and Auriga galaxy formation simulations, we find that satellites surrounding the massive and radially extended stellar discs formed in Auriga have similar $\beta(r)$ profiles to that of the MW, while the weaker discs in APOSTLE produce profiles that are similar to those from DMO simulations. This suggests that the central stellar disc affects the $\beta(r)$ profile of the MW satellite system by preferentially destroying radial orbits that pass near the disc, as discussed by @2017MNRAS.467.4383S and @2017MNRAS.471.1709G. - However, when matching the radial distributions of simulated subhaloes to that of the MW satellites, some of the inferred $\beta(r)$ profiles for APOSTLE and even DMO haloes also can match the MW data. This implies that the partial sky coverage and the increasing incompleteness with distance of the currently available satellite sample significantly impair the ability of our scheme to robustly estimate the true $\beta(r)$ profile. The difficulty in interpreting the inferred $\beta(r)$ profile may also be alleviated by more fully exploring the Milky Way’s virial volume. @2018MNRAS.479.2853N expect that – assuming a MW halo mass of $1.0\times10^{12}$ M$_\odot$ – there are $46_{-8}^{+12}$ ultra-faint [($-8 < M_V \leq -3$)]{} satellites and $61_{-23}^{+37}$ hyper-faint ($-3 < M_V \leq 0$) satellites within 300 kpc that are detectable. At least half of these satellites should be found by LSST within the next decade. Obtaining proper motions for these faint and distant objects will be challenging but clearly possible, given the results already obtained for 7 satellites fainter than $M_V = -5$ and farther than $d_\odot = 100$ kpc. Furthermore, since the precision in proper motion measurements grows as the 1.5 power of the time baseline, the satellite proper motions from [*Gaia*]{} should be a factor 4.5 more precise after the nominal mission and possibly a factor 12 more precise after the extended mission . [Artificially scaling the observed proper motion errors by a factor of 4.5 results in a $\beta(r)$ profile that has a narrower confidence interval by a factor of $\sim$1.5 and provides better agreement with the Auriga $\beta(r)$ profiles.]{} With this improved dataset, future studies will be less limited by observational selection effects and be able to study in greater depth the impact of the central stellar disc on the $\beta(r)$ profile of the Milky Way satellite system. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Marius Cautun for useful discussions regarding this work [and an anonymous referee for helpful suggestions regarding its presentation]{}, as well as Emily Cunningham and Eugene Vasiliev for sharing their data. This research made use of the Python Programming Language, along with many community-developed or maintained software packages including Astropy [@astropy13; @astropy18], corner.py [@corner], emcee [@emcee], Jupyter [@jupyter], Matplotlib [@matplotlib], NumPy [@numpy], Pandas [@pandas], and SciPy [@scipy]. This research made extensive use of [arXiv.org](https://arxiv.org/) and NASA’s Astrophysics Data System for bibliographic information. AHR acknowledges support from a Texas A&M University Merit Fellowship. AF is supported by a European Union COFUND/Durham Junior Research Fellowship (under EU grant agreement no. 609412). ABP acknowledges generous support from the Texas A&M University and the George P. and Cynthia Woods Institute for Fundamental Physics and Astronomy. LES acknowledges support from DOE Grant de-sc0010813. CSF acknowledges a European Research Council (ERC) Advanced Investigator grant (DMIDAS; GA 786910). FAG acknowledges financial support from FONDECYT Regular 1181264, and funding from the Max Planck Society through a Partner Group grant. RG acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) Research Centre Sonderforschungsbereich (SFB)-881 ‘The Milky Way System’ through project A1. [This work partly used the computing and storage hardware provided by WestGrid ([www.westgrid.ca](https://www.westgrid.ca/)) and Compute Canada Calcul Canada ([www.computecanada.ca](https://www.computecanada.ca/)), as well as UK Research Data Facility ([http://www.archer.ac.uk/documentation/rdf-guide](http://www.archer.ac.uk/documentation/rdf-guide/)).]{} Part of the simulations of this paper used the SuperMUC system at the Leibniz Computing Centre, Garching, under the project PR85JE of the Gauss Centre for Supercomputing. This work was supported in part by the UK Science and Technology Facilities Council (STFC) ST/P000541/1. This work used the DiRAC Data Centric system at Durham University, operated by the ICC on behalf of the STFC DiRAC HPC Facility ([www.dirac.ac.uk](https://dirac.ac.uk/)). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. MW satellite properties ======================= Table \[tab:properties\] lists the observed properties of the MW satellites used throughout this analysis. Table \[tab:galactoprops\] lists the Galactocentric spherical positions and velocities, along with corresponding uncertainties, of each satellite obtained by the Monte Carlo sampling detailed in Section \[subsec:coords\]. The references in the last two columns of Table \[tab:properties\] are as follows: \[1\] @helmi; \[2\] @simon; \[3\] @kallivayalil; \[4\] @massari; \[5\] @pace; \[6\] @2016MNRAS.463..712T; \[7\] @2006ApJ...653L.109D; \[8\] @2008ApJ...684.1075M; \[9\] @2011ApJ...736..146K; \[10\] @2008ApJ...688..245W; \[11\] @2009ApJ...690..453K; \[12\] @2008ApJ...674L..81K; \[13\] @2007ApJ...670..313S; \[14\] @2008ApJ...675L..73G; \[15\] @2012ApJ...756...79S; \[16\] @2015AJ....150...90K; \[17\] @2014MNRAS.444.3139M; \[18\] @2009AJ....137.3100W; \[19\] @2018MNRAS.475.5085T; \[20\] @2018ApJ...857..145L; \[21\] @2010AJ....140..138M; \[22\] @2009ApJ...695L..83M; \[23\] @2017ApJ...839...20C; \[24\] @2016MNRAS.459.2370T; \[25\] @2015MNRAS.448.2717W; \[26\] @2004AJ....127..861B; \[27\] @2015ApJ...813...44L; \[28\] @2009AJ....138..459P; \[29\] @2015ApJ...805..130K; \[30\] @2016ApJ...819...53W; \[31\] @aden; \[32\] @2012ApJ...756..121M; \[33\] @2015ApJ...811...62K; \[34\] @2015ApJ...810...56K; \[35\] @2016AJ....151..118V; \[36\] @2015ApJ...804L...5M; \[37\] @2018MNRAS.479.5343K; \[38\] @2008ApJ...675..201M; \[39\] @2014PASP..126..616S; \[40\] @2017ApJ...836..202S; \[41\] @2005MNRAS.360..185B; \[42\] @2009ApJ...699L.125M; \[43\] @2017MNRAS.467..573C; \[44\] @2017ApJ...845L..10M; \[45\] @2012AJ....144....4M; \[46\] @2015ApJ...808...95S; \[47\] @2018ApJ...863...25M; \[48\] @2015MNRAS.454.1509M; \[49\] @2007ApJ...654..897B; \[50\] @2011ApJ...733...46S; \[51\] @2013AJ....146...94B; \[52\] @2013ApJ...770...16K; \[53\] @2017MNRAS.467..208O; \[54\] @2017ApJ...838...83K; \[55\] @2017AJ....154..267C; \[56\] @2015ApJ...807...50B; \[57\] @2017ApJ...838...11S; \[58\] @2015ApJ...813..109D; \[59\] @2013ApJ...767...62G; \[60\] @2012ApJ...752...42D; \[61\] @2002AJ....124.3222B; \[62\] @2011AJ....142..128W. ------------------- --------- --------- --------- ------------------ ------------------- -------------------------------- --------------------------- --------------------------------- ---------------------------- ------------------ ---------------------- Satellite RA Dec $M_V$ $d_\odot$ $v_\odot$ $\mu_{\alpha^\ast}^\text{F18}$ $\mu_{\delta}^\text{F18}$ $\mu_{\alpha^\ast}^\text{gold}$ $\mu_{\delta}^\text{gold}$ PM$_\text{gold}$ Refs. \[deg\] \[deg\] \[mag\] [\[kpc\]]{} \[km s$^{-1}$\] \[mas yr$^{-1}$\] \[mas yr$^{-1}$\] \[mas yr$^{-1}$\] \[mas yr$^{-1}$\] Aquarius II 338.481 -9.327 -4.36 $107.9 \pm 3.3$ $-71.1 \pm 2.5$ $-0.252 \pm 0.53$ $0.011 \pm 0.452$ $-0.491 \pm 0.306$ $-0.049 \pm 0.266$ \[3\] \[6\] Bo[ö]{}tes I 210.015 14.512 -6.3 $66 \pm 3.0$ $102.2 \pm 0.8$ $-0.554 \pm 0.098$ $-1.111 \pm 0.076$ $-0.459 \pm 0.041$ $-1.064 \pm 0.029$ \[1\] \[7\] \[8\] \[9\] Bo[ö]{}tes II 209.521 12.859 -2.3 $42 \pm 1.6$ $-117.1 \pm 7.6$ $-2.686 \pm 0.393$ $-0.53 \pm 0.292$ $-2.517 \pm 0.325$ $-0.602 \pm 0.235$ \[2\] \[10\] \[11\] Canes Venatici I 202.016 33.559 -8.6 $210 \pm 6$ $30.9 \pm 0.6$ $-0.159 \pm 0.1$ $-0.067 \pm 0.064$ … … … \[8\] \[12\] \[13\] Canes Venatici II 194.292 34.321 -4.6 $160 \pm 7$ $-128.9 \pm 1.2$ $-0.342 \pm 0.238$ $-0.473 \pm 0.178$ … … … \[13\] \[14\] \[15\] Carina I 100.407 -50.966 -8.6 $105.6 \pm 5.4$ $222.9 \pm 0.1$ $0.485 \pm 0.038$ $0.131 \pm 0.038$ $0.495 \pm 0.015$ $0.143 \pm 0.014$ \[1\] \[16\] \[17\] \[18\] Carina II 114.107 -57.999 -4.5 $37.4 \pm 0.4$ $477.2 \pm 1.2$ $1.867 \pm 0.085$ $0.082 \pm 0.08$ $1.79 \pm 0.06$ $0.01 \pm 0.05$ \[4\] \[19\] \[20\] Carina III 114.630 -57.900 -2.4 $27.8 \pm 0.6$ $284.6 \pm 3.25$ $3.046 \pm 0.132$ $1.565 \pm 0.147$ $3.065 \pm 0.095$ $1.567 \pm 0.104$ \[3\] \[19\] \[20\] Coma Berenices I 186.746 23.908 -3.8 $42 \pm 1.5$ $98.1 \pm 0.9$ $0.471 \pm 0.113$ $-1.716 \pm 0.11$ $0.546 \pm 0.092$ $-1.726 \pm 0.086$ \[2\] \[13\] \[21\] \[22\] Crater II 177.310 -18.413 -8.2 $117.5 \pm 1.1$ $87.5 \pm 0.4$ $-0.184 \pm 0.07$ $-0.106 \pm 0.068$ $-0.246 \pm 0.052$ $-0.227 \pm 0.026$ \[3\] \[23\] \[24\] Draco I 260.060 57.965 -8.75 $76 \pm 6$ $-291.0 \pm 0.1$ $-0.012 \pm 0.013$ $-0.158 \pm 0.038$ $-0.019 \pm 0.009$ $-0.145 \pm 0.01$ \[1\] \[25\] \[26\] Draco II 238.198 64.565 -2.9 $20 \pm 3.0$ $-347.6 \pm 1.75$ $1.242 \pm 0.282$ $0.845 \pm 0.291$ $1.165 \pm 0.26$ $0.866 \pm 0.27$ \[3\] \[24\] \[27\] Fornax 39.962 -34.511 -13.4 $147 \pm 9$ $55.3 \pm 0.1$ $0.374 \pm 0.035$ $-0.401 \pm 0.035$ $0.376 \pm 0.003$ $-0.413 \pm 0.003$ \[1\] \[18\] \[28\] Grus I 344.176 -50.163 -3.4 $120.2 \pm 11.1$ $-140.5 \pm 2.0$ $-0.261 \pm 0.178$ $-0.437 \pm 0.242$ $-0.25 \pm 0.16$ $-0.47 \pm 0.23$ \[5\] \[29\] \[30\] Hercules 247.763 12.787 -6.6 $132 \pm 6$ $45.2 \pm 1.09$ $-0.297 \pm 0.123$ $-0.329 \pm 0.1$ … … … \[31\] \[32\] Horologium I 43.882 -54.119 -3.5 $79 \pm 7$ $112.8 \pm 2.55$ $0.891 \pm 0.105$ $-0.55 \pm 0.099$ $0.95 \pm 0.07$ $-0.55 \pm 0.06$ \[5\] \[33\] Hydra II 185.425 -31.985 -4.8 $151 \pm 8$ $303.1 \pm 1.4$ $-0.416 \pm 0.523$ $0.134 \pm 0.426$ $-0.417 \pm 0.402$ $0.179 \pm 0.339$ \[3\] \[34\] \[35\] \[36\] Hydrus I 37.389 -79.309 -4.71 $27.6 \pm 0.5$ $80.4 \pm 0.6$ $3.733 \pm 0.052$ $-1.605 \pm 0.05$ $3.761 \pm 0.029$ $-1.371 \pm 0.027$ \[3\] \[37\] Leo I 152.122 12.313 -12.03 $258.2 \pm 9.5$ $282.5 \pm 0.1$ $-0.086 \pm 0.069$ $-0.128 \pm 0.071$ $-0.097 \pm 0.056$ $-0.091 \pm 0.047$ \[1\] \[38\] \[39\] Leo II 168.370 22.152 -9.6 $233 \pm 15$ $78.5 \pm 0.6$ $-0.025 \pm 0.087$ $-0.173 \pm 0.09$ $-0.064 \pm 0.057$ $-0.21 \pm 0.054$ \[1\] \[40\] \[41\] Leo IV 173.233 -0.540 -4.97 $154 \pm 5$ $132.3 \pm 1.4$ $-0.59 \pm 0.534$ $-0.449 \pm 0.362$ … … … \[13\] \[42\] Leo V 172.784 2.222 -4.4 $173 \pm 5$ $172.1 \pm 2.2$ $-0.097 \pm 0.56$ $-0.628 \pm 0.307$ … … … \[43\] \[44\] LMC 80.894 -69.756 -18.1 $51.0 \pm 2.0$ $262.2 \pm 3.4$ … … $1.85 \pm 0.03$ $0.24 \pm 0.03$ \[1\] \[45\] Pisces II 344.634 5.955 -4.1 $183 \pm 15$ $-226.5 \pm 2.7$ $-0.108 \pm 0.647$ $-0.586 \pm 0.502$ … … … \[15\] \[34\] Reticulum II 53.949 -54.047 -3.6 $31.4 \pm 1.4$ $64.8 \pm 0.5$ $2.398 \pm 0.053$ $-1.319 \pm 0.059$ $2.36 \pm 0.05$ $-1.32 \pm 0.06$ \[5\] \[46\] \[47\] Sagittarius I 283.831 -30.545 -13.5 $26 \pm 2.0$ $140.0 \pm 2.0$ $-2.736 \pm 0.036$ $-1.357 \pm 0.036$ $-2.692 \pm 0.001$ $-1.359 \pm 0.001$ \[1\] \[45\] Sculptor 15.039 -33.709 -10.7 $83.9 \pm 1.5$ $111.4 \pm 0.1$ $0.084 \pm 0.036$ $-0.133 \pm 0.0356$ $0.082 \pm 0.005$ $-0.131 \pm 0.004$ \[1\] \[18\] \[48\] Segue 1 151.763 16.074 -1.5 $23 \pm 2$ $208.5 \pm 0.9$ $-1.697 \pm 0.198$ $-3.501 \pm 0.178$ $-1.867 \pm 0.11$ $-3.282 \pm 0.102$ \[2\] \[49\] \[50\] Segue 2 34.817 20.175 -2.5 $36.6 \pm 2.45$ $-40.2 \pm 0.9$ $1.656 \pm 0.161$ $0.135 \pm 0.113$ $1.01 \pm 0.14$ $-0.48 \pm 0.18$ \[4\] \[51\] \[52\] Sextans 153.268 -1.620 -9.3 $92.5 \pm 2.2$ $224.2 \pm 0.1$ $-0.438 \pm 0.045$ $0.055 \pm 0.045$ $-0.496 \pm 0.025$ $0.077 \pm 0.02$ \[1\] \[18\] \[53\] SMC 13.187 -72.829 -16.8 $64.0 \pm 4.0$ $145.6 \pm 0.6$ … … $0.797 \pm 0.03$ $-1.22 \pm 0.03$ \[1\] \[45\] Triangulum II 33.322 36.172 -1.2 $28.4 \pm 1.6$ $-381.7 \pm 1.1$ $0.588 \pm 0.194$ $0.554 \pm 0.169$ $0.588 \pm 0.187$ $0.554 \pm 0.161$ \[3\] \[54\] \[55\] Tucana II 343.060 -58.570 -3.9 $57.5 \pm 5.3$ $-129.1 \pm 3.5$ $0.91 \pm 0.069$ $-1.159 \pm 0.082$ $0.91 \pm 0.06$ $-1.16 \pm 0.08$ \[5\] \[30\] \[56\] Tucana III 359.150 -59.600 -2.4 $25 \pm 2$ $-102.3 \pm 0.4$ $-0.025 \pm 0.049$ $-1.661 \pm 0.049$ $-0.03 \pm 0.04$ $-1.65 \pm 0.04$ \[5\] \[57\] \[58\] Ursa Major I 158.685 51.926 -6.75 $97.3 \pm 5.85$ $-55.3 \pm 1.4$ $-0.683 \pm 0.1$ $-0.72 \pm 0.135$ $-0.659 \pm 0.093$ $-0.635 \pm 0.131$ \[2\] \[13\] \[59\] Ursa Major II 132.874 63.133 -3.9 $34.7 \pm 2.1$ $-116.5 \pm 1.9$ $1.691 \pm 0.064$ $-1.902 \pm 0.075$ $1.661 \pm 0.053$ $-1.87 \pm 0.065$ \[2\] \[13\] \[60\] Ursa Minor 227.242 67.222 -8.4 $76 \pm 4$ $-246.9 \pm 0.1$ $-0.184 \pm 0.044$ $0.082 \pm 0.042$ $-0.182 \pm 0.01$ $0.074 \pm 0.008$ \[1\] \[61\] Willman 1 162.341 51.053 -2.7 $38 \pm 7$ $-12.8 \pm 1.0$ $0.199 \pm 0.194$ $-1.342 \pm 0.37$ $0.382 \pm 0.119$ $-1.152 \pm 0.216$ \[2\] \[8\] \[62\] ------------------- --------- --------- --------- ------------------ ------------------- -------------------------------- --------------------------- --------------------------------- ---------------------------- ------------------ ---------------------- \[tab:properties\] ------------------- ------------------------- ----------------------- ------------------------ -------------------------- ---------------------------- ---------------------------- -- Satellite $r$ $\theta$ $\phi$ $v_r$ $v_\theta$ $v_\phi$ \[kpc\] \[deg\] \[deg\] \[km s$^{-1}$\] \[km s$^{-1}$\] \[km s$^{-1}$\] Aquarius II $105.2_{-3.2}^{+3.3}$ $145.0_{-0.1}^{+0.1}$ $61.7_{-0.2}^{+0.2}$ $43.2_{-20.2}^{+19.5}$ $-282.3_{-239.9}^{+222.8}$ $29.0_{-252.0}^{+268.5}$ Bo[ö]{}tes I $63.6_{-2.8}^{+2.9}$ $13.6_{-0.3}^{+0.3}$ $-2.9_{-0.1}^{+0.1}$ $97.5_{-3.4}^{+3.3}$ $110.7_{-25.4}^{+26.2}$ $-126.0_{-33.9}^{+32.4}$ Bo[ö]{}tes II $39.8_{-1.5}^{+1.6}$ $10.4_{-0.5}^{+0.4}$ $-13.3_{-0.7}^{+0.6}$ $-48.1_{-16.2}^{+16.5}$ $-311.1_{-68.3}^{+70.7}$ $-246.9_{-67.4}^{+70.5}$ Canes Venatici I $209.8_{-5.8}^{+5.9}$ $9.8_{-0.0}^{+0.0}$ $86.0_{-0.4}^{+0.5}$ $83.5_{-3.4}^{+3.4}$ $93.1_{-85.8}^{+80.0}$ $83.8_{-86.1}^{+79.6}$ Canes Venatici II $160.9_{-7.0}^{+6.7}$ $8.8_{-0.1}^{+0.1}$ $130.4_{-0.6}^{+0.7}$ $-93.2_{-8.9}^{+8.4}$ $-144.2_{-137.4}^{+127.7}$ $135.0_{-181.1}^{+171.6}$ Carina I $106.9_{-5.2}^{+5.8}$ $111.8_{-0.0}^{+0.0}$ $-104.6_{-0.2}^{+0.3}$ $-4.6_{-2.9}^{+3.0}$ $-165.0_{-24.4}^{+22.0}$ $-22.5_{-18.2}^{+17.4}$ Carina II $38.3_{-0.4}^{+0.3}$ $106.7_{-0.0}^{+0.0}$ $-103.0_{-0.2}^{+0.2}$ $204.4_{-4.3}^{+4.3}$ $-228.6_{-15.6}^{+15.5}$ $195.6_{-13.6}^{+14.3}$ Carina III $29.0_{-0.6}^{+0.6}$ $106.1_{-0.0}^{+0.0}$ $-107.2_{-0.4}^{+0.4}$ $46.6_{-6.6}^{+6.4}$ $-383.6_{-20.1}^{+18.9}$ $36.1_{-18.1}^{+16.9}$ Coma Berenices I $43.2_{-1.4}^{+1.5}$ $14.9_{-0.4}^{+0.4}$ $-158.2_{-0.6}^{+0.6}$ $28.9_{-5.1}^{+4.7}$ $-252.6_{-24.0}^{+21.9}$ $104.5_{-25.3}^{+25.6}$ Crater II $116.4_{-1.1}^{+1.1}$ $47.5_{-0.0}^{+0.0}$ $-82.3_{-0.1}^{+0.1}$ $-83.7_{-3.6}^{+3.4}$ $-77.0_{-37.3}^{+38.5}$ $-24.2_{-39.0}^{+40.5}$ Draco I $75.9_{-5.6}^{+6.0}$ $55.3_{-0.0}^{+0.0}$ $93.8_{-0.6}^{+0.6}$ $-88.5_{-2.9}^{+2.8}$ $124.1_{-5.7}^{+5.7}$ $-50.7_{-13.5}^{+14.2}$ Draco II $22.4_{-2.7}^{+2.8}$ $52.5_{-1.0}^{+1.2}$ $125.4_{-3.0}^{+3.5}$ $-154.4_{-12.1}^{+12.4}$ $300.5_{-23.8}^{+26.0}$ $-68.8_{-32.9}^{+31.6}$ Fornax $149.5_{-9.0}^{+8.6}$ $153.9_{-0.1}^{+0.1}$ $-129.1_{-0.4}^{+0.3}$ $-40.9_{-1.5}^{+1.5}$ $-104.5_{-32.9}^{+30.8}$ $112.5_{-28.3}^{+28.4}$ Grus I $116.3_{-10.6}^{+11.5}$ $151.6_{-0.3}^{+0.3}$ $-24.5_{-0.4}^{+0.3}$ $-203.2_{-7.0}^{+7.0}$ $-187.5_{-135.2}^{+133.9}$ $123.7_{-113.6}^{+110.8}$ Hercules $126.3_{-6.0}^{+6.1}$ $51.3_{-0.1}^{+0.1}$ $30.9_{-0.1}^{+0.1}$ $150.5_{-3.2}^{+3.3}$ $-10.0_{-70.7}^{+73.4}$ $-54.8_{-68.2}^{+73.4}$ Horologium I $79.4_{-7.0}^{+7.0}$ $144.4_{-0.1}^{+0.1}$ $-99.1_{-1.0}^{+0.9}$ $-33.7_{-5.5}^{+5.2}$ $-193.4_{-49.0}^{+46.6}$ $0.6_{-40.1}^{+42.0}$ Hydra II $148.3_{-8.3}^{+7.9}$ $58.9_{-0.0}^{+0.0}$ $-67.6_{-0.2}^{+0.2}$ $129.3_{-21.2}^{+21.1}$ $-164.7_{-287.8}^{+281.8}$ $-187.8_{-396.2}^{+392.9}$ Hydrus I $25.7_{-0.5}^{+0.5}$ $129.9_{-0.0}^{+0.0}$ $-84.4_{-0.5}^{+0.5}$ $-57.2_{-3.3}^{+3.2}$ $-328.9_{-9.9}^{+9.6}$ $-161.9_{-8.6}^{+9.0}$ LMC $50.3_{-1.9}^{+2.0}$ $123.3_{-0.0}^{+0.0}$ $-90.7_{-0.5}^{+0.4}$ $63.1_{-4.3}^{+4.3}$ $-310.3_{-18.0}^{+18.4}$ $-40.9_{-9.3}^{+8.3}$ Leo I $261.9_{-9.3}^{+9.2}$ $41.7_{-0.0}^{+0.0}$ $-135.8_{-0.1}^{+0.1}$ $168.6_{-3.1}^{+3.1}$ $24.4_{-74.3}^{+66.4}$ $-71.4_{-97.0}^{+101.2}$ Leo II $235.2_{-14.5}^{+15.2}$ $24.1_{-0.1}^{+0.1}$ $-142.7_{-0.2}^{+0.2}$ $18.5_{-4.0}^{+3.8}$ $-72.0_{-88.7}^{+86.1}$ $-14.0_{-101.7}^{+112.0}$ Leo IV $154.7_{-4.9}^{+5.1}$ $33.8_{-0.0}^{+0.0}$ $-99.8_{-0.2}^{+0.2}$ $13.8_{-21.5}^{+20.8}$ $321.4_{-270.9}^{+265.2}$ $-183.6_{-393.5}^{+372.5}$ Leo V $174.0_{-5.0}^{+4.6}$ $31.9_{-0.0}^{+0.0}$ $-102.9_{-0.2}^{+0.2}$ $40.5_{-18.9}^{+19.9}$ $225.1_{-351.4}^{+373.5}$ $236.3_{-381.6}^{+369.4}$ Pisces II $181.8_{-14.5}^{+14.6}$ $137.4_{-0.0}^{+0.0}$ $83.1_{-0.3}^{+0.3}$ $-79.7_{-24.1}^{+24.4}$ $173.6_{-471.3}^{+475.1}$ $-356.8_{-533.7}^{+537.6}$ Reticulum II $32.8_{-1.3}^{+1.4}$ $136.9_{-0.2}^{+0.2}$ $-115.3_{-0.9}^{+0.8}$ $-99.8_{-3.1}^{+3.0}$ $-215.8_{-19.1}^{+18.7}$ $56.4_{-9.8}^{+10.7}$ SMC $61.3_{-3.8}^{+4.2}$ $136.9_{-0.1}^{+0.1}$ $-66.8_{-0.7}^{+0.6}$ $-5.6_{-2.4}^{+2.3}$ $-245.3_{-27.0}^{+26.3}$ $-67.5_{-17.1}^{+16.2}$ Sagittarius I $18.3_{-2.0}^{+2.0}$ $110.6_{-0.6}^{+0.8}$ $8.2_{-0.3}^{+0.3}$ $140.0_{-2.6}^{+2.3}$ $-275.2_{-17.0}^{+17.2}$ $-53.3_{-21.4}^{+21.6}$ Sculptor $84.0_{-1.5}^{+1.5}$ $172.5_{-0.1}^{+0.1}$ $-119.7_{-0.8}^{+0.9}$ $75.0_{-1.6}^{+1.6}$ $169.6_{-14.4}^{+13.9}$ $-72.8_{-15.1}^{+16.2}$ Segue 1 $27.9_{-1.9}^{+1.9}$ $50.4_{-0.8}^{+0.9}$ $-153.8_{-0.9}^{+0.8}$ $116.8_{-5.7}^{+5.9}$ $142.1_{-31.3}^{+34.5}$ $142.0_{-30.2}^{+34.0}$ Segue 2 $42.4_{-2.3}^{+2.4}$ $121.9_{-0.4}^{+0.3}$ $156.1_{-0.3}^{+0.4}$ $72.8_{-4.5}^{+4.6}$ $-214.7_{-25.2}^{+26.2}$ $9.5_{-30.1}^{+28.7}$ Sextans $95.5_{-2.2}^{+2.3}$ $49.3_{-0.0}^{+0.0}$ $-122.2_{-0.1}^{+0.1}$ $79.2_{-2.6}^{+2.6}$ $-12.2_{-18.3}^{+17.6}$ $-239.5_{-21.6}^{+23.1}$ Triangulum II $34.7_{-1.6}^{+1.6}$ $109.2_{-0.3}^{+0.2}$ $150.0_{-0.4}^{+0.5}$ $-255.2_{-5.0}^{+4.9}$ $-175.7_{-24.0}^{+23.6}$ $-122.6_{-25.5}^{+24.9}$ Tucana II $54.0_{-5.2}^{+5.2}$ $148.1_{-0.5}^{+0.6}$ $-40.9_{-1.1}^{+0.9}$ $-187.6_{-4.1}^{+4.6}$ $48.6_{-18.1}^{+18.3}$ $-208.0_{-43.1}^{+40.5}$ Tucana III $23.0_{-1.9}^{+1.9}$ $154.5_{-0.3}^{+0.2}$ $-80.4_{-4.2}^{+3.4}$ $-228.1_{-2.2}^{+2.3}$ $28.2_{-21.2}^{+19.3}$ $48.3_{-13.6}^{+11.0}$ Ursa Major I $102.1_{-5.9}^{+5.7}$ $39.0_{-0.2}^{+0.2}$ $161.9_{-0.1}^{+0.1}$ $11.5_{-3.8}^{+3.7}$ $165.7_{-54.7}^{+54.0}$ $206.1_{-60.8}^{+62.6}$ Ursa Major II $40.9_{-2.0}^{+2.2}$ $58.8_{-0.3}^{+0.3}$ $158.6_{-0.3}^{+0.3}$ $-57.7_{-2.8}^{+2.8}$ $-280.3_{-24.1}^{+23.7}$ $32.6_{-17.0}^{+19.9}$ Ursa Minor $78.2_{-4.0}^{+4.0}$ $46.5_{-0.1}^{+0.1}$ $112.9_{-0.4}^{+0.4}$ $-71.4_{-2.8}^{+2.7}$ $136.9_{-12.3}^{+12.8}$ $-11.5_{-18.4}^{+18.3}$ Willman 1 $42.5_{-6.5}^{+6.8}$ $41.9_{-1.3}^{+1.6}$ $164.5_{-0.7}^{+0.9}$ $17.8_{-6.4}^{+6.3}$ $-106.4_{-59.3}^{+47.6}$ $-55.5_{-59.6}^{+75.0}$ ------------------- ------------------------- ----------------------- ------------------------ -------------------------- ---------------------------- ---------------------------- -- \[tab:galactoprops\] \[lastpage\] [^1]: E-mail: <alexriley@tamu.edu>. Code for this work is available [on Github](https://github.com/ahriley/beta-MW-dwarfs). [^2]: Mitchell Astronomy Fellow [^3]: [The satellites that are in the full sample that are excluded from the “gold” sample are: Canis Venatici I, Canis Venatici II, Hercules, Leo IV, Leo V, and Pisces II.]{} [^4]: Defined to be the mass inside a sphere in which the mean matter density is 200 times the critical density $\rho_\text{crit} = 3H^2(z)/8\pi G$. [Virial quantities are defined at that radius and are identified by a ‘200’ subscript.]{} [^5]: The gas content of these galaxies is sub-dominant compared to the stellar component at small radii. [^6]: We note that the host galaxies in low and medium resolution APOSTLE runs have disky morphologies and higher stellar masses compared to the high resolution runs used here.

--- abstract: | Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $\phi_H$ of a Hamiltonian $H\in C^{\infty}(M)$ on a symplectic manifold $(M,\omega)$. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures and we construct an example showing that their support can be extremely unstable when $H$ fails to be convex, even for nearly integrable $H$. Parts of these results extend work by Viterbo [@Viterbo10], and Vichery [@Vichery14]. Using ideas due to Entov-Polterovich [@EntovPolterovich17], [@Polterovich14] we also detect interesting invariant measures for $\phi_H$ by studying a generalization of the *symplectic shape* of sublevel sets of $H$. This approach differs from the first one in that it works also for $(M,\omega)$ in which every compact subset can be displaced. We present applications to Hamiltonian systems on ${\mathbb R}^{2n}$ and twisted cotangent bundles. address: 'Mads Bisgaard, D-MATH, ETH Z[ü]{}rich, R[ä]{}mistrasse 101, 8092 Z[ü]{}rich, Switzerland' author: - 'Mads R. Bisgaard' bibliography: - 'BIB.bib' title: Mather theory and symplectic rigidity --- Introduction ============ Consider a symplectic manifold $(M^{2n},\omega)$ and a Hamiltonian $H\in C^{\infty}(M)$ with complete flow $\phi_H=\{\phi_H^t\}_{t\in {\mathbb R}}$. This paper concerns the question of how to systematically determine interesting invariant sets for $\phi_H$. The celebrated KAM theorem asserts (loosely speaking) that, if $H$ is non-degenerate and sufficiently regular, an invariant Lagrangian torus, on which the dynamics of $\phi_H$ has a Diophantine rotation vector, will persist under small (sufficiently regular) perturbations of $H$. Mather [@Mather91] discovered that one can loosen the regularity assumptions and the Diophantine condition in the KAM theorem and still find plenty of interesting invariant sets which persist perturbations of $H$, if one pays the price of replacing the non-degeneracy condition with a stronger *convexity* assumption. One of the aims of the current paper is to study, using methods coming from symplectic topology, what happens when one relaxes the convexity assumption. It is interesting to understand the connection between Aubry-Mather theory and symplectic topology. Here we hope to provide some understanding of how pseudo-holomorphic curve techniques can explain phenomena from Aubry-Mather theory. A first indication that these theories can be connected was obtained by Bernard [@Bernard07]. The approach we take here builds heavily on Viterbo’s symplectic homogenization paper [@Viterbo08] as well as the Floer-homological version studied by Monzner-Vichery-Zapolsky [@ZapolskyMonznerVichery12] and Vichery [@Vichery14]. The second part of the paper obtains interesting invariant measures by studying a generalization of the notion of symplectic shape (due to Sikorav [@Sikorav89]) of sublevel sets of a Hamiltonian. It employs ideas due to Buhovsky-Entov-Polterovich [@BuhovskyEntovPolterovich12], Entov-Polterovich [@EntovPolterovich17] and Polterovich [@Polterovich14]. The contents of the paper are as follows: In Section \[MatherFloer\] we present our results on existence of invariant measures using homogenized Lagrangian spectral invariants. Section \[Mathersec\] compares properties of these measures to Mather measures. Section \[secC0\] contains our results on invariant measures using the $C^0$-techniques developed by Buhovsky-Entov-Polterovich. Finally, Section \[seqprem\] and \[secproof\] contain preliminaries and proofs. *Setting and notation:* $\mathcal{M}(X)$ will denote the space of Borel probability measures which are *compactly* supported on $X\subset M$. We use the convention that the symplectic gradient $X_{H_t}$ associated to $H\in C^{\infty}({\mathbb R}\times M)$ is defined by $\iota_{X_{H_t}}\omega=-dH_t$. The flow generated by $X_{H_t}$ is denoted by $\phi_H=\{\phi_H^t\}_{t\in {\mathbb R}}$. We denote by $\mathcal{M}(X;\phi_H)\subset \mathcal{M}(X)$ the subset of $\phi_H$-invariant measures and write $\mathcal{M}=\mathcal{M}(M)$ as well as $\mathcal{M}(\phi_H)=\mathcal{M}(M;\phi_H)$. Symplectic “Mather-Floer theory” {#MatherFloer} ================================ Throughout this section $(M^{2n},\omega)$ will denote a closed monotone symplectic manifold.[^1] We consider a closed *monotone* Lagrangian submanifold $L\subset (M,\omega)$ which is non-narrow in the sense that its quantum homology $QH_*(L;{\mathbb Z}_2)$ doesn’t vanish (see Section \[seqprem\]). In this setting Leclercq-Zapolsky [@LeclercqZapolsky15] recently developed a theory of *Lagrangian spectral invariants*. We will denote by $$l_L:\widetilde{\operatorname{Ham}}(M,\omega) \to {\mathbb R}.$$ the Leclercq-Zapolsky spectral invariant associated to the unity in $QH_*(L;{\mathbb Z}_2)$.[^2] An idea due to Viterbo [@Viterbo08] says that homogenizing $l_L$ gives rise to an analogue of Mather’s $\alpha$-function. The approach we consider here was first studied by Monzner-Vichery-Zapolsky [@ZapolskyMonznerVichery12] and Vichery [@Vichery14]. Denote by $\mathcal{H}:=\{H\in C^{\infty}(M)\ |\ \int H \omega^n=0\}$ the space of *normalized* Hamiltonians. \[def2\] Associated to $H\in \mathcal{H}$ we define a function $\sigma_{H:L}:H^1(M;{\mathbb R}) \to {\mathbb R}$ by $$\label{eq2} \sigma_{H:L}(c)=\lim_{{\mathbb N}\ni k\to \infty}\frac{l_L(\psi_1^{-1}\tilde{\phi}^k_H \psi_1)}{k},$$ where $\psi:[0,1]\times M \to M$ is any smooth symplectic isotopy with $\psi_0=\operatorname{id}$ and $\operatorname{Flux}(\psi)=c\in H^1(M;{\mathbb R})$, and $\tilde{\phi}_H^k\in \widetilde{\operatorname{Ham}}(M,\omega)$ denotes the element represented by the path $[0,1]\ni t\mapsto \phi_H^{tk}$ in $\operatorname{Ham}(M,\omega)$. For more details on the construction and properties of $\sigma_{H:L}$ we refer to Section \[seqprem\] (see in particular Theorem \[prop1\] on page ). It turns out that $\sigma_{H:L}$ is locally Lipschitz, so at every point $c\in H^1(M;{\mathbb R})$ it has a well-defined non-empty set of *Clarke subdifferentials* (see Section \[secmeas\]) $$\partial \sigma_{H:L}(c)\subset H_1(M;{\mathbb R})=H^1(M;{\mathbb R})^*.$$ The dynamical information contained in $\sigma_{H:L}$ is that it guarantees existence of analogues of *Mather measures* (from Aubry-Mather theory) to the present setting, i. e. $\mathcal{M}(\phi_H)$-measures whose rotation vector (or asymptotic cycle, see Section \[secmeas\]) is prescribed by $\partial \sigma_{H:L}(c)$. \[thm1\] Let $H\in \mathcal{H}$. For any $c\in H^1(M;{\mathbb R})$ and any Clarke subdifferential $h\in \partial \sigma_{H:L}(c)\subset H_1(M;{\mathbb R})$ of $\sigma_{H:L}$ at $c$, there exists a $\mu \in \mathcal{M}(\phi_H)$ which has rotation vector $$\rho(\mu)=h.$$ For a more detailed discussion of how Theorem \[thm1\] relates to classical Aubry-Mather theory we refer to Section \[Mathersec\]. After having proved the main part of Theorem \[thm1\] we learned about Vichery’s [@Vichery14], where a result similar to Theorem \[thm1\] is proved in the case where $L$ is the zero-section of a cotangent bundle as well as Viterbo’s [@Viterbo10] where results from [@Viterbo08] are applied to yield a result similar to Theorem \[thm1\] for $T^*\mathbb{T}^n$. The main difference between our result and Vichery’s is greater generality. Moreover, our approach detects the measures directly using pseudo-holomorphic curve techniques and avoids the use of generating functions. \[rem101\] Fix $c=0$ and consider a measure $\mu \in \mathcal{M}(\phi_H)$ with $\rho(\mu)\in \partial \sigma_{H:L}(0)$ whose existence is guaranteed by Theorem \[thm1\]. As the proof of Theorem \[thm1\] will show, $\mu$ arises as a convex combination of weak$^*$-limits of sequences $(\mu_{\varsigma})_{\varsigma \in {\mathbb N}}\subset \mathcal{M}$ characterized by $$\label{equa12} \int f\ d\mu_{\varsigma}=\frac{1}{k_{\varsigma}}\int_0^{k_{\varsigma}}f\phi_H^t(x_{\varsigma})\ dt \quad \forall \ f\in C^0_{bd}(M).$$ Here, $(x_{\varsigma})_{\varsigma \in {\mathbb N}}$ is a sequence of points with $x_{\varsigma}\in \psi_{\varsigma}(L)$ for a sequence of symplectomorphisms $(\psi_{\varsigma})_{\varsigma \in {\mathbb N}}\subset \operatorname{Symp}(M,\omega)$ satisfying $$\psi_{\varsigma}\stackrel{\varsigma \to \infty}{\longrightarrow}\operatorname{id}$$ in the Whitney $C^{\infty}$-topology, and $(k_{\varsigma})_{\varsigma \in {\mathbb N}}\subset (0,\infty)$ is a sequence of times with $k_{\varsigma}\uparrow \infty$. In addition, $\phi_H^{k_{\varsigma}}(x_{\varsigma})\in \psi_{\varsigma}(L)$ for all $\varsigma \in {\mathbb N}$ and to each orbit $\gamma_{\varsigma}=\{\phi_H^t(x_{\varsigma})\}_{t\in [0,k_{\varsigma}]}$ is associated an action $\mathcal{A}_{H:\psi_{\varsigma}(L)}(\gamma_{\varsigma})\in {\mathbb R}$ such that[^3] $$\frac{\mathcal{A}_{H:\psi_{\varsigma}(L)}(\gamma_{\varsigma})}{k_{\varsigma}} \stackrel{\varsigma \to \infty}{\longrightarrow}\sigma_{H:L}(0).$$ We will denote by $\mathfrak{M}_{H:L}\subset \mathcal{M}(\phi_H)$ the set of measures $\mu \in \mathcal{M}(\phi_H)$ with $\rho(\mu)\in \partial \sigma_{H:L}(0)$ arising as convex combinations of weak$^*$-limits of sequences $(\mu_{\varsigma})_{\varsigma}$ given by (\[equa12\]), such that $(x_{\varsigma})_{\varsigma}$ and $(k_{\varsigma})_{\varsigma}$ satisfy the criteria mentioned above. This notation is justified by results, first due to Viterbo [@Viterbo08 Proposition 13.3] and later Monzner-Vichery-Zapolsky [@ZapolskyMonznerVichery12 Theorem 1.11], saying that $\sigma_{H:L}$ coincides with Mather’s $\alpha$-function when $L\subset T^*L$ is the zero-section and $H$ is Tonelli. In particular, $\mathfrak{M}_{H:L}$-measures are Mather measures in this setting (see Remark \[Matrem901\] below). For a discussion about the extent to which $\mathfrak{M}_{H:L}$ coincides with the set of all Mather measures associated to $0\in H^1(M;{\mathbb R})$ we refer to Remark \[Matrem902\] below. Under certain circumstances, one can reduce the domain of $\sigma_{H:L}$ further, thus getting further restrictions on its subdifferentials. In the following we will denote by $r_L:H^1(M;{\mathbb R}) \to H^1(L;{\mathbb R})$ and by $i_L:H_1(L;{\mathbb R}) \to H_1(M;{\mathbb R})$ the (co)homological maps induced by the inclusion $L\hookrightarrow M$. \[lem1\] Suppose $r_L$ is surjective. Then $\sigma_{H:L}$ descends to a locally Lipschitz function $$\alpha_{H:L}: H^1(L;{\mathbb R}) \to {\mathbb R}.$$ In particular, for every $c\in H^1(L;{\mathbb R})$ and every subdifferential $h\in \partial \alpha_{H:L}(c)\subset H_1(L;{\mathbb R})$ there exists a $\mu \in \mathcal{M}(\phi_H)$ with rotation vector $$\rho(\mu)=i_L(h)\in H_1(M;{\mathbb R}).$$ Of course, the notation is meant to suggest that $\alpha_{H:L}$ should play the role of Mather’s $\alpha$-function on a general symplectic manifold. Polterovich [@Polterovich14] used Poisson bracket invariants to study invariant measures which have “large” rotation vectors. In certain situations Corollary \[lem1\] allows us to sharpen his result in the sense that we can detect that the rotation vector of the constructed measure is contained in the subspace $i_L(H_1(L;{\mathbb R})) \leq H_1(M;{\mathbb R})$. A smooth map $\psi:[0,1]\times L\to M$ is said to be a *Lagrange isotopy* if each $\psi_t:L\to (M,\omega)$ is a Lagrange embedding. We use the terminology that $\psi$ *starts at $L$* if $\psi_0:L\to M$ is the inclusion $L\hookrightarrow M$. To a Lagrange isotopy $\psi$ starting at $L$ is associated a class $\operatorname{Flux}_L(\psi)\in H^1(L;{\mathbb R})$ as follows: A smooth loop $\gamma:S^1\to L$ gives rise to a cylinder $\Gamma:S^1 \times [0,1]\to M$ via $\Gamma(s,t)=\psi_t(\gamma(s))$ and $\operatorname{Flux}_L(\psi)$ is characterised by $$\label{Meq4} \langle \operatorname{Flux}_L(\psi),[\gamma]\rangle=\int_{S^1 \times [0,1]} \Gamma^*\omega.$$ For further details on Lagrange Flux ($\operatorname{Flux}_L$) we refer to [@Solomon13 Section 6]. \[cor2\] Suppose $r_L$ is surjective. For every $c\in H^1(L;{\mathbb R})$ there exists an $h\in H_1(L;{\mathbb R})$ such that $i_L(h)\in H_1(M;{\mathbb R})$ is realized as the rotation vector of a $\mathcal{M}(\phi_H)$-measure. Moreover, $$\langle c,h\rangle \geq \sup_{\psi}\left( \min_{\psi_1(L)}(H)-\max_{L}(H) \right)$$ where the supremum runs over all Lagrange isotopies $\psi:[0,1]\times L\to M$ starting at $L$ and $\operatorname{Flux}_L(\psi)=c$. The non-compact setting {#noncompact} ----------------------- Here we will discuss the analogue of the above results in the setting of a non-compact $(M,\omega)$. These can be stated for very general $(M,\omega)$, but the best results are available for Liouville manifolds. This setting in particular generalizes the case of cotangent bundles covered by Vichery [@Vichery14]. Following [@EntovPolterovich17] we will use the terminology that an *exact* symplectic manifold $(M,\omega)$ is *Liouville* if the following two conditions are met: (a) $M$ is equipped with a distinguished 1-form $\lambda$ such that $d\lambda=\omega$ and the Liouville vector field $Z\in \mathfrak{X}(M)$ defined by $i_Z \omega=\lambda$ is complete. (b) \[Mcondb\] There exists a closed connected hypersurface $\Sigma \subset M$, which is transverse to $Z$ and bounds a domain $U\subset M$ whose closure is compact, such that $$M=U \sqcup \left( \bigcup_{t\geq 0}\theta_t(\Sigma) \right),$$ where $(\theta_t)_{t\in {\mathbb R}}$ denotes the Liouville flow generated by $Z$. The last condition implies that $(\Sigma,\lambda_0:=\lambda|_{\Sigma})$ is a contact manifold, called the *ideal contact boundary of $M$*. For a Liouville $(M,\omega=d\lambda)$, there is a diffeomorphism $\cup_{t\in {\mathbb R}}\theta_t(\Sigma)\cong (0,\infty)\times \Sigma$ which identifies $\lambda$ with $s\lambda_0$ on $(0,\infty)\times \Sigma$ (here $s$ denotes the coordinate on $(0,\infty)$). In the following we will tacitly use the identification $\cup_{t\in {\mathbb R}}\theta_t(\Sigma)\cong (0,\infty)\times \Sigma$ and think of $(0,\infty)\times \Sigma$ as a subset of $M$ with the property that $\lambda|_{(0,\infty)\times \Sigma}=s\lambda_0$. Let now $(M,\omega=d\lambda)$ be a Liouville manifold and denote by $L\subset (M,\omega)$ a closed $\lambda$-exact Lagrangian submanifold.[^4] We will denote by $\mathcal{H}=\mathcal{H}(M,\omega)$ the class of autonomous Hamiltonians $H\in C^{\infty}(M)$ whose Hamiltonian flow $\phi_H=\{\phi_H^t\}_{t\in {\mathbb R}}$ is complete and satisfies the condition that $\mathcal{O}_+(X)\subset M$ is compact for every compact $X\subset M$. Here $$\mathcal{O}_+(X):=\overline{\bigcup_{t\geq 0}\phi_H^t(X)}.$$ Note that $\mathcal{H}$ contains all proper Hamiltonians. Given $H\in \mathcal{H}$, every measure $\mu \in \mathcal{M}$ has a well-defined action $\mathcal{A}_{H,\lambda}(\mu)\in {\mathbb R}$ given by[^5] $$\label{eq8} \mathcal{A}_{H,\lambda}(\mu):=\int H- \langle \lambda, X_H \rangle \ d\mu.$$ In order to define $$\sigma_{H:L}:H^1(M;{\mathbb R}) \to {\mathbb R}$$ for $H\in \mathcal{H}$ in this setting we make use of \[Mprop1\] If $(M,\omega=d\lambda)$ is Liouville, then every class $c\in H^1_{dR}(M;{\mathbb R})$ admits a representative $\eta$ such that $$\label{Meq1} \eta|_{\{s\}\times \Sigma}=\eta|_{\{1\}\times \Sigma}\quad \forall \ s\in [1,\infty).$$ Here the equality means the two 1-forms on $\Sigma$ are the same. In particular, the vector field $X\in \mathfrak{X}(M)$ characterized by $i_X\omega=\eta$ is complete. The proof is more or less immediate, but is carried out in Section \[seqprem\]. Given a class $c\in H^1_{dR}(M;{\mathbb R})=H^1(M;{\mathbb R})$ we define $\sigma_{H:L}(c)$ by the formula (\[eq2\]), where $(\psi_t)_{t\in {\mathbb R}}$ is the flow generated by a vector field $X$ satisfying $i_X\omega=\eta$ for a closed 1-form $\eta \in c$ meeting condition (\[Meq1\]). In Section \[seqprem\] below we argue that the value of $\sigma_{H:L}(c)$ depends neither on the choice of $\eta$ in the class $c$ satisfying (\[Meq1\]), nor on the choice of ideal contact boundary $\Sigma$ meeting condition (b) on page . The non-compact version of Theorem 3 now reads as follows: \[thm2\] Let $H\in \mathcal{H}(M,d\lambda)$. Then, for every $c\in H^1(M;{\mathbb R})$ and every Clarke subdifferential $h\in \partial \sigma_{H:L}(c)\subset H_1(M;{\mathbb R})$, there exists a measure $\mu \in \mathcal{M}(\phi_H)$ whose action and rotation vector satisfy $$\mathcal{A}_{H,\lambda}(\mu)=\sigma_{H:L}(c)-\langle c,\rho(\mu) \rangle \quad \text{and} \quad \rho(\mu)=h.$$ The following is the non-compact version of Corollary \[lem1\]. \[cor3\] Let $H\in \mathcal{H}(M,d\lambda)$ and suppose $r_{L}$ is surjective. Then $\sigma_{H:L}$ descends to a locally Lipschitz function $$\alpha_{H:L}:H^1(L;{\mathbb R})\to {\mathbb R}.$$ In particular, for every $c\in H^1(L;{\mathbb R})$ and every Clarke subdifferential $h\in \partial \alpha_{H:L}(c)$, there exists a measure $\mu \in \mathcal{M}(\phi_H)$ satisfying $$\label{Mateq902} \mathcal{A}_{H,\lambda}(\mu)=\alpha_{H:L}(c)-\langle c,\rho(\mu) \rangle \quad \text{and} \quad \rho(\mu)=i_L(h)\in H_1(M;{\mathbb R}).$$ \[Matrem901\] Suppose now that $M=T^*L$ with $L$ a closed manifold viewed as the zero-section $L\subset T^*L$ and $\omega=d\lambda$ with $\lambda=pdq$ being the canonical Liouville one-form. A Tonelli Lagrangian $l\in C^{\infty}(TN)$ (see [@Sorrentino15 Definition 1.1.1]) sets up a Legendre transform $$\mathcal{L}:TN\to T^*N,$$ so that the associated Tonelli Hamiltonian $H$ and $l$ satisfy the equation $$l(\mathcal{L}^{-1}(q,p))=\langle \lambda, X_H \rangle (q,p) -H(q,p) \quad \forall \ p\in T_q^*N.$$ In particular, in this setting one sees that the condition on the action in (\[Mateq902\]) can be written $$\label{Mateq903} -\alpha_{H:L}(c)=\int l\circ \mathcal{L}^{-1}\ d\mu-\langle c,\rho(\mu) \rangle.$$ As mentioned in Remark \[rem101\], it is known that $\alpha_{H:L}$ coincides with Mather’s $\alpha$-function in this setting so (by [@Sorrentino15 Section 3.1]) (\[Mateq903\]) implies that the measures detected in Corollary \[cor3\] are Mather measures in this setting. \[Matrem902\] In the Tonelli setting from the above Remark \[Matrem901\], a Mather measure associated to $0\in H_1(L;{\mathbb R})$ is a measure $\mu \in \mathcal{M}(\phi_H)$ which satisfies (\[Mateq902\]) with $c=0$. So why don’t we define $\mathfrak{M}_{H:L}$ as the set of measures satisfying a condition similar to (\[Mateq902\]) with $c=0$? The reason is quite simple: In the Tonelli setting all ergodic components of a Mather measure associated to $0\in H_1(M;{\mathbb R})$ are themselves Mather measures associated to $0\in H_1(M;{\mathbb R})$. In the general setting which we consider here this need not be true. E.g. we have the following pathological example: Fix $p_0\in {\mathbb R}^n \backslash \{0\}$ and choose $r>0$ such that $0\notin \overline{B_{2r}(p_0)}$. Pick any function $h\in C^{\infty}(B_r(p_0))$ with $h(p_0)=0$ and $dh(p_0)\cdot p_0=0$, but $dh(p_0)\neq 0$. Extend $h$ to $B_r(p_0)\cup (-B_r(p_0))$ by $h(p)=h(-p)$ for all $p\in -B_r(p_0)$. Now extend $h$ to a function on all of ${\mathbb R}^n$ by cutting it off outside $\overline{B_{2r}(p_0)} \cup (-\overline{B_{2r}(p_0)})$. The Hamiltonian $H\in C^{\infty}({\mathbb T}^n \times {\mathbb R}^n)$ given by $H(q,p)=h(p)$ is integrable with $\alpha_{H:{\mathbb T}^n}(0)=0$ and $\partial \alpha_{H:L}(0)=\{0\}$. Moreover, we have $$\alpha_{H:{\mathbb T}^n}(p_0)=\alpha_{H:{\mathbb T}^n}(-p_0)=0 \quad \& \quad \partial \alpha_{H:{\mathbb T}^n}(p_0)= -\partial \alpha_{H:{\mathbb T}^n}(-p_0)=\{dh(p_0)\}.$$ In particular Corollary \[cor3\] guarantees the existence of $\mathcal{M}(\phi_H)$-measures $\mu_1$ and $\mu_2$ such that $\rho(\mu_1)=dh(p_0)=-\rho(\mu_2)$ and $$\begin{aligned} \mathcal{A}_{H,\lambda}(\mu_1)=\alpha_{H:{\mathbb T}^n}(p_0)-\langle p_0,\rho(\mu) \rangle =0 \\ \mathcal{A}_{H,\lambda}(\mu_2)=\alpha_{H:{\mathbb T}^n}(-p_0)+\langle p_0,\rho(\mu_2) \rangle =0. \end{aligned}$$ Moreover, from Remark \[rem101\] it is not hard to deduce that $\operatorname{Supp}(\mu_1)\subset {\mathbb T}^n \times \{p_0\}$ and $\operatorname{Supp}(\mu_2)\subset {\mathbb T}^n \times \{-p_0\}$. In particular the measure $\mu:=\frac{\mu_1+\mu_2}{2}$ is $\phi_H$-invariant and satisfies $$\mathcal{A}_{H,\lambda}(\mu)=\frac{\mathcal{A}_{H,\lambda}(\mu_1)+\mathcal{A}_{H,\lambda}(\mu_2)}{2}=0=\alpha_{H:{\mathbb T}^n}(0)-\langle 0,\rho(\mu) \rangle$$ and $\rho(\mu)=0\in \partial \alpha_{H:{\mathbb T}^n}(0)$, so $\mu$ formally satisfies (\[Mateq902\]) with $c=0$. However, $\operatorname{Supp}(\mu)$ clearly carries no information about the “dynamics close to the zero-section ${\mathbb T}^n\times \{0\}$”. This example shows that in the non-Tonelli setting the requirement (\[Mateq902\]) is not enough to guarantee that the support of $\mu$ carries information about the dynamics of $\phi_H$ close to $L$. The definition of $\mathfrak{M}_{H:L}$ in Remark \[rem101\] guarantees that $\mathfrak{M}_{H:L}$-measures do carry such information. We do not know if $\mathfrak{M}_{H:L}$ contains all Mather measures when $H$ is Tonelli. \[Rem1rev2\] In Aubry-Mather theory, when $H\in C^{\infty}(T^*L)$ is Tonelli, the associated function $\alpha_{H:L}:H^1(L;{\mathbb R})\to {\mathbb R}$ is also convex and superlinear. In particular, the set of all subdifferentials of $\alpha_{H:L}$ is all of $H_1(L;{\mathbb R})$. Hence, all vectors in $H_1(L;{\mathbb R})$ are realized as rotation vectors of some $\mathcal{M}(\phi_H)$-measure. In the setting of Corollary \[cor3\], if $H\in C^{\infty}(M)$ is a Hamiltonian for which $$\label{rev2eq2} \frac{\alpha_{H:L}(c)}{|c|} \stackrel{|c|\to \infty}{\longrightarrow} \infty$$ (with respect to some norm on $H^1(L;{\mathbb R})$), then the same conclusion holds true.[^6] It is interesting to understand which conditions should be imposed on $H$ to achieve (\[rev2eq2\]). Since $\rho:\mathcal{M}(\phi_H)\to H_1(M;{\mathbb R})$ is continuous (with respect to the weak$^*$-topology on $\mathcal{M}(\phi_H)$) it is easy to see that its image is compact if $M$ is compact (use weak$^*$-compactness of $\mathcal{M}(\phi_H)$). Hence, to achieve (\[rev2eq2\]) one will need that $M$ be non-compact. However, even on non-compact $(M,\omega)$ one might not be able to achieve (\[rev2eq2\]) for any $H$ as the following example shows: Consider the two symplectic manifolds $(T^*N,d\lambda)$ and $({\mathbb T}^2=S^1\times S^1,\omega_0)$ for $N$ a closed manifold and $\omega_0$ the standard area form on ${\mathbb T}^2$ with $\int_{{\mathbb T}^2}\omega_0=1$. Now take $(M,\omega):=(T^*N\times {\mathbb T}^2, d\lambda \oplus \omega_0)$ and consider the Lagrangian $L:=N\times (S^1 \times \{0\}) \subset M$. Denote by $\eta \in \Omega^1(M)$ the closed 1-form obtained by pulling back the angle 1-form $d\varphi \in \Omega(S^1)$ (with $\int_{S^1}d\varphi=1$) by the projection $M\to {\mathbb T}^2\to S^1\times \{0\}$ and denote by $\psi$ the isotopy generated by the vector field $X\in \mathfrak{X}(M)$ satisfying $\iota_X \omega=\eta$. Then the flow $\psi$ is $1$-periodic, so $L=\psi_{{\mathbb Z}}(L)$ and c) in Theorem \[prop1\] implies $\alpha_{H:L}({\mathbb Z}[\eta])=\alpha_{H:L}(0)$ for all admissible $H\in C^{\infty}(M)$. In particular $\alpha_{H:L}(t[\eta])$ will *not* tend to $\infty$ for $t\to \infty$. Of course one could consider this a non-example: In fact $\alpha_{H:L}$ naturally descends to a function on the “cylinder” $H^1(L;{\mathbb R})/{\mathbb Z}[\eta]\cong H^1(N;{\mathbb R})\times S^1$ and, viewed as such a function, (\[rev2eq2\]) will certainly be true for some Hamiltonians $H\in C^{\infty}(M)$. On the other hand, if $H\in C^{\infty}(T^*N)$ satisfies $$\frac{H(q,p)}{|p|} \stackrel{|p|\to \infty}{\longrightarrow}\infty$$ and $H^1(N;{\mathbb R})$ admits a basis represented by closed 1-forms, all of which are nowhere vanishing, then it is easy to check that (\[rev2eq2\]) follows. This is in particular the case on $T^*{\mathbb T}^n$. See also [@Viterbo10] for an in-depth discussion of this case. Comparison with Mather’s theory {#Mathersec} =============================== In this section we compare properties of the support of the measures whose existence is guaranteed by Theorem \[thm1\] and \[thm2\] to those which arise in Mather’s theory and place our results in a historical context. To accomplish this, it will suffice to consider $(M,\omega)=(T^*{\mathbb T}^2,d\lambda)$. We will construct an $H$ on $T^*{\mathbb T}^2$ for which the support of $\mathfrak{M}_{H:L}$-measures becomes extremely “wild”. This phenomenon is closely related to diffusion phenomena such as Arnold’ diffusion and superconductivity channels [@BounemouraKaloshin14]. Studying $\mathfrak{M}_{H:L}$-measures in this setting was generously suggested to me by Vadim Kaloshin. Consider ${\mathbb R}^2(p)$ as well as ${\mathbb T}^2={\mathbb R}^2 /{\mathbb Z}^2$ equipped with the (mod ${\mathbb Z}^2$) coordinate $q$. Given sufficiently smooth functions $H_0:{\mathbb R}^2 \to {\mathbb R}$ and $F:{\mathbb T}^2 \times {\mathbb R}^2 \to {\mathbb R}$ as well as $\epsilon \geq 0$ we consider the Hamiltonian $$H_{\epsilon}(q,p):=H_0(p)+\epsilon F(q,p)$$ on $(T^*{\mathbb T}^2={\mathbb T}^2 \times {\mathbb R}^2, dp\wedge dq)$. Throughout this section we make use of the canonical identification $H^1(T^*{\mathbb T}^2,{\mathbb R})\cong {\mathbb R}^2$. The flow $\phi_{H_0}$ is integrable in the sense that it leaves every Lagrangian torus of the form $T(p):={\mathbb T}^2 \times \{ p \}$ invariant. KAM theory guarantees that, if $H_0$ is non-degenerate in the sense that its Hessian satisfies $$\det(\operatorname{Hess}(H_0))(p)\neq 0 \quad \forall \ p\in {\mathbb R}^2,$$ then, for all small enough $\epsilon>0$, many Lagrangian tori $T(p)$ persist. More precisely, if the rotation vector of $\phi_{H_{0}}|_{T(p)}$ is Diophantine and $\epsilon >0$ is small enough, then there is a Lagrangian torus which is invariant for $\phi_{H_{\epsilon}}$ close to $T(p)$. We are interested in what happens to the invariant torus $T(p)$ for $\epsilon>0$ when $\phi_{H_0}|_{T(p)}$ does *not* have Diophantine rotation vector. Does it (at least partially) persist or does it disappear? Mather [@Mather91] studied the case $$\label{equa11} \det(\operatorname{Hess}(H_0))(p)>0 \quad \forall \ p\in {\mathbb R}^2.$$ In this case all $T(p)$ *partially* persist. The invariant set studied by Mather, which is to be thought of as an “avatar” of $T(p)$, is the $p$-*Mather set* $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})\subset {\mathbb T}^2 \times {\mathbb R}^2$.[^7] This interpretation of the Mather set relies on two key properties exhibited by $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})$ under assumption (\[equa11\]): - Graph property: $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})\subset {\mathbb T}^2 \times {\mathbb R}^2$ is the graph of a Lipschitz function defined on a subset of $T(p)$. - Localization property: In a suitable sense $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})$ “is close” to (a subset of) $T(p)$ when $\epsilon >0$ is small. The precise statement of the localization property takes different forms, depending on the dynamics of $\phi_{H_0}|_{T(p)}$. Regardless of the dynamics of $\phi_{H_0}|_{T(p)}$ it is not hard to see that, under assumption \[equa11\], one will always have $$\label{Rev2eq1} \operatorname{Supp}(\mathfrak{M}_{H_{0}:T(p)})\cap \liminf_{\epsilon \downarrow 0}\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})\neq \emptyset,$$ where $\liminf$ is to be understood in the sense of Kuratowski. This follows simply because, under assumption (\[equa11\]), $\mathfrak{M}_{H_{\epsilon}:T(p)}$-measures minimize a Lagrange functional. Mather [@Mather91 Section 5] studied the case when the rotation vector of $\phi_{H_0}|_{T(p)}$ meets a Diophantine condition (i.e. when $T(p)$ is a KAM torus). He found that, under this assumption, $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})$ converges to $T(p)$ in the Hausdorff distance as $\epsilon \to 0$. The Diophantine condition was later relaxed by Bernard [@Bernard00; @Bernard10]. He studied the case when the dynamics $\phi_{H_0}|_{T(p)}$ is resonant, but $T(p)$ is foliated by invariant subtori on which the dynamics of $\phi_{H_0}$ are “sufficiently irrational”. In this case he found that $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})$ Hausdorff converges to one of the invariant subtori of $T(p)$ as $\epsilon \to 0$ [@Bernard00 Theorem 1]. Summing up, Mather’s theory (morally speaking) guarantees that, if (\[equa11\]) is satisfied, then the system cannot be *too* unstable in the sense that every $T(p)$ will partially survive small perturbations of $H_0$. In a different direction, Arnold’ conjectured [@Arnold64], [@Arnold94] that the Hamiltonian flow generated by a generic Hamiltonian $H\in C^{\infty}(T^*{\mathbb T}^n)$, with $n\geq 3$, will exhibit diffusing orbits (this phenomenon is today known as *Arnold’ diffusion*). Hence, if Arnold’s conjecture holds true, then all systems with more than two degrees of freedom will exhibit *some* instability. When $$\label{equa2} \det(\operatorname{Hess}(H_0))(p)<0 \quad \forall \ p\in {\mathbb R}^2$$ the candidate for a perturbation of $T(p)$ is again $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})$. Do these sets satisfy the graph property or the localization property? To our knowledge Herman [@Herman88] was the first to observe that, in general, in the case (\[equa2\]) $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})$ will *not* be a Lipschitz graph over $T(p)$ (see also [@Chen92] for an English exposition of Herman’s example). Below we construct an example showing that not only can the graph property be violated, but so can the localization property. In fact the example violates even the simplest localization property (\[Rev2eq1\]) in the worst possible way. The dynamics responsible for the failure of this kind of convergence is a very fast type of diffusion arising from so-called *superconductivity channels*. We learned about this phenomenon from Bounemoura-Kaloshin’s [@BounemouraKaloshin14]. A first indication that symplectic methods can be used to study this phenomenon was found in [@EntovPolterovich17]. The concept of superconductivity channels goes back to the work of Nekoroshev [@Nekoroshev77], [@Nekoroshev79] who discovered that for every $x=(q,p)\in {\mathbb T}^2 \times {\mathbb R}^2$, $\phi_{H_{\epsilon}}^t(x)$ remains close to $\phi_{H_0}^t(x)\in T(p)$ for $|t|\leq e^{c\epsilon^{-a}}$ for some positive constants $c,a>0$, as long as $\epsilon>0$ is sufficiently small and $H_{0}$ is *steep*.[^8] By a result due to Ilyashenko [@Ilyashenko86], the steepness condition can be phrased as requiring that the restriction of $H_0$ to any $1$-dimensional linear subspace of ${\mathbb R}^2$ has only isolated critical points. Hence, the philosophy we follow is that, in order to find diffusion which is fast enough to “push” $\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(p)})$ far away from $T(p)$, we should choose $H_0$ *non*-steep. The example {#secex} ----------- Consider in symplectic coordinates $\theta=(\theta_1,\theta_2)\in {\mathbb T}^2$ and $I=(I_1,I_2)\in {\mathbb R}^2$ the Hamiltonian $$H_{\epsilon}(\theta,I) =I_1I_2 +\epsilon \varphi(I_1)\sin(2\pi \theta_1),$$ where $\varphi:{\mathbb R}\to [0,1]$ is a smooth function with $\varphi(t)=\varphi(-t)$ and $$\varphi(s)= \begin{cases} 1 & \text{if} \ |s|\leq K \\ 0 & \text{if} \ |s|>K+1 \end{cases}, \quad \varphi'(s) \begin{cases} \geq 0 & \text{if} \ s\leq 0 \\ \leq 0 & \text{if} \ s\geq 0 \end{cases}$$ for some large constant $K>0$ (see Figure \[Mfig1\]). Note that, if we view $H_0$ as a function on ${\mathbb R}^2$, then $\det(\operatorname{Hess}(H_0))(I)<0$ for all $I\in {\mathbb R}^2$, so $H_{\epsilon}$ fits into the framework we discussed above. The resulting equations of motion are given by $$\label{equa4} \begin{cases} \dot{I}_1=-2\pi \epsilon \varphi (I_1)\cos(2\pi \theta_1)\\ \dot{I}_2=0 \\ \dot{\theta}_1=I_2+\epsilon \varphi'(I_1)\sin(2\pi \theta_1) \\ \dot{\theta}_2=I_1. \end{cases}$$ In particular, we see that $\mathcal{O}_+(X)$ is compact for every compact $X$, so Corollary \[cor3\] applies to this system. \[rem000\] The superconductivity channels $\{I_1=0\}\cup \{I_2=0\}$ give rise to diffusion: Consider initial conditions $$(\theta_1(0),\theta_2(0),I_1(0),I_2(0))$$ such that $I_2(0)=0$, $|I_1(0)|<K$ and $\sin(2\pi \theta_1(0))=0$. Then $t\mapsto \theta_1(t)$ is constant and $$\dot{I}_1=\pm 2\pi \epsilon \varphi(I_1),$$ so the function $t\mapsto |I_1(t)-I_1(0)|$ is strictly increasing (assuming $\epsilon>0$). We denote by $$\mathcal{D}:=\{ (\theta_1,\theta_2,I_1,I_2)\ |\ \sin(2\pi \theta_1)=0, \ I_2=0, \ |I_1|<K \}$$ the set of initial conditions with $|I_1(0)|<K$ which diffuse. Poincar[é]{} recurrence implies $$\label{equa9} \mu(\mathcal{D})=0 \quad \forall \ \mu \in \mathcal{M}(\phi_{H_{\epsilon}}),$$ for every $\epsilon>0$. Note that if $\epsilon=0$, then (by Remark \[rem101\] and the fact that each $T(I_1,I_2)$ is $\phi_{H_0}$-invariant) $$\operatorname{Supp}(\mathfrak{M}_{H_0:T(I_1,I_2)})\subset T(I_1,I_2).$$ In contrast, when $\epsilon >0$ we have \[propdiff\] Fix $I\in (-K,K)$ and denote by $B_K\subset {\mathbb R}^2$ the open $K$-ball centered at $0$. For every $\epsilon >0$ we have $$\operatorname{Supp}(\mathfrak{M}_{H_{\epsilon}:T(I,0)})\cap ({\mathbb T}^2 \times B_K)=\emptyset.$$ This proposition exhibits the violation of the localization property for $\mathfrak{M}_{H_{\epsilon}:L}$ when $H_0$ fails to meet the convexity condition (\[equa11\]) in the following sense: If we impose the convexity hypothesis on $H_0$, a result due to Bernard [@Bernard00] implies that $\mathfrak{M}_{H_{\epsilon}:T(I,0)}$ Hausdorff-converges to the circle $$\{(\tfrac{1}{4},\theta_2,I,0) \ |\ \theta_2\in S^1\}\subset T(I,0)$$ as $\epsilon \downarrow 0$ for all $I\in {\mathbb R}\backslash \mathbb{Q}$ with $|I|<K$. Clearly, Proposition \[propdiff\] implies that this is not the case for $\mathfrak{M}_{H_{\epsilon}:T(I,0)}$ in our setting. To present how our example fits into Bernard’s theory [@Bernard00] we will from now on impose the assumption that $H_0$ be convex[^9] and check that, after adding this hypothesis to our example, it formally fits into Bernard’s framework. In Bernard’s terminology our $\phi_{H_{0}}^1$ is denoted by $\Phi_0$, and its lift to ${\mathbb R}^2\times {\mathbb R}^2$ is denoted by $\phi_0$. Bernard’s <span style="font-variant:small-caps;">Hypothesis 1</span> requires that there is a diffeomorphism $w:{\mathbb R}^2\to {\mathbb R}^2$ such that[^10] $$\phi_0(\Theta,I)=(\Theta+w(I),I).$$ Clearly, in our case $w(I_1,I_2)=(-I_2,-I_1)$. A generating function $S_0:{\mathbb R}^2 \times {\mathbb R}^2 \to {\mathbb R}$ for $\phi_0$ is a smooth function such that $\phi_0(\Theta(0),I(0))=(\Theta(1),I(1))$ if and only if $$I(0)=\partial_{1}S_0(\Theta(0),\Theta(1)) \quad \text{and} \quad I(1)=-\partial_{2}S_0(\Theta(0),\Theta(1)).$$ In our example we can take $$S_0(\Theta(0),\Theta(1))=H_0(\Theta(1)-\Theta(0))=(\Theta_1(1)-\Theta_1(0))(\Theta_2(1)-\Theta_2(0)),$$ so our imposed convexity condition on $H_0$ amounts to Bernard’s <span style="font-variant:small-caps;">Hypothesis 2</span>. Bernard studies the Aubry-Mather set for the diffeomorphism $\Phi_{\epsilon}:T^*{\mathbb T}^2 \to T^*{\mathbb T}^2$ whose pull-back $\phi_{\epsilon}$ to ${\mathbb R}^2\times {\mathbb R}^2$ has as a generating function $$S_{\epsilon}(\Theta(0),\Theta(1))=S_0(\Theta(0),\Theta(1))+\epsilon P(\Theta(0),\Theta(1)),$$ for a function $P:{\mathbb R}^2 \times {\mathbb R}^2\to {\mathbb R}$ which satisfies an exactness condition (Bernard’s <span style="font-variant:small-caps;">Hypothesis 3</span>). Moreover, as he explains, one can assume that $S_{\epsilon}$ is “standard at $\infty$” (<span style="font-variant:small-caps;">Hypothesis 4</span>). Under our imposed convexity assumption on $H_0$, $\phi_{H_{\epsilon}}^1$ would also admit a generating function $S_{\epsilon}$ of the type above, and the two latter assumptions would also hold true in our setting (after perhaps adapting $S_{\epsilon}$ “at $\infty$”). Fix now $I\in {\mathbb R}\backslash \mathbb{Q}$ such that $|I|<K$. Then the torus $T(I,0)$ is foliated by an $S^1$-family of $\phi_{H_0}^1$-invariant circles $S^1(\theta_1)=\{(\theta_1,\theta_2,I,0) \ | \ \theta_2 \in S^1\},\ \theta_1\in S^1$. Bernard introduces the *averaged perturbation* which in our situation is given by $$\mathcal{P}(\Theta_1)=\int_{S^1}P(\Theta_1,\Theta_2,\Theta_1,\Theta_2-I) d\Theta_2.$$ $\mathcal{P}$ descends to a function $\mathcal{P}:S^1 \to {\mathbb R}$ and Bernard’s <span style="font-variant:small-caps;">Hypothesis 5</span> requires that this function have a unique non-degenerate minimum $\theta_1^*\in S^1$. Once we have checked that this is the case in our example (under our imposed convexity assumption) Bernard’s [@Bernard00 Theorem 1] would imply that $\mathfrak{M}_{H_{\epsilon}:T(I,0)}$ Hausdorff-converges to the subset $S^1(\theta_1^*)\subset T(I,0)$ as $\epsilon \downarrow 0$. To see that $\mathcal{P}$ would have to have a unique non-degenerate minimum on $S^1$ in our example, note that $S_0$ vanishes on $T(I,0)$, so in fact we have $S_{\epsilon}=\epsilon P$ on $T(I,0)$. Recall [@McDuffSalamon17 Section 9.1] that the generating function $S_{\epsilon}$ of the lift $\phi_{\epsilon}$ of $\phi_{H_{\epsilon}}^1$ is (up to addition by a constant) given by the symplectic action in the following sense: Given $(\Theta(0),\Theta(1))\in {\mathbb R}^2 \times {\mathbb R}^2$ there exist (due to the convexity assumption) unique $(I(0),I(1))\in {\mathbb R}^2 \times {\mathbb R}^2$ such that $\phi_{\epsilon}(\Theta(0),I(0))=(\Theta(1),I(1))$ and $$\begin{aligned} S_{\epsilon}(\Theta(0),\Theta(1))&=\int_0^1 \langle \lambda ,X_{H_{\epsilon}}\rangle (\Theta(t),I(t))-H_{\epsilon}(\Theta(t),I(t))\ dt\\ &=-\epsilon \int_0^1 (\varphi(I_1(t))-I_1(t)\varphi'(I_1(t)))\sin(2\pi \Theta_1(t)) \ dt \end{aligned}$$ Now from (\[equa4\]) it follows that, if $\epsilon >0$ is small enough then $|I_1(t)|<K$ for all $t\in [0,1]$, so that in particular $\Theta_1(t)=\Theta_1(0)$ for all $t\in [0,1]$ (since $I_2(t)=0$). Hence, for such $\epsilon>0$ we have $$\begin{aligned} S_{\epsilon}(\Theta(0),\Theta(1))=-\epsilon \sin(2\pi \Theta_1(0)), \end{aligned}$$ and thus $$\mathcal{P}(\theta_1)=-\sin(2\pi \theta_1), \quad \theta_1\in S^1.$$ So clearly $\mathcal{P}$ would have a unique non-degenerate minimum at $\theta_1^*=\frac{1}{4}$. This finishes the demonstration that our example, formally speaking, fits into Bernard’s framework if only one imposes a convexity condition on $H_0$. The rest of this section will be spent proving Proposition \[propdiff\]. To make the notation a little less heavy we will write $\alpha_{H_{\epsilon}}(I_1,I_2)=\alpha_{H_{\epsilon}:T(0,0)}(I_1,I_2)$ for $(I_1,I_2)\in {\mathbb R}^2$. The conclusion of Proposition \[propdiff\] resonates well with classical insight from symplectic topology. For example, it is nowadays well understood that $\dim_{{\mathbb R}}H_*(M;{\mathbb R})\leq \# \text{Fix}(\phi)$ for non-degenerate $\phi \in \operatorname{Ham}(M,\omega)$ (this is one of the celebrated Arnol’d conjectures). Symplectic topology can verify this inequality, but it cannot say anything about where in $M$ the fixed points are located. \[lemmat1\] For every $\epsilon \geq 0$ we have $$\alpha_{H_{\epsilon}}(I,0)=0 \quad \forall \ I\in {\mathbb R}.$$ The proof of this lemma (which uses nothing but standard properties of Lagrangian spectral invariants) is presented in Section \[secproof2\]. The proof of Proposition \[propdiff\] consists in studying carefully the solutions to (\[equa4\]). From Remark \[rem101\] and the proof of Theorem \[thm1\] we know that, in order to study the support of $\mathfrak{M}_{H_{\epsilon}:T(I,0)}$-measures for $\epsilon>0$, it suffices to study the support of $\mu \in \mathcal{M}(\phi_{H_{\epsilon}})$ which arise as the weak$^*$-limits of sequence $(\mu_{\varsigma})_{\varsigma\in {\mathbb N}}\subset \mathcal{M}$ characterized by (\[equa12\]). In the current setting we can write $(x_{\varsigma})_{\varsigma \in {\mathbb N}}=(\theta^{\varsigma}_1(0),\theta_2^{\varsigma}(0),I_1^{\varsigma}(0),I_2^{\varsigma})_{\varsigma \in {\mathbb N}}$ ($I_2$ is an integral of motion). The conditions described in Remark \[rem101\] and in the proof of Theorem \[thm1\] now amount to $$\begin{aligned} &(I_1^{\varsigma}(0),I_2^{\varsigma})\stackrel{\varsigma \to \infty}{\longrightarrow} (I,0) \label{equa5}\\ &\frac{\mathcal{A}_{H_{\epsilon}:T(I_1^{\varsigma}(0),I_2^{\varsigma}(0))}(\gamma_{\varsigma})}{k_{\varsigma}}\stackrel{\varsigma \to \infty}{\longrightarrow}\alpha_{H_{\epsilon}}(I,0)=0. \label{equa7}\end{aligned}$$ From the computation in Remark \[Mrem1\] on page below, it follows that $$\frac{\mathcal{A}_{H_{\epsilon}:T(I_1^{\varsigma}(0),I_2^{\varsigma}(0))}(\gamma_{\varsigma})}{k_{\varsigma}}=\mathcal{A}_{H_{\epsilon}:\lambda}(\mu_{\varsigma})+\int \langle \eta_{\varsigma},X_{H_{\epsilon}} \rangle d\mu_{\varsigma},$$ where the closed 1-form $\eta_{\varsigma}$ is given by $$\eta_{\varsigma}=I_1^{\varsigma}(0)d\theta_1+I_2^{\varsigma}(0)d\theta_2.$$ Hence, as a consequence of the following claim we conclude that (\[equa7\]) amounts to saying $$\label{MEQ2} \mathcal{A}_{H_{\epsilon}:\lambda}(\mu)=\lim_{\varsigma \to \infty} \mathcal{A}_{H_{\epsilon}:\lambda}(\mu_{\varsigma})=0.$$ $\lim_{\varsigma \to \infty}\int \langle \eta_{\varsigma},X_H \rangle d\mu_{\varsigma}=0$ Suppose for contradiction that this were not the case. Then the rotation vector $\rho(\mu)\in {\mathbb R}^2$ of the weak$^*$-limit $\mu$ of $(\mu_{\varsigma})_{\varsigma \in {\mathbb N}}$ satisfies $ \langle Id\theta_1,\rho(\mu) \rangle \neq 0$. Since $\operatorname{Supp}(\mu)\subset \{I_2=0\}$, this implies the existence of an ergodic measure $\nu \in \mathcal{M}(\phi_{H_{\epsilon}})$ with $\operatorname{Supp}(\nu)\subset \{I_2=0\}$ such that $\langle Id\theta_1,\rho(\nu) \rangle \neq 0$. By Birkhoff’s ergodic theorem $\nu$-almost every initial condition $x$ would then satisfy $$\label{MEQ1} \frac{1}{T}\int_0^T \langle Id\theta_1,X_{H_{\epsilon}} \rangle \phi_{H_{\epsilon}}^t(x)\ dt\stackrel{T\to \infty}{\longrightarrow}C$$ for some constant $C\neq 0$. However, as pointed out in Remark \[rem000\], if $I_2=0$ and $\sin(2\pi \theta_1(t_0))=0$ for some $t_0$ then $\sin(2\pi \theta_1(t))=0$ for all $t$. In particular we have that either $\sin(2\pi \theta_1(t))\leq 0$ for all $t$ or $\sin(2\pi \theta_1(t))\geq 0$ for all $t$. In either case $d\theta_1$ is an exact 1-form on a neighbourhood of $\{\phi_{H_{\epsilon}}^t(x)\ |\ t\in {\mathbb R}\}$, which contradicts (\[MEQ1\]). This contradiction finishes the proof of the claim. Since $$\begin{aligned} \mathcal{A}_{H_{\epsilon}:\lambda}(\mu_{\varsigma})= \int H_{\epsilon}-\langle \lambda,X_{H_{\epsilon}}\rangle \ d\mu_{\varsigma}=\int -I_1I_2+\epsilon (\varphi(I_1)-I_1\varphi'(I_1))\sin(2\pi \theta_1) \ d\mu_{\varsigma}, \end{aligned}$$ (\[MEQ2\]) can be written $$\begin{aligned} \label{MEQ3} 0=\mathcal{A}_{H_{\epsilon}:\lambda}(\mu)=& \epsilon \int (\varphi(I_1)-I_1\varphi'(I_1))\sin(2\pi \theta_1) \ d\mu. \end{aligned}$$ To further understand $\operatorname{Supp}(\mu)$, note that energy-preservation (i.e. $H_{\epsilon}(x)=H_{\epsilon}(\phi_{H_{\epsilon}}^t(x))$) implies the estimate $$\label{Meq6} |\varphi(I_1^{\varsigma}(t))\sin(2\pi \theta_1^{\varsigma}(t))-\sin(2\pi \theta_1^{\varsigma}(0))|\leq \frac{2K}{\epsilon}|I_2^{\varsigma}|.$$ For this computation we assume that $|I_1^{\varsigma}(0)|<K$, which will be true for large enough $\varsigma$ by (\[equa5\]) and the assumption that $|I|<K$. We will now use the estimates (\[MEQ3\]) and (\[Meq6\]) to prove Proposition \[propdiff\].[^11] We first claim that these two estimates together imply that $$\label{MEQ4} \sin(2\pi \theta_1^{\varsigma}(0))\stackrel{\varsigma \to \infty}{\longrightarrow} 0.$$ Indeed, if we could extract a subsequence with $\sin(2\pi \theta_1^{\varsigma}(0)) \to C\neq 0$ then (\[Meq6\]) would imply that the functions $I_1\mapsto \varphi(I_1)$ and $\theta_1 \mapsto \sin(2\pi \theta_1)$ were bounded away from $0$ on $\operatorname{Supp}(\mu)$, and that the latter had a constant sign (equal to $\text{Sign}(C)$) on $\operatorname{Supp}(\mu)$. In particular (since $-I_1\varphi'(I_1)\geq 0 \ \forall \ I_1$) the function $I_1\mapsto (\varphi(I_1)-I_1\varphi'(I_1))$ would be positive and bounded away from $0$ on $\operatorname{Supp}(\mu)$, so one would arrive at a contradiction to (\[MEQ3\]). Together, (\[Meq6\]) and (\[MEQ4\]) now imply $$\operatorname{Supp}(\mu)\cap \{\varphi_1(I_1)>0\}\subset \{\sin(2\pi \theta_1)=0\},$$ which by Remark \[rem000\] implies $\operatorname{Supp}(\mu)\cap \{\varphi_1(I_1)>0\}=\emptyset$. Hence, Proposition \[propdiff\] follows. Symplectic “Mather theory” in the absence of intersections {#secC0} ========================================================== From the point of view of the previous sections, it is tempting to say that the existence of Mather-like measures is a consequence of symplectic intersection phenomena. In this section we study invariant measures in the “absence of intersections”, e.g. when every compact subset of $(M,\omega)$ can be displaced by a Hamiltonian diffeomorphism. More precisely, we develop a $C^0$-approach to studying invariant measures of (autonomous) Hamiltonian systems using ideas due to Entov-Polterovich [@EntovPolterovich17] and Polterovich [@Polterovich14]. Our main application is to the study of Hamiltonian systems on twisted cotangent bundles and ${\mathbb R}^{2n}$. The setup we consider here is slightly different from the one in the previous section. Here $(M,\omega)$ will denote a symplectic manifold which is either closed or geometrically bounded (see e.g. [@AudinLafontaine94]) and $L\subset (M,\omega)$ will denote any closed (i.e. compact without boundary) connected Lagrangian submanifold. Consider an autonomous Hamiltonian $H\in C^{\infty}(M)$ with complete flow $\phi_H=\{\phi_H^t\}_{t\in {\mathbb R}}$. Since $H$ is an integral of motion, sublevel sets $\Sigma_k=\Sigma_k(H):=\{H<k\}$ are invariant. As discussed previously, every $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ has a well-defined rotation vector or asymptotic cycle $$\rho(\mu)\in H_1(\Sigma_k;{\mathbb R})$$ defined by (\[eq10\]). In the following we denote by $e(X)\in [0,\infty]$ the *displacement energy* of a subset $X\subset M$ and by $e_S(X)$ the *stable displacement energy* of $X$. We recall that $$e(X):=\inf \left\{ || H || :\ H\in C^{\infty}_c([0,1]\times M) \ \text{ s.t.} \ \phi_H^1(\overline{X})\cap \overline{X}=\emptyset \right\},$$ where we use the convention that the infimum over $\emptyset$ equals $\infty$ and $$| | H | |:=\int_0^1 \max_M(H_t)-\min_M(H_t)\ dt$$ denotes the Hofer norm of $H$. By definition, $e_S(X)=e(X\times S^1)$, where $S^1\subset T^*S^1$ denotes the 0-section and $X\times S^1$ is viewed as a subset of the symplectic manifold $(M\times T^*S^1,\omega \oplus \omega_0)$, with $\omega_0$ the canonical symplectic form on $T^*S^1$. Clearly, $e_S(X)\leq e(X)$ for all $X\subset M$ with strict inequality in several important examples [@Schlenk06]. \[cor00\] Suppose $(M,\omega)$ is weakly exact and let $H\in C^{\infty}(M)$ be proper and bounded from below. Fix an energy value $k\in {\mathbb R}$ such that $\Sigma_k$ is stably displaceable. Suppose $L\subset \Sigma_k \subset (M,\omega)$ is a Lagrangian with Abelian $\pi_1(L)$ and fix $\tilde{c}\in H^1(L;{\mathbb R})$ such that $\partial \tilde{c}|_{\pi_2(M,L)}\equiv [\omega]|_{\pi_2(M,L)}$. Then for every $a\in H^1(\Sigma_k;{\mathbb R})$ satisfying $$\label{Meq5} a|_L\in (e_S(\Sigma_k),\infty)\cdot H^1(L;{\mathbb Z})-\tilde{c} \quad (\subset H^1(L;{\mathbb R}))$$ and every $\epsilon \in (0,k-\max_{L}H)$ there exists a measure $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ satisfying the two conditions $$\begin{aligned} \langle a,\rho(\mu) \rangle &\leq \max_L H-k+\epsilon \\ H(\operatorname{Supp}(\mu))&\subset [\max_L(H)+\tfrac{\epsilon}{2},k-\tfrac{\epsilon}{2}]. \end{aligned}$$ In this statement, $(e_S(\Sigma_k),\infty)\cdot H^1(L;{\mathbb Z})-\tilde{c}$ denotes the subset $\{t\cdot c-\tilde{c}\ |\ c\in H^1(L;{\mathbb Z}),\ t\in (e_S(\Sigma_k),\infty)\}$ of $H^1(L;{\mathbb R})$ and $\partial :H^1(L;{\mathbb R})\to H^2(M,L;{\mathbb R})$ denotes the cohomological boundary map. Lemma \[lem00\] on page below guarantees the existence of a $\tilde{c}\in H^1(L;{\mathbb R})$ such that $\partial \tilde{c}|_{\pi_2(M,L)}=[\omega]|_{\pi_2(M,L)}$ when $(M,\omega)$ is weakly exact (i.e. $\omega|_{\pi_2(M)}\equiv 0$) and $\pi_1(L)$ is Abelian. Of course, a class $a\in H^1(\Sigma_k;{\mathbb R})$ satisfying (\[Meq5\]) might not exist. However, it does exist if the restriction map $H^1(\Sigma_k;{\mathbb R})\to H^1(L;{\mathbb R})$ is surjective. Theorem \[cor00\] is in fact an application of a general existence result obtained in Theorem \[thm00\] below. Before discussing the abstract setting, we present another consequence of Theorem \[thm00\] (Theorem \[thm01\]) as well as an example. To state Theorem \[thm01\] we need \[def00\] Fix $c\in H^1(L;{\mathbb R})$. The *(Lagrangian) $(L,c)$-shape* of a subset $X\subset M$ is defined as the subset of $H^1(X;{\mathbb R})$ consisting of those classes $a$ for which there exists a Lagrange isotopy $\psi:[0,1]\times L\to M$ starting at $L$ and ending in $X$ in the sense that $\psi_1(L)\subset X$ and which in addition satisfies the condition $$\operatorname{Flux}_{L}(\psi)-c=\psi_1^*a\in H^1(L;{\mathbb R}).$$ We will denote the $(L,c)$-shape of $X$ by $\operatorname{Sh}(X;L,c)\subset H^1(X;{\mathbb R})$. By the *$L$-shape of $X$* we simply mean the $(L,0)$-shape of $X$ and use the shorthand notation $\operatorname{Sh}(X;L):=\operatorname{Sh}(X;L,0)$. Of course it might be that $\operatorname{Sh}(X;L,c)=\emptyset$. The notion of *symplectic shape* was introduced by Sikorav [@Sikorav89] [@Sikorav91] and later studied by Eliashberg [@Eliashberg91]. The above definition is “modelled” on this classical concept, but also makes sense for closed symplectic manifolds. In case $(T^*N,\omega=d\lambda)$ is a cotangent bundle, our definition of $(N,c)$-shape of $X\subset T^*N$ should be thought of as corresponding to the shape of $X$ in the sense of [@Eliashberg91] computed with respect to the primitive $\lambda-\eta_c$ of $\omega$, where $\eta_c$ is a closed $1$-form representing $c\in H^1(N;{\mathbb R})=H^1(T^*N;{\mathbb R})$. For the next result we will use the following notation: Given our closed Lagrangian $L\subset (M,\omega)$ and a class $c\in H^1(L;{\mathbb R})$, we denote by $\gamma_L(c)$ the positive generator of the subgroup $$\label{eqrev2eq1} \mathcal{G}_L(c):=\langle [\omega]-\partial c, \pi_2(M,L)\rangle \leq {\mathbb R},$$ if this subgroup is discrete and non-trivial. If $\mathcal{G}_L(c)$ is trivial (i.e. $\mathcal{G}_L(c)=\{0\}$) we set $\gamma_L(c)=+\infty$ and we set $\gamma_L(c)=0$ if $\mathcal{G}_L(c)$ is *not* discrete. We say that $L$ is *rational* if $\gamma_L(0)\in (0,\infty)$. \[thm01\] Let $L\subset (M,\omega)$ be a closed Lagrangian and let $H\in C^{\infty}(M)$ be proper and bounded from below. Suppose $k\in {\mathbb R}$ is an energy value and $c\in H^1(L;{\mathbb R})$ such that $$\label{ref2eq2} e_S(\Sigma_k)<\gamma_L(c).$$ Then for every $a\in \operatorname{Sh}(\Sigma_k;L,c)\subset H^1(\Sigma_k;{\mathbb R})$ there exists an ergodic $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ which satisfies $$\label{eq05} \langle a,\rho(\mu) \rangle <0.$$ Again, this is in fact a corollary of a more general result given below. \[ref2rem1\] Note that if $L$ is rational and $e_S(\Sigma_k)<\gamma_L(0)$, then (\[ref2eq2\]) holds for every element $c$ in the lattice $\gamma_L(0)\cdot H^1(L;{\mathbb Z}) \leq H^1(L;{\mathbb R})$: $$\mathcal{G}_L(c)=\langle [\omega]-\partial c,\pi_2(M,L)\rangle \subset \gamma_L(0)\cdot {\mathbb Z}-\langle c,\pi_1(L)\rangle \subset \gamma_L(0)\cdot {\mathbb Z}.$$ Theorem \[thm01\] deduces information about a Hamiltonian system using Lagrange isotopies which *cannot* be realized by globally defined symplectic isotopies. This should be compared with information about invariant measures coming from Lagrange isotopies induced by globally defined symplectic isotopies as in [@Polterovich14]. More precisely, the Lagrange isotopies $\psi$ which are the source of information in Theorem \[thm01\] are those for which $\operatorname{Flux}_L(\psi)$ does *not* lie in the image of the restriction map $H^1(M;{\mathbb R}) \to H^1(L;{\mathbb R})$. To see this, note that, by the main result in Chekanov’s beautiful paper [@Chekanov98], no stable Lagrange isotopy with flux in the image of $H^1(M \times S^1;{\mathbb R})\to H^1(L \times S^1;R)$ can take $L\times S^1$ into $\Sigma_k \times S^1$ if $e_S(\Sigma_k)<\gamma(L)$. \[ref2rem2\] Let’s explain how Theorem \[cor00\] follows from Theorem \[thm01\]. Consider an energy level $k\in {\mathbb R}$ such that $\Sigma_k$ is stably displaceable as well as a $\tilde{c}\in H^1(\Sigma_k;{\mathbb R})$ such that $[\omega]|_{\pi_2(M,L)}=\partial \tilde{c}|_{\pi_2(M,L)}$.[^12] If $a\in H^1(\Sigma_k;{\mathbb R})$ satisfies (\[Meq5\]) then $a|_L\in H^1(L;{\mathbb R})$ can be written $a|_L=Tc'-\tilde{c}$ with $T\in (e_S(\Sigma_k),\infty)$ and $c'\in H^1(L;{\mathbb Z})$. In particular $$\mathcal{G}_L(-a|_L)=\langle [\omega]+\partial(Tc'-\tilde{c}),\pi_2(M,L)\rangle \subset T \langle c',\pi_1(L)\rangle \subset T\cdot {\mathbb Z},$$ so $\gamma_L(-a|_L)>e_S(\Sigma_k)$. Moreover, since $L\subset \Sigma_k$, we have $a \in \operatorname{Sh}(\Sigma_k;L,-a|_L)$. To see this, consider the constant Lagrange isotopy $\psi:[0,1]\times L \to M$ given by $\psi_t(x)=x$. Then $\psi_1(L)=L\subset \Sigma_k$ and $\operatorname{Flux}_L(\psi)=0$, so $$\operatorname{Flux}_L(\psi)-(-a|_L)=a|_L=\psi_1^*a$$ and $a \in \operatorname{Sh}(\Sigma_k;L,-a|_L)$ as claimed. Hence, by Theorem \[thm01\] there exists a $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ such that $$\langle a, \rho(\mu) \rangle <0.$$ This is essentially the statement of Theorem \[cor00\] (the additional estimates are a consequence of Remark \[rem2\] below). \[ex01\] Let’s illustrate the phenomenon captured in Theorem \[thm01\] by a very simple example. Denote by $h:[0,\infty)\to {\mathbb R}$ the function whose graph is illustrated in Figure \[fig01\] and consider the Hamiltonian $H(x,y)=h(\sqrt{x^2+y^2})$ on the plane. The symplectic structure we use on ${\mathbb R}^2(x,y)$ is $dx \wedge dy$. The sublevel set $\Sigma_{3/2}$ is an annulus containing periodic orbits of $\phi_H$ which represent both positive and negative classes in $H_1(\Sigma_{3/2};{\mathbb R})\cong {\mathbb R}$. This is exactly what our theory detects: We choose the Lagrangian $L:=\{x^2+y^2=16\}$, so that $e_S(\Sigma_{3/2})\leq e(\Sigma_{3/2})<16\pi=\gamma(L)$. Denote by $d\varphi$ the angle 1-form on $L$ (so that $\int_{l}d\varphi=2\pi$ for $l(t)=(4\cos(t),4\sin(t)),\ t\in [0,2\pi]$). According to (\[Meq4\]), the Lagrange isotopy $\psi$ which “shrinks” $L$ to the Lagrangian $\{x^2+y^2=4\}=\{H=0\}\subset \Sigma_{3/2}$ satisfies $$\langle \operatorname{Flux}_{L}(\psi),[l] \rangle =16\pi-4\pi=12\pi,$$ so $\operatorname{Flux}_{L}(\psi)=6[d\varphi]\in H^1(L;{\mathbb R})$. Hence, $6[d\varphi]\in \operatorname{Sh}(\Sigma_{3/2};L)\subset H^1(\Sigma_{3/2};{\mathbb R})$ and Theorem \[thm01\] guarantees the existence of a $\mu \in \mathcal{M}(\Sigma_{3/2};\phi_H)$ such that $$\langle 6[d\varphi],\rho(\mu) \rangle <0 \Leftrightarrow \langle [d\varphi],\rho(\mu) \rangle <0,$$ I.e. the motion detected by $\mu$ is that which is indicated by arrow $b$ in Figure \[fig01\]. Moreover, $8[d\varphi]\in \gamma(L)\cdot H^1(L;{\mathbb Z})$ and $$-2[d\varphi]=\operatorname{Flux}_L(\psi)-8[d\varphi]\in \operatorname{Sh}(\Sigma;L,8[d\varphi]),$$ so Theorem \[thm01\] also detects the existence of a $\nu \in \mathcal{M}(\Sigma_{3/2};\phi_H)$ such that $$-\langle 2[d\varphi],\rho(\nu) \rangle <0 \Leftrightarrow \langle [d\varphi],\rho(\nu) \rangle >0.$$ I.e. the motion detected by $\nu$ is that which is indicated by arrow $a$ in Figure \[fig01\]. For more sophisticated applications of the above results, see Section \[secexamples\]. The theorems above are in fact corollaries of a general theorem based on the study of a $\kappa$-function which we now define. Given a Hamiltonian $H\in C^{\infty}(M)$ which is bounded from below we associate to it a function $\kappa_{H:L}:H^1(L;{\mathbb R})\to {\mathbb R}\cup \{+\infty \}$ defined by $$\kappa_{H:L}(c):=\inf_{\psi}\max_{\psi_1(L)}(H), \quad c\in H^1(L;{\mathbb R}),$$ where the infimum runs over all Lagrange isotopies $\psi:[0,1]\times L\to M$ starting at $L$ and satisfying $\operatorname{Flux}_{L}(\psi)=c$. Again we use the convention that the infimum over $\emptyset$ equals $+\infty$. The motivation for this function comes from Aubry-Mather theory where it was discovered by Contreras-Iturriaga-Paternain [@ContrerasIturriagaPaternain98 Corollary 1] that the Mather $\alpha$-function, associated to a convex Hamiltonian on the cotangent bundle of a closed manifold, can be defined in a similar way. \[thm00\] Consider an autonomous Hamiltonian $H\in C^{\infty}(M;{\mathbb R})$ which is proper and bounded from below. Let $c\in H^1(L;{\mathbb R})$ and suppose $k\in {\mathbb R}$ is an energy value such that $k\leq \kappa_{H:L}(c)$. Then for every $$a\in \operatorname{Sh}(\Sigma_k;L,c)$$ there exists an ergodic $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ which satisfies $$\label{eq06} \langle a,\rho(\mu) \rangle <0.$$ \[rem2\] In fact the proof of Theorem \[thm00\] (presented in Section \[secproofthm00\]) provides information both about $\langle a,\rho(\mu) \rangle$ as well as $\operatorname{Supp}(\mu)$. More precisely, if $\psi:[0,1]\times L\to (M,\omega)$ is a Lagrange isotopy with $\operatorname{Flux}_{L}(\psi)-c=\psi_1^*a$ and $\psi_1(L)\subset \Sigma_k$ then for a given $\epsilon>0$ one can find an ergodic $\mu \in \mathcal{M}(\phi_H)$ achieving $$\begin{aligned} \langle a,\rho(\mu) \rangle &\leq \max_{\psi_1(L)}(H)-k+\epsilon \\ H(\operatorname{Supp}(\mu))&\subset [\max_{\psi_1(L)}(H)+\tfrac{\epsilon}{2},k-\tfrac{\epsilon}{2}]. \end{aligned}$$ The proof of Theorem \[thm00\] relies heavily on beautiful ideas due to Buhovsky-Entov-Polterovich [@BuhovskyEntovPolterovich12], Entov-Polterovich [@EntovPolterovich17] and Polterovich [@Polterovich14]. Of course this result is only useful if we can find reasonable lower bounds for $\kappa_{H:L}$. The main source of such estimates come from rigidity results in symplectic topology, which is where $\mathcal{G}_L(c)$ from (\[eqrev2eq1\]) comes into play. This happens for the following reason: If $\psi:[0,1]\times L\to M$ is a Lagrange isotopy starting at $L$ with $\operatorname{Flux}_L(\psi)=c\in H^1(L;{\mathbb R})$, then the period group of $\psi_1(L)\subset (M,\omega)$ is exactly $\mathcal{G}_L(c)$: $$\label{eq33} \langle [\omega], \pi_2(M,\psi_1(L))\rangle =\mathcal{G}_L(c).$$ The following Proposition was the original idea employed in [@EntovPolterovich17]. For more quantitative versions of this idea, see below. Suppose $c\in H^1(L;{\mathbb R})$ satisfies $\partial c|_{\pi_2(M,L)}=[\omega]|_{\pi_2(M,L)}$ and suppose that $(M,\omega)$ does not admit any weakly exact Lagrangian submanifolds. Then $\kappa_{H,L}(c)=+\infty$ for all Hamiltonians $H$. There exists no Lagrange isotopy $\psi$ starting at $L$ with $\operatorname{Flux}_L(\psi)=c$. This follows from (\[eq33\]) and our assumption. This proposition can be refined in several ways. Suppose $(M,N, \omega)$ is a *subcritical polarized K[ä]{}hler manifolds* in the sense of [@BiranCieliebak01]. I.e. $(M,\omega)$ is a closed K[ä]{}hler manifold and $N\subset M$ is a subset such that $(M\backslash N,\omega)$ admits the structure of a subcritical Stein manifold. In this situation we have the following estimate of $\kappa_{H:L}$. Let $(M,N, \omega)$ be as above and suppose $H\in C^{\infty}(M)$. If $L\subset (M,\omega)$ is a closed Lagrangian and $c\in H^1(L;{\mathbb R})$ satisfies the condition that $\partial c|_{\pi_2(M,L)}=[\omega]|_{\pi_2(M,L)}$ then $$\kappa_{H:L}(c)\geq \min_{N}H.$$ Suppose $\psi:[0,1]\times L \to (M,\omega)$ is a smooth Lagrange isotopy starting at $L$ with $\operatorname{Flux}_L(\psi)=c\in H^1(L;{\mathbb R})$. It suffices to show that $\psi_1(L)\cap N\neq \emptyset$. Suppose for contradiction that $\psi_1(L)\subset (M \backslash N,\omega)$. Since $(M \backslash N,\omega)$ is subcritical Stein it follows from [@BiranCieliebak02 Lemma 3.2] that $\psi_1(L)$ can be displaced by a compactly supported Hamiltonian diffeomorphism. However, $\mathcal{G}_L(c)=\{0\}$, which means that $\psi_1(L)$ is weakly exact, so it cannot be displaced by a Hamiltonian diffeomorphism [@Polterovich93]. This finishes the proof. The following Proposition connects Theorem \[thm00\] to the main result in [@Polterovich14]. Suppose that $M$ is closed, that the restriction map $r_L:H^1(M;{\mathbb R})\to H^1(L;{\mathbb R})$ is surjective and that $L$ is non-displaceable. I.e. $\phi(L)\cap L\neq \emptyset$ for all $\phi \in \operatorname{Ham}(M,\omega)$. Then $$\label{eq34} \kappa_{H:L}(0)\geq \min_L H \quad \forall \ H\in C^{\infty}(M).$$ Let $\psi:L \times [0,1]\to M$ be a Lagrange isotopy starting at $L$ with $\operatorname{Flux}_L(\psi)=0\in H^1(L;{\mathbb R})$. By [@Solomon13 Lemma 6.6] there exists a symplectic isotopy $\theta:M\times [0,1]\to M$ with $\theta_0=\operatorname{id}$ such that $\theta_t(L)=\psi_t(L)$ for all $t\in [0,1]$ and such that $\operatorname{Flux}(\theta)=0\in H^1(M;{\mathbb R})$. In particular $\theta_1\in \operatorname{Ham}(M,\omega)$. Hence, $\psi_1(L)\cap L=\theta_1(L)\cap L\neq \emptyset$ which implies (\[eq34\]). The guiding philosophy above is that symplectic rigidity prevents a given Lagrangian from being (Lagrangian) isotoped with a given (Lagrange) flux into a fixed sublevel set of our Hamiltonian $H$. Perhaps the most fundamental situation where this idea can be used to estimate $\kappa_{H:L}$ from below is the following: \[prop01\] Let $H\in C^{\infty}(M;{\mathbb R})$ be bounded from below. Suppose $c\in H^1(L;{\mathbb R})$ and $k\in {\mathbb R}$ meet the condition $$\label{eq000} e_S(\Sigma_k)<\gamma_L(c).$$ Then $$\kappa_{H:L}(c)\geq k.$$ Consider a Lagrange isotopy $\psi:[0,1]\times L\to M$ starting at $L$ and satisfying $$\label{eq00} \operatorname{Flux}_L(\psi)=c.$$ We need to show that $\max_{\psi_1(L)}(H)\geq k$. Since $S^1\subset T^*S^1$ is exact, $\mathcal{G}_L(c)$ coincides with the period group of $\psi_1(L)\times S^1 \subset (M\times T^*S^1,\omega \oplus \omega_0)$. Hence, a fundamental result first due to Polterovich [@Polterovich93] and later Chekanov [@Chekanov98] says that $\gamma_L(c) \leq e(\psi_1(L)\times S^1)=e_S(\psi_1(L))$ (i.e. $\psi_1(L)$ is stably non-displaceable if $\gamma_L(c)=\infty$). It now follows from (\[eq000\]), that $e_S(\Sigma_k)<e_S(\psi_1(L))$ which in turn gives $$\psi_1(L)\times S^1 \nsubseteq \Sigma_k\times S^1,$$ or $\psi_1(L) \nsubseteq \Sigma_k=\{H<k\}$. Thus, $\max_{\psi_1(L)}(H)\geq k$ which finishes the proof. Together Theorem \[thm00\] and Proposition \[prop01\] establish a link between the fundamental notion of *displacement energy* in symplectic topology and the fundamental idea of looking for invariant measures with a prescribed rotation vector in Aubry-Mather theory. We hope to explore this idea further in future research. More precisely, we hope to find conditions where one can make sense of thinking of the measures in Theorem \[thm00\] as Mather measures and derive more precise properties of these measures. Other results linking the idea of Hofer geometry/displacement energy with Aubry-Mather theory known to the author are [@Siburg98] and [@SorrentinoViterbo10]. Recall that we consider an energy value $k\in {\mathbb R}$ and a class $c\in H^1(L;{\mathbb R})$ such that $e_S(\Sigma_k)<\gamma_L(c)$. By Theorem \[thm00\] it suffices to show that $\kappa_{H:L}(c)\geq k$. This is the content of Proposition \[prop01\]. Note that in combination with Remark \[ref2rem2\] on page this in fact also proves Theorem \[cor00\]. Examples {#secexamples} -------- \[ex02\] Consider ${\mathbb R}^{2n}$ equipped with the standard symplectic structure $\omega:=\sum_{l=1}^n dx_l\wedge dy_l$ and denote by $p:{\mathbb R}^{2n}\to {\mathbb R}^2$ the projection onto the first factor of ${\mathbb R}^{2n}={\mathbb R}^2\times \cdots \times {\mathbb R}^2$. Choose a Hamiltonian $H\in C^{\infty}({\mathbb R}^{2n})$ which is proper and bounded from below and fix an energy value $k\in {\mathbb R}$ such that $\Sigma_k \neq \emptyset$. Choose $r>0$ such that $p(\Sigma_k)\subset {\mathbb R}^2$ is contained in the open disc of radius $r$ centered at the origin. Denote by $L\subset ({\mathbb R}^{2n},\omega)$ the split Lagrangian $L=S^1(r)\times \cdots \times S^1(r)$ which is the product of circles in ${\mathbb R}^2$ of radius $r$, centered at the origin. Then $e_S(\Sigma_k)\leq e(\Sigma_k)<r^2\pi=\gamma(L)$, so Theorem \[thm01\] (in combination with Remark \[ref2rem1\]) gives the following statement: For every $c\in r^2\pi \cdot H^1(L;{\mathbb Z})\subset H^1(L;{\mathbb R})$ and every $a\in \operatorname{Sh}(\Sigma_k;L,c)$ there exists $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ such that $$\langle a,\rho(\mu) \rangle <0.$$ Suppose in the above example that $n=2$ and that $\Sigma_k$ contains a Lagrangian $2$-torus $L'\subset \Sigma_k$ such that the restriction map $H^1(\Sigma_k;{\mathbb R})\to H^1(L';{\mathbb R})$ is surjective. Then it follows from [@RizellGoodmanIvrii16 Theorem A] that there exists a Lagrange isotopy $\psi:[0,1]\times L\to M$ such that $\psi_1(L)=L'$. In particular $\operatorname{Sh}(\Sigma_k;L,c)\neq \emptyset$ for all $c\in H^1(L;{\mathbb R})$. Of course in this case, one could also apply Theorem \[cor00\] instead of Theorem \[thm01\]. Apart from ${\mathbb R}^{2n}$, the main example we have in mind is that of Hamiltonian dynamics on a twisted cotangent bundle. Consider the two-torus $\mathbb{T}^2={\mathbb R}^2 / {\mathbb Z}^2$ and denote by $\Omega$ a closed $2$-form on $\mathbb{T}^2$ such that $\int_{\mathbb{T}^2} \Omega=1$. The symplectic structure on the *twisted cotangent bundle* $(T^*\mathbb{T}^2,\omega)$ is defined by $$\omega:=d\lambda_{\mathbb{T}^2} +\pi^*\Omega,$$ where $\pi:T^*\mathbb{T}^2\to \mathbb{T}^2$ denotes the footpoint map and $\lambda_{\mathbb{T}^2}$ denotes the Liouville $1$-form. Note that $(T^*\mathbb{T}^2,\omega)$ is weakly exact. Let $H\in C^{\infty}(T^*\mathbb{T}^2)$ be a Hamiltonian which is proper and bounded from below. By [@GinzburgKerman99] every sublevel set $\Sigma_k=\{H<k\}$ is displaceable. We want to apply this fact to study the dynamics of $\phi_H|_{\Sigma_k}$. Suppose $L\subset \Sigma_k \subset (T^*\mathbb{T}^2,\omega)$ is a Lagrangian 2-torus.[^13] Since $\pi_1(L)$ is Abelian, Lemma \[lem00\] on page guarantees the existence of a $\tilde{c}\in H^1(L;{\mathbb R})$ such that $\partial \tilde{c}|_{\pi_2(M,L)}\equiv [\omega]|_{\pi_2(M,L)}$. Applying Theorem \[cor00\] we then conclude that for any $a\in H^1(\Sigma_k;{\mathbb R})$ satisfying $$a|_L\in (e_S(\Sigma_k),\infty)\cdot H^1(L;{\mathbb Z})-\tilde{c}$$ and any $\epsilon>0$ there exists a $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ such that $$\langle a,\rho(\mu) \rangle \leq \max_L H-k+\epsilon.$$ \[ex03\] Consider the 2-sphere $S^2$ equipped with a closed $2$-form $\Omega$ such that $\int_{S^2}\Omega=1$. Denote by $\pi:T^*S^2 \to S^2$ the footpoint map. We want to apply our results above to study Hamiltonian systems on the twisted cotangent bundle $(T^*S^2,\omega)$, where $$\omega:=d\lambda_{S^2} + \pi^*\Omega.$$ Clearly $(T^*S^2,\omega)$ is *not* weakly exact. There exists a rational Lagrangian 2-torus $L\subset (T^*S^2,\omega)$ with $\gamma_L(0)=1$. Assuming this fact for now, let $H\in C^{\infty}(T^*S^2)$ denote an autonomous Hamiltonian which is proper and bounded from below and let $k\in {\mathbb R}$ denote an energy value such that $e_S(\Sigma_k)<1$. Then by Theorem \[thm01\] (and the following remark), for every $c\in H^1(L;{\mathbb Z})$ and every $a\in \operatorname{Sh}(\Sigma_k;L,c)$ there exists a $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ such that $$\langle a,\rho(\mu) \rangle <0.$$ For concreteness, consider $H(q,p)=\frac{|p|^2}{2}+U(q)$ for some Riemannian metric on $T^*S^2\to S^2$ and $U\in C^{\infty}(S^2)$. By [@Schlenk06 Proposition 1.4] there exists a $\delta>0$ such that $e_S(\{(q,p)\ |\ |p|\leq \delta\})<1$. In particular we conclude that, for all sufficiently small $k>\min(H)=\min(U)$, we have $e_S(\Sigma_k)<1$. Of course, if $U$ is constant then $\Sigma_k$ is simply connected for all $k\in {\mathbb R}$, but this is not the case if $U$ is not constant. In order to see the existence of $L$, let’s be explicit and view $$S^2=\{(x,y,z)\in {\mathbb R}^3 \ |\ x^2+y^2+z^2=1\}.$$ Since $\Omega|_{S^2 \cap \{z>-1\}}$ is exact we can denote by $\xi$ a $1$-form on $S^2 \cap \{z>-1\}$ such that $\Omega=d\xi$ and choose a smooth function $\rho:S^2\to [0,1]$ such that $$\rho(x,y,z)= \begin{cases} 1, & \text{if}\ z\geq -1/2 \\ 0,& \text{if}\ z\leq -2/3. \end{cases}$$ By Moser’s argument the symplectic manifold $(T^*S^2,\widetilde{\omega})$, where $$\widetilde{\omega}=\omega - d\pi^*(\rho \xi),$$ is symplectomorphic to $(T^*S^2,\omega)$. Denote by $f:B^2(2)\to S^2$ an embedding which identifies the closed $2$-disc $B^2(2):=\{(x_1,x_2)\in {\mathbb R}^2 \ |\ x_1^2+x_2^2 \leq 4\}$ with the upper hemisphere $S^2 \cap \{z \geq 0\}$. Since $\widetilde{\omega}|_{S^2 \cap \{z \geq 0\}}=d\lambda_{S^2}|_{S^2 \cap \{z \geq 0\}}$ $f$ induces a symplectic embedding $$F=(df^*)^{-1}:(T^*B^2(2),\lambda_{B^2(2)}) \to (T^*S^2,\widetilde{\omega})$$ Denote by $T\subset B^4(2)\subset B^2(2)\times {\mathbb R}^2=T^2B^2(2)$ a rational Lagrangian $2$-torus with $\gamma(T)=1$. We claim that $F(T)\subset (T^*S^2,\widetilde{\omega})$ too is a rational Larangian $2$-torus with $\gamma(F(T))=1$. In order to see this, let $u:(B^2(1),\partial B^2(1))\to (T^*S^2,F(T))$ denote a smooth topological disc on $F(T)$. It suffices to check that $\int_u \widetilde{\omega} \in {\mathbb Z}$. Since we clearly have $\int_u d\lambda_{S^2}\in {\mathbb Z}$ it in fact suffices to check that $\int_{v} (\Omega-d(\rho \xi)) \in {\mathbb Z}$, where $v=\pi \circ u$. To see this we note that, since $\pi(F(T))\subset S^2 \cap \{z\geq 0 \}$ and $S^2 \cap \{z\geq 0 \}$ is contractible, we can extend $v$ to a map $\widetilde{v}:B^2(2) \to S^2$ such that $\widetilde{v}(\partial B^2(2))=\{(0,0,1)\}$. Since the $2$-form $(\Omega-d(\rho \xi))$ vanishes on $S^2 \cap \{z\geq 0\}$ we have $\int_{\widetilde{v}}(\Omega-d(\rho \xi))=\int_{v}(\Omega-d(\rho \xi))$ and since $\widetilde{v}(\partial B^2(2))=\{(0,0,1)\}$, $\widetilde{v}$ induces a map $\widehat{v}:S^2 \to S^2$. Now $\int_{\widetilde{v}}(\Omega-d(\rho \xi))$ is simply the degree of $\widehat{v}$. In particular it is an integer. This proves the claim that $F(T)\subset (T^*S^2,\widetilde{\omega})$ is a rational Lagrangian torus with $\gamma(F(T))=1$. Since $(T^*S^2,\widetilde{\omega})\cong (T^*S^2,\omega)$ we conclude that there exists a rational Lagrangian torus $L\subset (T^*S^2,\omega)$ with $\gamma(L)=1$. Of course the above examples can also be carried out for twisted cotangent bundles of closed manifolds $N\neq {\mathbb T}^2,S^2$. The main point is that if $[\omega]\neq 0$ then $N\subset (T^*N,\omega)$ has stable displacement energy 0 by [@Schlenk06 Proposition 1.4], so if $H\in C^{\infty}(T^*N)$ is *mechanical*[^14] then $\Sigma_k$ has small displacement energy for all small enough $k>\min(H)$. However, the choice of $S^2$ in the above example is not coincidental: One could ask if it is possible to study Hamiltonian systems on the twisted $T^*S^2$ from the previous example using Lagrangian Floer homology. However, this approach seems unlikely to succeed. Indeed, as pointed out to me by MathOverflow-user Nikolaki [@Nikolaki], if one manages to find a Lagrangian $L\subset (T^*S^2,\omega)$ for which there exists a well-defined Floer homology $HF(L)$, then probably $HF(L)=0$. This follows from the fact that (if well-defined) $HF(L)$ should be a module over the symplectic homology $SH(T^*S^2)$ of $(T^*S^2,\omega)$ and a result due to Benedetti and Ritter [@BenedettiRitter18 Theorem 1.1] says that $SH(T^*S^2)=0$ (see also [@Ritter14] as well as Benedetti’s thesis [@Benedetti15] for applications of symplectic homology to Hamiltonian systems on twisted cotangent bundles). Preliminaries {#seqprem} ============= Here we discuss some of the background material used frequently in the above sections. Recall that a Lagrangian submanifold $L\subset (M,\omega)$ is said to be *monotone* if there exists a constant $\tau_L\geq 0$ such that $$\omega|_{\pi_2(M,L)}=\tau_L\cdot \mu|_{\pi_2(M,L)},$$ where $\omega|_{\pi_2(M,L)}$ denotes integration of $\omega$ and $\mu|_{\pi_2(M,L)}$ denotes the Maslov index. If in addition the minimal Maslov number of $L$ is $\geq 2$, then the quantum homology $QH_*(L;{\mathbb Z}_2)$ and Floer homology $HF(L;{\mathbb Z}_2)$ of $L$ are well-defined [@BiranCornea07], [@Zapolsky15]. Our main reference for $QH_*(L;{\mathbb Z}_2)$ and $HF_*(L;{\mathbb Z}_2)$ is Zapolsky’s excellent [@Zapolsky15]. In order to set the notation we use here we will discuss a few details about $HF_*(L;{\mathbb Z}_2)$ and the construction of spectral invariants. Recall that, given a Hamiltonian $H\in \mathcal{H}$ such that $\phi_H^1(L)\pitchfork L$ and a generic path $\{J_t\}_{t\in [0,1]}$ of $\omega$-compatible almost complex structures, $HF_*(L{:}H,J)$ is the homology of a Morse-type chain complex $(CF_*(L{:}H,J),d)$ generated by critical points of the action functional $$\mathcal{A}_{H:L}(\widetilde{\gamma}=[\gamma,\widehat{\gamma}])=\int_0^1 H_t(\gamma(t))\ dt-\int \widehat{\gamma}^*\omega.$$ Here the input $\widetilde{\gamma}$ consists of a smooth path $\gamma{:}([0,1],\{0,1\})\to (M,L)$ as well as an equivalence class of *cappings* $\widehat{\gamma}$ of $\gamma$ in the sense of [@Zapolsky15]. The equivalence relation identifies cappings which have the same symplectic area. As a result, generators of $CF_*(L{:}H,J)$ have an associated action and we denote by $(CF^a_*(L{:}H,J),d)$ $(a\in {\mathbb R})$ the subcomplex generated by elements whose action is $<a$. This is indeed a subcomplex as $d$ decreases action, and its homology is denoted by $HF^a_*(L{:}H,J)$. The inclusion $i_a:CF^a_*(L{:}H,J)\to CF_*(L{:}H,J)$ induces maps $i_*^a:HF_*^a(L{:}H,J)\to HF_*(L{:}H,J)$. By standard arguments $HF_*(L{:}H,J)$ is independent of the data $(H,J)$ and identifying all such groups via canonical continuation isomorphisms we obtain a ring $HF(L;{\mathbb Z}_2)$, which is isomorphic to $QH(L;{\mathbb Z}_2)$ via a $\operatorname{PSS}$-isomorphism $$\operatorname{PSS}:QH_*(L;{\mathbb Z}_2)\stackrel{\cong}{\to}HF_*(L;{\mathbb Z}_2).$$ Under the assumption $QH_*(L;{\mathbb Z}_2)\neq 0$, the Leclercq-Zapolsky spectral invariant $l_L:\widetilde{\operatorname{Ham}}(M,\omega)\to {\mathbb R}$ which we consider is defined by $$l_L(\tilde{\phi}_H):=\inf\{a\in {\mathbb R}\ |\ \operatorname{PSS}([L])\in \operatorname{Image}(i^a_*)\subset HF_*(L;{\mathbb Z}_2) \},$$ where $i_*^a:HF_*^a(L{:}H,J)\to HF_*(L{:}H,J)\cong HF_*(L;{\mathbb Z}_2)$ and $[L]\in QH(L;{\mathbb Z}_2)$ denotes the “fundamental class”, i.e. the unity for the ring-structure on $QH_*(L;{\mathbb Z}_2)$. For further details on $l_L$ and the properties satisfied by $l_L$ we refer to [@LeclercqZapolsky15] and [@ZapolskyMonznerVichery12]. An important property for us is that $l_L$ satisfies the *triangle inequality*: $$l_L(\tilde{\phi}_H\tilde{\phi}_K)\leq l_L(\tilde{\phi}_H)+l_L(\tilde{\phi}_K) \quad \forall \ H,K\in \mathcal{H}.$$ As a consequence of this, the limit (\[eq2\]) exists and $\sigma_{H:L}$ is well-defined. The following theorem is due to [@ZapolskyMonznerVichery12] in a slightly different setting. In fact, [@ZapolskyMonznerVichery12] presents many more properties of $\sigma_{H:L}$ in the setting of a cotangent bundle. Here we only list the properties which will be useful to us. \[prop1\] Given any $H\in \mathcal{H}$, the function $\sigma_{H:L}:H^1(M;{\mathbb R})\to {\mathbb R}$ from Definition \[def2\] is well-defined and it satisfies the following properties a) $\sigma_{H:L}$ is locally Lipschitz with respect to any norm on $H^1(M;{\mathbb R})$. In particular $\sigma_{H:L}$ is differentiable almost everywhere. b) Suppose $\psi=\{\psi_t \}_{t\in [0,1]}$ is a symplectic isotopy with $\psi_0=\operatorname{id}$. Then $$\min_{x\in \psi_1(L)}H(x)\leq \sigma_{H:L}(c)\leq \max_{x\in \psi_1(L)}H(x),$$ where $c:=\operatorname{Flux}(\psi)\in H^1(M;{\mathbb R})$. c) If $\psi=\{\psi_t\}_{t\in [0,1]} \subset \operatorname{Symp}(M,\omega )$ is a path of symplectomorphisms with $\psi_0=\operatorname{id}$ and $c':=\operatorname{Flux}(\psi)\in H^1(M;{\mathbb R})$ then $$\sigma_{H:\psi_1(L)}(c)=\sigma_{H:L}(c+c') \quad \forall \ c\in H^1(M;{\mathbb R}).$$ d) For every $\phi \in \operatorname{Ham}(M,\omega)$ we have $\sigma_{H:L}=\sigma_{H:\phi(L)}$. In [@Bisgaard16] we obtained an alternative version of the last property: If $L'\subset (M,\omega)$ is another monotone Lagrangian submanifold which is monotone Lagrangian cobordant to $L$ then $\sigma_{H:L}=\sigma_{H:L'}$. We now make a few remarks concerning the definition of $\sigma_{H:L}$ from Section \[noncompact\]. For this purpose, let $(M,\omega=d\lambda)$ be a Liouville manifold with ideal contact boundary $(\Sigma,\lambda_0)$ (see page ). We first prove Proposition \[Mprop1\]. Fix a closed 1-form $\eta_0$ on $M$ satisfying $[\eta_0]=c\in H^1_{dR}(M;{\mathbb R})$. Define $\eta_1$ on $(0,\infty)\times \Sigma$ as the pull-back of $\eta_0|_{\{1\}\times \Sigma}$ via the projection $(0,\infty)\times \Sigma \to \Sigma$. By homotopy invariance $[\eta_0|_{(0,\infty)\times \Sigma}]=[\eta_1]$ in $H^1_{dR}((0,\infty)\times \Sigma;{\mathbb R})$, so there exists $f\in C^{\infty}((0,\infty)\times \Sigma)$ such that $\eta_1=\eta_0|_{(0,\infty)\times \Sigma}+df$. Choose a function $\chi \in C^{\infty}((0,\infty))$ such that $\chi(s)=0$ for all $s\in (0,\tfrac{1}{4})$ and $\chi(s)=1$ for all $s\in (\tfrac{3}{4},\infty)$. Then $$\eta:=\eta_0+d(\chi f)$$ defines a closed 1-form on all of $M$ in the class $c$. Since $\eta=\eta_1$ on $(\tfrac{3}{4},\infty)\times \Sigma$ we have the wanted property: $$\label{Meq2} \eta|_{\{s\}\times \Sigma}=\eta|_{\{1\}\times \Sigma} \quad \forall \ s\in [1,\infty).$$ An easy computation using that $\omega=ds\wedge \lambda_0+sd\lambda_0$ on $(0,\infty)\times \Sigma$ shows that, if $\eta$ satisfies (\[Meq2\]), then the vectorfield $X\in \mathfrak{X}(M)$ characterized by $\eta =i_X\omega$ satisfies $ds(X)|_{(s,x)}=\eta_{(s,x)}(R(x))=\eta_{(1,x)}(R(x))$ for all $(s,x)\in [1,\infty)\times \Sigma$ (here $R$ denotes the Reeb vector field on $(\Sigma,\lambda_0)$). Since $\Sigma$ is compact, this implies that $X$ is complete. Suppose now that $\eta_1$ and $\eta_2$ are two closed 1-forms in the class $c$ satisfying (\[Meq2\]) for (potentially) different ideal contact boundaries $(\Sigma_1,\lambda_0^1)$ and $(\Sigma_2,\lambda_0^2)$ of $(M,d\lambda)$. Denote by $(\psi^k_t)_{t\in {\mathbb R}}$ the flows generated by the vectorfields $X_k\in \mathfrak{X}(M)$ characterized by $i_{X_k}\omega=\eta_k$, $k=1,2$. The symplectic isotopy $t\mapsto \psi_{-t}^1\psi^2_t$ is generated by the time-dependent vectorfield $(\psi^1_{-t})_*X_2-X_1$. Since $$[i_{(\psi^1_{-t})_*X_2-X_1}\omega]=[i_{(\psi^1_{-t})_*X_2}\omega]-[\eta_1]=[(\psi_t^1)^*\eta_2]-[\eta_1]=c-c=0$$ it follows that $\psi_{-t}^1\psi^2_t=\phi_t$ for a (possibly non-compactly supported) Hamiltonian isotopy $t\mapsto \phi_t$. Since $\psi_1^2=\psi^1_1\phi_1$ it follows from arguments completely analogous to those in [@ZapolskyMonznerVichery12] that for $H\in \mathcal{H}$ we have $$\label{Meq3} \lim_{{\mathbb N}\ni k\to \infty}\frac{l_L((\psi^1_1)^{-1}\widetilde{\phi}_H^k\psi^1_1)}{k}=\lim_{{\mathbb N}\ni k\to \infty}\frac{l_L((\psi^2_1)^{-1}\widetilde{\phi}_H^k\psi^2_1)}{k}.$$ The crux of this argument (like in [@ZapolskyMonznerVichery12]) lies in the fact that, since $\mathcal{O}_+(L)$ is compact, all Hamiltonians involved in the above computation can be cut-off outside a compact set without changing the spectral invariants. Hence, ultimately one only has to work with compactly supported Hamiltonians (for which (\[Meq3\]) is standard). This concludes the reasoning that $\sigma_{H:L}(c)$ is well-defined in the non-compact setting. Moreover, it is Lipschitz by exactly the same arguments as in [@ZapolskyMonznerVichery12], so it has a non-empty set of Clarke subdifferentials at every point $c\in H^1(M;{\mathbb R})$. Subdifferentials and rotation vectors {#secmeas} ------------------------------------- In this section we recall a few basic facts about the Clarke subdifferential $\partial f$ of a function $f$ as well as about the approximate subdifferential $\partial_A f$ which is needed in the proofs below. For further references on the Clarke subdifferential we refer to [@Clarke83] and for further references on the approximate subdifferential we refer to [@Ioffe84]. Denote by $V$ a finite dimensional vector space over ${\mathbb R}$ and by $f:V\to {\mathbb R}$ a function which is locally Lipschitz (with respect to any norm on $V$). The generalized directional derivative $f^{\circ}(x,v)$ of $f$ at $x\in V$ in direction $v\in V$ is by definition the number $$f^{\circ}(x,v):=\limsup_{\substack{y\to x \\ \tau \downarrow 0}}\frac{f(y+\tau v)-f(y)}{\tau}.$$ $f^{\circ}(x,v)$ is a real number because $f$ is locally Lipschitz. The *Clarke subdifferential* (or *generalized derivative*) of $f$ at $x\in V$ is by definition the non-empty subset $$\partial f(x)=\{v^*\in V^*\ |\ \langle v^*,v\rangle \leq f^{\circ }(x,v) \ \forall \ v\in V\}$$ of $V^*$. Note that since $f$ is Locally Lipschitz it is differentiable almost everywhere.[^15] A key property of $\partial f$ is [@Clarke83 Theorem 2.5.1], according to which, for any Lebesgue 0-set $S\subset V$ containing the non-differentiability points of $f$ we have $$\label{eq99} \partial f(x)=\operatorname{conv}( \{\lim_{\varsigma \to \infty}df(x_{\varsigma})\ |\ (x_{\varsigma})_{\varsigma \in {\mathbb N}}\subset V\backslash S \ \text{s.t.} \ x_{\varsigma}\to x \} ),$$ where $\operatorname{conv}(X)\subset V$ denotes the convex hull of the subset $X\subset V$. I.e. $\partial f(x)$ is the convex hull of the set of all limit points of sequences of differentials $df(x_{\varsigma})$ of $f$ with $(x_{\varsigma})_{\varsigma \in {\mathbb N}}\subset V\backslash S$ a sequence such that $x_{\varsigma}\to x$. Another important property which will be used below is Lebourg’s mean value theorem for $\partial f$ [@Clarke83 Theorem 2.3.7]: For every pair $x,y\in V$ there exists a $t\in (0,1)$ such that for some $$v^*\in \partial f(ty+(1-t)x)$$ we have $$\langle v^*,x-y \rangle =f(x)-f(y).$$ We will now discuss some of the basic properties of the approximate subdifferential $\partial_Af$ of $f$ due to Ioffe [@Ioffe84]. We point out that the $\partial_Af$ makes sense also if $f$ is not locally Lipschitz, but here we restrict ourselves to considering the case when $f$ is locally Lipschitz. For $x,v\in V$, the *lower Dini directional derivative* of $f$ at $x$ in direction $v$ is defined by $$d^-f(x;v):=\liminf_{\tau \downarrow 0}\frac{f(x+\tau v)-f(x)}{\tau }\in {\mathbb R}$$ and the *Dini subdifferential* $\partial^-f(x)\subset V^*$ of $f$ at $x$ is the subset $$\label{eq1} \partial ^-f(x):=\{ v^*\in V^*\ |\ \langle v^*,v\rangle \leq d^-f(x;v) \ \forall \ v\in V \}$$ of $V^*$. Finally the approximate subdifferential of $f$ at $x$ is defined as the subset[^16] $$\label{eq222} \partial_Af(x):=\bigcap_{\delta>0}\bigcup_{\ ||z-x||<\delta}\partial^-f(z),$$ of $V^*$. A key property of $\partial_Af$ (when $f$ is locally Lipschitz) is [@Ioffe84 Theorem 2], according to which the Clarke subdifferential and the approximate subdifferential are related by $$\label{eq111} \partial f(x)=\overline{\operatorname{conv}(\partial_Af(x))} \quad \forall \ x\in V.$$ Our main interest in $\partial_A$ is that it behaves well under uniform approximation: By a result due to Jourani [@Jourani99 Theorem 3.2], if $(f_{\varsigma})_{\varsigma \in {\mathbb N}}$ is a sequence of locally Lipschitz functions $V\to {\mathbb R}$ such that $$f_{\varsigma} \stackrel{\varsigma \to \infty}{\longrightarrow}f$$ uniformly, then $$\label{eq333} \partial_Af(x)\subset \limsup_{\substack{ \varsigma \to \infty \\ y\to x}}\partial_Af_{\varsigma}(y) \quad \forall \ x\in V,$$ where $$\limsup_{\substack{\varsigma \to \infty\\ y\to x}}\partial_Af_{\varsigma}(y):=\bigcap_{\varsigma =1}^{\infty}\bigcap_{\delta>0} \overline{\bigcup_{\substack{k\geq \varsigma \\ 0<|x-y|<\delta}}\partial_Af_k(y)}.$$ We point out that this result is a generalization of Ioffe’s result [@Ioffe84 Theorem 3]. We are interested in the subdifferential of $\sigma_{H:L}:H^1(M;{\mathbb R})\to {\mathbb R}$ because its elements are rotation vectors of $\mathcal{M}(\phi_H)$-measures (Theorem \[thm1\]). We recall that, under the canonical identification $H_1(M;{\mathbb R})=H^1_{dR}(M;{\mathbb R})^*$, the rotation vector $\rho(\mu)\in H_1(M;{\mathbb R})$ of a $\phi_H$-invariant measure $\mu \in \mathcal{M}(\phi_H)$ is given by $$\label{eq10} \langle [\eta],\rho(\mu)\rangle =\int \langle \eta,X_H \rangle \ d\mu, \quad [\eta]\in H^1_{dR}(M;{\mathbb R}).$$ For further details and in-depth explanations of rotation vectors we refer to [@Schwartzman57] or [@Sorrentino15]. Proofs of results {#secproof} ================= Before proving the results from Section \[MatherFloer\] we present the remaining proofs of results from Section \[secC0\]. This means proving Theorem \[thm01\]. Before doing that we prove a lemma which was implicitely used in the statement of Theorem \[cor00\]. \[lem00\] Let $(M,\omega)$ be weakly exact (i.e. $\omega|_{\pi_2(M)}\equiv 0$) and denote by $L\subset (M,\omega)$ a Lagrangian satisfying either the condition that $\pi_1(L)$ is Abelian or that the map $\pi_1(L)\to \pi_1(M)$ is trivial. Then there exists $\tilde{c}\in H^1(L;{\mathbb R})$ such that $$\partial \tilde{c}|_{\pi_2(M,L)}\equiv [\omega]|_{\pi_2(M,L)}.$$ Since $(M,\omega)$ is weakly exact we may view $[\omega]$ as a homomorphism $$\pi_2(M,L)/\pi_2(M)\to {\mathbb R}.$$ By the long exact sequence for homotopy groups we may view $\pi_2(M,L)/\pi_2(M)$ as a subgroup $\pi_2(M,L)/\pi_2(M) \leq \pi_1(L)$. If $\pi_1(L)\to \pi_1(M)$ is trivial then $$\pi_2(M,L)/\pi_2(M) \cong \pi_1(L),$$ so $[\omega]$ gives rise to a homomorphism $\pi_1(L)\to {\mathbb R}$ which is given by an element $\tilde{c}\in H^1(L;{\mathbb R})$. In case $\pi_1(L)$ is Abelian (so $\pi_1(L)=H_1(L;{\mathbb Z})$) $[\omega]$ descends to a homomorphism $G\to {\mathbb R}$, where $G$ denotes the image of $\pi_2(M,L)/\pi_2(M)$ under the quotient map $$\pi_1(L)\to \pi_1(L)/\text{Torsion}.$$ Any homomorphism $G\to {\mathbb R}$ extends to a homomorphism $\pi_1(L)/\text{Torsion}\to {\mathbb R}$ by elementary algebra (use e.g. [@Hungerford80 Chapter II, Theorem 1.6]). The composition $$\pi_1(L)\to \pi_1(L)/\text{Torsion}\to {\mathbb R}$$ extends $[\omega]$ and corresponds to an element $\tilde{c}\in H^1(L;{\mathbb R})$. Proof of Theorem \[thm00\] {#secproofthm00} -------------------------- The proof relies heavily on the work of Entov-Polterovich [@EntovPolterovich17] as well as the work of Polterovich [@Polterovich14]. Given a 1-form $\eta$ on $M$ we denote by $X_{\eta}$ the vector field defined by requiring $$\iota_{X_{\eta}}\omega =\eta.$$ Given an autonomous Hamiltonian $H\in C^{\infty}(M)$, the Poisson bracket $\{H,\eta\}$ is the function given by $$\{H,\eta\}:=\omega(X_{\eta},X_H)=\eta(X_H)=dH(X_{\eta}).$$ Given an open subset $U\subset M$ we consider the following quantity which is a slight modification of a quantity appearing in [@Polterovich14]. $$pb^+_{H,U}(c):=\inf_{\eta \in c}\max_{x\in U}\{H,\eta\}(x), \quad c\in H^1(U;{\mathbb R}).$$ If $X\subset U$ is a compact subset we also define $$pb_{H,U}^+(c;X):=\inf_{\eta \in c}\max_{x\in X}\{H,\eta\}(x), \quad c\in H^1(U;{\mathbb R}).$$ The following theorem which is due to Polterovich is the main tool which allows us to construct invariant measures with given rotation vectors using $pb^+_{H,U}$. In [@Polterovich14] the statement appears in a slightly different form, but the version presented here follows directly from the theory developed in [@Polterovich14]. \[thm02\] Let $H\in C^{\infty}(M)$ be an autonomous Hamiltonian which is proper and bounded from below. Given $k\in {\mathbb R}$ and $\delta>0$ we set $X:=\{H\leq k-\delta\}$. Suppose for some $c\in H^1(U;{\mathbb R})$ there exists $\epsilon >0$ such that $$pb^+_{H,\Sigma_k}(c;X)\geq \epsilon .$$ Then there exists a $\mu \in \mathcal{M}(\Sigma_k;\phi_H)$ with $\operatorname{Supp}(\mu) \subset X$, which satisfies $$\langle c,\rho(\mu) \rangle \geq \epsilon.$$ The following proof is a small variation of the proof of the main technical result (Proposition 5.1) in Entov-Polterovich’s [@EntovPolterovich17]. Recall that we consider $c\in H^1(L;{\mathbb R})$ and $k\in {\mathbb R}$ such that $$k\leq \kappa_{H:L}(c).$$ Fix now $$a\in \operatorname{Sh}(\Sigma_k;L,c)\subset H^1(\Sigma_k;{\mathbb R}).$$ This means (by definition) that there exists a Lagrange isotopy $\psi:[0,1]\times L \to M$, starting at $L$, which satisfies $$\label{eq02} L_1:=\psi_1(L)\subset \Sigma_k \quad \text{and} \quad \operatorname{Flux}_L(\psi)-c=\psi_1^*a \in H^1(L;{\mathbb R}).$$ Choose $\delta>0$ so small that $\max_{L_1}(H)+\delta<k-\delta$. For every $\epsilon >0$ we can find a smooth function $u:{\mathbb R}\to {\mathbb R}$ such that $$u(t)=\left\{ \begin{array}{ll} \max_{L_1}(H)+\delta,& \text{if} \ t\leq \max_{L_1}(H)+\delta \\ k-\delta,& \text{if} \ t\geq k-\delta, \end{array} \right.$$ and $0\leq u'(t)\leq 1+\epsilon$. Consider now the Hamiltonian $$F:=u\circ H -(k-\delta).$$ Setting $X:=\{H\leq k-\delta\}$ it suffices (by Theorem \[thm02\]) to show that $$pb^+_{H,\Sigma_k}(-a,X)\geq \Delta:=k-\max_{L_1}(H)-2\delta>0.$$ Since $\max_X\{F,\eta\}\leq (1+\epsilon)\max_X\{H,\eta\}$ for every 1-form $\eta$ it suffices (by letting $\epsilon \downarrow 0$) to show that $pb^+_{F,\Sigma_k}(-a,X)\geq \Delta$. To see that this inequality holds we choose a closed $1$-form $\eta$ on $\Sigma_k$ with $[\eta]=-a$. Our task is to show that $\max_{X}\{F,\eta\} \geq \Delta$. Suppose for contradiction that this were not the case, i.e. that $$\max_{X}\{F,\eta\} < \Delta.$$ Exactly as in [@EntovPolterovich17] we then have that the closed $2$-form $$\omega_s:=\omega-sdF\wedge \eta$$ is symplectic for all $s\in [0,T]$, where $T:=\frac{1}{\Delta}$. Note that $\omega_s$ is a well-defined 2-form on all of $M$ because $dF=0$ on $M\backslash X$. Hence, we can define a family of vectorfields $(Y_s)_{s\in [0,T]}$ by requiring $$\label{eq01} \iota_{Y_s}\omega_s =F\eta.$$ Since $F$ is compactly supported we see that $Y_s$ is compactly supported, so it integrates to an isotopy $(\rho_s)_{s\in [0,T]}$ which by Moser’s argument [@Moser65] satisfies $\rho_s^*\omega_s=\omega$ for all $s\in [0,T]$. Since $L_1\subset (M,\omega)$ is Lagrangian and $F$ is constant on $L_1$ we have $\omega_s|_{L_1}\equiv 0$, so $\rho_s^{-1}(L_1)\subset (M,\omega)$ is also Lagrangian for all $s\in [0,T]$. Hence, we obtain a Lagrange isotopy $\tilde{\psi}:[0,1]\times L \to M$ by defining $$\tilde{\psi}_t:= \left\{ \begin{array}{ll} \psi_{2t}, & \text{if} \ t\in [0,\tfrac{1}{2}] \\ \rho_{(2t-1)T}^{-1}\circ \psi_1, & \text{if} \ t\in [\tfrac{1}{2},1], \end{array} \right.$$ which starts at $L$ and is smooth after a reparametrisation. Now note that because $\operatorname{Supp}(F)\subset X$ it follows from (\[eq01\]) that $\operatorname{Supp}(Y_s)\subset X$ for all $s\in [0,T]$. In particular $\rho_s|_{M\backslash X}=\operatorname{id}_{M\backslash X}$ for all $s\in [0,T]$. Since $L_1\subset X$ it follows that $\tilde{\psi}_1(L)=\rho_{T}^{-1}(L_1)\subset X$, meaning $$\label{eq04} \max_{\tilde{\psi}_1(L)}H\leq k-\delta.$$ Since the family of vectorfields $(Z_s)_{s}$ generating $(\rho_s^{-1})_s$ is given by $$Z_s(x)=-d\rho_s^{-1}(\rho_s(x))\cdot Y_s(\rho_s(x))$$ it is easy to check that $$\omega(Z_s,v)=-\rho_s^*(F\eta)(v) \quad \forall \ v\in T_xM.$$ From $F|_{L_1}\equiv -\Delta$ it now follows that $$\operatorname{Flux}_{L_1}((\rho^{-1}_{s})_{s\in [0,T]})=\int_0^T\Delta [\eta|_{L_1}]ds=-a|_{L_1}\in H^1(L_1;{\mathbb R}),$$ and thus (using (\[eq02\])) $$\operatorname{Flux}_L(\tilde{\psi})=\operatorname{Flux}_L(\psi)-\psi_1^*a=c.$$ In particular we can estimate $$\max_{\tilde{\psi}_1(L)}H\geq \kappa_{H:L}(c)\geq k.$$ But this contradicts (\[eq04\]). This contradiction shows that $\max_{M}\{F,\eta\} \geq \Delta$ which by Theorem \[thm02\] finishes the proof of the existence of $\mu$. In order to see that $\mu$ can be chosen to be ergodic we consider the function $$\begin{aligned} R:\mathcal{M}(X;\phi_H)&\to {\mathbb R},\quad \nu \mapsto R(\nu):=\langle a,\rho(\nu)\rangle. \end{aligned}$$ $\operatorname{Image}(R)$ is a closed interval: $\operatorname{Image}(R)=[A,B]$. The above proof shows that $A<0$. By [@BiranPolterovichSalamon03 Lemma 2.2.1] there exists an ergodic $\nu \in \mathcal{M}(X;\phi_H)$ such that $R(\nu)=A$. Since $\mathcal{M}(X;\phi_H)\subset \mathcal{M}(\Sigma_k;\phi_H)$ this finishes the proof of Theorem \[thm00\]. Combining the proof of Theorem \[thm02\] and the above proof, it is easy to deduce that the constructed measure $\mu$ satisfies $H(\operatorname{Supp}(\mu))\subset [\max_{L_1}H+\delta,k-\delta]$. Proofs of results from Section \[MatherFloer\] {#secproof2} ---------------------------------------------- We first prove that $\sigma_{H:L}$ descends to a Lipschitz function $$\alpha_{H:L}:H^1(L;{\mathbb R}) \to {\mathbb R}$$ under the assumption that $r_L:H^1(M;{\mathbb R}) \to H^1(L;{\mathbb R})$ is onto. Suppose $c,c' \in H^1(M;{\mathbb R})$ satisfy $r_L(c)=r_L(c')\in H^1(L;{\mathbb R})$. Fix a closed 1-form $\eta$ on $M$ with $[\eta]=c-c'$ and denote by $\psi=\{\psi_t\}_{t\in {\mathbb R}}$ the symplectic isotopy generated by the vector field $X_{\eta}$ characterised by $$\iota_{X_{\eta}}\omega=\eta.$$ By property c) of Theorem \[prop1\] we can now compute $$\sigma_{H:L}(c)=\sigma_{H:L}(c'+(c-c'))=\sigma_{H:\psi_1(L)}(c').$$ We can view $\psi$ as a Lagrange isotopy starting at $L$ $$\label{eq27} \psi|_L:[0,1]\times L \to M,$$ with *Lagrangian Flux path* $s\mapsto \operatorname{Flux}_L(\{ \psi_t|_L\}_{0\leq t\leq s})\in H^1(L;{\mathbb R})$. We compute that $$\operatorname{Flux}_L(\{ \psi_t|_L\}_{0\leq t\leq s})=\int_0^s r_L[\iota_{X_{\eta}}\omega] dt=r_L(c-c')s=0\in H^1(L;{\mathbb R})$$ for all $s\in [0,1]$. Hence, (\[eq27\]) is an exact Lagrange isotopy, so there exists $\phi \in \operatorname{Ham}(M,\omega)$ such that $\phi(L)=\psi_1(L)$. Applying property d) of Theorem \[prop1\] we conclude that $\sigma_{H:\psi_1(L)}(c')=\sigma_{H:\phi(L)}(c')=\sigma_{H:L}(c')$. It follows that, under the assumption that $r_L:H^1(M;{\mathbb R}) \to H^1(L;{\mathbb R})$ is onto, $\alpha_{H:L}:H^1(L;{\mathbb R}) \to {\mathbb R}$, $$\alpha_{H:L}(r_L(c)):=\sigma_{H:L}(c), \quad c\in H^1(M;{\mathbb R})$$ is well-defined. Clearly $\alpha_{H:L}$ is Lipschitz. The second part of Corollary \[lem1\] follows from Theorem \[thm1\] if only we prove that $$\partial \sigma_{H:L}(c)=i_L(\partial \alpha_{H:L}(r_L(c)))\quad \forall \ c\in H^1(M;{\mathbb R}),$$ which can be seen as follows.[^17] Note first that, under the canonical identification $H_1(-;{\mathbb R})\cong H^1(-;{\mathbb R})^*$ this equality can be written $$\partial \sigma_{H:L}(c)=r_L^*(\partial \alpha_{H:L}(r_L(c)))\quad \forall \ c\in H^1(M;{\mathbb R}).$$ The “$\supset$” inclusion is not hard to see: By definition, we have $h'\in \partial \alpha_{H:L}(r_L(c))$ if and only if $$\forall \ c'\in H^1(L;{\mathbb R}): \ \langle c',h'\rangle \leq \alpha_{H:L}^{\circ}(r_L(c),c').$$ In particular, if $h'\in \partial \alpha_{H:L}(r_L(c)))$ then $$\forall \ c'\in H^1(M;{\mathbb R}): \ \langle c',r_L^*(h')\rangle=\langle r_L(c'),h'\rangle \leq \alpha_{H:L}^{\circ}(r_L(c),r_L(c'))=\sigma_{H:L}^{\circ}(c,c'),$$ where in the last equality we use that $r_L$ is linear. This shows $$\partial \sigma_{H:L}(c)\supset r_L^*(\partial \alpha_{H:L}(r_L(c))).$$ For the “$\subset$” inclusion, note that $$h\in \partial \sigma_{H:L}(c)\Rightarrow \left( \forall \ c'\in \ker(r_L): \langle c',h \rangle \leq \sigma^{\circ}_{H:L}(c,c')= \alpha_{H:L}^{\circ}(r_L(c),0)=0 \right).$$ I.e. $\partial \sigma_{H:L}(c)\subset \ker(r_L)^{\perp}$, where the latter denotes the annihilator of $\ker(r_L)$. Since the image of $r^*_L$ coincides with the annihilator of $\ker(r_L)$ (because $H^1(M;{\mathbb R})$ and $H^1(L;{\mathbb R})$ are finite dimensional), this is equivalent to $\partial \sigma_{H:L}(c)\subset \operatorname{Image}(r_L^*)$. I.e. if $h\in \partial \sigma_{H:L}(c)$ then $h=r_L^*(h')$ for some $h'\in H^1(L;{\mathbb R})^*$ and $$\forall \ c'\in H^1(M;{\mathbb R}): \langle r_L(c'),h' \rangle = \langle c',h \rangle \leq \sigma_{H:L}^{\circ}(c,c')=\alpha_{H:L}^{\circ}(r_L(c),r_L(c')).$$ Since $r_L$ is assumed surjective we conclude that $h'\in \partial \alpha_{H:L}(r_L(c))$, which shows $$\partial \sigma_{H:L}(c)\subset r_L^*(\partial \alpha_{H:L}(r_L(c))).$$ Consider the path $\gamma:[0,1]\to H^1(L;{\mathbb R})$ given by $\gamma(t)=tc$. By Lebourg’s mean value theorem (see Section \[secmeas\]) there exists $s\in (0,1)$ and $h\in \partial \alpha_{H:L}(\gamma(s))$ such that $$\langle c,h\rangle=\alpha_{H:L}(c)-\alpha_{H:L}(0).$$ The existence of $\mu \in \mathcal{M}(\phi_H)$ with $\rho(\mu)=i_L(h)$ now follows from Corollary \[lem1\]. To get the estimate note that, given a Lagrange isotopy $\psi:[0,1]\times L\to M$ with $\psi_0$ the inclusion and $\operatorname{Flux}_{L}(\psi)=c$ there exists (by a result due to Solomon [@Solomon13 Lemma 6.6]) a symplectic isotopy $\tilde{\psi}:[0,1]\times M \to M$ such that $\tilde{\psi}_t(L)=\psi_t(L)$ and such that $r_L(\operatorname{Flux}(\tilde{\psi}))=c$. Using property b) of Theorem \[prop1\] we can therefore estimate $$\langle c,h\rangle \geq \min_{x\in \tilde{\psi}_1(L)}H(x)-\max_{x\in L}H(x)=\min_{x\in \psi_1(L)}H(x)-\max_{x\in L}H(x).$$ Recall that we need to show $\alpha_{H_{\epsilon},T(0,0)}(I,0)=0$ for $I\in {\mathbb R}$, where $$H_{\epsilon}(\theta,I)=H_0(I)+\epsilon F(\theta,I),$$ for $$H_0(I)=I_1I_2 \quad \& \quad F(\theta,I)=\varphi(I_1)\sin(2\pi \theta_1).$$ By Theorem \[prop1\] we have $$\alpha_{H_{\epsilon},T(0,0)}(I_1,I_2)=\alpha_{H_{\epsilon},T(I_1,I_2)}(0,0) \quad \forall \ (I_1,I_2)\in {\mathbb R}^2.$$ Now fix $I\in {\mathbb R}$. We need to show that $\alpha_{H_{\epsilon},T(I,0)}(0,0)=0$. We will think of ${\mathbb T}^2\times {\mathbb R}^2=(S^1\times {\mathbb R})\times (S^1\times {\mathbb R})$ with coordinates $(\theta_1,I_1)$ on the first factor and $(\theta_2,I_2)$ on the second. Choose an embedded circle $S\subset (S^1\times {\mathbb R})$ such that $S$ is Hamiltonian isotopic to $\{I_1=I\}$ and $$S\cap \{I_1\leq K+2\}=\{(\theta_1,I_1)\ |\ \theta_1\in \{ 0,\tfrac{1}{2}\}\ \& \ |I_1|\leq K+2 \}.$$ Then there exists $\phi \in \operatorname{Ham}({\mathbb T}^2 \times {\mathbb R}^2,d\lambda)$ such that $\phi(T(I,0))=S\times \{I_2=0\}\subset (S^1\times {\mathbb R})\times (S^1\times {\mathbb R})$ and applying Theorem \[prop1\] gives $$\alpha_{H_{\epsilon},T(I,0)}(0,0)=\alpha_{H_{\epsilon},S\times \{I_2=0\}}(0,0).$$ Since $H_{\epsilon}|_{S\times \{I_2=0\}}\equiv 0$ for all $\epsilon \geq 0$ it follows from point b) of Theorem \[prop1\] that $\alpha_{H_{\epsilon},S\times \{I_2=0\}}(0,0)=0$ for all $\epsilon \geq 0$. This finishes the proof. The next several pages take up the proof of Theorem \[thm1\]. First let’s consider the setup: Set $m=\dim_{{\mathbb R}}H^1(M;{\mathbb R})$ and fix once and for all a basis $e_1,\ldots ,e_m$ for $H^1(M;{\mathbb R})$. Choose closed one forms $\eta_1,\ldots ,\eta_m$ on $M$ such that $e_l=[\eta_l]$ and define vector fields $X_{\eta_1},\ldots X_{\eta_m}$ by requiring $$\iota_{X_{\eta_l}}\omega=\eta_l \quad \forall \ l=1,\ldots ,m.$$ Denote the symplectic isotopy generated by $X_{\eta_l}$ by $\psi^l=\{ \psi_t^l \}_{t\in {\mathbb R}}$. For each $k\in {\mathbb N}$ we define a function $a_k:H^1(M;{\mathbb R})\to {\mathbb R}$ by $$a_k(c)=\frac{l_L(\psi^m_{-\kappa_m} \cdots \psi^1_{-\kappa_1} \tilde{\phi}_H^k\psi^1_{\kappa_1} \cdots \psi^m_{\kappa_m})}{k}$$ if $c=\sum_{l=1}^{m}\kappa_le_l$. The following lemma is due to Vichery [@Vichery14]. In [@Vichery14] the case of the zero-section in a cotangent bundle is considered. But in the general case the same proof applies. \[lemVichery\] The sequence $(a_k)_{k\in {\mathbb N}}$ converges uniformly to $\sigma_{H:L}$ on compact subsets. \[prop2\] Let $h\in H_1(M;{\mathbb R})$ be a Dini subdifferential of $a_k$ at the point $c=\sum_{l=1}^{m}\kappa_l e_l\in H^1(M;{\mathbb R})$.[^18] We associate to any $c'=\sum_{l=1}^{m}\kappa'_l e_l\in H^1(M;{\mathbb R})$ the symplectic isotopy $$\label{eq11} \psi_{\tau}:=\psi^1_{\kappa_1+\tau \kappa'_1} \cdots \psi^m_{\kappa_m+\tau \kappa'_m}$$ and denote by $X_{\tau}(\psi_{\tau}(x)):=\left. \frac{d}{ds}\right|_{s=0}\psi_{s+\tau}(x)$ its infinitesimal generator. Then there exists a point $x\in \psi_0(L)$ such that $\phi_H^k(x)\in \psi_0(L)$ and $$\label{eq12} \langle c',h \rangle \leq \frac{1}{k}\int_{0}^{k}\langle dH,X_0\rangle \phi_H^t(x)\ dt.$$ Moreover, the curve $\gamma:([0,k],\{0,k\})\to (M,L)$ given by $\gamma(t)=\phi_H^t(x)$ represents the trivial element in $\pi_1(M,\psi_0(L))$ and there exists a capping $\widehat{\gamma}$ of $\gamma$ such that $$\begin{aligned} \label{eq5} a_k(c)&=\frac{1}{k}\int_0^kH(\gamma(t))\ dt-\frac{1}{k}\int \widehat{\gamma}^*\omega.\end{aligned}$$ Let us first prove the result under the non-degeneracy assumption $$\label{eq13} \phi_{H\psi_0}^k(L)\pitchfork L.$$ The Hamiltonian $kH$ generates the isotopy $t\mapsto \phi_H^{tk}$. Choose a generic path $J=\{J_t\}_{t\in [0,1]}$ of $\omega$-compatible almost complex structures so that we have a well-defined Floer chain complex $(CF(L{:}kH\psi_0,J),d)$. Choose a function $\chi \in C^{\infty}({\mathbb R};[0,1])$ such that $$\chi(s)= \left\{ \begin{array}{ll} 0, & \text{if} \ s\leq 0 \\ 1, & \text{if} \ s\geq 1 \end{array} \right.$$ and $\chi' \geq 0$. Now fix $\tau >0$ so small that $\phi^k_{H\psi_{\tau}}(L)\pitchfork L$ and consider the $s$-dependent Hamiltonian[^19] $$\label{eq7} F^{\tau}_s(x)=kH\psi_0(x)+\chi(s)(kH\psi_{\tau}(x)-kH\psi_0(x)).$$ There exists a regular homotopy of Floer data $(K_{\tau, s},J^{\tau, s})_{s\in {\mathbb R}}$ which is stationary for $s\notin (0,1)$ such that $$\begin{aligned} \label{eq6} |F^{\tau}-K^{\tau}|_{C^{\infty}(M\times [0,1] \times [0,1])}&\leq \tau^2 \\ |J-J^{\tau, s}|_{C^{\infty}(M\times [0,1])}&\leq \tau^2 \quad \forall \ s\in (0,1), \nonumber\end{aligned}$$ The Floer continuation map $$\Phi: CF_*(L{:}kH\psi_0,J)\to CF_*(L{:}kH\psi_{\tau},J^{\tau,1})$$ is defined “by counting” finite-energy solutions $u\in C^{\infty}({\mathbb R}\times [0,1];M)$ to the problem $$\label{eq3} \left\{ \begin{array}{ll} \partial_su+J_t^{\tau, s}(u)(\partial_tu-X_{K_s}(u))=0 \\ u({\mathbb R}\times \{0,1\})\subset L, \end{array} \right.$$ where the *energy of $u$* denotes the non-negative quantity $$E_{J^{\tau}}(u)=\int_{-\infty}^{\infty}\int_0^1 \omega(\partial_su,J^{\tau,s}(u)\partial_su)\ dtds.$$ By a very general result due to Usher [@Usher08 Theorem 1.4], there exists a cycle $$\mathfrak{c}\in CF_*(L{:}kH\psi_0,J),$$ all of whose elements have action $\leq l_L(\psi_0^{-1}\tilde{\phi}_H^k\psi_0)$ and which satisfies $[\mathfrak{c}]=\operatorname{PSS}([L])\in HF_*(L{:}kH\psi_0,J)$.[^20] Since we also have $\Phi(\mathfrak{c})=\operatorname{PSS}([L])$, at least one of the Hamiltonian chords of the cycle $\Phi(\mathfrak{c})$ must have action $\geq l_L(\psi_{\tau}^{-1}\tilde{\phi}_H^k\psi_{\tau})$. We hence deduce the existence of a finite-energy solution $u^{\tau}_s(t)=u^{\tau}(s,t)$ to (\[eq3\]) such that $$\label{eq4} \mathcal{A}_{kH\psi_0:L}(u^{\tau}_{-\infty}) \leq l_L(\psi_{0}^{-1}\tilde{\phi}_H^k\psi_{0}) \quad \text{and} \quad l_L(\psi_{\tau}^{-1}\tilde{\phi}_H^k\psi_{\tau}) \leq \mathcal{A}_{kH\psi_{\tau}:L}(u^{\tau}_{\infty}).$$ Here $u_{-\infty}^{\tau}$ (respectively $u_{\infty}^{\tau}$) denotes one of the elements of the chain $\mathfrak{c}$ (respectively $\Phi(\mathfrak{c})$). Integrating by parts reveals that $u^{\tau}$ satisfies the energy identity $$\begin{aligned} \label{eq30} 0\leq E_{J^{\tau}}(u^{\tau})=\mathcal{A}_{kH\psi_0:L}(u^{\tau}_{-\infty})-\mathcal{A}_{kH\psi_{\tau}:L}(u^{\tau}_{\infty})+\int_{-\infty}^{\infty}\int_0^1 \partial_sK_s(u^{\tau}(s,t))dtds.\end{aligned}$$ Using (\[eq6\]) and (\[eq4\]), this identity implies the estimate $$\begin{aligned} \label{eq9} \frac{a_k(c+\tau c')-a_k(c)}{\tau}&= \frac{l_L(\psi_{\tau}^{-1}\tilde{\phi}_H^k\psi_{\tau})-l_L(\psi_{0}^{-1}\tilde{\phi}_H^k\psi_{0})}{k\tau} \nonumber \\ &\leq \frac{\mathcal{A}_{kH\psi_{\tau}:L}(u^{\tau}_{\infty})-\mathcal{A}_{kH\psi_0:L}(u^{\tau}_{-\infty})}{k\tau} \nonumber \\ &\leq \frac{1}{k\tau}\int_{-\infty}^{\infty}\int_0^1 \partial_sK_s(u^{\tau}(s,t))dtds \\ &\leq \frac{1}{k\tau}\int_{-\infty}^{\infty}\int_0^1 \partial_sF^{\tau}_s(u^{\tau}(s,t))dtds +\frac{\tau}{k} \nonumber \\ &= \int_{-\infty}^{\infty}\int_0^1 \chi'(s)\left( \frac{H\psi_{\tau}(u^{\tau}(s,t))-H\psi_{0}(u^{\tau}(s,t))}{\tau} \right) dtds +\frac{\tau}{k}. \nonumber\end{aligned}$$ Moreover, we can use the energy identity together with (\[eq4\]) to estimate the energy of $u^{\tau}$ from above: $$\begin{aligned} \label{eq15} E_{J^{\tau}}(u^{\tau})&\leq l_L(\psi_{0}^{-1}\tilde{\phi}_H^k\psi_{0})-l_L(\psi_{\tau}^{-1}\tilde{\phi}_H^k\psi_{\tau})+k\max_{M}|H\psi_{\tau}-H\psi_0|+\tau^2 \\ &\leq 2k\max_{M}|H\psi_{\tau}-H\psi_0|+\tau^2. \nonumber \end{aligned}$$ The last inequality makes use of the Lipschitz property of Lagrangian spectral invariants (see [@LeclercqZapolsky15]). Choose now a sequence $(\tau_{\varsigma})_{\varsigma \in {\mathbb N}}\subset (0,1)$ such that $\tau_{\varsigma} \downarrow 0$ and $$\label{eq17} \frac{a_k(c+\tau_{\varsigma} c')-a_k(c)}{\tau_{\varsigma}}\stackrel{\varsigma \to \infty}{\longrightarrow} \liminf_{\tau \downarrow 0}\frac{a_k(c+\tau c')-a_k(c)}{\tau}$$ For each $\tau_{\varsigma}$ we find a Floer trajectory $u^{\varsigma}:=u^{\tau_{\varsigma}}$ meeting the requirement (\[eq4\]) with $\tau=\tau_{\varsigma}$. Applying Gromov convergence to the sequence $(u^{\varsigma})_{\varsigma \in {\mathbb N}}$ we obtain a subsequence (still denoted by $(u^{\varsigma})_{\varsigma \in {\mathbb N}}$) which $C^{\infty}_{loc}$-converges. This follows from a result due to Hofer [@Hofer88 Proof of Proposition 2] saying that, in our setup, the obstruction to $C^{\infty}_{loc}$-precompactness is “bubbling off” of a non-constant holomorphic disc or sphere. However, this cannot occur to $(u^{\varsigma})_{\varsigma \in {\mathbb N}}$ since the estimate (\[eq15\]) implies $$E_{J^{\tau_{\varsigma}}}(u^{\varsigma})\stackrel{\varsigma \to \infty}{\longrightarrow} 0.$$ By Fatou’s lemma, this also implies that the pointwise limit $u$ of $(u^{\varsigma})_{\varsigma \in {\mathbb N}}$ is an $s$-independent map given by $u(s,t)=u(t)=\phi_{kH\psi_0}^t(y)$ for some $y\in L$. Now, using $\chi' \geq 0$ and the mean value theorem, we find for every $(s,t)\in {\mathbb R}\times [0,1]$ a $\tau'_{\varsigma}(s,t)\in (0,\tau_{\varsigma})$ such that $$\begin{aligned} \left| \chi'(s)\left( \frac{H\psi_{\tau_{\varsigma}}(u^{\varsigma}(s,t))-H\psi_{0}(u^{\varsigma}(s,t))}{\tau_{\varsigma}} \right) \right|&= \chi'(s)|\langle dH,X_{\tau_{\varsigma}'(s,t)} \rangle|(\psi_{\tau_{\varsigma}'(s,t)}(u^{\varsigma}(s,t))) \\ &\leq \chi'(s)\max_{(\tau,z)\in [0,1]\times M}|\langle dH,X_{\tau} \rangle|(z)\end{aligned}$$ Since $h$ is a Dini subdifferential of $a_k$ at $c$ and $$\int_{-\infty}^{\infty} \int_{0}^{1}\chi'(s)\max_{(\tau,z)\in [0,1]\times M}|\langle dH,X_{\tau} \rangle|(z) \ dtds=\max_{(\tau,z)\in [0,1]\times M}|\langle dH,X_{\tau} \rangle|(z)<\infty,$$ we can apply Lebesgue dominated convergence in the estimates (\[eq9\]) to conclude that $$\begin{aligned} \langle c', h \rangle &\leq \lim_{\varsigma \to \infty} \int_{-\infty}^{\infty}\int_0^1 \chi'(s)\left( \frac{H\psi_{\tau_{\varsigma}}(u^{\varsigma}(s,t))-H\psi_{0}(u^{\varsigma}(s,t))}{\tau_{\varsigma}} \right) dtds \\ &= \int_{-\infty}^{\infty}\int_0^1 \chi'(s)\lim_{\varsigma \to \infty}\left( \frac{H\psi_{\tau_{\varsigma}}(u^{\varsigma}(s,t))-H\psi_{0}(u^{\varsigma}(s,t))}{\tau_{\varsigma}} \right) dtds \\ &=\int_{-\infty}^{\infty}\int_0^1 \chi'(s) \langle dH,X_0 \rangle \psi_0(u(t)) dtds =\int_0^1 \langle dH,X_0 \rangle \phi_{H}^{kt}\psi_0(y)dt,\end{aligned}$$ where we use the fact that $\psi_0\phi_{kH\psi_0}^{t}(y)=\phi_{kH}^{t}\psi_0(y)=\phi_{H}^{kt}\psi_0(y)$. By changing variables to $s=kt$ one now concludes that (\[eq12\]) holds for $x:=\psi_0(y)\in \psi_0(L)$. In order to find a capping $\widehat{\gamma}$ for $\gamma(t)=\phi_H^t(x),\ t\in [0,k]$ such that (\[eq5\]) holds, we first note that $$\label{eq16} \mathcal{A}_{kH\psi_0:L}(u^{\varsigma}_{-\infty}),\mathcal{A}_{kH\psi_{\tau_{\varsigma}}:L}(u^{\varsigma}_{\infty}) \stackrel{\varsigma \to \infty}{\longrightarrow} l_L(\psi_{0}^{-1}\tilde{\phi}_H^k\psi_{0}),$$ which is easily seen using the estimates (\[eq4\]) and (\[eq30\]). By $C^{\infty}_{loc}$-convergence we may choose $\varsigma$ so large that the curve $[0,1]\ni t \mapsto u^{\varsigma}_s(t)=u^{\varsigma}(s,t)$ is $C^0$-close to $u$ for every $s\in [-2,2]$. In particular we can choose $\varsigma$ so large that the areas of the two cappings of $u$, obtained by adjusting the cappings of $u^{\varsigma}_{-2}$ and $u^{\varsigma}_{2}$ given by $u^{\varsigma}$ slightly, differ by less than $\tau_L$. Hence, by monotonicity of $L$ they have the same area and thus define a class of cappings $\widetilde{u}$ of $u$. Given $\epsilon >0$ we can in addition achieve $$|\mathcal{A}_{kH\psi_0}(u^{\varsigma}_{-2})-\mathcal{A}_{kH\psi_0}(\widetilde{u})|<\epsilon \quad \text{and}\quad |\mathcal{A}_{kH\psi_{\tau_{\varsigma}}}(u^{\varsigma}_{2})-\mathcal{A}_{kH\psi_{\tau_{\varsigma}}}(\widetilde{u})|<\epsilon.$$ Since $\mathcal{A}_{kH\psi_{\tau_{\varsigma}}}(\widetilde{u}) \to \mathcal{A}_{kH\psi_{0}}(\widetilde{u})$ for $\varsigma \to \infty$ these estimates together with $$\begin{aligned} 0&\leq \mathcal{A}_{kH\psi_{\tau_{\varsigma}}:L}(u^{\varsigma}_2)-\mathcal{A}_{kH\psi_{\tau_{\varsigma}}:L} (u^{\varsigma}_{\infty})\leq E_{J^{\tau_{\varsigma}}}(u^{\varsigma}) \stackrel{\varsigma \to \infty}{\longrightarrow} 0 \\ 0&\leq \mathcal{A}_{kH\psi_{\tau_0}:L}(u^{\varsigma}_{-\infty})-\mathcal{A}_{kH\psi_{0}:L} (u^{\varsigma}_{-2})\leq E_{J^{\tau_{\varsigma}}}(u^{\varsigma}) \stackrel{\varsigma \to \infty}{\longrightarrow} 0\end{aligned}$$ and (\[eq16\]) imply that $$\mathcal{A}_{kH\psi_{0}}(\widetilde{u})=l_L(\psi_{0}^{-1}\tilde{\phi}_H^k\psi_{0}).$$ Hence, the cappings of $\widetilde{u}$ give rise to a capping $\widehat{\gamma}$ of $\gamma$ such that (\[eq5\]) holds. Let’s now discuss how to deal with the case when assumption (\[eq13\]) is violated. Then we proceed as follows: Again we choose a sequence $(\tau_{\varsigma})_{\varsigma \in {\mathbb N}}$ such that $\tau_{\varsigma} \downarrow 0$ and (\[eq17\]) is satisfied as well as a path $J=\{ J_{t} \}_{t \in [0,1]}$ of $\omega$-compatible almost complex structures. For each $\varsigma$ we choose $H^{\varsigma}\in C^{\infty}([0,1]\times M)$ satisfying $$\label{eq18} |H^{\varsigma}-kH|_{C^{\infty}([0,1]\times M)}\leq \tau_{\varsigma}^2$$ as well as $$\phi_{H^{\varsigma}\psi_0}^1(L)\pitchfork L \quad \text{and} \quad \phi_{H^{\varsigma}\psi_{\tau_{\varsigma}}}^1(L)\pitchfork L.$$ Now choose paths of compatible almost complex structures $J^{\varsigma}=\{ J^{\varsigma}_t \}_{\tau \in [0,1]}$ which are regular in the sense that we have well-defined Floer chain complexes $$(CF(L{:}H^{\varsigma}\psi_0, J^{\varsigma}),d), \quad (CF(L{:}H^{\varsigma}\psi_{\tau_{\varsigma}}, J^{\varsigma}),d)$$ and such that $J^{\varsigma}$ satisfies $$|J^{\varsigma}-J|_{C^{\infty}([0,1]\times M)}\leq \tau_{\varsigma}^2.$$ For each $\varsigma$ we obtain by the above procedure a Floer trajectory $u^{\varsigma}:=u^{\tau_{\varsigma}}$ such that (\[eq4\]) is satisfied with $H=H^{\varsigma}$ and $\tau=\tau_{\varsigma}$. In particular (\[eq15\]) follows with $H=H^{\varsigma}$ and $\tau=\tau_{\varsigma}$. Using (\[eq9\]) and (\[eq18\]) as well as the Lipschitz property of Lagrangian spectral invariants [@LeclercqZapolsky15], we can estimate $$\begin{aligned} \frac{a_k(c+\tau_{\varsigma} c')-a_k(c)}{\tau_{\varsigma}}&=\frac{l_L(\psi_{\tau}^{-1}\tilde{\phi}_H^k\psi_{\tau})-l_L(\psi_{0}^{-1}\tilde{\phi}_H^k\psi_{0})}{k\tau} \nonumber \\ &\leq \frac{l_L(\psi_{\tau}^{-1}\tilde{\phi}_{H^{\varsigma}}^1\psi_{\tau})-l_L(\psi_{0}^{-1}\tilde{\phi}_{H^{\varsigma}}^1\psi_{0})}{k\tau_{\varsigma}}+\frac{2\tau_{\varsigma}}{k} \label{eq19} \\ &\leq \frac{1}{k}\int_{-\infty}^{\infty}\int_0^1 \chi'(s)\left( \frac{H_t^{\varsigma}\psi_{\tau_{\varsigma}}(u^{\varsigma}(s,t))-H_t^{\varsigma}\psi_{0}(u^{\varsigma}(s,t))}{\tau_{\varsigma}} \right) dtds +\frac{3\tau_{\varsigma}}{k} \nonumber \\ &\leq \int_{-\infty}^{\infty}\int_0^k \chi'(s)\left( \frac{H\psi_{\tau_{\varsigma}}(u^{\varsigma}(s,t))-H\psi_{0}(u^{\varsigma}(s,t))}{\tau_{\varsigma}} \right) dtds +\frac{5\tau_{\varsigma}}{k}. \nonumber\end{aligned}$$ Again, by applying Hofer’s compactness argument [@Hofer88 Proof of Proposition 2], we may assume that $(u_{\varsigma})_{\varsigma \in {\mathbb N}}$ $C^{\infty}_{loc}$-converges. Exactly as above the pointwise limit $u$ is a $s$-constant map given by $u(t)=\phi^{kt}_{H\psi_0}(y)$ for some $y\in L$. Via dominated convergence the estimate (\[eq19\]) now gives $$\langle c', h \rangle \leq \frac{1}{k}\int_{0}^{k}\langle dH,X_0 \rangle \phi_H^t(x)\ dt$$ for $x=\psi_0(y)\in \psi_0(L)$ and the existence of a capping is obtained exactly as before. This finishes the proof. \[Mrem1\] Suppose now that $\omega=d\lambda$ is exact with $\lambda|_L=df$ for some $f\in C^{\infty}(L)$. In this case we need to relate the action appearing in (\[eq5\]) to the one appearing in (\[eq8\]). Choose a closed one form $\eta$ in the class $c\in H^1(M;{\mathbb R})$. An easy computation shows that $[\lambda|_{\psi_0(L)}]=[\eta|_{\psi_0(L)}]\in H^1(\psi_0(L);{\mathbb R})$, so in fact $\eta$ can be chosen to satisfy $(\lambda-\eta)|_{\psi_0(L)}\equiv 0$. In particular we then have $$\int \widehat{\gamma}^*\omega=\int d\widehat{\gamma}^*(\lambda-\eta)=\int_0^k \langle \lambda, X_H \rangle \gamma(t)dt - \int_0^k \langle \eta, X_H \rangle \gamma(t)dt,$$ so (\[eq5\]) reads $$\label{Mateq901} a_k(c)=\mathcal{A}_{H,\lambda}(\mu_{\gamma})+\int \langle \eta, X_H \rangle d\mu_{\gamma},$$ where $\mu_{\gamma}$ is the Borel probability measure obtained by pushing forward the normalized Lebesgue measure on $[0,k]$ to $M$ via $\gamma$. Though we are mainly interested in the Clarke subdifferentials of $\alpha_{H:L}$, the proof of Theorem \[thm1\] goes via the so-called *approximate subdifferential* due to Ioffe [@Ioffe84]. \[prop3\] Let $h\in \partial_A a_k(c)\subset H_1(M;{\mathbb R})$ be an approximate subdifferential of $a_k$ at $c=\sum_{l=1}^{m}\kappa_l e_l\in H^1(M;{\mathbb R})$.[^21] Given a fixed $c'=\sum_{l=1}^{m}\kappa'_l e_l\in H^1(M;{\mathbb R})$ we consider the symplectic isotopy (\[eq11\]) and denote its infinitesimal generator by $X_{\tau}$. Then there exists a point $x\in L$ such that the curve $[0,k]\ni t\mapsto \gamma(t)=\phi_{H\psi_0}^t(x)$ satisfies (\[eq12\]) and (\[eq5\]). By (\[eq222\]) there exists a sequence $(c_{\varsigma})_{\varsigma \in {\mathbb N}}\subset H^1(M;{\mathbb R})$ such that $h$ is a Dini subdifferential of $a_k$ at $c_{\varsigma}$ and $$\begin{aligned} \label{eq14} c_{\varsigma}\stackrel{\varsigma \to \infty}{\longrightarrow}c\end{aligned}$$ Writing $c_{\varsigma}=\sum_{l=1}^m \xi_l^{\varsigma} e_l$ we define the symplectic isotopy $$\upsilon^{\varsigma}_{\tau}:=\psi^1_{\xi_1^{\varsigma}+\tau \kappa_1'}\cdots \psi^m_{\xi_m^{\varsigma}+\tau \kappa_m'}$$ and denote by $X^{\varsigma}_{\tau}$ the infinitesimal generator of $\tau \mapsto \upsilon^{\varsigma}_{\tau}$. By Proposition \[prop2\] there exists a sequence of points $(x_{\varsigma})_{\varsigma \in {\mathbb N}}$ with $x_{\varsigma}\in \upsilon_{0}^{\varsigma}(L)$ such that the curves $[0,k]\ni t\mapsto \gamma_{\varsigma}(t):=\phi_{H}^t(x_{\varsigma})$ have cappings $\widehat{\gamma}_{\varsigma}$ for which $$\begin{aligned} \langle c',h \rangle &\leq \frac{1}{k}\int_{0}^{k}\langle dH,X^{\varsigma}_0\rangle (\gamma_{\varsigma}(t))dt \\ a_k(c_{\varsigma})&=\frac{1}{k}\int_0^kH(\gamma_{\varsigma}(t))dt-\frac{1}{k}\int \widehat{\gamma}_{\varsigma}^*\omega.\end{aligned}$$ By compactness of $M$, we may assume that $x_{\varsigma}\to x$ for some $x\in M$, after passing to a subsequence. The claim now follows by passing to the limit, using the fact that $a_k$ is locally Lipschitz with respect to any norm on $H^1(M;{\mathbb R})$. Consider the set $$K:=\{ \rho(\mu)\in H_1(M;{\mathbb R}) \ |\ \mu \in \mathcal{M}(\phi_H)\}$$ of rotation vectors realized by $\mathcal{M}(\phi_H)$-measures. Clearly $K\subset H_1(M;{\mathbb R})$ is non-empty, convex and compact. Hence, $K$ is characterized by its *support function* [@Schneider14 Section 1.7.1] $\chi_K:H^1(M;{\mathbb R}) \to {\mathbb R}$, defined by $$\chi_K(c'):=\max_{h\in K}\langle c',h \rangle, \quad c'\in H^1(M;{\mathbb R}).$$ The characterization is that $h\in K$ if and only if $$\label{eq25} \langle c',h \rangle \leq \chi_K(c') \quad \forall \ c'\in H^1(M;{\mathbb R}).$$ We need to show that $\partial \sigma_{H:L}(c)\subset K$ for all $c\in H^1(M;{\mathbb R})$. In fact, by property c) of Theorem \[prop1\], it suffices to show that $\partial \sigma_{H:L}(0)\subset K$. Moreover, since $\sigma_{H:L}$ is locally Lipschitz and $K$ is convex and closed, it suffices (by (\[eq111\])) to show that $$\label{eq24} \partial_A \sigma_{H:L}(0) \subset K.$$ Fix some $h\in \partial_A \sigma_{H:L}(0)$. By the above discussion we need to show that (\[eq25\]) holds, so choose some $c'=\sum_{l=1}^m \kappa_l' e_l \in H^1(M;{\mathbb R})$. By Lemma \[lemVichery\] $a_k \to \sigma_{H:L}$ uniformly on compact subsets. Hence, applying (\[eq333\]) we can find an increasing sequence $(k_{\varsigma})_{\varsigma \in {\mathbb N}} \subset {\mathbb N}$ as well as sequences $(c_{\varsigma})_{\varsigma \in {\mathbb N}}\subset H^1(M;{\mathbb R})$ and $(h_{\varsigma})_{\varsigma \in {\mathbb N}}\subset H_1(M;{\mathbb R})$ such that $h_{\varsigma}\in \partial_A a_{k_{\varsigma}}(c_{\varsigma})$ and $$c_{\varsigma} \stackrel{\varsigma \to \infty}{\longrightarrow} 0 \quad \text{and} \quad h_{\varsigma} \stackrel{\varsigma \to \infty}{\longrightarrow} h.$$ Writing $c_{\varsigma}=\sum_{l=1}^m\kappa_l^{\varsigma}e_l$ we define a symplectomorphism $\psi_{\varsigma}$ by $$\psi_{\varsigma}:=\psi^1_{\kappa_1^{\varsigma}} \cdots \psi^m_{\kappa_m^{\varsigma}}$$ and denote by $X^{\varsigma}_{\tau}$ the infinitesimal generator of the symplectic isotopy $$\tau \mapsto \psi^1_{\kappa_1^{\varsigma}+\tau \kappa_1'} \cdots \psi^m_{\kappa_m^{\varsigma} +\tau \kappa_m'}$$ Applying Proposition \[prop3\] we obtain for each $\varsigma$ the existence of $x_{\varsigma}\in \psi_{\varsigma}(L)$ such that $$\langle c', h_{\varsigma} \rangle \leq \frac{1}{k_{\varsigma}} \int_0^{k_{\varsigma}}\langle dH,X_{0}^{\varsigma}\rangle \phi_H^t(x_{\varsigma})\ dt.$$ Denote by[^22] $\mu_{\varsigma}\in \mathcal{M}$ the unique probability measure characterized by $$\int f\ d\mu_{\varsigma}=\frac{1}{k_{\varsigma}} \int_0^{k_{\varsigma}}f\phi_H^t(x_{\varsigma})\ dt \quad \forall \ f\in C^0(M),$$ so that $$\label{eq26} \langle c', h_{\varsigma} \rangle \leq \int \langle dH,X_{0}^{\varsigma}\rangle \ d\mu_{\varsigma}.$$ By weak$^*$-compactness of $\mathcal{M}$ we may after passing to a subsequence (still denoted by $(\mu_{\varsigma})_{\varsigma \in {\mathbb N}}$) assume that $$\mu_{\varsigma}\stackrel{w^*}{\rightharpoonup} \mu \in \mathcal{M}.$$ By the classical Krylloff-Bogoliouboff argument [@KryloffBogoliouboff37] $\mu$ is $\phi_H$-invariant (see also [@Sorrentino15 Proposition 3.1.1]). Passing to the limit in (\[eq26\]) we conclude that $$\label{eq31} \langle c',h \rangle \leq \int \langle dH,Y\rangle \ d\mu,$$ where $$Y=\sum_{l=1}^m \kappa_l'X_{\eta_l}.$$ In particular (using $\langle dH,X_{\eta_l} \rangle =\langle \eta_l,X_H \rangle$) we obtain $$\langle c',h \rangle \leq \sum_{l=1}^m \kappa_l' \int \langle \eta_l,X_H \rangle d\mu=\sum_{l=1}^m \kappa_l' \langle e_l,\rho(\mu) \rangle=\langle c',\rho(\mu) \rangle \leq \chi_K(c').$$ Since $c'\in H^1(M;{\mathbb R})$ was arbitrary we conclude that (\[eq25\]) holds, so $h\in K$ and the proof is done. We will now discuss the adaptions required to prove the results in the non-compact setting from Section \[noncompact\]. Recall that in the non-compact setting we consider a Liouville manifold $(M,\omega=d\lambda)$ and assume that $\lambda|_L=df$ for some $f\in C^{\infty}(M)$. One then defines each $a_k$ exactly as above, except that $\eta_1,\ldots , \eta_m$ are closed 1-forms which represent a basis for $H^1_{dR}(M;{\mathbb R})$ and satisfy condition (\[Meq1\]). Now Proposition \[prop2\] and \[prop3\] hold exactly as in the compact setting. Fix some $c=\sum_{l=1}^m\kappa_l [\eta_l]\in H^1_{dR}(M;{\mathbb R})$ and denote by $C\subset M$ the closure of the smallest $\phi_H$-invariant set containing the subset $$\bigcup_{(s_1,\ldots, s_m)\in [0,1]^m}\! \! \! \! \! \psi^1_{\kappa_1+s_1}\cdots \psi^m_{\kappa_m+s_m}(L).$$ Since $H\in \mathcal{H}$, $C$ is a compact subset of $M$. Now define $$K:=\{\rho(\mu)\in H_1(M;{\mathbb R})\ |\ \mu \in \mathcal{M}(C;\phi_H) \ \text{with}\ \mathcal{A}_{H,\lambda}(\mu)+\langle c, \rho(\mu) \rangle =\sigma_{H:L}(c) \}.$$ $K$ is again a convex compact subset of $H_1(M;{\mathbb R})$, so it is characterized by its support function $\chi_K:H^1(M;{\mathbb R})\to {\mathbb R}$. Given $h\in \partial_A\sigma_{H:L}(c)$ and $c'\in H^1(M;{\mathbb R})$, an argument similar to that of the proof of Theorem \[thm1\] shows the existence of a $\mu \in \mathcal{M}(C;\phi_H)$ satisfying $$\label{Mateq32} \langle c',h\rangle \leq \langle c',\rho(\mu) \rangle.$$ Moreover, applying the statement of Proposition \[prop3\][^23] together with the weak$^*$-convergence it follows that $$\sigma_{H:L}(c)=\mathcal{A}_{H,\lambda}(\mu)+\langle c,\rho(\mu) \rangle,$$ which implies that $\rho(\mu)\in K$ and (\[Mateq32\]) now implies $\langle c',h\rangle \leq \chi_K(c')$. Since $c'\in H^1(M;{\mathbb R})$ was arbitrary this shows that $h\in K$, finishing the proof of Theorem \[thm2\]. The proof of Corollary \[cor3\] is identical to that of Corollary \[lem1\]. \[rem1\] Suppose each $a_k$ is $C^1$, $\alpha_{H:L}$ is $C^1$ and that $(a_k)_{k\in {\mathbb N}}$ $C^1$-converges to $\alpha_{H:L}$. In this setting one can avoid applying Jourani’s result and “upgrade” the proof of Theorem \[thm1\] to show the following: For every $c\in H^1(L;{\mathbb R})$ there exists a measure $\mu \in \mathcal{M}(\phi_H)$ with rotation vector $\rho(\mu)=d\alpha_{H:L}(c)$, whose support satisfies $$\operatorname{Supp}(\mu)\subset \overline{\bigcup_{t\in {\mathbb R}}\phi_H^t(L)}.$$ In particular, if $L$ is $\phi_H$-invariant then $\operatorname{Supp}(\mu)\subset L$. **Acknowledgments** I am very grateful to Vadim Kaloshin for teaching an excellent Nachdiplom lecture on Arnold Diffusion at the ETH Z[ü]{}rich. Everything I know about Aubry-Mather theory I learned from him, and the insight that invariant measures should be “wild” in the presence of superconductivity channels was generously suggested to me by him. I am also very grateful to Leonid Polterovich. Several of the ideas (in particular the applications to twisted cotangent bundles) were generously suggested by him. I thank Paul Biran and Will Merry for helpful discussions and I am grateful to Nicolas Vichery, Vincent Humili[è]{}re and Claude Viterbo for their interest in the paper. Last (but certainly not least) I thank the extremely careful anonymous referee for lots of high-quality input. Part of this research was carried out during visits to Tel Aviv University and University of Maryland at College Park. I thank Leonid and Vadim for the opportunities to make these visits as well as both universities for truly excellent research atmospheres. [^1]: In order to avoid certain compactness issues, $M$ will be assumed compact in this section. See Section \[noncompact\] for the non-compact (exact) case. [^2]: In [@LeclercqZapolsky15] our $l_L$ goes under the name $l_+$. [^3]: See Section \[seqprem\] for a precise definition of $\mathcal{A}_{H:\psi_{\varsigma}(L)}(\gamma_{\varsigma})$. [^4]: Recall that a Lagrangian $L\subset (M,d\lambda)$ is said to be *$\lambda$-exact* if $\lambda|_L=df$ for some $f\in C^{\infty}(L)$. [^5]: Recall that all $\mathcal{M}$-measures are required to have *compact* support. [^6]: Use e.g. Lebourg’s mean value theorem from Section \[secmeas\] below. [^7]: It is a consequence of Mather’s theory (or Theorem \[thm2\]) that this set is always $\neq \emptyset$. [^8]: The “closeness” in this Nekoroshev estimate is in fact quantitative. For precise details we refer to [@BounemouraKaloshin14 Section 1.1.2] [^9]: Which, of course, it clearly is *not*. [^10]: We denote by $\Theta=(\Theta_1,\Theta_2)$ standard coordinates on ${\mathbb R}^2$ which project to $(\theta_1,\theta_2)$ on ${\mathbb T}^2$. [^11]: We thank the careful referee for suggesting the following elegant proof as a drastic simplifications of our original (admittedly rather clumsy) proof. [^12]: By Lemma \[lem00\] on page such a $\tilde{c}$ exists if $(M,\omega)$ is weakly exact and $\pi_1(L)$ is Abelian. [^13]: There are several ways of finding Lagrangian 2-tori in $(T^*\mathbb{T}^2,\omega)$. One can find “small” ones in Darboux charts but there are also several other approaches. See e.g. Example \[ex03\]. Note that, since every closed Lagrangian in $(T^*\mathbb{T}^2,\omega)$ is displaceable, it follows from the adjunction formula, that every closed orientable Lagrangian in $(T^*\mathbb{T}^2,\omega)$ is topologically a 2-torus. [^14]: I.e. $H(q,p)=\frac{|p|^2}{2}+U(q)$ for some Riemannian metric and $U\in C^{\infty}(N)$. [^15]: This should be understood in the following sense: If we identify $V\cong {\mathbb R}^k$ by choosing a basis, then the set of points in ${\mathbb R}^k$ at which $f$ isn’t differentiable is a Lebesgue 0-set. [^16]: $||\cdot ||$ is any norm on $V$. [^17]: I am grateful to the anonymous referee whose careful advice significantly simplified this part of the proof. [^18]: See (\[eq1\]). [^19]: Here and in the rest of the proof one should *not* think of $\tau$ as a continuous variable, but as a fixed parameter. [^20]: See Section \[seqprem\] for the notation used here. [^21]: See (\[eq222\]). [^22]: Recall that $\mathcal{M}$ denotes the space of compactly supported Borel probability measures on $M$. [^23]: More precisely we make use of the fact that (\[Mateq901\]) holds.

--- bibliography: - './../../references/references.bib' --- Introduction {#sec:beer_introduction} ============ The beer game consists of a serial supply chain network with four agents—a retailer, a warehouse, a distributor, and a manufacturer—who must make independent replenishment decisions with limited information. The game is widely used in classroom settings to demonstrate the [*bullwhip effect*]{}, a phenomenon in which order variability increases as one moves upstream in the supply chain, as well as the importance of communication and coordination in the supply chain. The bullwhip effect occurs for a number of reasons, some rational [@LePaWh97ms] and some behavioral [@sterman1989modeling]. It is an inadvertent outcome that emerges when the players try to achieve the stated purpose of the game, which is to minimize costs. In this paper, we are interested not in the bullwhip effect but in the stated purpose, i.e., the minimization of supply chain costs, which underlies the decision making in every real-world supply chain. For general discussions of the bullwhip effect, see, e.g., @LePaWh04 [@GeDiTo06], and @snyder2018fundamentals. The agents in the beer game are arranged sequentially and numbered from 1 (retailer) to 4 (manufacturer), respectively. (See Figure \[fig:beer\_game\].) The retailer node faces a stochastic demand from its customer, and the manufacturer node has an unlimited source of supply. There are deterministic transportation lead times ($l^{tr}$) imposed on the flow of product from upstream to downstream, though the actual lead time is stochastic due to stockouts upstream; there are also deterministic information lead times ($l^{in}$) on the flow of information from downstream to upstream (replenishment orders). Each agent may have nonzero shortage and holding costs. In each period of the game, each agent chooses an order quantity $q$ to submit to its predecessor (supplier) in an attempt to minimize the long-run system-wide costs, $$\sum_{t=1}^{T} \sum_{i=1}^{4} c^i_h (IL_t^i)^{+} + c^i_p (IL_t^i)^{-}, \label{eq:beer_cost}$$ where $i$ is the index of the agents; $t=1,\dots,T$ is the index of the time periods; $T$ is the time horizon of the game (which is often unknown to the players); $c^i_h$ and $c^i_p$ are the holding and shortage cost coefficients, respectively, of agent $i$; and $IL_t^i$ is the inventory level of agent $i$ in period $t$. If $IL_t^i > 0$, then the agent has inventory on-hand, and if $IL_t^i < 0$, then it has backorders.The notation $x^+$ and $x^-$ denotes $\max\{0,x\}$ and $\max\{0,-x\}$, respectively. The standard rules of the beer game dictate that the agents may not communicate in any way, and that they do not share any local inventory statistics or cost information with other agents until the end of the game, at which time all agents are made aware of the system-wide cost. In other words, each agent makes decisions with only partial information about the environment while also cooperates with other agents to minimize the total cost of the system. According to the categorization by [@claus1998dynamics], the beer game is a decentralized, independent-learners (ILs), multi-agent, cooperative problem. The beer game assumes the agents incur holding and stockout costs but not fixed ordering costs, and therefore the optimal inventory policy is a [*base-stock policy*]{} in which each stage orders a sufficient quantity to bring its inventory position (on-hand plus on-order inventory minus backorders) equal to a fixed number, called its base-stock level [@clark1960optimal]. When there are no stockout costs at the non-retailer stages, i.e., $c_p^i = 0,~i \in \{2,3,4\}$, the well known algorithm by [@clark1960optimal] provides the optimal base-stock levels. To the best of our knowledge, there is no algorithm to find the optimal base-stock levels for general stockout-cost structures.More significantly, when some agents do not follow a base-stock or other rational policy, the form and parameters of the optimal policy that a given agent should follow are unknown. In this paper, we propose an extension of deep Q-networks (DQN) to solve this problem. Our algorithm is customized for the beer game, but we view it also as a proof-of-concept that DQN can be used to solve messier, more complicated supply chain problems than those typically analyzed in the literature. The remainder of this paper is as follows. Section \[sec:lit\_review\] provides a brief summary of the relevant literature and our contributions to it. The details of the algorithm are introduced in Section \[sec:beer\_dqn\_for\_beer\_game\]. Section \[sec:beer\_numerical\_experiment\] provides numerical experiments, and Section \[sec:beer\_conclusion\] concludes the paper. Literature Review {#sec:lit_review} ================= Current State of Art {#sec:Current_State_of_Art} -------------------- The beer game consists of a serial supply chain network. Under the conditions dictated by the game (zero fixed ordering costs, no ordering capacities, linear holding and backorder costs, etc.), a base-stock policy is optimal at each stage [@LePaWh97ms]. If the demand process and costs are stationary, then so are the optimal base-stock levels, which implies that in each period (except the first), each stage simply orders from its supplier exactly the amount that was demanded from it. If the customer demands are i.i.d. random and if backorder costs are incurred only at stage 1, then the optimal base-stock levels can be found using the exact algorithm by [@clark1960optimal]. There is a substantial literature on the beer game and the bullwhip effect. We review some of that literature here, considering both independent learners (ILs) and joint action learners (JALs) [@claus1998dynamics]. (ILs have no information about the other agent’s current states, whereas JALs may share such information.) For a more comprehensive review, see [@devika2016optimizing]. See [@martinez2014beergame] for a thorough history of the beer game. In the category of ILs, [@mosekilde1988deterministic] develop a simulation and test different ordering policies, which are expressed using a formula that involves state variables such as the number of anticipated shipments and unfilled orders. They assume the customer demand is 4 in each of the first four periods, and then 7 per period for the remainder of the horizon. [@sterman1989modeling] uses a similar version of the game in which the demand is 8 after the first four periods. (Hereinafter, we refer to this demand process as $C(4,8)$ or the [*classic*]{} demand process.) Also, he do not allow the players to be aware of the demand process. He proposes a formula (which we call the [*Sterman formula*]{}) to determine the order quantity based on the current backlog of orders, on-hand inventory, incoming and outgoing shipments, incoming orders, and expected demand. His formula is based on the anchoring and adjustment method of [@TverskyKahneman1979]. In a nutshell, the Sterman formula attempts to model the way human players over- or under-react to situations they observe in the supply chain such as shortages or excess inventory. Note that Sterman’s formula is not an attempt to optimize the order quantities in the beer game; rather, it is intended to model typical human behavior. There are multiple extensions of Sterman’s work. For example, [@strozzi2007beer] considers the beer game when the customer demand increases constantly after four periods and proposes a genetic algorithm (GA) to obtain the coefficients of the Sterman model. Subsequent behavioral beer game studies include [@CrosonDonohue2003] and [@CrosonDonohue2006]. Most of the optimization methods described in the first paragraph of this section assume that every agent follows a base-stock policy. The hallmark of the beer game, however, is that players do not tend to follow such a policy, or [*any*]{} policy. Often their behavior is quite irrational. There is comparatively little literature on how a given agent should optimize its inventory decisions when the other agents do not play rationally [@sterman1989modeling; @strozzi2007beer]—that is, how an individual player can best play the beer game when her teammates may not be making optimal decisions. Some of the beer game literature assumes the agents are JALs, i.e., information about inventory positions is shared among all agents, a significant difference compared to classical IL models. For example, [@kimbrough2002computers] propose a GA that receives a current snapshot of each agent and decides how much to order according to the [*$d+x$ rule*]{}. In the $d+x$ rule, agent $i$ observes $d_t^i$, the received demand/order in period $t$, chooses $x_t^i$, and then places an order of size $a_t^i = d_t^i + x_t^i$. In other words, $x_t^i$ is the (positive or negative) amount by which the agent’s order quantity differs from his observed demand. [@giannoccaro2002inventory] consider a beer game with three agents with stochastic shipment lead times and stochastic demand. They propose a RL algorithm to make decisions, in which the state variable is defined as the three inventory positions, which each are discretized into 10 intervals. The agents may use any actions in the integers on $[0,30]$. [@chaharsooghi2008reinforcement] consider the same game and solution approach except with four agents and a fixed length of 35 periods for each game. In their proposed RL, the state variable is the four inventory positions, which are each discretized into nine intervals. Moreover, their RL algorithm uses the $d+x$ rule to determine the order quantity, with $x$ restricted to be in $\{0,1,2,3\}$. Note that these RL algorithms assume that real-time information is shared among agents, whereas ours adheres to the typical beer-game assumption that each agent only has local information. Reinforcement Learning {#sec:RL} ---------------------- Reinforcement learning [@sutton1998reinforcement] is an area of machine learning that has been successfully applied to solve complex sequential decision problems. RL is concerned with the question of how a software agent should choose an action to maximize a cumulative reward. RL is a popular tool in telecommunications, robot control, and game playing, to name a few (see [@li2017deep]). Consider an agent that interacts with an environment. In each time step $t$, the agent observes the current state of the system, $s_t \in \mathbb{S}$ (where $\mathbb{S}$ is the set of possible states), chooses an action $a_t \in \mathbb{A}(s_t)$ (where $\mathbb{A}(s_t)$ is the set of possible actions when the system is in state $s_t$), and gets reward $r_t \in \mathbb{R}$; and then the system transitions randomly into state $s_{t+1} \in S$. This procedure is known as a [*Markov decision process*]{} (MDP) (see Figure \[fig:RL\]), and RL algorithms can be applied to solve this type of problem. The matrix $P_a(s,s{'})$, which is called the [*transition probability matrix*]{}, provides the probability of transitioning to state $s'$ when taking action $a$ in state $s$, i.e., $P_a(s,s') = \Pr(s_{t+1}=s' \mid s_t = s, a_t=a)$. Similarly, $R_{a} (s,s')$ defines the corresponding reward matrix. In each period $t$, the decision maker takes action $a_t = \pi_t(s)$ according to a given policy, denoted by $\pi_t$. The goal of RL is to maximize the expected discounted sum of the rewards $r_t$, when the systems runs for an infinite horizon. In other words, the aim is to determine a policy $\pi: \mathbb{S} \rightarrow \mathbb{A}$ to maximize $\sum_{t=0}^{\infty} \gamma^t E\left[R_{a_t} (s_t,s_{t+1})\right]$, where $a_t = \pi_t(s_t)$ and $0 \le \gamma < 1$ is the discount factor. For given $P_a(s,s{'})$ and $R_{a} (s,s')$, the optimal policy can be obtained through dynamic programming or linear programming [@sutton1998reinforcement]. Another approach for solving this problem is [*Q-learning*]{}, a type of RL algorithm that obtains the [*Q-value*]{} for any $s \in S$ and $a = \pi(s)$, i.e. $Q(s,a) = \mathbb{E} \left[r_{t} + \gamma r_{t+1} + \gamma^2 r_{t+2} + \dots \mid s_t = s, a_t = a; \pi \right].$ The Q-learning approach starts with an initial guess for $Q(s,a)$ for all $s$ and $a$ and then proceeds to update them based on the iterative formula $$\label{eq:q-learning} Q(s_t,a_t) = (1- \alpha_t)Q(s_t,a_t) + \alpha_t \left( r_{t+1} + \gamma \max \limits_{a} Q(s_{t+1},a)\right), \forall t = 1,2,\dots,$$ where $\alpha_t$ is the learning rate at time step $t$. In each observed state, the agent chooses an action through an $\epsilon$-greedy algorithm: with probability $\epsilon_t$ in time $t$, the algorithm chooses an action randomly, and with probability $1-\epsilon_t$, it chooses the action with the highest cumulative action value, i.e., $a_{t+1} = \text{argmax}_{a} Q(s_{t+1},a)$. The random selection of actions, called exploration, allows the algorithm to explore the solution space and gives an optimality guarantee to the algorithm if $\epsilon_t \rightarrow 0$ when $t \rightarrow \infty$ [@sutton1998reinforcement]. After finding optimal $Q^*$, one can recover the optimal policy as $\pi^* (s) = \argmax_a Q^*(s,a)$. Both of the algorithms discussed so far (dynamic programming and Q-learning) guarantee that they will obtain the optimal policy. However, due to the curse of dimensionality, these approaches are not able to solve MDPs with large state or action spaces in reasonable amounts of time. Many problems of interest (including the beer game) have large state and/or action spaces. Moreover, in some settings (again, including the beer game), the decision maker cannot observe the full state variable. This case, which is known as a [*partially observed MDP*]{} (POMDP), makes the problem much harder to solve than MDPs. In order to solve large POMDPs and avoid the curse of dimensionality, it is common to approximate the Q-values in the Q-learning algorithm [@sutton1998reinforcement]. Linear regression is often used for this purpose [@melo2007q]; however, it is not powerful or accurate enough for our application. Non-linear functions and neural network approximators are able to provide more accurate approximations; on the other hand, they are known to provide unstable or even diverging Q-values due to issues related to non-stationarity and correlations in the sequence of observations [@mnih2013playing]. The seminal work of [@mnih2015human] solved these issues by proposing [*target networks*]{} and utilizing [*experience replay memory*]{} [@lin1992self]. They proposed a [*deep Q-network*]{} (DQN) algorithm, which uses a deep neural network to obtain an approximation of the Q-function and trains it through the iterations of the Q-learning algorithm while updating another target network. This algorithm has been applied to many competitive games, which are reviewed by [@li2017deep]. Our algorithm for the beer game is based on this approach. The beer game exhibits one characteristic that differentiates it from most settings in which DQN is commonly applied, namely, that there are multiple agents that cooperate in a decentralized manner to achieve a common goal. Such a problem is called a decentralized POMDP, or Dec-POMDP. Due to the partial observability and the non-stationarity of the local observations of agents, Dec-POMDPs are hard to solve and are categorized as NEXP-complete problems [@bernstein2002complexity]. The beer game exhibits all of the complicating characteristics described above—large state and action spaces, partial state observations, and decentralized cooperation. In the next section, we discuss the drawbacks of current approaches for solving the beer game, which our algorithm aims to overcome. Drawbacks of Current Algorithms {#sec:drawbacks} ------------------------------- In Section \[sec:Current\_State\_of\_Art\], we reviewed different approaches to solve the beer game. Although the model of [@clark1960optimal] can solve some types of serial systems, for more general serial systems neither the form nor the parameters of the optimal policy are known. Moreover, even in systems for which a base-stock policy is optimal, such a policy may no longer be optimal for a given agent if the other agents do not follow it. The formula-based beer-game models by @mosekilde1988deterministic [@sterman1989modeling], and @strozzi2007beer attempt to model human decision-making; they do not attempt to model or determine optimal decisions. A handful of models have attempted to optimize the inventory actions in serial supply chains with more general cost or demand structures than those used by [@clark1960optimal]; these are essentially beer-game settings. However, these papers all assume full observation or a centralized decision maker, rather than the local observations and decentralized approach taken in the beer game. For example, @kimbrough2002computers use a genetic algorithm (GA), while @chaharsooghi2008reinforcement [@giannoccaro2002inventory] and [@jiang2009case] use RL. However, classical RL algorithms can handle only a small or reduced-size state space. Accordingly, these applications of RL in the beer game or even simpler supply chain networks also assume (implicitly or explicitly) that size of the state space is small. This is unrealistic in the beer game, since the state variable representing a given agent’s inventory level can be any number in $(-\infty, +\infty)$. Solving such an RL problem would be nearly impossible, as the model would be extremely expensive to train. Moreover, [@chaharsooghi2008reinforcement] and [@giannoccaro2002inventory], which model beer-game-like settings, assume sharing of information. Also, to handle the curse of dimensionality, they propose mapping the state variable onto a small number of tiles, which leads to the loss of valuable state information and therefore of accuracy. Thus, although these papers are related to our work, their assumption of full observability differentiates their work from the classical beer game and from our paper. Another possible approach to tackle this problem might be classical supervised machine learning algorithms. However, these algorithms also cannot be readily applied to the beer game, since there is no historical data in the form of “correct” input/output pairs. Thus, we cannot use a stand-alone support vector machine or deep neural network with a training data-set and train it to learn the best action (like the approach used by [@oroojlooyjadid2016applying; @oroojlooyjadid2017stock] to solve some simpler supply chain problems). Based on our understanding of the literature, there is a large gap between solving the beer game problem effectively and what the current algorithms can handle. In order to fill this gap, we propose a variant of the DQN algorithm to choose the order quantities in the beer game. Our Contribution {#sec:beer_our_contribution} ---------------- We propose a Q-learning algorithm for the beer game in which a DNN approximates the Q-function. Indeed, the general structure of our algorithm is based on the DQN algorithm [@mnih2015human], although we modify it substantially, since DQN is designed for single-agent, competitive, zero-sum games and the beer game is a multi-agent, decentralized, cooperative, non-zero-sum game. In other words, DQN provides actions for one agent that interacts with an environment in a competitive setting, and the beer game is a cooperative game in the sense that all of the players aim to minimize the total cost of the system in a random number of periods. Also, beer game agents are playing independently and do not have any information from other agents until the game ends and the total cost is revealed, whereas DQN usually assumes the agent fully observes the state of the environment at any time step $t$ of the game. For example, DQN has been successfully applied to Atari games [@mnih2015human], but in these games the agent is attempting to defeat an opponent and observes full information about the state of the systems at each time step $t$. One naive approach to extend the DQN algorithm to solve the beer game is to use multiple DQNs, one for each agent. However, using DQN as the decision maker of each agent results in a competitive game in which each DQN agent plays independently to minimize its own cost. For example, consider a beer game in which players 2, 3, and 4 each have a stand-alone, well-trained DQN and the retailer (stage 1) uses a base-stock policy to make decisions. If the holding costs are positive for all players and the stockout cost is positive only for the retailer (as is common in the beer game), then the DQN at agents 2, 3, and 4 will return an optimal order quantity of 0 in every period, since on-hand inventory hurts the objective function for these players, but stockouts do not. This is a byproduct of the independent DQN agents minimizing their own costs without considering the total cost, which is obviously not an optimal solution for the system as a whole. Instead, we propose a unified framework in which the agents still play independently from one another, but in the training phase, we use a feedback scheme so that the DQN agent learns the total cost for the whole network and can, over time, learn to minimize it. Thus, the agents in our model play smartly in all periods of the game to get a near-optimal cumulative cost for any random horizon length. In principle, our framework can be applied to multiple DQN agents playing the beer game simultaneously on a team. However, to date we have designed and tested our approach only for a single DQN agent whose teammates are not DQNs, e.g., they are controlled by simple formulas or by human players. Enhancing the algorithm so that multiple DQNs can play simultaneously and cooperatively is a topic of ongoing research. Another advantage of our approach is that it does not require knowledge of the demand distribution, unlike classical inventory management approaches [e.g., @clark1960optimal]. In practice, one can approximate the demand distribution based on historical data, but doing so is prone to error, and basing decisions on approximate distributions may result in loss of accuracy in the beer game. In contrast, our algorithm chooses actions directly based on the training data and does not need to know, or estimate, the probability distribution directly. The proposed approach works very well when we tune and train the DQN for a given agent and a given set of game parameters (e.g., costs, lead times, action spaces, etc.). Once any of these parameters changes, or the agent changes, in principle we need to tune and train a new network. Although this approach works, it is time consuming since we need to tune hyper-parameters for each new set of game parameters. To avoid this, we propose using a [*transfer learning*]{} approach [@pan2010survey] in which we transfer the acquired knowledge of one agent under one set of game parameters to another agent with another set of game parameters. In this way, we decrease the required time to train a new agent by roughly one order of magnitude. To summarize, our algorithm is [*a variant of the DQN algorithm for choosing actions in the beer game*]{}. In order to attain near-optimal cooperative solutions, we develop [*a feedback scheme as a communication framework*]{}. Finally, to simplify training agents with new settings, we use [*transfer learning*]{} to efficiently make use of the learned knowledge of trained agents. In addition to playing the beer game well, we believe our algorithm serves as a proof-of-concept that DQN and other machine learning approaches can be used for real-time decision making in complex supply chain settings. Finally, we note that we have integrated our algorithm into a new online beer game developed by Opex Analytics (<http://beergame.opexanalytics.com/>); see Figure \[fig:opex\_BG\]. The Opex beer game allows human players to compete with, or play on a team with, our DQN agent. \[fig:opex\_BG\] {width="40.00000%"} The DQN Algorithm {#sec:beer_dqn_for_beer_game} ================= In this section, we first present the details of our DQN algorithm to solve the beer game, and then describe the transfer learning mechanism. DQN for the Beer Game --------------------- In our algorithm, a DQN agent runs a Q-learning algorithm with DNN as the Q-function approximator to learn a semi-optimal policy with the aim of minimizing the total cost of the game. Each agent has access to its local information and considers the other agents as parts of its environment. That is, the DQN agent does not know any information about the other agents, including both static parameters such as costs and lead times, as well as dynamic state variables such as inventory levels. We propose a feedback scheme to teach the DQN agent to work toward minimizing the total system-wide cost, rather than its own local cost. The details of the scheme, Q-learning, state and action spaces, reward function, DNN approximator, and the DQN algorithm are discussed below. [**State variables:**]{} Consider agent $i$ in time step $t$. Let $OO_t^i$ denote the on-order items at agent $i$, i.e., the items that have been ordered from agent $i+1$ but not received yet; let $AO_t^i$ denote the size of the arriving order (i.e., the demand) received from agent $i-1$; let $AS_t^i$ denote the size of the arriving shipment from agent $i+1$; let $a_t^i$ denote the action agent $i$ takes; and let $IL_t^i$ denote the inventory level as defined in Section \[sec:beer\_introduction\]. We interpret $AO_t^1$ to represent the end-customer demand and $AS_t^4$ to represent the shipment received by agent 4 from the external supplier. In each period $t$ of the game, agent $i$ observes $IL_t^i$, $OO_t^i$, $AO_t^i$, and $AS_t^i$. In other words, in period $t$ agent $i$ has historical observations $o^i_t =\left [((IL_1^i)^{+},IL_1^i)^{-},OO_1^i,AO_1^i,RS_1^i), \dots, ((IL_t^i)^{+},IL_t^i)^{-},,OO_t^i,AO_t^i,AS_t^i)\right].$ In addition, any beer game will finish in a finite time horizon, so the problem can be modeled as a POMDP in which each historic sequence $o^i_t$ is a distinct state and the size of the vector $o^i_t$ grows over time, which is difficult for any RL or DNN algorithm to handle. To address this issue, we capture only the last $m$ periods (e.g., $m=3$) and use them as the state variable; thus the state variable of agent $i$ in time $t$ is $s^i_t =\left[((IL_j^i)^{+},IL_j^i)^{-},OO_j^i,AO_j^i,RS_j^i)\right]_{j=t-m+1}^t.$ [**DNN architecture:**]{} In our algorithm, DNN plays the role of the Q-function approximator, providing the Q-value as output for any pair of state $s$ and action $a$. There are various possible approaches to build the DNN structure. The natural approach is to provide the state $s$ and action $a$ as the input of the DNN and then get the corresponding $Q(s,a)$ from the output. Thus, we provide as input the $m$ previous state variables into the DNN and get as output $Q(s,a)$ for every possible action $a \in \mathbb{A}$ (since in beer game $\mathbb{A}(s)$ is fixed for any $s$, we use $\mathbb{A}$ hereinafter). [**Action space:**]{} In each period of the game, each agent can order any amount in $[0,\infty)$. Since our DNN architecture provides the Q-value of all possible actions in the output, having an infinite action space is not practical. Therefore, to limit the cardinality of the action space, we use the $d+x$ rule for selecting the order quantity: The agent determines how much more or less to order than its received order; that is, the order quantity is $d+x$, where $x$ is in some bounded set. Thus, the output of the DNN is $x \in [a_l , a_u]$ ($a_l, a_u \in {\mathbb Z}$), so that the action space is of size $a_u-a_l + 1$. [**Experience replay:**]{} The DNN algorithm requires a mini-batch of input and a corresponding set of output values to learn the Q-values. Since we use DQN algorithm as our RL engine, we have the new state $s_{t+1}$, the current state $s_t$, the action $a_t$ taken, and the observed reward $r_t$, in each period $t$. This information can provide the required set of input and output for the DNN; however, the resulting sequence of observations from the RL results in a non-stationary data-set in which there is a strong correlation among consecutive records. This makes the DNN and, as a result, the RL prone to over-fitting the previously observed records and may even result in a diverging approximator [@mnih2015human]. To avoid this problem, we follow the suggestion of [@mnih2015human] and use [*experience replay*]{} [@lin1992self].In this way, agent $i$ has experience memory $E^i$ that in iteration $t$ of the algorithm, agent $i$’s observation $e^i_t=(s^i_t,a^i_t,r^i_t,s^i_{t+1})$ is added in, so that $E^i$ includes $\{e^i_1, e^i_2, \dots, e^i_t\}$ in period $t$. Then, in order to avoid having correlated observations, we select a random mini-batch of the agent’s experience replay to train the corresponding DNN (if applicable). This approach breaks the correlations among the training data and reduces the variance of the output [@mnih2013playing]. Moreover, as a byproduct of experience replay, we also get a tool to keep every piece of the valuable information, which allows greater efficiency in a setting in which the state and action spaces are huge and any observed experience is valuable. [**Reward function:**]{} In iteration $t$ of the game, agent $i$ observes state variable $s^i_t$ and takes action $a^i_t$; we need to know the corresponding reward value $r^i_t$ to measure the quality of action $a^i_t$. The state variable, $s^i_{t+1}$, allows us to calculate $IL^i_{t+1}$ and thus the corresponding shortage or holding costs, and we consider the summation of these costs for $r^i_t$. However, since there are information and transportation lead times, there is a delay between taking action $a^i_t$ and observing its effect on the reward. Moreover, the reward $r^i_t$ reflects not only the action taken in period $t$, but also those taken in previous periods, and it is not possible to decompose $r_t^i$ to isolate the effects of each of these actions. However, defining the state variable to include information from the last $m$ periods resolves this issue to some degree; the reward $r^i_t$ represents the reward of state $s^i_t$, which includes the observations of the previous $m$ steps. On the other hand, the reward values $r^i_t$ are the intermediate rewards of each agent, and the objective of the beer game is to minimize the total reward of the game, $\sum_{i=1}^{4} \sum_{t=1}^{T} r_t^i,$ which the agents only learn after finishing the game. In order to add this information into the agents’ experience, we use reward shaping through a feedback scheme. [**Feedback scheme:**]{} When any episode of the beer game is finished, all agents are made aware of the total reward. In order to share this information among the agents, we propose a penalization procedure in the training phase to provide feedback to the DQN agent about the way that it has played. Let $\omega=\sum_{i=1}^{4} \sum_{t=1}^{T} \frac{r_t^i}{T}$ and $\tau^i = \sum_{t=1}^{T} \frac{r_t^i}{T}$, i.e., the average reward per period and the average reward of agent $i$ per period, respectively. After the end of each episode of the game (i.e., after period $T$), for each DQN agent $i$ we update its observed reward in all $T$ time steps in the experience replay memory using $r_t^i = r_t^i + \frac{\beta_i}{3}(\omega - \tau^i)$, $\forall t \in \{1, \dots, T\}$, where $\beta_i$ is a regularization coefficient for agent $i$. With this procedure, agent $i$ gets appropriate feedback about its actions and learns to take actions that result in minimum total cost, not locally optimal solutions. [**Determining the value of [***m***]{}:**]{} As noted above, the DNN maintains information from the most recent $m$ periods in order to keep the size of the state variable fixed and to address the issue with the delayed observation of the reward. In order to select an appropriate value for $m$, one has to consider the value of the lead times throughout the game. First, when agent $i$ takes action $a^i_t$ at time $t$, it does not observe its effect until at least $l^{tr}_i + l^{in}_i$ periods later, when the order may be received. Moreover, node $i+1$ may not have enough stock to satisfy the order immediately, in which case the shipment is delayed and in the worst case agent $i$ will not observe the corresponding reward $r^i_t$ until $\sum_{j=i}^{4} (l^{tr}_j + l^{in}_j)$ periods later. However, one needs the reward $r_t^i$ to evaluate the action $a^i_t$ taken. Thus, ideally $m$ should be chosen at least as large as $\sum_{j=1}^{4} (l^{tr}_j + l^{in}_j)$. On the other hand, this value can be large and selecting a large value for $m$ results in a large input size for the DNN, which increases the training time. Therefore, selecting $m$ is a trade-off between accuracy and computation time, and $m$ should be selected according to the required level of accuracy and the available computation power. In our numerical experiment, $\sum_{j=1}^{4} (l^{tr}_j + l^{in}_j) = 15$ or $16$, and we test $m\in\{5,10\}$. [**The algorithm:**]{} Our algorithm to get the policy $\pi$ to solve the beer game is provided in Algorithm \[alg:dqn\_beer\_game\]. The algorithm, which is based on that of [@mnih2015human], finds weights $\theta$ of the DNN network to minimize the Euclidean distance between $Q(s,a;\theta)$ and $y_j$, where $y_j$ is the prediction of the Q-value that is obtained from target network $Q^{-}$ with weights $\theta^{-}$. Every $C$ iterations, the weights $\theta^{-}$ are updated by $\theta$. Moreover, the actions in each training step of the algorithm are obtained by an $\epsilon$-greedy algorithm, which is explained in Section \[sec:RL\]. Initialize Experience Replay Memory $E_i = \left[~\right],~ \forall i$ Reset $IL$, $OO$, $d$, $AO$, and $AS$ for each agent $\text{With probability } \epsilon \text{ take random action } a_t$, otherwise set $a_t = \underset{a}{\text{argmin }} Q \left(s_t,a;\theta \right)$ Execute action $a_t$, observe reward $r_t$ and state $s_{t+1}$ Add $(s^i_t,a^i_t,r^i_t,s^i_{t+1})$ into the $E_i$ Get a mini-batch of experiences $(s_j,a_j,r_j,s_{j+1})$ from $E_i$ Set $y_j = \begin{cases} r_j & \text{if it is the terminal state} \\ r_j + \min \limits_{a} Q(s_j,a;\theta^{-}) & \text{otherwise} \end{cases}$ Run forward and backward step on the DNN with loss function $\left (y_j - Q \left (s_j,a_j;\theta \right) \right)^2$ Every $C$ iterations, set $\theta^{-} = \theta$ Run feedback scheme, update experience replay of each agent In the algorithm, in period $t$ agent $i$ takes action $a^i_t$, satisfies the arriving demand/order $AO^i_{t-1}$, observes the new demand $AO^i_t$, and then receives the shipments $AS^i_t$. This sequence of events, which is explained in detail in online supplement \[sec:appnd:sudocode\], results in the new state $s_{t+1}$. Feeding $s_{t+1}$ into the DNN network with weights $\theta$ provides the corresponding Q-value for state $s_{t+1}$ and all possible actions. The action with the smallest Q-value is our choice. Finally, at the end of each episode, the feedback scheme runs and distributes the total cost among all agents. [**Evaluation procedure:**]{} In order to validate our algorithm, we compare the results of our algorithm to those obtained using the optimal base-stock levels (when possible) in serial systems by [@clark1960optimal], as well as models of human beer-game behavior by [@sterman1989modeling]. (Note that none of these methods attempts to do exactly the same thing as our method. The methods by [@clark1960optimal] optimizes the base-stock levels assuming all players follow a base-stock policy—which beer game players do not tend to do—and the formula by [@sterman1989modeling] models human beer-game play, but they do not attempt to optimize.) The details of the training procedure and benchmarks are described in Section \[sec:beer\_numerical\_experiment\]. Transfer Learning {#sec:transfer_learning} ----------------- Transfer learning [@pan2010survey] has been an active and successful field of research in machine learning and especially in image processing. In transfer learning, there is a [*source*]{} dataset [S]{} and a trained neural network to perform a given task, e.g. classification, regression, or decisioning through RL. Training such networks may take a few days or even weeks. So, for similar or even slightly different [*target*]{} datasets [T]{}, one can avoid training a new network from scratch and instead use the same trained network with a few customizations. The idea is that most of the learned knowledge on dataset [S]{} can be used in the target dataset with a small amount of additional training. This idea works well in image processing (e.g. [@rajpurkar2017chexnet]) and considerably reduces the training time. In order to use transfer learning in the beer game, assume there exists a source agent $i \in \{1,2,3,4\}$ with trained network $S_i$ (with a fixed size on all agents), parameters $P_1^i = \{|\mathbb{A}_1^j|, c^j_{p_1}, c^j_{h_1}\}$, observed demand distribution $D_1$, and co-player policy $\pi_1$. The weight matrix $W_i$ contains the learned weights such that $W_i^q$ denotes the weight between layers $q$ and $q+1$ of the neural network, where $q \in \{0,\dots,nh\}$, and $nh$ is the number of hidden layers. The aim is to train a neural network $S_j$ for target agent $j \in \{1,2,3,4\}$, $j\ne i$. We set the structure of the network $S_j$ the same as that of $S_i$, and initialize $W_j$ with $W_i$, making the first $k$ layers not trainable. Then, we train neural network $S_j$ with a small learning rate. Note that, as we get closer to the final layer, which provides the Q-values, the weights become less similar to agent $i$’s and more specific to each agent. Thus, the acquired knowledge in the first $k$ hidden layer(s) of the neural network belonging to agent $i$ is transferred to agent $j$, in which $k$ is a tunable parameter. Following this procedure, in Section \[sec:results:transfer\_learning\], we test the use of transfer learning in six cases to transfer the learned knowledge of source agent $i$ to: - Target agent $j \neq i$ in the same game. - Target agent $j$ with $\{|\mathbb{A}_1^j|, c^j_{p_2}, c^j_{h_2}\}$, i.e., the same action space but different cost coefficients. - Target agent $j$ with $\{|\mathbb{A}_2^j|, c^j_{p_1}, c^j_{h_1}\}$, i.e., the same cost coefficients but different action space. - Target agent $j$ with $\{|\mathbb{A}_2^j|, c^j_{p_2}, c^j_{h_2}\}$, i.e., different action space and cost coefficients. - Target agent $j$ with $\{|\mathbb{A}_2^j|, c^j_{p_2}, c^j_{h_2}\}$, i.e., different action space and cost coefficients, as well as a different demand distribution $D_2$. - Target agent $j$ with $\{|\mathbb{A}_2^j|, c^j_{p_2}, c^j_{h_2}\}$, i.e., different action space and cost coefficients, as well as a different demand distribution $D_2$ and co-player policy $\pi_2$. Unless stated otherwise, the demand distribution and co-player policy are the same for the source and target agents. Transfer learning could also be used when other aspects of the problem change, e.g., lead times, state representation, and so on. This avoids having to tune the parameters of the neural network for each new problem, which considerably reduces the training time. However, we still need to decide how many layers should be trainable, as well as to determine which agent can be a base agent for transferring the learned knowledge. Still, this is computationally much cheaper than finding each network and its hyper-parameters from scratch. Numerical Experiments {#sec:beer_numerical_experiment} ===================== In Section \[sec:uniform\_0\_2\_result\], we discuss a set of numerical experiments that uses a simple demand distribution and a relatively small action space: - $d_0^t \in \mathbb{U}[0,2]$, $\mathbb{A} = \{-2,-1,0,1,2\}$. After exploring the behavior of our algorithm under different co-player policies, in Section \[sec:uniform8\_normal\_classic\_result\] we test the algorithm using three well-known cases from the literature, which have larger possible demand values and action spaces: - $d_0^t \in \mathbb{U}[0,8]$, $\mathbb{A} = \{-8,\dots, 8\}$ [@croson2006behavioral] - $d_0^t \in \mathbb{N}(10,2^2)$, $\mathbb{A} = \{-5,\dots, 5\}$ [adapted from @chen2000stationary, who assume $\mathbb{N}(50,20^2)$] - $d_0^t \in C(4,8)$, $\mathbb{A} = \{-8,\dots, 8\}$ [@sterman1989modeling]. As noted above, we only consider cases in which a single DQN plays with non-DQN agents, e.g., simulated human co-players. In each of the cases listed above, we consider three types of policies that the non-DQN co-players follow: (i) base-stock policy, (ii) Sterman formula, (iii) random policy. In the random policy, agent $i$ also follows a $d+x$ rule, in which $a_i^t \in \mathbb{A}$ is selected randomly and with equal probability, for each $t$. After analyzing these cases, in Section \[sec:results:transfer\_learning\] we provide the results obtained using transfer learning for each of the six proposed cases. We test values of $m$ in $\{5,10\}$ and $C \in \{5000, 10000\}$. Our DNN network is a fully connected network, in which each node has a ReLU activation function. The input is of size $5m$, and there are three hidden layers in the neural network. There is one output node for each possible value of the action, and each of these nodes takes a value in ${\mathbb R}$ indicating the Q-value for that action. Thus, there are $a_u - a_l + 1$ output nodes, and the neural network has shape $[5m, 180, 130, 61, a_u - a_l + 1]$. In order to optimize the network, we used the Adam optimizer [@kingma2014adam] with a batch size of $64$. Although the Adam optimizer has its own weight decaying procedure, we used exponential decay with a stair of $10000$ iterations with rate $0.98$ to decay the learning rate further. This helps to stabilize the training trajectory. We trained each agent on at most 60000 episodes and used a replay memory $E$ equal to the one million most recently observed experiences. Also, the training of the DNN starts after observing at least $500$ episodes of the game. The $\epsilon$-greedy algorithm starts with $\epsilon = 0.9$ and linearly reduces it to $0.1$ in the first $80\%$ of iterations. In the feedback mechanism, the appropriate value of the feedback coefficient $\beta_i$ heavily depends on $\tau_j$, the average reward for agent $j$, for each $j\neq i$. For example, when $\tau_i$ is one order of magnitude larger than $\tau_j$, for all $j\ne i$, agent $i$ needs a large coefficient to get more feedback from the other agents. Indeed, the feedback coefficient has a similar role as the regularization parameter $\lambda$ has in the lasso loss function; the value of that parameter depends on the $\ell$-norm of the variables, but there is no universal rule to determine the best value for $\lambda$. Similarly, proposing a simple rule or value for each $\beta_i$ is not possible, as it depends on $\tau_i$, $\forall i$. For example, we found that a very large $\beta_i$ does not work well, since the agent tries to decrease other agents’ costs rather than its own. Similarly, with a very small $\beta_i$, the agent learns how to minimize its own cost instead of the total cost. Therefore, we used a similar cross validation approach to find good values for each $\beta_i$. Basic Cases {#sec:uniform_0_2_result} ----------- In this section, we test our approach using a beer game setup with the following characteristics. Information and shipment lead times, $l^{tr}_j$ and $l^{in}_j$, equal 2 periods at every agent. Holding and stockout costs are given by $c_h=[2,2,2,2]$ and $c_p = [2,0,0,0]$, respectively, where the vectors specify the values for agents $1,\ldots,4$. The demand is an integer uniformly drawn from $\{0,1,2\}$. Additionally, we assume that agent $i$ observes the arriving shipment $AS_{t}^i$ when it chooses its action for period $t$. We relax this assumption later. We use $a_l = -2$ and $a_u =2$; so that there are 5 outputs in the neural network. i.e., each agent chooses an order quantity that is at most 2 units greater or less than the observed demand. (Later, we expand these to larger action spaces.) We consider two types of simulated human players. In Section \[sec:results:dnn\_vs\_base\_stock\], we discuss results for the case in which one DQN agent plays on a team in which the other three players use a base-stock policy to choose their actions, i.e., the non-DQN agents behave rationally. See [<https://youtu.be/gQa6iWGcGWY>](https://youtu.be/gQa6iWGcGWY) for a video animation of the policy that the DQN learns in this case. Then, in Section \[sec:results:dnn\_vs\_formula\], we assume that the other three agents use the Sterman formula (i.e., the formula by [@sterman1989modeling]), which models irrational play. For the cost coefficients and other settings specified for this beer game, it is optimal for all players to follow a base-stock policy, and we use this policy as a benchmark and a lower bound on the base stock cost. The vector of optimal base-stock levels is $[8,8,0,0]$, and the resulting average cost per period is $2.0705$, though these levels may be slightly suboptimal due to rounding. This cost is allocated to stages 1–4 as $[2.0073, 0.0632, 0.03, 0.00]$,\[opt\_cost\]. In the experiments in which one of the four agents is played by DQN, the other three agents continue to use their optimal base-stock levels. ### DQN Plus Base-Stock Policy {#sec:results:dnn_vs_base_stock} We consider four cases, with the DQN playing the role of each of the four players and the co-players using a base-stock policy. We then compare the results of our algorithm with the results of the case in which all players follow a base-stock policy, which we call [BS]{} hereinafter. The results of all four cases are shown in Figure \[fig:dqn\_vs\_three\_optimal\]. Each plot shows the training curve, i.e., the evolution of the average cost per game as the training progresses. In particular, the horizontal axis indicates the number of training episodes, while the vertical axis indicates the total cost per game. After every 100 episodes of the game and the corresponding training, the cost of 50 validation points (i.e., 50 new games) each with 100 periods, are obtained and their average plus a 95% confidence interval are plotted. (The confidence intervals, which are light blue in the figure, are quite narrow, so they are difficult to see.) The red line indicates the cost of the case in which all players follow a base-stock policy. In each of the sub-figures, there are two plots; the upper plot shows the cost, while the lower plot shows the normalized cost, in which each cost is divided by the corresponding [BS]{} cost; essentially this is a “zoomed-in” version of the upper plot. We trained the network using values of $\beta \in \{5, 10, 20, 50, 100, 200\}$, each for at most 60000 episodes. Figure \[fig:dqn\_vs\_three\_optimal\] plots the results from the best $\beta_i$ value for each agent; we present the full results using different $\beta_i, m$ and $C$ values in Section \[sec:appdx:base\_stock\_beta\] of the online supplement. The figure indicates that DQN performs well in all cases and finds policies whose costs are close to those of the [BS]{} policy. After the network is fully trained (i.e., after 60000 training episodes), the average gap between the DQN cost and the [BS]{} cost, over all four agents, is 2.31%. [0.47]{} ![DQN plays manufacturer \[fig:4Agent:vs\_optm:A\_DNN\_Manufacturer\]](3_10_10000_40.png) [0.47]{} ![DQN plays manufacturer \[fig:4Agent:vs\_optm:A\_DNN\_Manufacturer\]](4_10_10000_10.png) [0.47]{} ![DQN plays manufacturer \[fig:4Agent:vs\_optm:A\_DNN\_Manufacturer\]](5_10_10000_5.png) [0.47]{} ![DQN plays manufacturer \[fig:4Agent:vs\_optm:A\_DNN\_Manufacturer\]](6_10_10000_20.png) Figure \[fig:4Agent:vs\_optm:play\_DNN\_Retailer\] shows the trajectories of the retailer’s inventory level ($IL$), on-order quantity ($OO$), order quantity ($a$), reward ($r$), and order up to level (OUTL) for a single game, when the retailer is played by the DQN with $\beta_1=50$, as well as when it is played by a base-stock policy ([BS]{}), and the Sterman formula ([Strm]{}). The base-stock policy and DQN have similar $IL$ and $OO$ trends, and as a result their rewards are also very close: [BS]{} has a cost of $[1.42, 0.00, 0.02, 0.05]$ (total 1.49) and DQN has $[1.43, 0.01, 0.02, 0.08]$ (total 1.54, or 3.4% larger). (Note that [BS]{} has a slightly different cost here than reported on page  because those costs are the average costs of 50 samples while this cost is from a single sample.) Similar trends are observed when the DQN plays the other three roles; see Section \[sec:appd:more\_play\_results\] of the online supplement. This suggests that the DQN can successfully learn to achieve costs close to [BS]{} when the other agents also play [BS]{}. (The OUTL plot shows that the DQN does not quite [*follow*]{} a base-stock policy, even though its costs are similar.) ### DQN Plus Sterman Formula {#sec:results:dnn_vs_formula} Figure \[fig:dqn\_vs\_three\_formula\] shows the results of the case in which the three non-DQN agents use the formula proposed by [@sterman1989modeling] instead of a base-stock policy. (See Section \[sec:appdx:sterman parameters\] of online supplement for the formula and its parameters.) For comparison, the red line represents the case in which the single agent is played using a base-stock policy and the other three agents continue to use the Sterman formula, a case we call [Strm-BS]{}. From the figure, it is evident that the DQN plays much [*better*]{} than [Strm-BS]{}. This is because if the other three agents do not follow a base-stock policy, it is no longer optimal for the fourth agent to follow a base-stock policy, or to use the same base-stock level. In general, the optimal inventory policy when other agents do not follow a base-stock policy is an open question. This figure suggests that our DQN is able to learn to play effectively in this setting. Table \[tb:4Agent:vs\_frmu:dqn\_result\] gives the cost of all four agents when a given agent plays using either DQN or a base-stock policy and the other agents play using the Sterman formula. From the table, we can see that DQN learns how to play to decrease the costs of the other agents, and not just its own costs—for example, the retailer’s and warehouse’s costs are significantly lower when the distributor uses DQN than they are when the distributor uses a base-stock policy. Similar conclusions can be drawn from Figure \[fig:dqn\_vs\_three\_formula\]. This shows the power of DQN when it plays with co-player agents that do not play rationally, i.e., do not follow a base-stock policy, which is common in real-world supply chains. Also, we note that when all agents follow the Sterman formula, the average cost of the agents is \[10.81, 10.76, 10.96, 12.6\], for a total of 45.13, much higher than when any one agent uses DQN. Finally, for details on $IL,OO,a,r,$ and [OUTL]{} on this case see Section \[sec:appd:more\_play\_results\] of the online supplement. -------------- --------------- ---------------- ---------------- ---------------- ---------------- DQN Agent Retailer Warehouse Distributer Manufacturer Total Retailer (0.89, 1.89) (10.87, 10.83) (10.96, 10.98) (12.42, 12.82) (35.14, 36.52) Warehouse (1.74, 9.99) (0.00, 0.13) (11.12, 10.80) (12.86, 12.34) (25.72, 33.27) Distributer (5.60, 10.72) (0.11, 9.84) (0.00, 0.14) (12.53, 12.35) (18.25, 33.04) Manufacturer (4.68, 10.72) (1.72, 10.60) (0.24, 10.13) (0.00, 0.07) (6.64, 31.52) -------------- --------------- ---------------- ---------------- ---------------- ---------------- : Average cost under different choices of which agent uses DQN or [Strm-BS]{}. \[tb:4Agent:vs\_frmu:dqn\_result\] [0.47]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_frmu:A_DNN_Manufacturer"}](7.png) [0.47]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_frmu:A_DNN_Manufacturer"}](8.png) [0.47]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_frmu:A_DNN_Manufacturer"}](9.png) [0.47]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_frmu:A_DNN_Manufacturer"}](10.png) Literature Benchmarks {#sec:uniform8_normal_classic_result} --------------------- We next test our approach on beer game settings from the literature. These have larger demand-distribution domains, and therefore larger plausible action spaces, and thus represent harder instances to train the DQN for. In all instances in this section, $l^{in}=[2,2,2,2]$ and $l^{tr}=[2,2,2,1]$. Shortage and holding cost coefficients and the base-stock levels for each instance are presented in Table \[tb:experimental\_parameters\]. Note that the Clark–Scarf algorithm assumes that stage 1 is the only stage with non-zero stockout costs, whereas the $\mathbb{U}[0,8]$ instance has non-zero costs at every stage. Therefore, we used a heuristic approach based on a two-moment approximation, similar to that proposed by [@Gr85], to choose the base-stock levels; see [@snyder2018meio]. In addition, the $C(4,8)$ demand process is non-stationary—4, then 8—but we allow only stationary base-stock levels. Therefore, we chose to set the base-stock levels equal to the values that would be optimal if the demand were 8 in every period. Finally, in the experiments in this section, we assume that agent $i$ observes $AS_{t}^i$ [*after*]{} choosing $a_{t}^i$, whereas in Section \[sec:uniform\_0\_2\_result\] we assumed the opposite. Therefore, the agents in these experiments have one fewer piece of information when choosing actions, and are therefore more difficult to train. demand $c_p$ $c_h$ BS level ---------------------- ---------------------- ------------------------- ----------------- $\mathbb{U}[0,8]$ \[1.0,1.0,1.0,1.0\] \[0.50,0.50,0.50,0.50\] \[19,20,20,14\] $\mathbb{N}(10,2^2)$ \[10.0,0.0,0.0,0.0\] \[1.00,0.75,0.50,0.25\] \[48,43,41,30\] $C(4,8)$ \[1.0,1.0,1.0,1.0\] \[0.50,0.50,0.50,0.50\] \[32,32,32,24\] : Cost parameters and base-stock levels for instances with uniform, normal, and classic demand distributions.[]{data-label="tb:experimental_parameters"} Tables \[tb:base\_stock\_results\], \[tb:sterman\_results\], and \[tb:random\_results\] show the results of the cases in which the DQN agent plays with co-players who follow base-stock, Sterman, and random policies, respectively. In each group of columns, the first column (“DQN”) gives the average cost (over 50 instances) when one agent (indicated by the first column in the table) is played by the DQN and co-players are played by base-stock (Table \[tb:base\_stock\_results\]), Sterman (Table \[tb:sterman\_results\]), or random (Table \[tb:random\_results\]) agents. The second column in each group (“[BS]{}”, “[Strm-BS]{}”, “[Rand-BS]{}”) gives the corresponding cost when the DQN agent is replaced by a base-stock agent (using the base-stock levels given in Table \[tb:experimental\_parameters\]) and the co-players remain as in the previous column. The third column (“Gap”) gives the percentage difference between these two costs. As Table \[tb:base\_stock\_results\] shows, when the DQN plays with base-stock co-players under uniform or normal demand distributions, it obtains costs that are reasonably close to the case when all players use a base-stock policy, with average gaps of 12.58% and 5.80%, respectively. These gaps are not quite as small as those in Section \[sec:uniform\_0\_2\_result\], due to the larger action spaces in the instances in this section. Since a base-stock policy is optimal at every stage, the small gaps demonstrate that the DQN can learn to play the game well for these demand distributions. For the classic demand process, the percentage gaps are larger. To see why, note that if the demand were to equal 8 in every period, the base-stock levels for the classic demand process will result in ending inventory levels of 0 at every stage. The four initial periods of demand equal to 4 disrupt this effect slightly, but the cost of the optimal base-stock policy for the classic demand process is asymptotically 0 as the time horizon goes to infinity. The absolute gap attained by the DQN is quite small—an average of 0.49 vs. 0.34 for the base-stock cost—but the percentage difference is large simply because the optimal cost is close to 0. Indeed, if we allow the game to run longer, the cost of both algorithms decreases, and so does the absolute gap. For example, when the DQN plays the retailer, after 500 periods the discounted costs are 0.0090 and 0.0062 for DQN and [BS]{}, respectively, and after 1000 periods, the costs are 0.0001 and 0.0000 (to four-digit precision). -- -------- -------- --------- -------- -------- --------- ------- -------- --------- DQN [BS]{} Gap (%) DQN [BS]{} Gap (%) DQN [BS]{} Gap (%) 904.88 799.20 13.22 881.66 838.14 5.19 0.50 0.34 45.86 960.44 799.20 20.18 932.65 838.14 11.28 0.47 0.34 36.92 903.49 799.20 13.05 880.40 838.14 5.04 0.67 0.34 96.36 830.16 799.20 3.87 852.33 838.14 1.69 0.30 0.34 -13.13 12.58 5.80 41.50 -- -------- -------- --------- -------- -------- --------- ------- -------- --------- : Results of DQN playing with co-players who follow base-stock policy. []{data-label="tb:base_stock_results"} Similar to the results of Section \[sec:results:dnn\_vs\_formula\], when the DQN plays with co-players who follow the Sterman formula, it performs far better than [Strm-BS]{}. As Table \[tb:sterman\_results\] shows, DQN performs 34% better than [Strm-BS]{} on average. Finally, when DQN plays with co-players who use the random policy, for all demand distributions DQN learns very well to play so as to minimize the total cost of the system, and on average obtains 8% better solutions than [Rand-BS]{}. -- -------- ------------- --------- -------- ------------- --------- -------- ------------- --------- DQN [Strm-BS]{} Gap (%) DQN [Strm-BS]{} Gap (%) DQN [Strm-BS]{} Gap (%) 6.88 8.99 -23.45 9.98 10.67 -6.44 3.80 13.28 -71.41 5.90 9.53 -38.10 7.11 10.03 -29.06 2.85 8.17 -65.08 8.35 10.99 -23.98 8.49 13.83 -38.65 3.82 20.07 -80.96 12.36 13.90 -11.07 13.86 15.37 -9.82 15.80 19.96 -20.82 -24.15 -20.99 -59.57 -- -------- ------------- --------- -------- ------------- --------- -------- ------------- --------- : Results of DQN playing with co-players who follow Sterman policy. []{data-label="tb:sterman_results"} -- ------- ------------- --------- -------- ------------- --------- ------- ------------- --------- DQN [Rand-BS]{} Gap (%) DQN [Rand-BS]{} Gap (%) DQN [Rand-BS]{} Gap (%) 31.39 28.24 11.12 13.03 28.39 -54.10 19.99 25.88 -22.77 29.62 28.62 3.49 27.87 35.80 -22.15 23.05 23.44 -1.65 30.72 28.64 7.25 34.85 38.79 -10.15 22.81 23.53 -3.04 29.03 28.13 3.18 37.68 40.53 -7.02 22.36 22.45 -0.42 6.26 -23.36 -6.97 -- ------- ------------- --------- -------- ------------- --------- ------- ------------- --------- : Results of DQN playing with co-players who follow random policy. []{data-label="tb:random_results"} To summarize, DQN does well regardless of the way the other agents play, and regardless of the demand distribution. The DQN agent learns to attain near-[BS]{} costs when its co-players follow a [BS]{} policy, and when playing with irrational co-players, it achieves a much smaller cost than a base-stock policy would. Thus, when the other agents play irrationally, DQN should be used. Faster Training through Transfer Learning {#sec:results:transfer_learning} ----------------------------------------- We trained a DQN network with shape $[50, 180, 130, 61, 5]$, $m=10$, $\beta=20$, and $C=10000$ for each agent, with the same holding and stockout costs and action spaces as in section \[sec:uniform\_0\_2\_result\], using 60000 training episodes, and used these as the base networks for our transfer learning experiment. (In transfer learning, all agents should have the same network structure to share the learned network among different agents.) The remaining agents use a [BS]{} policy. Table \[tb:transfer\_learning\_result\] shows a summary of the results of the six cases discussed in Section \[sec:transfer\_learning\]. The first set of columns indicates the holding and shortage cost coefficients, the size of the action space, as well as the demand distribution and the co-players’ policy for the base agent (first row) and the target agent (remaining rows). The “Gap” column indicates the average gap between the cost of the resulting DQN and the cost of a [BS]{} policy; in the first row, it is analogous to the 2.31% average gap reported in Section \[sec:results:dnn\_vs\_base\_stock\]. The average gap is relatively small in all cases, which shows the effectiveness of the transfer learning approach. Moreover, this approach is efficient, as demonstrated in the last column, which reports the average CPU times for all agent. In order to get the base agents, we did hyper-parameter tuning and trained 140 instances to get the best possible set of hyper-parameters, which resulted in a total of 28,390,987 seconds of training. However, using the transfer learning approach, we do not need any hyper-parameter tuning; we only need to check which source agent and which $k$ provides the best results. This requires only 12 instances to train and resulted in an average training time (across case 1-4) of 1,613,711 seconds—17.6 times faster than training the base agent. Additionally, in case 5, in which a normal distribution is used, full hyper-parameter tuning took 20,396,459 seconds, with an average gap of 4.76%, which means transfer learning was 16.6 times faster on average. We did not run the full hyper-parameter tuning for the instances of case-6, but it is similar to that of case-5 and should take similar training time, and as a result a similar improvement from transfer learning. Thus, once we have a trained agent $i$ with a given set $P_1^i$ of parameters, demand $D_1$ and co-players’ policy $\pi_1$, we can efficiently train a new agent $j$ with parameters $P_2^j$, demand $D_2$ and co-players’ policy $\pi_2$. -- -------- ---------- --------- ---------- ----- ----------------------- ---------- -------- ------------ Gap CPU Time (%) (sec) (2,2) (2,0) (2,0) (2,0) 5 ${\mathbb U}[0,2]$ [BS]{} 2.31 28,390,987 (2,2) (2,0) (2,0) (2,0) 5 ${\mathbb U}[0,2]$ [BS]{} 6.06 1,593,455 (5,1) (5,0) (5,0) (5,0) 5 ${\mathbb U}[0,2]$ [BS]{} 6.16 1,757,103 (2,2) (2,0) (2,0) (2,0) 11 ${\mathbb U}[0,2]$ [BS]{} 10.66 1,663,857 (10,1) (10,0) (10,0) (10,0) 11 ${\mathbb U}[0,2]$ [BS]{} 12.58 1,593,455 (1,10) (0.75,0) (0.5,0) (0.25,0) 11 ${\mathbb N}(10,2^2)$ [BS]{} 17.41 1,234,461 (1,10) (0.75,0) (0.5,0) (0.25,0) 11 ${\mathbb N}(10,2^2)$ [Strm]{} -38.20 1,153,571 (1,10) (0.75,0) (0.5,0) (0.25,0) 11 ${\mathbb N}(10,2^2)$ [Rand]{} -0.25 1,292,295 -- -------- ---------- --------- ---------- ----- ----------------------- ---------- -------- ------------ : Results of transfer learning when $\pi_1$ is [BS]{} and $D_1$ is $\mathbb{U}[0,2]$[]{data-label="tb:transfer_learning_result"} In order to get more insights about the transfer learning process, Figure \[fig:agents\_cp10ch1\_action5\] shows the results of case 4, which is a quite complex transfer learning case that we test for the beer game. The target agents have holding and shortage costs (10,1), (10,0), (10,0), and (10,0) for agents 1 to 4, respectively; and each agent can select any action in $\{-5,\dots,5\}$. Each caption reports the base agent (shown by [b]{}) and the value of $k$ used. Compared to the original procedure (see Figure \[fig:dqn\_vs\_three\_optimal\]), i.e., $k=0$, the training is less noisy and after a few thousand non-fluctuating training episodes, it converges into the final solution. The resulting agents obtain costs that are close to those of [BS]{}, with a $12.58\%$ average gap compared to the [BS]{} cost. (The details of the other cases are provided in Sections \[sec:results:transfer\_learning:same\_agents\]—\[sec:results:transfer\_learning:case\_6\] of the online supplement.) [0.47]{} ![Target agent = manufacturer ($b=4,k=2$)[]{data-label="fig:agent_cp10ch1_action5_6-6-2"}](3-5-1.png "fig:") [0.47]{} ![Target agent = manufacturer ($b=4,k=2$)[]{data-label="fig:agent_cp10ch1_action5_6-6-2"}](4-3-1.png "fig:") [0.47]{} ![Target agent = manufacturer ($b=4,k=2$)[]{data-label="fig:agent_cp10ch1_action5_6-6-2"}](5-5-2.png "fig:") [0.47]{} ![Target agent = manufacturer ($b=4,k=2$)[]{data-label="fig:agent_cp10ch1_action5_6-6-2"}](6-6-2.png "fig:") Finally, Table \[tb:transfer\_learning\_comparisons\] explores the effect of $k$ on the tradeoff between training speed and solution accuracy. As $k$ increases, the number of trainable variables decreases and, not surprisingly, the CPU times are smaller but the costs are larger. For example, when $k=3$, the training time is 46.89% smaller than the training time when $k=0$, but the solution cost is 17.66% and 0.34% greater than the [BS]{} policy, compared to 4.22% and -11.65% for $k=2$. $k=0$ $k=1$ $k=2$ $k=3$ ------------------------------------ --------- --------- --------- --------- Training time 185,679 126,524 118,308 107,711 Decrease in time compared to $k=0$ — 37.61% 41.66% 46.89% Average gap in cases 1–4 2.31% 4.39% 4.22% 17.66% Average gap in cases 1–6 — -15.95% -11.65% 0.34% : Savings in computation time due to transfer learning. First row provides average training time among all instances. Third row provides average of the best obtained gap in cases for which an optimal solution exists. Fourth row provides average gap among all transfer learning instances, i.e., cases 1–6. \[tb:transfer\_learning\_comparisons\] To summarize, transferring the acquired knowledge between the agents is very efficient. The target agents achieve costs that are close to those of the [BS]{} policy (when co-players follow [BS]{}) and they achieve smaller costs than [Strm-BS]{} and [Rand-BS]{}, regardless of the dissimilarities between the source and the target agents. The training of the target agents start from relatively small cost values, the training trajectories are stable and fairly non-noisy, and they quickly converge to a cost value close to that of the [BS]{} policy or smaller than [Strm-BS]{} and [Rand-BS]{}. Even when the action space and costs for the source and target agents are different, transfer learning is still quite effective, resulting in a 12.58% gap compared to the [BS]{} policy. This is an important result, since it means that if the settings change—either within the beer game or in real supply chain settings—we can train new DQN agents much more quickly than we could if we had to begin each training from scratch. Conclusion and Future Work {#sec:beer_conclusion} ========================== In this paper, we consider the beer game, a decentralized, multi-agent, cooperative supply chain problem. A base-stock inventory policy is known to be optimal for special cases, but once some of the agents do not follow a base-stock policy (as is common in real-world supply chains), the optimal policy of the remaining players is unknown. To address this issue, we propose an algorithm based on deep Q-networks. It obtains near-optimal solutions when playing alongside agents who follow a base-stock policy and performs much better than a base-stock policy when the other agents use a more realistic model of ordering behavior. Furthermore, the algorithm does not require knowledge of the demand probability distribution and uses only historical data. To reduce the computation time required to train new agents with different cost coefficients or action spaces, we propose a transfer learning method. Training new agents with this approach takes less time since it avoids the need to tune hyper-parameters and has a smaller number of trainable variables. Moreover, it is quite powerful, resulting in beer game costs that are close to those of fully-trained agents while reducing the training time by an order of magnitude. A natural extension of this paper is to apply our algorithm to supply chain networks with other topologies, e.g., distribution networks. Another important extension is having multiple learnable agents. Finally, developing algorithms capable of handling continuous action spaces will improve the accuracy of our algorithm. [ ]{} Sterman Formula Parameters {#sec:appdx:sterman parameters} ========================== The computational experiments that use [Strm]{} agents calculate the order quantity using formula , adapted from [@sterman1989modeling]. $$\begin{split} & q^i_t = \max \{0, AO^{i-1}_{t+1} + \alpha^i (IL^i_t - a^i) + \beta^i (OO_t^i - b^i)\} \\ \end{split} \label{eq:sterman_formula}$$ where $\alpha^i$, $a^i$, $\beta^i$, and $b^i$ are the parameters corresponding to the inventory level and on-order quantity. The idea is that the agent sets the order quantity equal to the demand forecast plus two terms that represent adjustments that the agent makes based on the deviations between its current inventory level (resp., on-order quantity) and a target value $a^i$ (resp., $b^i$). We set $a^i=\mu_d$, where $\mu_d$ is the average demand; $b^i=\mu_d(l_i^{fi} + l_i^{tr})$; $\alpha^i = -0.5$; and $\beta^i = -0.2$ for all agents $i=1,2,3,4$. The negative $\alpha$ and $\beta$ mean that the player over-orders when the inventory level or on-order quantity fall below the target value $a_i$ or $b_i$. Extended Numerical Results {#sec:appd:more_play_results} ========================== This appendix shows additional results on the details of play of each agent. Figure \[fig:4Agent:vs\_optm:b3:play\_DNN\_W\_D\_M\] provides the details of $IL$, $OO$, $a$, $r$, and OUTL for each agent when the DQN retailer plays with co-players who use the [BS]{} policy. Clearly, DQN attains a similar IL, OO, action, and reward to those of [BS]{}. Figure \[fig:4Agent:vs\_optm:b4:play\_DNN\_R\_W\_D\_M\] provides analogous results for the case in which the DQN manufacturer plays with three [Strm]{} agents. The DQN agent learns that the shortage costs of the non-retailer agents are zero and exploits that fact to reduce the total cost. In each of the figures, the top set of charts provides the results of the retailer, followed by the warehouse, distributor, and manufacturer. The Effect of $\beta$ on the Performance of Each Agent {#sec:appdx:base_stock_beta} ====================================================== Figure \[fig:beta:dqn\_vs\_three\_optimal\] plots the training trajectories for DQN agents playing with three [BS]{} agents using various values of $C$, $m$, and $\beta$. In each sub-figure, the blue line denotes the result when all players use a [BS]{} policy while the remaining curves each represent the agent using DQN with different values of $C$, $\beta$, and $m$, trained for 60000 episodes with a learning rate of $0.00025$. As shown in Figure \[fig:4Agent:vs\_optm:beta:A\_DNN\_Retailer\], when the DQN plays the retailer, $\beta_1 \in \{20,40\}$ works well, and $\beta_1 = 40$ provides the best results. As we move upstream in the supply chain (warehouse, then distributor, then manufacturer), smaller $\beta$ values become more effective (see Figures \[fig:4Agent:vs\_optm:beta:A\_DNN\_Warehouse\]–\[fig:4Agent:vs\_optm:beta:A\_DNN\_Manufacturer\]). Recall that the retailer bears the largest share of the optimal expected cost per period, and as a result it needs a larger $\beta$ than the other agents. Not surprisingly, larger $m$ values provide attain better costs since the DQN has more knowledge of the environment. Finally, larger $C$ works better and provides a stable DQN model. However, there are some combinations for which smaller $C$ and $m$ also work well, e.g., see Figure \[fig:4Agent:vs\_optm:beta:A\_DNN\_Manufacturer\], trajectory $5000$-$20$-$5$. [0.28]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_optm:beta:A_DNN_Manufacturer"}](brain_3_mult_100_dnnUpcnt_10000_betta_100.png){width="180.00000%"} [0.28]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_optm:beta:A_DNN_Manufacturer"}](brain_4_mult_100_dnnUpcnt_10000_betta_100.png){width="180.00000%"} [0.28]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_optm:beta:A_DNN_Manufacturer"}](brain_5_mult_100_dnnUpcnt_10000_betta_100.png){width="180.00000%"} [0.28]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_optm:beta:A_DNN_Manufacturer"}](brain_6_mult_100_dnnUpcnt_10000_betta_100.png){width="180.00000%"} [0.58]{} ![DQN plays manufacturer[]{data-label="fig:4Agent:vs_optm:beta:A_DNN_Manufacturer"}](legend.png){width="180.00000%"} Extended Results on Transfer Learning {#sec:appdx:extended_tl} ===================================== Transfer Knowledge Between Agents {#sec:results:transfer_learning:same_agents} --------------------------------- In this section, we present the results of the transfer learning method when the trained agent $i \in \{1,2,3,4\}$ transfers its first $k \in \{1,2,3\}$ layer(s) into co-player agent $j \in \{1,2,3,4\}$, $j \neq i$. For each target-agent $j$, Figure \[fig:agents\_all\_same\] shows the results for the best source-agent $i$ and the number of shared layers $k$, out of the 9 possible choices for $i$ and $k$. In the sub-figure captions, the notation $j$-$i$-$k$ indicates that source-agent $i$ shares weights of the first $k$ layers with target-agent $j$, so that those $k$ layers remain non-trainable. Except for agent 2, all agents obtain costs that are very close to those of the [BS]{} policy, with a $6.06\%$ gap, on average. (In Section \[sec:results:dnn\_vs\_base\_stock\], the average gap is $2.31\%$.) However, none of the agents was a good source for agent 2. It seems that the acquired knowledge of other agents is not enough to get a good solution for this agent, or the feature space that agent 2 explores is different from other agents, so that it cannot get a solution whose cost is close to the [BS]{} cost. [0.24]{} ![Case 4-2-1 \[fig:agent\_6-5-2\] ](3-6-1.png "fig:") [0.24]{} ![Case 4-2-1 \[fig:agent\_6-5-2\] ](4-6-2-.png "fig:") [0.24]{} ![Case 4-2-1 \[fig:agent\_6-5-2\] ](5-3-1.png "fig:") [0.24]{} ![Case 4-2-1 \[fig:agent\_6-5-2\] ](6-4-1.png "fig:") In order to get more insight, consider Figure \[fig:dqn\_vs\_three\_optimal\], which presents the best results obtained through hyper-parameter tuning for each agent. In that figure, all agents start the training with a large cost value, and after 25000 fluctuating iterations, each converges to a stable solution. In contrast, in Figure \[fig:agents\_all\_same\], each agent starts from a relatively small cost value, and after a few thousand training episodes converges to the final solution. Moreover, for agent 3, the final cost of the transfer learning solution is smaller than that obtained by training the network from scratch. And, the transfer learning method used one order of magnitude less CPU time than the approach in Section \[sec:results:dnn\_vs\_base\_stock\] to obtain very close results. We also observe that agent $j$ can obtain good results when $k=1$ and $i$ is either $j-1$ or $j+1$. This shows that the learned weights of the first DQN network layer are general enough to transfer knowledge to the other agents, and also that the learned knowledge of neighboring agents is similar. Also, for any agent $j$, agent $i=1$ provides similar results to that of agent $i=j-1$ or $i=j+1$ does, and in some cases it provides slightly smaller costs, which shows that agent $1$ captures general feature values better than the others. Transfer Knowledge for Different Cost Coefficients {#sec:results:transfer_learning:different_cp_ch} -------------------------------------------------- Figure \[fig:agents\_cp5\] shows the best results achieved for all agents, when agent $j$ has different cost coefficients, $(c_{p_2}, c_{h_2}) \neq (c_{p_1}, c_{h_1})$. We test target agents $j \in \{1,2,3,4\}$, such that the holding and shortage costs are (5,1), (5,0), (5,0), and (5,0) for agents 1 to 4, respectively. In all of these tests, the source and target agents have the same action spaces. All agents attain cost values close to the [BS]{} cost; in fact, the overall average cost is 6.16% higher than the [BS]{} cost. [0.24]{} ![Case 4-4-2[]{data-label="fig:agent_cp5_6-5-2"}](3-6-1---.png "fig:") [0.24]{} ![Case 4-4-2[]{data-label="fig:agent_cp5_6-5-2"}](4-5-3.png "fig:") [0.24]{} ![Case 4-4-2[]{data-label="fig:agent_cp5_6-5-2"}](5-3-1.png "fig:") [0.24]{} ![Case 4-4-2[]{data-label="fig:agent_cp5_6-5-2"}](6-6-2--.png "fig:") In addition, similar to the results of Section \[sec:results:transfer\_learning:same\_agents\], base agent $i=1$ provides good results for all target agents. We also performed the same tests with shortage and holding costs (10,1), (1,0), (1,0), and (1,0) for agents 1 to 4, respectively, and obtained very similar results. Transfer Knowledge for Different Size of Action Space {#sec:results:transfer_learning:different_action} ----------------------------------------------------- Increasing the size of the action space should increase the accuracy of the $d+x$ approach. However, it makes the training process harder. It can be effective to train an agent with a small action space and then transfer the knowledge to an agent with a larger action space. To test this, we test target-agent $j \in \{1,2,3,4\}$ with action space $\{-5,\dots,5\}$, assuming that the source and target agents have the same cost coefficients. Figure \[fig:agents\_action5\] shows the best results achieved for all agents. All agents attained costs that are close to the [BS]{} cost, with an average gap of approximately 10.66%. [0.24]{} ![Case 4-2-1[]{data-label="fig:agent_action5_6-6-1"}](3-5-1.png "fig:") [0.24]{} ![Case 4-2-1[]{data-label="fig:agent_action5_6-6-1"}](4-5-2---.png "fig:") [0.24]{} ![Case 4-2-1[]{data-label="fig:agent_action5_6-6-1"}](5-6-2.png "fig:") [0.24]{} ![Case 4-2-1[]{data-label="fig:agent_action5_6-6-1"}](6-4-1.png "fig:") Transfer Knowledge for Different Action Space, Cost Coefficients, and Demand Distribution {#sec:results:transfer_learning:case_5} ----------------------------------------------------------------------------------------- This case includes all difficulties of the cases in Sections \[sec:results:transfer\_learning:same\_agents\], \[sec:results:transfer\_learning:different\_cp\_ch\], \[sec:results:transfer\_learning:different\_action\], and \[sec:results:transfer\_learning\], in addition to the demand distributions being different. So, the range of demand, $IL$, $OO$, $AS$, and $AO$ that each agent observes is different than those of the base agent. Therefore, this is a hard case to train, and the average optimality gap is 17.41%; however, as Figure \[fig:case\_5\] depicts, the cost values decrease quickly and the training noise is quite small. [0.24]{} ![Case 4-3-2[]{data-label="fig:agent_case5_6-5-2"}](brain3.png "fig:") [0.24]{} ![Case 4-3-2[]{data-label="fig:agent_case5_6-5-2"}](brain4.png "fig:") [0.24]{} ![Case 4-3-2[]{data-label="fig:agent_case5_6-5-2"}](brain5.png "fig:") [0.24]{} ![Case 4-3-2[]{data-label="fig:agent_case5_6-5-2"}](brain6.png "fig:") Transfer Knowledge for Different Action Space, Cost Coefficients, Demand Distribution, and $\pi_2$ {#sec:results:transfer_learning:case_6} -------------------------------------------------------------------------------------------------- Figures \[fig:case\_6\_sterman\] and \[fig:case\_6\_random\] show the results of the most complex transfer learning cases that we tested. Although the DQN plays with non-rational co-players and the observations in each state might be quite noisy, there are relatively small fluctuations in the training, and for all agents after around 40,000 iterations they converge. [0.24]{} ![Case 4-1-1[]{data-label="fig:agent_case6_10-3-1"}](brain7.png "fig:") [0.24]{} ![Case 4-1-1[]{data-label="fig:agent_case6_10-3-1"}](brain8.png "fig:") [0.24]{} ![Case 4-1-1[]{data-label="fig:agent_case6_10-3-1"}](brain9.png "fig:") [0.24]{} ![Case 4-1-1[]{data-label="fig:agent_case6_10-3-1"}](brain10.png "fig:") [0.24]{} ![Case 4-1-1[]{data-label="fig:agent_case6_14-3-3"}](brain11.png "fig:") [0.24]{} ![Case 4-1-1[]{data-label="fig:agent_case6_14-3-3"}](brain12.png "fig:") [0.24]{} ![Case 4-1-1[]{data-label="fig:agent_case6_14-3-3"}](brain13.png "fig:") [0.24]{} ![Case 4-1-1[]{data-label="fig:agent_case6_14-3-3"}](brain14.png "fig:") Pseudocode of the Beer Game Simulator {#sec:appnd:sudocode} ===================================== The DQN algorithm needs to interact with the environment, so that for each state and action, the environment should return the reward and the next state. We simulate the beer game environment using Algorithm \[alg01\]. In addition to the notation defined earlier, the algorithm also uses the following notation:\ $d^{t}$: The demand of the customer in period $t$.\ $OS_i^t$: Outbound shipment from agent $i$ (to agent $i-1$) in period $t$.\ Set $T$ randomly, and $t = 0$, Initialize $IL_i^0$ for all agents, $AO_i^t = 0, AS_i^t = 0, \forall i,t$ $AO_i^{t+l_i^{fi}} += d^{t}$ get action $a_i^t$ $OO_i^{t+1} = OO_i^t + a_i^t$ $AO_{i+1}^{t+l_i^{fi}} += a_{i}^t$ $AS_4^{t+l_4^{tr}} += a_4^{t}$ $IL_i^{t+1} = IL_i^{t} + AS_i^t$ $OO_i^{t+1} -= AS_i^t$ current\_Inv = $\max \{0, IL_i^{t+1}\}$ current\_BackOrder = $\max \{0, -IL_i^{t}\}$ $OS_i^t = \min \{$ current\_Inv, current\_BackOrder + $AO_i^{t}$ $\}$ $AS_{i-1}^{t+l_i^{tr}} += OS_i^{t}$ $IL_{i}^{t+1} -= AO_i^{t}$ $c_i^t = c_i^p \max\{-IL_{i}^{t+1}, 0\} + c_i^h \max\{IL_{i}^{t+1}, 0 \}$ $t += 1$

--- abstract: 'We introduce a method to lower bound an entropy-based measure of genuine multipartite entanglement via nonlinear entanglement witnesses. We show that some of these bounds are tight and explicitly work out their connection to a framework of nonlinear witnesses that were published recently. Furthermore, we provide a detailed analysis of these lower bounds in the context of other possible bounds and measures. In exemplary cases, we show that only a few local measurements are necessary to determine these lower bounds.' author: - 'Jun-Yi Wu, Hermann Kampermann, Dagmar Bru[ß]{}' - Claude Klöckl - Marcus Huber date: date title: Determining lower bounds on a measure of multipartite entanglement from few local observables --- Introduction ============ Quantum entanglement is central to the field of quantum information theory. Due to its numerous applications in upcoming quantum technology much research has been devoted to its understanding (for a recent overview consider Ref. [@horodeckiqe]).Especially in systems comprised of many particles entanglement provides numerous challenges and of course potential applications, such as building quantum computers (see Ref. [@qc]), performing quantum algorithms (the connection to multipartite entanglement is demonstrated in Ref. [@qa]) and multi-party cryptography (see e.g. Ref. [@SHH3]).Furthermore, the understanding of the behavior of complex systems seems to be closely linked to the understanding of multipartite entanglement manifestations, demonstrated by the connection to phase transitions and ionization in condensed matter systems (e.g. [@cond]), the properties of ground states in relation to entanglement (as shown e.g. in Ref. [@spin; @ground]), or potentially even biological systems (such as e.g. bird navigation [@bird]).In order to judge the relevance of entanglement in such systems it is crucial to not only detect its presence, but also quantify the amount. The structure of entangled states, especially in multipartite systems [@acin], is very complex and the question whether a given state is entangled is even NP-hard [@gurvits]. Thus, in general, it will not be possible to derive a computable measure of entanglement that reveals all entangled states to be entangled and discriminates between different entanglement classes. Furthermore, full information about the state of the system requires a number of measurements that grows exponentially in the size of the system. For the detection of entanglement in multipartite systems most researchers have therefore made it a primary goal to develop entanglement witnesses, which via a limited amount of local measurements can detect the presence of entanglement, even in complex systems (for an overview of multipartite entanglement witnesses consider Ref. [@guehnetoth]).The expectation value of witness-operators are usually expressed in terms of inequalities, which if violated show the presence of entanglement. Nonlinear witnesses (first introduced in Ref. [@horodeckinonlinear] see also early discussions in e.g. Ref. [@nonlin]) provide a generalization that is no longer a linear function of density matrix elements, but a nonlinear one. Thus one cannot reformulate the criteria in terms of an expectation value of a hermitian operator (unless one considers coherent measurements on multiple copies of the state, which out of experimental infeasibility we do not discuss in our manuscript). We will henceforth refer to inequalities that involve nonlinear functions of density matrix elements as nonlinear entanglement witnesses.\ Recently some authors pointed out a connection between the possible amount of violation of these nonlinear inequalities and quantification of entanglement in multipartite systems (in Ref. [@Guehnetaming] and Ref. [@maetal]).The aims of this paper are twofold. First to systematically show the connection of numerous witnesses to a meaningful measure of genuine multipartite entanglement and second to use this established relation for the development of novel witnesses, which by construction give lower bounds on that measure. To that end we follow and generalize the approach from Ref. [@maetal].\ It turns out that only a small number of density matrix elements enters into our lower bounds, making the construction experimentally feasible even in larger systems of high dimensionality. A measure of multipartite entanglement and its lower bounds =========================================================== A measure of genuine multipartite entanglement (GME) ---------------------------------------------------- The entropy of subsystems has often been used, in order to quantify entanglement contained in multipartite pure states (e.g. see [@horodeckiqe; @Milburn; @Love; @HH2; @HHK1]). In this paper we will follow the definition first presented in Ref.[@Milburn] and define a measure of GME for multipartite pure states as $$\begin{aligned} E_{m}(|\psi\rangle\langle\psi|):=\min_{\gamma}\sqrt{S_{L}\left( \rho _{\gamma}\right) }=\min_{\gamma}\sqrt{2\left( 1-\text{Tr}(\rho_{\gamma}^{2})\right) }\,, \label{Def. GME measure}$$ where $S_{L}\left( \rho_{\gamma}\right) $ is the linear entropy of the reduced density matrix of subsystem $\gamma$, i.e. $\rho_{\gamma}:=\text{Tr}_{\bar{\gamma}}(|\psi\rangle\langle\psi|)$. The minimum is taken over all possible reductions $\gamma$ (where the complement is denoted as $\bar{\gamma}$), which corresponds to a bipartite split into $\gamma|\bar{\gamma}$. As any proper measure of multipartite entanglement for pure states can be generalized to mixed states via a convex roof, i.e. $$\label{definitionroof} E_{m}(\rho):=\inf_{\{p_{i},|\psi_{i}\rangle\}}\sum_{i}p_{i}E_{m}(|\psi _{i}\rangle\langle\psi_{i}|)\,.$$ Due to its construction this measure fulfills almost all desirable properties one would expect from measures of GME (see Ref. [@maetal] for details). Because computing all possible pure state decompositions of a density matrix is computationally impossible even if one is given the complete density matrix, we require lower bounds to be calculable for this expression.\ Also note that a lower bound on the linear entropy directly leads to a lower bound on the R[é]{}nyi $2$-entropy $S_R^{(2)}(\rho_\gamma)$ via the relation $S_R^{(2)}(\rho_\gamma)=-\log_2(\frac{2-S_L(\rho_\gamma)}{2})$, which also provides one of the physical interpretations of this measure. The R[é]{}nyi $2$-entropy in itself is a lower bound to the von Neumann entropy $S(\rho_\gamma)$ and the mutual information can be expressed as $I_{\gamma\bar{\gamma}}:=S(\rho_\gamma)+S(\rho_{\bar{\gamma}})-S(\rho)=2S(\rho_\gamma)$. Thus by our lower bound we gain a lower bound on the average minimal mutual information across all bipartitions of the pure states in the decomposition, minimized over all decompositions. Linear entropy and its convex roof ---------------------------------- The state vector of an $n$-partite qudit state can be expanded in terms of the computational basis $$|\psi\rangle=\sum_{i_{1},i_{2},\cdots,i_{n}=0}^{d-1}c_{i_{1},i_{2},\cdots,i_{n}}|i_{1},i_{2},\cdots,i_{n}\rangle=:\sum_{\eta\in\mathbb{N}_{d}^{\otimes n}}c_{\eta}|\eta\rangle\,,$$ where a basis vector is denoted by $\eta=\left( i_{1},i_{2},\cdots,i_{n}\right) \in\mathbb{N}_{d}^{\otimes n}$. This vector notation will facilitate the upcoming derivations. A crucial element of the notation in this paper will be the permutation operator acting upon two vectors, exchanging vector components corresponding to the set of indices. E.g. the permutation operator $P_{\{1,3\}}(\eta_1,\eta_2)$ will exchange the first and third component of the vector $\eta_{1}$ with the corresponding component of the vector $\eta_{2}$, i.e. $$P_{\left\{ \textcolor{red}{1},\textcolor{blue}{3}\right\} }(\mathbf{\textcolor{red}{0}}1\mathbf{\textcolor{blue}{2}}13,\mathbf{\textcolor{red}{3}}0\mathbf{\textcolor{blue}{1}}21)=(\mathbf{\textcolor{red}{3}}1\mathbf{\textcolor{blue}{1}}13,\mathbf{\textcolor{red}{0}}0\mathbf{\textcolor{blue}{2}}21).$$ Using this notation one can write down a very simple expression for the linear entropy of a reduced state $\rho_{\gamma}$ (derivation see section \[linear entropy of pure state\] in the appendix) $$S_{L}\left( \rho_{\gamma}\right) =\sum_{\eta_{1}\not =\eta_{2}}\left\vert c_{\eta_{1}}c_{\eta_{2}}-c_{\eta_{1}^{\gamma}}c_{\eta_{2}^{\gamma }}\right\vert ^{2}, \label{result. linear entropy}$$ where $\left( \eta_{1}^{\gamma},\eta_{2}^{\gamma}\right) =P_{\gamma}\left( \eta_{1},\eta_{2}\right) $. For pure states we can of course find lower bounds on $E_m(|\psi\rangle\langle\psi|)$ by lower bounding the linear entropy for all possible bipartitions. For mixed states we can then provide a lower bound for the convex roof $E_m(\rho)$. We now illustrate our method in one exemplary case and then continue to articulate the main theorem.\ Note that the linear entropy of subsystems has been widely used for lower bounding measures of entanglement due to the well known and simple structure of eq.(\[result. linear entropy\]). None of the previous methods, however, work for lower bounding the inherently multipartite measure $E_m(\rho)$, due to the additional minimization over all bipartitions in each decomposition element of the convex roof. W-states\[sec. W-state\] ------------------------ In order to demonstrate how our framework works let us start by deriving the explicit lower bound detecting the three-qubit $W$ state $|W\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$. For three-qubit states there are three bipartitions ($1|23,2|13,3|12$) and thus we have three linear entropies to look at in order to calculate $E_m(|\psi\rangle\langle\psi|)$, $$\begin{gathered} \sqrt{S_L(\rho_1)}=\\2\sqrt{|c_{001}c_{100}-c_{101}c_{000}|^2+|c_{010}c_{100}-c_{110}c_{000}|^2+(\cdots)}\,,\nonumber\\ \sqrt{S_L(\rho_2)}=\\2\sqrt{|c_{010}c_{100}-c_{110}c_{000}|^2+|c_{010}c_{001}-c_{011}c_{000}|^2+(\cdots)}\,,\nonumber\\ \sqrt{S_L(\rho_3)}=\\2\sqrt{|c_{001}c_{100}-c_{101}c_{000}|^2+|c_{010}c_{001}-c_{011}c_{000}|^2+(\cdots)}\nonumber\,.\end{gathered}$$ Now using $\sqrt{a^2+b^2}\geq\frac{1}{\sqrt{2}}(a+b)$ (which is a specific case of the inequality \[eq. ineq. sqrt\] in appendix \[appx. ineq.sqrt\]) and $|a-b|\geq|a|-|b|$ it is obvious that $$\begin{aligned} \sqrt{S_L(\rho_1)}\geq\frac{2(|c_{001}c_{100}|-|c_{101}c_{000}|+|c_{010}c_{100}|-|c_{110}c_{000}|)}{\sqrt{2}}\,,\\ \sqrt{S_L(\rho_2)}\geq\frac{2(|c_{010}c_{100}|-|c_{110}c_{000}|+|c_{010}c_{001}|-|c_{011}c_{000}|)}{\sqrt{2}}\,,\\ \sqrt{S_l(\rho_3)}\geq\frac{2(|c_{001}c_{100}|-|c_{101}c_{000}|+|c_{010}c_{001}|-|c_{011}c_{000}|)}{\sqrt{2}}\,.\end{aligned}$$ Then using $|ab|-\frac{1}{2}(a^2+b^2)\leq 0$ we can add one negative term for each entropy and it will still be a lower bound, i.e. we add $|c_{010}c_{001}|-\frac{1}{2}(|c_{010}|^2+|c_{001}|^2)$ in the first lower bound, $|c_{100}c_{001}|-\frac{1}{2}(|c_{100}|^2+|c_{001}|^2)$ in the second and $|c_{010}c_{100}|-\frac{1}{2}(|c_{010}|^2+|c_{100})|^2$ in the third. Then we can use that $\min[P-N_1,P-N_2,P-N_3]\geq P-N_1-N_2-N_3$ and end up with $$\begin{aligned} E_m(|\psi\rangle\langle\psi|)\geq\sqrt{2}(|c_{001}c_{100}|+|c_{001}c_{010}|+|c_{100}c_{010}|)-\nonumber\\ \frac{\sqrt{2}}{2}(|c_{010}|^2+|c_{100}|^2+|c_{001}|^2)-\nonumber\\ \sqrt{2}(|c_{101}c_{000}|+|c_{110}c_{000}|+|c_{011}c_{000}|)\,.\end{aligned}$$ Finally we can bound the convex roof using the following two relations $$\begin{aligned} \inf_{\{p_i,|\psi_i\rangle\}}\sum_ip_i|c^i_{\eta_1} c^i_{\eta_2}|&\geq|\langle\eta_1|\rho|\eta_2\rangle|\,,\\ \inf_{\{p_i,|\psi_i\rangle\}}\sum_ip_i|c^i_{\eta_1} c^i_{\eta_2}|&\leq\sqrt{\langle\eta_1|\rho|\eta_1\rangle\langle\eta_2|\rho|\eta_2\rangle}\,,\end{aligned}$$ and end up with a lower bound for mixed states as $$\begin{gathered} E_m(\rho)\geq\sqrt{2}(|\langle 001|\rho|100\rangle|+|\langle 001|\rho|010\rangle|+|\langle 100|\rho|010\rangle|)-\nonumber\\ \frac{\sqrt{2}}{2}(\langle 010|\rho|010\rangle+\langle 100|\rho|100\rangle+\langle 001|\rho|001\rangle)-\nonumber\\ \sqrt{2}\sqrt{\langle 101|\rho|101\rangle\langle 000|\rho|000\rangle}-\nonumber\\ \sqrt{2}\sqrt{\langle 110|\rho|110\rangle\langle 000|\rho|000\rangle}-\nonumber\\ \sqrt{2}\sqrt{\langle 011|\rho|011\rangle\langle 000|\rho|000\rangle}\,. \label{WWitness}\end{gathered}$$ Surprisingly this leads directly to the nonlinear entanglement witness inequality presented in Refs. [@Guehnewit; @hmgh1] up to a factor of $\sqrt{2}$. Using only simple algebraic relations we have thus shown how to lower bound the convex roof construction. The first apparent strength of this lower bound is the limited number of density matrix elements needed to compute it. E.g. in our exemplary three-qubit case only ten out of possibly sixty-four elements need to be measured. Obviously we can extend the analysis using the same techniques to systems beyond three qubits. A General Construction of lower bounds on the GME measure $E_{m}$ ================================================================= Now we can generalize the connection of the 3-qubit W state witness and the measure $E_{m}$. Just as for three qubits we can always get lower bounds by summing the coefficient pairs $c_{\eta_1}c_{\eta_2}$ that belong to a certain target pure state and appear in some or all reduced linear entropies. The construction of such general lower bounds also starts by selecting a subset of coefficient pairs that will be translated into off-diagonal elements $\rho_{\eta_1,\eta_2}$, where $(\eta_1,\eta_2)$ is the vector basis pair denoting the row and column of the element in density matrix $\rho$. We denote the selected vector basis pairs as $R:=\{(\eta_1,\eta_2)\}$. Then we can repeat the steps analogously to eq.(6-11) and arrive at a general lower bound on the measure as the following theorem: \[A general lower bound on the GME measure\]\[theorem. lower bound of GME measure\]For a set of row-column pairs $R=\{(\eta_1,\eta_2)\}$, the genuine multipartite entanglement measure $E_{m}$ has the following lower bound: $$E_{m}\geq 2\sqrt{\frac{1}{\left\vert R\right\vert -N_{R}}}\left[ \sum_{\left( \eta_{1},\eta_{2}\right) \in R}\left( \left\vert \rho_{\eta_{1}\eta_{2}}\right\vert -\sum_{\gamma\in\Gamma(\eta_{1},\eta_{2})}\sqrt{\rho_{\eta _{1}^{\gamma}\eta_{1}^{\gamma}}\rho_{\eta_{2}^{\gamma}\eta_{2}^{\gamma}}}\right) -\left( \frac{1}{2}\sum_{\eta\in I(R)}N_{\eta}\left\vert \rho _{\eta\eta}\right\vert \right) \right] \label{eq. theorem-result}\,.$$ The right-hand-side of eq.(\[eq. theorem-result\]) defines a GME witness $W_R(\rho)$, where $\rho_{\eta_{1},\eta_{2}}:=\langle\eta_{1}|\rho|\eta_{2}\rangle$, $(\eta_{1}^{\gamma},\eta_{2}^{\gamma}):=P_{\gamma}(\eta_{1},\eta_{2})$, $\Gamma(\eta_{1},\eta_{2}):=\left\{ \gamma:(\eta_{1}^{\gamma},\eta _{2}^{\gamma})\notin R\right\} $ and $I(R):=\left\{ \eta:\exists\eta ^{\prime}\text{ that }(\eta^{\prime},\eta)\text{ or }(\eta,\eta^{\prime})\in R\right\} $ is the set of basis vectors $\eta$, which appear in the set $R$. $N_{R}$ is the maximal (or minimal) value of $|R^{\gamma}|$ over all possible bipartitions $\gamma|\bar{\gamma}$, where $R^{\gamma}$ is the set of coefficient pairs $(c_{\eta_{1}},c_{\eta_{2}})\in R$, which do not contribute to the $\gamma$-subsystem entropy. $N_{\eta}$ are normalization constants given by the maximal value of $n_{\eta}^{\gamma}$ over all possible bipartitions $\gamma|\bar{\gamma}$, where $n_{\eta}^{\gamma}$ is the number of coefficients $c_\eta$ from some pairs in $R$, which are not counted in the $\gamma$-subsystem entropy (and how many are counted depends on whether one chooses $N_R$ to be maximal or minimal).\ See Appendix \[appx. proof of lower bound on GME measure\] for the full proof. It is evident that not every choice of coefficient pairs will yield a useful lower bound, because one really needs to select those that are actually contributing to multipartite entanglement. There is however always an obvious choice. The set of coefficient pairs $R$ must be chosen such that in every subsystem at least one of the elements of $R$ contribute to the linear entropy of the reduced state. E.g. in the case of GHZ states given in a specific basis $|GHZ\rangle=\frac{1}{\sqrt{2}}(|0\rangle^{\otimes n}+|1\rangle^{\otimes n})$ one would choose the pair $({00\cdots0},{11\cdots1})$, which contributes to all reduced entropies. In the general case however there is still some freedom of choice left to get a valid lower bound. For some sets $R$ it can happen, that the coefficients do not contribute to every subsystem entropy equally (which we show in an exemplary case in section \[sec. four-qubit-example\]). Then one can choose $N_R$ in different ways, but in all considered cases we found that choosing it maximal or minimal will produce the best bounds (where choosing it maximal usually yields the tightest bounds close to pure states, whereas choosing it minimal improves the noise resistance). Since these coefficients are in general basis dependent, so is also our witness construction. The prefactor $\sqrt{\frac{1}{\left\vert R\right\vert-N_R }}$ suggests that the optimal basis for constructing such a lower bound is given by the minimal tensor rank representation of the pure state. Applications and Examples ========================= Four-qubit singlet state {#sec. four-qubit-example} ------------------------ Let us illustrate how to apply Theorem \[theorem. lower bound of GME measure\] with an explicit example. In an experimental setting where one expects to produce a four-qubit singlet state (which was e.g. discussed in the context of solving the liar detection problem in Ref. [@cabello]), i.e. $$\begin{aligned} |S_4\rangle=&\frac{1}{2\sqrt{3}}(2|0011\rangle+2|1100\rangle-|0110\rangle\nonumber\\ &-|1001\rangle-|1010\rangle-|0101\rangle)\,,\end{aligned}$$ one is confronted with the following expected coefficients: $c_{0011},c_{1100},c_{0101},c_{1010},c_{0110},c_{1001}$. Following the recipe of theorem \[theorem. lower bound of GME measure\] we now select some coefficient pairs. We could choose e.g. $R_1=(0011,0101)$, $R_2=(0011,1010)$, $R_3=(0011,0110)$ and $R_4=(0011,1001)$, such that $R=\{R_1,R_2,R_3,R_4\}$. For this selection we use theorem \[theorem. lower bound of GME measure\] to bound the GME measure. We see that in every subsystem at least two of these pairs appear naturally. Although there are more coefficient pairs we now choose to only take into account two per subsystem entropy and thus choose $N_R$ to be the minimal number of coefficient pairs in every subsystem which gives $N_R=2$. Thus we need to add negative terms that compensate for the missing terms just as we did in the three-qubit case, but now we need to do it two times in every subsystem. This results in the following individual prefactors $N_\eta$ for the diagonal elements: $N_{0011}=2$ (as this coefficient appears in two missing pairs in every subsystem), $N_{0101}=1$, $N_{1001}=1$, $N_{1010}=1$ and $N_{0110}=1$ (as those appear maximally once per subsystem entropy). Inserting this in theorem \[theorem. lower bound of GME measure\] we end up with the lower bound as $$\begin{gathered} E_m(\rho)\geq\frac{2}{\sqrt{2}}(|\rho_{R_1}|+|\rho_{R_2}|+|\rho_{R_3}|+|\rho_{R_4}|\nonumber\\ -\sqrt{\rho_{0111,0111}\rho_{0001,0001}}-\sqrt{\rho_{0111,0111}\rho_{0010,0010}}\nonumber\\ -\sqrt{\rho_{1011,1011}\rho_{0001,0001}}-\sqrt{\rho_{1011,1011}\rho_{0010,0010}}\nonumber\\ -\frac{1}{2}(\rho_{0101,0101}+\rho_{1001,1001}+\rho_{1010,1010}+\rho_{0110,0110})\nonumber\\ -\rho_{0011,0011})\,.\end{gathered}$$ We have thus created a nonlinear witness function that lower bounds our measure. From an experimental point of view this is very favorable as few local measurement settings suffice to ascertain the needed thirteen density matrix elements (especially since the nine diagonal elements can be constructed from a single measurement setting). Of course we could also exploit the connection of our lower bound to the Dicke state witness $Q^{(2)}_2$ (which is discussed in section \[Dicke\]), which also detects GME in this state (although at the cost of more required measurements). In this case even the resistance to white noise is more favorable with our construction method, as for a state $\rho=p|S_4\rangle\langle S_4|+\frac{1-p}{16}\mathbbm{1}$ this exemplary lower bound detects GME until $p=\frac{21}{29}\approx 0.72$, whereas the old witness construction yields a worse resistance up to $p=\frac{27}{35}\approx 0.77$. This shows the versatility of our general approach. By choosing certain coefficients one can tailor these lower bounds to specific experimental situations. If one is confronted with a low noise system it is always beneficial to choose as few coefficients as possible, such that very few local measurements suffice (even a number that is linear in the size of the system is often sufficient). Every additional measurement can then be included in the lower bound and improves the bound and its noise resistance if necessary. Bipartite witnesses and lower bounds on the measure --------------------------------------------------- Although we have presented our theorem and measures in the general case of $n$-qudits, we can always apply the lower bounds also for $n=2$, as our theorem holds for any $n$ and $d$. Suppose we are given a bipartite qutrit system and want to lower bound the concurrence with only a few local measurements. If the expected state is e.g. $|\psi\rangle=\frac{1}{\sqrt{3}}(|00\rangle+|11\rangle+|22\rangle)$ we can use the lower bounding procedure outlined above, yielding $$\begin{aligned} E_m(\rho)\geq\frac{2}{\sqrt{3}}(\Re e[\left.\left\langle 00| \rho |11 \right\rangle\right.]-\sqrt{\left.\left\langle 01| \rho| 01 \right\rangle\left\langle| 10 \rho |10 \right\rangle\right.}+\nonumber\\ \Re e[\left.\left\langle 00| \rho |22 \right\rangle\right.]-\sqrt{\left.\left\langle 02| \rho |02 \right\rangle\left\langle 20| \rho| 20 \right\rangle\right.}+\nonumber\\ \Re e[\left.\left\langle 11| \rho |22 \right\rangle\right.]-\sqrt{\left.\left\langle 12| \rho |12\right\rangle\left\langle 21| \rho |21 \right\rangle\right. })\,. \end{aligned}$$ In order to determine the lower bound we have to measure nine different density matrix elements. Of course any density matrix element can always be obtained via local measurements. How these measurements can be performed in a basis consisting of a tensor product of the generalized Gell-Mann matrices we show explicitly in appendix \[GellMann\].\ It turns out that these nine different density matrix elements can be obtained via ten local measurement settings. Let us study the lower bound in the presence of noise. Suppose we have white noise in the system, i.e. $\rho=p|\psi\rangle\langle\psi|+\frac{1-p}{d}{\mathbbm{1}}$. Calculating the lower bound results in $E_m(\rho)\geq\frac{2(4p-1)}{\sqrt{27}}$, which is equivalent to the analytical expression of Wootter’s concurrence for these systems (as proven in Ref. [@hashemi; @caves]). In this case we have a necessary and sufficient entanglement criterion and a tight lower bound on the concurrence from ten local measurements for a special class of states. Indeed if one generalizes this example to arbitrary dimension $d$, we find that the bound is always tight for bipartite isotropic states. Dicke States {#Dicke} ------------ We will now continue to show how this construction relates to an entanglement witness for Dicke-state, which are multi-dimensional generalizations of the $W$ states(which were first introduced in the context of laser emission in Ref. [@Dicke]). In the original article [@maetal], where this approach was first introduced, the authors connected the violation of a witness suitable for GHZ states (first introduced in Ref. [@Guehnewit] and later presented in a more general framework in Ref. [@hmgh1]) with a lower bound on the measure $E_{m}$. We want to follow this approach and establish a general connection between a set of witnesses suitable for all generalized Dicke states introduced in Ref. [@hesgh1] and generalized in Ref. [@shgh1]. To that end let us first introduce a concise notation for those states.Let $\alpha$ be a set containing specific subsystems of a multipartite state. We then define the state $|\alpha^{l}\rangle$ as a tensor product of states $|l\rangle$ for all subsystems not contained in $\alpha$ and excited states $|l+1\rangle$ in the subsystems contained in $\alpha$. E.g. for the four-partite state $|\{1,3\}^{2}\rangle$ we have $|3232\rangle$. Using this abbreviated notation we can define a generalized set of Dicke states, consisting of $n$ $d$-dimensional subsystems, as $$\begin{aligned} |D_{m}^{d}\rangle=\frac{1}{\sqrt{{n\choose m}(d-1)}}\sum_{l=0}^{d-2}\sum_{\alpha :|\alpha|=m}|\alpha^{l}\rangle\,,\end{aligned}$$ where the parameter $m$ denotes the number of excitations, with $0<m<n$. Since the explicit form of the nonlinear witness from Ref. [@shgh1] will be used in the following considerations we will repeat it in appendix \[DickeWitness\]. For all biseparable states this witness $Q_{m}^{\left( d\right) }$ is strictly smaller equal zero, i.e. $$\begin{aligned} Q\left( \rho\right) & \leq0\Leftarrow\rho\text{ is biseparable}\\ Q\left( \rho\right) & >0\Rightarrow\rho\text{ is multipartite entangled}\,.\end{aligned}$$ Furthermore, the witness can also detect the “dimensionality” of GME, by which we mean the maximal number of degrees of freedom $f_\rho (f_\rho \le d)$ that occurs in the pure states of an ensemble constituting $\rho$, minimised over all ensembles (this is the natural generalization of the concept of Schmidt number [@schmidtnumber] to multipartite systems, further explored e.g. in Ref.[@shgh1]). I.e. the dimensionality is defined as $$\begin{aligned} f_\rho:=\inf_{\{p_i,|\psi_i\rangle\}}\max_i(\min_\gamma(\text{rank}(\rho_\gamma)))\end{aligned}$$ Since $$\begin{aligned} Q_{m}^{\left( d\right) }\left( \rho\right) & \leq f_{\rho}-1,\,\forall \rho\,,\end{aligned}$$we can directly infer that $$Q_{m}^{\left( d\right) }\left( \rho\right) >f-2\Rightarrow f_\rho\geq f$$ In fig.\[fig. f\_dimensional\_gme\_entanglement\] we show how $Q_{m}^{\left( d\right) }$ detects the GME dimensionality. The maximal violation of these inequalities is always achieved for $m$-excitation Dicke states, i.e. $Q_{m}^{\left( d\right) }(|D_{m}^{d} \rangle\langle D_{m}^{d}|)=d-1$. \[ptb\] [f\_dimensional\_GME\_entanglement]{} If we can find a proper $R$, as a result of theorem \[theorem. lower bound of GME measure\] that uses the Dicke state coefficients, we can connect a lower bound of the measure $E_{m}$ with the GME witness $Q_{m}^{\left( d\right) }\,\left( \rho\right) $. Indeed choosing the ordered subset $R_{\sigma}$ of the set of coefficients $\sigma$ used in (\[eq.: GME-witness formular\]), i.e.$$R_{\sigma}=\left\{ \left( \alpha^{a},\beta^{b}\right) \in\sigma:a\leq b\right\}\,,$$ we immediately arrive at a lower bound on $E_{m}$ as $$E_{m}\left( \rho\right) \geq m\sqrt{\frac{1}{\left\vert R_{\sigma} \right\vert - N_{R_{\sigma}} }}Q_{m}^{\left( d\right) }\,\left( \rho\right) \geq m\sqrt{\frac{1}{\left\vert R_{\sigma} \right\vert }}Q_{m}^{\left( d\right) }\,\left( \rho\right) ,\label{result. lower bound on GME measure}$$ where $\left\vert R_{\sigma}\right\vert =\frac{1}{2}\left( d-1\right) ^{2}{\binom{n}{m}m}(n-m)$. In this case $N_{\eta}\leq m\left( n-m-1\right) +\Theta\left( d-3\right) \left( n-m\right) $, where $\Theta$ is a Heaviside step function. PPT-Witness and Our Witness --------------------------- [ppt\_witness\_and\_vs\_witness]{} Using the result on entanglement across bipartitions from the previous section we can explore the relation of our lower bounds to other bipartite entanglement witnesses. In our witness construction, the permutation operator $P_{\gamma}$ acting on a pure state is a $\gamma|\bar{\gamma}$-partial transpose operator, i.e. $P_\gamma|\psi\rangle\langle\psi|=(|\psi\rangle\langle\psi|)^{T_{\gamma|\bar{\gamma}}}$ (in the sense that our permutation operator now acts upon the index pairs of the coefficients of the pure state). It is thus intuitive to believe that there is certain connection between our witness and a PPT-witness [@pptwitness]. Indeed our witnesses are related to a standard PPT-witness construction (where the witnesses separate the convex set of states that are positive under partial transpose (PPT) from its complement). E.g. for diagonal GHZ states we can use the standard PPT-witness construction which goes as follows. For $\left\vert \text{GHZ}_{\eta_{1},\eta_{2}}\right\rangle :=\frac{1}{\sqrt{2}}\left( \left\vert \eta_{1}\right\rangle +\left\vert \eta_{2}\right\rangle \right) $ with $\eta_{1}+\eta_{2}=\left( d-1,\cdots,d-1\right) $, we can use the eigenvector belonging to the negative eigenvalue of the $\gamma|\bar{\gamma}$-partial transposed $\left\vert \text{GHZ}_{\eta_{1},\eta_{2}}\right\rangle \left\langle \text{GHZ}_{\eta_{1},\eta_{2}}\right\vert ^{T_{\gamma}}$ which we denote as $\left\vert \lambda_{\eta_{1},\eta_{2}}^{-}\right\rangle =\frac{1}{\sqrt{2}}\left( \left\vert \eta_{1}^{\gamma}\right\rangle -\left\vert \eta_{2}^{\gamma}\right\rangle \right) $. One can then construct the PPT-witness and write its expectation value as $$\Omega_{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{\eta _{1},\eta_{2}}^{-}\right\rangle \right) =\text{Tr}\left( \left\vert \lambda _{\eta_{1},\eta_{2}}^{-}\right\rangle \left\langle \lambda_{\eta_{1},\eta_{2}}^{-}\right\vert ^{T_{\gamma|\bar{\gamma}}}\rho\right) ,\label{eq. ppt-witness construction}$$ For instance in the three-qubit case, $$\begin{aligned} \left\vert \lambda_{001,110}^{-}\right\rangle \left\langle \lambda_{001,110}^{-}\right\vert ^{T_{1|23}}=\nonumber\end{aligned}$$ $$\begin{aligned} \frac{1}{2}\left( \begin{array} [c]{cccccccc}0 & 0 & & \cdots & \cdots & & 0 & 0\\ & 0 & & & & & -1 & \\ & & 1 & & & 0 & & \\ \vdots & & & 0 & 0 & & & \vdots\\ \vdots & & & 0 & 0 & & & \vdots\\ & & 0 & & & 1 & & \\ & -1 & & & & & 0 & \\ 0 & & & \cdots & \cdots & & & 0 \end{array} \right) .\end{aligned}$$ With the PPT-witness construction in eq.(\[eq. ppt-witness construction\]) we end up with the following PPT-witness expectation value$$\begin{aligned} \Omega_{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{GHZ}^{-}\right\rangle \right) =\frac{1}{2}\left( \rho_{\eta_{1}^{\gamma}\eta _{1}^{\gamma}}+\rho_{\eta_{2}^{\gamma}\eta_{2}^{\gamma}}\right) -\operatorname{Re}\left( \rho_{\eta_{1}\eta_{2}}\right) .\end{aligned}$$ Under the fixed bipartition $\gamma|\bar{\gamma}$, we construct our witness by choosing $R=\left( \eta_{1},\eta_{2}\right) $ as$$-W_{\left( \eta_{1},\eta_{2}\right) }^{\gamma}\left( \rho\right) =\sqrt{\rho_{\eta_{1}^{\gamma}\eta_{1}^{\gamma}}\rho_{\eta_{2}^{\gamma}\eta_{2}^{\gamma}}}-\left\vert \rho_{\eta_{1}\eta_{2}}\right\vert\,.$$ It is obvious that $-W_{\left( \eta_{1},\eta_{2}\right) }^{\gamma}\left( \rho\right) $ $\leq\Omega_{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{GHZ}\right\rangle \right) $. Hence we say that the witness $W_{R}\left( \rho\right) $ is stronger than the PPT-witness $\Omega _{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{\eta_{1},\eta _{2}}^{-}\right\rangle \right) $. The relation between our witness, the PPT-witness and the PPT-convex set is illustrated in fig.\[fig. ppt\_witness\_and\_vs\_witness\]. For clearness we just draw two PPT-witnesses in the figure. For the $n$-qudit case there are $\frac{1}{2}d^{n}$ such eigenvectors $\left\vert \lambda_{\eta_{1},\eta_{2}}^{-}\right\rangle $, corresponding to negative eigenvalues. Every witness $\Omega _{\text{ppt}}^{\gamma|\bar{\gamma}}\left( \rho,\left\vert \lambda_{\eta_{1},\eta _{2}}^{-}\right\rangle \right) $ is tangent to the set of PPT states (i.e. there exists one PPT state for which the witness yields zero). However also our witness $W_{R}\left( \rho\right)$ is zero for all these PPT states, i.e. our new witness detects more states than the traditional PPT-witness. Conclusions =========== In conclusion we have presented a method to derive lower bounds on a measure of genuine multipartite entanglement. We show that in experimentally plausible scenarios (i.e. one knows which state one aims to produce) we can derive such lower bounds simply based on coefficients of the corresponding pure states. We also connected the lower bound construction to a framework of nonlinear entanglement witnesses developed in Refs. [@Guehnewit; @hmgh1; @Dicke; @hesgh1; @shgh1]. These witnesses are experimentally feasible in terms of required local measurement settings. We provide further evidence in the bipartite case, where we also show that for certain families of mixed states our lower bounds are tight.\ Some open questions remain, such as whether this general construction method will work for all kinds of states and how it can be generalized beyond just multi- and bipartite entanglement, but anything in between. 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Proofs\[appx. proof of result\] =============================== The Formulas Used in The Main Article ------------------------------------- ### Reduced linear entropy of pure states {#linear entropy of pure state} Let $\left\vert \psi\right\rangle =\sum_{\eta\in\mathbb{N}_{d}^{\otimes n}}c_{\eta}\left\vert \eta\right\rangle $ be an n-qudit pure state. The linear entropy of $\left\vert \psi\right\rangle $ can be written as$$S_{L}\left( \rho_{\gamma}\right) =\sum_{\eta_{1}\not =\eta _{2},\in\mathbb{N}_{d^{n}}} \left\vert c_{\eta_{1}}c_{\eta_{2}}-c_{\eta_{1}^{\gamma}}c_{\eta_{2}^{\gamma }}\right\vert ^{2},$$ where $\left( \eta_{1}^{\gamma},\eta_{2}^{\gamma}\right) =P_{\gamma}\left( \eta_{1},\eta_{2}\right) $. The linear entropy regarding a specific partition $\gamma|\bar{\gamma}$ is defined as $S_{L}\left( \rho_{\gamma}\right) =2(1-tr\left( \rho_{\gamma}^{2}\right))$, where $\rho_{\gamma}$ is the $\gamma$-reduced matrix of $\rho $. The trace of $\rho_{\gamma}$ is $tr\left( \rho_{\gamma}^{2}\right) =\sum_{\alpha_{1},\alpha_{2}\in H_{\gamma}}\left( \rho_{\gamma}\right) _{\alpha_{1}\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha_{2}\alpha_{1}}$, where $H_{\gamma}$ is the subspace of the reduction $\gamma$. We separate the summation into diagonal and off-diagonal parts. For the diagonal part we use the normalization condition to evaluate its value.$$\begin{aligned} & tr\left( \rho_{\gamma}^{2}\right) \nonumber\\ & =\sum_{\alpha_{1}=\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha _{1}\alpha_{1}}^{2}+\sum_{\alpha_{1}\not =\alpha_{2}}\left( \rho_{\gamma }\right) _{\alpha_{1}\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha _{2}\alpha_{1}}\nonumber\\ & =\left( \sum_{\alpha}(\rho_{\gamma})_{\alpha\alpha}\right) ^{2}-\sum_{\alpha_{1}\not =\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha _{1}\alpha_{1}}\left( \rho_{\gamma}\right) _{\alpha_{2}\alpha_{2}}\nonumber\\ & +\sum_{\alpha_{1}\not =\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha_{1}\alpha_{2}}\left( \rho_{\gamma}\right) _{\alpha_{2}\alpha_{1}}\nonumber\\ & =1-\sum_{\substack{\alpha_{1}\not =\alpha_{2}\in H_{\gamma}\\\beta _{1},\beta_{2}\in H_{\bar{\gamma}}}}\left\vert c_{\alpha_{1}\otimes\beta_{1}}\right\vert ^{2}\left\vert c_{\alpha_{2}\otimes\beta_{2}}\right\vert ^{2}\nonumber\\ & +\sum_{\substack{\alpha_{1}\not =\alpha_{2}\in H_{\gamma}\\\beta_{1},\beta_{2}\in H_{\bar{\gamma}}}}c_{\alpha_{1}\otimes\beta_{1}}c_{\alpha _{2}\otimes\beta_{1}}^{\ast}c_{\alpha_{2}\otimes\beta_{2}}c_{\alpha_{1}\otimes\beta_{2}}^{\ast}.\end{aligned}$$ By exchanging the indices $\alpha_{1}$ and $\alpha_{2}$ one has $$\begin{aligned} tr\left( \rho_{\gamma}^{2}\right) & =1-\frac{1}{2}\sum_{\substack{\alpha _{1}\not =\alpha_{2}\in H_{\gamma}\\\beta_{1},\beta_{2}\in H_{\bar{\gamma}}}}\left\vert c_{\alpha_{1}\otimes\beta_{1}}c_{\alpha_{2}\otimes\beta_{2}}-c_{\alpha_{1}\otimes\beta_{2}}c_{\alpha_{2}\otimes\beta_{1}}\right\vert ^{2}\nonumber\\ & =1-\frac{1}{2}\sum_{\eta_{1},\eta_{2}\in\mathbb{N}_{d^{n}}}\left\vert c_{\eta_{1}}c_{\eta_{2}}-c_{\eta_{1}^{\gamma}}c_{\eta_{2}^{\gamma}}\right\vert ^{2},\end{aligned}$$ where $\eta=\alpha\otimes\beta$ and $\left( \eta_{1}^{\gamma},\eta _{2}^{\gamma}\right) =P_{\gamma}\left( \eta_{1},\eta_{2}\right) $. The linear entropy is then calculated to$$S_{L}\left( \rho_{\gamma}\right) =\sum_{\eta_{1}\not =\eta_{2},\in \mathbb{N}_{d^{n}}}\left\vert c_{\eta_{1}}c_{\eta_{2}}-c_{\eta_{1}^{\gamma}}c_{\eta_{2}^{\gamma}}\right\vert ^{2}.$$ ### An Important Inequality {#appx. ineq.sqrt} The following is an inequality, which is crucial for derivation of the prefactor $\sqrt{\frac{1}{|R|-N_R}}$ in the theorem \[theorem. lower bound of GME measure\]: $$\left\vert I\right\vert \sum_{i\in I}\left\vert a_{i}\right\vert ^{2}\geq\left\vert \sum_{i\in I}a_{i}\right\vert ^{2}. \label{eq. ineq. sqrt}$$ We prove this inequality by constructing two vectors as follows (using $|I|=n$)$$\vec{x}=\left( \begin{array} [c]{c}\left. \begin{array} [c]{c}a_{1}\\ \vdots\\ a_{1}\end{array} \right\} \text{n times}\\ \vdots\\ \left. \begin{array} [c]{c}a_{n}\\ \vdots\\ a_{n}\end{array} \right\} \text{n times}\end{array} \right) ,\vec{y}=\left( \begin{array} [c]{c}\begin{array} [c]{c}a_{1}^{\ast}\\ \vdots\\ a_{n}^{\ast}\end{array} \\ \vdots\\\begin{array} [c]{c}a_{1}^{\ast}\\ \vdots\\ a_{n}^{\ast}\end{array} \end{array} \right) .$$ The right hand side of \[eq. ineq. sqrt\] can be written as the scalar product of $\vec{x}$ and $\vec{y}$.$$\left\vert \sum_{i\in I}a_{i}\right\vert ^{2}=\sum_{i,j\in I}a_{i}a_{j}^{\ast }=\left\vert \vec{x}\cdot\vec{y}\right\vert .$$ According to the Cauchy-Schwarz inequality, one can derive$$\left\vert I\right\vert \sum a_{i}^{2}=\left\vert \vec{x}\right\vert \cdot\left\vert \vec{y}\right\vert \geq\left\vert \vec{x}\cdot\vec {y}\right\vert =\left\vert \sum_{i\in I}a_{i}\right\vert ^{2} .$$ Proof of Theorem \[theorem. lower bound of GME measure\] and Approach of Construction of a GME Witness {#appx. proof of lower bound on GME measure} ------------------------------------------------------------------------------------------------------ Firstly one can estimate the lower bound on $S_{L}\left( \rho_{\gamma }\right) $ by summing its elements over a selected Region $R$, and dropping the other non-negative summands (i.e. lower bounding them with $0$),$$\begin{aligned} S_{L}\left( \rho_{\gamma}^{i}\right) & \geq4\sum_{\left( \eta_{1},\eta_{2}\right) \in R}\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}-c_{\eta_{1}^{\gamma}}^{i}c_{\eta_{2}^{\gamma}}^{i}\right\vert ^{2}\label{eq.: linear_entropy_lower_bound_1}\\ & =4\sum_{\left( \eta_{1},\eta_{2}\right) \in R\backslash R^{\gamma}}\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}-c_{\eta_{1}^{\gamma}}^{i}c_{\eta_{2}^{\gamma}}^{i}\right\vert ^{2}.\nonumber\end{aligned}$$ Here we add a prefactor $4$ in eq.(\[eq.: linear\_entropy\_lower\_bound\_1\]), since the symmetric factor of all $(\eta_{1},\eta_{2})$ equals $4$. That means for every $(\eta_{1},\eta_{2})$ there are three other $\left( \tilde{\eta }_{1},\tilde{\eta}_{2}\right) $ having the same value of $|c_{\tilde{\eta }_{1}}c_{\tilde{\eta}_{2}}-c_{\tilde{\eta}_{1}^{\gamma}}c_{\tilde{\eta}_{2}^{\gamma}}|$ as $(\eta_{1},\eta_{2})$. Here we choose a non-degenerate vector basis set $R$, and therefore need a prefactor $4$ in the lower bound. The set $R^{\gamma}$ is the subset of $R$, whose elements do not contribute to the linear entropy, i.e. $R^{\gamma}:=\left\{ \left( \eta_{1},\eta _{2}\right) \in R:(\eta_{1}^{\gamma},\eta_{2}^{\gamma})=\left( \eta_{1},\eta_{2}\right) \text{ or }\left( \eta_{2},\eta_{1}\right) \right\} $. Now we use the inequality (\[eq. ineq. sqrt\]) to bound the square root of $S_{L}\left( \rho_{\gamma}^{i}\right) $.$$\begin{aligned} S_{L}\left( \rho_{\gamma}^{i}\right) & \geq\frac{4}{\left\vert R\backslash R^{\gamma}\right\vert }\left( \sum_{\eta_{1},\eta_{2}\in R}\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}-P_{\gamma}c_{\eta_{1}}^{i}c_{\eta}^{i}\right\vert \right) ^{2}\label{eq. ineq._for_linear_entropy},\\ & \Downarrow\nonumber\\ \sqrt{S_{L}\left( \rho_{\gamma}^{i}\right) } & \geq2\sqrt{\frac {1}{\left\vert R\right\vert -\left\vert R^{\gamma}\right\vert }}\sum_{\left( \eta_{1},\eta_{2}\right) \in R}\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}-c_{\eta_{1}^{\gamma}}^{i}c_{\eta_{2}^{\gamma}}^{i}\right\vert .\label{result. lower bound on linear entropy}$$ According to eq.(\[definitionroof\]) together with eq.(\[result. lower bound on linear entropy\]), the lower bound reads where $\gamma_{i}$ is the partition in which the linear entropy $S_{L}\left( \left\vert \psi_{i}\right\rangle \left\langle \psi _{i}\right\vert _{\gamma}\right) $ of $\left\vert \psi_{i}\right\rangle \left\langle \psi_{i}\right\vert $ has its minimum. By defining the normalization factor $N_{R}:=\min_{\gamma}\left\vert R^{\gamma}\right\vert$ , which is the minimal value of $\left\vert R^{\gamma}\right\vert $ over all possible bipartitions $\left\{ \gamma|\bar{\gamma}\right\} $, we can extract the prefactor from the convex roof summation. The most difficult part of detecting entanglement of mixed states is a result of the mixing of the decomposition coefficients $c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}$. In the lab we have only the information about the mixed density matrix element $\rho_{\eta_{1}\eta_{2}}$ but not $c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}$, therefore we must exchange the two summations in eq.(\[eq. proof of general lower bound on GME measure\]), and mix the coefficients $c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}$ into density matrix elements. Therefore we estimate the summands with a bound, which is independent of the specific partition $\gamma_{i}|\bar{\gamma}_{i}$, by adding a summation of non-positive terms $\sum_{R^{\gamma_{i}}}\left[ \left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}\right\vert -\frac{1}{2}\left( \left\vert c_{\eta_{1}}^{i}\right\vert ^{2}+\left\vert c_{\eta_{2}}^{i}\right\vert ^{2}\right) \right] $ into the summands. where $I\left( R\right) :=\left\{ \eta\in\mathbb{N}_{d}^{\otimes n}:\exists\left( \eta,\eta^{\prime}\right) \text{ or }\left( \eta^{\prime },\eta\right) \in R\right\} $ is the set of indices contained in the set $R$, $\Gamma\left( \eta_{1},\eta_{2}\right) =\left\{ \gamma|P\left( \eta _{1},\eta_{2}\right) \not \in R\right\} $ and $n_{\eta}^{\gamma_{i}}$ is the number of vector pairs in $R^{\gamma_{i}}$ containing index $\eta$. In order to eliminate the dependence of the partition $\gamma^{i}$, we define the maximal value of $n_{\eta}^{\gamma}$ over all possible partitions $\left\{ \gamma |\bar{\gamma}\right\} $ as $N_{\eta}:=\max_{\gamma}n_{\eta}^{\gamma}$. Then one can estimate the GME measure with eq.(\[eq. proof of general lower bound on GME measure\] and \[eq. proof. lower bound on GME measure\]) as Now one can safely exchange the summation in eq.(\[eq. proof of general lower bound on GME measure-1\]) and lower bound it with the triangle inequality (i.e. $\sum_{p_{i}}p_{i}\left\vert c_{\eta_{1}}^{i}c_{\eta_{2}}^{i}\right\vert \geq\left\vert \rho_{\eta_{1}\eta_{2}}\right\vert $) and the Cauchy-Schwarz inequality (i.e. $\sum_{p_{i}}p_{i}\left\vert c_{\eta_{1}^{\gamma}}^{i}c_{\eta_{2}^{\gamma}}^{i}\right\vert \leq\sqrt{\rho_{\eta_{1}^{\gamma}\eta_{1}^{\gamma}}\rho_{\eta_{2}^{\gamma}\eta_{2}^{\gamma}}}$). Finally we arrive at the result where $\rho_{\eta_{1}\eta_{2}}:=\langle\eta_{1}|\rho|\eta_{2}\rangle$. Above is the proof of theorem \[theorem. lower bound of GME measure\] in the case of $N_{R}:=\min_{\gamma}\left\vert R^{\gamma}\right\vert$. For the choice of $N_{R}:=\max_{\gamma}\left\vert R^{\gamma}\right\vert$, one just needs to calculate $\max_{\gamma}\left\vert R^{\gamma}\right\vert$ at the first step, i.e. eq.(\[eq.: linear\_entropy\_lower\_bound\_1\]), then pick up $|R|-\max_{\gamma}\left\vert R^{\gamma}\right\vert$ elements from $R\backslash R^{\gamma}$ as summation region in the second line and then repeat the whole proof above. At the end we will attain the same expression for the lower bound on $E_{m}$ as eq.(\[eq. proof of witness constructions\]), but with different $N_\eta$ from the ones before $N_{R}=\min_{\gamma}\left\vert R^{\gamma}\right\vert$. $N_\eta$ in this maximum choice is greater or equal to the one derived in the minimal-case. In the four-qubit singlet example in sec.\[sec. four-qubit-example\], the value of $N_\eta$ is exactly the same for both choices. Therefore we choose the maximum, i.e. $N_R=2$, to get a tighter lower bound on $E_m$. Explicit decomposition of the bipartite witness into local observables {#GellMann} ---------------------------------------------------------------------- The measurements needed to ascertain the relevant density matrix elements in the bipartite scenario can be performed in a basis consisting of a tensor product of the generalized Gell-Mann matrices. We continue to provide for each of the density matrix elements above their respective coefficients. The density matrix elements are either off diagonal elements or diagonal elements. The off-diagonal elements can be obtained by expectation values of the symmetric and antisymmetric generalized Gell-Mann matrices: $$\begin{aligned} \Lambda_{s}^{12} & = & \left(\begin{smallmatrix} 0&1&0\\ 1&0&0 \\ 0&0&0 \end{smallmatrix}\right),& \Lambda_{s}^{13} & = & \left(\begin{smallmatrix} 0&0&1\\ 0&0&0 \\ 1&0&0 \end{smallmatrix}\right),& \Lambda_{s}^{23} & = & \left(\begin{smallmatrix} 0&0&0\\ 0&0&1 \\ 0&1&0 \end{smallmatrix}\right),&\\ \Lambda_{a}^{12} & = & \left(\begin{smallmatrix} 0&-i&0\\ i&0&0 \\ 0&0&0 \end{smallmatrix}\right),& \Lambda_{a}^{13} & = & \left(\begin{smallmatrix} 0&0&-i\\ 0&0&0 \\ i&0&0 \end{smallmatrix}\right),& \Lambda_{a}^{23} & = & \left(\begin{smallmatrix} 0&0&0\\ 0&0&-i \\ 0&-i&0 \end{smallmatrix}\right).& \end{aligned}$$ They can be written as follows: $$\begin{aligned} \Re e\left[\left\langle 00 |\rho |11 \right\rangle\right] & = \frac{1}{2}\left\langle\Lambda_{s}^{12} \otimes \Lambda_{s}^{12}-\Lambda_{a}^{12} \otimes \Lambda_{a}^{12}\right\rangle , \\ \Re e\left[\left\langle 00 |\rho| 22 \right\rangle\right] & = \frac{1}{2}\left\langle\Lambda_{s}^{13} \otimes \Lambda_{s}^{13}-\Lambda_{a}^{13} \otimes \Lambda_{a}^{13}\right\rangle , \\ \Re e\left[\left\langle 11 |\rho| 22 \right\rangle\right] & = \frac{1}{2}\left\langle\Lambda_{s}^{23} \otimes \Lambda_{s}^{23}-\Lambda_{a}^{23} \otimes \Lambda_{a}^{23}\right\rangle . \end{aligned}$$ We now consider the terms obtained via the diagonal generalized Gell-Mann matrices. $\Lambda_{d}^{0} = \left(\begin{smallmatrix} 1&0&0\\ 0&1&0 \\ 0&0&1 \end{smallmatrix}\right)$, $\Lambda_{d}^{1} = \left(\begin{smallmatrix} 1&0&0\\ 0&-1&0 \\ 0&0&0 \end{smallmatrix}\right)$, $\Lambda_{d}^{2} = \frac{1}{\sqrt{3}}\left(\begin{smallmatrix} 1&0&0\\ 0&1&0 \\ 0&0&-2 \end{smallmatrix}\right)$. We will expand the soughtafter terms into coefficients, utilizing the following basis: $$\begin{aligned} b=\left(\begin{matrix} \Lambda_{d}^{0} \otimes \Lambda_{d}^{0} \\ \Lambda_{d}^{0} \otimes \Lambda_{d}^{1} \\ \Lambda_{d}^{0} \otimes \Lambda_{d}^{2} \\ \Lambda_{d}^{1} \otimes \Lambda_{d}^{0} \\ \Lambda_{d}^{1} \otimes \Lambda_{d}^{1} \\ \Lambda_{d}^{1} \otimes \Lambda_{d}^{2} \\ \Lambda_{d}^{2} \otimes \Lambda_{d}^{0} \\ \Lambda_{d}^{2} \otimes \Lambda_{d}^{1} \\ \Lambda_{d}^{2} \otimes \Lambda_{d}^{2} \\ \end{matrix}\right). \end{aligned}$$ For further reference the coefficients are given as: $$\begin{aligned} \left\langle 01| \rho| 01 \right\rangle & = \left\langle b*\left( \frac{1}{9},\frac{1}{6},\frac{1}{6\sqrt{3}}, -\frac{1}{6},-\frac{1}{4},-\frac{1}{8\sqrt{3}},\frac{1}{6\sqrt{3}},\frac{1}{4\sqrt{3}},\frac{1}{12} \right)\right\rangle,\\ \left\langle 10 |\rho |10 \right\rangle & = \left\langle b*\left( \frac{1}{9},-\frac{1}{6},\frac{1}{6\sqrt{3}},\frac{1}{6},-\frac{1}{4},\frac{1}{8\sqrt{3}},\frac{1}{6\sqrt{3}},-\frac{1}{4\sqrt{3}},\frac{1}{12} \right)\right\rangle,\\ \left\langle 02 |\rho |02 \right\rangle & = \left\langle b*\left( \frac{1}{9},\frac{1}{6},\frac{1}{6\sqrt{3}},0,0,0,-\frac{1}{3\sqrt{3}},-\frac{1}{2\sqrt{3}},-\frac{1}{6} \right)\right\rangle,\\ \left\langle 20| \rho| 20 \right\rangle & = \left\langle b*\left( \frac{1}{9},0,-\frac{1}{3\sqrt{3}},\frac{1}{6},0,-\frac{1}{4\sqrt{3}},\frac{1}{6\sqrt{3}},0,-\frac{1}{6} \right)\right\rangle,\\ \left\langle 12 |\rho |12 \right\rangle & = \left\langle b*\left( \frac{1}{9},-\frac{1}{6},\frac{1}{6\sqrt{3}},0,0,0,-\frac{1}{3\sqrt{3}},\frac{1}{2\sqrt{3}},-\frac{1}{6} \right)\right\rangle,\\ \left\langle 21| \rho |21 \right\rangle & = \left\langle b*\left( \frac{1}{9},0,-\frac{3}{3\sqrt{3}},-\frac{1}{6},0,\frac{1}{4\sqrt{3}},\frac{1}{6\sqrt{3}},0,-\frac{1}{6} \right)\right\rangle. \end{aligned}$$ Explicit form of the GME witness $Q_m^{(d)}$ {#DickeWitness} -------------------------------------------- Here we recall the explicit form of the nonlinear witness from Ref. [@shgh1]. Using the notation for Dicke states introduced in section \[Dicke\] we arrive at the following lower bound $$Q_{m}^{\left( d\right) }=\frac{1}{m}\left[ \sum_{l,l^{\prime}=0}^{d-2}\sum_{\sigma}\left( \left\vert \left\langle \alpha^{l}\left\vert \rho\right\vert \beta^{l^{\prime}}\right\rangle \right\vert -\sum_{\delta \in\Delta}\sqrt{\left\langle \alpha^{l}\right\vert \otimes\left\langle \beta^{l^{\prime}}\right\vert P_{\delta}^{\dagger}\rho^{\otimes2}P_{\delta }\left\vert \alpha^{l}\right\rangle \otimes\left\vert \beta^{l^{\prime}}\right\rangle }\right) -N_{D}\sum_{l=0}^{d-2}\sum_{\alpha}\left\langle \alpha^{l}\left\vert \rho\right\vert \alpha^{l}\right\rangle \right], \label{eq.: GME-witness formular}$$ with$$\begin{aligned} m & \in\left\{ 1,\cdots,\left\lfloor n/2\right\rfloor \right\} ,N_{D}=\left( d-1\right) m\left( n-m-1\right) \nonumber,\\ \sigma & :=\left\{ \left( \alpha,\beta\right) :\left\vert \alpha\cap \beta\right\vert =m-1\right\} \nonumber,\\ \Delta & :=\left\{ \begin{array} [c]{cc}\alpha & ,l'=l\\ \left\{ \delta|\delta\subset\overline{\alpha\backslash\beta}\right\} & ,l'<l\\ \left\{ \delta|\delta\subset\overline{\beta\backslash\alpha}\right\} & ,l'>l \end{array} \right. .\label{def. GME witness - permutation set}$$ The properties of this witness are discussed in the main text.

--- abstract: 'A new algorithm for deciding the satisfiability of polynomial formulas over the reals is proposed. The key point of the algorithm is a new projection operator, called sample-cell projection operator, custom-made for Conflict-Driven Clause Learning (CDCL)-style search. Although the new operator is also a CAD (Cylindrical Algebraic Decomposition)-like projection operator which computes the cell (not necessarily cylindrical) containing a given sample such that each polynomial from the problem is sign-invariant on the cell, it is of singly exponential time complexity. The sample-cell projection operator can efficiently guide CDCL-style search away from conflicting states. Experiments show the effectiveness of the new algorithm.' author: - Haokun Li - Bican Xia bibliography: - 'samplecad.bib' title: 'Solving Satisfiability of Polynomial Formulas By Sample-Cell Projection ' --- Introduction ============ The research on SMT (Satisfiability Modulo Theories) [@deMoura+Dutertre+Shankar:cav2007; @DBLP:journals/cacm/MouraB11; @DBLP:series/faia/2009-185] in recent years brings us many popular solvers such as Z3 [@DBLP:conf/tacas/MouraB08], CVC4 [@BCD+11], Yices [@Dutertre:cav2014], MathSAT5 [@mathsat5], etc. Nevertheless, in theory and practice, it is important to design efficient SMT algorithms and develop tools (or improve existing ones) for many other theories, [*e.g.*]{} string [@DBLP:conf/cav/LiangRTBD14], linear arithmetic [@DBLP:conf/cav/DutertreM06; @DBLP:conf/cade/JovanovicM12] and non-linear arithmetic [@DBLP:journals/corr/abs-1905-09227; @DBLP:conf/smt/KorovinKS14] over the reals. A straightforward idea is to integrate Conflict-Driven Clause Learning (CDCL)-style search with theory solvers [@DBLP:series/faia/2009-185]. For example, integrating CDCL-style search with a theory solver for determining whether a basic semialgebraic set is empty can solve satisfiability in the theory of non-linear arithmetic over the reals. It is well-known that the problem whether a basic semialgebraic set is empty is decidable due to Tarski’s decision procedure [@10.1007/978-3-7091-9459-1_3]. Tarski’s algorithm cannot be a theory solver in practice because of its very high complexity. Cylindrical algebraic decomposition (CAD) algorithm [@DBLP:journals/cca/Collins76] is a widely used theory solver in practice though it is of doubly exponential time complexity. The idea of CAD algorithm is to decompose ${\mathbb{R}}^n$ into cells such that each polynomial from the problem is sign-invariant in every cell. A key concept in CAD algorithm is the projection operator. Although many improved projection operators have been proposed [@DBLP:conf/issac/Hong90; @DBLP:journals/jsc/McCallum88; @10.1007/978-3-7091-9459-1_12; @brown_improved_2001; @Han_Dai_Xia_2014; @Dai_Han_Hong_Xia_2015; @Xia_Yang_2016], the CAD method is still of doubly exponential time complexity. The main reason is that in order to carry enough information, projection of variables causes the number of polynomials grows rapidly. So the cost of simply using CAD as a theory solver is unacceptable. Jovanovic and de Moura [@DBLP:conf/cade/JovanovicM12] eased the burden of using CAD as a theory solver by modifying the CDCL-style search framework. They changed the sequence of search states by adding variable assignments to the sequence. The benefit of this is that they can use real-root isolation, which is of polynomial time complexity, to check consistency of literals for there will be only one unassigned variable in the literals of the current state. When a conflict of literals is detected, they explain the conflict by applying CAD to a polynomial set called conflicting core to find the cell where the sample of assignments belongs. But even using CAD only when explaining conflicts is a huge computational cost, as CAD is of doubly exponential time complexity. Furthermore, CAD will produce all cells in ${\mathbb{R}}^n$ other than the only one we need, making computation waste. In this paper, we propose a new custom-made CAD-like projection operator, called sample-cell projection operator. It only processes the cell containing a given sample, which is exactly what conflict explanation needs. The idea of our operator is trying to project polynomials related to the target cell and ignore irrelevant polynomials. We integrate our sample-cell projection operator with Jovanovic’s improved CDCL-style search framework. The new operator can efficiently guide CDCL-style search away from conflicting states. It is proved that the new algorithm is of singly exponential time complexity. We have implemented a prototype solver LiMbS which is base on Mathematica 12. Experiments show the effectiveness of the new algorithm. The rest of this paper is structured as follows: Section \[sec:pre\] introduces the background knowledge and notation. Section \[sec:sample\] defines sample-cell projection and presents the details of our approach. Section \[sec:cdcl\] describes the CDCL-style search framework which we adopt. We evaluate our approach on many well-known examples and analyze its performance in Section \[sec:exp\]. The paper is concluded in Section \[sec:conclusion\]. Notation {#sec:pre} ======== Let ${\mathbb{R}}$ denote the field of real numbers, ${\mathbb{Z}}$ denote the ring of integers and ${\mathbb{Q}}$ denote the field of rational numbers. Unless stated otherwise, we assume that all polynomials in this paper are in ${\mathbb{Z}}[\bar{x}]$, the ring of multivariate polynomials in variables $\bar{x}$ with integer coefficients. For a polynomial $f\in {\mathbb{Z}}[\bar{y},x]$: $$f(\bar{y},x)=a_mx^m+a_{m-1}x^{m-1}+\ldots+a_1x+a_0$$ where $a_m\neq0$ and $a_i\in {\mathbb{Z}}[\bar{y}]$ for $i=0,...,m$, the [*degree*]{} of $f$ with respect to (w.r.t.) $x$ is $m$, denoted by ${\mathtt{deg}}(f,x)$. The [*leading coefficient*]{} of $f$ w.r.t. $x$ is $a_m$, denoted by ${\mathtt{lc}}(f,x)$ and the [*leading term*]{} of $f$ w.r.t. $x$ is $a_mx^{m}$, denoted by ${\mathtt{lt}}(f,x)$. Let $${\mathtt{coeff}}(f, x)=\{a_i|0\leq i \leq m \land a_i\neq 0\}$$ denote the [*set of coefficients*]{} of $f$ w.r.t. $x$ and ${\mathtt{var}}(f)=\{\bar{y},x\}$ denote the variables appearing in $f$. Suppose $g\in {\mathbb{Z}}[\bar{y},x]$: $$g(\bar{y},x)=b_nx^{n}+b_{n-1}x^{n-1}+\ldots+b_1x+b_0$$ where $b_n\neq 0$ and $b_i\in {\mathbb{Z}}[\bar{y}]$ for $i=0,...,n$ . Let ${\mathtt{res}}(f,g,x)$ denote the Sylvester [*resultant*]{} of $f$ and $g$ w.r.t. $x$, [*i.e.*]{} the determinant of the following matrix $$\left(\begin{array}{cccccccc} a_m & a_{m-1}& a_{m-2}& \ldots& a_0 & 0 & \ldots & 0 \\ 0 & a_m &a_{m-1} & \ldots& a_1 & a_0 & \ldots & 0 \\ \vdots & \vdots &\ddots & \ddots&\ddots &\ddots& \ddots & \vdots \\ 0 & 0 &\ldots & a_m &a_{m-1}&\ldots& \ldots & a_0 \\ b_n & b_{n-1}& b_{n-2}& \ldots& b_0 & 0 & \ldots & 0 \\ 0 & b_n &b_{n-1} & \ldots& b_1 & b_0 & \ldots & 0 \\ \vdots & \vdots &\ddots & \ddots&\ddots &\ddots& \ddots & \vdots \\ 0 & 0 &\ldots & b_n &b_{n-1}&\ldots& \ldots & b_0 \\ \end{array}\right)$$ which has $n$ rows of $a_i$ and $m$ rows of $b_j$. The discriminant of $f$ w.r.t. $x$ is $${\mathtt{disc}}(f,x)=(-1)^\frac{m(m-1)}{2}{\mathtt{res}}(f,f',x).$$ An [*atomic polynomial constraint*]{} is $f\triangleright0$ where $f$ is a polynomial and $\triangleright\in \{\geq,>,=\}$. A [*polynomial literal*]{} (simply [*literal*]{}) is an atomic polynomial constraint or its negation. For a literal $l$, ${\mathtt{poly}}(l)$ denotes the polynomial in $l$ and ${\mathtt{var}}(l)={\mathtt{var}}({\mathtt{poly}}(l))$. A [*polynomial clause*]{} is a disjunction $l_1\lor\cdots\lor l_s$ of literals. Sometimes, we write a clause as $\lnot(\bigwedge_i l_i)\lor \bigvee_j l_j$. A [*polynomial formula*]{} is a conjunction of clauses. An [*extended polynomial constraint*]{} $l$ is $x\triangleright {\mathtt{Root}}(f,k)$ where $\triangleright\in \{\geq,>,=\}$, $f\in{\mathbb{Z}}[\bar{y},u]$ with $x\not\in {\mathtt{var}}(f)$ and $k (0\leq k\leq {\mathtt{deg}}(f,u))$ is a given integer. Notice the variable $u$ is an exclusive free variable that cannot be used outside the ${\mathtt{Root}}$ object. For a formula $\phi$, $\phi[a/x]$ denote the resulting formula via substituting $a$ for $x$ in $\phi$. For variables $\bar{x}=(x_1,\ldots,x_r)$ and $\bar{a}=(a_1,\ldots,a_r)\in{\mathbb{R}}^r$, a mapping $\alpha$ which maps $x_i$ to $a_i$ for $i=1,...,r$ is called a [*variable assignment*]{} of $\bar{x}$ and $\bar{a}$ is called a [*sample*]{} of $\alpha$ or a [*sample*]{} of $\bar{x}$ in ${\mathbb{R}}^r$. We denote $\phi[a_1/x_1,\ldots,a_r/x_r]$ by $\alpha(\phi)$. If $\alpha(\phi)=0$, we say $\phi$ vanishes under $\alpha$ or vanishes under $\bar{a}$. Suppose an extended polynomial constraint $l$ is of the form $x\triangleright {\mathtt{Root}}(f,k)$ and $\alpha$ is a variable assignment of $(\bar{y},x)$. If $\beta_k$ is the $k$th real root of $\alpha(f)$, $\alpha(l)$ is defined to be $\alpha(x)\triangleright \beta_k$. If $\alpha(f)$ has less than $k$ real roots, $\alpha(l)$ is defined to be [False]{}. Sample-Cell Projection {#sec:sample} ====================== In this section, we first introduce some well-known concepts and results concerning CAD and then define the so-called sample-cell projection operator. Let $f$ be an analytic function defined in some open set $U$ of $K^n$ where $K$ is a field. For a point $p\in U$, if $f$ or some partial derivative (pure and mixed) of $f$ of some order does not vanish at $p$, then we say that $f$ has [*order*]{} $r$ where $r$ is the least non-negative integer such that some partial derivative of total order $r$ does not vanish at $p$. Otherwise, we say $f$ has infinite order at $p$. The order of $f$ at $p$ is denoted by ${\mathtt{order}}_pf$. We say $f$ is [*order-invariant*]{} in a subset $S\subset U$ if ${\mathtt{order}}_{p_1}f={\mathtt{order}}_{p_2}f$ for any $p_1, p_2\in S$. Obviously, if $K={\mathbb{R}}$ and the analytic function $f$ is order-invariant in $S$, then $f$ is sign-invariant in $S$. An $r$-variable polynomial $f(\bar{x},x_r)$ where $\bar{x}=(x_1,\ldots,x_{r-1})$ is said to be [*analytic delineable*]{} on a connected $s$-dimensional submanifold $S\subset {\mathbb{R}}^{r-1}$ if 1. The number $k$ of different real roots of $f(a,x_r)$ is invariant for any point $a\in S$. And the trace of the real roots are the graphs of some pairwise disjoint analytic functions $\theta_1<\ldots<\theta_k$ from $S$ into ${\mathbb{R}}$ ([*i.e.*]{} the order of real roots of $f(a,x_r)$ is invariant for all point $a\in S$); 2. There exist positive integers $m_1,\ldots,m_k$ such that for every point $a\in S$, the multiplicity of the real root $\theta_i(a)$ of $f(a,x_r)$ is $m_i$ for $i=1,...,k$. Especially, if $f$ has no zeros in $S\times {\mathbb{R}}$, then $f$ is delineable on $S$ with $k=0$. The analytic functions $\theta_i$’s are called the [*real root functions*]{} of $f$ on $S$, the graphs of the $\theta_i$’s are called the [*$f$-sections*]{} over $S$, and the connected regions between two consecutive $f$-sections (for convenience, let $\theta_0=-\infty$ and $\theta_{k+1}=+\infty$) are called [*$f$-sectors*]{} over $S$. Each $f$-section over $S$ is a connected $s$-dimensional submanifold in ${\mathbb{R}}^r$ and each $f$-sector over $S$ is a connected $(s+1)$-dimensional submanifold in ${\mathbb{R}}^r$. \[thm:mc\] Let $r\geq 2$ and $f(\bar{x},x_r)$ be a polynomial in ${\mathbb{R}}[\bar{x},x_r]$ of positive degree where $\bar{x}=(x_1,...,x_{r-1})$. Let $S$ be a connect submanifold of ${\mathbb{R}}^{r-1}$ where $f$ is degree-invariant and does not vanish identically. Suppose that ${\mathtt{disc}}(f,x_r)$ is a nonzero polynomial and is order-invariant in $S$. Then $f$ is analytic delineable on $S$ and is order-invariant in each $f$-section over $S$. Suppose $a=(\bar{a},a_n)=(a_1,\ldots,a_n)$ is a sample of $(\bar{x},x_n)$ in ${\mathbb{R}}^n$ and $F=\{f_1(\bar{x},x_n)$ $,\ldots , f_r(\bar{x},x_n)\}$ is a polynomial set in ${\mathbb{Z}}[\bar{x},x_n]$ where $\bar{x}=(x_1,\ldots,x_{n-1})$. Consider the real roots of polynomials in $\{f_1(\bar{a},x_n),\ldots,f_r(\bar{a},x_n)\}\setminus \{0\}$. Denote the $k$th real root of $f_i(\bar{a},x_n)$ by $\theta_{i,k}$. We define two concepts: the [*sample polynomials set*]{} of $a$ in $F$ (denoted by ${\mathtt{s\_poly}}(F,x_n,a)$) and the [*sample interval*]{} of $a$ in $F$ (denoted by ${\mathtt{s\_interval}}(F,x_n,a)$) as follows. If there exists $\theta_{i,k}$ such that $\theta_{i,k}=a_n$ then $$\begin{aligned} {\mathtt{s\_poly}}(F,x_n,a)&=\{f_i\},\\ {\mathtt{s\_interval}}(F,x_n,a)&=(x_n={\mathtt{Root}}(f_i(\bar{x},u),k));\end{aligned}$$ If there exist two consecutive real roots $\theta_{i_1,k_1}$ and $\theta_{i_2,k_2}$ such that $\theta_{i_1,k_1}<a_n<\theta_{i_2,k_2}$ then $$\begin{aligned} {\mathtt{s\_poly}}(F,x_n,a)&=\{f_{i_1},f_{i_2}\},\\ {\mathtt{s\_interval}}(F,x_n,a)&={\mathtt{Root}}(f_{i_1}(\bar{x},u),k_1)<x_n<{\mathtt{Root}}(f_{i_2}(\bar{x},u),k_2);\end{aligned}$$ If there exists $\theta_{i',k'}$ such that $a_n>\theta_{i',k'}$ and for all $\theta_{i,k}$ $\theta_{i',k'}\geq\theta_{i,k}$ then $$\begin{aligned} {\mathtt{s\_poly}}(F,x_n,a)&=\{f_{i'}\},\\ {\mathtt{s\_interval}}(F,x_n,a)&=x_n>{\mathtt{Root}}(f_{i'}(\bar{x},u),k');\end{aligned}$$ If there exists $\theta_{i',k'}$ such that $a_n<\theta_{i',k'}$ and for all $\theta_{i,k}$ $\theta_{i',k'}\leq\theta_{i,k}$then $$\begin{aligned} {\mathtt{s\_poly}}(F,x_n,a)&=\{f_{i'}\},\\ {\mathtt{s\_interval}}(F,x_n,a)&=x_n<{\mathtt{Root}}(f_{i'}(\bar{x},u),k').\end{aligned}$$ Specially, if every polynomial in $\{f_1(\bar{a},x_n),\ldots,f_r(\bar{a},x_n)\}\setminus \{0\}$ does not have any real roots, define $$\begin{aligned} {\mathtt{s\_poly}}(F,x_n,a)&=\emptyset,\\ {\mathtt{s\_interval}}(F,x_n,a)&=\text{{\tt True}}.\end{aligned}$$ \[ex:s\] Let $F=\{f_1,f_2,f_3\}$ where $f_1=y+0.5x-10$, $f_2=y+0.01(x-9)^2-7$, $f_3=y-0.03x^2-1$ and $A=(4,9),B=(4,6.75),C=(4,4),D=(4,1)$. We have (see Figure \[fig:1\]) $$\begin{array}{ll} {\mathtt{s\_poly}}(F,y,A)=\{f_1\}, &{\mathtt{s\_interval}}(F,y,A)=y>{\mathtt{Root}}(f_1(x,u),1),\\ {\mathtt{s\_poly}}(F,y,B)=\{f_2\}, &{\mathtt{s\_interval}}(F,y,B)=y={\mathtt{Root}}(f_2(x,u),1),\\ {\mathtt{s\_poly}}(F,y,C)=\{f_2,f_3\},&{\mathtt{s\_interval}}(F,y,C)=\begin{array}{rl} & y>{\mathtt{Root}}(f_3(x,u),1) \\ \wedge & y<{\mathtt{Root}}(f_2(x,u),1) \end{array},\\ {\mathtt{s\_poly}}(F,y,D)=\{f_3\},&{\mathtt{s\_interval}}(F,y,D)=y<{\mathtt{Root}}(f_3(x,u),1).\\ \end{array}$$ (-0.5865830202854977,-0.6301322314049569) rectangle (12.486294515401967,10.113669421487607); plot(,[0-0.01\*(()-9)\^(2)+7]{}); plot(,[0-0.5\*()+10]{}); plot(,[0.03\*()\^(2)+1]{}); (5,-0.4) node\[scale=3\] [$x$]{} ; (-0.4,5) node\[scale=3\] [$y$]{} ; (1.4062674680691189,5.97677986476334) node\[scale=3\] [$f_2$]{} ; (1.0745740045078913,9.093459053343354) node\[scale=3\] [$f_1$]{}; (1.0062674680691189,1.5978918106686724) node\[scale=3\] [$f_3$]{}; (4,9) circle (2.5pt); (4.161726521412478,9.45251239669422) node\[scale=2\] [$A$]{}; (4.002201149761497,6.7502200665255465) circle (2.5pt); (4.161726521412478,7.198567993989485) node\[scale=2\] [$B$]{}; (4,4) circle (2.5pt); (4.161726521412478,4.44875582268971) node\[scale=2\] [$C$]{}; (4,1) circle (2.5pt); (4.161726521412478,1.4434966190833982) node\[scale=2\] [$D$]{}; Additionally, for a polynomial $$h=c_mx_n^{d_m}+c_{m-1}x_n^{d_{m-1}}+\ldots+c_{0}x_n^{d_0}$$ where $d_m>d_{m-1}>\cdots>d_0$, $c_i\in {\mathbb{R}}[\bar{x}]$ and $c_i\neq0$ for $i=0,...,m$. If there exists $j\ge 0$ such that $c_j(\bar{a})\neq 0$ and $c_i(\bar{a})=0$ for any $i>j$, then the [*sample coefficients*]{} of $h$ at $(\bar{a},a_n)$ is defined to be $\{c_m,c_{m-1},\ldots,c_j\}$, denoted by ${\mathtt{s\_coeff}}(h,x_n,(\bar{a},a_n))$. Otherwise ${\mathtt{s\_coeff}}(h,x_n,(\bar{a},a_n))=\{c_m,\ldots,c_0\}$. \[def:projsc\] Suppose $\bar{a}$ is a sample of $\bar{x}$ in ${\mathbb{R}}^n$ and $F=\{f_1,\ldots , f_r\}$ is a polynomial set in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_n)$. The [*sample-cell projection*]{} of $F$ on $x_n$ at $\bar{a}$ is $$\begin{split} {\mathtt{Proj}}_{sc}(F,x_n,\bar{a})= \bigcup_{f\in F}&{\mathtt{s\_coeff}}(f,x_n,\bar{a})\cup\\ \bigcup_{f\in F}&\{{\mathtt{disc}}(f,x_n)\}\cup\\ \bigcup_{\begin{subarray}{c}f\in F,g\in\\ {\mathtt{s\_poly}}(F,x_n,\bar{a}),\\f\neq g\end{subarray}}&\{{\mathtt{res}}(f,g,x_n)\} \end{split}$$ - If $f\in F$ and $x_n\not\in {\mathtt{var}}(f)$, $f$ is obviously an element of ${\mathtt{Proj}}_{sc}(F, x_n, \bar{a})$. - Computing ${\mathtt{Proj}}_{sc}(F,x_n,\bar{a})$ will produce $O(rn+3r)$ elements, so the time complexity of projecting all the variables by recursively using ${\mathtt{Proj}}_{sc}$ is $O((n+3)^nr)$. Now we prove the property of the new projection operator. A set of polynomials in ${\mathbb{Z}}[\bar{x}]$ is said to be a [*squarefree basis*]{} if the elements of the set have positive degrees, and are primitive, squarefree and pairwise relatively prime. For a connected submanifold $S$ of ${\mathbb{R}}^{n-1}$, we denote by $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$ $$\left\{(\alpha_1,\ldots,\alpha_n)\in {\mathbb{R}}^n~|~~\begin{array}{l} (\alpha_1,\ldots,\alpha_{n-1})\in S \\ \wedge~ {\mathtt{s\_interval}}(F,x_n,\bar{a})[\alpha_1/x_1,\ldots,\alpha_n/x_n] \end{array} \right\}.$$ \[th:sc\] Let $F$ be a finite squarefree basis in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_n)$ and $n\geq 2$. Let $\bar{a}=(a_1,\ldots,a_n)$ be a sample of $\bar{x}$ in ${\mathbb{R}}^n$ and $S$ be a connected submanifold of ${\mathbb{R}}^{n-1}$ such that $(a_1,\ldots,a_{n-1})\in S$. Suppose that each element of ${\mathtt{Proj}}_{sc}(F,x_n,\bar{a})$ is order-invariant in $S$. Then each element in $F$ either vanishes identically on $S$ or is analytic delineable on $S$, each section over $S$ of the element of $F$ which do not vanish identically on $S$ is either equal to or disjoint with $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$, and each element of $F$ either vanishes identically on $S$ or is order-invariant in $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$. For any $f\in F$, if $f$ vanishes identically on $S$, there is nothing to prove. So we may assume that any element in $F$ does not vanish identically on $S$. For any $f\in F$ such that $f\not\in{\mathtt{s\_poly}}(F,x_n,\bar{a})$, let $f'=f\cdot~\prod_{g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})} g$. Notice that $f'$ is degree-invariant on $S$ (each element of ${\mathtt{s\_coeff}}(f,x_n,\bar{a})$ is order-invariant, hence sign-invariant in $S$). And we have $$\begin{split} {\mathtt{disc}}(f',x_n)={\mathtt{disc}}(f,x_n)\cdot\prod_{g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})} {\mathtt{disc}}(g,x_n)\cdot\\ \prod_{g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})}{\mathtt{res}}(f,g,x_n)\cdot\\ \prod_{\begin{subarray}{c}g_1\in {\mathtt{s\_poly}}(F,x_n,\bar{a}),\\ g_2\in {\mathtt{s\_poly}}(F,x_n,\bar{a}),\\g_1\neq g_2\end{subarray}}{\mathtt{res}}(g_1,g_2,x_n). \end{split}$$ It follows from this equality that ${\mathtt{disc}}(f',x_n)\neq 0$ (because $f_i$’s are squarefree and pairwise relatively prime). Obviously, each factor of ${\mathtt{disc}}(f',x_n)$ is a factor of ${\mathtt{Proj}}_{sc}(F,x_n,\bar{a})$, so ${\mathtt{disc}}(f',x_n)$ is order-invariant in $S$. By Theorem \[thm:mc\], $f'$ is analytic delineable on $S$ and is order-invariant in each $f'$-section over $S$. So $f$ and $g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})$ are order-invariant in each $f'$-section over $S$. It follows that the sections over $S$ of $f$ and $g$ are pairwise disjoint. Therefore, $f$ and $g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})$ are analytic delineable on $S$, every section of them is either equal to or disjoint with $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$, and $f$ and $g$ are order-invariant in $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$. $\blacksquare$ \[rm:sc\] Notice that when $f$ vanishes identically on $S$, $f$ isn’t always order-invariant in $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$. This is avoidable by changing the ordering of variables and is negligible when the satisfiability set of formulas is full-dimensional. We find a way to handle this rare case: either to determine whether the coefficients of $f$ have finitely many common zeros, or to enlarge $F$ by adding partial derivatives of $f$ whose order is less than ${\mathtt{order}}(f)$ and one non-zero partial derivative whose order is exactly equal to ${\mathtt{order}}(f)$. When integrating the new projection operator with the CDCL-type search (see Section \[sec:cdcl\]), we need a traditional CAD projection operator [@DBLP:journals/jsc/McCallum88; @10.1007/978-3-7091-9459-1_12]. Suppose $F=\{f_1,\ldots , f_r\}$ is a polynomial set in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_n)$. The McCallum projection of $F$ on $x_n$ is $${\mathtt{Proj}}_{mc}(F)=\bigcup_{f\in F}\{{\mathtt{coeff}}(f),{\mathtt{disc}}(f,x_n)\}\cup\bigcup_{\begin{subarray}{c}f\in F,g\in F,\\f\neq g\end{subarray}} {\mathtt{res}}(f,g,x_n)$$ Notice that ${\mathtt{coeff}}$ can be replaced by ${\mathtt{s\_coeff}}$ when we have a sample of $n-1$ dimension. \[[@10.1007/978-3-7091-9459-1_12], Theorem 1\]\[th:mc\] Let $F$ be a finite squarefree basis in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_n)$ and $n\geq 2$ and $S$ be a connected submanifold of ${\mathbb{R}}^{n-1}$ such that each element of ${\mathtt{Proj}}_{mc}(F,x_n)$ is order-invariant in $S$. Then each element in $F$ either vanishes identically on $S$ or is analytic delineable on $S$, the sections over $S$ of the elements of $F$ which do not vanish identically on $S$ are pairwise disjoint, and each element of $F$ which does not vanish identically on $S$ is order-invariant in every such section. Now, let us use the following definition to describe the procedure of calculating sample cells. We denote by ${\mathtt{factor}}(A)$ the set of irreducible factors of all polynomials in $A$. Suppose $a=(a_1,\ldots,a_{n-1})$ is a sample of $(x_1,\ldots,x_{n-1})$ in ${\mathbb{R}}^{n-1}$ and $F=\{f_1,\ldots , f_r\}$ is a polynomial set in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_{n})$. The [*sample cell*]{} of $F$ at $a$ is $${\mathtt{s\_cell}}(F,a)={\mathtt{s\_interval}}(F_1,\alpha_1)\land\cdots\land{\mathtt{s\_interval}}(F_{n-1},\alpha_{n-1})$$ where $\alpha_{n-1}=a$, $F_{n-1}={\mathtt{factor}}({\mathtt{Proj}}_{mc}({\mathtt{factor}}(F)))$, $\alpha_i=(a_1,\ldots,a_i)$, and $F_i={\mathtt{factor}}({\mathtt{Proj}}_{sc}(F_{i+1},x_{i+1},\alpha_{i+1}))$ for $i=1,\ldots,{n-2}$. - It is a standard way to use ${\mathtt{factor}}$ to ensure that every $F_i$ is a finite squarefree basis. - Notice that the complexity of computing sample cell ${\mathtt{s\_cell}}$ depends on $\sum_{i=1}^{n-1}|F_i|$ where $|F_i|$ means the number of polynomials in $F_i$. From the recursive relationship $|F_{n-1}|=O(r^2+rn)$, $|F_i|<(3+i+1)|F_{i+1}|,i=1,\ldots,n-2$, it is not hard to know that the complexity of computing ${\mathtt{s\_cell}}$ is $O((r^2+rn)(2+n)^{n-1})$. Let $F=\{f_1(\bar{x},x_n),\ldots,f_r(\bar{x},x_n)\}$ be a polynomial set and $a\in {\mathbb{R}}^{n-1}$, where $\bar{x}=(x_1,\ldots,x_{n-1})$. If $$\forall b\in {\mathbb{R}}\;\bigvee_{i=1}^{r} f_i(a,b)\rhd_i 0,$$ where $\rhd_i\in \{>,\geq,=\}$, then $$\forall \alpha \in {\mathtt{s\_cell}}(\{f_1,\ldots,f_r\},a)\forall b \in {\mathbb{R}}\; \bigvee_{i=1}^{r} f_i(\alpha,b)\rhd_i 0.$$ It is a direct corollary of Theorem \[th:mc\] and Theorem \[th:sc\]. Suppose $f=ax^2+bx+c$ and $\alpha=(1,1,1)$ is a sample of $(a,b,c)$. Then $$\begin{aligned} &F_3={\mathtt{factor}}({\mathtt{Proj}}_{mc}(\{f\},x))={\mathtt{factor}}(\{b^2-4ac,a\})=\{b^2-4ac,a\},\\ &F_2={\mathtt{factor}}({\mathtt{Proj}}_{sc}(\{b^2-4ac,a\},c))={\mathtt{factor}}(\{1,a,-4a\})=\{a\},\\ &F_1={\mathtt{factor}}({\mathtt{Proj}}_{sc}(\{a\},b))=\{a\}.\end{aligned}$$ So $${\mathtt{s\_cell}}(\{f\},a)=c>{\mathtt{Root}}(b^2-4au,1)\land a>{\mathtt{Root}}(u,1),$$ and after simplification $${\mathtt{s\_cell}}(\{f\},a)=c>\frac{b^2}{4a}\land a>0.$$ CDCL-style search framework {#sec:cdcl} =========================== In this section, we introduce a search framework combined with the new projection operator proposed in the previous section. The main notation and concepts about the search framework are taken from Section 3 of [@DBLP:conf/cade/JovanovicM12] and Section 26.4.4 of [@DBLP:series/faia/2009-185]. Let $\bar{x}=(x_1,\ldots,x_n)$ and ${\mathtt{level}}(x_i)=i$. For a polynomial $f$, a literal $l$ and a clause $c$, we define ${\mathtt{level}}(f)=\max(\{{\mathtt{level}}(a)|a\in {\mathtt{var}}(f))\}$, ${\mathtt{level}}(l)={\mathtt{level}}({\mathtt{poly}}(l))$ and ${\mathtt{level}}(c)=\max(\{{\mathtt{level}}(l)|l \in c\})$. We describe the search framework by transition relations between search states as in [@DBLP:conf/cade/JovanovicM12]. The [*search states*]{} are indexed pairs of the form $M \| \zeta$, where $\zeta$ is a finite set of polynomial clauses and $M$ is a sequence of literals and variable assignments. Every literal is marked as a decision or a propagation literal. We denote a [*propagation literal*]{} $l$ by $c\rightarrow l$ if $l$ is propagated from $c$ and denote a [*decision literal*]{} $l$ by $l^\bullet$. We denote by $x_i\mapsto a_i$ a variable assignment. Let ${\mathtt{level}}(x_i\mapsto a_i)={\mathtt{level}}(x_i)$ and $v[M]=\{x_i\mapsto a_i|(x_i\mapsto a_i)\in M\}$. For a set $L$ of literals, $v[M](L)$ means the resulting set of $L$ after applying the assignments of $v[M]$. Next, we introduce transition relations between search states. Transition relations are specified by a set of transition rules. In the following, we use simple juxtaposition to denote the concatenation of sequences ([*e.g.*]{}, $M,M'$). We treat a literal or a variable assignment as one-element sequence and denote the empty sequence as $\emptyset$. We say the sequence $M$ is ordered when the sequence is of the form $$M=[N_1,x_1\mapsto a_1,\ldots,N_{k-1},x_{k-1}\mapsto a_{k-1},N_k]$$ where $N_j$ is a sequence of literals and each literal $l\in N_j$ satisfies ${\mathtt{level}}(l)=j$. Notice that $N_j$ might be $\emptyset$. We define ${\mathtt{level}}(M)=k$ even if $N_k=\emptyset$. We use ${\mathtt{sample}}(M)$ to denote the sample $(a_1,\ldots,a_{k-1})$ of $(x_1,\ldots,x_{k-1})$ in $M$ and ${\mathtt{feasible}}(M)$ to denote the feasible set of $v[M](N_k)$. For a new literal $l$ with $x_k\in {\mathtt{var}}(l)$, we say $l$ is consistent with $M$ if ${\mathtt{feasible}}([M,l])\neq\emptyset$. If $l$ is not consistent with $M$, we define ${\mathtt{core}}(l,M)$ to be a minimal set of literals $L$ in $M$ such that $v[M](L\cup\{l\})$ does not have a solution for $x_k$. Since there is only one unassigned variable $x_k$ in the polynomials in $N_k$, so ${\mathtt{feasible}}(M)$ can be easily calculated by real-root isolation. Suppose $l$ is a literal and $M$ is an ordered sequence which satisfies ${\mathtt{level}}(M)={\mathtt{level}}(l)$ and $\lnot l$ is not consistent with $M$. Define the [*explain clause*]{} of $l$ with $M$ as $${\mathtt{explain}}(l,M)=\lnot({\mathtt{s\_cell}}(F,{\mathtt{sample}}(M))\land {\mathtt{core}}(\lnot l,M))\lor l,$$ where $F=\{{\mathtt{poly}}(l')|l'\in {\mathtt{core}}(\lnot l,M)\}\cup \{{\mathtt{poly}}(l)\}$. Meanwhile, we define the [*state value*]{} of a literal $l$ as $${\mathtt{value}}(l,M)=\left\{\begin{array}{lcl} v[M](l) & &{\mathtt{level}}(l)<k, \\ \text{{\tt True}} & &l\in M, \\ \text{{\tt False}} & &\lnot l\in M, \\ \text{{\tt undef}} & &\text{otherwise}. \end{array} \right.$$ And for a clause $c$, $${\mathtt{value}}(c,M)=\left\{\begin{array}{lcl} \text{{\tt True}} & &\exists l\in c ({\mathtt{value}}(l,M)=\text{{\tt True}}), \\ \text{{\tt False}} & &\forall l\in c ({\mathtt{value}}(l,M)=\text{{\tt False}}), \\ \text{{\tt undef}} & &\text{otherwise}. \end{array} \right.$$ Specially, ${\mathtt{value}}(\emptyset,M)=\text{{\tt False}}$. A set of rules for transition relations between search states are defined as follows where $c$ is a clause and $l$ is a literal. Decide-Literal : $$M\|\zeta,c\Longrightarrow M,l^\bullet\|\zeta,c$$ if $l,l'\in c$, ${\mathtt{value}}(l,M)={\mathtt{value}}(l',M)=\text{{\tt undef}}$, ${\mathtt{level}}(c)={\mathtt{level}}(M)$ and $l$ is consistent with $M$. Boolean-Propagation : $$M\|\zeta, c\lor l\Longrightarrow M,c\lor l\rightarrow l\|\zeta,c\lor l$$ if ${\mathtt{value}}(c,M)=\text{{\tt False}},{\mathtt{value}}(l,M)=\text{{\tt undef}}$, ${\mathtt{level}}(c\lor l)={\mathtt{level}}(M)$ and $l$ is consistent with $M$. Lemma-Propagation : $$M\|\zeta\Longrightarrow M,{\mathtt{explain}}(l,M)\rightarrow l\|\zeta$$ if $l\in \zeta$ or $\lnot l \in \zeta$, ${\mathtt{value}}(l,M)=\text{{\tt undef}}$, ${\mathtt{level}}(l)={\mathtt{level}}(M)$ and $\lnot l$ is not consistent with $M$. Up-Level : $$M\|\zeta\Longrightarrow M,x\mapsto a\|\zeta$$ if $\forall c\in \zeta\;({\mathtt{level}}(c)\neq{\mathtt{level}}(M)\lor {\mathtt{value}}(c,M)=\text{{\tt True}})$, ${\mathtt{level}}(x)={\mathtt{level}}(M)$ and $a\in {\mathtt{feasible}}(M)$. Sat : $$M\|\zeta\Longrightarrow(\text{sat},v[M])$$ if ${\mathtt{level}}(M)>n$. Conflict : $$M\|\zeta\Longrightarrow M\|\zeta\not\vdash c$$ if ${\mathtt{level}}(c)={\mathtt{level}}(M)$ and ${\mathtt{value}}(c,M)=\text{{\tt False}}$. backtrack-Propagation : $$M,E\rightarrow l\|\zeta \not\vdash c\Longrightarrow M\|\zeta\not\vdash R$$ if $\lnot l\in c,{\mathtt{value}}(c,[M,E])=\text{{\tt False}}$ and $R={\mathtt{resolve}}(c,E,l)$[^1]. backtrack-Decision : $$M,l^\bullet\| \zeta\not\vdash c \Longrightarrow M\|\zeta,c$$ if $\lnot l\in c$. Skip : $$\begin{aligned} M,l^\bullet\| \zeta\not\vdash c &\Longrightarrow M\|\zeta\not\vdash c\\ M,E\rightarrow l\| \zeta\not\vdash c &\Longrightarrow M\|\zeta\not\vdash c \end{aligned}$$ if $\lnot l\not\in c$ . Down-Level : $$\begin{array}{ll} M,x\mapsto a\|\zeta\not\vdash c \Longrightarrow M\|\zeta \not \vdash c,&\text{ if }{\mathtt{value}}(c,M)=\text{{\tt False}}, \\ M,x\mapsto a\|\zeta\not\vdash c \Longrightarrow M\|\zeta,c,&\text{ if }{\mathtt{value}}(c,M)=\text{{\tt undef}}. \end{array}$$ Unsat : $$M\|\zeta \not \vdash c \Longrightarrow \text{unsat}$$ if ${\mathtt{value}}(c,M)=\text{{\tt False}}$ and no assignment or decide literal in $M$. Forget : $$M\|\zeta,c\Longrightarrow M\|\zeta$$ if $c$ is a learnt clause. Note that in this framework we rely on the rule [*lemma-propagation*]{} to guide the search away from conflicting states. When applying lemma-propagation, the most important thing is the explain clause. We cannot simply use the conflicting core as the explain clause, as this will cause explain to be an incorrect lemma because it ignores assignments. Using full CAD to calculate explain is also costly. Thanks to the sample cell calculated by the novel sample-cell projection operator, we can now efficiently calculate an effective explain to achieve our purpose. Given a polynomial formula $\zeta$ with finitely many clauses, any transition starting from the initial state $\emptyset\|\zeta$ will terminate either in a state $(sat,v)$, where the assignment $v$ satisfies the formula $\zeta$, or in the $unsat$ state. In the later case, $\zeta$ is unsatisfiable in ${\mathbb{R}}$. By Theorem 1 in [@DBLP:conf/cade/JovanovicM12], if there is a finite set such that all the literals returned every time by calling ${\mathtt{explain}}$ are always contained in the set, then the above theorem holds. On the other hand, it is not hard to see that all literals that may be generated by ${\mathtt{s\_cell}}$ are determined by finitely many polynomials and their real roots and thus finite. That completes the proof. $\blacksquare$ Experiments {#sec:exp} =========== In order to better demonstrate the effectiveness of our algorithm, we have implemented a prototype solver LiMbS[^2] which is base on Mathematica 12. The solver is a clean translation of the algorithm in this paper. Our solver is compared to the following solvers that have been popular in SMT nonlinear competition: Z3 (4.8.7-1), CVC4 (1.6-2), Yices (2.6.1) and MathSAT5 (5.6.0). All tests were conducted on 6-Core Intel Core i7-8750H@2.20GHz with 32GB of memory and ARCH LINUX SYSTEM (5.5.4-arch1-1). The timeout is set to be 5 hours. The examples listed below, which we collect from several related papers, are either special or cannot be well-solved by existing SMT solvers. All results are listed in Table \[tb:exp\]. ([**Han**]{}\_$n$)[@Dai_Han_Hong_Xia_2015] Decide whether $$\exists x_1,\ldots,\exists x_n \;(\sum_{i=1}^nx_i^2)^2-4(\sum_{i=1}^nx_i^2x_{i+1}^2)<0$$ where $x_{n+1} = x_{1}$. $({\mathbf P})$\ $\exists a,\exists b,\exists c,\exists d,\exists e,\exists f (a^2 b^2 e^2+a^2 b^2 f^2+a^2 b^2-a^2 b c d e-a^2 b d f+a^2 c^4 f^4+2 a^2 c^4 f^2+a^2 c^4-3 a^2 c^3 e f^3-3 a^2 c^3 e f+3 a^2 c^2 e^2 f^2+a^2 c^2 e^2+a^2 c^2 f^2+a^2 c^2-a^2 c e^3 f-a^2 c e f-a b^2 d e-2 a b c^3 f^4-4 a b c^3 f^2-2 a b c^3+4 a b c^2 e f^3+4 a b c^2 e f+a b c d^2-2 a b c e^2 f^2+a b c f^2+a b c-a b e f+2 a c^3 d f^3+2 a c^3 d f-4 a c^2 d e f^2-2 a c^2 d e+2 a c d e^2 f+b^2 c^2 f^4+2 b^2 c^2 f^2+b^2 c^2-b^2 c e f^3-b^2 c e f-2 b c^2 d f^3-2 b c^2 d f+2 b c d e f^2+c^2 d^2 f^2+c^2 d^2+c^2 f^2+c^2-c d^2 e f-c e f<0)$ [@Hong91comparisonof] Hong\_$n$ : $$\exists x_1,\ldots,\exists x_n\;\sum_{i=1}^nx_i^2<1\land\prod_{i=1}^n x_i>1$$ Hong2\_$n$ : $$\exists x_1,\ldots,\exists x_n\;\sum_{i=1}^nx_i^2<2n\land\prod_{i=1}^n x_i>1$$ ([**C**]{}\_$n$\_$r$) Whether the distance between the ball $B_r(\bar{x})$ and the complement of $B_8(\bar{x})$ is less than $\frac{1}{1000}$? $$\exists_{i=1}^{n} x_i,\exists_{i=1}^{n} y_i \;\sum_{i=1}^nx_i^2<r\land\sum_{i=1}^ny_i^2>8^2\land \sum_{i=1}^n(x_i-y_i)^2<\frac{1}{1000^2}$$ ans LiMbS Z3 CVC4 MathSAT5 Yices ----------- ------- -------- ------- --------- ---------- ------- Han\_3 SAT 0.01s 0.01s 0.01s 0.01s 0.01s Han\_4 UNSAT 0.08s 0.01s $>5$h $>5$h 0.01s Han\_5 UNSAT 1.26s $>5$h $>5$h $>5$h $>5$h Han\_6 UNSAT 60s $>5$h $>5$h $>5$h $>5$h P SAT 1.06s 0.05s $>5$h $>5$h $>5$h Hong\_10 UNSAT 222s 2058s 0.01s 0.10s $>5$h Hong\_11 UNSAT 806s 6357s 0.01s 0.10s $>5$h Hong2\_11 SAT 30.43s 1997s 0.01s $>5$h 0.01s Hong2\_12 SAT 563s 6693s 0.01s $>5$h 0.01s C\_3\_1 UNSAT 0.44s $>5$h 0.62s 5811s $>5$h C\_3\_32 UNSAT 0.48s $>5$h unknown $>5$h $>5$h C\_3\_63 UNSAT 0.48s $>5$h unknown $>5$h $>5$h C\_3\_64 SAT 0.02s 4682s unknown $>5$h $>5$h C\_4\_1 UNSAT 1.31s $>5$h 2.28s $>5$h $>5$h C\_4\_32 UNSAT 1.42s $>5$h unknown $>5$h $>5$h C\_4\_63 UNSAT 1.42s $>5$h unknown $>5$h $>5$h C\_4\_64 SAT 0.02s $>5$h unknown $>5$h $>5$h C\_5\_1 UNSAT 5.48s $>5$h 19.33s $>5$h $>5$h C\_5\_32 UNSAT 5.73s $>5$h unknown $>5$h $>5$h C\_5\_63 UNSAT 5.68s $>5$h unknown $>5$h $>5$h C\_5\_64 SAT 0.02s $>5$h unknown $>5$h 1.75s : Comparison with other solvers on 21 examples\[tb:exp\] Our solver LiMbs solves all the $21$ examples shown in Table \[tb:exp\]. LiMbs is faster than the other solvers on 15 examples. Only LiMbs can solve 9 of the examples within a reasonable time while other solvers either run time out or return unknown state. From this we can see that our algorithm has great potential in solving satisfiability of polynomial formulas, especially considering that our prototype solver is a small program with less than 1000 lines of codes. For Hong$\_n$ and Hong2$\_n$, though our solver is much faster than Z3, CVC4 is the one that performs best. We note that the examples of Hong$\_n$ and Hong2$\_n$ are all symmetric. This reminds us it is worth exploiting symmetry to optimize our solver’s performance. Conclusions {#sec:conclusion} =========== A new algorithm for deciding the satisfiability of polynomial formulas over the reals is proposed. The key point is that we design a new projection operator, the sample-cell projection operator, which can efficiently guide CDCL-style search away from conflicting states. Preliminary evaluation of the prototype solver LiMbS shows the effectiveness of the new algorithm. We will further develop our algorithm, looking into problems with symmetry, equations or other special structures. We also hope to develop an easy-to-use, robust and concise open-source algorithm framework based on our prototype solver to achieve a wider range of applications. Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported partly by NSFC under grants 61732001 and 61532019. [^1]: ${\mathtt{resolve}}(c_1\lor l,c_2\lor \lnot l,l)=c_1\lor c_2.$ [^2]: https://github.com/lihaokun/LiMbS

--- abstract: 'Ordinary least-squares (OLS) estimators for a linear model are very sensitive to unusual values in the design space or outliers among $y$ values. Even one single atypical value may have a large effect on the parameter estimates. This article aims to review and describe some available and popular robust techniques, including some recent developed ones, and compare them in terms of breakdown point and efficiency. In addition, we also use a simulation study and a real data application to compare the performance of existing robust methods under different scenarios.' author: - | Chun Yu$^{1}$, Weixin Yao$^{1}$, and Xue Bai$^{1}$\ \ $^{1}$Department of Statistics,\ Kansas State University, Manhattan, Kansas, USA 66506-0802.\ title: 'Robust Linear Regression: A Review and Comparison' --- [**Key words**]{}: Breakdown point; Robust estimate; Linear Regression. Introduction ============ Linear regression has been one of the most important statistical data analysis tools. Given the independent and identically distributed (iid) observations $(\bx_{i},y_{i})$, $i = 1,\ldots,n$, in order to understand how the response $y_i$s are related to the covariates $\bx_i$s, we traditionally assume the following linear regression model $$y_{i}=\bx_{i}^T\bbeta+\varepsilon_{i},$$ where $\bbeta$ is an unknown $p\times 1$ vector, and the $\varepsilon_{i}$s are i.i.d. and independent of $\bx_{i}$ with ${\text{E}}(\varepsilon_{i}\mid\bx_i)=0$. The most commonly used estimate for $\bbeta$ is the ordinary least square (OLS) estimate which minimizes the sum of squared residuals $$\sum_{i=1}^{n}(y_{i}-\bx_{i}^T\bbeta)^{2}. \label{lse}$$ However, it is well known that the OLS estimate is extremely sensitive to the outliers. A single outlier can have large effect on the OLS estimate. In this paper, we review and describe some available robust methods. In addition, a simulation study and a real data application are used to compare different existing robust methods. The efficiency and breakdown point (Donoho and Huber 1983) are two traditionally used important criteria to compare different robust methods. The efficiency is used to measure the relative efficiency of the robust estimate compared to the OLS estimate when the error distribution is exactly normal and there are no outliers. Breakdown point is to measure the proportion of outliers an estimate can tolerate before it goes to infinity. In this paper, finite sample breakdown point (Donoho and Huber 1983) is used and defined as follows: Let $\textbf{z}_{i} = \left(\bx_{i},y_{i}\right)$. Given any sample $\bz = \left(\bz_{i},\ldots,\bz_{n}\right)$, denote $T(\bz)$ the estimate of the parameter $\bbeta$. Let $\bz'$ be the corrupted sample where any $m$ of the original points of $\bz$ are replaced by arbitrary bad data. Then the finite sample breakdown point $\delta^{*}$ is defined as $$\delta^{*}\left(\bz,T\right)=\min_{1\leq m\leq n}\left\{\frac{m}{n}: \sup_{\bz'}\left\|T\left(\bz'\right)- T\left( \bz\right)\right\| = \infty\right\},$$ where $\left\|\cdot\right\|$ is Euclidean norm. Many robust methods have been proposed to achieve high breakdown point or high efficiency or both. M-estimates (Huber, 1981) are solutions of the normal equation with appropriate weight functions. They are resistant to unusual $\textit{y}$ observations, but sensitive to high leverage points on $\textbf{x}$. Hence the breakdown point of an M-estimate is $1/n$. R-estimates (Jaeckel 1972) which minimize the sum of scores of the ranked residuals have relatively high efficiency but their breakdown points are as low as those of OLS estimates. Least Median of Squares (LMS) estimates (Siegel 1982) which minimize the median of squared residuals, Least Trimmed Squares (LTS) estimates (Rousseeuw 1983) which minimize the trimmed sum of squared residuals, and S-estimates (Rousseeuw and Yohai 1984) which minimize the variance of the residuals all have high breakdown point but with low efficiency. Generalized S-estimates (GS-estimates) (Croux et al. 1994) maintain high breakdown point as S-estimates and have slightly higher efficiency. MM-estimates proposed by Yohai (1987) can simultaneously attain high breakdown point and efficiencies. Mallows Generalized M-estimates (Mallows 1975) and Schweppe Generalized M-estimates (Handschin et al. 1975) downweight the high leverage points on $\textbf{x}$ but cannot distinguish “good" and “bad" leverage points, thus resulting in a loss of efficiencies. In addition, these two estimators have low breakdown points when $\textit{p}$, the number of explanatory variables, is large. Schweppe one-step (S1S) Generalized M-estimates (Coakley and Hettmansperger 1993) overcome the problems of Schweppe Generalized M-estimates and are calculated in one step. They both have high breakdown points and high efficiencies. Recently, Gervini and Yohai (2002) proposed a new class of high breakdown point and high efficiency robust estimate called robust and efficient weighted least squares estimator (REWLSE). Lee et al. (2011) and She and Owen (2011) proposed a new class of robust methods based on the regularization of case-specific parameters for each response. They further proved that the M-estimator with Huber’s $\psi$ function is a special case of their proposed estimator. The rest of the paper is organized as follows. In Section 2, we review and describe some of the available robust methods. In Section 3, a simulation study and a real data application are used to compare different robust methods. Some discussions are given in Section 4. Robust Regression Methods ========================= M-Estimates ----------- By replacing the least squares criterion (\[lse\]) with a robust criterion, M-estimate (Huber, 1964) of $\bbeta$ is $$\hat{\bbeta}=\arg\min_{\bbeta}\sum_{i=1}^{n}\rho\left(\frac{y_i-\bx_i^T \bbeta}{\hat{\sigma}}\right), \label{mest}$$ where $\rho(\cdot)$ is a robust loss function and $\hat{\sigma}$ is an error scale estimate. The derivative of $\rho$, denoted by $\psi(\cdot)=\rho'(\cdot)$, is called the influence function. In particular, if $\rho(t)$ = $ \frac{1}{2}t^{2}$, then the solution is the OLS estimate. The OLS estimate is very sensitive to outliers. Rousseeuw and Yohai (1984) indicated that OLS estimates have a breakdown point (BP) of BP = $1/n$, which tends to zero when the sample size $n$ is getting large. Therefore, one single unusual observation can have large impact on the OLS estimate. One of the commonly used robust loss functions is Huber’s $\psi$ function (Huber 1981), where $\psi_c(t)=\rho'(t)=\max\{-c,\min(c,t)\}$. Huber (1981) recommends using $c=1.345$ in practice. This choice produces a relative efficiency of approximately $95\%$ when the error density is normal. Another possibility for $\psi(\cdot)$ is Tukey’s bisquare function $\psi_c(t)=t\{1-(t/c)^2\}_{+}^2$. The use of $c=4.685$ produces $95\%$ efficiency. If $\rho(t)$ = $\left|t \right|$, then *least absolute deviation* (LAD, also called median regression) estimates are achieved by minimizing the sum of the absolute values of the residuals $$\hat{\bbeta}=\arg\min_{\bbeta}\sum_{i=1}^{n} \left|y_i-\bx_i^T\bbeta\right|.$$ The LAD is also called $L_{1}$ estimate due to the $L_{1}$ norm used. Although LAD is more resistent than OLS to unusual $y$ values, it is sensitive to high leverage outliers, and thus has a breakdown point of BP = $1/n$ $\rightarrow 0$ (Rousseeuw and Yohai 1984). Moreover, LAD estimates have a low efficiency of 0.64 when the errors are normally distributed. Similar to LAD estimates, the general monotone M-estimates, i.e., M-estimates with monotone $\psi$ functions, have a BP = $1/n$ $\rightarrow 0$ due to lack of immunity to high leverage outliers (Maronna, Martin, and Yohai 2006). LMS Estimates ------------- The LMS estimates (Siegel 1982) are found by minimizing the median of the squared residuals $$\hat{\bbeta}=\arg\min_{\bbeta}\text{Med}\{\left( y_i-\bx_i^T\bbeta\right)^{2}\}.$$ One good property of the LMS estimate is that it possesses a high breakdown point of near 0.5. However, the LMS estimate has at best an efficiency of 0.37 when the assumption of normal errors is met (see Rousseeuw and Croux 1993). Moreover, LMS estimates do not have a well-defined influence function because of its convergence rate of $n^{-\frac{1}{3}}$ (Rousseeuw 1982). Despite these limitations, the LMS estimate can be used as the initial estimate for some other high breakdown point and high efficiency robust methods. LTS Estimates ------------- The LTS estimate (Rousseeuw 1983) is defined as $$\hat{\bbeta}=\arg\min_{\bbeta}\sum_{i=1}^{q}r_{(i)}\left( \bbeta\right)^{2},$$ where $r_{(i)}(\bbeta)=y_{(i)}-\bx_{(i)}^T\bbeta$, $r_{\left(1\right)}\left(\bbeta\right)^{2} \leq\cdots\leq r_{\left(q\right)}\left(\bbeta\right)^{2}$ are ordered squared residuals, $q = \left[n \left(1-\alpha \right)+1 \right]$, and $\alpha$ is the proportion of trimming. Using q = $\left( \frac {n}{2} \right)$ +1 ensures that the estimator has a breakdown point of BP $= 0.5$, and the convergence rate of $n^{-\frac{1}{2}}$ (Rousseeuw 1983). Although highly resistent to outliers, LTS suffers badly in terms of very low efficiency, which is about 0.08, relative to OLS estimates (Stromberg, et al. 2000). The reason that LTS estimates call attentions to us is that it is traditionally used as the initial estimate for some other high breakdown point and high efficiency robust methods. S-Estimates ----------- S-estimates (Rousseeuw and Yohai 1984) are defined by $$\hat{\bbeta}=\arg\min_{\bbeta}\hat{\sigma}\left(r_{1}\left(\bbeta\right),\cdots,r_{n}\left(\bbeta\right)\right),$$ where $r_{i}\left(\bbeta\right)=y_i-\bx_i^T\bbeta$ and $\hat{\sigma}\left(r_{1}\left(\bbeta\right),\cdots,r_{n}\left(\bbeta\right)\right)$ is the scale M-estimate which is defined as the solution of $$\frac{1}{n}\sum_{i=1}^{n}\rho \left(\frac{r_{i}\left( \bbeta \right)}{\hat{\sigma}}\right)=\delta,$$ for any given $\bbeta$, where $\delta$ is taken to be ${\text{E}}_{\Phi}\left[\rho\left(r\right)\right]$. For the biweight scale, S-estimates can attain a high breakdown point of BP = 0.5 and has an asymptotic efficiency of 0.29 under the assumption of normally distributed errors (Maronna, Martin, and Yahai 2006). Generalized S-Estimates (GS-Estimates) -------------------------------------- Croux et al. (1994) proposed generalized S-estimates in an attempt to improve the low efficiency of S-estimators. Generalized S-estimates are defined as $$\hat{\bbeta} = \arg\min_{\bbeta}S_{n}(\bbeta),$$ where $S_{n}(\bbeta)$ is defined as $$S_{n}(\bbeta)=\sup\left\{S>0; \binom{n}{2}^{-1}\sum_{i<j}\rho\left(\frac{r_{i}-r_{j}}{S}\right) \geq k_{n,p}\right\},$$ where $r_{i} = y_{i} - \bx_{i}^T\bbeta$, $\textit{p}$ is the number of regression parameters, and $k_{n,p}$ is a constant which might depend on $n$ and $p$. Particularly, if $\rho(x)= I(\left|x\right|\geq 1)$ and $k_{n,p} = \left(\binom{n}{2}-\binom{h_{p}}{2}+1\right)/\binom{n}{2}$ with $h_{p} = \frac{n+p+1}{2}$, generalized S-estimator yields a special case, the least quartile difference (LQD) estimator, which is defined as $$\hat{\bbeta} = \arg\min_{\bbeta}Q_{n}(r_{1},\ldots,r_{n}),$$ where $$Q_{n} =\left\{\left|r_{i}-r_{j}\right|; i<j\right\}_{\binom{h_{p}}{2}}$$ is the $\binom{h_{p}}{2}$th order statistic among the $\binom{n}{2}$ elements of the set $\left\{\left|r_{i}-r_{j}\right|; i<j\right\}$. Generalized S-estimates have a breakdown point as high as S-estimates but with a higher efficiency. MM-Estimates ------------ First proposed by Yohai (1987), MM-estimates have become increasingly popular and are one of the most commonly employed robust regression techniques. The MM-estimates can be found by a three-stage procedure. In the first stage, compute an initial consistent estimate $\hat{\bbeta}_{0}$ with high breakdown point but possibly low normal efficiency. In the second stage, compute a robust M-estimate of scale $\hat{\sigma}$ of the residuals based on the initial estimate. In the third stage, find an M-estimate $\hat{\bbeta}$ starting at $\hat{\bbeta}_{0}$. In practice, LMS or S-estimate with Huber or bisquare functions is typically used as the initial estimate $\hat{\bbeta_{0}}$. Let $\rho_{0} (r) =\rho_{1}\left( r/k_{0}\right)$, $\rho (r) =\rho_{1}\left( r/k_{1}\right)$, and assume that each of the $\rho$-functions is bounded. The scale estimate $\hat{\sigma}$ satisfies $$\frac{1}{n}\sum_{i=1}^{n}\rho_{0} \left(\frac{r_{i}\left( \hat{\bbeta} \right)}{\hat{\sigma}}\right)=0.5.$$ If the $\rho$-function is biweight, then $k_{0} = 1.56$ ensures that the estimator has the asymptotic BP = 0.5. Note that an M-estimate minimizes $$L(\beta) = \sum_{i=1}^{n}\rho \left(\frac{r_{i}\left( \hat{\bbeta} \right)}{\hat{\sigma}}\right).$$ Let $\rho$ satisfy $\rho\leq \rho_{0}$. Yohai (1987) showed that if $\hat{\bbeta}$ satisfies $L(\hat{\bbeta})\leq (\hat{\bbeta}_{0})$, then $\hat{\bbeta}$’s BP is not less than that of $\hat{\bbeta}_{0}$. Furthermore, the breakdown point of the MM-estimate depends only on $k_{0}$ and the asymptotic variance of the MM-estimate depends only on $k_{1}$. We can choose $k_{1}$ in order to attain the desired normal efficiency without affecting its breakdown point. In order to let $\rho\leq \rho_{0}$, we must have $k_{1} \geq k_{0}$; the larger the $k_{1}$ is, the higher efficiency the MM-estimate can attain at the normal distribution. Maronna, Martin, and Yahai (2006) provides the values of $k_{1}$ with the corresponding efficiencies of the biweight $\rho$-function. Please see the following table for more detail. Efficiency 0.80 0.85 0.90 0.95 ------------ ------ ------ ------ ------ $k_{1}$ 3.14 3.44 3.88 4.68 However, Yohai (1987) indicates that MM-estimates with larger values of $k_{1}$ are more sensitive to outliers than the estimates corresponding to smaller values of $k_{1}$. In practice, an MM-estimate with bisquare function and efficiency 0.85 ($k_{1}$ = 3.44) starting from a bisquare S-estimate is recommended. Generalized M-Estimates (GM-Estimates) -------------------------------------- ### Mallows GM-estimate In order to make M-estimate resistent to high leverage outliers, Mallows (1975) proposed Mallows GM-estimate that is defined by $$\sum_{i=1}^{n}w_{i}\psi\left\{\frac{r_{i}\left(\hat{\bbeta}\right)}{\hat{\sigma}}\right\}\bx_{i}=0,$$ where $\psi(e)=\rho'(e)$ and $w_{i} = \sqrt{1-h_{i}}$ with $h_{i}$ being the leverage of the $\textit{i}th$ observation. The weight $w_{i}$ ensures that the observation with high leverage receives less weight than observation with small leverage. However, even “good" leverage points that fall in line with the pattern in the bulk of the data are down-weighted, resulting in a loss of effiency. ### Schweppe GM-estimate Schweppe GM-estimate (Handschin et al. 1975) is defined by the solution of $$\sum_{i=1}^{n}w_{i}\psi\left\{\frac{r_{i}\left(\hat{\bbeta}\right)}{w_{i}\hat{\sigma}}\right\}\bx_{i}=0,$$ which adjusts the leverage weights according to the size of the residual $r_{i}$. Carroll and Welsh (1988) proved that the Schweppe estimator is not consistent when the errors are asymmetric. Furthermore, the breakdown points for both Mallows and Schweppe GM-estimates are no more than $1/(p+1)$, where $\textit{p}$ is the number of unknown parameters. ### S1S GM-estimate Coakley and Hettmansperger (1993) proposed Schweppe one-step (S1S) estimate , which extends from the original Schweppe estimator. S1S estimator is defined as $$\hat{\bbeta}= \hat{\bbeta}_{0}+ \left[\sum_{i=1}^{n}\psi'\left(\frac{r_{i}\left(\hat{\bbeta}_{0}\right)}{\hat{\sigma}w_{i}}\right)\bx_{i}\bx_{i}'\right]^{-1}\times \sum_{i=1}^{n}\hat{\sigma}w_{i}\psi\left(\frac{r_{i}\left(\hat{\bbeta}_{0}\right)}{\hat{\sigma}w_{i}}\right)\bx_{i},$$ where the weight $w_{i}$ is defined in the same way as Schweppe’s GM-estimate. The method for S1S estimate is different from the Mallows and Schweppe GM-estimates in that once the initial estimates of the residuals and the scale of the residuals are given, final M-estimates are calculated in one step rather than iteratively. Coakley and Hettmansperger (1993) recommended to use Rousseeuw’s LTS for the initial estimates of the residuals and LMS for the initial estimates of the scale and proved that the S1S estimate gives a breakdown point of BP = 0.5 and results in 0.95 efficiency compared to the OLS estimate under the Gauss-Markov assumption. R-Estimates ----------- The R-estimate (Jaeckel 1972) minimizes the sum of some scores of the ranked residuals $$\sum_{i=1}^{n}a_{n}\left(R_{i} \right)r_{i}=min,$$ where $R_{i}$ represents the rank of the *i*th residual $r_{i}$, and $a_{n}\left(\cdot\right)$ is a monotone score function that satisfies $$\sum_{i=1}^{n}a_{n}\left(i\right) = 0.$$ R-estimates are scale equivalent which is an advantage compared to M-estimates. However, the optimal choice of the score function is unclear. In addition, most of R-estimates have a breakdown point of BP = $1/n \rightarrow 0$. The bounded influence R-estimator proposed by Naranjo and Hettmansperger (1994) has a fairly high efficiency when the errors have normal distribution. However, it is proved that their breakdown point is no more than 0.2. REWLSE ------ Gervini and Yohai (2002) proposed a new class of robust regression method called robust and efficient weighted least squares estimator (REWLSE). REWLSE is much more attractive than many other robust estimators due to its simultaneously attaining maximum breakdown point and full efficiency under normal errors. This new estimator is a type of weighted least squares estimator with the weights adaptively calculated from an initial robust estimator. Consider a pair of initial robust estimates of regression parameters and scale, $\hat{\bbeta}_0$ and $\hat{\sigma}$ respectively, the standardized residuals are defined as $$r_{i}=\frac{y_{i}-\bx_{i}^T\hat{\bbeta}_{0}}{\hat{\sigma}}.$$ A large value of $\left|r_{i}\right|$ would suggest that $(\bx_{i},y_{i})$ is an outlier. Define a measure of proportion of outliers in the sample $$d_{n} = \max_{i>i_{0}}\left\{F^{+}(\left|r\right|_{(i)})-\frac{(i-1)}{n}\right\}^{+},$$ where $\left\{\cdot\right\}^{+}$ denotes positive part, $F^{+}$ denotes the distribution of $\left|X\right|$ when $X \thicksim F$, $\left|r\right|_{(1)} \leq \ldots \leq \left|r\right|_{(n)}$ are the order statistics of the standardized absolute residuals, and $i_{0} = \max\left\{i: \left|r\right|_{(i)} < \eta\right\}$, where $\eta$ is some large quantile of $F^{+}$. Typically $\eta = 2.5$ and the cdf of a normal distribution is chosen for $F$. Thus those $\left\lfloor nd_{n}\right\rfloor$ observations with largest standardized absolute residuals are eliminated (here $\left\lfloor a\right\rfloor$ is the largest integer less than or equal to a). The adaptive cut-off value is $t_{n} = \left|r\right|_{(i_{n})}$ with $i_{n} = n - \left\lfloor nd_{n}\right\rfloor$. With this adaptive cut-off value, the adaptive weights proposed by Gervini and Yohai (2002) are $$w_{i}= \begin{cases} 1 &\text{if $\left|r_{i}\right| < t_{n}$}\\ 0 &\text{if $\left|r_{i}\right| \geq t_{n}.$} \end{cases}$$ Then, the REWLSE is $$\hat{\bbeta} = (\bX^T\bW\bX)^{-1}\bX^T\bW\by, $$ where $\bW ={\text{diag}}(w_{1},\cdots, w_{n}),\bX=(\bx_1,\ldots,\bx_n)^T,$ and $\textbf{y}=(y_{1},\cdots, y_{n})'$. If the initial regression and scale estimates with BP = 0.5 are chosen, the breakdown point of the REWLSE is also 0.5. Furthermore, when the errors are normally distributed, the REWLSE is asymptotically equivalent to the OLS estimates and hence asymptotically efficient. Robust regression based on regularization of case-specific parameters --------------------------------------------------------------------- She and Owen (2011) and Lee et al$.$ (2011) proposed a new class of robust regression methods using the case-specific indicators in a mean shift model with regularization method. A mean shift model for the linear regression is $$\by=\bX\bbeta + \bgamma +\bvarepsilon, \; \bvarepsilon \thicksim N(0,\sigma^{2}I)$$ where $\textbf{y}=(y_{1},\cdots, y_{n})^T$, $\bX=(\bx_1,\ldots,\bx_n)^T$, and the mean shift parameter $\gamma_{i}$ is nonzero when the *i*th observation is an outlier and zero, otherwise. Due to the sparsity of $\gamma_i$s, She and Owen (2011) and Lee et al$.$ (2011) proposed to estimate $\bbeta$ and $\bgamma$ by minimizing the penalized least squares using $L_1$ penalty: $$L(\bbeta,\bgamma) = \frac{1}{2}\left\{\by-(\bX\bbeta+\bgamma)\right\}^{T}\left\{\by-(\bX\bbeta+\bgamma)\right\}+\lambda\sum_{i=1}^{n}\left|\gamma_{i}\right|, \label{penlse}$$ where $\lambda$ are fixed regularization parameters for $\bgamma$. Given the estimate $\hat{\bgamma}$, $\hat{\bbeta}$ is the OLS estimate with $\textbf{y}$ replaced by $\textbf{y}-\bgamma$. For a fixed $\hat{\bbeta}$, the minimizer of (\[penlse\]) is $\hat{\gamma}_i= sgn(r_i)(\left|\gamma_i\right|-\lambda)_{+}$, that is, $$\hat{\gamma}_i= \begin{cases} 0 &\text{if $\left|r_{i}\right|\leq \lambda$;}\\ y_{i}-\bx_{i}^{T}\hat{\bbeta} &\text{if $\left|r_{i}\right|> \lambda$.} \end{cases}$$ Therefore, the solution of (\[penlse\]) can be found by iteratively updating the above two steps. She and Owen (2011) and Lee et al$.$ (2011) proved that the above estimate is in fact equivalent to the M-estimate if Huber’s $\psi$ function is used. However, their proposed robust estimates are based on different perspective and can be extended to many other likelihood based models. Note, however, the monotone M-estimate is not resistent to the high leverage outliers. In order to overcome this problem, She and Owen (2011) further proposed to replace the $L_1$ penalty in (\[penlse\]) by a general penalty. The objective function is then defined by $$L_{p}(\bbeta,\bgamma) = \frac{1}{2}\left\{\by-(\bX\bbeta+\bgamma)\right\}^{T}\left\{\by-(\bX\bbeta+\bgamma)\right\}+\sum_{i=1}^{n}p_{\lambda}(\left|\gamma_{i}\right|), \label{lsepen}$$ where $p_{\lambda}(\left|\cdot\right|)$ is any penalty function which depends on the regularization parameter $\lambda$. We can find $\hat{\bgamma}$ by defining thresholding function $\Theta(\bgamma;\lambda)$ (She and Owen 2009). She and Owen (2009, 2011) proved that for a specific thresholding function, we can always find the corresponding penalty function. For example, the soft, hard, and smoothly clipped absolute deviation (SCAD; Fan and Li, 2001) thresholding solutions of $\bgamma$ correspond to $L_{1}$, Hard, and SCAD penalty functions, respectively. Minimizing the equation (\[lsepen\]) yields a sparse $\hat{\bgamma}$ for outlier detection and a robust estimate of $\bbeta$. She and Owen (2011) showed that the proposed estimates of (\[lsepen\]) with hard or SCAD penalties are equivalent to the M-estimates with certain redescending $\psi$ functions and thus will be resistent to high leverage outliers if a high breakdown point robust estimates are used as the initial values. Examples ======== In this section, we compare different robust methods and report the mean squared errors (MSE) of the parameter estimates for each estimation method. We compare the OLS estimate with seven other commonly used robust regression estimates: the M estimate using Huber’s $\psi$ function ($M_H$), the M estimate using Tukey’s bisquare function ($M_T$), the S estimate, the LTS estimate, the LMS estimate, the MM estimate (using bisquare weights and $k_{1} = 4.68$), and the REWLSE. Note that we didn’t include the case-specific regularization methods proposed by She and Owen (2011) and Lee et al$.$ (2011) since they are essentially equivalent to M-estimators (She and Owen (2011) did show that their new methods have better performance in detecting outliers in their simulation study). [**Example 1.**]{} We generate $n$ samples $\{(x_1,y_1),\ldots,(x_n,y_n)\}$ from the model $$Y = X + \varepsilon,$$ where $X \sim N (0,1)$. In order to compare the performance of different methods, we consider the following six cases for the error density of $\varepsilon$: Case I: : $\varepsilon\sim N (0,1)$- standard normal distribution. Case II: : $\varepsilon\sim t_{3}$ - t-distribution with degrees of freedom 3. Case III: : $\varepsilon\sim t_{1}$ - t-distribution with degrees of freedom 1 (Cauchy distribution). Case IV: : $\varepsilon\sim 0.95N (0,1) + 0.05N (0,10^{2})$ - contaminated normal mixture. Case V: : $\varepsilon\sim$ N (0,1) with $10\%$ identical outliers in $y$ direction (where we let the first $10\%$ of $y's$ equal to 30). Case VI: : $\varepsilon\sim$ N (0,1) with $10\%$ identical high leverage outliers (where we let the first $10\%$ of $x's$ equal to 10 and their corresponding $y's$ equal to 50). Tables 1 and 2 report the mean squared errors (MSE) of the parameter estimates for each estimation method with sample size $n = 20$ and 100, respectively. The number of replicates is 200. From the tables, we can see that MM and REWLSE have the overall best performance throughout most cases and they are consistent for different sample sizes. For Case I, LSE has the smallest MSE which is reasonable since under normal errors LSE is the best estimate; $M_H$, $M_T$, MM, and REWLSE have similar MSE to LSE, due to their high efficiency property; LMS, LTS, and S have relative larger MSE due to their low efficiency. For Case II, $M_H$, $M_T$, MM, and REWLSE work better than other estimates. For Case III, LSE has much larger MSE than other robust estimators; $M_H$, $M_T$, MM, and REWLSE have similar MSE to S. For Case IV, M, MM, and REWLSE have smaller MSE than others. From Case V, we can see that when the data contain outliers in the y-direction, LSE is much worse than any other robust estimates; MM, REWLSE, and $M_T$ are better than other robust estimators. Finally for Case VI, since there are high leverage outliers, similar to LSE, both $M_T$ and $M_H$ perform poorly; MM and REWLSE work better than other robust estimates. In order to better compare the performance of different methods, Figure 1 shows the plot of their MSE versus each case for the slope (left side) and intercept (right side) parameters for model 1 when sample size $n=100$. Since the lines for LTS and LMS are above the other lines, S, MM, and REWLSE of the intercept and slopes outperform LTS and LMS estimates throughout all six cases. In addition, the S estimate has similar performance to MM and REWLSE when the error density of $\varepsilon$ is Cauchy distribution. However, MM and REWLSE perform better than S-estimates in other five cases. Furthermore, the lines for MM and REWLSE almost overlap for all six cases. It shows that MM and REWLSE are the overall best approaches in robust regression. [**Example 2.**]{} $$Y=X_{1}+X_{2}+X_{3}+\varepsilon,$$ where $X_i\sim N(0,1), i=1,2,3$ and $X_i$’s are independent. We consider the following six cases for the error density of $\varepsilon$: Case I: : $\varepsilon\sim N (0,1)$- standard normal distribution. Case II: : $\varepsilon\sim t_{3}$ - t-distribution with degrees of freedom 3. Case III: : $\varepsilon\sim t_{1}$ - t-distribution with degrees of freedom 1 (Cauchy distribution). Case IV: : $\varepsilon\sim 0.95N (0,1) + 0.05N (0,10^{2})$ - contaminated normal mixture. Case V: : $\varepsilon\sim N (0,1)$ with $10\%$ identical outliers in $y$ direction (where we let the first $10\%$ of $y's$ equal to 30). Case VI: : $\varepsilon\sim N (0,1)$ with $10\%$ identical high leverage outliers (where we let the first $10\%$ of $x's$ equal to 10 and their corresponding $y's$ equal to 50). Tables 3 and 4 show the mean squared errors (MSE) of the parameter estimates of each estimation method for sample size $n = 20$ and $n = 100$, respectively. Figure 2 shows the plot of their MSE versus each case for three slopes and the intercept parameters with sample size $n =100$. The results in Example 2 tell similar stories to Example 1. In summary, MM and REWLSE have the overall best performance; LSE only works well when there are no outliers since it is very sensitive to outliers; M-estimates ($M_H$ and $M_T$) work well if the outliers are in $y$ direction but are also sensitive to the high leverage outliers. **Example 3:** Next, we use the famous data set found in Freedman et al$.$ (1991) to compare LSE with MM and REWLSE. The data set are shown in Table 5 which contains per capita consumption of cigarettes in various countries in 1930 and the death rates (number of deaths per million people) from lung cancer for 1950. Here, we are interested in how the death rates per million people from lung cancer (dependent variable $y$) dependent on the consumption of cigarettes per capita (the independent variable $x$). Figure \[figure3\] is a scatter plot of the data. From the plot, we can see that USA $(x=1300,y=200)$ is an outlier with high leverage. We compare different regression parameters estimates by LSE, MM, and REWLSE. Figure \[figure3\] shows the fitted lines by these three estimates. The LSE line does not fit the bulk of the data, being a compromise between USA observation and the rest of the data, while the fitted lines for the other two estimates almost overlap and give a better representation of the majority of the data. Table 6 also gives the estimated regression parameters of these three methods for both the complete data and the data without the outlier USA. For LSE, the intercept estimate changes from 67.56 (complete data set) to 9.14 (without outlier) and the slope estimate changes from 0.23 (complete data set) to 0.37 (without outlier). Thus, it is clear that the outlier USA strongly influences LSE. 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(1987), High Breakdown-point and High Efficiency Robust Estimates for Regression. *The Annals of Statistics*, 15, 642-656. \[tab1\] 0.05in TRUE OLS $M_{H}$ $M_{T}$ LMS LTS S MM REWLSE ---------------- ----------- --------- --------- -------- -------- -------- -------- -------- $\beta_{0}: 0$ 0.0497 0.0532 0.0551 0.2485 0.2342 0.1372 0.0564 0.0645 $\beta_{1}: 1$ 0.0556 0.0597 0.0606 0.2553 0.2328 0.1679 0.0643 0.0733 $\beta_{0}: 0$ 0.1692 0.0884 0.0890 0.3289 0.3076 0.1637 0.0856 0.0982 $\beta_{1}: 1$ 0.1766 0.1041 0.1027 0.4317 0.3905 0.2041 0.1027 0.1189 $\beta_{0}: 0$ 1003.8360 0.2545 0.2146 0.3215 0.2872 0.1447 0.1824 0.1990 $\beta_{1}: 1$ 1374.0645 0.4103 0.3209 0.3659 0.3496 0.1843 0.2996 0.3164 $\beta_{0}: 0$ 0.3338 0.0610 0.0528 0.2105 0.2135 0.1228 0.0523 0.0538 $\beta_{1}: 1$ 0.4304 0.0808 0.0644 0.3149 0.2908 0.1519 0.0636 0.0691 $\beta_{0}: 0$ 9.3051 0.1082 0.0697 0.2752 0.2460 0.1430 0.0671 0.0667 $\beta_{1}: 1$ 5.5747 0.1083 0.0762 0.2608 0.2029 0.1552 0.0746 0.0801 $\beta_{0}: 0$ 0.8045 0.8711 0.8857 0.2161 0.1984 0.1256 0.0581 0.0598 $\beta_{1}: 1$ 13.4258 13.7499 13.8487 0.3377 0.3019 0.1695 0.0749 0.0749 : MSE of Point Estimates for Example 1 with $n = 20$ \[tab2\] 0.05in TRUE OLS $M_{H}$ $M_{T}$ LMS LTS S MM REWLSE ---------------- --------- --------- --------- -------- -------- -------- -------- -------- $\beta_{0}: 0$ 0.0113 0.0126 0.0125 0.0755 0.0767 0.0347 0.0125 0.0131 $\beta_{1}: 1$ 0.0096 0.0102 0.0103 0.0693 0.0705 0.0312 0.0103 0.0112 $\beta_{0}: 0$ 0.0283 0.0154 0.0153 0.0596 0.0659 0.0231 0.0153 0.0170 $\beta_{1}: 1$ 0.0255 0.0157 0.0164 0.0652 0.0752 0.0356 0.0163 0.0185 $\beta_{0}: 0$ 40.8454 0.0416 0.0310 0.0550 0.0392 0.0201 0.0323 0.0354 $\beta_{1}: 1$ 39.5950 0.0469 0.0387 0.0607 0.0476 0.0274 0.0402 0.0447 $\beta_{0}: 0$ 0.0650 0.0119 0.0107 0.0732 0.0737 0.0296 0.0107 0.0110 $\beta_{1}: 1$ 0.0596 0.0126 0.0123 0.0696 0.0775 0.0353 0.0122 0.0134 $\beta_{0}: 0$ 8.9470 0.0465 0.0107 0.0674 0.0658 0.0283 0.0106 0.0108 $\beta_{1}: 1$ 0.7643 0.0146 0.0120 0.0611 0.0704 0.0338 0.0119 0.0120 $\beta_{0}: 0$ 0.2840 0.2999 0.2983 0.0575 0.0595 0.0234 0.0107 0.0106 $\beta_{1}: 1$ 13.2298 13.5907 13.7210 0.0624 0.0790 0.0310 0.0127 0.0131 : MSE of Point Estimates for Example 1 with $n = 100$ \[tab3\] 0.05in TRUE OLS $M_{H}$ $M_{T}$ LMS LTS S MM REWLSE ---------------- ---------- --------- --------- -------- -------- -------- -------- -------- $\beta_{0}: 0$ 0.0610 0.0659 0.0744 0.3472 0.2424 0.1738 0.0679 0.0800 $\beta_{1}: 1$ 0.0588 0.0664 0.0752 0.4066 0.3247 0.2299 0.0709 0.1051 $\beta_{2}: 1$ 0.0620 0.0653 0.0725 0.3557 0.2724 0.2018 0.0716 0.0880 $\beta_{3}: 1$ 0.0698 0.0719 0.0758 0.3444 0.2657 0.1904 0.0751 0.0999 $\beta_{0}: 0$ 0.1745 0.1125 0.1168 0.3799 0.3040 0.2326 0.1177 0.1210 $\beta_{1}: 1$ 0.1998 0.1332 0.1364 0.4402 0.3404 0.2539 0.1311 0.1485 $\beta_{2}: 1$ 0.1704 0.1203 0.1272 0.4868 0.3831 0.2118 0.1242 0.1461 $\beta_{3}: 1$ 0.2018 0.1520 0.1732 0.5687 0.4964 0.3145 0.1649 0.2049 $\beta_{0}: 0$ 248.0170 0.3492 0.2579 0.7935 0.4657 0.3615 0.2630 0.2957 $\beta_{1}: 1$ 209.8339 0.4503 0.3713 1.2482 0.9701 0.4355 0.3784 0.4443 $\beta_{2}: 1$ 93.1344 0.4089 0.2936 1.0517 0.6203 0.5086 0.2965 0.3365 $\beta_{3}: 1$ 374.7307 0.4387 0.3206 1.0829 0.7704 0.4717 0.3123 0.4023 $\beta_{0}: 0$ 0.3245 0.0853 0.0837 0.2820 0.2433 0.1873 0.0785 0.0924 $\beta_{1}: 1$ 0.3391 0.1026 0.1001 0.4609 0.2875 0.2328 0.0996 0.1047 $\beta_{2}: 1$ 0.3039 0.0898 0.0938 0.4077 0.3053 0.1887 0.0900 0.1170 $\beta_{3}: 1$ 0.2618 0.0846 0.0941 0.4560 0.3023 0.2054 0.0900 0.1007 $\beta_{0}: 0$ 9.9455 0.1442 0.0706 0.3127 0.2334 0.1759 0.0680 0.0713 $\beta_{1}: 1$ 5.1353 0.1015 0.0636 0.3638 0.2769 0.1508 0.0617 0.0654 $\beta_{2}: 1$ 5.1578 0.1245 0.0730 0.4647 0.2796 0.1759 0.0690 0.0722 $\beta_{3}: 1$ 6.0662 0.1273 0.0612 0.3922 0.2733 0.1797 0.0597 0.0654 $\beta_{0}: 0$ 1.0096 1.0733 1.1334 0.3339 0.2491 0.1716 0.0821 0.0840 $\beta_{1}: 1$ 13.6630 14.0715 14.1688 0.4698 0.3126 0.2500 0.1467 0.1031 $\beta_{2}: 1$ 0.9201 0.9684 1.0108 0.4088 0.2681 0.2064 0.0899 0.1088 $\beta_{3}: 1$ 0.8538 0.9316 0.9937 0.4411 0.3373 0.2077 0.0709 0.0957 : MSE of Point Estimates for Example 2 with $n = 20$ \[tab4\] 0.05in TRUE OLS $M_{H}$ $M_{T}$ LMS LTS S MM REWLSE ---------------- --------- --------- --------- -------- -------- -------- -------- -------- $\beta_{0}: 0$ 0.0097 0.0108 0.0109 0.0743 0.0690 0.0359 0.0108 0.0119 $\beta_{1}: 1$ 0.0111 0.0120 0.0121 0.0736 0.0778 0.0399 0.0119 0.0130 $\beta_{2}: 1$ 0.0100 0.0106 0.0107 0.0713 0.0715 0.0404 0.0107 0.0114 $\beta_{3}: 1$ 0.0110 0.0116 0.0118 0.0662 0.0712 0.0388 0.0118 0.0121 $\beta_{0}: 0$ 0.0294 0.0145 0.0159 0.0713 0.0655 0.0330 0.0158 0.0179 $\beta_{1}: 1$ 0.0464 0.0198 0.0180 0.0651 0.0674 0.0368 0.0181 0.0195 $\beta_{2}: 1$ 0.0375 0.0183 0.0181 0.0727 0.0733 0.0352 0.0181 0.0195 $\beta_{3}: 1$ 0.0365 0.0176 0.0167 0.0646 0.0736 0.0344 0.0167 0.0175 $\beta_{0}: 0$ 36.7303 0.0388 0.0287 0.0681 0.0590 0.0317 0.0289 0.0326 $\beta_{1}: 1$ 31.6433 0.0499 0.0351 0.0624 0.0618 0.0262 0.0367 0.0372 $\beta_{2}: 1$ 41.4547 0.0422 0.0337 0.0788 0.0613 0.0321 0.0344 0.0369 $\beta_{3}: 1$ 29.7017 0.0476 0.0317 0.0714 0.0506 0.0320 0.0332 0.0362 $\beta_{0}: 0$ 0.0591 0.0109 0.0100 0.0656 0.0625 0.0281 0.0100 0.0109 $\beta_{1}: 1$ 0.0492 0.0122 0.0112 0.0558 0.0643 0.0349 0.0110 0.0115 $\beta_{2}: 1$ 0.0640 0.0123 0.0110 0.0635 0.0683 0.0337 0.0109 0.0118 $\beta_{3}: 1$ 0.0696 0.0135 0.0122 0.0573 0.0608 0.0333 0.0122 0.0128 $\beta_{0}: 0$ 9.1058 0.0560 0.0118 0.0631 0.0579 0.0322 0.0118 0.0120 $\beta_{1}: 1$ 0.8544 0.0186 0.0137 0.0738 0.0814 0.0377 0.0136 0.0143 $\beta_{2}: 1$ 0.9538 0.0189 0.0141 0.0672 0.0717 0.0379 0.0140 0.0146 $\beta_{3}: 1$ 0.8953 0.0193 0.0121 0.0652 0.0696 0.0363 0.0120 0.0123 $\beta_{0}: 0$ 0.2673 0.2869 0.2901 0.0632 0.0596 0.0300 0.0114 0.0114 $\beta_{1}: 1$ 13.2587 13.6355 13.6754 0.0590 0.0658 0.0305 0.0123 0.0127 $\beta_{2}: 1$ 0.1817 0.1889 0.1922 0.0660 0.0727 0.0344 0.0139 0.0144 $\beta_{3}: 1$ 0.1546 0.1607 0.1643 0.0668 0.0710 0.0344 0.0107 0.0108 : MSE of Point Estimates for Example 2 with $n = 100$ \[tab6\] 0.05in Country Per capita consumption of cigarette Deaths rates -------------- ------------------------------------- -------------- Australia 480 180 Canada 500 150 Denmark 380 170 Finland 1100 350 GreatBritain 1100 460 Iceland 230 060 Netherlands 490 240 Norway 250 090 Sweden 300 110 Switzerland 510 250 USA 1300 200 : Cigarettes data \[tab7\] 0.05in ------------ ----------- -------- ----------- -------- Estimators Intercept Slope Intercept Slope LS 67.5609 0.2284 9.1393 0.3687 MM 7.0639 0.3729 5.9414 0.3753 REWLSE 9.1393 0.3686 9.1393 0.3686 ------------ ----------- -------- ----------- -------- : Regression estimates for Cigarettes data \[tab8\] 0.05in Estimator Breakdown Point Asymptotic Efficiency --------- -------------------------------- ----------------- ----------------------- High BP LMS 0.5 0.37 LTS 0.5 0.08 S-estimates 0.5 0.29 GS-estimates 0.5 0.67 MM-estimates 0.5 0.85 REWLSE 0.5 1.00 Low BP GM-estimates(Mallows,Schweppe) $1/(p+1)$ 0.95 Bounded R-estimates $<0.2$ 0.90-0.95 Monotone M-estimates $1/n$ 0.95 LAD $1/n$ 0.64 OLS $1/n$ 1.00 : Breakdown Points and Asymptotic Efficiencies of Various Regression Estimators ![Plot of MSE of intercept (left) and slope (right) estimates vs. different cases for LMS, LTS, S, MM, and REWLSE, for model 1 when $n=100$.[]{data-label="figure1"}](myfig1.pdf){width="6in" height="5in"} ![Plot of MSE of different regression parameter estimates vs. different cases for LMS, LTS, S, MM, and REWLSE, for model 2 when $n=100$. []{data-label="figure2"}](myfig2.pdf){width="6in" height="6in"} ![Fitted lines for Cigarettes data[]{data-label="figure3"}](myfig3.pdf){width="6in" height="5in"}

--- abstract: | Recent data on the reaction $pd \to (pp) n$ with a fast forward $pp$ pair with very small excitation energy is analyzed within a covariant approach based on the Bethe-Salpeter formalism. It is demonstrated that the minimum non-relativistic amplitude is completely masked by relativistic effects, such as Lorentz boost and the negative-energy $P$ components in the $^1S_0$ Bethe-Salpeter amplitude of the $pp$ pair.\ [*PACS:*]{} 13.75.Cs, 25.10.+s author: - 'L.P. Kaptari' - 'B. Kämpfer' - 'S.S. Semikh' - 'S.M. Dorkin$^\ddag$' title: 'Relativistic effects in proton-induced deuteron break-up at intermediate energies with forward emission of a fast proton pair' --- The investigation of hadronic processes at high energies, such as reactions of protons scattering off deuterons, provides a refinement of information about strong interaction at short distances. Nowadays, large research programs of experimental studies of processes with polarized particles are in progress. Important are setups with deuteron targets or beams [@kox0; @preliminar; @cosyproposal], since the deuteron serves as a unique source of information on the neutron properties at high transferred momenta, the knowledge of which allows, e.g. to check a number of QCD predictions and sum rules. Additionally, the hadron-deuteron processes can be considered as complementary tool in investigating short-distance phenomena and also as a source of information unavailable in electromagnetic reactions. Of interest is the study of nucleon resonances, checking non-relativistic effective models, meson-nucleon theory, $NN$ potentials etc. In this line is the investigation of the deuteron break-up reaction with a fast $pp$ pair at low excitation energy, proposed in [@cosyproposal] and with first results reported in [@preprint_komarov]. One motivation for the experiment [@preprint_komarov] was the possibility to investigate the off-mass shell effects in $NN$ interactions. As predicted in [@preprint_komarov; @preprint_uzikov], at a certain initial energy of the beam protons, the cross section should exhibit a deep minimum, corresponding to the node of the non-relativistic $^1S_0$ wave function of the two outgoing protons, provided the non-relativistic picture holds and the off-mass shell effects can be neglected. The recent data [@preprint_komarov] exhibits, however, a completely different behavior: the cross section is smoothly decreasing; there is no sign of a pronounced minimum. Accounting for corrections beyond the one-nucleon exchange mechanism improve the agreement with data, however a quantitative description have not achieved [@preprint_komarov]. It is clear, that the non-relativistic treatment of the process becomes inadequate because of the high virtuality of the proton in the deuteron at the considered kinematics. More realistic approaches which take into account relativistic effects and the off-mass shellness of the interacting nucleons are desired. The Bethe-Salpeter (BS) formalism can serve as an appropriate approach to the problem because the off-mass shellness of the nucleons is an intrinsic feature of the BS equation. Moreover, the solution of the BS equation, being manifestly covariant, incorporate genuine relativistic effects (Lorenz boosts, negative-energy components etc.), hardly accessible within the Schrödinger formalism. In the present note we use the BS approach to analyze the data [@preprint_komarov] on deuteron break-up with the emission of a fast forward $pp$ pair [@foonote_1]. We pay particular attention on relativistic effects in the wave function of two nucleons in the continuum. The model is based on our solution of the BS equation for the deuteron with a realistic one-boson exchange kernel [@solution]. The final state interaction of the two protons is treated also within the BS formalism by solving the BS equation for the $t$-matrix within the one-iteration approximation [@ourfewbody; @nashi]. In doing so, a big deal of off-mass shell effects and relativistic corrections are taken into account already within the spectator mechanism. Let us consider the process $$p\,+ d \,=\,(p_1p_2)(0^0 ) +n(180^0) \label{reaction}$$ at low excitation energy of the pair ($E_x \sim 0 - 3$ MeV) and intermediate initial kinetic energies $T_p \sim 0.6 - 2.0$ GeV corresponding to the conditions at the Cooler Synchrotron COSY in the experiment [@cosyproposal; @preprint_komarov]. In the one-nucleon exchange approximation this reaction can be represented by the diagram depicted in Fig. \[pict1\], where the following notation is adopted: $p = (E_p,\bp)$ and $n=(E_n,\bn)$ are the four-momenta of the incoming (beam) proton and outgoing (not registered) neutron, $P_f$ is the total four-momentum of the $pp$ pair, which is a sum of the corresponding four-momenta of the detected protons, $p_1=(E_1,\bp_1)$, $p_2=(E_2,\bp_2)$. The invariant mass of the $pp$-pair is $M_{pp}^2=P_f^2=(2m+E_x)^2$, where $m$ stands for the nucleon mass and $E_x$ is the excitation energy. Our calculations are performed in the laboratory system where the deuteron is at rest. For specific purposes, the center of mass of the pair will be considered as well, where all relevant quantities are superscripted with asterisks. A peculiarity of the processes (\[reaction\]) is that the transferred three-momentum from the initial proton to the second proton in the pair is rather high. Moreover, from the kinematics one finds that the momentum of the neutron is also high ($|\bn| \sim 0.3-0.5$ GeV/c), which implies that, since the outgoing neutron is on-mass shell, the proton inside the deuteron was essentially off-mass shell before the interaction. Correspondingly, it becomes clear that the process of $NN$ interaction in the upper part of the diagram is by far more complicate than the elastic interaction. For instance, let us consider a typical kinematical situation, say $|\bp |=1.22$ GeV/c, $\theta_1'\sim 4^0$, excitation energy $E_x = 3$ MeV and $|\bp_1|=0.765$ GeV/c. This means that the neutron three-momentum is $|\bn |\simeq 0.5$ GeV/c, i.e., the four-momentum of the off-mass shell proton was $q = (M_d-E_n,-\bn)$. Now, if one supposes that in the upper vertex there was an elastic process of two on-mass shell protons, then only one kinematical quantity would be necessary to describe the process, e.g., at given $|\bp_1 |=0.765$ GeV/c the scattering angle would be $\sim 28^0$ in the elastic kinematics (instead of $4^0$ in the full reaction); or at given scattering angle $4^0$, the momentum of the elastically scattered proton would correspond to $|\bp_1 |=1.21$ GeV/c ($|\bp_2| = 0.334$ GeV/c) instead of the detected momentum $|\bp_1|\simeq |\bp_2|\simeq 0.765$ GeV/c. This demonstrates that the $NN$ interaction in the upper part of the diagram has a quite complicate nature. It can be considered as consisting at least of two steps: (i) an inelastic process which puts the target nucleon on mass-shell, and (ii) an elastic interaction in the $pp$ pair in the $^1S_0$ final state. Since a large amount of the transferred energy is needed to locate the second proton on mass-shell, the relativistic corrections can play a crucial role here. The invariant cross section of the reaction (\[reaction\]) reads $$d^6\sigma\,=\,\frac{1}{(4\pi)^5\ \lambda(p,d)}\, |M_{fi}|^2 \frac{d\bp_1\ d\bp_2}{E_1E_2}\, \delta\left( E_0+E_d-E_1-E_2-E_n\right), \label{cross2}$$ where $\lambda(p,d)$ is the flux factor, and $M_{fi}$ the invariant amplitude; the statistical factor $1/2$ due to two identical particles (protons) in the final state has been already included. The covariant matrix element corresponding to the diagram in Fig. \[pict1\] can be written in the form $$M_{fi} = \bar u(s,\bn)(\hat n-m)\Psi_d(n) \left[ (\hat p_2+m) \overline{\Psi}_{^1S_0}(p) (\hat p_1-m) u(r,\bp) \right], \label{ampl}$$ where $u(r,\bp)$ ($\bar u(s,\bn)$) stands for the Dirac spinor of the incident proton (outgoing neutron) with the spin projection $r$ ($s$) and 3-momentum $\bp$ ($\bn$), $\Psi_{D(^1S_0)}$ denote the BS amplitudes for the deuteron and the $pp$-pair in the continuum, respectively. By using the spin angular basis to obtain the partial decomposition of the deuteron BS amplitude and the covariant form for the four partial components for the $^1S_0$-state [@nashi] of the $NN$ pair [@footnote_2], the invariant amplitude $M_{fi}$ is $$M_{fi}=(-1)^{\frac12 -r}\ \CK(^1S_0)\frac{1}{\sqrt{8\pi}(M_d-2E_n)}\left\{ \sqrt{2}C_{\frac12 s\frac12 -r}^{1\CM} \left (G_S-\frac{G_D}{\sqrt{2}}\right )+ 3\delta_{\CM,0}\delta_{s,r}\frac{G_D}{\sqrt{2}} \right\}, \label{am1}$$ where the contribution $\CK(^1S_0)$ from the upper part of the diagram in Fig. \[pict1\] is $$\CK(^1S_0) = \sqrt{\frac{E_p+m}{E_n+m}} \left[ h_1\left( E_n+m-\frac{|\bn||\bp|}{E_p+m}\right) -h_3\frac{M_d-2E_n}{m}\left(E_n+m+\frac{|\bn||\bp|}{E_p+m}\right) \right]. \label{k1s0}$$ $G_{S,D}$ denote the BS vertices for the deuteron, $h_1,h_3$ are the non-vanishing covariant partial components of the $^1S_0$ configuration in the continuum. The relation of the amplitudes $h_i\ (i=1\ldots 4)$ to the partial solutions of the BS equation in the $NN$ center of mass $^1S^{++}_0, ^1S^{--}_0, ^3P^{+-}_0, ^3P^{-+}_0$ can be found, e.g., in ref. [@nashi]. In what follows we neglect the contribution of the extremely small $^1S^{--}$ component, keeping only the $^1S^{++}$ component as the main one and the $P$ components as the ones providing purely relativistic corrections. It is easy to check that for the unpolarized particles the cross section factorizes in two independent parts, i.e., $ \frac16 \sum\limits_{s,r,\CM} |M_{fi}|^2 \simeq \left| \CK(^1S_0)\right |^2\ \left( u_S(\bn)^2 + u_D(\bn)^2 \right ),$ as it should be within the spectator mechanism with $^1S_0$ $(L_f=0)$ in the final state (see also discussion in ref. [@ourdphi]); $u_S$ and $u_D$ are the BS deuteron $S$ and $D$ waves [@ourfewbody] In our numerical calculations we use the deuteron BS amplitude from [@solution]. For $^1S_0$ state one should solve the inhomogeneous BS equation in the $NN$ center of mass to obtain the covariant amplitudes $h_{1,3}$. Note that in such a way the effects of the Lorentz boost are automatically taken into account (see, e.g., [@ourfewbody]). The partial “$++$” components of the BS vertices in the $NN$ center of mass, for both the bound and the scattering states have a direct analogue with the corresponding non-relativistic wave functions. Moreover, at low intrinsic relative momenta the BS and non-relativistic wave functions basically coincide. Hence, due to the low excitation energy of the $pp$-pair, in expressing $h_{1,3}$ via the partial amplitudes at rest one may safely replace the “$++$” vertex by its non-relativistic analogue. Relativistic effects are then included in boosting the $^1S^{++}_0$ component to the deuteron center of mass system and also by taking into account the contributions of $P$ components in $h_{1,3}$. To find the $P$ waves we solve the BS equation for the pair in its center of mass within the one iteration approximation [@ourfewbody; @ourcharge], with the trial function as the exact solution for the $t$ matrix within a separable potential [@plessas]. By defining new partial components $g_i$ ($i=1\ldots 4$) as “connected amplitudes”, i.e., the partial amplitudes without the free terms, we obtain for the $P$-waves $$\begin{aligned} g_3(k) = i\, g_{\pi NN}^2\,\, \left[ \frac{M_{pp}}{\sqrt{\pi}}\,V_{31}(k,p^*)- i\,\int\limits_0^\infty\, \frac{dp\,p^2}{(2\pi)^3}\,V_{31}(k,p)\,\frac{g_1(p)}{M_{pp}-2\,E_p+i\epsilon} \right], \label{oia}\end{aligned}$$ where the partial kernel $V_{31}$ is $V_{31}(k,p^*)$ $=\displaystyle\frac{\pi m}{|\bp^*||\bk|E_{\bp^*}E_{\bk}} \left\{ |\bp|^*Q_1(y)-|\bk|Q_0(y) \right\}$ with $Q_L(y)$ as Legendre function with the argument $y=(|\bp^*|^2+|\bk|^2)+\mu_\pi^2-k_0^2)/(2|\bp^*||\bk|)$ [@ourcharge; @nashi]. In the first iteration the trial function $g_1(k)$ is expressed via the non-relativistic $t$ matrix [@plessas] as $$\begin{aligned} g_1(k)=i\, (4\pi)^{5/2}\,\frac{m}{2}\,\,t_{NR}(k,p^*), \label{g1}\end{aligned}$$ where the normalization of $t_{NR}$ corresponds to $ t_{NR}(p^*,p^*)=-\displaystyle\frac{2}{\pi m |\bp^*|}\, {\rm e}^{i\delta_0}\,\sin\, \delta_0, $ where $\delta_0$ denotes the experimentally known phase-shift of the elastic $pp$ scattering in the $^1S_0$ state. By using the Sokhotsky-Weierstrass formula for the Cauchy type integrals one finally obtains $$\begin{aligned} %\begin{equation} g_3(k) %& = & =\frac{i g_{\pi NN}^2}{\sqrt{\pi}} \left[M_{pp} V_{31}(k,p^*) \left\{ 1 - \frac{i\pi m|\bp^*|}{2} t_{NR}(p^*, p^*)\right\} %\right. \nonumber\\ %& + &\left. + m {\cal P} \int\limits_0^\infty dp p^2 V_{31}(k,p) \frac{t_{NR}(p, p^*)}{E_{p^*}-E_p} \right], \nonumber \label{oiafinal}\end{aligned}$$ where, due to the separable stricture of the chosen $t_{NR}$, further calculations of principal values of the relevant integrals can be carried out analytically. In Figs. \[pict2\] and \[pict3\] we present results of numerical calculations of the five-fold cross section ${d\sigma}/{d\Omega_1d\Omega_2d|\bp_1|}$ and the two-fold cross section $ {d\sigma}/{d\Omega_{n}}$ (with $\Omega_{n}$ as the solid angle of the momentum of the neutron in the center of mass of the reaction), integrated over the excitation energy in a range $E_x = 0 - 3$ MeV. The calculations have been performed with our numerical solution for the deuteron BS amplitude (inclusion of the $P$ components in the deuteron amplitude leads to negligibly small corrections, therefore here we do not discus these contributions). The dotted curves are results of non-relativistic calculations, while the dashed curves include pure Lorentz boos effects, i.e., relativistic calculations with including the $^1S^{++}_0$ component only. It is clearly seen that the boost corrections are fairly visible: They cause a shift of the minimum of the cross section. The agreement at low initial energies becomes better, however, the cross section is still too small at large values of $T_p$. Fig. \[pict3\] reveals that an account for only $"++"$ components is not sufficient to describe data. An excellent description is achieved by taking into account all the relativistic effects, including the contribution of the negative-energy $P$ waves. Note, that as demonstrated in refs. [@ourfewbody; @nashi] in reactions of $pd$ and in threshold-near $ed$ disintegration, the inclusion of $P$ waves exactly recovers the non-relativistic calculation with taking into account the $N\bar N$ pair production effects. Hence, in our case this is a hint that covariant calculations within the relativistic spectator mechanism contains already some contributions beyond the one-nucleon exchange mechanism. In summary we analyze the recent data [@preprint_komarov] of the reactions $p d(p,n) \to (pp) n$ within a covariant approach based on the Bethe-Salpeter formalism. Relativistic effects (Lorentz boost, negative-energy $P$ components) are important and responsible for the smooth decline of the cross section, in contrast to predictions of non-relativistic models. We are grateful to H.W. Barz for valuable discussions. L.P.K. and S.S.S. acknowledge the warm hospitality of the nuclear theory group in the Research Center Rossendorf. The work has been supported by Heisenberg - Landau program, BMBF grant 06DR921 and RFBR 00-15-96737. [99]{} S. Kox, E.J. Beise (spokespersons), TJNAF experiments 94-018 “Measurement of the Deuteron Polarization at Large Momentum Transvers in $D(e,e')D$ Scattering”; http://www.jlab.org/ex\_prog; Phys. Rev. Lett. [**84**]{}, 5053, (2000); Nucl. Phys. [**A684**]{} 521, (2001). E. Tomasi-Gustafsson, in Proc. of [*XIV International Seminar On High Energy Physics Problems*]{}, Preprint JINR No. E1,2-2000-166 (Dubna, 2000). V.I. Komarov (spokesman) et al., COSY proposal \#20 (updated 1999), “Exclusive deuteron break-up study with polarized protons and deuterons at COSY”, http://ikpd15.ikp.kfa-juelich.de:8085/doc/Proposals.html. V.I. Komarov et al., e-Print Archive: nucl-ex/0210017. Yu.N. Uzikov, V.I. Komarov, F. Rathmann, H. Seyfarth, e-Print Archive: nucl-th/0211001;\ Yu.N. Uzikov, J.Phys. [**G28**]{}, B13 (2002). The considered reaction $p d \to (pp) n$ must not be confused with the charge exchange reaction $p d \to n (pp)$ with a slow pair in the final state. A.Yu. Umnikov, L.P. Kaptari, F.C. Khanna, Phys. Rev. [**C56**]{}, 1700 (1997);\ A.Yu. Umnikov, L.P. Kaptari, K.Yu. Kazakov, F.C. Khanna, Phys. Lett. [**B334**]{}, 163 (1994);\ A.Yu. Umnikov, Z. Phys. [**A357**]{}, 333 (1997). L.P. Kaptari, B. Kämpfer, S.M. Dorkin, S.S. Semikh, Few-Body Systems [**27**]{}, 189 (1999); Phys. Rev. [**C57**]{}, 1097 (1998); Phys. Lett. [**B404**]{}, 8 (1997);\ P. Kaptari, B. Kämpfer, A.Yu. Umnikov, F.C. Khanna, Phys. Lett. [**B351**]{}. 400 (1995). S.G. Bondarenko, V.V. Burov, M. Beyer, S.M. Dorkin, Phys. Rev. [**C58**]{}, 3143 (1998); e-Print Archive: nucl-th/9612047. Actually, when one particle is on mass shell (the incident proton in our case), only two partial amplitudes differ from zero corresponding to positive values of the $\rho$-spin of the on mass shell particle. L.P. Kaptari, B. Kämpfer, S.S. Semikh, e-Print Archive: nucl-th/0212066. L.P. Kaptari, B. Kämpfer, S.S. Semikh, S.M. Dorkin, arXiv:nucl-th/021107;\ L.P. Kaptari, B. Kämpfer, S.S. Semikh, S.M. Dorkin, Phys. Atom. Nucl.[**65**]{}, 442 (2002). J. Haidenbauer, Y. Koike, W. Plessas, Phys. Rev. [**C33**]{}, 439 (1986).  2.7in  -6mm  4.5in  4.5in

--- author: - 'Oleksiy Kashuba[^1]' - Herbert Schoeller - Janine Splettstoesser title: Nonlinear adiabatic response of interacting quantum dots --- Introduction ============ Adiabatic transport through quantum dots associated with a slow cyclic time dependence of the system parameters has generated a lot of interest in recent years, particularly in connection with quantum pumps [@buttiker0; @brouwer; @pothier; @giazzotto] and mesoscopic capacitors [@buttiker1; @gabelli], see also Ref. [@review_nt]. For noninteracting systems the scattering formalism is a powerful tool to describe the adiabatic response [@buttiker0; @brouwer]. A central challenge in this field is the understanding of the influence of strong interactions as they typically occur in small quantum dots. Although general current formulas have been derived in terms of Green’s functions [@splett0; @selaoreg; @fiorettosilva], their evaluation is quite difficult in the coherent regime at low temperature. Progress has been achieved in the perturbative regime of high temperature [@splett1], where it was shown that pure interaction effects can be revealed by the adiabatic response which would be covered by more dominant effects in the steady state [@splett1; @reckermann10]. These studies also included the properties of the $RC$-time in linear response [@splett2]. In contrast, for interacting quantum dots at low temperature, a generic formalism providing the adiabatic time evolution in response to any parameter in linear or nonlinear response is not yet available. So far, quantum pumping has been studied for special models, like e.g. the 2-channel Kondo model in the strong coupling regime [@selaoreg], the Kondo model at the exactly solvable Toulouse point [@schiller2], and the single-impurity Anderson model within slave-boson mean-field approximation [@aono]. In addition, the research on interaction effects in mesoscopic capacitors has concentrated on the special case of linear charge response to an external AC gate voltage by using the standard relation to the equilibrium density-density correlation function. Here, the main object of interest was the charge relaxation resistance $R$ defined by expanding the charge response $\delta Q$ in the external frequency $\omega$ via $$\delta Q = ( C + i\omega RC^{2}) \delta \mathcal{V} \quad, \label{eq:RC}$$ where $Q(t)=\delta Q e^{-i\omega t}$ denotes the charge, $C$ is the static quantum capacitance and $\mathcal{V}(t)=\delta \mathcal{V}e^{-i\omega t}$ defines the external AC gate voltage. For a single transport channel and provided that the Coulomb interaction is weak, a universal relaxation resistance $R=h/2e^2$ was found in the coherent regime at zero temperature [@buttiker1; @buttiker2]. For interacting metallic dots and the single-impurity Anderson model the Shiba relation was shown to be a powerful tool to analyze $R$ and its universality [@rodionov; @lehur-ondotint; @lehur_preprint]. Using bosonization, the influence of Luttinger-liquid leads on $R$ has also been studied [@kato-inleadint] and a numerical approach has been used away from the Fermi-liquid regime [@leelopez]. In this Letter we develop a general approach to deal with the adiabatic dissipative response, where the time scale of the external modulation $\propto\omega^{-1}$ is much larger than the inverse of typical relaxation rates $\Gamma_{c}$, of a coherent quantum dot at low temperature including Coulomb interactions. We show that the adiabatic response can be calculated very efficiently by using quantum field theoretical methods in Liouville space developed in Refs. [@hs_epj09; @leijnse_wegewijs_prb08] and generalized here for the case of time-dependent Hamiltonians (for approaches within Keldysh formalism see the review articles [@review_nt; @hs_epj09] and the recent development [@kennes_meden]). We provide a general relationship of the adiabatic response to effective Liouvillians and vertices known from real-time renormalization group (RTRG) in the stationary limit with instantaneous time parametrization, based on powerful techniques for the calculation of Laplace-variable derivatives, recently used within the $E$-flow scheme of RTRG [@pletyukhov_hs_preprint]. As a consequence, our formalism is suitable to any model which can be treated by RTRG, which is applicable for many generic models with charge and spin fluctuations [@hs_epj09]. Recent applications of RTRG cover the Kondo model for both weak [@hs_reininghaus_prb09; @pletyukhov_schuricht_hs_prl10; @schuricht_hs_prb09; @pletyukhov_schuricht_prb11; @hoerig_etal_prb12] and strong [@pletyukhov_hs_preprint] coupling, and the interacting resonant level model (IRLM) [@rtrg_irlm]. Most importantly, in contrast to previous research, our formalism allows for the adiabatic variation of [*any*]{} parameter in [*nonlinear*]{} response, where no relation to equilibrium density-density correlation functions is possible and where certain identities like e.g. the Shiba relation are no longer applicable. Therefore, going beyond previous studies of the linear response to an external gate voltage, we also study the response to other parameters, like the tunneling coupling or the Coulomb interaction, which experimentally can be either realized intentionally, or indirectly induced by the gate voltage. We even cover the regime of nonlinear response, motivated by recent works on mesoscopic capacitors in the nonlinear driving regime [@nonlinear]. Instead of the linear response formula (\[eq:RC\]) for the charge variation by an external gate voltage, we decompose the dynamics of any observable $A$, with a nonvanishing instantaneous contribution and its adiabatic correction in response to any parameter, as $A(t)=A^{(i)}(t)+A^{(a)}(t)$. The central quantity of our interest is the delay time scale $\tau_A$ for the expectation value $A$, defined by $$\label{eq:delay_time} \tau_A\,=\, |A^{(a)}/\dot{A}^{(i)}|\quad,$$ which describes the delay of the full solution comparing to the instantaneous one. For $A\equiv \dot{Q}$ in linear response to a time-dependent gate voltage, it is equivalent to the $RC$-time. In general $\tau_A$ can be quite different from typical relaxation times, depending on the observable, the type of excitation and its amplitude, and it is of fundamental interest to understand its dependence on interactions. We use our method to consider the IRLM with a single lead, which constitutes a minimal model for the mesoscopic capacitor with one single-particle level, where strong correlations are induced by a local Coulomb interaction between the dot and the lead. Recently, the IRLM has been extensively used to study nonequilibrium transport through interacting quantum dots [@mehta; @saleur1; @saleur2; @rtrg_irlm; @karrasch10], including the dynamics of the time evolution into the stationary state [@rtrg_irlm]. We calculate the nonlinear adiabatic charge response and the delay time $\tau_Q$, including renormalization effects of the tunneling enhanced by correlations. Importantly, we find that the functional form of the charge delay time $\tau_Q$ is robust against the choice of the time-dependent parameter even in nonlinear response, whereas the capacitance $C$ or the relaxation resistance $R$ get a more complex form when the tunneling or the Coulomb interaction are varied. Finally, we analyze further possible experimental implementations of the predicted results for the IRLM with time-dependent parameters, namely via the Anderson-Holstein model in molecular electronics or via the spin-boson model in cold-atom setups. Method ====== We start from a general Hamiltonian $H(t)=H_{\rm res}+\sum_\alpha\mu_\alpha(t)\hat{N}_\alpha+H_{\rm dot}(t)+V(t)$ of an interacting quantum dot coupled to noninteracting fermionic reservoirs with time-dependent chemical potentials $\mu_\alpha(t)$ and a flat d.o.s. of width $2D$ via the coupling $V(t)$. Generalizing the Liouvillian approach of Ref. [@hs_epj09] to the case of time-dependent Hamiltonians, one finds that the dissipative dynamics of the reduced density matrix $\rho(t)$ of the dot can be described by the effective Liouvillian equation $$i\partial_{t}\rho(t) = \int_{t_{0}}^{t} L(t,t') \rho(t') dt', \label{eq:rhoeq}$$ where $L(t,t')$ is the effective dot Liouvillian obtained by integrating out the reservoirs ($\hbar=1$). At the initial time $t_{0}$ the total density matrix factorizes into an arbitrary dot and an equilibrium reservoir part. Since we are only interested in the asymptotic dynamics we set $t_0=-\infty$ below. Following Ref. [@hs_epj09], the effective Liouvillian can be calculated diagrammatically, where each diagram of order $O(V^n)$ consists of a product of vertices $G(t_i)$, $t_1>\dots >t_n$, with effective dot propagators $\Pi(t_i,t_{i-1})$ in between. In addition, the information of the Fermi distribution and the d.o.s. of the reservoirs is contained in time-independent reservoir contractions between the vertices. Using the formal definition $G(t,t')=G(t)\delta(t-t'-0^+)$, we find that each term can be written in terms of a generalized convolution in time space as $(G\circ\Pi\circ\dots\circ\Pi\circ G)(t,t')$, where $(A\circ B)(t,t')\equiv \int_{t'}^{t} d\tau A(t,\tau)B(\tau,t')$. Introducing the partial Laplace transform $A(t;E)=\int_{-\infty}^{t} dt' e^{i(E+i0)(t-t')}A(t,t')$, we get $$\begin{aligned} \nonumber &(A_1\circ A_2\circ \dots \circ A_n)(t;E) =e^\mathcal{D} A_1(t;E)\dots A_n(t;E)=\\ \label{eq:gradient_expansion} &\left.e^{i\sum_{j>k}\partial_{E_{j}}\partial_{t_{k}}} A_1(t_{1},E_{1}) \dots A_n(t_{n},E_{n})\right|_{\scriptsize E_{j}=E,t_{k}=t}\,.\end{aligned}$$ The special differential operator $\mathcal{D}=i\partial_E^{\text{left}}\partial_t^{\text{right}}$ prescribes the energy derivative to act left to the time derivative. This rule is a natural generalization to Laplace space of analog identities in Fourier space used for gradient expansions in the Keldysh formalism [@rammer]. Formally, it allows for the straightforward application of the Liouvillian approach to time-dependent Hamiltonians, with the difference that the exponential differential operator has to be taken beforehand. In the adiabatic case, the exponential can be expanded in $\partial_E\partial_t\sim \frac{\omega}{\Gamma_{c}}\ll 1$, leading to an expansion of the effective Liouvillian, $$\label{eq:L_adiabatic_expansion} L(t;E) \,=\, L^{(i)}(t;E) \,+\, L^{(a)}(t;E) \,+\,\dots\quad.$$ Here, $L^{(i)}(t;E)$ denotes the instantaneous part, where the time $t$ enters only parametrically via the external parameters, and $L^{(a)}(t;E)$ is the first adiabatic correction, which is linear in the time derivatives of the external parameters. Once the effective Liouvillian $L(t;E)$ is known up to the adiabatic correction, one can use it in eq. (\[eq:rhoeq\]) which reads $$i\partial_{t}\rho(t) = (L\circ\rho)(t;0) = \left.e^{i \partial^{L}_{E} \partial^{\rho}_{t}} L(t;E) \rho(t)\right|_{E=0} \label{eq:tE_kinetic_equation}$$ in the mixed $(t;E)$-representation. Expanding $\rho(t)=\rho^{(i)}(t)+\rho^{(a)}(t)+\dots$ analogously to eq. (\[eq:L\_adiabatic\_expansion\]), we find by comparing equal powers in the external frequency $$\begin{aligned} \label{eq:inst} L^{(i)}\rho^{(i)}=0 \quad,\quad \mathrm{Tr}\,\rho^{(i)}=1 \quad,\quad \mathrm{Tr}\,\rho^{(a)}=0 \quad, \\ \label{eq:adia} L^{(i)}\rho^{(a)} + L^{(a)}\rho^{(i)} - i(1-\partial_{E}L^{(i)})\partial_{t}\rho^{(i)} =0 \quad .\end{aligned}$$ In all arguments of $L^{(i/a)}$ and $\partial_{E}L^{(i)}$, $E=0$ has to be taken. From these equations the instantaneous density matrix $\rho^{(i)}(t)$ and the first adiabatic correction $\rho^{(a)}(t)$ can be determined. We emphasize that this approach is even applicable in nonlinear response in the amplitude of the external perturbations, i.e. only the time scale of the external modulation needs to be large enough. Furthermore, it allows for an adiabatic modulation of any parameter of the Hamiltonian and is not restricted to a time-dependent gate voltage. The algebra of (\[eq:inst\]) and (\[eq:adia\]) can be easily evaluated for quantum dots with two accessible states. If additional conservation laws are present (as, *e.g.*, charge conservation in the IRLM or spin-$S_z$ conservation in the Kondo model), the nonvanishing matrix elements of the Liouvillian can be written as $$\begin{aligned} \label{eq:Liouvillian} L_{\bar{s}\bar{s},ss}=-L_{ss,ss}=i\Gamma_{s}=i\Gamma/2+is\Gamma' \,\,,\,\, L_{s\bar{s},s\bar{s}}=\epsilon_s\,,\end{aligned}$$ where $s\equiv\pm$ denotes the two states and $\bar{s}=-s$. At $E=0$ we get from (\[eq:inst\]) and (\[eq:adia\]) that the instantaneous density matrix is diagonal $\rho^{(i)}_s=\Gamma^{(i)}_s/\Gamma^{(i)}= 1/2+s \Gamma^{\prime(i)}/\Gamma^{(i)}$ and the adiabatic correction fulfills $\rho^{(a)}_+=-\rho^{(a)}_-$ with $$\begin{aligned} \label{eq:rho_adiab} \rho^{(a)}_+=\frac{1}{\Gamma^{(i)}}\left\{\Gamma^{\prime (a)}- \frac{\Gamma^{\prime(i)}}{\Gamma^{(i)}}\Gamma^{(a)}- (1+i\partial_E\Gamma^{(i)})\partial_t\left(\frac{\Gamma^{\prime(i)}}{\Gamma^{(i)}}\right)\right\}\end{aligned}$$ Below we use this result to evaluate the adiabatic response for the IRLM. Calculation of $L^{(a)}$ ======================== We now turn to the central issue of how to relate the adiabatic correction $L^{(a)}(t;E)$ to the instantaneous quantities known from RTRG. Using (\[eq:gradient\_expansion\]), we can formally write $L^{(a)}(t;E)=\mathcal{D} L^{(i)}(t;E)=i\partial_E^{\text{left}}\partial_t^{\text{right}} L^{(i)}(t;E)$. The representation of $L^{(i)}(t;E)$ by its diagrammatic expansion specifies what “left” and “right” means for the derivatives with respect to $E$ and $t$. As a first step, we represent the derivative $\partial_E L^{(i)}$ by effective vertices and propagators, using a method developed in Ref. [@pletyukhov_hs_preprint]. For generic models with two types of vertices, namely single (e.g. tunneling) and double (e.g. Coulomb interaction, exchange, etc.), we decompose it into two contributions in leading order and find $$\begin{aligned} \nonumber \partial_E L^{(i)}(t;E)&=\quad\partial_E L^{(i)}_\Gamma(t;E)+\partial_E L^{(i)}_U(t;E)\\ &= \, % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{red} \qbezier(-3,-2.5)(0,0)(3,2.5) \end{picture} + \frac{1}{2} \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{red} \qbezier(-3,-2.5)(0,0)(3,2.5) \end{picture}\quad. \label{eq:L_E_der}\end{aligned}$$ The diagrammatic rules are explained in detail in Refs. [@hs_epj09; @pletyukhov_hs_preprint]. The single (double) circles represent full effective single (double) vertices with effective propagators $\Pi^{(i)}(t;E)=\frac{1}{E-L^{(i)}(t;E)}$ in between (the Laplace variable is shifted by the frequencies and chemical potentials of all reservoir contractions crossing over the propagator). The left slash indicates $\partial_E$ and the grey (green, color online) line represents the reservoir contraction given by the antisymmetric part $f(\omega)-\frac{1}{2}$ of the Fermi distribution function. All possible diagrams for $\partial_{E}L^{(i)}$ can be classified by the number of lines over the propagator containing a derivative. In the next step we perform the time derivative $i\partial_t$ right to the energy derivative. The energy derivative is then shifted by partial integration to the reservoir contraction (indicated by a (blue) cross) [@pletyukhov_hs_preprint]. This yields $$\begin{aligned} \nonumber L^{(a)}_\Gamma(t;E)\quad &= \quad\quad\! % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{red} \qbezier(-3,-2.5)(0,0)(3,2.5) \qbezier(19.5,-2.5)(17,0)(14.5,2.5) \end{picture} + \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{red} \qbezier(-4,-2.5)(-2,0)(0,3) \qbezier(0,3)(2,0)(4,-2.5) \end{picture} \\ \label{eq:LG_Et_der} &= \quad - \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \color{red} \qbezier(19.5,-2.5)(17,0)(14.5,2.5) \end{picture} + \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{red} \qbezier(-4,-2.5)(-2,0)(0,3) \qbezier(0,3)(2,0)(4,-2.5) \end{picture}\quad, \\ \label{eq:LU_Et_der} L^{(a)}_U(t;E)\quad &= \quad \frac{1}{2} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{red} \qbezier(-3,-2.5)(0,0)(3,2.5) \qbezier(23,-2.5)(17,0)(11,2.5) \end{picture} +\,\,\frac{1}{2} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{red} \qbezier(-4,-2.5)(-2,0)(0,3) \qbezier(0,3)(2,0)(4,-2.5) \end{picture}\quad, %\end{aligned}$$ where the right slash represents $i\partial_t$ and the hat indicates the differential operator $\mathcal{D}=i\partial_E^{\text{left}}\partial_t^{\text{right}}$. The frequency integral in both diagrams of (\[eq:LG\_Et\_der\]) is well-defined in the wide-band limit, so (\[eq:LG\_Et\_der\]) provides an explicit expression for the adiabatic correction containing the tunneling vertices in terms of renormalized vertices and propagators. In contrast, the frequency integrals in (\[eq:LU\_Et\_der\]) are logarithmically divergent. We therefore take a second derivative with respect to $E$, yielding an RG equation for the adiabatic part, $L^{(a)}_U(t;E)$, after partial integration. This contains the double vertices $$\begin{aligned} \label{eq:LU_EEt_der} \partial_E\,L^{(a)}_U(t;E)\, &= \, \frac{1}{2} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \qbezier(-2.5,3.5)(0,6)(2.5,8.5) \qbezier(-2.5,8.5)(0,6)(2.5,3.5) \color{red} \qbezier(23,-2.5)(17,0)(11,2.5) \end{picture} -\,\,\frac{1}{2} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \color{red} \qbezier(-4,-2.5)(-2,0)(0,3) \qbezier(0,3)(2,0)(4,-2.5) \end{picture}\quad. %\end{aligned}$$ Eqs. (\[eq:LG\_Et\_der\]) and (\[eq:LU\_EEt\_der\]) are the final results for the evaluation of adiabatic corrections of the Liouvillian in leading order, based on the instantaneous values of the renormalized vertices and Liouvillian, which are obtained from RTRG[^2]. For the adiabatic part of the propagator, appearing in the second diagram of (\[eq:LG\_Et\_der\]) and (\[eq:LU\_EEt\_der\]) each, we insert $\Pi^{(a)}=\Pi^{(i)} L^{(a)}\Pi^{(i)} + (\partial_E\Pi^{(i)})(i\partial_t L^{(i)})\Pi^{(i)}$. The first term does however not contribute to the adiabatic propagator in leading order. An interesting question is whether derivatives with respect to the Laplace and time variable commute in leading order, i.e. whether the adiabatic correction to the effective Liouvillian, eqs. (\[eq:LG\_Et\_der\]) and (\[eq:LU\_EEt\_der\]), can be written as $$L^{(a)}_{\Gamma/U}(t;E) \stackrel{?}{=} \frac{1}{2}i\partial_{E}\partial_{t}L^{(i)}_{\Gamma/U}(t;E)\, . \label{eq:L_a_approx}$$ A similar relation was investigated so far only for noninteracting systems [@moskalets]. To analyze its validity we introduce the complementing differential operator $\mathcal{D}'=i\partial_E^{\text{right}}\partial_t^{\text{left}}$, where the energy derivative is taken [*right*]{} to the time derivative. Analogously to (\[eq:LG\_Et\_der\]) and (\[eq:LU\_EEt\_der\]) one finds in leading order $$\begin{aligned} \label{eq:LG_Et_der_c} \mathcal{D}'L^{(i)}_\Gamma(t;E)\,\, &= \,\, - % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \color{red} \qbezier(-14.5,-2.5)(-17,0)(-19.5,2.5) \end{picture} + % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{red} \qbezier(0,-2.5)(-2,0)(-4,3) \qbezier(4,3)(2,0)(0,-2.5) \end{picture}\quad, % \\ \label{eq:LU_EEt_der_c} \partial_E\,\mathcal{D}'L^{(i)}_U(t;E)\, &= \, \frac{1}{2} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \qbezier(-2.5,3.5)(0,6)(2.5,8.5) \qbezier(-2.5,8.5)(0,6)(2.5,3.5) \color{red} \qbezier(-11,-2.5)(-17,0)(-23,2.5) \end{picture} -\,\,\frac{1}{2} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \color{red} \qbezier(0,-2.5)(-2,0)(-4,3) \qbezier(4,3)(2,0)(0,-2.5) \end{picture}\,\,.\end{aligned}$$ The inverted hat represents the differential operator $\mathcal{D}'$. Using $i\partial_E\partial_t = \mathcal{D}+\mathcal{D}'$, we can write $\mathcal{D}=\frac{1}{2}i\partial_E\partial_t + \frac{1}{2}(\mathcal{D}-\mathcal{D}')$ and, thus, the correction to eq. (\[eq:L\_a\_approx\]) for $L^{(a)}_{\Gamma}$ ($\partial_E L^{(a)}_U$) is given by half the difference of (\[eq:LG\_Et\_der\]) and (\[eq:LG\_Et\_der\_c\]) ((\[eq:LU\_EEt\_der\]) and (\[eq:LU\_EEt\_der\_c\])). We first address the second diagrams on the r.h.s. of these equations: their differences involve the expression $$\begin{aligned} \label{eq:prop} \frac{1}{2}(\mathcal{D}-\mathcal{D}')\Pi^{(i)}&=\Pi^{(i)}\left(\frac{1}{2}(\mathcal{D}-\mathcal{D}')L^{(i)}\right)\Pi^{(i)}\\ \nonumber &\hspace{-2cm} +\frac{1}{2}\left\{(\partial_E\Pi^{(i)})(i\partial_t L^{(i)})\Pi^{(i)}- \Pi^{(i)}(i\partial_t L^{(i)})(\partial_E \Pi^{(i)})\right\}\end{aligned}$$ for the propagator. Here, the first term on the r.h.s. can be neglected in leading order, whereas the second one is only zero if the Liouvillian and its time and energy derivative commute. For special cases this is indeed possible: it follows trivially for blocks where the Liouvillian is diagonal, as e.g. the $2\times 2$-block $L^{(i)}_{s\bar{s},s'\bar{s}'}=\delta_{ss'}\epsilon^{(i)}_s$ of eq. (\[eq:Liouvillian\]). For $2$-level systems with conservation laws, see eq. (\[eq:Liouvillian\]), it holds also for the block $L^{(i)}_{ss,s's'}$ since the zero eigenvalue of the Liouvillian can be omitted in a propagator standing left to a vertex averaged over the Keldysh indices [@hs_epj09]. Therefore, for this block one can replace the Liouvillian by its nonzero eigenvalue $-i\Gamma^{(i)}(t;E)$ and the second term on the r.h.s. of (\[eq:prop\]) is again zero. If this is given (or if the term can be neglected in leading order for certain models), we can write the correction to eq. (\[eq:L\_a\_approx\]) generically as $$\begin{aligned} \nonumber L^{(a)}_\Gamma(t;E)\,\, &= \,\,\frac{1}{2}i\partial_E\partial_t L^{(i)}_\Gamma(t;E) \\ \label{eq:LG_final} % &+\,\,\frac{1}{2} \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \color{red} \qbezier(-14.5,-2.5)(-17,0)(-19.5,2.5) \end{picture} -\,\,\frac{1}{2} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dlgamma}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \color{red} \qbezier(19.5,-2.5)(17,0)(14.5,2.5) \end{picture}\quad, % \\ \nonumber \partial_E L^{(a)}_U(t;E)\,\, &= \,\, \partial_E\left\{\frac{1}{2}i\partial_E\partial_t L^{(i)}_U(t;E) \right\} \\ \label{eq:LU_final} % &+\,\,\frac{1}{4} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \qbezier(-2.5,3.5)(0,6)(2.5,8.5) \qbezier(-2.5,8.5)(0,6)(2.5,3.5) \color{red} \qbezier(23,-2.5)(17,0)(11,2.5) \end{picture} -\,\,\frac{1}{4} % \begin{picture}(50,15)(-25,0) \put(0,0){\usebox{\dluu}} \thicklines \color{blue} \qbezier(-2.5,9.5)(0,12)(2.5,14.5) \qbezier(-2.5,14.5)(0,12)(2.5,9.5) \qbezier(-2.5,3.5)(0,6)(2.5,8.5) \qbezier(-2.5,8.5)(0,6)(2.5,3.5) \color{red} \qbezier(-11,-2.5)(-17,0)(-23,2.5) \end{picture}\quad. %\end{aligned}$$ From this result we observe another condition for the validity of (\[eq:L\_a\_approx\]), namely that it should not matter whether the right or the left vertex is differentiated with respect to time, i.e. the two vertices should be equivalent. Whether this is the case, depends on the algebra of the model under consideration. For noninteracting systems one can take bare vertices and the reservoir indices of the two vertices have to be the same due to the reservoir contractions connecting them. In this case the condition is fulfilled if the vertices do not depend on the level index of the dot states, e.g. through differently time-dependent coupling to different leads [@Moskalets08]. For interacting systems, the validity of (\[eq:L\_a\_approx\]) is more restrictive. The renormalized vertices can be quite different from the bare ones and the vertices get an additional dependence on the Laplace variable $E$ which is shifted by the chemical potentials of the reservoir lines crossing over the propagator standing left to that vertex. As a consequence, the two vertices are never equivalent in the presence of a bias voltage and correction terms definitely occur for time-dependent voltages. As discussed below, for the particular case of the IRLM with one single reservoir, correction terms to eq. (\[eq:L\_a\_approx\]) do not appear in leading order. Results ======= We use the above developed method to analyze the response of a mesoscopic capacitor at zero temperature, described by the IRLM, where $H_{\rm res}=\sum_k \epsilon_k a^\dagger_k a_k$ describes a noninteracting reservoir with flat d.o.s. $\nu$ of band width $2D$, $H_{\rm dot}(t)=\epsilon(t) c^\dagger c$ denotes a spinless single-level quantum dot with time-dependent level position $\epsilon(t)$, and $$\begin{aligned} \nonumber V(t)&\quad=\quad\sqrt{\frac{\Gamma_0(t)}{2\pi\nu}}\,\sum_k\,(c^\dagger a_k+h.c.)\\ \label{eq:interaction} &\hspace{0.8cm} +\quad\frac{U(t)}{\nu}\,\sum_{kk'}\,(c^\dagger c-1/2)\,a^\dagger_k a_{k'}\end{aligned}$$ is the dot-reservoir coupling with the bare time-dependent tunneling rate $\Gamma_0(t)$ and the time-dependent dimensionless Coulomb interaction $U(t)$. As shown above, we can evaluate the adiabatic response from eq. (\[eq:rho\_adiab\]), where $\Gamma^{(a)}$ and $\Gamma^{\prime (a)}$ can be extracted from eqs. (\[eq:LG\_final\]) and (\[eq:LU\_final\]) together with the RTRG results for the instantaneous vertices and the Liouvillian derived in Ref. [@rtrg_irlm]. For $E=0$ and leading order in $U$, the results of Ref. [@rtrg_irlm] read $$\begin{aligned} \label{eq:gamma} \Gamma&=\Gamma_0\left(\frac{D}{|\epsilon-i\Gamma/2|}\right)^{2U} \,,& \partial_E\Gamma&=i\frac{U\Gamma^2}{\epsilon^2 + (\frac{\Gamma}{2})^2}\,,\\ \label{eq:gamma'} \Gamma'&=-\frac{\Gamma}{\pi}\arctan \frac{\epsilon}{\Gamma/2} \,\,,& \partial_E\Gamma'&=-\frac{i}{\pi}\frac{\Gamma\epsilon}{\epsilon^2+(\frac{\Gamma}{2})^2}\,,\end{aligned}$$ where we have omitted the index $(i)$ for the instantaneous quantities. Furthermore, the analysis in Ref. [@rtrg_irlm] shows that the Coulomb vertex is zero in leading order for the Liouvillian elements containing $\Gamma$ and $\Gamma'$. Therefore eq. (\[eq:LG\_final\]) is sufficient to evaluate $\Gamma^{(a)}$ and $\Gamma^{\prime (a)}$. Inserting the algebra for the instantaneous tunneling vertices into eq. (\[eq:LG\_final\]), one finds that eq. (\[eq:L\_a\_approx\]) is valid for the calculation of $\Gamma^{\prime (a)}$, whereas for $\Gamma^{(a)}$ a correction term occurs proportional to $\partial_{t} U$. This yields the total result $$\begin{aligned} \label{eq:gamma_a} \Gamma^{(a)}&=-\frac{U}{2}\partial_t\frac{\Gamma^2}{\epsilon^2+(\frac{\Gamma}{2})^2} \,,& \Gamma^{\prime(a)}&=\frac{1}{2\pi}\partial_t\frac{\Gamma\epsilon}{\epsilon^2+(\frac{\Gamma}{2})^2} \,.\end{aligned}$$ Since $\partial_E\Gamma,\Gamma^{(a)}\sim O(U)$ we can neglect them in leading order in eq. (\[eq:rho\_adiab\]) and, by inserting (\[eq:gamma\]) to (\[eq:gamma\_a\]) into (\[eq:rho\_adiab\]), we find after a straightforward analysis for the charge response given by $Q=e\rho_+$, $$\dot{Q}^{(i)}=C_0 \Gamma \partial_t \frac{\epsilon}{e\Gamma},\qquad Q^{(a)}=-R_0 C_0^2 \Gamma \partial_t \frac{\epsilon}{e\Gamma}, \label{eq:response}$$ where $R_0=\frac{h}{2e^2}$ and $C_0=\frac{e^2}{2\pi}\frac{\Gamma}{\epsilon^2+(\Gamma/2)^2}$. In the special case of linear response and when only $\epsilon$ is varied with time, $C=C_0$ is the static capacitance and $R=R_0$ the universal relaxation resistance, in agreement with (\[eq:RC\]). In contrast, when $\Gamma$ is varied with intent or via an accidental (but experimentally unavoidable) gate voltage dependence of $\Gamma_0$ or $U$, we obtain in linear response eq. (\[eq:RC\]) with $$C=C_0\left(1-\frac{\epsilon}{\Gamma}\frac{\partial\Gamma}{\partial\epsilon}\right),\qquad R=\frac{R_0 C_0}{C},$$ where $\frac{\partial\Gamma}{\partial\epsilon}\approx \frac{\Gamma}{\Gamma_0}\frac{\partial\Gamma_0}{\partial\epsilon}+ 2\Gamma\frac{\partial U}{\partial\epsilon}\log\frac{D}{|\epsilon-i\Gamma/2|}$. As a consequence, $C$ and $R$ are very sensitive to the variation of other parameters, and logarithmic terms due to renormalization effects occur, if the Coulomb interaction $U$ varies with time. In this general case, where also the renormalized $\Gamma$ varies with time, we propose to analyze the time scale $\tau_Q$. From (\[eq:delay\_time\]) and (\[eq:response\]) we get $$\tau_Q = \left|\frac{Q^{(a)}}{\dot{Q}^{(i)}}\right|=\frac{\Gamma/2 }{ \epsilon^2 + (\Gamma/2)^2} = R_0 C_0 \quad, \label{eq:tau_Q}$$ which is of $O(\Gamma^{-1})$ close to resonance $\epsilon\sim\Gamma$ and of $O(\Gamma/\epsilon^2)$ away from resonance. This result holds for [*any*]{} variation of $\epsilon$, $\Gamma_0$ and $U$ and is also valid in nonlinear response. Interaction effects enter only weakly via the renormalized $\Gamma$ given by (\[eq:gamma\]). Importantly, $\tau_Q$ reveals the static capacitance $C_0$ for a pure change of the gate voltage in linear response, with the advantage that $\tau_Q$ can be determined in the presence of the variation of any parameter. The experimentally accessible time scale $\tau_Q$ is thus an interesting quantity, which, for the case of the IRLM, is stable for the variation of any parameter in linear or nonlinear response. We note that this time scale can vary quite drastically if other observables or other models are studied. E.g., the time scales $\tau_{Q}$ and $\tau_I$, when $Q$ is replaced by the current $I=\dot{Q}$, are in general the same only in linear response. For the IRLM, the time scale $\tau_{I}$ shows similar logarithmic renormalizations in nonlinear response as they occur in $C$ and $R$ for time varying $U$. Realizations ============ Several experimental realization of the IRLM exist, where the different parameters can be modulated in a controlled way. As we outline here, the applicability of the IRLM is not limited to the description of an interacting quantum dot, but allows the observation of the predicted effects for various physical systems. First, we show that the low-energy physics of the IRLM is equivalent to the one of the Anderson-Holstein model, as first predicted in Ref. [@schiller1]. This model is widely used in molecular electronics [@schoeller:vib] and describes a single-level molecular quantum dot, having a vibrational degree of freedom with frequency $\Omega$ coupled linearly to the charge of the dot $$\begin{aligned} H_\mathrm{dot} &=& \epsilon_M c^\dagger c + \Omega b^\dagger b - \lambda\Omega (b+b^\dagger)c^\dagger c \,,\\ V &=& \sqrt{\frac{\Gamma_M}{ 2\pi\nu}}\sum_k(c^\dagger a_k+h.c.)\,.\end{aligned}$$ The parameters $\epsilon_M$, $\Gamma_M$, $\lambda$ and $\Omega$ can be related to the effective parameters $\epsilon$, $\Gamma_0$ and $U$ of the IRLM. Applying a Lang-Firsov transformation [@langfirsov], the coupling to the vibrational mode can be incorporated into the tunneling, leading to an effective level position, $\epsilon=\epsilon_M-\lambda^{2}\Omega$, and tunneling rate, $\Gamma_0= \Gamma_M e^{-\lambda^2}$ [@langfirsov; @koch-lf]. If the vibration frequency $\Omega$ is large compared to the other energy scales, the virtual intermediate states between the tunneling sequences with one or more bosons can be integrated out. This produces terms with $n\geqslant 2$ lead operators in the effective Hamiltonian. At large $\lambda$, all cotunneling processes with $n>2$ are exponentially suppressed, while the two-particle processes enter as an effective interaction. Hence, by integrating out all vibrational modes the Anderson-Holstein model can be mapped onto the IRLM with effective interaction, $U=\frac{\Gamma_M}{2\pi\lambda^2 \Omega}$, with $\Omega\sim D$. This is in agreement with Ref. [@schiller1], where it was shown numerically that this formula has even a broader range of applicability. A modulation of the tunneling barriers is always accompanied by a modulation of the effective interaction $U$, since it is proportional to the tunneling rate $\Gamma_M$. In this case, our results predict that logarithmic corrections appear for $C$ and $R$, whereas the time scale $\tau_Q$ only depends on $\epsilon$ and $\Gamma$ via eqs. (\[eq:tau\_Q\]) and (\[eq:gamma\]). The Holstein coupling in the Hamiltonian allows for the observation of the dot charge via the displacement of the dot $\sim\langle b+b^{\dagger}\rangle$. Finally, our results can be used to extract information on the relaxation behavior of systems described by the spin-boson model, namely, two-level dissipative systems connected to a large ensemble of oscillators $$H= \frac{\epsilon}{2}\sigma_z-\frac{\Delta}{2}\sigma_x+\sum_q \omega_q b_q^\dagger b_q + \frac{\sigma_z}{2}\sum_q g_q (b_q+b_q^\dagger).$$ The spin-boson model can be implemented by a Bose condensate of atoms trapped by a focused laser beam [@recati]. Such ultracold gases in optical lattices provide experimental realizations for theoretical models with remarkably independent tunability of parameters including the interaction strength, in contrast to usual semiconductor quantum dot setups. The system’s behavior depends crucially on the spectral coupling function. For the ohmic case, i.e. when the coupling constant obeys $\sum_q g_q^2\delta(\omega-\omega_q) = 2\alpha \omega e^{-\omega/D}$, the spin-boson model can be mapped onto the IRLM if $\alpha\approx 1/2$ (close to the Toulouse limit) [@tsvelick-sbm; @zwerger-sbm]. The effective IRLM parameters are $U=1-\sqrt{2\alpha}$ and $\Gamma_0=\Delta^2/D$. Changing the coupling of the Bose condensate to the spin via $\alpha$ one generates a time-dependent effective interaction $U$. The resulting response $\langle S_z\rangle$ of the spin, identified with $(\rho_+-\rho_-)/2$ in the effective IRLM, allows for the determination of the interesting time scale $\tau_{S_z}$, given by (\[eq:tau\_Q\]). Especially in the biased case, where $\tau_{S_z}\sim\Gamma/\epsilon^2$, this time scale is expected to differ significantly from typical relaxation rates $\Gamma$ and $\Gamma/2$ for the diagonal and nondiagonal components of the density matrix [@weiss; @rtrg_irlm]. Conclusions =========== In this Letter we provide a generic relation of the adiabatic response to real-time RG results for the stationary case. The presented scheme allows for the variation of any parameter in linear or nonlinear response and provides criteria when the adiabatic correction to the Liouvillian can be calculated directly via energy and time derivatives of the instantaneous one. We suggest a delay time as an interesting time scale and show for the IRLM that its expression is robust against the choice of time-dependent parameters and their amplitude. We confirm the universality of the AC relaxation resistance, unless a time dependence of tunneling and interaction is present, revealing logarithmic renormalizations due to charge fluctuations. We proposed different setups in molecular electronics and cold-atom systems to observe the effects experimentally. We acknowledge valuable discussion with S. Andergassen and M. Pletyukhov, and useful comments by M. Büttiker, as well as financial support from the Ministry of Innovation NRW and DFG-FG 723. [99]{} . . . . . . . . . . . . . . . . . . . . . . . D. Kennes and V. Meden, . . . . . . . ; . ; . ; (Erratum). . . . . . . in preparation. ; . . . . . . . [^1]: E-mail: [^2]: Provided that the frequency integrals converge, we note that our results can even be applied to a frequency-dependent d.o.s. in the leads. Otherwise, another derivative with respect to $E$ may be required.

--- abstract: 'We prove a multiparameter version of a classical theorem of Jones and Journé on weak-star convergence in the Hardy space $H^1$.' address: - 'Jill Pipher, Department of Mathematics, Brown University, 151 Thayer Str./Box 1917, Providence, RI 02912, USA ' - 'Sergei Treil, Department of Mathematics, Brown University, 151 Thayer Str./Box 1917, Providence, RI 02912, USA ' author: - Jill Pipher - Sergei Treil title: 'Weak-star convergence in multiparameter Hardy spaces' --- [^1] [^2] Introduction ============ It is a well known and classical result that the commutator of a singular integral with the operator of multiplication by a function in ${{\rm BMO}}$, the space of bounded mean oscillation, is bounded in $L^p$, for $1<p< \infty$. The first proof appeared in [@CRW] (see also [@Ja]), and there are now generalizations of this result to the bidisc and polydisc ([@FL], [@LT], [@LPPW]). Since $BMO$ is the dual of the Hardy space $H^1$ of functions whose Poisson maximal function (or square function) belongs to $L^1$, one can formulate a dual version of the commutator result. This dual formulation asserts that certain quantities involving products and sums of Riesz transforms (or more general singular integrals) belong to $H^1$. For example, if $R_j$ denotes the $j$th Riesz transform, the quantity $gR_jf + fR_jg$ belongs to $H^1({\mathbb{R}}^d)$, whenever $f,g \in L^2({\mathbb{R}}^d)$. Each of the summands in this quantity clearly belongs to $L^1$, but it is the special form of this sum which puts it into $H^1$. The space ${{\rm VMO}}$ is the predual of $H^1$, and this gives $H^1$ a richer structure than $L^1$. In [@CLMS], a much more general approach was developed. There, the authors showed that a variety of expressions with a special form of cancellation (the div-curl quantities) belong to some Hardy space $H^p$. Their approach paved the way for a striking collection of extensions of the theory of compensated compactness in partial differential equations. A result of P. Jones and J.-L. Journé concerning weak convergence in the Hardy space $H^1$ ([@JJ]) was essential. In this paper we prove the multiparamater analog of this theorem. That is, if ${\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$ denotes a $d$-parameter product space, where $n_i \geq 1$, we have the following: \[thm:weak\] Suppose that $\{f_n\}$ is a sequence of $H^1({\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})$ functions such that $\Vert f_n \Vert_{H^1} \leq 1$, for all $n$, and such that $f_n(x) \rightarrow f(x)$ for almost every $x \in {\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$. Then $f \in H^1({\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})$, $\Vert f \Vert_{H^1} \leq 1$, and $f_n \xrightarrow{w*} f$, i.e. for any ${\varphi}\in {{\rm VMO}}({\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})$, $$\int_{{\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}} f_n {\varphi}dx \rightarrow \int_{{\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}} f{\varphi}dx.$$ The inherent difficulty in working with the multiparameter ${{\rm BMO}}$ and ${{\rm VMO}}$ spaces is that the definitions require one to deal with arbitrary open sets, as opposed to intervals or products of intervals. The paper is organized as follows. We recall (in section \[sec:definitions\]) some definitions and the results from prior work which are required in the proof. The proof in Section 3 follows the template provided in [@JJ], but some new ideas, see Lemmas \[Lemma:LemmaA\] and \[Lemma:LemmaB\] below, are needed to get from the one-parameter to multiparameter case. Definitions {#sec:definitions} =========== A real-valued function $f\in L^1_{\text{loc}}({\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})$ is in the space $bmo$ (called “little bmo" in the literature), if its $bmo$ norm is finite: $$\label{equation:defbmo} \Vert f \Vert_{bmo} := \sup_R \, \frac{1}{|R|} \, \int_R \, |f(x) - (f)_R| \, dx < \infty.$$ Here $(f)_R = \frac{1}{|R|}\int_R f(x) \, dx$ is the average value of $f$ on the rectangle $R=Q_1 \times ... \times Q_d \subset {\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$, where $Q_i \subset {\mathbb{R}}^{n_i}$. When $d=1$, this is the classical ${{\rm BMO}}$ space, the dual of the Hardy space $H^1$. In the multiparamater setting, $bmo$ is one of several possible generalizations of the one-parameter ${{\rm BMO}}$ space. It is not hard to see that a function $f$ belongs to $bmo({\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})$ if and only if, for all indices $i$, $f$ belongs to the classical one-parameter spaces ${{\rm BMO}}(R^{n_i})$ (with the uniform estimates on the norms), with the other variables fixed. Moreover, by the John-Nirenberg theorem for the classical one-parameter ${{\rm BMO}}$ space, the $L^1$ norm in can be replaced by the $L^p$ norm, $1\le p<\infty$. In this paper we will use the equivalent norm with $p=2$, $$\label{equation:defbmo2} \Vert f \Vert_{bmo} := \sup_R \,\left( \frac{1}{|R|} \, \int_R \, |f(x) - (f)_R|^2 \, dx \right)^{1/2} < \infty.$$ However, the true analog of ${{\rm BMO}}$ - in the sense of duality with the multiparameter Hardy space, and boundedness of singular integrals - is the product ${{\rm BMO}}$ space which was defined and characterized by S.-Y. Chang and R. Fefferman ([@C] and [@CF]). The space $bmo$ defined above is strictly smaller than this product ${{\rm BMO}}$ space. (See [@FS].) The dyadic lattice ${\mathcal{D}}({\mathbb{R}}^n)$ in ${\mathbb{R}}^n$ is constructed as follows: for each $k\in {\mathbb{Z}}$ consider the cube $[0,2^k)^n$ and all of its shifts by elements of ${\mathbb{R}}^n$ whose coordinates are $j2^k$, $j\in {\mathbb{Z}}$; then take the union over all $k\in {\mathbb{Z}}$. Let $E_k$ denote the averaging operator over cubes $Q \in {\mathcal{D}}(R^n)$ of side length $2^k$: $E_kf(x) = 1/|Q|\int_Q f(y) dy$, if $Q$ has side length $2^k$ and contains $x$. If $Q$ has side length $2^k$, then $E_Qf(x) = E_kf(x)\chi_E(x)$. Set $\Delta_k = E_{k-1} - E_k$, and $\Delta_Qf(x) = \Delta_kf(x)\chi_Q(x)$, when $Q$ has side length $2^k$. For a “dyadic rectangle” $R=Q_1 \times ... \times Q_d$, $ Q_i\in {\mathcal{D}}({\mathbb{R}}^{n_i})$ define the multiparameter difference operator $\Delta_R = \Delta_{Q_1} \otimes ... \otimes \Delta_{Q_d}$. We use the symbol $\otimes$ to emphasize that the difference operators $\Delta_{Q_i}$ act on independent variables $x_i\in {\mathbb{R}}^{n_i}$. Here we use the same notation for the one-parameter difference operator and for the multiparameter one; cubes are always subsets of the “building blocks” ${\mathbb{R}}^{n_i}$, and the “rectangles” are the Cartesian products of cubes. Even if the size of all cubes $Q_i$ is the same, we will call the product $Q_1\times Q_2\times\ldots\times Q_d$ a “rectangle”. Denote by ${\mathcal{R}}={\mathcal{R}}( {\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})$ the collection of all “dyadic rectangles”. The (multiparameter) square function of $f$ in ${\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$ is defined as $$Sf(x) = \left(\sum_{R\in{\mathcal{R}}} |\Delta_Rf(x)|^2\right)^{1/2}$$ A function $f$ belongs to the Hardy space $H^1$ if its norm $\Vert f \Vert_{H^1} := \Vert Sf \Vert_{L^1}$ is finite. \[r1\] Similarly to the one-parameter case for $f\in L^2({\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})$ $$\|f\|^2_2 = \sum_{R\in {\mathcal{R}}} \|\Delta_R f \|^2_2 .$$ This fact is well-known in one-parameter situation: the general case can be easily obtained by iterating the one-parameter case. A function $f$ belongs to $ {{\rm BMO}}_d$ if there exists a constant $C$ such that for every open set $\Omega \subset {\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$, $$\label{eqn:bmodyadic} \sum_{R \subset \Omega} \,\Vert \Delta_Rf \Vert_2^2 \leq C|\Omega|.$$ (See [@B].) The dyadic $H^1$ and ${{\rm BMO}}$ spaces can be defined in terms of a Carleson packing condition using the product system of Haar wavelets. (See [@BP] for example.) The same Carleson packing condition, but using a basis of smooth wavelets, such as the Meyer wavelets, defines the product ${{\rm BMO}}$ space, which is the dual of product $H^1$. We refer to [@CF] for the precise definition, and the duality theorem. Here, we shall only need the following relationship between product ${{\rm BMO}}$ and its dyadic counterpart, ${{\rm BMO}}_d$: \[prop:averaging\] If ${\varphi}$ and all its translates belong to the product ${{\rm BMO}}_d$, with uniform bounds on their ${{\rm BMO}}_d$ norms, then ${\varphi}$ belongs to the product space ${{\rm BMO}}$. This statement is trivial in one-parameter settings. In multiparameter situation, it can be treated as a special case of the so-called “BMO from dyadic BMO” result (which is a significantly stronger statement), see [@PW], [@Tr Remark 0.5]. Namely, let us consider all translations ${\mathcal{D}}_\omega$ of the standard dyadic lattice ${\mathcal{D}}$. If we have a measurable family of functions ${\varphi}_\omega$, such that each ${\varphi}_\omega$ belongs to ${{\rm BMO}}_d$ with respect to the corresponding lattice ${\mathcal{D}}_\omega$ (with the uniform estimate of the norm), then the average (over all $\omega$) of ${\varphi}_\omega$ is a BMO function. Here we do not explain how the average over all $\omega$ is computed, since in our situation ${\varphi}_\omega={\varphi}$, so the average is also ${\varphi}$; see [@PW], [@Tr Remark 0.5] for more details. Note, that the “BMO from dyadic BMO” statement is non-trivial even in one-parameter setting, see [@D; @GJ] for the proof in this case. The product ${{\rm VMO}}$ space is the closure of the $C^{\infty}$ functions in the product ${{\rm BMO}}$ norm. [**Remark**]{}. As in the classical one-parameter setting, the product ${{\rm VMO}}$ space is the predual of $H^1$. (See [@LTW].) Proof of the main result. ========================= If ${\Omega}\subset {\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$ is an open set, and $x_1 \in {\mathbb{R}}^{n_1}$, the “slice" $\Omega_{x_1}$ is the $(n_2+...+n_d) -$dimensional set: $$\{x' \in {\mathbb{R}}^{n_2} \times ... \times {\mathbb{R}}^{n_d}: (x_1,x') \in {\Omega}\}.$$ The slices $\Omega_{x_i}$, for $i=2,...,n$ are defined similarly. Let ${\mathcal{F}}\subset{\mathcal{R}}$ be a family of “dyadic rectangles” $R=Q_1 \times... \times Q_d$ , $Q_i \in {\mathcal{D}}( {\mathbb{R}}^{n_i})$. For $x_1\in {\mathbb{R}}^{n_1}$ let ${\mathcal{F}}_{x_1}$ denote the $x_1$ “slice” of the family ${\mathcal{F}}$, i.e. the set of all “rectangles” $R'= Q_2\times Q_3\times \ldots \times Q_d\subset {\mathbb{R}}^{n_2}\times{\mathbb{R}}^{n_3}\times\ldots\times {\mathbb{R}}^{n_d}$ for which there exists a cube $Q_1\subset {\mathbb{R}}^{n_1}$ such that $x_1\in Q_1$ and $$R = Q_1\times R' = Q_1\times Q_2 \times \ldots \times Q_{d} \in{\mathcal{F}}.$$ \[Lemma:LemmaA\] Let ${\mathcal{F}}\subset {\mathcal{R}}$. Then $$\label{eq:A} \sum_{R \in {\mathcal{F}}} \Vert\Delta_Rf\Vert^2 \le \int_{{\mathbb{R}}^{n_1}} \sum_{R' \in {\mathcal{F}}_{x_1}} \Vert\Delta_{R'}f(x_1,\,.\,)\Vert^2 dx_1.$$ For $x_1\in {\mathbb{R}}^{n_1}$ and $x' \in {\mathbb{R}}^{n_2}\times{\mathbb{R}}^{n_3}\times\ldots\times {\mathbb{R}}^{n_d}$ set $$\tilde{f}(x_1,\,.\,) = \sum_{R' \in {\mathcal{R}}_{x_1}} \Delta_{R'}f(x_1,\, . \,) .$$ Then, for fixed $x_1\in {\mathbb{R}}^{n_1}$, we observe that when $Q_1 \times R' \in {\mathcal{F}}$ and $x_1 \in Q_1$, then $$\label{eq:relate} \Delta_{R'}f(x_1,\,.\,) = \Delta_{R'}\widetilde{f}(x_1,\,.\,)..$$ This is because our assumptions $x_1 \in Q_1$ and $Q_1 \times R' \in {\mathcal{F}}$ mean exactly that $R' \in {\mathcal{F}}_{x_1}$. Thus, using the fact that $\Delta_{Q_1\times R'} =\Delta_{Q_1} \otimes \Delta_{R'}$ we get that $$\Delta_{Q_1\times R'} f = \Delta_{Q_1\times R'}\widetilde f$$ and so $$\begin{aligned} \sum_{Q_1 \times R' \in {\mathcal{F}}} \Vert\Delta_{Q_1\times R'}f\Vert^2_2 &= \sum_{Q_1 \times R' \in {\mathcal{F}}} \Vert \Delta_{Q_1\times R'}\tilde{f}\Vert^2_2\\ \leq \Vert \tilde{f} \Vert^2_2 &= \int_{{\mathbb{R}}^{n_1}} \Vert \sum_{R' \in {\mathcal{F}}_{x_1}} \Delta_{R'}\widetilde f(x_1,.)\Vert^2_2 dx_1\\ &= \int_{{\mathbb{R}}^{n_1}} \Vert \sum_{R' \in {\mathcal{F}}_{x_1}} \Delta_{R'}f(x_1,.)\Vert^2_2 dx_1\\ &= \int_{{\mathbb{R}}^{n_1}} \sum_{R' \in {\mathcal{F}}_{x_1}} \Vert \Delta_{R'}f \Vert^2_2 dx_1.\end{aligned}$$ \[Lemma:LemmaB\] Suppose ${\varphi}\in C^{1}({\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})$, $\|{\varphi}\|_\infty\le1$, $\|\nabla_{x_i} {\varphi}(x)\|_{\ell^1}\le1$, $i=1, 2, \ldots, d$, and $b$ is a bounded function with $\Vert b \Vert_{\infty} \leq 1.$ Then, for any $\alpha < 1$, and any open $\Omega\subset {\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$, $$\label{equation:induction} \sum_{R\in {\mathcal{R}}: R \subset {\Omega}, |R| \leq \alpha} \Vert\Delta_R({\varphi}b)\Vert^2_2 \leq 2d!(\Vert b \Vert^2_{bmo} + \alpha^{2/n})|{\Omega}|,$$ where $n=n_1+ .. + n_d$, and $\Vert f \Vert_{bmo}$ is defined by . The proof is by induction on $d$: the base case for one-parameter BMO ($d=1$) was proven in [@JJ], but we’ll give the short argument here for the sake of completeness. When $d=1$, it suffices to prove (\[equation:induction\]) for $\Omega = Q_0$, where $Q_0 \subset {\mathbb{R}}^{n_1}$ is a dyadic cube. The “rectangles"’ $R$ are themselves dyadic cubes (which we now denote by $Q$), and by subdividing $Q_0$ into smaller dyadic cubes if necessary, we may without loss of generality assume that $|Q_0| \leq \alpha$. Then we see that $$\begin{aligned} \sum_{Q \subset Q_0} \Vert\Delta_Q({\varphi}b)\Vert^2_2 & = & \int_{Q_0} \, |{\varphi}(x) b(x) - ({\varphi}b)_{Q_0}|^2 \, dx \\ & \le & \int_{Q_0} \, |{\varphi}(x) b(x) - {\varphi}_{Q_0} b_{Q_0}|^2 \, dx \\ & \le & 2\left( \int_{Q_0} \, |{\varphi}(x) b(x) - {\varphi}(x) b_{Q_0}|^2 \, dx + \int_{Q_0} \, |{\varphi}(x) b_{Q_0} - {\varphi}_{Q_0} b_{Q_0}|^2 \, dx \right)\end{aligned}$$ On the one hand, by the pointwise bound on ${\varphi}$, $$\int_{Q_0} \, |{\varphi}(x) b(x) - {\varphi}(x) b_{Q_0}|^2 \, dx \leq \Vert b \Vert_{bmo}^2 |Q_0|,$$ and using the pointwise bounds on $b$ (so $|b_{Q-0}|\le 1$) and on derivatives of ${\varphi}$ (so $|{\varphi}(x) - {\varphi}_{Q_0}|\le (\alpha)^{1/n_1}$ for $x\in Q_0$), we get $$\int_{Q_0} \, |b_{Q_0}({\varphi}(x) - {\varphi}_{Q_0})|^2 \, dx \leq (\alpha)^{2/n_1}|Q_0|.$$ Combining the above 3 estimates we get with $d=1$. For the induction step, we’ll use the notation ${\mathbb{R}}^{n_1} \times ... \widehat{{\mathbb{R}}}^{n_i} \times ... {\mathbb{R}}^{n_d}$ to denote the $d-1$ fold product of the ${\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$ with ${\mathbb{R}}^{n_i}$ missing, and similar notation for a $d-1$ fold product of cubes with one cube $Q_i$ missing. Suppose now that $R = Q_1 \times ... \times Q_d$ is a rectangle in ${\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$ with $|R| < \alpha$. Then there exists an $i$ such that the $d-1$ dimensional rectangle $R_i'=Q_1 \times ...\times...\hat{Q_i}... \times Q_d$ has volume $|R'_i| < \alpha^{N_i}$, where $N_i = (n_1+...\hat{n_i}+...+n_d)/(n_1+...+n_d)$. Indeed, if not, $$|R|^{d-1} = \prod_{i=1}^d |R'_i| > \prod_{i=1}^d \alpha^{N_i} = \alpha^{d-1}$$ contradicting the assumption $|R|<\alpha$. Thus each “rectangle” $R \subset {\Omega}$, $|R|<\alpha$ satisfies this condition for at least one index $i =1,...,d$. Therefore, the collection ${\mathcal{F}}=\{R\in {\mathcal{R}}: R\subset \Omega, |R|<\alpha\}$ can be represented as a union ${\mathcal{F}}= \cup_{i=1}^d {\mathcal{F}}^i$, where ${\mathcal{F}}^i:= \{ R\in {\mathcal{F}}: R_i'< \alpha^{N_i} \}$. (Note, that ${\mathcal{F}}^i$s are not necessarily disjoint.) Applying Lemma \[Lemma:LemmaA\] (with $x_1$ replaced by $x_i$) to each collection ${\mathcal{F}}^i$, we see that $$\sum_{R \in {\mathcal{F}}} \Vert\Delta_R({\varphi}b)\Vert^2_2 \le \sum_{i=1}^{d}\int_{{\mathbb{R}}^{n_i}} \sum_{R' \in {\mathcal{F}}^i_{x_i}} \Vert\Delta_{R'}({\varphi}b)(x_i,\,.\,)\Vert^2_2 dx_i .$$ Note that ${\mathcal{F}}^i_{x_i} \subset \{R'\in {\mathcal{R}}({\mathbb{R}}^{n_1} \times ... \widehat{{\mathbb{R}}}^{n_i} \times ... {\mathbb{R}}^{n_d}): R'\subset \Omega_{x_i}, |R'|\le \alpha^{N_i} \}$, so by the induction step with $\tilde{n_i} = n_1+...n_{i-1}+n_{i+1}+...+n_d)$ instead of $n$ and $d-1$ instead of $d$, we get $$\begin{aligned} \int_{{\mathbb{R}}^{n_i}} \sum_{R' \in {\mathcal{F}}^i_{x_i}, |R'| < \alpha^{N_i}} \Vert \Delta_{R'}({\varphi}b)(x_i,.)\Vert^2_2 &\le & 2 (d-1)! \int_{x_i} (\Vert b \Vert_{bmo}^2 + (\alpha^{N_i})^{2/\tilde{n_i}}) |E_{x_i}| dx_i\\ &=& 2(d-1)!(\Vert b \Vert_{bmo}^2 + \alpha^{2/n}) |{\Omega}|.\end{aligned}$$ Here we have also used the (trivial) fact that the $bmo({\mathbb{R}}^{n_1} \times ... \widehat{{\mathbb{R}}}^{n_i} \times ... {\mathbb{R}}^{n_d})$ norm of $b$ is bounded by $\Vert b \Vert_{bmo({\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}})}$. Adding estimates for all $i=1, 2, \ldots, d$ we get the conclusion of the lemma. We will require the following fact about $bmo$ functions. \[lem:max\] If $f$ and $g$ belong to $bmo$, then $\text{max}\{f,g\}$ also belongs to $bmo$. The proof is exactly as in the one-parameter setting, since the space $bmo$ is defined by averages over rectangles. That is, for any rectangle $R$, we have $$\frac{1}{|R|} \int_R |\, |f(x)| - |f|_R \,|dx \leq \frac{1}{|R|} \int_R \,|f(x) - f_R| \, dx$$ and $\max\{f,g\} = (|f-g| + f + g)/2$. \[lem:littlebmo\] Let $E \subset {\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$ be a set of finite measure, and let $\delta > 0$ be a given parameter. Then there exists a function $\tau \in bmo$ such that $\tau = 1$ on $E$, $\Vert \tau \Vert_{bmo} < C_1 \delta$, and $|\operatorname{supp}\tau | < C_2 e^{2/\delta} |E|$, where $C_1$ and $C_2$ are some absolute constants. Recall that a weight $w$ belongs to the $A_1$ class if there exists a constant $C$ such that for all $x$, $Mw(x) \leq Cw(x)$. Here, if $M$ is the Hardy-Littlewood maximal function then this is the usual class of (one-parameter) Muckenhoupt weights. And if $M$ is the strong maximal function where the averages are taken over arbitrary rectangles in ${\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$, then this is the multiparamater $A_1$ class. (See [@FP] for some basic facts about product $A_p$ weights.) We define the following $A_1$ weight, with $M^{(k)}$ denoting the k-fold iteration of the strong maximal function: $$m(x) = K^{-1}\sum_{k=0}^{\infty} c^k M^{(k)}\chi{_{ {}_{\scriptstyle E}}}(x).$$ where $K=\sum_k c^k$, and $c>0$ is chosen to insure the convergence of the series. Namely, we chose $c$ such that $\|cM\|<1$, i.e. that for some $q<1$, we have $c \|M f \|_2 \le q\|f\|_2$ for all $f\in L^2$. Then $\Vert m \Vert{_{ {}_{\scriptstyle L^2}}} \leq C \|\chi{_{ {}_{\scriptstyle E}}}\|{_{ {}_{\scriptstyle L^2}}} = C |E|^{1/2}$. Observe that $m=1$ a.e. on $E$, and $m \leq 1$ a.e. outside of $E$. Define, as in [@JJ], following [@CR], the function $$\tau(x) = \max\{0, 1 + \delta \log m(x) \}.$$ The function $\tau$ belongs to $bmo$ and also satisfies $\tau = 1$ a.e. on $E$. However, $\tau$ has small $bmo$ norm: $\Vert \tau \Vert_{bmo} \lesssim \delta$. This follows from Lemma \[lem:max\] and the fact that for any $A_1$ weight $w$, $\log w$ belongs to $bmo$, which is proved exactly as in the one-parameter setting. (See, for example, [@G].) The estimate for the size of the support of $\tau$ follows from Tchebychev’s theorem and the estimate $\Vert m \Vert{_{ {}_{\scriptstyle L^2}}} \leq C |E|^{1/2}$. We now prove Theorem \[thm:weak\]. First notice, that since $C^{\infty}_0$ is dense in VMO, it is sufficient to prove Theorem \[thm:weak\] for ${\varphi}\in C^{\infty}_0$. Because $f_n \rightarrow f$ a.e., and $\Vert f_n \Vert_{H^1} \leq 1$, we have $\Vert f \Vert|_{L^1} \leq 1$ by Fatou’s Lemma. Choose a ${\varphi}\in C^{\infty}_0$, normalized to have $|{\varphi}| \leq 1$, $\Vert {\varphi}\Vert_{L^1} \leq 1$. Let ${\varepsilon}> 0$ be fixed. We need to show that, for $n$ sufficiently large, $$\label{eqn:estimate} \Bigl|\int f_n {\varphi}dx - \int f {\varphi}dx \Bigr| < C{\varepsilon},$$ where $C$ is some absolute constant. We now fix an $\eta$ to be determined, and define $$E_n = \{x \in \text{supp}\,{\varphi}: |f_n(x) - f(x)| > \eta \}$$ Choose $n$ sufficiently large that $|E_n| < \eta$. Define $\tau$ as in Lemma \[lem:littlebmo\], relative to the set $E_n$. Then, if $\eta$ is chosen sufficiently small, since $\|f\|_{L^1}\le 1$, we will have $$\left|\int_{\operatorname{supp}\tau} f dx < {\varepsilon}\right| .$$ Then we break up the integral in (\[eqn:estimate\]) as $$\label{eqn:sum} \left| \int (f-f_n) dx \right| \leq \left| \int (f-f_n){\varphi}(1-\tau) dx \right| + \left|\int(f-f_n){\varphi}\tau dx\right| .$$ In the complement of $E_n$, $|f-f_n| < \eta$, so the first integral on the left hand side of (\[eqn:sum\]) is bounded by $\eta \Vert {\varphi}\Vert_{L^1}$, which in turn is less than ${\varepsilon}$ if $\tau$ is appropriately small. The second integral in (\[eqn:sum\]) is bounded by $$\int_T |f{\varphi}| dx + \left| \int f_n {\varphi}\tau dx\right|,$$ and since $$\int_T |f{\varphi}| dx < {\varepsilon},$$ the proof is completed by showing that $\Vert {\varphi}\tau \Vert_{{{\rm BMO}}} \lesssim {\varepsilon}$. We will show that the dyadic ${{\rm BMO}}$ norm of ${\varphi}\tau$ has the required estimate, and observe that the same proof shows that any translate of ${\varphi}\tau$ is in dyadic ${{\rm BMO}}$ with the same bound. The estimate on the product ${{\rm BMO}}$ norm will follow from Proposition \[prop:averaging\]. Fix an arbitrary open set ${\Omega}\subset {\mathbb{R}^{n_1} \times...\times \mathbb{R}^{n_d}}$ and consider two cases: \(i) $|{\Omega}| \leq \alpha$, where $\alpha>0$ will be chosen in a moment. In this case, all rectangles contained in ${\Omega}$ have size less than $\alpha$, and Lemma \[Lemma:LemmaB\] gives $$\sum_{R \subset {\Omega}, |R| \leq \alpha} \Vert\Delta_R({\varphi}\tau)\Vert^2_2 \leq C(\Vert {\varphi}\Vert^2_{bmo} + \alpha^{2/n})|{\Omega}|,$$ for $n=n_1+...+n_d$. With the appropriate choice of $\alpha$ and $\delta$ from Lemma \[lem:littlebmo\] (so $\Vert {\varphi}\Vert^2_{bmo} \le \delta^2$), this will be smaller than ${\varepsilon}|{\Omega}|$. Note that $\eta$ does not appear in the above estimate, so it holds for all $\eta$ ($\eta$ appears in the estimate of $|\operatorname{supp}\tau|$, but we do not use this quantity in the estimate). \(ii) $|{\Omega}| > \alpha$ ($\alpha$ and $\delta$ are already chosen). In this case, using the estimates $\| {\varphi}\tau\|_\infty\le 1$ and $|\operatorname{supp}\tau| \le C_2 e^{2/\delta} \eta $, we get $$\begin{aligned} \sum_{R \subset {\Omega}} \Vert\Delta_R({\varphi}\tau)\Vert^2_2 &\leq& \int |{\varphi}\tau|^2 dx\\ &\leq& \Vert {\varphi}\tau \Vert_{\infty}^2 |\operatorname{supp}\tau|\\ &\leq& C_2 \eta e^{2/\delta} \\ &\leq& C_2 \eta e^{2/\delta} \frac{|{\Omega}|}{\alpha}, $$ and the last quantity is bounded by ${\varepsilon}|{\Omega}|$ if $\eta$ is small enough. Note that the multiparameter version of the Jones-Journé theorem cannot be obtained by *trivial* iteration of the original one-parameter version. Namely, the function ${\varphi}\tau$ in the proof does not have small [*bmo*]{} norm, so we need to use the norm of the product ${{\rm BMO}}$. Lemmas \[Lemma:LemmaA\] and \[Lemma:LemmaB\] are necessary to perform this iteration. As it was mentioned in [@JJ], it is easy to see that the analogue of the main result does not hold for $L^1$ functions: it is easy to construct a sequence of $L^1({\mathbb{R}}^N)$ functions $f_n$ converging (in the weak\* topology of the space of measures $M({\mathbb{R}}^N)$) to a singular measure and such that $f_n\to0$ a.e. Moreover, picking a sequence of discrete measures $\mu_n$, converging (in the weak\* topology of $M({\mathbb{R}}^N)$) to a given $f\in L^1({\mathbb{R}}^N)$, and then approximating the measures $\mu_n$ by absolutely continuous measures with densities $f_n$ (recall that the weak\* topology of $M({\mathbb{R}}^N)$ is metriziable on any bounded set), we get that $f_n \xrightarrow{w*} f$ in $M({\mathbb{R}}^N)$. One can definitely pick a sequence $f_n$ such that $f_n\to0$ a.e., which gives us even more striking counterexample. On the other hand, the analogue of Theorem \[thm:weak\] holds for any reflexive function space $X$ of locally integrable functions (so convergence in $X$ implies the convergence in $L^1{_{\scriptstyle \text{\rm loc}}}$). Namely, if $\sup_n\Vert{f_n}\Vert<\infty$ and $f_n\to f$ a.e., then $f_n\to f$ in the weak (which is the same as weak\*) topology of $X$. This is a simple exercise in basic functional analysis, we leave the details to the reader. The space $H^1$ however is not reflexive: it is only a dual (of VMO). So, maybe an analogue of Theorem \[thm:weak\] is true for any space of function which is dual to some space. It would be interesting to prove or disprove the following conjecture. Let $X$ be a Banach function space $X$ of locally integrable functions (so convergence in $X$ implies the convergence in $L^1{_{\scriptstyle \text{\rm loc}}}$), which is dual to some Banach function space $X_*$ (with respect to the natural duality). If $f_n\in X$ such that $\sup_n \|f_n\|<\infty$ and $f_n\to f$ a.e., then $f\in X$ and $f_n\to f$ in weak\* topology of $X$. If it helps to prove the conjecture, one can assume that $X$ is a “reasonable” space: for example that $C_0^\infty$ is dense in $X$ and/or $X^*$, etc. The result under these (or similar) additional assumptions will still be extremely interesting. [999999]{} A.Bernard, *Espaces $H\sp{1}$ de martingales à deux indices. Dualité avec les martingales de type “BMO”*, Bull. Sci. Math. (2) **103** (1979), no. 3, 297–303. O. Blasco, S. Pott *Dyadic BMO on the bidisc*, Rev. Mat. Iberoamericana 21, No. 2, (2005), p. 483-510 S.-Y.A.Chang, *Carleson measure on the bi-disc*, Ann. of Math. (2) **109** (1979), no. 3, 613–620. 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--- author: - | **[Pintu Mandal$^{\ast}$, Manas Mukherjee$^{\dag}$]{}\ \ Raman Center for Atomic, Molecular and Optical Sciences\ Indian Association for the Cultivation of Science\ 2A & 2B Raja S. C. Mullick Road, Kolkata 700 032\ \ $^{\dag}$ *Present address:* Centre for Quantum Technologies\ National University of Singapore, Singapore - 117543\ \ $^{\ast}$Email: *pintuphys@gmail.com*** title: '**Dynamics of ion cloud in a linear Paul trap**' --- **Abstract**\ A linear ion trap setup has been developed for studying the dynamics of trapped ion cloud and thereby realizing possible systematics of a high precision measurement on a single ion within it. The dynamics of molecular nitrogen ion cloud has been investigated to extract the characteristics of the trap setup. The stability of trap operation has been studied with observation of narrow nonlinear resonances pointing out the region of instabilities within the broad stability region. The secular frequency has been measured and the motional spectra of trapped ion oscillation have been obtained by using electric dipole excitation. It is applied to study the space charge effect and the axial coupling in the radial plane. Introduction {#section1} ============ “A single atomic particle forever floating at rest in free space" [@Dehmelt88] is an ideal system for precision measurement and a single trapped ion provides the closest realization to this ideal. A single or few ions can be trapped within a small region of space in an ion trap and they are free from external perturbations. Such a system has been used for the precision measurement of electron’s g - factor [@Dyck87], various atomic properties like the lifetime of atomic states [@Yu97], the quadrupole moment [@Barwood04; @Oskay05; @Roos06] *etc*. Precision table-top experiments of fundamental physics in the low energy sector like the atomic parity violation measurement, nuclear anapole moment measurement, electron’s electric dipole moment measurement are either in progress in different laboratories worldwide or proposed [@Fortson93; @Mandal10; @Sahoo11; @Versolato10]. Any high-precision experiment appears with systematics which are required to be tracked or removed and hence a systematic investigation on the system itself is essential at the initial stage. In order to prepare for measuring atomic parity violation with trapped ions, a series of experiments have been performed in a linear ion trap to fully understand its behaviour and associated systematics. In this colloquium, the results of some experiments will be presented that are of preeminent interest to an audience coming from a variety of physics disciplines. It is organised with a brief overview on the physics of ion trapping in a linear Paul trap, description of the experimental setup and followed by results. Physics of ion trapping {#section2} ======================= An electrostatic field can not produce a potential minimum in three dimensional space as is required for trapping the charged particles. It is therefore, either a combination of static magnetic field and an electric field is used (Penning trap) or a combination of a time-varying and an electrostatic field is used (Paul trap). In Paul trap a radio-frequency (rf) potential superposed with a dc potential is applied on electrodes of hyperbolic geometry to develop quadrupolar potential in space. The geometry of the electrodes evolved over the decades for ease in machining, smooth optical access to the trapped ions *etc*. Figure \[lintrap\] shows one of the most frequently used trap geometries with four three-segmented rods placed symmetrically at four corners of a square and is commonly called a linear Paul trap. The four rods at each end are connected together and a common dc potential ($V_{e}$) is applied so as to produce an axial trapping potential. The diagonally opposite rods at the middle are connected and a rf ($V_{0}\cos\Omega t$) in addition to a dc potential ($U$) is applied on one pair with respect to the other pair for providing a dynamic radial confinement. The radial potential inside the trap is $$\label{eqn1} \Phi(x,y,t) = (U-V_{0}\cos\Omega t)\left(\frac{x^{2}-y^{2}}{2r_{0}^{2}}\right),$$ where $2r_{0}$ is the separation between the surfaces of the diagonal electrodes as depicted in figure \[lintrap\](b). The equipotential lines are rectangular hyperbolae in the $xy$ plane having four-fold symmetry about the $z$ axis. The equation of motion of an ion of charge $e$ and mass $m$ under the potential $\Phi(x,y,t)$ (eqn. \[eqn1\]) can be represented as $$\begin{aligned} \label{eqn2} \frac{d^{2}x}{dt^{2}} &=& -\frac{e}{mr_{0}^{2}}(U-V_{0}\cos\Omega t)x \\ \nonumber \frac{d^{2}y}{dt^{2}} &=& \frac{e}{mr_{0}^{2}}(U-V_{0}\cos\Omega t)y.\end{aligned}$$ ![(a) Schematic of the linear ion trap used in the experiment. (b) End view of the four middle electrodes with relevant electrical connections. Various dimensions as marked by $l_{e}$, $l_{m}$, $l$, $r_{e}$ and $r_{0}$ are described in section \[section3\]. []{data-label="lintrap"}](lintrap.eps "fig:"){width="90.00000%"}\ These equations (eqn. \[eqn2\]) can be rewritten as $$\label{eqn3} \frac{d^{2}u}{d\zeta^{2}} + (a_{u}-2q_{u}\cos2\zeta)u = 0,$$ with $u=x,y$, where $$\begin{aligned} \label{eqn4} a_{x}&=&-a_{y}=\frac{4eU}{mr_{0}^{2}\Omega^{2}}, \nonumber \\ q_{x}&=&-q_{y}=\frac{2eV_{0}}{mr_{0}^{2}\Omega^{2}},\end{aligned}$$ and $\zeta = \Omega t/2$. Eqn. \[eqn3\] is standard Matheiu differential equation and its solution provides stability or instability of the ion motion [@Dawson76] depending on the values of the parameters $a$ and $q$ as defined in eqn. \[eqn4\]. There exists a region in $a$ vs. $q$ diagram for which the ion-motion is stable along a particular direction, for example along $x$. A similar stability region exists for the motion along $y$ direction. An intersection between these two stability regions thus signifies a stable motion in $xy$ plane. For stable ion motion the trap should be operated at $q<0.908$. The stable solutions of Mathieu differential equation show that the trapped ion oscillates with different frequencies given by [@Ghosh95] $$\label{eqn5} \omega_{n}=\frac{(2n\pm\beta)\Omega}{2}, n = 0, 1, 2, 3...$$ Here $\beta$ is a function of the trap operating parameters $a$, $q$ and for their small values, $\beta=\sqrt{a+q^{2}/2}$. The fundamental frequency $\omega_{0}$ (that corresponds to $n=0$) of secular motion and other micromotion frequencies are given by $$\begin{aligned} \label{eqn6} \omega_{0}&=&\frac{\beta\Omega}{2}, \\ \nonumber \omega_{1\pm}&=&\Omega\pm\omega_{0}, \\ \omega_{2\pm}&=&\Omega\pm2\omega_{0} \nonumber\end{aligned}$$ and so on. A large spectra of the motional frequency have been obtained in our experiment by using electric dipole excitation technique. Though in ideal case the trap potential is quadrupolar, real traps appear with misalignment, defect in machining, truncation and holes in the electrodes to have optical access. In addition, there are space charge developed by the trapped ions themselves. All these result in deviation from pure quadrupole trap potential contributing to other higher order terms and make the ion motion unstable for certain values of the trapping parameters, for which the stability exists in ideal case. The ions gain energy from the rf trapping field and their motional amplitudes get enhanced resulting loss from the trap. The condition of such nonlinear resonances is given by [@Dawson69] $$\label{eqn7} n_{x}\omega_{0x}+n_{y}\omega_{0y}=\Omega, n_{x}, n_{y}=0, 1, 2, 3...$$ where $\omega_{0x}$ and $\omega_{0y}$ are the secular frequencies for the motion along $x$ and $y$ respectively. Here $n_{x}+n_{y}=k$ is the order of the multipole. If one of the trap parameters is varied, a parametric resonance appears at a definite value subjected to the condition defined by eqn. \[eqn7\] and it gives rise to instabilities called “black canyons” [@March05] within the stability diagram. ![Schematic of the experimental setup. The trap, filament and the CEM with other ion optics (extraction cylinder) are housed in a vacuum chamber. The functioning and control of the signal processing devices are explained in the text.[]{data-label="setup_schematic"}](setup_schematic.eps "fig:"){width="95.00000%"}\ Experimental setup {#section3} ================== The schematic of the whole experimental setup is presented in figure \[setup\_schematic\]. It consists of a linear Paul trap as shown in figure \[lintrap\], an ionization setup, extraction and detection setup. The linear trap is assembled from four three-segmented electrodes each placed at four corners of a square of side ($l$) $12.73$ mm \[figure \[lintrap\]\](b). Each of twelve rods are of diameter ($2r_{e}$) $10$ mm. The four middle rods are of length ($l_{m}$) $25$ mm while all others are $15$ mm long ($l_{e}$) \[figure \[lintrap\](a)\]. The separation between the surfaces of the diagonally opposite rods ($2r_{0}$) is $8$ mm. The middle electrode is separated from the end electrodes by a gap of $2$ mm. The molecular nitrogen ions (N$_{2}^{+}$) are created by electron impact ionization. The ions are dynamically trapped for few hundreds ms before they are extracted by lowering the axial potential in one direction. The extracted ions are detected by a channel electron multiplier (CEM). The CEM produces one pulse corresponding to each ion and the pulse is successively processed through an amplifier, a discriminator, a TTL converter before it is fed into a multichannel scalar (MCS) card which ultimately counts the number of ions reaching the CEM. This time-of-flight (TOF) technique provides a detection efficiency around $10\%$. The time sequences are generated by National Instruments’ Data Acquisition (DAQ) hardware which is controlled by Labview and monitored by a personal computer (PC). The trap is operated at a rf frequency of $1.415$ MHz and no dc potential is applied to the middle electrodes ($U=0$, $a_{u}=0$). The end electrodes are kept at $+20$ V while trapping. At the time of extraction, the end electrodes at the ion-exit-side are switched fast (within $75~ns$) from $+20$ V to $-45$ V. Experimental results {#section4} ==================== ![Number of trapped ions ($N$) as a function of $q$ ($a=0$). Sudden fall of $N$ about some specific values of $q$ corresponds to nonlinear resonances as explained in the text. The numbers $6$, $7$, $8$ describe the order of the multipoles to which the resonances are assigned.[]{data-label="q_scan"}](q_scan_auto.eps "fig:"){width="60.00000%"}\ Stability characteristics ------------------------- The stability behaviour of the trap is studied by varying the trap-operating-parameter $q$ while keeping the other parameter $a$ at zero. The $q$ is varied in steps of $0.0008$ by changing the rf amplitude at small intervals of $0.35$ V while the number of trapped ions ($N$) is plotted in figure \[q\_scan\] as a function of $q$. It shows that $N$ grows with $q$ initially but decreases above $q\approx0.5$. It remains almost constant and shows a plateau for $0.3<q<0.5$. The q scanning is restricted to $0.6$ due to the presence of heavier masses which can not be resolved in the TOF spectra. One of the significant observations within this stability diagram is the appearance of narrow nonlinear resonances for specific values of $q$. These are due to the existence of higher order multipoles within the trap potential as explained in section \[section2\]. The resonances appear at $q=0.3461, 0.4073$ and $0.4885$ are assigned to the $8$th,$7$th and $6$th order multipoles respectively. The $7$th order multipole is unlikely as the symmetry of trap setup forbids non-zero perturbations due to odd order multipole. However, such a nonlinear observation has been observed previously [@Drakoudis06]. It could result from any misalignment of the setup that partially breaks the radial symmetry or due to some electrical connection wires near the trap center. The nonlinear resonance at $q=0.5163$ in our experiment could not be assigned. It may result from other atmospheric species, or some molecules produced by charge-transfer-reactions inside the trap. As can be seen from figure \[q\_scan\], the depth of the resonance appearing at $q=0.4885$ is maximum and hence it can be concluded that the $6$th order multipole is the strongest one in our trap setup. ![Schematic of the circuit used for dipole excitation of trapped ions. The dipole excitation signal $v_{i}\cos\omega t$ is applied between the electrodes marked as I and III.[]{data-label="circuit_dipole"}](circuit_dipole.eps "fig:"){width="75.00000%"}\ While operating the trap for a single ion, the region of instabilities should be avoided as the ion gains energy from the time varying trapping field corresponding to these operating regions and its motional amplitude increases. It can add to systematics in precision measurement on the ion. Dipole excitation of trapped ions --------------------------------- Electric dipole excitation of the trapped ions has been employed to measure their secular frequency and to obtain motional spectra. An electric dipole field has been applied on one of the middle electrodes as shown schematically in figure \[circuit\_dipole\]. The amplitude of the excitation potential ($v_{i}$) is kept fixed while its frequency is tuned so as to match with the secular frequency of the trapped ions. The trap operating parameters are kept fixed during the experiment. After the ions are loaded into the trap, the dipole excitation field is applied for few hundreds of ms. After a short waiting time, the ions are released and detected. The frequency of the excitation signal ($\omega$) is varied and the total number of ions is detected in each step. ![Dipole excitation resonance of trapped ions. Solid line shows a fit to the data with model function described in eqn. \[eqn8\].[]{data-label="dipole_scan"}](dipole_scan.eps "fig:"){width="60.00000%"}\ ### Measurement of secular frequency The experimentally obtained ion counts ($N$) have been normalised after dividing by the maximum ion count ($N_{max}$) during a particular experiment. The normalised ion count ($N_{n}=N/N_{max}$) with associated uncertainty, has been plotted as a function of the frequency ($\omega/2\pi$) of the dipole excitation signal. Figure \[dipole\_scan\] shows such a dipole excitation resonance plot obtained with an excitation amplitude $v_{i}=50$ mV. The frequency is scanned from $165$ kHz to $205$ kHz in steps of $500$ Hz. The excitation signal is applied during $150$ ms in each step. The experimental data points have been fitted with the following function, $$\label{eqn8} N_{n}=N_{0}+A\exp\left[-\exp(-\omega')-\omega'+1\right],$$ with $\omega'=(\omega-\omega_{0})/\sigma$. Here $\omega_{0}$ is the resonant frequency and is equal to the secular frequency of the trapped ions. $N_{0}$ is an offset, $A$ is a scaling factor and $\sigma$ is the full-width at half-maxima (FWHM) of the resonance. The secular frequency of the trapped ions obtained from the fit is $182.730$($76$) kHz and it is in good agreement with theoretically calculated value. ### Motional spectra The motional spectra of the trapped ions as described in section 2 have been measured by varying the dipole excitation signal frequency over long range. Figure \[motional\_spectra\] shows the motional spectra in the radial plane. The fundamental or the first harmonic frequency of oscillation is observed at $\omega_{0}=2\pi\times184$ kHz that corresponds to the trap operating parameter $a=0$, $q=0.39$ and it is the strongest one. The second and third harmonics are observed at $386$ kHz and $577$ kHz respectively. The other motional spectra as described in eqn. \[eqn6\] are observed at $\omega_{2-}=2\pi\times915$ kHz, $\omega_{1-}=2\pi\times1.109$ MHz, $\omega_{1+}=2\pi\times1.492$ kHz and $\omega_{2+}=2\pi\times1.685$ MHz. ![The normalized ion count $N$ plotted as a function of the dipole excitation frequency (in kHz) presenting the motional spectra of the trapped ion cloud. The amplitude of the excitation voltage is $v_{i}=100$ mV and the trap operating parameters are set at $a=0$, $q=0.39$ for N$_{2}^{+}$. The frequency of the trap supply voltage is $\Omega=2\pi\times1.3$ MHz.[]{data-label="motional_spectra"}](motional_spectra_dipole.eps "fig:"){width="65.00000%" height="0.8\textheight"}\ ### Application The accurate measurement of the motional frequency of the trapped ions is essential for different studies on them [@Mandal13th]. In a real linear Paul trap the radial motion is coupled with the axial motion and hence a variation in the axial potential affects the secular frequency of the ions [@Drakoudis06]. The motional frequency of the trapped ions for different axial potentials has been measured with the technique described in section 4.2.1 and from this measurement the geometrical radial-axial coupling constant has been determined. This is important for any precision spectroscopic study on a single ion confined in this setup. The dipole excitation technique is also applied to study the shift in the motional frequency due to space charge created by the trapped ions. It is observed that the frequency decreases while they oscillate collectively with increasing space charge [@Mandal13]. Detailed discussion on these topics can be found elsewhere [@Mandal13th; @Mandal13]. Conclusions {#section5} =========== This colloquium paper describes the development of an ion trap facility at IACS and the results of some experiments fundamentally based on the dynamics of a trapped ion cloud. It presents a demonstration of some first principles of ion-trap-physics that are of common interest to an audience coming from wide variety of physics and participating in this colloquium. The results are also some significant feeds to the precision measurement based on a single ion in a linear Paul trap. Acknowledgement {#section6} =============== The authors thank S. Das, D. De Munshi and T. Dutta, presently at the Centre for Quantum Technologies, National University of Singapore, for their support in developing the experimental setup at IACS, and beyond it. The machining support from Max-Planck Institute, Germany is gratefully acknowledged. H Dehmelt *Physica Scripta* **T22** 102 (1988) R S Van Dyck, P B Schwinberg and H G Dehmelt *Phys. Rev. Lett.* **59** 26 (1987) N Yu, W Nagourney and H Dehmelt *Phys. Rev. Lett.* **78** 4898 (1997) G P Barwood *et al.* *Phys. Rev. Lett.* **93** 133001 (2004) W H Oskay *et al.* *Phys. Rev. Lett.* **94** 163001 (2005) C F Roos *et al.* Nature **443** 316 (2006) N Fortson *Phys. Rev. Lett.* **70** 2383(1993) P Mandal and M Mukherjee *Phys. Rev. A* **82** 050101(R) (2010) B K Sahoo, P Mandal and M Mukherjee *Phys. Rev. A* **83** 030502(R) (2011) O O Versolato *et al.* *Phys. Rev. A* **82** 010501(R) (2010) P H Dawson *Quadrupole Mass Spectrometry and Its Applications* Elsevier (1976) P K Ghosh *Ion Traps* Oxford University Press (1995) P H Dawson and N R Whetten *Int. J. Mass Spectrom. Ion Phys.* **2** 45 (1969) R E March and J F J Todd *Quadrupole Ion Trap Mass Spectrometry* John Wiley & Sons (2005) A Drakoudis, M Söllner and G Werth *Int. J. Mass Spectrom.* **252** 61 (2006) P. Mandal *PhD Thesis* University of Calcutta *submitted* (2013) P. Mandal *et al.* *arXiv* 1305.7081v1 \[physics-atom-ph\] (2013)

--- abstract: 'Deep learning has shown that learned functions can dramatically outperform hand-designed functions on perceptual tasks. Analogously, this suggests that learned optimizers may similarly outperform current hand-designed optimizers, especially for specific problems. However, learned optimizers are notoriously difficult to train and have yet to demonstrate wall-clock speedups over hand-designed optimizers, and thus are rarely used in practice. Typically, learned optimizers are trained by truncated backpropagation through an unrolled optimization process resulting in gradients that are either strongly biased (for short truncations) or have exploding norm (for long truncations). In this work we propose a training scheme which overcomes both of these difficulties, by dynamically weighting two unbiased gradient estimators for a variational loss on optimizer performance, allowing us to train neural networks to perform optimization of a specific task faster than tuned first-order methods. We demonstrate these results on problems where our learned optimizer trains convolutional networks faster in wall-clock time compared to tuned first-order methods and with an improvement in test loss.' bibliography: - 'main.bib' --- Introduction ============ Gradient based optimization is a cornerstone of modern machine learning. A large body of research has been targeted at developing improved gradient based optimizers. In practice, this typically involves analysis and development of hand-designed optimization algorithms [@nesterov1983method; @duchi2011adaptive; @tieleman2012lecture; @kingma2014adam]. These algorithms generally work well on a wide variety of tasks, and are tuned to specific problems via hyperparameter search. On the other hand, a complementary approach is to *learn* the optimization algorithm [@bengio1990learning; @schmidhuber1995learning; @hochreiter2001learning; @andrychowicz2016learning; @wichrowska2017learned; @li2017learning; @lv2017learning; @Bello17]. That is, to learn a function that performs optimization, targeted at particular problems of interest. In this way, the algorithm may learn *task specific* structure, enabling dramatic performance improvements over more general optimizers. However, training learned optimizers is notoriously difficult. Existing work in this vein can be classified into two broad categories. On one hand are black-box methods such as evolutionary algorithms [@goldberg1988genetic; @bengio1992optimization], random search [@bergstra2012random], reinforcement learning [@Bello17; @li2016learning; @li2017learning], or Bayesian optimization [@snoek2012practical]. However, these methods scale poorly with the number of optimizer parameters. {width="5.5in"}\ -------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- **Term** **Definition** \[-0.2cm\] \[-0.2cm\] $\mathcal{D}$ Dataset consisting of train and validation split, ${\mathcal}D_\text{train}$ and ${\mathcal}D_\text{valid}$. ${\mathcal}T$ The set of tasks, where each task is a dataset (e.g., a subset of Imagenet classes). $w^{{\left( t \right)}}$ Parameters of inner-problem at iteration $t$. These are updated by the learned optimizer, and depend implicitly on $\theta$ and ${\mathcal}D_{\text{train}}$. $\ell{\left( x; w^{{\left( t \right)}} \right)}$ Loss on inner-problem, for mini-batch $x$. $\theta$ Parameters of the optimizer. $u{\left( \cdot; \theta \right)} Function defining the learned optimizer. The inner-loop update is $w^{{\left( t+1 \right)}} = u{\left( w^{{\left( t \right)}}, x, \nabla_{w} \ell, \ldots; \theta \right)}$, for $x \sim {\mathcal{D}_\text{train}}$. $ $L_\text{train}{\left( \theta \right)}$ Outer-level objective targeting training loss, $\mathbb{E}_{\mathcal{D} \sim {\mathcal}T} {\mathbb{E}_{x \sim \mathcal{D}_\text{train}}\left[ \frac{1}{T} \sum_{t=1}^{T} \ell{\left( x; w^{{\left( t \right)}} \right)} \right]}$. $L_\text{valid}{\left( \theta \right)}$ Outer-level objective targeting validation loss, $\mathbb{E}_{\mathcal{D} \sim {\mathcal}T} {\mathbb{E}_{x \sim \mathcal{D}_\text{valid}}\left[ \frac{1}{T} \sum_{t=1}^{T} \ell{\left( x; w^{{\left( t \right)}} \right)} \right]}$. ${\mathcal}L{\left( \theta \right)}$ The variational (smoothed) outer-loop objective, ${\mathbb{E}_{\tilde{\theta} \sim {\mathcal}N{\left( \theta, \sigma^2 I \right)}}\left[ L{\left( \tilde{\theta} \right)} \right]}$. -------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The other approach is to use first-order methods, by computing the gradient of some measure of optimizer effectiveness with respect to the optimizer parameters. Computing these gradients is costly as we need to iteratively apply the learned update rule, and then backpropagate through these applications, a technique commonly referred to as “unrolled optimization” [@bengio2000gradient; @maclaurin2015gradient]. To address the problem of backpropagation through many optimization steps (analogous to many timesteps in recurrent neural networks), many works make use of truncated backpropagation though time (TBPTT) to partition the long unrolled computational graph into separate pieces [@werbos1990backpropagation; @domke2012generic; @tallec2017unbiasing]. This not only yields computational savings, at the cost of increased bias [@tallec2017unbiasing], but also limits exploding gradients which emerge from too many iterated non-linear function applications [@pascanu2013difficulty; @parmas2018pipps]. Existing methods have been proposed to address the bias of TBPTT but come at the cost of increased variance or computational complexity [@williams1989learning; @ollivier2015training; @tallec2017unbiasing]. Previous techniques for training RNNs via TBPTT have thus far not been effective for training optimizers. In this paper, we analytically and experimentally explore the debilitating role of bias and exploding gradients on training optimizers ([§\[sec bias explosion\]]{}). We then show how these pathologies can be remedied by optimizing the parameters of a distribution over the optimizer parameters, known as variational optimization [@staines2012variational] ([§\[sec more stable\]]{}). We define two unbiased gradient estimators for this objective: a reparameterization based gradient [@kingma2013auto], and evolutionary strategies [@rechenberg1973evolutionsstrategie; @nesterov2011random]. By dynamically reweighting the contribution of these two gradient estimators [@fleiss1993review; @parmas2018pipps; @buckman2018sample], we are able to avoid exploding gradients and unroll longer, to stably and efficiently train learned optimizers. We demonstrate the utility of this approach by training a learned optimizer to target optimization of small convolutional networks on image classification ([§\[sec:experiments\]]{}). With our method, we are able to outer-train on more inner steps (10k inner-parameter updates) with more complex inner-problems than prior work. Additionally, we can simplify the parametric form of the optimizer, utilizing a small MLP without any complex tricks such as extensive use of normalization[@metz2018learning], or annealing training from existing algorithms[@houthooft2018evolved] previously needed for stability. On the targeted task distribution, this learned optimizer achieves better test loss, and is faster in *wall-clock time*, compared to hand-designed optimizers such as SGD+Momentum, RMSProp, and ADAM (Figure \[fig:unroll\_on\_problem\]). To our knowledge, this is the first instance of a learned optimizer performing comparably to existing methods on wall-clock time, as well as the first parametric optimizer outer-trained against validation loss. While not the focus of this work, we also find that the learned optimizer demonstrates promising generalization ability on out of distribution tasks (Figure \[fig:outofdomain\]). Unrolled optimization for learning optimizers ============================================= Problem Framework ----------------- Our goal is to learn an optimizer which is well suited to some set of target optimization tasks. Throughout the paper, we will use the notation defined in Figure \[tab:my\_label\]. Learning an optimizer can be thought of as a bi-level optimization problem [@franceschi2018bilevel], with [*inner*]{} and [*outer*]{} levels. The inner minimization consists of optimizing the weights ($w$) of a target problem $\ell(w)$ by the repeated application of an update rule ($u{\left( \cdot \right)}$). The update rule is a parameterized function that defines how to map the weights at iteration $t$ to iteration $t+1$: $w^{(t+1)} = u(w^{(t)}, x, \nabla_{w} \ell, ...; \theta)$. Here, $\theta$ represents the parameters of the learned optimizer. In the outer loop, these optimizer parameters ($\theta$) are updated so as to minimize some measure of optimizer performance, the outer-objective ($L(\theta)$). Our choice for $L$ will be the average value of the target loss ($\ell{\left( \cdot \right)}$) measured over either training or validation data. Throughout the paper, we use *inner-* and *outer-* prefixes to make it clear when we are referring to applying a learned optimizer on a target problem (inner) versus training a learned optimizer (outer). {width="6.65in"} {width="6.65in"} Unrolled optimization --------------------- In order to train an optimizer, we wish to compute derivatives of the outer-objective $L$ with respect to the optimizer parameters, $\theta$. Doing this requires unrolling the optimization process. That is, we can form an unrolled computational graph that consists of iteratively applying an optimizer ($u$) to optimize the weights ($w$) of a target problem (Figure \[tab:my\_label\]). Computing gradients for the optimizer parameters involves backpropagating the outer loss through this unrolled computational graph. This is a costly operation, as the entire inner-optimization problem must be unrolled in order to get a single outer-gradient. Partitioning the unrolled computation into separate segments, known as truncated backpropagation, allows one to compute multiple outer-gradients over shorter segments. That is, rather than compute the full gradient from iteration $t=0$ to $t=T$, we compute gradients in windows from $t=a$ to $t=a+\tau$. The gradients from these segments can be used to update $\theta$ *without* unrolling all $T$ iterations, dramatically decreasing the computation needed for each update to $\theta$. The choice for the number of inner-steps per truncation is challenging. Using a large number of steps per truncation can result in exploding gradients making outer-training difficult, while using a small number of steps can produce biased gradients resulting in poor performance. In the following sections we analyze these two problems. Exponential explosion of gradients with increased sequence length {#sec bias explosion} ----------------------------------------------------------------- We can illustrate the problem of exploding gradients analytically with a simple example: learning a learning rate. Following the notation in Figure \[tab:my\_label\], we define the optimizer as: $$w^{(t+1)} = u(w^{(t)}; \theta) = w^{(t)} - \theta \nabla \ell{\left( w^{(t)} \right)},$$ where $\theta$ is a scalar learning rate that we wish to learn for minimizing some target problem $\ell(w^{(t)})$. For simplicity, we assume a deterministic loss ($\ell(\cdot)$) with no batch of data ($x$). The quantity we are interested in is the derivative of the loss after $T$ steps of gradient descent with respect to $\theta$. We can compute this gradient (see Appendix \[app:derivation\]) as: $$\frac{d \ell(w^{(T)})}{d\theta} = \left\langle g^{(T)}, -\sum_{i = 0}^{T-1} \left(\prod_{j=i+1}^{T-1} (I - \theta H^{(j)}) \right) g^{(i)}\right\rangle,$$ where $g^{(i)}$ and $H^{(j)}$ are the gradient and Hessian of the target problem $\ell(w)$ at iteration $i$ and $j$, respectively. We see that this equation involves a sum of products of Hessians. In particular, the first term in the sum involves a product over the *entire sequence* of Hessians observed during training. When optimizing a quadratic loss, the Hessian is constant, and the outer-gradient becomes a matrix polynomial of degree $T$, where $T$ is the number of gradient descent steps. Thus, the outer-gradient can grow exponentially with $T$ if the maximum eigenvalue of $(I-\theta H^{(j)})$ is greater than $1$. In general, the Hessian is not a constant, but in practice we still see an exponential growth in gradient norm across a variety of settings. We can see another problem with long unrolled gradients empirically. Consider the task of optimizing a loss surface with two local minima defined as $\ell(w) = (w-4)(w-3)w^2$ with initial condition $w^{(0)}=-1.2$ using a momentum based optimizer with a parameterized momentum value $\theta$ (Figure \[fig:2minimum\]a). At low momentum values the optimizer converges in the first of the two local minima, whereas for larger momentum values the optimizer settles in the second minimum. With even larger values of momentum, the iterate oscillates between the two minima before settling. We visualize both the trajectory of $w^{{\left( t \right)}}$ over training and the final loss value for different momentum values in Figure \[fig:2minimum\]b and \[fig:2minimum\]c. With increasing unrolling steps, the loss surface as a function of the momentum $\theta$ becomes less and less smooth, and develops near-discontinuities at some values of the momentum resulting in extremely large gradient norms. This behavior is not unique to momentum optimizer parameters. In Appendix \[app:toy\_lr\] we perform additional experiments modifying learning rates instead and show similar behavior. Although these are toy systems, we see similar pathologies when optimizing more complex inner-models with both hand designed and learned optimizers. Trajectories taken during optimization often change dramatically as a result of only small changes in optimizer parameters. To illustrate this, in Figure \[fig:2minimum\]d we train a 2 layer MLP with ReLU activations for 40 iterations using Adam. We vary the learning rate between 0.1469 to 0.1484 with 100 samples spaced uniformly in log scale, but keep all sources of randomness fixed. We plot 2D random projections of the MLP’s parameters during training, using color to denote different learning rates. Early in training the trajectories are similar, but as more steps are taken the trajectories diverge, producing drastically different trained models with only small changes in the learning rate. In the case of both neural network inner-problems and neural network optimizers, the outer-loss surface can grow even more complex with increasing number of unrolling steps. We illustrate this in Figure \[fig:real\_loss\_surface\]e and \[fig:real\_loss\_surface\]f for slices through the loss landscape $L{\left( \theta \right)}$ of the outer-problem for a neural network optimizer. {width="6.5in"} Increasing bias with truncated gradients {#sec:increase_bias} ---------------------------------------- Existing work on learned optimizers often avoids exploding gradients ([§\[sec bias explosion\]]{}) by using a short truncation window. Here, we demonstrate the bias short truncation windows can introduce in unrolled optimization. These results are similar to those presented in @wuunderstanding, except that we utilize *multiple* truncations rather than a single, shortened unroll. First, consider outer-learning the learning rate of Adam when optimizing a small two layer neural network on MNIST [@lecun1998mnist]. We initialize Adam with a learning rate of $0.001$ and outer-train using increasing truncation amounts (Figure \[fig:meta\_train\_learning\_rate\]ab). Adam is used as the outer-optimizer. Other outer-optimizers can be found in \[app:trunc\_bias\]. Despite initializing close to the optimal learning rate, when outer-training with severely truncated backprop the resulting learning rate decreases, increasing the outer-loss. The sum of truncated outer-gradients are anti-correlated with the true outer-gradient. Towards stable training of learned optimizers {#sec more stable} ============================================= To perform outer-optimization of a loss landscape with high frequency structure like that in Figure \[fig:real\_loss\_surface\], one might intuitively want to smooth the outer-objective loss surface. To do this, instead of optimizing $L(\theta)$ directly we instead optimize a smoothed outer-loss ${\mathcal}L{\left( \theta \right)}$, $${\mathcal}L{\left( \theta \right)} = {\mathbb{E}_{\tilde{\theta} \sim {\mathcal}N{\left( \theta, \sigma^2 I \right)}}\left[ L{\left( \tilde{\theta} \right)} \right]},$$ where $\sigma^2$ is a fixed variance (set to 0.01 in all experiments) which determines the degree of smoothing. This is the same approach taken in variational optimization [@staines2012variational]. We can construct two different unbiased gradient estimators for ${\mathcal}L{\left( \theta \right)}$: one via the reparameterization trick [@kingma2013auto]; and one via the “log-derivative trick”, similarly to what is done in evolutionary strategies (ES) and REINFORCE [@williams1992simple; @wierstra2008natural]. We denote the two estimates as $g_\text{rp}$ and $g_\text{es}$ respectively, $$\begin{aligned} {4} g_\text{rp} &= \frac{1}{S}\sum_s {\nabla_{\theta} L{\left( \theta+\sigma n_s \right)}},\\ \qquad & n_s \sim N{\left( 0, I \right)},& \\ g_\text{es} &= \frac{1}{S}\sum_s L{\left( \tilde{\theta}_s \right)} \nabla_{\theta} \left[\text{log}{\left( N{\left( \tilde{\theta}_s; \theta, \sigma^2I \right)} \right)}\right],\\ \qquad & \tilde{\theta}_s \sim N{\left( \theta, \sigma^2 I \right)},&\end{aligned}$$ where $N{\left( \tilde{\theta}_s; \theta, \sigma^2I \right)}$ is the probability density of the given ES sample, $\tilde{\theta}_s$, $S$ is the sample count, and in implementation the same samples can be reused for $g_{rp}$ and $g_{es}$. Following the insight from [@parmas2018pipps] in the context of reinforcement learning[^1], we combine these estimates using inverse variance weighting [@fleiss1993review], $$\label{eq:combine} g_\text{merged} = \frac{g_\text{rp}\sigma_\text{rp}^{-2} + g_\text{es}\sigma_\text{es}^{-2}}{\sigma_\text{rp}^{-2}+\sigma_\text{es}^{-2}},$$ where $\sigma_\text{rp}^2$ and $\sigma_\text{es}^2$ are empirical estimates of the variances of $g_{rp}$ and $g_{es}$ respectively. When outer-training learned optimizers we find the variances of $g_\text{es}$ and $g_\text{rp}$ can differ by as many as 20 orders of magnitude (Figure \[fig:grad\_var\]). This merged estimator addresses this by having at most the lowest of the two variances. To further reduce variance, we employ antithetic sampling. Each normal distribution draw is used twice, both positive and negative, when computing $g_\text{rp}$ and $g_\text{es}$. ![As outer-training progresses, the variance of the reparameterization gradient estimator grows, while the variance of the ES estimator remains constant. The variance of the reparameterization gradient progresses from approximately five orders of magnitude better than that of the ES gradient, to approximately twenty orders of magnitude worse. \[fig:grad\_var\] ](figures/grad_variance_v2.pdf){width="3.2in"} The cost of computing a single sample of $g_\text{es}$ and $g_\text{rp}$ is thus two forward and two backward passes of an unrolled optimization. To compute the empirical variance, we leverage data parallelism to compute multiple samples of $g_\text{es}$ and $g_\text{rp}$. In theory, to prevent bias the samples used to evaluate $\sigma_\text{rp}^2$ and $\sigma_\text{es}^2$ must be independent of those used to estimate $g_\text{es}$ and $g_\text{rp}$, but in practice we found good performance using the same samples for both. This gradient estimator fixes the exploding gradients problem when computing gradients over a long truncation, and longer truncations enable lower bias gradient estimates. In practice, these longer truncations are computationally expensive, and early in outer-training shorter truncations are sufficient. The full outer-training algorithm is described in Appendix \[app:algo\]. Experiments {#sec:experiments} =========== As a proof of principle, we use the training algorithm described in [§\[sec more stable\]]{} to train a simple learned optimizer. For this work, we focus on training an optimizer to target a specific architecture. In the following sections we describe the optimizer architecture used, the task distribution on which we outer-train, as well as outer training details. We then discuss the performance of our learned optimizer on both in and out of distribution target problems. We finish with an ablation study showing the importance of our gradient estimator, as well as aspects of the optimizer’s architecture. ![ Performance after 10k iterations of inner-training for different learned optimizers over the course of outer-training. Each line represents a different random initialization of outer-parameters ($\theta$). Optimizers are trained targeting the train outer-objective (left), and the validation outer-objective (right). Dashed lines indicate performance of learning rate tuned Adam. Models in orange are the best performing and used in [§\[sec:experiments\]]{}. \[fig:meta-training-curves\] ](figures/meta_training_curves_v2.pdf){width="3.4"} Optimizer architecture {#sec:architecture} ---------------------- The optimizer architecture used in all experiments consists of a small, fast to compute, fully connected neural network, with one hidden layer containing 32 ReLU units ($\sim$1k parameters). This network is applied to each target problem inner-parameter independently. The outputs of the MLP consist of an un-normalized update direction and a per parameter log learning rate which gets exponentiated. These two quantities are multiplied and subtracted from the previous inner-parameter value to form the next inner-parameter value. The MLP for each weight takes as input: the gradient with respect to that weight, the parameter value, exponentially weighted moving averages of gradients at multiple time scales [@lucas2018aggregated], as well as a representation of the current iteration number. Many of these input features were motivated by [@wichrowska2017learned]. We conduct ablation studies for these inputs in [§\[sec:ablations\]]{}. See Appendix \[app:arch\_details\] for further architectural details. {width="6.8in"} Optimizer target problem {#sec:target_problem} ------------------------ The problem that each learned optimizer is trained against ($\ell{\left( \cdot \right)}$) consists of training a three layer convolutional neural network (32 units per layer,  20k parameters) inner-trained for ten thousand inner-iterations on 32x32x3 image classification tasks. Due to the weight sharing in convolutions, the compute per parameter is high and thus relatively less computation is needed for each inner-iteration. We split the Imagenet dataset [@ILSVRC15] by class into 700 training and 300 test classes, and sample training and validation problems by sampling 10 classes at random using all images from each class. This experimental design lets the optimizer learn problem specific structure (e.g. convolutional networks trained on object classification), but does not allow the optimizer to memorize class-specific weights for the base problem. This task is modeled after the fact that standard architectures, e.g. ResNet [@he2016identity], are often applied to a variety of different datasets. See Appendix \[app:arch\_details\] for further details. Outer-training {#sec:curriculum} -------------- To train the optimizer, we linearly increase the number of unrolled steps from 50 to 10,000 over the course of 5,000 outer-training weight updates. The number of unrolled steps is additionally jittered by a small percentage (sampled uniformly up to 20%).Due to the heterogeneous, small iterated computations, we train with asynchronous, batched SGD using 128 CPU workers. Figure \[fig:meta-training-curves\] shows the performance of the optimizer (averaged over 40 randomly sampled outer-train and outer-test inner-problems) while outer-training. Despite the stability improvements described in the last section, there is still variability in optimizer performance over random initializations of the optimizer parameters. As expected given our optimizer parameterization, there is very little outer-overfitting. Nevertheless, we use outer-training loss to select the best model and use this in the remainder of the evaluation. Learned optimizer performance {#sec:learned_opt_performance} ----------------------------- Figure \[fig:unroll\_on\_problem\] shows performance of the learned optimizer, after outer-training, compared against other first-order methods on a sampled validation task (classes not seen during outer-training). For “Adam”, “RMSProp”, and “Momentum”, we report the best performance after tuning the learning rate by grid search using 11 values over a logarithmically spaced range from $10^{-4}$ to 10 on a per task basis. Searching over fixed learning rates is the baseline most commonly used in other learned optimizer work [@andrychowicz2016learning; @wichrowska2017learned]. However, practitioners often adjust many more hyperparameters. Therefore, we provide an additional baseline consisting of tuning: all Adam optimizer parameters (beta1, beta2, epsilon, learning rate); learning rate decay (exponential decay coefficient, linear decay coefficient); and regularization parameters (l1 regularization, l2 regularization). We outer-optimize these parameters against the distribution of tasks, evaluating &gt;2k parameter combinations totalling 100k total inner-problem training runs (performance is averaged over 80 inner runs for a lower variance estimate of $L$). As with the learned optimizer, we outer-optimize with respect to $L_\text{train}$, and $L_\text{test}$. We outer-optimize our baseline using uniform random search over all 8 hyperparameters. Outer-optimizing the baseline with the framework we present in this paper yields similar results both in compute cost, and final performance achieved. We performed a similar optimization for RMSProp and SGD+Momentum and find similar performance to that of Adam. Results are not shown here for clarity but can be found in Appendix \[app:inner\_loop\_figure\]. When outer-trained against the training outer-objective, $L_\text{train}$, our learned optimizer achieves faster convergence on training loss (Figure \[fig:unroll\_on\_problem\]a), but poor performance on test loss (Figure \[fig:unroll\_on\_problem\]b). This is expected, as our outer-training procedure never sees validation loss, and thus only minimizes training loss, causing overfiting. When outer-trained against the validation outer-objective, $L_\text{valid}$, we also achieve fast optimization and reach a lower test loss in the given time interval (Figure \[fig:unroll\_on\_problem\]b). We suspect further gains in validation performance could be obtained with the addition of more regularization techniques into both the learned optimizer, and the tuned Adam baseline. Figure \[fig:histogram\]c summarizes the performance of the learned optimizer across 100 sampled outer-test tasks (tasks not seen during outer-training). It shows the difference in loss (averaged over the first 10k iterations of training) between the learned optimizer and the 8 parameter Adam (Adam+Reg+Decay) baseline tuned against the corresponding loss. Our learned optimizer outperforms this baseline on the majority of tasks. {width="5.5in"} Although the focus of our approach was *not* generalization, we find that our learned optimizer nonetheless generalizes to varying degrees to dissimilar datasets, different numbers of units per layer, different number of layers, and even to fully connected networks. In Figure \[fig:outofdomain\] we show performance on a six layer convolutional neural network trained on MNIST. Despite the different number of layers, dissimilar dataset, and different input size, the learned optimizers still reduces the loss, and in the case of the validation outer-objective trains faster and generalizes well. We further explore the limits of generalization of our learned optimizer on additional tasks in Appendix \[app:outofdist\]. Ablations {#sec:ablations} --------- {width="5.3in"} To assess the importance of the gradient estimator discussed in [§\[sec more stable\]]{}, the unrolling curriculum [§\[sec:curriculum\]]{}, as well as the features fed to the optimizer enumerated in [§\[sec:architecture\]]{}, we re-trained the learned optimizer removing each of these additions. In particular, we trained optimizers with: only the reparameterization gradient estimator (Gradients), only with evolutionary strategies (ES), a fixed number unrolled steps per truncation (10, 100, 1000) as opposed to a schedule keeping while keeping the same total inner-weight updates, no momentum terms (No Mom), and without the current iteration (No Time). To account for variance, each configuration is repeated with multiple random seeds. Figure \[fig:ablations\] summarizes these findings, showing the learned optimizer performance for each of these ablations. We find that the gradient estimator (in [§\[sec more stable\]]{}) and an increasing schedule of unroll steps are critical to performance, along with including momentum as an input to the optimizer. Discussion ========== In this work we demonstrate two difficulties when training learned optimizers: “exploding” gradients, and a bias introduced by truncated backpropagation through time. To combat this, we construct a variational bound of the outer-objective and minimize this via a combination of reparameterization and ES style gradient estimators. By using our combined estimator and a curriculum over truncation step we are able to train learned optimizers that are faster in wall-clock time compared to existing optimizers. In this work, we focused on applying optimizers to a restricted family of tasks. While useful in its own right (e.g. rapid retraining of models on new data), future work will explore the limits of “no free lunch”  [@wolpert1997no] in the context of optimizers, to understand how and when learned optimizers generalize across tasks. We are also interested in using these methods to better understand what problem structure our learned optimizers exploit. By analyzing the trained optimizer, we hope to develop insights that may transfer back to hand-designed optimizers. Outside of meta-learning, we believe the outer-gradient estimator presented here can be used to train other long time dependence recurrent problems such as neural turning machines [@graves2014neural], or neural GPUs [@kaiser2015neural]. Much in the same way deep learning has replaced feature design for perceptual tasks, we see meta-learning as a tool capable of learning new and interesting algorithms, especially for domains with unexploited problem-specific structure. With better outer-training stability, we hope to improve our ability to learn interesting algorithms, both for optimizers and beyond. ### Acknowledgments {#acknowledgments .unnumbered} We would like to thank Madhu Advani, Alex Alemi, Samy Bengio, Brian Cheung, Chelsea Finn, Sam Greydanus, Hugo Larochelle, Ben Poole, George Tucker, and Olga Wichrowska, as well as the rest of the Brain Team for conversations that helped shape this work. Derivation of the unrolled gradient {#app:derivation} =================================== For the case of learning a learning rate, we can derive the unrolled gradient to get some intuition for issues that arise with outer-training. Here, the update rule is given by: $$w \leftarrow w - \theta \nabla \ell(w),$$ where $w$ are the inner-parameters to train, $\ell$ is the inner-loss, and $\theta$ is a scalar learning rate, the only outer-parameter. We use superscripts to denote the iteration, so $w^{(t)}$ are the parameters at iteration $t$. In addition, we use $g^{(t)} = \nabla \ell(w^{(t)})$ and $H^{(t)} = \nabla^2 \ell(w^{(t)})$ to denote the gradient and Hessian of the loss at iteration $t$, respectively. We are interested in computing the gradient of the loss after $T$ steps of gradient descent with respect to the learning rate, $\theta$. This quantity is given by $\frac{\partial d \ell}{\partial \theta} = \langle g^{(T)}, \frac{d w^{(T)}}{d \theta}\rangle $. The second term in this inner product tells us how changes in the learning rate affect the final parameter value after $T$ steps. This quantity can be defined recursively using the total derivative: $$\begin{aligned} \frac{d w^{(T)}}{d\theta} &=& \frac{\partial w^{(T)}}{\partial w^{(T-1)}} \frac{d w^{(T-1)}}{d\theta} - \frac{\partial w^{(T)}}{\partial \theta} \\ &=& \left (I - H^{(T-1)} \right)\frac{d w^{(T-1)}}{d\theta} - g^{(T-1)}\end{aligned}$$ By expanding the above expression from $t=1$ to $t=T$, we get the following expression for the unrolled gradient: $$\frac{d \ell(w^{(T)})}{d\theta} = \left\langle g^{(T)}, -\sum_{i = 0}^{T-1} \left(\prod_{j=i+1}^{T-1} (I - \theta H^{(j)}) \right) g^{(i)}\right\rangle.$$ This expression highlights where the exploding outer-gradient comes from: the recursive definition of $ \frac{d w^{(T)}}{d \theta} $ means that computing it will involve a product of the Hessian at every iteration. This expression makes intuitive sense if we restrict the number of unrolled steps to one. In this case, the unrolled gradient is the negative inner product between the current and previous gradients: $\frac{d \ell(w^{(T)})}{d\theta} = -\langle g^{(T)}, g^{(T-1)} \rangle$. This means that if the current and previous gradients are correlated (have positive inner product), then updating the learning rate in the direction of the negative unrolled gradient means that we should *increase* the learning rate. This makes sense as if the current and previous gradients are correlated, we expect that we should move faster along this direction. Exploding Gradients on 1D Loss Surfaces {#app:toy_lr} ======================================= In addition to varying momentum, shown in Figure \[fig:2minimum\] we also explore the effects of varying learning rate in Figure \[fig:2dmin\_lr\]. ![ Extension of \[fig:2minimum\] showing varying learning rate with a fixed momentum (top), and varying learning rate with no momentum (bottom). Outer-problem optimization landscapes can become increasingly pathological with increasing inner-problem step count. **(a)** A toy 1D inner-problem loss surface with two local minimum. Initial parameter value ($w^{(0)}$) is indicated by the star. **(b)** Final inner-parameter value ($w^{(T)}$) as a function of the number of inner problem training steps $T$. The top row SGD+momentum. Color denotes different values of the optimizer’s learning rate parameter. In both settings, low learning rates converge to the first local minimum at $w=0$. Slightly larger learning rates escape this minimum to settle at the global minimum ($w\approx3.5$). Even larger values (purples) oscillate before eventually diverging. **(c)** The final loss after some number steps of optimization as a function of the learning rate. Larger values of $T$ result in near discontinuous loss surfaces around the transition points between the two minima. \[fig:2dmin\_lr\] ](figures/2minimum_lr.pdf "fig:"){width="6.5in"} ![ Extension of \[fig:2minimum\] showing varying learning rate with a fixed momentum (top), and varying learning rate with no momentum (bottom). Outer-problem optimization landscapes can become increasingly pathological with increasing inner-problem step count. **(a)** A toy 1D inner-problem loss surface with two local minimum. Initial parameter value ($w^{(0)}$) is indicated by the star. **(b)** Final inner-parameter value ($w^{(T)}$) as a function of the number of inner problem training steps $T$. The top row SGD+momentum. Color denotes different values of the optimizer’s learning rate parameter. In both settings, low learning rates converge to the first local minimum at $w=0$. Slightly larger learning rates escape this minimum to settle at the global minimum ($w\approx3.5$). Even larger values (purples) oscillate before eventually diverging. **(c)** The final loss after some number steps of optimization as a function of the learning rate. Larger values of $T$ result in near discontinuous loss surfaces around the transition points between the two minima. \[fig:2dmin\_lr\] ](figures/2minimum_lr_nomom.pdf "fig:"){width="6.5in"} Outer-Training Algorithm {#app:algo} ======================== Initialize outer-parameters ($\theta$). Sample a dataset $\mathcal{D}$, from task distribution $\mathcal{T}$. Initialize the inner loop parameters $w^{{\left( 0 \right)}}$ randomly. Sample ES perturbation: $e\sim N(0, \sigma^2I)$. Sample a number of steps per truncation, $k$, based on current outer-training iteration. Compute a positive, and negative sequence starting from $w^{{\left( t \right)}}$ by iteratively applying (for $k$ steps), $u(\cdot; \theta+e)$, to $w^{{\left( t \right)}}$ Compute a pair of outer-objectives with both a positive, and negative antithetic sample ($L^+$, $L^-$) using the 2 sequences of $w$ from $t$ to $t+k$ using either the train or validation inner-problem data. Compute a single sample of $g^{rp} = \nabla_{\theta} \frac{1}{2}(L^++L^-)$. Compute a single sample of $g^{es} = \frac{1}{2}(L^+ - L^-)\nabla_{\theta}log(N(e;\theta,\sigma^2I))$ Store the sample of $(g_{rp}, g_{es})$ in a buffer until a batch of samples is ready. Assign the current inner-parameter $w$ from one of the two sequences with the inner-parameter value from the end of the truncation ($w^{{\left( t+k \right)}}$). When a batch of gradients is available, compute empirical variance and empirical mean of each weight for each estimator. Use equation \[eq:combine\] to compute the combined gradient estimate. Update outer-parameters with SGD: $\theta \leftarrow \theta - \alpha g_{combined}$ where $\alpha$ is a learning rate. Architecture details {#app:arch_details} ==================== Architecture ------------ In a similar vein to diagonal preconditioning optimizers, and existing learned optimizers our architecture operates on each parameter independently. Unlike other works, we do not use a recurrent model as we have not found applications where the performance gains are worth the increased computation. We instead employ a single hidden layer feed forward MLP with 32 hidden units. This MLP takes as input momentum terms at a few different decay values: \[0.5, 0.9, 0.99, 0.999, 0.9999\]. A similar idea has been explored in [@lucas2018aggregated]. The current gradient as well as the current weight value are also used as features (2 additional features). By passing in weight values, the optimizer can learn to do arbitrary norm weight decay. To emulate learning rate schedules, the current training iteration is fed in transformed via applying a tanh squashing functions at different timescales: $\text{tanh}(t/\eta-1)$ where $\eta$ is the timescale. We use 9 timescales logarithmicly spaced from (3, 300k). All non-time features are normalized by the second moment with regard to other elements in the “batch” dimension (the other weights of the weight tensor). We choose this over other normalization strategies (e.g. batch norm) to preserve directionality. These activations are then passed the into a hidden layer, 32 unit MLP with ReLU activations. Many existing optimizer hyperparameters (such as learning rate) operate on an exponential scale. As such, the network produces two outputs, and we combine them in an exponential manner: $\text{exp}(\lambda_{exp} o_1)\lambda_{lin}o_2$ making use of two scaling parameters $\lambda_{\text{exp}}$ and $\lambda_{\text{lin}}$ which are both set to $1e-3$. Without these scaling terms, the default initialization yields steps on the order of size 1 – far above the step size of any known optimizer and result in highly chaotic regions of $\theta$. It is still possible to optimize given our estimator, but training is slow and the solutions found are quite different. Code for this optimizer can be found at <https://github.com/google-research/google-research/tree/master/task_specific_learned_opt>. Inner-problem ------------- The optimizer targets a 3 layer convolutional neural network with 3x3 kernels, and 32 units per layer. The first 2 layers are stride 2, and the 3rd layer has stride 1. We use ReLU activations and glorot initializations [@glorot2010understanding]. At the last convolutional layer, an average pool is performed, and a linear projection is applied to get the 10 output classes. Outer-Training {#outer-training} -------------- We train using the algorithm described in Appendix \[app:algo\] using a linear schedule on the number of unrolling steps from 50 - 10k over the course of 5k outer-training iterations. To add variation in length, we additionally shift this length by a percentage uniformly sampled between (-20%, 20%). We optimize the outer-parameters, $\theta$, using Adam [@kingma2014adam] with a batch size of 128 and with a learning rate of 0.003 for the training outer-objective and 0.0003 for the validation outer-objective, and $\beta_1=0.5$(following existing literature on non-stationary optimization [@arjovsky2017wasserstein]). While both values of learning rate work for both outer-objectives, we find the validation outer-objective to be *considerably* harder, and training is more stable with the lower learning rate. Additional inner loop problem learning curves {#app:inner_loop_figure} ============================================= We plot additional learning curves from both the outer-train task distribution and the outer-validation task distribution. See Figure \[fig:unroll\_on\_problem\]. Additionally, we supply 4 additional baselines: RMSProp+Reg+Decay hyper parameter searched for validation and training loss over learning rate, learning rate schedule (both exponential and linear), epsilon, and l1/l2 regularization using 1000 random configurations, and SGDMom+Reg+Decay hyper parameter searched for validation and training loss over learning rate, learning rate schedule, momentum, to use Nesterov momentum, and l1/l2 regularization using 1000 random configurations. The search procedure and search space matches the existing Adam+Reg+Decay procedure described in \[sec:learned\_opt\_performance\]. ![Additional outer-validation problems. \[fig:my\_label\] ](figures/unroll_on_problems_test.pdf){width="6.5in"} ![Outer-training problems. \[fig:my\_label\] ](figures/unroll_on_problems_train.pdf){width="6.5in"} Out of domain generalization {#app:outofdist} ============================ In this work, we focus our attention to learning optimizers over a specific task distribution (3 layer convolutional networks trained on ten class subsets of 32x32 Imagenet). In addition to testing on these in domain problems (Appendix \[app:inner\_loop\_figure\]), we test our learned optimizer on a variety of out of domain target problems. Despite little variation in the outer-training task distribution, our models show promising generalization when transferred to a wide range of different architectures (fully connected, convolutional networks) depths (2 layer to 6 layer) and number of parameters (models roughly 16x more parameters). We see these as promising sign that our learned optimizer has a reasonable (but not perfect) inductive bias. We leave training with increased variation to encourage better generalization as an area for future work. {width="5.5in"} {width="5.5in"} ![ Inner problem: 6 convolutional layer network. 32 units per layer, strides: \[2,1,2,1,1,1\] with ReLU activations on 28x28 MNIST.](figures/mnist_conv_2x_imagenet.pdf){width="5.5in"} ![ Inner problem: 3 convolutional layer network. 32 units per layer, strides: \[2,2,1\] with ReLU activations on 28x28 MNIST.](figures/mnist_conv_same_as_imagenet.pdf){width="5.5in"} ![ Inner problem: 3 convolutional layer network. 128 units per layer, strides: \[2,2,1\] with ReLU activations trained on a ten-way classification sampled from 32x32 Imagenet (using holdout classes).](figures/conv_128_imagenet.pdf){width="5.5in"} Inner-loop training speed {#app:wallclock} ========================= When training models, often one cares about taking less wall-clock time as compared to loss decrease per weight update. Much like existing first order optimizers, the computation performed in our learned optimizer is linear in terms of number of parameters in the model being trained and smaller than the cost of computing gradients. The bulk of the computation in our model consists of two batched matrix multiplies of size *features*x32, and 32x2. When training models that make use of weight sharing, e.g. RNN or CNN, the computation performed per weight often grows super linearly with parameter count. As the learned optimizer methods are scaled up, the additional overhead in performing more complex weight updates will vanish. For the specific models we test in this paper, we measure the performance of our optimizer on CPU and GPU. We re-implement Adam, SGD, and our learned optimizer in TensorFlow (no fused ops) for this comparison. Given the small scale of problem we are working at, we implement training in graph in a *tf.while\_loop* to avoid TensorFlow Session overhead. We use random input data instead of real data to avoid any data loading confounding. On CPU the learned optimizer executes at 80 batches a second where Adam runs at 92 batches a second and SGD at 93 batches per second. The learned optimizer is 16% slower than both. On a GPU (Nvidia Titan X) we measure 177 batches per second for the learned and 278 batches per second for Adam, and 358 for sgd. This is or 57% slower than Adam and 102% slower than SGD. Overhead is considerably higher on GPU due to the increased number of ops, and thus kernel executions, sent to the GPU. We expect a fused kernel can dramatically reduce this overhead. Despite the slowdown in computation, the performance gains exceed the slowdown, resulting in an optimizer that is still considerably faster when measured in wall-clock time. For the wall-clock figures presented in this paper we rescale the step vs performance curves by the steps per second instead of directly measuring wall-clock time. When running evaluations, we perform extensive logging which dominates the total compute costs. Ablation learning curves {#app:ablation} ======================== ![Training curves for ablations described in [§\[sec:ablations\]]{}. The thick line bordered in black is the median performance, with the shaded region containing the 25% and 75% percentile. Thinner solid lines are individual runs. ](figures/ablation_all_updated.pdf){width="5.5in"} Additional Truncation Bias Experiments {#app:trunc_bias} ====================================== In [§\[sec:increase\_bias\]]{} we show the effect of truncation bias when learning Adam hyper parameters using the Adam outer-optimizer. When using truncated gradients, the directions passed into this outer-optimizer are not well behaved, and are not even guaranteed to be a conservative vector field. As such, different outer optimizers might behave differently. To test this, we test multiple configurations of outer-optimizer in figure \[fig:different\_opt\_trunc\]. ![Additional experiments showing truncation bias when attempting to learn Adam hyper-parameters with different outer-optimizers. We show 4 configurations of Adam (2 learning rates, with and without beta1), and 2 configurations of SGD (2 different learning rates) with outer-loss and learning rate plotted for each. In all experiments we see similar, and significant truncation bias when using low amounts of steps per truncation. For some of the higher learning rate experiments (e.g. SGD with lr=0.01), we see diverging loss.\[fig:different\_opt\_trunc\] ](figures/different_opt_trunc.pdf){width="5.5in"} Batch-Normed Base Model ======================= In this section, we present experiments targeting a different task family. In particular, we add batch normalization to the convolution layers. We employ the same meta-training procedure and same learned optimizer architecture as used in the rest of this paper and target the validation loss outer-objective. Meta-training curves can be found in Figure \[fig:bn\_curves\]. We find we outperform learning rate tuned Adam, but do not out perform the 8 parameter tuned Adam baseline. At this point, we are unsure the source of this gap but suspect hyper parameter tuning would improve this result. ![ Outer-training curves for 5 different random seeds. The learned optimizers outperform LR tuned Adam on 4/5 random seeds but do not yet out perform the 8 parameter Adam baseline. \[fig:bn\_curves\] ](figures/bn_training_curve2.pdf){width="3.5in"} Longer Unrolls -------------- All of the learned optimizers presented in this paper are outer-trained using 10k inner-iterations. In Figure \[fig:longer\_unroll\] we show an application of a learned optimizer outside of the outer-training regime – up to 100k inner iterations using the learned optimizer shown in orange in figure \[fig:bn\_curves\]. Unlike hand designed optimizers, our learned optimizer does not completely minimize the training loss. As a result, the test performance remains consistent far outside the outer-training regime. ![ Example unrolls on held out outer-tasks when inner-trained for 10x more inner steps. Each panel represents a different held out task. The dashed vertical line denotes the max number of steps seen at outer-training time. The solid line shows inner-test performance, where as the dashed denotes inner-train. We find that our learned optimizers keep a consistent test and train loss even after the 10k iterations used for outer-training. \[fig:longer\_unroll\] ](figures/longer_unroll.pdf){width="5.5in"} [^1]: [@parmas2018pipps] go on to propose a more sophisticated gradient estimator that operates on a per iteration level. While this should result in an even lower variance estimator in our setting, we find that the simpler solution of combing both terms at the end is easier to implement and works well in practice.

--- abstract: '[The hadron resonance gas (HRG) model with Tsallis distribution has been used to explain the energy dependence of the product of the moments, $S\sigma$ and $\kappa\sigma^2$ of net-proton multiplicity distributions of recently published STAR data at RHIC energies. While excellent agreements are found between model predictions and measurements of $S\sigma$ and $\kappa \sigma^2$ of most peripheral collisions and $S\sigma$ of most central collisions, the $\kappa \sigma^2$ of most central collisions deviates significantly from the model predictions particularly at $\sqrt{s_{NN}}=19.6$ GeV and $27$ GeV. This could be an indication of the presence of additional dynamical fluctuations not contained in the HRG-Tsallis model.]{}' author: - 'D. K. Mishra$^1$, P. Garg$^2$, P. K. Netrakanti$^1$, A. K. Mohanty$^1$' title: ' Net-baryon number fluctuations with HRG model using Tsallis distribution ' --- Recently, the STAR collaboration has reported the energy dependence of the moments (mean $M$, variance $\sigma$, skewness $S$ and kurtosis $\kappa$) and their products for net-proton multiplicity distribution at RHIC energies  [@Adamczyk:2013dal]. The product of the moments $S\sigma$ and $\kappa\sigma^{2}$ can be linked to the ratios of susceptibilities $(\chi)$ associated with the baryon number conservation [@Bazavov:2012vg; @Aggarwal:2010wy; @Gupta:2011wh]. For example, the product $S\sigma$ can be written as the ratio of third order ($\chi_B^3$) to second order ($\chi^2_B$) and the product $\kappa\sigma^{2}$ as the ratio of fourth order ($\chi_B^4$) to second order ($\chi_B^2$) baryon susceptibilities. The recent STAR measurements of $S\sigma$ and $\kappa\sigma^2$ show significant deviations from the predictions of the Skellam distribution (where $\kappa \sigma^2$ should be unity) at all energies indicating presence of large non-statistical fluctuations [@skellam]. The particle production in heavy ion collisions at relativistic energies are well described in terms of the hadron resonance gas (HRG) model where fermions and bosons follow Fermi-Dirac (FD) and Bose-Einstein (BE) distributions respectively. The success of HRG model would mean that thermal system which might have gone through a possible phase transition has (nearly) equilibrated both thermally and chemically at freeze out. It is believed that if the thermal system has retained some memory of the phase transition with finite correlation length at freeze out, it must be reflected in the higher moments of the conserved quantities although not so obvious in the study of thermal abundance of the individual species [@Karsch:2010ck; @Stephanov:2011pb; @Cheng:2008zh]. Therefore, the study of fluctuations in various conserved quantities like: net-charge, net-strangeness and net-baryon number through the higher moments using HRG model is expected to provide a base line to observe the deviation in experimental observables, which may indicate the presence of non statistical fluctuations, if any. The HRG model in Boltzmann approximation follows an exponential behavior of particle production corresponding to Boltzmann-Gibbs (BG) statistics. Recently, it has been realized that particle production both in heavy ion and proton-proton collisions at RHIC and LHC energies can be described successfully using a power law type distribution rather than the exponential one [@Tang:2008ud; @Wong:2012zr; @Cleymans:2011in]. Therefore, the Tsallis distribution function is being used for particle production with nonextensive parameter $q$ such that it approaches Boltzmann distribution in the limit $q\rightarrow 1$. In the context of particle production in heavy ion collisions, Tsallis distribution has been interpreted as the superposition of Boltzmann distributions with different temperatures [@Wilk:1999dr]. Such a situation can occur when the hadronic fireball is not homogeneous in temperature $T$ but fluctuates from point to point around some equilibrium value $T_f$  [@Wilk:2012zn]. The temperature fluctuation which exists in small part of the phase space with respect to the whole is another source of non-statistical fluctuation. This is different from the statistical fluctuations of event by event type and should be properly accounted in the model. In general, Tsallis nonextensive statistics is supposed to include all situations characterized by long range interactions, long range microscopic memory and space time fractal structure of the process [@tsallis; @Biro:2005uv]. Therefore, we consider a hadron resonance gas model using Tsallis nonextensive distribution to study fluctuations of net-baryon production in Au+Au collisions at RHIC energies. In the limit $q\rightarrow 1$, we recover the HRG results. In the present work, we show that the HRG-Tsallis model with a temperature dependent nonextensive $q$ parameter reproduces the energy dependence of $S\sigma$ and $\kappa\sigma^2$ for most peripheral collisions as well as $S\sigma$ for central collisions. However, the energy dependence of $\kappa\sigma^2$ of central collision deviate significantly from the HRG-Tsallis model predictions particularly at energies $19.6$ GeV and $27$ GeV. We argue here that the predictions of HRG-Tsallis characterized by a temperature dependent $q$ parameter should be taken as the base line to study (experimentally) fluctuations of dynamical origin if any, which is still not contained in the Tsallis non-extensive thermodynamics. The Tsalli’s form of FD and BE distribution can be written as, $$f = \frac{1}{\mathrm{exp}_{q}\frac{(E-\mu)}{T}\pm 1}\, \label{eqnt}$$ where ”+” and ”-” signs are used for fermions and bosons respectively and $\mathrm{exp}_{q}(x)$ is given by, $$\exp_q(x) = \left\{ \begin{array}{l l} \left[1+(q-1)x\right]^{1/(q-1)}&~~\mathrm{if}~~~x > 0 \\ \left[1+(1-q)x\right]^{1/(1-q)}&~~\mathrm{if}~~~x \leq 0 \\ \end{array} \right. \label{eq:tsallis}$$ where $x=(E-\mu)/T$. The above distribution approaches standard FD and BE distributions in the limit $q\rightarrow 1$. Using above nonextensive distribution function, the average number density can be written as, $$\begin{aligned} n_q &=& \sum_i g_{i} X_i\int\frac{d^3k}{(2\pi)^3} f_i^q(E_i,T_f,\mu_i), \label{nq0} \end{aligned}$$ where $T_f$ is the chemical freeze-out temperature, $\mu_{i}$ is the chemical potential and $g_i$ is the degeneracy factor of the $i^{th}$ particle. The total chemical potential $\mu_{i}$ = $B_{i}\mu_{B}$ + $Q_{i}\mu_{Q}$ + $S_{i}\mu_{S}$, where $B_{i}$, $Q_{i}$ and $S_{i}$ are the baryon, electric charge and strangeness number of the $i^{th}$ particle, with corresponding chemical potentials $\mu_{B}$, $\mu_{Q}$ and $\mu_{S}$, respectively. The term $X_i$ represents either $B$, $Q$ or $S$ of the $i^{th}$ particle depending on whether the computed $n_q$ represents baryon density, electric charge density or strangeness density respectively [@Cleymans:2011in]. Note that the exponent $q$ in $f_i$ has been introduced as a constraint for thermodynamical consistency [@Cleymans:2012ya]. The factor $d^3k$ can be expressed in terms of phase space variables, $d^{3}k=p_T\sqrt{p_T^2 + m^2}\cosh\eta~d{p_{T}} d\eta d\phi$ where energy $E$ is expressed as, $ E= \sqrt{p_T^2 + m^2} \cosh\eta$. Since the average density and the pressure $P$ are shown to be thermodynamically consistent i.e. $n_q=\frac{\partial P}{\partial \mu}$, we can define a generalized susceptibility as, $$\begin{aligned} \chi_{q}^{n} &=& \frac{\partial^n [P(T_f,\mu)]}{\partial \mu^n}|_T= \frac{\partial^{n-1} [n_q(T_f,\mu)]} {\partial \mu^{n-1}}. \label{eq:chi_q}\end{aligned}$$ Using Eq.\[nq0\] and Eq.\[eq:chi\_q\], we have estimated $S\sigma$ ($\chi^3/\chi^2$) and $\kappa\sigma^2$ ($\chi^4/\chi^2$) for net-baryon density within STAR detector acceptance. For comparison with experiments, it would have been more appropriate to estimate net-proton density which includes primordial as well as yields from resonance decays. However, we have noticed from an earlier study that within STAR acceptance, the difference between two methods are negligible (less than $2\%$) [@Garg:2013ata]. Therefore, in the present study we consider net-baryon density within STAR acceptance. We parametrized the freeze-out temperatures and chemical potentials using the relations, $\mu_B(\sqrt{s_{NN}}) $= $\frac{d}{1+e\sqrt{s_{NN}}}$ and $T(\mu_B)$ = a - b$\mu_B^2$ - c$\mu_B^4$. The parameters are taken from Ref. [@Cleymans:2005xv] for most central collision and are listed in Table \[tab:mut\_cent\]. For peripheral collision, we extract these parameters from STAR preliminary data [@Das:2012yq] for most peripheral (70-80)% centrality collisions and the extracted parameters are listed in Table \[tab:mut\_peri\]. We use similar parametrization for $\mu_S$ and $\mu_Q$ as that of $\mu_B$ and the corresponding parameters are listed in the table. We set $\mu_Q$ to zero for peripheral collision as its contribution is very small as compared to $\mu_B$ and $\mu_S$. The non-extensive parameter $q$ characterizes the degree of non-equilibrium in the system which depends on both collision energy as well as on the centrality of collisions. In fact, $q$ depends on $T_f$, $\mu$ as well as on the type of identified particles [@Urmossy:2012pb]. From the previous study of STAR data, it is noticed that $q$ value decreases with increasing centrality indicating an evolution from a highly non-equilibrated system towards a more thermalized one [@Tang:2008ud]. It is found that $(q-1)$ has a parabolic dependence on temperature [@Wilk:1999dr]. Similarly, from the analysis of SPS and RHIC data , it is noticed that $q$ value is maximum at SPS energy of 80 $A$GeV (which corresponds to 12.3 GeV in center of mass energy) and decreases on either side i.e. at 158 $A$GeV as well as at 40 $A$GeV [@Cleymans:2008mt]. The $q$ value is also found nearly unity at 200 GeV RHIC energy. In term of freeze out temperature, these two observations suggest that the $q$ parameter may have a maximum value at a particular temperature $T_m$ and will decrease on either side. The maximum temperature $T_m$ is fixed at $0.154$ GeV corresponding to the freeze out temperature (for central collision) at $\sqrt {s_{NN}}$=12.3 GeV which corresponds to 80 $A$GeV SPS collision energy. Therefore, we use the parametrization $q= 1+ [\alpha (T_{0} - T_f )]^{1/2}$ for $T_m \le T_f \le T_0$ and allow it to decrease to unity linearly with a slope $(q_m-1)/(T_m-T_l)$ for $T_f < T_m$ where $q_m$ is the $q$ value at $T_m$. We set $q=1$ for $T\le T_l$. In the above, $T_{0}$ is a reference temperature fixed at $0.167$ GeV which is close to the freeze-out temperature for $\sqrt {s_{NN}}=200$ GeV so that $q=1$ for $T_f >T_0$. The parameter $T_l$ is taken $0.120$ GeV corresponding to $\sqrt {s_{NN}}=5$ GeV. The inset in Fig.\[fig1\], shows a typical dependence of $q$ on freeze-out temperature. The parameter $\alpha$ is kept free and is adjusted to fit the data. When $\alpha=0$ ($q\rightarrow 1$), we get back the HRG results. Note that we use the same parametrization irrespective of whether the collision is central or peripheral except different centrality will have different chemical freeze-out parameters. ![(Color online) The energy dependence of product of moments, $S\sigma$ and $\kappa\sigma^2$ of net-baryon distribution calculated using standard HRG (solid curve) and Tsallis-HRG with $\alpha=0.015$ (dotted curve). The filled circles are STAR data points for most peripheral collision corresponding to $(70-80)\%$ collision centrality in Au+Au collisions [@Adamczyk:2013dal]. The bottom panel shows the ratio of data and our model calculation for $S\sigma$ and $\kappa\sigma^2$. The figure inset at the top shows the temperature dependence of the $q$ parameter.[]{data-label="fig1"}](netbar_peri.eps) ![(Color online) The energy dependence of product of moments, $S\sigma$ and $\kappa\sigma^2$ of net-baryon distribution calculated using standard HRG (solid curve) and Tsallis-HRG with $\alpha=0.015$ (dotted curve). The dot-dashed curve shows the corresponding values for $\alpha=0.3$. The filled circles are STAR data points for most central collision corresponding to $(0-5)\%$ collision centrality in Au+Au collisions [@Adamczyk:2013dal]. The bottom panel shows the ratio of data to model calculation for $S\sigma$ only. []{data-label="fig2"}](netbar_cent.eps) Figure \[fig1\], shows the energy dependence of $S\sigma$ (top panel) and $\kappa\sigma^2$ (middle panel) estimated using parameters as listed in Table \[tab:mut\_peri\] for most peripheral (70-80)% centrality collisions. As can be seen, the HRG results ($\alpha$ = 0) do not explain the experimental values [@Adamczyk:2013dal], and $\kappa\sigma^2$ in HRG model is always close to unity where as data points are about $20\%$ below the HRG values. On the other hand, HRG-Tsallis with a nominal $\alpha=0.015$ (corresponds to a maximum $q$ value of $1.02$) can explain both $S\sigma$ and $\kappa\sigma^2$ extremely well. The excellent agreement between data and model predictions can be seen from the ratio plot in the bottom panel of Fig.\[fig1\]. The corresponding results for most central (0-5)% centrality collisions are shown in Fig.\[fig2\]. Also for the central collisions, the HRG predictions do not explain the experimental data. It is also interesting to note that $\alpha=0.015$ explains energy dependence of $S\sigma$ rather well except at $200$ GeV (Although small, the disagreement between measurement and prediction for $S\sigma$ is also found for peripheral collisions at $200$ GeV). However, the model fails to explain $\kappa\sigma^2$ as the deviations are more significant for lower energy data ($\sqrt {s_{NN}} <$ 39 GeV). We have also increased $\alpha$ from $0.015$ to $0.3$ which results in the dashed-dot curves in Fig.\[fig2\]. Although use of a higher $\alpha$ parameter can explain the general energy dependence of $\kappa\sigma^2$, it fails to explain the $S\sigma$ data. In short, it is not possible to explain both $S\sigma$ and $\kappa\sigma^2$ simultaneously by varying $\alpha$. In the frame work of Tsallis non-extensive thermodynamics, the definition of generalized susceptibility as defined in Eq.\[eq:chi\_q\] can be interpreted as follows. Assuming $f^q \approx\mathrm{exp}_q$(x) which is true when $q$ is close to 1 (maximum $q$ value used in this work is about $1.02$), the Tsallis-Boltzmann distribution Eq.\[eq:tsallis\] can be written as superposition of Boltzmann distribution and the average baryon number density as $$\begin{aligned} n_{q}^{B}(T_f,\mu)=\int_0^\infty f(T,T_f,q) n^{B}(T,\mu) d(1/T), \label{eq:m_q}\end{aligned}$$ where the weight factor $f(T,T_f,q)$ represents a gamma distribution as defined in [@Wilk:1999dr] and $n^{B}$ is the average number density obtained using Boltzmann statistics. Using above relation, Eq.\[eq:chi\_q\] can be expressed as weighted sum of the susceptibilities estimated using standard Boltzmann statistics and can be interpreted as an average taken over the whole phase space which is inhomogeneous in temperature. Probably, this is what will be measured experimentally if the temperature fluctuation exists in the hadronizing medium corresponding to a nonextensive value of $q$ which deviates from unity. Therefore, it is reasonable to argue that Eq.\[eq:chi\_q\] can be used to estimate temperature average higher moments. It may be mentioned here that the above argument is strictly correct when quantum statistics is not important and Tsallis distribution can be written as the superposition of Boltzmann distributions. This is mostly true for baryons where quantum statistics are not important. It may be mentioned here that, we have not considered the van der Waals (VDW) type excluded volume effect which could be another source of deviation of $S\sigma$ and $\kappa\sigma^2$ from the HRG predictions [@Fu:2013gga]. However, the VDW type deviation increases with decreasing energies where as the STAR experimental values have maximum deviation at $\sqrt{s_{NN}}=19.6$ GeV and less deviation at the two lowest RHIC energies. Therefore, we have not considered VDW type effect in the present calculation. The transport model like UrQMD within STAR acceptance also produces similar results for $S\sigma$ and $\kappa\sigma^2$ which decrease with decreasing energies. In the frame work of grand canonical HRG model, the $n$-$th$ order cumulant for net-proton density can be written as $C_n=C_n^1+(-1)^n C_n^2$ where $C_n^1$ and $C_n^2$ are the $n$-$th$ cumulants for proton and anti-proton respectively. This is the same expression which has been used in Ref. [@Adamczyk:2013dal] except for the fact that proton and anti-proton productions are independent. Since $C_n^1$ and $C_n^2$ are taken from the experiments, it is not surprising that the independent production model explains the STAR data. However, in the HRG model, the proton and anti-proton productions are not independent rather constraints through baryon number conservation. Further, we would like to add here that the present HRG-Tsallis model is different from the grand canonical HRG model as $C_n^i$ in our model represents a weighted average over the distribution of temperatures. The primary motivation of the STAR measurements of $S\sigma$ and $\kappa \sigma^2$ is to look for the existence of a critical end point in the QCD phase diagram where the product of these moments may show large non monotonic variations [@Stephanov:2008qz]. The Tsallis distribution also includes quantum effect which is important at lower collision energies. Therefore, in this work we explored, how much non-statistical fluctuations inherently present in the Tsallis non-extensive distribution, which can explain the experimental observations without considering other physics effects. Another important aspect which has not been considered here is the effect of non-extensivity on the freeze-out parameters which are generally extracted from the experiments using HRG model. It is observed in Ref.[@Cleymans:2008mt] that while freeze-out temperature decreases, chemical potential increases with increasing $q$ parameter. However, for $q<1.02$ which has been used in the present study, we notice that the increase in $\mu_B$ is not significant although $T_f$ decreases by about $10\%$ from the value when $q=1$. As argued in [@Cleymans:2008mt], since Tsallis distribution is broader than the Boltzmann distribution, temperature needs to be decreased in order to conserve the particle density. Therefore, we have estimated the moments keeping $\mu_B$ unchanged but allowing freeze-out temperature to decrease up to $10\%$. Interestingly, we notice that while $S\sigma$ increases slightly, $k\sigma^2$ remains unchanged. This suggests that $S\sigma$ is sensitive to both temperature and chemical potential while $\kappa \sigma^2$ is sensitive to $q$ parameter. In conclusion, we have studied the energy dependence of the fluctuations of net-baryon productions through higher moments namely $S\sigma$ and $\kappa\sigma^2$ using HRG with Tsallis non-extensive distribution function. When the non-extensive parameter $q$ is close to unity, the moments obtained using HRG-Tsallis model can be interpreted as the weighted average of the moments estimated using many Boltzmann distribution corresponding to the distribution of temperatures over the whole phase space. It is shown that this HRG-Tsallis model can explain the energy dependence of $S\sigma$ and $\kappa\sigma^2$ measurements of recently published STAR data corresponding to the most peripheral collisions which is otherwise impossible to explain using normal HRG model which predicts $\kappa\sigma^2$ close to unity for net-baryon productions. The HRG-Tsallis model also explains the energy dependence of $s\sigma$ data of central collision. However, the model can not explain the corresponding $k\sigma^2$ values of the central collision particularly at energies $19.6$ GeV and $27$ GeV. This deviation is so significant that it is an indication of the presence of additional fluctuations at around $20$ GeV which may have some dynamical origin not contained in the Tsallis HRG model. $a$ (GeV) $b$ (GeV$^{-1}$) $c$ (GeV$^{-3}$) ----------- -- ------------------- -- ------------------- ---------------------- -- T 0.166 $\pm$ 0.002 0.139 $\pm$ 0.016    0.053 $\pm$ 0.021 $d$ (GeV) $e$ (GeV$^{-1}$) $\mu_{B}$ 1.308$\pm 0.028$ 0.273$\pm 0.008$ $\mu_{S}$ 0.214 0.161 $\mu_{Q}$ 0.0211 0.106 : Parametrization of chemical potentials and freeze-out temperature for most central collisions taken from [@Cleymans:2005xv]. \[tab:mut\_cent\] $a$ (GeV) $b$ (GeV$^{-1}$) $c$ (GeV$^{-3}$) ----------- -- ------------------- -- ------------------- --------------------- -- T 0.158 $\pm$ 0.002 0.159 $\pm$ 0.034   0.500 $\pm$ 0.001 $d$ (GeV) $e$ (GeV$^{-1}$) $\mu_{B}$ 0.900 $\pm$ 0.059 0.251 $\pm$ 0.008 $\mu_{S}$ 0.239 $\pm$ 0.001 0.300 $\pm$ 0.001 : Parametrization of chemical potentials and freeze-out temperature extracted from the STAR experimental data for (70-80)% centrality [@Das:2012yq]. \[tab:mut\_peri\] Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank B. 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Introduction and Main Result ============================ The Navier-Stokes equations in two space dimensions are particularly well-studied and the existence of a unique, global strong solution is well-known. Perturbations with a Gaussian noise are also covered, for example using the weak martingale or the variational approach, cf. [@FlaGatMartingale; @LiuGeneralCoercive]. For an overview on randomly forced $2$D fluids we refer to [@KuksinLecture]. However, all these results need a smooth noise which does not include the case of so-called space-time white noise. Such a perturbation has some technical drawbacks but also a very reasonable legitimation. It has been observed in several articles [@AlbeverioGibbsMeasureOlder; @AlbeverioGibbsMeasureOld; @AlbeverioGibbsMeasure] that the periodic Euler flow on the torus ${{\mathbb T}}^2$, which is the inviscid limit of the $2$D-Navier-Stokes equations, has a family of infinitesimally invariant measures ${\mu_{\sigma,\nu}}$, the so-called enstrophy measures. These are exactly the unique invariant measures of the Ornstein-Uhlenbeck processes corresponding to the purely linear problem, i.e. omitting the convection term, and is explicitly given as an infinite product measure. However, such a rough noise has technical drawbacks in terms of regularity issues making a pathwise interpretation difficult. For example the Ornstein-Uhlenbeck process mentioned above takes values in a Sobolev space of negative order, hence merely distributions, since the convolution with the Stokes semigroup is not regularizing enough. The nonlinear problem is not expected to have more regularity, thus the main difficulty is the appropriate definition of the convection term for such distributions. In [@DPD2DNavierStokes], Da Prato and Debussche prove the existence of a strong solution (in the probabilistic sense) with values in a certain Besov space of full measure ${\mu_{\sigma,\nu}}$ for every initial condition within that space. Moreover, uniqueness is proven using an additional condition involving the stationary Ornstein-Uhlenbeck process. The problem with the convection term is tackled with a so-called renormalization technique. The, in some sense, unnatural notion of uniqueness is improved by Albeverio and Ferrario in [@AlbFerUniqueness2DNSE] to a pathwise uniqueness result in the same space, where existence holds. In this article we are concerned with the associated Kolmogorov operator to these equations and its Cauchy problem in $L^1({\mu_{\sigma,\nu}})$. This is related to the uniqueness of the corresponding Martingale problem, in particular a weaker formulation concerning stationary solutions. We use the concept of $L^1$-uniqueness, i.e. the closure of the Kolmogorov operator (with appropriate domain) generates a $C_0$-semigroup on $L^1({\mu_{\sigma,\nu}})$. $L^1$-uniqueness for the stochastic Navier-Stokes equations perturbed by space-time white noise has been shown by Stannat in [@Stannat2DNSE] for large viscosity $\nu$. Similar, but weaker results have been obtained chronologically in [@FlandoliGozzi; @AlbFerUniquenessGenerator; @StannatRegularizedEuler; @AlbFerUniquenessGenerator2]. The regularity issues from above translate into poor support properties of ${\mu_{\sigma,\nu}}$ containing only distributions. This implies a poor convergence of the Galerkin approximations of the convection term, which is the major difficulty in this approach. Despite the vast literature on $2$D fluids, such equations are not a very realistic setting. In most cases, they are used as an example of approximations for fluid flows, where the vertical length scale is negligible compared to the horizontal ones. Such applications often appear in the studies of atmospheric or oceanic flows. On these huge length scales, the rotation of the earth cannot be neglected and fictitious forces appear in the equations. The fictitious forces concerning rotation are the centrifugal force and the Coriolis force. We incorporate these forces to obtain a toy model for geophysical flows. One of the intriguing observations is that the additional forces still have the same invariants, i.e. energy and enstrophy, thus supposedly keep the enstrophy measure as an invariant measure. We consider the following equations for the velocity field $u$ and hydrodynamic pressure $\pi$ on the two-dimensional torus ${{\mathbb T}}^2 \df (0,2\pi)^2$ with periodic boundary conditions. [equation]{}\[eq:2DSNSCE\] \_t u &= u - (u )u- l e\_3 u - + &\[0,) [[T]{}]{}\^2,\ u &= 0 &\[0,) [[T]{}]{}\^2,\ u(0)&= u\_0 & [[T]{}]{}\^2, where $\eta$ is the so-called space-time white noise. The centrifugal force is of gradient type and can be hidden in the pressure $\pi$, whereas the Coriolis force is modeled in the so-called $\beta$-plane model, for a motivation we refer to [@Pedlosky]. In this model $l = \omega + \beta \xi_2$ with $\omega, \beta > 0$ denoting the angular velocity and its fluctuation around the equatorial line. Here, some part of the earth’s surface is approximated by a rectangle and $\xi_2$ denotes the longitudinal component. We set $e_3 \times u = u^\bot$ where $u^\bot = (-u_2, u_1)^T$. As usual we consider this equation in the function space $$\textstyle H \df \Big\{ u \in L^2({{\mathbb T}}^2;\R^2): \Div u = 0, \int u(\xi)\dxi = 0, u\cdot n \text{ is periodic}\Big\},$$ where $n$ denotes the outward normal. The abstract evolution equation on $H$ is obtained after applying the (orthogonal) Helmholtz projection $\mathcal{P}: L^2({{\mathbb T}}^2;\R^2) \to H$. This equation is given by [equation]{}\[eq:SNSE-SDE\] u(t) &= (A u(t) - B(u(t)) - C(u(t)))+\ u(0)&= u\_0 H, where $A \df \mathcal{P}\Delta$ is the Stokes operator, $B(u) \df B(u,u) \df \mathcal{P}(u \cdot \nabla)u$ and $C(u) \df \mathcal{P} l u^\bot$. The noise is represented by a cylindrical Wiener process $W(t)$ on $H$. We then consider the Kolmogorov operator associated to defined by $$\label{def:Kolmogorov2DSNSCE} \big({K_{\sigma,\nu}}\phi\big)(u) = \frac{\sigma^2}{2} \tr \big(D^2 \phi(u)\big) + \scp{\nu A u - B(u) - C(u)}{D\phi(u)}$$ for $\phi \in \mathcal{F}C^2_b$, the space of all cylindrical functions on $H$. In detail $${\mathcal{F}C_b}^m := \Big\{ \phi (u) = {\tilde{\phi}}(u_{k_1}, \dots, u_{k_n}) : n\in \N, {\tilde{\phi}}\in C_b^m(\R^n), u_{k_i} = \scp{u}{e_{k_i}}\Big\}.$$ In particular, we are interested in the the well-posedness of its Cauchy problem in $L^1({\mu_{\sigma,\nu}})$. These enstrophy measures are explicitly known and given by $${\mu_{\sigma,\nu}}\df \mathcal{N} \Big(0, \tfrac{\sigma^2}{2\nu} A^{-1}\Big),$$ i.e. the invariant measures of the linear problem with drift $\nu A$ and diffusion $\sigma I$. Due to the invariance of the enstrophy for both vector fields $B$ and $C$, these measures are indeed infinitesimally invariant for ${K_{\sigma,\nu}}$, in formula $\int {K_{\sigma,\nu}}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}= 0$ for all $\phi \in {\mathcal{F}C_b}^2$, and thus a reasonable candidate for a reference measure, see Section 3 for more details. In order to state the assumptions for our main result we need the following notation. Let $S(s) := \sum_{k \in {\Z^2_\ast}} \abs{k}^{-2 s}$ denote the value of the convergent infinite series for $s > 1$. \[assum:ViscosityCondition\] Assume that $\sigma, \nu > 0$ satisfy $\nu^3 > 40 S(2) \pi^{-2} \sigma^2$. With the assumption above, the main result of this article is stated as follows. \[thm:UniquenessCoriolis\] Let ${\mu_{\sigma,\nu}}$ be the Gaussian measure related to the enstrophy, then $({K_{\sigma,\nu}}, {\mathcal{F}C_b}^2)$ is dissipative in $L^1({\mu_{\sigma,\nu}})$, hence closable. Now suppose Assumption \[assum:ViscosityCondition\] holds. Then the operator $({K_{\sigma,\nu}}, {\mathcal{F}C_b}^2)$ is $L^1$-unique. In particular, the closure $(\overline{K}_{\sigma,\nu}, D(\overline{K}_{\sigma,\nu}))$ generates a $C_0$-semigroup of contractions $P_t$ in $L^1({\mu_{\sigma,\nu}})$ and ${\mu_{\sigma,\nu}}$ is invariant for $P_t$. Such a result has some implications concerning uniqueness of the associated martingale problem. In particular, the semigroup $P_t$ is Markovian and yields the transition probabilities of a stationary martingale solution of . We refer to the monograph [@Eberle] for a detailed discussion on this subject. Furthermore, we can obtain the following corollary to Theorem \[thm:UniquenessCoriolis\] for the system without rotation. Set $\omega = \beta = 0$, i.e. the reference frame is fixed. Then, under Assumption \[assum:ViscosityCondition\], the operator $({K_{\sigma,\nu}}, {\mathcal{F}C_b}^2)$ is $L^1$-unique. Although we are able to extend the result of [@Stannat2DNSE] regarding a smaller lower bound for $\nu$, the limit for small viscosity parameter still remains a challenge. This lower bound for $\nu$ is due to the techniques used in the proofs that, more or less, absorb the nonlinear contributions of the generator ${K_{\sigma,\nu}}$ by the Stokes part. Note that in the pathwise formulation [@AlbFerUniqueness2DNSE; @DPD2DNavierStokes] this assumption is not needed, thus it is somehow artificial. The proof of Theorem \[thm:UniquenessCoriolis\] is contained in the following sections. At first, we derive a spectral representation of which together with the product structure of ${\mu_{\sigma,\nu}}$ favors the use of finite dimensional spectral Galerkin approximations of ${K_{\sigma,\nu}}$. Sharp convergence results for the approximated vector fields $B$ and $C$ are contained in Lemmas \[lem:energyB\] and \[lem:energyC\] and in particular the definition of $B(u)$ and $C(u)$ via an $L^2$-limit for distributions $u$ from the support of ${\mu_{\sigma,\nu}}$. The integration by parts formula in Lemma \[lem:PartialIntegrationGaussian\] is essential for the main ingredient of proof, the a priori gradient estimate for the solution of the finite dimensional resolvent problem for ${K_{\sigma,\nu}}$ in Proposition \[prop:W1+sEstimate\]. We seize this idea due to Stannat in [@Stannat2DNSE], however suitable modifications for both the convection and Coriolis term are necessary. Note that this a priori estimate is not uniform in the approximation but introduces some logarithmic growth that is sufficiently small. With this approach we are able to weaken the smallness condition on the viscosity $\nu$ to some extent. A Spectral Representation ========================= In the following, we expand the vector field $u$ into its Fourier series. We use the complete orthonormal system of $H$ given by $$e_k(\xi) \df \frac{1}{\sqrt{2}\pi} \frac{k^\bot}{\abs{k}} \phi_k(\xi) \quad\text{with}\quad \phi_k(\xi) \df \begin{cases} \sin(k \xi), \quad k \in \Z^2_+,\\ \cos(k \xi), \quad k \in -\Z^2_+,\\ \end{cases}$$ where $\Z^2_+ \df \{ k \in \Z^2: k_1 > 0 \text{ or } (k_1 =0 \text{ and } k_2>0)\}$. Furthermore, let ${\Z^2_\ast}\df \Z \setminus \{0\} = \Z^2_+ \cup -\Z^2_+$. It is an easy task to verify that $(e_k)\subset H$ and that these are eigenvectors of the Stokes operator $A$ with corresponding eigenvalues $-\abs{k}^2$. In addition to that, we can write the cylindrical Wiener process as a formal sum $W(t) = \sum_{k \in {\Z^2_\ast}} \beta_k(t) e_k$ with a family $\beta_k$ of independent real-valued Brownian motions. Expanding w.r.t. the orthonormal system $(e_k)$ yields a spectral representation of , which is $$\label{eq:specNSCE} \mathrm{d} u_k(t) = \Big(-\nu \abs{k}^2 u_k(t) - B_k\big(u(t)\big)-C_k\big(u(t)\big) \Big)\dt + \sigma \,\mathrm{d}\beta_k(t), \quad k \in {\Z^2_\ast}.$$ In a similar fashion the associated Kolmogorov operator reads as $$\big({K_{\sigma,\nu}}\phi\big)(u) = \frac{\sigma^2}{2} \sum_{k \in {\Z^2_\ast}} {\partial_k}^2 \phi(u) - 2 \bigl(\nu \abs{k}^2 u_k + B_k(u) + C_k(u) \bigr) {\partial_k}\phi(u).$$ Here and in the following, we denote by ${\partial_k}$ the derivative w.r.t. the $u_k$ variable. It remains to identity the Fourier coefficients of the convection and the Coriolis term, compare [@FlandoliGozzi] for similar calculations concerning the convection term. One easily verifies that $$\scp{B(e_l, e_m)}{e_k} = - \scp{B(e_l,e_k)}{e_m} = \frac{\sqrt{2}}{4 \pi} \frac{(k^\bot \cdot l) (k \cdot m)}{\abs{k} \abs{l} \abs{m}} \frac{1}{\pi^2} \int_{{{\mathbb T}}^2} \bigl( \phi_{-k} \phi_l \phi_m \bigr) (\xi) \dxi \fd \beta^k_{l,m},$$ hence $$\label{eq:fourierB} B_k(u) \df \scp{B(u)}{e_k} = \sum_{l,m \in {\Z^2_\ast}} u_l u_m \scp{B(e_l, e_m)}{e_k} = \sum_{l,m \in {\Z^2_\ast}} \beta^k_{l,m} u_l u_m$$ The integral in the definition of $\beta^k_{l,m}$ essentially yields some Kronecker deltas, $\delta_k = 1$ if $k=0$ or $\delta_k = 0$ otherwise. In detail $$\delta_{k,l,m} \df \frac{1}{\pi^2} \int_{{{\mathbb T}}^2} \bigl( \phi_{-k} \phi_l \phi_m \bigr) (\xi) \dxi = \begin{cases} \delta_{k-l-m} - \delta_{k-l+m} - \delta_{k+l-m}, & k,l,m \in {\Z^2_+},\\ \delta_{k+l-m} + \delta_{k-l+m} + \delta_{k+l+m}, & k \in {\Z^2_+}, l,m \in - {\Z^2_+},\\ \delta_{k-l-m} - \delta_{k-l+m} - \delta_{k+l+m}, & k,l \in -{\Z^2_+}, m \in {\Z^2_+},\\ \delta_{k-l-m} - \delta_{k+l-m} - \delta_{k+l+m}, & k,m \in -{\Z^2_+}, l \in {\Z^2_+},\\ 0 &\text{otherwise.} \end{cases}$$ By similar calculations we can obtain the formula for the Coriolis forcing term. Note that only the second summand remains after applying the Helmholtz projection since $\mathcal{P} u^\bot = 0$. Analogue to the above, we see that $$C_k(u) \df \scp{C(u)}{e_k} = \beta \sum_{l \in {\Z^2_\ast}} u_l \frac{k^\bot l}{\abs{k}\abs{l}} \frac{1}{2\pi^2}\int_{{{\mathbb T}}^2} \xi_2 \phi_k(\xi) \phi_l(\xi) \dxi$$ and with straightforward calculations conclude that $$\label{eq:fourierC} C_k(u) = -\beta \sum_{l \in {\Z^2_\ast}} u_l \frac{k^\bot l}{\abs{k}\abs{l}} \frac{1}{k_2+l_2}\delta_{k_1+l_1} (1 - \delta_{k_2+l_2}) = -\beta \sum_{l \in {\Z^2_\ast}} \gamma^k_l u_l.$$ Furthermore, let us introduce some notation on the function spaces used in this article. With the complete orthonormal system $(e_k)$ the periodic, divergence free Sobolev spaces following [@BerghLoefstroem] can be identified with $$H^s := \Biggl\{ u \in \R^{{\Z^2_\ast}}: \snorm{u}{s}^2 := \sum_{k \in {\Z^2_\ast}} \abs{k}^{2 s} u_k^2 < \infty \Biggr\}, \quad s \in \R, \quad \text{ with }H^0 = H.$$ Recall the complex interpolation of these Sobolev spaces which states that for $s_0 < s < s_1$ and $u \in H^{s_1}$ it holds that $$\label{eq:Interpolation} \norm{u}_s \leq \norm{u}_{s_0}^{\frac{s_1-s}{s_1-s_0}} \norm{u}_{s_1}^{\frac{s-s_0}{s_1-s_0}}.$$ The Gaussian Invariant Measure Given by the Enstrophy ===================================================== In the coordinates of $(e_k)$ the measure ${\mu_{\sigma,\nu}}$ is simply an infinite product of centered Gaussian measures on $\R$, i.e. $${\mu_{\sigma,\nu}}= \mathcal{N} \Bigl(0, \tfrac{\sigma^2}{2 \nu} A^{-1} \Bigr) = \bigotimes_{k \in {\Z^2_\ast}} \mathcal{N} \Bigl(0, \tfrac{\sigma^2}{2 \nu \abs{k}^2} \Bigr).$$ This measure is usually called the enstrophy measure, because the enstrophy associated to the vector field $u$ appears in the exponent of the heuristic density of ${\mu_{\sigma,\nu}}$. It is well-known that $H$ does not have full measure w.r.t. ${\mu_{\sigma,\nu}}$ and one even has ${\mu_{\sigma,\nu}}(H) = 0$, cf. [@AlbeverioGibbsMeasureOlder]. This is due to the elementary calculation $\int u_k^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) = \frac{\sigma^2}{2 \nu \abs{k}^2}$, hence for $S_n(s) = \frac{2\nu}{\sigma^2} \sum_{k=1}^n \abs{k}^{-2s} u_k^2$ follows $$\lim_{n \to \infty} \int S_n(s) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) = \lim_{n \to \infty} \sum_{k=1}^n \abs{k}^{-2-2s} < \infty$$ if and only if $s > 0$. This implies ${\mu_{\sigma,\nu}}(H^{-s}) = 1$ if and only if $s>0$. The support of ${\mu_{\sigma,\nu}}$ also contains all Sobolev spaces of negative order with any integrability parameter $1 \leq p < \infty$ and the Besov spaces $B^{-s}_{pq}$ for all $s >0$, $2 \leq p \leq q < \infty$, see [@AlbFerUniqueness2DNSE] for detailed computations. As mentioned before, it has been shown that this measure is infinitesimally invariant for the Euler flow, see [@AlbeverioGibbsMeasure], and also invariant for the Ornstein-Uhlenbeck process $$\mathrm{d} u(t) = \nu A u(t) \dt + \sigma \dwt,$$ see for example [@dpzErgo Theorem 6.2.1]. In the following, we want to prove that ${\mu_{\sigma,\nu}}$ is in fact infinitesimally invariant for $({K_{\sigma,\nu}}, {\mathcal{F}C_b}^2)$. This is mostly due to the two invariants $\scp{B(u)+C(u)}{u} = 0$ and $\scp{B(u) + C(u)}{Au} = 0$ for all smooth $u \in H$, see for example [@Titi]. This implies, at least for $u$ with $u_k \neq 0$ only for a finite number of $k \in {\Z^2_\ast}$, that $$\label{eq:InvarianceBC} \sum_{k \in {\Z^2_\ast}} \abs{k}^{2 i} (B_k (u) + C_k(u)) u_k = 0, \quad i=0,1.$$ Now fix a function $\phi \in {\mathcal{F}C_b}^2$, hence there exists $k_1, \dots, k_n$, $n \in \N$ such that $\phi$ has an admissible representative ${\tilde{\phi}}$ in $C_b^2(H_n)$ and for $\tilde{u} = (u_{k_1}, \dots, u_{k_n})$ follows $$\label{eq:InvarianceBCproof} \int {K_{\sigma,\nu}}\phi (u) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) = \sum_{i=1}^n \int \bigl( B_{k_i}(u) + C_{k_i}(u) \bigr) \partial_{k_i} {\tilde{\phi}}(\tilde{u}) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u).$$ On the right hand side we do an integration by parts w.r.t. the Gaussian density, more precisely we have the following lemma. \[lem:PartialIntegrationGaussian\] Let $\phi \in {\mathcal{F}C_b}^1$. Then $$\int {\partial_k}\phi (u) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) = \frac{2 \nu}{\sigma^2} \abs{k}^2 \int u_k \phi (u) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u),$$ in particular $$\int {\partial_k}\phi (u) (B_k (u) + C_k(u)) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) = \frac{2 \nu}{\sigma^2} \abs{k}^2 \int u_k \phi (u) (B_k(u) + C_k(u)) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u).$$ We can use the product structure of the measure ${\mu_{\sigma,\nu}}$ and obtain $$\begin{aligned} \int {\partial_k}\phi (u) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) &= \int {\partial_k}{\tilde{\phi}}(\tilde{u}) \, \mathrm{d} \Bigl( \bigotimes_{i=1}^n \mathcal{N} \bigl(0, \tfrac{\sigma^2}{2\nu \abs{k_i}^2}\bigr) \Bigr).\\ \intertext{If there exists $i^\ast$ with $k_{i^\ast} = k$ we do an integration by parts in this coordinate.} &= \int \int_{-\infty}^\infty {\partial_k}{\tilde{\phi}}(\tilde{u}) e^{- \frac{\nu \abs{k}^2 u_k^2}{\sigma^2}} \, \mathrm{d}u_k \, \mathrm{d} \Bigl( \bigotimes_{i \neq i^\ast} \mathcal{N} \bigl(0, \tfrac{\sigma^2}{2\nu \abs{k_i}^2}\bigr) \Bigr)\\ &= - \int \int_{-\infty}^\infty {\tilde{\phi}}(\tilde{u}) \Bigl( -\frac{2 \nu \abs{k}^2}{\sigma^2} u_k \Bigr) e^{- \frac{\nu \abs{k}^2 u_k^2}{\sigma^2}} \, \mathrm{d}u_k \, \mathrm{d} \Bigl( \bigotimes_{i \neq i^\ast} \mathcal{N} \bigl(0, \tfrac{\sigma^2}{2\nu \abs{k_i}^2}\bigr) \Bigr)\\ &= \frac{2 \nu \abs{k}^2}{\sigma^2} \int u_k \phi (u) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u).\end{aligned}$$ Equations and imply that ${\partial_k}B_k = 0$, ${\partial_k}C_k = 0$ for all $k \in {\Z^2_\ast}$, hence the second assertion follows easily if we replace $\phi$ by $\phi (B_k + C_k)$. Going back to , we can apply this integration by parts formula and obtain $$\int {K_{\sigma,\nu}}\phi (u) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) = \sum_{i=1}^n \frac{2 \nu \abs{k_i}^2}{\sigma^2} \int u_{k_i} \bigl( B_{k_i}(u) + C_{k_i}(u) \bigr) {\tilde{\phi}}(\tilde{u}) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u).$$ Furthermore, we can use the invariance for $i=1$, which reads as $\sum_{i=1}^n u_{k_i} ( B_{k_i}(u) + C_{k_i}(u) ) = 0$ pointwise for all $u$ with $u_k = 0$ if $k \notin \{ k_1, \dots, k_n \}$. We conclude that $$\label{eq:InvarianceKol} \int {K_{\sigma,\nu}}\phi(u) {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) = 0 \quad \text{for all } \phi \in {\mathcal{F}C_b}^2.$$ Since the invariant measure is a product measure, it is reasonable to use the usual finite dimensional spectral Galerkin approximations in order to obtain an approximating equation for . Define $I_n := \{ k \in {\Z^2_\ast}: \abs{k} \leq n \}$ and let $\iota_n$, $\pi_n$ be the canonical embedding and projection from and onto the subspace $H_n := \spann \{ e_k: k \in I_n \}$, respectively. Associated to $\iota_n$ and $\pi_n$ define $B^n(u) \df \sum_{k \in I_n} B_k^n(u)e_k$ and $C^n(u) \df \sum_{k \in I_n} C_k^n(u) e_k$ with $B_k^n(u) \df \sum_{l,m \in I_n} \beta^k_{l,m} u_l u_m$ and $C_k^n(u) \df -\beta \sum_{l \in I_n} \gamma^k_l u_l$, respectively. The approximating Kolmogorov operator ${K_{\sigma,\nu}}^n$ is defined in the canonical way by replacing all parts by the approximations $${K_{\sigma,\nu}}^n \phi(u) = \frac{\sigma^2}{2} \sum_{k \in I_n} {\partial_k}^2 \phi(u) - 2 \bigl(\nu \abs{k}^2 u_k + B^n_k(u) + C^n_k(u) \bigr) {\partial_k}\phi(u).$$ As a suitable domain we consider $C_b^2(H_n)$. It is clear that ${\mu_{\sigma,\nu}}^n := {\mu_{\sigma,\nu}}\circ \pi_n^{-1}$ is infinitesimally invariant for $({K_{\sigma,\nu}}^n, C_b^2(H_n))$. For the proof of Theorem \[thm:UniquenessCoriolis\] it is essential in which sense $B^n$ and $C^n$ converge to $B$ and $C$. In the next two lemmas we obtain convergence in $L^2({\mu_{\sigma,\nu}}; H^{-s})$ for $s>1$ and $s>0$, respectively. As a byproduct, this allows to define the unique measurable extensions of the vector fields $B$ and $C$ to $L^2({\mu_{\sigma,\nu}})$, i.e. an extension for distributions $u \in H^{-s}$ given any $s > 0$. Moreover, $L^2({\mu_{\sigma,\nu}})$-convergence implies ${\mu_{\sigma,\nu}}$-a.s. convergence along some subsequence, hence the limits $B(u)$ and $C(u)$ are in fact elements of $H^{-s}$ for $s>1$ and $s>0$, respectively. \[lem:energyB\] Let $\sigma, \nu > 0$ be arbitrary. Then, $\snorm{B(u)}{-s} \in L^2({\mu_{\sigma,\nu}})$ if and only if $s >1$. In particular, for all $0<\eps < \min \{s-1, 1\}$ it holds that $$\int \snorm{\pi_n (B - B^n)(u)}{-s}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) \leq c \log (n) n^{-2\eps} \xrightarrow{n \to \infty} 0$$ with a constant $c$ uniform in $n$. Moreover, $B(u)$ is an element of $H^{-s}$ for ${\mu_{\sigma,\nu}}$-a.e. u. The first and last part of the statement have already been considered in the literature, see e.g. [@AlbFerUniqueness2DNSE Proposition 3.2]. A crucial part in the proof is the dependence of $\int \abs{B_k(u)}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u)$ on the index $k$. On can show that it is of order $\log (\abs{k})$. Essential for us however, is the explicit convergence rate of the approximations as $n \to \infty$. Note that $$\snorm{\pi_n (B - B^n)(u)}{-s}^2 = \sum_{k \in I_n} \abs{k}^{-2s} \abs{B_k(u) - B_k^n(u)}^2,$$ thus we have to consider the difference of the $k$th Fourier coefficients in $L^2({\mu_{\sigma,\nu}})$ for $k \in I_n$. Straightforward calculations yield $$\begin{aligned} &\int \abs{B_k(u) - B_k^n(u)}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) = \sum_{\{l,m \in I_n\}^C} \bigl((\beta^k_{l,m})^2 + \beta^k_{l,m} \beta^k_{m,l} \bigr) \int u_l^2 u_m^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u)\\ &\quad= \frac{\sigma^4}{\nu^2} \sum_{\{l,m \in I_n\}^C} \bigl((\beta^k_{l,m})^2 + \beta^k_{l,m} \beta^k_{m,l} \bigr) \abs{l}^{-2} \abs{m}^{-2} \leq c \abs{k}^2 \sum_{l \in {\Z^2_\ast}, \abs{l}>n} \frac{1}{\abs{l}^2 \abs{k-l}^2}, \end{aligned}$$ with a uniform constant $c$ independent of $k$ and $\{l,m \in I_n\}^C \df \{ l,m \in {\Z^2_\ast}\setminus I_n\} \cup \{l \in I_n, m\in {\Z^2_\ast}\setminus I_n\} \cup \{ l \in {\Z^2_\ast}\setminus I_n, m \in I_n \}$. We can bound this sum by an integral and make explicit calculations. At first, let $\abs{l} > 2n > 2\abs{k}$, then this part of the sum is bounded up to a uniform constant by $$\int_{ \{y \in \R^2: \abs{y} > 2n \}} \frac{1}{\abs{y}^2 \abs{k-y}^2}\dy \leq 8 \pi \int_{2n}^\infty \frac{1}{r^3}\dr = \frac{\pi}{n^2}.$$ We do the same for the part where $n < \abs{l} < 2n$: $$\int_{ \{y \in \R^2: n< \abs{y} < 2n \}} \frac{1}{\abs{y}^2 \abs{k-y}^2}\dy \leq \frac{8 \pi}{n^2} \int_{\frac12}^{3n} \frac{1}{r}\dr \leq \frac{c}{n^2} \log (n).$$ Now choose $0< \eps < \min \{s-1,1\}$ and estimate $n^{-2} \leq n^{-2\eps} \abs{k}^{2\eps-2}$, hence $$\int \snorm{\pi_n (B - B^n)(u)}{-s}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) \leq c \sum_{k \in I_n} \abs{k}^{2-2s} \frac{\log (n)}{n^2} \leq c \sum_{k \in I_n} \abs{k}^{-2s+2\eps} \frac{\log (n)}{n^{2\eps}}.$$ The sum is convergent as $n \to \infty$ therefore the statement is proven. \[lem:energyC\] Let $\sigma, \nu > 0$ be arbitrary. Then, $\snorm{C}{-s} \in L^2({\mu_{\sigma,\nu}})$ if and only if $s > 0$. In particular, there exists a constant $c$ uniform in $n$ such that $$\int \snorm{\pi_n (C - C^n)(u)}{-s}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) \leq c n^{-1} \xrightarrow{n\to \infty} 0.$$ Moreover, $C(u)$ is an element of $H^{-s}$ for ${\mu_{\sigma,\nu}}$-a.e. u. The first and last part are stated for a similar presentation to Lemma \[lem:energyB\]. However, $C$ is only linear in $u$, hence these points are obvious and one can easily verify that $\int \abs{C_k^n(u)}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u) \leq c \abs{k}^{-2}$ with a constant $c$ uniform in $n$. This immediately yields the summability in $H^{-s}$ for $s>0$. The convergence rate can be obtained similar to Lemma \[lem:energyB\]. For $k \in I_n$ $$\begin{aligned} &\int \abs{C_k(u) - C_k^n(u)}^2 d{\mu_{\sigma,\nu}}(u) = \beta^2 \sum_{l \in {\Z^2_\ast}\setminus I_n} \big(\gamma_l^k\big)^2 \int u_l^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}(u)\\ &\quad= \frac{\beta^2 \sigma^2}{\nu} \sum_{l \in {\Z^2_\ast}\setminus I_n} \frac{(k^\bot l)^2}{\abs{k}^2 \abs{l}^4} \frac{1}{(k_2 + l_2)^2} \delta_{k_1+l_1} ( 1-\delta_{k_2+l_2}) = \frac{\beta^2 \sigma^2}{\nu} \frac{k_1^2}{\abs{k}^2} \sum_{j \in \Z, j^2 > n^2-k_1^2} \frac{1}{(k_1^2 + j^2)^2}.\end{aligned}$$ The sum is again bounded up to a uniform constant by $$\int^\infty_{\sqrt{n^2-k_1^2}} \frac{1}{(k_1^2 + y^2)^2}\dy \leq \frac{1}{n^2} \int^\infty_{\sqrt{n^2-k_1^2}} \frac{1}{(k_1^2 + y^2)}\dy \leq \frac{1}{n^2 \abs{k_1}}.$$ Therefore we have $$\int \abs{C_k(u) - C_k^n(u)}^2 d{\mu_{\sigma,\nu}}(u) \leq \frac{c}{n^2 \abs{k}} \leq \frac{c}{n \abs{k}^2}$$ and this proves the lemma. Estimates for the Solution of the Resolvent Problem =================================================== In this section we prove integrated gradient estimates for the solution of the resolvent problem $(\lambda - {K_{\sigma,\nu}}^n) \psi = \phi$, $\lambda > 0$. Lemma \[lem:energyB\] suggests that these a priori estimates have to be done in the space $H^{1+s}$, $s>0$. To simplify notation, we introduce the spaces $W^{1,2}_s$ as the closure of ${\mathcal{F}C_b}^1$ in $L^2({\mu_{\sigma,\nu}})$ w.r.t. the bilinear form $$\mathcal{E}^s(\phi,\psi) \df \sum_{k \in {\Z^2_\ast}} \abs{k}^{2s} \int {\partial_k}\phi {\partial_k}\psi {\, \mathrm{d}{\mu_{\sigma,\nu}}}, \quad \phi, \psi \in {\mathcal{F}C_b}^1$$ and denote by $\snorm{\cdot}{W^{1,2}_s}$ the corresponding norm. We will use this norm for functions on $H_n$ via the canonical embedding $\iota_n$ without explicit mention. The following proposition is a conclusion of the results in [@StannatDirichlet] and yields a first a priori estimate. \[prop:L1UniqueFinite\] The closure $(\overline{K}_{\sigma,\nu}^n, D(\overline{K}_{\sigma,\nu}^n))$ of $({K_{\sigma,\nu}}^n, C_b^2(H_n))$ in $L^1({\mu_{\sigma,\nu}}^n)$ generates a Markovian $C_0$-semigroup of contractions $(\overline{T}_t^n)_{t \geq 0}$. Thus, the operator $({K_{\sigma,\nu}}, C_b^2(H_n))$ is $L^1$-unique. Moreover, $$D\big(\overline{K}_{\sigma,\nu}^n\big)_b \df D\big(\overline{K}_{\sigma,\nu}^n\big) \cap L^\infty\big({\mu_{\sigma,\nu}}^n\big) \subset \pi_n\big(W^{1,2}_0\big)$$ and $$\label{eq:L2estimate} \frac{\sigma^2}{2} \sum_{k \in I_n} \int \abs{{\partial_k}\phi}^2 d{\mu_{\sigma,\nu}}^n \leq - \int \overline{K}_{\sigma,\nu}^n \phi \phi d{\mu_{\sigma,\nu}}^n, \quad \phi \in D\big(\overline{K}_{\sigma,\nu}^n\big)_b.$$ [@StannatDirichlet Theorem I.1.5] implies the existence of a closed extension on $L^1({\mu_{\sigma,\nu}}^n)$ generating a sub-Markovian semigroup of contractions $(\overline{T}_t^n)_{t \geq 0}$. In particular $D(\overline{K}_{\sigma,\nu}^n)_b \subset D(\mathcal{E}^0|_{H_n}) = \pi_n(W^{1,2}_0)$ and inequality holds. By [@StannatDirichlet Proposition I.1.10] the measure ${\mu_{\sigma,\nu}}^n$ is $(\overline{T}_t^n)_{t \geq 0}$-invariant because $B_k^n$ and $C_k^n \in L^1({\mu_{\sigma,\nu}}^n)$ and therefore $\overline{T}_t^n 1 = 1$ holds. Hence the semigroup is Markovian and [@StannatDirichlet Corollary I.2.2] implies $L^1$-uniqueness. Inequality implies the following a priori estimate for the corresponding resolvent ${\overline{R}_{\sigma,\nu}^n(\lambda)}= (\lambda - \overline{K}_{\sigma,\nu}^n)^{-1}$, $\lambda >0$. \[cor:W0estimate\] Let $\psi \in \mathcal{B}_b(H_n)$ and $\lambda >0$. Then ${\overline{R}_{\sigma,\nu}^n(\lambda)}\psi \in D(\overline{K}_{\sigma,\nu}^n)_b$ and $$\snorm{{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}{W^{1,2}_0}^2 \leq \frac{2}{\lambda \sigma^2} \snorm{\psi}{L^\infty}.$$ Clearly ${\overline{R}_{\sigma,\nu}^n(\lambda)}\psi \in D(\overline{K}_{\sigma,\nu}^n)$ because $\rg ({\overline{R}_{\sigma,\nu}^n(\lambda)}) \subset D(\overline{K}_{\sigma,\nu}^n)$ and of course $\mathcal{B}_b(H_n) \subset L^1({\mu_{\sigma,\nu}}^n)$. Furthermore, the boundedness follows from the Markovianity of $\lambda {\overline{R}_{\sigma,\nu}^n(\lambda)}$. Therefore, we can use and conclude $$\begin{aligned} &\frac{\sigma^2}{2} \sum_{k \in I_n} \int \abs{{\partial_k}{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}^2 d{\mu_{\sigma,\nu}}^n + \lambda \int \abs{{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}^2 d{\mu_{\sigma,\nu}}^n\\ &\quad\leq - \int \overline{K}_{\sigma,\nu}^n {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi d{\mu_{\sigma,\nu}}^n + \lambda \int \abs{{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}^2 d{\mu_{\sigma,\nu}}^n\\ &\quad= \int \psi {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi d{\mu_{\sigma,\nu}}^n \leq \snorm{\psi}{L^\infty} \snorm{{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}{L^\infty} \leq \frac{1}{\lambda} \snorm{\psi}{L^\infty}^2.\qedhere\end{aligned}$$ However, this integrated gradient estimate is not enough to show $L^1$-uniqueness of the operator $({K_{\sigma,\nu}}, \mathcal{F}C_b^2)$ and we need the following improvement which is the essential part of the proof of Theorem \[thm:UniquenessCoriolis\]. \[prop:W1+sEstimate\] Suppose Assumption \[assum:ViscosityCondition\] holds. Let $s \in (0,1]$ and $\psi \in C_b^1(H_n)$. Then there exists $\delta > 0$ and $c(\delta)$ independent of $n$ such that $$\snorm{{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}{W^{1,2}_{1+s}}^2 \leq \frac{1}{4 \delta \lambda} \snorm{\psi}{W^{1,2}_{s}}^2 + c(\delta) \big(\log (n)\big)^{\frac{1+s}{s}} \snorm{\psi}{L^\infty}^2.$$ The proof of this proposition will be divided into several technical lemmas. The first one is standard and identifies the commutator of $D$ and ${K_{\sigma,\nu}}^n$. \[lem:Commutator\] Let $s \in \R$ and $\phi \in C_b^3(H_n)$. Then $$\begin{aligned} &\sum_{k \in I_n}{\kern -3pt} \abs{k}^{2s} {\kern -3pt}\int{\kern -3pt} {\partial_k}({K_{\sigma,\nu}}^n \phi) {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n = -\frac{\sigma^2}{2} {\kern -3pt}\sum_{k,l \in I_n}{\kern -3pt} \abs{k}^{2s} {\kern -3pt}\int{\kern -3pt} \abs{{\partial_l}{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n -\nu {\kern -2pt}\sum_{k \in I_n}{\kern -3pt} \abs{k}^{2+2s} {\kern -3pt}\int{\kern -3pt} \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\\ &\quad - \sum_{k,l \in I_n} \abs{k}^{2 s} (\beta^l_{\pm k \pm l,k} + \beta^l_{k,\pm l \pm k}) \int u_{\pm l \pm k} {\partial_l}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + \beta \sum_{k,l \in I_n} \abs{k}^{2 s} \gamma^l_k \int {\partial_l}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\end{aligned}$$ The useful terms are both negative summands on the right hand side. The second one is exactly the one needed for the gradient estimate in Proposition \[prop:W1+sEstimate\] and small viscosity $\nu$ results in worse estimates. Let $k \in I_n$ be fixed, then $$\big({\partial_k}{K_{\sigma,\nu}}^n \phi\big)(u) = \big({K_{\sigma,\nu}}^n {\partial_k}\phi\big)(u) - \nu \abs{k}^2 {\partial_k}\phi(u) - \sum_{l \in I_n} {\partial_k}\big( B_l^n(u) + C_l^n(u) \big) {\partial_l}\phi(u).$$ The derivatives of the Fourier coefficients of $B$ and $C$ can be given explicitly. $$\begin{aligned} {\partial_k}B_l^n(u) =& {\partial_k}\Big(\sum_{i,j \in I_n} \beta^l_{i,j} u_i u_j \Big) = \sum_{i \in I_n} \big(\beta^l_{i,k} + \beta^l_{k,i}\big) u_i\\ = &\big(\beta^l_{l-k,k} + \beta^l_{k,l-k}\big) u_{l-k} + \big(\beta^l_{k-l,k} + \beta^l_{k,k-l}\big) u_{k-l} \\ &+ \big(\beta^l_{k+l,k} + \beta^l_{k,k+l}\big) u_{k+l} + \big(\beta^l_{-k-l,k} + \beta^l_{k,-k-l}\big) u_{-k-l}\end{aligned}$$ as long as all indices are in $I_n$ and ${\partial_k}C_l^n(u) = -\beta \gamma^l_k$. The identity ${K_{\sigma,\nu}}^n (\psi^2) = 2 \psi {K_{\sigma,\nu}}^n(\psi) + \sigma^2 \sum_{l \in I_n} \abs{{\partial_l}\psi}^2$ for all $\psi \in C_b^2(H_n)$ together with the invariance of ${\mu_{\sigma,\nu}}^n$ implies $$\sum_{k \in I_n} \abs{k}^{2s}\int {K_{\sigma,\nu}}^n({\partial_k}\phi) {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n = - \frac{\sigma^2}{2} \sum_{k,l \in I_n} \abs{k}^{2s} \int \abs{{\partial_l}{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n.$$ Consequently, for any $s \in \R$ $$\begin{aligned} &\sum_{k \in I_n} \abs{k}^{2s} \int {\partial_k}({K_{\sigma,\nu}}^n \phi) {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n = -\frac{\sigma^2}{2} \sum_{k,l \in I_n} \abs{k}^{2s} \int \abs{{\partial_l}{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\\ &\qquad - \nu \sum_{k \in I_n} \abs{k}^{2+2s} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n - \sum_{k,l \in I_n} \abs{k}^{2s} \int ( {\partial_k}B_l^n(u) + {\partial_k}C_l^n(u) ) {\partial_l}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n.\qedhere\end{aligned}$$ In the course of the proof of Proposition \[prop:W1+sEstimate\] we will replace $\phi$ by the resolvent. In particular, the additional commutator terms have to be estimated in terms of the two negative ones. Because of its linear structure, the Coriolis term is easier to handle and we get an estimate independent of $\sigma$ and $\nu$. One key tool is the following. Let $\phi \in {\mathcal{F}C_b}^1$ and $0 < s_0 < 1 + s$. Then, for any $\delta > 0$ $$\label{eq:SobolevInterpolation} \log (n) \snorm{D \phi}{s_0}^2 \leq \delta \snorm{D\phi}{1+s}^2 + c(s, s_0, \delta) \big(\log (n)\big)^{\frac{1+s}{1+s-s_0}} \snorm{D\phi}{0}^2.$$ This relation follows from the interpolation inequality applied pointwise for fixed $u$ and Young’s inequality. \[lem:CommCoriolis\] Let $s \in (0,1]$ and $\phi \in C_b^1(H_n)$. Then, for every $\delta > 0$ there exists $c(\delta) < \infty$ independent of $n$ such that $$\beta \sum_{k,l \in I_n} \abs{k}^{2s} \gamma_k^l \int {\partial_l}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n \leq \delta \sum_{k \in I_n} \abs{k}^{2 + 2s} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + c(\delta) \sum_{k \in I_n} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n.$$ Obviously, it holds that $$\beta \sum_{k,l \in I_n} \abs{k}^{2s} \gamma_k^l \int {\partial_l}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n \leq \beta \sum_{k,l \in I_n} \frac{\abs{k}^{2s}}{\abs{k+l}} \delta_{k_1 + l_1} (1 - \delta_{k_2 + l_2}) \int \abs{{\partial_l}\phi} \abs{{\partial_k}\phi} {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n.$$ Note that the constraints on the indices imply $\abs{k+l} \neq 0$ and also yield a summation over only a one dimensional subset of ${\Z^2_\ast}$. With $\abs{k}^s \leq \abs{k+l}^s + \abs{l}^s$ for $s \in (0,1]$ follows $$\frac{\abs{k}^{2s}}{\abs{k+l}} \leq \frac{\abs{k}^s}{\abs{k+l}^{1-s}} + \frac{\abs{k}^s \abs{l}^s}{\abs{k+l}}$$ and Young’s inequality with $p=q=2$ implies $$\begin{aligned} \frac{\abs{k}^{2s}}{\abs{k+l}} \abs{{\partial_l}\phi} \abs{{\partial_k}\phi} &\leq \frac12 \frac{\abs{k}^{\frac32 + 2s}}{\abs{k+l}^{1-s} \abs{l}^{\frac32}} \abs{{\partial_k}\phi}^2 + \frac12 \frac{\abs{l}^{\frac32}}{\abs{k+l}^{1-s} \abs{k}^{\frac32}} \abs{{\partial_l}\phi}^2\\ &\quad+ \frac12 \frac{\abs{k}^{1 + 2s}}{\abs{k+l} \abs{l}} \abs{{\partial_k}\phi}^2 + \frac12 \frac{\abs{l}^{1+2s}}{\abs{k+l}\abs{k}}\abs{{\partial_l}\phi}^2.\end{aligned}$$ For fixed $k$ and $l$, all denominators are summable in $l$ and $k$, respectively. Thus, we just derived $$\sum_{k,l \in I_n} \frac{\abs{k}^{2s}}{\abs{k+l}} \delta_{k_1 + l_1} (1 - \delta_{k_2 + l_2}) \abs{{\partial_l}\phi} \abs{{\partial_k}\phi}\leq c \Bigl( \snorm{D{\tilde{\phi}}}{\frac34 + s}^2 + \snorm{D\phi}{\frac34}^2 + \snorm{D{\tilde{\phi}}}{\frac12 + s}^2 \Bigr),$$ where ${\tilde{\phi}}\in {\mathcal{F}C_b}^1$ is the extension of $\phi$ via $\iota_n$ to ${\mathcal{F}C_b}^1$. An application of the interpolation inequality yields the desired result. Of course, we want to achieve a similar result for the convection term. The critical step is the integration by part formula in Lemma \[lem:PartialIntegrationGaussian\]. This eliminates $u_{\pm k \pm l}$ but yields a second derivative of $\phi$. \[lem:CommConvection\] Let $s \in (0,1]$ and $\phi \in C_b^2(H_n)$. Then, for every $\eps, \delta > 0$ there exists $c(\eps,\delta)< \infty$ independent of $n$ such that $$\begin{aligned} &\sum_{k,l \in I_n} \abs{k}^{2 s} (\beta^l_{\pm k \pm l,k} + \beta^l_{k,\pm l \pm k}) \int u_{\pm l \pm k} {\partial_l}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n \leq 4 \eps \sigma^2 \sum_{k,l \in I_n} \abs{k}^{2s} \int \abs{{\partial_l}{\partial_k}\phi} {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\\ &\qquad + c(\eps, \delta) \big(\log (n)\big)^{\frac{1+s}{s}} \sum_{k \in I_n} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + \frac{\sigma^2}{\nu^2}\frac{5(\delta + S(2))}{ \pi^2 \eps} \int \sum_{k \in I_n} \abs{k}^{2+2s} \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n.\end{aligned}$$ Recall Lemma \[lem:PartialIntegrationGaussian\] and apply this to the convection part of the commutator which yields $$\int u_{\pm k \pm l} {\partial_l}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n(u) = \frac{\sigma^2}{\nu \abs{ k \pm l}^2} \int \bigl(\partial_{\pm k \pm l} {\partial_l}\phi\bigr) {\partial_k}\phi + {\partial_l}\phi \bigl(\partial_{\pm k \pm l} {\partial_k}\phi \bigr) {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n.$$ As a next step we need an estimate for $\abs{ \beta^l_{\pm l \pm k,k} + \beta^l_{k,\pm l \pm k}}$, namely $$\frac{\sqrt{2}}{4 \pi} \Biggl\lvert \frac{\pm (l^\bot\cdot (l \pm k)) ( l \cdot k)}{\abs{l}\abs{l \pm k}\abs{k}} \delta_{l, \pm l \pm k, k}+ \frac{\pm (l^\bot\cdot k) ( l \cdot (l \pm k))}{\abs{l}\abs{l \pm k}\abs{k}} \delta_{l,k, \pm l \pm k}\Biggr\rvert \leq \frac{\sqrt{2}}{2 \pi} \abs{l}.$$ Combining the last two estimates yields the following upper bound, $$\begin{aligned} &\sum_{k,l \in I_n} \abs{k}^{2 s} (\beta^l_{\pm k \pm l,k} + \beta^l_{k,\pm l \pm k}) \int u_{\pm l \pm k} {\partial_l}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\\ &\quad\leq \frac{\sigma^2}{\nu} \frac{\sqrt{2}}{2\pi} \sum_{k,l \in I_n} \abs{k}^{2 s} \frac{\abs{l}}{\abs{\pm k \pm l}^2}\int \bigl( \abs{\partial_{\pm k \pm l} {\partial_l}\phi} \abs{{\partial_k}\phi} + \abs{{\partial_l}\phi} \abs{ \partial_{\pm k \pm l} {\partial_k}\phi}\bigr)d{\mu_{\sigma,\nu}}^n.\end{aligned}$$ The main task is to control all second derivatives such that they vanish in the final estimate. This is similar to the proof of Lemma \[lem:CommCoriolis\] for the Coriolis part and for a shorter notation we consider only the case $\pm k \pm l = l-k$ in the following. The other three cases are done in the same way. We also have to remark, that we cannot use the estimates in [@Stannat2DNSE] to derive the result, since our modified proof involves the a priori estimate from Corollary \[cor:W0estimate\] and some logarithmic growth in $n$. It is matched by the sharp convergence results in Lemmas \[lem:energyB\] and \[lem:energyC\]. Essentially, we have to take care of two terms. The first one is estimated as follows, using $\abs{k}^s \leq \abs{l}^s + \abs{l-k}^s$ for $s \in (0,1]$. $$\frac{\sigma^2}{\nu} \frac{\sqrt{2}}{2 \pi} \sum_{k,l \in I_n} \frac{\abs{k}^{2s} \abs{l}}{\abs{l-k}^2} \abs{\partial_{l-k} {\partial_l}\phi} \abs{{\partial_k}\phi} \leq \frac{\sigma^2}{\nu} \frac{\sqrt{2}}{2\pi} \sum_{k,l \in I_n} \frac{\abs{k}^s \abs{l}}{\abs{l-k}^2} \bigl( \abs{l}^s + \abs{l-k}^s \bigr) \abs{\partial_{l-k} {\partial_l}\phi} \abs{{\partial_k}\phi}$$ An application of Young’s inequality with $p=q=2$ and a coefficient $\eps > 0$ together with $\abs{l}^2 \leq 2\abs{k}^2 + 2 \abs{l-k}^2$ yields $$\begin{aligned} &\leq \frac{\eps}{2} \sigma^2 \sum_{k,l \in I_n}\abs{l}^{2s} \abs{\partial_{l-k} {\partial_l}\phi}^2 + \frac{\eps}{2} \sigma^2 \sum_{k,l \in I_n}\abs{l-k}^{2s} \abs{\partial_{l-k} {\partial_l}\phi}^2 + \frac{\sigma^2}{\nu^2} \frac{1}{2 \pi^2 \eps} \sum_{k,l \in I_n} \frac{\abs{k}^{2 s}\abs{l}^2 }{\abs{l-k}^4} \abs{ {\partial_k}\phi}^2\\ &\leq \eps \sigma^2 \sum_{k,l \in I_n}\abs{l}^{2s} \abs{\partial_{l-k} {\partial_l}\phi}^2 + \frac{\sigma^2}{\nu^2} \frac{1}{\pi^2 \eps} \sum_{k,l \in I_n} \Bigl( \abs{k}^{2+ 2 s}\frac{1}{\abs{l-k}^4} + \abs{k}^{2 s} \frac{1}{\abs{l-k}^2} \Bigr) \abs{ {\partial_k}\phi}^2.\end{aligned}$$ Clearly $\abs{l-k}^{-4}$ is summable over $l \in {\Z^2_\ast}$. It follows $$\sum_{k,l \in I_n} \abs{k}^{2+ 2 s}\frac{1}{\abs{l-k}^4} \abs{ {\partial_k}\phi}^2 \leq S(2) \sum_{k \in I_n} \abs{k}^{2+ 2s} \abs{ {\partial_k}\phi}^2 = S(2) \snorm{D {\tilde{\phi}}}{1+s}.$$ Again, denote by ${\tilde{\phi}}$ the extension of $\phi$ to ${\mathcal{F}C_b}^2$ via $\iota_n$. Similarly, $$\sum_{k,l \in I_n} \abs{k}^{2 s} \frac{1}{\abs{l-k}^2} \abs{{\partial_k}\phi}^2 \leq c \log (n) \snorm{D \tilde{\phi}}{s}.$$ The logarithmic growth in $n$ is sufficiently small and we use, as in Lemma \[lem:CommCoriolis\], the interpolation inequality to obtain $$\sum_{k,l \in I_n} \abs{k}^{2 s} \frac{1}{\abs{l-k}^2} \abs{{\partial_k}\phi}^2 \leq \delta \snorm{D \tilde{\phi}}{1+s}^2 + c(s,\delta) \big(\log (n)\big)^{(1+s)} \snorm{D \tilde{\phi}}{0}^2.$$ Essentially, we just found the estimate $$\label{proof:estimate1} \begin{split} &\frac{\sigma^2}{\nu} \frac{\sqrt{2}}{2 \pi} \sum_{k,l \in I_n} \frac{\abs{k}^{2s} \abs{l}}{\abs{l-k}^2} \abs{\partial_{l-k} {\partial_l}\phi} \abs{{\partial_k}\phi} \leq \eps \sigma^2 \sum_{k,l \in I_n}\abs{l}^{2s} \abs{\partial_{l-k} {\partial_l}\phi}^2\\ &\qquad+ \frac{\sigma^2}{\nu^2}\frac{\delta + S(2)}{\pi^2 \eps} \sum_{k \in I_n} \abs{k}^{2+2s} \abs{{\partial_k}\phi}^2 + \frac{\sigma^2}{\nu^2}\frac{c(s, \delta)}{\pi^2 \eps} \big(\log (n)\big)^{(1+s)} \sum_{k \in I_n} \abs{{\partial_k}\phi}^2. \end{split}$$ As the next step, we have to estimate the remaining terms in similar ways. Note that the roles of $k$ and $l$ are not symmetric, thus the estimates differ. With $\abs{k}^{2s} \leq 2 \abs{l-k}^{2s} + 2 \abs{l}^{2s}$ it follows that $$\begin{aligned} &\frac{\sigma^2}{\nu} \frac{\sqrt{2}}{2 \pi} \sum_{k,l \in I_n} \frac{\abs{k}^{2s} \abs{l}}{\abs{l-k}^2} \abs{{\partial_l}\phi}\abs{\partial_{l-k} {\partial_k}\phi}\\ &\quad\leq \eps \sigma^2 \sum_{k,l \in I_n}\abs{k}^{2s} \abs{\partial_{l-k} {\partial_k}\phi}^2 + \frac{\sigma^2}{\nu^2} \frac{1}{8 \pi^2 \eps} \sum_{k,l \in I_n} \frac{\abs{k}^{2 s}\abs{l}^2 }{\abs{l-k}^4} \abs{ {\partial_l}\phi}^2\\ &\quad\leq \eps \sigma^2 \sum_{k,l \in I_n} \abs{k}^{2s} \abs{\partial_{l-k} {\partial_k}\phi}^2 + \frac{\sigma^2}{\nu^2} \frac{1}{4 \pi^2 \eps}\sum_{k,l \in I_n} \left( \frac{\abs{l}^2}{\abs{l-k}^{4-2s}} + \frac{\abs{l}^{2+2s} }{\abs{l-k}^4}\right) \abs{ {\partial_l}\phi}^2\end{aligned}$$ Again, $\abs{l-k}^{-4}$ is summable in $k$, so it is exactly treated like above. $\abs{l-k}^{-4+2s}$ is summable in $k$ if $s < 1$ and of order $\log n$ if $s=1$. So we bound it similarly to the case above by $$\sum_{k,l \in I_n} \frac{\abs{l}^2}{\abs{l-k}^{4-2s}} \abs{ {\partial_l}\phi}^2 \leq c \log (n) \snorm{D \tilde{\phi}}{1} \leq \delta \snorm{D \tilde{\phi}}{1+s}^2 + c(s,\delta) \big(\log (n)\big)^{\frac{1 + s}{s}} \snorm{D \tilde{\phi}}{0}^2.$$ Thus, we arrive at an estimate as in . It is clear, that the other three cases of $\pm k \pm l$ can be estimated in the exact same way. In our next step we derive an overall estimate by combining Lemmas \[lem:Commutator\], \[lem:CommCoriolis\] and \[lem:CommConvection\]. Note that the second derivatives in this equation would not appear if the fluid was not perturbed by a random noise – this can be interpreted as a regularizing effect of the noise. \[lem:OverallEstimate\] Let $s \in (0,1]$, $\phi \in C_b^3(H_n)$ and $\lambda > 0$. Then, there exists $\delta > 0$ such that $$\label{eq:OverallEstimate} \begin{split} &\lambda \sum_{k \in I_n} \abs{k}^{2s} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + \delta \sum_{k \in I_n} \abs{k}^{2+2s} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n \\ &\quad\leq \sum_{k \in I_n} \abs{k}^{2s} \int {\partial_k}((\lambda - {K_{\sigma,\nu}}^n) \phi) {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + c(\delta) \big(\log (n)\big)^{\frac{1+s}{s}} \sum_{k \in I_n} \int \abs{ {\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n, \end{split}$$ where $c(\delta) < \infty$ independent of $n$. This lemma is an improvement in comparison to [@Stannat2DNSE]. We weaken the smallness condition for $\nu$ by trading this to some growth in $n \in \N$ in front of a weaker norm. This is sufficiently small to be matched by the convergence of $B^n$ and $C^n$ later on. Also, note that the parameters $\omega, \beta$ of the Coriolis force do not appear in the smallness condition. In the preceding lemmas we deduced $$\begin{aligned} &\sum_{k \in I_n} \abs{k}^{2s} \int {\partial_k}({K_{\sigma,\nu}}^n \phi) {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n \leq \Bigl(-\frac{\sigma^2}{2} + 4 \eps \sigma^2 \Bigr) \sum_{k,l \in I_n} \abs{k}^{2s} \int \abs{{\partial_l}{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\\ &+ \Bigl(-\nu + \frac{\sigma^2}{\nu^2} \frac{6 \delta + 5 S(2)}{\pi^2 \eps} \Bigr) \sum_{k \in I_n} \abs{k}^{2+2s} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + c(\eps, \delta) \big(\log (n)\big)^{\frac{1+s}{s}} \sum_{k \in I_n} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\end{aligned}$$ Choosing $\eps = \frac{1}{8}$ will provide that all the second derivatives of $\phi$ vanish. Now by Assumption \[assum:ViscosityCondition\], set $\delta \df \nu^3 \pi^2/(7 \sigma^2) - (40/7) S(2) > 0$ and the assertion follows immediately. \[lem:Extension\] Inequality extends to all $\phi = {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi$ with $\psi \in C_b^1(H_n)$. The proof of this lemma follows [@Stannat2DNSE Lemma 2.6], which appears to be slightly inaccurate since there the identity (20) does not hold. However, the remaining proof can be modified, as done below. In particular, the statement in [@Stannat2DNSE Lemma 2.6] is also valid. In a first step, we need a different uniqueness result for ${K_{\sigma,\nu}}^n$, in particular [@Eberle Theorem 2.5, Chapter 2.F]. The statement says that $({K_{\sigma,\nu}}^n, C_0^\infty(H_n) )$ is $L^2$-unique, hence $C_0^\infty(H_n)$ is a core for ${K_{\sigma,\nu}}^n$, i.e. dense w.r.t. the graph norm. This implies that for fixed $\psi \in C_b^1(H_n)$, we can find a sequence $(\phi_m) \subset C_0^\infty(H_n)$ such that $$\lim_{m \to \infty} \Bigl( \snorm{\phi_m - {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}{L^2(H_n, {\mu_{\sigma,\nu}}^n)} + \snorm{{K_{\sigma,\nu}}^n \phi_m - {K_{\sigma,\nu}}^n {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}{L^2(H_n, {\mu_{\sigma,\nu}}^n)} \Bigr) = 0.$$ Now consider $$L_s^n \phi (u) \df \frac{\sigma^2}{2} \sum_{k \in I_n} \abs{k}^{2s} {\partial_k}^2 \phi (u) - \nu \sum_{k \in I_n} \abs{k}^{2+2s} u_k {\partial_k}\phi(u),$$ which is the generator associated to the bilinear form $\mathcal{E}^s$, i.e. $$\int L_s^n \phi \phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n = - \mathcal{E}^s(\phi,\phi),$$ since with the integration by parts from Lemma \[lem:PartialIntegrationGaussian\] it follows that $$\frac{\sigma^2}{2} \sum_{k \in I_n} \abs{k}^{2s} \int {\partial_k}^2 \phi \phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n = -\frac{\sigma^2}{2} \sum_{k \in I_n} \abs{k}^{2s} \int {\partial_k}\phi \Bigl( {\partial_k}\phi - \frac{2\nu \abs{k}^2}{\sigma^2} \phi \Bigr) {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n.$$ The bilinear form is used in the gradient estimates in Lemma \[lem:OverallEstimate\] and in the following we want to prove that $L_s^n \phi_m \to L_s^n {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi$ weakly along some subsequence. For this purpose consider $$\begin{aligned} \int \bigl( L_s^n \phi \bigr)^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n &= - \mathcal{E}^s (L_s^n \phi, \phi)\notag\\ &= -\frac{\sigma^2}{2} \sum_{k \in I_n} \abs{k}^{2s} \int L_s^n {\partial_k}\phi {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + \frac{\sigma^2 \nu}{2} \sum_{k \in I_n} \abs{k}^{2 + 4s} \int \abs{{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\notag\\ &\leq \frac{\sigma^4}{4} \sum_{k,l \in I_n} \abs{k}^{2s} \abs{l}^{2s} \int \abs{{\partial_l}{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + c(n) \mathcal{E}^s(\phi,\phi),\label{proof:UpperLowerBound}\end{aligned}$$ with some constant $c(n)$. A trivial lower bound is given by the above with $c(n) = 0$. These inequalities immediately imply that we can switch between different values of $s$, because $$\label{proof:Different_s} \int \bigl( L_{s_1}^n \phi \bigr)^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n \leq c(s_1, s_2, n) \int \bigl( L_{s_2}^n \phi \bigr)^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n.$$ Thus, we set $s= 0$ in the following and the proof of Lemma \[lem:OverallEstimate\] with the choice $\eps = \frac{1}{16}$ instead of $\frac{1}{8}$ yields $$\frac{\sigma^4}{4} \sum_{k,l \in I_n} \int \abs{{\partial_l}{\partial_k}\phi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n \leq - \sigma^2 \mathcal{E}^0 ({K_{\sigma,\nu}}^n \phi, \phi) + C(n) \mathcal{E}^0(\phi,\phi).$$ Now we use the lower bound obtained in , and the fact that $L_0^n$ is associated to $\mathcal{E}^0$, together with Hölder’s and Young’s inequality. $$\begin{aligned} \int \bigl( L_0^n \phi \bigr)^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n &\leq 2 \int L_0^n \phi {K_{\sigma,\nu}}^n \phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n - c(n) \int {K_{\sigma,\nu}}^n \phi \phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\notag\\ &\leq \frac12 \int \bigl( L_0^n \phi \bigr)^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + 2 \snorm{{K_{\sigma,\nu}}^n \phi}{L^2}^2 + c(n) \snorm{{K_{\sigma,\nu}}^n \phi}{L^2} \snorm{\phi}{L^2},\notag\\ \intertext{hence} \int \bigl( L_0^n \phi \bigr)^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n &\leq 4 \snorm{{K_{\sigma,\nu}}^n \phi}{L^2}^2 + c(n) \snorm{{K_{\sigma,\nu}}^n \phi}{L^2} \snorm{\phi}{L^2}.\label{proof:EstimateL0}\end{aligned}$$ Recall and we deduce $$\begin{aligned} &\Bigl| \frac{\sigma^2}{2} \sum_{k \in I_n} \abs{k}^{2s} \int {\partial_k}((\lambda - {K_{\sigma,\nu}}^n)\phi) {\partial_k}\phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}\Bigr| \\ &\quad= \abs{\mathcal{E}^s((\lambda - {K_{\sigma,\nu}}^n)\phi, \phi)} = \Bigl| \int (\lambda - {K_{\sigma,\nu}}^n)\phi L_s^n \phi {\, \mathrm{d}{\mu_{\sigma,\nu}}}\Bigr|\\ &\quad\leq \snorm{ (\lambda - {K_{\sigma,\nu}}^n)\phi}{L^2}^2 + \snorm{L_s^n \phi}{L^2}^2 \leq \snorm{ (\lambda - {K_{\sigma,\nu}}^n)\phi}{L^2}^2 + c(s, n) \snorm{L_0^n \phi}{L^2}^2\phantom{\Big|}\\ &\quad\leq \snorm{ (\lambda - {K_{\sigma,\nu}}^n)\phi}{L^2}^2 + c(s, n) \bigl(\snorm{{K_{\sigma,\nu}}^n \phi}{L^2}^2 + c(n) \snorm{{K_{\sigma,\nu}}^n \phi}{L^2} \snorm{\phi}{L^2}\bigr).\phantom{\Big|}\end{aligned}$$ Now we turn back to the sequence $(\phi_m)$ and due to we know that $$\sup_m \int \bigl( L_0^n \phi_m \bigr)^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n < \infty,$$ hence ${\overline{R}_{\sigma,\nu}^n(\lambda)}\psi \in D(L_0^n) = D(L_s^n)$ for all $s \in \R$. In particular, the boundedness implies weak convergence in $L^2({\mu_{\sigma,\nu}}^n)$ of $L_s^n \phi_{m_l} \to L_s^n {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi$ along some subsequence $(m_l)$, thus $$\Bigl| \frac{\sigma^2}{2} \sum_{k \in I_n} \abs{k}^{2s} \int {\partial_k}((\lambda - {K_{\sigma,\nu}}^n)\phi_{m_l}) {\partial_k}\phi_{m_l} - {\partial_k}\psi {\partial_k}{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi {\, \mathrm{d}{\mu_{\sigma,\nu}}}\Bigr| \to 0.$$ Inequality holds for all $\phi_{m_l}$ and the assertion follows by Lebesgue’s dominated convergence theorem. The rest of the proof is a simple manipulation. We have shown that for $\psi \in C_b^1(H_n)$ $$\begin{aligned} &\lambda \sum_{k \in I_n} \abs{k}^{2 s} \int \abs{{\partial_k}{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + \delta \sum_{k \in I_n} \abs{k}^{2+2 s} \int \abs{{\partial_k}{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n \\ &\leq \sum_{k \in I_n} \abs{k}^{2 s} \int {\partial_k}\psi {\partial_k}{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + c(\delta) \big(\log (n)\big)^{\frac{1+s}{s}} \sum_{k \in I_n} \int \abs{ {\partial_k}{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n\\ &\leq \frac{1}{4 \lambda} {\kern -2pt}\sum_{k \in I_n} {\kern -2pt}\abs{k}^{2 s} {\kern -2pt}\int {\kern -2pt}\abs{{\partial_k}\psi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + \lambda {\kern -2pt}\sum_{k \in I_n} {\kern -2pt}\abs{k}^{2 s} {\kern -2pt}\int {\kern -2pt}\abs{{\partial_k}{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}^n + \frac{2 c(\delta)}{\lambda \sigma^2} \big(\log (n)\big)^{\frac{1+s}{s}}\snorm{\psi}{L^\infty}^2.\end{aligned}$$ Rearranging the terms yields the result. Proof of Theorem \[thm:UniquenessCoriolis\] =========================================== The remaining part of the proof is fairly standard. By general arguments $({K_{\sigma,\nu}}, {\mathcal{F}C_b}^2)$ is dissipative, hence closable in $L^1({\mu_{\sigma,\nu}})$. Thus, it remains to check the range condition $(\lambda - {K_{\sigma,\nu}}) ({\mathcal{F}C_b}^2) \subset L^1({\mu_{\sigma,\nu}})$ dense for some $\lambda >0$, see e.g. [@Eberle]. Let us fix a function $\psi \in C_b^1(H_{n_0})$ for some finite $n_0$. Clearly, $\psi$ has its representative $\tilde{\psi} \in {\mathcal{F}C_b}^1$ and can be considered as a function on $H_n$ for arbitrary $n \geq n_0$. Thus, the resolvent ${\overline{R}_{\sigma,\nu}^n(\lambda)}\psi \in D(\overline{K}_{\sigma,\nu})_b$ for all $n$ and $$\begin{aligned} (\lambda - \overline{K}_{\sigma, \nu}) {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi &= (\lambda - \overline{K}^n_{\sigma, \nu}) {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi + (\overline{K}^n_{\sigma, \nu} - \overline{K}_{\sigma, \nu}) {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi \\ &= \psi + \sum_{k \in I_n} \bigl(B^n_k - B_k + C^n_k - C_k \bigr){\partial_k}{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi.\end{aligned}$$ Now, we combine the convergence of the Galerkin approximations with the integrated gradient estimated for the resolvent. Let $s \in (0,1]$, then for any $0 < \eps < s$ $$\begin{aligned} &\snorm{ (\lambda - \overline{K}_{\sigma, \nu}) {\overline{R}_{\sigma,\nu}^n(\lambda)}\psi - \psi}{L^1}\\ &\quad\leq \snorm{{\overline{R}_{\sigma,\nu}^n(\lambda)}\psi}{W^{1,2}_{1 + s}} \cdot \Biggl(\int \snorm{\pi_n (B-B^n)}{-1-s}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}+ \int \snorm{\pi_n (C-C^n)}{-1-s}^2 {\, \mathrm{d}{\mu_{\sigma,\nu}}}\Biggr)^{\frac12}\\ &\quad\leq c(\psi) (1 + \log n)^{\frac{1+s}{s} + \frac12} \cdot c n^{-\eps} \xrightarrow{n \to \infty} 0,\end{aligned}$$ which implies the denseness of the range $(\lambda - {K_{\sigma,\nu}}) \big({\mathcal{F}C_b}^2\big) \subset L^1({\mu_{\sigma,\nu}})$, since ${\mathcal{F}C_b}^1 \subset L^1({\mu_{\sigma,\nu}})$ dense. Acknowledgement {#acknowledgement .unnumbered} =============== During the prepatation of this article the author was supported by the DFG and JSPS as a member of the International Research Training Group Darmstadt-Tokyo IRTG 1529. 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--- abstract: 'Motivated by applications such as recommendation systems, we consider the estimation of a binary random field $\mathbf X$ obtained by [*unknown*]{} row and column permutations of a block constant random matrix. The estimation of $\mathbf X$ is based on observations $\mathbf Y$, which are obtained by passing entries of $\mathbf X$ through a binary symmetric channel (BSC) (representing noisy user behavior) and an erasure channel (representing missing data). We analyze an estimation algorithm based on local popularity. We study the bit error rate (BER) in the limit as the matrix size approaches infinity and the erasure rate approaches unity at a specified rate. Our main result identifies three regimes characterized by the cluster size and erasure rate. In one regime, the algorithm has asymptotically zero BER, in another regime the BER is bounded away from 0 and 1/2, while in the remaining regime, the algorithm fails and BER approaches 1/2. Numerical results for the Movielens dataset and comparison with earlier work is also given.' author: - - bibliography: - 'myBib.bib' title: Local Popularity Based Collaborative Filters --- Introduction ============ Recommendation systems are commonly used in e-commerce to suggest relevant content to users. One approach considers the user-item rating matrix, predicts the missing entries, and recommends items based on the predicted values (for example, see [@netflixprize]). Recently, a number of researchers have considered mathematical models for this problem and studied fundamental limits. One model assumes the rating matrix to be a low-rank random matrix ([@Candes1; @Montanari1; @Bresler1]), and then bounds on the number of samples needed to recover the [*complete*]{} matrix with high probability are obtained. In another model ([@Aditya1; @Aditya2]), the rating matrix is assumed to be obtained from a block constant matrix by applying unknown row and column permutations, a noisy discrete memoryless channel representing noisy user behavior, and an erasure channel denoting missing entries. The goal for such a model is not matrix completion, but estimation of the underlying “noiseless” matrix. In [@Aditya1; @Aditya2], the probability of error in recovering the [*entire*]{} matrix for fixed erasure rate is considered, and threshold results reminiscent of the channel coding theorem (but with different scaling) are established. In this paper, we consider the model in [@Aditya2], but we allow the erasure rate to approach unity, and focus on the BER - the probability of error that a specific recommendation fails. We analyze the BER for a specific algorithm, which makes recommendations based on “local popularity”. Such an analysis is of interest for two reasons: - It gives an upper bound on achievable BER; - The local popularity algorithm used is motivated by algorithms used in practice [@Linden1], and has lower complexity compared to those in the above mentioned references. - The algorithm has competitive empirical performance on real datasets such as the Movielens data [@MovieLens]. For example, next we compare the algorithm with OptSpace [@Montanari1] on Movielens data. While OptSpace uses ratings on the scale 1-5 given by Movielens, in our algorithm we quantize the ratings as follows: 4,5 are mapped to 1, while 1-3 are mapped to 0. (Similarly, the output of OptSpace is quantized to $\{0,1\}$.) We find that the local algorithm yields a BER of 0.091, while on the same test data, OptSpace gives a BER of 0.107. Thus the performance of both algorithms is similar. (More detailed simulation results will be presented in a future publication.) In this paper, we seek to understand the reason for the competitive performance of the relatively simple local algorithm by analyzing its BER for the model proposed in [@Aditya1]. Suppose that the matrix is of size $n \times n$ and the erasure probability $\epsilon = 1-c/n^{\alpha}$. If $\alpha \in [0,1/2)$, then our main result says that if the cluster size is greater than $n^{\alpha -\gamma_n}$ where $\gamma_n \rightarrow 0$, then the BER approaches 0, but if the cluster size is less than $n^{\alpha -\gamma}$, $\gamma >0$, the BER is bounded away from zero and a lower bound is obtained in terms of the observation noise and $\gamma$. For $\alpha >1/2$, BER always approaches 1/2. Due to space constraints we only provide an outline of the proofs; the details with additional results will be reported in a journal submission. The rest of the paper is organized as follows. In Section \[sec:basic\_setup\], we describe our model, the local popularity algorithm, and establish notation. The main results are stated and discussed in Section \[sec:main\]. The proof of the main result is given in \[sec:proof\_main\] and some related lemmas are established in \[sec:proof\_lemma\]. The conclusion of given in Section \[sec:con\]. Basic Setup {#sec:basic_setup} =========== In Section \[subsec:model\] we describe our model, and discuss a local popularity based algorithm in Section \[subsec:local\]. The Model {#subsec:model} --------- We consider an $n\times n$ rating matrix $\mathbf X$ whose entries are binary. The rows of the matrix represent users and the columns represent items. Suppose $\mathcal A=\{A_i\}_{i=1}^r$ and $\mathcal B=\{B_i\}_{i=1}^r$ are row and column partitions respectively, representing sets of similar users and items. We assume that for all $i=1,\dots,r$ we have $|A_i|=|B_i|=k$. The sets $A_i\times B_j$ are the clusters of the matrix and they are unknown. If $(p,q)\in A_i\times B_j$, then $\mathbf X(p,q)=\xi_{ij}$ where $\xi_{ij}$ are i.i.d. Bernoulli(1/2). This matrix $\mathbf X$ is passed through a memoryless binary symmetric channel (BSC) with parameter $p$, and then through an erasure channel with each entry being erased independently with probability $\epsilon$. The erasures characterize the missing entries in a rating matrix, while the BSC characterizes the noisy behaviour of the users. The entries of the observed matrix $\mathbf Y$ are from $\{0,1,*\}$, where $*$ denotes an erased entry. We consider the case of binary entries and uniform cluster size is for simplicity, and like in [@Aditya2], these can be relaxed. For more detailed motivation of this model, we refer to [@Aditya1],[@Aditya2]. A Local Popularity Algorithm {#subsec:local} ---------------------------- Without loss of generality suppose the first row belongs to $A_1$. Upon observing $\mathbf Y$, we want to recommend an item (a column) to the user 1. In this paper we study a particular “local” algorithm, which only uses pairwise row correlations. Let the number of commonly sampled entries between two rows (similarity) $s_{ij}:=\sum_{k=1}^n\mathbf 1_{\{\mathbf Y(i,k)\neq *\}}\cdot \mathbf 1_{\{\mathbf Y(j,k)\neq *\}}\cdot\mathbf 1_{\{\mathbf Y(i,k)=\mathbf Y(j,k)\}},$ where $\mathbf 1_{\{.\}}$ denotes the indicator function. We use the following local algorithm ($\texttt{local\_algo}(T)$) to recommend an item $j_0$ to user 1. \ Suppose we represent each row by a vertex in a graph with an edge between vertex $i$ and $j$ iff $s_{ij}>0$. Then to recommend an item to user 1, the above algorithm depends only on the rows neighboring to user 1, and chooses the most popular item among the top few neighbors. Hence we use the adjective “local popularity”. We study the probability of error for this algorithm, denoted as $P_e[\texttt{local\_algo}(T)]:=Pr[\mathbf X(1,j_0)=0]$. Main Result {#sec:main} =========== From the results in [@Aditya2], it follows that for $k > c_1n^\alpha \log n$, $\alpha \in [0,1/2)$, with high probability we can recover the entire matrix $\mathbf X$ using a “local” algorithm, and hence the BER also approaches zero. In the following theorem, we establish a stronger result for $\texttt{local\_algo}(T)$. \[thm:01\] Suppose $\alpha\in [0,1/2)$ and $c>0$. Assume that the erasure probability $\epsilon =1-\frac{c}{n^\alpha}$, the BSC error probability $p\in [0,1/2)$, and $r$ goes to infinity with $n$. - (**Large cluster size**) If there exists a sequence $\gamma_n\ge 0$ such that $\gamma_n \rightarrow 0$ and $k \ge n^{\alpha-\gamma_n}$, then $P_e[\texttt{local\_algo}(k)]\rightarrow 0$ as $n\rightarrow \infty$. - (**Small cluster size**) If there is a constant $\gamma > 0$ such that $k \le n^{\alpha -\gamma}$, then $$\lim \inf_{n\rightarrow \infty} P_e[\texttt{local\_algo}(k)] \ge \frac{p^{\left\lfloor \frac{1}{\gamma}\right\rfloor}}{p^{\left\lfloor \frac{1}{\gamma}\right\rfloor} + (1-p)^{\left\lfloor \frac{1}{\gamma}\right\rfloor}}.$$ In Theorem \[thm:01\] we restrict ourselves to $\alpha \in [0, 1/2)$. For $\alpha < 1/2$, as we show in Section \[sec:proof\_main\], all the rows picked by Step 1 of the algorithm are from $A_1$ (“good”) with high probability. But, for $\alpha >1/2$, most of the rows picked are from outside $A_1$ (“bad”), and hence the algorithm breaks down. Due to lack of space, the results for $\alpha > 1/2$ will be presented in subsequent publications. In the rest of this paper, we present a proof of Theorem \[thm:01\]. Proof of Theorem \[thm:01\] {#sec:proof_main} =========================== In this section we present the proof of Theorem \[thm:01\]. To begin with, we introduce some notation. [**Notation:**]{} By $X\sim B(n,p)$ we mean that a random variable $X$ is binomially distributed with parameters $n$ and $p$. For a real valued function $f(n)$, by $\Omega(f(n)), \Theta(f(n))$ and $o(f(n))$ we represent the standard asymptotic order notation (see for example[@motwani95 p. 433]). We say that $f(n) \doteq g(n)$ if $\lim_{n\rightarrow \infty} \frac{f(n)}{g(n)}=1$. For a matrix $\mathbf X$, $\mathbf X(:,j)$ denotes the $j$th column of $\mathbf X$. For a vector $\bar y\in \{0,1,*\}^n$, $|\bar y|_0$, $|\bar y|_1$ and $|\bar y|$ represent number of 0’s, number of 1’s and the total number of 0’s and 1’s respectively. For a sequence of events $\{E_n\}$, if $P[E_n]\rightarrow 1$ with $n$, then we say that $E_n$ occurs w.h.p.. [**Analysis of Step 1 of the algorithm:**]{} We show that w.h.p. the top $k$ rows are all from $A_1$. We observe that for $i\in A_1\backslash\{1\}$, $s_{1i} \sim B(n, p_g)$ with $p_g:=(1-\epsilon)^2[(1-p)^2+p^2]$. For $i\not \in A_1$ we observe that $s_{1i}$ is a mixture of binomials with $\mathbb E[s_{1i}]=np_b$ for $p_b:=\frac{(1-\epsilon)^2}{2}< p_g$. We omit the proofs of the following two lemmas, which are consequences of the Chernoff bound [@Dubhashi1 Theorem 1.1] together with a union bound. \[lemma:01\] For $\delta\in (0,1)$, we have $$Pr\big[\min_{i\in A_1} s_{1i} \le np_g(1-\delta)\big] \le ke^{-np_g\delta^2/3}=:p_1.$$ \[lemma:02\] For $\delta\in (0,1)$, we have $$Pr\big[\max_{i\not\in A_1}s_{1i} \ge np_b(1+\delta)^2\big] \le(n-k)e^{-\frac{np_b \delta^2}{3}}+2re^{-\frac{r\delta^2}{6}}=:p_2.$$ Since $p_g> p_b$, we can choose a small enough constant $\delta_0$ such that $np_g(1-\delta_0) > np_b(1+\delta_0)^2 $. Let $E_1$ denote the event that there is an error in Step 1 of the algorithm, i.e., we choose some rows from outside $A_1$ in the top $k$ users. Using Lemma \[lemma:01\] and Lemma \[lemma:02\] we obtain $$\begin{aligned} \label{eq:1} Pr[E_1] & \le Pr\left[ \min_{i\in A_1} s_{1i} \le \max_{i\not\in A_1} s_{1i}\right] \le p_1 +p_2 \buildrel (a)\over = o(1). %\nonumber & \le k e^{-np_g\delta_0^2/3} + (n-k) e^{-np_b\delta_0^2/3} + 2r e^{-r\delta_0^2/6}\\ %\nonumber & \buildrel (a) \over= k e^{-c_1n^{2(1-2\alpha)}} + (n-k) e^{-c_2 n^{2(1-2\alpha)}} + 2r e^{-c_3 r}\\ %\label{eq:1} &\buildrel (b) \over =o(1). \end{aligned}$$ Here (a) follows since $np_g=\Theta(np_b)=\Theta(n^{1-2\alpha})$, and $r$ increases to infinity with $n$. This implies that w.h.p. Step 1 of $\texttt{local\_algo}$ does not contribute to the error. [**Analysis of Step 2 of the algorithm:**]{} We assume that Step 1 picks all the $k$ “good” neighbors. (i.e., we condition on the event $E_1^c$.) [**Large cluster size:**]{} Suppose $k\ge n^{\alpha -\gamma_n}$ for $\gamma_n=o(1)$. Let $j_{max}$ denote the most popular column chosen by `local_algo`($k$), and suppose $\mathbf X_k$ and $\mathbf Y_k$ denotes the matrices $\mathbf X$ and $\mathbf Y$ respectively, restricted to the top $k$ rows. Since we have conditioned on $E_1^c$, we observe that for a column $j$ such that $\mathbf X(1,j)=1$, we have $|\mathbf Y_k(:,j)|_1\sim B(k,(1-\epsilon)(1-p))$. Define $\mu_Y:=\mathbb E[|\mathbf Y_k(:,j)|_1 ]$ and $\sigma_Y^2:=Var(|\mathbf Y_k(:,j)|_1 )$ to obtain the following two lemmas. \[lemma:101\] For different values of $k$, we have the following lower bounds on $|\mathbf Y_k(:,j_{max})|_1$. 1. If $k=n^{\alpha-\gamma_n}$ such that $\gamma_n\ge 0$ and $\gamma_n\rightarrow 0$, then w.h.p. $|\mathbf Y_k(:,j_{max})|_1\ge \min\{\sqrt{\log n}, \frac{1}{2\gamma_n}\} =:t_1(n)$. 2. If $k=n^\alpha g_n$ for $g_n \ge 1$, then w.h.p. $|\mathbf Y_k(:,j_{max})|_1\ge \max\{\mu_Y+\min\{\sigma_Y^{1/4},\sqrt{\log n}\} \sigma_Y,\sqrt{\log n}\}=:t_2(n)$. The proof is given in Section \[proof:lemma:101\] \[lemma:102\] Let $j_{max}$ be the most popular column. Then w.h.p. $|\mathbf Y_k(:,j_{max})|_1 -|\mathbf Y_k(:,j_{max})|_0$ increases to $\infty$ with $n$. The proof is given in Section \[proof:lemma:102\] Now we use Lemma \[lemma:101\] and Lemma \[lemma:102\] to prove that the local algorithm makes vanishingly small probability of error. We define $t(k,n):=t_1(n)$ if $k=n^{\alpha-\gamma_n}$ for $\gamma_n\rightarrow 0$, and $t(k,n):=t_2(n)$ if $k=n^\alpha g_n$ for $g_n \ge 1$ (here $t_1(n)$ and $(t_2(n)$ are as defined in Lemma \[lemma:101\]). Suppose $$M:=\{\bar y\in \{0,1,*\}^k: (|\bar y|_1-|\bar y|_0)\rightarrow \infty, \text{ and }|\bar y|_1 \ge t(k,n)\}.$$ We also observe that for a column $j$, $$\begin{aligned} \label{eq:103} \mathbf X_k(:,j) \longrightarrow \mathbf Y_k(:,j)\longrightarrow \{j_{max}=j\},\end{aligned}$$ i.e., the random variables $\{\mathbf X_k(:,j), \mathbf Y_k(:,j), \{j_{max}=j\}\}$ form a Markov chain. We are interested in finding the overall probability of error. In the following, by $p_{k,j}(\bar y)$ we mean $Pr[\mathbf Y_k(:,j)=\bar y|j_{max}=j, E_1^c]$. Then we have $$\begin{aligned} \nonumber & P_e[\texttt{local\_algo}(k)] = Pr[\mathbf X(1,j)=0|j_{max}=j]\\ \nonumber \buildrel(a)\over = & Pr[\mathbf X(1,j)=0|j_{max}=j, E_1^c] +o(1)\\ %\nonumber =& \sum_{\substack{\bar y \in \{0,1, *\}^k}} \hspace{-0.2in}Pr[\mathbf X(1,j)=0, \mathbf Y_k(:,j)=\bar y|j_{max}=j, E_1^c]+o(1)\\ \nonumber \buildrel (b)\over = & \sum_{\substack{\bar y \in \{0,1,*\}^k\\ \bar y \in M}} \hspace{-0.2in}Pr[\mathbf X(1,j)=0, \mathbf Y_k(:,j)=\bar y|j_{max}=j, E_1^c] + o(1)\\ \nonumber \buildrel (c)\over = &\sum_{\substack{\bar y \in \{0,1,*\}^k\\\bar y \in M}} Pr[\mathbf X(1,j)=0\big|\mathbf Y_k(:,j)=\bar y, E_1^c]\cdot p_{k,j}(\bar y) + o(1)\\ \nonumber \buildrel (d)\over = &\sum_{\substack{\bar y \in \{0,1,*\}^k\\\bar y \in M}} \frac{Pr[\mathbf Y_k(:,j)=\bar y\big|\mathbf X(1,j)=0, E_1^c]}{2Pr[\mathbf Y_k(:,j)=\bar y|E_1^c]} p_{k,j}(\bar y) + o(1)\\ \nonumber =&\sum_{\substack{\bar y \in \{0,1,*\}^k\\\bar y \in M}} \frac{p^{|\bar y|_1}(1-p)^{|\bar y|_0}}{p^{|\bar y|_1}(1-p)^{|\bar y|_0} + p^{|\bar y|_0}(1-p)^{|\bar y|_1}} p_{k,j}(\bar y) + o(1)\end{aligned}$$ $$\begin{aligned} %\nonumber = &\sum_{\substack{\bar y \in \{0,1,*\}^k\\\bar y \in M}} \frac{p^{|\bar y|_1-|\bar y|_0}}{p^{|\bar y|_1-|\bar y|_0} + (1-p)^{|\bar y|_1-|\bar y|_0}}p_{k,j}(\bar y)+ o(1)\\ \nonumber \le &\max_{\substack{\bar y \in \{0,1,*\}^k\\\bar y \in M}} \frac{p^{|\bar y|_1-|\bar y|_0}}{p^{|\bar y|_1-|\bar y|_0} + (1-p)^{|\bar y|_1-|\bar y|_0}}+ o(1)\\ \label{eq:2} & \buildrel (e) \over = o(1),\end{aligned}$$ where (a) follows from , (b) is true because of Lemma \[lemma:101\] and Lemma \[lemma:102\], (c) is due to the Markov property and the notation of $p_{k,j}(\bar y)$, (d) is the Bayes’ expansion, and (e) is true since for $\bar y\in M$, $|\bar y|_1-|\bar y|_0$ goes to infinity with $n$, and the fact that $\frac{p^x}{p^x+(1-p)^x}=o(x)$ for $p<1/2$. This proves the first part of Theorem \[thm:01\]. [**Small cluster size:**]{} Now suppose $k\le n^{\alpha -\gamma}$ for a constant $\gamma >0$. We show that in this case the most popular column has a finite number of unerased entries. This allows us to find a lower bound on the probability of error. \[lemma:103\] W.h.p. $$\max_j |\mathbf Y_k(:,j)| \le \lfloor 1/\gamma\rfloor.$$ The proof is based on bounding the tail of $\mathbf Y_k(:,j)$ and is not given here due to space restrictions. Suppose $$I:=\{\bar y \in \{0,1,*\}^k : |\bar y|\le \lfloor 1/\gamma\rfloor \}.$$ We want to find a lower bound for the total probability of error. By following the steps as in and replacing the event $M$ by the event $I$ (this replacement is justified due to Lemma \[lemma:103\]), we have $$\begin{aligned} & P_e[\texttt{local\_algo}(k)] \\ = &\sum_{\substack{\bar y \in \{0,1,*\}^k\\\bar y \in I}} \frac{p^{|\bar y|_1-|\bar y|_0}}{p^{|\bar y|_1-|\bar y|_0} + (1-p)^{|\bar y|_1-|\bar y|_0}}p_{k,j}(\bar y)+ o(1)\\ \ge &\min_{\substack{\bar y \in \{0,1,*\}^k\\\bar y \in I}} \frac{p^{|\bar y|_1-|\bar y|_0}}{p^{|\bar y|_1-|\bar y|_0} + (1-p)^{|\bar y|_1-|\bar y|_0}}+ o(1)\\ \buildrel (a) \over\ge &\frac{p^{\lfloor \frac{1}{\gamma}\rfloor}}{p^{\lfloor \frac{1}{\gamma}\rfloor} + (1-p)^{\lfloor \frac{1}{\gamma}\rfloor}}+o(1) % & = \frac{p^{\lfloor \frac{1}{\gamma}\rfloor}}{p^{\lfloor \frac{1}{\gamma}\rfloor} + (1-p)^{\lfloor \frac{1}{\gamma}\rfloor}}+o(1),\\\end{aligned}$$ where (a) is trues since $|\bar y|_1 - |\bar y|_0 \le |\bar y| \le \lfloor 1/\gamma\rfloor$ for $\bar y \in I$, and for $x\in \mathbb R$, $\frac{p^{x}}{p^{x}+ (1-p)^{x}}$ is a decreasing function of $x$ for $p<1/2$. Taking $\lim \inf$ to both the sides proves the claim. Proofs of lemmas {#sec:proof_lemma} ================ To prove Lemma \[lemma:101\] and Lemma \[lemma:102\], we need the following theorem. Suppose $Q(t)$ denotes the upper tail of a standard normal distribution, i.e., $Q(t):=\frac{1}{\sqrt{2\pi}} \int_t^\infty e^{-t^2/2} dt$. \[thm:large\] Suppose $X_n\sim B(n,p_n)$. If $t_n\rightarrow \infty$ in such a way that $t_n^6=o\left(Var(X_n)\right)=o(np_n(1-p_n))$, then $$Pr\big[X_n > np_n + t_n \sqrt{np_n(1-p_n)}\big] \doteq Q(t_n).$$ The above theorem is an adaptation of a theorem about moderate deviations of binomials when $p_n$ is a constant [@Feller1 p. 193]. The proof is very similar to the one presented in [@Feller1] for the constant probability case, and is omitted here due to lack of space. Proof of Lemma \[lemma:101\] {#proof:lemma:101} ---------------------------- 1\) Recall that we have conditioned on the event that all the rows in the top $k$ neighbors chosen by $\texttt{local\_algo}(k)$ are “good”. Suppose $k=n^{\alpha-\gamma_n}$. Let $S$ be the set of columns $j$ such that $\mathbf X(1,j)=1$. Thus $|S|\sim B(n,1/2)$ and due to Chernoff bound we have w.h.p. $|S|\ge n/3$. For a column $j\in S$ we see that $|\mathbf Y_k(:,j)|_1\sim B(k,(1-\epsilon)(1-p))$, and they are independent for different values of $j$. Thus for $j\in S$, $$\begin{aligned} \nonumber & Pr\big[|\mathbf Y_k(:,j)|_1\ge t\big] \ge Pr\big[|\mathbf Y_k(:,j)|_1=t\big]\\ \nonumber \buildrel (a)\over \ge &{k \choose t} ((1-\epsilon)(1-p))^t \epsilon^{k-t}\\ \label{eq:222} \buildrel(b)\over \ge &\left(\frac{k}{t}\right)^t \left(\frac{c(1-p)}{n^\alpha}\right)^t e^{-2\ln (2) c/n^{\gamma_n}}, \text{ for large $n$}\\ \nonumber \buildrel (c) \over \ge &\left(\frac{c(1-p)}{tn^{\gamma_n}}\right)^t e^{-2\ln (2) c}. \end{aligned}$$ where (a) is true since $1-(1-\epsilon)(1-p) \ge \epsilon$, (b) follows since $\epsilon=1-c/n^\alpha$, $1-x \ge e^{-2\ln (2) x}$ for $x\in [0,1/2]$, and ${k \choose t} \ge \left(\frac{k}{t}\right)^t$ (see [@motwani95 p. 434]), and (c) is true because $\gamma_n \ge 0$. Since w.h.p. $|S|\ge n/3$, we now have $$\begin{aligned} \nonumber & Pr[|\mathbf Y_k(:,j_{max})|_1 < t] \\ \nonumber \le & Pr\left[\max_{j\in S} |\mathbf Y_k(:,j)|_1 < t \big| |S|\ge n/3 \right]+o(1)\\ \nonumber \le & \left( 1- \left(\frac{c(1-p)}{tn^{\gamma_n}}\right)^t e^{-2\ln (2) c}\right)^{n/3}+o(1)\\ \label{eq:201} \le & e^{-\frac{n}{3}\left(\frac{c(1-p)}{tn^{\gamma_n}}\right)^t e^{-2\ln (2) c} }+o(1) \end{aligned}$$ Suppose we put $t= t_0:=\min\{\sqrt{\log n}, \frac{1}{2\gamma_n}\}$. Then $$\left(\frac{tn^{\gamma_n}}{c(1-p)}\right)^t = \frac{ n^{\gamma_n t} t^t}{(c(1-p))^t} \buildrel (a) \over \le \sqrt n \left(\frac{\sqrt{\log n}}{c(1-p)}\right)^{\sqrt{\log n}}\hspace{-0.1in}=o(n),$$ where (a) follows since $\gamma_nt \le 1/2$ and $t\le \sqrt{\log n}$. Thus from we obtain $$\begin{aligned} & Pr[|\mathbf Y_k(:,j_{max})|_1 < t_0] \le e^{-\frac{1}{o(1)}}+o(1)=o(1).\end{aligned}$$ This proves the first part of the lemma. 2\) Recall that we have assumed $k=n^\alpha g_n$ for $g_n \ge 1$. By following a very similar analysis as in the first part, we see that w.h.p. $|\mathbf Y_k(:,j_{max})|_1 \ge \sqrt{\log n}$. In particular for $g_n=1$ (or equivalently for $k=n^\alpha$), becomes $$\begin{aligned} \nonumber Pr\big[|\mathbf Y_k(:,j)|_1\ge t\big] \ge & \left(\frac{k}{t}\right)^t \left(\frac{c(1-p)}{n^\alpha}\right)^t e^{-2\ln (2) c}\\ \label{eq:555} & =\left(\frac{c(1-p)}{t}\right)^t e^{-2\ln (2) c}. \end{aligned}$$ Observe that for two random variables $X$ and $Y$ such that $X\sim B(n_1,p)$ and $Y\sim B(n_2,p)$ with $n_1 \ge n_2$, we have $Pr[X\ge t] \ge Pr[Y\ge t]$. Thus using we have $$\begin{aligned} Pr\left[|\mathbf Y_k(:,j)|_1\ge t\big|g_n\ge 1\right] &\ge Pr\left[|\mathbf Y_k(:,j)|_1\ge t\big|g_n=1\right] \\ &\ge \left(\frac{c(1-p)}{t}\right)^t e^{-2\ln (2) c}.\end{aligned}$$ Hence for $t=\sqrt{\log n}$, has the following counterpart, $$\begin{aligned} Pr[|\mathbf Y_k(:,j_{max})|_1 < t] \le & e^{-\frac{n}{3}\left(\frac{c(1-p)}{t}\right)^t e^{-2\ln (2) c} }+o(1)\\ &=e^{-\frac{1}{o(1)}}+o(1)=o(1). \end{aligned}$$ But in Lemma \[lemma:102\] we need better bounds for $g_n \rightarrow \infty$, and we consider this case now. Recall that for $j\in S$, $\mu_Y=\mathbb E[|\mathbf Y_k(:,j)|_1 ]=c(1-p)g_n$ and $\sigma_Y^2=Var(|\mathbf Y_k(:,j)|_1 )=g_nc(1-p)(1-(1-\epsilon)(1-p))$. We define $t_n:=\min\{\sigma_Y^{1/4}, \sqrt{\log n}\}$. Since $\sigma_Y\rightarrow \infty$, we have $t_n^6=o(\sigma_Y^2)$, and then Theorem \[thm:large\] implies that for a column $j\in S$, $$\begin{aligned} Pr[|\mathbf Y_k(:,j)|_1 > \mu_Y +t_n \sigma_Y] \doteq& Q(t_n) \buildrel (a)\over \doteq \frac{1}{\sqrt{2\pi}t_n}e^{-t_n^2/2}\\ \ge &\frac{1}{2}\frac{1}{\sqrt{2\pi}t_n}e^{-t_n^2/2}, \text{ for large $n$}\\ &\buildrel (b)\over = \Omega\left(\frac{1}{\sqrt{n \log n}}\right).\end{aligned}$$ where (a) is true because $Q(t)\doteq \frac{1}{\sqrt{2\pi}t}e^{-t^2/2}$ [@Feller1 Lemma 1.2], and (b) is true since $t_n \le \sqrt{\log n}$. Since w.h.p. $|S|\ge n/3$, we have $$\begin{aligned} \nonumber & Pr[|\mathbf Y_k(:,j_{max})|_1 \le \mu_Y +t_n \sigma_Y]\\ \nonumber \le & Pr\left[\max_{j\in S} |\mathbf Y_k(:,j)|_1 \le \mu_Y +t_n \sigma_Y \big| |S|\ge n/3\right]+o(1)\\ \nonumber \le &\left(1-\Omega\left(\frac{1}{\sqrt{n \log n}}\right)\right)^{n/3}+o(1)=o(1). %\nonumber \le &e^{-\Omega\left(\frac{n}{\sqrt{n\log n}}\right)}+o(1) %= o(1).\end{aligned}$$ Thus w.h.p. $|\mathbf Y_k(:,j_{max})|_1 \ge \mu_Y +t_n \sigma_Y$, if $g_n \rightarrow \infty$. We have already observed that w.h.p. $|\mathbf Y_k(:,j_{max})|_1 \ge \sqrt{\log n}$. Thus the lemma is implied. Proof of Lemma \[lemma:102\] {#proof:lemma:102} ---------------------------- Lemma \[lemma:101\] gives us a lower bound for $|\mathbf Y_k(:,j_{max})|_1$ that holds w.h.p.. Next we find an upper bound for $|\mathbf Y_k(:,j_{max})|_0$ to prove Lemma \[lemma:102\]. First we condition on the event that $\mathbf X(1,j_{max})=1$. We observe that $$|\mathbf Y_k(:,j)|_0 \longrightarrow |\mathbf Y_k(:,j)|_1 \longrightarrow \{j_{max}=j\}.$$ Then conditioned on the value of $|\mathbf Y_k(:,j_{max})|_1=t$, the distribution of $|{\mathbf Y_k(:,j_{max})}|_0$ does not depend on the fact that $j_{max}$ is the most popular column chosen by the algorithm, and hence $|{\mathbf Y_k(:,j_{max})}|_0 \sim B\left(k-t, p_0\right)$, where $p_0:=\frac{p(1-\epsilon)}{p(1-\epsilon)+\epsilon}$. This is because for a given column $j$ of $\mathbf Y_k$, upon observing that there are exactly $t$ 1’s, the other $k-t$ entries are i.i.d. with probability of 0 being $p_0$. 1\) Suppose $k=n^{\alpha - \gamma_n}$ such that $\gamma_n\rightarrow 0$. We define $b(k,p,i):={ k\choose i} p^i (1-p)^{n-i}$ to be the $i$th binomial term, and observe that $b(k,p,i)\le \left(kpe/i\right)^i$, since ${k\choose i}\le (ke/i)^i$ (see [@motwani95 p. 434]). We see that $$\begin{aligned} & Pr\left[|{\mathbf Y_k(:,j_{max})}|_0 \ge \frac{\sqrt{\log n}}{2}\right] =\sum_{i=\frac{\sqrt{\log n}}{2}}^{k-t} b(k-t, p_0,i)\\ =& \sum_{i=\frac{\sqrt{\log n}}{2}}^{2\log n}b(k-t, p_0,i) + \sum_{i=2\log n+1}^{k-t} b(k-t, p_0,i)\\ \buildrel (a)\over \le & 2\log n \cdot b\left(k-t, p_0, \frac{\sqrt{\log n}}{2}\right) + k\cdot b(k-t, p_0, 2\log n+1)\\ %& \buildrel (b)\over \le 2\log n \cdot b\left(k, p_0, \frac{\sqrt{\log n}}{2}\right) + k\cdot b(k, p_0, 2\log n+1)\\ \buildrel (b) \over \le &2\log n \left(\frac{(k-t)p_0e}{\sqrt{\log n}/2}\right)^{\frac{\sqrt{\log n}}{2}} \hspace{-0.1in}+\hspace{-0.05in} (k-t) \left(\frac{(k-t)p_0e}{2\log n+1}\right)^{2\log n+1}\\ %\le & 2\log n \left(\frac{kp_0e}{\sqrt{\log n}/2}\right)^{\frac{\sqrt{\log n}}{2}} + k \left(\frac{kp_0e}{2\log n+1}\right)^{2\log n+1}\\ \buildrel (c)\over \le &2 \log n \left(\frac{2c'}{n^{\gamma_n}\sqrt{\log n}}\right)^{\frac{\sqrt{\log n}}{2}} \hspace{-0.1in} + k \left(\frac{c'}{n^{\gamma_n} (2\log n+1)}\right)^{2\log n+1}\\ & = o(1).\end{aligned}$$ where (a) is true since $b(k,p,i)$ is a decreasing function of $i$ for $i$ more than $kp$ and we have $(k-t)p_0=o(1)$, (b) is due to the fact that $b(k,p,i)\le (kpe/i)^i$ , and (c) follows by observing that $kp_0e\le \left(\frac{c'}{n^{\gamma_n}}\right)$ for a constant $c'>0$. Thus w.h.p. we have $|{\mathbf Y_k(:,j_{max})}|_0 < \frac{\sqrt{\log n}}{2}$. Now suppose $\gamma_n > \frac{1}{2\sqrt{\log n}}$. Then we see that $$\begin{aligned} &Pr\left[|{\mathbf Y_k(:,j_{max})}|_0 \ge \frac{1}{4\gamma_n}\right] =\sum_{i=\frac{1}{4\gamma_n}}^{k-t} b(k-t, p_0,i) \\ \le & \sum_{i=\frac{1}{4\gamma_n}}^\infty b(k-t, p_0,i) \buildrel (a)\over \le \sum_{i=\frac{1}{4\gamma_n}}^\infty ((k-t) p_0e/i)^i\\ \buildrel (b)\over \le & \sum_{i=\frac{1}{4\gamma_n}}^\infty \left( \frac{4c' \gamma_n}{n^{\gamma_n}}\right)^i \buildrel (c)\over = \Theta\left(\left( \frac{4c' \gamma_n}{n^{\gamma_n}}\right)^{1/4\gamma_n}\right)\\ \buildrel (d) \over < & \Theta\left(\frac{(4c' \gamma_n)^{\sqrt{\log n}/2}}{n^{1/4}}\right)= o(1),\end{aligned}$$ where (a) is true since $b(k,p,i)\le (kpe/i)^i$, (b) follows because $kp_0e\le \left(\frac{c'}{n^{\gamma_n}}\right)$ for a constant $c'$, (c) is true by observing that for $x=o(1)$, we have $\sum_{i=m}^\infty x^i = \Theta(x^m)$, and (d) follows since $\frac{1}{4\gamma_n} < \frac{\sqrt{\log n}}{2}$ whenever $\gamma_n > \frac{1}{2\sqrt{\log n}}$. Thus we have proved that w.h.p. $|{\mathbf Y_k(:,j_{max})}|_0 < \min \{ \frac{\sqrt{\log n}}{2}, \frac{1}{4\gamma_n}\}$. This together with the observation in Lemma \[lemma:101\] that w.h.p. $|{\mathbf Y_k(:,j_{max})}|_1 \ge \min \{ \sqrt{\log n}, \frac{1}{2\gamma_n}\}$, proves that w.h.p. $|{\mathbf Y_k(:,j_{max})}|_1 -|{\mathbf Y_k(:,j_{max})}|_0$ increases to $\infty$ with $n$. 2\) Now we consider the other case of $k=n^\alpha g_n$ for $g_n\ge 1$. If $g_n$ is upper bounded by a constant, then arguments very similar to those used in the first part tell us that w.h.p. $|{\mathbf Y_k(:,j_{max})}|_0 <\sqrt{\log n}/2$. In the remaining part of the proof, we assume that $g_n\rightarrow \infty$. Recall that for a column $j$ such that $\mathbf X(1,j)=1$, we have $\mu_Y=\mathbb E[|\mathbf Y_k(:,j)|_1 ]=k(1-\epsilon)(1-p)$ and $\sigma_Y^2=Var(|\mathbf Y_k(:,j)|_1 )=k(1-p)(1-\epsilon)(1-(1-\epsilon)(1-p))$. Conditioned on the value of $|\mathbf Y_k(:,j_{max})|_1=t$, suppose $\mu_{\bar Y}$ and $\sigma_{\bar Y}^2$ denote the conditional mean and variance of $|\mathbf Y_k(:,j_{max})|_0$. We observe that for $t\ge \mu_Y$ and large enough $n$, $$\mu_{\bar Y}=(k-t)p_0\le \mu_Y,\text{ and } \sigma_{\bar Y}^2 =(k-t)p_0(1-p_0)\le 2 \sigma_Y^2.$$ Suppose $t_n:=\min\{\sigma_Y^{1/4},\sqrt{\log n}\} $. Since $\sigma_Y\rightarrow \infty$, we have $t_n^6=o(\sigma_Y^2)$, and since w.h.p. $y_1:=|\mathbf Y_k(:,j_{max})|_1\ge \mu_Y$ (see Lemma \[lemma:101\]), using Theorem \[thm:large\] we obtain $$\begin{aligned} & Pr\left[|{\mathbf Y_k(:,j_{max})}|_0 > \mu_{Y}+ \frac{t_n}{2} \sigma_{Y}\right] \\ \le &Pr\left[|{\mathbf Y_k(:,j_{max})}|_0 > \mu_{\bar Y}+ \frac{1}{2\sqrt 2} t_n \sigma_{\bar Y}\big| y_1\ge \mu_{Y}\right] +o(1)\\ & \doteq Q\left(\frac{t_n}{2\sqrt 2 }\right) = o(1). \end{aligned}$$ Thus w.h.p. $|{\mathbf Y_k(:,j_{max})}|_0 \le \max\{\sqrt{\log n}/2, \mu_Y+\frac{t_n}{2}\sigma_Y\}$. This together with the observation made in Lemma \[lemma:101\] that w.h.p. $|{\mathbf Y_k(:,j_{max})}|_1 \ge\max\{\sqrt{\log n}, \mu_Y+t_n\sigma_Y\}$, proves that w.h.p. $|{\mathbf Y_k(:,j_{max})}|_1 - |{\mathbf Y_k(:,j_{max})}|_0$ increases to $\infty$ with $n$. [**Remark:** ]{} In the above proof, we had conditioned on the event that $\mathbf X(1, j_{max})=1$. When we condition on $\mathbf X(1,j_{max})=0$, we have $p_0=\frac{(1-p)(1-\epsilon)}{(1-p)(1-\epsilon)+\epsilon}$, and a similar set of steps prove the claim. Conclusion {#sec:con} ========== We have considered estimation of a binary random field obtained by permuting rows and columns of a block constant matrix, by observing a sub-sampled and noisy version. It would be interesting to analyze the performance of “local” algorithms on a more general class of matrices obtained from realizations of a “smooth” stochastic process. Further, non-uniform sampling models are also of interest. Acknowledgment {#acknowledgment .unnumbered} ============== The work of Kishor Barman was supported by the Infosys fellowship and the Microsoft Research India Travel Grants Program.. The work of Onkar Dabeer was supported by the XI plan funding.

--- author: - | Juan R. Sánchez\ \ Facultad de Ingeniería\ Universidad Nacional de Mar del Plata\ J.B. Justo 4302, Mar del Plata, Argentina.\ *jsanchez@fi.mdp.edu.ar* title: 'A modified one-dimensional Sznajd model' --- [ The Sznajd model is an Ising spin model representing a simple mechanism of making up decisions in a closed community. In the model each member of the community can take two attitudes A or B represented by a spin up or spin down state respectively. It has been shown that, in one-dimension starting from a totally random initial state, three final fixed points can be obtained; all spins up, all spins down or an [*antiferromagnetic*]{} state in which each site take a state which is opposite from its two nearest neighbors. Here, a modification of the updating rule of the Sznajd model is proposed in order to avoid such antiferromagnetic state since it is considered to be an [*unrealistic*]{} state in a real community. ]{} The Sznajd model is a successful Ising spin model describing a simple mechanism of making up decisions in a closed community. The model allow each member of the community to have two attitudes, to vote for option A or to vote for option B. These two attitudes are identified with the state of spins variables up or down respectively. A dynamic is established in the model in which a selected pair of adjacent spins influence their nearest neighbors through certain rules, applied in a random sequential manner. In several votes (units of evolution time) some difference $m$ of voters for A and against is expected. The dynamic rules of the Sznajd model are [@1] – if $S_iS_{i+1}=1$ then $S_{i-1}$ and $S_{i+2}$ take the direction of the selected pair \[i,i+1\], ($r1$) – if $S_iS_{i+1}=$ -$1$ then $S_{i-1}$ takes the direction of $S_{i+1}$ and $S_{i+2}$ the direction of $S_i$, ($r2$) being $S_i$ the state of the spin variable at site $i$. These rules describe the influence of a given pair of members of the community on the decision of its nearest neighbours. In one dimension, the original rules give rise to three limiting cases in the evolution of the system \(i) all members of the community vote for $A$ (all spins up), \(ii) all members of the community vote for $B$ (all spins down), \(iii) $50$% vote for $A$ and $50$% vote for $B$ ([*alternating*]{} state). Here, attention is paid to the last limiting antiferromagnetic case (iii). This antiferromagnetic case, although posible in other spins systems, can be considered to be quite [*unrealistic*]{} in a model trying to represent the behavior of a community. To achieve [**exactly**]{} a $50$-$50$ final state in a community is almost impossible, specially if it is composed by more than a few dozens of members. [@2] On the other hand such antiferromagnetic state implies that each member of the community is surrounded by a neighbor which has an opposite opinion. A quite “uncomfortable" situation, certainly. From a simulational point of view, if the evolution of a one-dimensional Sznajd model is [**started**]{} from an antiferromagnetic state, i.e., a chain of neighbors with opposite opinions, the original dynamic rules does not give rise to any evolution at all. In order to avoid the unrealistic $50$-$50$ alternating final state, new dynamic rules are proposed: – if $S_i S_{i+1} = 1$ then $S_{i-1}$ and $S_{i+2}$ take the same direction of the pair $[i,i+1]$, ($r1$) – if $S_i S_{i+1} =$ -$1$ then $S_{i}$ take the direction of $S_{i-1}$ and $S_{i+1}$ take the direction of $S_{i+2}$. ($r2$) Using the new rules, in case of disagreement of the pair $S_i$-$S_{i+1}$, rule $r2$ make the spin $i$ to “feel more confortable" since it ends up with at least one neighbor having its own opinion. Two samples of evolution of a system following the new rules and starting from an antiferromagnetic state are shown in Fig. 1, for a $N=100$ lattice size. It can be seen that the $50$-$50$ final state in completely avoided and that the other two types of total agreement (ferromagnetic) final states can be achieved, with equal probability, starting the systems from the same initial condition. Time $t$ is advanced by one when each spin of the lattice has had one (probabilistic) opportunity to be updated. Finally the scaling properties of the new model are tested by calculating the scaling exponent of the number of spins that does change their state with time. The value of this exponent has been shown to be $3/8$ for the original Szanjd model. [@3; @4] In Fig. 2 a log-log plot of the evolution of the number of spins in remaining the same state at time $t$ is shown for the original Sznajd model and for the new model proposed here. Plots of Fig. 2 were obtained from simultaneous simulations of both models, using the same random numbers for update each lattice starting from the same initial condition. See figure caption for the parameters used in simulations. It can be seen that the model proposed here share the same type of scaling features as the original Sznajd model, but the value for the scaling exponent seems to be different. Although, more detailed simulations would be needed in order to verify exactly this last statement. The author appreciate the critical review of the manuscript by Prof. D. Stauffer. [000]{} Sznajd-Weron K. and Sznajd J., arXiv:cond-mat/0101130. The skeptic reader is referred to the year 2000 USA Presidential election. Stauffer D., Int. J. Mod. Phys. C [**11**]{}, 1157 (2000) Stauffer D. and d Oliveira P.M.C., Eur. J. Phys. [**30**]{}, 587 (2002)

--- abstract: 'Using a series of three-dimensional, hydrodynamic simulations on an adaptive grid, we have performed a systematic study on the effect of bubble-induced motions on metallicity profiles in clusters of galaxies. In particular, we have studied the dependence on the bubble size and position, the recurrence times of the bubbles, the way these bubbles are inflated and the underlying cluster profile. We find that in hydrostatic cluster models, the resulting metal distribution is very elongated along the direction of the bubbles. Anisotropies in the cluster or ambient motions are needed if the metal distribution is to be spherical. In order to parametrise the metal transport by bubbles, we compute effective diffusion coefficients. The diffusion coefficients inferred from our simple experiments lie at values of around $\sim 10^{29}$ cm$^2$s$^{-1}$ at a radius of 10 kpc. The runs modelled on the Perseus cluster yield diffusion coefficients that agree very well with those inferred from observations.' author: - | E. Roediger$^{1}$, M. Brüggen$^{1}$, P. Rebusco$^{2}$, H. Böhringer$^{3}$, E. Churazov$^{2,4}$\ $^1$ International University Bremen, P.O. Box 750561, 28725 Bremen, Germany\ $^2$ Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Strasse 1, 85741 Garching, Germany\ $^3$ MPI für Extraterrestrische Physik, P.O. Box 1603, 85740 Garching, Germany\ $^4$ Space Research Institute (IKI), Profsoyuznaya 84/32, Moscow 117810, Russia bibliography: - | % ../BIBLIOGRAPHY/radio.bib - | % ../BIBLIOGRAPHY/metals%.bib date: 'Accepted. Received; in original form ' --- \[firstpage\] Acknowledgements {#acknowledgements .unnumbered} ================ We thank Mateusz Ruszkowski and Matthias Hoeft for helpful discussions. The anonymous referee has made a number of very useful suggestions to improve the paper. Furthermore, we acknowledge the support by the DFG grant BR 2026/3 within the Priority Programme “Witnesses of Cosmic History” and the supercomputing grants NIC 1927 and 1658 at the John-Neumann Institut at the Forschungszentrum Jülich. Some of the simulations were produced with STELLA, the LOFAR BlueGene/L System in Groningen. The results presented were produced using the FLASH code, a product of the DOE ASC/Alliances-funded Center for Astrophysical Thermonuclear Flashes at the University of Chicago. \[lastpage\]

--- author: - Thomas Peternell - 'Andrew J. Sommese' date: 'Dedicated to [*Robin Hartshorne*]{} on his 60th birthday' title: Ample Vector Bundles and Branched Coverings --- 22.5cm =-14mm =-13mm =msbm10 =msbm7 =msbm5 ===\#1[[\#1]{}]{} ptsize=0 ptsize=2 \[section\] \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Question]{} \[theorem\][Remark]{} \[theorem\][Setup]{} \[theorem\][Basic Observation]{} \[theorem\][Definition]{} \[theorem\][Conjecture]{} \[theorem\][The Exceptional Cases]{} \[theorem\][Problem]{} Introduction {#introduction .unnumbered} ============ Let $f: X \to Y$ be a finite surjective morphism of degree $d$ between $n$-dimensional connected projective manifolds $X$ and $Y$. Let ${{\cal E}}:= [(f_*{{\cal O}}_X)/{{\cal O}}_Y]^*$. It is a beautiful theorem of Lazarsfeld [@La80] that when $Y$ is ${{\Bbb P}^{n}}$, then for any such $f$ and $X$, the vector bundle ${{\cal E}}(-1)$ is spanned. There have been many partial generalizations of this result for homogeneous $Y$, e.g., [@De95; @De96; @Ki96a; @Ki96b; @Ma97; @KM98]. In this article we investigate spannedness and ampleness properties for not necessarily homogeneous $Y$. ${{\cal E}}$ has a number of tendencies towards positivity. For example, using relative duality ${{\cal E}}\cong f_*\omega_{X/Y}/{{\cal O}}_Y$, and thus ${{\cal E}}$ is weakly positive in the sense of Viehweg, e.g., [@Vi95]. In particular this implies the nefness of ${{\cal E}}$ for $Y$ a curve. We also prove this for $Y$ an abelian surface. Moreover, ${{\cal E}}\otimes{{\cal E}}$ always has a nontrivial section, coming from a natural linear map from ${{\cal E}}^*$ to ${{\cal E}}$ induced by $f$, which is an isomorphism over a dense Zariski open set of $Y$, see Lemma (\[genericIso\]). Though it follows from the weak positivity that ${{\cal E}}$ is nef when $\dim Y=1$, it is easy to construct examples with ${{\cal E}}$ not even nef. Indeed, if $f$ expresses $X$ as a double cover of $Y$, then ${{\cal E}}\cong L$ where the discriminant locus is an element of $|2L|$. Thus if $Y$ possesses a nonnef line bundle with some even power having a smooth section, then we can construct a covering with ${{\cal E}}$ not nef. Even when ${{\cal E}}$ is nef, ampleness can fail in a number of different ways. For example, if $Y$ possesses an unramified finite cover $f:X\to Y$ of degree $d>1$, then ${{\cal E}}$ is nef with no sections, but with all Chern classes zero (see Theorem (\[unramifiedCase\])). Thus the best possible result for ampleness would be, “if ${{\cal E}}$ is nef, and $f$ does not factor through an unramified covering of $Y$, then ${{\cal E}}$ is ample.” The first place to check this result is for curves, since in this case ${{\cal E}}$ is always nef. It is, in fact, true for $Y={{\Bbb P}^{1}}$ by Lazarsfeld’s result, and for $Y$ an elliptic curve by a result of Debarre [@De96]. Our Theorem (\[curveTheorem\]) shows that this result holds for arbitrary smooth curves. Unfortunately, in dimensions $\ge 2$, the double cover construction shows that ${{\cal E}}$ can fail to be ample, even “if ${{\cal E}}$ is nef, and $f$ does not factor through an unbranched covering of $Y$”. Some of the restrictions imposed by the hypothesis that ${{\cal E}}$ is ample for all covers are given in Theorem (\[rarelyAmple\]) and Corollary (\[rarelyAmpleCor\]). For example, the only smooth connected projective surfaces $S$, which stand any chance of having the property that for all finite surjective morphisms from connected projective surfaces to $S$, the associated bundle ${{\cal E}}$ is ample, are ${{\Bbb P}^{2}}$; a minimal $K3$-surface with no $-2$ curves; or a surface with an ample canonical bundle, for which there are neither nontrivial unramified covers over it, nor nonfinite morphisms from it onto positive dimensional varieties. In §\[surfaceSection\], we show in Theorem (\[delPezzoSurfaceTheorem\]) that ${{\cal E}}$ is always spanned for $Y$ a smooth Del Pezzo surface with $K_Y^2\ge 5$. We show in Theorem (\[spannedOverHirz\]) that ${{\cal E}}$ is spanned over the Hirzebruch surfaces ${\Bbb F}_r$, under the added condition that if $r\ge 2$, then the section $E$, of the projection $p: Y\to {{\Bbb P}^{1}}$, with negative self-intersection does not lie in the discriminant locus of the covering. We also show in Theorem (\[ampleOverHirz\]) that ${{\cal E}}$ is ample over the Hirzebruch surfaces ${\Bbb F}_r$, under the further condition that the general fiber $F$, of the projection $p: Y\to {{\Bbb P}^{1}}$, has connected inverse image under the given covering. The conditions are also shown to be necessary. In §\[delPezzoSection\], we show in Theorem (\[delPezzoManifoldTheorem\]) that ${{\cal E}}$ is always spanned for $Y$ a smooth Del Pezzo manifold of dimension $n\ge3$ when $b:=H^n\ge 5$. Here by definition $-K_Y=(n-1)H$ for some ample line bundle $H$ on $Y$. We also show Theorem (\[weakExistenceTheorem\]), which states that $h^0({{\cal E}})\not = 0$ under very mild conditions, i.e., if $b$ times the degree of the covering is at least five. Finally we ask when $h^0({{\cal E}}\otimes (-H))\not =0$. Using adjunction theory, we show in Theorem (\[minuHExistenceTheorem\]) and (\[exceptions\]) that this is so for branched covers of Del Pezzo manifolds, except in very rare circumstances. For our induction arguments, we were forced to consider branched covers over manifolds $Y$ with domains $X$ merely normal Cohen-Macaulay with finite irrational locus, i.e., for each point $x$ of $X$, the local rings of germs of holomorphic (or algebraic) functions at $x$ are Cohen-Macaulay and normal, with the set of nonrational singularities of $X$ finite. One interesting aspect of our work is that many results, e.g., Lazarsfeld’s original result, hold at this level of generality. We dedicate this paper to Robin Hartshorne. His work has significantly enriched algebraic geometry. It has also been a stimulus for our work: indeed, a main result of this paper is based primarily on Robin Hartshorne’s beautiful characterization [@Ha71] of ample vector bundles on curves. We thank the Department of Mathematics and the Duncan Chair of the University of Notre Dame for making our collaboration possible. We also thank Meeyoung Kim and Robert Lazarsfeld for helpful comments on this paper. In particular, Robert Lazarsfeld suggested using Castelnuovo-Mumford regularity in Theorem \[old1.17\] to improve our original result about nefness of ${{\cal E}}$ off the branch locus. He also suggested Problem \[lazarsfeldProblem\]. The second author thanks the Department of Mathematics of the K.T.H. (Royal Institute of Technology) of Stockholm, Sweden, where this paper was finished. General results on branched coverings ===================================== In this paper we work over ${{\Bbb C}}$. By a variety we mean a complex analytic space, which might be neither reduced or irreducible. Let $f: X\to Y$ be a surjective finite morphism between $n$-dimensional projective manifolds. Let ${{\cal E}}:= [(f_*{{\cal O}}_X)/{{\cal O}}_Y]^*$. Here are some of the known results on the structure of ${{\cal E}}$. 1. If $Y = {{\Bbb P}^{n}},$ then ${{\cal E}}$ is ample and spanned by Lazarsfeld [@La80]. 2. If $Y$ is a quadric, then ${{\cal E}}$ is spanned by [@Ki96b] and ${{\cal E}}$ is ample if $3 \leq n \leq 6$ by [@KM98]. 3. If $Y$ is a Grassmannian, then ${{\cal E}}$ is spanned, and $k$-ample [@Ki96a], and in fact ample [@Ma97]. 4. By [@KM98], it follows that if $Y$ is a Lagrangian Grassmannian, then ${{\cal E}}$ is ample; and that for some other homogeneous $Y$, ${{\cal E}}$ is spanned. 5. ${{\cal E}}$ is ample if $Y$ is an elliptic curve and if $f$ does not factor through an étale map [@De96]. We need generalizations of Lazarsfeld’s theorem for mildly singular $X$. The general setup of these results carries over to Cohen-Macaulay $X$. Note that reduced curves have at worst Cohen-Macaulay singularities. \[setup\]Let $f:X\to Y$ be a finite, degree $d$, morphism from a reduced pure $n$-dimensional projective variety $X$ onto an $n$-dimensional projective manifold $Y$. Assume that $X$ has at worst Cohen-Macaulay singularities. Since $f$ is flat, $f_*({{\cal O}}_X)$ is locally free of rank $d.$ By the trace map, $f_*({{\cal O}}_X)$ has a canonical direct summand ${{\cal O}}_Y$; therefore we can define a locally free sheaf ${{\cal E}}_f$ of rank $d-1$ by setting $$f_*({{\cal O}}_X) = {{\cal E}}_f^* \oplus {{\cal O}}_Y.$$ When the map $f$ is obvious from the context, we usually denote the sheaf ${{\cal E}}_f$ as ${{\cal E}}$. We make the above singularity assumption on $X$, not because we want to state the most general theorems, but rather because we will need the more general choice of $X$ in induction procedures. We will often make the added assumption that $X$ is normal with the irrational locus ${{\cal I}}(X)$ finite, i.e., with the locus of nonrational singularities finite. This assumption guarantees that the crucial vanishing theorems are still valid [@So85]. We need the following slight variant of the results in [@So85]. \[almostKodaira\]Let $X$ be a normal $n$-dimensional projective variety with at worst Cohen-Macaulay singularities. Assume that the irrational locus of $X$ is finite. Assume that $L$ is ample and $D$ is an effective (possibly empty) reduced Cartier divisor. Assume that either: 1. $n\le 2$; or 2. $D$ is normal with finite irrational locus. Then $h^i(-L-D)=0$ for $i<n$. [[**Proof. **]{}]{}If $D$ is empty, this is part of the Kodaira vanishing theorem of [@So85]. For $n=1$, the result follows immediately from Serre duality. Consider the sequence: $$0\to \omega_X\otimes L\to \omega_X\otimes L\otimes {{\cal O}}_X(D)\to \omega_D\otimes L_D\to 0.$$ If $n=2$ then by Serre duality we have $h^1(\omega_D\otimes L_D)=0$ and thus using [@So85], $h^i(\omega_X\otimes L\otimes {{\cal O}}_X(D))=0$ for $i\ge 1$. This shows the lemma when $n=2$. If $n>2$ then we obtain the result by using the results of [@So85] on $D$ and $X$. [[Q.E.D. plus 1pt]{}]{} It is conjectured that ${{\cal E}}$ is always ample if $Y$ is rational homogeneous with $b_2 = 1$ [@KM98] or if $Y$ is simple abelian and $f$ does not factor through an étale map [@De96]. First we make a technically very useful generalization of Lazarsfeld’s theorem to the Cohen-Macaulay case. \[newLazarsfeld\] Let $f : X \to {{\Bbb P}^{n}}$ be a finite surjective morphism from a pure dimensional reduced projective variety $X$ with at worst Cohen-Macaulay singularities. Assume that either $n=1$ or that $X$ is normal with finite irrational locus. Then ${{\cal E}}$ is spanned. If further $X$ is either irreducible or a connected curve, then ${{\cal E}}(-1)$ is spanned, and, in particular, ${{\cal E}}$ is ample. [[**Proof. **]{}]{}First assume that $n=1$. Then the spannedness of ${{\cal E}}$ will follow if $h^1({{\cal E}}(-1))=0$. We have $$h^1({{\cal E}}(-1))=h^0({{\cal E}}^*(-1))=h^0((f_*{{\cal O}}_X)(-1))= h^0(f^*{{{\cal O}}_{{\Bbb P}^{1}}({-1})})=0.$$ The spannedness of ${{\cal E}}(-1)$ will follow from $h^1({{\cal E}}(-2))=0$. We have $$h^1({{\cal E}}(-2)=h^0({{\cal E}}^*)=h^0(f_*{{\cal O}}_X)-1=h^0({{\cal O}}_X)-1.$$ Now note that if $X$ is connected, then $h^0({{\cal O}}_X)=1$. To show that ${{\cal E}}$ is spanned for a given $n>1$, we assume by induction that the result is true for dimensions $\le n-1$. Choose an arbitrary point $y\in {{\Bbb P}^{n}}$. Choose a general $D\in |{{{\cal O}}_{{\Bbb P}^{n}}({1})}|$ passing through $y\in {{\Bbb P}^{n}}$. ${{\overline {D}}}:=f^{-1}(D)$ is reduced, with at worst Cohen-Macaulay singularities. By induction the sheaf ${{\cal E}}_{f_{{{\overline {D}}}}}$ is spanned. Moreover the irrational locus is finite. Then by [@So85] (see Lemma (\[almostKodaira\])) it follows that $$H^1({{\cal E}}(-1)) =H^{n-1}(X,f^*{{{\cal O}}_{{\Bbb P}^{n}}({-n})}) = 0,$$ and thus the map $H^0({{\cal E}})\to H^0({{\cal E}}_D)\to 0$. Finally note that ${{\cal E}}_{f_{{{\overline {D}}}}}\cong {{\cal E}}_D$ (see (\[simpleFact\]). To show that ${{\cal E}}(-1)$ is spanned for a given $n>1$, we assume by induction that the result is true for dimensions $\le n-1$. Choose an arbitrary point $y\in {{\Bbb P}^{n}}$. Choose a general $D\in |{{{\cal O}}_{{\Bbb P}^{n}}({1})}|$ passing through $y\in {{\Bbb P}^{n}}$. ${{\overline {D}}}:=f^{-1}(D)$ is reduced, with at worst Cohen-Macaulay singularities. Moreover [@So86], ${{\overline {D}}}$ is irreducible if $\dim D\ge 2$ and connected if $\dim {{\overline {D}}} =1$. Then by [@So85] it follows that $$H^1({{\cal E}}(-2)) =H^{n-1}(X,f^*{{{\cal O}}_{{\Bbb P}^{n}}({-n+1})}) = 0.$$ Thus the map $H^0({{\cal E}}(-1))\to H^0({{\cal E}}_D(-1))\to 0$. Now use ${{\cal E}}_{f_{{{\overline {D}}}}}(-1)\cong {{\cal E}}_D(-1)$. [[Q.E.D. plus 1pt]{}]{} By the same argument as in Theorem (\[newLazarsfeld\]) we obtain Let $f : X \to Q_n$ be a finite surjective morphism from a pure dimensional reduced projective variety $X$ with at worst Cohen-Macaulay singularities onto a smooth quadric $Q_n$. Assume further that $X$ is normal with finite irrational locus. Then ${{\cal E}}$ is spanned. We will use (and have already used) the following easy fact several times. \[simpleFact\] Let $f : X \to Y$ be a finite surjective morphism from a pure dimensional reduced projective variety $X$, which is locally a complete intersection, onto a smooth projective manifold $Y$. Let $C \subset Y$ be a smooth (or a locally complete intersection) curve. Then, letting $X_C := f^{-1}(C)$, ${{\cal E}}\vert C = (f\vert X_C)_*(\omega_{X_C/C}) / {{\cal O}}_C.$ [[**Proof. **]{}]{}Notice that $X_C$ might not be smooth, however $X_C \subset X$ is locally a complete intersection, so that ${\omega_{X/Y}}$ makes good sense. Then our claim follows immediately from the adjunction formulas for $C \subset Y$ and $X_C \subset X$ and from the relation for the normal bundles $ N_{X_C\vert X} = (f\vert C)^*N_{C\vert Y}$. [[Q.E.D. plus 1pt]{}]{} Let ${{\cal G}}{{\cal S}}$ (respectively ${{\cal G}}{{\cal S}}'$) denote the set of all projective manifolds $Y$ with the property that given a finite surjective morphism $f:X\to Y$ from a pure dimensional projective manifold $X$ (respectively normal Cohen-Macaulay variety $X$ with finite irrational locus) to $Y$, $H^0(Y,{{\cal E}}_f)$ spans ${{\cal E}}_f$ over a dense Zariski open set of $Y$. Then it follows that given $Y\in {{\cal G}}{{\cal S}}$ (respectively ${{\cal G}}{{\cal S}}'$), $A\times Y\in {{\cal G}}{{\cal S}}$ (respectively ${{\cal G}}{{\cal S}}'$) for $A$ either projective space or the smooth quadric. This follows by the same induction used above, and the fact that given a semiample bundle $L$ on a pure $n$-dimensional normal irreducible Cohen-Macaulay projective variety $X$ with ${{\cal I}}(X)$ finite, we have the vanishing $H^j(K_X+L)=0$ for $j>n- \kappa(L)$, where $\kappa(L)$ is the Kodaira dimension of $L$. Let $f:X\to Y$ be a finite morphism from a reduced pure $n$-dimensional Cohen-Macaulay variety $X$ onto a connected projective manifold $Y$. There are many instances when ${{\cal E}}_f$ is spanned. In this case, what can we say about the $k$-ampleness of ${{\cal E}}_f$ in the sense of [@So78]? Clearly ${{\cal E}}$ is $(n-1)$-ample if and only if $X$ is connected. To see this, just note that since ${{\cal E}}_f$ is assumed spanned, it follows that if it is not $k$-ample, then there exists an irreducible subvariety $Z\subset Y$ of dimension at least $k+1$ and a trivial summand of ${{\cal E}}_{f|Z}$. Thus $h^0([f_*{{\cal O}}_X]_Z)\ge 2$. So if ${{\cal E}}_f$ is not $(n-1)$-ample, we have $h^0({{\cal O}}_X)=h^0(f_*{{\cal O}}_X)\ge 2$, which combined with $X$ being reduced, shows that $X$ is not connected. The same argument implies the following “criterion” for ampleness of ${{\cal E}}_f$ when $\dim Y=2$: “if ${{\cal E}}_f$ is spanned, then ${{\cal E}}_f$ is ample if the restriction of ${{\cal E}}_f$ to the branch locus of $f$ is ample and if inverse images under $f$ of irreducible curves are always connected.” The key point of the argument is that if $\dim Y=2$, then an irreducible and reduced curve $C$ on $Y$ is Cartier. Thus $C'$, the inverse image of $C$ under $f$, is Cartier. Therefore, $C'$ is Cohen-Macaulay (since effective Cartier divisors on a Cohen-Macaulay variety are Cohen-Macaulay). If $C$ is not contained in the ramification locus of $f$, it follows that $C'$ is generically reduced, and therefore (since $C'$ is Cohen-Macaulay) it is reduced. Thus we conclude that $h^0({{\cal E}}^*_{f|C})=0$ if and only if $C'$ is connected. Here is a general result giving “generic semipositivity" of ${{\cal E}}$, under the assumptions in (\[setup\]) plus the assumption that $X$ is smooth. Let $f : X \to Y$ be a finite surjective morphism from a pure dimensional reduced projective variety $X$, which is locally a complete intersection, onto a smooth projective manifold $Y$. $f_*(\omega_{X/Y}) \simeq {{\cal E}}\oplus {{\cal O}}_Y.$ [[**Proof. **]{}]{}Notice first that $f_*(\omega_{X/Y}) $ is locally free of rank $d,$ which follows immediately from the flatness of $f$ via standard cohomology theorems. Then by relative duality $$f_*({\omega_{X/Y}})^* \simeq f_*({{\cal O}}_X)$$ which gives the claim. [[Q.E.D. plus 1pt]{}]{} Our purpose is to generally investigate how “positive" the vector bundle ${{\cal E}}$ could be. To state a general result, we recall the followingdefinition due to Viehweg. [Let $Y$ be a projective manifold and ${{\cal F}}$ a locally free sheaf on $Y.$ Then ${{\cal F}}$ is [*weakly positive*]{} if and only if for all ample line bundles $H$ and all positive integers $a$ the bundle $S^{ab}{{\cal F}}\otimes H^b$ is generically spanned for $b \gg 0.$ ]{} To make this definition a little more suggestive we weaken this notion in the following form [We say that ${{\cal F}}$ is [*generically nef,*]{} if there exists a countable union $A = \bigcup _i A_i$ of proper analytic sets in $X$ such that ${{\cal F}}\vert C$ is nef for all curves $C \not \subset A.$]{} For a discussion of generically nef bundles we refer to [@DPS99]. We clearly have the following result. If ${{\cal F}}$ is weakly positive, then ${{\cal F}}$ is generically nef. Let $f:X\to Y$ be a finite surjective morphism between projective manifolds. Then the bundle ${{\cal E}}$ is weakly positive, in particular it is generically nef. [[**Proof. **]{}]{}Having in mind (1.5), this is just a special case of a result in [@Vi82]. [[Q.E.D. plus 1pt]{}]{} We shall see below that “in general" ${{\cal E}}$ will be neither ample nor spanned nor even nef, even if $Y$ is a surface, e.g., due to the existence of $(-1)$-curves. Nevertheless there is some tendency towards ${{\cal E}}$ having sections. In (1.22) below we see by a very simple argument that ${{\cal E}}\vert C$ is nef fo the general smooth complete intersection curve $C.$ \[genericIso\]Let $f:X\to Y$ be a finite surjective morphism between projective manifolds. Then, there is an exact sequence $$0\to {{\cal E}}^*\to {{\cal E}}\to {{\cal S}}\to 0,$$ with the reduced support of ${{\cal S}}$ equal to the image under $f$ of the ramification divisor $B\subset X$. Thus $h^0({{\cal E}}\otimes {{\cal E}}) \ge 1$ and there is a section of $2\det{{\cal E}}$ with zero set equal set theoretically to $f(B)$. [[**Proof. **]{}]{}Since the higher direct images under $f$ of coherent sheaves are $0$, the direct image of the exact sequence $$0\to {{\cal O}}_X\to \omega_{X/Y} \to {{\cal O}}_X(B)\to 0,$$ gives the desired exact sequence. Since the map from ${{\cal E}}^*$ to ${{\cal E}}$ is an isomorphism over a Zariski dense open set, we conclude that we have a sheaf map from $-\det {{\cal E}}$ to $\det {{\cal E}}$, with cokernel having support equal to the image under $f$ of the ramification divisor $B\subset X$. Thus $2\det{{\cal E}}$ has a section which vanishes on the set theoretic image $f(B)$ under $f$ of the ramification divisor $B\subset X$. [[Q.E.D. plus 1pt]{}]{} There is some hope for ampleness if ${\rm Pic}(Y) = {\Bbb Z}.$ Let $f:X\to Y$ be a finite surjective morphism between $n$-dimensional projective manifolds. Assume that $f$ is not an unramified cover. If the Picard number $\rho(Y) = 1,$ then $\det {{\cal E}}$ is ample. [[**Proof. **]{}]{}By Lemma (\[genericIso\]), $2\det{{\cal E}}$ has a section which vanishes on the set theoretic image $f(B)$ under $f$ of the ramification divisor $B\subset X$. Since $f$ is not an unramified cover, $f(B)$ is nonempty, and $\det {{\cal E}}$ is ample. [[Q.E.D. plus 1pt]{}]{} Using the relative Riemann-Roch theorem it follows that given a finite degree $d$ surjective morphism between $n$-dimensional projective manifolds, $$c_1({{\cal E}})=\frac{f_*(\omega_{X/Y})}{2}=\frac{f_*(B)}{2},$$ where $B$ is the branch locus of $f$. Similarly, the higher Chern classes of ${{\cal E}}$ can be computed. For example, when $n=2$, $\displaystyle c_2({{\cal E}})= d\chi({{\cal O}}_Y)-\chi({{\cal O}}_X)+\frac{(\omega_Y+\det{{\cal E}})\cdot\det{{\cal E}}}{2}.$ Next we collect some trivial but basic results on the cohomology of ${{\cal E}}$ which will be used again and again. Let $f: X\to Y$ be a finite morphism from a pure dimensional projective variety $X$ with at worst Cohen-Macaulay singularities onto a projective manifold $Y$. Then 1. $ h^q(X,{\omega_{X/Y}}) = h^q(Y,{{\cal E}}) \oplus H^q(Y,{{\cal O}}_Y).$ 2. $h^0(Y,{{\cal E}}) = h^0(X,{\omega_{X/Y}}) - 1 $ 3. $H^q(X,{{\cal O}}_X) = H^q(Y,{{\cal E}}^*) \oplus H^q(Y,{{\cal O}}_Y).$ 4. If $Y$ is Fano and $X$ satisfies the additional conditions of being normal with finite irrational locus, then $H^q(Y,{{\cal E}}) = 0$ for $q \geq 1.$ 5. If $Y$ is Fano, and $X$ satisfies the additional conditions of being normal with finite irrational locus, and $-\omega_X = rH$ with $r \geq 2$ (so $X$ has index at least 2), then $H^q(Y,{{\cal E}}(-H)) = 0$ for $q \geq 1.$\ [[**Proof. **]{}]{}(1), (2) and (3) being clear, (4) and (5) follow by Kodaira’s vanishing theorem. [[Q.E.D. plus 1pt]{}]{} Notice that (5) implies that if $h^q({{\cal O}}_X) > h^q({{\cal O}}_Y)$ for some $1 \leq q \leq n-1,$ then ${{\cal E}}$ cannot be Nakano positive. We will investigate ${{\cal E}}$ mostly on Fano manifolds of large index and in low dimensions. Here is one result of more general nature. \[old1.17\] Let $f:X\to Y$ be a finite surjective morphism from the projective manifold $X$ of dimension $n$ to the abelian variety $Y$. Then ${{\cal E}}_f$ is nef. [[**Proof. **]{}]{}This is an easy adoption of the proof of [@DPS94 3.21]. Adopting the arguments of (ibid.(ii)) word by word, we need to prove the following.\ Let $G$ be a very ample line bundle on $Y.$ Then $f_*(\omega_X) \otimes G^{n+1}$ is spanned.\ This follows immediately from noting that $f_*(\omega_X) \otimes G^{n+1}$ is Castelnuovo-Mumford $n$-regular. Indeed, let $y$ be a point of $Y$. Choose $n$ sections $s_1,\ldots,s_n$ of $G$, whose scheme-theoretic zero set includes $y$ as a nonsingular component. Let $K^\bullet$ denote the tensor the Koszul complex associated to the section $s_1\oplus\cdots\oplus s_n$ of $\displaystyle G^{\oplus n}$ with $f_*(\omega_X) \otimes G$. Applying Kodaira vanishing to the hypercohomology of $K^\bullet$ shows that $f_*(\omega_X) \otimes G^{n+1}$ is spanned. Then introduce $\lambda : Y \to Y, z \to 2z $, make the base change $\lambda^p$ and argue as in the second part of [@DPS94].\ [[Q.E.D. plus 1pt]{}]{} If $Y$ is the product of an abelian variety and projective spaces, then ${{\cal E}}$ is nef. \[lazarsfeldProblem\] Let $f:X\to Y$ be a finite surjective morphism between $n$-dimesnional projective manifolds. Is it true that ${{\cal E}}_f$ is always nef modulo the discriminant locus D, i.e., ${{\cal E}}\vert C$ is nef for all curves $C \not \in D$? The following summarizes the general things that we know in the unramified case. \[unramifiedCase\] Let $f : X \to Y$ be a finite unramified morphism from a connected $n$-dimensional projective manifold $X$ onto a connected $n$-dimensional projective manifold $Y$. Then 1. ${{\cal E}}\cong{{\cal E}}^*$ and $h^0({{\cal E}})=0$; 2. ${{\cal E}}$ is nef; 3. the $i$-th Chern class $c_i({{\cal E}})$ vanishes for $i\ge 1$; and 4. ${{\cal E}}$ has a filtration by hermitian flat vector bundles. [[**Proof. **]{}]{}Since $\omega_X\cong f^*\omega_Y$, we have that $\omega_{X/Y}\cong {{\cal O}}_X$. Taking direct images we conclude that $${{\cal E}}^*\oplus {{\cal O}}_Y\cong f_*({{\cal O}}_X)\cong f_*(\omega_{X/Y})\cong {{\cal E}}\oplus {{\cal O}}_Y,$$ and therefore that ${{\cal E}}^*\cong{{\cal E}}$. To see that $h^0({{\cal E}})=0$, simply note that $$h^0({{\cal E}})=h^0({{\cal E}}^*)=h^0({{\cal O}}_X)-h^0({{\cal O}}_Y)=0.$$ To see nefness of ${{\cal E}}$, it is equivalent to show nefness of $(f_*{{\cal O}}_X)^*$. By definition this comes down to showing that there is no finite morphism $g:C\to Y$ of an irreducible and reduced projective curve $C$ to $Y$ with the property that there is a surjection $\displaystyle g^*[V^*]\to {{\cal L}}^*\to 0$ of the pullback under $g$ of the vector bundle $V^*$ associated $(f_*{{\cal O}}_X)^*$ onto ${{\cal L}}^*$ a line bundle on $C$ with $\deg{{\cal L}}>0$. We argue by contradiction. Assume otherwise that there is such a surjection, or equivalently that there is an injection $$0\to {{\cal L}}\to g^*(f_*{{\cal O}}_X).$$ We have an inner product defined on $f_*{{\cal O}}_X$ by using the trace. Given two elements $s,t$ of the fiber of $V$ at $y\in Y$, define $(s,t)={\rm tr}(s\overline t)$, where, letting ${{\cal I}}_{f^{-1}(y)}$ denote the ideal sheaf of the fiber of $f$ over $y$, we identify $s,t$ with the corresponding elements of $\displaystyle{{\cal O}}_X/{{\cal I}}_{f^{-1}(y)}$. Define the function $h:V\to {{\Bbb R}}$ by sending $s\in V$ to $(s,s)$. This function is a plurisubharmonic exhaustion of $V$, since $f$ is unramified. In particular the restriction of $h$ to ${{\cal L}}$ is a nonconstant plurisubharmonic function on ${{\cal L}}$. But, since ${{\cal L}}$ has positive degree, any plurisubharmonic function on ${{\cal L}}$ is constant. This contradiction shows the nefness of ${{\cal E}}$. Since the metric $(s,t)$ is flat, i.e., its curvature form vanishes, we conclude that the Chern classes $c_i({{\cal E}})$ of ${{\cal E}}$ with $i>0$ vanish. For the last statement we refer to \[DPS94\]. [[Q.E.D. plus 1pt]{}]{} Let $f : X \to Y$ be a finite morphism from a connected $n$-dimensional projective manifold $X$ onto a connected $n$-dimensional projective manifold $Y$. Then $f$ is unramified if and only if $c_1({{\cal E}})=0$. [[**Proof. **]{}]{}By Theorem (\[unramifiedCase\]), we can assume without loss of generality that $f$ is ramified. If $n=1$, then the result follows from the following Lemma. \[detEonCurve\] Let $f:X\to Y$ be a finite morphism between smooth connected curves. Let $\tau$ denote the degree of the ramification divisor of $f.$ Then $f_*({\omega_{X/Y}})$ and ${{\cal E}}$ are nef, and $\deg f_*({\omega_{X/Y}})= {{\tau} \over {2}};$ in particular $\det ({{\cal E}})$ is ample unless $f$ is an unramified cover. [[**Proof. **]{}]{}The nefness follows from (1.12). Let $c := \deg f_*({\omega_{X/Y}}) = \deg {{\cal E}}.$ By Riemann-Roch we have $$\chi (f_*({\omega_{X/Y}})) = d(1-g(Y)) + c,$$ where $g$ denotes the genus. On the other hand Riemann-Roch on $X$ gives $$\begin{aligned} \chi(f_*{\omega_{X/Y}}) &=& \chi({\omega_{X/Y}}) = \chi(\omega_X \otimes f^*(\omega_Y^{-1}))\\ &=& - \chi(f^*(\omega_Y^{-1})) = g(X)-1 -d(2g(Y)-2). \end{aligned}$$ Putting things together and using the Riemann-Hurwitz formula, the claim follows. [[Q.E.D. plus 1pt]{}]{} Thus we can assume that $f$ is ramified and also that $n\ge 2$. In this case let $C$ be a smooth curve on $Y$ obtained as the intersection of $n-1$ general elements of $|L|$, where $L$ is a very ample line bundle on $Y$. By Bertini’s theorem, $C$ and $X_C:=f^{-1}(C)$ are smooth. By Kodaira’s vanishing theorem, $X_C$ is connected. Since $X_C$ is the intersection of ample divisors, it meets the ramification divisor nontrivially. This gives that $\deg(f_*(\omega_{X_C/C})/{{\cal O}}_C)>0$. Since $f_*(\omega_{X_C/C})/{{\cal O}}_C\cong {{\cal E}}_C$, we conclude the contradiction that $c_1({{\cal E}})\not=0$. [[Q.E.D. plus 1pt]{}]{} \[unf\] Suppose that $f: X \to Y$ factors through an unramified cover: $f = b \circ a$ with $b: Z \to Y$ unramified of degree at least 2. Then ${{\cal E}}$ cannot be ample and moreover ${{\cal E}}$ is not spanned at any point. [[**Proof. **]{}]{}In fact, ${{\cal E}}_b = (b_*({{\cal O}}_Z)/{{\cal O}}_Y)^*$ is numerically flat with no sections by Theorem (\[unramifiedCase\]). Since ${{\cal E}}_b$ is a direct summand of ${{\cal E}}$, the claims follow. [[Q.E.D. plus 1pt]{}]{} Let $f:X\to Y$ be a finite surjective morphism between smooth connected $n$-dimensional projective manifolds. Let $r$ denote the rank of the subsheaf of ${{\cal E}}$ generated by the images of elements of $H^0({{\cal E}})$. If $f$ factors through an unramified cover $b:Z\to Y$ from a connected projective manifold $Z$ onto $Y$, then the sheet number of $b$ is bounded by $r+1$. In particular, if global sections of ${{\cal E}}$ span ${{\cal E}}$ at some point, then 1. $f$ does not factor through any nontrivial unramified cover of $Y$; and 2. given a smooth $C$ obtained as the intersection of general elements $A_i\in |L_i|$, for $n-1$ ample and spanned line bundles $L_1,\ldots,L_{n-1}$, we have ${{\cal E}}_C$ is ample. Probably the generic spannedness of ${{\cal E}}$ is unneeded in the next result. Let $f:X\to Y$ be a finite surjective morphism between smooth connected $n$-dimensional projective manifolds. Assume that global sections of ${{\cal E}}$ span ${{\cal E}}$ at some point, and let $C$ be the smooth curve obtained as the intersection of general elements $A_i\in |L_i|$, for $n-1$ ample and spanned line bundles $L_1,\ldots,L_{n-1}$. Then the fundamental group of $f^{-1}(C)$ surjects onto the fundamental group of $C$. [[**Proof. **]{}]{}First note that $C':=f^{-1}(C)$ is smooth by Bertini’s theorem, and connected by the Kodaira vanishing theorem. Since ${{\cal E}}_{C'}\cong f_{C*}(\omega_{C'/C})/{{\cal O}}_C$, by Lemma (\[simpleFact\]), $f_{C'}$ does not factor through an unramified cover of $C$. This is equivalent to having the fundamental group of $C'$ surject onto the fundamental group of $C$. [[Q.E.D. plus 1pt]{}]{} \[badExamples\] We would like to give some examples that show that ${{\cal E}}$, even when ample, may have very few or no sections. Let $L$ be a line bundle on a projective manifold $Y$. Assume that there exists a section $s$ of $L$ with a smooth zero set $D$. Assume that $A$ is a line bundle on $Y$ with $rA=L$. Let $f:X\to Y$ be the $r$-sheeted cover obtained by taking the $r$-th root of $s$. In this case $\displaystyle {{\cal E}}\cong \bigoplus_{j=1}^{r-1}jA$. Thus if we choose $A$ and $L$ such that $A$ is not spanned, we have an example with ${{\cal E}}$ not spanned. Such examples are plentiful. For example, 1. We could take $L$ as the line bundle associated to two distinct points on a curve $Y$. If $h^1({{\cal O}}_Y)>0$, then any line bundle $A$ with $2A=L$, is of degree one, and thus has at most one section. A dimension count shows that if $h^1({{\cal O}}_Y)\ge 2$, we can choose the two points so that in fact $h^0(A)=0$; 2. If $Y$ is a Del Pezzo surface, i.e., if $-\omega_Y$ is ample, then it follows by a straightforward argument using Reider’s theorem, that if $kA=L$ with $h^0(L)\ge 1$, $A$ not spanned, and $k\ge 2$, then $L=-k\omega_Y$, $A=-\omega_Y$, and $\omega_Y\cdot \omega_Y=1$. 3. There is a similar example for Del Pezzo threefolds. Let $Y$ denote a Del Pezzo threefold with $-\omega_Y=2H$ with $H^3=1$. Then $-\omega_Y$ is spanned, but for the double cover associated to $2H=-\omega_Y$, ${{\cal E}}=H$ is spanned except at one point. Similar examples show that ${{\cal E}}$ does not have to be nef. Let $Y$ be a Hirzebruch surface $F_r$ with $r\ge 2$, i.e., a ${{\Bbb P}^{1}}$-bundle over ${{\Bbb P}^{1}}$ with a section $E$ satisfying $E\cdot E=-r\le -2$. Let $f$ denote a fiber of the tautological surjection of $F_r$ to ${{\Bbb P}^{1}}$. Take $L$ as the line bundle $rE+r(r-1)f$. Note that since $(r-1)[E+rf]$ is spanned we can choose a smooth divisor $D'\in |(r-1)[E+rf]|$. Since $E\cdot D'=0$, $D:=E+D'$ is a smooth divisor $\in |rE+r(r-1)f|$. Taking the $r$-sheeted cyclic cover associated to the $r$-th root of $D$, we have $${{\cal E}}\cong \bigoplus_{j=1}^{r-1}jA,$$ where $A$ the line bundle associated to $E+(r-1)f$. Note that this bundle is negative restricted to $E$, and so ${{\cal E}}$ is not even nef in this case. It is worth emphasizing that it is hard for ${{\cal E}}$ to be ample for all branched covers over a fixed base. \[rarelyAmple\]Let $Y$ be an $n$-dimensional connected projective manifold. Assume that for all finite surjective morphisms $f:X\to Y$ from connected projective manifolds $X$, the associated bundle ${{\cal E}}$ is ample. Then: 1. the profinite completion of the fundamental group of $Y$ is $0$, and thus in particular the first betti number of $Y$ is $0$; and 2. there exist no surjective morphisms $g: Y\to Z$ with $\dim Z\ge 1$, $Z$ projective, and $g$ having at least one positive dimensional fiber. [[**Proof. **]{}]{}The first assertion is clear. To see the second assertion, let $L$ be a very ample line bundle on $Z$. Choosing the double cover associated to a smooth $D\in |2g^*L|$, we have ${{\cal E}}\cong g^*L$, which, by the hypothesis on $g$, cannot be ample. [[Q.E.D. plus 1pt]{}]{} \[rarelyAmpleCor\]Let $Y$ be an $n$-dimensional connected projective manifold. Assume that for all finite surjective morphisms $f:X\to Y$ from connected projective manifolds $X$, the associated bundle ${{\cal E}}$ is ample. Then either $-\omega_X$ is ample with [Pic]{}$(Y)={{\Bbb Z}}$ or $\omega_Y$ is nef. Moreover $\omega_Y$ is ample if $\omega_Y$ is big. Using the prediction of the Abundance Conjecture that $mK_Y$ is spanned for suitable large $m$ whenever $K_Y$ is nef, it follows in (\[rarelyAmpleCor\]), that either $-\omega_Y$ is ample or $\omega_Y$ is ample or $\omega_Y \equiv 0.$ In the last case $Y$ is an irreducible Calabi-Yau or symplectic manifold. In particular, if $\dim Y = 2,$ then $Y$ is the projective plane or $\omega_Y$ is ample or $Y$ is K3 without $(-2)-$curves. If $\dim Y = 3,$ then $Y$ is Fano with $b_2 = 1$ or $\omega_Y$ is ample or $Y$ is Calabi-Yau without any contraction. Coverings of curves {#curveSection} =================== In this section we prove a result generalizing the theorems of Lazarsfeld [@La80] for ${{\Bbb P}^{1}}$ and Debarre [@De96] for elliptic curves. \[curveTheorem\] Let $f: X\to Y$ be a finite morphism from a smooth connected projective curve $X$ onto a smooth connected projective curve $Y$. The bundle ${{\cal E}}=[(f_*{{\cal O}}_X)/{{\cal O}}_Y]^*$ is ample if and only if $f$ does not factor through a nontrivial unramified covering of $Y$. As is pointed out in (\[badExamples\]), even when ${{\cal E}}$ is ample, 1. if $h^1({{\cal O}}_Y)>0$, then ${{\cal E}}$ does not have to be spanned; and 2. if $h^1({{\cal O}}_Y)\ge 2$, then $h^0({{\cal E}})$ can equal $0$. When $h^1({{\cal O}}_Y)=1$, then $h^1({{\cal E}})=h^0({{\cal E}}^*)=0$ and hence by the Riemann-Roch theorem, and Lemma (\[detEonCurve\]), $$h^0({{\cal E}})=\deg({{\cal E}})=h^1({{\cal O}}_X)-1.$$ The rest of the section is devoted to the proof of Theorem (\[curveTheorem\]). By (1.23) we may assume that $f$ does not factor through an unramified covering. We assume that ${{\cal E}}$ is not ample, and argue by contradiction. We have the following simple corollary of Hartshorne’s characterization of ampleness of vector bundles on curves [@Ha71]. \[hartshorne\] Let $V$ be a nef vector bundle on a smooth connected curve $C$. There exists a unique maximal ample vector subbundle $A$. The quotient $V/A$ has degree zero. [[**Proof. **]{}]{}If $V$ is ample the Lemma is trivially true. Thus we can assume that $V$ is not ample. By Hartshorne’s characterization of ampleness, if $V$ is not ample, then there exists a vector subbundle $A$ of$V$ with $\deg V/A\le 0$. Since $V$ is nef, $V/A$ is nef also, and we conclude that $\deg V/A =0$. If $A$ is the $0$ dimensional subbundle, we have $\deg V=0$, and the Lemma is proven. Indeed, if there was a nontrivial ample subbundle $A'$ in this case, we would have $\deg V/A'=\deg V-\deg A' <0$, which contradicts the nefness of $V$. Thus we can assume that the degree of $V$ is positive. Thus let $A$ be a vector subbundle of $V$ of minimal rank with the property that $V/A$ has degree zero. We will be done if we show that 1. $A$ is ample; and 2. $A$ contains any other ample subbundle of $V$. To see the first assertion, note that if $A$ is not ample, then by Hartshorne’s theorem, there is a vector subbundle $A'$ of $A$ with $\deg A/A'\le 0$. Thus $\deg V/A'=\deg V/A+\deg A/A'\le 0$ contradicting the choice of $A$ with minimal rank. To see the second assertion, assume that there was an ample subbundle $B$ of $V$ with $B\not\subset A$. Then the saturation of the image of the sheaf of germs of sections of $B$ in the sheaf of germs of sections of $V/A$ is an ample subbundle of $V/A$. The inverse image of this bundle in $V$ is an ample vector subbundle of $V$ containing $A$, but the quotient of $V$ by this bundle has negative degree. This contradicts the nefness of $V$. [[Q.E.D. plus 1pt]{}]{} Let ${{\cal F}}$ be the ample subbundle of ${{\cal E}}$ given by Lemma (\[hartshorne\]). Clearly this bundle is also the maximal ample subbundle of ${{\cal O}}_X\oplus {{\cal E}}$. Let $T:={{\cal E}}/{{\cal F}}$. We claim that ${{\cal O}}_Y\oplus T^*$ is a subring of $f_*{{\cal O}}_X = {{\cal O}}_Y\oplus {{\cal E}}^*$. Indeed, we have that $T^*$ is nef since $T$ is nef and of degree zero. Therefore, we have that $({{\cal O}}_Y\oplus T^*)\otimes ({{\cal O}}_Y\oplus T^*)$ is nef, and that $B$, the saturation of its image in ${{\cal O}}_Y\oplus {{\cal E}}^*$, is nef. Since $ {{\cal O}}_Y\oplus T^*\subset B$, the quotient $B/({{\cal O}}_Y\oplus T^*)$ is a nef subbundle of the bundle ${{\cal F}}^*$. But since ${{\cal F}}$ is ample, we conclude that $B=T^*$ and ${{\cal O}}_Y \oplus T^* \subset f_*(\O_X)$ is a subring. Now consider the analytic spectrum $$Z := {\rm Specan}({{\cal O}}_Y \oplus T^*).$$ It comes along with a finite map $b: Z \to Y$, and the inclusion ${{\cal O}}_Y \oplus T^* \to {{\cal O}}_Y \oplus {{\cal E}}^*$ gives a map $$a: {\rm Specan}({{\cal O}}_Y \oplus {{\cal E}}^*) = X \to Z$$ such that $f = b \circ a.$ Since $T^*$ equals its saturation in ${{\cal E}}^*$, we conclude that $Z$ is normal and hence smooth. Since $\deg T = 0,$ we conclude that $b$ is unramified. [[Q.E.D. plus 1pt]{}]{} Coverings over surfaces {#surfaceSection} ======================= Let $f: X\to Y$ be a finite morphism from an irreducible normal Cohen-Macaulay projective variety $X$, with a finite irrational locus, onto a projective manifold $Y$ of dimension $n\ge 2$. Let $H$ be a line bundle on $Y$ such that $-\omega_Y-H$ is nef and big. Then $h^1({{\cal E}}\otimes H^*)=0$. \[delPezzoSurfaceTheorem\] Let $f: X\to Y$ be a finite morphism from an irreducible normal Gorenstein projective surface $X$ onto a smooth projective surface $Y$. Let $Y$ be a Del Pezzo surface with $\omega_Y^2 =9-r \geq 5$ (i.e., $Y$ is a smooth quadric, or the plane blown up in $r\le 4$ points in sufficiently general position). Then ${{\cal E}}$ is spanned. [[**Proof. **]{}]{}Since we already know this for the quadric, we can assume that we have a blowing up map $p : Y\to {{\Bbb P}^{2}}$. We have $$\omega_Y=-3p^*H+\sum_{i=1}^rE_i$$ the $E_i$ are smooth ${{\Bbb P}^{1}}$s on $Y$ with self-intersection $-1$. We will identify $H$ with $p^*(H).$ First let $r=1$. Let $A=2H-E_1$. Note that $-\omega_Y-A=H$ is nef and big, and thus that $h^1({{\cal E}}(-A))=0$ (3.1). We thus have $\displaystyle H^0({{\cal E}})\to H^0({{\cal E}}_C)\to 0$. Since $A$ is very ample, a general $C \in |A-y|$ for any $y\in Y$ is smooth. For a general $C \in |A-y|$, we conclude that $f^{-1}(C)$ is reduced. Since smooth $C\in |A|$ have genus $0$, we conclude by Theorem (\[newLazarsfeld\]) that ${{\cal E}}_C$ is spanned, hence ${{\cal E}}$ is spanned. Now let $r\ge 2$. Let $H_i=H-E_i$. Note that $-\omega_Y-H_i=2H-\sum_{j\not=i}E_j$ is nef and big. Since $H_i$ is spanned with smooth $C\in |H_i|$ having genus $0$ we conclude that ${{\cal E}}$ is spanned by global sections over a generic smooth $C\in |H_i|$. Thus ${{\cal E}}$ is spanned at all points $y\in Y$ where there is a smooth $C\in |H_i-y|$ for some $i$. The set of $y\in Y$ for which this is not true for a fixed $i$ is $B_i:=\cup_{j\not= i}\left(E_j\cup E_{ij}\right)$ where $E_{ij}$ is the proper transform under $p$ of the line through $\{p(E_i),p(E_j)\}$. Note that $\cap_i B_i=\emptyset$ unless $r=2$. In the case $r=2$ we have that $B_1\cap B_2=E_{12}$. Note that $-\omega_Y-H=2H-E_1-E_2$ is nef and big, and therefore we can use $H$ in place of $H_i$. This shows spannedness except at the points $y_1:=E_1\cap E_{12}$ and $y_2:=E_2\cap E_{12}$. Thus if we show that $$H^0({{\cal E}})\to H^0({{\cal E}}_{E_{12}})\to 0,$$ then we will be done. To see this note that $-\omega_Y-(E_i+E_{12})$ is nef and big. Therefore using the Koszul complex and (3.1) $$0\to {{\cal E}}\otimes(-E_1-E_{12}-E_2)\to ({{\cal E}}\otimes(-E_1-E_{12}))\oplus ({{\cal E}}\otimes(-E_2-E_{12}))\to {{\cal E}}\otimes(-E_{12})\to 0,$$ we are reduced to showing that $h^2({{\cal E}}\otimes(-E_1-E_{12}-E_2)=0$. But we have $$h^2({{\cal E}}\otimes(-E_1-E_{12}-E_2))=h^0({{\cal E}}^*\otimes (-2H+E_1+E_2))=h^0(f^*(-2H+E_1+E_2))=0.$$ [[Q.E.D. plus 1pt]{}]{} To get spannedness over Hirzebruch surfaces, it is enough to assume that the branch locus does not contain any components of the inverse image of $E$. \[spannedOverHirz\]Let $f:X\to Y$ be finite surjective morphism from a pure dimensional, normal variety $X$ with at worst Cohen-Macaulay singularities onto a smooth Hirzebruch surface $Y:={{\Bbb F}_{r}}$ (with surjection $ p: Y\to{{\Bbb P}^{1}}$). Let $E$ denote a section of $ p$ with $E\cdot E=-r$. If $r\le 1$; or $r>1$ and $E$ does not belong to the discriminant variety of $f$, then ${{\cal E}}$ is spanned. [[**Proof. **]{}]{}By Theorem (\[delPezzoSurfaceTheorem\]), we may assume without loss of generality that $r\ge 2$. Let ${{\overline {E}}} :=f^{-1}(E)$. Since $E$ does not belong to the discriminant variety of $f$, it follows that ${{\overline {E}}}$ is reduced. Let $d:=\deg(f)$ be the degree of $f.$ Note that global sections of ${{\cal E}}$ span ${{\cal E}}_E$. To see this, observe that by Theorem (\[newLazarsfeld\]), we need only show that $h^1({{\cal E}}(-E))=0$. But this is equal to $h^1({{\cal E}}^*\otimes \omega_{Y}\otimes {{\cal O}}_Y(E))$. Since $-\omega_Y-E$ is ample, we conclude that $h^1(\omega_{Y}\otimes {{\cal O}}_Y(E))=0$ and $h^1(f^*(\omega_{Y}+E))=0$. Therefore $$h^1({{\cal E}}(-E))=h^1(f^*(\omega_{Y}+ E))=0.$$ Given an arbitrary fiber $F$ of $ p$ we see that ${{\cal E}}_{F}$ is spanned since global sections span it at $F\cap E$. Thus we will be done if we show that $h^1({{\cal E}}(-F))=0$. But this is equal to $h^1({{\cal E}}^*\otimes \omega_{Y}\otimes {{\cal O}}_Y(F))$. Since $h^1(\omega_Y+F)=0$ we are reduced to showing that $h^1(f^*(\omega_Y+F))=0$. Since $f^*(\omega_Y+F)=-f^*(E+(r+1)F)-f^*E$, $f^*(E+(r+1)F)$ is ample, and $f^{-1}(E)$ is reduced, we have vanishing by Lemma (\[almostKodaira\]). [[Q.E.D. plus 1pt]{}]{} Theorem (\[HirzWithTwist\]) is sharp for $r\ge 2$. Indeed consider the cyclic branched cover of ${{\Bbb F}_{r}}$ with discriminant locus given by $E+(r-1)\left(E+rF\right)$. We have not pursued the question of what is the “minimal” twist, which makes ${{\cal E}}$ spanned, but such questions can be answered for special classes of varieties by the same techniques. Here is a simple example. \[HirzWithTwist\] Let $f: X\to Y$ be a finite morphism from an irreducible normal projective surface $X$ onto smooth Hirzebruch surface $Y:={{\Bbb F}_{r}}$. If $r\ge 1$, then ${{\cal E}}\otimes {{\cal O}}_Y((r-1)F)$ is spanned. [[**Proof. **]{}]{}Let $p: Y\to{{\Bbb P}^{1}}$ be the ${{\Bbb P}^{1}}$-bundle projection of $Y$ onto ${{\Bbb P}^{1}}$. Let $F$ be a fiber of $ p$ and let $E$ be the section of $Y$ with $E\cdot E=-r$. To see that ${{\cal E}}\otimes {{\cal O}}_Y((r-1)F)$ is spanned, let $y\in Y$ be a point. A general $C\in |E+(r+1)F|$ passing through $y$ is smooth and does not belong to the branch locus. Therefore, it suffices by Theorem \[newLazarsfeld\] to show that $H^1({{\cal E}}\otimes{{\cal O}}_Y((r-1)F)\otimes{{\cal O}}_Y(-C))=0$. But the latter group is isomorphic to $H^1(f^*{{\cal O}}_Y(-E-rF))$, and is therefore zero since $E+rF$ is nef and big. [[Q.E.D. plus 1pt]{}]{} For higher dimensional projective bundles, it is not so easy to prove spannedness theorems under natural conditions. There is one special case for which the simple technique of splitting the anticanonical divisor works. Since we do not need this in the rest of the paper, we leave the proof to the reader. \[projectiveBundleOverP1\] Let $f:X\to Y$ be finite surjective morphism from a pure dimensional, normal variety $X$ with finite irrational locus and at worst Cohen-Macaulay singularities onto $Y$, a ${{\Bbb P}^{n-1}}$-bundle over ${{\Bbb P}^{1}}$. 1. If $-\omega_Y$ nef, then ${{\cal E}}$ is spanned except if $Y = {\Bbb P}({{\cal O}}_{{{\Bbb P}^{1}}}^{\oplus (n-1)} \oplus {{{\cal O}}_{{\Bbb P}^{1}}({2})}),$ in which case ${{\cal E}}$ is spanned outside the exceptional divisor ${\Bbb P}({{\cal O}}_{{{\Bbb P}^{1}}}^{\oplus n-1}).$ 2. If $Y={\Bbb P}(V)$ with $V:={{\cal O}}_{{{\Bbb P}^{1}}}\oplus {{{\cal O}}_{{\Bbb P}^{1}}({a_1})}\oplus \cdots\oplus{{{\cal O}}_{{\Bbb P}^{1}}({a_{n-1}})}$ and with $\displaystyle \deg V\le 1+a_{n-1}$, then ${{\cal E}}$ is generically spanned. The bundle ${{\cal E}}$ will fail to be ample if any curve on $Y$ has disconnected inverse image under $f$. The examples in (\[badExamples\]) show that the following is optimal. \[ampleOverHirz\] Let $f:X\to Y$ be a finite surjective morphism from a pure dimensional, normal variety $X$ with at worst Cohen-Macaulay singularities onto a smooth Hirzebruch surface $Y:={{\Bbb F}_{r}}$ (with surjection $ p: Y\to{{\Bbb P}^{1}}$). Let $E$ denote a section of $ p$ with $E\cdot E=-r$. and let $F$ denote a general fiber of $ p$. If $f^{-1}(F)$ is connected and $f^{-1}(E)$ is connected and reduced, then $H^0({{\cal E}}(-E))$ spans ${{\cal E}}(-E)$ on all of $Y$ and $H^0({{\cal E}}(-F-E))$ spans ${{\cal E}}(-F-E)$ on the complement of $E$. From this it follows that ${{\cal E}}$ is ample. [[**Proof. **]{}]{}We first show that ${{\cal E}}(-E)$ is spanned by global sections. Let $C$ be a general element of $|E+(r+1)F|$ that contains $y$. Since $E+(r+1)F$ is very ample, $C$ is smooth and does not belong to the discriminant variety of $f$. Thus ${{\overline {C}}} := f^{-1}(C)$ is reduced. It is also connected since it is an ample divisor on $X$. Thus ${{\cal E}}_C(-1)$ is spanned by Theorem (\[newLazarsfeld\]). The assertion that ${{\cal E}}(-E)$ is spanned by global sections, will therefore follow if we show that $h^1({{\cal E}}(-E-C))=0$. But, the usual computation shows that this equals $h^1(f^*(-F))$. Since $f^{-1}(F)$ is connected, we conclude that the fibers of $ p\circ f:X\to {{\Bbb P}^{1}}$ are all connected and thus $h^1(f^*(-F))=0$. Now we will show that ${{\cal E}}(-F-E)$ is spanned by global sections on the complement of $E$. Let $C$ be a general element of $|E+rF|$ that contains $y$. Since global sections of $E+rF$ embed $Y-E$, the curve $C$ is smooth and does not belong to the discriminant variety. Thus ${{\overline {C}}} := f^{-1}(C)$ is reduced. It is also connected since it is a big divisor on $X$. Thus ${{\cal E}}_C(-1)$ is spanned by Theorem (\[newLazarsfeld\]). The assertion that ${{\cal E}}(-F-E)$ is spanned by global sections, will follow if we show that $h^1({{\cal E}}(-F-E-C))=0$. But, the usual computation shows that this equals $h^1(f^*(-F))$. Since $f^{-1}(F)$ is connected, we conclude that the fibers of $ p\circ f:X\to {{\Bbb P}^{1}}$ are all connected and thus $h^1(f^*(-F))=0$. Since ${{\cal E}}$ is spanned by Theorem (\[spannedOverHirz\]), for ${{\cal E}}$ to fail to be ample, there must exist an irreducible and reduced curve $D\subset Y$ such that ${{\cal E}}_D$ has a trivial line bundle as a direct summand. Since ${{\cal E}}(-E)$ is spanned, we conclude that $D\cdot E=0$ so that $D\in |a(E+rF)|$ for some integer $a>0$. Since ${{\cal E}}(-E-F)$ is spanned off $E$ and since $D\subset Y\setminus E$, we conclude that $D\cdot F=0$. Since $D\sim a(E+rF)$, we conclude that $a=0$ and thus that $D$ is the empty curve. [[Q.E.D. plus 1pt]{}]{} Hirzebruch [@Hi83] (see also, [@BHH87]), showed how to associate smooth projective surfaces to configurations of lines on ${{\Bbb P}^{2}}$. Let $\Lambda$ denote the configuration of six lines through four points in ${{\Bbb P}^{2}}$, no two of which are collinear. Associated to this configuration there is a smooth projective surfaces ${{\cal H}}(\Lambda,k)$ for each integer $k\ge 2$ with the following properties: 1. there are finite morphisms $f_k:{{\cal H}}(\Lambda,k)\to Y$, where $Y$ is ${{\Bbb P}^{2}}$ blown up at the four points; and 2. the covering $f_k$ is Galois with Galois group equal to the direct sum of five copies of ${{\Bbb Z}}_k$. Since $Y$ is Del Pezzo with $\omega_Y^2=5$, Theorem (\[delPezzoSurfaceTheorem\]) shows that the bundle ${{\cal E}}$ associated to $f_k$ is spanned. This is particularly interesting because of the beautiful result of Hirzebruch that $X:={{\cal H}}(\Lambda,5)$ is a quotient of the unit ball in ${{\Bbb C}}^2$ by a freely acting discrete group. It would be interesting to know in general what are the properties of the bundle ${{\cal E}}$ associated to the natural finite morphisms from line configuration surfaces to blowups of ${{\Bbb P}^{2}}$. In particular: For which line configurations $\Lambda$ and which integers $k$ are the bundles ${{\cal E}}$ spanned? If there are any points where more than two lines of the configuration meet, then the surface $Y$ fibers over ${{\Bbb P}^{1}}$ with the inverse image of the general fiber of this fibration disconnected in the line configuration surface, see, e.g., [@So84]. This shows that ${{\cal E}}$ will never be ample, except possibly for the trivial configuration where at most two lines meet in a point. Coverings over Del Pezzo Manifolds {#delPezzoSection} ================================== We always let $\rho(Y)$ denote the Picard number of $Y.$ The guideline of this section is the following Let $f: X \to Y$ be a finite surjective morphism of degree $d\ge 2$ between projective manifolds $X,Y$ of dimension $n$. If $Y$ is Fano with $\rho(Y) = 1,$ then ${{\cal E}}$ is ample (and spanned in most cases). First we consider the case that $Y$ is a Del Pezzo manifold of dimension $n = \dim Y \geq 3,$ i.e. $-\omega_Y = (n-1)H$ with some (ample) line bundle $H.$ Let $b := H^{n}.$ Then $1 \leq b \leq 8,$ moreover all Del Pezzo manifolds are classified (see [@Fu90]). In our standard situation (1.1) we obtain $$\begin{aligned} h^0({{\cal E}}(-H)) &=& h^0(X,{\omega_{X/Y}}\otimes f^*((n-2)H)) \label{star}\\ h^0({{\cal E}}) &=& h^0(X,{\omega_{X/Y}}\otimes f^*((n-1)H)) - 1. \label{starstar} \end{aligned}$$ As (\[badExamples\]) shows, ${{\cal E}}$ is not always spanned if $b=1$. Though we expect spannedness for the cases $b\ge 2$, we can only prove it for $b\ge 5$. \[delPezzoManifoldTheorem\] Let $f:X\to Y$ be a finite surjective morphism from an $n$-dimensional normal irreducible Gorenstein projective variety $X$ onto an $n$-dimensional projective manifold $Y$. Assume that the irrational locus of $X$ is finite. Let $Y$ be a Del Pezzo manifold with $b \geq 5.$ Then ${{\cal E}}$ is spanned. [[**Proof. **]{}]{}We proceed by induction on $n,$ noticing that $H$ is very ample. We denote a general member of $\vert H \vert $ again by $H.$ If $n = 2,$ then we conclude by (3.2). So suppose $n \geq 3.$ By (1.17)(4) we obtain the exact sequence $$0 \to H^0({{\cal E}}(-H)) \to H^0({{\cal E}}) \to H^0({{\cal E}}_H) \to 0.$$ By induction hypothesis, ${{\cal E}}\vert H $ is spanned, so it follows that ${{\cal E}}$ is spanned outside a finite set $Z$. But, $H$ being very ample, through a given point we can still find a smooth member of $\vert H \vert $, hence $Z = \emptyset.$ Notice that here we need to work in the Gorenstein category since the general $f^{-1}(H)$ need not be smooth! [[Q.E.D. plus 1pt]{}]{} In the cases $1 \leq b \leq 4,$ we do not know the spannedness of ${{\cal E}}$ on the corresponding Del Pezzo surfaces, we are only able to prove much weaker results. We start with a general result showing that almost always ${{\cal E}}$ has at least one section. \[weakExistenceTheorem\] Let $f:X\to Y$ be a finite surjective morphism of degree $d\ge 2$ from an $n$-dimensional normal irreducible Gorenstein projective variety $X$ onto an $n$-dimensional projective manifold $Y$. Assume that the irrational locus of $X$ is finite. Assume that $n\ge 2$ and that $Y$ is Del Pezzo. If $bd\ge 5$, i.e., if either $b\ge 3$; or $b=2$ and $d\ge 3$; or $b=1$ and $d\ge 5$; then $h^0({{\cal E}})\not = 0$. [[**Proof. **]{}]{}Write $-\omega_Y = (n-1)H$ with $H$ ample. Note that if $n\ge 3$, $$h^1({{\cal E}}(-H))=h^{n-1}(-f^*(n-2)H)=0.$$ Since $H$ is either spanned or has at most one base point, we can reduce the proof of the result to the case when $n=2$. So without loss of generality we assume that $n=2$. Let $L := f^*H$. We have $$h^0({{\cal E}})=h^2({{\cal E}}^*\otimes \omega_Y)=h^2(-L)-1=h^0(\omega_Y+L)-1.$$ Thus we have reduced to showing that $h^0(\omega_Y+L)\ge 2$. Note that since $h^0(\omega_Y+L)=h^0({{\cal E}})+1$, we know that $h^0(\omega_Y+L)\ge 1$. From this it follows that that $\omega_X+L$ is nef, see [@So85]. Again using the main result of [@So85], note in this case that $(\omega_X+L)\cdot L=0$ implies that $-\omega_X=L$, which implies the absurdity that $f$ is unramified. Thus we have that: 1. $L$ is spanned outside of a finite set; 2. $\omega_X+L$ is nef and $(\omega_X+L)\cdot L\ge 2$. Noting that $L\cdot L=b\deg f$ implies that $L\cdot L\ge 5,$ the following Lemma will complete the proof of the theorem. Let $L$ be an ample line bundle on an irreducible normal Gorenstein projective surface $X$. Assume that $L$ is spanned outside of a finite set; $h^0(\omega_X+L)\ge 1$; and $(\omega_X+L)\cdot L\ge 2$. Then either $h^0(\omega_X+L)\ge 2$ or $L\cdot L\le 4$. [**Proof of Lemma. **]{} We assume that $h^0(\omega_X+L)=1$ and then argue to a contradiction. Since $h^0(L)\ge 2$, we would have $h^0(\omega_X+L)\ge 2$ if $h^0(\omega_X)\ge 1$. Therefore without loss of generality we can assume that $$\label{pgIs0} h^0(\omega_X)=0.$$ Since $h^1(\omega_X+L)=0$, we conclude from $$0\to \omega_X\to \omega_X\otimes L\to \omega_C\to 0$$ that $$\label{qgRelation} g=q+1,$$ where $q=h^1({{\cal O}}_X) $ and $2g-2=(\omega_X+L)\cdot L$ defines the arithmetic genus $g$ of a reduced curve $C$ from $|L|$. Since $g\ge 2$ by hypothesis, we conclude further that $$\label{qlowerBound}q\ge 1.$$ Assume first that the irrational locus ${{\cal I}}(X)$ of $X$ is nonempty. Recall the Grauert-Riemenschneider canonical sheaf ${{\cal K}}_X$ is defined as $\pi_*(\omega_{{{\overline {X}}}})$ where $\pi : {{\overline {X}}}\to X$ is a desingularization. Following the argument of [@So85], we consider the sequence: $$0\to {{\cal K}}_X\otimes L\to \omega_X\otimes L \to {{\cal S}}\to 0,$$ where ${{\cal S}}$ is a skyscraper sheaf whose support exactly equals the set of nonrational singularities ${{\cal S}}(X)$. Since $h^1({{\cal K}}_X+L)=0$, $h^0(\omega_X+L)=1$, and the locus ${{\cal I}}(X)$ is nonempty, we conclude that $h^0({{\cal S}})=1$. Using this, equation (\[pgIs0\]), and the sequence: $$0\to {{\cal K}}_X\to \omega_X \to {{\cal S}}\to 0,$$ we conclude that: $$\label{onDeSing} h^0(\omega_{{{\overline {X}}}})=0;\ \ \ \ \ {{\overline {q}}} := h^1({{\cal O}}_{{{\overline {X}}}})=q+1.$$ Therefore $X$ has only one irrational singularity, an elliptic singularity $y_0.$ Consider the Albanese mapping $\alpha : {{\overline {X}}}\to Z$. By equation (\[onDeSing\]) we conclude first that ${{\overline {q}}}\ge 1$ and therefore that $\dim Z\ge 1$. Since $h^0(\omega_{{{\overline {X}}}})=0$, we conclude that $Z$ is a curve, and thus smooth with $\alpha$ having connected fibers. Since $q\ge 1$ by equation (\[qlowerBound\]), we conclude that ${{\overline {q}}}\ge 2$. Let $B$ denote the singular set of $X$, and note that $\alpha(\pi^{-1}(B))$ is finite. This is clear since $y_0$ is elliptic and all other singularities are rational. In particular, the strict transform $\overline C$ of $C$ cannot be a fiber of $\alpha.$ Thus a general fiber $F$ of $\alpha$ is mapped isomorphically by $\pi$ onto a curve $F'$ contained in the smooth points of $X$. Since $L$ is spanned off of a finite set, we conclude that global sections of $L$ give rise to at least two linearly independent sections of $L_{F'}$. Hence we deduce easily that $C \cdot F' \geq 2$ and $\overline C$ has degree at least 2 over $Z.$ Thus using that ${\overline {q}}\ge 2$, we have that $$g\ge 2{{\overline {q}}} \ge 2q+2=2g,$$ which is absurd since $g\ge 2$. Therefore we can assume without loss of generality that the singularities of $X$ are rational. Thus, e.g., [@BS95], the Albanese map $\alpha : X\to Z$ is well defined. Since $h^0(\omega_X)=0$ by equation (\[pgIs0\]), we conclude that by the same argument as above, but without the need of a desingularization, that if $q\ge 2$ then $$g\ge 2 q =2g-2,$$ which is absurd given that $g=q+1\ge 3$. Thus we conclude that $q=1$ and $g=2$. In this case we have that $$(\omega_X+L)^2 L^2\le [L\cdot (\omega_X+L)]^2=1.$$ Since $L\cdot L\ge 5$ we conclude using the nefness of $\omega_X+L$ that $(\omega_X+L)^2=0$. This gives $\omega_X^2\ge 1$. Since $q=1$, $h^0(\omega_X)=0$, and we have at worst rational Gorenstein singularities, we conclude from the Riemann-Roch formula for the Euler characteristic of the structure sheaf of the desingularization, that $\omega_X^2\le 0$. This proves the Lemma and the Theorem. [[Q.E.D. plus 1pt]{}]{} Note if $Y$ is an elliptic curve, which can be considered as the most appropriate definition of a Del Pezzo manifold of dimension $1$, then Theorem (\[delPezzoManifoldTheorem\]) is still true, unless $f$ is an unramified covering. We now show how adjunction theory can be used to give general conditions for an existence theorem, which would not follow even from knowing that ${{\cal E}}$ is spanned. Indeed, as shown in (\[exceptions\]), some of the possible exceptions to this result exist and in fact have ${{\cal E}}$ spanned. We do not need the precise classification of Del Pezzo manifolds with $1 \leq b \leq 4$ after Fujita [@Fu90], but just the following \[classification\] Let $Y$ be a Del Pezzo manifold of dimension at least 3. Then $\rho(Y) = 1,$ unless $X$ is $\left({{\Bbb P}^{1}}\right)^3$, ${{\Bbb P}^{2}} \times {{\Bbb P}^{2}}$, ${{\Bbb P}}(T_{{{\Bbb P}^{2}}})$ or ${{\Bbb P}^{3}}$ blown up in one point. Moreover $H $ is very ample for $b\ge 3$, and spanned unless $b = 1$, in which case we have one simple base point. In all cases $\vert H \vert $ contains a smooth element. \[minuHExistenceTheorem\] Let $f:X\to Y$ be a finite surjective morphism of degree $d\ge 2$ from an $n$-dimensional projective manifold $X$ onto an $n$-dimensional projective manifold $Y$. Assume that $n\ge 3$ and that $Y$ is Del Pezzo, i.e., $-\omega_Y=(n-1)H$ with $H$ ample. Let $Y$ be of degree $b:= H^n\ge 2$. Then $H^{0}({{\cal E}}(-H)) \ne 0$, except for the following possible cases: in the cases 3) and 4), $(X_0,L_0)$ denotes the first reduction of $(X,L)$ [@BS95]. 1. $(X,L)$ is a quadric fibration over a curve with $Y=\left({{\Bbb P}^{1}}\right)^3$; or 2. $(X,L)$ is a scroll over a surface with either $n = 3$; or 3. $(X_0,L_0)$ is a 3-dimensional quadric, $L_0 = {{\cal O}}(2)$, $Y ={{\Bbb P}^{3}}(x_0)$, $d=2$, and $f$ is induced by a covering $f_0: X_0 \to Y_0 = {{\Bbb P}^{3}}$; or 4. $n = 3$ and $X_0$ is a ${{\Bbb P}^{2}}$-bundle over a curve with $L_F = {{{\cal O}}_{{\Bbb P}^{2}}({2})}$ on a fiber $F$. Examples of dimension $3,4$, for the cases of the Theorem are given in (\[exceptions\]). [[**Proof. **]{}]{}Let $L = f^*(H).$ Suppose $H^0({{\cal E}}(-H)) = 0.$ Then by (\[star\]) $$H^0(\omega_X \otimes f^*((n-2)H)) = 0.$$ Then, since $b\ge 2$ and $H$ is spanned, [@So89] applies. Using the equation (\[starstar\]) that $h^0(\omega_X+(n-1)L)=h^0({{\cal E}})+1\ge 1$, we obtain that either $(X,L)$ is 1. a quadric fibration over a curve; or 2. a scroll over a surface; or,\   \ $(X_0,L_0)$, the first reduction of $(X,L)$ [@BS95], exists and is either: 3. $(X_0,L_0) = {({\Bbb P}^{n},{{\cal O}}_{{\Bbb P}^{n}}({2}))}$ with $n = 3,4$; or 4. $(X_0,L_0)$ is a 3-dimensional quadric and $L_0 ={{\cal O}}(2);$ or 5. $n = 3$ and $X_0$ is a ${{\Bbb P}^{2}}$-bundle over a curve with $L = {{\cal O}}(2)$ on the fibers. We will treat these cases separately.\ (a) If we have a quadric fibration $p: X \to C,$ let $F$ be a general fiber and $F' = f(F).$ Since $L^{n-1} \cdot F = 2,$ the map $f_F$ has degree 2 or 1. $H$ being ample and spanned (recall $b \geq 2$), it follows that in the first case $F'$ must be projective space, and the adjunction formula implies $N_{F' \vert Y} = {{\cal O}}(-1),$ which is absurd since $F'$ moves in $Y.$ In the second case, we obtain $H^{n-1} \cdot F' = 2$ and we conclude similarly that $F'$ is a quadric. By adjunction, $N_{F' \vert Y} = {{\cal O}},$ hence $\rho(Y) \geq 2.$ By the classification (\[classification\]) we obtain $Y = \left({{\Bbb P}^{1}}\right)^3.$ This is in one of the exceptions.\ (b) Now suppose that $p: X \to S$ is a scroll over the surface $S.$ Then, using the same notations as before, we conclude $F' = {{\Bbb P}^{n-2}}$ and $H \vert F' = {{\cal O}}(1).$ By adjunction, $ \det N_{F'} = {{\cal O}}.$ Notice that $N_{F'}$ is generically generated for [*general*]{} $F'$, hence $N_{F'}$ must be trivial. If $b \geq 3$ and $n \geq 5$, then $H$ is very ample, so that by Ein [@Ei85], $Y$ has a ${{\Bbb P}^{n-2}}$-bundle structure, contradicting (\[classification\]) unless $Y = {{\Bbb P}^{2}} \times {{\Bbb P}^{2}} $ giving rise to one of the exceptions (4.10). Therefore it remains to treat the case $b = 2$ or $n \le 4.$\ (b.1) In case $n = 3,$ then $F'$ is a line and we are in the exceptions (4.10), the cases $Y = {{\Bbb P}}(T_{{{\Bbb P}}_2}), {{\Bbb P}}_3(x), {{\Bbb P}}_1^3$ being obvious.\ (b.2) Concerning $b = 2,$ the linear system $\vert H \vert$ realizes $Y$ as a $2:1-$covering $h: Y \to {{\Bbb P}}_n,$ moreover $L = f^*(H).$ Since $H \vert F' = {{\cal O}}(1),$ it follows that $f \vert F$ is $1:1$ for all $F$. This is clearly absurd, since $f^{-1}(F')$ would always consist of 1 or 2 disjoint fibers of $p$, which implies $\rho(Y) \geq 2,$ a contradiction to the classification.\ (b.3) It remains to treat the case $b \geq 3$ and $n = 4.$\ ($\alpha$) Suppose $b = 3.$ Then $Y \subset {{\Bbb P}}_5$ has degree and we obtain (for general $F' = {{\Bbb P}}_2$) an exact sequence $$0 \to N_{{{\Bbb P}}_2 \vert Y} \to N_{{{\Bbb P}}_2\vert {{\Bbb P}}_5} \to N_{Y \vert {{\Bbb P}}_5} \vert {{\Bbb P}}_2 \to 0,$$ which reads $$0 \to {{\cal O}}\oplus {{\cal O}}\to {{\cal O}}(1)^{\oplus 3} \to {{\cal O}}(3) \to 0.$$ This sequence is of course absurd.\ ($\beta$) Suppose $b = 4.$ Then $Y$ is the intersection of 2 quadrics in ${{\Bbb P}}_6.$ Take one of the quadrics, say $Q$, to obtain a sequence $$0 \to N_{{{\Bbb P}}_2 \vert Y} \to N_{{{\Bbb P}}_2 \vert Q} \to N_{Y \vert Q} \vert {{\Bbb P}}_2 \to 0.$$ This sequence immediately implies $$N_{{{\Bbb P}}_2 \vert Q} = {{\cal O}}\oplus {{\cal O}}\oplus {{\cal O}}(2),$$ which clearly contradicts the 1-ampleness of the tangent bundle $T_Q.$\ ($\gamma$) Finally we treat $b = 5.$ First notice that every line $l \subset Y $ has a normal bundle either of the form ${{\cal O}}\oplus {{\cal O}}\oplus {{\cal O}}(1)$ or ${{\cal O}}(1) \oplus {{\cal O}}(1) \oplus {{\cal O}}(-1).$ This follows by choosing a smooth hyperplane section through $l$ (which is easily seen to exist, cp. \[Is77,p.505\]). We are going to show that [*every*]{} plane $F' \subset Y$ has normal bundle ${{\cal O}}\oplus {{\cal O}}.$ Once we know this, we can argue as in \[Ei85,1.7\] to obtain a ${{\Bbb P}}_2-$bundle structure on $Y.$ Since the planes $F'$ cover $Y,$ the normal bundle $N$ is generically generated by 2 sections $s_1,s_2.$ Since by adjunction $\det N = {{\cal O}},$ it follows that these sections generate $N$ everywhere so that $N$ is trivial. \(c) Suppose $(X_0,L_0) = {({\Bbb P}^{n},{{\cal O}}_{{\Bbb P}^{n}}({2}))}$ with $n = 3 $ or $4.$ We claim that always $X = X_0.$ In fact, if $X \not = X_0,$ then $X$ contains a divisor $E = {{\Bbb P}^{n-1}}$ with $N_E = {{\cal O}}(-1).$ Consider $E' = f(E).$ Since $L \vert E = {{\cal O}}(1),$ we have $H^{n-1} \cdot E' = 1,$ so that since $H$ is therefore spanned, we have that $E' = {{\Bbb P}^{n-1}}.$ Adjunction gives $N_{E'} = {{\cal O}}(-1),$ hence by classification $Y = {{\Bbb P}^{3}}(x_0).$ Therefore $X_0 = {{\Bbb P}^{3}}$. Now, introducing the ramification divisor $R$ of $f,$ we have $$\omega_X + f^*(H) = f^*(\omega_Y) + R + f^*(H) = f^*(-H) + R,$$ hence $$H^0(X,{{\cal O}}_X(R) \otimes f^*(-H)) = 0$$ by our assumption. If $\sigma: X \to X_0$ denotes the contraction of the exceptional divisors $E,$ we conclude that $$R' = \sigma_*(R) \in \vert {{\cal O}}_{X_0}(1) \vert.$$ Hence the induced covering $h: X_0 \to Y_0$ is ramified exactly over a plane, which is absurd since $X_0 = Y_0 = {{\Bbb P}^{3}}.$ So we shall assume $X = X_0.$ But then we have the contradiction that $Y = {{\Bbb P}^{n}}$. \(d) Suppose $X_0$ is a three dimensional quadric with $L={{\cal O}}_{X_0}(2)$. If $X=X_0$, then $Y$ is ${{\Bbb P}^{3}}$ or a quadric. Since $Y$ is Del Pezzo, we have $Y={{\Bbb P}^{3}}$ and $H={{{\cal O}}_{{\Bbb P}^{3}}({2})}$. Since $K_X+L$ has no sections, we conclude that $L={{\cal O}}_{X}(2)$, and therefore that $f$ has degree $2$. From this we conclude that ${{\cal E}}={{{\cal O}}_{{\Bbb P}^{3}}({1})}$. This gives an example. If $X \not = X_0,$ then we argue as in (c) to conclude that $Y = {{\Bbb P}^{(}}3)(x_0).$ Hence $Y_0 = {{\Bbb P}^{3}},$ see (4.10)(4). [[Q.E.D. plus 1pt]{}]{} In the preceding and following results we need $b\ge 2$ to get spannedness so that the main result of [@So89] holds. Actually, we only need that $2L$ is spanned and $|L|$ has a smooth element. This would be satisfied when $b=1$ under the extra condition that the point where $H$ is not spanned is not an element of the discriminant divisor of the covering . Using this the statement of the result is the same, except there is one case that cannot be ruled out: the first reduction $(X_0,L_0)$ of $(X,L)$ exists and is equal ${({\Bbb P}^{n},{{\cal O}}_{{\Bbb P}^{n}}({2}))}$ with $n = 3,4$. If $b\ge 2$, then $h^0({{\cal E}}) \geq h^0({{\cal O}}_Y(H)) $ unless $X$ is of the one of the exceptions in Theorem (\[minuHExistenceTheorem\]). In particular these inequalities hold if $n \geq 5.$ We now give examples of the exceptional cases. \[exceptions\] \(1) $Y = \left({{\Bbb P}^{1}}\right)^3$ and we fix a projection $q: \left({{\Bbb P}^{1}}\right)^3 \to {{\Bbb P}^{1}}.$ Let $h: C \to {{\Bbb P}^{1}}$ be a suitable ramified cover and put $X = Y \times_{ {{\Bbb P}^{1}}} C.$\ (2) Let $Y$ be a Del Pezzo 3-fold and $T$ a component of the Hilbert schemes of lines. Suppose that through every point of $X$ there are only finitely many lines. Let $S$ be a smooth surface and $S \to T$ be a surjective map. Let $X$ be the $ {{\Bbb P}^{1}}$-bundle over $S$ given by the graph of the family of line parameterized by $T.$\ (3) Let $Y = {{\Bbb P}^{2}} \times {{\Bbb P}^{2}}.$ Let $S$ be a surface and $S \to {{\Bbb P}^{2}}$ be a finite map. Put $X = S \times {{\Bbb P}^{2}} \to Y.$ Analogously we choose $S \to {{\Bbb P}^{1}} \times {{\Bbb P}^{1}}$ finite and let $X = S \times {{\Bbb P}^{1}}$ resp. $h: S \to {{\Bbb P}^{2}}$ finite and $X = {{\Bbb P}}(h^*(T_{ {{\Bbb P}^{2}}})).$\ (4) Suppose $X_0$ is a three dimensional quadric. We have a degree $2$ map $f: X_0\to {{\Bbb P}^{3}}$. In this case ${{\cal E}}={{{\cal O}}_{{\Bbb P}^{3}}({1})}$. Since ${{\Bbb P}^{3}}$ is Del Pezzo with $H={{{\cal O}}_{{\Bbb P}^{3}}({2})}$, we have ${{\cal E}}(-H)$ has no sections. Note $L_0={{\cal O}}_{X_0}(2)$. If $\pi : Y\to {{\Bbb P}^{3}}$ is the blowup of ${{\Bbb P}^{3}}$ at a point $y$. Then ${{\overline {H}}}:=\pi^*{{{\cal O}}_{{\Bbb P}^{3}}({2})}-E$ is very ample where $E$ is the exceptional fiber of $\pi$. Thus $-K_Y=\pi^*{{{\cal O}}_{{\Bbb P}^{3}}({4})}-2E=2{{\overline {H}}}$, and we obtain a Del Pezzo manifold $Y$. Letting $X$ denote the blowup of $X$ at the inverse image under $f$ of $y$, we have a degree $2$ finite morphism ${{\overline {f}}} : X\to Y$. We have $h^0(K_X+L)=h^0(K_{X_0}+L_0)=0$.\ (5) Let $p: X_0 \to C$ be a $ {{\Bbb P}^{2}}$-bundle over a smooth curve $C.$ Let $h: X_0 \to Y_0 = {{\Bbb P}^{3}}$ be a finite cover such that $$[h^*{{{{\cal O}}_{{\Bbb P}^{3}}({1})}}]_F = {{{\cal O}}_{{\Bbb P}^{2}}({1})},$$ where $F$ denotes a fiber of $p.$ Choose $y_0$ not in the image of the ramification divisor of $h.$ Then let $\tau : Y \to Y_0$ be the blow-up of $Y_0$ at $y_0$ and $\sigma: X \to X_0$ the blow-up of $X_0$ at the finite set $h^{-1}(y_0).$ Then $h$ lifts to a finite cover $f: X \to Y$ and it is easy to check that $H^0(X,\omega_X \otimes f^*(H)) = 0,$ where $-\omega_Y = 2H.$ In order to obtain a specific example, consider the family ${\cal C}$ of hyperplanes in ${{\Bbb P}^{3}}.$ The canonical map $$q: {\cal C} \to {{\Bbb P}^{3}} = Y_0$$ is a ${{\Bbb P}^{2}}$- bundle and so does the other projection $$p: {\cal C} \to {{{\Bbb P}^{3}}}^*.$$ Actually ${\cal C} = {{\Bbb P}}(T_{{{\Bbb P}^{3}}}).$ Now choose $C \subset {{{\Bbb P}^{3}}}^*$ to be a general smooth curve. Then $X_0 = p^{-1}(C)$ is smooth and moreover $h := q \vert X_0$ is finite. To see this last property, consider $T(x) \subset {{{\Bbb P}^{3}}}^*,$ the set of hyperplanes through a given point $x \in {{\Bbb P}^{3}}$. So it is sufficient to choose $C$ such that $C \not \subset T(x)$ for all $x$ which comes down to choose $C$ non-degenerate, since all $T(x)$ are linear in $({{{\Bbb P}^{3}}})^*.$ [999999]{} Barthel,G.; Hirzebruch,F.; Höfer,Th.: Geradenkonfigurationen und Algebraische Flächen. Aspects of Mathematics, D4. Friedr. Vieweg & Sohn 1987 Beltrametti,M.; Sommese,A.J.: The adjunction theory of complex projective varieties. de Gruyter 1995 Campana,F.;Peternell,Th.: Towards a Mori theory on compact Kähler threefolds, I. Math. Nachr. 187, 29–59 (1997) Cho,K.;Sato,E.: Smooth projective varieties dominated by smooth hyperquadrics in any characteristic. Math. Z. 217, 553–565 (1994) Debarre,O.: Théoremes de connexité et variétés abéliennes. Amer.J.Math. 117, 787–805 (1995) Debarre,O.: Sur un théoreme de connexité de Mumford pour les espace homogenes. Manuscr. Math. 89, 407-425 (1996) Demailly,J.P.;Peternell,Th.;Schneider,M.: Compact complex manifolds with numerically effective tangent bundles. J. Alg. Geom. 3, 295–345 (1994) Demailly,J.P.;Peternell,Th.;Schneider,M.: Pseudo-effective line bundles on projective varieties and Kähler manifolds. Preprint in prep. (1999) Ein,L.: Varieties with small dual varieties,II. Duke Math. J. 52, 895–907 (1985) Fujita,T.: Classification theories of polarized varieties. London Math. Soc. Lect. Notes Ser. 155 (1990) Hartshorne,R.: Ample vector bundles on curves. Nagoya Math. J. 43, 73–89 (1971) Hirzebruch,F.: Arrangements of lines and algebraic surfaces. Arithmetic and geometry, Vol. II, 113–140, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983 Iskovskih,V.A.: Fano 3-folds I. Math. USSR Izv. 11, 485-527 (1977) Kim,M.: Barth-Lefschetz type theorem for branched coverings of Grassmannians, J. Reine Angew. Math. 470, 109–122 (1996) Kim,M.: On branched coverings of quadrics. Arch. Math. 67, 76–79 (1996) Kim,M.;Manivel,L.: On branched coverings of some homogeneous spaces. Topology 38, 1141–1160 (1999) Manivel,L.: Vanishing theorems for ample vector bundles. Invent. Math. 127, 401–416 (1997) Lazarsfeld,R.: A Barth type theorem for branched coverings of projective spaces. Math. Ann. 249, 153–162 (1980) Okonek,C.;Schneider,M.;Spindler,H.: Vector bundles on complex projective spaces. Birkhäuser 1980 Paranjapé,K.H.;Srinivas,V.: Selfmaps of homogeneous spaces. Inv. Math. 98, 425–444 (1989) Sommese,A.J.: Submanifolds of Abelian varieties. Math. Ann. 233, 229–256 (1978) Sommese,A.J.: On the density of ratios of Chern numbers of algebraic surfaces. Math. Ann. 268, 207–221 (1984) Sommese,A.J.: Ample divisors on normal Gorenstein surfaces. Abh. Math. Sem. Univ. Hamburg 55, 151–170 (1985) Sommese,A.J.: On the adjunction theoretic structure of projective varieties. In: Proceedings of the Complex Analysis and Algebraic Geometry Conference, Göttingen, 1985, ed. by H. Grauert, Lecture Notes in Math. 1194, 175–213. Springer 1986 Sommese,A.J.: On the nonemptiness of the adjoint linear system of a hyperplane section of a threefold. J. reine u. angew. Math. 402, 211–220 (1989); Erratum, 411, 122–123 (1990) Viehweg,E.: Die Additivität der Kodairadimension für projektive Faserräume über Varieäten des allgemeinen Typs. J.reine u. angew. Math. 330, 132–142 (1982) Viehweg,E.: Quasi-projective moduli for polarized manifolds. Springer 1995 ---------------------------------- -- ----------------------------------- Thomas Peternell Andrew J. Sommese Mathematisches Institut Department of Mathematics Universit" at Bayreuth University of Notre Dame D-95440 Bayreuth, Germany Notre Dame, Indiana 46556, U.S,A, fax: Germany + 921–552999 fax: U.S.A. + 219–631-6579 thomas.peternell@uni-bayreuth.de sommese@nd.edu URL: [www.nd.edu/$\sim$sommese]{} ---------------------------------- -- -----------------------------------

--- abstract: 'We show that $K_1({\mathcal{E}})$ of an exact category ${\mathcal{E}}$ agrees with $K_1({\mathbb{D}}{{\mathcal{E}}})$ of the associated triangulated derivator ${\mathbb{D}}{{\mathcal{E}}}$. More generally we show that $K_1({\mathcal{W}})$ of a Waldhausen category ${\mathcal{W}}$ with cylinders and a saturated class of weak equivalences agrees with $K_1({\mathbb{D}}{{\mathcal{W}}})$ of the associated right pointed derivator ${\mathbb{D}}{{\mathcal{W}}}$.' address: 'Universitat de Barcelona, Departament d’Àlgebra i Geometria, Gran via de les corts catalanes 585, 08007 Barcelona, Spain' author: - Fernando Muro title: 'Maltsiniotis’s first conjecture for $K_1$' --- [^1] Introduction {#introduction .unnumbered} ============ For a long time there was an interest in defining a nice $K$-theory for triangulated categories such that Quillen’s $K$-theory of an exact category ${\mathcal{E}}$ agrees with the $K$-theory of its bounded derived category ${D^b}({\mathcal{E}})$. Schlichting [@kttc] showed that such a $K$-theory for triangulated categories cannot exist. It was then natural to ask about the definition of a nice $K$-theory for algebraic structures interpolating between ${\mathcal{E}}$ and ${D^b}({\mathcal{E}})$. The best known intermediate structure is ${C^b}({\mathcal{E}})$, the Waldhausen category of bounded complexes in ${\mathcal{E}}$, with quasi-isomorphisms as weak equivalences and cofibrations given by chain morphisms which are levelwise admissible monomorphisms. The derived category ${D^b}({\mathcal{E}})$ is the localization of ${C^b}({\mathcal{E}})$ with respect to weak equivalences. The Gillet-Waldhausen theorem[^2], relating Quillen’s $K$-theory to Waldhausen’s $K$-theory, states that the homomorphisms $$\tau_n\colon K_n({\mathcal{E}}){\longrightarrow}K_n({C^b}({\mathcal{E}})),\quad n\geq0,$$ induced by the inclusion ${\mathcal{E}}\subset{C^b}({\mathcal{E}})$ of complexes concentrated in degree $0$, are isomorphisms. The category ${C^b}({\mathcal{E}})$ is considered to be too close to ${\mathcal{E}}$ so one would still like to find an algebraic stucture with a nice $K$-theory interpolating between ${C^b}({\mathcal{E}})$ and ${D^b}({\mathcal{E}})$. The notion of a triangulated derivator [@derivateurs; @ktdt] seems to be a strong candidate. Maltsiniotis [@ktdt] defined a $K$-theory for triangulated derivators together with natural homomorphisms $$\rho_n\colon K_n({\mathcal{E}}){\longrightarrow}K_n({\mathbb{D}}{{\mathcal{E}}}),\quad n\geq0,$$ where ${\mathbb{D}}{{\mathcal{E}}}$ is the triangulated derivator associated to an exact category ${\mathcal{E}}$, constructed by Keller in the appendix of [@ktdt]. Cisinski and Neeman proved the additivity of triangulated derivator $K$-theory [@adkt]. Maltsiniotis also conjectured that $\rho_n$ is an isomorphism for all $n$. He succeeded in proving the conjecture for $n=0$. The following theorem is the main result of this paper. \[main\] Let ${\mathcal{E}}$ be an exact category. The natural homomorphism $$\rho_1\colon K_1({\mathcal{E}})\st{\cong}{\longrightarrow}K_1({\mathbb{D}}{{\mathcal{E}}})$$ is an isomorphism. In order to obtain Theorem \[main\] we use techniques introduced in [@1tk]. There we give a presentation of an abelian $2$-group ${\mathcal{D}}_*{\mathcal{W}}$ which encodes $K_0({\mathcal{W}})$ and $K_1({\mathcal{W}})$ of a Waldhausen category ${\mathcal{W}}$, and moreover the $1$-type of the $K$-theory spectrum $K({\mathcal{W}})$ whose homotopy groups are the $K$-theory groups of ${\mathcal{W}}$. This presentation is a higher dimensional analogue of the classical presentation of $K_0({\mathcal{W}})$. Here we similarly define an abelian $2$-group ${\mathcal{D}^{\mathrm{der}}}_*{\mathcal{W}}$ which models the $1$-type of the $K$-theory spectrum $K({\mathbb{D}}{\mathcal{W}})$ of the right[^3] pointed derivator ${\mathbb{D}}{\mathcal{W}}$ associated to a Waldhausen category ${\mathcal{W}}$ with cylinders and a saturated class of weak equivalences, such as ${\mathcal{W}}={C^b}({\mathcal{E}})$. The $K$-theory for this kind of derivators, more general than triangulated derivators, was defined by Garkusha [@sdckt1] extending the work of Maltsiniotis [@ktdt]. There are defined comparison homomorphisms $$\mu_n\colon K_n({\mathcal{W}}){\longrightarrow}K_n({\mathbb{D}}{\mathcal{W}}),\quad n\geq0.$$ These homomorphisms cannot be isomorphisms in general, as shown in [@ktsc]. Nevertheless we here prove the following result. \[main2\] Let ${\mathcal{W}}$ be a Waldhausen category with cylinders and a saturated class of weak equivalences. The natural homomorphism $$\begin{array}{c}\mu_0\colon K_0({\mathcal{W}})\st{\cong}{\longrightarrow}K_0({\mathbb{D}}{\mathcal{W}}),\\ \mu_1\colon K_1({\mathcal{W}})\st{\cong}{\longrightarrow}K_1({\mathbb{D}}{\mathcal{W}}), \end{array}$$ are isomorphisms. In Remark \[last\] we comment on the case where the hypothesis on the saturation of weak equivalences is replaced by the 2 out of 3 axiom, which is a weaker assumption. Theorem \[main\] is actually a corollary of the Gillet-Waldhausen theorem and Theorem \[main2\], since ${\mathbb{D}}{C^b}({\mathcal{E}})={\mathbb{D}}{\mathcal{E}}$ and the natural homomorphisms $\rho_n$ factor as $$\rho_n\colon K_n({\mathcal{E}})\st{\tau_n}{\longrightarrow}K_n({C^b}({\mathcal{E}}))\st{\mu_n}{\longrightarrow}K_n({\mathbb{D}}{{\mathcal{E}}}),\quad n\geq0.$$ We assume the reader certain familiarity with exact, Waldhausen and derived categories, with simplicial constructions and with homotopy theory. We refer to [@wiak; @mha; @gj] for the basics. Acknowledgements {#acknowledgements .unnumbered} ---------------- I am very grateful to Grigory Garkusha for suggesting the possibility of using [@1tk] in order to tackle Maltsiniotis’s first conjecture in dimension $1$. I also feel indebted to Denis-Charles Cisinski, who kindly indicated how to extend the results of a preliminary version of this paper to a broader generality. The bounded derived category of an exact category {#ho} ================================================= In this section we outline the two-step construction of the derived category ${D^b}({\mathcal{E}})$ of an exact category ${\mathcal{E}}$. This construction is a special case of the homotopy category $\ho {\mathcal{W}}$ of a Waldhausen category ${\mathcal{W}}$ with cylinders satisfying the 2 out of 3 axiom, ${D^b}({\mathcal{E}})=\ho{C^b}({\mathcal{E}})$. A *Waldhausen category* is a category ${\mathcal{W}}$ with a distinguished zero object $0$ and two distinguished subcategories $w{\mathcal{W}}$ and $c{\mathcal{W}}$, whose morphisms are called *cofibrations* and *weak equivalences*, respectively. A morphism which is both a weak equivalence and a cofibration is said to be a *trivial cofibration*. The arrow $\rightarrowtail$ stands for a cofibration and $\st{\sim}\r$ for a weak equivalence. - All morphisms $0\r A$ are cofibrations. All isomorphisms are cofibrations and weak equivalences. - The push-out of a morphism along a cofibration is always defined $$\xymatrix{A\ar@{>->}[r]\ar[d]\ar@{}[rd]|{\text{push}}&B\ar[d]\\X\ar@{>->}[r]&X\cup_AB}$$ and the lower map is also a cofibration. - Given a commutative diagram $$\xymatrix{X\ar[d]_\sim&A\ar[d]_\sim\ar[l]\ar@{>->}[r]&B\ar[d]_\sim\\X'&A'\ar[l]\ar@{>->}[r]&B'}$$ the induced map $X\cup_AB\st{\sim}\r X'\cup_{A'}B'$ is a weak equivalence. Notice that coproducts $A\vee B=A\cup_0B$ are defined in ${\mathcal{W}}$. A functor ${\mathcal{W}}\r{\mathcal{W}}'$ between Waldhausen categories is *exact* if it preserves cofibrations, weak equivalences, push-outs along cofibrations and the distinguished zero object. Recall that an *exact category* ${\mathcal{E}}$ is a full subcategory of an abelian category ${\mathcal{A}}$ such that ${\mathcal{E}}$ contains a zero object of ${\mathcal{A}}$ and ${\mathcal{E}}$ is closed under extensions in ${\mathcal{A}}$. A *short exact sequence* in ${\mathcal{E}}$ is a short exact sequence in ${\mathcal{A}}$ between objects in ${\mathcal{E}}$. A morphism in ${\mathcal{E}}$ is an *admissible monomorphism* if it is the initial morphism of some short exact sequence. The category ${\mathcal{E}}$ is a Waldhausen category with admissible monomorphisms as cofibrations and isomorphisms as weak equivalences. In order to complete the structure we fix a zero object $0$ in ${\mathcal{E}}$. We denote by ${C^b}({\mathcal{E}})$ the category of bounded complexes in ${\mathcal{E}}$, $$\cdots \r A^{n-1}\st{d}{\longrightarrow}A^n\st{d}{\longrightarrow}A^{n+1}\r\cdots, \quad d^2=0, \quad A^n=0\text{ for }|n|\gg0.$$ A chain morphism $f\colon A^{\estrella}\r B^{\estrella}$ in ${C^b}({\mathcal{E}})$ is a *quasi-isomorphism* if it induces an isomorphism in homology computed in the ambient abelian category ${\mathcal{A}}$. The category ${C^b}({\mathcal{E}})$ is a Waldhausen category. Weak equivalences are quasi-isomorphisms and cofibrations are levelwise admissible monomorphisms. There is a full exact inclusion of Waldhausen categories ${\mathcal{E}}\subset{C^b}({\mathcal{E}})$ sending an object $X$ in ${\mathcal{E}}$ to the complex $$\cdots\r 0\r X\r 0\r\cdots,$$ with $X$ in degree $0$. \[hotsat\] The *homotopy category* $\ho{\mathcal{W}}$ of a Waldhausen category is a category equipped with a functor $$\zeta\colon {\mathcal{W}}{\longrightarrow}\ho{\mathcal{W}}$$ sending weak equivalences to isomorphisms. Moreover, $\zeta$ is initial among all functors ${\mathcal{W}}\r{\mathcal{C}}$ sending weak equivalences to isomorphisms, so $\ho{\mathcal{W}}$ is well defned up to canonical isomorphism over ${\mathcal{W}}$. This category can be constructued as a category of fractions, in the sense of [@gz], by formally inverting weak equivalences in ${\mathcal{W}}$. The class of weak equivalences is *saturated* if any morphism $f\colon A\r B$ in ${\mathcal{W}}$ such that $\zeta(f)$ is an isomorphism in $\ho{\mathcal{W}}$ is indeed a weak equivalence $f\colon A\st{\sim}\r B$. Weak equivalences in ${C^b}({\mathcal{E}})$, i.e. quasi-isomorphisms, are saturated since they are detected by a functor $H^*\colon{C^b}({\mathcal{E}})\r{\mathcal{A}}^\Z$, the cohomology functor from bounded complexes in ${\mathcal{E}}$ to $\Z$-graded objects in ${\mathcal{A}}$, see [@scm Proposition 1.1]. The homotopy category always exists up to set theoretical difficulties which do not arise if ${\mathcal{W}}$ is a small category, for instance. This is not a harmful assumption if one is interested in $K$-theory since smallness may also be required in order to have well defined $K$-theory groups. The homotopy category can however be constructed in a more straighforward way if the Waldhausen category ${\mathcal{W}}$ satisfies further properties. A Waldhausen category ${\mathcal{W}}$ satisifies the *2 out of 3 axiom* provided given a commutative diagram in ${\mathcal{W}}$ $$\xymatrix@C=20pt{&C&\\A\ar[ru]\ar[rr]&&B\ar[lu]}$$ if two arrows are weak equivalences then the third one is also a weak equivalence. Given an object $A$ in ${\mathcal{W}}$ a *cylinder* $IA$ is an object together with a factorization of the folding map $(1,1)\colon A\vee A\r A$ as a cofibration followed by a weak equivalence, $$A\vee A\mathop{\rightarrowtail}\limits_i IA\mathop{{\longrightarrow}}\limits^\sim_p A.$$ We say that ${\mathcal{W}}$ has cylinders if all objects have a cylinder. The Waldhausen category ${C^b}({\mathcal{E}})$ has cylinders. The cylinder of a bounded complex $A$ can be functorially chosen as $$(IA)^n\;=\;A^n\oplus A^{n+1}\oplus A^n,\quad d=\left(\begin{array}{ccc}d&-1&0\\0&-d&0\\0&1&d\end{array}\right)\colon (IA)^n{\longrightarrow}(IA)^{n+1}.$$ The 2 out of 3 axiom is often called the saturation axiom. We do not use this terminology in this paper in order to avoid confusion with Definition \[hotsat\]. Usually one considers more structured cylinders in Waldhausen categories, compare [@wiak Definition IV.6.8]. For the purposes of this paper it is enough to consider cylinders as defined above. As one can easily check, a Waldhausen category with a saturated class of weak equivalences satisfies the 2 out of 3 axiom. This applies to ${C^b}({\mathcal{E}})$. A Waldhausen category with cylinders ${\mathcal{W}}$ satisfying the 2 out of 3 axiom is an example of a *right derivable category*, in the sense of [@ciscd], also called *precofibration category* in [@cht], see [@ciscd Example 2.23] or [@cht Proposition 2.4.2]. In particular any morphism in ${\mathcal{W}}$ can be factored as a cofibration followed by a weak equivalence which is left inverse to a trivial cofibration, see [@cht Proposition 1.3.1]. Moreover, one can define a homotopy relation in ${\mathcal{W}}$ and construct the homotopy category $\ho{\mathcal{W}}$ by a homotopy calculus of left fractions as we indicate below, see [@ciscd Section 1] or [@cht Section 5.4]. Let ${\mathcal{W}}$ be a Waldhausen category with cylinders satisfying the 2 out of 3 axiom. As usual we say that two morphisms $f,g\colon A\r B$ in ${\mathcal{W}}$ are *strictly homotopic* if there is a morphism $H\colon IA\r B$ with $Hi=(f,g)$. The maps $f,g$ are *homotopic* $f\simeq g$ if there exists a weak equivalence $h\colon B\st{\sim}\r B'$ such that $hf$ and $hg$ are strictly homotopic. ‘Being homotopic’ is a natural equivalence relation and the quotient category is denoted by $\pi{\mathcal{W}}$. The homotopy category $\ho{\mathcal{W}}$ is obtained by calculus of left fractions in $\pi{\mathcal{W}}$. Objects in $\ho{\mathcal{W}}$ are the same as in ${\mathcal{W}}$. A morphism $A\r B$ in $\ho{\mathcal{W}}$ is represented by a diagram in ${\mathcal{W}}$, $$A\mathop{{\longrightarrow}}\limits_{\alpha_1} X\mathop{\longleftarrow}\limits^\sim_{\alpha_2} B.$$ Another diagram $$A\mathop{{\longrightarrow}}\limits_{\alpha'_1} Y\mathop{\longleftarrow}\limits^\sim_{\alpha'_2} B$$ represents the same morphism if there is a diagram in ${\mathcal{W}}$ $$\xymatrix{&X^{\estrella}\ar@{<-}[ld]_{\alpha_1}\ar@{<-}[rd]_{\sim}^{\alpha_2}&\\ A^{\estrella}&Z^{\estrella}\ar@{<-}[l]\ar@{<-}[r]^\sim\ar@{<-}[u]\ar@{<-}[d]&B^{\estrella}\\&Y^{\estrella}\ar@{<-}[lu]^{\alpha'_1}\ar@{<-}[ru]^\sim_{\alpha'_2}&}$$ whose projection to $\pi{\mathcal{W}}$ is commutative. Notice that, by the 2 out of 3 axiom, the vertical arrows in this diagram are also weak equivalences. The composite of two morphisms $A\mathop{\r}\limits_{\alpha} B\mathop{\r}\limits_{\beta} C$ in $\ho {\mathcal{W}}$ represented by $$A\mathop{{\longrightarrow}}\limits_{\alpha_1} X\mathop{\longleftarrow}\limits^\sim_{\alpha_2} B \mathop{{\longrightarrow}}\limits_{\beta_1} Y\mathop{\longleftarrow}\limits^\sim_{\beta_2} C$$ is defined as follows. If $\beta_1$ is a cofibration then the push-out $$\xymatrix{B\ar@{}[rd]|-{\text{push}}\ar@{>->}[r]^-{\beta_1}\ar[d]^-\sim_-{\alpha_2}&Y\ar[d]_-\sim^-{\bar{\alpha}_2}\\ X\ar@{>->}[r]_-{\bar{\beta}_1}&X\cup_BY}$$ is defined, $\bar{\alpha}_2$ is a weak equivalence, and $\beta\alpha\colon A\r C$ is represented by $$A\mathop{{\longrightarrow}}\limits_{\bar{\beta}_1\alpha_1} X\cup_BY\mathop{\longleftarrow}\limits^\sim_{\bar{\alpha}_2\beta_2} C.$$ In general we can factor $\beta_1$ as cofibration followed by a weak equivalence $$\beta_1\colon B\mathop{\rightarrowtail}\limits_{\beta'_1} Z\mathop{{\longrightarrow}}\limits^\sim_{r} Y$$ such that there is a morphism $s\colon Y\st{\sim}\into Z$ with $rs=1_Y$. The diagram $$\xymatrix{&Y^{\estrella}\ar@{<-}[ld]_{\beta_1}\ar@{<-}[rd]_{\sim}^{\beta_2}&\\ B^{\estrella}&Z^{\estrella}\ar@{<-<}[l]^{\beta'_1}\ar@{<-}[r]^\sim_{s\beta_2}\ar[u]^-{r}&C^{\estrella}}$$ commutes in ${\mathcal{W}}$, so $\beta\colon B\r C$ is also represented by $$B\mathop{\rightarrowtail}\limits_{\beta'_1} Z\mathop{\longleftarrow}\limits^\sim_{s\beta_2} C,$$ where the first arrow is a cofibration, and we can use this representative to define the composite $\beta\alpha\colon A\r C$. The functor $$\zeta\colon {\mathcal{W}}{\longrightarrow}\ho{\mathcal{W}}$$ is the identity on objects and sends a morphism $f\colon A\r B$ to the morphism $\zeta(f)\colon A\r B$ represented by $$A\mathop{{\longrightarrow}}\limits_{f} B\mathop{\longleftarrow}\limits^\sim_{1_B} B.$$ If $f\colon A\st{\sim}\r B$ is a weak equivalence then $\zeta(f)$ is an isomorphism and $\zeta(f)^{-1}$ is represented by $$B\mathop{{\longrightarrow}}\limits_{1_B} B\mathop{\longleftarrow}\limits^\sim_{f} A,$$ hence a morphism $\alpha\colon A\r B$ in $\ho{\mathcal{W}}$ represented by $$A\mathop{{\longrightarrow}}\limits_{\alpha_1} X\mathop{\longleftarrow}\limits^\sim_{\alpha_2} B$$ coincides with $\zeta(\alpha_2)^{-1}\zeta(\alpha_1)=\alpha$. If $\alpha$ above is an isomorphism in $\ho{\mathcal{W}}$ then $\zeta(\alpha_1)=\zeta(\alpha_2)\alpha$ is also an isomorphism. In particular if ${\mathcal{W}}$ has a saturated class of weak equivalences then $\alpha_1\colon A\st{\sim}\r X$ is necessarily a weak equivalence. For ${\mathcal{W}}={C^b}({\mathcal{E}})$ the category $\pi{\mathcal{W}}={H^b}({\mathcal{E}})$ is usually termed the *bounded homotopy category*, while $\ho{\mathcal{W}}={D^b}({\mathcal{E}})$ is called the *bounded derived category* of ${\mathcal{E}}$. On Waldhausen and derived $K$-theory {#kt} ==================================== Recall that a *cofiber sequence* in a Waldhausen category ${\mathcal{W}}$ $$A\into B\onto B/A$$ is a push-out diagram $$\xymatrix{A\ar@{>->}[r]\ar[d]\ar@{}[rd]|{\text{push}}&B\ar@{->>}[d]\\{0}\ar@{>->}[r]&B/A}$$ Therefore the quotient $B/A$ is only defined up to canonical isomorphism over $B$, although the notation $B/A$ is standard in the literature. The $K$-theories we deal with in this paper are constructed by using the Waldhausen categories $S_n{\mathcal{W}}$ that we now recall. An object $A_{\bullet\bullet}$ in the category $S_n{\mathcal{W}}$, $n\geq0$, is a commutative diagram in ${\mathcal{W}}$ $$\label{escalera} \xymatrix{ &&&&A_{nn}\\ &&&&\vdots\ar[u]\\ &&A_{22}\ar[r]&\quad\cdots\quad\ar[r]&A_{2n}\ar[u]\\ &A_{11}\ar[r]&A_{12}\ar[r]\ar[u]&\quad\cdots\quad\ar[r]&A_{1n}\ar[u]\\ A_{00}\ar[r]&A_{01}\ar[r]\ar[u]&A_{02}\ar[r]\ar[u]&\quad\cdots\quad\ar[r]&A_{0n}\ar[u]}$$ such that $A_{ii}=0$ and $A_{ij}\into A_{ik}\onto A_{jk}$ is a cofiber sequence for all $0\leq i\leq j\leq k\leq n$. Notice that these conditions imply that the whole diagram is determined, up to canonical isomorphism, by the sequence of $n-1$ composable cofibrations $$\label{cadena} A_{01}\into A_{02}\into\cdots\into A_{0n}.$$ A morphism $A_{\bullet\bullet}\r B_{\bullet\bullet}$ in $S_n{\mathcal{W}}$ is a natural transformation between diagrams given by morphisms $A_{ij}\r B_{ij}$ in ${\mathcal{W}}$. The category $S_n{\mathcal{W}}$ is a Waldhausen category. A morphism $A_{\bullet\bullet}\st{\sim}\r B_{\bullet\bullet}$ is a weak equivalence if all morphisms $A_{ij}\st{\sim}\r B_{ij}$ are weak equivalences in ${\mathcal{W}}$. A cofibration $A_{\bullet\bullet}\into B_{\bullet\bullet}$ is a morphism such that $A_{ij}\into B_{ij}$ and $B_{ij}\cup_{A_{ij}}A_{ik}\into B_{ik}$ are cofibrations, $0\leq i\leq j\leq k\leq n$. The distinguished zero object is the diagram with $0$ in all entries. The categories $S_n{\mathcal{W}}$ assemble to a simplicial category $S.{\mathcal{W}}$. The face functor $d_i\colon S_n{\mathcal{W}}\r S_{n-1}{\mathcal{W}}$ is defined by removing the $i^{\text{th}}$ row and the $i^{\text{th}}$ column, and the degeneracy functor $s_i\colon S_n{\mathcal{W}}\r S_{n+1}{\mathcal{W}}$ is defined by duplicating the $i^{\text{th}}$ row and the $i^{\text{th}}$ column, $0\leq i\leq n$. Faces and degeneracies are exact functors. For the definition of the simplicial structure it is crucial to consider the whole diagram instead of just . One can obtain a pointed space out of the simplicial category $S.{\mathcal{W}}$ as follows. We restrict to the subcategories of weak equivalences $wS.{\mathcal{W}}$, then we take levelwise the nerve in order to get a bisimplicial set $\ner wS.{\mathcal{W}}$, we consider the diagonal simplicial set $\diag\ner wS.{\mathcal{W}}$, and its geometric realization $$|\diag\ner wS.{\mathcal{W}}|.$$ This pointed space, actually a reduced $CW$-complex, is the $1$-stage of the *Waldhausen $K$-theory* spectrum $K({\mathcal{W}})$ [@akts], which is an $\Omega$-spectrum, hence the $K$-theory groups of ${\mathcal{W}}$ are the homotopy groups $$\begin{aligned} K_n({\mathcal{W}})&=&\pi_{n+1}|\diag\ner wS.{\mathcal{W}}|,\quad n\geq0.\end{aligned}$$ We now assume that ${\mathcal{W}}$ has cylinders and satisifies the 2 out of 3 axiom, so that the associated right pointed derivator ${\mathbb{D}}{\mathcal{W}}$ is defined, see [@ciscd Corollary 2.24 and the duals of Lemmas 4.2 and 4.3]. Then the Waldhausen categories $S_n{\mathcal{W}}$ also have cylinders and satisfy the 2 out of 3 axiom. We will neither recall the notion of derivator nor the definition of the derivator ${\mathbb{D}}{\mathcal{W}}$ but just the $K$-theory of ${\mathbb{D}}{\mathcal{W}}$, we refer the interested reader to [@derivateurs; @ktdt; @sdckt1; @cht]. For this we consider the homotopy categories $\ho S_n{\mathcal{W}}$ and the subgroupoids of isomorphisms $i\ho S_n{\mathcal{W}}$. These groupoids form a simplicial groupoid $i\ho S.{\mathcal{W}}$ and we can consider the pointed space $$|\diag\ner i\ho S.{\mathcal{W}}|,$$ which is the $1$-stage of Garkusha’s *derived $K$-theory* $\Omega$-spectrum $DK({\mathcal{W}})$. Garkusha [@sdckt2] considers derived $K$-theory for ${\mathcal{W}}={C^b}({\mathcal{E}})$, and more generally for ${\mathcal{W}}$ a nice complicial biWaldhausen category, although the definition immediately extends to Waldhausen categories with cylinders satisfying the 2 out of 3 axiom, as indicated here. Moreover, Garkusha shows that there is a natural weak equivalence $DK({\mathcal{W}})\st{\sim}\r K({\mathbb{D}}{\mathcal{W}})$ between the derived $K$-theory spectrum of a nice complicial biWaldhausen category ${\mathcal{W}}$ and the $K$-theory spectrum of the associated derivator ${\mathbb{D}}{\mathcal{W}}$, compare [@sdckt2 Corollary 4.3]. Nevertheless [@sdckt2 Corollary 4.3] only uses the fact that all morphisms in ${\mathcal{W}}$ factor as a cofibration followed by a weak equivalence, compare also [@sdckt2 Lemmas 4.1 and 4.2], so we also have a natural weak equivalence $DK({\mathcal{W}})\st{\sim}\r K({\mathbb{D}}{\mathcal{W}})$ for ${\mathcal{W}}$ a Waldhausen category with cylinders satisfying the 2 out of 3 axiom, and therefore $$\begin{aligned} K_n({\mathbb{D}}{\mathcal{W}})&\cong&\pi_{n+1}|\diag\ner i\ho S.{\mathcal{W}}|,\quad n\geq0.\end{aligned}$$ The functors $\zeta\colon S_n{\mathcal{W}}\r\ho S_n{\mathcal{W}}$ restrict to $wS_n{\mathcal{W}}\r i\ho S_n{\mathcal{W}}$. These functors give rise to a map $$|\diag\ner wS.{\mathcal{W}}|{\longrightarrow}|\diag\ner i\ho S.{\mathcal{W}}|$$ which induces the comparison homomorphisms in homotopy groups, $$\mu_n\colon K_n({\mathcal{W}}){\longrightarrow}K_n({\mathbb{D}}{\mathcal{W}}),\quad n\geq0.$$ This map is actually the $1$-stage of a comparison map of spectra $$\label{cms} K(W){\longrightarrow}K({\mathbb{D}}{\mathcal{W}}).$$ In the rest of this paper we will be mainly concerned with the structure of the bisimplicial sets $X=\ner wS.{\mathcal{W}}$ and $Y=\ner i\ho S.{\mathcal{W}}$ in low dimensions, that we now review more thoroughly. A *bisimplicial set* $Z$ consists of sets $Z_{m,n}$, $m,n\geq 0$, together with horizontal and vertical face and degeneracy maps $$\begin{array}{cc} d^h_i\colon Z_{m,n}{\longrightarrow}Z_{m-1,n},&s^h_i\colon Z_{m,n}{\longrightarrow}Z_{m+1,n},\quad 0\leq i\leq m,\\ d^v_j\colon Z_{m,n}{\longrightarrow}Z_{m,n-1},&s^v_j\colon Z_{m,n}{\longrightarrow}Z_{m,n+1},\quad 0\leq j\leq n, \end{array}$$ satisfying some relations that we will not recall here, compare [@gj]. An element $z_{m,n}\in Z_{m,n}$ is a *bisimplex* of *bidegree* $(m,n)$ and *total degree* $m+n$. A generic bisimplex $z_{m,n}$ of bidegree $(m,n)$ can be depicted as the product of two geometric simplices of dimensions $m$ and $n$ with vertices labelled by the product set $${\left\{0,\dots,m\right\}}\times{\left\{0,\dots,n\right\}},$$ see Figs. 1 and 2. \[12gen\] $$\entrymodifiers={+0} \begin{array}{c}\xymatrix@!R=3pc@!C=3pc{\ar@{-}[r]^(0){(0,0)}^(1){(1,0)}&}\end{array}\; z_{1,0} \quad \begin{array}{c} \xymatrix@!=2pc{\ar@{-}[r]^(0){(0,1)}^(1){(1,1)}\ar@{-}[d]&\ar@{-}[d]\\ \ar@{-}[r]_(0){(0,0)}_(1){(1,0)}&} \end{array}\; z_{1,1} \quad \begin{array}{c} \xymatrix@!R=3.5pc@!C=3.5pc{&\\ \ar@{-}[r]_(0){(0,0)}_(1){(1,0)}\ar@{-}[ru]^(1){(2,0)} &\ar@{-}[u]} \end{array}\; z_{2,0}$$ The horizontal $i^{\mathrm{th}}$ face $d^h_i(z_{m,n})$ is the face obtained by removing the interior, the vertices $(i,j)$, for all $j$, and the incident faces of the boundary. Similarly the vertical $j^{\mathrm{th}}$ face $d^v_j(z_{m,n})$ is obtained by removing the interior, the vertices $(i,j)$, for all $i$, and the incident faces of the boundary. \[3gen\] $$\entrymodifiers={+0}\xymatrix@!R=2pc@!C=0.3pc{ \arr@{-}[rrr]^(0){(0,2)}^(1){(1,2)}&&&\\ &\arr@{.}[lu]&&&\arr@{-}[lu]\\ \arr@{-}[uu]\arr@{.}[ru]_(1){(0,1)}\arr@{-}[rrr]_(0){(0,0)}_(1){(1,0)}&&&\arr@{-}[ru]_(1){(1,1)}\arr@{-}[uu] \arr@{.}"2,2";"2,5"|!{"1,4";"3,4"}\hole } $$ z\_[1,2]{}z\_[2,1]{} @!C=1pc@!R=0.7pc[ @[-]{}\[rrr\]\^(0)[(0,1)]{}\^(1)[(2,1)]{} @[-]{}\[rd\]&&&\ & @[-]{}\[urr\]\_(0)[(1,1)]{}\ @[-]{}\[uu\] @[.]{}\[rrr\]|(0.3333333) @[-]{}\[rd\]\_(0)[(0,0)]{}&&&@[-]{}\[uu\]\ &@[-]{}\[uu\] @[-]{}\[urr\]\_(0)[(1,0)]{}\_(1)[(2,0)]{} ]{} $$\def\tetraco#1#2#3#4#5{{\entrymodifiers={+0}\xymatrix@!R=3.5pc@!C=0.8pc{ #5\\#5\\#5\\ z_{3,0} #5\\#5 \arr@{-}"#1";"#2"^(0){(0,0)\qquad\qquad} \arr@{-}"#1";"#4" \arr@{-}"#2";"#3"_(0){(1,0)} \arr@{-}"#2";"#4" \arr@{-}"#3";"#4"_(0){(2,0)}_(1){(3,0)} \arr@{.}"#1";"#3" |!{"#2";"#4"}\hole}}} \tetraco{3,1}{4,4}{3,6}{1,4}{&&&&&}\qquad\qquad$$ The bisimplicial sets $X$ and $Y$ are horizontally reduced, i.e. $X_{0,n}=Y_{0,n}$ are singletons for all $n\geq 0$, $X_{1,0}=Y_{1,0}$ is the set of objects in ${\mathcal{W}}$, $X_{1,1}$ is the set of weak equivalences in ${\mathcal{W}}$, $Y_{1,1}$ is the set of isomorphisms in $\ho{\mathcal{W}}$, and $X_{2,0}=Y_{2,0}$ is the set of cofiber sequences, see Fig. 3. $$\entrymodifiers={+0} \begin{array}{c}\xymatrix@!R=3pc@!C=3pc{\ar@{-}[r]|*+{\scriptstyle A^{\estrella}}&}\end{array}\quad x_{1,0}=y_{1,0} \qquad\qquad\qquad\qquad \begin{array}{c} \xymatrix@!=2pc{\ar@{-}[r]|*+{\scriptstyle {A'}^{\estrella}}="a"\ar@{-}[d]&\ar@{-}[d]\\ \ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="b"&\ar@{->}"b";"a"_{\sim}} \end{array}\quad x_{1,1}$$ $$\entrymodifiers={+0} \begin{array}{c} \xymatrix@!=2pc{\ar@{-}[r]|*+{\scriptstyle {A'}^{\estrella}}="a"\ar@{-}[d]&\ar@{-}[d]\\ \ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="b"&\ar@{->}"b";"a"_{\cong}} \end{array}\quad y_{1,1} \qquad\qquad\qquad \begin{array}{c} \xymatrix@!R=3.5pc@!C=3.5pc{&\\ \ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="a"\ar@{-}[ru]|*+{\scriptstyle B^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle B^{\estrella} /A^{\estrella}}="c"\ar@{>->}"a";"b"\ar@{->>}"b";"c"} \end{array}\quad x_{2,0}=y_{2,0}$$ The set $X_{1,2}$ consists of pairs of composable weak equivalences in ${\mathcal{W}}$, $Y_{1,2}$ is the set of composable isomorphisms in $\ho{\mathcal{W}}$, $X_{2,1}$ is the set of weak equivalences between cofiber sequences i.e. weak equivalences in $S_2{\mathcal{W}}$ which are commutative diagrams in ${\mathcal{W}}$ $$\label{wecs} \xymatrix{{A'}^{\estrella}\ar@{>->}[r]& {B'}^{\estrella}\ar@{->>}[r]& {B'}^{\estrella}/{A'}^{\estrella}\\ A^{\estrella}\ar@{>->}[r]\ar[u]^\sim& B^{\estrella}\ar@{->>}[r]\ar[u]^\sim& B^{\estrella}/A^{\estrella}\ar[u]^\sim}$$ $Y_{2,1}$ is the set of isomorphisms in $\ho S_2{\mathcal{W}}$, and $X_{3,0}=Y_{3,0}$ is the set of four cofiber sequences assicated to pairs of composable cofibrations $$\label{comcof} \xymatrix{&&C^{\estrella}/B^{\estrella}\\&B^{\estrella}/A^{\estrella}\;\ar@{>->}[r]&C^{\estrella}/A^{\estrella}\ar@{->>}[u]\\A^{\estrella}\;\ar@{>->}[r]&B^{\estrella}\;\ar@{>->}[r]\ar@{->>}[u]&C^{\estrella}\ar@{->>}[u]}$$ see Fig. 4. $$\entrymodifiers={+0}\xymatrix@!R=2pc@!C=0.3pc{ \arr@{-}[rrr]|*+{\scriptscriptstyle C^{\estrella}}="c"&&&\\ &\arr@{.}[lu]&&&\arr@{-}[lu]\\ \arr@{-}[uu]\arr@{.}[ru]\arr@{-}[rrr]|*+{\scriptscriptstyle A^{\estrella}}="a"&&&\arr@{-}[ru]\arr@{-}[uu] \arr@{.}"2,2";"2,5"|!{"c";"a"}\hole|*+{\scriptscriptstyle B^{\estrella}}="b"|!{"1,4";"3,4"}\hole \arr@*{}@{.>}"a";"b"_*{\scriptscriptstyle\!\!\!\sim} \arr@*{}@{.>}"b";"c"_*{\scriptscriptstyle\!\!\!\sim} \arr@*{}@{->}"a";"c"^(0.3)*{\scriptscriptstyle\sim\!} } \!\!\!\! x_{1,2}\qquad\qquad\qquad\qquad\quad \xymatrix@!R=2pc@!C=0.3pc{ \arr@{-}[rrr]|*+{\scriptscriptstyle C^{\estrella}}="c"&&&\\ &\arr@{.}[lu]&&&\arr@{-}[lu]\\ \arr@{-}[uu]\arr@{.}[ru]\arr@{-}[rrr]|*+{\scriptscriptstyle A^{\estrella}}="a"&&&\arr@{-}[ru]\arr@{-}[uu] \arr@{.}"2,2";"2,5"|!{"c";"a"}\hole|*+{\scriptscriptstyle B^{\estrella}}="b"|!{"1,4";"3,4"}\hole \arr@*{}@{.>}"a";"b"_*{\scriptscriptstyle\!\!\!\cong} \arr@*{}@{.>}"b";"c"_*{\scriptscriptstyle\!\!\!\cong} \arr@*{}@{->}"a";"c"^(0.3)*{\scriptscriptstyle\cong\!} } \!\!\!\! y_{1,2}$$ $$\entrymodifiers={+0} x_{2,1}\qquad \xymatrix@!C=1pc@!R=0.7pc{ \arr@{-}[rrr]|*+{\scriptscriptstyle {B'}^{\estrella}}="bb" \arr@{-}[rd]|*+{\scriptscriptstyle {A'}^{\estrella}}="aa"&&&\\& \arr@{-}[urr]|*+{\scriptscriptstyle {B'}^{\estrella}/{A'}^{\estrella}}="bbaa"\\\arr@{-}[uu] \arr@{.}[rrr]|(0.1666666){\hole}|(0.3333333){\hole} |*+{\scriptscriptstyle B^{\estrella}}="b"|(0.6666666){\hole} \arr@{-}[rd]|*+{\scriptscriptstyle A^{\estrella}}="a"&&&\arr@{-}[uu]\\&\arr@{-}[uu] \arr@{-}[urr]|*+{\scriptscriptstyle B^{\estrella}/A^{\estrella}}="ba" \arr@*{}@{>->} "aa";"bb" \arr@*{}@{>.>}"a";"b"|{\hole} \arr@*{}@{->>} "bb";"bbaa" \arr@*{}@{.>>} "b";"ba" \arr@*{}@{->} "a";"aa"^*{\scriptscriptstyle\sim} \arr@*{}@{->} "ba";"bbaa"^*{\scriptscriptstyle\sim} \arr@*{}@{.>} "b";"bb"^(0.3)*{\scriptscriptstyle\sim} |!{"2,2";"bbaa"}{\hole}} \qquad\qquad\qquad y_{2,1}\qquad \xymatrix@!C=1pc@!R=0.7pc{ \arr@{-}[rrr]|*+{\scriptscriptstyle {B'}^{\estrella}}="bb" \arr@{-}[rd]|*+{\scriptscriptstyle {A'}^{\estrella}}="aa"&&&\\& \arr@{-}[urr]|*+{\scriptscriptstyle {B'}^{\estrella}/{A'}^{\estrella}}="bbaa"\\\arr@{-}[uu] \arr@{.}[rrr]|(0.3333333){\hole} |*+{\scriptscriptstyle B^{\estrella}}="b" \arr@{-}[rd]|*+{\scriptscriptstyle A^{\estrella}}="a"&&&\arr@{-}[uu]\\&\arr@{-}[uu] \arr@{-}[urr]|*+{\scriptscriptstyle B^{\estrella}/A^{\estrella}}="ba" \arr@*{}@{>->} "aa";"bb" \arr@*{}@{>.>}"a";"b"|{\hole} \arr@*{}@{->>} "bb";"bbaa" \arr@*{}@{.>>} "b";"ba" \arr@*{}@{} "a";"ba"|{}="c" \arr@*{}@{} "aa";"bbaa"|{}="cc" \arr@*{}@{.>} "c";"cc"_{\text{ in }\ho(S_2{\mathcal{W}})}_(.6)\cong|(.12){\hole}|(.2){\hole}|(.28){\hole}|(.78){\hole}|(.85){\hole}}$$ $$\entrymodifiers={+0} \def\tetra#1#2#3#4#5{{\entrymodifiers={+0}\xymatrix@!R=3.5pc@!C=0.8pc{ #5\\#5\\#5\\ x_{3,0}=y_{3,0} #5\\#5 \arr@{-}"#1";"#2"|*+{\scriptscriptstyle A^{\estrella}}="a" \arr@{-}"#1";"#4"|*+{\scriptscriptstyle C^{\estrella}}="c" \arr@{-}"#2";"#3"|*+{\scriptscriptstyle B^{\estrella}/A^{\estrella}}="ba" \arr@{-}"#2";"#4"|*+{\scriptscriptstyle C^{\estrella}/A^{\estrella}}="ca" \arr@{-}"#3";"#4"|*+{\scriptscriptstyle C^{\estrella}/B^{\estrella}}="cb" \arr@{.}"#1";"#3"|!{"a";"c"}\hole|*+{\scriptscriptstyle B}="b"|!{"#2";"#4"}\hole|!{"ba";"ca"}\hole \arr@{.>>}@*{}"b";"ba"|!{"#2";"#4"}\hole \arr@{.>>}@*{}"c";"cb"|!{"#2";"#4"}\hole \arr@{>.>}@*{}"a";"b" \arr@{>->} @*{}"a";"c" \arr@{>.>}@*{}"b";"c" \arr@{>->} @*{}"ba";"ca" \arr@{->>} @*{}"c";"ca" \arr@{->>} @*{}"ca";"cb"}}} \tetra{3,1}{4,4}{3,6}{1,4}{&&&&&}$$ Suppose that ${\mathcal{W}}$ has a saturated class of weak equivalences. Then the categories $S_n{\mathcal{W}}$ inherit this property. Therefore the isomorphism $y_{2,1}$ in $\ho S_2{\mathcal{W}}$ is represented by a commutative diagram in ${\mathcal{W}}$ $$\label{wes2} \xymatrix{{A'}^{\estrella}\ar@{>->}[r]& {B'}^{\estrella}\ar@{->>}[r]& {B'}^{\estrella}/{A'}^{\estrella}\\ X^{\estrella}\ar@{>->}[r]\ar@{<-}[u]^\sim_{\alpha_2}\ar@{<-}[d]_\sim^{\alpha_1}& Y^{\estrella}\ar@{->>}[r]\ar@{<-}[u]^\sim_{\beta_2}\ar@{<-}[d]_\sim^{\beta_1}& Y^{\estrella}/X^{\estrella}\ar@{<-}[u]^\sim_{\gamma_2}\ar@{<-}[d]_\sim^{\gamma_1}\\ A^{\estrella}\ar@{>->}[r]& B^{\estrella}\ar@{->>}[r]& B^{\estrella}/A^{\estrella}}$$ where the horizontal lines are cofiber sequences and the vertical arrows are weak equivalences. The face $d_1^v(y_{2,1})$ is a cofiber sequence in ${\mathcal{W}}$ which is the source of the isomorphism in $\ho S_2{\mathcal{W}}$, and the face $d_0^v(y_{2,1})$ is the target. The faces $d_2^h(y_{2,1})$, $d_1^h(y_{2,1})$, $d_0^h(y_{2,1})$ correspond, in this order, to the isomorphisms $\alpha$, $\beta$, $\gamma$ in $\ho S_2{\mathcal{W}}$ represented by the vertical lines in the previous diagram. Notice that the representative of $y_{2,1}$ corresponds to the pasting of two bisimplices of bidegree $(2,1)$ in $X$ through a common face, see Fig. 5. $$\entrymodifiers={+0} \xymatrix@!C=1pc@!R=0.7pc{ \arr@{-}[rrr]|*+{\scriptscriptstyle {B'}^{\estrella}}="bb" \arr@{-}[rd]|*+{\scriptscriptstyle {A'}^{\estrella}}="aa"&&&\\& \arr@{-}[urr]|*+{\scriptscriptstyle {B'}^{\estrella}/{A'}^{\estrella}}="bbaa"\\\arr@{-}[uu] \arr@{.}[rrr]|(0.1666666){\hole}|(0.3333333){\hole} |*+{\scriptscriptstyle Y^{\estrella}}="y"|(0.6666666){\hole} \arr@{-}[rd]|*+{\scriptscriptstyle X^{\estrella}}="x"&&&\arr@{-}[uu]\\&\arr@{-}[uu] \arr@{-}[urr]|*+{\scriptscriptstyle Y^{\estrella}/X^{\estrella}}="yx" \arr@*{}@{>.>}"x";"y"|{\hole} \arr@*{}@{->>} "bb";"bbaa" \arr@*{}@{.>>} "y";"yx" \arr@*{}@{<-} "x";"aa"^*{\scriptscriptstyle\sim} \arr@*{}@{<-} "yx";"bbaa"^*{\scriptscriptstyle\sim} \arr@*{}@{<.} "y";"bb"^(0.3)*{\scriptscriptstyle\sim} |!{"2,2";"bbaa"}{\hole}\\ \arr@{-}[uu] \arr@{.}[rrr]|(0.1666666){\hole}|(0.3333333){\hole} |*+{\scriptscriptstyle B^{\estrella}}="b"|(0.6666666){\hole} \arr@{-}[rd]|*+{\scriptscriptstyle A^{\estrella}}="a"&&&\arr@{-}[uu]\\&\arr@{-}[uu] \arr@{-}[urr]|*+{\scriptscriptstyle B^{\estrella}/A^{\estrella}}="ba" \arr@*{}@{>->} "aa";"bb" \arr@*{}@{>.>}"a";"b"|{\hole} \arr@*{}@{.>>} "b";"ba" \arr@*{}@{->} "a";"x"^*{\scriptscriptstyle\sim} \arr@*{}@{->} "ba";"yx"^*{\scriptscriptstyle\sim} \arr@*{}@{.>} "b";"y"^(0.3)*{\scriptscriptstyle\sim}|(0.63){\hole} |!{"2,2";"bbaa"}{\hole}}$$ The degenerate bisimplices of total degree $1$ and $2$ in $X$ and $Y$ are depicted in Fig. 6. $$\entrymodifiers={+0} \begin{array}{c}\xymatrix@!R=3pc@!C=3pc{\ar@{-}[r]|*+{\scriptstyle 0}&}\end{array}\quad s^h_0(0) \qquad\qquad \begin{array}{c} \xymatrix@!=2pc{\ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="a"\ar@{-}[d]&\ar@{-}[d]\\ \ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="b"&\ar@{=}"b";"a"} \end{array}\quad s^v_0(A)$$ $$\entrymodifiers={+0} \begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="a"\ar@{-}[ru]|*+{\scriptstyle A^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle 0}="c"\ar@{=}"a";"b"\ar@{->>}"b";"c"} \end{array}\quad s^h_1(A) \qquad\qquad \begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle 0}="a"\ar@{-}[ru]|*+{\scriptstyle A^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle A^{\estrella}}="c"\ar@{>->}"a";"b"\ar@{=}"b";"c"} \end{array}\quad s^h_0(A)$$ The choice of binary coproducts $A\vee B$ in ${\mathcal{W}}$ gives rise to a biexact functor $\vee\colon{\mathcal{W}}\times{\mathcal{W}}\r{\mathcal{W}}$ which induces maps of bisimplical sets [@akts; @sdckt2] $$\begin{array}{c}X\times X\st{\vee}{\longrightarrow}X,\\{}\\ Y\times Y\st{\vee}{\longrightarrow}Y,\end{array}$$ in the obvious way. These maps induce co-$H$-multiplications in $|\diag X|$ and $|\diag Y|$, which come from the fact that they are infinite loop spaces. Abelian $2$-groups ================== In this section we recall the definition of stable quadratic modules, introduced in [@ch4c Definition IV.C.1]. Related structures are stable crossed modules [@2cm] and symetric categorical groups [@ccscg; @sccscg]. All these algebraic structures yield equivalent $2$-dimensional extensions of the theory of abelian groups. Among them stable quadratic modules are specially convenient since they are as small and strict as possible. \[ob\] A *stable quadratic module* $C_*$ is a diagram of group homomorphisms $$C_0^{ab}\otimes C_0^{ab}\st{{\langle \cdot,\cdot\rangle}}\longrightarrow C_1\st{\partial}\longrightarrow C_0$$ such that given $c_i,d_i\in C_i$, $i=0,1$, 1. $\partial{\langle c_0,d_0\rangle}=[d_0,c_0]$, 2. ${\langle \partial (c_1),\partial (d_1)\rangle}=[d_1,c_1]$, 3. ${\langle c_0,d_0\rangle}+{\langle d_0,c_0\rangle}=0$. Here $[x,y]=-x-y+x+y$ is the commutator of two elements $x,y\in K$ in a group $K$, and $K^{ab}$ is the abelianization of $K$. A *morphism* $f\colon C_*\rightarrow D_*$ of stable quadratic modules is given by group homomorphisms $f_i\colon C_i\rightarrow D_i$, $i=0,1$, compatible with the structure homomorphisms of $C_*$ and $D_*$, i.e. $f_0\partial=\partial f_1$ and $f_1{\langle \cdot,\cdot\rangle}={\langle f_0,f_0\rangle}$. \[crom\] It follows from Definition \[ob\] that the image of ${\langle \cdot,\cdot\rangle}$ and ${\operatorname{Ker}}\partial$ are central in $C_1$, the groups $C_0$ and $C_1$ have nilpotency class $2$, and $\partial(C_1)$ is a normal subgroup of $C_0$. There is a natural right action of $C_0$ on $C_1$ defined by $$\begin{aligned} c_1^{c_0}&=c_1+{\langle c_0,\partial(c_1)\rangle}.\end{aligned}$$ The axioms of a stable quadratic module imply that commutators in $C_0$ act trivially on $C_1$, and that $C_0$ acts trivially on the image of ${\langle \cdot,\cdot\rangle}$ and on ${\operatorname{Ker}}\partial$. The action gives $\partial\colon C_1\to C_0$ the structure of a crossed module. Indeed a stable quadratic module is the same as a commutative monoid in the category of crossed modules such that the monoid product of two elements in $C_0$ vanishes when one of them is a commutator, see [@1tk Lemma 4.18]. The forgetful functor from stable quadratic modules to pairs of sets $${\mathbf{squad}}{\longrightarrow}{\mathbf{Set}}\times{\mathbf{Set}}\colon C_*\mapsto (C_0,C_1)$$ has a left adjoint. This makes possible to define a free stable quadratic module with generating set $E_0$ in dimension $0$ and $E_1$ in dimension $1$. One can more generally define a stable quadratic module by a presentation with generators and relations in degrees $0$ and $1$. The explicit construction of a stable quadratic module with a given presentation can be found in the appendix of [@1tk]. For the purposes of this paper it will be enough to assume the existence of this construction, satisfying the obvious universal property as in the case of groups. We now recall the connection of stable quadratic modules with stable homotopy theory. The *homotopy groups* of a stable quadratic module $C_*$ are $$\begin{aligned} \pi_0 C_*&=& C_0/\partial(C_1),\\ \pi_1 C_*&=& {\operatorname{Ker}}[\partial\colon C_1\r C_0].\end{aligned}$$ Notice that these groups are abelian. Homotopy groups are obviously functors in the category ${\mathbf{squad}}$ of stable quadratic modules. A morphism in ${\mathbf{squad}}$ is a *weak equivalence* if it induces isomorphisms in $\pi_0$ and $\pi_1$. The *$k$-invariant* of $C_*$ is the natural homomorphism $$k\colon \pi_0C_*\otimes \mathbb{Z}/2{\longrightarrow}\pi_1 C_*$$ defined as $k(x\otimes 1)={\langle x,x\rangle}$. Weak equivalences in the Bousfield-Friedlander category ${\mathbf{Spec}}_0$ of connective spectra of simplicial sets [@htgamma] are also morphisms inducing isomorphisms in homotopy groups. Extending Definition \[hotsat\], if ${\mathbf{C}}$ is a category endowed with a class of weak equivalences we denote by $\ho{\mathbf{C}}$ the localization of ${\mathbf{C}}$ with respect to weak equivalences in the sense of [@gz]. [[@1tk Lemma 4.22]]{}\[llave\] There is defined a functor $$\lambda_0\colon\ho{\mathbf{Spec}}_0{\longrightarrow}\ho{\mathbf{squad}}$$ together with natural isomorphisms $$\begin{aligned} \pi_0 \lambda_0X&\cong& \pi_0X,\\ \pi_1 \lambda_0X&\cong& \pi_1X.\end{aligned}$$ The $k$-invariant of $\lambda_0 X$ corresponds to the action of the stable Hopf map in the stable homotopy groups of spheres $0\neq\eta\in \pi_1^s\cong\mathbb{Z}/2$, $$\pi_0X\otimes\mathbb{Z}/2{\longrightarrow}\pi_1X\colon x\otimes 1\mapsto x\cdot\eta.$$ Moreover, $\lambda_0$ restricts to an equivalence of categories on the full subcategory of spectra with homotopy groups concentrated in dimensions $0$ and $1$. We interpret this lemma as follows. Chain complexes of abelian groups $$\cdots\r 0\r B_1\st{\partial}{\longrightarrow}B_0\r 0\r\cdots$$ do not model all spectra with homotopy groups concentrated in dimensions $0$ and $1$ since these complexes neglect the stable Hopf map. However these spectra are modelled by stable quadratic modules, which can be regarded as non-abelian chain complexes $$\cdots\r 0\r C_1\st{\partial}{\longrightarrow}C_0\r 0\r\cdots$$ endowed with an extra map $$C_0^{ab}\otimes C_0^{ab}\st{{\langle \cdot,\cdot\rangle}}{\longrightarrow}C_1$$ which keeps track of the behaviour of commutators in $C_1$ and $C_0$. The homology of this non-abelian chain complex are the homotopy groups of the corresponding spectrum. Moreover, squaring the bracket ${\langle \cdot,\cdot\rangle}$ we recover the action of the stable Hopf map. In Section \[kt\] we recalled that $K$-theory spectra are spectra of topological spaces. In this section we have stated Lemma \[llave\] for spectra of simplicial sets. The geometric realization functor from simplicial sets to spaces induces an equivalence between the the homotopy categories of spectra of simplicial sets and spectra of topological spaces. Therefore in the next section we feel free to apply the functor $\lambda_0$ in Lemma \[llave\] to $K$-theory spectra. Algebraic models for lower $K$-theory ===================================== In [@1tk] we define a stable quadratic module ${\mathcal{D}}_*{\mathcal{W}}$ for any Waldhausen category ${\mathcal{W}}$ which is naturally isomorphic to $\lambda_0K({\mathcal{W}})$ in the homotopy category of stable quadratic modules, therefore ${\mathcal{D}}_*{\mathcal{W}}$ is a model for the $1$-type of the Waldhausen $K$-theory of ${\mathcal{W}}$. The stable quadratic module ${\mathcal{D}}_*{\mathcal{W}}$ is defined by a presentation with as few generators as possible. We now recall this presentation. We define ${\mathcal{D}}_*{\mathcal{W}}$ as the stable quadratic module generated in dimension zero by the symbols 1. $[A^{\estrella}]$ for any object in ${\mathcal{W}}$, and in dimension one by 1. $[A^{\estrella}\st{\sim}\rightarrow {A'}^{\estrella}]$ for any weak equivalence, 2. $[A^{\estrella}\into B^{\estrella}\onto B^{\estrella}/A^{\estrella}]$ for any cofiber sequence, such that the following relations hold. 1. $\partial[A^{\estrella}\st{\sim}\rightarrow {A'}^{\estrella}]=-[{A'}^{\estrella}]+[A^{\estrella}]$. 2. $\partial[A^{\estrella}\into B^{\estrella}\onto B^{\estrella}/A^{\estrella}]=-[B^{\estrella}]+[B^{\estrella}/A^{\estrella}]+[A^{\estrella}]$. 3. $[0]=0$. 4. $[A^{\estrella}\st{1}\rightarrow A^{\estrella}]=0$. 5. $[A^{\estrella}\st{1}\into A^{\estrella} \onto 0]=0$, $[0\into A^{\estrella}\st{1}\onto A^{\estrella}]=0$. 6. For any pair of composable weak equivalences $ A^{\estrella}\st{\sim}\rightarrow B^{\estrella}\st{\sim}\rightarrow C^{\estrella}$, $$[A^{\estrella}\st{\sim}\rightarrow C^{\estrella}]=[B^{\estrella}\st{\sim}\rightarrow C^{\estrella}]+[A^{\estrella}\st{\sim}\rightarrow B^{\estrella}].$$ 7. For any weak equivalence between cofiber sequences in ${\mathcal{W}}$, given by a commutative diagram , we have $$\begin{aligned} [A^{\estrella}\st{\sim}\rightarrow {A'}^{\estrella}]+[B^{\estrella}/A^{\estrella}\st{\cong}\rightarrow {B'}^{\estrella}/{A'}^{\estrella}]^{[A^{\estrella}]}=& -[{A'}^{\estrella}\into {B'}^{\estrella}\onto {B'}^{\estrella}/{A'}^{\estrella}]\\& +[B^{\estrella}\st{\sim}\rightarrow {B'}^{\estrella}]+[A^{\estrella}\into B^{\estrella}\onto B^{\estrella}/A^{\estrella}].\end{aligned}$$ 8. For any commutative diagram consisting of four cofiber sequences in ${\mathcal{W}}$ associated to a pair of composable cofibrations we have $$\begin{aligned} &[B^{\estrella}\into C^{\estrella}\onto C^{\estrella}/B^{\estrella}]+[A^{\estrella}\into B^{\estrella}\onto B^{\estrella}/A^{\estrella}] \\ &\quad =[A^{\estrella}\into C^{\estrella}\onto C^{\estrella}/A^{\estrella}]+[B^{\estrella}/A^{\estrella}\into C^{\estrella}/A^{\estrella}\onto C^{\estrella}/B^{\estrella}]^{[A^{\estrella}]}.\end{aligned}$$ 9. For any pair of objects $A^{\estrella}, B^{\estrella}$ in ${\mathcal{W}}$ $${\langle [A],[B]\rangle}= -[B^{\estrella}\st{i_2}\into A^{\estrella}\vee B^{\estrella}\st{p_1}\onto A^{\estrella}] +[A^{\estrella}\st{i_1}\into A^{\estrella}\vee B^{\estrella}\st{p_2}\onto B^{\estrella}] .$$ Here $$\xymatrix{A\ar@<.5ex>[r]^-{i_1}&A\vee B\ar@<.5ex>[l]^-{p_1}\ar@<-.5ex>[r]_-{p_2}&B\ar@<-.5ex>[l]_-{i_2}}$$ are the inclusions and projections of a coproduct in ${\mathcal{W}}$. \[semip\] The presentation of the stable quadratic module ${\mathcal{D}}_*{\mathcal{W}}$ is completely determined by the bisimplicial structure of $X=\ner wS.{\mathcal{W}}$ and the map $\vee\colon X\times X\r X$ in total degree $\leq 3$, see Section \[kt\]. More precisely, ${\mathcal{D}}_*{\mathcal{W}}$ is generated in degree $0$ by the bisimplices of total degree $1$ and in degree $1$ by the bisimplices of total degree $2$, see Fig. 3. Relations (1) and (2) identify the image by $\partial$ of a degree $1$ generator with the summation, in an appropriate order, of the faces of the boundary of the corresponding bisimplex of total degree $2$, see again Fig. 3. Relations (3)–(5) say that degenerate bisimplices of total degree $1$ or $2$ are trivial in ${\mathcal{D}}_*{\mathcal{W}}$, see Fig. 6. Relations (6)–(8) tell us that the summation, in an appropriate order, of the faces of the boundary of a bisimplex of total degree $3$ is zero in ${\mathcal{D}}_*{\mathcal{W}}$, see Fig 4. Finally (9) says that the bracket ${\langle [A],[B]\rangle}$ coincides with $$-[s_0^h(A)\vee s_1^h(B)]+[s_1^h(A)\vee s_0^h(B)],$$ i.e. it is obtained as follows. We first take the two possible degenerate bisimplices of bidegree $(2,0)$ associated to $A$ and $B$ in the following order. $$\entrymodifiers={+0} \begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle 0}="a"\ar@{-}[ru]|*+{\scriptstyle A^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle A^{\estrella}}="c"\ar@{>->}"a";"b"\ar@{=}"b";"c"} \end{array} \qquad \begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle B^{\estrella}}="a"\ar@{-}[ru]|*+{\scriptstyle B^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle 0}="c"\ar@{=}"a";"b"\ar@{->>}"b";"c"} \end{array} \qquad \begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="a"\ar@{-}[ru]|*+{\scriptstyle A^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle 0}="c"\ar@{=}"a";"b"\ar@{->>}"b";"c"} \end{array} \qquad \begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle 0}="a"\ar@{-}[ru]|*+{\scriptstyle B^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle B^{\estrella}}="c"\ar@{>->}"a";"b"\ar@{=}"b";"c"} \end{array}$$ We then take the coproduct of the first and the second pair of degenerate bisimplices. $$\entrymodifiers={+0} \begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle B^{\estrella}}="a"\ar@{-}[ru]|*+{\scriptstyle A^{\estrella}\vee B^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle A^{\estrella}}="c"\ar@{>->}"a";"b"\ar@{->>}"b";"c"} \end{array} \qquad\qquad\qquad \begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="a"\ar@{-}[ru]|*+{\scriptstyle A^{\estrella}\vee B^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle B^{\estrella}}="c"\ar@{>->}"a";"b"\ar@{->>}"b";"c"} \end{array}$$ Finally we take the difference between the corresponding generators in ${\mathcal{D}}_1{\mathcal{W}}$ $$\entrymodifiers={+0} -\left[\begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle B^{\estrella}}="a"\ar@{-}[ru]|*+{\scriptstyle A^{\estrella}\vee B^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle A^{\estrella}}="c"\ar@{>->}"a";"b"\ar@{->>}"b";"c"} \end{array}\right] \qquad+\qquad \left[\begin{array}{c} \xymatrix@!R=3pc@!C=3pc{&\\ \ar@{-}[r]|*+{\scriptstyle A^{\estrella}}="a"\ar@{-}[ru]|*+{\scriptstyle A^{\estrella}\vee B^{\estrella}}="b" &\ar@{-}[u]|*+{\scriptstyle B^{\estrella}}="c"\ar@{>->}"a";"b"\ar@{->>}"b";"c"} \end{array}\right].$$ There is a non-abelian Eilenberg-Zilber theorem behind this formula, compare [@1tk Theorem 4.10 and Example 4.13]. The main result of [@1tk] is the following theorem. [[@1tk Theorem 1.7]]{}\[main1tk\] Let ${\mathcal{W}}$ be a Waldhausen category. There is a natural isomorphism in $\ho{\mathbf{squad}}$ $${\mathcal{D}}_*{\mathcal{W}}\st{\cong}{\longrightarrow}\lambda_0K({\mathcal{W}}).$$ This result is meaningful since $\lambda_0K({\mathcal{W}})$ is huge compared with ${\mathcal{D}}_*{\mathcal{W}}$, while ${\mathcal{D}}_*{\mathcal{W}}$ is directly defined in terms of the basic structure of the Waldhausen category ${\mathcal{W}}$. As a consequence we have an exact sequence of groups $$K_1({\mathcal{W}})\hookrightarrow{\mathcal{D}}_1{\mathcal{W}}\st{\partial}{\longrightarrow}{\mathcal{D}}_0{\mathcal{W}}\twoheadrightarrow K_0({\mathcal{W}}).$$ We now extend Theorem \[main1tk\] to derived $K$-theory. We define ${\mathcal{D}^{\mathrm{der}}}_*{\mathcal{W}}$ as the stable quadratic module generated in dimension zero by the symbols 1. $[A]$ for any object in ${\mathcal{W}}$, i.e. the same as (1), and in dimension one by 1. $[A\st{\cong}\rightarrow A']$ for any isomorphism in $\ho {\mathcal{W}}$, 2. $[A\into B\onto B/A]$ for any cofiber sequence in ${\mathcal{W}}$, i.e. the same as (3), such that the following relations hold. 1. $\partial[A\st{\cong}\rightarrow A']=-[A']+[A]$. 2. $=$ (2). 3. $=$ (3). 4. $[A\st{1}\rightarrow A]=0$. 5. $=$ (5). 6. For any pair of composable isomorphisms $A\st{\cong}\rightarrow B\st{\cong}\rightarrow C$ in $\ho{\mathcal{W}}$, $$[A\st{\cong}\rightarrow C]=[B\st{\cong}\rightarrow C]+[A\st{\cong}\rightarrow B].$$ 7. For any commutative diagram in ${\mathcal{W}}$ as we have $$\begin{aligned} [\alpha\colon A\st{\cong}\rightarrow A']+[\gamma\colon B/A\st{\cong}\rightarrow B'/A']^{[A]}=& -[A'\into B'\onto B'/A']\\& +[\beta\colon B\st{\cong}\rightarrow B'] +[A\into B\onto B/A].\end{aligned}$$ Here $\alpha=\zeta(\alpha_2)^{-1}\zeta(\alpha_1)$, $\beta=\zeta(\beta_2)^{-1}\zeta(\beta_1)$ and $\gamma=\zeta(\gamma_2)^{-1}\zeta(\gamma_1)$. 8. $=$ (8). 9. $=$ (9). If ${\mathcal{W}}$ is a Waldhausen category with cylinders and a saturated class of weak equivalences then the presentation of the stable quadratic module ${\mathcal{D}^{\mathrm{der}}}_*{\mathcal{W}}$ is determined by the bisimplicial structure of $Y=\ner i\ho S.{\mathcal{W}}$ and the map $\vee\colon Y\times Y\r Y$ in total degree $\leq 3$, see Section \[kt\], exactly in the same way as ${\mathcal{D}}_*{\mathcal{W}}$ is determined by $X=\ner wS.{\mathcal{W}}$ and $\vee\colon X\times X\r X$, see Remark \[semip\]. Therefore replacing $X$ by $Y$ in the proof of [@1tk Theorem 1.7] we obtain the following result. \[main3\] Let ${\mathcal{W}}$ be a Waldhausen category with cylinders and a saturated class of weak equivalences. There is a natural isomorphism in $\ho{\mathbf{squad}}$ $${\mathcal{D}^{\mathrm{der}}}_*{\mathcal{W}}\st{\cong}{\longrightarrow}\lambda_0K({\mathbb{D}}{\mathcal{W}}).$$ As a consequence we have an exact sequence of groups $$K_1({\mathbb{D}}{{\mathcal{W}}})\hookrightarrow{\mathcal{D}^{\mathrm{der}}}_1{\mathcal{W}}\st{\partial}{\longrightarrow}{\mathcal{D}^{\mathrm{der}}}_0{\mathcal{W}}\twoheadrightarrow K_0({\mathbb{D}}{{\mathcal{W}}}).$$ \[clave\] As one can easily check, taking $\lambda_0$ in the comparison map of spectra which induces $\mu_n\colon K_n({\mathcal{W}})\r K_n({\mathbb{D}}{{\mathcal{W}}})$ in homotopy groups corresponds to the natural morphism in ${\mathbf{squad}}$, $$\bar{\mu}\colon{\mathcal{D}}_*{\mathcal{W}}{\longrightarrow}{\mathcal{D}^{\mathrm{der}}}_*{\mathcal{W}},\;\;$$ $$\begin{array}{rcl} {[}A{]}&\mapsto& {[}A{]},\\ {[}f\colon A\st{\sim}\r A'{]}&\mapsto& {[}\zeta(f)\colon A\st{\cong}\r A'{]},\\ {[}A\into B\onto B/A{]}&\mapsto& {[}A\into B\onto B/A{]}, \end{array}$$ under the natural isomorphisms of Theorems \[main1tk\] and \[main3\]. In particular taking $\pi_0$ and $\pi_1$ in this morphism of stable quadratic modules we obtain $\mu_0$ and $\mu_1$, respectively. This fact will be used below in the proof of Theorem \[main2\]. Proof of Theorem \[main2\] ========================== Theorem \[main2\] is a consequence of the following result. \[el\] Let ${\mathcal{W}}$ be a Waldhausen category with cylinders and a saturated class of weak equivalences. The natural morphism in ${\mathbf{squad}}$ $$\bar{\mu}\colon{\mathcal{D}}_*{\mathcal{W}}{\longrightarrow}{\mathcal{D}^{\mathrm{der}}}_*{\mathcal{W}},$$ defined in Remark \[clave\], is an isomorphism. The key for the proof of Theorem \[el\] is the following lemma. \[la\] Let ${\mathcal{W}}$ be a Waldhausen category with cylinders satisfying the 2 out of 3 axiom. Two weak equivalences $f,g\colon A\st{\sim}\r A'$ which are homotopic $f\simeq g$ represent the same element in ${\mathcal{D}}_1{\mathcal{W}}$, $$[f\colon A\st{\sim}\r A']\;=\;[g\colon A\st{\sim}\r A'].$$ Let $IA$ be a cylinder of $A$ and $$A\vee A\mathop{\rightarrowtail}\limits_i IA\mathop{{\longrightarrow}}\limits^\sim_p A$$ a factorization of the folding map, i.e. if $i=(i_0,i_1)$ then $pi_0=pi_1=1_A$. Since both $p$ and $1_A$ are weak equivalences we deduce from the 2 out of 3 axiom that $i_0$ and $i_1$ are also weak equivalences. Moreover, for $j=0,1$, $$\begin{aligned} 0&\st{\text{\scriptsize ({{\textrm{\rm R}}}4)}}=&[A\st{1_A}\r A]\\ &=&[pi_j\colon A\st{\sim}\r A]\\ \text{\scriptsize ({{\textrm{\rm R}}}6)}\quad&=&[p\colon IA\st{\sim}\r A]+[i_j\colon A\st{\sim}\r IA],\end{aligned}$$ therefore $$[i_0\colon A\st{\sim}\r IA]\;\;=\;\; -[p\colon IA\st{\sim}\r A] \;\;=\;\;[i_1\colon A\st{\sim}\r IA].$$ Furthermore, $f\simeq g$, so there is a weak equivalence $h\colon A'\st{\sim}\r A''$ and a morphism $H\colon IA\r A''$ such that $Hi_0=hf$ and $Hi_1=hg$. Again by the 2 out of 3 axiom $H$ is a weak equivalence, and $$\begin{aligned} {[}h\colon A'\st{\sim}\r A''{]}+{[}f\colon A\st{\sim}\r A'{]}&\st{\text{\scriptsize ({{\textrm{\rm R}}}6)}}=&{[}hf=Hi_0\colon A\st{\sim}\r A''{]}\\ \text{\scriptsize({{\textrm{\rm R}}}6)}\quad&=&{[}H\colon IA\st{\sim}\r A''{]}+{[}i_0\colon A\st{\sim}\r IA{]}\\ &=&{[}H\colon IA\st{\sim}\r A''{]}+{[}i_1\colon A\st{\sim}\r IA{]}\\\text{\scriptsize ({{\textrm{\rm R}}}6)}\quad&=& {[}hg=Hi_1\colon A\st{\sim}\r A''{]}\\ \text{\scriptsize({{\textrm{\rm R}}}6)}\quad&=& {[}h\colon A'\st{\sim}\r A''{]}+{[}g\colon A\st{\sim}\r A'{]},\end{aligned}$$ hence we are done. We are now ready to prove Theorem \[el\]. We are going to define the inverse of $\bar{\mu}$, $$\bar{\nu}\colon{\mathcal{D}^{\mathrm{der}}}_*{\mathcal{W}}{\longrightarrow}{\mathcal{D}}_*{\mathcal{W}}.$$ We first show that $$\begin{aligned} \bar{\nu}_0[A]&=&[A],\\ \bar{\nu}_1[\alpha\colon A\st{\cong}\r A']&=&-[\alpha_2\colon A'\st{\sim}\r X]+[\alpha_1\colon A\st{\sim}\r X],\\ \bar{\nu}_1{[}A\into B\onto B/A{]}&=&{[}A\into B\onto B/A{]},\end{aligned}$$ defines a stable quadratic module morphism $\bar{\nu}$. Here $$A\mathop{{\longrightarrow}}\limits^\sim_{\alpha_1} X\mathop{\longleftarrow}\limits^\sim_{\alpha_2} A'$$ is a representative of the isomorphism $\alpha$. For this we are going to prove that the image of $[\alpha\colon A\st{\cong}\r A']$ does not depend on the choice of a representative. Suppose that $$A\mathop{{\longrightarrow}}\limits_{\alpha'_1}^\sim Y\mathop{\longleftarrow}\limits^\sim_{\alpha'_2} A'$$ also represents $\alpha$. Then there is a diagram in ${\mathcal{W}}$ $$\xymatrix{&X^{\estrella}\ar@{<-}[ld]_{\alpha_1}\ar@{<-}[rd]^{\alpha_2}&\\ A^{\estrella}&Z^{\estrella}\ar@{<-}[l]|{f_1}\ar@{<-}[r]|{f_2}\ar@{<-}[u]|{g}\ar@{<-}[d]|{g'}&A'^{\estrella}\\&Y^{\estrella}\ar@{<-}[lu]^{\alpha'_1}\ar@{<-}[ru]_{\alpha'_2}&}$$ where all arrows are weak equivalences and the four triangles commute up to homotopy, so $$\begin{aligned} -[\alpha_2\colon A'\st{\sim}\r X]+[\alpha_1\colon A\st{\sim}\r X]&=&-[\alpha_2\colon A'\st{\sim}\r X]-[g\colon X\st{\sim}\r Z]\\&&+[g\colon X\st{\sim}\r Z]+[\alpha_1\colon A\st{\sim}\r X]\\ \text{\scriptsize ({{\textrm{\rm R}}}6)}\quad&=&-[g\alpha_2\colon A'\st{\sim}\r Z]+[g\alpha_1\colon A\st{\sim}\r Z]\\ \text{\scriptsize Lemma \ref{la}}\quad&=&-[f_2\colon A'\st{\sim}\r Z]+[f_1\colon A\st{\sim}\r Z]\\ \text{\scriptsize Lemma \ref{la}}\quad&=&-[g'\alpha'_2\colon A'\st{\sim}\r Z]+[g'\alpha'_1\colon A\st{\sim}\r Z]\\ \text{\scriptsize ({{\textrm{\rm R}}}6)}\quad&=&-[\alpha'_2\colon A'\st{\sim}\r Y]-[g'\colon Y\st{\sim}\r Z]\\&&+[g'\colon Y\st{\sim}\r Z]+[\alpha_1'\colon A\st{\sim}\r Y]\\ &=&-[\alpha_2'\colon A'\st{\sim}\r X]+[\alpha_1'\colon A\st{\sim}\r X].\end{aligned}$$ Now we check that the definition of $\bar{\nu}$ on generators is compatible with the defining relations. The only non-trivial part concerns relations (1), (6) and (7). Compatibility with (1) follows from $$\begin{aligned} \bar{\nu}_0\partial[\alpha\colon A\st{\cong}\r A']&=&\partial\bar{\nu}_1[\alpha\colon A\st{\cong}\r A']\\&=&-\partial[\alpha_2\colon A'\st{\sim}\r X]+\partial[\alpha_1\colon A\st{\sim}\r X]\\ {\scriptstyle ({{\textrm{\rm R}}}1)}\quad &=&-(-[X]+[A'])+(-[X]+[A])\\ &=&-[A']+[A]\\ &=&-\bar{\nu}_0[A']+\bar{\nu}_0[A].$$ In order to check compatibility with (6) we consider two composable isomorphisms in $\ho{\mathcal{W}}$ $$A\mathop{{\longrightarrow}}\limits^{\cong}_{\alpha} B\mathop{{\longrightarrow}}\limits^{\cong}_{\beta} C$$ and we take representatives of $\alpha$, $\beta$ and $\beta\alpha$ as in the following commutative diagram of weak equivalences in ${\mathcal{W}}$ $$\xymatrix@C=15pt{&&X\cup_BY\ar@{<-}[rd]^{\bar{\alpha}_2}\ar@{<-<}[ld]_{\bar{\beta}_1}&&\\&X\ar@{<-}[rd]_{\alpha_2}\ar@{<-}[ld]_{\alpha_1}\ar@{}[rr]|{\text{push}}&&Y\ar@{<-}[rd]^{\beta_2}&\\A&&B\ar@{>->}[ru]_{\beta_1}&&C}$$ Then $$\begin{aligned} \bar{\nu}_1[\beta\alpha\colon A\st{\cong}\r C] &=&-[\bar{\alpha}_2\beta_2\colon C\st{\sim}\r X\cup_BY]+[\bar{\beta}_1\alpha_1\colon A\st{\sim}\r X\cup_BY]\\ &=&-[\bar{\alpha}_2\beta_2\colon C\st{\sim}\r X\cup_BY]+[\bar{\alpha}_2\beta_1\colon B\st{\sim}\r X\cup_BY]\\ &&-[\bar{\beta}_1\alpha_2\colon B\st{\sim}\r X\cup_BY]+[\bar{\beta}_1\alpha_1\colon A\st{\sim}\r X\cup_BY]\\ {\scriptstyle ({{\textrm{\rm R}}}6)}\qquad&=&-([\bar{\alpha}_2\colon Y\st{\sim}\r X\cup_BY]+[\beta_2\colon C\st{\sim}\r Y])\\ &&+[\bar{\alpha}_2\colon Y\st{\sim}\r X\cup_BY]+[\beta_1\colon B\st{\sim}\r Y]\\ &&-([\bar{\beta}_1\colon X\st{\sim}\r X\cup_BY]+[\alpha_2\colon B\st{\sim}\r X])\\ &&+[\bar{\beta}_1\colon X\st{\sim}\r X\cup_BY]+[\alpha_1\colon A\st{\sim}\r X]\\ &=&-[\beta_2\colon C\st{\sim}\r Y]+[\beta_1\colon B\st{\sim}\r Y]\\ &&-[\alpha_2\colon B\st{\sim}\r X]+[\alpha_1\colon A\st{\sim}\r X]\\ &=&\bar{\nu}_1[\beta\colon B\st{\cong}\r C]+\bar{\nu}_1[\alpha\colon A\st{\cong}\r B].\end{aligned}$$ Let us now check compatibility with (7). $$\begin{aligned} -\bar{\nu}_1[A'\into B'\onto B'/A']&&\\ +\bar{\nu}_1[\beta\colon B\st{\cong}\r B']&&\\ +\bar{\nu}_1[A\into B\onto B/A]&=&-[A'\into B'\onto B'/A']-[\beta_2\colon B'\st{\sim}\r Y]\\ &&+[X\into Y\onto Y/X]-[X\into Y\onto Y/X]\\ &&+[\beta_1\colon B\st{\sim}\r Y]+[A\into B\onto B/A]\\ {\scriptstyle ({{\textrm{\rm R}}}7)}\quad&=&-([\alpha_2\colon A'\st{\sim}\r X]+[\gamma_2\colon B'/A'\st{\sim}\r Y/X]^{[A']})\\ &&+[\alpha_1\colon A\st{\sim}\r X]+[\gamma_1\colon B/A\st{\sim}\r Y/X]^{[A]}\\ \text{\scriptsize Rem. \ref{crom}}\quad&=&-[\gamma_2\colon B'/A'\st{\sim}\r Y/X]-[\alpha_2\colon A'\st{\sim}\r X]\\ &&+[\alpha_1\colon A\st{\sim}\r X]+[\gamma_1\colon B/A\st{\sim}\r Y/X]\\ &&-{\langle [A'],\partial[\gamma_2]\rangle}+{\langle [A],\partial[\gamma_1]\rangle}\\ \text{\scriptsize Defn. \ref{ob} (2) and Rem. \ref{crom}}\quad&=&-[\alpha_2\colon A'\st{\sim}\r X]+[\alpha_1\colon A\st{\sim}\r X]\\ &&-[\gamma_2\colon B'/A'\st{\sim}\r Y/X]+[\gamma_1\colon B/A\st{\sim}\r Y/X]\\ &&+{\langle -\partial[\alpha_2]+\partial[\alpha_1],-\partial[\gamma_2]\rangle} -{\langle [A'],\partial[\gamma_2]\rangle}+{\langle [A],\partial[\gamma_1]\rangle}\\ \text{\scriptsize ({{\textrm{\rm R}}}1)}\quad&=&-[\alpha_2\colon A'\st{\sim}\r X]+[\alpha_1\colon A\st{\sim}\r X]\\ &&-[\gamma_2\colon B'/A'\st{\sim}\r Y/X]+[\gamma_1\colon B/A\st{\sim}\r Y/X]\\ &&+{\langle -(-[X]+[A'])+(-[X]+[A]),-\partial[\gamma_2]\rangle} \\&& +{\langle [A'],-\partial[\gamma_2]\rangle}+{\langle [A],\partial[\gamma_1]\rangle}\\ &=&-[\alpha_2\colon A'\st{\sim}\r X]+[\alpha_1\colon A\st{\sim}\r X]\\ &&-[\gamma_2\colon B'/A'\st{\sim}\r Y/X]+[\gamma_1\colon B/A\st{\sim}\r Y/X]\\ &&+{\langle [A],-\partial[\gamma_2]\rangle}+{\langle [A],\partial[\gamma_1]\rangle}\\ &=&+\bar{\nu}_1[\alpha\colon A\st{\cong}\r A']+\bar{\nu}_1[\gamma\colon B/A\st{\cong}\r B'/A']\\ &&+{\langle \bar{\nu}_0[A],\partial\bar{\nu}_1[\gamma\colon B/A\st{\cong}\r B'/A']\rangle}\\ \text{\scriptsize Rem. \ref{crom}}\quad&=&\bar{\nu}_1[\alpha\colon A\st{\cong}\r A'] +\bar{\nu}_1[\gamma\colon B/A\st{\cong}\r B'/A']^{\bar{\nu}_0[A]}.\end{aligned}$$ This establishes that $\bar{\nu}$ is a well defined morphism of stable quadratic modules. Let us now check that $\bar{\mu}\bar{\nu}=1_{{\mathcal{D}^{\mathrm{der}}}_*{\mathcal{W}}}$ and $\bar{\nu}\bar{\mu}=1_{{\mathcal{D}}_*{\mathcal{W}}}$. Both equations are obvious on generators $({{\textrm{\rm G}}}1)=({{\textrm{\rm DG}}}1)$ and $({{\textrm{\rm G}}}3)=({{\textrm{\rm DG}}}3)$. For $({{\textrm{\rm G}}}2)$ $$\begin{aligned} \bar{\nu}_1\bar{\mu}_1[f\colon A\st{\sim}\r A']&=&\bar{\nu}_1[\zeta(f)\colon A\st{\cong}\r A']\\ &=&-[1_{A'}\colon A'\st{\sim}\r A']+[f\colon A\st{\sim}\r A']\\ {\scriptstyle ({{\textrm{\rm R}}}4)}\quad&=&[f\colon A\st{\sim}\r A'].\end{aligned}$$ If $\alpha\colon A\st{\cong}\r A'$ is an isomorphism in $\ho{\mathcal{W}}$ we have the following equation in ${\mathcal{D}^{\mathrm{der}}}_1{\mathcal{W}}$, $$\begin{aligned} 0&\st{\text{\scriptsize ({{\textrm{\rm DR}}}4)}}=&[A\st{1_A}\r A]\\ &=&[\alpha^{-1}\alpha\colon A\st{\cong}\r A]\\ \text{\scriptsize ({{\textrm{\rm DR}}}6)}\quad&=&[\alpha^{-1}\colon A'\st{\cong}\r A] + [\alpha\colon A\st{\cong}\r A'],\end{aligned}$$ so $[\alpha^{-1}\colon A'\st{\cong}\r A]=-[\alpha\colon A\st{\cong}\r A']$. Now for $({{\textrm{\rm DG}}}2)$ $$\begin{aligned} \bar{\mu}_1\bar{\nu}_1[\alpha\colon A\st{\cong}\r A']&=&-\bar{\mu}_1[\alpha_2\colon A'\st{\sim}\r X]+\bar{\mu}_1[\alpha_1\colon A\st{\sim}\r X]\\ &=&-[\zeta(\alpha_2)\colon A'\st{\cong}\r X]+[\zeta(\alpha_1)\colon A\st{\cong}\r X]\\ &=&[\zeta(\alpha_2)^{-1}\colon X\st{\cong}\r A']+[\zeta(\alpha_1)\colon A\st{\cong}\r X]\\ \text{\scriptsize ({{\textrm{\rm DR}}}6)}\quad&=&[\alpha=\zeta(\alpha_2)^{-1}\zeta(\alpha_1)\colon A\st{\cong}\r A'].\end{aligned}$$ The proof of Theorem \[el\] is now finished. \[last\] Let ${\mathcal{W}}$ be a Waldhausen category with cylinders satisfying the 2 out of 3 axiom. We do not assume that ${\mathcal{W}}$ has a saturated class of weak equivalences. However we can endow the underlying category with a new Waldhausen category structure which does have a saturated class of weak equivalences. We consider the Waldhausen category $\overline{{\mathcal{W}}}$ with the same underlying category as ${\mathcal{W}}$. Cofibrations in $\overline{{\mathcal{W}}}$ are also de same as in ${\mathcal{W}}$. Weak equivalences in $\overline{{\mathcal{W}}}$ are the morphisms in ${\mathcal{W}}$ which are mapped to isomorphisms in $\ho{\mathcal{W}}$ by the canonical functor $\zeta\colon{\mathcal{W}}\r\ho{\mathcal{W}}$. Therefore weak equivalences in ${\mathcal{W}}$ are also weak equivalences in $\overline{{\mathcal{W}}}$ but the converse need not hold. This indeed defines a Waldhausen category $\overline{{\mathcal{W}}}$ with cylinders and a saturated class of weak equivalences, and the obvious exact functor ${\mathcal{W}}\r\overline{{\mathcal{W}}}$ induces an isomorphism on the associated derivators ${\mathbb{D}}{\mathcal{W}}\cong{\mathbb{D}}\overline{{\mathcal{W}}}$, compare [@ciscd dual of Proposition 3.16] and [@cht Theorem 6.2.2]. Hence we have a commutative diagram for $n=0,1$, $$\xymatrix{K_n({\mathcal{W}})\ar[r]^{\mu_n}\ar[d]&K_n({\mathbb{D}}{\mathcal{W}})\ar[d]^\cong\\ K_n(\overline{{\mathcal{W}}})\ar[r]_{\mu_n}^\cong&K_n({\mathbb{D}}\overline{{\mathcal{W}}})}$$ Here the lower arrow is an isomorphism by Theorem \[main2\]. Now we can use the ‘fibration theorem’, [@akts 1.6.7] and [@nktdc Theorem 11], to embed the morphisms $\mu_n\colon K_n({\mathcal{W}})\r K({\mathbb{D}}{\mathcal{W}})$, $n=0,1$, in an exact sequence. More precisely, let ${\mathcal{W}}_0$ be the full subcategory of ${\mathcal{W}}$ spanned by the objects which are isomorphic to $0$ in $\ho{\mathcal{W}}$. The category ${\mathcal{W}}_0$ is a Waldhausen category where a morphism is a cofibration, resp. a weak equivalence, if and only if it is a cofibration, resp. a weak equivalence, in ${\mathcal{W}}$. There is an exact sequence $$K_1({\mathcal{W}}_0){\longrightarrow}K_1({\mathcal{W}})\st{\mu_1}{\longrightarrow}K_1({\mathbb{D}}{\mathcal{W}})\st{\delta}{\longrightarrow}K_0({\mathcal{W}}_0){\longrightarrow}K_0({\mathcal{W}})\st{\mu_0}{\longrightarrow}K_0({\mathbb{D}}{\mathcal{W}})\r 0.$$ The group $K_0({\mathcal{W}}_0)$ has also been considered by Weiss in [@hlwc]. Weiss defines the Whitehead group of ${\mathcal{W}}$ as $\operatorname{Wh}({\mathcal{W}})=K_0({\mathcal{W}}_0)$. Moreover, for any morphism $f\colon A\r A'$ which becomes an isomorphism in $\ho{\mathcal{W}}$ he defines the Whitehead torsion $\tau(f)\in\operatorname{Wh}({\mathcal{W}})$, which is the obstruction for $f$ to be a weak equivalence in ${\mathcal{W}}$. 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This hypothesis is not strictly necessary, since the general case follows then from cofinality arguments, see [@tckt]. [^3]: The references [@derivateurs; @ciscd] and [@sdckt1; @cht] follow a different convention with respect to sides. Here we follow the convention in [@derivateurs; @ciscd], so what we call a ‘right pointed derivator’ is the same as a ‘left pointed derivator’ in [@sdckt1].

--- author: - Dan Segal title: 'Defining $\mathbb{A}$ in $G(\mathbb{A})$' --- It is shown in the papers [@NST] and [@ST] that for many integral domains $R,$ the ring is bi-intepretable with various Chevalley groups $G(R)$. The model theory of adèle rings and some of their subrings has attracted some recent interest ([@DM], [@D], [@AMO]), and it seemed worthwhile to extend the results in that direction. Let $\mathbb{A}$ denote the adèle ring of a global field $K$, with $\mathrm{char}(K)\neq2,3,5$. We consider subrings of $\mathbb{A}$ of the following kind:$$\begin{aligned} R & =\mathbb{A},\\ R & =~\prod_{\mathfrak{p}\in\mathcal{P}}\mathfrak{o}_{\mathfrak{p}}$$ where $\mathfrak{o}$ is the ring of integers of $K$ and $\mathcal{P}$ may be any non-empty set of primes (or places) of $K$. For example, $R$ could be the whole adèle ring of $\mathbb{Q}$, or $\widehat{\mathbb{Z}}=\prod _{p}\mathbb{Z}_{p}.$ \[sl2\]The ring $R$ is bi-interpretable with each of the groups $\mathrm{SL}_{2}(R)$, $\mathrm{SL}_{2}(R)/\left\langle -1\right\rangle ,$ $\mathrm{PSL}_{2}(R)$. \[high\_rank\]Let $G$ be a simple Chevalley-Demazure group scheme of rank at least $2$. Then $R$ is bi-interpretable with the group $G(R)$. The special cases where $R=\mathfrak{o}_{\mathfrak{p}}$ were established in [@NST], §4 and [@ST]. For a rational prime $p$ we write $R_{p}=\prod_{\mathfrak{p}\in\mathcal{P},~\mathfrak{p}\mid p}\mathfrak{o}_{\mathfrak{p}}$. \[S-lemma\]$R$ has a finite subset $S$ such that every element of $R$ is equal to one of the form$$\xi^{2}-\eta^{2}+s \label{formula}$$ with $\xi,\eta\in R^{\ast}$ and $s\in S$. In any field of characteristic not $2$ and size $>5$, every element is the difference of two non-zero squares. It follows that the same is true for each of the rings $\mathfrak{o}_{\mathfrak{p}}$ with $N(\mathfrak{p})>5$ and odd. If $N(\mathfrak{p})$ is $3$ or $5$ then every element of $\mathfrak{o}_{\mathfrak{p}}$ is of the form (\[formula\]) with $\xi,\eta\in \mathfrak{o}_{\mathfrak{p}}^{\ast}$ and $s\in\{0,\pm1\}.$ If $\mathfrak{p}$ divides $2$, the same holds if $S$ is a set of representatives for the cosets of $4\mathfrak{p}$ in $\mathfrak{o}$. Now by the Chinese Remainder Theorem (and Hensel’s lemma) we can pick a finite subset $S_{1}$ of $R_{2}\times R_{3}\times R_{5}$ such that every element of $R_{2}\times R_{3}\times R_{5}$ is of the form (\[formula\]) with $\xi ,\eta\in\mathfrak{o}_{\mathfrak{p}}^{\ast}$ and $s\in S_{1}$. Finally, let $S$ be the subset of elements $s\in R$ that project into $S_{1}$ and have $\mathfrak{o}_{\mathfrak{p}}$-component $1$ for all $\mathfrak{p}\nmid30$ (including infinite places if present). **Remark** If $K=\mathbb{Q}$ one could choose $S\subset\mathbb{Z}$ (diagonally embedded in $R$). The plethora of parameters in the following argument can then be replaced by just three - $h(\tau),~u(1),$ $v(1)$ - or even two when $R=\mathbb{A}$, in which case we replace $h(\tau)$ by $h(2),$ which can be expressed in terms of $u(1)$ and $v(1)$ by the formula (\[h-identity\]) below. Also the formula (\[P-formula\]) can be replaced by the simpler one: $y_{2}=u^{x}u^{-y}u^{s}\wedge y_{3}=y_{1}^{x}y_{1}^{-y}y_{1}^{s}$. For a finite subset $T$ of $\mathbb{Z}$ let $$R_{T}=\left\{ r\in R~\mid~r_{\mathfrak{p}}\in T\text{ for every }\mathfrak{p}\right\} .$$ This is a definable set, since $r\in R_{T}$ if and only if $f(r)=0$ where $f(X)=\prod_{t\in T}(X-t)$. Choose $S$ as in Lemma \[S-lemma\], with $0,~1\in S$, and write $S^{2}=S.S.$ Let $\Gamma=\mathrm{SL}_{2}(R)/Z$ where $Z$ is $1$, $\left\langle -1\right\rangle $ or the centre of $\mathrm{SL}_{2}(R)$.  For $\lambda\in R$ write$$u(\lambda)=\left( \begin{array} [c]{cc}1 & \lambda\\ 0 & 1 \end{array} \right) ,~v(\lambda)=\left( \begin{array} [c]{cc}1 & 0\\ -\lambda & 1 \end{array} \right) ,~~h(\lambda)=\left( \begin{array} [c]{cc}\lambda^{-1} & 0\\ 0 & \lambda \end{array} \right) ~~(\lambda\in R^{\ast})$$ (matrices interpreted modulo $Z$; note that $\lambda\longmapsto u(\lambda)$ is bijective for each choice of $Z$). Fix $\tau\in R^{\ast}$ with $\tau_{\mathfrak{p}}=2$ for $\mathfrak{p}\nmid2,$ $\tau_{\mathfrak{p}}=3$ for $\mathfrak{p}\mid2$. It is easy to verify that$$\mathrm{C}_{\Gamma}(h(\tau))=h(R^{\ast}):=H. \label{Hdef}$$ \[P1\]The ring $R$ is definable in $\Gamma.$ We take $h:=h(\tau)$ and $\{u(c)~\mid~c\in S^{2}\}$ as parameters, and put $u:=u(1)$. ‘Definable’ will mean definable with these parameters. For $\lambda\in R$ and $\mu\in R^{\ast}$ we have$$u(\lambda)^{h(\mu)}=u(\lambda\mu^{2}).$$ Now (\[Hdef\]) shows that $H$ is definable. If $\lambda=\xi^{2}-\eta^{2}+s$ and $x=h(\xi),~y=h(\eta)$ then$$u(\lambda)=u^{x}u^{-y}u(s).$$ It follows that$$U:=u(R)=\bigcup_{s\in S}\{u^{x}u^{-y}u(s)~\mid~x,~y\in H\}$$ is definable. The map $u:R\rightarrow U$ is an isomorphism from $(R,+)$ to $U$. It becomes a ring isomorphism with multiplication $\ast$ if one defines$$u(\beta)\ast u(\alpha)=u(\beta\alpha). \label{thing}$$ We need to provide an $L_{\mathrm{gp}}$ formula $P$ such that for $y_{1},~y_{2},~y_{3}\in U$,$$y_{1}\ast y_{2}=y_{3}\Longleftrightarrow\Gamma\models P(y_{1},y_{2},y_{3}). \label{nextthing}$$ Say $\alpha=\xi^{2}-\eta^{2}+s$, $\beta=\zeta^{2}-\rho^{2}+t$. Then$$u(\beta\alpha)=u(\beta)^{x}u(\beta)^{-y}u(s)^{z}u(s)^{-r}u(st)$$ where $x=h(\xi),~y=h(\eta),~z=h(\zeta)~$and $r=h(\rho)$. So we can take $P(y_{1},y_{2},y_{3})$ to be a formula expressing the statement: there exist $x,~y,z,r\in H$ such that for some $s,t\in S$$$\begin{aligned} y_{1} & =u^{z}u^{-r}u(t),~y_{2}=u^{x}u^{-y}u(s),\label{P-formula}\\ y_{3} & =y_{1}^{x}y_{1}^{-y}u(s)^{z}u(s)^{-r}u(st).\nonumber\end{aligned}$$ The group $\Gamma$ is interpretable in $R$. When $\Gamma=\mathrm{SL}_{2}(R)$,  clearly $\Gamma$ is definable as the set of $2\times2$ matrices with determinant $1$ and group operation matrix multiplication. For the other cases, it suffices to note that the equivalence relation ‘modulo $Z$’ is definable by $A\thicksim B$ iff there exists $Z\in\{\pm1_{2}\}$ with $B=AZ$, resp. $Z\in H$ with $Z^{2}=1$ and $B=AZ$. To complete the proof of Theorem \[sl2\] it remains to establish **Step 1** and **Step 2** below. We take $v=v(1)$ as another parameter, and set $w=uvu=\left( \begin{array} [c]{cc}0 & 1\\ -1 & 0 \end{array} \right) .$ Then $u(\lambda)^{w}=v(\lambda)$, so $V:=v(R)=U^{w}$ is definable. Note the identity (for $\xi\in R^{\ast}$):$$h(\xi)=v(\xi)u(\xi^{-1})v(\xi)w^{-1}=w^{-1}u(\xi)w.u(\xi^{-1}).w^{-1}u(\xi).\label{h-identity}$$ **Step 1:** The ring isomorphism from $R$ to $U\subset \mathrm{M}_{2}(R)$ is definable. Indeed, this is just the mapping$$r\longmapsto\left( \begin{array} [c]{cc}1 & r\\ 0 & 1 \end{array} \right) .$$ **Step 2:** The map $\theta$ sending $g=(a,b;c,d)$ to $(u(a),u(b);u(c),u(d))\in\Gamma^{4}$ is definable; this is a group isomorphism when $U$ is identified with $R$ via $u(\lambda)\longmapsto\lambda$. Assume for simplicity that $\Gamma=\mathrm{SL}_{2}(R)$. We start by showing that the restriction of $\theta$ to each of the subgroups $U,~V\,,~H$ is definable. Recall that $u(0)=1$ and $u(1)=u$. If $g\in U$ then $g\theta=(u,g;1,u)$. If $g=v(-\lambda)\in V$ then $g^{-w}=u(\lambda)\in U$ and $g\theta=(u,1;g^{-w},u).$ Suppose $g=h(\xi)\in H$. Then $g=w^{-1}xwyw^{-1}x$ where $x=u(\xi),$ $y=u(\xi^{-1}),$ and $g\theta=(y,1;1,x)$. So $g\theta=(y_{1},y_{2};y_{3},y_{4})$ if and only if $$\begin{aligned} ~y_{4}\ast y_{1} & =u,~y_{2}=y_{3}=1,\\ g & =w^{-1}y_{4}wy_{1}w^{-1}y_{4}.\end{aligned}$$ Thus the restriction of $\theta$ to $H$ is definable. Next, set$$W:=\left\{ x~\in\Gamma\mid x_{\mathfrak{p}}\in\{1,w\}\text{ for every }\mathfrak{p}~\right\} .$$ To see that $W$ is definable, observe that an element $x$ is in $W$ if and only if there exist $y,z\in u(R_{\{0,1\}})$ such that $$x=yz^{w}y\text{ and }x^{4}=1.$$ Note that $u(R_{\{0,1\}})$ is definable by (the proof of) Proposition \[P1\]. Put$$\Gamma_{1}=\{g\in\Gamma\mid g_{11}\in R^{\ast}\}.$$ If $g=(a,b;c,d)\in\Gamma_{1}$ then $g=\widetilde{v}(g)\widetilde{h}(g)\widetilde{u}(g)$ where$$\begin{aligned} \widetilde{v}(g) & =v(-a^{-1}c)\in V\\ \widetilde{h}(g) & =h(a^{-1})\in H\\ \widetilde{u}(g) & =u(a^{-1}b)\in U.\end{aligned}$$ This calculation shows that in fact $\Gamma_{1}=VHU$, so $\Gamma_{1}$ is definable; these three functions on $\Gamma_{1}$ are definable since$$\begin{aligned} x & =\widetilde{v}(g)\Longleftrightarrow x\in V\cap HUg\\ y & =\widetilde{u}(g)\Longleftrightarrow y\in U\cap HVg\\ z & =\widetilde{h}(g)\Longleftrightarrow z\in H\cap VgU.\end{aligned}$$ Let $g=(a,b;c,d)$. Then $gw=(-b,a;-d,c)$. We claim that there exists $x\in W$ such that $gx\in\Gamma_{1}$. Indeed, this may be constructed as follows: If $a_{\mathfrak{p}}\in\mathfrak{o}_{\mathfrak{p}}^{\ast}$ take $x_{\mathfrak{p}}=1$. If $a_{\mathfrak{p}}\in\mathfrak{po}_{\mathfrak{p}}$ and $b_{\mathfrak{p}}\in\mathfrak{o}_{\mathfrak{p}}^{\ast}$ take $x_{\mathfrak{p}}=w$. If both fail, take $x_{\mathfrak{p}}=1$ when $a_{\mathfrak{p}}\neq0$ and $x_{\mathfrak{p}}=w$ when $a_{\mathfrak{p}}=0$ and $b_{\mathfrak{p}}\neq0$. This covers all possibilities since for almost all $\mathfrak{p}$ at least one of $a_{\mathfrak{p}}$, $b_{\mathfrak{p}}$ is a unit in $\mathfrak{o}_{\mathfrak{p}},$ and $a_{\mathfrak{p}}$, $b_{\mathfrak{p}}$ are never both zero. As $gx\in\Gamma_{1},$ we may write$$gx=\widetilde{v}(gx)\widetilde{h}(gx)\widetilde{u}(gx)\text{.}$$ We claim that the restriction of $\theta$ to $W$ is definable. Let $x\in W$ and put $P=\{\mathfrak{p}~\mid~x_{\mathfrak{p}}=1\},$ $Q=\{\mathfrak{p}~\mid~x_{\mathfrak{p}}=w\}$. Then $(u^{x})_{\mathfrak{p}}$ is $u$ for $\mathfrak{p}\in P$ and $v$ for $\mathfrak{p}\in Q$, so $u^{x}\in\Gamma_{1}$ and $$\widetilde{u}(u^{x})_{\mathfrak{p}}=\left\{ \begin{array} [c]{ccc}u & & (\mathfrak{p}\in P)\\ 1 & & (\mathfrak{p}\in Q) \end{array} \right. .$$ Recalling that $u=u(1)$ and $1=u(0)$ we see that$$x\theta=\left( \begin{array} [c]{cc}\widetilde{u}(u^{x}) & \widetilde{u}(u^{x})^{-1}u\\ u^{-1}\widetilde{u}(u^{x}) & \widetilde{u}(u^{x}) \end{array} \right) .$$ We can now deduce that $\theta$ is definable. Indeed, $g\theta=A$ holds if and only if there exists $x\in W$ such that $gx\in\Gamma_{1}$ and $$A.x\theta=\widetilde{v}(gx)\theta.\widetilde{h}(gx)\theta.\widetilde{u}(gx)\theta$$ (of course the products here are matrix products, definable in the language of $\Gamma$ in view of Proposition \[P1\]). This completes the proof of Theorem \[sl2\] for $\Gamma=\mathrm{SL}_{2}(R)$. When $\Gamma=\mathrm{SL}_{2}(R)/Z$, the same formulae now define $\theta$ as a map from $\Gamma$ into the set of $2\times2$ matrices with entries in $U$ modulo the appropriate definable equivalence relation. $\blacksquare$ Now we turn to the proof of Theorem \[high\_rank\]. This largely follows [@ST], §§3, 4, but is simpler because we are dealing here with ‘nice’ rings. Henceforth $G$ denotes a simple Chevalley-Demazure group scheme of rank at least $2$. The root subgroup associated to a root $\alpha$ is denoted $U_{\alpha}$, and $Z$ denotes the centre of $G$. Put $\Gamma=G(R)$. Let $S$ be any integral domain with infinitely many units. According to [@ST], Theorem 1.5 we have$$U_{\alpha}(S)Z(S)=\mathrm{Z}\left( C_{G(S)}(v)\right)$$ whenever $1\neq v\in U_{\alpha}(S).$ This holds in particular for the rings $S=\mathfrak{o}_{\mathfrak{p}}$. Take $u_{\alpha}\in U_{\alpha}(R)$ to have $\mathfrak{p}$-component $x_{\alpha}(1)$ for each $\mathfrak{p}\in\mathcal{P}$ (or every $\mathfrak{p}$ when $R=\mathbb{A}$); then$$U_{\alpha}(R)Z(R)=\mathrm{Z}\left( C_{G(R)}(u_{\alpha})\right) .$$ Given this, the proof of Corollary 1.6 of [@ST] now shows that $U_{\alpha }(R)$ is a definable subgroup of $\Gamma$; the result is stated for integral domains but the argument remains valid, noting that in the present case $R/2R$ is finite. Associated to each root $\alpha$ there is a morphism $\varphi_{\alpha }:\mathrm{SL}_{2}\rightarrow G$ sending $u(r)=\left( \begin{array} [c]{cc}1 & r\\ 0 & 1 \end{array} \right) $ to $x_{\alpha}(r)$ and $v(r)=\left( \begin{array} [c]{cc}1 & 0\\ r & 1 \end{array} \right) $ to $x_{-\alpha}(r)$ ([@S], Chapter 3). This morphism is defined over $\mathbb{Z}$ and satisfies $$K_{\alpha}:=\mathrm{SL}_{2}(R)\varphi_{\alpha}\leq G(R).$$ \[Klemma\]$K_{\alpha}=U_{-\alpha}(R)U_{\alpha}(R)U_{-\alpha}(R)U_{\alpha }(R)U_{-\alpha}(R)U_{\alpha}(R)U_{-\alpha}(R)U_{\alpha}(R).$ This follows from the corresponding identity in $\mathrm{SL}_{2}(R),$ which in turn follows from (\[h-identity\]) and the fact that $w=uvu$. We may thus infer that each $K_{\alpha}$ is a definable subgroup of $G(R)$. Fixing a root $\gamma$, we identify $R$ with $U_{\gamma}(R)$ by $r\longmapsto r^{\prime}=x_{\gamma}(r).$ Proposition \[P1\] now shows that $R$ is definable in $G(R)$. As above, $G(R)$ is $R$-definable as a set of $d\times d$ matrices that satisfy a family of polynomial equations over $\mathbb{Z}$, with group operation matrix multiplication. To complete the proof we need to establish **1.** The ring isomorphism $R\rightarrow U_{\gamma }(R);~r\longmapsto r^{\prime}=x_{\gamma}(r)\in\mathrm{M}_{d}(R)$ is definable in ring language. This follows from the definition$$x_{\gamma}(r)=\exp(rX_{\gamma})=1+rM_{1}(\gamma)+\ldots+r^{q}M_{q}(\gamma)$$ where each $M_{i}(\gamma)$ is a matrix with integer entries ([@S], Chaps. 2, 3). **2.** The group isomorphism $\theta:G(R)\rightarrow G(R^{\prime}) \subseteq\mathrm{M}_{d}(U_{\gamma}(R))$ is definable in group language. To begin with, Lemma 3.5 of [@ST] shows that for each root $\alpha$, the restriction of $\theta$ to $U_{a}(R)$ is definable (this is established for $R$ an integral domain, but the proof is valid in general). Next, we observe that $G(R)$ has ‘finite elementary width’ in the sense of [@ST]: There is is finite sequence of roots $\beta_{i}$ such that$$G(R)=\prod_{i=1}^{N}U_{\beta_{i}}(R).$$ This relies on results from Chapter 7 of [@S]. Specifically, Corollary 2 to Theorem 18 asserts that if $R$ is a PID, then (in the above notation) $G(R)$ is generated by the groups $K_{\alpha}$. It is clear from the proof that each element of $G(R)$ is in fact a product of bounded length of elements from various of the $K_{\alpha}$; an upper bound is given by the sum $N_{1}$, say, of the following numbers: the number of positive roots, the number of fundamental roots, and the maximal length of a Weyl group element as a product of fundamental reflections. If the positive roots are $\alpha_{1},\ldots,\alpha_{n}$ it follows (if $R$ is a PID) that $$G(R)=\left( \prod_{j=1}^{n}K_{\alpha_{j}}\right) \cdot\left( \prod _{j=1}^{n}K_{\alpha_{j}}\right) \cdot\ldots\cdot\left( \prod_{j=1}^{n}K_{\alpha_{j}}\right) ~\ \text{(}N_{1}\text{ factors).}$$ As each of the rings $\mathfrak{o}_{\mathfrak{p}}$ is a PID (or a field), the same holds for our ring $R$ in general. The result now follows by Lemma \[Klemma\], taking $N=8nN_{1}$. Thus $\theta$ is definable as follows: for $g\in G(R)$ and $A\in\mathrm{M}_{d}(U_{\gamma}(R)),$ $g\theta=A$ if and only if there exist $v_{i}\in U_{\beta_{i}}(R)$ and $A_{i}\in\mathrm{M}_{d}(U_{\gamma}(R))$ such that $g=v_{1}\ldots v_{N}$, $A=A_{1}\cdot\ldots\cdot A_{N}$ and $A_{i}=v_{i}\theta$ for each $i$. Here $A_{1}\cdot A_{2}$ etc denote matrix products, which are definable in the language of $G$ because the ring operations on $R^{\prime }=U_{\gamma}(R)$ are definable in $G$. This completes the proof. *Acknowledgment.*  Thanks to Jamshid Derakhshan for references and advice. [99999]{} P. D’Aquino, A. J. Macintyre and M. Otero , Some model-theoretic perspectives on the structure sheaves of $\widehat{\mathbb{Z}}$ and the ring of finite adèles over $\mathbb{Q}$, `arXiv: 2002.06660 [math. AC]` J. Derakhshan, Model theory of adeles and number theory, `arXiv:2007.09237 [math.LO]`, 2020 J. Derakhshan and A. Macintyre, Model theory of adeles I, `arXiv: 1603.09698 [math.LO]` A. Nies, D. Segal and K. Tent, Finite axiomatizability for profinite groups, `arXiv:1907.02262v4` `(math.GR)`, 2020 D. Segal and K. Tent, Defining $R$ and $G(R)$, `arXiv:2004.13407v3` `(math.GR),` 2020 R. Steinberg, *Lectures on Chevalley groups*, A. M. S. University Lecture Series **66**, 2016.

--- abstract: 'We propose new ergodic interference alignment techniques for $K$-user interference channels with delayed feedback. Two delayed feedback scenarios are considered – delayed channel information at transmitter (CIT) and delayed output feedback. It is proved that the proposed techniques achieve total $2K/(K+2)$ DoF which is higher than that by the retrospective interference alignment for for the delayed feedback scenarios.' author: - '[^1]' title: Ergodic Interference Alignment with Delayed Feedback --- Interference channel, degrees of freedom (DoF), ergodic interference alignment, delayed feedback. Introduction ============ In these days, interference management is one of the most important issues in wireless communication systems. In order to obtain high spectral efficiency, many interference management techniques have been proposed and studied. For the two-user interference channel, the capacity region is already known for weak and strong interference regions in [@lowic] and [@costa]. For the moderate region, the capacity region is still unknown, but there are some works that the capacity regions can be achieved by rate-splitting within one bit [@Tse]. The authors in [@Tse] also proved that the optimal generalized degrees of freedom are achievable using the rate-splitting scheme. Compared to the two-user interference channel, the general $K$-user interference channel have not been much known yet. Many researchers have studied degrees of freedom (DoF) to understand the asymptotic capacity because of the difficulty of finding the exact capacity region. For the $K$-user interference channel, the DoF was shown to be upper bounded by $\frac{K}{2}$ in [@Host]. The authors in [@Jafar] showed that this upper bound can be achieved by the interference alignment (IA) scheme that all interfering signals from other transmitters are aligned in the same dimension to independently decode the desired signals at the receivers. This scheme operates in high SNR to guarantee independence between the desired signal dimension and the interference aligned dimension. In order to operate in any finite SNR, [@EIA] proposed ergodic IA that all interfering signals are perfectly cancelled out by properly choosing two time indices. Using ergodic IA, each user can achieve half the interference-free ergodic capacity. The IA schemes generally require perfect channel state information (CSI). In rapidly time varying channels, however, channel state information becomes outdated due to feedback delay. In other words, it is impractical to assume that transmitters have perfect knowledge of current channel state information. In order to solve this problem, recent studies [@HJSV; @BIA; @RIA; @RIAK] focus on exploiting imperfect channel state information – no channel state information at transmitter (CSIT) or delayed feedback information. It was shown in [@RIA] that $9/8$ DoF is achievable for the three-user interference channel with delayed CSIT and $6/5$ DoF is achievable for the three-user interference channel with delayed output feedback without CSIT. More generally, [@RIAK] showed that $K^2/(K^2-1)$ DoF is achievable for the $K$-user interference channel $(K\ge3)$ with delayed CSIT and $\lceil K/2\rceil K/(\lceil K/2\rceil(K-1)+1)$ DoF is achievable for the $K$-user interference channel with delayed output feedback without CSIT. In this paper, we assume two delayed feedback scenarios as follows. (i) *Delayed channel information at transmitter*: in this scenario, nothing but the past channel information is given at the transmitter. The channel information implies either channel state information or time index information. Output feedback is not assumed in this case. (ii) *Delayed output feedback without CSIT*: in this scenario, nothing but the past output feedback information is given at the transmitter and the channel information is not available at the transmitter. We devise effective interference management strategies in the $K$-user interference channel for these two scenarios. The proposed schemes are developed in the framework of ergodic IA [@EIA] and enables interference-free decoding of the desired message at the receiver. It is shown that the proposed strategies achieve $\frac{2K}{K+2}$ DoF in the $K$-user interference channel for the scenarios of the delayed channel information and the delayed output feedback without CSIT. The proposed schemes achieve higher DoF than the retrospective IA [@RIA; @RIAK] in the $K$ user interference channel with the same assumptions of delayed feedback. System model and preliminary ============================ Interference channel model -------------------------- The received signal in the $K$-user interference channel is given by $$\begin{aligned} \mathbf{Y}(t) = \mathbf{H}(t)\mathbf{X}(t)+\mathbf{Z}(t)\end{aligned}$$ where $\mathbf{Y}(t) = [Y_1(t)~Y_2(t)~\cdots~Y_K(t)]^T$, the transmitted signal vector $\textbf{X}(t) = [X_1(t)~X_2(t)$ $~\cdots~X_K(t)]^T\in \mathbb{C}^{K \times 1} $ with power constraint $P$, $\mathbf{H}(t) \in \mathbb{C}^{K \times K}$ represents the time varying fading channel matrix and is given by $$\begin{aligned} \mathbf{H}(t) = \left[ \begin{array}{ccc} H_{11}(t) & \cdots & H_{1K}(t) \\ \vdots & \ddots & \vdots~ \\ H_{K1}(t) & \cdots & H_{KK}(t) \\ \end{array} \right].\end{aligned}$$ where $H_{ji}$ denotes the channel coefficient from transmitter $i$ to receiver $j$ and is an independent and identically distributed (i.i.d.) complex Gaussian random variable with distribution $\sim \mathcal{CN}(0,1)$. At receivers, full channel state information is assumed to be available, i.e., CSIR. The element of the additive white Gaussian noise vector $\textbf{Z}(t) = [Z_1(t)~Z_2(t)~\cdots~Z_K(t)]^T$ is assumed to follow complex Gaussian distribution $\sim\mathcal{CN}(0,N_0)$. Preliminary: Ergodic IA with Full CSIT -------------------------------------- The ergodic IA [@EIA] requires perfect knowledge of channel state information at the transmitter (CSIT). For a $K$-user interference channel, $K/2$ total DoF can be achieved in an ergodic sense if the channel is time varying. Contrary to other IA techniques, the ergodic IA works for infinite SNR as well as any finite SNR since interfering signals are canceled out when the channel matrices at two different time instants satisfy a certain condition. Specifically, let $t_1$ and $t_2$ be the time instants (or time indices) at which the the channel matrices satisfy the following relationship: $$\begin{aligned} \textbf{H}(t_1) &= \left[ \begin{array}{ccc} H_{11}(t_1) & \cdots & H_{1K}(t_1) \\ \vdots & \ddots & \vdots~ \\ H_{K1}(t_1) & \cdots & H_{KK}(t_1) \\ \end{array} \right]\label{eq:channel1}\\ \textbf{H}(t_2) &= c(t_2)\cdot\left[ \begin{array}{ccc} H_{11}(t_1) & \cdots & -H_{1K}(t_1) \\ \vdots & \ddots & \vdots~ \\ -H_{K1}(t_1) & \cdots & H_{KK}(t_1) \\ \end{array} \right] \label{eq:channel2}\end{aligned}$$ where $c(t_2)$ is a complex valued constant and $H_{kk}(t_2) = c(t_2)H_{kk}(t_1)$, $H_{kj}(t_2) = -c(t_2)H_{kj}(t_1)$, $k\neq j$, $k,j\in{1,\ldots,K}$. At the time $t_2$, the message which was previously sent at the time $t_1$ is again sent from the transmitter. In other words, the transmitted signal vector $\mathbf{X}(t_1)$ is equal to $\mathbf{X}(t_2)$. To decode the message, receiver $k$ adds the received signals at $t_1$ and $t_2$ and constructs a sufficient statistics for the message $X_k(t_1)$ as $$\begin{aligned} Y_k(t_1)+Y_k(t_2)/c(t_2) = 2H_{kk}(t_1)X_k(t_1)+Z_k(t_1)+Z_k(t_2)/c(t_2).\end{aligned}$$ Then, the achievable rate is determined by $$\begin{aligned} R_k = \frac{1}{2}\log(1+\frac{2|H_{kk}|^2}{(1+1/c(t_2)^2)}SNR)-\epsilon\end{aligned}$$ where $SNR = \frac{P}{N_0}$, $\epsilon > 0$. Correspondingly, the total $\frac{K}{2}$ DoF is achievable [@EIA]. Ergodic interference alignment with delayed CIT =============================================== Contrary to the existing ergodic IA, we assume only full CSIR and imperfect or partial CSIT by feedback delay. Specifically, all receivers feed either channel state information or time indices back to the transmitters. Each transmitter cannot use the current channel information but can use the outdated channel information due to feedback delay. Three-user interference channel with delayed CSIT ------------------------------------------------- In this subsection, we propose a new strategy to achieve high DoF when the receivers feed CSI back to the transmitters. To effectively establish the concept of the proposed scheme, we start from the three-user interference channel, i.e., $K =3$. Three-user interference channel with delayed CSIT can achieve total $\frac{6}{5}$ DoF by ergodic IA. The proposed ergodic IA is carried out over two phases: *Transmission phase 1*: Transmission phase 1 for data transmission is continued until $t_2$. At each time until $t_2$, new messages are continuously transmitted. For the time $t_1$ and $ t_2$ at which the channel condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied, the received signals are given by $$\begin{aligned} \mathbf{Y}(t_1)=\mathbf{H}(t_1)\mathbf{X}(t_1)+\mathbf{Z}(t_1),\label{rs1}\\ \mathbf{Y}(t_2)=\mathbf{H}(t_2)\mathbf{X}(t_2)+\mathbf{Z}(t_2).\label{rs2}\end{aligned}$$ Due to delayed CSIT, the transmitters cannot recognize what the current channel states are so that they cannot send the same message of $t_1$ at $t_2$ as in the conventional ergodic IA. Thus, they just send independent messages at $t_1$ and $t_2$. After the channel changes, the transmitters can figure out that the channel condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied at the previous time $t_2$ due to delayed CSI feedback. Then, transmission phase 2 is entered. *Transmission phase 2*: If the previous time was $t_2$, the transmitters send the following signals at time $t_2+1,t_2+2,t_2+3$, respectively: $$\begin{aligned} &\mathrm{Transmitter~1~at~time}~t_2+1:~X_1(t_1)-X_1(t_2)\nonumber\\ &\mathrm{Transmitter~2~at~time}~t_2+2:~X_2(t_1)-X_2(t_2)\nonumber\\ &\mathrm{Transmitter~3~at~time}~t_2+3:~X_3(t_1)-X_3(t_2).\nonumber\end{aligned}$$ After transmission phase 2 is completed, the transmission mode goes back to transmission phase 1. Each receiver adds its received signals at $t_1$ and $t_2$ and constructs the following variables. $$\begin{aligned} Y_1(t_1)+Y_1(t_2)/c(t_2) &= H_{11}(t_1)(X_1(t_1)+X_1(t_2))+H_{12}(t_1)(X_2(t_1)-X_2(t_2))+H_{13}(t_1)(X_3(t_1)-X_3(t_2))\nonumber\\ &~+Z_1(t_1)+Z_1(t_2)/c(t_2),\label{eq:rx1}\\ Y_2(t_1)+Y_2(t_2)/c(t_2) &= H_{22}(t_1)(X_2(t_1)+X_2(t_2))+H_{21}(t_1)(X_1(t_1)-X_1(t_2))+H_{23}(t_1)(X_3(t_1)-X_3(t_2))\nonumber\\ &~+Z_2(t_1)+Z_2(t_2)/c(t_2),\label{eq:rx2}\\ Y_3(t_1)+Y_3(t_2)/c(t_2) &= H_{33}(t_1)(X_3(t_1)+X_3(t_2))+H_{31}(t_1)(X_1(t_1)-X_1(t_2))+H_{32}(t_1)(X_2(t_1)-X_2(t_2))\nonumber\\ &~+Z_3(t_1)+Z_3(t_2)/c(t_2)\label{eq:rx3}.\end{aligned}$$ *Decoding at receiver 1*: Using the received signal at $t_2+2$ and $t_2+3$, receiver 1 removes the interfering signals from the other senders in (\[eq:rx1\]). Then, we have an equation for $X_1(t_1)+X_1(t_2)$. Using another equation for $X_1(t_1)-X_1(t_2)$ received at $t_2+1$, receiver 1 can decode both $X_1(t_1)$ and $X_1(t_2)$. *Decoding at receiver 2*: Similarly, receiver 2 removes the interfering signals in (\[eq:rx2\]) using the received signal at $t_2+1$ and $t_2+3$ and decodes $X_2(t_1)$ and $X_2(t_2)$ using the received signal at $t_2+2$ and (\[eq:rx2\]). *Decoding at receiver 3*: Similarly, receiver 2 removes the interfering signals in (\[eq:rx3\]) using the received signal at $t_2+1$ and $t_2+2$ and decodes $X_3(t_1)$ and $X_3(t_2)$ using the received signal at $t_2+3$ and (\[eq:rx3\]). According to the decoding procedure, the proposed scheme enables each receiver to decode its 2 messages in 5 symbol times. That is, total 6 messages are decodable over 5 symbol times and hence total 6/5 DoF is achievable. $K$-user interference channel with delayed CSIT {#kudcsit} ----------------------------------------------- Total $\frac{2K}{K+2}$ DoF is achievable in a $K$-user interference channel with delayed CSIT by ergodic IA. For a $K$-user interference channel, two independent messages sent at time $t_1$ and $t_2$ are decoded at each receiver over $K+2$ symbol times. As in the three-user interference channel, the time $t_1$ and $t_2$ correspond to transmission phase 1. If the transmitters realize the channel matrix at time $t_2$ satisfies the condition in (\[eq:channel1\]) and (\[eq:channel2\]), transmission phase 2 starts. Then, transmitter $k$, $k\in\{1,\ldots,K\}$, sends the signal $X_k(t_1)-X_k(t_2)$ at time $t_2+k$. Similarly to the three-user interference channel, each receiver can decode its two messages over $K+2$ symbol times. Therefore, total $\frac{2K}{K+2}$ DoF is achievable in a $K$-user interference channel by the proposed ergodic IA. Fig. \[Fig:P1\] shows the achievable DoF by the proposed ergodic IA (solid line) and the retrospective IA [@RIAK] (dashed line) with delay CSIT according to the number of users in a $K$-user interference channel. The total achievable DoF by the retrospective IA is $\frac{K^2}{K^2-1}$. It starts from $\frac{9}{8}$ for a three-user case and converges to $1$ as $K$ goes to infinity. On the other hand, the total achievable DoF by the proposed ergodic IA starts from $\frac{6}{5}$ and converges to $2$. $K$-user interference channel with delayed time index feedback -------------------------------------------------------------- If the receivers feed the time indices at which the condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied, total achievable DoF is the same as the case that delay CSIT is used. Total achievable DoF by the proposed ergodic IA with delay time index feedback in a $K$-user interference channel is $\frac{2K}{K+2}$. The strategy in Section \[kudcsit\] can be applied. Instead of using the channel state information (i.e., channel matrix) to find the time $t_1$ and $t_2$ at which the condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied, the receivers send the time indices $t_1$ and $t_2$ since receivers can find them by the assumption of full CSIR. After receiving the time indices, the transmitters realize that the previous time was $t_2$ and enter into transmission phase 2. Ergodic interference alignment with delayed output feedback without CSIT ======================================================================== In this section, we assume full CSIR and delayed output feedback. The received signals themselves are fed back to the transmitters. Each transmitter cannot use the channel information but can use only the delayed received output feedback. It is also assumed that the receivers can reform the output signals and feed them back to the transmitters if necessary. Three-user interference channel ------------------------------- Our new proposed strategy is first applied to a three-user interference channel for better explanation of the proposed idea. Total $\frac{6}{5}$ DoF is achievable by the proposed ergodic IA in a three-user interference channel when only delayed output feedback information is available. The operation of the proposed ergodic IA is classified into two phases: *Transmission phase 1*: The transmission phase 1 for data transmission is continued until $t_2$. At each time until $t_2$, new messages are continuously transmitted. For the time $t_1$ and $ t_2$ at which the channel condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied, the received signals are given by (\[rs1\]) and (\[rs2\]). Similarly to the case of delayed CIT in Section III, the transmitters cannot send the same message at $t_1$ and $t_2$ because they cannot realize that $t_2$ is the time instant at which the channel condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied due to the absence of CSIT. Therefore, the transmitters continue to send independent messages at $t_2$. However, the receivers know that the channel condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied at $t_2$ owing to full CSIR so that they construct the following output feedback information and send them back to the transmitters after receiving the signals at $t_2$. $$\begin{aligned} &\mathrm{Receiver~1}:~(Y_1(t_1)+Y_1(t_2)/c(t_2))/H_{11}(t_1)\nonumber\\ &\mathrm{Receiver~2}:~(Y_2(t_1)+Y_2(t_2)/c(t_2))/H_{22}(t_1)\nonumber\\ &\mathrm{Receiver~3}:~(Y_3(t_1)+Y_3(t_2)/c(t_2))/H_{33}(t_1).\nonumber\end{aligned}$$ Then, the transmitters can figure out that the previous time $t_2$ was the time instant at which the channel condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied after receiving the delayed output feedback information. However, note that they do not know the time instant $t_1$ as well as the message sent at $t_1$. Once after receiving the delayed output feedback information, transmission phase 2 is entered. *Transmission phase 2*: After the output feedback signals are received, the transmitters send their signals at time $t_2+1,t_2+2,t_2+3$, respectively: $$\begin{aligned} &\mathrm{Transmitter~1~at~time}~t_2+1:~(Y_1(t_1)+Y_1(t_2)/c(t_2))/H_{11}(t_1)-2X_1(t_2)=\nonumber\\ &(X_1(t_1)-X_1(t_2))+(H_{12}(t_1)(X_2(t_1)-X_2(t_2))+H_{13}(t_1)(X_3(t_1)-X_3(t_2))+Z_1(t_1)+Z_1(t_2)/c(t_2))/H_{11}(t_1)\nonumber\\ &\mathrm{Transmitter~2~at~time}~t_2+2:~(Y_2(t_1)+Y_2(t_2)/c(t_2))/H_{22}(t_1)-2X_1(t_2)=\nonumber\\ &(X_2(t_1)-X_2(t_2))+(H_{21}(t_1)(X_1(t_1)-X_1(t_2))+H_{23}(t_1)(X_3(t_1)-X_3(t_2))+Z_2(t_1)+Z_2(t_2)/c(t_2))/H_{22}(t_1)\nonumber\\ &\mathrm{Transmitter~3~at~time}~t_2+3:~(Y_3(t_1)+Y_3(t_2)/c(t_2))/H_{33}(t_1)-2X_1(t_2)=\nonumber\\ &(X_3(t_1)-X_3(t_2))+(H_{31}(t_1)(X_1(t_1)-X_1(t_2))+H_{32}(t_1)(X_2(t_1)-X_2(t_2))+Z_3(t_1)+Z_3(t_2)/c(t_2))/H_{33}(t_1).\nonumber\end{aligned}$$ After transmission phase 2 is completed, the transmission mode goes back to transmission phase 1. *Decoding at receiver 1*: Linearly combining the received signals at $t_2+1$, $t_2+2$ and $t_2+3$, receiver 1 can obtain the values of $X_1(t_1)-X_1(t_2)$, $X_2(t_1)-X_2(t_2)$ and $X_3(t_1)-X_3(t_2)$. By substituting the values of $X_2(t_1)-X_2(t_2)$ and $X_3(t_1)-X_3(t_2)$ to $Y_1(t_1)+Y_1(t_2)/c(t_2)$, the value of $X_1(t_1)+X_1(t_2)$ can also be obtained. Then, receiver 1 can decode both $X_1(t_1)$ and $X_1(t_2)$ because it has two independent equations on $X_1(t_1)$ and $X_1(t_2)$ – one is given in terms of $X_1(t_1)-X_1(t_2)$ and the other is given in terms of $X_1(t_1)+X_1(t_2)$. *Decoding at receiver 2*: Similarly, receiver 2 can decode $X_2(t_1)$ and $X_2(t_2)$ using linear combination and substitution. *Decoding at receiver 3*: Similarly, receiver 3 can decode $X_3(t_1)$ and $X_3(t_2)$ using linear combination and substitution. In this way, all 6 messages are decoded in 5 symbol times so that total 6/5 DoF is achievable by the proposed ergodic IA. K-user interference channel --------------------------- When only delayed output feedback information is available at the transmitters, total $\frac{2K}{K+2}$ DoF is achievable in a $K$-user interference channel by the proposed ergodic IA For a $K$-user interference channel, two independent messages sent at time $t_1$ and $t_2$ are decoded at each receiver over $K+2$ symbol times, where the time time $t_1$ and $t_2$ correspond to transmission phase 1. After the receivers receive the signals at $t_2$, receiver $k$, $k\in\{1,\ldots,K\}$, feed the output $(Y_k(t_1)-Y_k(t_2)/c(t_2))/H_{kk}(t_1)$ back to its own transmitter. After the output feedback signals are received, the transmitters realize that the previous time instant was $t_2$ at which the channel condition in (\[eq:channel1\]) and (\[eq:channel2\]) is satisfied and enters into transmission phase 2. In transmission phase 2, transmitter $k$ sends the signal $(Y_k(t_1)-Y_k(t_2)/c(t_2))/H_{kk}(t_1)-2X_k(t_2)$ at only time $t_2+k$. As in the three-user interference channel, each receiver decode its two messages over $K+2$ symbol times. Therefore, total $\frac{2K}{K+2}$ DoF is achievable in a $K$-user interference channel by the proposed ergodic IA. Fig. \[Fig:P2\] shows total achievable DoF by the proposed ergodic IA (solid line) and the retrospective IA [@RIAK] (dashed line) with delayed output feedback without CSIT according to the number of users. The total achievable DoF of the retrospective IA is $\frac{\lceil K/2\rceil K}{\lceil K/2\rceil(K-1)+1}$. It starts from $\frac{6}{5}$ for the three-user case and converges to $1$ as $K$ goes to infinity. On the other hand, the total achievable DoF by the proposed ergodic IA starts from $\frac{6}{5}$ and approaches to $2$ as $K$ goes to infinity. Conclusion ========== In this paper, we proposed new ergodic IA techniques in $K$-user interference channels with delayed feedback. Total achievable DoF by the proposed ergodic IA is derived for two scenarios of delayed feedback – delayed channel information and delayed output feedback information. We showed that total $2K/(K+2)$ DoF is achievable by the proposed schemes for both scenarios. The proposed ergodic IA schemes achieve higher DoF than the retrospective IA when the feedback information is outdated. [1]{} V. S. Annapureddy and V. V. Veeravalli, “Gaussian interference networks: sum capacity in the low-interference regime and new outer bounds on the capacity region,” *IEEE Trans. Inform. Theory*, vol. 55, no. 7, pp. 3032-3050, Jul. 2009. M. Costa and A. Gamal, “The capacity region of the discrete memoryless interference channel with strong interference," *IEEE Trans. Info. Theory*, vol. 33, no. 5, pp.710-711, Sept. 1987. R. H. Etkin, D. N. C. Tse and H. Wang, “Gaussian interference channel capacity to within one bit,” *IEEE Trans. Info. Theory*, vol. 54, no. 12, pp. 5534-5562, Dec. 2008. A. Host-Madsen and A. Nosratinia, “The multiplexing gain of wireless networks,” in *Proc. IEEE Int. Symp. Info. Theory*, Adelaide, SA, Sep. 2005, pp. 2065-2069. V. R. Cadambe and S. A. Jafar, “Interference alignment and degrees of freedom of the K-user interference channel,” *IEEE Trans. Info. Theory*, vol. 54, no. 8, pp. 3425-3441, Aug. 2008. B. Nazer, M. Gastpar, S. A. Jafar and S. Vishwanath, “Ergodic interference alignment,” in *Proc. IEEE Int. Symp. Info. Theory*, Seoul, Korea, Jun. 2009. C. Huang, S. A. Jafar, S. Shamai and S. Vishwanath, “On degrees of freedom region of MIMO networks without channel state information at transmitters,” *IEEE Trans. Info. Theory*, vol. 58, no. 2, pp. 849-857, Feb. 2012. S. A. Jafar, “Blind interference alignment,” *IEEE Trans. Info. Theory*, vol. 6, no. 3, pp. 216-227 , Jun. 2012. H. Maleki, S. A. Jafar and S. Shamai, “Retrospective interference alignment,” in *Proc. IEEE Int. Symp. Info. Theory*, Saint Petersburg, Russia, Jul. 2011. L. Maggi and L. Cottatellucci, “Retrospective interference alignment for interefence channels with delayed feedback,” in *Proc. IEEE Wireless Communications and Networking Conference (WCNC)*, Paris, France, Apr. 2012. [^1]: M. G. Kang and W. Choi are with Department of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701, Korea (e-mail: casutar@kaist.ac.kr, wchoi@ee.kaist.ac.kr).

--- abstract: 'An extreme-mass-ratio system composed of a white dwarf (WD) and a massive black hole can be observed by the low-frequency gravitational wave detectors, such as the Laser Interferometer Space Antenna (LISA). When the mass of the black hole is around $10^4 \sim 10^5 M_\odot$, the WD will be disrupted by the tidal interaction at the final inspiraling stage. The event position and time of the tidal disruption of the WD can be accurately determined by the gravitational wave signals. Such position and time depend upon the mass of the black hole and especially on the density of the WD. We present the theory by using LISA-like gravitational wave detectors, the mass-radius relation and then the equations of state of WDs could be strictly constrained (accuracy up to $0.1\%$). We also point out that LISA can accurately predict the disruption time of a WD, and forecast the electromagnetic follow-up of this tidal disruption event.' author: - 'Wen-Biao Han' - 'Xi-Long Fan' title: 'Determining the nature of white dwarfs from low-frequency gravitational waves' --- Introduction {#sec:intro} ============ The era of gravitational wave (GW) astronomy arrived when the advanced Laser Interferometer Gravitational-Wave Obser-vatory (LIGO) observed the first gravitational wave event GW150914 [@gw15]. The latest observation of a double neutron star merger gives us a special chance to explore the universe using multi-message astronomy observations [@bn17a; @bn17b]. The planning space-based gravitational wave detectors, such as Laser Interferometer Space Antenna (LISA),[^1] China’s Taiji [@taiji] and Tianqin [@tianqin], will provide an opportunity to observe extreme-mass-ratio inspirals (EMRIs) and intermediate-mass-ratio inspirals (IMRIs), which are composed by a stellar compact object and a massive black hole [@lisal3]. White dwarf (WD) mass-radius relation, which is determined by the physics of WDs, is a fundamental tool in modern astrophysics [@tool; @tool2]. People usually believe that the theory of the equation of state (EoS) for WDs is clear. However, the varied compositions of WDs still yield different mass-radius relations @book [and references inside]. From Hipparcos data, the measurement accuracy of the mass-radius relation of WDs is very low ($\gtrsim 10\%$), and in pareicular some WDs fall onto the iron track – a result that does not follow from standard stellar evolution theory (see Fig. 5.17 in [@book]). This situation asks for a more accurate measurement of the mass-radius relation of WDs, for determining the composition (C/O or Fe et. al.) and confirming the standard theory of WDs. WDs inspiralling into MBHs (supermassive or intermediate massive), known as one type of EMRIs or IMRIs, are potential GW sources for space-based gravitational wave detectors. If the mass of black hole is less than $10^6 M_\odot$, the WD will be very probably disrupted by the tidal force from the black hole before merger (i.e., before the innermost stable circular orbit, ISCO). Note that the burst of GWs generated by the disruption event [@rosswog07] and GWs emitted by a star colliding with an MBH [@east14], are distinguishable from GWs in the inspiralling phase. Other signatures from tidal disruption of WDs by BHs, such as nucleosynthesis and the relativistic jet, have also been theoretically analyzed or numerically simulated in the literature (see [@rosswog09; @macleod16; @kawana17], among others). Rate predictions for such events are extremely uncertain at present. LISA likely detects several events during its lifetime[@sesana08]. Since GW interferometers (like LISA) are very sensitive to the phase of the signal, this phase difference is crucial for distinguishing the disruption position and time of the WD. Such position and time are determined by the mass and radius of WD and the mass of the black hole. LISA can estimate the masses of the WD and MBH with accuracies of $10^{-3}$ and $10^{-4}$ ,respectively, with SNR $>20$ [@lisal3]. Therefore, by observing the GW waveform cutoff caused by the WD tidal disruption, one in principle can constrain the radius of WD accurately. Once the mass and radius of a WD are determined, then EoS or the composition of WD will be constrained rigidly too. We point out that, with such kind of WD disruption events, LISA will constrain the mass-radius relation of WDs much better than current astronomical methods [@m-r; @m-r2; @m-r3]. Method {#sec:intro} ====== The tidal radius for a WD can be easily estimated. Ignoring the rotation of WD and taking it as a rigid body, the Roche limit is $$\begin{aligned} \nonumber r_{\rm tidal} &= \sqrt[3]{2} R_{\rm WD} \left(\frac{M_{\rm BH}}{m_{\rm WD}}\right)^{1/3} \\ &= \sqrt[3]{2} M_{\rm BH} \left(\frac{\rho_{\rm BH}}{\rho_{\rm WD}}\right)^{1/3} \,, \label{rtidal} $$ where $R_{\rm WD}$ is the radius of WD. We find that the tidal radius is independent of the spin of the black hole (BH), but relates with the density of WD and mass of BH. A tidal disruption event (TDE) will happen at the position of tidal radius. Due to the TDE, the GW signals of this EMRI event disappears once the WD enter the tidal radius. Therefore, the GW frequency at this moment is the cutoff frequency, which can be calculated easily. At the radii of tidal disruption, the orbital frequency is $$\begin{aligned} \tilde{\Omega}_{\rm tidal} = \frac{1}{\tilde{r}_{\rm tidal}^{3/2}+q} \,, \label{omega}\end{aligned}$$ where the variables with tildes mean dimensionless, and $q$ is the dimensionless spin parameter of the black hole defined as $q \equiv J/M^2$ ($J$ is the rotating angular momentum). Due to the tidal disruption, there is a cutoff frequency of GWs for observation. From Eq. (\[omega\]), changing to the SI units, the cutoff frequency of the dominant (2, 2) mode is $$\begin{aligned} f^{\rm c}_{22} =0.64625\left[\left(\frac{r_{\rm tital}}{1.47663\times10^5 {\rm km}}\frac{10^5 m_\odot}{M}\right)^{3/2}+q\right]^{-1} \left(\frac{10^5 m_\odot}{M}\right) {\rm Hz} \,, \label{cutoff}\end{aligned}$$ or $$\begin{aligned} r_{\rm tidal} = 1.47663\times10^5 \left[\left(\frac{f^{\rm c}_{22}}{0.64625}\frac{M}{10^5 m_\odot}\right)^{-1}+q\right]^{2/3} \left(\frac{M}{10^5 m_\odot}\right) {\rm km} \,. \label{rtidal2}\end{aligned}$$ The $r_{\rm tidal}$ is decided by the cutoff frequency, the mass and spin of MBH. It is well known that LISA can measure the masses of the binary system and spin of MBH precisely from the insprialling waveforms [@lisal3]. By using the TDE, in principle, one can independently obtain the tidal radius of WDs from the GW’s cutoff frequency. The error of $r_{\rm tidal}$ depends on the measurement accuracy of $M, ~q$ and frequency resolution. Once we get the value of $r_{\rm tidal}$, with Eq. (\[rtidal\]), the radius of WD can be obtained at the same accuracy of the tidal radius, $$\begin{aligned} R_{\rm WD} &= r_{\rm tidal} \left(\frac{1}{2}\frac{M}{m_{\rm WD}}\right)^{-1/3} \label{rwd2} \,. \end{aligned}$$ However, since we do not know the frequency resolution of LISA data, we replace it with a waveform dephasing resolution. LISA is very sensitive to the phase of gravitational wave signals. Using matched filtering, LISA will be able to determine the phase of an EMRI to an accuracy of half a cycle [@babak14]. Correspondingly, we assume one-cycle (2$\pi$) waveform dephasing will be recognized by LISA, and one-cycle waveform dephasing just needs $\delta t = 1/{f}_{22}^{\rm c}$ s. We have [@hughes00] $$\begin{aligned} \dot{r} = -\frac{c_{21}}{d}\dot{E}-\frac{c_{22}}{d}\dot{L_z} \,, \label{rdot}\end{aligned}$$ where the coefficients are listed as $$\begin{aligned} \nonumber c_{21} \equiv \,& 2Er^{5}-6EMr^{4}+4a^{2}Er^{3}+2a(L_{z}-2aE)Mr^{2}+2a^{4}Er-2a^{3}(L_{z}-aE)M\,, \\ \nonumber c_{22} \equiv \,&2aEMr^{2}-2a^{2}L_{z}r+2a^{2}(L_{z}-aE)M , \, \\ \nonumber d \equiv \, &-2(1-E^{2})Mr^{4}+8(1-E^{2})M^{2}r^{3}+[Q+L_{z}^{2}-5a^{2}(1-E^{2})- \\ \nonumber & \,6M^{2}]Mr^{2}+2[a^{2}(3-E^{2})+2aEL_{z}-(L_{z}^{2}+Q)]M^{2}r+2(L_{z}^{2}+Q)M^{3}- \\ \nonumber & \,4aEL_{z}M^{3}+a^{2}(2E^{2}M^{2}-L_{z}^{2}-Q)M-a^{4}(1-E^{2})M \,,\end{aligned}$$ where $E, ~L_z$, and $Q$ are energy, angular momentum, and Carter constant respectively. If we constrain the particle on the equatorial plane of the Kerr black hole, then $Q=0$. $E, ~L_z$ can be analytical obtained for the circular orbit cases. The gravitational fluxes $\dot{E}$ and $\dot{L_z}$ can be calculated very accurately by solving the Teukolsky equations in frequency domain [@han10; @han11]. From Eq. (\[rdot\]), one can calculate the uncertainty of radii $r$ with $\delta t$ allowed for one-cycle dephasing and $r_{\rm tidal}$ from Eq. (\[rtidal2\]). An analytical approximation for the mass-radius relation for nonrotating WDs [@nauenberg72] is $$\begin{aligned} \frac{R_{\rm WD}}{R_\odot} = \frac{0.0225}{\mu}\frac{[1-(m_{\rm WD}/m_{\rm max})^{4/3}]^{1/2}}{(m_{\rm WD}/m_{\rm max})^{1/3}}\,, \label{rwd} \end{aligned}$$ where $\mu$ is the mean molecular weight. It is usually set equal to 2, corresponding to helium and heavier elements, which is appropriate for most astrophysical WDs. The Chandrasekhar’s limit of mass of WD is $m_{\rm max} = 5.816m_\odot/\mu^2$. However, as explained in Nauenberg’s paper, Eq. (\[rwd\]) only takes into account electron degeneracy. This induces an uncertainty on the order of 3%. And WD binary system simulations have shown us that WDs heat up a lot due to tidal effects, making Eq. (\[rwd\]) even less accurate. Thus, with the measured $R_{\rm WD}$ by GWs from Eq. (\[rwd2\]), we can validate the accuracy of Eq. (\[rwd\]) or, alternatively, determine the value of $\mu$. In addition, with Eq. (\[rwd\]), one can predict the TDE time by using the mass and spin parameters from the insprialling waveform. At a given GW frequency, the total remaining time until the end of the inspiral is [@finn00] $$\begin{aligned} T_{\rm rem} = \frac{1.41\times10^6 {\rm sec}}{(f_{\rm GW}/0.01 {\rm Hz})^{8/3}} \left(\frac{10 m_\odot}{m_{\rm WD}}\right)\left(\frac{10^6 m_\odot}{M}\right)^{2/3} \cal{T} \label{tremain} \,, \end{aligned}$$ where $\cal{T}$ is the relativistic correction, which can be calculated with accurate Teukolsky-based fluxes. From the radius-mass relation (\[rwd\]) and Eq. (\[rtidal\]), we can determine the tidal radius and calculate the remaining time of TDE to ISCO. At any given $r > r_{\rm tidal}$, one can calculate the remaining time to ISCO. The difference of two remaining times is just the expected time of TDE relative to the position at radius $r$. WD’s EoS constraint results {#sec:res} =========================== In Fig. \[simple1\], we consider five WDs with different masses and densities. Three of them (WD1: 0.4 $m_\odot$ & $\rho = 1.4648\times 10^8$ kg/m$^{3}$, WD2: 0.6 $m_\odot$ & $\rho = 2.1512\times 10^8$ kg/m$^3$, and WD3: 0.8 $m_\odot$ & $\rho = 1.4714\times 10^8$ kg/m$^3$) are inspiralling into an MBH with mass $10^5~m_\odot$ and spin $q = 0.9$. The tidal disruption turns up at $r_{\rm tidal} = 5.8612, 5.1565$， and $5.8524$$M_{\rm BH}$ respectively. The radii of ISCO is 2.32 $M_{\rm BH}$, so all the three WDs will be disrupted before merger and inside the sensitive band of GW detectors. ![Noise curves of LISA, Taiji, and Tianqin; the root of the power-spectrum-density (PSD) curves of GWs from five WD tidal disruption events. All these five disruption events happen before the WDs arrive at ISCO, and three of the TDEs are in the sensitive band of detectors. These three TDEs take place, respectively 21, 16, and 42 days (corresponding 0.4, 0.6, and 0.8 solar mass WDs and $10^5$ solar mass MBH, the dimensionless spin parameter of the MBH is $q = 0.9$.) before they arrive at ISCO. Two of the five TDEs happen out of the sensitive band ( WD4 and WD5, densities are 4.2525 $\times 10^8$ and 1.0712 $\times 10^9$ kg/m$^3$ respectively). The black points represent the time of 1 year before the tidal disruption events. []{data-label="simple1"}](signalnew4.pdf){height="4.0in"} The cutoff point (\*) in Fig. \[simple1\] represents the disruption event, and the GW frequency at the moment of TDE is just the cutoff frequency. In this paper, we request $f^{\rm c}_{22} < f^{\rm isco}_{22}$ so that the WD can be tidally disrupted by the black hole. If using this relation and Eq. (\[rwd\]) to Eq. (\[rtidal\]), we have $$\begin{aligned} r_{\rm tidal} &= 0.05098 \mu^{-5/3} \left[1-\left(\frac{\mu^2}{5.816} \frac{m_{\rm WD}}{m_\odot}\right)^{4/3}\right]^{1/2}\left(\frac{m_\odot}{m_{\rm WD}}\right)^{1/3}\left(\frac{M_{\rm BH}}{m_{\rm WD}}\right)^{1/3} R_\odot \,. \label{rtmu} \end{aligned}$$ Applying this relation to Eq. (\[cutoff\]), we find that the cutoff frequency is not sensitive to the mass of the black hole, but very sensitive to the mass of the WD and the parameter $\mu$ (or the density). The cutoff frequencies of the first three systems in Fig. \[simple1\] are 0.0428 Hz, 0.0513 Hz, and 0.0429 Hz for the WD1, WD2, and WD3, respectively, and these three TDEs happen inside the sensitive band of detectors. The phase of the EMRI waveform will stop evolving at the moment of TDE. Thus the value of the phase at the cutoff frequency is decided by the moment of TDE or tidal radius. From Eq. (\[rtidal\]), the tidal radius is decided by the masses of the binary and the radius of WD. From the inspiralling GWs, LISA can accurately determine the masses. Therefore, the phase of the cutoff waveform decides the radius of a given WD based on our analysis. The measurement accuracy of the waveform phase by LISA can be a fraction of $2\pi$ during $10^5$ cycles, i.e., a fractional phase accuracy of up to $10^6$. All information about the EMRI is encoded in the GW phase and thus we can expect to make measurements of the intrinsic parameters to this same fractional accuracy [@babak14]. Here, even if we assume a $2\pi$ phase accuracy of the waveform, we will see that the radius or density of a WD can be totally constrained from the observation of GWs of a disruption phenomenon. At $r_{\rm tidal} = 5.8612, 5.1565$, and $5.8524$ $M_{\rm BH}$, the energy fluxes are $6.2603 \times 10^ {-4}, ~1.1286 \times 10^ {-3}$, and $6.3040 \times 10^{-4} (m_{\rm WD}/M_{\rm BH})^2$ respectively, for the mentioned three WDs. From Eq. (\[rdot\]), $\dot{r} = -2.0093\times10^{-7}, ~-4.3275\times10^{-7}$ and $-4.0357\times10^{-7}$ at the moment of tidal disruption. If LISA can measure the phase of GWs in a precision of one cycle ($2\pi$), then the $\delta t =1/f_{22}^{\rm c} \approx 47.4, 39.6, {\rm and} ~ 47.3 M_{\rm BH}$ (23.7, 19.8, and 23.7 s for a $10^5$ solar MBH) for the above three WDs respectively. In a result, the tidal disruption position can be determined in a relative uncertainty $\delta r_{\rm tidal}/r_{\rm tidal}$ only $1.6252 \times 10^{-6}, ~3.3245 \times 10^{-6}$ and $3.2621 \times 10^{-6}$ respectively, for the above three cases. From Eq. (\[rtidal\]), it means LISA can determine the radii of WD with the same relative uncertainty if we can totally determine the masses of the WD and the black hole. Equivalently, the density of WDs can be constrained in relative uncertainties $4.8756 \times 10^{-6}, ~9.9735 \times 10^{-6}$ and $9.7863 \times 10^{-6}$ respectively. From the masses of the system, and a theoretical model of WD’s mass-radius relation such as in Eq. (\[rtmu\]), we can predict the tidal disruption time by Eq. (\[tremain\]). In Fig. \[simple1\], we plot a black dot to represent the moment of 1 year before TDE. A small error (0.1%) of the radius will induce an incorrect prediction time of about a few hours. In practice, one cannot know the exact spin and mass of a system. The measurement uncertainties of the mass of a WD and the mass and spin of a black hole will also contribute to the measurement error of the tidal disruption position. LISA can estimate the masses of the WD and MBH at an accuracy of $10^{-3}$ and $10^{-4}$ respectively, and the spin of the BH at a level of $10^{-3}$ with SNR $>20$ [@lisal3]. From Eq. (\[rtidal\]), one can immediately determine that the error of the black hole’s mass will induce an error of estimation $\delta R_{\rm WD}/R_{\rm WD} = 1/3 \delta M_{\rm BH}/M_{\rm BH} \sim 3.3\times10^{-5}$. At the same time, the error of spin of the BH will induce $\delta R_{\rm WD}/R_{\rm WD} < 2/3 \delta q/q < 10^{-3}$ and $\delta m_{\rm WD}/m_{\rm WD}$ will produce an error of WD radius $\sim 3.3\times10^{-4}$. In this way, the constraint accuracy of the WD radius is mainly influenced by the precision of the WD mass and the BH’s spin. Despite this, while observing a WD tidal disruption with LISA, the mass-radius ratio of a WD can still be constrained with an accuracy $ \sim 0.1\%$ level, which will be much better than the current results from astrophysical observations [@m-r; @m-r2; @m-r3](around 10% level). A few of these types of events will totally decide the composition and EoS of WDs. WD mass density \[kg/m$^3$\] $r_{\rm td}$ $T_{\rm td}$ $f^{\rm c}_{22}$ $|\delta R_{\rm WD}|/R_{\rm WD}$ (\*) ----- --------------- ---------------------- -------------------- -------------- ------------------ --------------------------------------- WD1 $0.4 m_\odot$ $1.4648\times 10^8$ 5.8612$M_{\rm BH}$ 42 day 0.0428 Hz $1.6252 \times 10^{-6} $ WD2 $0.6 m_\odot$ $2.1512\times 10^8$ 5.1565$M_{\rm BH}$ 16 day 0.0513 Hz $3.3245 \times 10^{-6} $ WD3 $0.8 m_\odot$ $1.4714\times 10^8$ 5.8524$M_{\rm BH}$ 21 day 0.0429 Hz $3.2621 \times 10^{-6} $ : The ideal constraint results of WD’s radius. The third column is the tidal radius for different WDs, and the forth one is the TDE time before the compact object arrives at ISCO. The last column is the constraint accuracy of the radius with one-cycle-dephasing measurement precision. (\*) means that the estimation is obtained by assuming that the other parameters, such as mass and spin of the system, are known exactly. The mass of black hole is $10^5$ solar masses and spin $q = 0.9$.[]{data-label="table"} Not all WD-MBH systems can be used to constrain the equations of state of WDs by TDEs with only GW signals. If the mass of the black hole is too large (something like a million solar masses), WDs will survive until merger. If the mass is too small ($ < 10^4 ~m_{\odot}$), WDs will be disrupted with a high cutoff frequency, which is not in the sensitive band of LISA. Two EMRIs (0.6 and 0.8 solar massive WDs with a $10^4 m_\odot$ BH) in Fig. \[simple1\] show that their TDEs are out of the sensitive band of LISA. In this way, we cannot use only gravitational waves to constrain the WD’s mass-radius relation. However, the GW signal will tell us the exact moment of TDE based on a given mass-radius relation of WD. Therefore, by using a multi-message astronomical observations, i.e., a GW observation of the inspiral waveforms and electronic-magnetic (EM) observation of TDEs will also offer an opportunity to constrain the EoS of WDs. We can estimate that a 0.1% WD radii difference will induce 0.57 and 2.59hr TDE’s EM signals arrival difference for the above two EMRIs. In Fig. \[simple2\], we plot the range of the mass-radius relation of arbitrary compact objects with TDEs, which can be observed by LISA, and the mass-radius relation of different kinds of WDs. For a typical WD ($\mu = 2$), we find that LISA can observe the tidal disruption of a small mass WD ($\lesssim 0.5 m_\odot$). ![Different colored shadows represent the range of masses and radii of compact objects with tidal disruption events seen by LISA. The larger the BH mass is, the smaller the range of mass-radius seen by LISA will be. The solid lines represent the mass-radius relations of astrophysical WDs with different compositions. []{data-label="simple2"}](massradius.pdf){height="4.0in"} Discussion {#sec:res} ========== In this paper, we assume a WD-MBH system with appropriate masses, i.e., a low massive WD and a $10^4 \sim 10^5 m_\odot$ black hole. This makes sure the tidal disruption happens inside the LISA band. By observing the waveforms and its cutoff frequencies of such kind of EMRIs, we can independently obtain the radii of WDs. Our method is independent of any WD models. Our analysis shows that the uncertainty of a WD’s radius with GWs can be as good as 0.1%. Therefore, we can constrain the mass-radius relation (equivalently, EoS) of WDs in a very high precision. The mass-radius relation obtained from GWs can be used to validate the theoretical models of WDs. In addition, one can use GWs and the theoretical WD model to predict the TDE time. This makes sure that the telescopes can be ready in advance to monitor the electromagnetic counterpart due to the TDE. The comparison of the real TDE time and the predicted one can also be used to constrain the EoS of WDs. In this way, it is interesting that even if a TDE happens outside of the sensitive band of LISA, we can also predict the TDE time based on the GW observations. when the mass of a WD is a little large ($\approx 1 m_\odot$), or the mass of a BH is around $10^3$, the TDE will happen with a higher cut-off frequency, which is outside of the LISA’s band. For example, for the $0.6/10^4 m_\odot$ and $0.8/10^4 m_\odot$ WD-MBH systems, though the TDEs happen outside of the sensitive band of LISA, GW observation can still predict the TDE time very accurately. Assuming the TDE is finally observed by telescopes at the prediction time and coincides with the position and distance of the GW source, it will also be very meaningful for the constraint of the mass-radius relation of WDs. Acknowledgement {#acknowledgement .unnumbered} =============== This work is supported by the National Natural Science Foundation of China, No. 11773059, No. 11673008, No. U1431120, and No. 11690023, and by the Key Research Program of Frontier Sciences, CAS, No. QYZDB-SSW-SYS016. W.H. is also supported by the Youth Innovation. We thank the anonymous referees for valuable comments and suggestions that helped us to improve the manuscript. Abbott B. P. et al. 2016, PhRvL, 116, 061102 Abbott B. P. et al. 2017, PhRvL, 119, 161101 Abbott B. P. et al. 2017, , 848, L12 Babak S., Gair J.R., R.H., Cole, 2014, in proceedings of EOM 2013 “Equations of Motion in Relativistic Gravity", arXiv: 1411.5253. Camenzind M., 2007, in “Compact Objects in Astrophysics: White Dwarfs, Neutron Stars and Black Holes", p. 137-185, Springer-Verlag Berlin Heidelberg 2007. Gong X., Xu S., Bai S. et al., 2011, Class. Quantum Grav. 28, 094012 Luo J.,Chen L.-S.,Duan H.-Z.et al., 2016, Class. 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--- abstract: 'We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well-known that in the primitive case the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive, and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize $\sig$-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported $\sig$-finite invariant measure, which is locally finite and unique up to scaling.' address: - 'María Isabel Cortez, Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Av. Libertador Bernardo O’Higgins 3363, Santiago, Chile.' - 'Boris Solomyak, Box 354350, Department of Mathematics, University of Washington, Seattle WA 98195' author: - 'M. I. Cortez' - 'B. Solomyak' title: 'Invariant measures for non-primitive tiling substitutions' --- [^1] Introduction ============ We consider self-affine substitution tilings of $\R^d$. A tile is a compact subset of $\R^d$ which is a closure of its interior. A tiling is a set of tiles with disjoint interiors whose union is all of $\R^d$. We restrict ourselves to tilings which satisfy the (translational) [*finite pattern condition*]{}, abbreviated FPC, i.e. for any $R>0$, there are finitely many patterns, or patches, of diameter less $R$, up to translation. In particular, there are finitely many tiles up to translation; we call some representatives of equivalence classes the [*prototiles*]{}. Sometimes tiles of the same shape need to be distinguished; this is achieved by considering a tile as a pair $T=(F,j)$ where $F=\supp(T)$ is a compact set (the [*support*]{} of the tile), and $j\in \{1,\ldots,\ell\}$ for some $\ell\ge 1$, is a [*label*]{} (or “type”, or “color”) of the tile. When translating a tile or a patch, the labels of the tiles are preserved. Let $\Ak$ be a finite set of prototiles and $\Ak^+$ the set of patches whose every tile is a translate of some $A\in \Ak$. Given an expansive linear map $\varphi:\,\R^d\to \R^d$, we say that $\om:\,\Ak \to \Ak^+$ is a [*tile substitution with expansion*]{} $\varphi$ if the union of tiles in $\om(A)$ equals $\varphi(\supp(A))$. In other words, every “inflated tile” can be subdivided into translates of the prototiles. This property allows us to iterate the substitution and obtain a family of patches $\om^n(A),\, A\in \Ak,\, n\ge 1$. The [*substitution tiling space*]{} $\Xa$ is defined as the set of all tilings of $\R^d$ whose every patch is a translate of a subpatch of $\om^n(A)$ for some $A\in \Ak$ and $n\in \Nat$. We assume that this space has the FPC property. We also assume that every prototile $A\in \Ak$ is [*admissible*]{}, that is, there exists a tiling $\Tk\in \Xa$ with $A\in \Tk$. The space $\Xa$ is compact in the usual “local” metric, in which two tilings are considered to be close if they agree on a large ball around the origin up to a small translation (see the next section for precise definitions), and $\R^d$ acts on $\Xa$ continuously by translations. This is the [*tiling dynamical system*]{} associated with $\om$. To a tiling substitution $\om$ we associate the [*substitution matrix*]{} $M$ whose entry $M(i,j)$ equals the number of tiles of type $i$ which appear in the substitution $\om$ applied to a tile of type $j$. The substitution is [*primitive*]{} if $M^k>0$ for some $k\in \Nat$. Substitution tiling dynamical systems have been studied almost exclusively in the primitive case, when they are minimal and uniquely ergodic. Here we begin the investigation of non-primitive tiling substitutions. A tiling substitution can be viewed as a generalization of a symbolic substitutions, see [@Queff; @Pytheas; @Robi]. Recently, a systematic investigation of non-primitive (one-dimensional) symbolic substitutions has been started in [@BKM; @BKMS] (see also [@Fi1; @Yuasa1; @Yuasa2]). Our work builds on [@BKMS]; however, we have to introduce many new ingredients. The substitution matrix $M$ is non-negative, and we can consider its irreducible components, see [@LM 4.4]. We will show that $\Xa = X_{\Ak,\om^k}$ for $k\in \Nat$; thus, by raising the substitution to a positive power we can assume, without loss of generality that all irreducible components are primitive or equal to $[0]$. Suppose that $M$ has irreducible components $M_1,\ldots,M_\ell$, and let $\A_1,\ldots,A_\ell$ be the corresponding subsets of the prototile set. By reordering the prototiles, we can assume that $M$ has a block upper-triangular form, with the diagonal blocks $M_i,\,i\le \ell$. In terms of the substitution, this means that $\om(A),\ A\in \Ak_j$, contains only translated of the tiles from $\bigcup_{i=1}^j \Ak_i$. Let $m\ge 1$ be the number of “minimal” irreducible components having the property that $\om(A) \seq \A_i^+$ for all $A\in \A_i$. Then $\om_i:=\om|_{\Ak_i}$ is the usual primitive substitution for $i\le m$. It turns out that $X_i:=X_{\Ak_i,\om_i}$, $i\le m$, are precisely the minimal components of the tiling dynamical system. Our first main result is the following. [**Theorem A.**]{} [*All ergodic invariant probability measures for the substitution tiling system are supported on minimal components.*]{} The proof uses the pointwise ergodic theorem and the fact that only the patches from minimal components may have a positive frequency in a tiling. However, this is only the beginning of the story, as there are natural and interesting [*infinite*]{} invariant measures. In order to characterize them, we need the property of [*recognizability*]{}, namely, the invertibility of the substitution map $\om$ extended to the tiling space $\Xa$. We prove that it holds whenever the tiling substitution is non-periodic, namely $\Tk+\bv\ne \Tk$ for $\Tk\in \Xa$ and $\bv \ne \b0$, generalizing [@So] from the primitive case. Even more, we establish recognizability of non-periodic tilings when the tiling space does contain periodic ones, under a mild geometric condition (the “non-periodic border” condition, see Section 4 for details). Applying recognizability, we construct a sequence of [*nested Kakutani-Rokhlin partitions*]{} of the transversal which allows us to determine the natural $\sig$-finite measures, first on the transversal, and then on the tiling space. (Actually, in general it is only a covering, but it becomes a partition when restricted to the set of non-periodic tilings, which is enough for our purposes.) The [*transversal*]{} is defined as the set of tilings which have a tile with a “puncture” at the origin; this is consistent with a view of the tiling space as a lamination. [*Transverse measures*]{} are in 1-to-1 correspondence with invariant measures. Although the use of transverse measures in tiling dynamics is by now standard, see [@BBG], we have to extend the theory to our setting, namely, to the non-primitive case, and to include $\sigma$-finite measures; this is done in the Appendix. In order to state our results on $\sig$-finite measures, we need to introduce some terminology. There are irreducible components $M_i$ and the corresponding prototile subsets $\Ak_i$, which we call “maximal”: they are characterized by the property [*$$\om(A)\ \ \mbox{contains a tile of type $\Ak_i$}\ \ \Longrightarrow\ \ A\in \Ak_i.$$* ]{} Let $p \ge 1$ be the number of maximal components; they correspond to prototile subsets $\Ak_{\ell-p+1},\ldots,$ $\Ak_\ell$. Denote by $Y_i$, for $i\ge \ell-p+1$, the set of tilings $\Tk\in \Xa$ which contain at least one tile of type $\Ak_i$. The subsets $Y_i\seq \Xa$ are non-empty (by the admissibility assumption), open, and invariant. We will show that $(Y_i,\R^d)$ is a non-compact minimal tiling dynamical system for $i\ge \ell-p+1$, $Y_i\cap Y_j=\es$ for $i\ne j$, and $\bigcup_{i=1}^p Y_i$ is dense in $\Xa$. We call $Y_i$ the [*maximal*]{} components of the tiling dynamical system. Now we can state our second main result. [**Theorem B.**]{} [*Suppose that the tiling substitution satisfies the non-periodic border condition (in particular, it is satisfied if the substitution is non-periodic). Then for each $i=\ell-p+1,\ldots,\ell$, there is a unique, up to scaling, ergodic invariant measure supported on $Y_i$, such that every point has an open neighborhood with positive finite measure.*]{} [**Remarks.**]{} 1. It is often the case that $\Xa \ne \bigcup_{i=1}^m X_i \cup \bigcup_{i=\ell-p+1}^\ell Y_i$, and there are other infinite invariant measures, but those described in Theorem B are the most natural ones. In Section 5 we classify all ergodic invariant measures which are positive and finite on a “cylinder set”; they correspond to some irreducible components of $M$. 2\. There is, in general, a greater variety of invariant measures for (one-dimensional) symbolic substitutions than for tile substitutions considered here; in particular, there are sometimes ergodic invariant probability measures of full support, see [@BKMS]. The reason is that in our case the vector of volumes of the prototiles is always a strictly positive left eigenvector of the substitution matrix, whereas for a symbolic substitution such an eigenvector may fail to exist. The paper is organized as follows. Section 2 contains preliminaries, including the topological results about minimal and maximal components. Theorem A is proved in Section 3. We obtain recognizability results, which are of independent interest, in Section 4, and in Section 5 we investigate $\sig$-finite invariant measures and prove Theorem B. (We would like to point out that Sections 4 and 5 can be read independently.) Section 6 is devoted to examples and concluding remarks. For specific examples the recognizability properties can usually be checked directly. In Section 7 (the Appendix) we treat transverse measures. [**Acknowledgment.**]{} We are grateful to Karl Petersen for his interest, helpful comments, and suggestion to consider the “Sierpiński carpet” and “gasket” tilings. Preliminaries ============= Tilings. -------- Fix a set of “types” labeled by $\{1,\ldots,N\}$, with $N\ge 1$. A [*tile*]{} in $\RR^d$ is defined as a pair $T = (F,i)$ where $F = \supp(T)$ (the support of $T$) is a compact set in $\R^d$ which is the closure of its interior, and $i = \ell(T)\in \{1,\ldots,N\}$ is the type of $T$. A [*tiling*]{} of $\R^d$ is a set $\Tk$ of tiles such that $\R^d = \bigcup \{\supp(T):\ T\in \Tk\}$ and distinct tiles (or rather, their supports) have disjoint interiors. A patch $P$ is a finite set of tiles with disjoint interiors. The [*support of a patch*]{} $P$ is defined by $\supp (P) = \bigcup\{\supp(T):\ T\in P\}$. The [*diameter of a patch*]{} $P$ is $\diam(P)=\diam(\supp(P))$. The [*translate*]{} of a tile $T=(F,i)$ by a vector $\bg\in \R^d$ is $T+\bg = (F+\bg, i)$. The translate of a patch $P$ is $P+\bg = \{T+\bg:\ T\in P\}$. We say that two patches $P_1,P_2$ are [*translationally equivalent*]{} if $P_2 = P_1+\bg$ for some $\bg\in \R^d$. Finite subsets of $\Tk$ are called $\Tk$-patches. For a set $B\seq \RR^d$ we write $$[B]^\Tk = \{T\in \Tk:\ \supp(T)\cap B \ne \es\}.$$ \[def-fpc\] [ We say that a tiling $\Tk$ has (translational) [*finite patch complexity*]{} (FPC), or satisfies the [*finite pattern condition*]{}, if for any $R>0$ there are finitely many $\Tk$-patches of diameter less than $R$ up to translation equivalence. This definition naturally extends to any collection of tilings. ]{} \[def-rep\] [A tiling $\Tk$ is [*repetitive*]{} if for any patch $P\subset \Tk$ there is $R>0$ such that for any $\bx\in \R^d$ there is a $\Tk$-patch $P'$ such that $\supp(P')\subset B_R(\bx)$ and $P'$ is a translate of $P$. ]{} Tile substitutions, self-affine tilings. ---------------------------------------- A linear map $\varphi : \R^d \rightarrow \R^d$ is [*expansive*]{} if all its eigenvalues lie outside the unit circle. \[def-subst\] [Let $\Ak = \{A_1,\ldots,A_N\}$ be a finite set of tiles in $\R^d$ such that for $i\neq j$ the tiles $A_i$ and $A_j$ are not translationally equivalent; we will call them [*prototiles*]{}. We assume that every prototile is “centered at the origin”, in the sense that $\b0\in \Int(\supp(A_j))$ for all $j$. Denote by $\Ak^+$ the set of patches made of tiles each of which is a translate of one of the prototiles. A map $\omega: \Ak \to \Ak^+$ is called a [*tile substitution*]{} with expansion $\varphi$ if $$\label{def-sub} \supp(\om(A_j)) = \varphi (\supp(A_j)) \ \ \ \mbox{for} \ j\le N.$$ ]{} In plain language, every expanded prototile $\varphi (A_j)$ can be decomposed into a union of tiles (which are all translates of the prototiles) with disjoint interiors. The substitution $\om$ is extended to all translates of prototiles by $\om(\bx+A_j)= \varphi(\bx) + \om(A_j)$, and to patches by $\om(P)=\bigcup\{\om(T):\ T\in P\}$. This is well-defined due to (\[def-sub\]). Denote by $X_\Ak$ the set of tilings whose tiles belong to $\Ak$ up to translation; note that $\om$ acts on $X_\Ak$ as well. \[def-tsp\] [ Let $\om$ be a tile substitution. A patch $P$ is said to be [*legal*]{} if there exists $n\ge 1$, $A_j\in \Ak$, and $\bx\in \R^d$, such that $P+\bx \subseteq \om^n(A_j)$. Denote by $\Xa\seq X_\A$ the set of all tilings of $\R^d$ whose every patch is legal. The set $X_{\Ak,\om}$ is called the [*tiling space*]{} corresponding to the substitution. We say that the substitution $\om$ has FPC if the space $X_{\Ak,\om}$ has FPC. The substitution $\om$ is [*admissible*]{} if for every prototile $A_j$ there exists $\Tk\in \Xa$ such that $A_j\in \Tk$. ]{} The additive group $\R^d$ acts on $X_{\Ak,\om}$ by translations; this action $(X_{\Ak,\om},\R^d)$ is called the [*tiling dynamical system*]{} or the [*self-affine tiling dynamical system*]{} associated to $\om$. It is clear from the definitions that $\om(\Xa)\seq\Xa$. We use a [*tiling metric*]{} on $\Xa$, in which two tilings are close if after a small translation they agree on a large ball around the origin. To make it precise, we say that two tilings $\Tk_1,\Tk_2$ [*agree*]{} on a set $K \subset \R^d$ if $$\supp(\Tk_1\cap \Tk_2) \supseteq K.$$ For $\Tk_1,\Tk_2 \in \Xa$ let $$\widetilde{\varrho}(\Tk_1,\Tk_2) := \inf\{r \in (0,2^{-1/2}): \ \exists\,\bg,\ \|\bg\| \le r\ \mbox{such that}$$ $$\Tk_1-\bg \mbox{ \ agrees with $\Tk_2$ on $B_{1/r}(\b0)$} \}.$$ Then $$\varrho(\Tk_1,\Tk_2): = \min\{2^{-1/2},\widetilde{\varrho}(\Tk_1,\Tk_2)\}.$$ [@Rud] [*(see also [@Robi]).*]{} $(\Xa,\varrho)$ is a complete metric space. It is compact, whenever the space has FPC. The action of $\R^d$ by translations on $\Xa$, given by $\bg(\Sk)= \Sk-\bg$, is continuous. \[def-submat\] [ To the tile substitution $\om$ we associate its $N \times N$ [*substitution matrix*]{} $M=M_\om$, where $M(i,j)$ is the number of tiles of type $A_i$ in the patch $\om(A_j)$. The substitution $\om$ is called [*primitive*]{} if the substitution matrix is primitive, that is, if there exists $k\in \NN$ such that $M^k$ has only positive entries.]{} [@Gott] [*(see also [@Robi Sec.5]).*]{} \[th-min\] 1. An FPC tiling system is repetitive if and only if it is minimal, that is, every orbit $\{\Sk-\bg:\ \bg\in \R^d\}$ is dense in $X$. 2. An FPC substitution tiling system is minimal if and only if the substitution is primitive. So far, much of the theory has focused on primitive tile substitutions $\om$. In this paper we investigate what happens in the absence of primitivity and repetitivity, in the context of tiling systems which satisfy FPC. \[onto\] Let $\omega$ be an admissible substitution on the set of prototiles $\A$. Then $\omega: X_{\A, \omega}\to X_{\A, \omega}$ is onto. Let $\T\in X_{\A,\omega}$. For $n\geq 1$, we define $$P_n=[B_n(\b0)]^{\T}.$$ Since the patches of $\T$ are legal, there exist $k_n\geq 1$, $\bx_n\in \RR^d$ and $A^{(n)}\in \A$ such that $$P_n+\bx_n\subseteq \omega^{k_n}(A^{(n)}).$$ Because $|\A|<\infty$, there exist an infinite subset $I\subseteq \NN$ and $A\in \A$ such that $$P_n+\bx_n\subseteq \omega^{k_n}(A) \mbox{ for every } n\in I.$$ Taking subsequences if it is necessary, we can suppose that $k_n<k_{n+1}$. For $n\in I$, let $Q_n=\omega^{k_n-1}(A)$. Observe that $P_n+\bx_n\subseteq \omega(Q_n)$. Let $\T_{A}\in \Xa$ be such that $A\in \T_A$ (the tiling $\T_A$ exists by the definition of admissible substitution), and let $\T_n=\omega^{k_n-1}(\T_A)$. We have $$Q_n=\omega^{k_n-1}(A)\subseteq \omega^{k_n-1}(\T_A)=\T_n,$$ which implies $P_n+\bx_n\subseteq \omega(\T_n)$. By compactness of $X_{\A,\om}$, there exist a subsequence $(n_j)_{j\geq 0}$ and $\T'\in \Xa$ such that $$\lim_{j\to \infty} (\T_{n_j}-\varphi^{-1}(\bx_{n_j}))=\T'.$$ Since for every $n\in I$, $$P_n \subseteq \omega(\T_n - \varphi^{-1} (\bx_n)),$$ we get $\omega(\T')=\T.$ \[lemma.power\] We have $X_{\A,\om} = X_{\A,\om^k}$ for $k\ge 2$. The inclusion $\supseteq$ is clear. For the other inclusion, let $P$ be a legal patch for $\om$. Then $P$ occurs in some $\om^n(A),\ A\in \A$. By assumption, there is a tiling $\S$ in $X_{\A,\om}$ which contains a tile of type $A$. By Lemma \[onto\], there exists a tiling $\S'\in X_{\A,\om}$ such that $\om^{n(k-1)} (\S') = \S$. Then a tile of type $A$ is in some $\om^{n(k-1)}(A')$ for some $A'\in \A$, hence $P$ occurs in $\om^{nk}(A')$ which implies that $P$ is legal for $\om^k$. Substitution matrix. -------------------- Let $M\in \M_{N\times N}(\ZZ_+)$. The graph $G(M)$ associated to $M$ is the directed graph whose set of vertices is $\{1,\ldots,N\}$, such that there is an edge from $i$ to $j$ if and only if $M(i,j)>0$. An equivalence relation is defined on the set of vertices of $G(M)$ as follows: $i\thicksim j$ if and only if $j=i$ or there is a path in $G(M)$ from $i$ to $j$ as well as a path from $j$ to $i$. The matrix is [*irreducible*]{} if and only if all the vertices of $G(M)$ are equivalent, otherwise, it is [*reducible*]{}. We call the equivalence classes of $G(M)$ the [*irreducible components*]{}, see [@LM 4.4].\ We say that an equivalence class $\alpha$ [*has access*]{} to the equivalence class $\beta$, or that $\beta$ is accessible from $\alpha$, if and only if $\alpha=\beta$ or if there exists a path in $G(M)$ from a vertex in $\alpha$ to a vertex in $\beta$. This relation is denoted by $\alpha \succeq \beta$. In a similar way we say that the vertex $i$ has access to $\beta$ if there is a path in $G(M)$ from $i$ to a vertex in $\beta$.\ For an equivalence class $\alpha$ we denote by $M_{\alpha}$ the irreducible submatrix (diagonal block) of $M$ corresponding to the restriction of $M$ to $\alpha$.\ Now we return to our tiling substitution $\om$ and let $M=M_\om$ be the substitution matrix. We identify the vertex set of the graph $G(M)$ with the prototile set $\Ak$. Since the tiling dynamical system is completely determined by the space $\Xa$, we can, in view of Lemma \[lemma.power\], replace $\om$ by $\om^k$ for $k\ge 2$. The substitution matrix for $\om^k$ is clearly $M^k$. By raising a reducible non-negative matrix to a power we can get rid of the “cyclic structure” of the irreducible components (this will increase the number of irreducible components if there is nontrivial cyclic structure), see [@LM 4.5] for details. Thus, we can (and will) assume, without loss of generality, that $$\label{eq-irre} \mbox{\em every irreducible block of $M$ is either primitive or equals $[0]$.}$$ Minimal and maximal components. ------------------------------- Consider the irreducible components of the graph $G(M)$ which are [*maximal*]{} in the partial order $\succeq$. In other words, a component $\alpha$ is maximal if it is not accessible from any other component. Denote by $\A_1,\ldots,\A_m$ the subsets of the prototile set $\A$ corresponding to these maximal components. Observe that $m\ge 1$. By the definition of the substitution matrix, we have $\om(\A_i) \seq \A_i^+$, so that $\om_i:=\om|_{\A_i}$ is a tile substitution on the prototile set $\A_i$. By assumption (\[eq-irre\]), this substitution is primitive, so $(X_{\A_i,\om_i},\R^d)$ is a minimal dynamical system by Theorem \[th-min\]. Let $X_i = X_{\A_i,\om_i}$ for $i \le m$. In the next lemma we show that these are precisely the minimal components of the tiling dynamical system. (It seems counter-intuitive that minimal components correspond to maximal irreducible components of the graph, but this is just the consequence of definitions. One can also consider the graph $G(M^T)$ for the transpose of the substitution matrix; this reverses the direction of edges, so the minimal components of the dynamical system correspond to minimal irreducible components of $G(M^T)$.) \[lem-minim\] Suppose that $\om$ is an admissible FPC tile substitution satisfying (\[eq-irre\]). Then 1. the minimal components of the tiling dynamical system $(\Xa,\R^d)$ are $(X_i,\R^d)$, for $i=1,\ldots,m$; they satisfy $\om(X_i) \seq X_i$; 2. for any tiling $\Tk\in \Xa$ and a prototile $A\in \A_j$ which occurs in $ \Tk$, the orbit closure $\clos\{\Tk-\bg:\ \bg\in \R^d\}$ contains every minimal component $X_i$ such that $\A_j$ is accessible from $\A_i$. We already know that $(X_i,\R^d)$ is minimal and $\om(X_i) \seq X_i$. Part (ii) will imply that there are no other minimal components; thus, it remains to prove (ii). Let $A\in \A_j$ and $A+\bx \in \Tk$ for some $\bx\in \R^d$ and $j\geq 1$. By Lemma \[onto\], there exists a sequence $k_n\uparrow \infty$ and prototiles $A^{(n)}$ such that $$A+\bx\in \om^{k_n}(A^{(n)})+\bx_n \seq \Tk\ \ \mbox{for some}\ \bx_n \in \R^d.$$ Passing to a subsequence, we can assume that $A^{(n)} = B$ for all $n$. Let $1\leq i\leq m$ be such that $\A_j$ is accessible from $\A_i$, then $B$ is accessible from $\A_i$, hence $\om^s(B)$ contains a prototile of type $\A_i$ for some $s\in \NN$. Since $\T$ contains a translates of $\omega^{n+s}(B)$ for arbitrarily large $n$, the closure of the orbit of $\Tk$ contains $X_{\A,\om_i}=X_i$. \[components\] There are at most $|\A|$ minimal components in $X_{\A,\omega}$. Suppose that the matrix $M$, and the graph $G(M)$, have $\ell$ irreducible components. It is also useful to consider the [*minimal*]{} irreducible components of the graph $G(M)$ (or maximal irreducible components of $G(M^T)$). The corresponding subsets $\A_j$ of the prototile set are characterized by the property that for any $i\ne j$ and $A\in \A_i$, the substitution $\om(A)$ does not contain tiles of type $\A_j$. Suppose there are $p$ such components; the corresponding prototile sets are $\A_{\ell-p+1},\ldots,\A_\ell$. Note that if $j\ge \ell-p+1$ and $A\in \A_j$, then the substitution $\om(A)$ necessarily contains a tile of type $\A_j$, since otherwise the substitution is not admissible. Thus, the corresponding matrices $M_j$ are nonzero and hence primitive by (\[eq-irre\]). Let $$Y_j:= \{\Tk \in \Xa:\ \Tk\ \mbox{contains a tile of type}\ \A_j\}.$$ We call $Y_j,\, j={\ell-p+1},\ldots,\ell$, the [*maximal components*]{} of the tiling space $\Xa$. \[lem-maxim\] Suppose that $\om$ is an admissible FPC tile substitution satisfying (\[eq-irre\]). Then 1. The subsets $Y_j,\, j={\ell-p+1},\ldots,\ell$, are mutually disjoint, open in $\Xa$, and invariant under the translation $\R^d$-action and the substitution $\Z_+$-action; 2. for any $\Tk \in Y_j,\, j={\ell-p+1},\ldots,\ell$, we have $Y_j\seq\clos\{\Tk-\bg:\,\bg\in \R^d\}$; 3. $\bigcup_{j=\ell-p+1}^\ell Y_j$ is dense in $\Xa$. \(i) This is immediate; $Y_j$ is invariant under $\om$ because the irreducible component $M_j$ is non-zero.\ (ii) Exactly as in the proof of Lemma \[onto\], we obtain $A\in \A$, an infinite set $I_\Tk$ and $\bx_n \in \R^d$ for $n\in I_\Tk$ such that $$\label{vspom} [B_n(\b0)]^{\Tk} + \bx_n \seq \om^{k_n}(A)\ \ \mbox{ for}\ n\in I_\Tk.$$ The assumption $\Tk \in Y_j$ implies that $A\in \A_j$. But $M_j$ is primitive, so we can use the same $A\in \A_j$ for any $\Sk \in Y_j$. This implies that the orbit of $\Tk$ contains $\Sk$ in its closure, as desired.\ (iii) Let $\Tk \in \Xa$. Again we find $A$ and $I_\Tk$ satisfying (\[vspom\]). By the definition of the graph $G(M)$, the vertex (prototile) $A$ has access to one of the maximal components $\A_j$, with $\ell-p+1 \le j \le \ell$. This means $A+ \bz \in \om^{k_0}(A')$ for some $A' \in \A_j$ and $k_0 \in \NN$. Let $\Tk'\in \Xa$ be a tiling containing $A'$, which exists by admissibility. Then $\om^{k_0+k_n}(\Tk')-\varphi^{k_n}(\bz) - \bx_n \in Y_j$, and this sequence converges to $\Tk$. Non-negative matrices. ---------------------- Let $M\in \M_{k\times k}(\ZZ_+)$ and let $\alpha$ be an equivalence class of $G(M)$. Denote by $\rho_{\alpha}$ its spectral radius. The class $\alpha$ is [*distinguished*]{} if $\rho_{\alpha}>\rho_{\beta}$ for every class $\beta\neq \alpha$ which has access to $\alpha$. In particular, if $\alpha$ is not accessible from any other class, then $\alpha$ is distinguished. A real number $\lambda$ is called a [*distinguished eigenvalue*]{} of $M$ if there exists a non-zero vector $\bx\geq \b0$ such that $M\bx=\lambda \bx$. The following theorem extends the Perron-Frobenius Theorem to reducible matrices, see [@S], [@TS2] and [@Vic] for the proof. \[Perron-Frobenius\] Let $M\in \M_{k\times k}(\ZZ_+)$. 1. A real number $\lambda$ is a distinguished eigenvalue if and only if there exists a distinguished class $\alpha$ for $M$ such that $\rho_{\alpha}=\lambda$. 2. If $\alpha$ is a distinguished class of $G(M)$, then there exists a unique (up to scaling) non-negative eigenvector $\bv_{\alpha}=(\bv_1,\cdots, \bv_k)$ corresponding to $\rho_{\alpha}$ with the property that $\bv_i>0$ if and only if the class $\alpha$ is accessible from the vertex $i$. We call $\bv_{\alpha}$ the [*distinguished eigenvector*]{} of $M$ corresponding to $\alpha$. [Let $M\in \M_{k\times k}(\ZZ_+)$. The [*core*]{} of $M$ is $core(M)=\bigcap_{n\geq 1}M^n(\RR_+^k).$ ]{} The next theorem can be found in [@TS1]. \[generated-core\] Let $M$ be a non-negative integer square matrix verifying (\[eq-irre\]). Then the core of $M$ is the cone generated by the distinguished eigenvectors of $M$. Core of the substitution matrix. -------------------------------- Let $\A$ be a finite set of prototiles and let $\omega:\A\to \A^+$ be a substitution with expansion map $\varphi:\RR^d\to \RR^d$ and substitution matrix $M_{\omega}=M\in\M_{\A\times \A}(\ZZ_+)$. (Here we abuse notation a little and label the rows and columns of $M$ by the elements of the prototile set $\A$.) Suppose that $\alpha_1,\cdots,\alpha_l$ are the equivalence classes (irreducible components) for the matrix $M$. For every $1\leq i\leq l$ we denote by $M_i$ the restriction of $M$ to the class $\alpha_i$. Let $X_1,\cdots, X_m$ be the minimal components of $X_{\A,\omega}$, and let $\A_i\subseteq \A$ be the minimal subset such that $X_i\subseteq X_{\A_i}$. As already mentioned, we may assume without loss of generality that $X_i=X_{\A_i,\omega}$ and $\omega|_{\A_i}:\A_i\to \A_i^{+}$ is primitive, for every $1\leq i\leq m$. Observe that for every $1\leq i\leq m$, the set of prototiles $\A_i$ is equal to an equivalence class $\alpha_i$ of $M$. The restriction $M_i$ of $M$ to $\A_i$ is the matrix associated to the substitution $\omega|_{\A_i}$. The equivalence classes of the matrix $M$ can be ordered in such a way that $M$ has the block upper-triangular form $$M= \left( \begin{array}{ccccccc} M_1 & 0 & \cdots & 0 & \ast & \cdots & \ast \\ 0 & M_2 & \cdots & 0 & \ast & \cdots & \ast \\ \vdots & \vdots & \ddots & \vdots & \vdots & \cdots& \vdots \\ 0 & 0 & \cdots & M_m & * & \cdots & * \\ 0 & 0 & \cdots & 0 & M_{m+1} & \cdots & \ast \\ \vdots & \vdots & \cdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 0 & \cdots & M_l \\ \end{array} \right)$$ where $\ast$ stand for arbitrary non-negative integer matrices. Observe that for each $m+1 \le i \le l$ there must be at least one non-zero off-diagonal block, because those $M_i$ correspond to non-minimal prototile subsets. The number $\lambda_0=|\det(\varphi)|$ is a left eigenvalue of $M$ with $\bv_0=(\vol(A))_{A\in \A}\in\RR_+^{\A}$ as an associated eigenvector. Indeed, the $A$-coordinate of $\bv_0^TM$ is equal to $\vol(\omega(A))$ and $\vol(\omega(A))=\lambda_0\vol(A)$, for every $A\in\A$ by (\[def-sub\]). The next proposition shows that $\lam_0$ is the unique distinguished eigenvalue of $M_{\omega}$ and characterizes the core of $M_{\omega}$. \[core\] Let $M\in \M_{\A\times\A}(\ZZ_+)$ be the substitution matrix associated to the substitution $\omega:\A\to\A^+$ with expansion map $\varphi:\RR^d\to\RR^d$. Assume that (\[eq-irre\]) holds. Let $X_1, \cdots, X_m$ be the minimal components of $X_{\A,\omega}$, and let $\A_i$ be the corresponding prototile sets, so that $X_i = X_{\A_i}$. Then 1. $\lam_0 = |\det(\varphi)|$ is the unique distinguished eigenvalue of $M$. 2. $\A_1,\cdots, \A_m$ are all the different distinguished classes of $M$, and $\rho(M_i)=|\det(\varphi)|$ for every $1\leq i\leq m$. 3. Let $1\leq i\leq m$. If $\bv_{i}$ is the distinguished eigenvector of $M$ corresponding to the class $\A_i$, then $\bv_i(p)>0$ if and only if $p\in \A_i$. 4. $core(M)$ is the cone generated by the vectors $\bv_1,\cdots, \bv_m$. We know that $M_i$, for $i=1,\ldots,m$, is the substitution matrix for the primitive tile substitution $\om|_{\A_i}$, hence $\rho(M_i)$ are distinguished eigenvalues of $M$. However, $\rho(M_i) = \rho(M_i^T) = |\det(\varphi)|=\lam_0$ for $i=1,\ldots,m$, by the definition of tile substitution (\[def-sub\]). On the other hand, the fact that $M^T$ has a strictly positive eigenvector implies, by [@gant Theorem 13.4.6] that $\rho(M_i) = \rho(M_i^T)< \rho(M^T) =\rho(M)$ for $i=m+1,\ldots,l$. (This is also easy to see directly: every non-minimal class $i$ “loses entropy” under the substitution, in view of $\om(\A_i) \not \seq \A_i$.) Therefore, the classes $i=m+1,\ldots,l$ are not distinguished by definition. Now all the claims of the proposition immediately follow from Theorem \[Perron-Frobenius\] and Theorem \[generated-core\]. Invariant measures and transverse measures. {#transversal} ------------------------------------------- Let $(X,\RR^d)$ be a tiling dynamical system (it need not be a substitution system, although for our purposes we can take $X = \Xa$). An [*invariant*]{} measure of $(X,\RR^d)$ is a measure $\mu:\B(X)\to \overline{\RR}_+$ such that $\mu(U)=\mu(U-\bv)$, for every $U\in \B(X)$ and $\bv\in\RR^d$.\ Recall that we consider every prototile $A\in\A$ centered at $\b0$. The [*center of the tile*]{} $t\in\T\in X$ is the point $\bx_t\in\RR^d$ for which there exists $A\in\A$ such that $t=A+\bx_t$. Let $\eta>0$ be such that the interior of the support of $A$ contains the closure of $B_\eta(\b0)$, for every $A\in\A$.\ By the [*transversal*]{} of $X$ we mean the set $\Gamma\subseteq X$ of all the tilings $\T$ in $X$ for which there exists a prototile $A\in \A$ such that $A\in \T$. In other words, if $L_{\T}\subseteq\RR^d$ is the set of all the points $\bx\in \RR^d$ for which there exist $\T\in X$ and $A\in\A$ such that $A+\bx$ is a tile of $\T$, then $$\Gamma=\{ \T\in X: \b0\in L_{\T} \}.$$ We say that a patch $P$ is [*$X$-admissible*]{} if there exists $\T\in X$ such that $P\subseteq \T$. We denote by $\Lambda_X$ the collection of all the $X$-admissible patches $P$ for which there exists a prototile $A\in\A$ such that $A\in P$. In other words, this is the collection of patches such that $\b0\in \Int(\supp(P))$ and $\b0$ coincides with the center of some tile $t\in P$. Every $X$-admissible patch is a translate of some patch in $\Lambda_X$. For every $P\in\Lambda_X$ we define $$C_{P}=\{\T\in X: P\subseteq \T \}.$$ The sets $C_{P}$ are subsets of $\Gamma$.\ Equipped with the induced topology, the space $\Gamma$ is compact and totally disconnected, with a countable base of clopen sets (the collection of the sets $C_P$ is a base of the topology). The collection of Borel sets $\B(\Gamma)$ of $\Gamma$ is equal to $\B(X)\cap\Gamma$. [If $U\in \B(\Gamma)$ and $\Theta\subseteq \RR^d$ is an open set, then $U+\Theta\in \B(X)$. To verify that, observe that this is true for the sets $C_P$, with $P\in \Lambda_X$. Verifying that the collection $\M=\{U\in \B(\Gamma): U+\Theta\in \B(X)\}$ is a $\sig$-algebra, we get the result. ]{} A [*transverse measure* ]{} on $\B(\Gamma)$ is a measure $\nu: \B(\Gamma)\to \overline{\RR}_+$ such that $\nu(A)=\nu(A-\bv)$, for every $A\subseteq \B(\Gamma)$ and $\bv\in\RR^d$ for which $A-\bv\subseteq \Gamma$.\ Tiling spaces have been studied as [*laminations*]{}, or [*translation surfaces*]{}, see [@BG; @BBG]. Our definition agrees with the notion of transverse measure for laminations. There is a one-to-one correspondence between finite invariant measures and finite transverse measures (see [@BBG Section 5]), however, in all the existing literature it is assumed that the tiling system is minimal. We need this correspondence in the non-minimal case, and also for $\sig$-finite positive measures. Therefore, we include details about this in the Appendix (Section \[Apendix A\]) for the reader’s convenience. If $\mu$ is an invariant measure, we denote by $\mu^T$ the associated transverse measure. Finite invariant measures. {#finmeas} ========================== \[th-finite\] Let $(X_{\A,\om},\R^d)$ be a self-affine tiling dynamical system. Then all finite invariant measures are supported on the minimal components. [*Scheme of the proof.*]{} Let $\mu$ be a finite invariant measure. We can assume that $\mu$ is ergodic. It is easy to see that the sets $C_P$, where $P$ is an admissible patch, generate the Borel $\sig$-algebra on $\Gam$. Therefore, the Borel $\sig$-algebra on $\Xa$ is generated by the sets $C_P+\Theta$, with $\Theta\seq B_r(\b0)$, and their translates. The claim will follow once we show that $\mu(C_P+\Theta)=0$ for every patch $P$ which does not occur in $\bigcup_{i=1}^m X_i$, the union of minimal components. To this end, we will use the pointwise ergodic theorem and show that the frequency of such a patch $P$ in any tiling $\T\in X_{\A,\om}$ equals zero. We have two cases to consider: (a) $P$ contains a tile from $\Ak':= \Ak \setminus \bigcup_{i=1}^m \Ak_i$; (b) $P$ contains tiles from two distinct minimal components. The following example shows why case (b) is needed. \[ex3\][*All the tiles have the unit square as its support and are distinguished only by the labels. Let $\Ak = \left\{\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\,, \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\,,\begin{tabular}{|c|} \hline 2 \\ \hline \end{tabular}\right\}$; the substitution $\om$ is given by $$\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|c|} \hline 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 1 & 0 \\ \hline 0 & 1 & 2 & 0 \\ \hline 0 & 0 & 0 & 0 \\ \hline \end{tabular} \,, \ \ \ \ \ \ \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\ \to\ {\begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}}^{\,\,4\times 4}\,, \ \ \ \ \ \ \begin{tabular}{|c|} \hline 2 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|} \hline 2 \\ \hline \end{tabular}^{\,\,4\times 4}.$$ where ${\begin{tabular}{|c|} \hline a \\ \hline \end{tabular}}^{\,\, k\times k}$ stands for the $k\times k$ square filled with identical prototiles labelled $a$. We have two minimal components, corresponding to the prototiles labelled $1$ and $2$. The tiling space $X_{\A,\om}$ contains tilings which have a half-plane filled by $1$’s and another half-plane filled by $2$’s. As we show, such tilings have zero measure for any finite invariant measure $\mu$.* ]{} Now we turn to the details. We use the pointwise ergodic theorem for $\R^d$-actions, which was first proved by Wiener [@wiener], with averaging over balls centered at the origin. For us, another averaging sequence is more convenient. Let \[eq-vanh\] F\_n:= \^n(B\_1(0)). It is well-known that the pointwise ergodic theorem holds with $F_n$ used instead of the balls (see e.g. [@tempel; @chatard]). \[thm-pet\] (Pointwise Ergodic Theorem for $\R^d$-actions) Let $\mu$ be an ergodic invariant probability measure for the system $(\Xa,\R^d)$. Then for any $f\in L^1(X_{\A,\om},\mu)$, \[th-pet\] \_[F\_n]{} f(-)df(§)d(§),  n, for $\mu$-a.e. $\T \in X_{\A,\om}$. For a bounded set $F\seq \R^d$ and $r>0$ let $$F^{+r} := \{\bx\in \R^d:\ \dist(\bx,F) \le r\}.$$ \[def-vanHove\] [A van Hove (F[ø]{}lner) sequence for $\R^d$ is a sequence $\{\Theta_n\}_{n\ge 1}$ of bounded measurable subsets of $\R^d$ satisfying $$\label{eq-vanHove} \lim_{n\to \infty} \frac{\vol((\partial \Theta_n)^{+r})}{\vol(\Theta_n)} = 0,\ \ \mbox{for all}\ r>0.$$ ]{} It is easy to see that our sequence $\{F_n\}_{n\ge 1}$ defined in (\[eq-vanh\]), is van Hove. Moreover, $\{\varphi^n(\supp(t))\}_{n\ge 1}$ is van Hove for any prototile $t\in \A_i,\ i\le m$. Indeed, the latter follows from the fact that $\vol(\partial(\supp(t)))=0$ for a prototile in a minimal component, which was proved for primitive tile substitutions in [@prag Prop.1.1].\ Given a set $F\seq \R^d$, patch $P$, and a tiling $\T$, denote by $N(F,P,\T)$ the number of patches in $\T$ equivalent to $P$ such that $\supp(P)\seq F$. Note that for any set $F$, \[eq-puga\] N(F,P,) \^[-1]{}(F), where $\delta$ is the volume of a ball contained in the interior of every prototile. For $\T\in X_{\A,\omega}$ and a patch $P$, the [*frequency*]{} of $P$ in $\T$ with respect to $F_n$ (which is our default averaging sequence) is defined by $$\freq_{\T}(P)=\lim_{n\to\infty}\frac{N(F_n,P,\T)}{\vol(F_n)}\,,$$ whenever the limit exists.\ For $f$ the indicator function of the set $C_P+\Theta$ and $\beta = \diam(P)+\diam(\Theta)$ we obtain $$\begin{aligned} 0\le \frac{1}{\vol(F_n)}\int_{F_n} f(\T - \bt)\,d\bt & \le & \frac{N((F_n)^{+\beta},P,\T)}{\vol(F_n)} \cdot \vol(\Theta) \\ & \le & \left( \frac{N(F_n,P,\T)}{\vol(F_n)} + \delta^{-1} \frac{\vol((\partial F_n)^{+\beta})}{\vol(F_n)} \right)\cdot \vol(\Theta)\end{aligned}$$ hence $\freq_{\T}(P)=0$ for $\mu$-a.e. $\T$ will imply $\mu(C_P+\Theta)=0$ by (\[th-pet\]), in view of $\{F_n\}$ being van Hove. \[lem-asymp\] Let $M$ be the substitution matrix for a tile substitution with expansion map $\varphi$. Then for every $A\in \A'$ and $B\in \A$, $$\lim_{n\to \infty} \frac{M^n(A,B)}{|\det(\varphi)|^n} = 0.$$ This follows from the structure of the matrix $M$ and Proposition \[core\], using that all classes in $\A'$ are non-distinguished. A direct reference is [@S Theorem 9.4]. \[lem-fre1\] Let $A$ be a tile in $\A'$ and $\T\in \Xa$. Then $\freq_\T(A)=0$. Fix $n\in \NN$. Recall that $\om:\ \Xa\to \Xa$ is onto, hence there exists $\T'\in \Xa$ such that $\om^n(\T') = \T$. Observe that $$F_n= \varphi^n(B_1(\b0)) \seq \supp (\om^n([B_1(\b0)]^{\T'})).$$ Let $[B_1(\b0)]^{\T'} = \{t_1+\bx_1,\ldots,t_k+\bx_k\}$ where $t_i$ are prototiles (not necessarily distinct) and $\bx_i\in \R^d$. Note that $k\le K$ where $K$ is a uniform constant by the FPC property. We have $$\frac{N(F_n,A,\T)}{\vol(F_n)} \le \frac{\sum_{i=1}^k N(\om^n(t_i + \bx_i),A,\T)}{\vol(F_n)} = \frac{\sum_{i=1}^k M^n(A,t_i)}{|\det(\varphi)|^n \,\vol(B_1(\b0))}\,,$$ and the claim follows from Lemma \[lem-asymp\]. \[lem-fre2\] Let $P$ be an $\Xa$-admissible patch such that all its prototiles belong to $\A\setminus \A'$ but $P$ is not admissible for any of the minimal components $X_i$. Then $\freq_\T(P)=0$ for any $\T\in \Xa$. Let $n=s+\ell$ and find $\T'\in \Xa$ such that $\om^n(\T') = \T$. Consider $$C:= \om^s([B_1(\b0)]^{\T'})=\{t_{1,s}+\by_{1,s},\ldots,t_{k_s,s} + \by_{k_s,s}\},$$ where $t_{i,s}$ are prototiles (not necessarily distinct) and $\by_{i,s}\in \R^d$. Then $F_n \seq \supp (\om^\ell(C))$ and we have $$N(F_n,P,\T) \le N(\om^{\ell}(C),P,\T) = N(\bigcup_{i=1}^{k_s} \om^\ell(t_{i,s} + \by_{i,s}), P, \T).$$ Observe that if $P$ has nonempty intersection with $\om^\ell(t_{i,s}+\by_{i,s})$, then either $t_{i,s} \in \A'$ or $\supp(P)$ intersects the boundary $\partial(\supp(\om^\ell(t_{i,s}+\by_{i,s})))$ for $t_{i,s} \not\in\A'$. Thus, in view of (\[eq-puga\]), \[eq-nuga\] N(F\_n,P,) \^[-1]{} \_[ik\_s: t\_[i,s]{} ’]{} ((\^(t\_[i,s]{}))) + \^[-1]{} \_[ik\_s: t\_[i,s]{} ’]{} ((((\^(t\_[i,s]{})))\^[+]{})) where $\alpha = \diam(P)$. Fix ${\varepsilon}>0$. By Lemma \[lem-fre1\], there exists $s_0$ such that for $s\ge s_0$ we have $$\#\{i\le k_s:\ t_{i,s} \in \A'\} \le {\varepsilon}|\det\varphi|^s.$$ Denoting by $V_{\max}$ the maximal volume of a prototile we obtain $$\label{eq-new1} \sum_{i:\ t_{i,s} \in \A'} \vol(\supp (\om^\ell(t_{i,s}))) \le {\varepsilon}|\det\varphi|^s \max_{i,s} \vol(\supp(\om^\ell(t_{i,s}))) \le {\varepsilon}|\det\varphi|^{s+\ell} V_{\max}.$$ On the other hand, $\{\supp(\om^\ell(t))\}_{\ell\ge 1}$ is a van Hove sequence for any prototile $t\not\in \A'$, hence there exists $\ell_0$ such that for any $t\not\in \A'$, for $\ell \ge \ell_0$, $$\vol((\partial(\supp (\om^\ell(t))))^{+\alpha}) \le {\varepsilon}\,\vol(\supp (\om^\ell(t))).$$ Combining this with (\[eq-nuga\]), (\[eq-new1\]), and using that $$\sum_{i=1}^{k_s} \vol(\om^{\ell}(t_{i,s})) = \vol(\om^{s+\ell}([B_1(\b0)]^{\T'}) \le |\det\varphi|^{s+\ell} KV_{\max}$$ yields $$N(F_{s+\ell},P,\T) \le {\varepsilon}\delta^{-1} |\det\varphi|^{s+\ell} (1+K) V_{\max}\ \ \ \mbox{for}\ s\ge s_0, \ell\ge \ell_0,$$ and the claim follows. Now Theorem \[th-finite\] is proved by the scheme given after its statement. We also obtain the following \[pi1\] There is an affine bijection between the set of finite invariant measures of $X_{\A,\omega}$ and $core(M)$. The finite ergodic measures of $X_{\A,\omega}$ are in one-to-one correspondence with the distinguished eigenvectors of $M$. Let $\mu$ be a finite ergodic measure. Theorem \[th-finite\] implies that $\mu$ is supported on a minimal component $X_i$. Since $X_i$ is a minimal substitution system, [@So1 Theorem 3.1]), [@So1 theorem 3.3], [@So1 Corollary 3.5] and [@HJ 8.2.11] imply that $\mu$ is determined by the Perron eigenvalue of the matrix $M_i$, where $M_i$ is the restriction of $M_{\omega}$ to the minimal component $X_i$. Indeed, $(\mu^T(C_A))_{A\in\A_i}$ is a Perron eigenvector of $M_i$. Since $\mu^T(C_D)=0$ for every $D\in\A'$, Theorem \[Perron-Frobenius\] implies that $(\mu^T(C_A))_{A\in\A}$ is a distinguished eigenvector of $M$ associated to the class $\A_i$. Since $core(M)$ is the cone generated by its distinguished eigenvectors, we get that $(\mu^T(C_A))_{A\in \A}$ is in $core(M)$. It is straightforward to show that the function $\mu\mapsto (\mu^T(C_A))_{A\in\A}$, defined on the set of finite invariant measures of $(X_{\A,\omega},\R^d)$ to $core(M)$, is affine and bijective. Recognizability =============== A substitution $\om$ is [*non-periodic*]{} if the dynamical system $(\Xa,\R^d)$ has no periodic points, that is, if $\Tk\in \Xa$ and $\Tk+\bv=\Tk$ for $\bv\in \RR^d$, then $\bv=\b0$. In this section we show the following theorem: \[th-recog\] Let $\omega:\A\to \A^+$ be an admissible tiling substitution. The function $\omega: X_{\A,\omega}\to X_{\A,\omega}$ is one-to-one if and only if $\omega$ is non-periodic. It is straightforward to show that a periodic substitution is not one-to-one. Indeed, if $\T\in X_{\A,\omega}$ is such that $\T=\T+\bv$ for some $\bv\in \RR^d\setminus\{\b0\}$, then Proposition \[onto\] implies that for every $i\geq 1$ there exists $\T_i\in X_{\A,\omega}$ such that $\omega^i(\T_i)=\omega^i(\T_i+\varphi^{-i}(\bv))=\T$. Observe that for $i\geq 1$ sufficiently large, $\T_i\neq \T_i+\varphi^{-i}(\bv)$ since $\varphi^{-i}(\bv)$ is close to zero. This proves that $\omega$ is not one-to-one.\ Theorem \[th-recog\] was already proved for primitive substitutions in [@So], so here we focus on the non-primitive case. We also obtain “partial recognizability” for a class of substitutions with periodic points which we now define. We can assume, without loss of generality, that (\[eq-irre\]) holds, and $X_1,\ldots, X_m$ are the minimal components of $X_{\A,\omega}$. We say that $X_j$ is [*periodic*]{} if there exists $\Tk\in X_j$ and $\bv\ne \b0$ such that $\Tk+\bv = \Tk$. Then $\bv$ is a translational period for all tilings in $X_j$. Denote by $\Aper$ the set of prototiles which occur in periodic minimal components and $\Anonp:= \A \setminus \Aper$. For any legal patch $P$ let $P|_{\rm nonp}$ be the subpatch of all $\Anonp$ tiles in $P$. \[def-border\] We say that a substitution $\om$ satisfies the [*non-periodic border condition*]{} if $$\label{eq-bord1} \forall\,\A\in \Anonp,\ \partial (\supp(\om(A)))\seq \supp(\om(A)|_{\rm nonp}).$$ \[part-recog\] A tile substitution $\om$ is said to be [*partially recognizable*]{} if for every $\T\in \Xa$ which contains a tile of $\Anonp$ type there is a unique tiling $\T'\in \Xa$ such that $\om(\Tk') = \Tk$. \[th-recog2\] An admissible tile substitution $\om$ satisfying the non-periodic border condition is partially recognizable. \[cor-period\] If $\om$ has non-periodic border and $\T\in \Xa$ contains a tile in $\Anonp$, then $\T$ is non-periodic. The non-periodic border condition is not necessary for the claim of Theorem \[th-recog2\] to hold (see Example \[ex-saf\] below), but the following example shows that some assumption on the substitution $\om$ is needed. It is plausible that, without any additional assumptions, a [*non-periodic*]{} tiling $\Tk$ has a unique preimage under $\om$; however, this remains an open question. [*Let $\Ak = \left\{\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\,, \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\right\}$; $$\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular}\, ,\ \ \ \ \ \ \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 0 & 1 & 0 \\ \hline 0 & 1 & 0 \\ \hline 0 & 1 & 0 \\ \hline \end{tabular}$$ Note that all tilings in $\Xa$ are periodic under the vertical shift, but according to our definition, only the prototile labelled 0 (which is in the minimal component) is periodic. Thus, the conclusion of Corollary \[cor-period\] is violated (of course, the non-periodic border condition does not hold).* ]{} Now we start preparation for the proofs. Clearly, a non-periodic substitution has non-periodic border, so Theorem \[th-recog2\] implies Theorem \[th-recog\]. Our argument is based on the method of [@HRS] where a new proof of recognizability for primitive tile substitutions was given (it applied to a more general class of tilings, not just translationally FPC). We should note that a direct proof of Theorem \[th-recog\] is simpler, and we will indicate which parts can be skipped if one is only interested in non-periodic substitutions. Recall that $\eta>0$ is such that the support of every prototile in $\A$ contains the closed ball $\ov{B}_\eta(\b0)$ in its interior. Let $\gamma=\max_{A\in \A}\{ \diam(A) \}$, and $1<\lambda_1\leq \lambda_2<\infty$ are positive numbers such that $$\label{eq-norm} \lambda_1\|\bx\|\leq \|\varphi(\bx)\|\leq \lambda_2\|\bx\|, \mbox{ for every } \bx\in \RR^d.$$ Since $\varphi$ is expansive, we can find a norm in $\RR^d$ with this property, and balls in this section will always be considered with respect to this norm. Then for every $n\geq 1$ and $\by\in \RR^d$, $$\label{eq-expand} B_{\lambda_1^nr}(\varphi^n(\by))\subseteq \varphi^n(B_r(\by))\subseteq B_{\lambda_2^nr}(\varphi^n(\by))$$ The next definition was introduced in [@HRS]. Let $W$ be an $X_{\A,\omega}$-admissible patch. For every $n\geq 0$, let $\P_n(W)$ be the set of $X_{\A,\omega}$-admissible patches $P$ satisfying: 1. $\omega^n(W)\subseteq \omega^n(P)$; 2. $\omega^n(W)$ is not included in $\omega^n(P')$, for any proper subpatch $P'$ of $P$. Actually, [@HRS] used only $\P_n(t)$ for a single tile $t$; this would be sufficient if we were to restrict ourselves to non-periodic tilings. \[lema1-p\] Let $W$ be an $X_{\A,\omega}$-admissible patch. 1. Let $n\geq 0$. For every $P\in \P_n(W)$, we have $ \supp(P)\subseteq B_{2\gam}(\supp(W)). $ 2. $\{W\}=\P_0(W)\subseteq \P_1(W)\subseteq \P_2(W)\subseteq\cdots$. \(i) Let $n\geq 0$ and $P\in \P_n(W)$. Since $\omega^n(W)\subseteq \omega^n(P)$, we have that $\supp(W) \subseteq \supp(P)$. Let $P'$ be the set of all tiles in $P$ whose supports intersect $\supp(W)$. Then $P'\subseteq P$, and we have $\om^n(W) \subseteq \om^n(P') \subseteq \om^n(P)$. By the definition of $\P_n(W)$ we must have $P'=P$, and the desired property follows, since $\gam$ is the maximal diameter of a $X_{\A,\om}$-tile. \(ii) It is clear that $\{W\}=\P_0(W)$. Let $n\geq 0$ and $P\in \P_n(W)$. The definition of the set $\P_n(W)$ implies that $\omega^{n+1}(W)\subseteq \omega^{n+1}(P)$. If $P'$ is a subpatch of $P$ for which $\omega^{n+1}(W)\subseteq \omega^{n+1}(P')$, then the support of $\omega^n(W)$ is included in the support of $\omega^n(P')$. This implies that $\omega^n(W)\subseteq \omega^n(P')$, and from definition of $\P_n(W)$, we get $P'=P$. This shows that $\P_n(W)\subseteq \P_{n+1}(W)$. Let $\P(W)=\bigcup_{n\geq 0}\P_n(W)$. The FPC assumption and part (i) of Lemma \[lema1-p\] imply that $\P(W)$ is finite up to translation. In the non-periodic case one can show that $\P(t)$ is finite, for any tile $t$. In the non-periodic border case, this is no longer true, and we have to work with “first coronas” or “collared tiles” containing at least one non-periodic tile. More precisely, consider all legal patches of the form $[\supp(t)]^\T,\ t\in \T$, for some $\T\in X_{\A,\om}$; there are finitely many of them, up to translation. We choose a representative for each of the translation-equivalent classes, and denote their collection by $\F$. Denote by $\Fnonp$ the set of patches in $\F$ which contain a tile of type $\Anonp$. Now we can state the key proposition needed in the proof of Theorem \[th-recog2\]. \[prop-principal2\] There exists $M\in \NN$ such that for any $\T\in X_{\A,\omega}$, $n\geq 0$ and $\bx,\by\in \RR^d$, if $P=\om^n(Z)$, with $Z - \by \in \Fnonp$, such that $$P\subseteq \T, \,\,\,\, P+\bx \subseteq \T,$$ then $$\bx\in \varphi^{n-M}(B_\eta(\b0))\ \mbox{implies}\ \bx=\b0.$$ The proof will be based on several lemmas. \[lem-claim1\] There exists $R_0>0$ such that for every $\T\in X_{\A,\omega}$ and $\bx\in \RR^d$, the ball $B_{R_0}(\bx)$ contains an $X_i$-admissible $\T$-patch, for some $1\leq i\leq m$. Suppose that for every $R>0$ there exist $\T_R\in X_{\A,\omega}$ and $\bx_R\in \RR^d$, such that the ball $B_{R}(\bx_R)$ does not contain patches belonging to any minimal component. Compactness of $X_{\A,\omega}$ implies there exists a sequence $R_n \uparrow\infty$ such that $(\T_{R_n}-\bx_{R_n})_{n\geq 0}$ converges to some tiling $\T\in X_{\A,\omega}$. It follows that $\T$ does not contain patches from any minimal component, which is not possible because $\clos\{\T-\bg:\,\bg\in \R^d\}$ must contain at least one minimal component of $X_{\A,\omega}$. The strategy of the proof of Proposition \[prop-principal2\] is to find a large sub-patch of $\om^n(t)\seq \om^n(Z)$, with $t-\bx_t \in \Fnonp$, which belongs to a minimal component $X_i$. This component may be non-periodic or periodic. The former case is treated with the following two lemmas. The first one is a special case of [@So Lemma 3.2]. \[lema5\][*([@So Lemma 3.2])*]{} Let $\omega:\A\to \A^+$ be a primitive non-periodic substitution, and let $\eta>0$ be such that the support of every prototile in $\A$ contains the ball $B_\eta(\b0)$. Then there exists $N\in \NN$ such that, for any $\T\in X_{\A,\om}$, $l>0$, and $\bx,\by\in \RR^d$, if $$P\subseteq \T, \,\,\,\, P+\bx \in \T, \,\,\,\, \varphi^l(B_\eta(\b0))+\by\subseteq \supp(P),$$ then $$\bx\in \varphi^{l-N}(B_\eta(\b0)) \mbox{ implies } \bx=\b0.$$ \[lem-period\] There exists $N\in \NN$ such that for any $\T\in X_{\A,\omega}$, $n\geq 0$ and $\bx,\by\in \RR^d$, if $P$ is an $X_i$-admissible patch, where $X_i$ is a non-periodic minimal component, such that $$P\subseteq \T, \,\,\,\, P+\bx \subseteq \T, \,\,\,\, \varphi^n(B_\eta(\b0))+\by\subseteq \supp(P),$$ then $$\bx\in \varphi^{n-N}(B_\eta(\b0))\ \mbox{implies}\ \bx=\b0.$$ Let $k\in \NN$ be such that $2B_\eta(\b0) = B_\eta(\b0)+B_\eta(\b0) \seq \varphi^k(B_\eta(\b0))$. We can take $k =\lceil \log 2/\log \lam_1 \rceil$ by (\[eq-norm\]). By Lemma \[lema5\], for $j\in \{1,\cdots, m\}$, there exists $N_j\in \NN$ such that if $Q$ is an $X_j$-admissible patch satisfying $$Q\subseteq \T', \,\,\,\, Q+\bw \seq \T', \,\,\,\, \varphi^n(B_\eta(\b0))+\bv\subseteq \supp(Q),$$ for some $\bw\in \RR^d\setminus\{\b0\}$, $\bv\in \RR^d$ and $\T'\in X_j$, then $$\bw\notin \varphi^{n-N_j}(B_\eta(\b0)).$$ We claim that the desired statement holds for $N=k+\max\{N_j: 1\leq j\leq m \}$. Let $\T\in X_{\A,\omega}$, $\bx\in \RR^d\setminus \{\b0\}$ and $P$ be an $X_i$-admissible patch such that $$P\subseteq \T, \,\,\,\, P+\bx \seq \T, \,\,\,\, \varphi^n(B_\eta(\b0))+\by\subseteq \supp(P),$$ for some $n\geq 0$ and $\by\in \RR^d$. Further, suppose that $\bx\in \varphi^{n-N}(B_\eta(\b0))\setminus \{\b0\}$. Clearly, $n>N$, since every tile contains a ball of radius $r$, so shifting a tile by a vector in $B_\eta(\b0)$ will result in a tile intersecting the original one, making $P, P+\bx\seq \T$ impossible. Since $P$ is $X_i$-admissible, there exists $\T'\in X_i$ such that $P\seq \T'$. Consider $$P':= [\varphi^{n-k}(B_\eta(\b0))+\by]^{\Tk} \seq \Tk'.$$ We want to apply Lemma \[lem-period\] to $P'$ and $\T'$. The only thing we need to check is that $P'+\bx \seq \T'$. This will follow if we show that $P'+\bx\seq P$, and to verify the latter it suffices to check that $$\varphi^{n-k}(B_\eta(\b0))+\by+\bx \seq \varphi^n(B_\eta(\b0)) + \by.$$ However, $$\varphi^{n-k}(B_\eta(\b0))+ \bx \seq \varphi^{n-k}(B_\eta(\b0))+ \varphi^{n-N}(B_\eta(\b0)) \seq \varphi^{n-k}(B_\eta(\b0))+ \varphi^{n-k}(B_\eta(\b0)) \seq \varphi^n(B_\eta(\b0))$$ by the definition of $k$. The proof is complete. Suppose that $\Tk, P$, and $\bx$ are as in the statement of the proposition, with some $M\ge 1$, which will be determined below. Suppose that $\bx\ne \b0$. Let $t$ be a tile of the patch $Z$ of type $\Anonp$. Its support contains the ball $B_\eta(\by)$ for some $\by\in \R^d$. Then $$\supp(P) = \supp(\om^n(Z))\supseteq \supp(\om^n(t))\supseteq \varphi^n(B_\eta(\by)).$$ Let $n_0\ge 0$ be the smallest integer satisfying $ \eta\lam_1^{n_0} \ge R_0 $, that is, $ n_0=\left \lceil \log_{\lambda_1}\left( \frac{R_0}{\eta}\right)\right \rceil. $ Here $R_0$ is the constant from Lemma \[lem-claim1\]. Let $\Sk$ be any tiling in $X_{\A,\om}$ satisfying $\om^{n-n_0}(\Sk) = \Tk$. Then $\varphi^{n_0}(B_\eta (\by))$ contains a ball of radius $R_0$, hence an $X_i$-admissible tile $t'\in \Sk$, for some minimal component $X_i$, $1 \le i \le m$, by Lemma \[lem-claim1\]. Therefore, $\varphi^n(B_\eta (\by))$, and hence the patch $\om^n(t)\seq P$, contains an $X_i$-admissible patch $P'=\om^{n-n_0}(t')\subseteq \Tk$. Now there are two cases. If $X_i$ is non-periodic, then we apply Lemma \[lema5\] to conclude that $\bx=\b0$, provided $M \ge N + n_0$ (where $N$ is from Lemma \[lem-period\]). [**This concludes the proof in the case when the substitution $\om$ is non-periodic.**]{} It remains to treat the case when $X_i$ is periodic. The idea is the following: since $P,P+\bx\seq \Tk$, we have that $P'+ \bx\seq P$ as long as $\supp(P'+ \bx)\seq \supp(P)$. Then $P'+ \bx$ is also $X_i$-admissible. We can continue in this manner as long as the translates of $P'$ by a multiple of $\bx$ remain in $\supp(P)$, and this works for individual tiles as well, not necessarily for the entire $P'$. If $\bx$ is small relative to the size of $P'$, we will obtain an entire “tube” from $P'$ to the border of $\om^n(t)$ which is $X_i$-admissible. But this will lead to a contradiction with the non-periodic border assumption. Now for the details. A slight complication arises because of the possibility that the interior of a tile is disconnected, so we actually take the “connected component" of $P'$. Let us continue with the proof of the proposition. We can assume that $M\ge n_0$. Clearly, the assumptions imply $n\ge M$ (since $\bx\in \varphi^{n-M}(B_\eta(\b0))$ is a non-zero translation between two tiles in $\Tk$, and every prototile contains $B_\eta(\b0)$ in the interior of its support). Recall that $P'=\varphi^{n-n_0}(t')$ is $X_i$-admissible, where $X_i$ is a periodic minimal component, $P'\seq \om^n(t)\seq P$, and $t$ is a tile of type $\Anonp.$ It follows by induction from (\[eq-bord1\]) that $$\label{eq-bord2} \partial (\supp(\om^n(t)))\seq \supp(\om^n(t)|_{\rm nonp}).$$ The tile $t'$ contains a ball $B_\eta(\bz)$ for some $\bz\in \R^d$, hence $\varphi^{n-n_0}(B_\eta(\bz))\seq\supp(P')$. Consider $$V:= \mbox{\ the component of \ \ $\Int(\supp(\om^{n-n_0}(t')))$ \ \ containing $\varphi^{n-n_0}(B_\eta(\bz))$}.$$ Clearly $[V]^\Tk\seq P'$, so all its tiles are of type $\Anonp$. Note that $V\cap (V+\bx)\ne \es$ because $$\varphi^{n-n_0}(B_\eta(\bz))\cap (\varphi^{n-n_0}(B_\eta(\bz))+\bx)\ne \es\ \ \mbox{for}\ \bx\in \varphi^{n-M}(B_\eta(\b0))\seq \varphi^{n-n_0}(B_\eta(\b0)).$$ Let $k\ge 0$ be the largest integer such that $V+k\bx\seq\supp(\om^n(t))$. Then $[V]^\Tk+(k+1)\bx\seq \Tk$, because $\om^n(t)\seq P$ and $P+\bx\seq \Tk$. Moreover, $$V+(k+1)\bx\not\seq \supp(\om^n(t))\ \ \mbox{and}\ \ (V+k\bx)\cap (V+(k+1)\bx)\ne \es.$$ It follows that $V+(k+1)\bx$ contains a point $\bz_1$ in the interior of a tile in $\Tk\setminus \om^n(t)$ and $V + k\bx$ contains a point $\bz_2$ in the interior of a tile in $\om^n(t)$. The set $$W:=(V+k\bx)\cup(V+(k+1)\bx)$$ is open and connected, being a union of two open connected sets with non-empty intersection. Thus it is arcwise connected. An arc in $W$ with the endpoints $\bz_1$ and $\bz_2$ must intersect the boundary of $\supp(\om^n(t))$. The point of intersection belongs to a tile, say, $t''$, of type $\Anonp$ by (\[eq-bord2\]). This tile is in $[W]^\Tk$, but the types of all tiles in $[W]^\Tk$ are those of the tiles in $[V]^\Tk$, hence they are $X_i$-admissible. Thus, $t''$ is of type $\Aper$, which is a contradiction. \[lem-finite-p\] There exists $N\in \NN$ such that for any patch $Z$ and $\by\in \R^d$ such that $Z-\by\in \Fnonp$, we have $\P_{n+1}(Z) = \P_n(Z)$ for all $n\ge N$. Let $n\ge 0$ and $P \in \P_n(Z)$. Suppose that $\bx\in \R^d\setminus \{\b0\}$ is such that $P+\bx \in \P_n(Z)$. We have $$\om^{n}(Z) \subseteq \om^n(P)\ \ \ \mbox{and}\ \ \ \om^n(Z) - \varphi^n(\bx) \subseteq \om^n(P).$$ We conclude from Lemma \[lem-period\] that $\varphi^n(\bx) \not\in \varphi^{n-M}(B_\eta(\b0))$, hence $\bx\not\in \varphi^{-M}(B_\eta(\b0))$, which implies $$\|\bx\| \ge \eta \lam_2^{-M}.$$ From part (i) of Lemma \[lema1-p\] it follows that the supports of translated copies of $P$ which belong to $\P_n(Z)$, are contained in $B_{2\gam}(\supp(Z))$. Recall that $Z-\by\in \Fnonp$ is a “collared tile” hence $\diam(Z) \le 3\gam$. Thus, there are at most $$\frac{\vol(B_{5\gam+\eta \lam_2^{-M}/2}(\b0))}{\vol(B_{\eta \lam_2^{-M}/2})(\b0)}=:N_1$$ copies of the patch $P$ in $\P_n(Z)$. The FPC ensures that there are finitely many patches in $\Fnonp$, up to translation. Also by FPC, there are at most $C$ patches, up to translation, whose support is contained in $B_{2\gam}(\supp(Z))$ for some $Z\in \Fnonp$. From this we deduce that $\P_n(Z)$ has at most $CN_1$ elements. Since this is valid for every $n\ge 0$, from part (ii) of Lemma \[lema1-p\] it follows that $|\P(Z)| \le CN_1.$ We continue with the scheme of [@HRS]. \[lem-easy1\] Suppose $\P_n(W) = \P_{n+1}(W)$, where $W$ is a legal patch. If $\S\in X_{\A,\om}$ is such that $\om^{n+1}(W) \subseteq \om(\S)$, then $\om^n(W) \seq \S$. Let $\S'\in X_{\A,\om}$ be such that $\om^n(\S') = \S$. Then $\om^{n+1}(W)\seq \om(\S) = \om^{n+1}(\S')$, hence there exists $P \in \P_{n+1}(W)$ such that $P\seq \S'$. Since $P\in \P_n(W)$ we have $\om^n(W) \seq \om^n(P) \seq \om^n(\S') = \S$. [*Proof of Theorem \[th-recog2\].*]{} Let $\T_1 \in X_{\A,\om}$ be such that $\om(\T_1) =\T$, and further, suppose $\T_n \in X_{\A,\om}$ are such that $\om(\T_n) = \T_{n-1}$ for $n\ge 2$. Let $t_n \in \T_n$ be such that $\supp(t_n) \ni \b0$, and let $Z_n = [\supp(t_n)]^{\T_n}$. Then $\om^n(Z_{n+1}) \seq \T_1$ and $$\bigcup_{n\ge 1} \supp(\om^n(Z_{n+1})) = \R^d,$$ hence $Z_k$ are of type $\Fnonp$ for $k$ sufficiently large (otherwise, $\T_1$ and $\T = \om(\T_1)$ contain only tiles from $\Aper$ contradicting our assumption), that is, there exists $k_0$ such that $Z_k - \bz_k \in \Fnonp$ for $k\ge k_0$. We want to show that $\T_1$, with $\om(\T_1) = \T$, is uniquely determined. To this end, consider any $\T'$, with $\om(\T') = \T$. We have for $n\ge \max\{k_0,N\}$, by Lemma \[lem-finite-p\], that $\P_{n+1}(Z_{n+1})= \P_n(Z_{n+1})$, hence by Lemma \[lem-easy1\], $$\om^{n+1}(Z_{n+1}) \seq \T=\om(\T')\ \Longrightarrow\ \om^n(Z_{n+1}) \seq \T'.$$ Therefore, $\T'$ contains the patches $\om^n(Z_{n+1})\seq \T_1$ for all $n$ sufficiently large, and these patches exhaust the entire tiling. Thus, $\T'=\T_1$, as desired. Infinite invariant measures. ============================ Non-negative matrices revisited ------------------------------- We use the notation and results from Sections 2.5 and 2.6. [ Let $M\in \M_{k\times k}(\ZZ_+)$. The [*infinite core*]{} of $M$ is the set of all the vectors in $core_{\infty}(M)=\bigcap_{n\geq 1}M^n(\overline{\RR}_+^k)$ where $\ov{\RR}_+ := \RR_+\cup \{\infty\}$.]{} We saw in Section 3 that $core(M)$ is isomorphic to the set of finite invariant measures. Here we will show that $core_{\infty}(M)$ is closely related to the set of “nice” invariant $\sig$-finite measures, under some mild assumptions. The goal of this subsection is to describe the infinite core. \[help-lemma\] Let $M_1$ and $M_2$ be two non-negative square matrices of dimensions $n_1$ and $n_2$ respectively, such that $M_1$ is primitive and $M_2$ has a positive eigenvalue $\rho_2>0$ associated to a positive eigenvector $\bv_2$. Let $C\neq 0$ be a non-negative $n_1\times n_2$-dimensional matrix and let $\rho_1$ be the Perron eigenvalue of $M_1$. If there exists a vector $\bx\in \overline{\RR}_+^{n_1}$ such that $$\left(\begin{array}{c} \bx \\ \bv_2 \\ \end{array}\right)\mbox{ is in the infinite core of } M= \left(\begin{array}{cc} M_1 & C \\ 0 & M_2 \\ \end{array}\right),$$ then $\bx=\infty$ whenever $\rho_1\geq \rho_2$. Let $n>0$, $C_n$ and $\bx_n\in \overline{\RR}^{n_1}_+$ be such that $$\left(\begin{array}{cc} M_1 & C \\ 0 & M_2 \\ \end{array}\right)^n = \left(\begin{array}{cc} M_1^n & C_n \\ 0 & M_2^n \\ \end{array}\right )$$ and $$\left(\begin{array}{c} \bx \\ \bv_2 \\ \end{array}\right)=M^n\left(\begin{array}{c} \bx_n \\ \frac{\bv_2}{\rho_2^n} \\ \end{array}\right).$$ Using the symbol “$\ge$” to denote the natural (component-wise) partial order on vector spaces, we have $$\label{Section6-eq1} \bx=M_1^{n}\bx_n+C_n\frac{\bv_2}{\rho_2^n}\geq C_n\frac{\bv_2}{\rho_2^n},$$ and $$\begin{aligned} C_{n+1}\frac{\bv_2}{\rho_2^{n+1}} & = &\frac{1}{\rho_2^{n+1}}M_1^nC\bv_2+C_n\frac{\bv_2}{\rho_2^{n}}\\ &=& \frac{1}{\rho_2}\left(\frac{\rho_1}{\rho_2}\right)^n\left(\frac{M_1}{\rho_1}\right)^nC\bv_2+ C_n\frac{\bv_2}{\rho_2^n}\\ &=& \frac{1}{\rho_2}\left[\sum_{k=0}^n\left(\frac{\rho_1}{\rho_2}\right)^k\left(\frac{M_1}{\rho_1}\right)^k\right]C\bv_2.\end{aligned}$$ Thus if $\rho_1\geq \rho_2$, we have $$\label{Section6-eq2} C_{n+1}\frac{\bv_2}{\rho_2^{n+1}}\geq \frac{1}{\rho_2}\left[\sum_{k=0}^n\left(\frac{M_1}{\rho_1}\right)^k\right]C\bv_2,$$ which tends to $\infty$ with $n$, because $C\bv_2\neq \b0$ and $\lim_{n\to \infty}(M_1/\rho_1)^n=\bw\bv>0$, where $\bw$ and $\bv$ are left and right Perron eigenvectors of $M_1$ respectively (see [@HJ Theorem 8.5.1]). Then from equations (\[Section6-eq1\]) and (\[Section6-eq2\]) we conclude that $\bx=\infty$ when $\rho_1\geq \rho_2$. Suppose that $\alpha_1,\cdots,\alpha_l$ are the equivalence classes associated to the matrix $M$. For every $1\leq i\leq l$ we denote by $M_i$ the restriction of $M$ to the class $\alpha_i$. We assume that $M_i$ is primitive or equal to $[0]$. When $M_i$ is primitive we denote by $\rho_i$ the Perron eigenvalue of $M_i$; if $M_i=[0]$ then $\rho_i=0$. \[definition1\] Let $M$ be a non-negative integer square matrix with irreducible components $M_1,\cdots, M_l$. For every $1\leq i\leq l$ such that $M_i$ is primitive we consider 1. $\J_i$—the set of indices $j\in\{1,\cdots, l\}\setminus\{i\}$ such that the class $\alpha_j$ has access to a class $\alpha_k$, where $\alpha_k$ has access to $\alpha_i$ and $\rho_k\geq \rho_i$. 2. $\I_i$—the set containing $i$ and all the indices $j\in\{1,\cdots,l\}\setminus \J_i$ such that the class $\alpha_j$ has access to the class $\alpha_i$ (then necessarily $\rho_j<\rho_i$). [The classes $\alpha_j$, with $j\in\I_i$ do not have access to the classes with indices in $\J_i$. The complement of $(\I_i\cup \J_i)$ is the set of $j$ such that $\alpha_j$ does not have access to $\alpha_i$.]{} Let $1\leq i\leq l$ be such that $M_i$ is primitive. The class $\alpha_i$ is distinguished with respect to $M|_{\I_i}$, the restriction of $M$ to the set of indices in $\I_i$. Then Theorem \[Perron-Frobenius\] implies that there exists a unique $|\I_i|$-dimensional normalized positive vector $\bw_i$ such that $M|_{\I_i}\bw_i=\rho_i\bw_i$. The restriction of $\bw_i$ to $\alpha_i$ is an eigenvector of $M_i$ associated to $\rho_i$. \[definition2\] For every $1\leq i\leq l$ such that $M_i$ is primitive, we define $\by_i$ in $\overline{\RR}_+^k$ as follows: - The restriction of $\by_i$ to $\I_i$ is equal to $\bw_i$. - The restriction of $\by_i$ to $\J_i$ is $\infty$ in every component. - The restriction of $\by_i$ to $(\I_i\cup \J_i)^c$ is zero. For every $1\leq i\leq l$, we define $\bz_i$ in $\overline{\RR}_+^k$ as the vector whose restriction to $\I_i\cup\J_i$ is infinite, and $\bz_i$ restricted to $(\I_i\cup\J_i)^c$ is zero. If $M_i=[0]$ we define $\by_i=\b0$. \[infinite-distinguished\] Let $M$ be a non-negative integer $k\times k$ matrix with irreducible components $M_1,\cdots, M_l$. For every $1\leq i\leq l$ we have $\by_i\in core_\infty(M)$. If $M_i=[0]$ then $\by_i=\b0$ is in $core_\infty(M)$. Then we can assume that $M_i$ is primitive. For every $n\geq 1$, we define $\by_{n,i}\in \overline{\RR}_+^k$, the vector such that $$\by_{n,i}|_{\I_i}=\frac{\bw_i}{\rho_i^n}, \hspace{4mm} \by_{n,i}|_{\J_i}=\infty, \hspace{4mm}\mbox{ and }\hspace{4mm} \by_{n,i}|_{(\I_i\cup\J_i)^c}=\b0.$$ For $1\leq r\leq k$, the $r$-coordinate of $M^n\by_{n,i}$ is equal to $$\sum_{s=1}^kM^n(r,s)\by_{n,i}(s)=\sum_{s\in\I_i}M^n(r,s)\by_{n,i}(s)+ \sum_{s\in\J_i}M^n(r,s)\by_{n,i}(s).$$ Thus we have the following:\ If $r\in (\I_i\cup \J_i)^c$, then $M^n(r,s)=0$ for every $s\in \I_i\cup \J_i$. This implies the $r$-coordinate of $M^n\by_{n,i}$ is equal to $\b0=\by_i(r)$.\ If $r\in \I_i$, then $M^n(r,s)=0$ for every $s\in \J_i$. This implies that the $r$-coordinate of $M^n\by_{n,i}$ is equal to $$\sum_{s\in\I_i}M^n(r,s)\by_{n,i}(s)=\bw_i(r)=\by_{i}(r).$$ If $r\in \J_i$, then for every $n\geq 1$ there exists $s_n\in \J_i$ such that $M^n(r,s_n)>0$. This implies that $r$-coordinate of $M^n\by_{n,i}$ is equal to $M^n(r,s_n)\by_{n,i}(s_n)=\infty=\by_i(r)$.\ The statements [**1**]{}, [**2**]{}, and [**3**]{} imply that $\by_i=M^n\by_{n,i}$, for every $n\geq 1$, which shows that $\by_i$ is in the infinite core of $M$. \[relation-between-eigenvalues\] Let $M$ be a non-negative integer $k\times k$ matrix with primitive irreducible components $M_1,\cdots, M_l$. Then every vector $\bx \in core_{\infty}(M)$ can be written as $$\bx=\sum_{j=1}^l\lambda_j\by_j+ \sum_{j=1}^l\delta_j\bz_j,$$ where $\lambda_1,\cdots, \lambda_l\geq 0$ and $\delta_1,\cdots,\delta_l\in\{0,1\}$. Conversely, every vector written in this way is in $core_{\infty}(M)$. Let $\bx\in core_{\infty}(M)$. Theorem \[generated-core\] implies that when $\bx$ is finite, this vector is in the cone generated by the vectors $\by_i$, for every $1\leq i\leq l$ such that $\alpha_i$ is distinguished for $M$. If $\bx$ has all its coordinates equal to $\infty$ or $zero$, it can be written as $\sum_{i=1}^l\delta_i\bz_i$, for some $\delta_1,\cdots, \delta_l\in\{0,1\}$. Thus we can assume that $\bx$ has a finite positive coordinate and an infinite coordinate. For $1\leq j\leq l$, let $\bx_j$ be the $|\alpha_j|$-dimensional vector given by the restriction of $\bx$ to $\alpha_j$. If some coordinate of $\bx_j$ is positive and finite, then all the coordinates of $\bx_j$ are positive and finite. Let $1\leq i\leq l$ be such that $$i=\max\{1\leq k \leq l: 0<\bx_k<\infty\}.$$ If there exists $i<k\leq l$ such that $\bx_{k}=\infty$ then for every $1\leq j \leq l$ such that $\alpha_j$ has access to $\alpha_k$ we have $\bx_j=\infty$. Then we can write $$\bx=\bu +\sum_{j=1}^l\delta_j\bz_j,$$ where $\bu_j=\bx_j$ for every $1\leq j \leq i$, $\bu_j=0$ for every $i<j\leq l$, and $\delta_1,\cdots, \delta_l\in \{0,1\}.$ After rearranging the coordinates of $\bx$ if it is necessary, we can suppose that there exists $1\leq s\leq i$ such that $0\leq \bx_j<\infty$ for every $s\leq j\leq i$, and $\bx_j=\infty$ for every $1\leq l<s$. The vector $\bx|_{\alpha_s,\cdots,\alpha_i}=(\bx_s,\cdots, \bx_i)$ is in the core of the restriction $M'$ of $M$ to the classes $\alpha_s,\cdots, \alpha_i$. Then by Theorem \[generated-core\], $\bx|_{\alpha_s,\cdots,\alpha_i}$ is in the cone generated by the distinguished eigenvectors of $M'$. Observe that $\bv$ is a distinguished eigenvector of $M'$ if and only if $\bv$ is the restriction to $\alpha_s,\cdots,\alpha_i$ of a scalar multiple of the vector $\by_j$, for some $s\leq j\leq i$ such that the class $\alpha_j$ is distinguished in $M'$. Thus, using Lemma \[help-lemma\], we can write $$\bx=\sum_{j=1}^l\lambda_j\by_j+\delta_j\bz_j,$$ where $\lambda_1,\cdots,\lambda_l\geq 0$ (with $\lambda_i>0$), and $\delta_1,\cdots,\delta_l\in\{0,1\}$. The converse holds by Lemma \[infinite-distinguished\]. Clopen nested partitions of the transversal. -------------------------------------------- As in the previous sections, we consider a substitution $\omega$ defined on a set of prototiles $\A\subseteq \RR^d$. We denote by $M\in\M_{\A\times\A}(\ZZ_+)$ the substitution matrix of $\omega$, and $\A_1,\cdots, \A_l$ the equivalence classes associated to $M$. We denote by $M_i$ the restriction of $M$ to the indices in $\A_i$, and we assume that $M_i$ is primitive or equal to $[0]$. We suppose there are equivalence classes which are not associated to minimal components, namely, $\A_{m+1},\cdots,\A_l$, for some $1\leq m< l$. We denote $\A'=\A\setminus (\A_1\cup\cdots\cup\A_m)$. Recall that we denote by $\Gamma$ the transversal of $X_{\A,\omega}$, and for every $A\in \A$, we set $$C_A=\{\T\in \Gamma: A\in\A\}.$$ The collection $\P_0=\{C_A: A\in \A \}$ is a clopen partition of $\Gamma$. For $n\geq 1$ and $A,B\in \A$, we define $$D_{n,A}=\supp(\omega^n(A)),$$ $$J_{A,B}^{(n)}=\{\bv\in D_{n,A}: B+\bv\subseteq \omega^n(A) \},$$ $$J_{A}^{(n)}=\bigcup_{B\in \A}J_{A,B}^{(n)},$$ and $$\P_n=\{\omega^n(C_A)-\bv:\, \bv\in J_A^{(n)}, A\in \A\}.$$ (These $\P_n$ have nothing to do with $\P_n(W)$ from Section 4.) [**For the rest of this section we assume that the admissible substitution $\om$ is partially recognizable, see Definition \[part-recog\], and also $$\label{eq-added} \mbox{For every prototile $A\in \Anonp$, the patch $\om(A)$ contains a tile of type $\Anonp$.}$$** ]{} Note that the latter condition is satisfied if $M$ has no components $[0]$, or if the non-periodic border condition holds. \[partition\] For every $n\geq 0$, the collection $\P_n$ is a covering of $\Gamma$. Furthermore, 1. For each $A\in \A$ and $n\geq 1$, $$\label{partition-1} C_A=\bigcup_{B\in \A}\bigcup_{\bv\in J_{B,A}^{(n)}}(\omega^n(C_B)-\bv).$$ 2. If $A,B\in \A$, $n\geq 1$, $\bv\in J_A^{(n)}$ and $\bu\in J_B^{(n)}$ are such that $$(\omega^n(C_A)-\bv)\cap (\omega^n(C_B)-\bu)\neq \emptyset,$$ then $A=B$ and $\bv=\bu$, or $A, B\in \A_{\per}$. 3. Let $\ell>n$, $A\in \A$ , $B\in \Anonp$, $\bv\in J_A^{(\ell)}$ and $\bu\in J_B^{(n)}$. We have $$(\omega^\ell(C_A)-\bv)\cap (\omega^n(C_B)-\bu)\neq \emptyset,$$ if and only if $\omega^\ell(C_A)-\bv\subseteq \omega^n(C_B)-\bu$ and $B+\varphi^{-n}(\bv-\bu)\in \omega^{\ell-n}(A)$. \(i) Proposition \[onto\] ensures that $\P_n$ is a covering of $\Gamma$ and implies (\[partition-1\]).\ (ii) By partial recognizability, if $A\neq B$ or $\bv\neq \bu$ then the tilings in $(\omega^n(C_A)-\bv)\cap (\omega^n(C_B)-\bu)$ contain only tiles from $\Aper$. Hence the patches $\omega^n(A)$ and $\omega^n(B)$ only contain tiles in $\A_{\per}$. Condition (\[eq-added\]) implies that $A$ and $B$ are in $\A_{\per}$.\ (iii) Let $D_1,\cdots, D_k\in \A$ and $\bx_1,\cdots, \bx_k\in \RR^d$ be such that $\omega^{\ell-n}(A)$ is the disjoint union $\bigcup_{i=1}^k(D_i+\bx_i)$. Then $\omega^\ell(A)$ is equal to the disjoint union $\bigcup_{i=1}^k(\omega^n(D_i)+\varphi^n(\bx_i))$. This implies that there exists $1\leq i\leq k$ such that $\bz=\bv-\varphi^{n}(\bx_i)\in J^{(n)}_{D_i}$ and $$\omega^\ell(C_A)-\bv= \omega^\ell(C_A)-\varphi^n(\bx_i)-\bz\subseteq \omega^n(C_{D_i})-\bz.$$ Then by hypothesis we have $(\omega^n(C_B)-\bu)\cap (\omega^n(C_{D_i})-\bz)\neq \emptyset$. Since $B\in \Anonp$, from part (ii) it follows that $B=D_i$, $\bz=\bu$, and then $\omega^\ell(C_A)-\bv\subseteq \omega^n(C_B)-\bu$ and $B+\varphi^{-n}(\bv-\bu)\in \omega^{\ell-n}(A)$. The other direction of the equivalence is immediate. \[rem-inf\] [1. If the substitution is non-periodic, then $(\P_n)_{n\geq 0}$ is a nested sequence of clopen partitions of $X_{\A,\omega}$. In the minimal non-periodic one-dimensional symbolic case, the sequences $(\P_n)_{n\geq 0}$ correspond to the sequence of Kakutani-Rohlin partitions for minimal substitution subshifts given in [@DHS]. In general, it is only a covering, but we will see below that it becomes a partition if we intersect it with the set of non-periodic tilings.\ 2. It is useful to give an informal interpretation of the covering $(\P_n)_{n\ge 0}$. Given a tiling $\T\in \Gamma$, we have a sequence of tilings $(\T_n)_{n\ge 0}$, with $\T_0= \T$, such that $\om(\T_n) = \T_{n-1}$ for $n\ge 1$. This defines a sequence of “supertilings” obtained by composing the tiles of $\T$, with “supertiles” that are translates of $\om^n(A)$ for $A\in \A$. This sequence is uniquely defined if $\T$ contains a non-periodic tile. We consider the supertile of order $n$ whose support contains the origin (it is uniquely defined since $\T$ is in the transversal). This determines the element of $\P_n$ to which $\T$ belongs. ]{} Necessary conditions for transverse measures. --------------------------------------------- Let $\mu^T$ be a transverse measure on $\B(\Gamma)$. For every $A\in \A$ and for every $n\geq 0$, we define $$\mu^T_{n,A}=\mu^T(\omega^n(C_A)-\bv),$$ where $\bv$ is a vector in $J_A^{(n)}$. The number $\mu^T_{n,A}$ does not depend on $\bv$ because $\mu$ is transverse. We denote by $\mu^T_{n}$ the vector $(\mu^T_{n,A})_{A\in \A}$ and by $\wt{\mu}^T_n$ the vector $(\mu^T_{n,A})_{A\in \A'}$. \[coreN\] Let $\mu$ be a transverse measure on $\B(\Gamma)$. Then for every $A\in \A'$ and $\ell>n\geq 0$, $$\mu^T_{n,A}=\sum_{B\in \A'}M^{\ell-n}(A,B)\mu^T_{\ell,B}.$$ Thus, $$\wt{\mu}^T_n = (M')^{\ell-n} \wt{\mu}^T_\ell,$$ where $M'$ is the restriction of $M$ to the set of indices in $\A'$. Let $A\in \A'$ and $\bu\in J_A^{(n)}$. From (iii) of Lemma \[partition\] we have $$\omega^n(C_A)-\bu=\bigcup_{B\in\A}\bigcup_{\bv\in I^{(n)}_B}(\omega^\ell(C_B)-\bv),$$ where $I_B^{(n)}$ is the set of $\bv$ in $J_B^{(n)}$ such that $A+\varphi^{-n}(\bv-\bu)\in \omega^{\ell-n}(B)$. Since the minimal components are $\omega$-invariant we can restrict the outer union to $\Ak'$: $$\omega^n(C_A)-\bu=\bigcup_{B\in\A'}\bigcup_{\bv\in I^{(n)}_B}(\omega^\ell(C_B)-\bv).$$ Observe that $|I_B^{(n)}|=M^{\ell-n}(A,B)$. Thus from (ii) of Lemma \[partition\] we obtain the desired equality. [If $\A_{per}=\emptyset$ then the same proof shows that $$\mu^T_{n,A}=\sum_{B\in\A}M^{\ell-n}(A,B)\mu^T_{\ell,B},$$ for every $A\in \A$ and $\ell> n\ge 0$. It follows that $\mu^T_0=M^n\mu^T_n$ for every $n>0$, hence this vector belongs to $core_\infty(M)$. Thus Lemmas \[help-lemma\], \[relation-between-eigenvalues\] and \[coreN\] imply that if $\mu^T_{0,A}>0$ for some $A\in \A'$, then the restriction of $\mu^T_0$ to every minimal component which has access to $A$ is infinity (because the component of $A$ is not distinguished). In the next lemma we show the same for the general case.]{} \[finite\] Let $\mu^T$ be a transverse measure on $\B(\Gamma)$ and let $1\leq p\leq m$. If there is $A\in \A_p$ for which there exist $D\in \A'$ and $n>0$ verifying $\mu^T(C_D)>0$ and $M^n(A,D)>0$, then $\mu^T(C_E)=\infty$, for every $E\in \A_p$. Let $$\Xnonp:= \{\T\in \Xa:\,\T\ \mbox{contains a tile of type}\ \Anonp\}.$$ This set is open and invariant under the $\R^d$ translation action. Using partial recognizability, we obtain, exactly as in the proof of Lemma \[partition\], the following statements: 1. If $A,B\in \A$, $n\geq 1$, $\bv\in J_A^{(n)}$ and $\bu\in J_B^{(n)}$ are such that $$(\omega^n(C_A)-\bv)\cap (\omega^n(C_B)-\bu)\cap \Xnonp\neq \emptyset,$$ then $A=B$ and $\bv=\bu$. 2. Let $\ell>n$, $A,B\in \A$, $\bv\in J_A^{(\ell)}$ and $\bu\in J_B^{(n)}$. We have $$(\omega^\ell(C_A)-\bv)\cap (\omega^n(C_B)-\bu)\cap \Xnonp \neq \emptyset,$$ if and only if $\omega^\ell(C_A)-\bv\subseteq \omega^n(C_B)-\bu$ and $B+\varphi^{-n}(\bv-\bu)\in \omega^{\ell-n}(A)$. In other words, by intersecting $\P_n$ with $\Xnonp$ we recover the nested partition properties even in the presence of periodic minimal components. Next we can argue as in Lemma \[coreN\] to deduce that the vector $$(\mu^T(\om^n(C_A) \cap \Xnonp))_{A\in \A}$$ belongs to the infinite core of $M$ for every $n\ge 0$. Let $D\in \A_i\seq \A'$ be such that $\mu(C_D) >0$. Then $D\in \Anonp$, so $$\mu^T(C_D\cap \Xnonp) = \mu^T(C_D) >0.$$ The assumption of the lemma implies that the class $\A_p$ has access to $\A_i$, and since $\A_p$ corresponds to a minimal component, we have $\rho_p> \rho_i$. Now it follows by Lemma \[relation-between-eigenvalues\] that $$\mu^T(C_E)\ge \mu^T(C_E\cap \Xnonp) = \infty\ \ \ \mbox{for all}\ E\in \A_p.$$ Constructing infinite transverse measures. ------------------------------------------ In Section \[finmeas\] we proved that finite invariant measures of the substitution tiling system $(X_{\A,\omega},\RR^d)=(X,\RR^d)$ are supported on its minimal components. Therefore, if $\mu$ is a finite invariant measure and $\mu^T$ is the associated transverse measure, then $\mu^T(C_D)=0$ for every prototile $D\in\A'$. In this section we characterize the infinite $\sig$-finite invariant measures $\mu$ for which there exists $D\in\A'$ such that $0<\mu^T(C_D)<\infty$. It follows from Lemmas \[coreN\] and \[finite\] that the values of $\mu^T$ on the elements of $\P_n$ belong to the infinite core of $M$ (at least, if we exclude periodic components). Lemma \[relation-between-eigenvalues\] suggests that those which correspond to ergodic measures should come from the vectors $\by_i$. We will show that this is indeed the case, under some mild assumptions. Recall that $\A_1,\cdots,\A_m$ are the equivalence classes of $M$ associated to the minimal components, and $\A'=\A\setminus\cup_{i=1}^m\A_i=\A_{m+1}\cup\cdots\cup\A_l$. Let $m+1\leq i\leq l$ be such that $M_i$ is primitive. Let $\A_{\J_i^c}$ be the set of prototiles $A\in\A$ such that if $A\in \A_j$ then $j\in \J_i^c$ (see Definition \[definition1\] for $\J_i$). Equivalently, $\AJC$ is the set of prototiles with indices in $\I_i$ and those which have no access to the class $\A_i$. By definition, $\A_i$ is a distinguished class for the restriction of $M$ to the indices in $\AJC$. A class $\A_j$, with $j\le m$, corresponding to a minimal component, is in $\AJC$ if and only if it has no access to $\A_i$. We define $$\Gamma_i=\bigcup_{n\geq 0}\bigcup_{A\in\AJC}\bigcup_{\ \ \bv\in J_A^{(n)}}(\omega^n(C_A)-\bv),$$ and $\F_i$, the collection of subsets of $\Gamma_i$ given by $$\F_i=\{\omega^n(C_A)-\bv: A\in\AJC,\ \bv\in J_A^{(n)}, n\geq 0\}.$$ Recalling the informal description of partition elements from Remark \[rem-inf\], we note that the tilings in $\Gam_i$ are those for which the “supertiles” of order $n$ containing the origin are of “type” $\AJC$ for $n$ sufficiently large. Observe that if this is the case for some $n_0$, then this is also true for $n > n_0$, since if a tile of type $\A_j$ occurs in $\om^n(B)$ for $B\in \A_k$, then $j$ has access to $k$, and so either $j=k$, or $k$ does not have access to $j$. Let $\by_i\in\overline{\RR}^{\A}_+$ be the vector given in Definition \[definition2\] for the class $\A_i$. For $n\geq 0$, let $\by_{n,i}\in\overline{\RR}^{\A}_+$ be such that $\by_i=M^n\by_{n,i}$ (that is, $\by_{n,i}=\by_{i}/\rho^n_i$, where $\rho_i$ is the Perron eigenvalue of $M_i$). We define the function $\phi_i: \F_i\cup\{\emptyset\}\to \RR_+$ by $\phi_i(\emptyset)=0$ and \[def-phi\] \_i(\^n(C\_A)-)= \_[n,i]{}(A), A n0. Since $\by_i|_{\J_i^c}<\infty$, this function is well-defined. A standard argument shows that the function $\phi_i^*: 2^{\Gam_i}\to \overline{\RR}_+$ given by $$\phi_i^*(U)=\inf\left\{\sum_{n\in\NN}\phi_i(C_n): (C_n)_{n\in\NN}\subseteq \F_i\cup\{\emptyset\}, \ U\subseteq \bigcup_{n\in\NN}C_n\right\}$$ is an outer measure. Observe that $\phi_i^*$ is well-defined because $\F_i$ is countable and the union of all the sets in $\F_i$ is equal to $\Gamma_i$. The collection $$\eta_i^*=\{U\subseteq \Gamma_i:\ \forall E\subseteq \Gamma_i,\ \phi_i^*(E)\geq \phi_i^*(E\cap U)+\phi_i^*(E\setminus U)\}$$ is a $\sig$-algebra and the restriction of $\phi_i^*$ to $\eta_i^*$ is a complete measure (every negligible set with respect to $\phi_i^*$ is in $\eta^*$). \[lem-meas1\] $\F_i \subseteq \eta_i^*$. Let $U=\omega^n(C_A)-\bv\in \F_i$, with $A\in \AJC$ and $\bv\in J_A^{(n)}$. We need to show \[eq-want1\] \_i\^\*(E)\_i\^\*(EU)+\_i\^\*(EU) for $E\seq \Gam_i$.\ Suppose first that $E=\omega^m(C_B)-\bw$ is also in $\F_i$. If $A$ (resp. $B$) is in a minimal component, then $\phi_i(U)=0$ (resp. $\phi_i(E)=0$). This immediately implies (\[eq-want1\]). Thus we can assume that $A$ and $B$ are in $\Anonp$. The inequality (\[eq-want1\]) is also clear if $U\cap E=\emptyset$ or $E\setminus U=\emptyset$. If $m\geq n$, then $U\cap E\neq \emptyset$ implies that $E\subseteq U$ by Lemma \[partition\](iii); then $E\setminus U=\emptyset$ and we are done. If $m<n$ and $U\cap E\neq \emptyset$, then $U\cap E=U$. In this case we have $$E\setminus U=\bigcup_{D\in \AJC}\bigcup_{\footnotesize{ \begin{array}{c} \bu\in J_D^{(n)} \\ \omega^n(C_D)-\bu\subseteq E \\ \bu\neq\bv \mbox{ if }D=A \end{array} }}(\omega^n(C_D)-\bu),$$ which implies that $$\begin{aligned} \phi_i^*(E\setminus U) & \leq & \sum_{D\in \AJC}\sum_{\footnotesize{ \begin{array}{c} \bu\in J_D^{(n)} \\ \omega^n(C_D)-\bu\subseteq E \\ \bu\neq\bv \mbox{ if }D=A \end{array} }}\phi_i^*(\omega^n(C_D)-\bu)\\ &=& \sum_{D\in \AJC}M^{n-m}(B,D)\by_{n,i}(D)-\phi_i^*(U).\end{aligned}$$ Therefore, $$\begin{aligned} \phi_i^*(E) = \by_{m,i}(B) & = & \sum_{D\in \AJC}M^{n-m}(B,D)\by_{n,i}(D) \\ & \ge & \phi_i^*(E\setminus U)+\phi_i^*(U) = \phi_i^*(E\setminus U)+\phi_i^*(U\cap E),\end{aligned}$$ as desired. If $E$ is any subset of $\Gamma_i$, then we have two cases: \(a) if $\phi_i^*(E)=\infty$, then (\[eq-want1\]) is clear. \(b) If $\phi_i^*(E)<\infty$, then given $\varepsilon>0$, there exists $(U_n)_{n\in\NN}\subseteq \F_i$ such that $$\sum_{n\in\NN} \phi_i^*(U_n)\leq \phi^*_i(E)+\varepsilon.$$ Using Step 1 and the fact that $\phi^*_i$ is an outer measure, we get $$\begin{aligned} \phi_i^*(E\cap U)+\phi^*_i(E\setminus U)& \leq & \phi^*_i(\bigcup_{n\in\NN} U_n\cap U)+\phi^*_i(\bigcup_{n\in\NN} U_n \setminus U)\\ & \leq & \sum_{n\in\NN}\left( \phi^*_i(U_n\cap U)+ \phi^*_i(U_n\setminus U)\right)\\ & \leq & \sum_{n\in\NN}\phi^*_i(U_n)\\ & \leq & \phi_i^*(E)+\varepsilon,\end{aligned}$$ concluding the proof. In the sequel we will prove that under some conditions, the Borel sets of $\Gamma_i$ (with respect to the induced topology) are contained in $\eta^*$. We define $$Y_i=\{\T\in \Xa: \T \mbox{ has a tile equivalent to some } A\in \A_i\}$$ and $$\wt{\Gam}_i = \bigcup_{m\ge 0}\bigcap_{n\geq m}\bigcup_{A\in\A_i}\,\bigcup_{\bv\in J_A^{(n)}}(\omega^n(C_A)-\bv)).$$ Recall (Section 2.4) that in the special case when $\A_i$ is a maximal irreducible component of the graph $G(M^T)$, the set $Y_i$ is a maximal component of the tiling space; here we consider the same kind of set for an arbitrary non-minimal component. \[supported-on-i\] We have [*(i)*]{} $\phi_i^*(\Gamma_i\setminus \wt{\Gam}_i) = 0$; [*(ii)*]{} $\phi_i^*(\Gam_i \setminus Y_i) = 0$. Note that (ii) follows from (i). Indeed, $\T\in \wt{\Gam}_i$ if and only if eventually all “supertiles” containing the origin (see Remark \[rem-inf\]) have a type from $\A_i$. Since we assumed that $M_i \ne [0]$, any substitution of a tile in $\A_i$ must contain a tile in $\A_i$, so $\wt{\Gam}_i \seq Y_i$.\ It remains to verify (i). For every $j\in \J_i^c$ and $m\geq 0$, we define $$\Gamma_{i,j,m}=\bigcap_{n\geq m}\bigcup_{A\in\A_j}\bigcup_{\bv\in J_A^{(n)}}(\omega^n(C_A)-\bv) \hspace{3mm} \mbox{ and }\hspace{3mm} \Gamma_{i,j}=\bigcup_{m\geq 0}\Gamma_{i,j,m}.$$ We have $$\label{eq-new2} \Gamma_i\setminus \wt{\Gam}_i = \bigcup_{j\in \J_i^c\setminus \{i\}}\Gamma_{i,j}.$$ Indeed, $\Gamma_{i,j}$ is the set of tilings for which the supertiles containing the origin are eventually of type from $\A_j$. Since we assumed that all non-zero components of $M$ are primitive, the types of the supertiles containing the origin must stabilize into types from one of the components, hence the claim (\[eq-new2\]). Let $j\in \J_i^c$, $j\ne i$. If $j$ has no access to $i$, then the definition of $\phi_i^*$ implies that $\phi_i^*(\Gamma_{i,j})=0$. If $M_j=[0]$ then $\Gamma_{i,j}=\emptyset$. Thus we can assume that $j\in \I_i$ and $M_j$ is primitive. Let $A\in \A_j$, $m\geq 0$ and $\bv\in J_A^{(m)}$. For every $n\geq m$ we have $$\Gamma_{i,j,m}\cap(\omega^m(C_A)-\bv)\subseteq \bigcup_{B\in\A_j}\bigcup_{\footnotesize{\begin{array}{c} \bw\in J_B^{(n)} \\ \omega^n(C_B)-\bw\subseteq \omega^m(C_A)-\bv \\ \end{array}}}\!\!\!\!(\omega^n(C_B)-\bw),$$ which implies that $$\begin{aligned} \phi_i^*(\Gamma_{i,j,m}\cap(\omega^m(C_A)-\bv)) & \leq & \sum_{B\in\A_j}M^{n-m}(A,B)\frac{\by_i(B)}{\rho_i^n}\\ &= &\rho_j^{-m}\sum_{B\in\A_j}\frac{M_j^{n-m}(A,B)}{\rho_j^{n-m}}\left(\frac{\rho_j}{\rho_i}\right)^n\by_i(B)\end{aligned}$$ Since $\lim_{n\to \infty}(M_j^{n-m}(A,B)/\rho_j^{n-m})$ exists and is finite, and since $\rho_j<\rho_i$, we get $$\phi_i^*(\Gamma_{i,j,m}\cap(\omega^m(C_A)-\bv))=0.$$ This implies that $\phi_i^*(\Gamma_{i,j,m})=0$, and then $\phi_i^*(\Gamma_i\setminus \wt{\Gamma}_i)=0$. For $n\geq 0$ and $A\in \A$ we define $$I_{n,r,A}=\bigcup_{B\in \A}\{\bv\in J^{(n)}_{A,B}: (\supp(B)+\bv)\cap (\partial D_{n,A})^{+r}\neq \emptyset \}.$$ In other words, $I_{n,r,A}$ is the set of $\bv$ such that $B+\bv$ occurs in $\om^n(A)$ for some $B$, within distance $r$ from the boundary. It is straightforward to show that the tilings $\T\in \Gamma_i$ for which there exists $\bv\in\RR^d$ such that $\T-\bv\in \Gamma\setminus \Gamma_i$ are in \[def-C\] C=\_[r]{}\_[k]{}\_[nk]{}\_[A]{}\_[I\_[n,r,A]{}]{}(\^n(C\_A)-). The set $C$ is a “bad set” for us: it is the set of tilings in $\Gam_i$ which “belong to the border” in the following sense: the union of supertiles containing the origin, discussed in Remark \[rem-inf\], is not the entire space $\R^d$. Observe that \[eq-goodint\] \_i C   \_[n]{} (\^n(C\_[A\_n]{}) - \_n) = {}, where $\omega^n(C_{A_n})-\bv_n\in\P_n$ is the set containing $\T$, for every $n\in\NN$. Note also that $C$ is translation-invariant. The next lemma gives a sufficient condition for $C$ to be negligible with respect to $\phi_i^*$. \[hip\] If there exist $D\in \A_i$ and $n>0$ such that a translate of $D$ appears in the interior of $\omega^n(D)$, then $C$ is negligible with respect to $\phi_i^*$. In view of Lemma \[supported-on-i\], it is enough to show that for every $m\ge 0$, $$C_{i,m}:=C\cap \bigcap_{n\geq m}\bigcup_{A\in\A_i}\bigcup_{\bv\in J_A^{(n)}}(\omega^n(C_A)-\bv)$$ is negligible with respect to $\phi_i^*$. It is clear that $$C_{i,m} = \bigcup_{r\in \NN} \bigcap_{n\ge m} \bigcup_{A\in \A_i} \bigcup_{\bv\in I_{n,r,A}} (\omega^n(C_A) - \bv).$$ We have $C_{i,m} = \bigcup_{r\in \NN} \bigcap_{n\ge m} C_{i,n,r}$ where $$C_{i,n,r} = \bigcup_{A\in \A_i} \bigcup_{\bv \in I_{n,r,A}} (\om^n(C_A) - \bv).$$ Fix $r\in \NN$. It is enough to show that $\phi^*(\bigcap_{n\ge m} C_{i,n,r}) = 0$ for $m$ sufficiently large, and we will do this by estimating $\phi^*(C_{i,n,r} \cap C_{i,m,r})$ for $n > m$. For $A \in \A_i$ consider the decomposition of $\om^n(A)$ into supertiles of order $m$, which is the inflated decomposition of $\om^{n-m}(A)$ into tiles. By the “border” of $\om^{n-m}(A)$ we mean the patch of tiles whose supports intersect the boundary of $\supp(\om^{n-m}(A))$. Applying $\om^m$ to the border increases its width, hence we can choose $m$ sufficiently large, so that any tile in $\om^n(A)$ within distance $r$ from $\partial D_{n,A} = \partial(\supp(\om^n(A)))$ belongs to a supertile $\om^m(B)$ in the inflated border. Then we have $$\phi_i^*(C_{i,n,r} \cap C_{i,m,r}) \le \sum_{A\in \A_i} \sum_{B\in \A_i} b_{n-m,B,A} |I_{m,r,B}| \frac{\by_i(A)}{\rho_i^n}\,,$$ where $b_{k,B,A}$ is the number of different translates of $B$ which appear in the border of $\om^{k}(A)$. On the other hand, the hypothesis implies that there exists $n_0>0$ such that for every $A,B\in \A_i$ there exists a translate of $B$ in the interior of $\om^{n_0}(A)$. Thus, there exists $0\leq \delta<1$ such that $b_{n_0,B,A}\leq \delta M^{n_0}(B,A)$ for every $A,B\in \A_{i}$. Inductively, we deduce $b_{kn_0,B,A}\leq \delta^k M^{kn_0}(B,A)$ for every $k>0$ and for every $A,B\in \A_{i}$. Hence we get $$\begin{aligned} \phi_i^*(C_{i,m+nn_0,r} \cap C_{i,m,r}) & \leq & \delta^n \sum_{A\in \A_{i}}\sum_{B\in\A_{i}}M^{nn_0}(B,A)|I_{m,r,B}|\frac{\by_i(A)}{\rho_i^{m+nn_0}}\\ & = & \delta^n\sum_{B\in\A_{i}}|I_{m,r,B}| \frac{\by_i(B)}{\rho_i^m},\end{aligned}$$ which implies that $\lim_{n\to\infty}\phi_i^*(C_{i,m+nn_0,r} \cap C_{i,m,r})=0$ and then $\phi_i^*(C)=0$. In the sequel we will suppose that the hypothesis of Lemma \[hip\] holds. That is, we will assume that $$\label{hyp-hip} \exists \,A\in \A_i, \, \exists\, n>0 \mbox{ such that a translate of } A \mbox{ appears in the interior of } \omega^n(A).$$ [If $\A_i$ is one of the maximal components, that is, $A\in \A_i$ does not appear in the substitution of any prototile from another component, then (\[hyp-hip\]) holds automatically, because the admissibility assumption implies that $A$ must appear in the interior of $\om^n(E)$ for some tile $E$, which can only be from $\A_i$. Lemma \[hip\] implies that in this case, the set $C$ is always negligible with respect to $\phi_i^*$.]{} \[medida0\] Suppose that $\omega$ verifies (\[hyp-hip\]). Then for every non-empty open set $U\subseteq \Gamma_i$ there exists a countable collection $(C_n)_{n\in\NN}\subseteq \F_i$ of disjoint sets such that $\bigcup_{n\in\NN}C_n\subseteq U$ and $\phi_i^*(U\setminus \bigcup_{n\in\NN}C_n)=0$. Thus $\B(\Gamma_i)\subseteq \eta_i^*$. Let $U_1\subseteq U$ the set of all $\T\in U$ for which there exist $n_{\T}\in\NN$, $A_{\T}\in \A_{\J^c_i}$ and $\bv_{\T}\in J_{A_{\T}}^{(n_{\T})}$ such that $$\T\in \omega^{n_{\T}}(C_{A_{\T}})-\bv_{\T}\subseteq U.$$ Note that $U\setminus U_1\seq C$ by (\[eq-goodint\]). We have $$U_1\subseteq \bigcup_{\T\in U_1}\omega^{n_{\T}}(C_{A_{\T}})-\bv_{\T}\subseteq U,$$ and since the collection $\F_i$ is countable, there exists a sequence $(\T_n)_{n\in\NN}\subseteq U_1$ such that $$U_1\subseteq\bigcup_{\T\in U_1}\left(\omega^{n_{\T}}(C_{A_{\T}})-\bv_{\T}\right)=\bigcup_{k\in\NN}\left(\omega^{n_{\T_k}}(C_{A_{\T_k}})-\bv_{\T_k}\right)\subseteq U.$$ Moreover, thanks to Lemma \[partition\], the sets $(\omega^{n_{\T_k}}(C_{A_{\T_k}})-\bv_{\T_k})_{k\in\NN}$ can be chosen disjoint. Since $$U\setminus \bigcup_{k\in\NN}\left(\omega^{n_{\T_k}}(C_{A_{\T_k}})-\bv_{\T_k}\right)\subseteq U\setminus U_1\subseteq C,$$ Lemma \[hip\] implies that $U$ is in $\eta_i^*$, because it is a countable union of sets in $\F_i$ up to a negligible set with respect to $\phi_i^*$. Now define $\mu^T_i: \B(\Gamma)\to \overline{\RR}_+$ by $$\mu^T_i(U)=\phi_i^*(U\cap \Gamma_i) \mbox{ for every } U\in \B(\Gamma).$$ \[medida1\] Suppose that $\omega$ verifies (\[hyp-hip\]). Then $\mu^T_i$ is a $\sigma$-finite transverse measure on $\B(\Gamma)$ supported on $Y_i\cap \Gam$. Furthermore, $$\mu^T_i(C_A)=\by_i(A) \mbox{ for every } A\in\A.$$ Lemma \[medida0\] ensures that the restriction of $\phi_i^*$ to $\B(\Gamma_i)$ is a measure. Then $\mu^T_i$ is a measure on $\B(\Gamma)$. It is $\sig$-finite because $\mu_i^T(\Gamma\setminus \Gamma_i)=0$ and $\Gamma_i$ is a countable union of sets of finite measure. Lemma \[supported-on-i\] implies that $\mu_i^T$ is supported on $Y_i\cap \Gam$. Next we prove that $\mu_i^T$ is transverse. Let $U\in \B(\Gamma)$ and $\bv\in\RR^d$ be such that $U-\bv\subseteq \Gamma$. First we suppose that $U=\omega^n(C_A)-\bu$, for some $A\in \AJC\cap \A'$, $\bu\in J_A^{(n)}$ and $n\geq 0$. If $\bu+\bv\in J_A^{(n)}$ then by definition of $\mu_i^T$ we have $\mu^T_i(U)=\mu^T_i(U-\bv)$. If not, for $m>n$ consider the sets $\omega^m(C_{A_{m,1}})-\bu_{m,1}, \cdots, \omega^m(C_{A_{m,k_m}})-\bu_{m,k_m}$ in $\F_i$ whose union is equal to $U$ (this corresponds to looking at $m$-level supertiles and finding translates of $\om^n(A)$ in them). This union is disjoint since $A\in \A'$. Let $J_m=\{1\leq i \leq k_m: u_{m,i}+\bv \in J_{A_{m,i}}^{(m)}\}$ and $U_m=\bigcup_{i\notin J_m}(\omega^m(C_{A_{m,i}})-\bu_{m,i})$. We have $$\begin{aligned} \mu_i^T(U)& = &\sum_{i\in J_m}\mu_i^T(\omega^m(C_{A_{m,i}})-\bu_{m,i}) +\mu_i^T(U_m),\\ \mu_i^T(U-\bv) & = & \sum_{i\in J_m}\mu_i^T(\omega^m(C_{A_{m,i}})-\bu_{m,i})+\mu_i^T(U_m-\bv),\end{aligned}$$ hence $$\mu_i^T(U)-\mu_i^T(U-\bv) = \mu_i^T(U_m) - \mu_i^T(U_m-\bv).$$ Note that $U_{m+1} \seq U_m$ and $\bigcap_{m} U_m \seq C,\ \ \bigcap_{m} (U_m -\bv)\seq C$, so Lemma \[hip\] implies $\mu^T_i(U-\bv)=\mu^T_i(U)$. If $A\in \AJC \setminus \A'$, then $A$ is in a minimal component which has no access to $\A_i$ and we have $\mu_i^T(U) = 0$. Then a similar argument yields $\mu_i^T(U - \bv) \le \mu_i^T(U_m-\bv)$, whence $\mu_i^T(U-\bv)=0$. Now we suppose that $U\subseteq \Gamma_i$ is an open set. Let $(C_n)_{n\in\NN}\subseteq \F_i$ be a disjoint collection of sets such that $\bigcup_{n\in\NN}C_n\subseteq U$ and $\mu^T_i(U)=\sum_{n\in\NN}\mu_i^T(C_n)$. This collection exists due to Lemma \[medida0\]. On the one hand, Step 1 implies that $\mu^T_i(U)=\sum_{n\in\NN}\mu^T_i(C_n-\bv)=\mu^T_i(\bigcup_{n\in\NN}(C_n-\bv))$. On the other hand, $(U-\bv)\setminus \bigcup_{n\in\NN}(C_n-\bv)\subseteq C$ which implies that $\mu^T_i(U-\bv)=\mu_i^T(U)$. Now let $U$ be any set in $\B(\Gamma)$. Since the elements in $\F_i$ are clopen sets in $\Gamma_i$, we have $$\begin{aligned} \phi_i^*(U\cap\Gamma_i)& = & \inf\left\{\sum_{n\in\NN}\mu_i^T(C_n): (C_n)_{n\in\NN}\subseteq \F_i\cup\emptyset, U\cap\Gamma_i\subseteq \bigcup_{n\in\NN}C_n\right\}\nonumber\\ &\geq & \inf\{\mu^T_i(V): U\cap\Gamma_i\subseteq V, \mbox{ and } V \mbox{ is open in } \Gamma_i\}. \label{eq-new3} $$ Let $V$ be an open set in $\Gamma_i$ which contains $U\cap \Gamma_i$. We have $$\mu_i^T((U\cap\Gamma_i)-\bv)\leq \mu_i^T(V-\bv)=\mu_i^T(V),$$ which implies, by (\[eq-new3\]), $$\mu_i^T((U\cap\Gamma_i)-\bv)\leq \phi_i^*(U\cap \Gamma_i) = \mu_i^T(U).$$ We claim that $\mu_i^T((U\cap\Gamma_i)-\bv)=\mu_i^T(U-\bv)$. Indeed, $$\mu_i^T(U-\bv) = \mu_i^T((U\cap \Gamma_i) - \bv) + \mu_i^T((U\cap \Gamma_i^c) - \bv),$$ but $$\mu_i^T((U\cap \Gamma_i^c) - \bv) = \mu_i^T(((U\cap \Gamma_i^c) - \bv)\cap \Gamma_i) = 0,$$ because $((U\cap \Gamma_i^c) - \bv)\cap \Gamma_i\seq C$. The claim is proved, and we obtain $\mu_i^T(U - \bv) \le \mu_i^T(U)$. Finally, replacing $U$ by $U-\bv$ and $\bv$ by $-\bv$ we get $\mu_i^T(U)=\mu_i^T(U-\bv)$. This concludes the proof that $\mu_i^T$ is a transverse measure. It remains to verify the formula. By definition, $\mu_i^T(C_A)=\by_i(A)$ for every $A\in\AJC$. If $A\in \J_i$ then Lemma \[coreN\], Lemma \[help-lemma\] and Lemma \[finite\] imply that $\mu_i^T(C_A)=\infty$, which is equal to $\by_i(A)$. \[conditions-zero-measure\] Let $\mu$ be an invariant $\sigma$-finite measure of the tiling system $(X,\RR^d)$, and let $U\in \B(X)$ be an invariant set such that $\mu(U)=0$. Then $\mu^T(U\cap \Gamma)=0$. Since $U$ is invariant, for every patch $P$ we have $U\cap (C_P+B_{\varepsilon}(\b0))=(U\cap C_P)+B_{\varepsilon}(\b0)$. Thus if $P$ is centered at $\b0$ and $\varepsilon$ is sufficiently small, we get $$0=\mu(U\cap (C_P+B_{\varepsilon}(\b0)))=\mu^T(U\cap C_P)\vol(B_{\varepsilon}(\b0))$$ by Lemmas \[invariant-transversal\] and \[transversal-0\]. Since $\Gamma=\bigcup_{A\in\A}C_A$, we deduce $\mu^T(U\cap \Gamma)=0$. \[ergodic\] Let $\mu$ and $\nu$ be two $\sig$-finite ergodic invariant measures for the tiling system $(X,\RR^d)$ for which there exists $A\in \A$ such that $0<\mu^T(C_A)=\nu^T(C_A)<\infty$ and $\mu^T|_{\B(C_A)}=\nu^T|_{\B(C_A)}$. Then $\mu=\nu$. Let $U$ be the subset of $X$ of all the tilings $\T$ containing a tile equivalent to $A$. This set is open and invariant, and contains the set $C_A+B_{\varepsilon}(\b0)$, for every $\varepsilon>0$. Thus, for $\varepsilon$ sufficiently small we get $\mu(U),\nu(U)\geq \mu^T(C_A)\vol(B_{\varepsilon}(\b0))>0$. This implies, by the ergodicity of $\mu$ and $\nu$, that $\mu(U^c)=\nu(U^c)=0$. Since $U^c$ is invariant, Lemma \[conditions-zero-measure\] implies that $\mu^T(U^c\cap\Gamma)=\nu^T(U^c\cap\Gamma)=0$. Then for every $V\in \B(\Gamma)$, $\mu^T(V)=\mu^T(V\cap U)$ and $\nu^T(V)=\nu^T(V\cap U)$. Let $\Lambda=\{\bv\in\RR^d: (\Gamma-\bv)\cap \Gamma\neq \emptyset\}$. This set is countable because $X$ satisfies the FPC; let $\Lambda=\{\bv_n: n\geq 0\}$. For $V\in\B(\Gamma)$, we have $$U\cap V=\bigcup_{n\geq 0}(C_A+\bv_n)\cap V=\bigcup_{n\geq 0} V_n,$$ where $V_0=V\cap(C_A+\bv_0)$ and $V_n=(V\cap(C_A+\bv_n))\setminus(V_0\cup\cdots\cup V_{n-1})$, for $n>0$. Then $$\mu^T(V)=\mu^T(V\cap U)=\sum_{n\geq 0}\mu^T(V_n) \hspace{2mm} \mbox{ and } \hspace{2mm} \nu^T(V)=\nu^T(V\cap U)=\sum_{n\geq 0}\nu^T(V_n).$$ Since for every $n\geq 0$ we have $V_n-\bv_n\subseteq C_A$, we get $$\mu^T(V_n)=\mu^T(V_n-\bv_n)=\nu^T(V_n-\bv_n)=\nu^T(V_n),$$ which implies that $\mu^T(V)=\nu^T(V)$. Theorem \[correspondence\] implies $\mu=\nu$. \[final-theorem-infinite\] Let $\om$ be an admissible tile substitution, which is partially recognizable and satisfies (\[eq-added\]) and (\[eq-irre\]). 1. Let $m+1\leq i\leq l$ be such that $M_i$ is primitive, and suppose that (\[hyp-hip\]) holds for $\A_i$. Then the measure $\mu^T_i$ is the unique transverse measure on $\B(\Gamma)$ supported on $Y_i\cap \Gamma$ such that $$\mu^T_i(C_A)=\by_i(A) \mbox{ for every } A\in \A.$$ Moreover, the associated invariant measure $\mu_i$ is $\sig$-finite and ergodic. 2. Suppose that (\[hyp-hip\]) holds for all $\A_i \seq \A'$. Then any $\sig$-finite ergodic measure $\mu$ with the property that $0<\mu^T(C_A)<\infty$ for some $A\in\A'$, is equal to some measure $\mu_i$ up to scaling. 3. If $Y_i$ is a maximal component, then any $\sig$-finite ergodic measure on $Y_i$ which is positive and finite on some open set, is equal to $\mu_i$ up to scaling. (Note that (\[hyp-hip\]) holds for maximal components by admissibility.) \(i) The uniqueness of $\mu_i^T$ follows from the fact that if $\nu^T$ is another transverse measure satisfying $\nu^T|_{\F_i}=\mu_i^T|_{\F_i}$ then $\nu^T|_{\B(\Gamma_i)}=\mu^T_i|_{\B(\Gamma_i)}$. Let $\mu_i$ be the invariant measure of $(X_{\A,\omega},\RR^d)$ associated to $\mu_i^T$. It is $\sig$-finite and invariant by Theorem \[correspondence\]. It remains to verify that it is ergodic. Let $U\in \B(X_{\A,\omega})$ be an invariant set, that is, $U-\bv=U$ for every $\bv\in\RR^d$. Since the set $U$ is invariant the measure $\mu_{U} =\mu_i|_{U}$ is invariant. Let $\mu_{U}^T$ be the transverse measure on $\B(\Gamma)$ associated to $\mu_{U}$. By Lemma \[invariant-transversal\], we get $$\mu_{U}^T(C)=\mu_i^T(C\cap U) \mbox{ for every } C\in\B(\Gamma).$$ The measure $\mu_{U}^T$ verifies $\mu_{U}^T\leq \mu_i^{T}$, which ensures that $\mu_{U}^T$ is supported on $\Gamma_i$, and that $\mu_U(C_B)<\infty$ for every $B\in \J_i^c$. In particular, $\mu_U^T(C_B)=0$ for every $B\in (\J_i\cup \I_i)^c.$ [*Case 1:*]{} there exists $A$ in the class $\A_i$ such that $\mu_{U}^T(C_A)>0$. Then $\mu_{U}^T(C_B)>0$ for every $B$ in the class $\A_i$. Thus Lemma \[help-lemma\] implies that $\mu_U^T(C_B)=\infty$ for every $B\in \J_i$, and since $\mu_U^T(C_B)=0$ for every $B\in (\J_i\cup \I_i)^c,$ the vector $(\mu_U^T(C_B))_{B\in\I_i}$ is in the core of the restriction of $M$ to $\I_i$. This implies that $(\mu^T_{U}(C_B))_{B\in\A}=\alpha \by_i$, for some $0<\alpha\leq 1$. This vector determines $\mu_U^T(\omega^n(C_B)-\bv)$, for every $B\in\J_i^c$, $\bv\in J_B^{(n)}$ and $n\geq 0$. Then $\alpha\mu^T_i|_{\F_i}=\mu^T_{U}|_{\F_i}$, and since $\mu^T_{U}$ is supported on $\Gamma_i$, we get that $\mu^T_{U}=\alpha\mu^T_i$. This implies that $\mu^T_i(U^c\cap \Gamma)=0$, because of $\mu^T_U(U^c\cap \Gamma)=0$. This shows that $\alpha=1$ and then $\mu_U^T=\mu_i^T$. From Theorem \[correspondence\] we obtain that $\mu_U=\mu_i$, which implies that $\mu_i(U^c)=0$. [*Case 2:*]{} If $\mu^T_{U}(C_A)=0$ for every $A$ in the class $\A_i$ then $\mu_{U^c}^T(C_A)=\mu_i^{T}(C_A)$ for every $A$ in the class $\A_i$. As in the previous case, but replacing $U$ by $U^c$, we get that $\mu^T_{U^c}=\mu_i^T$ and $\mu_i(U)=0$. This shows that $\mu_i$ is ergodic. \(ii) Let $\mu$ be a $\sig$-finite ergodic measure such that $0<\mu^T(C_A)<\infty$ for some $A\in\A'$. Let $i=\max\{1\leq j\leq l: 0<\mu^T(C_A)<\infty, A\in\A_j\}.$ By Lemma \[coreN\], the vector $(\mu^T(C_A))_{A\in\A_i}$ is in $core(M_i)$. Then there exists $\lambda>0$ such that for every $A\in\A_i$, $\bv\in J_A^{(n)}$ and $n\geq 0$, $$\mu^T(\omega^n(C_A)+\bv)=\lambda\mu_i^T(\omega^n(C_A)+\bv).$$ A standard argument shows that this equation implies that $\mu_T$ and $\lambda\mu_i^T$ coincide on each Borel set contained in $\bigcup_{A\in\A_i}C_A$. From Lemma \[ergodic\] we obtain that $\mu=\lambda\mu_i$. \(iii) Let $\mu$ be an ergodic $\sig$-finite measure on a maximal component $Y_i$, such that $\mu(U)$ is positive and finite for some open set $U\seq Y_i$. Let $\mu^T$ be the corresponding transverse measure; then $\mu^T(U\cap \Gamma)$ is positive and finite. The topology on $Y_i\cap \Gamma$ is generated by the sets $C_P$ for $P \in \Lambda_X$ (see Section 2.7), with $P$ containing at least one tile from $\A_i$. Decomposing $C_P$ as a disjoint union, we can find $A\in \A_i$ and $n\in \NN$ such that $\mu^T(\om^n(C_A)-\bv)$ is positive and finite for some $\bv\in \R^d$ such that $\om^n(C_A)-\bv\in \Gamma$. Then Lemma \[coreN\] implies that $\mu^T(C_A)$ is positive and finite, and we can conclude by applying part (ii). [*Proof of Theorem B.*]{} This follows from Theorem \[final-theorem-infinite\]. We only need to note that every point in a maximal component $Y_i$ has a neighborhood of the form $(\om^n(C_A)-\bv)+B_\varepsilon(\b0)$, with $A\in \A_i$, which has positive and finite $\mu_i$ measure. Examples and concluding remarks =============================== \[nonperiodic-substitution\][*All the tiles have the unit square as its support and are distinguished only by the labels. Let $\Ak = \left\{\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\,, \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\,,\begin{tabular}{|c|} \hline 2 \\ \hline \end{tabular}\right\}$; $$\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 0 & 0 & 1 \\ \hline 0 & 0 & 1 \\ \hline 1 & 1 & 1 \\ \hline \end{tabular}\,, \ \ \ \ \ \ \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 1 & 1 & 0 \\ \hline 1 & 1 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular}\,, \ \ \ \ \ \ \begin{tabular}{|c|} \hline 2 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 1 & 1 & 1 \\ \hline 1 & 2 & 2 \\ \hline 1 & 2 & 2 \\ \hline \end{tabular}$$ The substitution matrix is $M = \left(\begin{array}{ccc} 4 & 5 & 0 \\ 5 & 4 & 5 \\ 0 & 0 & 4 \end{array}\right)$. The tiling dynamical system has one minimal and one maximal component. It is easy to check that it is non-periodic. The restriction of the substitution matrix to the minimal components is $ \left(\begin{array}{cc} 4 & 5 \\ 5 & 4 \\ \end{array}\right). $ Then the unique probability measure $\mu$ is given by $\mu^T(C_2)=0$ and $$\mu^T(\omega^n(C_0)-\bv)=\mu^T(\omega^n(C_1)-\bv)=\frac{1}{2\cdot 9^n}, \mbox{ for every } \bv\in J_0^ {(n)}=J_1^{(n)} \mbox{ and } n\geq 0.$$ This substitution satisfies (\[hyp-hip\]); then applying Theorem \[final-theorem-infinite\] we get that every $\sig$-finite ergodic measure $\mu$ such that $0<\mu^T(C_{2})<\infty$, is a constant multiple of the unique measure $\mu_2$ such that $\mu_2^T$ is supported on $\bigcup_{n\geq 0}\bigcup_{\bv\in J_2^{(n)}}(\omega^n(C_2)-\bv)$ and that verifies $$\mu_2^{T}(C_2)=1 \mbox{ and } \mu_2^T(C_0)=\mu_2^T(C_1)=\infty.$$ Since the restriction of the substitution matrix to $\A'$ is equal to $[4]$, we get $$4^{-n}\mu_2^T(C_2)=\mu_2^T(\omega^n(C_2)-\bv),$$ for every $\bv\in J_2^{(n)}$ and $n\geq 0$.*]{} \[Cantor-substitution\][*All the tiles have the unit square as its support and are distinguished only by the labels. Let $\Ak = \left\{\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\,, \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\right\}$; $$\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular}\,, \ \ \ \ \ \ \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 1 & 1 & 1 \\ \hline 1 & 0 & 1 \\ \hline 1 & 1 & 1 \\ \hline \end{tabular}$$ The substitution matrix is $M = \left(\begin{array}{cc} 9 & 1 \\ 0 & 8 \end{array} \right)$. Here the minimal component is periodic; it consists of periodic tilings with only one tile type, labeled $0$. However, the “non-periodic border” condition holds (see Section 4 for details). This example, which we call the “integer Sierpiński carpet” tiling, is a generalization of the 1-dimensional symbolic substitution $0\to 000,\ 1\to 101$, which was analyzed by A. Fisher [@Fi1]. The intersection of the transversal with the unique minimal component contains only one element $\{\T_{0}\}$. Then the transverse measure associated to the unique invariant probability measure $\mu_0$ supported on the minimal component is the atomic measure $\mu_0^T(\{\T_0\})=1$. The measure $\mu_0$ corresponds to the Lebesgue measure on the torus (the minimal component is conjugate to the $\RR^2$-translations on the torus). This substitution satisfies (\[hyp-hip\]), then applying Theorem \[final-theorem-infinite\] we get that every $\sig$-finite ergodic measure $\mu$ such that $0<\mu^T(C_{1})<\infty$, is a constant multiple of the unique measure $\mu_1$ such that $\mu_1^T$ is supported on $\bigcup_{n\geq 0}\bigcup_{\bv\in J_1^{(n)}}(\omega^n(C_1)-\bv)$ and verifies $$\mu_1^{T}(C_1)=1 \mbox{ and } \mu_1^T(C_0)=\infty.$$ Since the restriction of the substitution matrix to $\A'$ is equal to $[8]$, we get $$8^{-n}\mu_1^T(C_1)=\mu_1^T(\omega^n(C_1)-\bv),$$ for every $\bv\in J_1^{(n)}$ and $n\geq 0$.*]{} *Consider $\A=\{\vartriangle, \triangledown, \blacktriangle\}$ and the substitution given below. $$\blacktriangle\ \to\ \begin{array}{ccc} & \!\!\!\!\!\!\!\!\blacktriangle & \\[-9pt] \blacktriangle & \!\!\!\!\!\!\!\!\triangledown & \!\!\!\!\!\!\!\!\!\!\!\!\blacktriangle \end{array},\ \ \ \vartriangle\ \to\ \begin{array}{ccc} & \!\!\!\!\!\!\!\!\vartriangle & \\[-9pt] \vartriangle & \!\!\!\!\!\!\!\!\triangledown & \!\!\!\!\!\!\!\!\!\!\!\!\vartriangle \end{array},\ \ \ \triangledown\ \to \begin{array}{ccc} \triangledown & \!\!\!\!\!\!\!\!\vartriangle & \!\!\!\!\!\!\!\!\!\!\!\!\triangledown \\[-9pt] & \!\!\!\!\!\!\!\!\triangledown & \end{array}$$ The substitution matrix of $\omega$ is $ M=\left(\begin{array}{ccc} 3 & 1 & 0 \\ 1 & 3 & 1\\ 0 & 0 & 3 \end{array}\right). $ The system $X_{\A,\omega}$ has a unique minimal component, and the submatrix of $M$ associated to the unique minimal component is $\left(\begin{array}{cc} 3 & 1 \\ 1 & 3 \\ \end{array}\right).$ This implies that the unique probability transversal measure is given by $$\mu^T(\omega^n(C_{\vartriangle})-\bv)=2^{-2n-1} \mbox{ and } \mu^T(\omega^n(C_{\triangledown})-\bu)=2^{-2n-1},$$ where $\bv\in J_{\vartriangle}^{(n)}$, $\bu\in J_{\triangledown}^{(n)}$ and $n\geq 0$.* The tile substitution satisfies the non-periodic border condition, so it is partially recognizable (this is also easy to verify directly) and satisfies (\[hyp-hip\]). By Theorem B, there is a unique, up to scaling, ergodic $\sig$-finite measure $\mu_\blacktriangle$, for which every point containing a tile $\blacktriangle$ has a neighborhood with positive and finite measure. For this measure we have $$3^{-n}\mu_\blacktriangle^T(C_{\blacktriangle})=\mu_\blacktriangle^T(\omega^n(C_{\blacktriangle})-\bv),$$ for every $\bv\in J_{\blacktriangle}^{(n)}$ and $n\geq 0$, and $\mu_\blacktriangle^T(C_{\vartriangle})=\mu_\blacktriangle^T(C_{\triangledown})=\infty$. \[ex-saf\] [*Let $\Ak = \left\{\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}, \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\right\}$; $$\begin{tabular}{|c|} \hline 0 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ \hline \end{tabular}\, ,\ \ \ \ \ \ \begin{tabular}{|c|} \hline 1 \\ \hline \end{tabular}\ \to\ \begin{tabular}{|c|c|c|} \hline 0 & 1 & 0 \\ \hline 1 & 0 & 1 \\ \hline \end{tabular}$$ This substitution is [*self-affine*]{}, rather than self-similar, and generates the integer analog of the “Bedford-McMullen carpet”, see [@Bedf; @McM]. Note that here the non-periodic border condition does not hold; however, partial recognizability (i.e. every tiling containing a prototile labeled by 1 has a unique pre-image under the substitution) is easy to verify directly. Thus Theorem \[final-theorem-infinite\] applies, and we get a conclusion similar to the above examples.* ]{} Concluding remarks. ------------------- 1. One can draw an (admittedly vague) analogy between substitution tiling flows and horocycle flows on manifolds of negative curvature. Moreover, the dynamics of the substitution $\omega$ is analogous to the geodesic flow. The compact manifold case corresponds to the case of minimal/primitive substitution systems, for which the horocycle flow, respectively, the tiling flow, is uniquely ergodic. The non-primitive substitutions then loosely correspond to the non-compact (but, perhaps, geometrically finite) case, where often the only “natural” invariant measure is $\sig$-finite, see e.g. [@Burger]. 2. A. Fisher [@Fi1] obtained a “second-order ergodic theorem” for his “integer Cantor set” substitution system. Is it possible to obtain similar results for our systems? We believe that it is, at least for examples such as the “Sierpiński gasket and carpet” tilings. What about more general non-primitive substitutions? One should probably stick to the self-similar case, or else use averaging over the F[ø]{}lner sets $\varphi^n (B_1(\b0))$. In general, there will be a [*graph-directed Iterated Function System*]{} associated to the tiling system. Objects analogous to the tilings, such as the “Sierpiński gasket and carpet” tilings, were considered by Strichartz, in the framework of [*Reverse Iterated Function Systems*]{} [@Str]. 3. All our examples have tiles of very simple geometry, but there exist tile substitutions with very complicated tiles: e.g. tiles with a fractal boundary, disconnected tiles, connected tiles with disconnected interior, etc. This is known for primitive substitution tilings, see e.g. [@Vince], and it is easy to construct a non-primitive substitution tiling using the same shapes as for a primitive one. For example, suppose we have a primitive substitution tiling $\om: \A\to \A$, with $\om(A) = \bigcup_{B\in \A} (B + D_B)$ where $D_B$ is a finite set for every prototile $B$. Let $\wt{\A} = \A \times \{0,1\}$, in other words, we consider “old” prototiles with an additional label. We assume that $\supp(A,j) = \supp(A)$ for $j=1,2$. Consider a non-trivial partition $\A = \A_1 \cup \A_2$ and define the substitution $\wt{\omega}:\,\wt{\A} \to \wt{\A}$ by $$\wt{\om}(A,0) = \bigcup_{B\in \A} ((B,0) + D_B),\ \ \ \wt{\om}(A,1) = \bigcup_{B\in \A_1} ((B,0) + D_B) \cup \bigcup_{B\in \A_2} ((B,1) + D_B).$$ This is a non-primitive tile substitution, and one can refine this construction to satisfy any additional properties, such as admissibility, non-periodic border, etc. Appendix: Invariant measures versus transverse measures. {#Apendix A} ======================================================== We use the notation and terminology from Section \[transversal\]. Recall that a [*transverse measure* ]{} on $\B(\Gamma)$ is a measure $\mu: \B(\Gamma)\to \overline{\RR}_+$ such that $\mu(A)=\mu(A-\bv)$, for every $A\subseteq \B(\Gamma)$ and $\bv\in\RR^d$ for which $A-\bv\subseteq \Gamma$ (see [@BBG Definition 5.1] for a definition of transverse measure in the context of laminations). Recall that $\eta>0$ is such that the closure of $B_\eta(\b0)$ is contained in the interior of every prototile. Observe that \[eq-inter\] , +, B\_[2]{}(0)  =0. We write $X:= \Xa$ to simplify the notation. From invariant measures to transverse measures. ----------------------------------------------- \[invariant-transversal\] Let $\mu$ be an invariant measure of $(X,\RR^d)$. For every $U\in\B(\Gamma)$, there exists $\mu^T(U)\in \overline{\RR}_+$ such that for every open set $\Theta$ contained in the ball $B_{\eta}(\b0)$, we have $$\frac{\mu(U+\Theta)}{\vol(\Theta)}=\mu^T(U).$$ Fix $U\seq \B(\Gam)$. Observe that $\mu_U:\, E\mapsto \mu(U+E)$ is a Borel measure on the ball $\Bl$. This follows from (\[eq-inter\]), which implies $$E_1, E_2 \seq \Bl,\ E_1\cap E_2 = \es\ \Rightarrow\ (U+E_1) \cap (U+E_2) = \es.$$ Moreover, by the invariance of $\mu$ we have $$E \seq \Bl,\ E-\bv \seq \Bl\ \Rightarrow \ \mu(E-\bv) = \mu(E).$$ It is easy to see that if for some open $\Th_1\seq \Bl$ we have $\mu(U+\Th_1)=0$, then $\mu(U+\Th)=0$ for all open subsets of $\Bl$ and we can set $\mu^T(U)=0$. Similarly, if for some open $\Th_1\seq \Bl$ we have $\mu(U+\Th_1)=\infty$, then $\mu(U+\Th)=\infty$ for all open subsets of $\Bl$ and we can set $\mu^T(U)=\infty$. So we can suppose that $\mu_U$ is positive and finite on open subsets of $\Bl$. Consider the restriction of $\mu_U$ to a cube $\prod_{i=1}^d [a_i, a_i+h)$ contained in $\Bl$ and extend it to $\R^d$ by periodicity, i.e. let $$\nu_U(E) = \sum_{\bx \in \ZZ^d} \mu\left(U + \left(E \cap \left(\prod_{i=1}^d [a_i, a_i+h) + h\bx\right)\right)\right).$$ This is a Borel measure on $\R^d$ which is translation-invariant and positive and finite on open subsets. It follows that $\nu_U(E) = c_U \vol(E)$ and we can set $\mu^T(U) = c_U$. \[transversal-0\] Let $\mu^T:\B(\Gamma)\to \overline{\RR}_+$ be the function obtained in Lemma \[invariant-transversal\]. Then $\mu^T$ is a transverse measure. It is clear that $\mu^T$ is a measure. Indeed, if $(U_n)_{n\in\NN}$ is a collection of disjoint sets in $\B(\Gamma)$, and $\varepsilon>0$ is small enough, then the sets $(U_n+B_{\varepsilon}(\b0))_{n\in\NN}$ are disjoint. It follows from the definition of $\mu^T$ that $\mu^T(\bigcup_{n\in\NN}U_n)=\sum_{n\in\NN}\mu^T(U_n)$. If $U\in\B(\Gamma)$ and $\bv\in\RR^d$ is such that $U-\bv\subseteq \Gamma$, then for $\varepsilon>0$ we have $\mu(U-\bv+B_{\varepsilon}(\b0))= \mu(U-B_{\varepsilon}(\b0))$, which implies that $\mu^T$ is transverse. Let $\mu$ be an invariant measure of $(X,\RR^d)$. We denote by $\mu^T$ the transverse measure associated to $\mu$. From transverse measures to invariant measures {#transversal-to-invariant} ---------------------------------------------- Let $\nu$ be a $\sigma$-finite transverse measure on $\Gamma$. We write $\lambda_d$ for the Lebesgue measure on $\RR^d$, and $\nu\otimes\lambda_d$ for the product measure on $\Gamma\times \RR^d$. For every $\bw, \bv\in \RR^d$, we define $\psi^{(\bw,\bv)}:X\times \RR^d\to X\times \RR^d$ by $$\psi^{(\bw,\bv)}(\T,\bu)=(\T-\bw,\bu-\bv), \mbox{ for every } \T\in X \mbox{ and } \bu\in\RR^d.$$ This function is a homeomorphism (with respect to the product topology). \[transverse-invariant-invariance\] Let $U$ be a Borel set in $\Gamma\times \RR^d$. Then $$\nu\otimes\lambda_d(\psi^{(\bw,\bv)}(U))=\nu\otimes\lambda_d(U),$$ for every $(\bw,\bv)\in\RR^{2d}$ such that $\psi^{(\bw,\bv)}(U)\subseteq \Gamma\times \RR^d$. For every $U$ in $X\times \RR^d$ and $\bx\in\RR^d$ we set $$(U)^{1}(\bx)=\{\T\in X: (\T,\bx)\in U\}.$$ Let $U$ be a Borel set in $\Gamma\times \RR^d$ and let $(\bw,\bv)\in\RR^d$ be such that $\psi^{(\bw,\bv)}(U)\subseteq \Gamma\times\RR^d$. We have $$\begin{aligned} \nu\otimes\lambda_d(\psi^{(\bw,\bv)}(U))& =& \int\nu((\psi^{(\bw,\bv)}(U))^{1}(\bx))\,d\lambda_d(\bx)\\ &=&\int\nu(U^1(\bx+\bv)-\bw)\,d\lambda_d(\bx)\\\end{aligned}$$ Since $\nu$ is transverse and $U^1(\by)-\bw\subseteq \Gamma$, for every $\by\in\RR^d$, we get $$\int\nu(U^1(\bx+\bv)-\bw)\,d\lambda_d(bx) = \int\nu(U^1(\bx+\bv))\,d\lambda_d(\bx).$$ The invariance under translations of the Lebesgue measure implies that $$\int\nu(U^1(\bx+\bv))\,d\lambda_d(\bx)=\int\nu(U^1(\bx))\,d\lambda(\bx)=\nu\otimes\lambda_d(U).$$ Since $X$ verifies FPC, $X$ is a finite union of sets $C_P+\Theta$, with $P\in\Lambda_X$ and $\Theta$ open in $\RR^d$ with diameter smaller than $\eta$. Namely, $X=\bigcup_{i=1}^nU_i$, where $U_i=C_{P_i}+\Theta_i$, for every $1\leq i\leq n$. From (\[eq-inter\]), for each $1\leq i\leq n$ the function $h_i:U_i\to C_i\times \Theta_i$ given by $h_i(\T+\bv)=(\T,\bv)$ is well-defined. Moreover, $h_i$ is a homeomorphism. For every $1\leq i\leq n$, let $\ba_i\in\RR^d$ be a vector such that $\b0\in \Theta_i-\ba_i$ and $\Theta_i-\ba_i$ is contained in the ball $B_{\eta}(\b0)$. Since for every $R>0$, the set $C_{P_i}$ is a finite and disjoint union of sets $C_P$, with $P\in\Lambda_X$ whose support contains the ball $B_R(\b0)$, we can assume that the support of $C_{P_i}$ contains the vectors $\ba_k-\ba_j$, for every $1\leq k,j\leq n$. This implies that for every $1\leq i,j\leq n$, there exist $\ba_{i,j}, \bB_{i,j}\in\RR^d$ such that $$\label{eq-transverse-invariant-1} h_j\circ h_i^{-1}(\T,\bv)=(\T+\ba_{i,j},\bv+\bB_{i,j}), \mbox{ for every } \T\in C_{P_i}, \bv\in\Theta_i.$$ Equation (\[eq-transverse-invariant-1\]) implies that $X$ has a [*$d$-lamination structure*]{} (see [@BBG] for details). The collection $\{U_i,h_i\}_{i=1}^n$ is called an [*atlas*]{} of $X$. For every $1\leq i\leq n$ define $\mu_i:\B(U_i)\to \overline{\RR}_+$ by $ \mu_i(U)=\nu\otimes\lambda_d(h_i(U))$. It is clear that $\mu_i$ is a measure. We define $\wt{U}_1=U_1$ and $\wt{U}_i=U_i\setminus(\bigcup_{j=1}^{i-1}U_j)$, for every $2\leq i\leq n$. The function $\mu=\sum_{i=1}^n\mu_i|_{\wt{U}_i}$ is a measure on $\B(X)$. Since $\nu\otimes\lambda_d$ is $\sig$-finite, $\mu$ is $\sig$-finite too. We will show that $\mu$ is invariant and that $\mu^T=\nu$. \[transverse-invariant-independence\] The measure $\mu$ does not depend on the atlas. Since $X$ verifies FPC, the set $\Lambda=\{\bv\in\RR^d: (\Gamma-\bv)\cap \Gamma\neq \emptyset\}$ is countable. Let $\Lambda=\{\bv_n: n\in\NN\}$. Let $\{V_i,f_i\}_{i=1}^m$ be another atlas of $X$. For every $1\leq i\leq m$, let $\wt{\mu}_i$ be the measure on $V_i$ defined as $\wt{\mu}_i=(\nu\otimes\lambda_d)\circ f_i$. We set $\wt{V}_1=V_1$ and $\wt{V}_i=V_i\setminus(\bigcup_{j=1}^{i-1}V_j)$, for every $2\leq i\leq m$. Denote by $\wt{\mu}$ the measure on $X$ defined by $\wt{\mu}=\sum_{i=1}^m\wt{\mu}_i|_{\wt{V}_i}$. Let $\T$ be a tiling in $U_i\cap V_j$, for some $1\leq i\leq n$ and $1\leq j\leq m$. If $h_i(\T)=(\T_i,\bv_i)$ then there exists $\bv(\T)\in\Lambda$ such that $f_j(\T)=(\T_i+\bv(\T), \bv_i-\bv(\T))$. Since $\bv(\T)$ is in $\Lambda$, there exists $n\in\NN$ such that $\bv(\T)=\bv_n$. Thus if $U$ is a Borel set in $U_i\cap V_j$, it can be written as $$U=\bigcup_{n\in\NN} U_n, \mbox{ where } U_n=\{\T\in U: \,f_j(\T)=\psi^{\bv_n}(h_i(\T))\},$$ where $\psi^{\bv_n}$ abbreviates $\psi^{(\bv_n,-\bv_n)}$. The sets $U_n$ are disjoint and measurable. From Lemma \[transverse-invariant-invariance\], for every $n\in\NN$ we have $$\begin{aligned} \mu_i(U_n)&=& \nu\otimes\lambda_d(h_i(U_n))\\ &=& \nu\otimes\lambda_d(\psi^{\bv_n}(h_i(U_n)))\\ &=&\nu\otimes\lambda_d(f_j(U_n))\\ &=&\wt{\mu}_j(U_n),\end{aligned}$$ which implies that $\mu_i(U)=\wt{\mu}_j(U)$, and hence $\mu=\wt{\mu}$. [From the proof of Lemma \[transverse-invariant-independence\] we also deduce that $\mu(U)=\mu_i(U)$, for every Borel set $U\subseteq U_i$. ]{} \[transverse-to-invariant\] The measure $\mu$ is invariant and $\mu^T=\nu$. Let $\bv\in\RR^d$. For every $1\leq i\leq n$, let $V_i=U_i-\bv$ and $f_i:V_i\to C_{P_i}\times(\Theta_i-\bv)$ be defined as $f_i(\T)=\psi^{(\b0,\bv)}(h_i(\T+\bv))$. The collection $\{V_i,f_i\}$ is an atlas of $X$. Let $U$ be a Borel set in $U_i$. We have $U-\bv\subseteq V_i$. Then from Lemma \[transverse-invariant-invariance\] and Lemma \[transverse-invariant-independence\] we get $$\begin{aligned} \mu(U-\bv) &=& \nu\otimes\lambda_d(f_i(U-\bv))\\ &=&\nu\otimes\lambda_d(\psi^{(\b0,\bv)}(h_i(U)))\\ &=&\nu\otimes\lambda_d(h_i(U)))\\ &=&\mu_i(U)\\ &=&\mu(U).\end{aligned}$$ This shows that $\mu$ is invariant. Let $C\in\B(\Gamma)$ and $\Theta\subseteq B_{\eta}(\b0)$ an open set. We can assume that $C+\Theta$ is the disjoint union of sets $C_i+\Theta$, where $C_i\subseteq C_{P_i}$ and $\Theta\subseteq \Theta_i$, for every $1\leq i\leq n$. We have $$\mu(C_i+\Theta)=\nu^T\otimes\lambda_d(C_i\times \Theta)=\nu(C_i)\lambda_d(\theta).$$ Hence $\mu(C+\Theta)=\nu(C)\lambda_d(\Theta)$, which implies that $\mu^T=\nu$. \[correspondence\] There is a linear one-to-one correspondence between the set of $\sigma$-finite invariant measures and the set of $\sigma$-finite transverse measures of $(X,\RR^d)$. Lemma \[transverse-to-invariant\] shows that the function that associates to every invariant measure $\mu$ its transverse measure $\mu^T$ is onto. Let $\nu$ be a transverse measure and let $\mu$ be an invariant measure such that $\mu^T=\nu$. Let $\{U_i,h_i\}_{i=1}^n$ be an atlas of $X$. The measure $\mu\circ h_i^{-1}$ is defined on $h_i(U_i)$ and verifies $$\mu\circ h_i^{-1}(C\times\Theta)=\nu\otimes\lambda_d(C\times \Theta),$$ for every $C\times \Theta\in (\B(\Gamma)\times \B(\RR^d))\cap h_i(U_i)$. The uniqueness of the product measure implies that $\mu$ is the invariant measure of Lemma \[transverse-to-invariant\]. Therefore, the function that associates to every invariant measure $\mu$ its transverse measure $\mu^T$ is one-to-one. The linearity is clear. [99]{} Ph.D. Thesis, University of Warwick (1984). Comm. Math. Phys. 261 (2006), no. 1, 1–41. Ergodic Theory Dynam. Systems 23(3) (2003), 673–691. Ergodic Theory Dynam. Systems 29(1) (2009), 37–72. . To appear. Duke Math. J. 61 (1990), no. 3, 779–803. . [*Sur une généralisation du théorème de Birkhoff*]{}, C.R. Acad. Sc. Paris, 275 (1972), Serie A, 1135–1138. Ergodic Theory Dynam. Systems 28(3) (2008), 739–747. Ergodic Theory Dynam. Systems 19(4) (1999), 953–993. . [*Ergodic Theory & Dynam. Syst.*]{}, 13(1) (1992), 45–64. . Chelsey, 1959. . Bull. Amer. Math. Soc. 50 (1944), 915–919. Preprint, March 2009. . [*Matrix analysis*]{}. Cambridge University Press, Cambridge, 1985. Commun. Math. Phys. 254 (2005), 343–359. , Cambridge University Press, Cambridge, 1995. Nagoya Math. J. 96 (1984), 1–9. Trans. Amer. Math. Soc. 351 (8) (1999), 3315–3349. , Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. Lecture Notes in Math. 1794, Springer-Verlag, Berlin, 2002. . Lecture Notes in Math. 1294, Springer-Verlag, Berlin, 1987. Symbolic dynamics and its applications, Proc. Sympos. Appl. Math., Vol. 60, Amer. Math. Soc., Providence, RI, 2004, 81–119. Measure and measurable dynamics (Rochester, NY, 1987), Contemp. Math., Vol. 94, Amer. Math. Soc., Providence, RI, 1989, 271–290. Proceedings of the symposium on operator theory (Athens, 1985). Linear Algebra Appl. 84 (1986), 161–189. Ergod. Th and Dynam. Sys. 17 (1997), 695–738. Discrete Comput. Geom. 20 (1998), 265–279. Canad. J. Math. 50 (1998), no. 3, 638–657. Trans. Amer. Math. Soc. 343 (1994), no. 2, 479–524. Trans. Amer. Math. Soc. 353 (2001), no. 1, 209–245. Soviet Math. Dokl. 8 (1967), no. 5, 1213–1216. SIAM J. Algebraic Discrete Methods 6 (1985), no. 3, 406–412. Discrete Comput. Geom. 21 (1999), 463–476. Duke Math. J. 5 (1939), no. 1, 1–18. . [*J. D’Analyse Math.*]{}, 102:143–180, 2007. [^1]: M. I. Cortez acknowledges financial support from proyecto Fondecyt 1100318. B. Solomyak was partially supported by NSF grant DMS-0654408.

--- abstract: | The inspiral of binary black holes is governed by gravitational radiation reaction at binary separations $r \lesssim 1000 M$, yet it is too computationally expensive to begin numerical-relativity simulations with initial separations $r \gtrsim 10 M$. Fortunately, binary evolution between these separations is well described by post-Newtonian equations of motion. We examine how this post-Newtonian evolution affects the distribution of spin orientations at separations $r \simeq 10 M$ where numerical-relativity simulations typically begin. Although isotropic spin distributions at $r \simeq 1000 M$ remain isotropic at $r \simeq 10 M$, distributions that are initially partially aligned with the orbital angular momentum can be significantly distorted during the post-Newtonian inspiral. Spin precession tends to align (anti-align) the binary black hole spins with each other if the spin of the more massive black hole is initially partially aligned (anti-aligned) with the orbital angular momentum, thus increasing (decreasing) the average final spin. Spin precession is stronger for comparable-mass binaries, and could produce significant spin alignment before merger for both supermassive and stellar-mass black hole binaries. We also point out that precession induces an intrinsic accuracy limitation ($\lesssim 0.03$ in the dimensionless spin magnitude, $\lesssim 20^\circ$ in the direction) in predicting the final spin resulting from the merger of widely separated binaries. author: - Michael Kesden - Ulrich Sperhake - Emanuele Berti bibliography: - 'Dec7.bib' date: February 2010 title: Final spins from the merger of precessing binary black holes --- Introduction {#S:intro} ============ The existence of black holes is a fundamental prediction of general relativity. Isolated individual black holes are stationary solutions to Einstein’s equations, but binary black holes (BBHs) can inspiral and eventually merge. BBH mergers offer a unique opportunity to test general relativity in the strong-field limit, and as such are a primary science target for current and future gravitational-wave (GW) observatories like LIGO, VIRGO, LISA, and the Einstein telescope. BBH mergers are also important for cosmology, as they can serve as standard candles to help determine the geometry and hence energy content of the universe [@Schutz:1986gp; @Holz:2005df]. Astrophysical BBHs are found on at least two very different mass scales. Compact objects believed to be stellar-mass black holes have been observed in binary systems with more luminous companions. These black holes are the remnants of massive main-sequence stars, and binary systems with two such stars may ultimately evolve into BBHs. On larger scales, supermassive black holes (SBHs) with masses $10^6 \lesssim M/M_\odot \lesssim 10^9$ reside in the centers of most galaxies. They can be observed through their dynamical influence on surrounding gas and stars, and when accreting as active galactic nuclei (AGN). SBHs will form binaries as well, following the merger of two galaxies which each host an SBH at their center. In order to merge, BBHs must find a way to shed their orbital angular momentum. At large separations, binary SBHs will be escorted inwards by dynamical friction between their host galaxies [@Begelman:1980vb]. The BBHs become gravitationally bound when the sum of their masses $M \equiv m_1 + m_2$ exceeds the mass of gas and stars enclosed by their orbit. The binary hardens further by scattering stars on “loss-cone” orbits that pass within a critical radius [@Frank:1976uy], though this scattering may stall at separations $r \simeq 0.01 - 1$ pc unless these orbits are refilled by stellar diffusion [@Milosavljevic:2001vi]. Unlike stars, gas can cool to form a circumbinary disk about the BBHs. A circumbinary disk of mass $M_d$ and radius $r_d$ will exert a tidal torque $$\label{E:Td} T_d \sim \frac{q^2 M_d M}{r} \left( \frac{r}{r_d - r} \right)^3$$ on the binary in the limit that the BBH mass ratio $q \equiv m_2/m_1 \leq 1$ is small and $|r_d - r| \ll r$ [@Lin:1979; @Goldreich:1980wa; @Chang:2008]. Throughout this paper we use relativists’ units in which Newton’s constant $G$ and the speed of light $c$ are unity. At a sufficiently small separation $r_{\rm GW}$, the magnitude of this tidal torque will fall below that of the radiation-reaction torque [@Peters:1964zz] $$\label{E:TGW} T_{\rm GW} = \frac{32 \eta^2 M^{9/2}}{5r^{7/2}}~,$$ where $\eta \equiv m_1m_2/M^2$ is the symmetric mass ratio. Once $T_{\rm GW} > T_d$, the inspiral of the BBH is dominated by radiation reaction. The precise value of $r_{\rm GW}$ depends on the properties of the circumbinary disk, but an order-of-magnitude estimate is given by [@Begelman:1980vb] \[E:rGW\] $$\begin{aligned} \label{E:rGWcgs} r_{\rm GW} &=& (5 \times 10^{16}~{\rm cm}) q^{1/4} M_{8}^{3/4} \left[ \frac{{\rm min}(t_h, t_{\rm gas})}{10^8~{\rm yr}} \right]^{1/4} \\ \label{E:rGWrel} &=& (3000 M) \left( \frac{q}{M_8} \right)^{1/4} \left[ \frac{{\rm min}(t_h, t_{\rm gas})}{10^8~{\rm yr}} \right]^{1/4} \end{aligned}$$ where $M_8$ is the mass of the larger black hole in units of $10^8 M_\odot$, $t_h$ is the dynamical friction timescale for a hard binary, and $t_{\rm gas}$ is the evolution timescale from gaseous tidal torques. General relativity completely determines the inspiral of BBH systems from separations less than $r_{\rm GW}$. These systems are fully specified by 7 parameters: the mass ratio $q$ and the 3 components of each dimensionless spin $\boldsymbol{\chi}_{1,2} \equiv {\bf S}_{1,2}/m_{1,2}^2$. To a good approximation the individual masses and spin magnitudes $\chi_{1,2} \equiv |\boldsymbol{\chi}_{1,2}|$ remain constant during the inspiral, so only the precession of the two spin directions needs to be calculated. At an initial separation $r_i = 1000 M \sim r_{\rm GW}$, the binary’s orbital speed $v/c \ll 1$ and the spin-precession equations can therefore be expanded in this small post-Newtonian (PN) parameter. The PN expansion remains valid until the BBHs reach a final separation $r_f = 10 M$, after which their evolution can only be described by fully nonlinear numerical relativity (for more precise assessments of the validity of the PN expansion for spinning precessing binaries, see e.g. [@Buonanno:2005xu; @Campanelli:2008nk]). Numerical relativists can simulate BBH mergers from separations $r_{\rm NR} \simeq r_f$ [@Pretorius:2005gq; @Campanelli:2005dd; @Baker:2005vv], but these simulations are too computationally expensive to begin when the binaries are much more widely separated. The GWs produced in the merger and the mass, spin, and recoil velocity of the final black hole depend sensitively on the orientation of the BBH spins at $r_{\rm NR}$, so it is important to determine what BBH spin orientations are expected at $r_i$ and whether these orientations are modified by the PN evolution between $r_i$ and $r_f$. The answer to the first of these questions comes from astrophysics, not general relativity. At very large separations, the two black holes are unaffected by each other and one would therefore expect an isotropic distribution of spin directions. However, an isotropic distribution of spins at $r_f$ would imply that most mergers would result in a gravitational recoil of $\sim 1000$ km/s for the final black hole [@Gonzalez:2007hi; @Campanelli:2007ew; @Dotti:2009vz]. Recoils this large would eject SBHs from all but the most massive host galaxies [@Merritt:2004xa], in seeming contradiction to the observed tight correlations between SBHs and their hosts [@Magorrian:1997hw; @Ferrarese:2000se; @Tremaine:2002js]. This problem can be avoided if Lense-Thirring precession and viscous torques align the spins of the BBHs with the accretion disk responsible for their inwards migration [@BP; @Bogdanovic:2007hp; @Berti:2008af]. The efficiency of this alignment depends on the properties of the accretion disk, but $N$-body simulations using smoothed-particle hydrodynamics (SPH) suggest that the residual misalignment of the BBH spins with their accretion disk at $r_i$ could typically be $\sim 10^\circ (30^\circ)$ for cold (hot) accretion disks [@Dotti:2009vz]. The second question, does the distribution of spin directions change as the BBHs inspiral from $r_i$ to $r_f$, can be answered by evolving this distribution over this interval using the PN spin-precession equations. We will describe these PN equations and our numerical solutions to them in Sec. \[S:PN\]. The precession of a given spin configuration in the PN regime can be understood in terms of the proximity of that configuration to the nearest spin-orbit resonance. Schnittman [@Schnittman:2004vq] identified a set of equilibrium spin configurations in which both black hole spins and the orbital angular momentum lie in a plane, along with the total angular momentum ${\bf J} = {\bf L} + m_{1}^2 {\boldsymbol{\chi}_1}+ m_{2}^2 {\boldsymbol{\chi}_2}$. In the absence of radiation reaction, ${\bf J}$ is conserved. For these equilibrium configurations, the spins and orbital angular momentum remain coplanar and precess jointly about ${\bf J}$ with the angles $\theta_{1,2}$ between ${\bf L}$ and $\boldsymbol{\chi}_{1,2}$ remaining fixed. The equilibrium configurations can thus be understood as spin-orbit resonances since the precession frequencies of ${\bf L}$ and $\boldsymbol{\chi}_{1,2}$ about ${\bf J}$ are all the same. Once radiation reaction is added, the spins and orbital angular momentum remain coplanar as the BBHs inspiral, although $\theta_1$ and $\theta_2$ slowly change on the inspiral timescale. Not only do resonant configurations remain resonant, but configurations near resonance can be captured into resonance during the inspiral. The resonances are thus very important for understanding the evolution of generic BBH systems, although the resonances themselves only occupy a small portion of the 7-dimensional parameter space characterizing generic mergers. We shall review these spin-orbit resonances in more detail in Sec. \[S:res\]. Bogdanović [*et al.*]{} [@Bogdanovic:2007hp] briefly considered whether spin-orbit resonances could effectively align SBH spins with the orbital angular momentum following the merger of gas-poor galaxies. They found that for a mass ratio $q = 9/11$ and maximal spins $\chi_1 = \chi_2 = 1$, an isotropic distribution of spins at $r_i = 1000 M$ remains isotropically distributed when evolved to $r_f = 10 M$. They therefore concluded that an alternative mechanism, such as the accretion torques considered later in their paper, is needed to align the BBH spins with ${\bf L}$. This conclusion is supported by a much larger set of PN inspirals presented by Herrmann [*et al.*]{} [@Herrmann:2009mr] who found that for equal-mass BBHs, an isotropic distribution of spins at $40 M$ yields a flat distribution in $\cos \theta_{12}$ at $7.4 M$. Here and in this paper $\theta_{12}$ is the angle between the two spins ${\boldsymbol{\chi}_1}$ and ${\boldsymbol{\chi}_2}$. In the final plot of their paper, Herrmann [*et al.*]{} [@Herrmann:2009mr] revealed their discovery of an anti-correlation between the initial and final values of $\cos \theta_{12}$ for $q = 2/3$ BBHs with equal dimensionless spins $\chi_1 = \chi_2 = 0.05$. Investigation of this anti-correlation was left to future work. Lousto [*et al.*]{} [@Lousto:2009ka] also found indications that an initially isotropic distribution of spins can become non-isotropic during the PN stage of the inspiral. For a range of mass ratios $1/16 \leq q \leq 1$ and equal spins $\chi_1 = \chi_2 = (0.485, 0.686, 0.97)$, they found that an isotropic spin distribution at $50 M$ develops a slight but statistically significant tendency towards anti-alignment with the orbital angular momentum ${\bf L}$. This amplitude of anti-alignment scales linearly in the BBH spin magnitudes and appears to decrease as $q \to 0$. We perform our own study of PN spin evolution from $r_i$ to $r_f$ for several reasons. BBHs get locked into spin-orbit resonances at a separation $$\label{E:rlock} r_{\rm lock} \propto \left( \frac{\chi_1 \cos \theta_1 - q^2 \chi_2 \cos \theta_2 }{1 - q^2} \right)^2 M~,$$ which can become large in the equal-mass $(q \to 1)$ limit [@Schnittman:2004vq]. This limit is important, as the largest recoil velocities occur for nearly equal-mass mergers. Numerical integration of the PN equations has shown that for a mass ratio $q = 9/11$, spin-orbit resonances affect spin orientations at separations $r \simeq 1000 M$. This is a much larger separation than was considered in previous studies [@Herrmann:2009mr; @Lousto:2009ka] of spin alignment, which may therefore have failed to capture the full magnitude of the effect. These studies also focused on whether an initially isotropic distribution of spins becomes anisotropic just prior to merger. However, as discussed above, tidal torques from a circumbinary disk partially align spins with the orbital angular momentum at separations $r \gg r_{\rm GW}$ before relativistic effects become important. As we will show in Sec. \[S:align\], such partially aligned distributions can be strongly affected by spin-orbit resonances despite the fact that isotropic distributions remain nearly isotropic. We will consider how spin precession affects the final spin magnitudes and directions in Sec. \[S:dist\]. The evolution of the distribution of BBH spin directions between $r_i$ and $r_f$ changes the distribution of final spin magnitudes and directions from what it would have been in the absence of precession. In addition, spin precession introduces a fundamental uncertainty in predicting the final spin of a given BBH system. At large separations, a small uncertainty in the separation leads to an uncertainty in the predicted time until merger that exceeds the precession time. In this case, one cannot predict at what phase of the spin precession the merger will occur and thus the resulting final spin. We will explore this uncertainty in Sec. \[S:err\]. A brief discussion of the chief findings of this paper is given in Sec. \[S:disc\]. Post-Newtonian Evolution {#S:PN} ======================== We evolve spinning BBH systems along a sequence of quasi-circular orbits according to the PN equations of motion for precessing binaries first derived by Kidder [@Kidder:1995zr], and later used by Buonanno, Chen and Vallisneri to build matched-filtering template families for GW detection [@Buonanno:2002fy]. The adiabatic evolution of the binary’s orbital frequency is described including terms up to 3.5PN order, and spin effects are included up to 2PN order. These evolution equations were chosen for consistency with previous work, in particular with the study by Barausse and Rezzolla [@Barausse:2009uz] of the final spin resulting from the coalescence of BBHs and with the statistical investigation of spinning BBH evolutions using Graphics Processing Units by Herrmann [*et al.*]{} [@Herrmann:2009mr]. Lousto [*et al.*]{} [@Lousto:2009ka] evolved a large sample of spinning BBH systems using a non-resummed, PN expanded Hamiltonian. The convergence properties of non-resummed Hamiltonians for spinning BBH systems are somewhat problematic (see e.g. Fig. 1 of Ref. [@Buonanno:2005xu]), and it will be interesting to repeat these statistical investigations of precessing BBH systems using the effective-one-body resummations of the PN Hamiltonian recently proposed by Barausse [*et al.*]{} [@Barausse:2009aa; @Barausse:2009xi]. In our simulations, the spins evolve according to \[E:SP\] $$\begin{aligned} \label{E:SP1} {\dot{\mathbf{S}}_1}&=& {\bar{\boldsymbol{\Omega}}_1}\times {\mathbf{S}_1}\, , \\ \label{E:SP2} {\dot{\mathbf{S}}_2}&=& {\bar{\boldsymbol{\Omega}}_2}\times {\mathbf{S}_2}\, , \end{aligned}$$ where \[E:oaP\] $$\begin{aligned} \lefteqn{{\bar{\boldsymbol{\Omega}}_1}= } \label{E:oa1}\\ && \frac{1}{2r^3}\left[ \left( 4 + 3q - \frac{3({\mathbf{S}_2}+q{\mathbf{S}_1}) \cdot {\mathbf{L}_N}}{L_{N}^2} \right) {\mathbf{L}_N}+ {\mathbf{S}_2}\right]\,,\nn \\ \lefteqn{{\bar{\boldsymbol{\Omega}}_2}= } \label{E:oa2}\\ && \frac{1}{2r^3}\left[ \left( 4 + \frac{3}{q} - \frac{3({\mathbf{S}_1}+q^{-1}{\mathbf{S}_2}) \cdot {\mathbf{L}_N}}{L_{N}^2} \right) {\mathbf{L}_N}+ {\mathbf{S}_1}\right]\nn \end{aligned}$$ are the spin precession frequencies averaged over a circular orbit, including the quadrupole-monopole interaction [@Racine:2008qv], $$\label{E:LNewt} {\mathbf{L}_N}= \eta M \mathbf{r} \times \mathbf{v} = \frac{\eta M^2}{(M \omega)^{1/3}} {\hat{\mathbf{L}}_{N}}$$ is the Newtonian orbital angular momentum, and = ( )\^[1/2]{} is the orbital frequency. In the absence of gravitational radiation, $\mathbf{J}$ and $|{\mathbf{L}_N}|$ are constant, implying that the direction of the orbital angular momentum evolves according to \[E:hLNdot\] [\_[N]{}]{}=- where ${\mathbf{S}}={\mathbf{S}_1}+{\mathbf{S}_2}$. Once radiation reaction is included, the orbital frequency slowly evolves as $$\begin{aligned} \label{omegadot} \dot\omega&=& \omega^2\f{96}{5}\eta (M\omega)^{5/3} \left\{ 1-\f{743+924\eta}{336}(M\omega)^{2/3} +\left[ \left(\f{19}{3}\eta-\f{113}{12}\right){\boldsymbol{\chi}_s}\cdot {\hat{\mathbf{L}}_{N}}-\f{113\delta}{12} {\boldsymbol{\chi}_a}\cdot {\hat{\mathbf{L}}_{N}}+4\pi \right]\right.(M\omega)\\ &+& \Bigl\{ \left( \f{34103}{18144}+\f{13661}{2016}\eta+\f{59}{18}\eta^2 \right) -\f{\eta\chi_1\chi_2}{48} \left(247{\hat{\mathbf{S}}_1}\cdot{\hat{\mathbf{S}}_2}-721({\hat{\mathbf{L}}_{N}}\cdot {\hat{\mathbf{S}}_1})({\hat{\mathbf{L}}_{N}}\cdot {\hat{\mathbf{S}}_2})\right) \nn\\ &+& \sum_{i=1}^2\frac{(m_i\chi_i)^2}{M^2} \left[ \frac{5}{2}\left( 3({\hat{\mathbf{L}}_{N}}\cdot {\hat{\mathbf{S}}_i})^2-1 \right) + \frac{1}{96}\left( 7-({\hat{\mathbf{L}}_{N}}\cdot {\hat{\mathbf{S}}_i})^2 \right) \right] \Bigr\} (M\omega)^{4/3} \nn\\ &-&\f{4159+15876\eta}{672}\pi(M\omega)^{5/3} +\left[ \left( \f{16447322263}{139708800}-\f{1712\gamma_E}{105} +\f{16\pi^2}{3} \right)+\left( -\f{273811877}{1088640}+\f{451\pi^2}{48} -\f{88}{3}\hat \theta \eta \right)\eta \right.\nn\\ &+&\f{541}{896}\eta^2 -\f{5605}{2592}\eta^3 -\left. \left. \f{856}{105}\log[16(M\omega)^{2/3}] \right](M\omega)^2 +\left( -\f{4415}{4032}+\f{358675}{6048}\eta+\f{91495}{1512}\eta^2 \right)\pi (M\omega)^{7/3} \right\} \nn\end{aligned}$$ where $\gamma_E \simeq 0.577$ is Euler’s constant, $\hat\theta \equiv 1039/4620$, and we have defined $$\begin{aligned} {\boldsymbol{\chi}_s}&\equiv&\f{1}{2}({\boldsymbol{\chi}_1}+{\boldsymbol{\chi}_2})\,,\\ {\boldsymbol{\chi}_a}&\equiv&\f{1}{2}({\boldsymbol{\chi}_1}-{\boldsymbol{\chi}_2})\,.\end{aligned}$$ The two terms in square parentheses on the third line of Eq. (\[omegadot\]) are due to the quadrupole-monopole interaction [@Poisson:1997ha] and to the spin-spin self interaction [@Mikoczi:2005dn], respectively, and they were neglected in the statistical study of Ref. [@Herrmann:2009mr]. Their sum agrees with Eq. (5.17) of Ref. [@Racine:2008kj]. The numerical integration of this system of ordinary differential equations is performed using the adaptive stepsize integrator [StepperDopr5]{} [@NR]. The evolution of any given BBH system is specified by the following parameters: the initial orbital frequency $\omega_i$, the binary’s mass ratio $q \equiv m_2/m_1$, the dimensionless magnitude of each spin $\chi_i$, and the relative orientation $(\theta_i,\phi_i)$ of each spin with respect to the orbital angular momentum at time $t=0$ ($i=1,\,2$). To monitor the variables along the whole evolution we output all quantities using a constant logarithmic spacing in the orbital frequency at low frequencies, and the stepsize as used in the integrator at high frequencies. Typically this results in a total of about $64,000$ points in the range $M\omega\in [M\omega_i,\,M\omega_f]$, where $M\omega_i=3.16\times 10^{-5}$ and $M\omega_f=0.1$. Numerical experimentation indicates that a tolerance parameter [atol]{}$=2\times 10^{-8}$ in the adaptive stepsize integrator is sufficient for a pointwise accuracy of order $1\%$ or better in the final quantities. Therefore the error induced by the numerical integrations of the PN equations of motion is subdominant with respect to the errors induced by precessional effects and by fits of the numerical simulations, which will be one of the main topics of this paper. Spin-orbit Resonances {#S:res} ===================== In this Section, we review the equilibrium configurations of BBH spins first presented in Schnittman [@Schnittman:2004vq] for which the Newtonian orbital angular momentum ${\mathbf{L}_N}$ and individual spins $\mathbf{S}_{1,2}$ all precess at the same resonant frequency. As discussed briefly in the Introduction, at a given binary separation $r$ fully general quasi-circular BBHs are described by 7 parameters: the mass ratio $q$ and the 3 components of each black hole spin. In spherical coordinates with ${\mathbf{L}_N}$ defining the $z$-axis, each spin is given by its magnitude $S_{i} = m_{i}^2 \chi_{i}$ and direction $(\theta_{i},\,\phi_{i})$ $(i=1,\,2)$. In the PN limit for which this analysis is valid, a clear hierarchy $$\label{E:thier} t_{\rm orb} \ll t_p \ll t_{\rm GW}$$ exists between the orbital time $t_{\rm orb} \propto r^{3/2}$, the precession time $t_p \sim \Omega_{1,2}^{-1} \propto r^{5/2}$, and the radiation time $t_{\rm GW} \sim \dot{E}_{\rm GW}/E \propto r^4$. This hierarchy implies that the BBH spins will precess many times before merger leaving only their [ *relative*]{} angular separation ${\Delta \phi}\equiv \phi_2 - \phi_1$ in the orbital plane well defined. This reduces the BBH parameter space to 6 dimensions. Since the mass ratio and individual spin magnitudes are preserved during the inspiral, a given BBH evolves through the 3-dimensional parameter space $(\theta_1, \theta_2, {\Delta \phi})$ on the precession timescale $t_p$. This evolution is governed by the spin precession equations (\[E:SP\]). Schnittman [@Schnittman:2004vq] discovered a one-parameter family of equilibrium solutions to these equations for which $(\theta_1, \theta_2, {\Delta \phi})$ remain fixed on the precession timescale $t_p$. These solutions have ${\Delta \phi}= 0^\circ$ or $180^\circ$, implying that ${\mathbf{L}_N}$, ${\mathbf{S}_1}$ and ${\mathbf{S}_2}$ all lie in a plane and precess at the same resonant frequency about the total angular momentum $\mathbf{J}$, which remains fixed in the absence of gravitational radiation. The values of $\theta_{1,2}$ for these resonances can be determined by requiring the first and second time derivatives of ${\mathbf{S}_1}\cdot {\mathbf{S}_2}$ to vanish. This is equivalent to satisfying the algebraic constraint $$\begin{gathered} \label{E:rescon} ({\bar{\boldsymbol{\Omega}}_1}\times {\mathbf{S}_1}) \cdot [{\mathbf{S}_2}\times ({\mathbf{L}_N}+ {\mathbf{S}_1})] \\ = ({\bar{\boldsymbol{\Omega}}_2}\times {\mathbf{S}_2}) \cdot [{\mathbf{S}_1}\times ({\mathbf{L}_N}+ {\mathbf{S}_2})]~.\end{gathered}$$ Since ${\mathbf{L}_N}$ appears in Eq. (\[E:rescon\]) both explicitly and implicitly through $\bar{\boldsymbol{\Omega}}_{1,2}$, the resonant values of $\theta_{1,2}$ vary with the binary separation. This is crucial, as otherwise these one-parameter families of resonances would affect only a small portion of the 3-dimensional parameter space $(\theta_1, \theta_2, {\Delta \phi})$ through which generic BBH configurations evolve. As gravitational radiation slowly extracts angular momentum from the binary on the radiation time $t_{\rm GW}$, the resonances sweep through a significant portion of the $(\theta_1, \theta_2)$ plane. The angular separation ${\Delta \phi}$ of a generic BBH is varying on the much shorter precession time $t_p$, and thus has a significant chance to closely approach the resonant values ${\Delta \phi}= 0^\circ$ or $180^\circ$ at some point during the long inspiral. Such generic BBHs will be strongly influenced or even captured by the spin-orbit resonances, as we will see in detail in Sec. \[S:align\]. ![Spin-orbit resonances for maximally spinning BBHs with a mass ratio of $q = 9/11$. The dotted black diagonal indicates where $\theta_1 = \theta_2$. Solid black curves below (above) this diagonal show $(\theta_1, \theta_2)$ for the one-parameter families of equilibrium spin configurations with ${\Delta \phi}= 0^\circ (180^\circ)$ at different fixed binary separations. Approaching the diagonal from below, these curves correspond to separations $r = 1000, 500, 250, 100, 50, 10 M$. The curves approaching from above correspond to separations $r = 250, 50, 20, 10 M$. The long-dashed red curves show how $\theta_{1,2}$ evolve as members of these resonant families inspiral from $r_i = 1000 M$ to $r_f = 10 M$. The projection $\mathbf{S} \cdot {\hat{\mathbf{L}}_{N}}$ of the total spin $\mathbf{S}$ onto the orbital angular momentum ${\mathbf{L}_N}$ is constant along the short-dashed blue lines, while the projection $\mathbf{S}_0 \cdot {\hat{\mathbf{L}}_{N}}$ of the EOB spin $\mathbf{S}_0$ is constant along the dot-dashed green lines.[]{data-label="F:resq1.22"}](Figure1.eps){width="3.5in"} ![Spin-orbit resonances for maximally spinning BBHs with a mass ratio of $q = 1/3$. Other than the different mass ratio, this figure is very similar to Fig. \[F:resq1.22\]. The solid black curves approaching the diagonal from below correspond to the families of resonant spin configurations at $r = 50, 20, 10, 5 M$, while those approaching from above correspond to separations $r = 20, 10, 5 M$.[]{data-label="F:resq3"}](Figure2.eps){width="3.5in"} We show the dependence of the spin-orbit resonances on $r$ for maximally spinning BBHs in Figs. \[F:resq1.22\] and \[F:resq3\]. Those resonances with ${\Delta \phi}= 0^\circ$ (shown in Fig. 2 of [@Schnittman:2004vq]) always have $\theta_1 < \theta_2$, and thus appear below the diagonal $\cos \theta_1 = \cos \theta_2$ in our Figs. \[F:resq1.22\] and \[F:resq3\]. Those resonances with ${\Delta \phi}= 180^\circ$ (shown in Fig. 3 of [@Schnittman:2004vq]) have $\theta_1 > \theta_2$ and therefore appear above the diagonal in our Figs. \[F:resq1.22\] and \[F:resq3\]. We plot $(\cos \theta_1, \cos \theta_2)$ rather than $(\theta_1, \theta_2)$ like [@Schnittman:2004vq] because isotropically oriented spins should have a flat distribution in these variables. In the limit $r \to \infty$, so that also $|{\mathbf{L}_N}|\to \infty$, the resonant configurations either have ${\mathbf{S}_1}$ or ${\mathbf{S}_2}$ aligned or anti-aligned with ${\mathbf{L}_N}$ (either $\theta_1$ or $\theta_2$ equals to $0^\circ$ or $180^\circ$). This corresponds to the four edges of the plot in Fig. \[F:resq1.22\]. For smaller fixed values of $|{\mathbf{L}_N}|$, the values $(\theta_1, \theta_2)$ for the one-parameter families of resonant configurations approach the diagonal $\theta_1 = \theta_2$. BBHs in spin-orbit resonances at large values of $|{\mathbf{L}_N}|$ (large $r$) remain resonant as they inspiral. As gravitational radiation carries away angular momentum, $r$ decreases and $\theta_{1,2}$ for individual resonant BBHs evolves towards this diagonal along the red long-dashed curves in Fig. \[F:resq1.22\]. For resonances with ${\Delta \phi}= 0^\circ$ (those below the diagonal), this evolution aligns the two spins with each other. Symmetry implies that aligning the spins with each other will lead to larger final spins and smaller recoil velocities [@Boyle:2007sz; @Boyle:2007ru]. The projection $$\label{E:Spar} \mathbf{S} \cdot {\hat{\mathbf{L}}_{N}}= S_1 \cos \theta_1 + S_2 \cos \theta_2$$ of the total spin $\mathbf{S} \equiv {\mathbf{S}_1}+ {\mathbf{S}_2}$ parallel to the orbital angular momentum is constant along the short-dashed blue lines in Figs. \[F:resq1.22\] and \[F:resq3\]. These blue lines have steeper slopes than the red lines along which the resonant binaries inspiral. This implies that the total spin $\mathbf{S}$ becomes anti-aligned (aligned) with the orbital angular momentum for resonant configurations with ${\Delta \phi}= 0^\circ (180^\circ)$, leading to smaller (larger) final spins. The families of resonances with ${\Delta \phi}= 0^\circ$ (below the diagonal) sweep through a larger area of the $(\cos \theta_1, \cos \theta_2)$ plane as the BBHs inspiral, and approach the diagonal more closely. This implies that anti-alignment may be more effective than alignment, which might explain the “small but statistically significant bias of the distribution towards counter-alignment” in $\mathbf{S} \cdot {\hat{\mathbf{L}}_{N}}$ noted in Lousto [*et al.*]{} [@Lousto:2009ka]. However, Table IV of [@Lousto:2009ka] indicates that both ${\mathbf{S}_1}$ and ${\mathbf{S}_2}$ individually become anti-aligned with ${\hat{\mathbf{L}}_{N}}$, whereas the spin-orbit resonances would align one black hole while anti-aligning the other. All of the PN evolutions in Lousto [*et al.*]{} [@Lousto:2009ka] begin at separations of $r = 50 M$, which corresponds to the ${\Delta \phi}= 0^\circ$ curve in Fig. \[F:resq1.22\] that is second closest to the diagonal. The resonances sweep through most of the plane below the diagonal at larger separations, suggesting that these short-duration PN evolutions may have failed to capture the full magnitude of the anti-alignment. We will investigate this possibility in Sec. \[S:align\]. Another interesting feature of Figs. \[F:resq1.22\] and \[F:resq3\] is that the red long-dashed curves along which the BBHs inspiral are nearly parallel to the dot-dashed green lines along which the projection ${\mathbf{S}_0}\cdot {\hat{\mathbf{L}}_{N}}$ of the effective-one-body (EOB) spin [@Damour:2001tu] $$\label{E:S0} {\mathbf{S}_0}\equiv (1 + q){\mathbf{S}_1}+ (1 + q^{-1}){\mathbf{S}_2}$$ is constant. The conservation of this quantity at 2PN order was first noted in Ref. [@Racine:2008qv] and follows directly from Eqs. (\[E:SP\]), (\[E:oaP\]), and (\[E:hLNdot\]). The conservation of ${\mathbf{S}_0}\cdot {\hat{\mathbf{L}}_{N}}$ rather than $\mathbf{S} \cdot {\hat{\mathbf{L}}_{N}}$ itself allows for the possible alignment of the total spin $\mathbf{S}$ discussed in the previous paragraph. We conclude this Section by briefly discussing how the spin-orbit resonances vary with the mass ratio $q$, as can be seen by comparing the $q = 9/11$ resonances in Fig. \[F:resq1.22\] with the $q = 1/3$ resonances in Fig. \[F:resq3\]. The most pronounced differences are that the $q = 1/3$ resonances sweep away from the edges of the $(\cos \theta_1, \cos \theta_2)$ plane at much smaller values of the separation $r$, and do not approach the diagonal as closely. This is consistent with the decreasing value of $r_{\rm lock}$ in Eq. (\[E:rlock\]) as $q \to 0$. In this limit both $t_p$ and $t_{\rm GW}$ are proportional to $q^{-1}$, implying that generic BBHs will be less likely to be affected by the resonances as they sweep through the plane over a smaller range in $r$. BBHs already in a resonant configuration will also be less affected since the resonant curves do not approach the diagonal as closely. The red long-dashed curves showing the inspiral of resonant configurations have steeper slopes for $q = 1/3$, consistent with the larger black hole being immune to its smaller companion in the limit $q \to 0$. This seems to contradict the puzzling result presented in Table IV of Lousto [*et al.*]{} [@Lousto:2009ka] that it is the [*smaller*]{} companion that remains randomly distributed during the inspiral. We will examine this behavior as well in the next Section. Spin Alignment {#S:align} ============== ![Distributions of $(\cos \theta_1, \cos \theta_2)$ at different separations $r$ for 1000 initially isotropic maximally spinning BBHs with a mass ratio $q = 9/11$. The top left panel shows the initial $10 \times 10 \times 10$ grid, evenly spaced in $(\cos \theta_1, \cos \theta_2, \Delta \phi)$. The dotted vertical lines show $\cos \theta_1 = \pm 0.4$. The 300 blue squares initially have $\cos \theta_1 > 0.4$, the 400 green triangles initially have $-0.4 < \cos \theta_1 < 0.4$, and the 300 red circles initially have $\cos \theta_1 < -0.4$. The values of $(\theta_1, \theta_2)$ for these BBHs are shown in the top right, bottom left, and bottom right panels after they have inspiraled to separations of $r = 1000, 100$ and $10 M$ respectively.[]{data-label="F:t1t2"}](Figure3.eps){width="3.5in"} ![Distributions of $(\cos \theta_1, \cos \theta_2)$ at different separations $r$ for 1000 initially isotropic maximally spinning BBHs with a mass ratio $q = 1/3$. The different panels, points, and lines are the same as those given for $q = 9/11$ in Fig. \[F:t1t2\].[]{data-label="F:q3t1t2"}](Figure4.eps){width="3.5in"} In this Section, we examine the extent to which the spins of [*generic*]{} (i.e. misaligned) BBH configurations become aligned with the orbital angular momentum and each other as the BBHs inspiral from $r_i = 1000 M$ to $r_f = 10 M$. Although we use maximally spinning BBHs to demonstrate this alignment, the magnitude of the alignment is comparable for all BBHs with $\chi_{1,2} \gtrsim 0.5$ as shown in Fig. 11 of [@Schnittman:2004vq]. We first consider initial spin configurations given by a uniform $10 \times 10 \times 10$ grid evenly spaced in $(\cos \theta_1, \cos \theta_2, \Delta \phi)$. This distribution is isotropic, and would be expected in the absence of an astrophysical mechanism to align the spins. BBHs with isotropically oriented spins might form in gas-poor mergers of SBHs and mergers of stellar-mass black holes in dense clusters. In Fig. \[F:t1t2\], we show how the distribution of $(\cos \theta_1, \cos \theta_2)$ evolves as maximally spinning BBHs with a mass ratio $q = 9/11$ inspiral from slightly beyond $r_i = 1000 M$ to $r_f = 10 M$. The top left panel shows our initial evenly spaced $10 \times 10 \times 10$ grid. The points are colored to indicate their [*initial*]{} value of $\cos \theta_1$: blue squares begin with $\cos \theta_1 > 0.4$ ($\theta_1\lesssim 66^\circ$), green triangles with $-0.4 < \cos \theta_1 < 0.4$, and red circles with $\cos \theta_1 < -0.4$. The dotted vertical lines $\cos \theta = \pm 0.4$ denote these boundaries. Only 100 points are visible in the top left panel, as the different values of ${\Delta \phi}$ cannot be distinguished in this two-dimensional projection. Spin precession reveals all 1000 points after the BBHs have inspiraled to $r_i = 1000 M$ as seen in the top right panel. Notice that the spins of all 1000 BBHs precess in a way that conserves the projection of ${\mathbf{S}_0}$ onto ${\hat{\mathbf{L}}_{N}}$ (parallel to the dot-dashed green lines in Fig. \[F:resq1.22\]). This is not a special feature of the spin-orbit resonances, but occurs for generically oriented spins as well. These generic spin configurations do not individually preserve $(\cos \theta_1, \cos \theta_2)$ over a precession time $t_p$ like the resonant configurations do, but they do preserve the combination ${\mathbf{S}_0}\cdot {\hat{\mathbf{L}}_{N}}$. This precession continues as the BBHs inspiral to $r = 100 M$ and $r_f = 10 M$ as shown in the bottom left and bottom right panels of Fig. \[F:t1t2\]. By the time they reach $r_f = 10 M$ the green points have diffused to fill most of the $(\cos \theta_1, \cos \theta_2)$ plane, while the blue (red) points have diffused into the upper right (lower left) portion of the middle $-0.4 < \cos \theta_1 < 0.4$ region. The bottom right panel, if the points had not been colored, would reproduce Fig. 1 of Bogdanovic [*et al.*]{} [@Bogdanovic:2007hp] and therefore support their conclusion that isotropically distributed spins remain isotropic as they inspiral. However, the colors reveal that PN evolution can drastically alter spin distributions that have been partially aligned by a circumbinary disk. For example, if the spin of the more massive black hole was aligned so that $\cos \theta_1 > 0.4$ at $r_i = 1000 M$ (shown by our blue points), by the time the binary reached $r_f = 10 M$ the larger spin could easily lie in the orbital plane and thus give rise to a smaller final spin and potentially large “superkick” [@Gonzalez:2007hi; @Campanelli:2007ew]. For comparison, we show the inspiral of the same $10 \times 10 \times 10$ grid of maximally spinning BBHs with a mass ratio $q = 1/3$ in Fig. \[F:q3t1t2\]. The points diffuse along the steeper lines that preserve ${\mathbf{S}_0}\cdot {\hat{\mathbf{L}}_{N}}$ for this less equal mass ratio. This inhibits their ability to diffuse across the $\cos \theta_1 = \pm 0.4$ boundaries, again shown by the vertical dotted lines. Even at $r_f = 10 M$ only a few points have trickled between the three regions. Since the spin of the more massive black hole remains aligned with the orbital angular momentum, one would expect a large final spin and an absence of superkicks for such small mass ratios. We will examine in detail how spin alignment affects recoil-velocity distributions in future work. ![Distributions of $(\Delta \phi, \cos \theta_{12})$ at different separations $r$ for 1000 initially isotropic maximally spinning BBHs with a mass ratio $q = 9/11$. The top left panel shows the initial $10 \times 10 \times 10$ grid of BBH spin configurations, evenly spaced in $(\cos \theta_1, \cos \theta_2, \Delta \phi)$. This distribution is peaked about the curve $\cos \theta_{12} = \cos \Delta \phi$ shown by the dotted curve. The points are colored according to their initial values of $\cos \theta_1$ as in Fig. \[F:t1t2\]. The top right, bottom left, and bottom right panels show the distribution evolves after the BBHs have inspiraled to $r = 1000, 100$ and $10 M$ respectively, also as in Fig. \[F:t1t2\].[]{data-label="F:DPt12"}](Figure5.eps){width="3.5in"} ![Distributions of $(\Delta \phi, \cos \theta_{12})$ at different separations $r$ for 1000 initially isotropic maximally spinning BBHs with a mass ratio $q = 1/3$. The different panels, points, and lines are the same as those given for $q = 9/11$ in Fig. \[F:DPt12\].[]{data-label="F:q3DPt12"}](Figure6.eps){width="3.5in"} ![Histograms of $\cos \theta_{12}$ and ${\Delta \phi}$ for BBHs with initially isotropic spins. The two left panels are for the mass ratio $q = 9/11$, while the two right panels are for $q = 1/3$. The two top panels give the distribution of $\cos \theta_{12}$, while the two bottom panels give the distribution of ${\Delta \phi}$. The black curves are for all 1000 BBHs in the $10 \times 10 \times 10$ grid discussed in the text, while the blue (red) curves correspond to the blue (red) points in Figs. \[F:t1t2\]-\[F:q3DPt12\] with initial values $\cos \theta_1 > 0.4$ $(\cos \theta_1 < -0.4)$. The horizontal dotted lines show the initially flat distributions, while the solid lines show the distributions at $r = 10 M$.[]{data-label="F:pdft12DP"}](Figure7.eps){width="3.5in"} In Fig. \[F:DPt12\] we show how the joint probability distribution function for $\Delta \phi$ and $\cos \theta_{12}$ evolves for our evenly spaced $10 \times 10 \times 10$ grid of initially isotropic BBH spin configurations. As defined in the Introduction, $\cos \theta_{12}$ is the cosine of the angle between ${\mathbf{S}_1}$ and ${\mathbf{S}_2}$. It can be expressed in terms of the individual spin angles as \[E:cos12\] \_[12]{} = \_1 \_2 + \_1 \_2 , and has a flat distribution between -1 and 1 for isotropic, uncorrelated spins such as those given by our $10 \times 10 \times 10$ grid. However, as seen in Eq. (\[E:cos12\]), the values of $\cos \theta_{12}$ and $\cos \Delta \phi$ are correlated; for a given value of $\Delta \phi$ the distribution of $\cos \theta_{12}$ is peaked about $\cos \Delta \phi$ for flat distributions of $\cos \theta_1$ and $\cos \theta_2$. This can be seen in Fig. \[F:DPt12\] from the clustering of points about the curve $\cos \theta_{12} = \cos \Delta \phi$. Although $\cos \theta_{12}$ and $\cos \Delta \phi$ are correlated even for isotropic spins, geometry implies that both are initially uncorrelated with the value of $\cos \theta_1$. This is revealed by the identical distributions of the red, green, and blue points in the top left panel of Fig. \[F:DPt12\] to within the resolution of our grid. These distributions do not remain identical as the BBHs inspiral from $r_i = 1000 M$ to $r_f = 10 M$. Influenced by the $\Delta \phi = 0^\circ$ spin-orbit resonances below the diagonal in Fig. \[F:resq1.22\], the blue points become concentrated about $\Delta \phi = 0^\circ, \cos \theta_{12} = 1$ by the time they reach $r_f$. The red points, similarly influenced by the $\Delta \phi = \pm 180^\circ$ resonances above the diagonal in Fig. \[F:resq1.22\], become concentrated about $\Delta \phi = \pm 180^\circ, \cos \theta_{12} = -1$. The effect of this spin alignment on the spin of the final black hole will be explored in detail in the next Section, while the effect on recoil velocities will be examined in future work. Qualitatively, alignment of the spins with each other ($\cos \theta_{12} \to 1$) increases the final spin and reduces the recoil velocity, while anti-alignment ($\cos \theta_{12} \to -1$) does the opposite. The magnitude of this spin alignment is greatly reduced for smaller mass ratios as seen in Fig. \[F:q3DPt12\] for the case $q = 1/3$. Although the clustering of all the points about $\cos \theta_{12} = \cos \Delta \phi$ is again apparent, the distributions of the red, green, and blue points remain similar all the way down to $r_f = 10 M$ as seen in the lower right panel. The weaker influence of the spin-orbit resonances for $q = 1/3$ follows from the smaller value of $r_{\rm lock}$ in Eq. (\[E:rlock\]), and is similarly reflected by the smaller fraction of the $(\cos \theta_1, \cos \theta_2)$ plane occupied by the resonant curves in Fig. \[F:resq3\]. We have provided histograms of $\cos \theta_{12}$ and ${\Delta \phi}$ in Fig. \[F:pdft12DP\] to clarify the differences between Figs. \[F:DPt12\] and \[F:q3DPt12\]. We see that the distributions of $\cos \theta_{12}$ and $\Delta \phi$ are initially flat for both mass ratios, but evolve considerably for $q = 9/11$ while remaining nearly flat for $q = 1/3$ within the limits set by Poisson fluctuations. The open blue (red) curves in the left panels of Fig. \[F:pdft12DP\] clearly show distributions peaked at $\cos \theta_{12} = 1, \Delta \phi = 0^\circ$ ($\cos \theta_{12} = -1, {\Delta \phi}= \pm 180^\circ$). Such trends are barely noticeable in the right panels. We will explore the implications of these findings for the final spins in the next Section. Final Spin Distributions {#S:dist} ======================== Several attempts have been made to predict the final dimensionless spin ${\boldsymbol{\chi}_f}$ of the black hole resulting from a BBH merger. Initial attempts focused on finding simple phenomenological fitting formulae for the final spin resulting from non-spinning, unequal-mass BBH merger simulations [@Buonanno:2006ui; @Berti:2007fi; @Buonanno:2007pf]. A group at the Albert Einstein Institute (AEI) developed a fitting formula that provides the magnitude and direction of ${\boldsymbol{\chi}_f}$ in terms of the initial spins ${\boldsymbol{\chi}_1}$, ${\boldsymbol{\chi}_2}$ and the mass ratio $q$ [@Rezzolla:2007xa; @Rezzolla:2007rd; @Rezzolla:2007rz]. They assumed that the final spin magnitude could be expressed as a polynomial in $\chi_1$, $\chi_2$, and the symmetric mass ratio $\eta$, then made some additional assumptions about the symmetries of this polynomial dependence and how energy and angular momentum are radiated to reduce the number of terms in their expression. The coefficients of the remaining terms were calibrated using numerical-relativity (NR) simulations of BBH mergers in which the initial spins were either aligned or anti-aligned with the orbital angular momentum. We shall refer to this older AEI formula as “AEIo”. A more recent paper [@Barausse:2009uz] by members of this group uses newer NR simulations to recalibrate their coefficients, and replaces earlier assumptions with the conjecture that the final spin points in the direction of the total angular momentum of the initial BBH at [*any*]{} separation. For consistency, this requires the further assumption that angular momentum is always radiated in the direction of the total angular momentum. We shall refer to this newer AEI formula as “AEIn”. An alternative fitting formula was proposed by a group at Florida Atlantic University (FAU) [@Tichy:2008du]. Following the procedure outlined in [@Boyle:2007sz; @Boyle:2007ru], the FAU group performed 10 equal-mass misaligned simulations to calibrate the coefficients of fitting formulae for the Cartesian components of ${\boldsymbol{\chi}_f}$. They then made additional assumptions about the mass-ratio dependence of these formulae, and found good agreement between their predictions and independent NR simulations with mass ratios as small as $q = 5/8$. We shall refer to the formula of this group as “FAU”. The Rochester Institute of Technology (RIT) group proposed yet another fitting formula during the preparation of this paper [@Lousto:2009mf]. This formula includes higher-order terms in the initial spins that may ultimately be needed to describe future high-accuracy NR simulations. However, current simulations are inadequate to calibrate all the terms appearing in the RIT formula, so we will not consider its predictions in this paper. Other groups have predicted final spins by extrapolating analytical test-particle calculations to finite mass ratios, rather than calibrating fitting formulae with NR simulations. Buonanno, Kidder, and Lehner (BKL) [@Buonanno:2007sv] derived a formula for the final spin by assuming, as is true in the test-particle limit, that the angular momentum radiated during the inspiral stage of a BBH merger exceeds that radiated during the plunge and ringdown. Using this assumption, they equated the final spin with the total angular momentum $\mathbf{J} = \mathbf{L}_{\rm ISCO} + {\mathbf{S}_1}+ {\mathbf{S}_2}$, where $\mathbf{L}_{\rm ISCO}$ is the orbital angular momentum at the innermost stable circular orbit (ISCO) of a test particle of mass $\eta M$ orbiting a black hole of mass $M$ and dimensionless spin ${\boldsymbol{\chi}_f}$ equal to that of the [*final*]{} black hole. This counterintuitive but inspired choice correctly provides ${\boldsymbol{\chi}_f}\to {\boldsymbol{\chi}_1}$ in the $q \to 0$ limit and respects the symmetry of BBH mergers under exchange of the labels of the two black holes. Though derived only from test-particle calculations, the BKL formula is remarkably successful at predicting final spins even for equal-mass BBH mergers. Kesden [@Kesden:2008ga] slightly modified the BKL spin formula to account for the energy radiated during the inspiral stage of the merger. This change makes the formula accurate to linear order in $q$ in the test-particle limit. It generically increases the magnitude of the predicted dimensionless final spin by reducing the predicted final mass $m_f$ below $M$ in the denominator of the expression ${\boldsymbol{\chi}_f}= S_f/m_{f}^2$. This increase improves the agreement with NR simulations of non-spinning BBH mergers, but leads to somewhat larger final spins than the other formulae for mergers of maximally spinning BBHs, such as those considered in this paper. The predictions of this formula are refered to as “Kes” in this paper. We now present the predictions of the spin formulae summarized above for various distributions of BBH spins that are allowed to inspiral from $r_i = 1000 M$ to $r_f = 10 M$. Spin Magnitudes {#SS:mag} --------------- In the top panel of Fig. \[F:mag1.22\], we show the final spin magnitude $\chi_f$ predicted by the AEIn formula for the evenly spaced $10 \times 10 \times 10$ grid of maximally spinning BBHs with $q = 9/11$ described in Sec. \[S:align\]. The other spin formulae give very similar results; the mean and variance of the final spin distributions predicted by the other formulae for some of the initial distributions described below are provided in Table \[T:fspin\]. As in Figs. \[F:t1t2\]-\[F:pdft12DP\], the black curves in Fig. \[F:mag1.22\] refer to all 1000 BBHs, the blue curves to the subset of 300 BBHs with the lowest values of $\theta_1$, and the red curves to the subset of 300 BBHs with the highest values of $\theta_1$. The dotted curves give the final spin distribution predicted for the BBH spin configurations at their initial separation $r_i = 1000 M$, while the solid curves give the final spin distribution predicted when these [*same*]{} BBHs are allowed to inspiral to $r_f = 10 M$ according to the PN evolution described in Sec. \[S:PN\]. The AEIn formula is unique in that it claims to accurately predict final spins at all separations; separations as large as $r = 2 \times 10^4 M$ were considered in [@Barausse:2009uz]. The other fitting formulae were intended to apply at $r_{\rm NR} \simeq 10 M$, the starting point for the NR simulations with which their coefficients were calibrated. The BKL and Kes formulae were designed for use at the ISCO. Although strictly speaking the formulae other than AEIn cannot be applied to widely separated BBHs, one can imagine that the BBHs inspiral to $r_f = 10 M$ without spin precession where these formulae are valid. It is in this sense that we consider the predictions of these other formulae when we claim in this Section to apply them to BBH spin configuration at $r_i = 1000 M$. ![[*Top panel:*]{} Histogram of the final spin $\chi_f$ predicted by the AEIn formula for 1000 BBHs with mass ratio $q = 9/11$ and isotropically distributed spins at $r_i = 1000 M$. The blue curves show the subset of 300 BBHs with the lowest initial values of $\theta_1$, while the red curves show the subset of 300 BBHs with the highest initial values of $\theta_1$. The solid curves show the predicted spins if the AEIn formula is applied at $r_f = 10 M$ after the BBHs have inspiraled to this separation according to the equations of Sec. \[S:PN\]. The dotted curves show the predicted spins if the AEIn formula is applied to the initial distribution at $r_i = 1000 M$. [*Bottom panel:*]{} Histograms of the predicted final spins for 6 sets of BBH mergers with $q = 9/11$, and flat distributions in $\cos \theta_2$ and $\Delta \phi$ at $r_i = 1000 M$. The red, orange, yellow, green, blue, and purple curves have $\theta_1 = 170^\circ, 160^\circ, 150^\circ, 30^\circ, 20^\circ,$ and $10^\circ$ respectively. As in the top panel the final spins predicted by applying the AEIn formula at $r_i = 1000 M$ are shown by dotted curves, while allowing the BBHs to inspiral to $r_f = 10 M$ before applying the formula leads to the spins shown by the solid curves.[]{data-label="F:mag1.22"}](Figure8.eps){width="3.5in"} ![Histograms of the final spins $\chi_f$ predicted by the AEIn formula for the same sets of BBHs presented in Fig. \[F:mag1.22\], but with the mass ratio $q = 1/3$ instead of $q = 9/11$. As in that figure, the predictions made at $r_i = 1000 M$ are shown with dotted curves, those made at $r_f = 10 M$ are shown with solid curves. The black curves in the top panel show the full set of 1000 BBHs, while the blue (red) curves show the subset of 300 BBHs with the lowest (highest) initial values of $\theta_1$. In the lower panel, the red, orange, and yellow curves show BBHs with ${\boldsymbol{\chi}_1}$ initially anti-aligned with ${\mathbf{L}_N}$ ($\theta_1 = 170^\circ, 160^\circ, 150^\circ$). The green, blue, and purple curves show BBHs with ${\boldsymbol{\chi}_1}$ initially aligned with ${\mathbf{L}_N}$ ($\theta_1 = 30^\circ, 20^\circ, 10^\circ$).[]{data-label="F:mag3"}](Figure9.eps){width="3.5in"} The dotted and solid black curves in the top panel of Fig. \[F:mag1.22\] are identical to within the Poisson noise of our limited number of BBH inspirals, confirming the finding of Refs. [@Bogdanovic:2007hp; @Herrmann:2009mr; @Lousto:2009ka] that isotropic distributions of BBH spins remain nearly isotropic as they inspiral. Even at $r_i = 1000 M$, the blue (red) subset of spin configurations yields the largest (smallest) predicted final spins, because for these configurations the spin of the more massive black hole is aligned (anti-aligned) with the orbital angular momentum. The spin-orbit resonances further enhance (reduce) the final spins predicted for these subsets by aligning (anti-aligning) the BBH spins [*with each other*]{} during the inspiral for small (large) initial values of $\theta_1$. As a result, the solid blue (red) distribution at $r_f = 10 M$ has a larger (smaller) mean final spin than the initial dotted distribution at $r_i = 1000 M$. This can be seen in the displacement of predicted final spins for the colored subsets away from $\chi_f \simeq 0.75$ towards larger and smaller values. To clarify the magnitude of this effect, we have performed 6 additional sets of BBH inspirals, each of which consists of a fixed value of $\theta_1$ and a $30 \times 30$ grid evenly spaced in $\cos \theta_2$ and $\Delta \phi$. Three of these sets have the spin of the more massive black hole nearly aligned with the orbital angular momentum ($\theta_1 = 10^\circ, 20^\circ, 30^\circ$), while the other 3 sets have ${\boldsymbol{\chi}_1}$ nearly anti-aligned with ${\mathbf{L}_N}$ ($\theta_1 = 150^\circ, 160^\circ, 170^\circ$). The choice of aligned distributions was partly motivated by the finding of Ref. [@Dotti:2009vz] that accretion torques will align BBH spins to within $10^\circ$ ($30^\circ$) of the orbital angular momentum for a cold (hot) disk. The predicted final spins for these distributions, both at $r_i = 1000 M$ and $r_f = 10 M$, are shown in the bottom panel of Fig. \[F:mag1.22\]. The final spins for the initially aligned ($\theta_1 \leq 30^\circ$) BBH distributions are significantly larger when predicted at $r_f = 10 M$ than at $r_i = 1000 M$, undermining the claim of [@Barausse:2009uz] that the AEIn formula can accurately predict final spins at large separations without the need for PN evolutions. The predicted final spins for the initially anti-aligned ($\theta_1 \geq 150^\circ$) BBH distributions conversely shift to lower values as the predictions are made later in the inspiral. We provide the mean and standard deviation of the final spins predicted for these 6 new sets of partially aligned BBH distributions for all 5 formulae in Table \[T:fspin\]. To explore the dependence of these effects on the mass ratio, we have provided histograms of the predicted final spins for these same BBH spin distributions with $q = 1/3$ in Fig. \[F:mag3\]. The discrete peaks at low values of $\chi_f$ in the histograms in the top panel are an artifact of the 10 discrete values of $\cos \theta_1$ in our $10 \times 10 \times 10$ grid. Each peak contains 100 points with the same initial value of $\theta_1$. The decrease in the width of each peak as the BBHs inspiral from $r_i = 1000 M$ to $r_f = 10 M$ is a consequence of the anti-alignment of the BBH spins for large $\theta_1$, but the gaps between the peaks would be filled in if we used a finer grid. The shifts in the mean values of the peaks should be robust with respect to the grid spacing. These shifts for the initially aligned BBH distributions are provided in Table \[T:fspin\] for all 5 formulae for $q = 1/3$, as well as for the intermediate mass ratio $q = 2/3$. Spin Directions {#SS:dir} --------------- Before providing quantitative results, we need to clarify what is meant by the [*direction*]{} of the spin of the final black hole. In what reference frame is this direction defined? Most of the fitting formulae calibrated with NR simulations attempt to predict the angle \[E:thf\] \_f between the BBH orbital angular momentum ${\mathbf{L}_N}$ at the separation $r_f = r_{\rm NR}$ where the NR simulations were performed and the final spin ${\boldsymbol{\chi}_f}$ predicted from the BBH spin configuration at this [*same*]{} separation. The analytical predictions of BKL and Kes were designed to apply to BBH spin configurations at $r_f = r_{\rm ISCO}$. If one assumed that neither the orbital angular momentum nor the BBH spins (upon which the prediction ${\hat{\boldsymbol{\chi}}_f}(r_f)$ depends) precessed during the inspiral, one could insert these quantities [*at any separation*]{} into the right-hand side of Eq. (\[E:thf\]) to predict $\vartheta_f$. The angle $\vartheta_f$ is physically interesting because it quantifies the post-merger alignment between ${\boldsymbol{\chi}_f}$ and the inner edge of the accretion disk if one assumes that torques have aligned the circumbinary disk with ${\mathbf{L}_N}$. However, one might also be interested in the alignment between ${\boldsymbol{\chi}_f}$ and a feature like the galactic disk that is assumed to be aligned with ${\mathbf{L}_N}$ at some larger scale $r_i$. In that case, one would need to compute the angle \[E:thi\] \_i between ${\mathbf{L}_N}$ at this larger separation and the final spin ${\boldsymbol{\chi}_f}(r_i)$ predicted from the BBH spins at this same separation. The proper way to predict ${\boldsymbol{\chi}_f}$ from the BBH spins at $r_i$ would be to use PN equations like those specified in Sec. \[S:PN\] to propagate those spins and ${\mathbf{L}_N}$ to down to $r_f$, then insert them into the fitting formula of one’s choice. The AEIn formula is based on the conjecture that ${\boldsymbol{\chi}_f}$ points in the direction of the total angular momentum $\mathbf{J}$ at [*any*]{} separation, since angular momentum is always radiated parallel to $\mathbf{J}$, thus preserving its direction. This conjecture is plausible because at large separations, the precession time $t_p$ is much shorter than the inspiral time $t_{\rm GW}$. If the vectors associated with the BBHs precess rapidly enough, all components except those parallel to $\mathbf{J}$ (which varies on the longer timescale $t_{\rm GW}$) will average to zero. The AEIn conjecture is very useful because it allows $\vartheta_i$ to be computed without solving any PN equations. However, the approximation $t_p \ll t_{\rm GW}$ upon which it depends breaks down at small separations. This may lead to incomplete cancellation of the angular momentum radiated perpendicular to $\mathbf{J}$. ![[*Top panel:*]{} Histogram of the angle $\theta_J$ (in degrees) between the total angular momentum $\mathbf{J}$ at $r_i = 1000 M$ and that at $r_f = 10 M$ for our set of 1000 BBHs with $q = 9/11$ and initially isotropic spins. As in previous figures, the blue (red) curve shows the subset of 300 BBHs with the lowest (highest) initial values of $\theta_1$. [*Bottom panel:*]{} Histograms of $\theta_J$ for the 6 sets of 900 BBH mergers with flat distributions in $\cos \theta_2$ and $\Delta \phi$ at $r_i = 1000 M$. The red, orange, yellow, green, blue, and purple curves show BBHs that have $\theta_1 = 170^\circ, 160^\circ, 150^\circ, 30^\circ, 20^\circ,$ and $10^\circ$ respectively at this initial separation.[]{data-label="F:J1.22"}](Figure10.eps){width="3.5in"} ![[*Left panel:*]{} Histogram of the angle $\theta_J$ (in degrees) between the total angular momentum $\mathbf{J}$ at $r_i = 1000 M$ and that at $r_f = 10 M$ for our set of 1000 initially isotropically spinning BBHs with $q = 1/3$. As in previous figures, the blue (red) curve shows the subset of 300 BBHs with the lowest (highest) initial values of $\theta_1$. [*Middle panel:*]{} Histograms of $\theta_J$ for the 3 sets of 900 BBH mergers initially with $\theta_1 = 10^\circ$ (purple), $20^\circ$ (blue), and $30^\circ$ (green). [*Right panel:*]{} Histograms of $\theta_J$ for the 3 sets of 900 BBH mergers initially with $\theta_1 = 150^\circ$ (yellow), $160^\circ$ (orange), and $170^\circ$ (red).[]{data-label="F:J3"}](Figure11.eps){width="3.5in"} We test this possibility by calculating \[E:thetaJ\] \_J  , the angle between the total angular momentum at $r_i = 1000 M$ and that after the BBHs have inspiraled to $r_f = 10 M$. If the direction of $\mathbf{J}$ really was preserved during the inspiral, $\theta_J$ would vanish. We present histograms of $\theta_J$ for mass ratio $q = 9/11$ in Fig. \[F:J1.22\]. The upper panel shows the $10 \times 10 \times 10$ grid of BBH spin configurations evenly spaced in ($\cos \theta_1, \cos \theta_2, {\Delta \phi}$) that we have discussed previously. The direction of $\mathbf{J}$ changes by $\theta_J \lesssim 2^\circ$ during most of the inspirals, though a tail extends to larger values for large initial values of $\theta_1$. This tail can be seen more clearly in the bottom panel for the BBHs with ${\boldsymbol{\chi}_1}$ initially anti-aligned with ${\mathbf{L}_N}$ ($\theta_1 \geq 150^\circ$). We agree with [@Barausse:2009uz] that these large changes in the direction of $\mathbf{J}$ are likely a consequence of the transitional precession first identified in Ref. [@Apostolatos:1994mx]. This transitional precession occurs to an even greater extent for smaller mass ratios, as can be seen in Fig. \[F:J3\] for $q = 1/3$. As in the upper panel of Fig. \[F:mag3\], discrete peaks resulting from the grid spacing in $\cos \theta_1$ can be seen in the left panel of Fig. \[F:J3\]. The middle panel shows that the direction of $\mathbf{J}$ remains nearly constant ($\theta_J \lesssim 0.5^\circ$) when ${\boldsymbol{\chi}_1}$ in closely aligned with ${\mathbf{L}_N}$ ($\theta_1 \leq 30^\circ$). However, the right panel shows that the assumption of constant ${\hat{\mathbf{J}}}$ fails badly for the BBHs with $\theta_1 \geq 150^\circ$, that comprise $\sim 7\%$ of isotropically distributed BBH mergers. The mass ratio $q = 1/3$ is not extreme compared to the majority of astrophysical mergers, so caution should be taken when assuming that ${\boldsymbol{\chi}_f}$ points in the direction of $\mathbf{J}$ such as in Eq. (\[E:thi\]). ![[*Top panel:*]{} Histogram of the angle $\vartheta_f$ (in degrees) between the orbital angular momentum ${\mathbf{L}_N}$ at $r_f = 10 M$ and the final spin ${\boldsymbol{\chi}_f}$ predicted by the AEIn formula from the BBH spins at that separation. The BBHs have a mass ratio $q = 9/11$. As in previous figures, the black curves show 1000 mergers with initially isotropic BBH spins, while the blue (red) curves show the subset of 300 BBHs with the lowest (highest) initial values of $\theta_1$. The dotted curves show predictions in the absence of spin precession, while the solid curves show how these predictions change when the BBH spins precess from $r_i = 1000 M$ to $r_f = 10 M$ according to the PN equations of Sec. \[S:PN\]. [*Middle panel:*]{} Histograms of $\vartheta_f$ for the 3 sets of 900 BBH mergers with ${\boldsymbol{\chi}_1}$ initially aligned with ${\mathbf{L}_N}$ \[$\theta_1 = 10^\circ$ (purple), $20^\circ$ (blue), $30^\circ$ (green)\]. [*Bottom panel:*]{} Histograms of $\vartheta_f$ for the 3 sets of 900 BBH mergers with ${\boldsymbol{\chi}_1}$ initially anti-aligned with ${\mathbf{L}_N}$ \[$\theta_1 = 150^\circ$ (yellow), $160^\circ$ (orange), $170^\circ$ (red)\].[]{data-label="F:thf1.22"}](Figure12.eps){width="3.5in"} ![Histograms of the angle $\vartheta_f$ predicted by the AEIn formula for the same sets of BBHs shown in Fig. \[F:thf1.22\] but with a mass ratio $q = 1/3$. As in that figure, the top panel shows BBHs with initially isotropic spins with the blue (red) curves indicating those BBHs with the lowest (highest) initial values of $\theta_1$. Dotted curves show predictions without spin precession, while the solid curves show how these predictions change if the BBHs spins precess from $r_i = 1000 M$ to $r_f = 10 M$ according to the PN equations of Sec. \[S:PN\]. The bottom panel shows distributions with flat initial distributions of $\cos \theta_2$ and $\Delta \phi$, but with $\theta_1$ now initially set to $170^\circ, 160^\circ, 150^\circ, 30^\circ, 20^\circ,$ and $10^\circ$ respectively for the red, orange, yellow, green, blue, and purple curves.[]{data-label="F:thf3"}](Figure13.eps){width="3.5in"} What about the less ambitious predictions of $\vartheta_f$ from BBH spins at $r_f = 10 M$, assuming that NR simulations correctly describe spin precession from this separation until merger? Spin-orbit resonances have significant implications for these predictions as well. We show predictions of $\vartheta_f$ by the AEIn formula for a mass ratio of $q = 9/11$ in Fig. \[F:thf1.22\]. The other formulae predict very similar results. As in Figs. \[F:mag1.22\] and \[F:mag3\], the dotted curves show predictions assuming that the initial BBH spin distribution is preserved down to $r_f = 10 M$. The solid curves include spin precession from $r_i = 1000 M$ to $r_f = 10 M$ according to the PN equations of Sec. \[S:PN\]. The difference between the dotted and solid black curves in the top panel is below the Poisson fluctuations, another consequence of the finding of Refs. [@Bogdanovic:2007hp; @Herrmann:2009mr; @Lousto:2009ka] that isotropically oriented BBH spins remain nearly isotropic as they inspiral. Careful examination of the upper panel reveals that spin precession has shifted the BBHs with ${\boldsymbol{\chi}_1}$ initially aligned with ${\mathbf{L}_N}$ (blue distribution) to larger $\vartheta_f$, while the anti-aligned BBHs have conversely shifted to smaller $\vartheta_f$. This trend is much more pronounced in the middle and bottom panels of Fig. \[F:thf1.22\]. Spin precession actually results in the initially aligned BBHs ($\theta_1 \leq 30^\circ$) having larger values of $\vartheta_f$ at $r_f = 10 M$ than the anti-aligned BBHs ($\theta_1 \geq 150^\circ$), a reversal of what would be predicted from the initial spin distributions shown by the dotted curves. The spin-orbit resonances explain this highly counterintuitive result. The BBHs initially with $\theta_1 \leq 30^\circ$ are influenced by the ${\Delta \phi}= 0^\circ$ resonances which align the BBH spins with each other and anti-align $\mathbf{S} = {\mathbf{S}_1}+ {\mathbf{S}_2}$ with ${\mathbf{L}_N}$. Both effects lead to larger predicted values of $\vartheta_f$. Conversely, the BBHs initially with $\theta_1 \geq 150^\circ$ are influenced by the ${\Delta \phi}= 180^\circ$ resonances, which greatly decrease the magnitude of $\mathbf{S}$ and align it with ${\mathbf{L}_N}$. This explains the reduced values of $\vartheta_f$ for these BBHs seen in the bottom panel of Fig. \[F:thf1.22\]. This same effect can be seen for a mass ratio of $q = 1/3$ in Fig. \[F:thf3\], albeit with less significance owing to the weaker resonances at this smaller mass ratio. Figs. \[F:thf1.22\] and \[F:thf3\] again illustrate the importance of accounting for spin precession between $r_i = 1000 M$ and $r_f = 10 M$ when attempting to predict final spins. Spin Precession Uncertainty {#S:err} =========================== So far, we focused on how spin precession between $r_i$ and $r_f$ alters the expected distribution of final spins. In this Section, we show that spin precession introduces a fundamental uncertainty in predicting the final spin. An uncertainty $\Delta r$ in the BBH separation leads to an uncertainty $\Delta t_{\rm GW}$ in the time until merger. If this uncertainty is comparable to the precession time $t_p$, the phase of the spin precession at which the merger occurs will be uncertain as well. This new uncertainty is independent of and may exceed that associated with the NR simulations themselves. Readers only interested in astrophysical distributions of final spins may wish to proceed to the discussion in Sec. \[S:disc\]. ------------------------- -------------------------     ------------------------- ------------------------- It is often useful to define the final spin direction relative to the orbital angular momentum ${\mathbf{L}_N}$ at different separations. We therefore generalize the angles defined in Eqs. (\[E:thf\]) and (\[E:thi\]) to the separation-dependent quantities $$\begin{aligned} \label{E:vt} \vartheta_f(r) &\equiv& \arccos [{\hat{\mathbf{L}}_{N}}(r) \cdot {\hat{\boldsymbol{\chi}}_f}(r)]\,, \label{E:vtf} \\ \vartheta_i(r) &\equiv& \arccos [{\hat{\mathbf{L}}_{N}}(r_i) \cdot {\hat{\boldsymbol{\chi}}_f}(r)]\,. \label{E:vti}\end{aligned}$$ Note that these quantities reduce to the previously defined angles in the appropriate limit: $\vartheta_f(r_f) = \vartheta_f$, $\vartheta_i(r_i) = \vartheta_i$. These definitions address two ambiguities; (i) the choice of the reference orbital angular momentum and (ii) the separation at which a given fitting formula is evaluated. ------------------------- -------------------------     ------------------------- ------------------------- Before we discuss the uncertainties in determining these angles and the final spin magnitude, we illustrate the evolution of these quantities during the PN inspiral for a few characteristic examples. In Fig. \[fig: illustration\_winif\] we display the final spin magnitude $\chi_f(r)$ and the angle $\vartheta_f(r)$ as predicted by the AEIn and the Kesden formulae for a binary with mass ratio $q=9/11$, extremal spins, and initial spin orientation specified by the angles $\theta_1=120^{\circ}$, $\theta_2=60^{\circ}$, ${\Delta \phi}= 288^{\circ}$. The behavior of the AEIo, FAU and BKL formulae is quite similar to the Kesden formula. The different curves in each panel correspond to slightly different initial frequencies or separations, $M\omega_i=3.16\times 10^{-5}$, $3.17\times 10^{-5}$, $3.18\times 10^{-5}$ and $3.19\times 10^{-5}$. The spin precession generically manifests itself in the oscillatory character of the curves; these oscillations would be absent for the resonant configurations described in Sec. \[S:res\]. The thin solid lines represent [*envelope*]{} functions obtained by fitting fourth-order polynomials to the maxima and minima, respectively, of the evolutions starting with $M\omega_i=3.16\times 10^{-5}$. Note that these fits contain no information on the results obtained by using different values of $M\omega_i$, and yet they still provide excellent envelopes in all cases. This figure illustrates two ambiguities in predicting ${\boldsymbol{\chi}_f}$: (i) the [ *initial*]{} frequency $\omega_i$ at which the BBH parameters are specified, and (ii) the [*final*]{} separation $r_f$ at which the given formula for ${\boldsymbol{\chi}_f}$ should be applied. Uncertainty in the separation at which the binary decouples from external interactions could lead to ambiguity in $\omega_i$ in theoretical studies, while uncertainty in the observed distance, projected separation, or line-of-sight velocity could lead to uncertainty in $\omega_i$ for models of particular systems. Gauge-dependent definitions of $r_f$ could lead to uncertainty in the separation at which fitting formulae should be applied. Our task in evaluating the resulting uncertainties for the fitting formulae AEIn, AEIo, FAU, BKL and Kes introduced in Sec. \[S:dist\] is somewhat simplified because both ambiguities are rooted in the rapid variations of the phase and in the resulting oscillations in the final quantities. These precession-induced oscillations are a clear manifestation of the hierarchy of time scales introduced in Eq. (\[E:thier\]): $t_p \ll t_{\rm GW}$. In the upper panels of Fig. \[fig: illustration\_winii\] we show the angle $\vartheta_i(r)$ for the same binary configuration illustrated in Fig. \[fig: illustration\_winif\]. In the lower panel of Fig. \[fig: illustration\_winii\] we consider instead, for comparison, a system with lower mass ratio $q=1/3$ and initial spin orientation $\theta_1=154^\circ$, $\theta_2=124^\circ$, $\Delta \phi=216^\circ$. As before, different curves correspond to different initial frequencies. The predicted spin direction as described by $\vartheta_i(r)$ shows little variation with $\omega_i$. On the other hand, the figure demonstrates a strong dependence of $\vartheta_i(r)$ on the separation $r$ at which we apply the fitting formulae. ------------------------- -------------------------   ------------------------- ------------------------- In the remainder of this Section, we discuss the uncertainties caused by the rapid spin precession of the following quantities: - $\chi_f(r_f)$: The magnitude of the final spin as predicted by applying a given fitting formula at small binary separation $r_f=10M$, i.e. shortly before merger. - $\vartheta_f(r_f)$: The angle between the orbital angular momentum at $r_f=10M$ and the final spin (as predicted using the binary parameters at $r_f=10M$). - $\vartheta_i(r_f)$: The angle between the orbital angular momentum of the binary at large separation and the final spin (as predicted using the binary parameters at $r_f=10M$). - $\vartheta_i(r_i)$: The angle between the orbital angular momentum of the binary at large separation and the final spin predicted using the binary parameters at this same large separation. We investigate the claim that the AEIn formula, unlike the others, can determine this angle without evolving the BBH parameters down to $r_f$. These quantities are important for modeling the assembly of supermassive black holes in the context of cosmological structure formation (see e.g. [@Volonteri:2002vz; @Berti:2008af; @Tanaka:2008bv; @Lagos:2009xr; @Sijacki:2009mn; @Fanidakis:2009ct]). They are also relevant for electromagnetic counterparts of gravitational-wave sources [@Dotti:2006zn], especially when the invoked mechanism producing the counterparts depends on the recoil velocity of the remnant black hole [@Lippai:2008fx; @Schnittman:2008ez; @Shields:2008va]. We determine the precession-induced uncertainties as follows. Individual evolutions, such as those considered in Fig. \[fig: illustration\_winif\], suggest that the width of the envelopes or the dispersion induced by varying the initial frequency provide very similar estimates for the uncertainty in $\chi_f(r_f)$ and $\vartheta_f(r_f)$. We have verified this conjecture by evolving the evenly spaced $10\times 10\times 10$ grid of initially isotropic, maximally spinning BBH configurations introduced in Sec. \[S:align\] for mass ratio $q=2/3$ and several slightly different initial frequencies. When we estimate uncertainties by varying $M\omega_i$ from $3.16\times 10^{-5}$ to $3.22\times 10^{-5}$ in steps of $0.005\times 10^{-5}$ we obtain the red dashed histograms in Fig. \[fig: q15\_wini\_vs\_env\]. These histograms are in good agreement with the black solid histograms, where the uncertainty was estimated from the width of the envelopes. In order to reduce computational cost, in the remainder of this Section we determine the uncertainties $\Delta \chi_f(r_f)$ and $\Delta \vartheta_f(r_f)$ by evolving an ensemble of binaries from a [*single*]{} initial frequency ($M\omega_i=3.16\times 10^{-5}$) and using the envelope method. Fig. \[fig: illustration\_winii\] shows that the envelope method does not adequately describe the uncertainty in $\vartheta_i(r)$. Why does this angle behave so differently from $\vartheta_f(r)$ as illustrated in Fig. \[fig: illustration\_winif\]? The direction of ${\hat{\mathbf{L}}_{N}}(r_i)$ is fixed, while according to the AEIn formula ${\hat{\boldsymbol{\chi}}_f}(r)$ points in the direction of ${\hat{\mathbf{J}}}(r)$. The total angular momentum $\mathbf{J}$ only varies on the radiation timescale $t_{\rm GW}$, so according to Eq. (\[E:vti\]) the AEIn prediction of $\vartheta_i(r)$ should only vary on this slower timescale as well. The left panels of Fig. \[fig: illustration\_winii\], at least at small orbital frequencies $M\omega$ where $t_p \ll t_{\rm GW}$, indeed lack the high-frequency oscillations characteristic of spin precession. In contrast, the Kesden predictions for $\vartheta_i(r)$ shown in the right panels of Fig. \[fig: illustration\_winii\] are varying in a [*more*]{} complicated way than the predicted values of $\vartheta_f(r)$. Changes in the angle $\vartheta_i(r)$ between the fixed ${\hat{\mathbf{L}}_{N}}(r_i)$ and varying ${\hat{\boldsymbol{\chi}}_f}(r)$ reflect the full complexity of spin precession for misaligned, unequal-mass BBHs. The simpler variation in $\vartheta_f(r)$ occurs because both ${\hat{\mathbf{L}}_{N}}(r)$ and ${\hat{\boldsymbol{\chi}}_f}(r)$ are jointly precessing about ${\hat{\mathbf{J}}}(r)$, albeit on the same short timescale $t_p$. Since the envelope method fails for $\vartheta_i(r)$, we somewhat arbitrarily define the uncertainty $\Delta \vartheta_i(r_f)$ as the maximum deviation of $\vartheta_i(r)$ from $\vartheta_i(r_f=10M)$ in the window $r_f<r<2r_f$. This window covers approximately the range of initial separations within the reach of present and near-future numerical relativity simulations, while smaller separations must be excluded due to the breakdown of the PN expansion. The formulae other than AEIn do not claim to predict ${\hat{\boldsymbol{\chi}}_f}$ from the BBH parameters at large separations. To apply these formulae correctly, one must evolve the BBH parameters inwards to $r_f$ according to PN equations such as those in Sec. \[S:PN\] before applying the formulae. This evolution requires significant additional effort, but if performed properly would only increase $\Delta \vartheta_i(r_i)$ above $\Delta \vartheta_i(r_f)$ by the uncertainty in the PN equations themselves. The uncertainty coming from PN evolutions could be quantified by comparing different PN orders and pushing the calculation of spin contributions to higher order; such an analysis is beyond the scope of this paper. The AEIn formula is special in that it predicts $\vartheta_i(r_i)$ without this additional PN evolution. Since AEIn claims that both ${\hat{\mathbf{L}}_{N}}(r_i)$ and ${\hat{\boldsymbol{\chi}}_f}(r)$ are independent of $r$, the uncertainty $\Delta \vartheta_i(r_i)$ for this formula is the maximum deviation from $\vartheta_i(r_f=10M)$ over the [*entire*]{} interval $r_f<r<r_i$. Since the orbital angular momentum ${\mathbf{L}_N}$ increasingly dominates over spin contributions in the sum $\mathbf{J} = {\mathbf{L}_N}+ {\mathbf{S}_1}+ {\mathbf{S}_2}$ at large separations, ${\mathbf{L}_N}$ has little opportunity to precess at large separations and the uncertainty $\vartheta_i(r_i)$ asymptotes to a constant value in this limit. We have evolved the uniform $10\times 10\times 10$ grid of maximally spinning binaries introduced in Sec. \[S:align\] for three different mass ratios: $q=9/11$, $q=2/3$ and $q=1/3$. The average uncertainties (plus or minus their associated standard deviations) are summarized in Table \[tab:uncertainties\]. Errors in the final spin magnitudes due to the rapid spin precession are in the range $\Delta \chi_f \lesssim 0.03$ for all mass ratios. The FAU formula performs exceptionally well for nearly equal masses, although it deteriorates to the level of the other predictions for $q=1/3$. We suspect that this is because several of the higher-order terms in $\eta$ in the FAU formula are symmetric in the dimensionless spins ${\boldsymbol{\chi}_1}, {\boldsymbol{\chi}_2}$, while physically one would expect the spin of the more massive black hole to be more important in the limit $q \to 0$. Overall however, all formulae are able to predict the spin magnitude with rather good accuracy. The uncertainty $\Delta \vartheta_f(r_f)$ in the angle between the final spin and the orbital angular momentum shortly before merger is typically in the range of a few to 20 degrees. Investigation of the angular dependence of the spin uncertainties shows that the AEIn formula tends to behave better for initially aligned spins (small $\theta_1$ and $\theta_2$) and worse for anti-aligned cases. This is likely a consequence of anti-aligned binaries being closer to the limit ${\bf L}(r) \approx -{\bf S}(r)$ where transitional precession [@Apostolatos:1994mx] occurs, violating assumptions (iii) and (iv) of Ref. [@Barausse:2009uz]. All formulae are able to predict the angle $\vartheta_i(r_f)$ between the initial orbital angular momentum and the final spin with decent accuracy. The AEIn predictions are overall more accurate, but investigation of the angular dependence reveals that this accuracy deteriorates (as expected) when $q=1/3$ and the spin of the larger black hole is nearly anti-aligned. In this limit the uncertainties increase up to $\sim 20^\circ$. This is again a consequence of those configurations approaching the transitional precession regime, where ${\bf L}(r) \approx -{\bf S}(r)$. The AEIn prediction is unique in that it claims to predict $\vartheta_i(r_i)$ using the binary parameters at large separation without PN evolution. Our findings confirm (quite remarkably) that the majority of binaries in an initially isotropic ensemble result in a final spin which is nearly aligned with the orbital angular momentum at large binary separation. The values of $\theta_J$ shown in Figs. \[F:J1.22\] and \[F:J3\] suggest that this would not be the case for BBHs initially anti-aligned with ${\mathbf{L}_N}$. The accuracy of the AEIn predictions also decreases for unequal masses (as expected and verified by our results for $q=1/3$). More extreme mass ratios are expected to play a significant and possibly dominant role in the coalescence of SBH binaries [@Koushiappas:2005qz; @Sesana:2007sh; @Gergely:2007ny], so it will be crucial to test the robustness of the Barausse-Rezzolla predictions for $q=1/10$ and beyond. Accurate PN evolutions are more difficult in this regime, and we plan to investigate more extreme mass ratios in the future. Discussion {#S:disc} ========== In this paper, we examined how precession affects the distribution of spin orientations as BBHs inspiral from an initial separations $r_i \approx 1000 M$ where gravitational radiation begins to dominate the dynamics, all the way down to separations $r_f \simeq 10 M$ where numerical-relativity simulations typically begin. We confirmed previous findings that isotropic spin distributions at $r_i \simeq 1000 M$ remain isotropic at $r_f \simeq 10 M$ [@Bogdanovic:2007hp; @Herrmann:2009mr; @Lousto:2009ka]. However, torques exerted by circumbinary disks may partially align BBH spins with the orbital angular momentum at separations $r > r_i$ before gravitational radiation drives the inspiral [@Bogdanovic:2007hp]. Recent simulations suggest that the residual misalignment of the BBH spins with their accretion disk could typically be $\sim 10^\circ (30^\circ)$ for cold (hot) accretion disks, respectively [@Dotti:2009vz]. Partially motivated by these findings, we carried out a more careful analysis of spin distributions that are partially aligned with the orbital angular momentum at $r = r_i$. We found that spin precession efficiently aligns the BBH spins with each other when the spin of the more massive black hole is initially partially aligned with the orbital angular momentum, increasing the final spin. We found the opposite trend when the spin of the more massive black hole is initially anti-aligned with the orbital angular momentum. Long evolutions are necessary to capture the full magnitude of the spin alignment. This could explain why these trends were not observed in the PN evolutions by Lousto [*et al.*]{} [@Lousto:2009ka], which began at a fiducial binary separation $r = 50M$. Some models of BBH evolution (see e.g. [@Sesana:2007sh; @Koushiappas:2005qz]) suggest that SBH mergers might have comparable mass ratios ($q \lesssim 1$) at high redshift and more extreme mass ratios at low redshift. Since spin alignment is stronger for comparable-mass binaries, more alignment might be expected in SBH binaries at high redshifts. Observational arguments (see e.g. [@Volonteri:2007tu]) and magnetohydrodynamic simulations of accretion disks [@Tchekhovskoy:2009ba] provide some evidence that black hole spins are related to the radio loudness of quasars. If so, the inefficient alignment (and consequently smaller spins) produced by unequal-mass mergers at low redshift would at least be consistent with recent observational claims that the mean radiative efficiency of quasars decreases at low redshift [@Wang:2009ws; @Li:2010vv]. Stellar-mass black hole binaries should also have comparable mass ratios, so significant spin alignment could occur in such systems as well. We also pointed out that predictions of the final spin ${\boldsymbol{\chi}_f}$ usually suffer from two sources of uncertainty: (i) the uncertainty in the [*initial*]{} frequency $\omega_i$ at which the BBH parameters are specified, and (ii) the uncertainty in the [*final*]{} separation $r_f$ at which the given formula for ${\boldsymbol{\chi}_f}$ should be applied. Both ambiguities are rooted in the rapid precessional modulation of the orbital parameters, which in turn results from the precessional timescale $t_p$ being much shorter than the radiation timescale $t_{\rm GW}$. Spin precession induces an intrinsic inaccuracy $\Delta \chi_f \lesssim 0.03$ in the dimensionless spin magnitude and $\Delta \vartheta_f \lesssim 20^\circ$ in the final spin direction. The spin-orbit resonances studied in this paper should have significant effects on the distribution of gravitational recoil velocities resulting from BBH mergers, because the maximum recoil velocity has a strong dependence on spin alignment [@Gonzalez:2007hi; @Campanelli:2007ew; @Dotti:2009vz]. We plan to extend this study to investigate the predictions of different formulae for the recoil velocities that have been proposed in the literature. Acknowledgements {#acknowledgements .unnumbered} ---------------- We are particularly grateful to Vitor Cardoso for helping to test our numerical implementation of the PN evolution equations described in Sec. \[S:PN\], and to Étienne Racine for pointing out the possible relevance of the quadrupole-monopole interaction. We would also like to thank Enrico Barausse, Manuela Campanelli, Yanbei Chen, Pablo Laguna, Carlos Lousto, Samaya Nissanke, Evan Ochsner, Sterl Phinney and Manuel Tiglio for useful discussions. This work was supported by grants from the Sherman Fairchild Foundation to Caltech and by NSF grants No. PHY-0601459 (PI: Thorne) and PHY-090003 (TeraGrid). M.K. acknowledges support from NASA BEFS grant NNX07AH06G (PI: Phinney). E.B.’s research was supported by NSF grant PHY-0900735. U.S. acknowledges support from NSF grant PHY-0652995. ------- ------ ---------------------- ---------------------- ---------------------- ---------------------- ---------------------- ---------------------- model $q$ $\theta_1=10^\circ$ $\theta_1=20^\circ$ $\theta_1=30^\circ$ $\theta_1=10^\circ$ $\theta_1=20^\circ$ $\theta_1=30^\circ$ AEIn 9/11 $0.867\pm 0.064$ $0.863\pm 0.065$ $0.857\pm 0.066$ $0.914\pm 0.034$ $0.905\pm 0.036$ $0.892\pm 0.038$ AEIo 9/11 $0.866\pm 0.063$ $0.863\pm 0.064$ $0.856\pm 0.065$ $0.912\pm 0.034$ $0.904\pm 0.036$ $0.891\pm 0.038$ FAU 9/11 $0.873\pm 0.059$ $0.868\pm 0.060$ $0.861\pm 0.061$ $0.909\pm 0.035$ $0.901\pm 0.037$ $0.888\pm 0.039$ BKL 9/11 $0.862\pm 0.067$ $0.858\pm 0.068$ $0.851\pm 0.070$ $0.905\pm 0.037$ $0.898\pm 0.039$ $0.884\pm 0.042$ Kes 9/11 $0.901\pm 0.072$ $0.896\pm 0.073$ $0.889\pm 0.075$ $0.950\pm 0.038$ $0.941\pm 0.041$ $0.927\pm 0.044$ AEIn 2/3 $0.886\pm 0.052$ $0.882\pm 0.053$ $0.875\pm 0.054$ $0.922\pm 0.030$ $0.914\pm 0.031$ $0.900\pm 0.034$ AEIo 2/3 $0.886\pm 0.052$ $0.882\pm 0.052$ $0.876\pm 0.054$ $0.922\pm 0.030$ $0.914\pm 0.031$ $0.900\pm 0.034$ FAU 2/3 $0.901\pm 0.043$ $0.895\pm 0.044$ $0.886\pm 0.046$ $0.924\pm 0.029$ $0.915\pm 0.030$ $0.901\pm 0.031$ BKL 2/3 $0.882\pm 0.052$ $0.878\pm 0.053$ $0.870\pm 0.054$ $0.914\pm 0.031$ $0.906\pm 0.032$ $0.893\pm 0.035$ Kes 2/3 $0.921\pm 0.056$ $0.917\pm 0.057$ $0.909\pm 0.059$ $0.958\pm 0.031$ $0.949\pm 0.034$ $0.935\pm 0.037$ AEIn 1/3 $0.950\pm 0.025$ $0.946\pm 0.025$ $0.938\pm 0.026$ $0.957\pm 0.023$ $0.951\pm 0.023$ $0.941\pm 0.022$ AEIo 1/3 $0.958\pm 0.025$ $0.953\pm 0.026$ $0.944\pm 0.026$ $0.964\pm 0.023$ $0.958\pm 0.023$ $0.947\pm 0.022$ FAU 1/3 $0.972\pm 0.013$ $0.964\pm 0.014$ $0.951\pm 0.016$ $0.975\pm 0.012$ $0.966\pm 0.012$ $0.953\pm 0.011$ BKL 1/3 $0.931\pm 0.020$ $0.927\pm 0.020$ $0.921\pm 0.021$ $0.936\pm 0.018$ $0.931\pm 0.018$ $0.923\pm 0.018$ Kes 1/3 $0.968\pm 0.021$ $0.965\pm 0.022$ $0.958\pm 0.023$ $0.974\pm 0.019$ $0.970\pm 0.019$ $0.962\pm 0.020$ model $q$ $\theta_1=150^\circ$ $\theta_1=160^\circ$ $\theta_1=170^\circ$ $\theta_1=150^\circ$ $\theta_1=160^\circ$ $\theta_1=170^\circ$ AEIn 9/11 $0.551\pm 0.080$ $0.527\pm 0.080$ $0.511\pm 0.079$ $0.535\pm 0.072$ $0.510\pm 0.076$ $0.493\pm 0.080$ AEIo 9/11 $0.551\pm 0.080$ $0.527\pm 0.080$ $0.512\pm 0.079$ $0.535\pm 0.072$ $0.510\pm 0.077$ $0.493\pm 0.080$ FAU 9/11 $0.542\pm 0.076$ $0.520\pm 0.076$ $0.506\pm 0.076$ $0.530\pm 0.070$ $0.507\pm 0.074$ $0.492\pm 0.076$ BKL 9/11 $0.514\pm 0.088$ $0.488\pm 0.087$ $0.471\pm 0.086$ $0.496\pm 0.078$ $0.468\pm 0.083$ $0.449\pm 0.087$ Kes 9/11 $0.531\pm 0.091$ $0.504\pm 0.090$ $0.486\pm 0.089$ $0.512\pm 0.081$ $0.483\pm 0.087$ $0.463\pm 0.091$ AEIn 2/3 $0.500\pm 0.067$ $0.467\pm 0.066$ $0.445\pm 0.065$ $0.490\pm 0.057$ $0.456\pm 0.062$ $0.432\pm 0.065$ AEIo 2/3 $0.499\pm 0.067$ $0.466\pm 0.067$ $0.444\pm 0.066$ $0.489\pm 0.057$ $0.455\pm 0.062$ $0.432\pm 0.065$ FAU 2/3 $0.490\pm 0.060$ $0.460\pm 0.060$ $0.441\pm 0.059$ $0.483\pm 0.053$ $0.452\pm 0.056$ $0.432\pm 0.059$ BKL 2/3 $0.465\pm 0.072$ $0.430\pm 0.071$ $0.405\pm 0.070$ $0.454\pm 0.060$ $0.416\pm 0.066$ $0.390\pm 0.070$ Kes 2/3 $0.480\pm 0.075$ $0.442\pm 0.071$ $0.417\pm 0.072$ $0.468\pm 0.063$ $0.428\pm 0.068$ $0.401\pm 0.072$ AEIn 1/3 $0.324\pm 0.034$ $0.233\pm 0.034$ $0.151\pm 0.032$ $0.323\pm 0.013$ $0.231\pm 0.015$ $0.145\pm 0.021$ AEIo 1/3 $0.321\pm 0.034$ $0.230\pm 0.034$ $0.146\pm 0.032$ $0.319\pm 0.012$ $0.227\pm 0.015$ $0.140\pm 0.021$ FAU 1/3 $0.301\pm 0.026$ $0.222\pm 0.026$ $0.154\pm 0.024$ $0.300\pm 0.016$ $0.220\pm 0.018$ $0.151\pm 0.020$ BKL 1/3 $0.315\pm 0.034$ $0.222\pm 0.034$ $0.136\pm 0.032$ $0.313\pm 0.011$ $0.219\pm 0.013$ $0.130\pm 0.019$ Kes 1/3 $0.322\pm 0.035$ $0.227\pm 0.035$ $0.139\pm 0.033$ $0.320\pm 0.012$ $0.224\pm 0.014$ $0.133\pm 0.019$ ------- ------ ---------------------- ---------------------- ---------------------- ---------------------- ---------------------- ---------------------- model $q$ $\Delta \chi_f(r=10M)$ $\Delta \vartheta_f(r=10M)$ $\Delta \vartheta_i(r=10M)$ $\Delta \vartheta_i(r=1000M)$ ------- -------- ------------------------ ----------------------------- ----------------------------- ------------------------------- AEIn $9/11$ $0.0159 \pm 0.0099$ $ 8.38 \pm 5.30$ $ 1.47 \pm 1.09$ $ 1.48 \pm 1.10$ AEIo $9/11$ $0.0155 \pm 0.0098$ $11.38 \pm 6.18$ $ 6.55 \pm 2.73$ $-$ FAU $9/11$ $0.0021 \pm 0.0035$ $ 8.51 \pm 4.75$ $ 3.67 \pm 1.68$ $-$ BKL $9/11$ $0.0153 \pm 0.0094$ $11.74 \pm 6.39$ $ 6.89 \pm 2.92$ $-$ Kes $9/11$ $0.0174 \pm 0.0105$ $11.99 \pm 6.51$ $ 7.04 \pm 2.96$ $-$ AEIn $2/3$ $0.0205 \pm 0.0127$ $11.96 \pm 6.17$ $ 1.81 \pm 1.21$ $ 1.83 \pm 1.24$ AEIo $2/3$ $0.0199 \pm 0.0124$ $14.10 \pm 6.99$ $ 7.37 \pm 2.80$ $-$ FAU $2/3$ $0.0034 \pm 0.0026$ $10.66 \pm 5.61$ $ 4.41 \pm 1.76$ $-$ BKL $2/3$ $0.0191 \pm 0.0108$ $14.52 \pm 7.05$ $ 7.79 \pm 2.98$ $-$ Kes $2/3$ $0.0217 \pm 0.0124$ $14.83 \pm 7.24$ $ 8.02 \pm 2.99$ $-$ AEIn $1/3$ $0.0165 \pm 0.0109$ $ 8.58 \pm 4.17$ $ 3.96 \pm 4.46$ $ 4.25 \pm 5.16$ AEIo $1/3$ $0.0156 \pm 0.0101$ $ 9.57 \pm 4.62$ $10.45 \pm 4.12$ $-$ FAU $1/3$ $0.0177 \pm 0.0090$ $ 7.80 \pm 3.84$ $ 6.60 \pm 2.75$ $-$ BKL $1/3$ $0.0148 \pm 0.0089$ $ 9.81 \pm 4.79$ $11.24 \pm 4.45$ $-$ Kes $1/3$ $0.0167 \pm 0.0105$ $10.01 \pm 5.06$ $11.49 \pm 4.48$ $-$

--- abstract: 'The Hesperos mission proposed in this paper is a mission to Venus to investigate the interior structure and the current level of activity. The main questions to be answered with this mission are whether Venus has an internal structure and composition similar to Earth and if Venus is still tectonically active. To do so the mission will consist of two elements: an orbiter to investigate the interior and changes over longer periods of time and a balloon floating at an altitude between 40 and 60km to investigate the composition of the atmosphere. The mission will start with the deployment of the balloon which will operate for about 25 days. During this time the orbiter acts as a relay station for data communication with Earth. Once the balloon phase is finished the orbiter will perform surface and gravity gradient mapping over the course of 7 Venus days.' address: - 'FOTEC Forschungs- und Technologietransfer GmbH, Wiener Neustadt, Austria' - 'Space Research Centre Polish Academy of Sciences, Warsaw, Poland' - 'Maynooth University, Department of Experimental Physics, Maynooth, Ireland' - 'Universidad Carlos III, Madrid, Spain' - 'Zentralanstalt für Meteorologie und Geodynamik, Vienna, Austria' - 'Royal Institute of Technology, Stockholm, Sweden' - 'Delft University of Technology, Delft, The Netherlands' - 'Institute of Geological Sciences, Polish Academy of Science, Wroclaw, Poland' - 'Institute for Space Research, Austrian Academy of Sciences' - 'HE Space Operations GmbH, Bremen, Germany' - 'Luleå University of Technology, Luleå, Sweden' - 'Instituto Superior Técnico, Lisbon, Portugal' - 'University of Oslo, Oslo, Norway' - 'University of Glasgow, Glasgow, United Kingdom' - 'Institute of Geophysics, ETH Zürich, Zürich, Switzerland' author: - 'Robert-Jan Koopmans' - Agata Białek - Anthony Donohoe - María Fernández Jiménez - Barbara Frasl - Antonio Gurciullo - Andreas Kleinschneider - Anna Łosiak - Thurid Mannel - Iñigo Muñoz Elorza - Daniel Nilsson - Marta Oliveira - 'Paul Magnus Sørensen-Clark' - Ryan Timoney - Iris van Zelst bibliography: - 'refs\_submission.bib' title: 'Hesperos: A geophysical mission to Venus' --- Venus ,Geophysics ,Balloon ,Tectonics ,Volcanoes ,Structure Introduction {#intro} ============ Since 1975, the Austrian *Research Promotion Agency* (FFG), organises yearly the Alpbach Summer School. During the ten-day course, an intensive in-depth programme around a central space related theme is taught to about 60 graduates, post-graduates and young scientists and engineers from ESA member and cooperating states. The taught programme is complemented with a design exercise, where students propose and work out in groups a space mission related to the central theme. At the end of the course, all student groups present their mission to a panel of experts as well as to all other students, tutors and lecturers. The panel then chooses one or more proposed missions to be further worked out by a selection of students during the post-Alpbach week held in Graz.\ The theme of the Alpbach Summerschool 2014 was “The geophysics of the terrestrial planets” and focussed, as the name suggests, on the inner planets, including the Moon, of the solar system. Independently, all four student teams in Alpbach decided to design a mission to Venus. Despite the fact that the same planet was chosen, four very different missions were presented at the end of the summer school. Two of these missions were combined to one and worked out further during a further week, called the post-Alpbach week.\ In this paper the result from the post-Alpbach week is presented. The aim of the mission to Venus is to investigate the tectonic activity and its associated time scale as well as study its core and mantle. The mission is named Hesperos, which refers to the evening star or planet Venus in the evening in Greek mythology. Hesperos was thought to be the brother of Phosphorus: the morning star or planet Venus as seen in the morning. It was only realised later that the two stars represented the same celestial body.\ This paper starts with a short review of past missions to Venus and what has been learned from these missions in context of the proposed mission. This forms the starting point of formulating the science objectives in the section that follows. These science objectives are then translated into observables. In the subsequent section the payload is presented that is required to fulfil the science objectives. The required spacecraft design is discussed next with special emphasis on those aspects that make a mission to Venus challenging. In the section after that the mission timeline is discussed and broken down in several phases. Finally, conclusions are drawn about a couple of key aspects of the mission.\ Past Missions {#past} ============= Recently, strong evidence was presented for present-day active volcanism on Venus in the form of local, transient bright spots measured by the Venus Express Monitoring Camera [@shmb15]. The transient spots were interpreted as an elevated temperature due to extrusive lava flows. They were identified in the relatively young Ganiki Chasma and are similar to locations on Earth that are associated with rift volcanism. This new finding strengthens the hypothesis that Venus is geologically as well as geodynamically active today.\ Active volcanism could be confirmed by the Akatsuki mission, which was inserted in an orbit around Venus in 2015 after a previous attempt failed [@nakamura2014]. Finding evidence of active volcanism is an important secondary objective of Akatsuki, although the main mission objectives are to study atmospheric dynamics and cloud physics.\ Volcanic activity on Venus could also explain the temporal variations in atmospheric SO$_2$ measured by Pioneer Venus (e.g., [@esposito1984]) and Venus Express [@mabm13]. Pioneer Venus performed in situ measurements in the atmosphere to determine, amongst others, its chemical composition. For this purpose, Pioneer Venus released four atmospheric entry probes, one large one and three smaller ones. The spatial variation of SO$_2$ in the Venusian troposphere was measured by the VEGA probes [@bertaux1996]. In addition to the probes measuring the chemistry of the atmosphere, both VEGA satellites made use of a meteorological balloon with an instrumented gondola to study the temporal behaviour and horizontal structure of the Venus atmosphere [@salb86].\ Volcanism itself is the surface expression of processes taking place in the interior of the planet of which very little is known. Other than that Venus has a distinct crust, as was concluded from the Venera 8 mission [@visk73], there is still a debate about the size and phase of the core.\ Pioneer Venus and Magellan were missions that contributed to a better understanding of the interior of Venus by constructing topographical maps, measuring surface characteristics and mapping the gravity field. The Pioneer Venus Orbiter mapped 93% of the surface of Venus with a surface resolution better than 150km [@peef80]. The vertical accuracy was typically 200m [@peef80]. NASA’s Magellan spacecraft was mainly dedicated to map the surface of Venus with a synthetic aperture radar and to measure the gravity field. The horizontal resolution of the radar imagery varied between 120 and 300m, depending on the altitude of the spacecraft [@pefj91]. The accuracy of the measurements varied between 50 and 100m, with a maximum accuracy of 5 m for individual measurements with a horizontal resolution of 10 km [@FordPenttengill1992]. A variety of deformational features was imaged by Magellan, such as families of graben, wrinkle ridges, ridge belts, mountain belts, quasi-circular coronae and broad rises with linear rift zones with dimensions of hundreds to thousands of kilometers [@sosb92]. These last two deformation types are not observed on Earth and appear to be unique for Venus.\ Science Objectives and Observables {#science} ================================== Although Venus and Earth are similar in size, mass and distance from the Sun, the chemical composition of the atmosphere, surface pressure and temperature and rotation reveal that they are very different worlds. The reason for these differences are not understood. The aim of the Hesperos mission is to gain insight in why and how Earth and Venus evolved so differently. For this purpose the geophysics of Venus is investigated by exploring the following scientific questions: Does Venus have a similar internal structure and composition as Earth? \[SciObj1\] Is Venus tectonically active and on what time scale? \[SciObj2\] Answering these questions will not only refine evolution models of Venus but also constrain planetary evolution models in general and conditions necessary for emergence of life on our planet as well as on others, including exoplanets.\ The importance of this scientific topic has been recognized in ESA’s and NASA’s strategic plans. To answer the first question stated by the Cosmic Vision (ESA, 2005) it is necessary to study what conditions are necessary for planet formation and emergence of life. Similar questions are raised in Visions and Voyages (National Research Council, 2011) in the Building New Worlds section. In the following, each scientific question will be elaborated on and supplemented by formulating sub-questions. Internal structure and composition {#subSciObj1} ---------------------------------- While the internal structure of Earth is relatively well known due to seismological studies (e.g., [@dzan81]), the interior of Venus is not well constrained. At present, the internal structure of Venus is often assumed to be similar to that of Earth, because of the comparable mass, radius and density. However, direct evidence for this assumption is not available. Therefore, the first science objective of the Hesperos mission is to provide more constraints on the internal structure and composition of Venus. To help achieve this objective, two sub-questions have been formulated: What is the size of the core and its phase? How do mantle processes drive surface activity? #### Core size and phase Estimates of the moment of inertia factor, which characterises the radial mass distribution in a planet, vary widely for Venus. Consequently, estimates of the size of the core of Venus have large uncertainties. From Doppler tracking data of the Magellan and Pioneer Venus Orbiter missions, the second harmonic potential $k_2$, or Love number, has been estimated to be $k_2 = 0.295\pm0.066$ [@koyo96]. Models predict a $k_2$ value of $0.23 \leq k_2 \leq 0.29$ for a liquid iron core and $k_2 = 0.17$ for a solidified iron core [@koyo96]. Hence, the Doppler data suggests that Venus’ core is liquid. However, these studies implicitly assumed that the viscosity of Venus’ core is similar to Earth’s [@bohp97].\ To estimate the moment of inertia factor of Venus and extrapolate from this the core size, variations in the spin state of the planet have to be determined accurately. We focus on measuring the spin rate of Venus, which has already been determined several times in the past. However, the measured values and their corresponding uncertainties are significantly different from each other [@muhp12] as shown in Table \[spin\_rate\]. The Magellan mission measured the spin rate with an uncertainty of $\pm0.0001$, which amounts to a precision of about 2100s. For meaningful results from which the interior of Venus can be better constrained, the length of a Venus day has to be measured with a precision of 10s or at least 200 times better than the current available data [@cottereau2011]. . \[spin\_rate\] **Observations** **Period of rotation (days)** ----------------------- ------------------------------- Goldstone 1972–1982 243.022 $\pm 0.003$ Earth based 1972–1988 243.022 $\pm 0.002$ Earth based 243.025 $\pm 0.002$ Magellan gravimetry 243.0200 $\pm 0.0002$ Magellan SAR 243.0185 $\pm 0.0001$ Venera & Magellan SAR 243.023 $\pm 0.001$ Earth based 1975–1983 243.026 $\pm 0.006$ : Measurements of the spin rate of Venus, modified from [@muhp12]. For additional references, the reader is referred to [@muhp12] A way to constrain the thermal evolution and core evolution of Venus is to study the magnetic field of Venus. If Venus has a magnetic field or if remnants of a magnetic field can be detected on Venus, (numerical) models of Venus’ thermal evolution and core growth can be used to determine the most likely phase of the core at present. So far, a magnetic field has not been measured on Venus. This is due to the difficulty of detecting the intrinsic magnetic field of Venus in the presence of the induced magnetosphere [@luhmann2015]. The results from the Pioneer Venus Orbiter are consistent with a  3nT equatorial dipole magnetosphere model, which is $10^4$ weaker than the magnetic field of the Earth. #### Mantle processes As the radius of Venus is known, constraints on the size of the core also provide constrains on the size of the mantle. However, this does not reveal anything about the structure or dynamics of the mantle. Mantle dynamics and structure are, at least on Earth, closely linked to activity on the surface. As such, knowledge of Venus’ mantle is essential in understanding its surface features.\ Studying the structure of the mantle from orbit mainly involves studying the mass distribution within the mantle. The density in the mantle varies, for example, when a hot plume causes an expansion of the mantle material which results in a decrease in density. These local variations in density result in local differences in the gravity gradient [@papg14]. The main advantage of measuring the gravity gradient over the simpler and more conventional gravity mapping, is that the gravity gradient is more localised to the source of the anomaly. This makes this method more suitable for linking the gravity gradient anomalies in the mantle to their surface expressions. Surface activity investigations are therefore a natural extension and are discussed in greater detail in the next section. Mapping of the gravity gradient to gain insight into mantle dynamics has already been successfully applied to Earth by using data from the GOCE mission [@papg14] and the authors speculate that this new method can contribute to constraints of the mantle density and viscosity on global and regional scales on Earth. GOCE measured six components of the Earth’s gravity gradient tensor, of which four were determined with an optimal accuracy of 1-2 $\cdot 10^{-11}$s$^{-2}$ that was reached at scales smaller than 750km [@papg14]. Tectonic activity and time scale {#subSciObj2} -------------------------------- Plate tectonics play a major role in recycling chemicals essential for life, increasing atmospheric pressure by degassing and creating diverse environments where organisms can live [@Valenciaetal.2007]. It has even been argued that life as we know it could not have arisen without plate tectonics. Mercury, Mars and the mMoon were tectonically active in the past, but this activity was not related to plate tectonics [@Harrison2000]. The only terrestrial planet in the solar system next to Earth that may still be volcanically active and may have plate tectonics is Venus.\ To investigate this, the second scientific question is divided into two sub-questions: Is there plate movement on Venus and what are its characteristics? Is there volcanic activity on Venus and to what extent? #### Plate movement There are currently several contrasting theories of Venus’ tectonics. The ‘stagnant lid theory’ [@Turcotte1993; @BasilevskyHead2000] states that no terrestrial-type plate tectonics occurs. Instead, heat accumulates in Venus’ mantle. This leads to episodic catastrophic resurfacing that most recently happened $\sim$500 Ma ago (e.g., [@scsm92] and [@Fegley1995]). During this time a large part of the surface was covered by thick lava flows. This theory assumes that there are no tectonic plates on Venus and surface activity between those catastrophic events is limited. Other theories assume that plate tectonic regimes analogous to that of Earth exist on Venus. For example, [@SchubertSandwell1995] suggest that subduction and rifting processes similar to those on Earth could account for the thermal evolution of Venus and could potentially explain the observations of the northern and southern margins of the Latona Corona as subduction sites. A lack of active volcanism in this latter region could corroborate this theory. [@ghail2015] suggests subcrustal lid rejuvenation as an analogue to Earth’s plate tectonics to account for the observed largely stable tectonic regime with subcrustal horizontal extension. The best opportunity to detect and measure these movements of Venus’ crust is to fly a mission capable of detecting change, for instance by using a radar, according to [@ghail2015]. Another end-member theory by [@JohnsonRichards2003] emphasizes the importance of (small-scale) mantle plumes to explain Venus’ tectonic regime. [@JohnsonRichards2003] mainly provide a mechanism for the formation of coronae, namely small-scale mantle upwellings. They hypothesized that the coronae in the lowland and plain regions are older on average than those in the Beta, Atla, and Themis Regiones, because of the interference beneath the plains of mantle downwellings with small-scale upwellings. Observations that could support this theory include the observation of partially buried coronae due to volcanism and global mapping of coronae to deduce their relative ages.\ The most obvious way of investigating tectonics from orbit is by determining the thickness of the crust and combine this information with detailed topographical maps. Obtaining this data allows to detect subsurface structures such as rift systems and subduction zones. Variations in the crustal thickness result in uneven mass distributions in the upper mantle [@papg14] and the lithosphere [@Boumanetal2013]. This causes small variations in the gravitational field. On Earth, such structures are associated to gravity anomalies of a magnitude of 10-30mgal and become visible at spatial resolutions of 50-100km.\ Rift-like features have been observed on Venus, and they are of similar size as terrestrial ones. They are thousands of kilometres long and tens of kilometres wide [@Ghail2002]. Unfortunately, current resolution of Venus’ gravity field is known at a spatial resolution of 700km [@Konoplivetal1999]. This does not allow to distinguish those features in the gravity field. In order to detect rifts and possible subduction (and/or obduction zones [@Ghail2002]) changes in gravitational field of 5mgal and with spatial resolution $<$100km need to be detected.\ Current global topographical maps obtained with Magellan have a horizontal resolution of 10-20km and vertical resolution of 5-100m [@FordPenttengill1992]. About 20% of the planet’s topography is known with a spatial resolution of 1-2km and a vertical accuracy of 50m [@Herricketal2012]. To achieve the science objective neither of these is sufficient. The minimum required spatial resolution is 40m and vertical accuracy on the order of meters.\ One of the most evident manifestations of tectonic activity is an earthquake. Usually earthquakes are studied with seismometers deployed carefully on the surface [@Khanetal2013]. However, the harsh conditions on the surface of Venus make this kind of investigation less feasible. However, it is known that due to seismic coupling with the atmosphere quakes can be detected in the atmosphere [@Kanamorietal1994]. Acoustic waves generated by Rayleigh surface waves have been observed during earthquakes [@Artruetal2004; @Dautermannetal2009]. On Venus, quakes with the same magnitude are predicted to generate infrasound waves with amplitudes 600 times larger than recorded on Earth at the same height [@Garciaetal2005]. This is due to the different atmospheric pressure and density. A difficulty is that the background noise in the atmosphere at Venus is largely unknown. Especially in the mid-cloud region the atmosphere is particularly turbulent. To measure quakes in the atmosphere, the pressure should be measured at two different locations with a vertical separation from each other of a couple of tens of meters. By comparing the signals one can discriminate between pressure waves from seismic origin and pressure waves generated by other mechanisms [@KISS2015].\ Another important feature of tectonic activity is degassing of the interior. The extent of degassing can be determined from measurements of the ratios of radiogenic noble gases in the atmosphere. Measurements of $^{40}$Ar/$^{36}$Ar ratio, performed by Venera 11 and 12, gave a value of 1.19 +/- 0.07 and were confirmed by Pioneer Venus to be 1.03 $\pm$0.04 [@Fegley1995]. Those results indicate that Venus is much less degassed than Earth, Mars or even Titan, where the ratios are in the range of 150-2000 [@Owen1992; @Atreyaetal2006; @Pujoletal2013]. The low degassing rate on Venus has strong implications for the thermal and tectonic evolution of Venus and is in favour of the stagnant lid theory [@SolomatovMoresi1996]. However, since lower amounts of $^{40}$Ar can also be explained by significantly lower amounts of $^{40}$K, from which radiogenic Argon is produced, an independent isotope ratio, such as $^3$He/$^4$He, needs to be measured as well to verify the extent of degassing of Venus. To do this an instrument is required capable of measuring with a resolution of 0.1ppb and a resolution of 0.1AMU at least. The measurement range should be in the range of 2 to 45AMU. #### Active volcanism Volcanic activity is one of the most prominent manifestations of the internal activity of a planet. Multiple lines of evidence suggest that Venus is a volcanically active planet. Venus is similar in size to Earth and its cooling rate should therefore be sufficient slow to still support volcanism. Geochemical composition of relatively fresh rocks measured by Venera landers is consistent with volcanic rocks [@Surkovetal1983]. Their age is not known, but dating performed by crater counting shows that the surface of Venus is young, likely less than 500Ma [@McKinnonetal1997]. Besides that it is covered by numerous landforms that resemble volcanoes [@BasilevskyHead1998]. In addition, the measured variation of the atmospheric abundance of SO$_2$ has been interpreted to be a result of volcanic activity [@esposito1984; @Marcqetal2013]. However, other explanations such as long term variation in the circulation within the atmosphere [@ClancyMuhleman1991] are also possible. Recently, short term heat pulses, in the order of a few days, were detected on the surface by the Venus Monitoring Camera on board of Venus Express [@shmb15]. They were interpreted as magma or hot volcanic release and thus a strong indication of volcanic activity. However the extent of this has not yet been determined.\ Determining the extent of active volcanism on Venus can be performed by investigating the different manifestations of volcanism. The most obvious one is the heat signature. Lava from active volcanoes produces a clear signal in the IR spectrum. The spectral range is governed by Venus’ atmospheric window, which only allows certain wavelengths (in the range 0.8 - 1.2$\mu$m [@taylor1997]) to pass through the optically thick clouds.\ Lava deposition results in small changes in the morphology of the surface. The extent of these changes is dependent on the size of the volcano. Morphology changes can be detected by comparing radar images of the surface from different moments in time. Previous missions such as Magellan have already conducted radar surveys whose results can be used as reference. Should activity be observed it is possible to determine the level of activity by repeating measurements. A resolution $\leq$ 50km is sufficient to resolve most shield volcanoes and all pancake volcanoes as well as pyroclastic flows and impact craters. A volcanic eruption that would be rated as VEI (volcanic explosivity index, [@Newhall1982]) 3 or higher would deposit enough material that a change in topography can be resolved with a vertical resolution of 25m. An instrument that meets these requirements should be able to detect new volcanic activity.\ Another indication of volcanic activity can be obtained by monitoring volcanic gases such as SO$_2$, H$_2$O and HCl abundance variations in the atmosphere over time. Measurements of SO$_2$ can be conducted in the spectral range of 0.2 to 0.3$\mu$m, which fits well in the Venus’ spectral window. However, the mentioned volcanic gasses do not necessarily originate from volcanic activity. Therefore, measuring isotopic patterns of sulphurous gases provides constrains to the identification of potential sources for those gases and their circulation patterns. The light stable isotopes, in this particular case hydrogen (H, D), oxygen ($^{16}$O, $^{18}$O) and sulphur($^{32}$S, $^{34}$S), provide information on composition of the volcanic gases, their distribution in the atmosphere, the source material, with implications for planet formation, as well as isotope fractionation processes within the atmosphere of Venus. One advantage measuring isotopes of the volcanic gases in addition to their abundance is that mass-dependent isotope fractionation is temperature sensitive and isotopic distribution offer more detailed information into the processes involved. Ideally, measurements are performed *in-situ* at different altitudes during extended periods of time. Especially measurements within the cloud layer are particularly interesting in order to test a hypothesis of sulphur circulation in the mesosphere [@ClancyMuhleman1991].\ Finally, if volcanic eruptions currently occur on Venus, volcanic ash may be present in the higher atmosphere. Volcanic ash is composed of fragmented volcanic glass and pulverized rock [@Bukowieckietal2011]. Under terrestrial conditions volcanic ash has a relatively short residence time in the troposphere - and most of the particles fall back on the surface within a day at most. The smallest particles, very fine ash $<$10$\mu$m in diameter, can be transported to the troposphere or lower stratosphere where they can stay there for several days. Volume fraction of grain size generally decreases with size in the fine ash range. In order to capture the grain size distribution of those airborne particles measurements in the range from 0.1-10$\mu$m and with accuracy of at least 0.1$\mu$m are required.\ Detecting volcanic ash in the atmosphere, complemented with the measurements of volcanic gas abundances and isotopes could give strong evidence for proof of very recent and/or current volcanic activity. Payload ======= As has become clear from the previous section, a wide range of different parameters have to be measured to fulfil the mission objectives. Some of the parameters can be measured by the same type of instrument while for others different types are required. Table \[observables\] summarises the required observables for each sub-question that has been discussed in the previous section. The last column indicates for each type of observable the required instrument type to perform the measurements with. They will be discussed in more detail below. **Objective** **Observable** **Instrument type** --------------- ------------------------------------- ------------------------- - spin rate radar (O) - magnetic field magnetometer (B) - gravity gradient gravity gradiometer (O) - topography radar (O) - gravity gradient gravity gradiometer (O) - topography radar (O) - acoustic waves infrasound detecto (B) - isotopic ratios of noble gases NMS/TLS (B) - topography radar (O) - surface heat signature spectrometer/camera (O) - amount of volcanic gases spectrometer (O) - isotopic ratios of volcanic gases NMS/TLS (B) - amount of volcanic ash nephelometer (B) : Summary of observables. (O): orbiter, (B): balloon[]{data-label="observables"} #### Radar A synthetic aperture radar (SAR) will be used to measure the surface topography and determine the spin rate of the planet. The resolution of a radar scan is determined by the length of the synthetic aperture [@Doerry2004], which can be adjusted by varying the frequency of the pulses, therefore it is possible for one radar to have several operational modes with differing resolution and therefore differing data-rate. For the primary objective of generating detailed topographical maps, a resolution of $\leq$ 50km is sufficient for the identification of volcanic features. As will become clear from section \[orbSciPhase\], the SAR will only be operated during part of the orbit. The resulting data rate is therefore low enough to send to the Earth.\ In measuring the spin rate, a reference point on the surface of Venus is required. The Venera landers, whose remains are still present on the surface of Venus and whose location is roughly known, are large and made of metal which is highly reflective to radar. This makes them a perfect candidate for reference points on the surface of Venus. Although no longer in operation, their remains will still be visible to a radar with a 20 m resolution. By measuring the position of the landers several times during the mission lifetime the variation of the spin rate over time can be determined. Only for those parts of the surface where the Venera landers are, the resolution of the SAR will be temporarily increased to 20 m. Alternatively, spin rate measurements can be performed by position measurement of surface features as well, as was done by Mueller et al.[@muhp12]. #### Gravity gradiometer Gravity gradient measurements will be performed with a gravity gradiometer. As was mentioned in section \[subSciObj2\] this instrument needs to have a spatial resolution of $<$100km and be able to measure changes as low as 5mgal. This can be obtained with a cold atom gradiometer on an orbiter [@Carrazetal2014]. An important requirement for operating this instrument is that it is in a vibration free environment.\ As a concept this atomic quantum sensor is beneficial due to the fact that it is not limited in the same ways as ordinary electrostatic gravity gradiometers. The new method have been proven to work reliable on Earth and, although it has not been used on any previous space mission, zero-g testing indicates the benefits of this technique. Reliable specifications for this instrument will be obtained after more thorough testing and development [@Muntinga2013]. #### Spectrometer/camera - in orbit Spectrometers are required for the determination of the amount and isotopic ratios of noble and volcanic gases and a camera to investigate the heat signatures on the surface of the planet. Because of the wide range of requirements for the different measurements, several spectrometers will be employed, each suitable for a different spectral range.\ An IR spectrometer and camera will be used for detecting spots with high thermal flux on the surface. This spectrometer is based on elements of the SPICAV and VIRTIS instruments on board Venus Express. To increase the chances of detecting hot spots the instrument should be operated as long as possible over the whole mission lifetime. Alongside the IR spectrometer an UV spectrometer will be employed similar to the SPICAV instrument on board Venus Express. #### Spectrometer/camera - in-situ In order to determine, more exactly, the amount of each chemical in the atmosphere as well as the isotopic ratios, a neutral mass spectrometer (NMS) and a tunable laser spectrometer (TLS) will be employed. The NMS includes a quadrupole mass analyser and a gas processing system which will be able to determine major and trace gas species with a high sensitivity (10$^{-6}$) [@Brinckerhoff2010]. The TLS uses infra-red lasers to analyse air samples in a so called multi-pass Herriot Cell with a senistivity of 2ppm for water. The unique signature of the absorbed laser light provides a measure of the concentration of various gas molecule species (in this case SO$_2$ and H$_2$O) and their isotopes. A TLS as part of a sample analysis unit was already used on multiple planetary missions, like the SAM (Sample Analysis at Mars unit) on Curiosity [@Webster2008].\ The NMS and TLS require to take samples of the atmospheric gases and need therefore be operated *in-situ*. These instruments need to perform multiple measurement cycles with a required measurement accuracy of 5-10% and 1-2% for SO$_2$ and H$_2$O and their isotopes, respectively, as well as 5-10% for noble gases (argon and helium isotopes). As there is a chance that the atmosphere contains solid particles the inlet of both instruments shall have an air filter to prevent solid particles from entering the instrument and clog and/or damage the system. Furthermore, ideally both spectrometers will sample at different altitudes to obtain a broader variability with along an atmospheric vertical profile and sample different atmospheric layers and their chemical composition. #### Nephelometer A nephelometer will be used to measure the size and amount of volcanic ash/dust as well as the amount of H$_2$SO$_4$ in the atmosphere. The instrument takes gas samples from the atmosphere and illuminates them with a laser. The scattering of light due to the particles in suspension, i.e. ashes and aerosols, in particular sulphuric acid, allows to compute for size, shape refractive indices and composition [@Banfield2004]. A nephelometer matching the requirements mentioned in section \[SciObj2\] is presented in the Venus Climate Mission [@VCM2010] and the Venus Flagship Mission [@VFM2010].\ The nephelometer requires sampling of the atmosphere and need therefore be operated *in-situ*. It is expected that ash is to be found in the lowest cloud layer at about 48 to 50km above the surface [@taylor2014]. #### Infrasound detector Earthquakes can be monitored by recording and cross-correlating signals from microbarometers [@Mutschlecner2005]. Infrasound Monitoring Stations (IMS), conceived to detect nuclear explosions, often record pressure waves originating from Earthquakes, but also from, for instance, volcanoes [@Dabrowa2011]. Monitoring quakes like this has the great advantage that a challenging landing mission is not required. However, a particular difficulty is that the environmental noise in the atmosphere of Venus is not known. Wind, pressure and temperature variations as well as turbulence all result in pressure fluctuations interfering with pressure oscillations resulting from quakes. Besides that, also the platform from which the measurements are taken gives rise to noise [@KISS2015].\ To reduce the noise from the platform the microbarometers should be placed far away from it. This can easily be achiever by means of a tether. As was discussed earlier, to distinguish between pressure waves originating from quakes and those from other sources, two microbarometers with a vertical separation of a couple of tens of meters should be employed. With currently available microbarometers measurements in the range from 0.01 to 1Hz and 1 to 5Hz are possible.\ It is estimated that for COTS microbarometers deployed in the Venusian atmosphere, quakes with a magnitude of 6 can be detected from a distance of 2200km and those of magnitude 7 from about 9000km. In case the platform happens to be above the epicentre of a quake, quakes with a magnitude down to 3 can easily be detected [@KISS2015]. Note, that one of the objectives of the Hesperos mission is to detect quakes. This is in stark contrast with the study described in [@KISS2015], which aims at using seismic activity to probe the planet’s interior structure. The payload proposed for the Hesperos mission is similar to the payload for the generation 1 missions, i.e. pathfinder missions, proposed in [@KISS2015]. It that report it was concluded that the required technology for such missions is already available. #### Magnetometer Up until now no magnetic field intrinsic to Venus has been detected. That means that the magnetometer has to be flown closer to the surface and/or be more sensitive. For any meaningful measurements the altitude at which the measurements should take place is bounded by the ionosphere. In the ionosphere and above the interaction of the solar wind with the atmosphere of Venus induces a strong, varying magnetic field [@zhang2016].\ Two triaxial fluxgate magnetometers of the type of the Magnetospheric Multiscale Magnetometers flying on the Magnetospheric Multiscale (MMS) mission [@Russell2014] shall be used. Their dynamic range lies between $\pm$650nT (low range) or $\pm$10.500nT (high range) with a noise density at 1 Hz of less than 8pT/$\sqrt{Hz}$ (low range) or 100pT/$\sqrt{Hz}$ (high range).\ To increase the accuracy of the measurements the magnetometers will be flown as close as possible to the surface. As every electric current will introduce a magnetic interfering field the duty cycle of the magnetometers must alternate with those of the other instruments on the balloon and the magnetometers will be placed on a boom. The length of the boom depends on the magnetic cleanliness expected for the balloon gondola which will be better than for a usual spacecraft due to the absence of orbital control units and solar panels. To further decrease the influence of disturbances a technique already carried out during the Venus Express mission is adopted [@pope2011], namely to position one magnetometer at 2/3 of a boom to measure the magnetic background. This magnetic background is then subtracted from the measurements taken at the end of the boom. Mission Design {#design} ============== As has become clear from the previous section, there is a wide variation in required altitude for the different instruments. The most obvious variation is that instruments such as the radar and gradiometer require an orbit with as little interference from the atmosphere as possible, while instruments such as the nephelometer and sounding device require positioning in the cloud layers of Venus. For this reason, the mission design consists of two components: an orbiter operating outside the atmosphere of Venus and a balloon performing measurements in the atmosphere of Venus. An overview of which instrument is carried by the orbiter and balloon is given in table \[tab:payloadAlloc\]. The estimated mass and power consumption of the payload is 271kg and 551W for the orbiter and 16kg and 69W for the balloon. **orbiter** **balloon** ----------------------- --------------------- SAR NMS cold atom gradiometer TLS IR spectrometer nephelometer UV spectrometer infrasound detector magnetometer : Payload for the orbiter and balloon.[]{data-label="tab:payloadAlloc"} Orbiter ------- The spacecraft design, and in particular the orbiter design, is driven by the high solar flux at Venus, the high required power and data rate for the SAR and the fact that the orbiter acts as a data relay station for the data returned by the balloon. Consequently, the thermal control, power and communications subsystems require special attention.\ Thermal control is a considerable challenge for spacecraft operating around Venus. The sun’s radiation (2.6kW/m$^2$), planetary albedo ($\sim0.8$), planetary IR radiation (0.15kW/m$^2$) [@VenusFactSheet] and internal power dissipation all contribute to heat load on the thermal control system. The orbiter uses a combination of multi-layered insulation (MLI), heater lines (during eclipse), louvers and radiating/reflecting surfaces for thermal control. The worst-case incoming heat flux is about 2.8kW/m$^2$. The heat load is rejected to space using two radiators of 8m$^2$ each. External MLI is composed of 23 layers of Kapton. High-reflectivity coating, heritage from CryoSat-2, is used on exposed surfaces. During eclipse, 16 redundant heater lines retain the temperature within the operational limits of the equipment and instruments.\ The spacecraft’s power is provided by solar arrays made of Gallium Arsenide cells, heritage of Venus Express. Based on the mission operation phases, as will be discussed in section \[timeline\], the maximum expected power consumption is 1.3kW, including margins. A simple simulation was performed to estimate the power generated as a function of the position in orbit with respect to the sun. Conservative estimates of the solar cell efficiency (0.2) and coverage ratio (0.85) result in a required solar panel area of about 3m$^2$. During eclipse, the spacecraft runs on batteries recharged in sunlit parts of the orbit. The surface area is such that enough power is generated to charge the batteries.\ The orbiter provides a link with ground stations on Earth. It also acts as a relay for the data generated by the balloon. Note that the balloon is only active in the first science phase of the mission, see section \[timeline\]. For the remaining phases no relay function is required.\ Telemetry and telecommand are supported by three antennas on the spacecraft. The science data is returned via a High-Gain Antenna (HGA) in X-band; a typical choice for deep-space ESA missions. As the scientific instruments, especially the SAR, produce a considerable data volume a large steerable 3m dish antenna is chosen for the HGA. Note that Ka-band might be well established by the time a Venus mission enters the detailed design phase, resulting in an increased data rate and decreased antenna diameter, at the cost of a more demanding requirement for precise pointing. Supplementary to the HGA, an S-band Medium-Gain Antenna (MGA) provides a link to the ground stations when the spacecraft is close to Earth, during transfer and during safe mode and other occurrences when the HGA cannot be pointed accurately and no high data rate is required. Lastly, an UHF link keeps contact with the balloon. As ground stations the 35m dishes of ESA’s Deep Space Antennas (DSA) are selected. An overview of the communication system elements is shown in Table \[tab:orbiter\_comm\]. **Element** **Dish Diameter** **Gain** -------------------- ------------------- ------------------------------ Deep Space Antenna 35 m $>$107 dB [@dsa_performance] HGA X-band 3 m 48 dB MGA S-band 0.5 m 20 dB 2x UHF on orbiter - $>$2 dB : Elements of the communication system.[]{data-label="tab:orbiter_comm"} The telecommunication subsystem, including steering equipment and electronics, have an estimated mass of about 68kg. Power requirements for the HGA X-band link are about 100W, varying with the distance between Venus and Earth. MGA and UHF require just a fraction of this power, and are not listed separately.\ The X-band link is the most critical one as the scientific data will be transferred to Earth via the HGA. Limiting parameter is the available data rate, which governs operating time and data storage. Figure \[fig:datarate\_with\_color\] shows the variation of available data rate in X-band as a function of time and throughout the different science phases, see section \[timeline\]. [c]{} Further design of the communication architecture of the mission should look into using the future European Data Relay System (EDRS) as an Earth-orbit relay, instead of transmitting directly through Earth’s atmosphere. Upcoming optical communication technology might be another way of increasing the data rate significantly.\ The attitude and orbit control system (AOCS) has a similar layout as for the Venus Express [@VenusExpress1636]. It consists of three star trackers, two sun acquisition sensors and 2 inertial measurement units for attitude determination. Spacecraft control is provided by five reaction wheels and twelve 10N hypergolic bi-propellant attitude control thrusters. The estimated dry mass of the AOCS is about 60kg and has a power consumption of about 160W.\ For insertion into Venus orbit a high-thrust engine is necessary. While the propulsion system does not have to bring the orbiter into a circular orbit - which is done using aerobraking instead - it does have to deliver the 1.1km/s required to bring the spacecraft into an elliptical orbit around Venus (Venus capture). See for further details section \[timeline\]. Apogee motors commonly used for GEO satellites are a feasible option. For simplicity the engine uses the same hypergolic propellant as the AOCS thrusters. Choosing Monomethyl Hydrazine (MMH) and Mixed Oxides of Nitrogen (MON) result in a total propellant mass for engine and AOCS thrusters of about 2100kg.\ The data management system (DMS) is similar to Venus Express as well [@VenusExpress1636]. It consists of an on board data handling (OBDH) computer for processing of science and housekeeping date, a solid state mass memory for data dumping and remote terminal units (RTU) forming the interface between the different instruments and the on board computer. The required storage capacity of the mass memory is driven by the data rate of the SAR and the data link capacity to Earth over time. A storage capacity of 20Tb is sufficient for the entire mission. The estimated mass of the DMS is about 20kg and requires about 70W of power. [c]{} The structural architecture of the spacecraft, see figure \[fig:orbiter\], is influenced by multiple factors. The proximity of Venus to the sun ensures that spacecraft orbiting Venus should have no issues generating sufficient power for instrumentation using arrays of solar cells. Having baselined the Ariane 5 as the launch vehicle of choice, it was determined that the payload fairing would have adequate volume to contain a spacecraft of sufficient surface area to accommodate the radiators and allow the use of body mounted solar arrays, not unlike that of CryoSat-2. As was mentioned before, a steerable X-band antenna will be used for the data transfer to Earth. Whilst the addition of such a mechanism is not without the risk associated with moving components, this has been traded against the need to preserve propellant and minimise perturbations incurred by frequent whole spacecraft attitude changes. Furthermore, the use of body mounted solar arrays was determined to be critical in order to avoid perturbing torques on the spacecraft caused by residual atmospheric drag, which would otherwise interfere with the operation of the spacecraft-mounted gradiometer instrumentation. The reduction in the overall drag coefficient of the spacecraft by minimising the frontal area would result in a favourable reduction in propellant required for attitude control, which, in an optimal case, could allow sufficient propellant for a primary mission extension. Balloon ------- A schematic overview of the balloon is given in figure \[fig:balloon\]. As shown in this figure the balloon consists of two envelopes, with the smaller one placed in the larger one. The outer shell, which has a diameter of 4.2m, contains helium and the inner one, which has a diameter of 1.05m, contains water vapour. The inner envelope is connected to a liquid water reservoir. Controlled heat transfer between the liquid water reservoir and Venus’ atmosphere results in either condensation of the water vapour in the inner shell or evaporation of water in the reservoir. By increasing or decreasing the amount of water vapour in the inner envelope the buoyancy of the balloon can be changed. With the current design, and based on previous studies by DiCicco et al., the oscillation cycle is estimated to take six hours [@DiCicco1995]. This type of balloon is called a phase change balloon with water as the working fluid. However, also ammonia or a combination of the two could be used [@Jones1995].\ [c]{} The gondola contains most of the payload and subsystems for command and data handling, communication with the orbiter, thermal control and power provision. A deployable scissors boom is attached to the gondola to which the magnetometers and sounding module are attached. In this way the requirement for both instruments to be as far away as possible from other instruments and structures is met, see section \[payload\].\ The material of the external envelope must withstand the harsh Venus environment. A possible candidate material is a composite consisting of Teflon-Mylar-Vectran layers. The Teflon outer layer has an excellent resistance to sulphuric acid and the overall envelope structure can withstand the temperature and wind atmospheric conditions [@website:VP] [@VenusFlagMission]. A full scale test over a period of two weeks has shown that the helium leakage is negligible [@Baines2007].\ The balloon will be operated at an altitude between 40 and 60km. The temperature at this altitude varies between -30$^{\circ}$C to 130$^{\circ}$C. The estimated solar flux is estimated to be about 600 W/m$^2$. The payload requires a temperature between -20$^{\circ}$C up to 50$^{\circ}$C, while the batteries require an operating temperature between 0$^{\circ}$C to 20$^{\circ}$C. To ensure that the temperature stays within the limits a passive thermal control system consisting of MLI and surface coatings and finishes is used. The estimated mass of the thermal control system is about 2kg.\ Power for the payload and other subsystems is provided by primary batteries only. A total of four Lithium-Monofluoride batteries with an energy density of around 300Wh/kg [@James2003] is used once the balloon has been fully deployed. A Lithium-Sulphur Dioxide battery with an energy density 300Wh/kg [@James2003] is selected to provide power during the descend phase in the atmosphere, providing power between ejection from the orbiter until the balloon is fully deployed. The mass of the batteries is estimated to be just over 9kg, including an uncertainty margin of 5%.\ The balloon will be communicating with the orbiter using the UHF-band as was proved practical through the atmosphere on Venus by the Venera landers [@Keldych1977]. The communication will utilize two 0.12m UHF antennas on the balloon. The whole system has an estimated mass of about 5kg and consumes and estimated 10W of power.\ The entry probe consists of a front shell and a back shell, a parachute system to decelerate the probe and deploy the balloon, an adapter to the spacecraft and the balloon itself, formed by the gondola, the aeroshells and the water and helium tanks. The latter will be released once the aeroshell is inflated. Both the front and the back shell are covered by a Carbon Phenolic ablative material layer. This TPS (themal protection system) technology can withstand heat fluxes up to 300MW/m2, which is higher than what is expected during entry [@Phipps2005]. Additionally, the probe structure and TPS materials are designed to withstand the high deceleration loads ($\sim$60g during entry with the configuration described above).\ The design of the entry probe follows the same philosophy as of several feasibility studies based on the Pioneer Venus heritage: a rigid aeroshell with a medium ballistic coefficient [@VenusFlagMission; @VenusEP; @VenusClimate]. The 45$^{\circ}$ sphere-cone, inspired by the Pioneer Venus entry probes, results in a reduced heat and pressure load on the probe [@Dutta2012]. Total mass and power consumption {#massPower} -------------------------------- The total mass of the entry probe is estimated to be about 206kg and requires about 103W of power. A detailed mass and power breakdown of the different subsystems is provided in table \[tab:balloon\_mbudget\] and table \[tab:balloon\_pbudget\]. Indicated in the same table is the contribution of each subsystem to the total mass or power budget. The last column of each table shows the assumed margin, i.e. the quoted number in the second column includes a margin of which the magnitude is quoted in the last column.\ **Subsystem** **Mass \[kg\]** **% of dry mass** **Margin \[%\]** ------------------------ ----------------- ------------------- ------------------ Payload 15.7 7.6 varying Communications 5.0 2.4 5 C&DH/OBDH 2.1 1.0 5 Thermal 1.7 0.8 10 Power 8.8 4.3 5 Structure & mechanisms 87.6 42.4 10 Entry probe 85.6 41.5 20 **total** **206.4** - - : Mass budget of the balloon.[]{data-label="tab:balloon_mbudget"} **Subsystem** **Power \[W\]** **% of total** **Margin \[%\]** ------------------------ ----------------- ---------------- ------------------ Payload 68.7 67.0 varying Communications 10.5 10.2 5 C&DH/OBDH 5.3 5.1 5 Thermal 0.0 - - Power 7.1 6.9 5 Structure & mechanisms 1.1 1.1 10 Entry probe 10.0 9.7 20 **total** **102.6** - - : Power budget of the balloon.[]{data-label="tab:balloon_pbudget"} The mass of the payload is based the mass of existing instruments. Depending on the required amount of adaptation of existing versions a margin was put on the mass. For instance, the NMS and TLS are well understood and relatively easy to adjust for the proposed mission, so a margin of only 5% is added to the mass. On the other hand the nephelometer has a lower readiness. For this reason the margin on the mass is set at 30%. A similar approach was used for estimating the power consumption.\ The total payload mass, including margins, was used to estimate the mass of the rest of the balloon and gondola. For this purpose the mass of gondola and balloon of the Venus Climate Mission [@VCM2010] was scaled according the payload mass. Any uncertainties in the mass estimation is covered by a margin as indicated in the last column of table \[tab:balloon\_mbudget\]. A similar approach was followed for estimation of the power requirements.\ The heaviest parts of the entry probe are the front and back shell, including parachute system and S/C adapter, ($\sim$88kg) as well as the gondola structure and boom ($\sim$86kg). Together, they form more than 84% of the total entry probe mass. Most of the power, about 2/3, is consumed by the payload.\ The Hesperos orbiter has an estimated dry mass of about 1490kg and requires about 1.5kW of power. The mass and power budget of each subsystem is provided in table \[tab:orbiter\_mbudget\] and table \[tab:orbiter\_pbudget\]. For each subsystem the mass and power consumption was estimated based on existing hardware sized to the needs of the proposed mission in a similar fashion as for the balloon. Note that the mass and power budget of the AOCS includes the mass and required power of the attitude control thrusters. The tanks in which the propellants are stored are part of the propulsion subsystem.\ About 50% of the mass comes from the propulsion system. The payload and power subsystem consume most of the power; each about 35% of the total.\ **Subsystem** **Mass \[kg\]** **% of dry mass** **Margin \[%\]** ------------------------ ----------------- ------------------- ------------------ Payload 271.2 18.2 varying Propulsion 732.0 49.1 5 AOCS 59.6 4.0 5 Communications 68.0 4.6 5 C&DH/OBDH 53.8 3.6 5 Thermal 5.3 0.4 5 Power 50.4 3.4 5 Structure & mechanisms 250.1 16.8 10 **total** **1490.4** - - : Mass budget of the orbiter.[]{data-label="tab:orbiter_mbudget"} **Subsystem** **Power \[W\]** **% of total** **Margin \[%\]** ------------------------ ----------------- ---------------- ------------------ Payload 551.4 36.4 varying Propulsion 50.4 3.3 5 AOCS 161.8 10.7 5 Communications 106.1 7.0 5 C&DH/OBDH 110.3 7.3 5 Thermal 0.0 - - Power 531.0 35.0 5 Structure & mechanisms 4.6 0.3 10 **total** **1515.5** - - : Power budget of the orbiter.[]{data-label="tab:orbiter_pbudget"} Mission Phases and Timeline {#timeline} =========================== Given the dry mass of the orbiter and entry probe of about 1700kg in total while keeping in mind the propellant mass needed, an Ariane 5 rocket is required for launch. A suitable launch window will open in December 2032. After launch the satellite is sent on an interplanetary Hohmann transfer trajectory to Venus. The transfer will take about 117 days. A burn with an apogee kick motor will bring the orbiter with entry probe in an elliptic orbit around Venus with an inclination of 85$^{\circ}$ and a periapsis and apoapsis of 250km and 19500km, respectively. In this orbit, some operations will take place before the final circular orbit is achieved, this is called the Balloon Phase. The balloon will be deployed in this phase in order to reduce the mass before the transference to a circular orbit and hence, reduce the propellant mass requirements. Once this phase is completed, an aerobraking manoeuvre will take place to circularise the orbit. This manoeuvre consists on taking advantage of the drag from the planet’s atmosphere at the perigee in order to reduce the apogee altitude with very low fuel requirements until the final desired altitude is reached. When this altitude is achieved, another $\Delta$V is given in order to enter and maintain the final circular orbit and once in this orbit, the Orbiter Phase will begin then. Balloon phase {#balloonPhase} ------------- The balloon is deployed as soon as the spacecraft is in its initial elliptical orbit and before the aerobraking manoeuvre. The entry probe will be released between the apoapsis and the periapsis from a highly elliptical orbit to reduce the entry velocity to about 8.6km/s. Given the kinetic energy, a steep entry flight path angles (EFPAs) of $\sim$40$^{\circ}$ can be used, while maintaining the g-load within acceptable limits.\ The correct attitude of the probe is crucial for the TPS to work as designed. To ensure static stability, the probe will be spin stabilised [@Lorenz2006]. The spin rate and the required velocity will be delivered by the spacecraft and thus reducing the complexity and weight of the entry probe [@Phipps2005; @VenusEP]. After release of the entry probe a small course correction of the orbiter is required to return it to its initial orbit.\ Once the probe reaches the outer edge of the atmosphere, assumed at an altitude of 200km, the induced drag causes rapid deceleration. At an altitude of 70km the front and back shell are separated by a pyrotechnic mechanism. The front shell will under influence of gravity accelerate towards the surface. The back shell acts as a drogue chute and pulls the main parachute for further deceleration of the probe. The parachute itself pulls the balloon, after which inflation of both balloon envelopes start. Once the balloon envelopes are fully inflated, both the parachute and the main helium tank are released. At this point the balloon is at an altitude of about 60km and checkout and commissioning of all systems will start.\ As was explained in section \[balloon\] the balloon will be oscillating between 40 and 60km above Venus’ surface in a period of about six hours. Given the energy available from the batteries and the power consumption of the payload and subsystems the balloon phase will last for about 25 days. Due to prevailing wind directions on Venus, the balloon will circumnavigate the planet a couple of times before the batteries run out. A secondary effect is the drift towards one of the poles during this time. As such, not only a vertical profile of Venus’ cloud-level atmosphere will be obtained but also as a function of latitude.\ The mass spectrometer measurements are considered as most important to understand the general properties of Venus’ atmosphere, thus they are conducted at the beginning of the balloon phase. As they are very power intensive only three measurements at highest balloon position and three measurements at lowest balloon position are foreseen. The other balloon instruments need far less power and are alternately operated in a continuous loop interrupted only for the spectrometer measurements. During one loop the magnetometer and sounding module collect data for 20% of the time each and the nephelometer for 60% of the time.\ During the balloon release and science phase the orbiter is in a highly elliptical orbit and acting as relay station for the science data returned by the balloon. At the same time the balloon will be followed by the IR and UV spectrometers on board the orbiter. This is designated as phase I of the orbiter.\ The end of the balloon phase is scheduled for May 2033. Shortly before the energy supply fails the gondola will be cut off. It will be in free fall for approximately 2 minutes before it hits the ground. During this time the magnetometer is switched on. As the measurement cycle of this instrument is very short it is possible to obtain data during this short period. Orbiter phase {#orbSciPhase} ------------- After finishing the balloon phase the orbiter will be put in a circular orbit around Venus, which will be achieved by aerobraking in the upper atmosphere of Venus as mentioned before. This technique has been successfully demonstrated by Venus Express [@VALSERRA2011]. The final orbit will have an altitude of 250km with an inclination of 85$^{\circ}$. The orbital parameters have been chosen such that a repeat track is achieved every two Venus days. After an initial checkout and commissioning phase the main science phase starts in June 2033 with a nominal duration of seven Venus days or about five Earth years. The nominal mission ends in February 2038. Although 5 years is a long period for the primary science phase of a planetary mission, experience with other missions shows that such long durations are not uncommon. For instance, the primary science phase of ESA’s Venus Express mission was planned for 2 years, but the mission was extended 5 times, lasting over 9 years [@Svedhem2007; @Svedhem2009]. Also this long duration would allow to tune the observations according to the requirements and constraints that would arise once the missions is operative. For instance, the extension of the Venus Express missions had the main objective of adding new fine-tuned observations from lessons learned over the ongoing mission that can only arise once the mission is operative [@Hoofs2009]. Depending on the resources left the Hesperos mission can also be extended.\ The main science phase of the orbiter consists of two phases. Phase II is dedicated to surface mapping and uses the SAR as well as the IR and UV spectrometers/cameras. Phase III is dedicated to gravity gradient mapping. During this phase only the cold atom gradiometer will be operated.\ The science mission of the orbiter starts with Phase II which lasts for four Venus days. To prevent the SAR from overheating it will be operated for half an orbit and then switched off to cool down. To ensure effective cooling, the SAR is operated on the day side and switched off on the night side of Venus. While cooling down, IR spectrometers/cameras are switched on. Note that the IR camera can only be used on the night side and as such is a good combination for the SAR.\ During the first two Venus days of phase II the whole surface is mapped with the SAR. This first sweep of the surface is used as reference. A second sweep, lasting two Venus days again, is required to establish a topographical map. The end of Phase II is on the night side of Venus with spectrometers/cameras switched on.\ Then Phase III starts which is fully dedicated to gravity gradient measurements with the cold atom gradiometer. This instrument requires a disturbance free environment, i.e. any noise in the form of sound, electric current or temperature above the background temperature should be avoided. For this reason, Phase II should end at the night side of Venus so the SAR has time to cool down sufficiently before the gradiometer is switched on. Phase III ends after one Venus day.\ After Phase III another radar mapping of the Venusian surface takes place, but this time for two Venus days only. This partial repetition of Phase II is to detect possible changes in the topography. Here the first full surface scan during the first two Venus days of Phase II is used as reference again. The obtained topographical map can then be compared with the topographical map established earlier and will show changes in topography over the course of one full Venus day.\ At several moments during the mission thruster firings are required for orbit insertion and orbit maintenance. The required $\Delta$V for each type of manoeuvre is shown in table \[deltaV\]. The numbers include a margin used for calculating the required propellant mass. **Manoeuvre** **$\Delta$V (km/s)** ------------------ ---------------------- Hohmann Transfer 2.84 Elliptical Orbit 1.47 Aerobreaking 0.001 Circular Orbit 0.001 Maintenance 0.11 **Total** **4.83** : $\Delta$V budget[]{data-label="deltaV"} Given the dry mass of the satellite, as mentioned in the previous paragraph, the propellant mass can be calculated. A margin of 20% is used for dry mass. It is further assumed that the specific impulse, I$_{sp}$, of the apogee kick motor is 321s and of the attitude control thrusters 291s. Combined with the $\Delta$V requirements as shown in table \[deltaV\] this results in a propellant mass of just over 2100kg. The total spacecraft mass is thus almost 4100 kg. **Mass \[kg\]** --------------------------- ----------------- Balloon total 206.4 Orbiter total 1490.4 S/C dry mass + 20% margin 1978.8 Propellant mass 2101.6 Launch mass 4080.4 : Overall mass budget[]{data-label="totalmbudget"} Conclusions =========== A comprehensive understanding of Venus, its evolution and why it is so different from Earth, despite a couple of striking resemblances, is still lacking. Clarifying some of the open questions not only helps understanding how Venus evolved to the planet as we know it, but will also help in constraining planetary models in general and help understand and interpret the variety of exoplanets discovered in the past decade.\ In order to obtain this understanding investigating the inner structure and the atmosphere as well as the interaction with each other need to be studied. To achieve this, monitoring of the planet from orbit as well as in-situ measurements are required. The required technology for such mission is for most part present already. However, a couple of instruments and subsystems require further development before it can be employed as envisaged for the mission. These includes the balloon and gondola as well as the cold atom gravity gradiometer. Acknowledgement {#acknowledge .unnumbered} =============== The authors would like to express their gratitude to FFG and ESA for organising the Alpbach Summer School as well as the post-Alpbach week. We would also like to thank our supervisors Günter Kargl and Olivier Baur from the Space Research Institute of the the Austrian Academy of Sciences, Richard Ghail from Imperial College London and Manuela Unterberger from the Technical University of Graz for their valuable advice and enthusiasm. The authors are also greatful for the many useful comments from Colin Wilson during the review process. Author A.Łosiak would further like to acknowledge the NCN grant, which made the participation in this study possible (2013/08/S/ST10/00586).

--- abstract: 'We propose a novel receding horizon planner for autonomous surface vehicle (ASV) path planning in urban waterways. The proposed planner is lightweight, as it requires no prior map and is suitable for deployment on platforms with limited computational resources. To find a feasible path in the presence of obstacles, the planner repeatedly generates a graph, which takes the dynamic constraints of the robot into account, using a global reference path. We also propose a novel method for multi-objective motion planning over the graph by leveraging the paradigm of *lexicographic optimization* and applying it to graph search within our receding horizon planner. The competing resources of interest are penalized hierarchically during the search. Higher-ranked resources cause a robot to incur non-negative costs over the paths traveled, which are occasionally zero-valued. This is intended to capture problems in which a robot must manage resources such as risk of collision. This leaves freedom for tie-breaking with respect to lower-priority resources; at the bottom of the hierarchy is a strictly positive quantity consumed by the robot, such as distance traveled, energy expended or time elapsed. We conduct experiments in both simulated and real-world environments to validate the proposed planner and demonstrate its capability for enabling ASV navigation in complex environments.' author: - 'Tixiao Shan, Wei Wang, Brendan Englot, Carlo Ratti, and Daniela Rus [^1]' bibliography: - 'CDC\_2020\_Tixiao.bib' title: | **A Receding Horizon Multi-Objective Planner\ for Autonomous Surface Vehicles in Urban Waterways** --- Introduction {#sec::introduction} ============ Great efforts haven been devoted to achieving autonomy for autonomous surface vehicles (ASVs) in the last few decades. Among them, the recently launched Roboat project seeks to explore the complex interactions between human society and ASVs [@johnsen2019roboat]. The Roboat project aims to provide water-based transportation to relieve the congestion of saturated road-based transportation in Amsterdam, the Netherlands. Such water-based transportation includes but is not limited to applications such as tourism and public transportation, waste collection, and package delivery. The deployment of Roboat seeks to contribute novel urban infrastructure that supports the development of a modern city. Deploying an autonomous boat in the busy canals of a major city involves designing systems for perception, localization, and path planning. Among them, path planning in particular plays a crucial role in enabling safe navigation of urban canals, as its outcome directly influences the interactions between the vehicle and its surrounding environment. Various challenges can be foreseen when deploying such a system. Cruising in urban waterways is subject to rules that are similar to driving on roadways. Autonomous vehicles may only be licensed to cruise in certain regions of the canal, while following a pre-defined route from the relevant authorities. In addition, the behavior of the vehicles shouldn’t disrupt the course of human-controlled boats. With these challenges in mind, this paper focuses on developing a path planning algorithm that enables ASV navigation in complex urban waterways. We model the various challenges to be solved during navigation as costs to be minimized. Then the planning problem can be treated as a multi-objective optimization problem. We propose a lexicographic search algorithm to solve this multi-objective planning problem quickly without parameter-tuning, by ranking the objectives hierarchically. The main contributions of our work can be summarized as follows: - A novel receding horizon planner that is suitable for autonomous navigation of ASVs in urban waterways. - An efficient, multi-objective search algorithm that enables real-time performance without iterative adjustment of constraints by hierarchically ranking the objectives. - The proposed framework is validated with tests in both simulated and real-world environments. Related Work {#sec::related-work} ============ The most relevant body of prior work is in multi-objective motion planning. In pursuit of solutions that can be produced quickly, preferably in real-time, and applied to problems of high dimension, sampling-based motion planning algorithms such as the probabilistic roadmap (PRM) [@kavraki1994], the rapidly-exploring random tree (RRT) [@lavalle2001], and their optimal variants PRM\*, RRT\*, and rapidly-exploring random graphs (RRG) [@RRT*] have been adapted to solve a variety of multi-objective motion planning problems. Such approaches have typically considered the tradeoff between a resource such as time, energy, or distance traveled and a robot’s information gathered [@hollinger2014], localization uncertainty [@bopardikar2015], [@shan2017belief], collision probability [@roy2011], clearance from obstacles [@kim2003], adherence to rules [@reyes2013], and exposure to threats [@clawson2015]. We consider problems in which two or more competing resources are penalized *hierarchically*. The higher-priority resources assume non-negative costs over robot paths, and are frequently zero-valued. This is intended to capture problems in which robots must manage resources such as collision risk, access to valuable measurements or following certain rules, which are present in some regions of the environment, and absent in others. For example, [@shan2015sampling] proposed a sampled-based planning algorithm for minimum risk planning. Risk is only penalized in the regions of the environment where collision is possible. This leaves freedom for tie-breaking with respect to a secondary resource, such as distance traveled. A min-max uncertainty planning algorithm is proposed in [@englot2016sampling] for planning under uncertainty. When the primary cost, localization uncertainty, is not increasing, a secondary and a tertiary cost are introduced to break ties. These planning problems fit nicely into the framework of *lexicographic optimization*. The lexicographic method [@stadler1988] is the technique of solving a multi-objective optimization problem by arranging each of several cost functions in a hierarchy reflecting their relative importance. The objectives are minimized in sequence, and the evolving solution may improve with respect to every subsequent cost function if it does not worsen in value with respect to any of the former cost functions. Use of this methodology has been prevalent in the civil engineering domain, in which numerous regulatory and economic criteria often compete with the other objectives of an engineering design problem. Variants of the lexicographic method have been used in land use planning [@veith2003], for vehicle detection in transportation problems [@sun1999], and in the solution of complex multi-objective problems, two criteria at a time [@engau2007]. Among the benefits of such an approach is the potential for the fast, immediate return of a feasible solution that offers globally optimal management of the primary resource, in addition to locally optimal management of secondary resources in areas where higher-ranked resources are zero-valued. Due to the fact that the spatial regions in which resources are penalized can often be intuitively derived from a robot’s workspace, using facts such as whether the robot is in an allowed operating region, or whether a robot is within range of communication or sensing resources, such an approach offers an intuitive means for managing the relative importance of competing cost functions, in which the user needs only to select the order in which the resources are penalized. This stands in contrast to methods that require tuning of additive weights on the competing cost functions [@reyes2013], and robot motion planning methods that manage the relative influence of competing cost criteria using constraints, [@hollinger2014], [@roy2011]. Avoiding any potential struggles to recover feasible solutions under such constraints, the lexicographic motion planning problem is *unconstrained* with respect to the resources of interest. Problem Definition {#sec::problem-definition} ================== Path Planning {#sec::problem-planning} ------------- Let $\mathcal{C}$ be a robot’s configuration space. $x \in \mathcal{C}$ represents the robot’s configuration. $\mathcal{C}_{obst} \subset \mathcal{C}$ denotes the set of configurations that are in collision with the obstacles, which are perceived from the sensor data $\mathcal{S}$. $\mathcal{C}_{free} = cl(\mathcal{C} \backslash \mathcal{C}_{obst})$, in which $cl()$ represents the closure of an open set, denotes the space that is free of collision in $\mathcal{C}$. We assume that given a current configuration $x_{c} \in \mathcal{C}_{free}$ and a global reference path $\mathcal{G}$, the robot must travel in $adj(\mathcal{G})$, which represents the neighboring regions of $\mathcal{C}$ adjacent to $\mathcal{G}$, and reach a goal state $x_{g}$, which is located at the end of $\mathcal{G}$. Let a *path* be a continuous function $\sigma : [ 0,1 ] \rightarrow \mathcal{C}$ of finite length. Let $\Sigma$ be the set of all paths $\sigma$ in a given configuration space. A path $\sigma$ is collision-free and feasible if $\sigma \in \mathcal{C}_{free}$, $\sigma(0) = x_{c}$ and $\sigma(1) = x_{g}$. A feasible path $\sigma$ is composed of two segments, $\sigma = \sigma_{G} \cup \sigma_{\mathcal{G}}$. $\sigma_{G}$, which exists in $adj(\mathcal{G})$, is obtained by searching a directed graph $G(V,E)$, with node set $V$ and edge set $E$. $\sigma_{\mathcal{G}} \in \mathcal{G}$ is directly obtained from $\mathcal{G}$. $\sigma_{G}$ and $\sigma_{\mathcal{G}}$ can be concatenated as $\sigma_{G}(1)=\sigma_{\mathcal{G}}(0)$. An edge $e_{ij} \in E$ is a path $\sigma_{i,j}$ for which $\sigma_{i,j}(0) = x_i \in V$ and $\sigma_{i,j}(1) = x_j \in V$. Two edges $e_{ij}$ and $e_{jk}$ are said to be linked if both $e_{ij}$ and $e_{jk}$ exist. A path $\sigma_{p,q} \in G$ is a collection of linked edges such that $\sigma_{p,q} = \{e_{p \; i_1},e_{i_1 i_2}, ..., e_{i_{n-1} i_n}, e_{i_n q}\}$. The problem of finding a feasible path may be specified using the tuple $({C}_{free},x_{c},\mathcal{G})$. Lexicographic Optimization {#sec::problem-lexico} -------------------------- We define cost functions $c_k(\sigma)$, where $c_k: \Sigma \rightarrow \mathbb{R}_{0}^{+}$ maps a path $\sigma$ to a $k$ th non-negative cost, $k \in \{1,2,...,K\}$, and $K$ is the total number of costs in a multi-objective planning problem. These $K$ cost functions are applied to the problem of lexicographic optimization [@marler2004], which may be formulated over collision-free paths as $$\begin{aligned} \label{eq::lexico-formula} &\sigma^* = \underset{\sigma_{k}\in \mathcal{C}_{free}}{\text{min}}{c_{k}(\sigma) }\;\;\;\; \\ &\text{subject to}: \; c_{j}(\sigma)\leq c_{j}(\sigma_{j}^{*})\;\; \nonumber \\ &\text{where}: j=1,2,...,k-1, k>1;\nonumber \\ &\;\;\;\;\;\;\;\;\;\;\;\; k=1,2,...,K. \nonumber\end{aligned}$$ The formulation of the lexicographic method is adapted here (we refer the reader to the description from [@marler2004], Section 3.3) to show cost functions that take collision-free paths as input. We also assume specifically that $c_K: \Sigma \rightarrow \mathbb{R}^{+}$, implying that ties never occur in the bottom level of the hierarchy. In one iteration of the procedure of Equation (\[eq::lexico-formula\]), a new solution $\sigma^*$ will be returned if it does not increase in cost with respect to any of the prior cost functions $j < k$ previously examined. Necessary conditions for optimal solutions of Equation (\[eq::lexico-formula\]) were first established by Rentmeesters [@rentmeesters1996]. Relaxed versions of this formulation have also been proposed, in which $c_{j}(\sigma_{k}) > c_{j}(\sigma_{j}^{*})$ is permitted, provided that $c_{j}(\sigma_{k})$ is no more than a small percentage greater in value than $c_{j}(\sigma_{j}^{*})$. This approach, termed the *hierarchical method* [@osyczka1984], has also been applied to multi-criteria problems in optimal control [@waltz1967]. Cost Function {#sec::problem-cost} ------------- We introduce three types of costs, which are inspired by our application of Roboat in urban waterways, to demonstrate the usage of lexicographic optimization in our planning problem. The three costs, which are ranked hierarchically, are risk cost, heading cost, and distance cost. The risk accumulated along a path $\sigma$ is derived using: $$\begin{aligned} \label{eq::cost-risk} c_{1}(\sigma) & := \int_{\sigma(0)}^{\sigma(1)} Risk(\sigma(s)) \;ds \\ Risk(x) & := \left\{ \begin{array}{lr} \mathcal{R}(x) &\text{if} \; \mathcal{R}(x) > Th_{risk} \\ \; \; \; 0 &\text{otherwise} \; \; \; \; \; \; \; \; \; \; \; \; \end{array} \right.,\end{aligned}$$ where the function $Risk()$ evaluates the risk at an individual robot state. Let us assume that $\mathcal{R}(x)$ is defined as the inverse of the distance between $x$ and the nearest obstacle to $x$. The $Risk()$ function is activated if $\mathcal{R}(x)$ is larger than a risk threshold $Th_{risk}$. For example, when we let $Th_{risk}=2$, $Risk()$ gives non-zero values when the robot is within 0.5 m of any obstacles. When we let $Th_{risk}=\infty$, $Risk()$ returns zero everywhere in $\mathcal{C}_{free}$. The logic behind employing this cost function is that we wish to place a *comfort zone* between the robot and other surrounding objects, especially human-driven boats. The ASV should try to avoid this zone to minimize its influence on other vessels. Minimizing the risk cost results in minimizing the travel distance of the ASV in these zones. Another approach to create such a zone is to naively inflate the obstacle regions. However, a naive inflation of obstacles may block the entire waterway if they are close, even though there is a feasible path passing through them. An example of the proposed comfort zones, which are colored gray, is shown in Figure \[fig::pipeline\](a). In addition, we define heading cost as the secondary cost, which penalizes the heading difference between the robot and the global reference path $\mathcal{G}$: $$\begin{aligned} \label{eq::cost-heading} c_{2}(\sigma) & := \int_{\sigma(0)}^{\sigma(1)} Heading(\sigma(s)) \;ds \\ Heading(x) & := \left\{ \begin{array}{lr} \mathcal{H}(x) &\text{if} \; \mathcal{H}(x) > Th_{head} \\ \; \; \; 0 &\text{otherwise} \; \; \; \; \; \; \; \; \; \; \;\;\; \end{array} \right.,\end{aligned}$$ where the function $\mathcal{H}(x)$ gives the heading difference between $x$ and the heading of the path segment that is the closest to $x$ on $\mathcal{G}$. Due to wind and wave interference, aligning the heading of the robot with $\mathcal{G}$ perfectly is practically impossible. To avoid exhaustive control effort, we define a heading difference threshold $Th_{head}$. $Heading(x)$ returns a non-zero value when the error $\mathcal{H}(x)$ is larger than $Th_{head}$. Incorporating this cost ensures the generated path is relatively smooth while adhering to the heading of $\mathcal{G}$. At last, we define the distance cost as the tertiary cost, which is strictly positive. This ensures that ties do not occur here as they do for the primary or secondary cost. The distance cost is defined as follows: $$\begin{aligned} \label{eq::cost-dist} c_{3}(\sigma) & := \int_{\sigma(0)}^{\sigma(1)} Distance(\sigma(s)) \;ds \;.\end{aligned}$$ The definition of these three costs is tightly-related to our deployment of Roboat in urban waterways. The risk cost minimizes the interference of Roboat with other objects to ensure a safe and comfortable ride for passengers on Roboat and other vehicles. The heading cost helps yield a smooth ride for the passengers while minimizes the control efforts. The monotonically increasing distance cost guarantees no ties exist in the bottom level of the hierarchy and minimizes the travel distance if possible. Note that without loss of generality, the user can also incorporate other types of costs or rules into the cost hierarchy based on their importance. Receding Horizon Planner {#sec::planner} ======================== To extend the application of the proposed planner to platforms with limited computational resources, we design our planner in a receding horizon manner. Given the global path $\mathcal{G}$ as the reference path, we only search for a new path in $adj(\mathcal{G})$ under necessary conditions, such as obstacles lying on $\mathcal{G}$ or the path to be executed. We also assume a prior map is not available for our planner, apart from the basic topological information required to formulate the reference path $\mathcal{G}$. This is because the environment of an urban waterway changes constantly due to human activities, such as canal maintenance and moving boats. A detailed understanding of the robot’s environment is achieved by using the real-time perceptual sensor data $S$. With no requirement for a prior map, the proposed planner may be readily deployed. The pipeline of the proposed receding horizon planner is introduced in Algorithm \[alg::pseudo-algorithm\]. The planner takes the robot’s current state $x_{c}$, a global reference path $\mathcal{G}$, and the perceptual sensor data $S$ as inputs. When the global path $\mathcal{G}$ is not fully executed by the robot, we check the feasibility of path $\sigma$ using the perceived sensor data $S$. Note that we let $\sigma = \mathcal{G}$ when $\mathcal{G}$ is received at the beginning of the planning process. An illustrative example of the proposed planner is shown in Figure \[fig::pipeline\]. If obstacles are detected on $\sigma$, we then generate candidate robot states $V$ using $\mathcal{G}$ as the reference. We adapt the method introduced in [@darweesh2017open] and propose a *roll-out* and *roll-in* generation approach for sampling $V$. The states from the roll-out generation need to satisfy the robot’s dynamic constraints while diverging from $x_{c}$ and spanning $adj(\mathcal{G})$. During the roll-in generation stage, the sampled states converge to $\mathcal{G}$ while remaining kinodynamically feasible. $adj(\mathcal{G})$ is defined by $d_{span}$ and $d_{roll}$. $d_{span}$ is the maximum distance between the sampled states and $\mathcal{G}$. The total distance between the roll-out and roll-in sections is denoted as $d_{roll}$. In practice, $d_{roll}$ is set to be larger than the robot’s sensor range $d_{sensor}$ in case the roll-in section converges on obstacles. Note that $V$ is sampled using a fixed density for the purpose of clear visualization. Figure \[fig::pipeline\](b) shows the generated $V$ with a fixed density of 0.1m. We then connect the states in $V$ and obtain a graph $G=(V,E)$. We only connect a state to its eight immediate neighboring states here for the purpose of visualization and show the resulting graph in Figure \[fig::pipeline\](c). $X_{queue} \leftarrow \{V\};$ $x_{init}.parent \leftarrow \{\}$ We adapt Dijkstra’s algorithm [@dijkstra1959] to perform a lexicographic graph search on $G$, which is detailed in Algorithm \[alg::lexicographic-search\]. Provided with a graph $G(V,E)$ and $x_{init}$ as inputs, a queue $X_{queue}$ is populated with the nodes of the graph (Line 1), and the algorithm initializes $K$ cost-to-come costs for each node (Lines 2-4). Each of these costs describes the $k$ th priority cost-to-come for a respective node, along the best path identified so far per the ranking of cost functions in Equation (\[eq::lexico-formula\]). In real-time applications of the search, $x_{init}$ is designated to be the closest configuration in the graph to the robot’s current state $x_{c}$, and $x_{goal}$ is the state in $\mathcal{G}$ that the graph $G$ converges to as roll-in occurs. In each iteration of the algorithm’s while loop, the $FindMinCost_k()$ operation returns the set of configurations that share the minimum $k$ th priority cost-to-come from among the nodes provided as input (Line 9). If $X_{min}$ contains more than one configuration, lower-priority costs for the nodes in this set are examined until the set $X_{min}$ contains a single node, whose neighbors are examined in detail. The selected node is designated $x_i$ (Line 11). Node $x_{i}$ is then used, if possible, to reduce the costs-to-come associated with neighboring nodes $x_{j}$, if edge $e_{ij}$ exists. In Line 16, if $c_{k}(x_{i},x_{j})$, which represents the $k$ th priority cost from $x_{init}$ to $x_{j}$ via $x_{i}$, is lower than the current cost, $x_{j}.c_{k}$, the costs-to-come of $x_{j}$ are updated by choosing $x_{i}$ as its new parent, per the function $UpdateCost()$ (Line 17). This function is detailed in Algorithm \[alg::cost-update\]; when it is called, the costs of node $x_j$ are updated. If, however, the $k$ th priority cost from $x_{init}$ to $x_{j}$ via $x_{i}$ is tied with the current cost, $x_{j}.c_{k}$ (Line 19), then Algorithm \[alg::lexicographic-search\] proceeds to the lower-priority cost $k+1$ and evaluate the potential $(k+1)$ th priority cost-to-come improvements at $x_j$ by traveling via $x_i$. To reduce the likelihood of end-stage ties, the lowest-priority cost $K$ is assumed to be strictly positive over all paths in the configuration space. Just as the problem formulation in Equation (\[eq::lexico-formula\]) only allows improvements to a solution’s lower-priority cost when it does not adversely impact a higher-priority cost, the proposed search method only allows improvements to be made in lower-priority costs when ties occur with respect to higher-priority costs. The single-source shortest paths solution produced by Algorithm \[alg::lexicographic-search\] would take on the same primary cost whether or not these improvements are performed, but the occurrence of ties allows us to opportunistically address auxiliary cost functions in the style of lexicographic optimization. Algorithm Complexity {#sec::algorithm-complexity} ==================== The proposed lexicographic search, per the pseudo-code provided in Algorithm \[alg::lexicographic-search\], takes on worst-case complexity $O(K|V|^2)$. For clarity and illustrative purposes, we have used a naive $O(|V|^2)$ implementation of Dijkstra’s algorithm, describing the lexicographic search using a basic queue that could be implemented using a linked list or similar. In the worst case, (1) finding the node(s) in the queue with the minimum cost (Line 9, costing $O(|V|^2)$ over the duration of the standard algorithm), and (2) expanding a node and inspecting its adjacent neighbors (Line 14, costing $O(|E|)$ over the duration of the standard algorithm) will each be repeated $K$ times, once for each cost function in the hierarchy, during every execution of the while loop. In the most time-efficient known implementation of Dijkstra’s algorithm, which uses Fibonacci heaps [@fredman1987], the complexity of the standard, single-objective algorithm is reduced from $O(|V|^2)$ to $O(|V|log|V| + |E|)$. Finding the minimum cost in the graph is trivial due to the maintenance of a priority queue, but deleting a node from the heap is a $O(log|V|)$ operation that must be repeated $|V|$ times over the duration of the algorithm. Expanding a node and inspecting its adjacent neighbors continues to cost $O(|E|)$ over the duration of the algorithm, since a worst case of $O(|E|)$ cost updates must be performed in the heap, each of which costs $O(1)$. To adapt this to a lexicographic search, the nodes in the heap must be prioritized per the lexicographic ordering of the graph nodes, so that the minimum cost reflects not only the minimum primary cost, but the optimum according to the formulation given in (1). Although only one node will be deleted from the heap in each iteration of the algorithm’s while loop, each of the $O(log|V|)$ comparisons required will take $O(K)$ time, and so the cost of node deletion over the duration of the algorithm will increase to $O(K|V|log|V|)$. The costs in the heap will also reflect the $K$ cost functions being considered. To maintain a lexicographic ordering among the nodes in the heap, all nodes undergoing cost changes during an iteration of the algorithm’s while loop may have their costs individually adjusted as many as $K$ times. Akin to the steps performed in lines 15-20 of Algorithm \[alg::lexicographic-search\], this is the worst-case number of times a node’s cost must be adjusted to establish the correct lexicographic ordering among a set of nodes with $K$ cost functions. Over the duration of the algorithm, this will result in a worst-case $O(K|E|)$ cost changes within the heap, each of which carries $O(1)$ complexity. As a result, the worst-case complexity of a lexicographic search using a Fibonacci heap will be improved to $O(K|V|log|V| + K|E|)$, from the original $O(K|V|^2)$. In the results to follow, we opt to implement and study the $O(K|V|^2)$ version of the algorithm in software, due to its ease of implementation and efficient memory consumption. We also note briefly that an adaptation of Dijkstra’s algorithm is selected in this application due to the fact that all graphs considered are characterized by non-negative, time-invariant edge weights. The consideration of negative edge weights would require an adaptation of the Bellman-Ford or Floyd-Warshall algorithm [@clrs], and the consideration of time-varying weights, such as those which might depend on the action or measurement history of a robot, as frequently occurs in belief space planning, may require search algorithms of exponential complexity [@shan2017belief]. Experiments {#sec::experiment} =========== We now describe a series of experiments to qualitatively and quantitatively validate our proposed receding horizon planner in both simulated and real-world environments. We compare the proposed planner with OpenPlanner [@darweesh2017open] and Timed-Elastic-Band (TEB) Planner [@rosmann2012trajectory]. All compared methods are implemented in C++ and executed on a laptop equipped with an i7-10710U CPU using the robot operating system (ROS) [@ROS-2009] in Ubuntu Linux. We note that only CPU is used for computation, without parallel computing enabled. For the parameters introduced in the previous sections, we let $Th_{risk}=2$, $Th_{head}=5^{\circ}$, $d_{sensor}=5.0\;m$, $d_{roll}=7.0\;m$ for all the tests. In the simulated tests, $d_{span}$ is chosen to be 1.0 m. In the real-world experiments, $d_{span}$ is set to be 1.5 m due to the large size of the dynamic obstacles. Simulated Experiments {#sec::experiment-sim} --------------------- ### Comparison with OpenPlanner OpenPlanner [@darweesh2017open] is a general planning algorithm that is developed for autonomous vehicles and integrated in Autoware [@kato2018autoware]. Upon receiving a goal location, OpenPlanner first finds a global path using a vector map. Then local candidate roll-out paths are generated while using the global path as a reference. As is shown in Figure \[fig::exp-openplanner\], the roll-out paths, which are colored green, start from the vehicle location and span to cover the neighborhoods adjacent to the global reference path. The aforementioned parameter $d_{span}$ is used for defining this adjacent neighborhood. The roll-out paths are designed to be parallel to the reference path eventually. At last, a path with the lowest cost among the candidate paths will be selected and executed. The planning scheme of OpenPlanner works well in environments with few obstacles. For a cluttered environment, it may fail to find a feasible path to execute. Such an example is shown in Figure \[fig::exp-openplanner\](c). All the candidates paths of OpenPlanner are blocked by three clusters of obstacles on the global reference path. On the other hand, the proposed receding horizon planner doesn’t have such a problem because we construct a graph and search for feasible paths. The returned path of our planner over this example is shown in Figure \[fig::pipeline\](c). ### Comparison with TEB TEB [@rosmann2012trajectory] is an optimization-based planner that also takes a global path as a reference. It generates an executable path by deforming the reference path while taking the dynamic constraints of the robot into account. The optimization problem of the TEB planner is formulated as a weighted-sum multi-objective problem, where the weights are manually adjusted by the user. In this test, we compare the proposed receding horizon planner with the TEB planner by following a U-shaped global path, which is shown in Figure \[fig::exp-teb-planner\](a). The environment is populated with randomly placed obstacles around the reference path. The path returned by the TEB planner is shown in Figure \[fig::exp-teb-planner\](b). Its path skips the entire operational region and leads the robot directly to the goal, which goes against its original intention of providing a reference path. When we test the proposed planner, the first path returned is shown in Figure \[fig::exp-teb-planner\](c). When the robot moves forward by following this path, new obstacles (shown in the mid-right and mid-top of Figure \[fig::exp-teb-planner\](d)) are detected on the path being executed. Thus the first path becomes invalid. Re-planning is performed and yields the second path (Figure \[fig::exp-teb-planner\](d)). As the robot explores the environment more, a safer path with lower risk cost is found and returned as the third path shown in Figure \[fig::exp-teb-planner\](e). During the entire run, our planner only re-plans three times and traverses safely amid the obstacles, while staying close to the reference path. ![Algorithm runtime as a function of the number of nodes in the graph when performing graph construction or search.[]{data-label="fig::benchmarking-time"}](Figures/experiment-2/exp-2.pdf){width=".99\columnwidth"} ### Benchmarking We show the algorithm runtime of the proposed planner in Figure \[fig::benchmarking-time\]. The graph construction time, which scales linearly as the number of the nodes increases, is plotted in black. In order to explore the influence of introducing new costs into the lexicographic ordering on the planning performance, we show benchmarking results using three cost combinations: (a) one criterion - distance only, (b) two criteria - a heading-distance ordering, and (c) three criteria - a risk-heading-distance ordering. The lexicographic search times when using these three cost combinations are depicted in cyan, orange, and pink respectively in Figure \[fig::benchmarking-time\]. Figure \[fig::benchmarking-path\] shows the corresponding solutions when applying three different cost combinations using the test environment described in Section \[fig::exp-teb-planner\](a). Table \[tab::benchmarking-cost\] shows the values of the corresponding costs of these three combinations. Note that the cost values marked in parentheses in Table \[tab::benchmarking-cost\] are calculated for reference and not used in the optimization process. When only distance cost is applied, the shortest path is found and shown in Figure \[fig::benchmarking-path\](a). This path starts with swinging to the left and ends with converging back to the global path. Though this path achieves the lowest distance cost of the three cost combinations, it yields very high risk cost, as it stays close to the obstacles. Figure \[fig::benchmarking-path\](b) shows the path when we apply two cost criteria. After adding the heading cost to the hierarchy, the obtained path possesses fewer heading differences from the reference trajectory even as the distance cost increases. As is shown in Figure \[fig::benchmarking-path\](c), the utilization of a risk-heading-distance hierarchy yields the safest path by keeping away from surrounding obstacles. Real-world Experiments {#sec::experiment-real} ---------------------- Finally, we implement the proposed receding horizon planner on an autonomous surface vehicle - the quarter-scale Roboat [@wang2019roboat]. As is shown in Figure \[fig::roboat-hardware\], Roboat has dimensions of 0.9 m $\times$ 0.45 m $\times$ 0.5 m (L$\times$W$\times$H) and weighs about 15 kg. It is outfitted with four thrusters as shown in Figure \[fig::roboat-hardware\] to enable omnidirectional maneuvering, and it’s equipped with a Velodyne VLP-16 for perception. Localization is performed using a modified lidar odometry adapted from [@shan2018lego] and [@shan2020lio]. We discard laser range returns that are more than 5 meters away from the robot, allowing a small-scale environment to produce varied outcomes. The laser range returns within 5 meters are considered to be readings from obstacles. We conduct an experiment in a 12.5 m $\times$ 6.5 m swimming pool. Three floating containers are placed in the pool to serve as obstacles. The containers move randomly in the pool due to water flow. The U-shaped global reference path is colored pink in Figure \[fig::exp-pool-2\]. Besides the containers, other structures, such as walls, in the environment are also considered as obstacles. The robot starts from the lower-right corner of the figure and tries to reach the goal at the lower-left corner. At the beginning of planning (Figure \[fig::exp-pool-2\](a)), only one container is within the sensor range and on the reference path. The returned path swings to the right to avoid it. In Figure \[fig::exp-pool-2\](b), the path changes accordingly when another container is detected by the robot. Due to the moving obstacles, the returned path from our planner changes several times (Figure \[fig::exp-pool-2\] (b)-(d)). Note that these paths strictly follow the cost-hierarchy we defined in Section \[sec::problem-cost\]. As is shown in Figure \[fig::exp-pool-2\](e), two obstacles completely block the forward path of the vehicle. The robot remains stationary and waits for the waterway to clear. In Figure \[fig::exp-pool-2\](f), a new path is found as one of the obstacles moves away. The robot follows this path and reaches the goal location eventually. Conclusions and Discussion ========================== We have proposed a receding horizon planner for path planning with a ASV in urban waterways. The receding horizon planner generates a graph from a global reference path to search for feasible paths in the presence of obstacles. We also propose a lexicographic search method intended for use with graphs in multi-objective robot motion planning problems, in which competing resources are penalized hierarchically. Over such problems, we have demonstrated that the proposed search method is capable of producing high-quality solutions with efficient runtime. The variant of Dijkstra’s algorithm proposed for performing the search offers appealing scalability, as its worst-case complexity scales linearly in the number $K$ of cost criteria. A key benefit of the approach is that, in contrast to planning methods that employ weight coefficients or constraints, minimal tuning is required, beyond the ordering of cost functions in the hierarchy. Since no constraints other than obstacle avoidance need be imposed, feasible solutions are obtained quickly. Real-world implementation of our method is also demonstrated on a ASV. Future work entails the extension of this method to time-varying costs that are history-dependent, for use in motion planning under uncertainty. Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported by Amsterdam Institute for Advanced Metropolitan Solutions, Amsterdam, the Netherlands. [^1]: T. Shan, W. Wang, and C. Ratti are with the Department of Urban Studies and Planning, Massachusetts Institute of Technology, USA, [{shant, wweiwang, ratti}@mit.edu]{}. B. Englot is with the Department of Mechanical Engineering, Stevens Institute of Technology, USA, [benglot@stevens.edu]{}. T. Shan, W. Wang and D. Rus are with the Computer Science & Artificial Intelligence Laboratory, Massachusetts Institute of Technology, USA, [{shant, wweiwang, rus}@mit.edu]{}.

--- abstract: 'The Louisiana Department of Education partnered with the Gordon A. Cain Center at LSU to pilot a Computing High School Graduation Pathway. The first course in the pathway, Introduction to Computational Thinking (ICT), is designed to teach programming and reinforce mathematical practice skills of nine-grade students, with an emphasis on promoting higher order thinking. In 2017-18, about 200 students and five teachers participated in the pilot, in 2018-2019 the participation increased to 400 students, and in the current 2019-2020 year about 800 students in 11 schools are involved. Professional development starts with a five-week intensive summer institute, which is complemented with follow-up Saturday sessions and coaching support during the academic year. After describing the course content and briefly the teacher training, we discuss the data we have collected in the last two years. The overall student reception of the course has been positive, but the course was categorized by most students as hard. However, the Computing Attitude Survey analysis indicates that the difficulty of the course did not demotivate the students. The pre-post test content assessments show that students learned not only the language, but also general principles of programming, logic and modeling, as well as use of variables, expressions and functions. Lessons learned during the pilot phase motivated changes, such as emphasizing during PD the need to provide timely feedback to students, provide detailed rubrics for the projects and reorganize the lessons to increase the initial engagement with the material. After two years of running pilots, the course is becoming student-centered, where most of the code and image samples provided in the lessons are based on code created by previous students.' author: - Fernando Alegre - John Underwoood - Juana Moreno - Mario Alegre title: 'Introduction to Computational Thinking: a new high school curriculum using CodeWorld' --- Introduction {#sec:overview} ============ Our project started in 2015, when we were contacted by the East Baton Rouge Parish School System (EBRPSS) to help develop computer science curricula for a new STEM magnet high school, to offer new opportunities to the under-served population of the district, which consists of 85% minority and 75% economically disadvantaged students. We were tasked with creating the curriculum, including its assessment and the delivery of the summer teacher training. Additionally, the curriculum had to be designed in such a way that teachers of other academic subjects could quickly learn it, since there were no computer science teachers available in the area. The first course in this set is Introduction to Computational Thinking (ICT), an introductory programming course offered to eighth or ninth graders who are concurrently taking an Algebra I. The course teaches the conceptual foundations of coding in a language syntax and semantics that follow closely the language of algebra. It is not intended to be a math remediation course, but rather to highlight the connections to algebra, geometry and science modeling. During the 2016-2017 academic year, we conducted several 3-month pilot tests of the course and developed an assessment instrument, the *Conceptual Foundations of Coding Test*, which was vetted with about 100 students. A full-course pilot was deployed in the 2017-2018 academic year. At that time, the Louisiana Department of Education (DoE) became interested in the ICT curriculum and partnered with the Cain Center to create and pilot a Computing High School Graduation Pathway, following the model pioneered by the EBRPSS STEM magnet high school. The Pathway offers a hybrid curriculum that prepares students both for college and to enter the workforce after graduation. During 2017-2018, the course was taken by more than 200 ninth grade students in four different schools. Approximately 400 students in ten schools in eight school districts were enrolled in ICT for the 2018-2019 academic year, and there are 800 students enrolled in the 2019-2020 academic year. In the summer of 2017, we conducted our first Professional Development program, which is an intensive five-week professional development summer institute. In 2017 we trained eight teachers, with an additional nine teachers in 2018, and most recently nine more teachers. The teachers were absolute novices with respect to programming. They were placed into student roles as the first part of their training, where they completed all the programming assignments, presented them to their peers, and practiced modifying their code according to the feedback received. The teachers were instructed in pedagogical techniques and lesson design throughout the summer. At the end of the summer PD, the teachers felt comfortable enough to teach the course and to modify the assignments to meet their school’s unique cultures and needs. The majority of the teachers participating were certified in either secondary math or science, but some were certified in other instructional areas. For example, in the 2017 training there was one social studies and one career and technical education teacher. In the 2018 training there were two social studies teachers, and in the 2019 training there two social studies and two computer science teachers. All the activities are programmed in CodeWorld[@codeworld], a web-based integrated development environment which was initially designed for middle school students, that uses a simplified variant of the Haskell language. The lessons are organized in units that follow the concepts of Computational Thinking, with the syntax of the language being presented at the beginning of each semester. However, very little emphasis is placed on teaching the language, whose features are introduced only when needed. In the first semester, only expressions, variables and functions are used. No conditionals, looping constructs or data structures are needed for the programming assignments. In the second semester, lists and tuples are the only new syntactic features needed, and looping constructs are based on a second-order function, called `foreach`, which is a regular function with no special syntax. [|p[1.in]{}|p[5.5in]{}|]{} Unit & Content\ The Software Development Cycle & Students learn how to use an IDE, how to draw basic shapes, how to overlay several pictures and move them around the screen. They also learn about design techniques, such as creating prototypes and using pseudo-code to plan a program, and practice collaboration with pair programming and a collaborative creation of a scene, where each team member is in charge of a character or prop.\ Abstraction and Decomposition & Students learn to map expressions to syntax trees, handling order of operations, and using trees to represent other aspects of code, such as dependencies between variables and organization of layout into nested layers. They also learn how to use an object dimension as a unit of measurement for other objects (e.g., 2.5 smileys wide, 3 monsters high) and how to combine rotations, translations and scalings to create complex mosaics or quilt patterns.\ Patterns and Regularity & Students use repetition to create regular polygons, regular stars and create recursive patterns. They also learn about generating random patterns and irregular grids, and use them to generate a procedural map of a neighborhood.\ Data and Calculations & Students learn to process lists to create bar charts and pie charts from scratch, create itemized bills including taxes and discounts, calculate weighted averages and compute areas of complex settings, such as the area occupied by chairs and tables in a dining hall.\ Models in Space and Time & Students create simple games (rock,paper,scissors; dice rolling games; tic-tac-toe) and simple animations (characters performing repetitive circular or linear motion; see-saws; slide shows; marquee messages)\ \[tab:ICTCurriculum\] Development Process {#sec:research} =================== Foundational Stages ------------------- The discipline of computer science currently has a large, if not traditionally recognized, impact on many other high school core subjects,and therefore it should not be studied in isolation [@heintz16]. Computer science must also have the integration of skills and content that reflect the real word connections it has to math and science [@bart14]. Learning computer science is not about learning a specific programming language. It is also not simply learning commands and techniques on how to program. The term *Computational Thinking* (CT) was introduced in education to convey this type of reasoning [@weintrop16CTMathSc; @angeli16CT-K6; @voogt15CTcore; @grover2015designing; @grover13ctk12; @wing2006computational]. Computational Thinking is not confined to strictly programming, but it is within the programming environment where CT manifests itself most prominently. It can be difficult to understand CT fully without exposure to programming. [@grover13ctk12; @denning17CT]. In our training materials, we try to provide insight about the meaning of computational thinking, so we explain to the teachers that when someone is thinking computationally, in our view, they do the following: 1. Use introspection to observe their own thought process as if it were performed by a machine and express their thoughts explicitly and without any ambiguity. 2. Imagine in their head a computer running a given program and anticipate the outcome without actually running the program. 3. Reason constructively, as the purpose of computing is to construct a solution. Computing works under a closed world assumption, where only those entities explicitly built are assumed to exist. 4. Invent a process to solve a problem as a series of mechanical steps, where each step requires no intelligence to perform. The intelligence contained in a program is an emergent feature and cannot be pinpointed to any particular line in the code. 5. Think in terms of causality. A function is not just a relationship between an input and an output, as it would be in mathematics. It is also a process that causes the computer to produce an output when the given input is consumed. This process occurs in time, and so the input must exist before the output can exist. Computations change the world. 6. Reason by proxy: Distinguish between what a concept is and how it is represented. For example, represent a polygon as a list of pairs of coordinates. 7. Establish relationships between concepts by writing equations between their corresponding representations. For example, move a polygon horizontally by adding the same number to each X coordinate in the corresponding list. Computational thinking is about expressing thoughts in a formal system, in a way that is actionable by an automated system. Programming languages are not the only possible formal systems in which computational thinking can be expressed, but they are the most accessible and prone to automation. Thus, using programming as a vehicle for computational thinking is a natural choice. Unfortunately, in many elementary and middle school settings, the term CT has become synonymous with either *computing with no programming* or *block-based programming*. This interpretation omits the central tenet of Computational Thinking, which is the *building of high-level abstractions that can be executed by a computer* [@wing08compthink]. Currently, there is a need to have a high school course that introduces CT within the context of substantial amounts of programming with clear connections to math and science. This CT course should depend as little as possible on the extensive knowledge of a particular language or technology. The CT course would be a natural progression for students to take along with Exploring Computer Science (ECS) and Computer Science Principles (CSP). For the most part, ECS, CSP and block-based programming courses rely on the teachers to act as facilitators of instruction provided by an online system. This instructional model is based on the idea that students will learn even if they are not being directly instructed by their teacher. However, a flaw of the model is the fact that many concepts, such as abstraction, are only developed through higher-order learning [@feathers19; @ramineni18]. A recognizable factor for why they are on the rise is due to a current scarcity of teachers who know how to program. Students need direct interaction with a teacher to master higher order thinking concepts. There is great value in having a teacher who can evaluate the students work, reflect on the student’s progress, offer guidance to the student on ways that they may correct habits, and examine unique work products from the perspective of the student’s intellectual evolution. These attributes have proven difficult to evaluate effectively in an automated way. Conceptual framework -------------------- ICT is an elective course in Louisiana, where a majority of the students have historically demonstrated weak mathematical skills for all grade levels. In designing the course, additional attention was given to ways to help students improve in their math skills as they learn computer science. This course was not intended to be strictly a math remediation or math intervention course, but rather an integrated part of a STEM elective pathway. The learning objectives were established and designed to be recurring throughout each of the units. The learning objectives are not isolated to specific lessons. The learning objectives include: - Develop a procedural understanding of the pillars of Computational Thinking: recognize patterns and regularity, decompose problems into smaller problems, formulate and solve simplified problems, generalize solutions and encapsulate solutions. - Acquire experience with algebraic manipulation of complex expressions - Use mathematical functions to model artifacts, such as diagrams or animations. - Transform many data items as if they were a single entity - Organize data hierarchically - Calculate totals, averages and quantities using rates, such as taxes and discounts. - Use random sampling to explore instances of relationships and find the general case Our approach is inspired by the work of [@felleisen04SchemeCS] and Bootstrap [@Schanzer13; @schanzer15transfer], due to their promising results concerning transfer between programming and mathematics  [@Schanzer13; @schanzer15transfer]. They introduced the *design recipe*, which is a series of steps for guiding students when they are trying to create a function: write a definition in English, then describe the inputs and outputs, then provide at least 3 examples, then look at what is common in those examples (the template) and what changes from example to example (the variables), and finally give names to those variables. However, we differ from Bootstrap in several different ways. We have developed a full-year curriculum centered on CT instead of a 17-hour intervention focused on math word problems. Our use of Haskell makes writing function definitions very lightweight, so students are encouraged to create lots of functions. Also, the lazy evaluation model relieves us from the need to have special syntax for program control. We have also extended the design recipe with the introduction of random variables, so that students create random samples of uses of a function after (or instead of) providing examples with fixed numbers. Finally, we put more emphasis on modeling techniques and using the software development cycle rather than on guided exercises based on code templates. ![\[fig:house\]A drawing of a house](./house.png){width="20.00000%"} CodeWorld activities -------------------- All the activities are programmed in CodeWorld[@codeworld], which uses a very limited set of graphical primitives to draw circles, rectangles, and text. It is then possible to apply translations, rotations, scalings and colors to them. Smaller elements can be combined into more complex shapes via the overlay operator (denoted by &). Animations are represented as functions that depend on a parameter, namely the time in seconds since the animation started. The language follows a syntax very similar to mathematical notation, and the evaluation semantics follows exactly the same rules as algebra. Here is a complete CodeWorld program to draw a house (see Fig. \[fig:house\]): program = drawingOf(house(red,yellow) & coordinatePlane) house(rcolor,fcolor) = colored(roof,rcolor) & windows & door & colored(facade,fcolor) & pathway roof = solidPolygon([ (-4,4),(4,4),(0,6) ]) windows = floor2 & floor3 floor2 = translated(window,-2,1) & translated(window,2,1) floor3 = translated(floor2,0,2) window = solidRectangle(1,1) door = translated(solidRectangle(1,2),0,-1) facade = translated(solidRectangle(8,6),0,1) pathway = overlays(tile,8) tile(n) = translated(stone,-(n-1)/2,-1.5-(n+1)/2) stone = colored(oval,translucent(grey(0.2))) oval = scaled(solidCircle(0.5),2,1) Practically all the syntax of the language is illustrated in the previous program, and all programs are written in exactly the same format (a list of lines that read `head = body`) with `program` being the starting point of the execution of the program. Functions are defined in the same way as variables, but the head includes parameters. No special constructions for loops or conditionals are necessary. Definite loops are provided by library functions, such as `overlays`, which works as follows: the expression `overlays(f,n)` is equivalent to `f(1) & f(2) & ... & f(n)`. Indefinite loops are created by recursive definitions. Conditionals are produced by having functions with special cases, which are created by adding a vertical bar and a condition to their definition. For example, the absolute value would be defined by the following two lines: absoluteValue(x) | x < 0 = -x absoluteValue(x) | x >= 0 = x In the second semester, lists and tuples are introduced. Basic list usage needs 3 additional symbols: `[`, `]` and `#`, to build a literal list, and to access the n-th element, respectively. The simplified version of Haskell we are using stops here. No advanced features of the language (such as typeclasses, IO or monads) are exposed to students. In a sense, our use of Haskell provides the same affordances that a block-based language would, because the key features of block-based languages are their simple, barebones syntax, as opposed to regular programming languages, and the avoidance of errors due to misspelling or misuse of variables and constructs placed in the wrong spot [@Weintrop15BlockProg; @weintrop15CommAss]. Haskell shares both of these features, because in addition to the simple syntax explained above, the advanced type-inference features of Haskell catch practically all misspellings and misuses of variables and functions. Curriculum content {#sec:content} ------------------ The ICT curriculum comprises five units, where the first two units take approximately eight weeks each, and the last three units take approximately five weeks each. In the 2019-2020 version of the curriculum, all the activities in the first four units include samples created by students who took the course in the previous two years. See Fig. 2 for the samples in an activity where students are asked to create a diagram of a cell. Assessment is based on a project at the end of each unit, plus a midterm project and a final project. Our team developed analytical rubrics that present the criteria and levels of performance for each assignment. The rubrics are tiered from minimal, to lower, to mid-level, and finally high attainment. Each tier contained descriptors with point values. Each attribute was aligned to the learning objectives, which are stated at the start of each lesson and integrated in the activities that build to the project. For example, in a project to create an analog clock face, students were evaluated on the following: whether they used expressions with variables (high score) or magic numbers (low score); whether they repeated the code for the hour hand and the minute hand (low score) or created a function to handle both (high score); whether they created different nested layers for the hour ticks, the minute ticks and other elements (high score) or they had a flat layout (low score); whether their code printed redundant elements, such as printing 12 o’clock twice (low score) or handled ranges properly, including only one end (high score); whether they used local variables (high score) or only global variables (low score); whether they followed good practices when naming, indenting and grouping parts of the code (high score) or not (low score); whether the calculations to convert hours and minutes to degrees of rotation were correct (high score) or not (low score); and whether the output showed an analog clock with hour ticks, minute ticks and numbers at each hour (high score) or only some of those elements (low score). Finally, extra credit was given for creativity in their aesthetic design. Table \[tab:ICTCurriculum\] lists the title and a brief summary of the content of each unit. {width="\linewidth"} {width="\linewidth"} {width="\linewidth"} {width="\linewidth"} {width="\linewidth"} {width="\linewidth"} {width="\linewidth"} {width="\linewidth"} {width="\linewidth"} {width="\linewidth"} Technical considerations ------------------------ Our choice of programming environment was also influenced by the following properties: 1) The programming language should make it easy for students to build high level abstractions; 2) The language should also have a syntax and semantics as similar to algebra as possible; 3) No prior or additional knowledge of coding or software should be needed by teachers to produce code for the lessons; 4) Execution of any component of the system should not depend on any third party service or product; and 5) The programming paradigm should preferably be functional. One technical restriction on our choice of programming environment was due to the fact that many Louisiana schools have policies concerning which software can be run on their computers. Often the computing environment is optimized for use on standardized testing platforms, which can prevent root access and local installation of software. In addition, policies in many schools prevent students from being required to register with third-party organizations and to submit student work to external web sites. Given these conditions, we elected to use a Web-based environment that required no local installation and could be used without restrictions and without the need for students to register or provide any personal information. The next requirement our team faced was that we were required by the Louisiana Department of Education, which partially funded our project, to rely upon fully open source software. Our final requirement was that as a team we wanted to do graphics-based programs rather than text-based programs. Given all of the aforementioned requirements the number of possibilities we considered was limited. For example, at the project’s onset there was no fully open source, fully online version of Python for graphics programming. Related and future work {#sec:related} ======================= The idea of using coding to help students learn mathematics and science has a long history. Early attempts to use coding as a tool were based on unguided discovery [@Papert80mindstorms]. This approach proved to be ineffective for transfer [@mayer04strikesrule]. Over the years, it has become clear that transfer between programming and mathematics is difficult to initiate, and whether it occurs or not depends strongly on the teaching methodology used [@Pea83Logo; @kurland86proghighschool; @butterfield89techtransfer]. Recent attempts to establish the link between programming and mathematics have been based on a modern framework of computational thinking [@wing2006computational; @wing08compthink; @grover13ctk12] and supported by modern theories such as convergent cognition [@rich13convergent]. One of the few cases in which a project targeted the learning of mathematics with coding and showed promising results is Bootstrap, a 17-hour curriculum designed to be used either standalone or embedded in a computer science or mathematics course. It is one of the few documented instances of transfer between programming and algebra. [@Schanzer13; @schanzer15transfer] attribute the favorable results of their intervention to their use of a functional language as the medium and to the absence of distracting features. Since our program also features a functional language and absence of distracting features, we will investigate in future publications the capacity of our program to promote transference between mathematics and computational thinking. As the ICT course is continuing to be adopted across Louisiana, a growing number of students will have taken ICT in eighth or ninth grade. With an increase in student participants our team will be able to study the impact of functional programming in the learning of math and science. Furthermore, as more math and science teachers are trained to teach ICT, we will be able to incorporate coding activities as essential components in math and science courses. Once complete, these activities will be accessible from a virtual bank of math and science programming activities. The goal will be for these activities to be adapted and incorporated by the teachers into their regular lessons that they currently do with their students. Using this population of students and teachers trained in our ICT course we can actually focus on the important question of whether students who know functional programming and have access to instructional material that requires use of their programming skills can reach a deeper understanding of math and science than students who do not have access to that kind of instruction. In this future work comparable control populations will be attainable from local area schools that have yet to be trained. Impact ====== ![\[fig:ICT\] Histogram comparing the results of the pre- and post-[*Conceptual Foundations of Coding Test*]{}. ](./StudentsCFC-1718-1819.png){width="45.00000%"} ![\[fig:ICTFactors\] Boxplots comparing results of the [*Conceptual Foundations of Coding Test*]{} by categories: variables, expressions and functions; CodeWorld specifics; modeling; and logic and programming. Boxes stretch from the 25th percentile to the 75th percentile of the distribution.](./StudentsCFC-Fact.png){width="45.00000%"} Results and Analysis -------------------- The data presented here corresponds to academic years 2017-2018 and 2018-2019. In 2017-2018 the course was deployed at four schools with five teachers and 208 students. Ten schools, 13 teachers and 395 students participated during the 2018-2019 academic year. We collected pre- and post-[*Conceptual Foundations of Coding Test*]{} results for 325 students. The test captured an overall average growth of 24% . Fig. \[fig:ICT\] displays a histogram comparing the pre- and post-test results, showing that the distribution of the post-test moves to higher scores and widens when compared with the pre-test scores. The initial average score was $29.5 \pm 0.6\%$, and it increased to $53.8 \pm 1.1\%$ at the end of the course. A Wilcoxon signed rank test pairing pre- and post-results determined that the difference between the distributions was statistically significant ($p < 2.2 \cdot 10^{-16}$), reinforcing that we can distinguish the pre- and post-distributions of correct answers. Fig. \[fig:ICTFactors\] displays the pre- and post-test results for the four categories included in the test: variables, expressions and functions, `CodeWorld` specifics, mathematical modeling, and logic and programming. The average of the post-results is higher than the average of the pre-results in all four categories. The pre- and post distributions for all the categories are very significantly different, with $p<10^{-12}$ as determined by the Wilcoxon signed rank test. In the analysis of the Computing Attitudes Survey (CAS) data we followed the prescribed guidelines of Dorn and Tew [@dorn15CAS] and analyzed the data utilizing the prescribed subcategories. The data demonstrated that the student attitudes did not change after completing the course, as can be determined by the fact that the shift in attitudes was minuscule ($0.02\pm0.02$), and the pre- and post-test results were very strongly correlated ($R=0.54$ and $p=1.3\cdot 10^{-10}$). Looking at the correlations between the results of the Computing Attitudes Survey (CAS) and the gains in the Conceptual Foundations of Coding Test (CFC), we find a positive correlation between the post results of CAS and the gains in CFC ($R=0.29$, $p=0.001$), indicating that those students who had a more positive attitude at the end of the course also tended to have the higher gains in learning. The overall reception of the course has been positive, but the course was uniformly categorized by most students as hard. Our observations indicate that students were not accustomed to having to use more than one mathematical idea in a single problem. They were also unsettled by the fact that the same image could be generated in many different ways, and there was not a canonical correct way to write code for it. Nevertheless, the Computing Attitude Survey analysis indicates that the difficulty of the course did not demotivate the students. The pre-post test analysis shows that the students learned not only the language, but also general principles of programming, logic and modeling, as well as use of variables, expressions and functions. Conclusion ---------- While the need for teaching computational thinking is already well established, there is still controversy about whether programming should be included or not, or, as Denning [@denning17CT] calls it, the clash between Traditional CT and New CT. Courses such as ECS or CSP are examples of New CT, but there is not much available in terms of courses that focus on Traditional CT. Due to its capacity for automation and formalization, programming is a natural vehicle for learning computational thinking. While Python and JavaScript courses are relatively available, they do not usually focus on CT. Instead, they follow traditional syntax-oriented approaches to teaching computing, with few connections to math and science. Those courses are more useful for students aspiring to be software developers than for the general student population. We have presented an alternative approach. We have described the design and implementation of a secondary Computational Thinking course based on programming with connections to science and math. This course provides a proof of concept for curricula halfway between traditional programming language courses and recent computational thinking courses with limited programming content. This course addresses the need for computational thinking courses intended not only for future software developers but for all students no matter what they do later in their lives. We find that a focus on programming content does not need to be discouraging to students. Our approach is highly student-centered, and has been proven to be suitable for traditionally underserved populations. We also build on Bootstrap ideas and techniques and have opened a way to investigate many interesting connections between the learning of programming and the learning of mathematics and science, and we are excited to delve into them. 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--- abstract: 'We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichmüller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theories and such that their traces define invariants of mapping tori.' author: - 'Stéphane Baseilhac$^1$, Riccardo Benedetti$^2$' title: 'On the quantum Teichmüller invariants of fibred cusped $3$-manifolds' --- $^1$ Institut Montpelliérain Alexander Grothendieck, CNRS, Université de Montpellier (sbaseilh@univ-montp2.fr) $^2$ Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy (benedett@dm.unipi.it) Introduction ============ Let $M$ be a [*fibred cusped $3$-manifold*]{}, that is, an oriented non compact finite volume complete hyperbolic $3$-manifold which fibres over the circle $S^1$. In this paper we describe the relationships between the quantum hyperbolic invariants of $M$ (QHI) ([@GT; @AGT; @NA]), and the intertwiners of finite dimensional representations of the quantum Teichmüller space of $S$ ([@B-L; @B-B-L], [@Filippo]), where $S$ is a fibre of some fibration of $M$ over $S^1$. Recall that $M$ is the interior of a compact manifold $\bar M$ with boundary made by tori, and that $S$ is the interior of a compact surface with boundary $\bar S$, properly embedded in $\bar M$, so that $\bar S \cap \bar M$ is a family of ‘meridian’ curves on $\partial \bar M$. The existence of such relationships has been expected for a long time, and their precise formulation often thought as depending only on the solution of a handful of technical problems. Let us recall the main arguments underlying this opinion. Let $q$ be a primitive $N$-th root of unity, where $N$ is odd. Then: - The quantum Teichmüller space ${{\mathfrak T}}_{S}^q$ of $S$ consists of the set of Chekhov-Fock algebras ${{\mathfrak T}}_{\lambda}^q$, considered up to a suitable equivalence relation (see Section \[QREP\]), where $\lambda$ varies among the ideal triangulations of $S$. A representation $\rho$ of ${{\mathfrak T}}_{S}^q$ is a suitable family of ‘compatible’ representations $\rho = \{\rho_\lambda:{{\mathfrak T}}_{\lambda}^q\to {\rm End}(V_\lambda)\}_\lambda$. The irreducible representations of ${{\mathfrak T}}_{S}^q$ were classified up to isomorphism in [@B-L], and so-called [*local representations*]{} were introduced and classified in [@B-B-L]. - The local representations are reducible, but more suited to perform the usual cut-and-paste operations of topological quantum field theories. In this framework, it is natural to restrict to a distinguished family of intertwiners, satisfying some functorial properties with respect to the inclusion of triangulated subsurfaces. Theorem 20 of [@B-B-L] proposed such a family, and argued that for every couple $(\rho,\rho')$ of isomorphic local representations of ${{\mathfrak T}}_{S}^q$ and every couple $(\lambda,\lambda')$ of ideal triangulations of $S$, there is a unique projective class of intertwiners in the family which are defined from the representation space $V_\lambda$ of $\rho_\lambda$ to the representation space $V_{\lambda'}$ of $\rho'_{\lambda'}$. Let us call the elements of these classes [*QT intertwiners*]{}. - The diffeomorphisms of $S$ act on the local representations of ${{\mathfrak T}}_{S}^q$. Denote by $\phi$ the monodromy of some fibration of $M$ over $S^1$. If a local representation $\rho$ is isomorphic to its pull-back $\phi^*\rho$ under $\phi$ (an instance is canonically associated to the hyperbolic holonomy of $M$ [@B-L; @B-B-L]), then the trace of a suitably normalized QT intertwiner which relates $\rho_{\lambda}$ to $\phi^*\rho_{\lambda'}$, $\lambda':=\phi(\lambda)$, should be an invariant of $M$ (possibly depending on $\phi$) and the isomorphism class of $\rho$. - The QT intertwiners can be constructed by composing elementary intertwiners, associated to the diagonal exchanges in a sequence relating two ideal triangulations of $S$, and these were identified with the Kashaev $6j$-symbols in [@Ba]. - The QHI of $M$ can be defined by [*QH state sums*]{} over “layered” triangulations of $M$, that is, roughly, $3$-dimensional ideal triangulations that realize sequences of diagonal exchanges relating the surface ideal triangulations $\lambda$ and $\phi(\lambda)$, in such a way that every diagonal exchange is associated to a tetrahedron of the triangulation. The main building blocks of the QH state sums are tensors called [*matrix dilogarithms*]{}, carried by the (suitably decorated) tetrahedra, which were derived in [@Top] from the Kashaev $6j$-symbols. In conclusion, the QHI of $M$ would coincide with the traces of QT intertwiners of local representations of ${{\mathfrak T}}_{S}^q$ (at least when $M$ is equipped with the hyperbolic holonomy). Only a suitable choice of normalization of the QT intertwiners and an explicit correspondence between formulas would be missing. Another fact pointing in the same direction is that both the isomorphism classes of local representations of the quantum Teichmüller spaces and the QHI depend on similar geometric structures. Every local representation $\rho$ of ${{\mathfrak T}}_{S}^q$ defines an [*augmented character*]{} of $\pi_1(S)$ in $PSL(2,{\mathbb{C}})$ [@B-B-L], called its [*holonomy*]{}, given by a system of [*(exponential) shear-bend*]{} coordinates on every ideal triangulation $\lambda$ of $S$. Moreover $\rho$ has a [*load*]{} $h_\rho$, defined as a $N$-th root of the product of the share-bend coordinates on any ideal triangulation $\lambda$ of $S$, and which does not depend on the choice of $\lambda$. The holonomy and the load classify $\rho$ up to isomorphism. On another hand, the QHI of any cusped $3$-manifold $M$ depend on a choice of augmented character ${{\mathfrak r}}$ of $\pi_1(M)$ in $PSL(2,{\mathbb{C}})$, lying in the irreducible component of the variety of augmented characters containing the hyperbolic holonomy. If $S$ is a fibre of a fibration of $M$ over $S^1$, ${{\mathfrak r}}$ can be recoved from its restriction to $\pi_1(S)$. [*However, other facts suggested that the relations between the QT intertwiners and the QHI would be subtler than expected*]{}: - The matrix dilogarithms are obtained from the Kashaev $6j$-symbols by rewriting them in a non trivial way as tensors depending on geometrically meaningful parameters (which are $N$-th roots of hyperbolic shape parameters), [*but also*]{} by applying a [*symmetrization*]{} procedure. This procedure is necessary to get the invariance of the QH state sums over any cusped $3$-manifold. It involves several additional choices, and eventually forces to multiply the QH state sums by a normalization factor. It implies furthermore that the QHI must actually depend not only on the choice of an augmented character ${{\mathfrak r}}$ of $\pi_1(M)$ in $PSL(2,{\mathbb{C}})$, [*but also*]{} on some cohomological classes, called [*weights*]{}, which cannot be recovered from the holonomy of local representations of the quantum Teichmüller spaces. The weights gauge out the additional choices. Surprisingly, both the normalization factor of the QH state sums and the weights did not appear in quantum Teichmüller theory. - An issue about the uniqueness of the projective classes of QT intertwiners has been fixed recently. By a careful analysis of the definition of local representations, and adapting the construction of the QT intertwiners from [@B-B-L], for every couple $(\rho_\lambda, \rho'_{\lambda'})$ as above, it is constructed in [@Filippo] a unique, minimal set $\mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ of projective classes of QT intertwiners which relate $\rho_\lambda$ to $\rho_{\lambda'}'$, and a free transitive action $$\label{action1} \psi_{\lambda\lambda'}^{\rho\rho'}\colon H_1(S;{\mathbb{Z}}/N{\mathbb{Z}}) \times \mathcal{L}_{\lambda\lambda'}^{\rho\rho'} \to \mathcal{L}_{\lambda\lambda'}^{\rho\rho'}.$$ In particular, the set $\mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ is far from being reduced to one point. The meaning of the normalization factor of the QH state sums has been better understood in [@NA]. In that paper we showed that for every cusped $3$-manifold $M$, the QH state sums of $M$ [*without the normalization factor*]{} define finer invariants ${{\mathcal H}}_N^{red}(M,{{\mathfrak r}},\kappa;\mathfrak{s})$, called [*reduced QH invariants*]{}, well-defined up to multiplication by $4N$-th roots of unity. They depend on an augmented $PSL(2,{\mathbb{C}})$-character ${{\mathfrak r}}$ of $\pi_1(M)$, a class $\kappa\in H^1(\partial \bar M;{\mathbb{C}}^*)$, ${\mathbb{C}}^*={\mathbb{C}}\setminus \{0\}$ being the multiplicative group, such that $\kappa^N$ coincides up to sign with the restriction of ${{\mathfrak r}}$ to $\pi_1(\partial \bar M)$, and an additional structure $\mathfrak{s}$ on $M$, called a [*non ambiguous structure*]{}. When $M$ fibres over $S^1$, a natural choice of $\mathfrak{s}$ is a fibration of $M$, say with monodromy $\phi$. Let us denote by $M_\phi$ the realization of $M$ as the mapping torus of $\phi$, equipped with the corresponding fibration, and the reduced QH invariant by ${{\mathcal H}}_N^{red}(M_\phi,{{\mathfrak r}},\kappa)$. The latter depends directly on the monodromy $\phi$, and hence is apparently more suited to investigate the relationships with the QT intertwiners. Note, however, that the class $\kappa$ is residual of the cohomological weights involved in the definition of the unreduced QHI, and so the above considerations about the weights still apply. Denote by $S$ a fibre of $M_\phi$. Rather than considering $M_\phi$ one can consider as well the mapping cylinder $C_\phi$ of $\phi$. In this case, instead of a scalar invariant there is a [*(reduced) QH-operator*]{} $${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})\in {\rm End}\left(({\mathbb{C}}^N)^{\otimes 2m}\right), \quad m:=-\chi(S),$$ which is such that $$\label{traceform} {{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa) = {\rm Trace}\left({{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})\right).$$ More precisely, in the spirit of topological quantum field theories, for every odd $N\geq 3$ the QHI define a contravariant functor from the category of (decorated) $(2 + 1)$-cobordisms to the category of vector spaces. In particular, it associates vector spaces $U_\lambda$ and $U_{\lambda'}$ to the triangulated surfaces $(S,\lambda)$ and $(S,\lambda')$ respectively, where $\lambda':=\phi(\lambda)$, and a QH-operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})\colon U_{\lambda'} {\rightarrow}U_\lambda$ to the cylinder $C_\phi$. The diffeomorphism $\phi$ gives an identification between $U_\lambda$ and $U_{\lambda'}$, so one can consider ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})$ as an element of ${\rm End}\left(U_\lambda\right)$. Moreover, the construction of the functor provides a natural identification of $U_\lambda$ with $({\mathbb{C}}^N)^{\otimes 2m}$. The operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})$ is defined by means of QH state sums over layered [*QH-triangulations*]{} $(T_{C_\phi},\tilde b,{{\bf w}})$ of $C_\phi$. Here, by ‘layered QH-triangulation’ we mean that: $T_{C_\phi}$ is a triangulation of $C_\phi$ that produces a layered triangulation of $M_\phi$ under the quotient map $C_\phi\rightarrow M_\phi$; ${{\bf w}}$ is a suitable system of $N$-th roots of shape parameters, solving the Thurston edge equations associated to the triangulation of $M_\phi$ induced by $T_{C_\phi}$, and which actually encodes both ${{\mathfrak r}}$ and $\kappa$; finally $\tilde b$ is a way of ordering the elements of ${{\bf w}}$ in each tetrahedron of $T_{C_\phi}$, called [ *weak-branching*]{}. The complete definition of $(T_{C_\phi},\tilde b,{{\bf w}})$ is given in Section \[quantum-hyp\]. Our results describe the relationships between the QH-operators ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})$ and the set $\mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ of QT intertwiners. First note that these are assigned to $(\lambda,\lambda')$ in a [*covariant*]{} way, as they go from the representation space $V_\lambda$ of $\rho_\lambda$ to the representation space $V_{\lambda'}$ of $\rho'_{\lambda'}$. For every finite dimensional vector space $V$, denote by $V'$ its dual space. The natural way to convert a QH operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})$ into a covariant one (preserving the value of its trace) is to consider its [*dual*]{}, or [*transposed operator*]{} $${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T \in {\rm End}\left(U_\lambda'\right).$$ Using the definitions of $U_\lambda'$ in the QHI and of $V_\lambda$ in the theory of local representation of ${{\mathfrak T}}_{\lambda}^q$ one gets natural identifications $U_\lambda' \cong \left(({\mathbb{C}}^N)'\right)^{\otimes 2m} \cong V_\lambda$. Under all these identifications $${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T\in {\rm End}\left(\left(({\mathbb{C}}^N)'\right)^{\otimes 2m}\right)$$ is a QT intertwiner. Clearly, the matrix elements of these tensors in the respective canonical bases of $\left(({\mathbb{C}}^N)'\right)^{\otimes 2m}$ and $({\mathbb{C}}^N)^{\otimes 2m}$ are related by $$\label{transposed} ({{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T)^{i_1\ldots i_{2m}}_{j_1\ldots j_{2m}} = {{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})_{i_1\ldots i_{2m}}^{j_1\ldots j_{2m}}.$$ We can now state our results precisely. Let $S$ be a punctured surface of negative Euler characteristic, $\phi$ a pseudo-Anosov diffeomorphism of $S$, $M_\phi$ the mapping torus of $\phi$ equipped with its fibration over $S^1$ with monodromy $\phi$, and consider triples $(M_\phi,{{\mathfrak r}},\kappa)$ and $(T_{C_\phi},\tilde b,{{\bf w}})$ as above. Denote by $\lambda$, $\lambda'$ the ideal triangulations of $S$ given by the restriction to $S\times \{0\}$ and $\phi(S)\times \{1\}$ of the (layered) triangulation $T_{C_\phi}$. \[MAINTEO\] (1) The QH-triangulation $(T_{C_\phi},\tilde b,{{\bf w}})$ determines representations $\rho_\lambda$ and $\rho_{\lambda'}$ of ${{\mathfrak T}}_\lambda^q$ and ${{\mathfrak T}}_{\lambda'}^q$ respectively, belonging to a local representation $\rho$ of ${{\mathfrak T}}_{S}^q$, such that $\rho_{\lambda}$ is isomorphic to $\phi^*\rho_{\lambda'}$, and acting on $V_\lambda= V_{\lambda'}=(\left(({\mathbb{C}}^N)'\right)^{\otimes 2m}$. Moreover the transposed operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T$, considered as an element of ${\rm Hom}(V_\lambda,V_{\lambda'})$, is a QT intertwiner which intertwins the representations $\rho_\lambda$ and $\rho_{\lambda'}$. \(2) For any other choice of weak branching $\tilde b'$, the operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b',{{\bf w}})^T$ intertwins local representations canonically isomorphic to $\rho_\lambda$, $\rho_{\lambda'}$ respectively. We call ${{\mathcal H}}^\rho_{\lambda,\lambda'}:={{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T$ a *QHI intertwiner*. Denote by $X_0(M)$ the (unique) irreducible component of the variety of augmented $PSL(2,{\mathbb{C}})$-characters of $M$ that contains the discrete faithful holonomy ${{\mathfrak r}}_h$. Denote by $X(S)$ the variety of augmented characters of $S$, and by $i^*\colon X_0(M) \rightarrow X(S)$ the restriction map. Recall the residual weight $\kappa\in H^1(\partial \bar M_\phi;{\mathbb{C}}^*)$ involved in the definition of the reduced QHI. \[MAINTEO2\] (1) There is a neighborhood of $i^*({{\mathfrak r}}_h)$ in $i^*(X_0(M)) \subset X(S)$ such that, for any isomorphism class of local representations of ${{\mathfrak T}}_S^q$ whose holonomy lies in this neighborhood, there is a representative $\rho$ of the class, representations $\rho_\lambda$, $\rho_{\lambda'}$ belonging to $\rho$, and a QHI intertwiner ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T$ intertwining $\rho_\lambda$ and $\rho_{\lambda'}$ as above. The load of $\rho$ is determined by the values of the weight $\kappa$ at the meridian curves that form $\bar S \cap \bar M$. \(2) The set of QHI intertwiners $\mathcal{H}^{\rho}_{\lambda,\lambda'}$ is the subset of QT intertwiners which intertwin the representation $\rho_\lambda$ to $\rho_{\lambda'}$, and whose traces are well defined invariants of the triples $(M_\phi,{{\mathfrak r}},\kappa)$ such that the restriction of ${{\mathfrak r}}$ to $\pi_1(S)$ is the holonomy of $\rho$. Let us make a few comments on Theorem \[MAINTEO\] and \[MAINTEO2\]. $\bullet$ A key point of Theorem \[MAINTEO\] (1) is to realize the transposed of the matrix dilogarithms as intertwiners between local representations of the two Chekhov-Fock algebras that one can associate to an ideal square. To this aim we consider an operator theoretic formulation of the matrix dilogarithms. As these are related to the Kashaev $6j$-symbols (in a non trivial way), one could alternatively develop a proof based on the result of Bai [@Ba] relating the Kashaev $6j$-symbols with intertwiners as above. However, it is not immediate. Bai’s result deals with local representations up to isomorphism, and does not provide explicit relations between actual representatives. Also the relation between the Kashaev $6j$-symbols and the intertwiners of local representations is expressed in abstract terms, using the cyclic representations of the Weyl algebra, and not in terms of the eventual geometrically relevant parameters, the $q$-shape parameters (see Section \[quantum-hyp\]). $\bullet$ Theorem \[MAINTEO2\] (1) holds true more generally for any isomorphism class of local representations whose holonomy lies in $i^*(B)$, where $B$ is a determined Zariski open subset of the [*eigenvalue*]{} subvariety $E(M)$ of $X_0(M)$, introduced in [@KT]. Let us note here that $E(M)=X_0(M)$ if $M$ has a single cusp, and in general, ${{\mathfrak r}}_h \in E(M)$, and dim$_{\mathbb{C}}E(M)$ equals the number of cusps of $M$. For simplicity, in this paper we restrict to characters in a neighborhood of $i^*({{\mathfrak r}}_h)$ in $i^*(X_0(M))$. The general case is easily deduced from the results of [@AGT]. $\bullet$ We can say that [*the invariance property of the QHI selects preferred elements in the set of all traces of QT intertwiners*]{} (which has no a priori basepoints). Note that, the action being transitive, it does not stabilizes the set of QHI intertwiners. As well as the reduced QHI of $M_\phi$ give rise to the QT intertwiners ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T$, with the mild ambiguity by $4N$-th roots of unity factors, the unreduced QHI of $M$ give rise to the QT intertwiners ${{\mathcal H}}_N(T_{C_\phi},\tilde b,{{\bf w}})^T$, in the same projective classes as the “reduced" ones. The unreduced QHI of $M$, hence the associated QT intertwiners, do not depend on the choice of a fibration of $M$. However they depend on the full set of cohomological weights, which dominates the classes $\kappa$. The ratio unreduced/reduced QHI is a simpler invariant called [*symmetry defect*]{}, which can be used to study the dependence of the reduced QHI with respect to the fibration of $M_\phi$ ([@NA]). Finally, the ambiguity of the invariants up to multiplication by $4N$-th roots of unity may not be sharp (see [@AGT] for some improvements in the case of the unreduced QHI). The theorems above have the following consequences. Recall that the reduced QH invariants satisfy the identity ${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa) = {\rm Trace}\big({{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})\big)$, and that $M_\phi$ is the interior of a compact manifold $\bar M_\phi$ with boundary made by tori. Let us call [*longitude*]{} any simple closed curve in $\partial \bar M_\phi$ intersecting a fibre of $\bar M_\phi$ in exactly one point. We have: \[cor0\] The reduced QH invariants ${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa)$ do not depend on the values of the weight $\kappa$ on longitudes. We can also use the QH-operators ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})$ to build finer invariants, associated to any irreducible representation of ${{\mathfrak T}}_S^q$. Namely, recall that every reducible representation is canonically the direct sum of its [*isotypical*]{} components (the maximal direct sums of isomorphic irreducible summands). Every intertwiner fixes globally each isotypical component. Then, we consider the *isotypical intertwiners* $$L^\phi_{\rho_\lambda(\mu)} := {{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T_{\vert \rho_\lambda(\mu)}$$ obtained by restriction to the isotypical components $\rho_\lambda(\mu)$ of $\rho_\lambda$, associated to irreducible representations $\mu$ of ${{\mathfrak T}}_{\lambda}^q$. Any irreducible representation of ${{\mathfrak T}}_\lambda^q$ can be embedded as a summand of some local representation ([@Tou]), and so has a corresponding isotypical component. We have: \[cor\](1) The trace of $L^\phi_{\rho_\lambda(\mu)}$ is an invariant of $(M_{\phi},{{\mathfrak r}},\kappa)$ and $\mu$, well-defined up to multiplication by $4N$-th roots of unity. It depends on the isotopy class of $\phi$ and satisfies $${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa) = \sum_{\rho_\lambda(\mu)\subset \rho_\lambda}{\rm Trace}\left(L^\phi_{\rho_\lambda(\mu)}\right).$$ \(2) The invariants ${\rm Trace}(L^\phi_{\rho_\lambda(\mu)})$ do not depend on the values of the weight $\kappa$ on longitudes. In perspective, another application of Theorem \[MAINTEO\] would be to study the representations of the Kauffman bracket skein algebras defined by means of the QHI intertwiners, by using the results of [@Tou]. The background material is recalled in Section \[top-comb\] and \[BACKRES\]. The proofs are in Section \[PF\]. [**Remark about the parameter $q$.**]{} In this paper we denote by $q$ an arbitrary primitive $N$-th root of unity, where $N\geq 3$ is odd. Our orientation conventions used to define the quantum Teichmüller space, in the relations and , imply that $q$ corresponds to $q^{-1}$ in some papers about quantum Teichmüller theory, eg. in [@B-L] or [@Filippo]. Our choice of $q$ is motivated by Thurston’s relations between shape parameters in hyperbolic geometry; in the quantum case, this choice gives the most natural form to the tetrahedral and edge relations between $q$-shape parameters (see Section \[quantum-hyp\]). Topological-combinatorial support {#top-comb} ================================= We fix a compact closed oriented smooth surface $S_0$ of genus $g$, and a subset $P=\{p_1, \dots, p_r\}\subset S_0$ of $r\geq 1$ marked points. We denote by $S$ the punctured surface S$_0\setminus P$. We assume that $$m:=-\chi(S)= (2g - 2) + r>0.$$ A diffeomorphism $\phi_0:S_0\to S_0$ such that $\phi_0(p_j)=p_j$ for every $j$ induces a diffeomorphism $\phi: S\to S$. Consider the cylinder $C_0:= S_0 \times [0,1]$ with the product orientation. Denote by $M_0$ the mapping torus of $\phi_0$ with the induced orientation. That is, $M_0:=C_0/\!\!\sim_{\phi_0}$, where $(x,0)\sim_{\phi_0} (y,1)$ if $y=\phi_0(x)$, $x, y\in S_0$. Let $\hat M_0$ be the space obtained by collapsing to one point $x_j$ the image in $ M_0$ of each line ${p_j}\times [0,1]$ in $C_0$. Then $\hat M_0$ is a pseudo-manifold; $X=\{x_1,\dots ,x_r\}$ is the set of singular (ie. non manifold) points of $\hat M_0$. Every singular point $x_j$ has a conical neighbourhood homeomorphic to the quotient of $T^2_j \times [0,1]$ by the equivalence relation identifying $T^2_j \times \{0\}$ to a point, where $T^2_j$ is a $2$-torus. The point $x_j$ corresponds to the coset of $T^2_j \times \{0\}$, and $T^2_j\times \{1\}$ is called the [*link*]{} of $x_j$ in $\hat M_0$. So $M= \hat M_0 \setminus X$ is the interior of a compact manifold $\bar M$ with boundary formed by $r$ tori. In fact $M$ is the mapping torus $M_\phi=C/\!\!\sim_\phi$, where $C=S\times [0,1]$. From now on, we denote by $M_\phi$ the manifold $M$ endowed with this fibration over the circle $S^1$, and by $C_\phi$ the mapping cylinder of $\phi$. Layered triangulations of $M_\phi$ can be constructed as follows (see for instance [@LA], where one can find also a proof of the well known fact that two ideal triangulations of $S$ are connected by a finite chain of diagonal exchanges). Consider the set of triangulations $\lambda$ of $S_0$ whose sets of vertices coincide with $P$. By removing $P$, such a triangulation $\lambda$ is also said an [*ideal triangulation*]{} of $S$; it has $3m$ edges and $2m$ triangles. Denote by $\lambda'= \phi_0(\lambda)$ the image ideal triangulation of $S$. Let us fix a chain of diagonal exchanges connecting $\lambda$ to $\lambda'$: $$\lambda = \lambda_0 \to \lambda_1 \to \dots \to \lambda_k= \lambda'.$$ Possibly by performing some additional diagonal exchanges followed by their inverses, we can assume that this chain is “full”, in the sense that every edge of $\lambda$ supports some diagonal exchange in the chain. Then the chain induces a $3$-dimensional triangulation $T$ of $\hat M_0$, whose tetrahedra are obtained by “superposing”, for every $j$, the two squares of $\lambda_j$, $\lambda_{j+1}$ involved in the diagonal exchange $\lambda_j\rightarrow \lambda_{j+1}$ (see Figure \[typefig\]). The set of vertices of $T$ coincides with $X$. The ideal triangulation $T\setminus X$ is called a [*layered triangulation*]{} of $M_\phi$. It lifts to a layered triangulation $T_{C_\phi}$ of the cylinder $C_\phi$. For each tetrahedron of $T$, we call *abstract tetrahedron* its underlying simplicial set considered independently of the face pairings in $T$. Similarly we call abstract edge or face of $T$ any edge or face of an abstract tetrahedra of $T$. Given any edge $e$ of $T$, we write $E \to e$ to mean that an abstract edge $E$ is identified to $e$ in the triangulation $T$. From now on we assume that the diffeomorphism $\phi$ of $S$ is [*pseudo-Anosov*]{}, so that $M_\phi$ is a hyperbolic manifold with $r$ cusps. By construction every (ideally triangulated) surface $(S,\lambda_j)$ is embedded in $T$ and $T_{C_\phi}$. The union of the surfaces $(S,\lambda_j)$ forms the $2$-skeletons of $T$ and $T_{C_\phi}$. Every surface $(S,\lambda_j)$ divides $M_\phi$ locally, and the given orientations of $S$ and $M_\phi$ determines the (local) [*positive side*]{} of $S$ in $M$. In terms of $T_{C_\phi}$, the positive side of $(S,\lambda)=(S,\lambda_0)$ lifts to a collar of $S\times \{0\}$, while the [*negative*]{} side of $(S,\lambda')=(S,\lambda_k)$ lifts to a collar of $S\times \{1\}$. The triangulation $T$ is naturally endowed with a [*taut pre-branching*]{} $\omega$. Let us recall briefly this notion (see [@NA], [@LA]). A taut pre-branching is a choice of transverse co-orientation for each $2$-face of $T$, so that for every tetrahedron, exactly two of its $2$-face co-orientations point inward/outward; moreover, it is required that there are exactly two abstract diagonal edges $E\to e$ for every edge $e$ of $T$. Here, we call *diagonal edges* the two edges of an abstract tetrahedron whose adjacent $2$-face co-orientations point both inward or both outward. The taut pre-branching $\omega$ of $T$ is defined by the system of transverse $2$-face co-orientations dual to the orientation of the embedded surfaces $(S,\lambda_j)$. We fix also a [*weak branching*]{} $\tilde b$ [*compatible with*]{} $\omega$ (see [@AGT]). Recall that $\tilde{b}$ is a choice of vertex ordering $b$ for each abstract tetrahedron of $T$, called [*(local) branching*]{}, satisfying the following constraint. The abstract $2$-faces of the tetrahedron have an orientation induced by $b$, given by their vertex orderings up to even permutations. Then, it is required that these orientations match, under the $2$-face pairings, with the orientations dual to $\omega$ (note that we do not require that the abstract vertex orderings match: this defines the stronger notion of [*global branching*]{}). Similarly, $\tilde{b}$ gives an orientation to every tetrahedron; without loss of generality, we can assume that it agrees with the orientation of $M$. A weak branching $\tilde b_j$ is also defined on every surface triangulation $\lambda_j$, by giving every triangle $\tau$ in $\lambda_j$ the branching induced by the tetrahedron lying on the positive side of $\tau$. In Figure \[typefig\] we show a typical [*local*]{} configuration occurring in $T$ or $T_{C_\phi}$, that is, a tetrahedron $\Delta$, represented under an orientation preserving embedding in ${\mathbb{R}}^3$, built by gluing squares $Q$, $Q'$ along the boundary, carrying triangulations $\lambda$, $\lambda'$ related by a diagonal exchange. The orientation of $\Delta$ agrees with the standard orientation of ${\mathbb{R}}^3$, and the $2$-face co-orientations that define the pre-branching $\omega$ are dual to the counter-clockwise orientation of the four faces of $\Delta$. We have chosen one branching $b$ of $\Delta$ (any edge being oriented from the lowest to the biggest endpoint) among the two which are compatible with $\omega$ and induce the given orientation of $\Delta$. The $2$-faces are ordered as the opposite vertices are, accordingly with $b$. We advertize that Figure \[typefig\] will be used to support all computations. In such a situation we will denote by $e_0$ the edge $[v_0,v_1]$, by $e_1$ the edge $[v_1,v_2]$, and by $e_2$ the edge $[v_0,v_2]$. ![\[typefig\] ](typefig.eps){width="7cm"} Background results {#BACKRES} ================== Classical hyperbolic geometry {#class-hyp} ----------------------------- Let $M$ be any cusped hyperbolic $3$-manifold with $r$ cusps, $r\geq 1$. We denote by $X(M)$ the variety of [*augmented $PSL(2,{\mathbb{C}})$-characters*]{} of $M$. Let us recall the definition. Consider the set of pairs $({{\bf r}},\{\xi_\Gamma\}_{\Gamma \in \Pi})$, where ${{\bf r}}\colon \pi_1(M)\rightarrow PSL(2,{\mathbb{C}})$ is a group homomorphism, $\Pi$ is the set of peripheral subgroups of $\pi_1(M)$ (associated to the boundary tori $T^2_i$ of $\bar M$, for all choices of base points of $\pi_1(T^2_i)$ and $\pi_1(M)$ and paths between them), and for every $\Gamma\in \Pi$, $\xi_\Gamma\in {\mathbb{C}}P^1$ is a fixed point of ${{\bf r}}(\Gamma)$ such that the assignment $\Gamma \rightarrow \xi_\Gamma$ is equivariant with respect to the action of $\pi_1(M)$ on $\Pi$ by conjugation and on ${\mathbb{C}}P^1$ via ${{\bf r}}$ by Moebius transformations. The set $\{({{\bf r}},\{\xi_\Gamma\}_{\Gamma \in \Pi})\}$ is a complex affine algebraic set $R(M)$, with an action of $PSL(2,{\mathbb{C}})$ defined on a pair $({{\bf r}},\{\xi_\Gamma\}_\Gamma)$ by conjugation on ${{\bf r}}$ and Moebius transformation on $\{\xi_\Gamma\}_\Gamma$. Then, $X(M)$ is the algebro-geometric quotient of $R(M)$ by $PSL(2,{\mathbb{C}})$, that is, the set of closed points of the ring of invariant functions ${\mathbb{C}}[R(M)]^{PSL(2,{\mathbb{C}})}$. Similarly we denote by $X(S)$ the variety of augmented characters of a surface $S$. In the setting of Section \[top-comb\], if $M=M_\phi$ and $S$ is a fibre of the fibration, then the inclusion map $i\colon S=S\times \{0\} \hookrightarrow M_\phi$ induces a regular (restriction) map $$i^*\colon X(M_\phi) \rightarrow X(S).$$ Recall that $\pi_1(M_\phi)$ is a HNN-extension of $\pi_1(S)$. That is, given a generating set $\gamma_1,\ldots,\gamma_u$ of $\pi_1(S)$ satisfying the relations $r_1,\ldots,r_v$, there is an isomorphism between $\pi_1(M_\phi)$ and the group generated by $\gamma_1,\ldots,\gamma_u$ and an element $t$ satisfying the relations $r_1,\ldots,r_v$ and $t\alpha t^{-1} = \phi_*(\alpha)$ for all $\alpha\in \pi_1(S)$, where $\phi_* \colon \pi_1(S){\rightarrow}\pi_1(S)$ is the isomorphism induced by $\phi$. Therefore, every representation ${{\bf r}}\colon \pi_1(S)\rightarrow PSL(2,{\mathbb{C}})$ that may be extended to $M_\phi$ is such that ${{\bf r}}(\pi_1(S))$ and ${{\bf r}}\circ \phi_*(\pi_1(S))$ are conjugate subgroups of $PSL(2,{\mathbb{C}})$. Any augmented character of $S$ that may be extended to $M_\phi$ is thus a fixed point of the map $\Phi_*\colon X(S) \rightarrow X(S)$ induced by the map ${{\bf r}}\mapsto {{\bf r}}\circ \phi_*$ on representations, and $\overline{i^*(X(M_\phi))}$ is a subvariety of ${\rm Fix}(\Phi_*)$. Again in the setting of Section \[top-comb\], let $(T,\tilde b)$ be a weakly branched layered triangulation of the fibred cusped manifold $M_\phi$. Let us recall a few facts about the [*gluing variety*]{} $G(T,\tilde{b})$, that is, the set of solutions of the Thurston gluing equations supported by $(T,\tilde{b})$. It is a complex affine algebraic set of dimension greater than or equal to $r$. The points of $G(T,\tilde{b})$ are classically called [*systems of shape parameters*]{}. Their coordinates, the shape parameters, are scalars in ${\mathbb{C}}\setminus \{0,1\}$ associated to the abstract edges of $T$. The shape parameters of opposite edges of a tetrahedron are equal, and the cyclically ordered triple of shape parameters of a tetrahedron encodes an isometry class of hyperbolic ideal tetrahedra. The set $G(T,\tilde{b})$ is defined by the following two sets of equations. For every branched tetrahedron, set $w_j := w(E_j)$, $j=0,1, 2$ (see before Figure \[typefig\] for the ordering of the edge $E_j$). For every edge $e$ of $T$, define the [*total shape parameter*]{} $W(e)$ as the product of the shape parameters $w(E)$, where $E \to e$. Then we have: - [*(Tetrahedral equation)*]{} For every tetrahedron and $j\in\{0,1,2\}$, $w_{j+1}=(1-w_j)^{-1}$ cyclically; hence $w_0w_1w_2=-1$. - [*(Edge equation)*]{} For every edge $e$ of $T$, $W(e)=1$. A point $w\in G(T,\tilde{b})$ determines a pseudo-developing map $F_w:\tilde{M}_\phi \rightarrow {\mathbb{H}}^3$, where $\tilde{M}_\phi$ is the universal cover of $M_\phi$, and $F_w$ is well-defined up to post-composition with an orientation-preserving isometry of ${\mathbb{H}}^3$. The map $F_w$ sends homeomorphically the edges of $\tilde T$ to complete geodesics, and it satisfies $F_w (g \tilde x) = {{\bf r}}_w(g) F_w (\tilde x)$ for all $\tilde x\in \tilde M_\phi$, $g\in \pi_1(M_\phi)$, where ${{\bf r}}_w\colon \pi_1(M_\phi)\to PSL(2,{\mathbb{C}})$ is a homomorphism. So $w$ encodes the conjugacy class of ${{\bf r}}_w$. In fact, $F_w$ determines also some ${{\bf r}}_w$-equivariant set $\{\xi_\Gamma\}_{\Gamma \in \Pi}$ as above, so that eventually the map $w\mapsto {{\bf r}}_w$ can be lifted to a regular “holonomy” map $$hol\colon G(T,\tilde{b})\rightarrow X(M_\phi).$$ We have (see Proposition 4.6 of [@AGT] when $M_\phi$ has a single cusp, and Remark 1.4 of [@NA] for the general case): \[gluing-var\] There is a subvariety $A$ of $G(T,\tilde{b})$ of dimension equal to the number of cusps of $M$, and a point $w_h\in A$, such that $hol(w_h)$ is the hyperbolic holonomy of $M_\phi$, and $hol_{\vert A}$ is a homeomorphism from a Zariski open subset of $A$ containing $w_h$ onto its image. In particular, every point $w\in A$ encodes an augmented character of $M_\phi$; the algebraic closure of $hol(A)$ is the [*eigenvalue*]{} subvariety $E(M_\phi)$ of $X_0(M_\phi)$, the irreducible component of $X(M_\phi)$ containing the discrete faithful holonomy ${{\bf r}}_h$ (see [@KT]). If $M_\phi$ has a single cusp, then $E(M_\phi)=X_0(M_\phi)$; in general $E(M_\phi)$ contains ${{\bf r}}_h$ and has complex dimension equal to the number of cusps of $M_\phi$. It follows from Proposition \[gluing-var\] that a point of $i^*(X_0(M_\phi))\subset X(S)$ close enough to $i^*({{\bf r}}_h)$ is encoded by a point $w\in A$. From now on, we consider only systems of shape parameters $w$ lying in the subvariety $A$ of $G(T,\tilde{b})$. Denote by $(T,\tilde{b},w)$ the layered triangulation $T$ of $M_\phi$ endowed with the weak branching $\tilde{b}$ and the labelling of the abstract edges of $T$ by a system of shape parameters $w$. For every $j=0,\dots, k$, consider the ideally triangulated surfaces $(S,\lambda_j)$ embedded into $(T,\tilde b, w)$ as in Section \[top-comb\]. For every edge $e$ of $\lambda_j$, define the [*lateral shape parameter*]{} $W^+_j(e)$ as the product of the shape parameters of the abstract edges $E \to e$ carried by the tetrahedra lying on the [*positive*]{} side of $(S,\lambda_j)$. \[shear-bend\] For any edge $e$ of $\lambda_j$, the (exponential) shear-bend coordinate of $e$ defined by any pleated surface $F_{w\vert \tilde S}$ (ie. any lift $(\tilde{S},\tilde \lambda_j)$ of $(S,\lambda_j)$ to $\tilde M_\phi$, and any pseudo-developing map $F_{w\vert \tilde S}\colon \tilde S \rightarrow {\mathbb{H}}^3$), coincides with the parameter $\textstyle x^j(e):= -W^+_j(e)$. The proof follows from the definitions (see eg. [@B-L]). Note that $x^0(e)=x^k(\phi_0(e))$ for every edge $e$ of $\lambda=\lambda_0$. Also, the shear-bend coordinate of any edge $\tilde e$ of $\tilde \lambda_j$ is eventually ‘attached’ to the corresponding edge $e$ of $\lambda_j$, because for different choices of $\tilde S$ or $F_w$ the images of $F_{w\vert \tilde S}$ differ only by an hyperbolic isometry. [The parameter $x^j(e)$ is the [*opposite*]{} of $W^+_j(e)$ because the (oriented) bending angle along an edge of $\lambda_j$ is traditionally measured by the external dihedral angle $\pi-\theta$ (see eg. [@B-L; @B-B-L]), whereas the shape parameters use the internal dihedral angle $\theta$. So $\theta=0$ when two adjacent ideal triangles $F (\tau)$ and $F(\tau')$ coincide, for triangles $\tau$, $\tau'$ of $(\tilde{S},\tilde{\lambda}_j)$.]{} Quantum hyperbolic geometry {#quantum-hyp} --------------------------- We keep the setting of the previous section. Recall that $N\geq 3$ is an odd integer, and $q$ a primitive $N$-th root of unity. We begin with a few qualitative, structural features of the [*reduced quantum hyperbolic state sum*]{} ${{\mathcal H}}^{red}_N(T,\tilde b,{{\bf w}})$ defined in [@AGT; @NA]. It is a regular rational function defined on a covering of the gluing variety $G(T,\tilde b)$. The points of this covering over a point $w\in G(T,\tilde b)$ are certain systems of $N$-th roots ${{\bf w}}(E)$ of the shape parameters $w(E)$, which label the abstract edges $E$ of $T$ and are called [ *quantum shape parameters*]{}. Alike the “classical" ones, opposite edges of an abstract tetrahedron are given the same quantum shape parameter. Moreover, the quantum shape parameters verify the following relations, which are “quantum" counterparts of the defining equations of the gluing variety. For every branched tetrahedron of $(T,\tilde b)$ put ${{\bf w}}_j := {{\bf w}}(E_j)$ (with the usual edge ordering fixed before Figure \[typefig\]). For every edge $e$ of $(T,\tilde b,{{\bf w}})$, define the [*total quantum shape parameter*]{} ${{\bf W}}(e)$ as the product of the quantum shape parameters of the abstract edges $E \to e$. Then we have: - [*(Tetrahedral relation)*]{} For every tetrahedron, ${{\bf w}}_0{{\bf w}}_1{{\bf w}}_2= -q$. - [*(Edge relation)*]{} For every edge $e$ of $T$, ${{\bf W}}(e)=q^{2}$. For simplicity we will work with systems of quantum shape parameters ${{\bf w}}$ such that the corresponding systems of “classical” shape parameters $w$ belong to a simply connected open neighborhood $A_0\subset A\subset G(T,\tilde b)$ of $w_h$ (the general case of systems ${{\bf w}}$ with arbitrary $w\in A$ is described in [@AGT]). Note that $A_0$ is chosen so that ${{\bf w}}(E)$ varies continuously with $w\in A_0$. \[qpar\] [(1) For every $z\in {\mathbb{C}}\setminus \{0\}$, denote by $\log(z)$ the determination of the logarithm which has the imaginary part in $]-\pi,\pi]$. There is a ${\mathbb{Z}}$-labelling $d$ of the abstract edges of $T$ such that, for every system ${{\bf w}}$ of quantum shape parameters over a point $w\in A_0$, and every abstract edge $E$, we have $$\label{qshapeconst} {{\bf w}}(E) = \exp\left(\frac{1}{N}\left(\log(w(E))+\pi i (N+1)d(E)\right)\right).$$ The above relations verified by the quantum shape parameters can be equivalently rephrased in terms of the ${\mathbb{Z}}$-labelling $d$. For any point $w\in A_0$, a ${\mathbb{Z}}$-labelling $d$ satisfying these relations defines a system of quantum shape parameters over $w$ by the formula . (2) In [@GT; @AGT; @NA] we solved (mainly in terms of the ${\mathbb{Z}}$-labellings $d$) the existence problem of triples $(T,\tilde b,{{\bf w}})$ for any cusped hyperbolic $3$-manifold in the special case $q=-\exp(-i\pi/N)$. The same method works for an arbitrary $q$, up to minor changes.]{} The following result summarizes in a qualitative way the invariance properties of ${{\mathcal H}}^{red}_N(T,\tilde b,{{\bf w}})$. Denote by $A_{0,N}$ the set of systems of quantum shape parameters over $A_0$. Recall from Section \[top-comb\] that $M_\phi$ can be considered as the interior of a compact manifold $\bar M_\phi$ bounded by tori. Fix a basis $(l_1,m_1), \ldots , (l_r,m_r)$ of $\pi_1(\partial \bar M_\phi)$, and use it to identify $H^1(\partial \bar{M}_\phi;{\mathbb{C}}^*)$ with $({\mathbb{C}}^*)^{2r}$. Any augmented character $[({{\bf r}},\{\xi_\Gamma\}_\Gamma)]\in X(M_\phi)$ determines the square of one of the two (reciprocally inverse) eigenvalues of ${{\bf r}}(\gamma)$, for any non trivial simple closed curve $\gamma$ on $\partial \bar M_\phi$ and representative ${{\bf r}}$ of the character. Namely, by taking ${{\bf r}}$ in its conjugacy class so that ${{\bf r}}(\pi_1(\partial \bar M_\phi))$ fixes the point $\infty$ on $\mathbb{C}P^1$, ${{\bf r}}([\gamma])$ acts on $\mathbb{C}$ as $w\mapsto \gamma_{{\bf r}}w + b$, where $\gamma_{{\bf r}}\in \mathbb{C}^*$ and $b \in \mathbb{C}$. The coefficient $\gamma_{{\bf r}}$ is that squared eigenvalue selected by $[({{\bf r}},\{\xi_\Gamma\}_\Gamma)]$. We have (see [@AGT] and [@NA]): \[SSum\] (1) There exists a determined regular rational map $\kappa_N\colon A_{0,N}{\rightarrow}({\mathbb{C}}^*)^{2r}$ such that the image $\kappa_N(A_{0,N})$ is the open subset of $({\mathbb{C}}^*)^{2r}$ made of all the classes $\kappa\in H^1(\partial \bar{M}_\phi;{\mathbb{C}}^*)$ such that $\kappa(l_j)^N$ and $\kappa(m_j)^N$ are equal up to a sign respectively to the squared eigenvalues selected by $[({{\bf r}},\{\xi_\Gamma\}_\Gamma)]$ at the curves $l_j$ and $m_j$, where ${{\mathfrak r}}:=[({{\bf r}},\{\xi_\Gamma\}_\Gamma)] =hol(w)$ for some $w\in A_0$. \(2) Given ${{\mathfrak r}}\in hol(A_0)$ and $\kappa\in H^1(\partial \bar{M}_\phi;{\mathbb{C}}^*)$ as above, the value of ${{\mathcal H}}^{red}_N(T,\tilde b,{{\bf w}})$ is independent, up to multiplication by $4N$-th roots of unity, of the choice of $(T,\tilde b,{{\bf w}})$ among all weakly branched layered triangulations of $M_\phi$ endowed with a system of quantum shape parameters ${{\bf w}}$ such that $hol(w) = {{\mathfrak r}}$ and $\kappa_N({{\bf w}}) =\kappa$. We denote the resulting invariant by ${{\mathcal H}}_N^{red}(M_\phi,{{\mathfrak r}},\kappa):= {{\mathcal H}}^{red}_N(T,\tilde b,{{\bf w}})$. By the results of [@NA], [*it actually depends on the fibration of $M_\phi$*]{}. The map $\kappa_N$ in Theorem \[SSum\] is a lift to $A_{0,N}$ of $j^*\circ hol_{\vert A_0}$, where $j^*$ is the restriction map $X(M_\phi) {\rightarrow}X(\partial M_\phi)$. Here is a concrete way to compute $\kappa_N({{\bf w}})(\alpha)$, for $\alpha \in H_1(\partial \bar{M}_\phi;{\mathbb{Z}})$. It is well known that the [*truncated tetrahedra*]{} of an ideal triangulation $T$ of $M_\phi$ provide a cell decomposition of $\bar M_\phi$ which restricts to a triangulation $\partial T$ of $\partial \bar M_\phi$. Every abstract vertex of $T$ corresponds to an abstract triangle of $\partial T$ (at which a tetrahedron has been “truncated”); every vertex of such a triangle is contained in one abstract edge of $T$. Represent the class $\alpha$ by an oriented simple closed curve $a$ on $\partial \bar M_\phi$, transverse to $\partial T$. The intersection of $a$ with a triangle $F$ of $\partial T$ is a collection of oriented arcs. We can assume that none of these arcs enters and exits $F$ by a same edge. Then, each one turns around a vertex of $F$. If $\Delta$ is a tetrahedron of $T$ and $F$ corresponds to a vertex of $\Delta$, for every vertex $v$ of $F$ we denote by $E_v $ the edge of $\Delta$ containing $v$, and write $a \rightarrow E_v$ to mean that some subarcs of $a$ turn around $v$. We count them algebraically, by using the orientation of $a$: if there are $s_+$ (resp. $s_-$) such subarcs whose orientation is compatible with (respectively, opposite to) the orientation of $\partial \bar M_\phi$ as viewed from $v$, then we set $ind(a,v):=s_+-s_-$. Then $$\label{cuspweightform2} \kappa_N({{\bf w}})(\alpha) = \prod_{a {\rightarrow}E_v} {{\bf w}}(E_v)^{ind(a,v)}.$$ In the case where $\alpha$ is the class of a positively oriented meridian curve $m_i$ of $\partial \bar M_\phi$, this gives $$\label{cuspweightform} \kappa_N({{\bf w}})(\alpha) = \prod_{m_i {\rightarrow}E_v} {{\bf w}}(E_v).$$ Let us give now more details about the definition of the reduced QH state sum ${{\mathcal H}}^{red}_N(T,\tilde b,{{\bf w}})$. We are going to do it in terms of the QH-operator associated to the cylinder $C_\phi$, already mentioned in the Introduction. By cutting $(T,\tilde b,{{\bf w}})$ along one of the surfaces $(S,\lambda_j)$ (that is, a triangulated fibre of $M_\phi$), we get a [*QH-triangulation*]{} $(T_{C_\phi},\tilde b,{{\bf w}})$ of the cylinder $C_\phi$ having as “source" boundary component $(S,\lambda)$, and as “target" boundary component $(S,\lambda')$, where $\lambda=\lambda_j$ and $\lambda'=\phi(\lambda_j)$. It carries in a [*contravariant way*]{} the QH-operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})\in {\rm End}\left(({\mathbb{C}}^N)^{\otimes 2m}\right)$, which is such that $${\rm Trace}\left({{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})\right)={{\mathcal H}}^{red}_N(T,\tilde b,{{\bf w}}).$$ We define the QH-operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})$ by means of elementary tetrahedral and $2$-face operators, as follows. We can regard the QH triangulation $(T,\tilde b,{{\bf w}})$ of $M_\phi$ as a [*network of abstract QH-tetrahedra*]{} $(\Delta,b,{{\bf w}})$, with gluing data along the $2$-faces. To each QH-tetrahedron $(\Delta,b,{{\bf w}})$ we associate a linear isomorphism called [*basic matrix dilogarithm*]{} (“basic” refers to the fact that we have dropped a symmetrization factor from the matrix dilogarithms that enter the definition of the unreduced QHI), $${{\mathcal L}}_N(\Delta,b,{{\bf w}})\colon V_2\otimes V_0 \rightarrow V_3 \otimes V_1$$ where $V_j$ is a copy of ${\mathbb{C}}^N$ associated to the $2$-face of $(\Delta,b)$ opposite to the vertex $v_j$. The basic matrix dilogarithms will be defined explicitly below. For the moment, note that $V_1$, $V_3$ correspond to the $2$-faces with pre-branching co-orientation pointing inside $\Delta$. Concerning the gluing data, every $2$-face $F$ of $T$ is obtained by gluing a pair of abstract $2$-faces. Denote by $F_s$ and $F_t$ the “source” and “target” $2$-face of the pair, with respect to the transverse co-orientation defined by the pre-branching $\omega_{\tilde b}$. Denote by $V_{F_s}$ and $V_{F_t}$ the copies of ${\mathbb{C}}^N$ associated to $F_s$ and $F_t$. The identification $F_s\to F_t$ is given by an even permutation on three elements, which encodes the image of the vertices of $F_s$ in $F_t$. So it can be encoded by an element $r(F)\in {\mathbb{Z}}/3{\mathbb{Z}}$. The triangulation $T_{C_\phi}$ of the cylinder $C_\phi$ has $2m$ free $2$-faces at both the source and target boundary components, $(S,\lambda)$ and $(S,\lambda')$. Note that at every $2$-face of $(S,\lambda)$ the pre-branching orientation points inside $C_\phi$, while it points outside at the $2$-faces of $(S,\lambda')$. For every gluing $F_s\to F_t$ occuring at an internal $2$-face $F$ of $T_{C_\phi}$, as well as to every $2$-face $F$ of $(S,\lambda')$, we associate an endomorphism (again in contravariant way) $$\label{Qop} Q_N^{r(F)} : V_{F_t} \to V_{F_s}.$$ By definition, the QH-operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}})$ is the total contraction of the network of tensors $\{{{\mathcal L}}_N(\Delta,b,{{\bf w}})\}_{\Delta}$ and $\{Q_N^{r(F)}\}_F$. In formulas: $$\label{Ssumformula} {{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}}) = \sum_s \prod_\Delta {{\mathcal L}}_N(\Delta,b,{{\bf w}})_s \prod_F Q_{N,s}^{r(F)}$$ where the sum ranges over all maps $s\colon\{$abstract $2$-faces of $T\} \cup \{$$2$-faces of $(S,\lambda')\} \to \{0,\ldots,N-1\}$ (the [*states*]{} of $(T_{C_\phi},\tilde b, {{\bf w}})$), and ${{\mathcal L}}_N(\Delta,b,{{\bf w}})_s$ and $Q_{N,s}$ denote the entries of the tensors ${{\mathcal L}}_N(\Delta,b,{{\bf w}})$ and $Q_{N}$ selected by $s$, when the tensors are written in the canonical basis $\{e_j\}$ of ${\mathbb{C}}^N$. Note that the domain of $s$ contains two copies of each $2$-face $F$ in the target boundary component of $C_\phi$. They correspond to the source and target spaces $V_{F_s}$, $V_{F_t}$ of $Q_N^{r(F)}$. Finally, we provide an operator theoretic definition of the basic matrix dilogarithms (it is a straightforward rewriting of formula (32) in [@GT]), as well as their entries and those of the endomorphisms $Q_N$. Since it is the transposed tensor ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}})^T$ mentioned in the Introduction (see formula ) that occurs in Theorem \[MAINTEO\], we consider the transposed endomorphisms of $Q_N$ and of the basic matrix dilogarithm instead: $$Q_N^{T} : V_{F_s}' \to V_{F_t}'\ ,\ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}): V_3' \otimes V_1' \to V_2' \otimes V_0'$$ where $V_{F_s}' = V_{F_t}' = V_j'=({\mathbb{C}}^N)'$, the dual space of ${\mathbb{C}}^N$, for every $j$. The matrix elements will be given with respect to the canonical basis of $\left(({\mathbb{C}}^N)'\right)^{\otimes 2}$. \[identrem\][If $\{e_j\}$ is the canonical basis of ${\mathbb{C}}^N$, and $\{e^j\}$ the dual basis of $({\mathbb{C}}^N)'$, the identification map $\iota: {\mathbb{C}}^N \to ({\mathbb{C}}^N)'$, $\iota(e_j)=e^j$, extends to a canonical identification, also denoted by $\iota$, between $({\mathbb{C}}^N)^{\otimes 2}$ and $\left( ({\mathbb{C}}^N)'\right)^{\otimes 2}$. Below we will rather deal with the endomorphisms $$\iota^{-1}\circ Q_N^{T} \circ \iota \in {\rm End}\left({\mathbb{C}}^N\right) \ ,\ \iota^{-1}\circ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) \circ \iota \in {\rm End}\left(({\mathbb{C}}^N)^{\otimes 2}\right).$$ Obviously, they have the same matrix elements as $Q_N^{T}$ and ${{\mathcal L}}_N^T(\Delta,b,{{\bf w}})$ with respect to the canonical basis of ${\mathbb{C}}^N$ and $({\mathbb{C}}^N)'$, and $({\mathbb{C}}^N)^{\otimes 2}$ and $\left(({\mathbb{C}}^N)'\right)^{\otimes 2}$, respectively. Similarly we will consider $$\iota^{-1}\circ {{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}})^T\circ \iota \in {\rm End}\left(({\mathbb{C}}^N)^{\otimes 2m}\right).$$ [*For simplicity we will systematically omit the maps $\iota$ from the notations*]{}.]{} The endomorphism $Q_N^T : V_{F_s} \to V_{F_t}$ is defined in the standard basis $\{e_j\}$ of (the copies $V_{F_s}$, $V_{F_t}$ of) ${\mathbb{C}}^N$ by $$\label{defQNT} Q_N^T(e_k) = \frac{1}{\sqrt{N}} \sum_{j=0}^{N-1}q^{2kj+j^2}e_j.$$ Define the endomorphisms $A_0$, $A_1$, $A_2 \in {\rm End}({\mathbb{C}}^N)$ by $$\label{standardmatrix} A_0e_k=q^{2k}e_k, \ A_1e_k=e_{k+1}, \ A_2e_k=q^{1-2k}e_{k-1}.$$ Note that $A_0A_1A_2 = q{\rm Id}_{{\mathbb{C}}^N}$. We stress that these operators arise in the representation of the [*triangle algebra*]{} (see Section \[standard\] below); their occurrence in the definition of the basic matrix dilogarithm is a key point to perform the computations at the end of the paper that will establish the bridge between the QHI and the QT intertwiners. Put $$g(x)=\prod_{j=1}^{N-1} (1-xq^{2j})^{j/N}, h(x)=\frac{g(x)}{g(1)}$$ $$\Upsilon = \frac{1}{N} \sum_{i,j=0}^{N-1} q^{-2ij-j} A_0^i \otimes (A_1A_0)^{-j}.$$ For any triple of quantum shape parameters ${{\bf w}}=({{\bf w}}_0,{{\bf w}}_1,{{\bf w}}_2)$ and any endomorphism $U$ such that $U^N=-{\rm Id}$, set $$\Psi_{{\bf w}}(U) = h({{\bf w}}_0) \sum_{i=0}^{N-1} U^i \prod_{j=1}^i \frac{-q^{-1}{{\bf w}}_2}{1-{{\bf w}}_0^{-1}q^{2j}}$$ where by convention we set the product equal to $1$ when $i=0$. Then $$\label{Stensor} {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) = \Psi_{{\bf w}}(-A_0A_1 \otimes A_1^{-1})\circ \Upsilon.$$ Later we will need the following elementary but crucial fact, which belongs (in a different form) to Faddeev-Kashaev [@FK]. Let ${{\bf w}}_0$, ${{\bf w}}_2$ be complex numbers such that ${{\bf w}}_2^N = 1-{{\bf w}}_0^{-N}$, and $U$ an endomorphism such that $U^N=-{\rm Id}$. Then $\Psi_{{\bf w}}(U)$ defined as above is the unique endomorphism up to scalar multiplication which is a solution of the functional relation $$\label{cycdil2} \Psi_{{\bf w}}(q^2U) = ({{\bf w}}_0+q^{-1}{{\bf w}}_0{{\bf w}}_2U)\Psi_{{\bf w}}(U).$$ [[*Proof. *]{}]{}The hypothesis on ${{\bf w}}_0$, ${{\bf w}}_2$ implies that the summands of $\Psi_{{\bf w}}(U)$ are periodic in $i$, with period $N$. Hence we get the same sum if $i$ ranges from $1$ to $N$. With this observation, we have $$\begin{aligned} q^{-1}{{\bf w}}_0{{\bf w}}_2U\Psi_{{\bf w}}(U) & = h({{\bf w}}_0) \sum_{i=0}^{N-1} U^{i+1} q^{-1}{{\bf w}}_0{{\bf w}}_2\prod_{j=1}^i \frac{-q^{-1}{{\bf w}}_2}{1-{{\bf w}}_0^{-1}q^{2j}} \\ & = h({{\bf w}}_0) \sum_{i=0}^{N-1} U^{i+1} q^{-1}{{\bf w}}_0{{\bf w}}_2 \left(\frac{1-{{\bf w}}_0^{-1}q^{2i+2}}{-q^{-1}{{\bf w}}_2} \right) \prod_{j=1}^{i+1} \frac{-q^{-1}{{\bf w}}_2}{1-{{\bf w}}_0^{-1}q^{2j}} \\ & = h({{\bf w}}_0) \left( - {{\bf w}}_0 \sum_{i=0}^{N-1} U^{i+1} \prod_{j=1}^{i+1} \frac{-q^{-1}{{\bf w}}_2}{1-{{\bf w}}_0^{-1}q^{2j}} + \sum_{i=0}^{N-1} (q^2U)^{i+1} \prod_{j=1}^{i+1} \frac{-q^{-1}{{\bf w}}_2}{1-{{\bf w}}_0^{-1}q^{2j}} \right)\\ & = - {{\bf w}}_0 \Psi_{{\bf w}}(U) + \Psi_{{\bf w}}(q^2U).\end{aligned}$$ Hence $\Psi_{{\bf w}}(U)$ solves the equation . Conversely, $U^N=-{\rm Id}$ implies that $U$ is diagonalizable with eigenvalues in the set $\{-q^{2i},i=0,\ldots,N-1\}$. Hence $\Psi_{{\bf w}}(U)$ is a polynomial in $U$ completely determined by its eigenvalues, which are of the form $\Psi_{{\bf w}}(-q^{2i})$. By they are given for $i\geq 1$ by $\textstyle \Psi_{{\bf w}}(-q^{2i}) = \prod_{j=1}^{i} ({{\bf w}}_0-q^{2j-3}{{\bf w}}_0{{\bf w}}_2)\Psi_{{\bf w}}(-1)$, and hence uniquely determined up to the choice of $\Psi_{{\bf w}}(-1)$. Therefore has a unique solution $\Psi_{{\bf w}}(U)$ up to scalar multiplication. $\Box$ [The normalization factor $h({{\bf w}}_0)$ is chosen so that det$\left(\Psi_{{\bf w}}(-A_0A_1 \otimes A_1^{-1})\right)=1$.]{} In particular, when $({{\bf w}}_0,{{\bf w}}_1,{{\bf w}}_2)$ is a triple of quantum shape parameters we have $$\label{cycdil} \Psi_{{\bf w}}(q^2U) = ({{\bf w}}_0-{{\bf w}}_1^{-1}U)\Psi_{{\bf w}}(U).$$ Finally the entries of ${{\mathcal L}}_N^T(\Delta,b,{{\bf w}})$ in the basis $\{e_k \otimes e_l\}_{k,l}$ of $\left({\mathbb{C}}^N\right)^{ \otimes 2}$ are as follows. Identify ${\mathbb{Z}}/N{\mathbb{Z}}$ with $\{0,1,\dots, N-1\}$. For every $a\in {\mathbb{Z}}$ set $\delta(a)=1$ if $a\equiv 0$ mod$(N)$, and $\delta(a)=0$ otherwise. For every $n\in {\mathbb{Z}}/N{\mathbb{Z}}$, consider the function $\omega(x,y|n)$ defined on the curve $\{(x,y\in {\mathbb{C}}^2\vert \ x^N+y^N=1\}$ by $$\omega(x,y|0)=1, \omega(x,y|n)=\prod_{j=1}^n \frac{y}{1-xq^{-2j}}, \ n\geq 1.$$ Then, for every $i,j,k,l\in {\mathbb{Z}}/N{\mathbb{Z}}$ we have $${{\mathcal L}}_N^T(\Delta, b, {{\bf w}})_{i,j}^{k,l}= h({{\bf w}}_0) q^{-2kj-k^2}\omega({{\bf w}}_0, {{\bf w}}_1^{-1}| i-k)\delta(i+j-l).$$ Representations of the quantum Teichmüller spaces {#QREP} ------------------------------------------------- Unless stated differently, the results recalled in this section are proved in [@B-L; @B-B-L] or [@Filippo]. Let $\lambda$ be an ideal triangulation of $S$, and $q$ as above. Recall that $m=-\chi(S)$. Put $n:=3m$ and fix an ordering $e_1,\ldots,e_{n}$ of the edges $\lambda$. For all distinct $i$, $j$ set $\sigma_{ij} := a_{ij}-a_{ji} \in\{0,\pm 1,\pm 2\}$, where $a_{ij}$ is the number of times $e_i$ is on the [*right*]{} of $e_j$ in a triangle of $\lambda$, using the orientation of $S$. The [*Chekhov-Fock algebra*]{} ${{\mathfrak T}}_\lambda^q$ is the algebra over ${\mathbb{C}}$ with generators $X_i^{\pm 1}$ associated to the edges $e_i$, and relations $$\label{or1} X_iX_j=q^{2\sigma_{ij}}X_jX_i .$$ When $q=1$, ${{\mathfrak T}}_\lambda^1$ is just the algebra of Laurent polynomials ${\mathbb{C}}[X_1^{\pm 1},\ldots,X_n^{\pm 1}]$, which is the ring of functions on the [*classical*]{} (enhanced) Teichmüller space generated by the exponential shear-bend coordinates on $\lambda$. The algebra ${{\mathfrak T}}_\lambda^q$ has a well-defined fraction algebra $\hat {{\mathfrak T}}_\lambda^q$, and any diagonal exchange $\lambda \to \lambda'$ induces an isomorphism of algebras $$\varphi_{\lambda\lambda'}^q: \hat {{\mathfrak T}}_{\lambda'}^q \rightarrow \hat {{\mathfrak T}}_\lambda^q.$$ The above definition of ${{\mathfrak T}}_\lambda^q$ works as well for any ideally triangulated punctured compact oriented surface $S$, possibly with boundary, where each boundary component is a union of edges. In particular, take $S$ the ideal triangulated squares $Q$, $Q'$ in Figure \[typefig\]. Number the edges of their triangulations $\lambda$, $\lambda'$ as in Figure \[typefig2\]. In such a situation $\varphi_{\lambda\lambda'}^q$ has the form $$\label{relq} \begin{array}{ll} \varphi_{\lambda\lambda'}^q(X_5') = X_5^{-1}\\ \varphi_{\lambda\lambda'}^q(X_1') = (1+qX_5)X_1& ,\ \varphi_{\lambda\lambda'}^q(X_3') = (1+qX_5)X_3\\ \varphi_{\lambda\lambda'}^q(X_2') = (1+qX_5^{-1})^{-1}X_2& ,\ \varphi_{\lambda\lambda'}^q(X_4') = (1+qX_5^{-1})^{-1}X_4. \end{array}$$ It is proved in [@Bai0] (see also Theorem 1.22 in [@Filippo]) that this case of the ideal square determines, for [*any*]{} punctured compact oriented surface $S$, a unique family $\{\varphi_{\lambda\lambda'}^q\}_{\lambda\lambda'}$ of algebra isomorphisms $\varphi_{\lambda\lambda'}^q: \hat {{\mathfrak T}}_{\lambda'}^q \rightarrow \hat {{\mathfrak T}}_\lambda^q$ defined for all ideal triangulations $\lambda$ and $\lambda'$ of $S$, if the family satisfies certain natural properties with respect to composition, diffeomorphisms of $S$, and decomposition into ideally triangulated subsurfaces. ![\[typefig2\] ](typefig2.eps){width="7cm"} Given any ideal triangulations $\lambda$, $\lambda'$ of $S$ and finite dimensional representation $\rho_\lambda: {{\mathfrak T}}_\lambda^q \rightarrow {\rm End}(V_\lambda)$, one says that $\rho_\lambda\circ \varphi_{\lambda\lambda'}^q$ [*makes sense*]{} if for every generator $X_i'\in {{\mathfrak T}}_{\lambda'}^q$ the element $\varphi_{\lambda\lambda'}^q(X_i')$ can be written as $P_iQ_i^{-1} \in \hat {{\mathfrak T}}_{\lambda}^q$, where $P_i$, $Q_i \in {{\mathfrak T}}_{\lambda}^q$ and $\rho_\lambda(P_i)$, $\rho_\lambda(Q_i)$ are invertible endomorphisms of $V_\lambda$. In such a case, $\rho_\lambda\circ \varphi_{\lambda\lambda'}^q(X_i') = \rho_\lambda(P_i)\rho_\lambda(Q_i)^{-1}$ is well-defined (ie. independent of the choice of the pair $(P_i,Q_i)$). By definition, the [*quantum Teichmüller space*]{} of $S$ is the quotient set $\textstyle {{\mathfrak T}}_S^q := (\coprod_{\lambda}\hat {{\mathfrak T}}_\lambda^q)/\sim$, where $\sim$ identifies the algebras $\hat {{\mathfrak T}}_\lambda^q$ and $\hat {{\mathfrak T}}_{\lambda'}^q$ by $\varphi_{\lambda\lambda'}^q$, for all ideal triangulations $\lambda$, $\lambda'$ of $S$. A [*representation of ${{\mathfrak T}}_S^q$*]{} is a family of representations $$\rho:=\{\rho_\lambda: {{\mathfrak T}}_\lambda^q \rightarrow {\rm End}(V_\lambda)\}_{\lambda}$$ indexed by the set of ideal triangulations of $S$, such that $\rho_\lambda\circ \varphi_{\lambda\lambda'}^q$ makes sense for every $\lambda$, $\lambda'$ and is isomorphic to $\rho_{\lambda'}$. In fact, it is enough to check the isomorphisms $\rho_{\lambda'}\cong \rho_\lambda\circ \varphi_{\lambda\lambda'}^q$ whenever $\lambda$ and $\lambda'$ differ by a flip. ### Local representations. {#localsection} The irreducible representations of ${{\mathfrak T}}_S^q$ were classified up to isomorphism in [@B-L]. The [*local*]{} representations of ${{\mathfrak T}}_S^q$ are special [*reducible*]{} representations defined as follows (see [@B-B-L; @Filippo]). Define the [*triangle algebra*]{} ${{\mathcal T}}$ as the algebra over ${\mathbb{C}}$ with generators $Y_0^{\pm 1}$, $Y_1^{\pm 1}$, $Y_2^{\pm 1}$ and relations $$\label{or2} Y_0Y_1=q^2Y_1Y_0,\ Y_1Y_2=q^2Y_2Y_1,\ Y_2Y_0=q^2Y_0Y_2.$$ Fix an ordering $\tau_1,\ldots,\tau_{2m}$ of the triangles of $\lambda$. Order the abstract edges $e_{0}^j,e_{1}^j,e_{2}^j$ of $\tau_j$ so that the induced cyclic ordering is counter-clockwise with respect to the orientation of $S$. Associate to $\tau_j$ a copy ${{\mathcal T}}_j$ of the algebra ${{\mathcal T}}$, with generators denoted by $Y_0^j$,$Y_1^j$,$Y_2^j$, so that $Y_i^j$ is associated to the edge $e_{i}^j$. There is an algebra embedding $$\label{embedi} \mathfrak{i}_\lambda : {{\mathfrak T}}_\lambda^q\rightarrow {{\mathcal T}}_1 \otimes \ldots \otimes {{\mathcal T}}_{2m}$$ defined on generators by (we denote a monomial $\textstyle \otimes_j A_j$ in $\otimes_j{{\mathcal T}}_j$ by omitting the terms $A_j=1$): - If $e_i$ is an edge of two distinct triangles $\tau_{l_i}$ and $\tau_{r_i}$, and $e_{a_i}^{l_i}$, $e_{b_i}^{r_i}$ are the edges of $\tau_{l_i}$, $\tau_{r_i}$ respectively identified to $e_i$, then $\mathfrak{i}_\lambda(X_i) := Y_{a_i}^{l_i}\otimes Y_{b_i}^{r_i}$. - If $e_i$ is an edge of a single triangle $\tau_{k_i}$, and $e_{a_i}^{k_i}$, $e_{b_i}^{k_i}$ are the edges of $\tau_{k_i}$ identified to $e_i$, with $e_{a_i}^{k_i}$ on the right of $e_{b_i}^{k_i}$, then $\mathfrak{i}_\lambda(X_i) := q^{-1}Y_{a_i}^{k_i}Y_{b_i}^{k_i} = qY_{b_i}^{k_i}Y_{a_i}^{k_i}$. A representation $\rho_\lambda$ of ${{\mathfrak T}}_\lambda^q$ is [*local*]{} if $$\rho_\lambda = (\rho_1\otimes \ldots \otimes \rho_{2m})\circ \mathfrak{i}_\lambda \colon {{\mathfrak T}}_\lambda^q \rightarrow {\rm End}(V_1\otimes \ldots \otimes V_{2m})$$ for some irreducible representations $\rho_j:{{\mathcal T}}\rightarrow {\rm End}(V_j)$ of ${{\mathcal T}}$, $j\in \{1,\ldots,2m\}$. So, a local representation is an equivalence class of tuples $(\rho_1,\ldots,\rho_{2m})$, where two tuples are equivalent if their restrictions to the subalgebra $\mathfrak{i}_\lambda({{\mathfrak T}}_\lambda^q)$ of ${{\mathcal T}}_1 \otimes \ldots \otimes {{\mathcal T}}_{2m}$ define the same representation. Two local representations $\rho_\lambda=(\rho_1\otimes \ldots \otimes \rho_{2m})\circ \mathfrak{i}_\lambda$ and $\rho_\lambda'=(\rho_1'\otimes \ldots \otimes \rho_{2m}')\circ \mathfrak{i}_\lambda$ of ${{\mathfrak T}}_\lambda^q$ are isomorphic if there are linear isomorphisms $L_j\colon V_j\to V_j'$ such that for every $j=1,\ldots,2m$ and $Y \in {{\mathcal T}}_j$ we have $$L_j \circ \rho_j(Y) \circ (L_j)^{-1} = \rho_j'(Y).$$ It is straightforward to check that this definition is independent of the choice of tuples $(\rho_1,\ldots,\rho_{2m})$, $(\rho_1',\ldots,\rho_{2m}')$. By definition a local representation $\rho$ of ${{\mathfrak T}}_S^q$ is a representation formed by local representations $\rho_\lambda$. ### Isomorphism classes and standard local representations {#standard} The isomorphism classes of irreducible representations of the triangle algebra ${{\mathcal T}}$ are parametrized by tuples of non zero scalars $(y_0,y_1,y_2,h)\in ({\mathbb{C}}^\bullet)^4$ such that $h^N=y_0y_1y_2$. The parameter $h$ is called the [*load*]{} of the class. The isomorphism class with parameters $(y_0,y_1,y_2,h)$ can be represented by [*standard*]{} representations $\rho:{{\mathcal T}}\to {\rm End}({\mathbb{C}}^N)$, which have the form $$\label{paramloc} \rho(Y_i)={{\bf y}}_iA_i, \ i\in \{0,1,2\}$$ where ${{\bf y}}_0,{{\bf y}}_1,{{\bf y}}_2\in {\mathbb{C}}^*$ satisfy $${{\bf y}}_i^N=y_i,\ i\in \{0,1,2\},\ h={{\bf y}}_0{{\bf y}}_1{{\bf y}}_2$$ and $A_0,A_1,A_2$ are the endomorphisms of ${\mathbb{C}}^N$ defined in . Any local representation of ${{\mathfrak T}}_\lambda^q$ has dimension $N^{2m}$. The isomorphism class of a local representation $\rho_\lambda$ of ${{\mathfrak T}}_\lambda^q$ is determined by: - a non zero complex [*weight*]{} $x_i$ associated to each edge of $\lambda$; - a $N$-th root $h$ of $x_1\ldots x_n$, called the [*load*]{}. The weights $x_i$ and the load $h$ are such that $$\label{relparamloc} \rho_\lambda(X_i^N) = x_i {\rm Id}_{V_1\otimes \ldots \otimes V_{2m}},\ \rho_\lambda(H) = h {\rm Id}_{V_1\otimes \ldots \otimes V_{2m}}$$ where $H$ is a central element of ${{\mathfrak T}}_\lambda^q$, called [*the principal central element*]{}, given by $$\label{punchdef} H = q^{-\sum_{l<l'}\sigma_{ll'}} X_1\ldots X_n.$$ Note that $h^N=x_1\ldots x_n$. It is straightforward to check that two local representations are isomorphic if and only if they are isomorphic as local representations. We call $(\{x_1,\ldots,x_n\},h)$ the [*parameters*]{} of $\rho_\lambda$. We say that a local representation $\rho_\lambda=(\rho_1\otimes \ldots \otimes \rho_{2m})\circ \mathfrak{i}_\lambda$ is standard if every $\rho_i$ is. This notion naturally extends to representations of ${{\mathfrak T}}_S^q$. Every point of $({\mathbb{C}}^*)^{n}$ can be realized as the $n$-tuple of parameters $x_i$, $i=1,\dots,n$, of a standard local representation of ${{\mathfrak T}}_\lambda^q$. There is a one-to-one correspondence between isomorphism classes of local representations $\{\rho_\lambda: {{\mathfrak T}}_\lambda^q \rightarrow {\rm End}(V_1\otimes \ldots \otimes V_{2m})\}_{\lambda}$ of ${{\mathfrak T}}_S^q$ and families of parameters $\{(\{x_1,\ldots,x_n\}_\lambda,h_\lambda)\}_{\lambda}$ such that $$\label{coordch} h_\lambda=h_{\lambda'}\ {\rm and}\ x_i=\varphi_{\lambda\lambda'}^1(x_i')$$ for every $i=1,\ldots,n$ and any two ideal triangulations $\lambda$, $\lambda'$ of $S$ with edge weights $\{x_1,\ldots,x_n\}$, $\{x_1',\ldots, x_n'\}$ and loads $h_\lambda$, $h_{\lambda'}$ respectively. The edge weights $x_i$ of a local representation $\rho_\lambda: {{\mathfrak T}}_\lambda^q \rightarrow {\rm End}(V_\lambda)$ define its “classical shadow” $$sh(\rho_\lambda):{{\mathfrak T}}_{\lambda}^1 \rightarrow {\rm End}({\mathbb{C}})$$ by taking the restriction of $\rho_\lambda$ to the subalgebra ${\mathbb{C}}[X_1^{\pm N},\ldots, X_n^{\pm N}]\cong {{\mathfrak T}}_{\lambda}^1$ of the center of ${{\mathfrak T}}_\lambda^q$. This notion extends immediately to local representations $\rho=\{\rho_\lambda: {{\mathfrak T}}_\lambda^q \rightarrow {\rm End}(V_\lambda)\}_{\lambda}$ of ${{\mathfrak T}}_S^q$; the classical shadow $$sh(\rho)=\{sh(\rho_\lambda): {{\mathfrak T}}_{\lambda}^1 \rightarrow {\rm End}({\mathbb{C}})\}_{\lambda}$$ is a representation of the coordinate ring ${{\mathfrak T}}_S^1$ of the Teichmüller space (considered as a rational manifold with transition functions the maps $\varphi^1_{\lambda\lambda'}$). Every representation of ${{\mathfrak T}}_S^1$ is the shadow of $N$ local representations of ${{\mathfrak T}}_S^q$. The shadows of local representations encode the Zariski open subset of $X(S)$ made of the so-called [*peripherically generic characters*]{}. These include all the augmented characters of injective representations. The load $h$ has the following geometric interpretation. Let $[({{\bf r}}, \{\xi_{\Gamma}\}_{\Gamma \in \Pi})]$ be the augmented character of $S$ determined by $sh(\rho)$. Each $\Gamma \in \Pi$ is a group generated by the class in $\pi_1(S)$ of a small loop $m_j$ in $S$ going once and counter-clockwise around the $j$-th puncture, for some $j\in \{1,\ldots, r\}$ and some choice of basepoints. Since $\pi_1(S)$ is a free group, ${{\bf r}}\colon \pi_1(S) \rightarrow PSL(2,{\mathbb{C}})$ can be lifted to a homomorphism $\hat {{\bf r}}\colon \pi_1(S) \rightarrow SL(2,{\mathbb{C}})$; the fixed point $\xi_\Gamma \in \mathbb{P}^1$ is then contained in an eigenspace of $\hat {{\bf r}}(m_j)$, corresponding to an eigenvalue $a_j\in {\mathbb{C}}$. Then $$\label{geomintpar} h^N = (-1)^r a_1^{-1}\ldots a_r^{-1}.$$ Finally, the decomposition into irreducible summands of a local representation $\rho_\lambda$ of ${{\mathfrak T}}_\lambda^q$ is ([@Tou]) $$\label{decompirr} \rho_\lambda = \oplus_{\rho_\lambda(\mu) \subset \rho_\lambda} \ \rho_\lambda(\mu)$$ where the sum ranges over the set of all spaces $\rho_\lambda(\mu)$ formed by intersecting one eigenspace for each of the so-called (central) [*puncture elements*]{} of ${{\mathfrak T}}_\lambda^q$. Each space $\rho_\lambda(\mu)$ is also the [*isotypical*]{} component of an irreducible representation $\mu$ of ${{\mathfrak T}}_\lambda^q$, that is, the direct sum of all irreducible summands of $\rho_\lambda$ isomorphic to $\mu$. Every irreducible representation of ${{\mathfrak T}}_\lambda^q$ has an isotypical summand appearing in some local representation, and has multiplicity $N^g$ in it. Intertwiners of local representations {#Filippowork} ------------------------------------- Let $(S,\lambda)$ and $q$ be as before. Any surface $R$ obtained by splitting $S$ along some (maybe all) edges of $\lambda$ inherits an orientation from $S$, and an ideal triangulation $\mu$ from $\lambda$. By gluing along edges backwards, one says that $(S,\lambda)$ is obtained [*by fusion*]{} from $(R,\mu)$. In such a case, every local representation $\eta_\mu =(\eta_1\otimes \ldots \otimes \eta_{2m})\circ \mathfrak{i}_\mu$ of ${{\mathfrak T}}_\mu^q$ determines a local representation $\rho_\lambda$ of ${{\mathfrak T}}_\lambda^q$ by setting $\rho_\lambda= (\eta_1\otimes \ldots \otimes \eta_{2m})\circ \mathfrak{i}_\lambda$. One says that $\eta_\mu$ [*represents*]{} $\rho_\lambda$. Given local representations $\eta = \{\eta_\mu \colon {{\mathfrak T}}_\mu^q \to {\rm End}(W_\mu)\}_\mu$, $\rho = \{\rho_\lambda \colon {{\mathfrak T}}_\lambda^q \to {\rm End}(V_\lambda)\}_\lambda$ of ${{\mathfrak T}}_R^q$ and ${{\mathfrak T}}_S^q$ respectively, one says that [*$\rho$ is obtained by fusion from $\eta$*]{} if $\eta_\mu$ represents $\rho_\lambda$ for all ideal triangulations $\mu$ of $R$, where $\lambda$ is the ideal triangulation of $S$ obtained by fusion from $\mu$. In [@Filippo], the following result is proved. The intertwiners $L_{\lambda\lambda'}^{\rho\rho'} \in \mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ in the statement are called [*QT intertwiners*]{}. \[Filippoteo\] There exists a collection $\{(\mathcal{L}_{\lambda\lambda'}^{\rho\rho'},\psi_{\lambda\lambda'}^{\rho\rho'})\}$, indexed by the couples of isomorphic local representations $\rho = \{\rho_\lambda \colon {{\mathfrak T}}_\lambda^q \to {\rm End}(V_\lambda)\}_\lambda$, $\rho' = \{\rho_\lambda' \colon {{\mathfrak T}}_\lambda^q \to {\rm End}(V_\lambda')\}_\lambda$ of ${{\mathfrak T}}_S^q$ and by the couples of ideal triangulations $\lambda$, $\lambda'$ of $S$, such that: \(1) $\mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ is a set of projective classes of linear isomorphisms $L_{\lambda\lambda'}^{\rho\rho'} \colon V_\lambda \to V_{\lambda'}$ such that for every $X'\in {{\mathfrak T}}_{\lambda'}^q$ we have $$L_{\lambda\lambda'}^{\rho\rho'} \circ (\rho_\lambda \circ \varphi^q_{\lambda\lambda'})(X')\circ (L_{\lambda\lambda'}^{\rho\rho'})^{-1} = \rho_{\lambda'}'(X').$$ \(2) $\psi_{\lambda\lambda'}^{\rho\rho'}\colon H_1(S;{\mathbb{Z}}/N{\mathbb{Z}}) \times \mathcal{L}_{\lambda\lambda'}^{\rho\rho'}\to \mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ is a free transitive action. \(3) Let $R$ be a surface such that $S$ is obtained by fusion from $R$. Let $\eta = \{\eta_\mu \colon {{\mathfrak T}}_\mu^q \to {\rm End}(W_\mu)\}_\mu$, $\eta' = \{\eta_\mu' \colon {{\mathfrak T}}_\mu^q \to {\rm End}(W_\mu')\}_\mu$ be two local representations of ${{\mathfrak T}}_R^q$ such that $\rho$, $\rho'$ are obtained respectively by fusion from $\eta$, $\eta'$. Then, for every ideal triangulations $\mu$, $\mu'$ of $R$, if $\lambda$, $\lambda'$ are the corresponding ideal triangulations of $S$, there exists an inclusion map $j\colon \mathcal{L}_{\mu\mu'}^{\eta\eta'}\to \mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ such that for every $L\in \mathcal{L}_{\mu\mu'}^{\eta\eta'}$ and every $c\in H_1(R;{\mathbb{Z}}/N{\mathbb{Z}})$ the following holds: $$(j\circ \psi_{\mu\mu'}^{\eta\eta'})(c,L) = \psi_{\lambda\lambda'}^{\rho\rho'}(\pi_*(c),j(L))$$ where $\pi\colon R\to S$ is the projection map. \(4) For every isomorphic local representations $\rho$, $\rho'$, $\rho''$ of ${{\mathfrak T}}_S^q$ and for every ideal triangulations $\lambda$, $\lambda'$, $\lambda''$ of $S$ the composition map $${ \begin{array}{ccll} \mathcal{L}_{\lambda\lambda'}^{\rho\rho'} \times \mathcal{L}_{\lambda'\lambda''}^{\rho'\rho''} & \longrightarrow & \mathcal{L}_{\lambda\lambda''}^{\rho\rho''} \\ (L_1,L_2) &\longmapsto & L_2\circ L_1 \end{array}}$$ is well-defined, and for all $c$, $d\in H_1(S;{\mathbb{Z}}/N{\mathbb{Z}})$ it satisfies $$\psi_{\lambda'\lambda''}^{\rho'\rho''}(d,L_2)\circ \psi_{\lambda\lambda'}^{\rho\rho'}(c,L_1) = \psi_{\lambda\lambda''}^{\rho\rho''}(c+d,L_2\circ L_1).$$ It is also proved in [@Filippo] that any collection of intertwiners satisfying a weak form of conditions (3) and (4) (not involving the actions $\psi_{\lambda\lambda'}^{\rho\rho'}$) contains the collection $\{\mathcal{L}_{\lambda\lambda'}^{\rho\rho'}\}$. So the latter is minimal with respect to these conditions. Note that property (3) describes the behaviour of the intertwiners in $\mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ under cut-and-paste of subsurfaces along edges of $\lambda$, $\lambda'$. In particular, it implies that they can be decomposed into elementary intertwiners as follows. Any two ideal triangulations $\lambda$, $\lambda'$ of $S$ can be connected by a sequence $\lambda=\lambda_0 \rightarrow \ldots \rightarrow \lambda_{k+1}=\lambda'$ consisting of $k$ diagonal exchanges followed by an edge reindexing $\lambda_k \rightarrow \lambda_{k+1}$ (with same underlying triangulation). Then, any projective class of intertwiners $[L^{\rho\rho'}_{\lambda\lambda'}] \in \mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ can be decomposed as $$\label{decompositionint} L^{\rho\rho'}_{\lambda\lambda'}= L^{\rho\rho'}_{\lambda'\lambda'}\circ L^{\rho\rho}_{\lambda_{k}\lambda_{k+1}} \circ \dots \circ L^{\rho\rho}_{\lambda_0\lambda_{1}}$$ where $L^{\rho\rho}_{\lambda_{i}\lambda_{i+1}} \in \mathcal{L}_{\lambda_i\lambda_{i+1}}^{\rho\rho}$ intertwins $\rho_{\lambda_{i}}$ and $\rho_{\lambda_{i+1}} \circ (\varphi^q_{\lambda_i\lambda_{i+1}})^{-1}$, related by the $i$-th diagonal exchange, for every $i\in \{0,\ldots,k-1\}$, and $L^{\rho\rho'}_{\lambda'\lambda'}$ intertwins $\rho_{\lambda'}$ and $\rho_{\lambda'}'$ on the triangulation $\lambda'$. In general the intertwiner $L^{\rho\rho'}_{\lambda\lambda'}$ depends on the choice of sequence $\lambda=\lambda_0 \rightarrow \ldots \rightarrow \lambda_{k+1}=\lambda'$, but the set $\mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ and the action $\psi_{\lambda\lambda'}^{\rho\rho'}$ do not. Proofs {#PF} ====== We are ready to prove our main Theorems \[MAINTEO\], \[MAINTEO2\], and Corollary \[cor\]. Let us reformulate them by using the background material recalled in the previous sections. Let $M_\phi$ be a fibred cusped hyperbolic $3$-manifold realized as the mapping torus of a pseudo Anosov diffeomorphism $\phi$ of a punctured surface $S$. Put $m=-\chi(S)>0$. Let $C_\phi$ be the associated cylinder. Let $T$ be a layered triangulation of $M_\phi$, and $T_{C_\phi}$ the induced layered triangulation of $C_\phi$, with source boundary component the ideally triangulated surface $(S,\lambda)$, and target boundary component $(S,\lambda')$, where $\lambda' = \phi(\lambda)$ (as in Section \[top-comb\]). For every odd $N\geq 3$ and every primitive $N$-th root of unity $q$, let $(T,\tilde b, {{\bf w}})$ be a QH layered triangulation of $M_\phi$, where ${{\bf w}}$ is a system of quantum shape parameters over $w\in A$, where $A$ is the subvariety of the gluing variety $G(T,\tilde{b})$ as in Proposition \[gluing-var\]. Let $(T_{C_\phi},\tilde b, {{\bf w}})$ be the induced QH triangulation of $C_\phi$, and ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}})\in {\rm End}\left(({\mathbb{C}}^N)^{\otimes 2m}\right)$ the QH operator defined by means of the QH state sum carried by $(T_{C_\phi},\tilde b, {{\bf w}})$. Associated to it we have the transposed operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}})^T\in {\rm End}\left((({\mathbb{C}}^N)')^{\otimes 2m}\right)$ and $\iota^{-1}\circ {{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}})^T\circ \iota \in {\rm End}\left(({\mathbb{C}}^N)^{\otimes 2m}\right)$ as in Section \[quantum-hyp\], still denoted by ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}})^T$ (see Remark \[identrem\]). Now we re-state and prove our first main theorem, Theorem \[MAINTEO\]. [**First Main Theorem.**]{} *(1) Every layered QH-triangulation $(T_{C_\phi},\tilde b,{{\bf w}})$ determines representations $\rho_\lambda$ and $\rho_{\lambda'}$ of ${{\mathfrak T}}_\lambda^q$ and ${{\mathfrak T}}_{\lambda'}^q$ respectively, belonging to a local representation $\rho$ of ${{\mathfrak T}}_{S}^q$ and such that $V_\lambda= V_{\lambda'}=({\mathbb{C}}^N)^{\otimes 2m}$, and $\rho_{\lambda}$ is isomorphic to $\phi^*\rho_{\lambda'}$. Moreover, the operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T$, considered as an element of ${\rm Hom}(V_\lambda,V_{\lambda'})$, is a QT intertwiner which intertwins the representations $\rho_\lambda$ and $\rho_{\lambda'}$.* \(2) For any other choice of weak branching $\tilde b'$, ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b',{{\bf w}})^T$ intertwins local representations canonically isomorphic to $\rho_\lambda$, $\rho_{\lambda'}$ respectively. [[*Proof. *]{}]{}We organize the proof in several steps: [**Step 1.**]{} Every transposed basic matrix dilogarithm ${{\mathcal L}}_N^T(\Delta,b,{{\bf w}})\in {\rm End}\left(({\mathbb{C}}^N)^{\otimes 2}\right)$ (see Section \[quantum-hyp\]) intertwins [*standard*]{} local representations of ${{\mathfrak T}}_\lambda^q$, ${{\mathfrak T}}_{\lambda'}^q$ (see Section \[standard\]), where $\lambda$, $\lambda'$ are the ideal triangulations of the squares $Q$, $Q'$ in Figure \[typefig3\]. ![\[typefig3\] ](typefig2.eps){width="7cm"} [**Step 2.**]{} Every transposed $2$-face operator $(Q_N^T)^{r(F)}: V_s \to V_t$ (see ) intertwins representations of the triangle algebra $\rho_s\colon {{\mathcal T}}\to {\rm End}\left({\mathbb{C}}^N\right)$, $\rho_t\colon {{\mathcal T}}\to {\rm End}\left({\mathbb{C}}^N\right)$, associated to the abstract $2$-faces $F_s$, $F_t$ so that for each one the generator $Y_0$ labels the edge joining the lowest vertex to the middle one (with respect to the vertex ordering induced by the branching), and $Y_1$ labels the edge joining the lowest vertex to the biggest one. [**Step 3.**]{} Every layered QH-triangulation $(T_{C_\phi},\tilde b,{{\bf w}})$ of the cylinder $C_\phi$, associated to a sequence of diagonal exchanges $\lambda=\lambda_0\to \lambda_1 \to \dots \to \lambda_k=\lambda'$, determines standard local representations $\rho_j\colon{{\mathfrak T}}^q_{\lambda_j}\to {\rm End}\left( ({\mathbb{C}}^N)^{\otimes 2m}\right)$ associated to the surfaces $(S,\lambda_j)$, for every $j\in \{0,\ldots,k\}$. These representations belong to a local representation $\rho$ of the quantum Teichmüller space ${{\mathfrak T}}_S^q$. By Step 1, Step 2, and its actual definition by means of a QH state sum, ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T$ intertwins the local representations $\rho_0$ and $\rho_k$. Step (2) proves the claim (2) in the First Main Theorem; the three steps together prove (1). The details follow. ### Proof of Step 1 {#localintert} Consider the squares $Q$, $Q'$ and the branched tetrahedron $(\Delta,b)$ in Figure \[typefig3\]. Use the branching to order as $e_0$, $e_1$, $e_2$ the edges of any triangle of the squares $Q$, $Q'$, so that $e_0$ goes from the lowest vertex to the middle one, and $e_1$ goes from the lowest vertex to the biggest one. In such a situation there are two embeddings , of the form $\mathfrak{i}_\lambda : {{\mathfrak T}}_\lambda^q\rightarrow {{\mathcal T}}_3 \otimes {{\mathcal T}}_{1}$, $\mathfrak{i}_{\lambda'} : {{\mathfrak T}}_{\lambda'}^q\rightarrow {{\mathcal T}}_2 \otimes {{\mathcal T}}_{0}$, where ${{\mathcal T}}_i$ is the copy of the triangle algebra associated to the $i$-th $2$-face (ie. opposite to the $i$-th vertex) of $(\Delta,b)$, and $$\mathfrak{i}_\lambda(X_1) = Y_0^3 \otimes 1, \mathfrak{i}_\lambda(X_2) = 1 \otimes Y_1^1, \mathfrak{i}_\lambda(X_3) = 1\otimes Y_2^1, \mathfrak{i}_\lambda(X_4) = Y_2^3\otimes 1, \mathfrak{i}_\lambda(X_5) = Y_1^3\otimes Y_0^1.$$ $$\mathfrak{i}_{\lambda'}(X_1') =Y_0^2 \otimes 1,\mathfrak{i}_{\lambda'}(X_2') = Y_1^2\otimes 1,\mathfrak{i}_{\lambda'}(X_3') = 1\otimes Y_2^0,\mathfrak{i}_{\lambda'}(X_4') = 1 \otimes Y_0^0,\mathfrak{i}_{\lambda'}(X_5') = Y_2^2 \otimes Y_1^0.$$ Note that $Y_j^i$ is associated to the edge $e_j$ of the $i$-th $2$-face of $(\Delta,b)$, that we have specified above. Recall the notion of standard local representation in Section \[standard\]. \[Sintertwins\] Let $\rho_\lambda = (\rho_3\otimes \rho_{1})\circ \mathfrak{i}_\lambda \colon {{\mathfrak T}}_\lambda^q \rightarrow {\rm End}(V_3\otimes V_{1})$ be a standard local representation, and ${{\bf y}}_j^i$ be such that $\rho_i(Y_j^i) ={{\bf y}}_j^iA_j$, for $i\in \{1,3\}$, $j\in \{0,1,2\}$. Let ${{\bf w}}$ be a triple of quantum shape parameters of the branched tetrahedron $(\Delta,b)$ such that ${{\bf w}}_2 = -q {{\bf y}}_1^3{{\bf y}}_0^1$. Then, for all $X\in {{\mathfrak T}}_\lambda^q$ we have $$\label{intprop} {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) \circ \rho_\lambda(X) = \rho_{\lambda'}((\varphi^q_{\lambda\lambda'})^{-1}(X)) \circ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}})$$ where $\rho_{\lambda'} = (\rho_2\otimes \rho_{0})\circ \mathfrak{i}_\lambda \colon {{\mathfrak T}}_{\lambda'}^q \rightarrow {\rm End}(V_2\otimes V_{0})$ is the standard representation given by $$\label{relexdiag} {{\bf y}}^2_0 = {{\bf w}}_0^{-1} {{\bf y}}^3_0\quad , \quad {{\bf y}}^2_1 = {{\bf w}}_1^{-1} {{\bf y}}^1_1\quad ,\quad {{\bf y}}^0_0 = {{\bf w}}_1^{-1}{{\bf y}}^3_2\quad, \quad {{\bf y}}^0_2 = {{\bf w}}_0^{-1} {{\bf y}}^1_2\quad , \quad {{\bf y}}^2_2{{\bf y}}^0_1 = -q{{\bf w}}_2^{-1}.$$ [Note that ${{\bf w}}_2$ and $w_1 = {{\bf w}}_1^N = (1-{{\bf w}}_2^N)^{-1}$ and $w_0= 1-w_1^{-1}$ are determined by $\rho_\lambda$, but the choice of $N$-th root ${{\bf w}}_1$ (or ${{\bf w}}_0$) is free; this choice determines ${{\bf w}}$, and hence $\rho_{\lambda'}$, completely by the relation ${{\bf w}}_0{{\bf w}}_1{{\bf w}}_2=-q$. Also, we have $\rho_\lambda(X_5) = {{\bf y}}_1^3{{\bf y}}_0^1 A_1 \otimes A_0 = -q^{-1}{{\bf w}}_2A_1 \otimes A_0$, $\rho_{\lambda'}(X_5') = -q{{\bf w}}_2^{-1}A_2 \otimes A_1$, so that $$\rho_\lambda(X_5^N) = -w_2 I_N \otimes I_N, \rho_{\lambda'}((X_5')^N)= -w_2^{-1} I_N \otimes I_N.$$ ]{} [[*Proof. *]{}]{}It is enough to check for $X\in \{X_1,\ldots,X_5\}$, using the relations . Let us do the cases $X=X_5$ and $X_4$, the other cases being respectively similar to this last. For $X=X_5$ recall that $\varphi_{\lambda\lambda'}^q(X_5')=X_5^{-1}$; in this case the identity reads $${{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) \circ ({{\bf y}}^1_0{{\bf y}}^3_1 A_1 \otimes A_0) = ({{\bf y}}^0_1{{\bf y}}^2_2 A_2 \otimes A_1)^{-1} \circ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}).$$ Consider the factorization formula . We have $(A_2 \otimes A_1)^{-1} = qA_1A_0 \otimes A_1^{-1}$, so it commutes with $\Psi_{{\bf w}}(- A_0A_1 \otimes A_1^{-1})$ and we get $$\begin{array}{l} ({{\bf y}}^0_1{{\bf y}}^2_2 A_2 \otimes A_1)^{-1} \circ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) = \\ \hspace*{0.5cm} = \Psi_{{\bf w}}(-A_0A_1 \otimes A_1^{-1})\circ (q({{\bf y}}^0_1{{\bf y}}^2_2)^{-1} A_1A_0 \otimes A_1^{-1}) \circ \left( \frac{1}{N} \sum_{i,j=0}^{N-1} q^{-2ij-j} A_0^i \otimes (A_1A_0)^{-j} \right)\\ \hspace*{0.5cm} = \Psi_{{\bf w}}(-A_0A_1 \otimes A_1^{-1})\circ \left( \frac{1}{N} \sum_{i,j=0}^{N-1} q^{-2(i+1)(j+1)-(j+1)+3} A_0^i \otimes (A_1A_0)^{-j} \right) \circ \\ \hspace*{11cm} \circ (q({{\bf y}}^0_1{{\bf y}}^2_2)^{-1} A_1A_0 \otimes A_1^{-1})\\ \hspace*{0.5cm} = {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) \circ (q^3 A_0^{-1}\otimes A_1A_0)\circ (q({{\bf y}}^0_1{{\bf y}}^2_2)^{-1} A_1A_0 \otimes A_1^{-1})\\ \\ \hspace*{0.5cm} = {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) \circ (({{\bf y}}^0_1{{\bf y}}^2_2)^{-1} A_1 \otimes A_0). \end{array}$$ Hence the identity holds true for $X=X_5$ whenever $$\label{firstrel} ({{\bf y}}^0_1{{\bf y}}^2_2)^{-1} = {{\bf y}}^1_0{{\bf y}}^3_1 = -q^{-1}{{\bf w}}_2.$$ For $X=X_4$ recall that $\varphi_{\lambda\lambda'}^q(X_4')= (1+qX_5^{-1})^{-1}X_4$; in this case the identity reads $$\label{intprop4} {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) \circ ({{\bf y}}^3_2 A_2 \otimes I_N) = (1+q {{\bf y}}^2_2{{\bf y}}^0_1 A_2\otimes A_1) \circ (I_N \otimes {{\bf y}}^0_0 A_0) \circ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}).$$ Now we have: $$\begin{aligned} {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) \circ & ({{\bf y}}^3_2 A_2 \otimes I_N) \\ & = \Psi_{{\bf w}}(-A_0A_1 \otimes A_1^{-1})\circ ({{\bf y}}^3_2 A_2 \otimes I_N) \circ \left( \frac{1}{N} \sum_{i,j=0}^{N-1} q^{-2i(j+1)-j} A_0^i \otimes (A_1A_0)^{-j} \right)\\ & = \Psi_{{\bf w}}(-A_0A_1 \otimes A_1^{-1})\circ (q{{\bf y}}^3_2 A_2 \otimes A_1A_0) \circ \Upsilon\\ & = (q{{\bf y}}^3_2 A_2 \otimes A_1A_0) \circ \Psi_{{\bf w}}(q^2(-A_0A_1 \otimes A_1^{-1}))\circ \Upsilon \\ & = (q{{\bf y}}^3_2 A_2 \otimes A_1A_0) \circ ({{\bf w}}_0+q^2{{\bf w}}_1^{-1} A_1A_0 \otimes A_1^{-1}) \circ \Psi_{{\bf w}}(-A_0A_1 \otimes A_1^{-1})\circ \Upsilon\\ & = (1+{{\bf w}}_0{{\bf w}}_1 (A_1A_0)^{-1} \otimes A_1) \circ ({{\bf w}}_1^{-1} A_1A_0 \otimes A_1^{-1}) \circ \\ & \hspace*{6cm} \circ (q{{\bf y}}^3_2 A_2 \otimes A_1A_0) \circ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}) \\ & = (1-q^2{{\bf w}}_2^{-1} A_2 \otimes A_1) \circ (I_N \otimes {{\bf w}}_1^{-1}{{\bf y}}^3_2 A_0) \circ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}}).\end{aligned}$$ Note that we used the relation in the fourth equality. Hence, using we see that holds true for $X=X_4$ whenever ${{\bf y}}^0_0 = {{\bf w}}_1^{-1}{{\bf y}}^3_2$. The other cases $X=X_1$, $X_2$ or $X_3$ are similar. $\Box$ ### Proof of Step 2 Consider QH-tetrahedra $(\Delta,b,{{\bf w}})$, $(\Delta',b',{{\bf w}}')$ glued along a $2$-face $F$. Denote as usual by $F_s$, $F_t$ the abstract $2$-faces of $\Delta$, $\Delta'$ corresponding to $F$. Recall that the vertices $v_0$, $v_1$, $v_2$ of $F_s$ and $F_t$ are ordered by the branchings $b$ and $b'$ respectively; the gluing $F_s \to F_t$ is encoded by an even permutation $p:=\sigma^{r(F)}$ of the vertices, where $\sigma$ is the $2$-cycle $(v_0v_1v_2)$ and $r(F) \in {\mathbb{Z}}/3{\mathbb{Z}}$. Assume that a standard representation of the triangle algebra $\rho_s\colon {{\mathcal T}}\to {\rm End}({\mathbb{C}}^N)$ is given on $F_s$, where the generators $Y_0$ labels the edge $[v_0,v_1]$, and $Y_1$ labels the edge $[v_0,v_2]$. Using the gluing, $\rho_s$ is the pull-back of a standard representation $\rho_t\colon {{\mathcal T}}\to {\rm End}({\mathbb{C}}^N)$ on $F_t$, where now $Y_0$ labels the edge $[v_{p(0)},v_{p(1)}]$, and $Y_1$ labels the edge $[v_{p(0)},v_{p(2)}]$. So $Y_i$ on $F_s$ corresponds to $Y_{p^{-1}(i)}$ on $F_t$. We have: \[changeb\] The endomorphism $(Q_N^T)^{r(F)}$ intertwins $\rho_s$ and $\rho_t$. Namely, for all $X$ in ${{\mathcal T}}$ we have $(Q_N^T)^{r(F)}\circ \rho_s(X) = \rho_t(X) \circ (Q_N^T)^{r(F)}$. [[*Proof. *]{}]{}It is enough to check this on generators, where it follows from $Q_N^TA_i(Q_N^T)^{-1} = A_{i-1}$ (indices [mod]{}$(3)$). This is straightforward to check.[$\Box$]{} Note that $Q_N$ has order $3$ up to multiplication by $4$-th roots of $1$ (see [@AGT], Lemma 7.3: if $N=2n+1$, we have $Q_N^3 = \phi_N^{-1}I_N$, where $\textstyle \phi_N = \left(\frac{n+1}{N}\right)$ if $N \equiv 1$ mod$(4)$, and $\textstyle \phi_N = \left(\frac{n+1}{N}\right) i$ if $N \equiv 3$ mod$(4)$). ### Proof of Step 3 Let $(T_{C_\phi},\tilde b,{{\bf w}})$ be a layered QH-triangulation of the cylinder $C_\phi$ as usual. Recall that for every $j\in \{0,\dots, k\}$, the triangulated surface $(S,\lambda_j)$ inherits a weak branching $\tilde b_j$ from its positive side in $(T,\tilde b,{{\bf w}})$. Order as $\tau_1^j,\ldots,\tau_{2m}^j$ the branched abstract triangles of $(\lambda_j,\tilde b_j)$. For every $i\in \{1,\dots, 2m\}$, order as $e_0^i$, $e_1^i$, $e_2^i$ the edges of the triangle $\tau_i^j$, as described before Proposition \[Sintertwins\] and Lemma \[changeb\]. Label $e_k^i$ with the generator $Y_k^i$ of a copy of the triangle algebra ${{\mathcal T}}$. Assume that for some $j\in \{0,\dots, k\}$ we are given a standard local representation $$\rho_j=(\rho_{1}^j\otimes\ldots \otimes \rho_{2m}^j) \circ i_{\lambda_j}\colon{{\mathfrak T}}^q_{\lambda_j}\to {\rm End}( ({\mathbb{C}}^N)^{\otimes 2m})$$ where the standard representations $\rho_{i}^j: {{\mathcal T}}\to {\rm End}({\mathbb{C}}^N)$ are associated to the triangles $\tau_i^j$. As in , put $\rho_{i}^j(Y_k) :={{\bf y}}_{i,k}^{j} A_k$. By the very definition of $\mathfrak{i}_{\lambda_j}$ (see ), for any edge $e_s$ of $\lambda_j$, with edge generator $X_{s}\in {{\mathfrak T}}^q_{\lambda_j}$, we have (as usual we denote a monomial $\textstyle \otimes_j A_j$ in ${{\mathcal T}}^{\otimes 2m}$ by omitting the terms $A_j=1$): - $\rho_j(X_s) = {{\bf y}}_{l_i,a_i}^{j}{{\bf y}}_{r_i,b_i}^{j} A_{a_i}^{l_i} \otimes A_{b_i}^{r_i}$, if $e_s$ is an edge of two distinct triangles $\tau_{l_i}^j$ and $\tau_{r_i}^j$, and $e_{a_i}^{l_i}$, $e_{b_i}^{r_i}$ are the edges of $\tau_{l_i}^j$, $\tau_{r_i}^j$ respectively identified to $e_s$. - $\rho_j(X_s) = q^{-1} {{\bf y}}_{k_i,a_i}^{j}{{\bf y}}_{k_i,b_i}^{j} A_{a_i}^{k_i}A_{b_i}^{k_i}$, if $e_s$ is an edge of a single triangle $\tau_{k_i}^j$, and $e_{a_i}^{k_i}$, $e_{b_i}^{k_i}$ are the edges of $\tau_{k_i}^j$ identified to $e_s$, with $e_{a_i}^{k_i}$ on the right of $e_{b_i}^{k_i}$. Consider the diagonal exchange $\lambda_j\to \lambda_{j+1}$, and the corresponding triangulated (abstract) squares $Q$ and $Q'$, as in Figure \[typefig3\]. The standard representations $\rho_i^j: {{\mathcal T}}\to {\rm End}({\mathbb{C}}^N)$ associated to the triangles of $Q$ define a standard local representation $\rho_{Q}$ of ${{\mathfrak T}}^q_{\lambda_j|Q}$. Denote by $\rho_{Q'}$ the standard local representation of ${{\mathfrak T}}^q_{\lambda_{j+1}|Q'}$ defined from $\rho_{Q}$ by the relations , where $(\Delta,b,{{\bf w}})$ is the QH-tetrahedron of $(T_{C_\phi},\tilde b,{{\bf w}})$ bounded by $Q$ and $Q'$; we describe below the relations between ${{\bf w}}$ and $\rho_j$ (and hence $\rho_Q$). The representation $\rho_{Q'}$ extends (by fusion) to a standard local representation $$\rho_{j+1}=(\rho_{1}^{j+1}\otimes\ldots \otimes \rho_{2m}^{j+1}) \circ i_{\lambda_j}\colon{{\mathfrak T}}^q_{\lambda_{j+1}}\to {\rm End}( ({\mathbb{C}}^N)^{\otimes 2m})$$ by stipulating that $\rho_{i}^{j+1} = \rho_{i}^{j}$ on the common triangles of $(S,\lambda_j)$ and $(S,\lambda_{j+1})$. Since $(S,\lambda_{j+1})$ is given the weak-branching $\tilde b_{j+1}$ induced by its positive side in $(T_{C_\phi},\tilde b,{{\bf w}})$, for every triangle $\tau_i^{j+1}$ of $(S,\lambda_{j+1})$ where $\tilde b_{j+1}$ differs from the weak-branching given by $(S,\lambda_j)$ or $Q'$, replace the corresponding representation $\rho_{i}^{j+1}$ by $(Q_N^T)^{r(F)} \circ \rho_i^{j+1}(X) \circ (Q_N^T)^{-r(F)}$, as in Lemma \[changeb\]. For simplicity, we keep however the same notation for $\rho_{j+1}$. Define $$L^{\rho_j\rho_{j+1}}_{\lambda_j\lambda_{j+1}}\colon {\rm End}(({\mathbb{C}}^N)^{\otimes 2m}) \rightarrow {\rm End}(({\mathbb{C}}^N)^{\otimes 2m})$$ as the operator acting as the identity on all factors except on the representation space ${\mathbb{C}}^N\otimes {\mathbb{C}}^N \hookrightarrow ({\mathbb{C}}^N)^{\otimes 2m}$ of $\rho_{Q}$, where it acts by $\left((Q_N^T)^{r(F_2)} \otimes (Q_N^T)^{r(F_0)}\right)\circ {{\mathcal L}}_N^T(\Delta,b,{{\bf w}})$. Here, as usual we denote by $F_2$ and $F_0$ the $2$-faces supporting the target space of ${{\mathcal L}}_N^T(\Delta,b,{{\bf w}})$. By construction, for every $X\in {{\mathfrak T}}^q_{\lambda_{j+1}}$ we have $$\label{intertwj} L^{\rho_j\rho_{j+1}}_{\lambda_j\lambda_{j+1}} \circ \rho_{j}\circ \varphi^q_{\lambda_j\lambda_{j+1}}(X')\circ (L^{\rho_j\rho_{j+1}}_{\lambda_j\lambda_{j+1}})^{-1} = \rho_{j+1}(X).$$ Finally, assuming that a standard local representation $\rho_j$ is given for $j=0$, by working as above we can define inductively a sequence of standard local representation $\rho_j$, $j=0,\dots , k$, and set $$L^{\rho_0,\rho_k}_{\lambda_0, \lambda_k}:= L^{\rho_{k-1}\rho_{k}}_{\lambda_{k-1}\lambda_{k}}\circ \dots \circ L^{\rho_{0}\rho_{1}}_{\lambda_{0}\lambda_{1}}.$$ Comparing with we see that $$L^{\rho_0,\rho_k}_{\lambda_0, \lambda_k} = {{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b, {{\bf w}})^T.$$ Now we have to specialize the choice of $\rho_0$, structurally related to the triangulation $(T_{C_\phi},\tilde b,{{\bf w}})$. Similarly to the “classical" lateral shape parameters $W_j^+(e)$ (Section \[class-hyp\]), put: \[qsbcoordinates\] [For every edge $e$ of $(S,\lambda_j)$, the [*lateral quantum shape parameter*]{} ${{\bf W}}_j^+(e)$ is the product of the quantum shape parameters ${{\bf w}}(E)$ of the abstract edges $E\rightarrow e$ carried by tetrahedra lying on the positive side of $(S,\lambda_j)$. The [*quantum shear-bend coordinates*]{} of $(S,\lambda_j)$ are the scalars $${{\bf x}}^j(e):=-q^{-1}{{\bf W}}_j^+(e).$$]{} Denote by $\lambda_j^{(1)}$ the set of edges of the ideal triangulation $\lambda_j$. By Proposition \[shear-bend\], the quantum shear-bend coordinates ${{\bf x}}^j(e)$, $e\in \lambda_j^{(1)}$, form a system of $N$-th roots of the shear-bend coordinates $x^j(e)$ of $\lambda_j$. By fixing $w$ and varying ${{\bf w}}$ over $w$ in the tuple $(T_{C_\phi},\tilde b,{{\bf w}})$, one obtains a family of systems $\{{{\bf x}}^j(e)\}_{e\in \lambda_j^{(1)}}$ such that ${{\bf x}}^0(e)={{\bf x}}^k(\phi(e))$ for every edge $e$ of $\lambda=\lambda_0$. The QH-triangulation $(T_{C_\phi},\tilde b,{{\bf w}})$ determines ${{\bf x}}^j(e_s)$ for all $j\in \{0,\ldots,k\}$ and every edge $e_s$ of $\lambda_j$. Consider the case $j=0$, and the regular map $$\pi_0\colon \{{{\bf y}}^0 = ({{\bf y}}_{i,k}^{0})_{i,k}\} \to \{{{\bf x}}^0 = ({{\bf x}}^0(e_s))_{e_s\in \lambda_0^{(1)}}\}$$ defined by $$\label{relloc} {{\bf x}}^0(e_s) = {{\bf y}}_{l_i,a_i}^{0}{{\bf y}}_{r_i,b_i}^{0}\quad {\rm or}\quad {{\bf y}}_{k_i,a_i}^{0}{{\bf y}}_{k_i,b_i}^{0}$$ according to the cases (a) or (b) described at the beginning of Step 3. A simple dimensional count shows that $\pi_0$ is surjective, with generic fibre isomorphic to $({\mathbb{C}}^*)^{3m}$. Then, take a point ${{\bf y}}^0\in \pi^{-1}_0({{\bf x}}^0)$. We get $$\label{relloc2} \rho_0(X_s) = {{\bf x}}^0(e_s) A_{a_i}^{l_i} \otimes A_{b_i}^{r_i} \quad {\rm or}\quad \rho_0(X_s) = q^{-1} {{\bf x}}^0(e_s) A_{a_i}^{k_i}A_{b_i}^{k_i}$$ again according to the cases (a) or (b) above. We have to check that the sequence of local representations $\rho_j$ constructed by starting from $\rho_0$ is consistent, that is, compatible with ${{\bf w}}$ under the diagonal exchanges. Using the notations of Figure \[typefig3\], by the tetrahedral and edge relations satisfied by the quantum shape parameters, for all $j\in \{0,\ldots,k\}$ we have $$\label{relqsb} {{\bf x}}^j(e_5)=-q^{-1}{{\bf w}}_2, {{\bf x}}^{j+1}(e_5)=-q{{\bf w}}_2^{-1}$$ and $$\label{relqsb2} \begin{array}{ccc} {{\bf x}}^{j+1}(e_1)={{\bf x}}^{j}(e_1){{\bf w}}_0^{-1}& , & {{\bf x}}^{j+1}(e_2)={{\bf x}}^{j}(e_2){{\bf w}}_1^{-1}\\ {{\bf x}}^{j+1}(e_3)={{\bf x}}^{j}(e_3){{\bf w}}_0^{-1}& , & {{\bf x}}^{j+1}(e_4)={{\bf x}}^{j}(e_4){{\bf w}}_1^{-1}. \end{array}$$ In particular, for $j=0$ is consistent with the relation ${{\bf w}}_2 = -q {{\bf y}}_1^3{{\bf y}}_0^1$ required by Proposition \[Sintertwins\] (where ${{\bf y}}_1^3 = {{\bf y}}_{r_i,b_i}^{0}$ and ${{\bf y}}_0^1 = {{\bf y}}_{l_i,a_i}^{0}$). Then, comparing and we see that for every $j\in \{0,\ldots,k\}$, $\rho_j$ is indeed defined by identities like , with ${{\bf x}}^0$ replaced by ${{\bf x}}^j$. By , the representations $\rho_0,\ldots,\rho_k$ belong to a local representation $\rho=\{\rho_\lambda: {{\mathfrak T}}_\lambda^q \rightarrow {\rm End}(V_\lambda)\}_{\lambda}$ of ${{\mathfrak T}}^q_S$. By , and the similar identities for $\rho_j$, we see that the parameters of $\rho_j$ are $$(\{x^j(e)\}_{e\in \lambda_j^{(1)}}, h)$$ where $x^j(e) = ({{\bf x}}^j(e))^N$ and $$\label{load2} h =\prod_{e\in \lambda_j^{(1)}} {{\bf x}}^j(e).$$ For this identity, first note that the right hand side, say $h^j$, does not depend on $j$. Indeed, by , and the tetrahedral and edge relations for ${{\bf w}}_0$, ${{\bf w}}_1$ and ${{\bf w}}_2$, we see that $$\textstyle \prod_{i=1}^5 {{\bf x}}^j(e_i) = \prod_{i=1}^5 {{\bf x}}^{j+1}(e_i)$$ under a diagonal exchange. Since the quantum shear-bend coordinates of edges not touched by a diagonal exchange are not altered, we deduce that $h^j=h^{j+1}$ for every $j=0,\dots, k-1$. Now we can check as follows. Consider a triangle $\tau^j_s$ of $\lambda_j$, with edges $X_i,X_j,X_k$, where $i<j<k$. The weak branching $\tilde b$ provides a bijection $p\colon \{i,j,k\}\mapsto \{0,1,2\}$. Then $\mathfrak{i}_{\lambda_j}(H)$ is a tensor product of monomials associated to the triangles of $\lambda_j$, and the monomial for $\tau^j_s$ is $q^{-\sigma_{\vert \tau^j_s}}Y_{p(i)}^s Y_{p(j)}^s Y_{p(k)}^s$ where $-\sigma_{\vert \tau^j_s}$ is the contribution to the summand $-\sigma_{ij}-\sigma_{ik}-\sigma_{jk}$ of $\textstyle -\sum_{l<l'}\sigma_{ll'}$ coming from the triangle $\tau^j_s$. Noting that the product $Y_0^sY_1^sY_2^s$ is invariant under cyclic permutations, one checks immediately that $q^{-\sigma_{\vert \tau_j^l}}Y_{p(i)}^s Y_{p(j)}^s Y_{p(k)}^s = q^{-1} Y_0^sY_1^sY_2^s$. So $$\mathfrak{i}_{\lambda_j}(H) = \otimes_{s=1}^{2m} (q^{-1}Y_0^sY_1^sY_2^s)$$ and hence $$\rho_j(H) = \otimes_{s=1}^{2m} ({{\bf y}}_0^s{{\bf y}}_1^s{{\bf y}}_2^sq^{-1}A_0^sA_1^sA_2^s) = {{\bf x}}^j(e_1)\ldots {{\bf x}}^j(e_n) {\rm Id}_{({\mathbb{C}}^N)^{\otimes 2m}}.$$ Summarizing our discussion, we have proved the following result. It concludes the proof of Step 3 of our First Main Theorem, and hence of Theorem \[MAINTEO\] in the introduction. \[QHIlocrep\] The QH triangulation $(T_{C_\phi},\tilde b,{{\bf w}})$ determines a sequence of standard local representations $\rho_j\colon{{\mathfrak T}}^q_{\lambda_j}\to {\rm End}( ({\mathbb{C}}^N)^{\otimes 2m})$ associated to the triangulated surfaces $(S,\lambda_j)$, with parameters $$\textstyle (\{x^j(e)\}_{e\in \lambda_j^{(1)}},h),\ h=\prod_{e\in \lambda_j^{(1)}}{{\bf x}}^j(e),$$ and belonging to a local representation $\rho$ of ${{\mathfrak T}}^q_{S}$. Moreover, ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T$ intertwins the representations $\rho_0$ and $\rho_k$. Let us consider now Theorem \[MAINTEO2\]. For the convenience of the reader we re-state it. We keep the usual setting. Recall that $M_\phi$ is the interior of a compact manifold $\bar M_\phi$ with boundary made by tori. [**Second Main Theorem.**]{} *There is a neighborhood of $i^*({{\mathfrak r}}_h)$ in $i^*(X_0(M_\phi)) \subset X(S)$ such that:* \(1) For any isomorphism class of local representations of ${{\mathfrak T}}_S^q$ whose holonomy lies in this neighborhood, there are: - A representative $\rho$ of the class, and representations $\rho_\lambda$, $\rho_{\lambda'}$ belonging to $\rho$ and acting on $({\mathbb{C}}^N)^{\otimes 2m}$, - a layered QH triangulation $(T,\tilde b, {{\bf w}})$ of $M_\phi$ such that ${{\mathcal H}}_N^{red}(T,\tilde b, {{\bf w}})={{\mathcal H}}_N^{red}(M_\phi,{{\mathfrak r}},\kappa)$, for some weight $\kappa\in H^1(\partial \bar M_\phi,{\mathbb{C}}^*)$ and augmented character ${{\mathfrak r}}$ of $\pi_1(M_\phi)$ such that the restriction of ${{\mathfrak r}}$ to $\pi_1(S)$ is the holonomy of $\rho$, such that the operator ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T$ is a QHI intertwiner which intertwins $\rho_\lambda$ and $\rho_{\lambda'}$ as in Proposition \[QHIlocrep\]. The load of $\rho$ is determined by the values of the weight $\kappa$ at the meridian curves that form $\bar S \cap \bar M_\phi$. \(2) The family of QHI intertwiners $\{{{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})^T\}$ consists of the QT intertwiners which intertwin $\rho_\lambda$ to $\rho_{\lambda'}$ and whose traces are well defined invariants of triples $(M_\phi,{{\mathfrak r}},\kappa)$ such that the restriction of ${{\mathfrak r}}$ to $\pi_1(S)$ is the holonomy of $\rho$. [[*Proof. *]{}]{}(1) Let $\rho\colon{{\mathfrak T}}^q_{\lambda}\to {\rm End}( ({\mathbb{C}}^N)^{\otimes 2m})$ be a local representation with parameters $(\{x(e)\}_{e\in \lambda^{(1)}},h)$. Recall Proposition \[gluing-var\] and the discussion that follows. Assume that the augmented character $[({{\bf r}},\{\xi_\Gamma\}_\Gamma)]$ associated to $\rho$ lies in $i^*(X_0(M_\phi))$, and can be encoded by a system of shape parameters $w\in A_0 \subset A$, the simply connected open neighborhood of $w_h$ in $G(T,\tilde b)$ chosen before Remark \[qpar\]. Then we can lift $w$ to a system of quantum shape parameters ${{\bf w}}\in A_{0,N}$, and using the QH-triangulation $(T_{C_\phi},\tilde b,{{\bf w}})$, we can define a local representation $\rho'\colon{{\mathfrak T}}^q_{\lambda}\to {\rm End}( ({\mathbb{C}}^N)^{\otimes 2m})$ in the same way as $\rho_0$ in the proof of Proposition \[QHIlocrep\] (see ). It has quantum shear-bend coordinates ${{\bf x}}(e)$, $e\in \lambda^{(1)}$, and parameters $(\{x(e)\}_{e\in \lambda^{(1)}},h')$, where $\textstyle h'=\prod_{e\in \lambda^{(1)}} {{\bf x}}(e)$. The system $\{x(e)\}_{e\in \lambda^{(1)}}$ is the same for $\rho$ and $\rho'$, so in order that $\rho'\cong \rho$, it remains to show that we can choose ${{\bf w}}$ so that $h'=h$. But $\textstyle (h')^N = \prod_{e\in \lambda^{(1)}} x(e) = h^N$, so $h'=\zeta h$ for some $N$-th root of unity $\zeta$. On the other hand, by Theorem \[SSum\] (1), at all the meridian curves $m_j$ of $\partial \bar M_\phi$ we have $$\kappa_N({{\bf w}})(m_j)^{N} = \pm a_j^{-2}$$ where $a_j$ is defined before the identity , and we note that $a_j^{-2}$ is the dilation factor of the similarity transformation associated to the conjugate of ${{\bf r}}(m_j)$ fixing $\infty$, whence the squared eigenvalue at $m_j$ selected by $[({{\bf r}},\{\xi_\Gamma\}_\Gamma)]$. Hence $$(h')^{2N} = a_1^{-2}\ldots a_r^{-2} = \pm \kappa_N({{\bf w}})(m_1)^{N}\ldots \kappa_N({{\bf w}})(m_r)^{N}.$$ Moreover, there is ${{\bf w}}'\in A_{0,N}$ so that $\kappa_N({{\bf w}}')(m_1)\ldots \kappa_N({{\bf w}}')(m_r)$ is any given $N$-th root of $a_1^{-2}\ldots a_r^{-2}$. Let $\zeta'$ be the $N$-th root of unity such that $(h')^{2} = \zeta'\kappa_N({{\bf w}})(m_1)\ldots \kappa_N({{\bf w}})(m_r)$. By using the formula it is easy to check that $$\label{formmj} \kappa_N({{\bf w}})(m_j) = (-q^{-1})^{t_j}\prod_{e_\vert p_j \in \partial e} {{\bf W}}^+(e)$$ where ${{\bf W}}^+(e)$ is the lateral quantum shape parameter of $(S,\lambda)$ at the edge $e$ of $\lambda$, $t_j$ is the number of spikes of the triangles of $\lambda$ at the $j$-th puncture $p_j$ of $S$, and the product is over all the edges $e$, counted with multiplicity, having $p_j$ as an endpoint. Recalling that there are $3m=-3\chi(S)$ edges in $\lambda$, we deduce $$\begin{aligned} \kappa_N({{\bf w}})(m_1)\ldots \kappa_N({{\bf w}})(m_r) & = (-q^{-1})^{3.2m} \prod_{e\in \lambda^{(1)}} {{\bf W}}^+(e)^2 \label{lastform0}\\ & = (-q^{-1})^{6m}(-q)^{6m} \prod_{e\in \lambda^{(1)}} {{\bf x}}(e)^2 = (h')^2.\label{lastform1} \end{aligned}$$ Hence $\zeta'=1$. Then take ${{\bf w}}'\in A_{0,N}$ so that $\kappa_N({{\bf w}}')(m_1)\ldots \kappa_N({{\bf w}}')(m_r) = (h')^{2}\zeta^{-2}$. The load $h''$ of the representation $\rho''$ associated to ${{\bf w}}'$ satisfies $(h'')^2 = (h')^{2}\zeta^{-2} = h^2$. Since $N$ is odd, and again $(h'')^N=h^N$, eventually $h=h''$. This achieves the proof of the claim (1) of the theorem. \(2) We need to describe the general form of the QT intertwiners associated to sequences of diagonal exchanges $\lambda\rightarrow \ldots \rightarrow \lambda'$, for arbitrary standard local representations $\rho_{\lambda}$, $\rho_{\lambda'}$ of ${{\mathfrak T}}_{\lambda}^q$, $ {{\mathfrak T}}_{\lambda'}^q$. This is done in Lemma \[allQT\]. For that, we reconsider the arguments of the proof of the claim (1) in the First Main Theorem. First, we look at the case of a diagonal exchange $\lambda \rightarrow \lambda'$, occuring in squares $Q$, $Q'$ as in Figure \[typefig3\]. We fix standard local representations $\rho_{\lambda}$, $\rho_{\lambda'}$ of ${{\mathfrak T}}_{\lambda}^q$, $ {{\mathfrak T}}_{\lambda'}^q$ such that $\rho_{\lambda'}\circ (\varphi_{\lambda\lambda'}^q)^{-1}$ is isomorphic to $\rho_{\lambda}$, and differs from it only on $Q$, $Q'$, where $\rho_{\lambda}$, $\rho_{\lambda'}$ are represented by irreducible representations $\rho_j$ of the triangle algebra with parameters ${{\bf y}}^j_k$, where $j\in \{0,1,2,3\}$ and $k\in\{0,1,2\}$. \[roughw\] With the above notations we have $$\label{relexdiag0} {{\bf y}}^3_0/{{\bf y}}^2_0 = {{\bf y}}^1_2/{{\bf y}}^0_2\ , \ {{\bf y}}^1_1/{{\bf y}}^2_1 = {{\bf y}}^3_2/{{\bf y}}^0_0\ ,\ {{\bf y}}_1^3{{\bf y}}_0^1 = ({{\bf y}}^2_2{{\bf y}}^0_1)^{-1}.$$ Moreover, denoting these scalars by $\tilde {{\bf w}}_0$, $\tilde {{\bf w}}_1$ and $\tilde {{\bf w}}_2$ respectively, $(w_0,w_1,w_2) := (\tilde{{\bf w}}_0^N,\tilde{{\bf w}}_1^N,\tilde{{\bf w}}_2^N)$ is a triple of shape parameters on $(\Delta,b)$, that is $w_{i+1} = (1-w_i)^{-1}$ for $i=0,1$. We say that $(\tilde {{\bf w}}_0,\tilde{{\bf w}}_1,\tilde{{\bf w}}_2)$ is a triple of *rough* $q$-shape parameters on $(\Delta,b)$. Each of the scalars $\tilde {{\bf w}}_j$ corresponds to a pair of opposite edges of $\Delta$. Note that, in general, the product $\tilde {{\bf w}}_0\tilde {{\bf w}}_1\tilde {{\bf w}}_2$ can be any $N$-th root of $-1$, the special case of $-q$ being achieved by the $q$-shape parameters. [[*Proof. *]{}]{}We prove first the second claim, that is, the $N$-th powers of the scalars set equal in are indeed equal, and form a triple of shape parameters $(w_0,w_1,w_2)$. By the relations we have $$\begin{array}{ll} \varphi_{\lambda\lambda'}^q((X_5')^N) = \varphi_{\lambda\lambda'}^q((X_5'))^N = X_5^{-N}\\ \varphi_{\lambda\lambda'}^q((X_1')^N) = \varphi_{\lambda\lambda'}^q(X_1')^N = (X_1+qX_5X_1)^N = X_1^N+X_5^NX_1^N\\ \varphi_{\lambda\lambda'}^q((X_2')^N) = \varphi_{\lambda\lambda'}^q(X_2')^N = (X_2^{-1}+qX_2^{-1}X_5^{-1})^{-N} = (X_2^{-N}+X_2^{-N}X_5^{-N})^{-1} \end{array}$$ and a relation like the second for $X_3'$, $X_3$, and like the fourth for $X_4'$, $X_4$. Note that we use the $q$-binomial formula and the fact that $q^2$ is a primitive $N$-th root of $1$, $N$ odd, to deduce the results. Hence the shear-bend parameters $x_i$, $x_i'$ at the edges $e_i$, $i\in \{1,\ldots,5\}$, satisfy $$x_5' = x_5^{-1}\ ,\ x_1' = x_1(1+x_5)\ ,\ x_3' = x_3(1+x_5)\ ,\ x_2' = x_2(1+x_5^{-1})^{-1}\ ,\ x_4' = x_4(1+x_5^{-1})^{-1}.$$ Set $x_5 = -w_2$. By a mere rewriting of formulas, it follows easily from this, and ${{\bf x}}(e)^N=x(e)$ for all edges $e$ of $\lambda$, that $$\label{2relexdiag} \begin{array}{c} - w_2 = ({{\bf y}}_1^3{{\bf y}}_0^1)^N = ({{\bf y}}^2_2{{\bf y}}^0_1)^{-N}\\ w_0=x_1/x_1' = x_3/x_3' = ({{\bf y}}^3_0/{{\bf y}}^2_0)^N = ({{\bf y}}^1_2/{{\bf y}}^0_2)^N\\ w_1 = x_2/x_2' = x_4/x_4' = ({{\bf y}}^1_1/{{\bf y}}^2_1)^N = ({{\bf y}}^3_2/{{\bf y}}^0_0)^N. \end{array}$$ and $w_{i+1} = (1-w_i)^{-1}$ for $i=0,1$. Now we prove . Let us reconsider Proposition \[Sintertwins\] in the present situation. Instead of $\rho_\lambda$ and the system $({{\bf w}}_0,{{\bf w}}_1,{{\bf w}}_2)$ of quantum shape parameters on $(\Delta,b)$, it is the pair of local representations $\rho_\lambda$, $\rho_{\lambda'}$ that are given on $Q$, $Q'$, and we are looking for $(\tilde{{\bf w}}_0,\tilde{{\bf w}}_1,\tilde{{\bf w}}_2)$ such that the identity holds true. Again yields the first equality of , which is the last relation of , without implying any constraints on $(\tilde{{\bf w}}_0,\tilde{{\bf w}}_1,\tilde{{\bf w}}_2)$. Then, use it to define $\tilde{{\bf w}}_2$ by the same formula. Replacing all uses of the relation by in the other computations defines $\tilde{{\bf w}}_0 = {{\bf y}}^3_0/{{\bf y}}^2_0$ and $\tilde{{\bf w}}_1 = {{\bf y}}^1_1/{{\bf y}}^2_1$, and eventually proves the first two identities in . For instance, in the case of $X=X_4$ the result becomes $${{\mathcal L}}_N^T(\Delta,b,\tilde {{\bf w}}) \circ ({{\bf y}}^3_2 A_2 \otimes I_N)= (1-q^2\tilde{{\bf w}}_2^{-1} A_2 \otimes A_1) \circ \left(I_N \otimes (-q^{-1}\tilde{{\bf w}}_0\tilde{{\bf w}}_2{{\bf y}}^3_2 A_0)\right) \circ {{\mathcal L}}_N^T(\Delta,b,\tilde{{\bf w}})$$ and hence holds true whenever $\tilde{{\bf w}}_0 = -q{{\bf y}}^0_0/{{\bf y}}^3_2\tilde{{\bf w}}_2$. Comparing with the result of the similar computation for $X=X_2$ we find the second identity in . The other cases are similar. $\Box$ Now we have: [(See [@B-L; @Filippo])]{} Assume that $S$ is an ideal polygon with $p\geq 3$ vertices. Then for every ideal triangulation $\lambda$ of $S$, every local representation of ${{\mathfrak T}}_\lambda^q$ is irreducible. In the situation of the lemma we have the ideal squares $Q$, $Q'$, and so the proposition implies that ${{\mathcal L}}_N^T(\Delta,b,\tilde {{\bf w}})$ is a representative of the unique projective class of intertwiners from $\rho_\lambda$ to $\rho_{\lambda'}\circ (\varphi_{\lambda\lambda'}^q)^{-1}$. More generally, let $\lambda$, $\lambda'$ be ideal triangulations of $S$ connected by a sequence of diagonal exchanges, and let $\rho$, $\rho'$ be isomorphic standard local representations of ${{\mathfrak T}}_S^q$. Then every QT intertwiner $L^{\rho\rho'}_{\lambda\lambda'}$ between $\rho_\lambda$ and $\rho_{\lambda'}'$ has a representative of the form , where $L^{\rho\rho'}_{\lambda_i\lambda_{i+1}} ={{\mathcal L}}_N^T(\Delta,b,\tilde {{\bf w}})$ for some system $\tilde {{\bf w}}:=(\tilde {{\bf w}}_0,\tilde{{\bf w}}_1,\tilde{{\bf w}}_2)$ of rough $q$-shape parameters (as in Lemma \[roughw\]) on the tetrahedron $(\Delta,b)$ associated to the $i$-th diagonal exchange. Hence, proceeding as in the proof of Proposition \[QHIlocrep\] we deduce that: \[allQT\] Every projective class of QT intertwiner in $\mathcal{L}_{\lambda\lambda'}^{\rho\rho'}$ is represented by a QH state sum ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,\tilde {{\bf w}})^T$, where $\tilde{{\bf w}}$ is a system of rough $q$-shape parameters on $T_{C_\phi}$. The proof of the claim (2) of the Second Main Theorem then follows by analysing the invariance properties of the QH state sums ${\rm Trace}\left({{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,\tilde {{\bf w}})\right)$. Such an analysis has been done in [@GT] and [@AGT] in the case of the unreduced QH state sums on arbitrary closed pseudo-manifold triangulations. It applies verbatim to the reduced QH state sums by combining the results of Section 5 of [@GT] with Proposition 6.3 (1) and 8.3 of [@NA]. The result is that ${\rm Trace}\left({{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,\tilde {{\bf w}})\right)$ is an invariant of the triple $(M_\phi,{{\mathfrak r}},\kappa)$ if and only if it is invariant under any layered ‘$2\leftrightarrow 3$ transit’ or ‘lune transit’ of $(T_{C_\phi},\tilde b,\tilde {{\bf w}})$; such transits enhance to $\tilde b$ and $\tilde {{\bf w}}$ the usual $2\leftrightarrow 3$ and lune moves between layered triangulations (which correspond to the pentagon and square relations for diagonal exchanges between surface triangulations). This invariance property happens if and only if $\tilde {{\bf w}}$ satisfies the tetrahedral and edge relations along every interior edge of $T_{C_\phi}$, that is, when $\tilde {{\bf w}}$ is a genuine system of quantum shape parameters. In that situation ${\rm Trace}\left({{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,\tilde {{\bf w}})\right)$ a QHI intertwiner. For instance, the relation implies that, modifying $T_{C_\phi}$ by introducing in the sequence $\lambda \rightarrow \ldots \rightarrow \lambda'$ two consecutive diagonal exchanges along an edge $e$, ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,\tilde {{\bf w}})$ is unchanged if and only if the total quantum shape parameter at $e$ satisfies ${{\bf W}}(e)=q^2$. This concludes the proof of our Second Main Theorem.[$\Box$]{} Finally, let us consider Corollary \[cor0\] and \[cor\]. For the reader’s convenience we re-state them below as Corollary 1 and 2. We keep the usual setting. Let us fix a QH-triangulation $(T_{C_\phi},\tilde b,{{\bf w}})$ of the cylinder $C_\phi$. Recall that ${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa) = {\rm Trace}\big({{\mathcal H}}_N^{red}\left(T_{C_\phi},\tilde b,{{\bf w}})\right)$, and that $M_\phi$ is the interior of a compact manifold $\bar M_\phi$ with boundary made by tori. Call [*longitude*]{} any simple closed curve in $\partial \bar M_\phi$ intersecting a fibre of $\bar M_\phi$ in exactly one point. [**Corollary 1.**]{} The reduced QH invariants ${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa)$ do not depend on the values of the weight $\kappa$ on the longitudes. [[*Proof. *]{}]{}Denote by $\rho_\lambda$ the local representation of ${{\mathfrak T}}_\lambda^q$ associated to $(T_{C_\phi},\tilde b,{{\bf w}})$ as in Proposition \[QHIlocrep\], where $\lambda=\lambda_0$. Since $\kappa = \kappa_N({{\bf w}})$ is given by its values on the longitudes and the meridian curves $m_j$ of $\partial \bar{M}_\phi$, it is enough to show that ${{\mathfrak r}}$ and the scalars $\kappa(m_j)$ determine the parameters $(\{x_1,\ldots,x_n\},h)$ of $\rho_\lambda$. We already know that $x_1,\ldots,x_n$ are determined by ${{\mathfrak r}}$. Now, by the identities and we have $h^2 = \kappa(m_1)\ldots \kappa(m_r)$. Among the two square roots of $h^2$, the load $h$ is the only one which is an $N$-th root of $(-1)^r a_1^{-1}\ldots a_r^{-1}$ (using the notations of ). The eigenvalues $a_i$ being determined by the augmented character ${{\mathfrak r}}$, the conclusion follows. [$\Box$]{} Recall that given a local representation $\rho_\lambda$ and an irreducible representation $\mu$ of ${{\mathfrak T}}_{\lambda}^q$, we denote by $\rho_\lambda(\mu)$ the isotypical component associated to $\mu$ in the direct sum decomposition of $\rho_\lambda$ into irreducible factors. Consider the [*isotypical intertwiners*]{} $L^\phi_{\rho_\lambda(\mu)} := {{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b',{{\bf w}})^T_{\vert \rho_\lambda(\mu)}$. [**Corollary 2.**]{} (1) The trace of $L^\phi_{\rho_\lambda(\mu)}$ is an invariant of $(M_{\phi},{{\mathfrak r}},\kappa)$ and $\mu$, well-defined up to multiplication by $4N$-th roots of unity. It depends on the isotopy class of $\phi$ and satisfies $${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa) = \sum_{\rho_\lambda(\mu)\subset \rho_\lambda}{\rm Trace}\left(L^\phi_{\rho_\lambda(\mu)}\right).$$ (2) The invariants ${\rm Trace}\left(L^\phi_{\rho_\lambda(\mu)}\right)$ do not depend on the values of the weight $\kappa$ on longitudes. [[*Proof. *]{}]{}(1) We take back the notations used for Corollary 1. Once again, ${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa)$ is determined by $\phi$ and the isomorphism class of the local representation $\rho_\lambda$, because it is the trace of ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})$. Each isotypical component of $\rho_\lambda$ is the intersection of one eigenspace of each of the so called [*puncture elements*]{} of ${{\mathfrak T}}_\lambda^q$ (see [@B-L] and [@Tou]). Hence it is determined by the parameters $(\{x_1,\ldots,x_n\},h)$ of $\rho_\lambda$, together with a system of [*puncture weights*]{} $p_j\in {\mathbb{C}}^*$, the eigenvalues of the puncture elements. These can be any $N$-th root of $\kappa(m_j)^N$, where $\kappa(m_j)$ is computed in . Hence ${{\mathfrak r}}$ and $\kappa$ determine one isotypical component of $\rho_\lambda$. The others correspond to all other systems of puncture weights obtained by multiplying each $p_j$ with some $N$-th root of unity. Since ${{\mathcal H}}_N^{red}(T_{C_\phi},\tilde b,{{\bf w}})$ intertwins $\rho_\lambda$ and $\rho_{\lambda'}$, it intertwins their isotypical components too. The usual invariance proof of the reduced QHI still apply, and we get that the traces of the isotypical intertwiners are singly invariant. By the results of [@AGT; @NA], the invariants ${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa)$ actually depend on the choice of mapping torus realization $M_\phi$ of $M$. Therefore the invariants ${\rm Trace}(L^\phi_{\rho_\lambda(\mu)})$ do as well. By the same arguments as for ${{\mathcal H}}_N^{red}(M_{\phi},{{\mathfrak r}},\kappa)$, they are also defined for any augmented character ${{\mathfrak r}}$ in a Zariski open subset of the eigenvalue subvariety of $X_0(M)$ (see the comments on Theorem \[MAINTEO2\] in the Introduction). The proof of (2) is the same as the one of the previous corollary. [$\Box$]{} [99]{} H. Bai, *A uniqueness property for the quantization of Teichmüller Spaces*, Geom. Dedicata 128:1 (2007) 1–16 H. Bai, *Quantum Teichmüller space and Kashaev’s $6j$-symbols*, Alg. Geom. Topol. 7 (2007) 1541–1560 S. Baseilhac, R. Benedetti, *Quantum Hyperbolic invariants of 3-manifolds with $PSL(2,{\mathbb{C}})$-characters*, Topology 43 (6) (2004) 1373–1423 S. Baseilhac, R. Benedetti, *Classical and quantum dilogarithmic invariants of flat $PSL(2,{\mathbb{C}})$-bundles over $3$-manifolds*, Geom. Topol. 9 (2005) 493–570 S. Baseilhac, R. Benedetti, *Analytic families of quantum hyperbolic invariants*, Alg. Geom. Topol. 15 (2015) 1983–2063 S. Baseilhac, R. Benedetti, *Non ambiguous structures on $3$-manifolds and quantum symmetry defects*, to appear in Quantum Topology, ArXiv:1506.01174v2 (2016) H. F. Bonahon, X. Liu, *Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms*, Geom. Topol. 11 (2007) 889–937 H. Bai, F. Bonahon, X. Liu, *Local representations of the quantum Teichmüller space*, ArXiv:0707.2151 (2007) L.D. Faddeev, R.M. Kashaev, *Quantum dilogarithm*, Mod. Phys. Lett. A 9 (5) (1994) 427–434 M. Lackenby, *Taut ideal triangulations of 3–manifolds*, Geom. Topol. 4 (2000) 369–-395 F. Mazzoli, *Intertwining operators of the quantum Teichmüller space*, ArXiv:1610.06056 (2016) B. Klaff, S. Tillmann, *A birationality result for character varieties*, Math. Res. Lett. (4) 23 (2016), 1099–1110 J. Toulisse, *Irreducible decomposition for local representations of quantum Teichmüller space*, Arxiv:1404.4938v2 (2016)

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--- abstract: 'We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and provide preliminaries results toward the full weak convergence rate for non-self-adjoint linear operator. Key part of the proof does not rely on Malliavin calculus. Depending of the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions.' address: - 'The African Institute for Mathematical Sciences(AIMS) and Stellenbosh University, 6-8 Melrose Road, Muizenberg 7945, South Africa' - 'Center for Research in Computational and Applied Mechanics (CERECAM), and Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa.' - | School of Mathematics, Statistics and Computer Science,\ University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa author: - Antoine Tambue - Jean Medard T Ngnotchouye bibliography: - 'weakconvergenceSPDE\_July6.bib' title: 'Weak convergence for a stochastic exponential integrator and finite element discretization of SPDE for multiplicative & additive noise' --- SPDE ,Finite element methods ,Exponential integrators,Weak convergence ,Strong convergence Introduction ============ The weak numerical approximation of an Itô stochastic partial differential equation defined in $\Omega\subset \mathbb{R}^{d}$ is analyzed. Boundary conditions on the domain $\Omega$ are typically Neumann, Dirichlet or Robin conditions. More precisely, we consider in the abstract setting the following stochastic partial differential equation $$\begin{aligned} \label{adr} dX=(AX +F(X))dt + B(X)d W, \qquad X(0)=X_{0},\qquad t \in [0, T],\qquad T>0\end{aligned}$$ on the Hilbert space $L^{2}(\Omega)$. Here the linear operator $A$ which is not necessarily selfadjoint, is the generator of an analytic semigroup $S(t):=e^{t A}, t\geq 0.$ The functions $F$ and $B$ are nonlinear functions of $X$ and the noise term $W(t)$ is a $Q$-Wiener process defined on a filtered probability space $(\mathbb{D},\mathcal{F},\mathbb{P},\left\lbrace F_{t}\right\rbrace_{t\geq 0})$, that is white in time. The filtration is assumed to fulfill the usual conditions (see e.g. [@PrvtRcknr Definition 2.1.11]). For technical reasons more interest will be on a deterministic initial value $X_0\in H$. The noise can be represented as a series in the eigenfunctions of the covariance operator $Q$ given by $$\begin{aligned} \label{eq:W} W(x,t)=\underset{i \in \mathbb{N}^{d}}{\sum}\sqrt{q_{i}}e_{i}(x)\beta_{i}(t),\end{aligned}$$ where $(q_i,e_{i})$, $i\in \mathbb{N}^{d}$ are the eigenvalues and eigenfunctions of the covariance operator $Q$ and $\beta_{i}$ are independent and identically distributed standard Brownian motions. Under some technical assumptions it is well known (see [@DaPZ; @PrvtRcknr; @Chw]) that the unique mild solution of is given by $$\begin{aligned} \label{eq1} X(t)=S(t)X_{0}+\displaystyle\int_{0}^{t}S(t-s)F(X(s))ds +\displaystyle\int_{0}^{t}S(t-s)B(X(s))dW(s).\end{aligned}$$ Equations of type arise in physics, biology and engineering [@shardlow05; @AtThesis; @SebaGatam] and in few cases, exact solutions exist. The study of numerical solutions of SPDEs is therefore an active research area and there is an extensive literature on numerical methods for SPDE [@Jentzen2; @Jentzen3; @Jentzen4; @AtThesis; @GTambueexpo; @allen98:_finit; @Stig1; @Yn:04]. Basicaly there are two types of convergence. The strong convergence or pathwise convergence studies the pathwise convergence of the numerical solution to true solution while the weak convergence aims to approximate the law of the solution at a fixed time. In many applications, weak error is more relevant as interest are usualy based on some functions of the solution i.e, $\mathbb{E} \Phi (X)$, where $ \Phi : H \rightarrow \mathbb{R}$ and $\mathbb{E}$ is the expectation. Strong convergence rates for numerical approximations of stochastic evolution equations of type with smooth and regular nonlinearities are well understood in the scientific literature (see [@Jentzen2; @Jentzen3; @Jentzen4; @allen98:_finit; @Stig1; @Yn:04; @AtThesis; @GTambueexpoM] and references therein). Weak convergence rates for numerical approximations of equation are far away from being well understood. For a linear SPDE with additive noise, the solution can be written explicitly and the weak error have been estimated in [@shardlow:2003; @stigweak] with implicit Euler method for time discretization. The space discretization have been performed with finite difference method [@shardlow:2003; @stigweak] and finite element method [@stigweak]. The weak error of the implicit Euler method is more complicated for nonlinear equation of type as the Malliavin calculus is usually used to handle the irregular term and the term involving the nonlinear operators $F$ and $B$ (see [@Debussche; @WangGan2013; @stignonlinearweak]). In almost all the literature for weak error estimation, the linear operator $A$ is assumed to be self adjoint. Furthermore no numerical simulations were made to sustain the theoretical results to the best of our knowledge. In this paper we consider a stochastic exponential scheme (called stochastic exponential Euler scheme) as in [@GTambueexpoM] and provide the weak error of the full discrete scheme ([[Theorem \[fullweak\]]{}]{}, [[Theorem \[opfullaplace\]]{}]{} and [[Remark \[oppta\]]{}]{}) where the space discretization is performed using finite element, following closely the works in [@Wang2014; @WangGan2013] on another exponential integrator scheme. Our weak convergence proof does not use Malliavin calculus. Furthermore we provide some preliminaries results (weak convergence of the semi discrete scheme in [[Theorem \[thm:mainadditive\]]{}]{} and [[Theorem \[thm:multiplicative\]]{}]{}) toward the weak convergence when the linear operator $A$ is not necessarily self-adjoint, and provide some numerical examples to sustain the theoretical results. Recent work in [@stignonlinearweak] is used to obtain optimal convergence order for additive noise when the linear operator is self adjoint in [[Theorem \[opfullaplace\]]{}]{} and [[Remark \[oppta\]]{}]{}. We also extend in [[Theorem \[strong\]]{}]{} the strong optimal convergence rate provided in [@kruse Theorem 1.1] to non-self-adjoint operator $A$. Note that as the operator $A$ is not necessarily sef-adjoint, our scheme here are based on exponential matrix computation. The deterministic part of this scheme have be proved to be efficient and robust comparing to standard schemes in many applications [@SebaGatam; @AtThesis; @TLG; @antoelisa] where the exponential matrix functions have been computed using the Krylov subspace technique [@kry] and fast Leja points technique [@LE1]. For convenience of presentation, we take $A$ to be a second order operator as this simplifies the convergence proof. Our results can be extended to high order semi linear parabolic SPDE. The paper is organized as follows. [[Section \[sec1\]]{}]{} provides abstract setting and the well posedness of . The stochastic exponential Euler scheme along with weak error representation are provided in [[Section \[sec2\]]{}]{}. The temporal weak convergence rate of the stochastic exponential Euler scheme is provided in [[Section \[sec3\]]{}]{} for additive noise and in [[Section \[sec4\]]{}]{} for multiplicative noise. Note that in this section the linear operator $A$ is not necessarily self-adjoint. [[Section \[sec5\]]{}]{} provides strong optimal convergence rate of the semi discrete solution for non-self-adjoint operator $A$ along with full weak convergence rate of the stochastic exponential Euler scheme for self-adjoint operator $A$. Numerical results to sustain some theoretical results are provided in [[Section \[sec6\]]{}]{}. The abstract setting and mild solution {#sec1} ====================================== Let us start by presenting briefly the notation for the main function spaces and norms that we use in the paper. Let $H $ be a separable Hilbert space with the norm $\Vert \cdot \Vert$ associated to the inner product $\langle,\rangle_H$. For a Banach space $U$ we denote by $\Vert\cdot\Vert_{U}$ the norm of $U$, $L(U, H)$ the set of bounded linear mapping from $U$ to $H$ and by $L_{2}(\mathbb{D},U)$[^1] the Hilbert space of all equivalence classes of square integrable $U-$valued random variables. For ease of notation $L(U,U)=L(U)$. Furthermore we denote by $\mathcal{L}_{1}(U,H)$ the set of nuclear operators from $U$ to $H$, $\mathcal{L}_{2}(U,H):=HS(U,H)$ the space of Hilbert Schmidt functions from $U$ to $H$ and $\mathcal{C}_{b}^{k}(U,H)$ the space of not neccessarily bounded mappings from $U$ to $H$ that have continuous and bounded Frechet derivatives up to order $k,\, \,k \in \mathbb{N}$. For simplicity we also write $\mathcal{L}_{1}(U,U)=\mathcal{L}_{1}(U)$ and $\mathcal{L}_{2}(U,U)=\mathcal{L}_{2}(U)$. For a given orthonormal basis $ (e_{i})$ of U, the trace of $ l\in \mathcal{L}_{1}(U)$ is defined by $$\begin{aligned} \label{trace} \text{Tr}(l):=\underset{i \in \mathbb{N}^{d}}{\sum}\langle l e_i, e_i\rangle_{U},\end{aligned}$$ while the norm of $ l\in \mathcal{L}_{2}(U)$ is defined by $$\begin{aligned} \label{HS} \Vert l\Vert_{\mathcal{L}_{2}(U)}^{2}:=\underset{i\in \mathbb{N}^{d}}{\sum}\Vert l e_{i} \Vert_{U} ^2 < \infty,\end{aligned}$$ Note that the trace in and the Hilbert Schmidt norm in are independent of the basis $ (e_{i})$. Let $Q: H\rightarrow H$ be an operator, we consider throughout this work the $Q$-Wiener process. We denote the space of Hilbert–Schmidt operators from $Q^{1/2}(H)$ to $H$ by $L_{2}^{0}:=\mathcal{L}_{2}(Q^{1/2}(H),H)= HS(Q^{1/2}(H),H)$ and the corresponding norm $\Vert . \Vert_{L_{2}^{0}}$ by $$\begin{aligned} \Vert l\,\Vert_{L_{2}^{0}} := \Vert l Q^{1/2}\Vert_{\mathcal{L}_{2}(H)}=\left( \underset{i \in \mathbb{N}^{d}}{\sum}\Vert l Q^{1/2} e_{i} \Vert^{2}\right)^{1/2},\qquad \ l\in L_{2}^{0}. \end{aligned}$$ Let $\varphi : [0,T] \times \Omega \rightarrow L_{2}^{0} $ be a $L_{2}^{0}-$valued predictable stochastic process with $\mathbb{P} \left[ \int_{0}^{T}\Vert \varphi \Vert_{L_{2}^{0}}^{2}ds < \infty \right]=1$. Then Ito’s isometry (see e.g. [@DaPZ Step 2 in Section 2.3.2]) gives $$\begin{aligned} \mathbb{E} \Vert \int_{0}^{t}\varphi dW \Vert^{2}=\int_{0}^{t} \mathbb{E} \Vert \varphi \Vert_{L_{2}^{0}}^{2}ds=\int_{0}^{t} \mathbb{E} \Vert \varphi Q^{1/2} \Vert^{2}_{\mathcal{L}_{2}(H)}ds,\qquad \qquad t\in [0,T]. \end{aligned}$$ Let us recall the following proprieties which will be used in our errors estimation. [@Chw] \[proposition\] Let $l, l_1, l_2$ be three operators in Banach spaces, the following proprieties hold - If $l \in \mathcal{L}_{1}(U)$ then $$\begin{aligned} \vert \text{Tr}(l) \vert \leq \Vert l \Vert_{\mathcal{L}_{1}(U)}. \end{aligned}$$ - If $l_1 \in L(H)$ and $ l_2 \in \mathcal{L}_{1}(H)$, then both $ l_1 l_2$ and $ l_2 l_1$ belong to $\mathcal{L}_{1}(H)$ with $$\begin{aligned} \label{eql} \text{Tr}(l_1 l_2)= \text{Tr}(l_2 l_1). \end{aligned}$$ - If $l_1 \in \mathcal{L}_{2}(U,H)$ and $l_2 \in \mathcal{L}_{2}(H,U)$, then $ l_1 l_2 \in \mathcal{L}_{1}(H)$ with $$\begin{aligned} \Vert l_1 l_2 \Vert_{\mathcal{L}_{1}(H)} \leq \Vert l_1 \Vert_{\mathcal{L}_{2}(U,H)} \Vert l_2\Vert_{\mathcal{L}_{2}(H,U)}. \end{aligned}$$ - If $l \in \mathcal{L}_{2}(U,H)$, then its adjoint $l^{*} \in \mathcal{L}_{2}(H,U)$ with $$\begin{aligned} \Vert l^{*}\Vert_{\mathcal{L}_{2}(H,U)} = \Vert l \Vert_{\mathcal{L}_{2}(U,H)}. \end{aligned}$$ - If $l \in L(U, H)$ and $l_j \in \mathcal{L}_{j}(U),\, j=1,2, $ then $ll_j \in \mathcal{L}_{j}(U,H)$ with $$\begin{aligned} \label{eqnn} \Vert l l_j \Vert_{\mathcal{L}_{j}(U,H)} \leq \Vert l \Vert_{L (U,H)} \Vert l_j\Vert_{\mathcal{L}_{j}(U)},\;\;\;\,\;\; j=1,2. \end{aligned}$$ For classical well posedness, some assumptions are required both for the existence and uniqueness of the solution of equation (\[adr\]). \[assumptionn\] The operator $A : \mathcal{D}(A) \subset H \rightarrow H$ is a negative generator of an analytic semigroup $S(t)=e^{t A}, \quad t\geq 0$. In the Banach space $\mathcal{D}((-A)^{\alpha/2})$, $\alpha \in \mathbb{R}$, we use the notation $ \Vert (-A)^{\alpha/2}. \Vert =:\Vert .\Vert_{\alpha} $. We recall some basic properties of the semigroup $S(t)$ generated by $A$. **\[Smoothing properties of the semigroup [@Henry]\]**\ \[prop1\] Let $ \alpha >0,\;\beta \geq 0 $ and $0 \leq \gamma \leq 1$, then there exist $C>0$ such that $$\begin{aligned} \Vert (-A)^{\beta}S(t)\Vert_{L(H)} &\leq& C t^{-\beta}\;\;\;\;\; \text {for }\;\;\; t>0\\ \Vert (-A)^{-\gamma}( \text{I}-S(t))\Vert_{L(H)} &\leq& C t^{\gamma} \;\;\;\;\; \text {for }\;\;\; t\geq 0.\end{aligned}$$ In addition, $$\begin{aligned} (-A)^{\beta}S(t)&=& S(t)(-A)^{\beta}\quad \text{on}\quad \mathcal{D}((-A)^{\beta} )\\ \text{If}\;\;\; \beta &\geq& \gamma \quad \text{then}\quad \mathcal{D}((-A)^{\beta} )\subset \mathcal{D}((-A)^{\gamma} ),\\ \Vert D_{t}^{l}S(t)v\Vert_{\beta}&\leq& C t^{-l-(\beta-\alpha)/2} \,\Vert v\Vert_{\alpha},\;\; t>0,\;v\in \mathcal{D}((-A)^{\alpha/2})\;\; l=0,1, \end{aligned}$$ where $ D_{t}^{l}:=\dfrac{d^{l}}{d t^{l}}$. We describe now in detail the standard assumptions usually used on the nonlinear terms $F$,$B$ and the noise $W$. \[assumption1\] **\[Assumption on the drift term $F$\]** There exists a positive constant $L> 0$ such that $F: H \rightarrow H$ satisfies the following Lipschitz condition $$\begin{aligned} \Vert F(Z)- F(Y)\Vert \leq L \Vert Z- Y\Vert \qquad \forall \quad Z, \; Y \in H. \end{aligned}$$ As a consequence, there exists a constant $C>0$ such that $$\begin{aligned} \Vert F(Z) \Vert &\leq& \Vert F (0)\Vert + \Vert F (Z) - F (0)\Vert \leq \Vert F (0)\Vert + L \Vert Z\Vert \leq C( 1 +\Vert Z \Vert )\qquad \qquad Z\in H.\\\end{aligned}$$ \[assumption2\] **\[Assumption on the diffusion term $B$\]**\ There exists a positive constant $L> 0$ such that the mapping $B: H\rightarrow \mathcal{L}_2(H) $ satisfies the following condition $$\begin{aligned} \Vert B(Z)- B(Y)\Vert_{\mathcal{L}_2(H)} \leq L \Vert Z- Y\Vert \qquad \forall Z, Y \in H.\end{aligned}$$ As a consequence, there exists a constant $C>0$ such that $$\begin{aligned} \Vert B(Z) \Vert_{\mathcal{L}_2(H)} &\leq& \Vert B(0)\Vert_{\mathcal{L}_2(H)} + \Vert B(Z) - B(0)\Vert_{\mathcal{L}_2(H)} \nonumber \\ &\leq &\Vert B(0)\Vert_{\mathcal{L}_2(H)} + L \Vert Z\Vert \leq C( 1 +\Vert Z \Vert )\qquad Z\in H.\label{noise}\end{aligned}$$ \[existth\] **\[Existence and uniqueness ([@DaPZ])\]**\ Assume that the initial solution $X_{0}$ is an $F_{0}-$measurable $H-$valued random variable and [[Assumption \[assumptionn\]]{}]{}, [[Assumption \[assumption1\]]{}]{}, [[Assumption \[assumption2\]]{}]{} are satisfied. There exists a mild solution $X$ to unique, up to equivalence among the processes, satisfying $$\begin{aligned} \mathbb{P} \left( \int_{0}^{T}\Vert X(s)\Vert^{2}ds < \infty\right)=1.\end{aligned}$$ For any $p\geq 2$ there exists a constant $C =C(p,T)>0 $ such that $$\begin{aligned} \label{ineq2} \underset{t\in [0,T]}{\sup}\mathbb{E} \Vert X(t)\Vert^{p} \leq C\left(1+\mathbb{E} \Vert X_{0} \Vert^{p}\right). \end{aligned}$$ For any $p>2$ there exists a constant $C_{1} =C_{1}(p,T)>0 $ such that $$\begin{aligned} \mathbb{E}\underset{t\in [0,T]}{\sup} \Vert X(t)\Vert^{p} \leq C_{1}\left(1+\mathbb{E} \Vert X_{0} \Vert^{p}\right). \end{aligned}$$ The following theorem proves a regularity result of the mild solution $X$ of . \[newtheo\] **\[Regularity of the mild solution ([@GTambueexpoM])\]**\ Assume that [[Assumption \[assumptionn\]]{}]{}, [[Assumption \[assumption1\]]{}]{} and [[Assumption \[assumption2\]]{}]{} hold. Let $X$ be the mild solution of (\[adr\]) given in (\[eq1\]). If $X_{0} \in L_{2}(\mathbb{D},\mathcal{D}((-A)^{\beta/2})),\, \beta \in [0,1)$ then for all $ t\in [0,T],\,X(t) \in L_{2}(\mathbb{D},\mathcal{D}((-A)^{\beta/2}))$ with $$\begin{aligned} \label{regsoluion} \left(\mathbb{E}\Vert X(t) \Vert_{\beta}^{2}\right)^{1/2}\leq C \,\left(1+\left(\mathbb{E}\Vert X_{0}\Vert_{\beta}^{2}\right)^{1/2}\right).\\\end{aligned}$$ For the weak error represention, we will need the following lemma. \[lemmaIto\] Let $(\mathbb{D}, \mathcal{F},P; \left\lbrace F_{t}\right\rbrace_{t\geq 0}$) be a filtered probability space. Let $\phi$ and $\Psi$ be $H-$valued predictable processes, Bochner integrable on $[0,T]$ P-almost surely (see [@KovacsLarsson2008]) , and $Y_0$ be an $F_0$-measurable $H-$valued random variable. Let $G:[0,T]\times H\rightarrow \mathbb{R}$ and assume that the Fréchet derivatives $G_t(t,x),\, G_x(t,x),$ and $G_{xx}(t,x)$ are uniformly continuous as functions of $(t,x)$ on bounded subsets of $[0;T]\times H.$ Note that, for fixed $t,\,\, G_x(t,x)\in L(H,\mathbb{R})$ and we consider $G_{xx}(t,x)$ as an element of $L(H)$. Let $W$ be the $Q$-Wiener process. If $Y$ satisfies $$Y(t) = Y(0) + \displaystyle\int_0^t\phi(s)ds + \displaystyle\int_0^t \Psi(s)dW(s),$$ then P-almost surely for all $t\in [0,T],$ $$\begin{array}{rcl}G(t,Y(t))&=& G(0,Y(0)) + \displaystyle\int_0^t G_x(s,Y(s))\Psi(s)dW(s) \\ && +\displaystyle\int_0^t\Big\{G_t(s,Y(s)) + G_x(s,Y(s))\phi(s) +\frac{1}{2}\text{Tr}\left(G_{xx}(s,Y(s))\Psi(s)Q^{\frac{1}{2}}(\Psi(s)Q^{\frac{1}{2}})^*\right)\Big\}\,ds \end{array}$$ A proof of this lemma can be found in [@DaPZ]. Application to the second order semi–linear parabolic SPDE {#sec2} ========================================================== We assume that $\Omega$ has a smooth boundary or is a convex polygon of $\mathbb{R}^{d},\;d=1,2,3$. In the sequel, for convenience of presentation, we take $A$ to be a second order operator as this simplifies the convergence proof. The result can be extended to high order semi linear parabolic SPDE. More precisely we take $H=L^{2}(\Omega)$ and consider the general second order semi–linear parabolic stochastic partial differential equation given by $$\begin{aligned} \label{sadr} dX(t,x)=\left(\nabla \cdot \textbf{D} \nabla X(t,x) -\mathbf{q} \cdot \nabla X(t,x) + f(x,X(t,x))\right) dt +b(x,X(t,x))dW(t,x),\end{aligned}$$ $x \in \Omega, t\in[0,T]$ where $f:\Omega \times\mathbb{R} \rightarrow \mathbb{R}$ is a globally Lipschitz continuous function and $b:\Omega \times\mathbb{R} \rightarrow \mathbb{R}$ is a continuously differentiable function with globally bounded derivatives. The abstract setting for second order semi–linear parabolic SPDE ---------------------------------------------------------------- In the abstract form given in , the nonlinear functions $F : H \rightarrow H$ and $ B : H\rightarrow HS(Q^{1/2}(H), H)$ are defined by $$\begin{aligned} \label{nemform} (F(v))(x)=f(x,v(x)),\qquad (B(v)u)(x)=b(x,v(x))\cdot u(x),\end{aligned}$$ for all $ x\in \Omega,\;v\in H,\; u \in Q^{1/2}(H)$, with $H=L^{2}(\Omega)$. Note that we can also define $ B : H\rightarrow \mathcal{L}_2(H)$ by $$\begin{aligned} (B(v)u)(x)=b(x,v(x))\cdot Q^{1/2}u(x),\end{aligned}$$ for all $ x\in \Omega,\;v\in H,\; u \in H$. In order to define rigorously the linear operator, let us set $$\begin{aligned} \label{eq:A} \mathcal{A}&=&\underset{i,j=1}{\sum^{d}}\dfrac{\partial }{\partial x_{i}}\left( D_{i,j}\dfrac{\partial }{\partial x_{j}}\right) - \underset{i=1}{\sum^{d}}q_{i}\dfrac{\partial }{\partial x_{i}},\end{aligned}$$ where we assume that $D_{i,j} \in L^{\infty}(\Omega),\,q_{i}\in L^{\infty}(\Omega)$ and that there exists a positive constant $c_{1}>0$ such that $$\begin{aligned} \label{ellipticity} \underset{i,j=1}{\sum^{d}}D_{i,j}(x)\xi_{i}\xi_{j}\geq c_{1}\vert \xi \vert^{2}, \;\;\;\;\;\;\forall \xi \in \mathbb{R}^{d},\;\;\; x \in \overline{\Omega},\;\;\; c_{1}>0.\end{aligned}$$ We introduce two spaces ${\mathbb{H}}$ and $V$ where ${\mathbb{H}}\subset V $ depends on the choice of the boundary conditions for the SPDE. For Dirichlet boundary conditions we let $$\begin{aligned} V= {\mathbb{H}}= H_{0}^{1}(\Omega)=\{v\in H^{1}(\Omega): v=0\;\; \text{on}\;\;\partial \Omega\}, \end{aligned}$$ and for Robin boundary conditions, Neumann boundary being a special case, we take $V= H^{1}(\Omega)$ and $$\begin{aligned} {\mathbb{H}}= \left\lbrace v\in H^{2}(\Omega): \partial v/\partial \nu_{\mathcal{A}}+\alpha_{0} v=0\quad \text{on}\quad \partial \Omega\right\rbrace, \qquad \alpha_{0} \in \mathbb{R}. \end{aligned}$$ See [@lions] for details. The corresponding bilinear form of $ -\mathcal{A}$ is given by $$\begin{aligned} \label{var} a(u,v)=\int_{\Omega}\left(\underset{i,j=1}{\sum^{d}} D_{i,j}\dfrac{\partial u}{\partial x_{j}} \dfrac{\partial v}{\partial x_{i}}+\underset{i=1}{\sum^{d}}q_{i} \dfrac{\partial u}{\partial x_{j}}v\right)dx,\;\;\;\;\;\;\; u, v \in V\end{aligned}$$ for Dirichlet and Neumann boundary conditions, and by $$\begin{aligned} \label{var1} a(u,v)=\int_{\Omega}\left(\underset{i,j=1}{\sum^{d}} D_{i,j}\dfrac{\partial u}{\partial x_{j}} \dfrac{\partial v}{\partial x_{i}}+\underset{i=1}{\sum^{d}}q_{i} \dfrac{\partial u}{\partial x_{j}}v\right)dx +\int_{\partial \Omega} \alpha_{0} u\,v\,dx, \;\;\;\;\;\;\; u, v \in V,\end{aligned}$$ for Robin boundary conditions. According to Gårding’s inequality (see [@AtThesis; @lions]), there exist two positive constants $c_{0}$ and $\lambda_{0}$ such that $$\begin{aligned} \label{coer} a(v,v)+c_{0}\Vert v\Vert^{2}\geq \lambda_{0}\Vert v\Vert_{H^{1}(\Omega)}^{2},\;\;\; \quad \quad \forall v\in V.\end{aligned}$$ By adding and subtracting $c_{0}X dt$ on the right hand side of (\[sadr\]), we have a new operator that we still call $\mathcal{A}$ corresponding to the new bilinear form that we still call $a$ such that the following coercivity property holds $$\begin{aligned} \label{ellip} a(v,v)\geq \; \lambda_{0}\Vert v\Vert_{H^{1}(\Omega)}^{2},\;\;\;\;\;\forall v \in V.\end{aligned}$$ Note that the expression of the nonlinear term $F$ has changed as we include the term $-c_{0}X$ in a new nonlinear term that we still denote by $F$. Note that $a(,)$ is bounded in $V\times V$, so the following operator $A: V\rightarrow V^{*}$ is well defined by the Riez’s representation theorem $$\begin{aligned} \label{opA} a(u,v)=-\langle Au, v\rangle,\,\,\,\, \forall u, v \in V,\end{aligned}$$ where $V^{*}$ is the adjoint space (or dual space) of $V$ and $\langle ,\rangle$ the duality pairing between $V^{*}$ and $V$. By identifying $H$ to its adjoint space $H^{*}$, we get the following continuous and dense inclusions $$\begin{aligned} \label{gerland} V\subset H \subset V^{*}.\end{aligned}$$ So, we have $$\begin{aligned} \label{gerland1} \langle u,v\rangle_H =\langle u, v \rangle \qquad \qquad \qquad \forall u\in H,\, \forall v\in V.\end{aligned}$$ The linear operator $A$ in our abstract setting is therefore defined by . The domain of $A$ denoted by $\mathcal{D}(A)$ is defined by $$\begin{aligned} \mathcal{D}(A)= \{ u \in V,\, Au \in H \}. \end{aligned}$$ We write the restriction of $A: V\rightarrow V^{*}$ to $\mathcal{D}(A)$ again by $A$, which is therefore regarded as an operator of $H$ (more precisely the $H$ realization of $\mathcal{A}$ [@lions p. 812]). The coercivity property (\[ellip\]) implies that $A$ is a sectorial on $L^{2}(\Omega)$ i.e. there exists $C_{1},\, \theta \in (\frac{1}{2}\pi,\pi)$ such that $$\begin{aligned} \Vert (\lambda I -A )^{-1} \Vert_{L(L^{2}(\Omega))} \leq \dfrac{C_{1}}{\vert \lambda \vert }\;\quad \quad \lambda \in S_{\theta},\end{aligned}$$ where $S_{\theta}=\left\lbrace \lambda \in \mathbb{C} : \lambda=\rho e^{i \phi},\; \rho>0,\;0\leq \vert \phi\vert \leq \theta \right\rbrace $ (see [@Henry; @lions]). Then $A$ is the infinitesimal generator of bounded analytic semigroups $S(t):=e^{t A}$ on $L^{2}(\Omega)$ such that $$\begin{aligned} S(t):= e^{t A}=\dfrac{1}{2 \pi i}\int_{\mathcal{C}} e^{ t\lambda}(\lambda I - A)^{-1}d \lambda,\;\;\;\;\;\;\; \;t>0,\end{aligned}$$ where $\mathcal{C}$ denotes a path that surrounds the spectrum of $A $. The condition property also implied that $-A$ is a positive operator and its fractional powers is well defined for any $\alpha>0,$ by $$\left\{\begin{array}{rcl} (-A)^{-\alpha} & =& \frac{1}{\Gamma(\alpha)}\displaystyle\int_0^\infty t^{\alpha-1}{\rm e}^{tA}dt,\\ (-A)^{\alpha} & = & ((-A)^{-1})^{-1}, \end{array}\right.$$ where $\Gamma(\alpha)$ is the Gamma function (see [@Henry]). Functions in ${\mathbb{H}}$ satisfy the boundary conditions and with ${\mathbb{H}}$ in hand we can characterize the domain of the operator $(-A)^{\alpha/2}$ and have the following norm equivalence ([@lions; @Stig; @ElliottLarsson]) for $\alpha\in \{1,2\}$ $$\begin{aligned} \Vert v \Vert_{H^{\alpha}(\Omega)} &\equiv &\Vert (-A)^{\alpha/2} v \Vert=:\Vert v \Vert_{\alpha}, \qquad \forall v\in \mathcal{D}((-A)^{\alpha/2}),\\ \mathcal{D}((-A)^{\alpha/2})&=& {\mathbb{H}}\cap H^{r}(\Omega)\,\,\qquad \qquad \,\text{ (Dirichlet boundary conditions)},\\ \mathcal{D}(-A)&=&{\mathbb{H}},\,\,\,\qquad\mathcal{D}((-A)^{1/2})= H^{1}(\Omega)\,\,\,\qquad\text{(Robin boundary conditions)}.\end{aligned}$$ Numerical scheme and weak error representation ---------------------------------------------- We consider the discretization of the spatial domain by a finite element method. Let $\mathcal{T}_{h}$ be a set of disjoint intervals of $\Omega$ (for $d=1$), a triangulation of $\Omega$ (for $d=2$) or a set of tetrahedra (for $d=3$) with maximal length $h$ satisfying the usual regularity assumptions [@EP; @Gyongy2]. Let $V_{h}\subset V$ denote the space of continuous functions that are piecewise linear over the triangulation $\mathcal{T}_{h}$. Note that for high order polynomial, high order accuracy in space can be achieved. To discretize in space we introduce the projection $P_h$ from $L^{2}(\Omega)$ onto $V_{h}$ defined for $u \in L^{2}(\Omega)$ by $$\begin{aligned} (P_{h}u,\chi)=(u,\chi),\qquad \forall\;\chi \in V_{h}.\end{aligned}$$ The discrete operator $A_{h}: V_{h}\rightarrow V_{h}$ is defined by $$\begin{aligned} ( A_{h}\varphi,\chi)=(A\varphi,\chi)=-a(\varphi,\chi),\qquad \varphi,\chi \in V_{h}.\end{aligned}$$ Like the operator $A$, the discrete operator $A_h$ is also the generator of an analytic semigroup $S_h:=e^{tA_{h}}$. Here we consider the following semi–discrete form of the problem (\[adr\]), which consists to find the process $X^{h}(t)=X^{h}(.,t) \in V_{h}$ such that for $t \in[0, T]$, $$\begin{aligned} \label{dadr} dX^{h}=(A_{h}X^{h} +P_{h}F(X^{h}))dt + P_{h} B(X^{h}) P_{h}d W,\qquad X^{h}(0)=:X_0^h=P_{h}X_{0}.\end{aligned}$$ Note that is a finite dimensional stochastic equation. The mild solution of (\[dadr\]) at time $t_{m}=m \Delta t $ is given by $$\begin{aligned} \label{dmild} X^{h}(t_{m})&=&S_{h}(t_{m})P_{h}X_{0}+\displaystyle\int_{0}^{t_{m}} S_{h}(t_{m}-s) P_{h}F(X^{h}(s))ds \nonumber \\ && + \displaystyle\int_{0}^{t_{m}} S_{h}(t_{m}-s)P_{h}B(X^{h}(s)) P_{h}d W(s),\end{aligned}$$ where $ \Delta t=T/M,\;\;\; m \in \{0,1,...,M \}, \;\,M \in \mathbb{N}$. Then, given the mild solution at the time $t_{m}$, we can construct the corresponding solution at $t_{m+1}$ as $$\begin{aligned} X^{h}(t_{m+1})&=&S_{h}(\Delta t)X^{h}(t_{m})+\displaystyle\int_{0}^{ \Delta t} S_{h}( \Delta t-s) P_{h}F(X^{h}(s+t_{m}))ds \\ && +\displaystyle\int_{t_{m}}^{t_{m+1}} S_{h}(t_{m+1}-s)P_{h}B(X^{h}(s)) P_{h}d W(s).\end{aligned}$$ To build the numerical scheme, we use the following approximations [@GTambueexpoM] $$\begin{aligned} P_{h} F(X^{h}( t_{m}+s))&\approx& P_{h}F(X^{h}( t_{m})),\;\;\;\;\; \;\;\;\;\;s \in [0,\; \Delta t],\\ S_{h}(t_{m+1}-s)P_{h}B(X^{h}(s))&\approx& S_{h}(\Delta t) P_{h}B(X^{h}(t_{m})),\;\;\; s\in [t_{m},t_{m+1}].\end{aligned}$$ We can define our approximation $X_{m}^{h}$ of $X(m \Delta t)$ by $$\begin{aligned} \label{new} X_{m+1}^{h}=e^{\Delta t A_{h}}X_{m}^{h}+A_{h}^{-1}\left( e^{\Delta t A_{h}}-I\right)P_{h}F(X_{m}^{h})+e^{\Delta t A_{h}}P_{h}B(X_{m}^{h}) P_{h}\left( W_{t_{m+1}}-W_{t_{m}}\right).\end{aligned}$$ We introduce for brevity the notations $$S_h(t) ={\rm e}^{tA_h},\quad S_h^{1}(t) = (tA_h)^{-1}({\rm e}^{tA_h}-I), \quad B_h: X \mapsto P_h B(X)P_h.$$ We can then rewrite the scheme (\[new\]) as $$\begin{aligned} \label{eq:fvscheme} X_{m+1}^{h}=S_h(\Delta t)X_{m}^{h}+\Delta t S_h^1(\Delta t )P_{h}F(X_{m}^{h})+ S_h(\Delta t)B_h(X_{m}^{h})\left( W_{t_{m+1}}-W_{t_{m}}\right).\end{aligned}$$ In order to study the weak convergence of the approximation of the solutions we define the functional $$\label{eq:mu} \mu^h(t,\psi) = \mathbb{E}\left[\Phi(X^h(t,\psi)\right],\,\,t\in[0,T],\,\psi\in V_h,$$ where $X^h(t,\psi)$ is defined by with the initial value $ X_0^h=\psi\in V_h.$ It can be shown (see Theorem 9.16 of \[9\]) that $\mu^h(t,\psi)$ defined by is differentiable with respect to $t$ and twice differentiable with respect to $\psi,$ and is the unique strict solution of $$\label{eq:Kolmogorov} \left\{\begin{array}{rcl} \frac{\partial \mu^h}{\partial t}(t,\psi) & = & \langle A_h\psi + P_hF(\psi) , D\mu^h(t,\psi)\rangle_H + \frac{1}{2}\text{Tr}\left[D^2\mu^h(t,\psi)B_h(\psi) B_h (\psi)^*\right]\\ \mu^h(0,\psi)&=& \Phi(\psi),\quad \psi\in V_h.\end{array}\right.$$ We can here, using the Riesz representation theorem, identify the first order derivative of $\mu^h(t,\psi)$ with respect to $\psi,$ denoted as $D\mu^h(t,\psi)$ with an element of $V_h$ and the second derivative denoted as $D^2\mu^h(t,\psi)$ with a linear operator in $V_h.$ More precisely $$\begin{aligned} \label{derivation} D\mu^h(t,\psi)(\phi_1)&=&\langle D\mu^h(t,\psi),\phi_1\rangle_H,\,\,\, \forall \,\psi, \phi_1 \in V_h,\\ D^2\mu^h(t,\psi)(\phi_1,\phi_2) &=& \langle D^2\mu^h(t,\psi)\phi_1,\phi_2\rangle_H, \,\,\forall \,\psi, \phi_1,\phi_2 \in V_h.\end{aligned}$$ The following theorem, which is similar to [@Wang2014 Theorem 2.2] is fundamental for our convergence proofs. Assume that all the conditions in the assumptions above are fulfilled and let $\{W(t)\}_{t\in[0,T]}$ be a cylindrical $H-$valued $Q-$Wiener process. Then for $\Phi\in\mathcal{C}_b^2(H,\mathbb{R})$ the weak approximation error of the scheme in has the representation $$\label{eq:errorrep}\displaystyle \begin{array}{l} \mathbb{E}[\Phi(X^h_M)]-\mathbb{E}[\Phi(X^h(T))] \\ = \sum\limits_{m=0}^{M-1}\left\{\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\left[\langle D\mu^h(T-s,\tilde X^h(s)),P_hF(X_m^h)-P_hF(\tilde X^h(s))\rangle_H \right]ds\right.\\ \hskip 1cm + \frac{1}{2}{\mathbb{E}}\displaystyle\int_{t_m}^{t_{m+1}}\text{Tr}\left[D^2\mu^h(T-s,\tilde X^h(s))\left(\left\{S_h(s-t_m) B_h(X_m^h)\right\}\left\{S_h(s-t_m)B_h(X_m^h)\right\}^*\right.\right.\\ \hskip 7cm\left. \left. \left.- B_h(X_m^h)B_h(X_m^h)^*\right)\right]ds \right. \Big\}, \end{array}$$ where $\tilde X^h(t)$ is a continuous extension of $X_m^h,$ defined by $$\label{eq:xtilde} \begin{array}{rcl} \tilde X^h(t) &= &S_h(t-t_m)X_m^h + (t-t_m)S_h^1(t-t_m)P_hF(X_m^h) \\ && \hskip 2cm + S_h(t-t_m)B_h(X_m^h)(W(t)-W(t_m)),\quad \text{ for } t\in [t_m, t_{m+1}].\end{array}$$ Introduce the process $\nu^h:[0,T]\times V_h\rightarrow \mathbb{R},$ given by $$\label{eq:nu} \nu^h(x,\psi) = \mu^h(t,S_h(-t)\psi),$$ which is twice differentiable with respect to $\psi$ and by previous identifications satisfies $$\begin{aligned} \label{deri} D\nu^h(t,\psi)\phi &=& \langle D\mu^h(t,S_h(-t)\psi),S_h(-t)\phi\rangle_H,\,\,\psi, \phi \in V_h\\ \label{iden} D^2\nu^h(t,\psi)(\phi_1,\phi_2) &=& \langle D^2\mu^h(t,S_h(-t)\psi)(S_h(-t)\phi_1), S_h(-t)\phi_2\rangle_H,\,\,\psi, \phi_1,\phi_2 \in V_h.\end{aligned}$$ One can then check that $\nu^h(t,\psi)$ solves the equation $$\label{eq:pdenu} \begin{array}{rcl}\frac{\partial\nu^h}{\partial t}(t,\psi) &=& \langle D\nu^h(t,\psi),S_h(t)P_hF(S_h(-t)\psi\rangle_H + \frac{1}{2}\text{Tr}[D^2\nu^h(t,\psi)S_h(t)B_h(\psi)(S_h(t)B_h(\psi))^*],\\ \nu^h(0,\psi) & =& \Phi(\psi),\quad \psi \in V_h. \end{array}$$ Indeed, since $\frac{\partial S_h(-t)\psi}{\partial t} = -A_h S_h(-t)\psi,$ we have that $$\begin{array}{rcl}\frac{\partial\nu^h}{\partial t}(t,\psi)&=& \frac{\partial\mu^h}{\partial t}(t,S_h(-t)\psi) + \langle D\mu^h(t,S_h(-t)\psi),-A_h S_h(-t)\psi\rangle_H. \end{array}$$ Using the Kolmogorov equation and the fact that $\langle, \rangle_H$ is symmetric yields $$\begin{array}{rcl}\frac{\partial\nu^h}{\partial t}(t,\psi) &= & \langle A_h S_h(-t)\psi + P_hF(S_h(-t)\psi) , D\mu^h(t,S_h(-t)\psi)\rangle_H\\ && + \frac{1}{2}\text{Tr}\left[D^2\mu^h(t,S_h(-t)\psi)S_h(-t)S_h(-t)S_h(t)S_h(t)B_h(\psi)^*\right]\\ && + \langle D\mu^h(t,S_h(-t)\psi),-A_h S_h(-t)\psi\rangle_H.\\ \end{array}$$ As $D^2\mu^h(t,\psi)$ is identified as linear operator in $V_h$, using [[Proposition \[proposition\]]{}]{}(mainly relation ), the definition of the trace , the expression and the fact that $\langle, \rangle_H$ is symmetric allow to have $$\begin{array}{rcl}\frac{\partial\nu^h}{\partial t}(t,\psi) &=& \langle D\mu^h(t,S_h(-t)\psi)S_h(-t)\psi,S_h(t) P_hF(S_h(-t)\psi)\rangle_H\\ && +\frac{1}{2}\text{Tr}\left[D^2\nu^h(t,\psi)S_h(t)B_h(\psi)(S_h(t)B_h(\psi))^*\right]\\ &&= \langle D\nu^h(t,\psi),S_h(t)P_hF(S_h(-t)\psi\rangle_H + \frac{1}{2}\text{Tr}[D^2\nu^h(t,\psi)S_h(t)B_h(\psi)(S_h(t)B_h(\psi))^*].\end{array}.$$ Now let $Z^h(t) = S_h(T-t)X^h(t),$ using and the notation $X_0^h = P_hX_0,$ we get $$\begin{aligned} \label{eq:zh}Z^h(t)&=&S_h(T)X_{0}^h+\displaystyle\int_{0}^{t}S_h(T-s) P_{h}F(X^{h}(s))ds \nonumber \\ && + \displaystyle\int_{0}^{t}S_h(T-s)B_h(X^{h}(s))d W(s), \,\,t\in[0,T].\end{aligned}$$ Also let $\tilde Z^h(t) = S_h(T-t)\tilde X^h(t),$ using , we get $$\label{eq:ztilde} \begin{array}{rcl} \tilde Z^h(t) &= &S_h(T-t_m)X_m^h + (t-t_m)S_h(T-t)S_h^1(t-t_m)P_hF(X_m^h) \\ && \hskip 2cm + S_h(T-t_m)B_h(X_m^h) (W(t)-W(t_m)),\quad \text{ for } t\in [t_m, t_{m+1}].\end{array}$$ which can also be written as $$\label{eq:ztilde2} \begin{array}{rcl} \tilde Z^h(t) &= &S_h(T-t_m)X_m^h + \displaystyle\int_{t_m}^{t}S_h(T-s)P_hF(X_m^h)ds \\ && \hskip 2cm + \displaystyle\int_{t_m}^tS_h(T-t_m)B_h(X_m^h) dW(s),\quad \text{ for } t\in [t_m, t_{m+1}].\end{array}$$ Consequently, we obtain $$\tilde Z^h(T) = \tilde X^h(t) = X_M^h,\quad Z^h(0) = S_h(T)\tilde X^h(0) = S_h(T)X_0^h.$$ Now the weak error is worked out as $$\begin{array}{rcl} {\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X^h(T,X_0^h))] &=& {\mathbb{E}}[\Phi(\tilde Z^h(T))] - {\mathbb{E}}[\mu^h(T,X_0^h)]\\ &=& {\mathbb{E}}[\nu^h(0,\tilde Z^h(T))]-{\mathbb{E}}[\nu^h(T,\tilde Z^h(0))]\\ &=&\sum\limits_{m=0}^{M-1}{\mathbb{E}}[\nu^h(T-t_{m+1},\tilde Z^h(t_{m+1}))]-{\mathbb{E}}[\nu^h(T-t_{m},\tilde Z^h(t_{m}))]. \end{array}$$ Now using Itô formula (see Lemma \[lemmaIto\]) applied to $G (t,x) = \nu^h(T-t,\tilde Z^h(t))$ in the interval $[t_m, t_{m+1}]$ and the fact that the Itô integral vanishes, we can write $$\begin{array}{l} {\mathbb{E}}[\nu^h(T-t_{m+1},\tilde Z^h(t_{m+1}))]-{\mathbb{E}}[\nu^h(T-t_{m},\tilde Z^h(t_{m}))]\\ = \displaystyle-\int_{t_m}^{t_{m+1}}{\mathbb{E}}\left[\frac{\partial \nu^h}{\partial t}(T-s,\tilde Z^h(s))\right]ds +\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\left[ D\nu^h(T-s,\tilde Z^h(s))S_h(T-s)P_hF(X_m^h) \right]ds\\ \hskip 1cm + \frac{1}{2}{\mathbb{E}}\displaystyle\int_{t_m}^{t_{m+1}}\text{Tr}\left[D^2\nu^h(T-s,\tilde Z^h(s))\left\{S_h(T-t_m) B_h(X_m^h)\right\}\left\{S_h(T-t_m)B_h(X_m^h)\right\}^*\right]ds. \end{array}$$ Using the fact that $D\nu^h(t,\psi)$ is identified to an element of $V_h$ (see the analogue representation at ) and , we finally have $$\begin{array}{l} {\mathbb{E}}[\nu^h(T-t_{m+1},\tilde Z^h(t_{m+1}))]-{\mathbb{E}}[\nu^h(T-t_{m},\tilde Z^h(t_{m}))]\\ = \displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\left[\langle D\nu^h(T-s,\tilde Z^h(s)),-S_h(T-s)P_hF(S_h(s-T)\tilde Z^h(s)) +S_h(T-s)P_hF(X_m^h)\rangle_H\right]ds\\ \hskip 1cm + \frac{1}{2}{\mathbb{E}}\Big\{\displaystyle\int_{t_m}^{t_{m+1}}\text{Tr}\Big[D^2\nu^h(T-s,\tilde Z^h(s))\Big(\left\{S_h(T-t_m) B_h(X_m^h)\right\}\left\{S_h(T-t_m)B_h(X_m^h)\right\}^*\\ \hskip 5cm - S_h(T-s)B_h(X_m^h)(S_h(T-s)B_h(X_m^h))^*\Big)\Big]\,ds\Big\} \\ = \displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\left[\langle D\mu^h(T-s,\tilde X^h(s)),P_hF(X_m^h)-P_hF(\tilde X^h(s))\rangle_H \right]ds\\ \hskip 1cm + \frac{1}{2}{\mathbb{E}}\Big\{\displaystyle\int_{t_m}^{t_{m+1}}\text{Tr}\Big[D^2\mu^h(T-s,\tilde X^h(s))\Big(\left\{S_h(s-t_m) B_h(X_m^h)\right\}\left\{S_h(s-t_m)B_h(X_m^h)\right\}^*\\ \hskip 7cm - B_h(X_m^h)B_h(X_m^h)^*\Big)\Big]ds. \\ \end{array}$$ Weak convergence for a SPDE with additive noise {#sec3} =============================================== In this section, we consider the additive noise where $B=Q^{1/2}$. In order to prove our weak error estimate the following weak assumptions [^2] [@Wang2014] will be used. \[ass:driftandB\] **\[Assumption on nonlinear function $F$, and $Q$ \]** We assume that $F:H\rightarrow H$ is Lipschitz and twice continuously differentiable and satisfies $$\begin{aligned} \label{asseq1} \|F(X)\|&\leq& L(1+\|X\|),\qquad X \in H,\\\ \|F'(Z)(X)\|&\leq& L\|X\|,\,\qquad\,X,\,Z \in H,\,\,\,\label{aq1}\,\hskip 4cm\\ \quad \|F''(Z)(X_1,X_2)&\leq& L\|X_1\|\|X_2\|,\quad Z,\,X,\,X_1,\,X_2 \in H, \label{aq2}\\ \|(-A)^{-\beta}F''(Z)(X_1,X_2)\|&\leq& L\|X_1\|\|X_2\|,\,\,Z,X_1, X_2\in H,\text{ for some }\beta\in[0,1),\label{asseq2}\\ \|(-A)^{-\frac{\gamma}{2}}F'(Z)(X)\|&\leq& L(1+\|Z\|_1)\|X\|_{-1}, \label{asseq3} \\ && \,\,X\in H,\,\,Z\in \mathcal{D}((-A)^{1/2}) \text{for some} \,\,\gamma\in[1,2)\nonumber. \end{aligned}$$ Furthermore, we assume that the covariance operator $Q$ satisfies $$\|(-A)^{\frac{\beta-1}{2}}Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)}<\infty,\text{ for some }\beta\in (0,1].\label{asseq4}$$ Note that the semigroup proprieties in [[Proposition \[prop1\]]{}]{} are satisfied for the discrete operator $A_h$. For our convergence proof, we add the following propriety to the discrete operator $A_h$. \[prop:discreteoperators1\] Under [[Assumption \[ass:driftandB\]]{}]{} and [[Assumption \[assumptionn\]]{}]{}, the following proprieties are satisfied for discrete operators $$\begin{aligned} \label{eq:maincond1} \|(-A_h)^{\frac{\beta-1}{2}}P_hQ^{\frac{1}{2}}\|_{\mathcal{L}_2 (H)}&<& C + C_\beta h^{1-\beta},\text{ for some }\beta\in (0,1], \label{eq:n1}\\ \|(-A_h)^{-\beta }P_hF''(Z)(X_1,X_2)\|&\leq& C(1+h^{2\beta}) \|X_1\|\|X_2\|,\,\,Z,X_1, X_2\in V_h;\,\,\beta \in[0,1),\label{eq:nn3}\\ \|(-A_h)^{-\frac{\gamma}{2}}P_hF'(Z)(X)\|&\leq& C (h^{\gamma} + (1+\|Z\|_1)\|X\|_{-1}),\,\,Z,X\in V_h;\,\,\gamma\in [1,2),\label{eq:n3}\end{aligned}$$ According to ([@lions], Theorem 5.2) for all $\alpha>0,$ the discrete operator $A_h$ and the continuous operator $A$ satisfy $$\label{eq:lions} \|A_h^{-\alpha}P_h-A^{-\alpha}\|_{L(H)}\leq C_\alpha h^{2\alpha},$$ where $C_\alpha$ is a constant dependent on $\alpha.$ For $\alpha=-\frac{\beta-1}{2},$ with $\beta\in (0,1]$, we have $$\|(A_h^{-\alpha}P_h-A^{-\alpha})Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)} = \|A_h^{-\alpha}P_hQ^{\frac{1}{2}}-A^{-\alpha}Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)}$$ Now let $\{e_i\}_{i=1}^\infty$ be any orthonormal basis of $H$, since $\|A^{-\alpha}Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)}<\infty, $ according to [[Assumption \[ass:driftandB\]]{}]{} (relation ), we have using $$\begin{array}{rcl} \|A_h^{-\alpha}P_hQ^{\frac{1}{2}}-A^{-\alpha}Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)} &=&\sum\limits_{i=1}^\infty \|(A_h^{-\alpha}P_h-A^{-\alpha})Q^{\frac{1}{2}}e_i\|\\ &\leq & \|A_h^{-\alpha}P_h-A^{-\alpha}\|_{L(H)}\|Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)}\\ &\leq & C_\alpha h^{2\alpha}.\end{array}$$ Now we can write, using and , that $$\begin{array}{rcl}\|A_h^{-\alpha}P_hQ^{\frac{1}{2}}\|_{\mathcal{L}_2(H)} &=& \|A_h^{-\alpha}P_hQ^{\frac{1}{2}}-A^{-\alpha}Q^{\frac{1}{2}}+A^{-\alpha}Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)}\\ &\leq & \|A_h^{-\alpha}P_hQ^{\frac{1}{2}}-A^{-\alpha}Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)} + \|A^{-\alpha}Q^{\frac{1}{2}}\|_{\mathcal{L}_2(H)}\\ &\leq & C + C_\alpha h^{2\alpha}\\ &=&C + C_\alpha h^{1-\beta}.\end{array}$$ This proves . Now to prove , we proceed as above and the fact that is satisfied. Indeed for $\beta \in[0,1)$, using , we have $$\|(A_h^{-\beta}P_h-A^{-\beta})F''(Z)(X_1,X_2)\| \leq \|A_h^{-\alpha}P_h-A^{-\beta}\|_{L(H)}\| \|F''(Z)(X_1,X_2) \|\leq CL \, h^{2\beta}\,\|X_1\|\|X_2\|.$$ Therefore, $$\begin{array}{rcl} \|A_h^{-\beta}P_hF''(Z)(X_1,X_2) \| &\leq& \|(A_h^{-\beta}P_h-A^{-\beta})F''(Z)(X_1,X_2)\|+\|A^{-\beta}F''(Z)(X_1,X_2) \|\\ &\leq& C(1+h^{2\beta}) \|X_1\|\|X_2\|. \end{array}$$ The proof of is done just like for using and . \ The estimates in Proposition \[prop:discreteoperators1\] are very tight and can influence the order of convergence in space and time when $\beta$ in is small. Indeed using the fact that $$h < C({\rm meas} (\Omega))^{\frac{1}{d}},$$ where ${\rm meas} (\Omega)$ is the either the length, the area or the volume of the domain $\Omega$ and $d$ is the dimension, we have the following corollary. \[estimatesAhb\] Under [[Assumption \[ass:driftandB\]]{}]{} the following discrete proprieties are satisfied $$\begin{aligned} \label{eq:maincond} \|(-A_h)^{\frac{\beta-1}{2}}P_hQ^{\frac{1}{2}}\|_{\mathcal{L}_2(H)}&<& C,\,\, \quad \quad \beta\in (0,1].\\ \|(-A_h)^{-\beta}P_hF''(Z)(X_1,X_2)\|&\leq& C \|X_1\|\|X_2\|,\,\,Z,X_1, X_2\in V_h;\,\,\beta\in[0,1),\label{eq:3}\\ \|(-A_h)^{-\frac{\gamma}{2}}P_hF'(Z)(X)\|&\leq& C (1+\|Z\|_1)\|X\|_{-1},\,\,Z, X\in V_h;\,\,\gamma \in[1,2),\label{eq:9}\end{aligned}$$ where the positive constant $C$ is independent of $h$. The following lemma will be helpful for the proofs of convergence. \[lem\] Assume that all the conditions above are fulfilled and that $\Phi\in \mathcal{C}_2^b(H;\mathbb{R}).$ For $\gamma\in[0,1],\,\gamma_1,\gamma_2\in[0,1)$ satisfying $\gamma_1 + \gamma_2<1,$ there exists constants $c_\gamma$ and $c_{\gamma_1,\gamma_2}$ such that $$\begin{aligned} \label{eq42} \|(-A_h)^{\gamma}D\mu^h(t,\psi)\|&\leq &c_{\gamma}t^{-\gamma},\\ \label{eq43}\|(-A_h)^{\gamma_2}D^2\mu^h(t,\psi)(-A_h)^{\gamma_1}\|_{L(H)}&\leq &c_{\gamma_1,\gamma_2}(t^{-(\gamma_1+\gamma_2)}+1),\end{aligned}$$ where $\mu^h(t,\psi)$ is defined by , and $\psi \in V_h$. The proof is the same as in [@Wang2014]. Note that although the linear operator in [@Wang2014] is assumed to be self adjoint, the proof don’t make use of that. The following results can be proven exactly along the line of [@Wang2014 Lemma 3.4 and Lemma 3.5]). \[lem:boundonxmh\] Suppose [[Assumption \[assumption1\]]{}]{} is fulfilled and $X_0\in \mathcal{D}((-A)^{1/2}),$ then it holds for $\gamma \in [0,\frac{\beta}{2})$ and arbitrary small $\epsilon$ that $$\label{eq44} \underset{0\leq m \leq M}{\sup} \|(-A_h)^{\gamma}X_m^h\|_{L_2(\mathbb{D}, H)} \leq C \text{ and } \|\tilde X^h(t)-X^h_m\|_{L_2(\mathbb{D}, H)} \leq C\Delta t^{\frac{\beta-\epsilon}{2}},$$ where $X_m^h$ is defined by and $\tilde X^h$ is given in . Furthermore $$\label{eqbound} \|(-A_h)^{\frac{1}{2}}X_m^h\|_{L_2(\mathbb{D}, H)}\leq C(1+\Delta t^{\frac{\beta-1-\epsilon}{2}}),$$ for $\beta \in (0,1]$. Before the proof of the lemma, we can prove exactly as in [@Wang2014 (3.3)] that $$\|X_{k}^{h}\|_{L_2(\mathbb{D}, H)} <\infty, \,\,\qquad 0 \leq k\leq M,\,\, \underset{\qquad 0 \leq k\leq M} {\sup} \| P_h F(X_{k}^{h})\|_{L_2(\mathbb{D}, H)} <\infty.$$ We concentrate on proving , the proof of the other assertion can be done just as in [@Wang2014]. Recall that $$\begin{aligned} X_{m}^{h}&=&e^{\Delta t A_{h}}X_{m-1}^{h}+A_{h}^{-1}\left( e^{\Delta t A_{h}}-I\right)P_{h}F(X_{m-1}^{h})+e^{\Delta t A_{h}}B_h\left( W_{t_{m}}-W_{t_{m}-1}\right),\\ B_h&=&P_h Q^{1/2}P_h.\end{aligned}$$ This can also be written as $$\begin{aligned} X_{m}^{h}=S_h(\Delta t)X_{m-1}^{h}+\displaystyle\int_0^{\Delta t} S_h(\Delta t-s)P_{h}F(X_{m-1}^{h})ds+S_h(\Delta t)B_h\left( W_{t_{m}}-W_{t_{m}-1}\right).\end{aligned}$$ Iterating gives $$\begin{aligned} X_{m}^{h}=S_h(t_m)X_{0}^{h}+\sum_{k=0}^{m-1}\displaystyle\int_{t_k}^{t_{k+1}} S_h(t_m-s)P_{h}F(X_{k}^{h})ds+\sum_{k=0}^{m-1}S_h(t_m-t_k)B_{h}\Delta W_k.\end{aligned}$$ Now $$\begin{array}{rcl}\|(-A_h)^{\frac{1}{2}}X_m^h\|_{L_2(\mathbb{D}, H)}&\leq& \|(-A_h)^{\frac{1}{2}}S_h(t_m)X_{0}^{h}\|+\sum\limits_{k=0}^{m-1}\displaystyle\int_{t_k}^{t_{k+1}}\|(-A_h)^{\frac{1}{2}} S_h(t_m-s)P_{h}F(X_{k}^{h})\|_{L_2(\mathbb{D},H)}ds \\ && +\left(\Delta t\sum\limits_{k=0}^{m-1}\|(-A_h)^{\frac{1}{2}}S_h(t_m-t_k)B_{h}\|_{\mathcal{L}_2(H)}^2\right)^{\frac{1}{2}}\\ = I_0 + I_1 + I_2.\end{array}$$ We can prove exactly as in ([@Wang2014], Lemma 3.5) that $$\label{eq:resultwang}I_2\leq C\Delta t^{\frac{\beta -1-\epsilon}{2} }\text{ and } I_0\leq \|X_0\|_1<\infty.$$ Now it remain to prove that $I_1$ is bounded above. We have $$\begin{array}{rcl}I_1 &\leq& \sum\limits_{k=0}^{m-1}\displaystyle\int_{t_k}^{t_{k+1}}\|(-A_h)^{\frac{1}{2}} S_h(t_m-s)P_{h}F(X_{k}^{h})ds\|_{L_2(\mathbb{D}, H)}\\ &\leq & C\sum\limits_{k=0}^{m-1}\displaystyle\int_{t_k}^{t_{k+1}}(t_m-s)^{-\frac{1}{2}}(1+\|X_{k}^{h}\|_{L_2(\mathbb{D}, H)})ds\\ & \leq & C \left(\sum\limits_{k=0}^{m-1} \displaystyle\int_{t_k}^{t_{k+1}}(t_m-s)^{-\frac{1}{2}}ds\right)\left( \underset{0 \leq k\leq m-1}{\sup}\left(1+\|X_{k}^{h}\|_{L_2(\mathbb{D}, H)}\right) \right)\\ &=&C \left(\displaystyle \int_{0}^{t_{m}}(t_m-s)^{-\frac{1}{2}}ds\right)\left( \underset{0 \leq k\leq m-1}{\sup}\left(1+\|X_{k}^{h}\|_{L_2(\mathbb{D}, H)}\right) \right)\\ &\leq& C T^{1/2} \left( \underset{0 \leq k\leq m-1}{\sup}\left(1+\|X_{k}^{h}\|_{L_2(\mathbb{D}, H)}\right) \right) < \infty. \end{array}$$ Combining with establishes the lemma. \ Now we can prove the convergence result for the semi disrete problem with non-self-adjoint operator $A$ as the time step goes to zero. \[thm:mainadditive\] Assume that [[Assumption \[assumption1\]]{}]{} is fulfilled $X_0\in \mathcal{D}((-A)^{1/2})$ and that $\Phi\in \mathcal{C}_2^b(H;\mathbb{R}).$ Then for arbitrary small $\epsilon>0,$ if the adjoint of the discrete operator $A_h^*$ satisfies the analogue inequality as $A_h$ in [[Corollary \[estimatesAhb\]]{}]{}, we have $$|{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X^h(T))]|\leq C\Delta t^{\beta -\epsilon},$$ where the constant $C$ depends on $\beta, \,\epsilon,\, ...,T,\,L$ and the initial data, but is independent of $h$ and $M.$ We derived in the following error representation $$\label{eq:estimate0} \mathbb{E}[\Phi(X^h_M)]-\mathbb{E}[\Phi(X^h(T))] = \sum\limits_{m=0}^{M-1}(b_m^1 + b_m^2)$$ where the decompositions $b_m^1$ and $b_m^2$ are as follows $$\begin{array}{rcl} b_m^1&=&\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\left[\langle D\mu^h(T-s,\tilde X^h(s)), P_hF(X_m^h)-P_hF(\tilde X^h(s))\rangle_{H} \right]ds \end{array}$$ and $$\begin{array}{rcl} b_m^2&=& \frac{1}{2}{\mathbb{E}}\Big\{\displaystyle\int_{t_m}^{t_{m+1}}\text{Tr}\Big[D^2\mu^h(T-s,\tilde X^h(s))\Big(\left\{S_h(s-t_m) B_h\right\}\left\{(S_h(s-t_m)-I)B_h\right\}^*\Big)\Big]ds\Big\}\\ && +\frac{1}{2}{\mathbb{E}}\Big\{\displaystyle\int_{t_m}^{t_{m+1}}\text{Tr}\Big[D^2\mu^h(T-s,\tilde X^h(s))(S_h(s-t_m)-I)B_h B_h^*\Big]ds\Big\}\\ &=&b_m^{2,1} + b_m^{2,2}. \end{array}$$ Note that our term $b_m^1 $ is more simple than the one in [@Wang2014]. Applying the Taylor’s formula to the drift term $F$ we get $$F(\tilde X^h(s)) - F(X_m^h) = F'(X_m^h)(\tilde X^h(s) - X_m^h) + \int_0^1F''(\chi(r))(\tilde X^h(s)-X_m^h,\tilde X^h(s)-X_m^h)(1-r)dr,$$ where $\chi(r) = X_m^h + r(\tilde X^h(s)-X_m^h),$ which allows to have $$\begin{array}{rcl} \lefteqn{|b_m^1|}\nonumber\\ & \leq & \Big|\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[\Big\langle_H D\mu^h(T-s,\tilde X^h(s)), P_hF'(X_m^h)(\tilde X^h(s) - X_m^h)\Big\rangle_H \Big]ds \Bigg|\\ && + \Bigg|\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[\Big\langle D\mu^h(T-s,\tilde X^h(s)), \int_0^1P_hF''(\chi(r))(\tilde X^h(s)-X_m^h,\tilde X^h(s)-X_m^h)(1-r)dr\Big\rangle_H \Big]ds \Bigg| \\ &=& J_m^1 + J_m^2. \end{array}$$ To estimate $J_m^2,$ we use [[Corollary \[estimatesAhb\]]{}]{},[[Lemma \[lem\]]{}]{}, [[Lemma \[lem:boundonxmh\]]{}]{} and the Holder inequality $$\begin{array}{rcl} J_m^2 & \leq & \displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[\Big|\Big\langle (-A_h)^{-\delta}(-A_h)^\delta D\mu^h(T-s,\tilde X^h(s)), \\ && \hskip 3cm \displaystyle\int_0^1P_hF''(\chi(r))(\tilde X^h(s)-X_m^h,\tilde X^h(s)-X_m^h)(1-r)dr\Big\rangle_H\Big| \Big]ds\\ &=& \displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[\Big|\Big\langle (-A_h)^\delta D\mu^h(T-s,\tilde X^h(s)),\\ && \hskip 2cm \displaystyle\int_0^1(-A_h^*)^{-\delta}P_hF''(\chi(r))(\tilde X^h(s)-X_m^h,\tilde X^h(s)-X_m^h)(1-r)dr\Big\rangle_H\Big| \Big]ds\\ &\leq& \displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[\Big\|(-A_h)^\delta D\mu^h(T-s,\tilde X^h(s))\Big\|\times \\ && \hskip 2cm \Big\| \displaystyle\int_0^1(-A_h^*)^{-\delta}P_hF''(\chi(r))(\tilde X^h(s)-X_m^h,\tilde X^h(s)-X_m^h)(1-r)dr\Big\| \Big]ds\\ &\leq & c_\delta \displaystyle\int_{t_m}^{t_{m+1}}\displaystyle\int_0^1{\mathbb{E}}\Big[\Big\|(-A_h^*)^{-\delta}P_hF''(\chi(r))(\tilde X^h(s)-X_m^h,\tilde X^h(s)-X_m^h)\Big\|\Big](T-s)^{-\delta}\,dr\,ds\\ &\leq & c_\delta L \displaystyle\int_{t_m}^{t_{m+1}}\displaystyle\int_0^1{\mathbb{E}}\Big[\Big\|\tilde X^h(s)-X_m^h\Big\|^2\big](T-s)^{-\delta}\,dr\,ds\\ &\leq & C\Delta t^{\beta-\epsilon}\displaystyle\int_{t_m}^{t_{m+1}}(T-s)^{-\delta}\,ds. \end{array}\label{eq:estimate2}$$ We now turn to estimate $J_m^1.$ Recall that from , $$\label{eqxtildemx} \begin{array}{rcl} \tilde X^h(s)-X_m^h &= &\big(S_h(s-t_m)-I\big)X_m^h + (s-t_m)\varphi_1^h(s-t_m)P_hF(X_m^h) \\ && \hskip 2cm + S_h(s-t_m) B_h(W(s)-W(t_m)).\end{array}$$ Since the expectation of the Brownian motion vanishes, we therefore have $$\begin{array}{rcl} J_m^1 &\leq & \Big|\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[\Big\langle D\mu^h(T-s,\tilde X^h(s)), P_hF'(X_m^h)\big(S_h(s-t_m)-I\big)X_m^h\Big\rangle_H \Big]ds \Bigg|\\ && +\Big|\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[\Big\langle D\mu^h(T-s,\tilde X^h(s)),P_hF'(X_m^h)(S_h^1(s-t_m)P_hF(X_m^h)(s-t_m)\Big\rangle_H \Big]ds \Bigg|\\ &= & I + II. \end{array}$$ Using [[Lemma \[lem\]]{}]{},[[Corollary \[estimatesAhb\]]{}]{}, [[Lemma \[lem:boundonxmh\]]{}]{} and [[Proposition \[prop:discreteoperators1\]]{}]{} yields $$\begin{array}{rcl} \lefteqn{I}\nonumber\\ &\leq & c_{\frac{\delta}{2}}\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[ \Big\|(-A_h^*)^{-\frac{\delta}{2}}P_hF'(X_m^h)\big(S_h(s-t_m)-I\big)X_m^h\Big\|\Big](T-s)^{-\frac{\delta}{2}}ds\\ &\leq & C\displaystyle\int_{t_m}^{t_{m+1}}\Big[ (1+\|X^h_m\|_1)|\big\|(S_h(s-t_m)-I)X_m^h\Big\|_1\Big](T-s)^{-\frac{\delta}{2}}ds\\ &\leq & \displaystyle\int_{t_m}^{t_{m+1}}\Big[ 1+ {\mathbb{E}}\|(-A_h)^{1/2}X^h_m\| \Big] {\mathbb{E}}\| (-A_h)^{-\frac{1+\beta-\epsilon}{2}}) S_h(s-t_m)-I)(-A_h)^{\frac{\beta-\epsilon}{2}} X_m^h \| (T-s)^{-\frac{\delta}{2}}ds \\ &\leq & \displaystyle\int_{t_m}^{t_{m+1}}\Big[1+\Delta t^{\frac{\beta-1-\epsilon}{2}}\Big] (s-t_m)^{\frac{1+\beta-\epsilon}{2}} {\mathbb{E}}\|(-A_h)^{\frac{\beta-\epsilon}{2}} X_m^h \| (T-s)^{-\frac{\delta}{2}}ds \\ &\leq & C\Delta t^{\beta -\epsilon}\displaystyle\int_{t_m}^{t_{m+1}}(T-s)^{-\frac{\delta}{2}}ds.\\ \end{array}\label{eq:estimate3}$$ Now we turn to approximate the term $II.$ Note that we can rewrite as $$\begin{array}{rcl} \tilde X^h(s)-X_m^h &= &\big(S_h(s-t_m)-I\big)X_m^h + \displaystyle\int_{t_m}^s S_h(t-s)P_hF(X_m^h)dt \\ && \hskip 2cm + S_h(s-t_m)B_h(W(s)-W(t_m)).\end{array}$$ Therefore $$\begin{array}{rcl} II&=& \Big|\displaystyle\int_{t_m}^{t_{m+1}}{\mathbb{E}}\Big[\Big\langle D\mu^h(T-s,\tilde X^h(s)), \displaystyle\int_{t_m}^s P_hF'(X_m^h)S_h(t-s)P_hF(X_m^h)dt\Big\rangle_H \Big]ds \Bigg|\\ &\leq & \displaystyle\int_{t_m}^{t_{m+1}}\displaystyle\int_{t_m}^s{\mathbb{E}}\Big[\|D\mu^h(T-s,\tilde X^h(s))\| \| P_hF'(X_m^h)S_h(t-s)P_hF(X_m^h)\|\Big]dtds\\ &\leq & C \displaystyle\int_{t_m}^{t_{m+1}}\displaystyle\int_{t_m}^s{\mathbb{E}}\Big[\|S_h(t-s)P_hF(X_m^h)\|\Big]dtds\\ &\leq & C{\mathbb{E}}[\|P_hF(X_m^h)\|]\displaystyle\int_{t_m}^{t_{m+1}}\displaystyle\int_{t_m}^sdtds\\ &\leq & C\Delta t^2. \end{array}\label{eq:estimate4}$$ Combining , and , we have $$\begin{aligned} |b_m^1| \leq C\Delta t ^{\beta-\epsilon}\displaystyle\int_{t_m}^{t_{m+1}}(T-s)^{-\frac{\delta}{2}}ds +C\Delta t^2\\ \sum\limits_{m=0}^{M-1} |b_m^1| \leq C\Delta t ^{\beta-\epsilon}\end{aligned}$$ The term $b_m^2$ can be approximated just as in [@Wang2014] to be $$\label{eq:estimatebm2} \sum\limits_{m=0}^{M-1} |b_m^2|\leq C \Delta t ^{\beta-\epsilon}.$$ Finally we therefore have $$|{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X^h(T))]|\leq C\Delta t^{\beta -\epsilon}.$$ Weak convergence for a SPDE with multiplicative noise {#sec4} ===================================================== Now we consider the case of a stochastic partial differential equation with multiplicative noise, that is For convergence proof, we make the following assumption on the noise term. \[noisemu\] We assume that there exists a constant $\alpha>0$ such that $$\|(-A)^{\frac{\beta-1}{2}}B(X)\|_{\mathcal{L}_2(H)}\leq C(1+\|X\|_{\beta-1}),\,\, \text{ for all } X\in H,\,\, \beta \in [0,1].$$ One can prove just as in the proof of Proposition \[prop:discreteoperators1\] that the discrete operators $A_h$ and $P_hB(X)$ satisfies $$\label{eq:condmultiplicative} \|(-A_h)^{\frac{\beta-1}{2}}P_hB(X)\|_{\mathcal{L}_2(H)}\leq Ch^{1-\beta}(1+\|X\|_{\beta-1})\leq C'(1+\|X\|_{\beta-1}),\,\, \text{ for all } X\in H.$$ For $\beta \in [0,1]$, we also have using [[Proposition \[proposition\]]{}]{} $$\begin{aligned} \label{eq:condmultiplicative1} \|(-A_h)^{\frac{\beta-1}{2}}B_h(X)\|_{\mathcal{L}_2(H)} &=&\|(-A_h)^{\frac{\beta-1}{2}}P_hB(X)P_h\|_{\mathcal{L}_2(H)}\nonumber\\ &\leq& C \|(-A_h)^{\frac{\beta-1}{2}}P_hB(X)\|_{\mathcal{L}_2(H)}\nonumber\\ &\leq& C'(1+\|X\|_{\beta-1}), \nonumber\\ &\leq& C'(1+\|X\|),\,\, \text{ for all } X\in H. \end{aligned}$$ Thanks to Lemma \[lem:boundonxmh\] and (or ) all the results presented above for additive noise also hold for multiplicative noise and we then have the following convergence result. \[thm:multiplicative\] Assume that all the conditions of Theorem \[thm:mainadditive\] are satisfied with the condition replaced by . For $X_0\in \mathcal{D}((-A)^{1/2}), \quad\Phi\in \mathcal{C}_2^b(H;\mathbb{R})$ and for arbitrary small $\epsilon>0,$ if the adjoint of the discrete operator $A_h^*$ satisfies the analogue inequality as $A_h$ in [[Corollary \[estimatesAhb\]]{}]{}, we have the following weak error convergence rate $$|{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X^h(T))]|\leq C\Delta t^{\beta-\epsilon},$$ where the constant $C$ depends on $\beta, \,\epsilon,\, ...,T,\,L$ and the initial data, but is independent of $h$ and $M.$ The proof follows the same lines as that of [[Theorem \[thm:mainadditive\]]{}]{} $$\mathbb{E}[\Phi(X^h_M)]-\mathbb{E}[\Phi(X^h(T))] = \sum\limits_{m=0}^{M-1}(b_m^{1} + b_m^{2}).$$ The term $b_m^{1}$ is the same as in [[Theorem \[thm:mainadditive\]]{}]{}. Here $ b_m^{2}$ is given by $$\begin{array}{rcl} b_m^2&=& \frac{1}{2}{\mathbb{E}}\Big[\displaystyle\int_{t_m}^{t_{m+1}} \text{Tr}\Big[D^2\mu^h(T-s,\tilde X^h(s))\left (S_h(s-t_m) B_h(X_m^h)\right) \\ && \hskip 5cm \times \left((S_h(s-t_m)-I)B_h(X_m^h)\right)^*\Big]ds\Big]\\ &&+ \frac{1}{2}{\mathbb{E}}\Big [\displaystyle\int_{t_m}^{t_{m+1}}\text{Tr}\Big[D^2\mu^h(T-s,\tilde X^h(s))(S_h(s-t_m)-I) \left(B_h(X_{m}^{h})\right) \left(B_h(X_{m}^{h})\right)^*\Big]ds\Big]\\ &=&b_m^{2,1} + b_m^{2,2}. \end{array}$$ Let us estimate $b_m^{2,1}$. [[Proposition \[proposition\]]{}]{} allows to have $$\begin{array}{rcl} \lefteqn{ b_m^{2,1}}\nonumber\\ &\leq& \frac{1}{2}{\mathbb{E}}\Big [\displaystyle\int_{t_m}^{t_{m+1}} \Big\|D^2\mu^h(T-s,\tilde X^h(s))\left [S_h(s-t_m) B_h(X_m^h)\right] \\ && \hskip 5cm\times \left[(S_h(s-t_m)-I)B_h(X_m^h)\right]^*\Big\|_{\mathcal{L}_1(H)}ds\Big]\\ &= & \frac{1}{2}{\mathbb{E}}\Big [\displaystyle\int_{t_m}^{t_{m+1}} \Big\|(-A_h)^{\frac{\beta+1}{2}-\epsilon}D^2\mu^h(T-s,\tilde X^h(s))(-A_h)^{\frac{1-\beta}{2}} (-A_h)^{\frac{\beta-1}{2}}\left [S_h(s-t_m) B_h(X_m^h)\right ]\\ && \hskip 5cm\times \left[(-A_h)^{-\frac{\beta+1}{2}+\epsilon}(S_h(s-t_m)-I)B_h(X_m^h)\right ]^*\Big\|_{\mathcal{L}_1(H)}ds\Big]\\ \end{array}$$ Using [[Lemma \[lem\]]{}]{},[[Proposition \[proposition\]]{}]{},[[Proposition \[prop1\]]{}]{} and yields the following estimations $$\begin{array}{rcl} \lefteqn{b_m^{2,1}} \nonumber\\ &\leq & C {\mathbb{E}}\Big [\displaystyle\int_{t_m}^{t_{m+1}} \Big\|(-A_h)^{\frac{\beta-1}{2}}S_h(s-t_m) B_h(X_m^h)\left[(-A_h)^{\frac{\beta+1}{2}-\epsilon}(S_h(s-t_m)-I)B_h(X_m^h)\right]^*\Big\|_{\mathcal{L}_1(H)} \\ && \hskip 5cm\times [(T-s)^{-1+\epsilon}+1]ds\Big ]\\ &\leq & C{\mathbb{E}}\Big [ \displaystyle\int_{t_m}^{t_{m+1}} \Big\|(-A_h)^{\frac{\beta-1}{2}}S_h(s-t_m) B_h(X_m^h)\Big\|_{\mathcal{L}_2(H)}\\ && \times \Big\|(-A_h)^{-\frac{\beta+1}{2}+\epsilon}(S_h(s-t_m)-I)B_h(X_m^h)\Big\|_{\mathcal{L}_2(H)} [(T-s)^{-1+\epsilon}+1]ds\Big]\\ &\leq & C{\mathbb{E}}\Big [\displaystyle\int_{t_m}^{t_{m+1}} \Big\|(-A_h)^{\frac{\beta-1}{2}}B_h(X_m^h)\Big\|_{\mathcal{L}_2(H)}\\ && \times \Big\|(-A_h)^{-\beta+\epsilon}(S_h(s-t_m)-I)(-A_h)^{\frac{\beta-1}{2}}B_h(X_m^h)\Big\|_{\mathcal{L}_2(H)} [(T-s)^{-1+\epsilon}+1]ds\Big]\\ &\leq & C\Big [\displaystyle\int_{t_m}^{t_{m+1}} \underset { 0 \le i \leq M }{\sup}{\mathbb{E}}(1+\|X_i^h\|)^2\cdot \Big\|(-A_h)^{-\beta+\epsilon}(S_h(s-t_m)-I)\Big\|_{L(H)} [(T-s)^{-1+\epsilon}+1]ds\Big]\\ &\leq & C\Delta t^{\beta-\epsilon}\displaystyle\int_{t_m}^{t_{m+1}} (T-s)^{-1+\epsilon}ds + C\Delta t^{\beta-\epsilon+1 }. \end{array}$$ Similar as for $b_m^{2,2}$, we have $$\begin{array}{rcl} \lefteqn{b_m^{2,2}} \nonumber\\ & =& \frac{1}{2}{\mathbb{E}}\Big [\displaystyle\int_{t_m}^{t_{m+1}}\text{Tr}\Big[ (-A_h)^{\frac{1-\beta}{2}}D^2\mu^h(T-s,\tilde X^h(s))(-A_h)^{\frac{\beta+1}{2}-\epsilon} \\ && \times (-A_h)^{-\frac{\beta+1}{2}+\epsilon}(S_h(s-t_m)-I) \left(B_h(X_{m}^{h}) \right) \left((-A_h)^{\frac{\beta-1}{2}} B_h(X_{m}^{h})\right)^*\Big]ds\Big]\\ &\leq & C {\mathbb{E}}\Big [\displaystyle\int_{t_m}^{t_{m+1}} \Big\|(-A_h)^{-\beta+\epsilon}\left(S_h(s-t_m)-I\right) (-A_h)^{\frac{\beta-1}{2}}B_h(X_m^h)\left[(-A_h)^{\frac{\beta-1}{2}}B_h(X_m^h)\right]^*\Big\|_{\mathcal{L}_1(H)} \\ && \hskip 5cm\times [(T-s)^{-1+\epsilon}+1]ds\Big ]\\ &\leq& C\Delta t^{\beta-\epsilon}\displaystyle\int_{t_m}^{t_{m+1}} (T-s)^{-1+\epsilon}ds + C\Delta t^{\beta-\epsilon+1}. \end{array}$$ So by summing up as in [[Theorem \[thm:mainadditive\]]{}]{}, the proof is ended. Strong convergence and toward full weak convergence results {#sec5} =========================================================== The goal here is to provide the space and time convergence proof of the exponential scheme. Before that we will provide strong convergence of the semi discrete solution. \[strong\] Let $X^h$ and $\overline{X}_h$ be the solutions respectively of and the following semi discrete problem $$\begin{aligned} \label{dadrnt} d\overline{X}_h&=&(A_{h}\overline{X}_h +P_{h}F(\overline{X}_h))dt + P_{h} B(\overline{X}_h)d W\\ \overline{X}_h(0)&=&P_{h}X_{0} \nonumber.\end{aligned}$$ Let $ \beta \in [0,2)$, assume that [[Assumption \[assumption1\]]{}]{} and [[Assumption \[assumption2\]]{}]{} are satisfied. For $\beta \in [0,1]$ assume that the relation of [[Assumption \[ass:driftandB\]]{}]{} (when dealing with additive noise) and [[Assumption \[noisemu\]]{}]{} (when dealing with multiplicative noise) are satisfied. For $\beta \in [1,2)$ assume that $B(\mathcal{D}((-A)^{\frac{\beta-1}{2}}))\subset HS\left(Q^{1/2}(H),\mathcal{D}((-A)^{\frac{\beta-1}{2}})\right)$ and $\Vert (-A)^{\frac{\beta-1}{2}}B(v)\Vert_{L_{0}^{2}}\leq c(1+\Vert v\Vert_{\beta-1})$ for $v \in \mathcal{D}((-A)^{\frac{\beta-1}{2}})$. If $X_0 \in L_2(\mathbb{D},\mathcal{D}((-A)^{\beta/2})) $, there exists a positive constant $C$ independent of $h$ such that the following estimations hold: $$\begin{aligned} \label{noseleq} \Vert X(t)-\overline{X}_{h}(t)\Vert_{L_2(\mathbb{D},H)} \leq Ch^{\beta},\;\;\,\, \beta \in [0,1]. \end{aligned}$$ Furthermore assume that the linear operator $A$ is self adjoint, the following estimation hold $$\begin{aligned} \label{seleq} \Vert X(t)-X^{h}(t)\Vert_{L_2(\mathbb{D},H)} \leq Ch^{\beta}, \,\;\; \beta \in [0,2). \end{aligned}$$ Before prove [[Theorem \[strong\]]{}]{} let us make some remarks and provide some preparatory results. [[Theorem \[strong\]]{}]{} extends [@kruse Theorem 1.1] for non-self-adjoint operator $A$ and also provide optimal convergence proof for $\beta \in [0,1)$ which was not studied in [@kruse]. \[troncate\] For additive noise, we can observe from [@stigstrong] that the semi discrete problem is equivalent to the following problem, find $X_h(t) \in V_h$ such that $$\begin{aligned} \label{dadrntt} d X_h&=&(A_{h} X^h +P_{h}F(X^h))dt + d W_h\\ X^h(0)&=&P_{h}X_{0} \nonumber.\end{aligned}$$ where $W_h(t)$ is a $P_{h}QP_{h}$-Wiener process on $V_h$ with the following representation $$\begin{aligned} \label{eq:Wh} W_h(t)=\underset{ i=1}{\sum^{N_h}}\sqrt{q_{h,i}}e_{h,i}\beta_{i}(t),\end{aligned}$$ where $(q_{h,i},e_{h,i})$ are the eigenvalues and eigenfunctions of the covariance operator $Q_h:=P_{h}QP_{h}$ and $\beta_{i}$ are independent and identically distributed standard Brownian motions. More precisely $(q_{h,i},e_{h,i})$ is the finite element solution of the eigenvalue problem $Qu=\gamma u$. If the exact eigenvalues and eigenfunctions $(q_{i},e_{i})$ of the covariance operator $Q$ are known, replacing in (or ) $W_h(t)$ by $W_h^{N_h}(t)$, defined by $$\begin{aligned} \label{eq:Whh} W_h^{N_h}(t)=\underset{ i=1}{\sum^{N_h}}\sqrt{q_{i}}e_{i}\beta_{i}(t),\end{aligned}$$ will not necessarily change the optimal convergence order in our scheme. From [@stigstrong], it is also proved that if the kernel of the covariance function $Q$ is regular and the mesh family is quasi-uniform, it is enough to take $M < N_h$ noise terms in (or ) without loss the optimal convergence order. Of course for multiplicative noise $P_h W$ can also be expanded on the basis of $V_h$ with $N_h$ terms (see [@Yn:05]). As we are also dealing in [[Theorem \[strong\]]{}]{} with non-self-adjoint operator in , let us provide some preparatory results before giving the proof of [[Theorem \[strong\]]{}]{}. We introduce the Riesz representation operator $R_{h}: V \rightarrow V_{h}$ defined by $$\begin{aligned} (-A R_{h}v,\chi)=(-A v,\chi)=a(v,\chi),\qquad \qquad v \in V,\; \forall \chi \in V_{h}.\end{aligned}$$ Under the usual regularity assumptions on the triangulation and in view of $V-$ellipticity , it is well known (see [@lions]) that the following error bounds holds $$\begin{aligned} \label{regulard} \Vert R_{h}v-v\Vert +h \Vert R_{h}v-v\Vert_{H^{1}(\Omega} \leq C h^{r} \Vert v\Vert_{H^{r}(\Omega)}, \qquad v\in V\cap H^{r}(\Omega),\; \; r \in \{1,2\}. \end{aligned}$$ By interpolation, we have $$\begin{aligned} \label{regulard} \Vert R_{h}v-v\Vert +h \Vert R_{h}v-v\Vert_{H^{1}(\Omega} \leq C h^{r} \Vert v\Vert_{r}, \qquad v \in \mathcal{D}(A^{r/2}),\; \; 1 \leq r \leq 2. \end{aligned}$$ Let us consider the following deterministic problem, which consists of finding $u \in V$ such that such that $$\begin{aligned} \label{homog} u'=Au \qquad \text{given} \quad u(0)=v,\qquad t\in (0,T] .\end{aligned}$$ The corresponding semi-discretization in space is : Find $u_{h} \in V_{h}$ such that $$u_{h}'=A_{h}u_{h}$$ where $u_{h}^{0}=P_{h}v$. Define the operator $$\begin{aligned} \label{form1} T_{h}(t) := S(t)-S_{h}(t) P_{h} = e^{tA} - e^{tA_h}P_h\end{aligned}$$ so that $u(t)-u_{h}(t)= T_{h}(t) v$. The following lemma will be important in our proof. \[lemme11\] The following estimates hold on the semi-discrete approximation of. There exists a constant $C>0$ such that - \(i) For $v \in \mathcal{D}((-A)^{\gamma/2})$ $$\begin{aligned} \label{form4} \Vert u(t)-u_{h}(t)\Vert &=&\Vert T_{h}(t) v\Vert \leq C h^{r} t^{-(r-\gamma)/2}\Vert v \Vert_{\gamma},\;1\leq r \leq 2, \, \;\; 0\leq \gamma \leq r.\end{aligned}$$ - \(ii) For $v \in \mathcal{D}((-A)^{(\gamma-1)/2})$ $$\begin{aligned} \left(\int_0^{t} \Vert T_{h}(s) v\Vert^{2}ds \right)^{\frac{1}{2}} \leq Ch^{\gamma}\Vert v\Vert_{\gamma-1},\,\, 0 \leq \gamma\leq 2.\end{aligned}$$ The proof of (i) can be found in [@GTambueexpoM] using . For self adjoint operator, the proof of (ii) is done as in [@Yn:04 Lemme 4.1] if $\gamma \in [0,1]$ and in[@kruse Lemma 4.2] if $\gamma \in [1,2)$ and the parameter $r$ used in [@kruse Lemma 4.2] is $r=\gamma-1$. Both proofs only uses general concepts and not spectral decomposition of the linear operator $A$, so can easily be generalized. Note that the non-self adjoint case should make use of [@vidar 4.17] or [@vidar Lemma 4.3] instead of [@vidar 2.29] used in [@kruse Lemma 4.2]. Let us now provide the proof of [[Theorem \[strong\]]{}]{}. The proof of the estimation for additive noise can be found in [@stigstrong] and can be updated to multiplicative noise following [@Yn:05]. For $\beta \in [1,2)$, the proof for the estimation when $A$ is self adjoint operator can be found in [@kruse] as the [[Assumption \[assumption1\]]{}]{} implies [@kruse Assumption 2.1] by taking $r=\beta-1$ in [@kruse Theorem 1.1]. Let us give more general proof by closely follow [@stigstrong; @kruse]. The corresponding mild solution of is given by $$\begin{aligned} \overline{X}_h(t)= S_h(t) P_hX_0+ \int_{0}^{t}S_h(t-s)P_hF(\overline{X}_h(s))ds +\int_{0}^{t} S_h(t-s)P_h B(\overline{X}_h(s))d W(s).\end{aligned}$$ Indeed, we have $$\begin{aligned} \lefteqn{\Vert X(t)-\overline{X}_{h}(t)\Vert_{L_2(\mathbb{D},H)}} \nonumber\\ &\leq& \Vert T_h(t)P_hX_0+ \int_{0}^{t}S(t-s)F(X(s))ds -\int_{0}^{t}S_h(t-s)P_hF(\overline{X}_h(s))ds\Vert_{L_2(\mathbb{D},H)} \nonumber\\ && + \Vert \int_{0}^{t} S(t-s)B(X(s))d W(s)-\int_{0}^{t} S_h(t-s)P_h B(\overline{X}_h(s))d W(s) \Vert_{L_2(\mathbb{D},H)}\nonumber\\ &=& I_1 +I_2.\end{aligned}$$ The estimation of $I_1$ is the same as in [@stigstrong] and we have $$\begin{aligned} I_1 \leq C \int_{0}^{t} \Vert X(s)-\overline{X}_{h}(s)\Vert_{L_2(\mathbb{D},H)} ds +C h^{\beta}.\end{aligned}$$ For the estimation of $I_2$, we follow closely [@kruse]. Indeed we have $$\begin{aligned} I_2 &\leq& C \Big ( \mathbb{E} \Big [ \int_{0}^{t} \Vert S(t-s)B(X(s))-S_h(t-s)P_h B(\overline{X}_{h}(s)) \Vert_{\mathcal{L}_2(H)}^2 ds \Big ] \Big)^{1/2}\\ &\leq& C \Big \Vert \Big ( \int_{0}^{t} \Vert S_h(t-s)P_h ( B(\overline{X}_{h}(s))-B(X(s)))\Vert_{\mathcal{L}_2(H)}^2 ds \Big )^{1/2} \Big \Vert_{L_2(\mathbb{D},\mathbb{R})}\\ &&+ C \Big \Vert \Big ( \int_{0}^{t} \Vert T_h(t-s)( B(X(s))-B(X(t)))\Vert_{\mathcal{L}_2(H)}^2 ds \Big )^{1/2} \Big \Vert_{L_2(\mathbb{D},\mathbb{R})}\\ &&+ C \Big \Vert \Big ( \int_{0}^{t} \Vert T_h(t-s)B(X(t)\Vert_{\mathcal{L}_2(H)}^2 ds \Big )^{1/2} \Big \Vert_{L_2(\mathbb{D},\mathbb{R})}\\ &=& I_2^{1}+ I_2^{2}+ I_2^{3}. \end{aligned}$$ The stability propriety of the semi group [[Proposition \[prop1\]]{}]{} and the Lipschitz condition in [[Assumption \[assumption1\]]{}]{} allow to have $$\begin{aligned} I_2^{1} \leq C \left( \int_{0}^{t} \Vert X(s)-\overline{X}_{h}(s)\Vert_{L_2(\mathbb{D},H)}^2 ds\right)^{\frac{1}{2}}.\end{aligned}$$ Following closely [@kruse], but with in [[Lemma \[lemme11\]]{}]{} with $r= \beta, \, \gamma=0$, allow to have $$\begin{aligned} I_2^{2} \leq C h^{\beta}.\end{aligned}$$ For $\beta \in [0,2)$, as in [@kruse], by using (ii) in [[Lemma \[lemme11\]]{}]{} gives $$\begin{aligned} I_2^{3} \leq C h^{\beta}.\end{aligned}$$ Coming back to $I_2$, we have $$\begin{aligned} &&\Vert \int_{0}^{t} S(t-s)B(X(s))d W(s)-\int_{0}^{t} S_h(t-s)P_h B(\overline{X}_h(s))d W(s) \Vert_{L_2(\mathbb{D},H)} \\ &\leq& C h^{2-\epsilon} +C \left( \int_{0}^{t} \Vert X(s)-\overline{X}_{h}(s)\Vert_{L_2(\mathbb{D},H)}^2 ds\right)^{\frac{1}{2}}\\ &\leq& C h^{\beta} +C \left( \int_{0}^{t} \Vert X(s)-\overline{X}_{h}(s)\Vert_{L_2(\mathbb{D},H)}^2 ds\right)^{\frac{1}{2}}.\end{aligned}$$ Combining $I_1$ and $I_2$ gives $$\begin{aligned} \Vert X(t)-\overline{X}_{h}(t)\Vert_{L_2(\mathbb{D},H)}^2 &\leq& Ch^{2 \beta} +C \int_{0}^{t} \Vert X(s)-\overline{X}_{h}(s)\Vert_{L_2(\mathbb{D},H)}^2 ds.\end{aligned}$$ Gronwall’s lemma is therefore applied to end the proof. The following theorem provide the full weak convergence when the solution is regular enough. \[fullweak\] Let $X$ and $X_M^h$ be respectively the solution of and the numerical solution from at the final time $T$. Let $ \beta \in [1,2)$, assume that [[Assumption \[assumption1\]]{}]{} (for multiplicative noise, all conditions except ) and [[Assumption \[assumption2\]]{}]{} (for multiplicative noise) are satisfied. For $\beta=1$ (trace class noise) assume of [[Assumption \[ass:driftandB\]]{}]{} (when dealing with additive noise) and [[Assumption \[noisemu\]]{}]{} (when dealing with multiplicative noise) are satisfied. For $\beta \in (1,2)$ assume that of [[Assumption \[ass:driftandB\]]{}]{} is also satisfied for additive noise, but $B(\mathcal{D}((-A)^{\frac{\beta-1}{2}}))\subset HS\left(Q^{1/2}(H),\mathcal{D}((-A)^{\frac{\beta-1}{2}/2})\right)$ and, $\Vert (-A)^{\frac{\beta-1}{2}}B(v)\Vert_{L_{0}^{2}}\leq c(1+\Vert v\Vert_{\beta-1})$ for $v \in \mathcal{D}((-A)^{\frac{\beta-1}{2}})$ for multiplicative noise. Furthermore assume that $X_0\in \mathcal{D}((-A)^{\beta/2})$ and the linear operator is selfadjoint. For $\Phi\in \mathcal{C}_2^b(H;\mathbb{R})$ and arbitrary small $\epsilon>0,$ the following estimation hold $$|{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X(T)]|\leq C(\Delta t^{1-\epsilon}+h^\beta),$$ where the constant $C$ depends on $\alpha, \,\epsilon,\, ...,T,\,L$ and the initial data, but is independent of $h$ and $M.$ Indeed we have the following decomposition $$\begin{aligned} |{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X(T)]| \leq|{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X^h(T)]|+|{\mathbb{E}}[\Phi(X^h(T)] - {\mathbb{E}}[\Phi(X(T)]|. \end{aligned}$$ Note that if $X_0\in \mathcal{D}((-A)^{\beta/2}),\, 1 \leq \beta< 2$, then $X_0\in \mathcal{D}((-A)^{1/2})$ and all the lemma used in [[Theorem \[thm:mainadditive\]]{}]{} and [[Theorem \[thm:multiplicative\]]{}]{} are still valid. From [[Theorem \[thm:mainadditive\]]{}]{} and [[Theorem \[thm:multiplicative\]]{}]{} with $\beta=1$ (as the noise is assumed to be trace class), we have $$\begin{aligned} |{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X^h(T)]|\leq C\Delta t^{1-\epsilon}.\end{aligned}$$ Note that this temporal order is the double of the strong convergence obtained in [@GTambueexpoM]. Remember that we are using low order finite element method for space discretization, where the optimal convergence is 2 (for deterministic case), therefore for $X_0\in \mathcal{D}((-A)^{\beta/2}),\, 1 < \beta< 2$, according to [[Theorem \[strong\]]{}]{} and [[Remark \[troncate\]]{}]{}, we cannot expect the order of weak convergence to double the strong order in space. The weak convergence order in space (of course, not necessarily optimal) can be the same as the strong convergence order when the solution is regular enough. Using the fact that $\Phi\in \mathcal{C}_2^b(H;\mathbb{R})$, so is Lipschitz, we therefore have $$\begin{aligned} |{\mathbb{E}}[\Phi(X^h(T)] - {\mathbb{E}}[\Phi(X(T)]|&\leq& |{\mathbb{E}}[\Phi(X^h(T))- \Phi(X(T))]| \\ &\leq& {\mathbb{E}}\Vert X^h(T)-X(T)\Vert \\ & \leq & C \Vert X^h(T)-X(T)\Vert_{L_2(\mathbb{D},H)}\\ & \leq & Ch^{\beta}. \end{aligned}$$ The following theorem using recent result in the literature shows that the space order of convergence in [[Theorem \[fullweak\]]{}]{} is far to be optimal for $\beta=1$. \[opfullaplace\] Assume that $A=\varDelta$ with Dirichlet boundary condition ($V=H_0^1(\Omega)$), assume that the noise is additive, [[Assumption \[ass:driftandB\]]{}]{} is satisfied with $\Vert (-A)^{\frac{\beta-1}{2}}Q^{1/2}\Vert_{\mathcal{L}_{2}(H)}<\infty$ for some $\beta \in [1/2,1]$. Furthermore assume that $X_0\in \mathcal{D}((-A)^{\beta})$. For $\Phi\in \mathcal{C}_2^b(H;\mathbb{R})$ and arbitrary small $\epsilon>0,$ the following estimation hold $$|{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X(T)]|\leq C(\Delta t^{\beta-\epsilon}+h^{2\beta-\epsilon}),$$ where the constant $C$ depends on $\alpha, \,\epsilon,\, ...,T,\,L$ and the initial data, but is independent of $h$ and $M.$ Let us prove [[Theorem \[opfullaplace\]]{}]{}. The proof follows the one for [[Theorem \[fullweak\]]{}]{} and we have $$\begin{aligned} |{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X(T)]| \leq|{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X^h(T)]|+|{\mathbb{E}}[\Phi(X^h(T)] - {\mathbb{E}}[\Phi(X(T)]|. \end{aligned}$$ If $X_0\in \mathcal{D}((-A)^{\beta}),\, 1/2 \leq \beta\leq 1$, then $X_0\in \mathcal{D}((-A)^{1/2})$ and all the lemma used in [[Theorem \[thm:mainadditive\]]{}]{} and [[Theorem \[thm:multiplicative\]]{}]{} are still valid. From [[Theorem \[thm:mainadditive\]]{}]{}, if $\Vert (-A)^{\frac{\beta-1}{2}}Q^{1/2}\Vert_{\mathcal{L}_{2}(H)}<\infty$ for some $\beta \in [1/2,1]$ we have $$\begin{aligned} |{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X^h(T)]|\leq C\Delta t^{\beta-\epsilon}.\end{aligned}$$ Note that [[Assumption \[ass:driftandB\]]{}]{} implies that $F \in \mathcal{C}_2^b(H;H)$. For $\Vert (-A)^{\frac{\beta-1}{2}}Q^{1/2}\Vert_{\mathcal{L}_{2}(H)}<\infty$ for some $\beta \in [1/2,1]$, [@stignonlinearweak Assumption A, Theorem 1.1] gives $$\begin{aligned} \label{spacedis} |{\mathbb{E}}[\Phi(X^h(T)] - {\mathbb{E}}[\Phi(X(T)]|\leq h^{2\beta-\epsilon}.\end{aligned}$$ The proof of in [@stignonlinearweak] uses some elements of Malliavin calculus. The following remark generalizes the [[Theorem \[opfullaplace\]]{}]{} for general selfadjoint operator with not necessarily Dirichet boundary condition. \[oppta\] For additive noise and under the same condition as in [[Theorem \[opfullaplace\]]{}]{}, if $A$ is selfadjoint, and $\Vert (-A)^{\frac{\beta-1}{2}}Q^{1/2}\Vert_{\mathcal{L}_{2}(H)}<\infty$ for some $\beta \in [k,1], \; k\geq 1/2$ and $X_0\in \mathcal{D}((-A)^{\beta})$. For $\Phi\in \mathcal{C}_2^b(H;\mathbb{R})$ and arbitrary small $\epsilon>0,$ the following estimation hold $$\label{optt} |{\mathbb{E}}[\Phi(X_M^h)] - {\mathbb{E}}[\Phi(X(T)]|\leq C(\Delta t^{\beta-\epsilon}+h^{2\beta-\epsilon}).$$ The proof is the same as in [@stignonlinearweak Assumption A, Theorem 1.1] where the Laplace operator is used just for simplicity. The difference comes from the set of the eigenvalues of the self adjoint operator $A$. Once the range of $\beta$ such that $\Vert (-A)^{\frac{\beta-1}{2}}Q^{1/2}\Vert_{\mathcal{L}_{2}(H)}<\infty$ is found, the proof of [@stignonlinearweak Assumption A, Theorem 1.1] is applied line by line. The following numerical simulations will confirm numerically estimation of [[Remark \[oppta\]]{}]{}. Numerical Simulations {#sec6} ===================== We consider the reaction diffusion equation $$\begin{aligned} \label{linear} dX=(D \varDelta X -0.5 X)dt+ dW \qquad \text{given } \quad X(0)=X_{0},\end{aligned}$$ on the time interval $[0,T]$ and homogeneous Neumann boundary conditions on the domain $\Omega=[0,L_{1}]\times [0,L_{2}]$. The noise is represented by . The eigenfunctions $\{ e_{i,j}\}_{i,j\geq 0}=\{e_{i}^{(1)}\otimes e_{j}^{(2)}\}_{i,j\geq 0} $ of the operator $-\varDelta$ here are given by $$\begin{aligned} e_{0}^{(l)}(x)=\sqrt{\dfrac{1}{L_{l}}},\qquad e_{i}^{(l)}(x)=\sqrt{\dfrac{2}{L_{l}}}\cos(\lambda_{i}^{(l)}x), \qquad \lambda_{0}^{(l)}=0,\qquad \lambda_{i}^{(l)}=\dfrac{i \pi }{L_{l}}\end{aligned}$$ where $l \in \left\lbrace 1, 2 \right\rbrace,\, x\in \Omega$ and $i \in \mathbb{N}$ with the corresponding eigenvalues $ \{\lambda_{i,j}\}_{i,j\geq 0} $ given by $\lambda_{i,j}= (\lambda_{i}^{(1)})^{2}+ (\lambda_{j}^{(2)})^{2}$. We take $L_1=L_2=1$. Notice that $A=D \varDelta $ does not satisfy [[Assumption \[assumptionn\]]{}]{} as $0$ is an eigenvalue. To eliminate the eigenvalue $0$ we use the perturbed operator $A=D \varDelta+\epsilon \mathbf{I},\, \epsilon>0$. The exact solution of is known . Indeed the decomposition of in each eigenvector node yields the following Ornstein-Uhlenbeck process $$\begin{aligned} \label{exact} dX_{i}=-(D \lambda_{i}+0.5)X_{i}dt+ \sqrt{q_{i}}d\beta_{i}(t)\qquad i \in \mathbb{N}^{2}.\end{aligned}$$ This is a Gaussian process with the mild solution $$\begin{aligned} X_{i}(t)= e^{-k_{i}t}X_{i}(0)+ \sqrt{q_{i}}\int_{0}^{t}e^{k_{i}(s-t)} d \beta_{i}(s),\quad k_{i}=D \lambda_{i}+0.5,\end{aligned}$$ which is therefore an Ornstein-Uhlenbeck process. Applying the Itô isometry yields the following exact variance of $X_{i}(t)$ $$\begin{aligned} \text{Var}(X_{i}(t))=\dfrac{q_{i}}{2 k_{i}}\left(1-e^{-2 \,k_{i}\,t}\right).\end{aligned}$$ During the simulations, we compute the exact solution recurrently as $$\begin{aligned} \label{exact} X_{i}^{m+1}&=& e^{-k_{i} \Delta t}X_{i}^m+ \sqrt{q_{i}}\int_{t_m}^{t_{m+1}}e^{k_{i}(s-t)} d \beta_{i}(s)\nonumber\\ &=& e^{-k_{i} \Delta t }X_{i}^m +\left(\dfrac{q_{i}}{2 k_{i}}\left(1-e^{-2 \,k_{i}\,\Delta t}\right)\right)^{1/2}R_{i,m},\end{aligned}$$ where $R_{i,m}$ are independent, standard normally distributed random variables with mean $0$ and variance $1$. The expression in allows to use the same set of random numbers for both the exact and the numerical solutions. Our function $F(u)=-0.5 u $ is linear and obviously satisfies the Lipschitz condition in [[Assumption \[assumption1\]]{}]{}. We assume that the covariance operator $Q$ and $A$ have the same eigenfunctions and we take the eigenvalues of the covariance operator to be $$\begin{aligned} \label{noise2} q_{i,j}=\left( i^{2}+j^{2}\right)^{-(\beta+\delta)}, \beta>0\end{aligned}$$ in the representation for some small $\delta>0$. Indeed for $ \beta \in [0,1]$ we have $$\begin{aligned} \Vert (-A)^{\frac{\beta-1}{2}}Q^{1/2}\Vert_{\mathcal{L}_{2}(H)}<\infty \Leftrightarrow \underset{(i,j) \in \mathbb{N}^{2}}{\sum}\lambda_{i,j}^{\beta-1}q_{i,j}< \pi^{2}\underset{(i,j) \in \mathbb{N}^{2}}{\sum} \left( i^{2}+j^{2}\right)^{-(1+\delta)} <\infty.\end{aligned}$$ We will consider the following two test functions $$\begin{aligned} \Phi_1: f \rightarrow \int_\Omega f(x) dx, \qquad \qquad \qquad \Phi_2: f \rightarrow \Vert f\Vert^2.\end{aligned}$$ which obviously belong to $\mathcal{C}_{b}^{2}(H,\mathbb{R})$. By setting $B=D \varDelta -0.5 I$, using the fact the the Ito’s integral vanishes, for $X_0 \in H$ we have $$\begin{aligned} \label{exact1} {\mathbb{E}}\Phi_1(X(t))&=& \int_{\Omega} e^{t B}X_0(x) dx+ {\mathbb{E}}\left[ \int_\Omega \left(\int_0^t e^{(t-s) B} dW(s,x)\right)dx \right ]\nonumber \\ &=& \int_{\Omega} e^{t B}X_0(x) dx+ \int_\Omega {\mathbb{E}}\left[\int_0^t e^{(t-s) B} dW(s,x) \right ]dx\nonumber \\ &=& \int_{\Omega} e^{t B}X_0(x) dx \nonumber\\ &=&\underset{i,j \geq 0}{\sum^{\infty}}e^{-k_{i j} t} \langle e_{i,j},X_0\rangle_H\int_{\Omega} e_{i j}(x)dx, \qquad k_{i j}=D\lambda_{i,j}+0.5. \end{aligned}$$ We also have $$\begin{aligned} \label{exact22} {\mathbb{E}}\Phi_2(X(t))&=&{\mathbb{E}}\Vert e^{t B}X_0+\int_0^t e^{(t-s) B} dW(s)\Vert^{2}\\ &=& {\mathbb{E}}\Vert e^{t B}X_0\Vert^2+ {\mathbb{E}}\Vert \int_0^t e^{(t-s) B} dW(s) \Vert^2+ 2{\mathbb{E}}\langle e^{t B}X_0, \int_0^t e^{(t-s) B} dW(s) \rangle_H \end{aligned}$$ Using the spectral decomposition, the fact that $X_0 \in H$ and the Ito isometry yields $$\begin{aligned} \label{exact2} {\mathbb{E}}\Phi_2(X(t))=\underset{i,j \geq 0}{\sum^{\infty}} e^{-2k_ij} \langle e_{i,j},X_0\rangle_{H}^2+ \underset{i,j \geq 0}{\sum^{\infty}} \langle \dfrac{q_{i j}}{2 k_{i j}}\left(1-e^{-2 \,k_{i j}\,\Delta t}\right)\rangle_H. \end{aligned}$$ To compute the exact solutions ${\mathbb{E}}\Phi_1(X(t))$ and ${\mathbb{E}}\Phi_2(X(t))$, we can either truncate and by using the first $N_h$ terms, $N_h$ being the number of finite element test basis functions, or use directly the Monte Carlo method with . Our code was implemented in Matlab 8.1. We use two different intial solutions with each test function $\Phi_1$ and $\Phi_2$. Indeed we use the initial solution $X_0=X_0^{(2)}=0$ for $\Phi_2$, and $$X_0=X_0^{(1)}= \underset{i,j \geq 1}{\sum^{\infty}} q^1_{i,j} e_{i,j},\;\;\;q^1_{i,j}=(i^2+j^2)^{-1.001}$$ for for $\Phi_1$. Our finite element triangulation has been done from rectangular grid of maximal length $h$. Fast Leja Points technique as presented in [@TLG; @SebaGatam] is used to compute the exponential matrix function $S_h^1= \varphi_1$. In the legends of our graphs, we use the following notations - ”TimeError1” denotes the weak error for fixed $h=1/150$ with $\Phi=\Phi_1$, $X_0=X_0^{(1)}$ and final time $T=1$. - ”TimeError2” denotes the weak error for fixed $h=1/50$ with $\Phi=\Phi_2$, $X_0=X_0^{(2)}$ and final time $T=1$. - ”SpaceError1” denotes the weak error for fixed time step ${\Delta t}=1/500$ with $\Phi=\Phi_1$, $X_0=X_0^{(1)}$ and final time $T=0.1$. - ”SpaceError2” denotes the weak error for fixed time step ${\Delta t}=1/20000$ with $\Phi=\Phi_2$ and $X_0=X_0^{(2)}$ and final time $T=0.1$. In all our graphs $D=0.1, \beta=1$, $\delta=0.001$, the weak errors are computed at the final time $T$ and $50$ realizations are used to estimate the weak errors with the Monte Carlo method. 0.01 [[Figure \[FIG01a\]]{}]{} shows the weak convergence in time of the exponential scheme for $\Phi=\Phi_1$ with $X_0=X_0^{(1)}$, and $\Phi=\Phi_2$ with $X_0=X_0^{(2)}$. The weak order of convergence in time is respectively $1.1$ for $\Phi=\Phi_1$ and $1.008$ for $\Phi=\Phi_2$. These orders of convergence are then closed to the optimal order $1$ obtained in [[Remark \[oppta\]]{}]{}. For space convergence, very small time steps are needed and the weak errors are performed at small final time $T=0.1$, [[Figure \[FIG01b\]]{}]{} shows the weak convergence in space of the exponential scheme for $\Phi=\Phi_1$ with $X_0=X_0^{(1)}$, and $\Phi=\Phi_2$ with $X_0=X_0^{(2)}$. The weak order of convergence in space is respectively $1.71$ for $\Phi=\Phi_1$ and $2.01$ for $\Phi=\Phi_2$. These orders of convergence are also closed to the optimal order $2$ obtained in [[Remark \[oppta\]]{}]{}. To sum up, our simulations confirm the theoretical results obtained in [[Remark \[oppta\]]{}]{}. Acknowledgements {#acknowledgements .unnumbered} ================ This project was supported by the Robert Bosch Stiftung through the AIMS ARETE chair programme. REFERENCES {#references .unnumbered} ========== [^1]: $\mathbb{D}$ is the sample space [^2]: This assumption is weak compared to [[Assumption \[assumption1\]]{}]{} and [[Assumption \[assumption2\]]{}]{}

--- abstract: 'The main objective of explanations is to transmit knowledge to humans. This work proposes to construct *informative* explanations for predictions made from machine learning models. Motivated by the observations from social sciences, our approach selects data points from the training sample that exhibit special characteristics crucial for explanation, for instance, ones contrastive to the classification prediction and ones representative of the models. Subsequently, semantic concepts are derived from the selected data points through the use of domain ontologies. These concepts are filtered and ranked to produce informative explanations that improves human understanding. The main features of our approach are that (1) knowledge about explanations is captured in the form of ontological concepts, (2) explanations include contrastive evidences in addition to normal evidences, and (3) explanations are user relevant.' author: - | Freddy Lécué[^1]\ INRIA, Sophia Antipolis, France\ Accenture Labs, Dublin, Ireland\ freddy.lecue@inria.fr\ Jiewen Wu$^\ast$\ A\*STAR Artificial Intelligence Initiative\ Institute for InfoComm Research\ Singapore\ jiewen.wu@acm.org bibliography: - 'exp18.bib' title: Semantic Explanations of Predictions --- Introduction ============ Machine learning, particularly deep learning, has attracted attentions from both industries and academia over the years. The algorithmic advancement has spurred near-human level accuracy applications such as neural machine translation [@googleMT], novel methods including Generative Adversarial Networks [@DBLP:conf/nips/SalimansGZCRCC16] and Deep Reinforcement Learning [@mnih-dqn-2015], among other things. Although highly scalable, accurate and efficient, most, if not all, of the machine learning models have exhibited limited interpretability [@DBLP:conf/kdd/LouCG12], which implies humans can hardly explain the final predictions [@shmueli2010explain]. The lack of meaningful explanations of prediction would become more problematic when the models are deployed in financial, medical, and public safety domains, among many others. Explanations are indispensable for building the *trust* relationship between human decision makers and intelligent systems making predictions. For instance, both the context and the [[rationale]{}]{} of any prediction result in medical diagnosis [@DBLP:conf/kdd/CaruanaLGKSE15] need to be understood as some of its consequences may be disastrous. In addition to trust, business owners can demand explanations for more informed decision making and developers can leverage explanations to debugging and maintenance. More stringent requirements have been dictated from legislation to safeguard fair and ethical decision making in general, notably the European Union General Data Protection Regulation (GDPR) warranting users the “right to explanation" in algorithmic decision-making [@DBLP:journals/corr/GoodmanF16]. Although there has been a lack of consensus on the definition of *explanations*, we have witnessed multiple avenues of research. As argued in [@mythos], these efforts generally fall into two (not necessarily disjoint) categories, i.e., one that aims at improving *transparency* of decision making by unveiling the internal mechanism of machine learning models and the other being *post hoc* explanations that justify the predictions generated by the models. The first category is sometimes referred to as *interpretability*. Our paper posits itself in the post hoc explanation category, in response to this specific question: $$\label{q:main} \emph{Why the input $x$ was labeled $y$?}$$ It is obvious that answers to such a question can be subjective. This paper, instead of deriving a complete solution to “correct" explanations, addresses the issue of *informativeness* [@mythos]. Towards informative explanations, we investigate certain salient properties on elucidating predictions to human users. In particular, we observe the following survey findings in social sciences [@millerXAI] towards explanations. - Human explanations imply social interaction [@conversationalExp]. The implication is that, for machine-generated explanations, it is indispensable to associate *semantic* information with an explanation (or the elements therein) for effective knowledge transmission to users. - Users favor contrastive explanations for understanding of causes [@contrastEQ; @contrastLipton]. That is, often implies the question: $$\label{q:contrast} \emph{Why the input $x$ was labeled $y$ \emph{instead of $y'$?}}$$ - Users *select* explanations. Due to the large space of possible explanations and a specific user’s understanding of the context, she selects the explanations based on what she believes to be the most *relevant* to her, rather than the most direct or probable causes [@conversationalExp]. The subjectivity of human choices implies that informative explanations may need to consider personalisation or contextualisation. This paper proposes a method that leverages semantic concepts drawn from data instances to characterize the aforementioned three observations for explanations, thus enabling more effective human understanding of predictions. Most of existing approaches focus on data-driven explanation and lack semantic interpretation, which defeats the objective of human-centric explanations. Instead, our proposed approach exploits the semantics of representative data points in the training samples. It works by (i) selecting representative data points and elaborating the decision boundary of classifiers, (ii) extracting and encoding the semantics of such data points using domain ontologies, and (iii) computing informative explanations based on optimizing certain criteria learned from humans’ daily explanations. The remainder of the paper subsequently reviews the basics and introduces the problem. Then, we describe how representative data points are extracted and show how semantics of data point is exploited to derive explanations. Conclusions and future research directions are given in the end. Related Work ============ Interpreting models or predictions dates back to at least twenty years ago. The resurgence of neural nets also attracted a lot of recent research into the area of interpretability of such deep models. To show the position of our proposed approach, we discuss a few representative work in the field of machine learning, from the angles outlined in $O_1$, $O_2$, and $O_3$. Decision trees and random forests have been studied to extract various levels of model interpretation [@treeNN; @DBLP:conf/iri/PalczewskaPRN13] together with some degrees of prediction explanation [@pang2006pathway]. Although their explanations are tuned to complex stochastic and uncertain rules they naturally expose high visibility on the decision process. [@DBLP:conf/icdm/WangRDLKM16] exploit the characteristics of classification models by exploiting and relaxing their decision boundary to approximate explanations. However explanations remain handcrafted from features and raw data, often as rules which remain very difficult to be generalized. [@DBLP:conf/naacl/LiCHJ16] targeted neural networks and observed the effects of erasing various parts of the data and its features on the model to derive a minimum but representative set, qualified as explanation. [@DBLP:conf/emnlp/LeiBJ16] study similar models and aim at identifying candidate [[rationales]{}]{} i.e., core elements of the model which aims at generalizing any prediction. Instead, [@DBLP:conf/nips/KimSD15] focused on placing interpretability criteria directly into the model to ensure fined-grained exploration and generation of explanations. [@kddexplain] elaborated a model-agnostic technique. To this end any test data is re-sampled and approximated using training data, which is then used as a view, or explanation of the predictions and model. Note that our work here is not model-agnostic and focuses more on the semantic interpretation to achieve higher level of informativeness. Leveraging *contrastive* information for explaining the predictions has seen its applications in image classification, e.g., [@Vedantam_2017_CVPR], which justifies why an image describes a particular, fine-grained concept as opposed to the distractor concept. Towards human-centric explanations, [@recsysdesign] designed some general properties of effective explanations in recommender systems. [@ijcai17justification] focuses on combining instance-level and feature-level information to provide a framework that generalizes several types of explanations. A more complete survey on human-centric explanations is also available in [@millerXAI], highlighting research findings from social sciences. Problem Statement {#sec:Background} ================= We focus on the predictions given by (w.l.o.g., binary) classifiers, where data points are partitioned into sets, each of which belonging to one class. The partition surface is a *decision boundary*. Before delving into the technical details, we first define the problem to be addressed. Ontology -------- An ontology ${\mathcal{O}}$ describes the concept hierarchy of domain knowledge. A concept, denoted by ${\mathbb{C}_{}}$, represents a type of objects. The most common relationship between concepts is subsumption (is-a), denoted $\sqsubseteq$. For instance, Human $\sqsubseteq$ Animal w.r.t. some ontology. An ontology can define many different types of semantic relationships beyond just is-a relationship, e.g., hasChild, hasParent, and so on. The hierarchical relations of concepts in ${\mathcal{O}}$ can be described as a graph for easy manipulation: each concept is a vertex, while the semantic relationship is a directed edge. An edge may be weighted to indicate how strong the semantic relation is between the concepts. Explanation Problem Statement ----------------------------- An informative explanation, the objective of this paper, can be defined based on observations $O_1$-$O_3$. Without loss of generality, consider a binary classifier, ${\mathbb{M}_{}}$, and a prediction $y_i$ of some given data point $x_i$, intuitively, the aim is to find a set of *human-understandable descriptions* of $y_i$ with respect to ${\mathbb{M}_{}}$ and $x_i$. To ease the presentation, a classifier is abused as a function, too. That is, ${\mathbb{M}_{}}(x_i) = y_i$ means that $x_i$ is predicted to be of label $y_i$ by the classifier ${\mathbb{M}_{}}$. A formal definition now follows. [**(Informative Explanation)**]{}[\[defn:explanation\]]{}\ Let ${\mathbb{M}_{}}$ be a binary classifier, $X$ be the set of training data points, $x_i$ be a test data point with $y_i$=${\mathbb{M}_{}}(x_i)$, and ${\mathcal{C}_{}}$ be a set of concepts. We define a data point selection function ${\mathfrak{F}_r}$:$X\rightarrow \{0,1\}$ and a semantic uplift function ${\mathfrak{F}_s}:2^X\rightarrow {2^{\mathcal{C}_{}}}$. An informative explanation is $e={\mathcal{C}_{}}^{+}\cup{\mathcal{C}_{}}^{-}$, where ${\mathcal{C}_{}}^{+}={\mathfrak{F}_s}(\{x_j~\mid~ j\ne i, {\mathfrak{F}_r}(x_j)=1, \text{ and }{\mathbb{M}_{}}(x_j) = y_i\})$ and ${\mathcal{C}_{}}^{-}={\mathfrak{F}_s}(\{x_k~\mid~ k\ne i, {\mathfrak{F}_r}(x_k)=1, \text{ and }{\mathbb{M}_{}}(x_k)\ne y_i\})$ . Observe that the definition of the semantic uplift function, ${\mathfrak{F}_s}$, implies that a set of data points can be assigned multiple ontological concepts. **Algorithmic considerations** In addition to defining functions ${\mathfrak{F}_r}$ and ${\mathfrak{F}_s}$, two more conditions are imposed on the algorithm design: (1) $e$ must be concise to meet observations $O_1$ and $O_2$. So, the size of $e$ needs to optimized using quantifiable measures. (2) the content of $e$ needs to show contrastive information and ranking is necessary to allow for user choices based on relevancy, as discussed in $O_3$. **Assumption** To semantically interpret data points, the raw feature must be (at least partially) semantically meaningful, so that semantics is available from the beginning. In practice most datasets have textual descriptions, and, in the rare cases where the raw features lack any descriptions, advices from dataset owners or domain experts can be sought. Note that the proposed approach assumes nominal features are expressed in one-hot encoding. [**(Explanation of Classification Prediction)**]{}\ The rest of the paper uses the dataset Haberman’s Survival from the UCI repository[^2] as an running example. The dataset contains cases from a study conducted 1958-1970 at the University of Chicago’s Billings Hospital on the survival of patients who had surgery for breast cancer. The task aims at classifying patients into those (1) survived 5 years or longer or (2) died within 5 year using the predictors: age, year of operation, and positive axillary nodes detected. We aim at identifying the informative explanations, as in Definition \[defn:explanation\], for any predicted data point w.r.t a classifier ${\mathbb{M}_{}}$, the $306$ training data points and a domain ontology ${\mathcal{O}}$. Organization ------------ The organization of the remainder is as follows. We first describe the data point selection function ${\mathfrak{F}_r}$, which chooses the most interesting data points for the defined problem. From the chosen data points, we then show how concepts can be drawn, enhanced by consulting ontologies, reduced for succinctness, and finally ranked for user choices. Identifying Representative Training Data {#sec:DB} ======================================== To generate an informative explanation, we need to first consider how to find representative data points, i.e., the function ${\mathfrak{F}_r}$. For the sake of clarity, we concentrate on binary classification (positive or negative) problems and two representative machine learning models, one is a linear classifier using [**L**]{}ogistic [**R**]{}egression ([**LR**]{}), while the other is the non-linear classifier using [**k**]{}-[**N**]{}earest [**N**]{}eighbour ([**k-NN**]{}). The approach can be easily extended to multi-class classifiers. Decision Boundary of Classification Models {#sec:DBCM} ------------------------------------------ We now review how to compute the decision boundaries of the two models. [**$\bullet$ [LR Models]{}**]{} are captured as follows: $$\begin{aligned} {\label{eq:LRM}} p = \frac{1}{1+e^{-Y}}, \text{ where } Y = \sum_{i=0}^{n}w_i\cdot X_i, \end{aligned}$$ where $p$ denotes the probability of being in the positive class. $w_i$ are the parameters for the $n$ given predictors $X_i$. In particular, $X_0$ is the constant $1$ and $w_0$ is the intercept. [**$\bullet$ [LR Decision Boundary]{}**]{} is computed using considering the positive class with $p\ge 0.5$ in . $$\begin{aligned} \label{eq:DBLR} \sum_{i=0}^{n}w_i\cdot X_i = 0. \end{aligned}$$ [**$\bullet$ [k-Nearest Neighbour (kNN) Models]{}**]{} are captured as: $$\begin{aligned} Y = \frac{1}{k}\cdot\sum_{x_i\in n(x)} y_i, \end{aligned}$$ where $n(x)$ is the neighbourhood of $x$ defined by the $k$ closest data points $x_i$ in the training sample and $y_i$ are the respective responses of $x_i$. [**$\bullet$ [k-NN Decision Boundary]{}**]{} for the data points belonging to the positive / negative class is computed by elaborating the convex hull [@chazelle1993optimal] of all points in the respective class. Given datasets of $n$ points in $d$ dimensions, the convex hull, also known as the smallest convex envelope that contains all $n$ points, can be efficiently computed in $O(n\log n)$ for $d\in\{2, 3\}$. However, in the case of high dimensions, the worse case complexity becomes $O(n^{\lfloor{d/2}\rfloor})$. Moreover, the convex hulls, represented by the points, have an average size of $O(n\log^{d-1}n)$ [@convexavg]. In practice, it is reasonable to obtain an approximation of the convex hulls for high dimensional datasets. For instance, the algorithm proposed in [@convexapp], of which the time complexity is insensitive to $d$ and the size of an approximate convex hull is user-specified. We will use a single dataset throughout this paper as an example to illustrate how the proposed approach is applied to a classifier to obtain explanations. [**(k-NN Decision Boundary)**]{}\ [[Among the $306$ data points of Haberman’s Survival dataset, a random point is chosen to be predicted, while the remaining $305$ are considered to be training sample.]{}]{} The convex hull, [[as decision boundary of the model]{}]{}, consists of 42 points, as blue triangle marks in Figure \[fig:ch\]. The convex hull, represented by blue triangle points, is computed all the training data points in Haberman’s Survival dataset. The red cross-squared point in the lower left is the test data point, and the rest of all training data points, the yellow rounded ones, are enclosed by the convex hull. ![k-NN Decision Boundary. []{data-label="fig:ch"}](chull.png){width="22.00000%"} Representative Data via Decision Boundary {#sec:CDPDB} ----------------------------------------- We illustrate our approach on identifying representative data [points]{} via the decision boundary. The set of *representative* data points ${\mathfrak{I}}$ is computed from the decision boundary of models, reflecting the *extreme* cases, and from the neighbours of the test data points, reflecting the *local* context. Technically, we define ${\mathfrak{I}}={\mathfrak{I}}_g\cup{\mathfrak{I}}_l$, and we compute the two subsets separately to obtain ${\mathfrak{I}}$. To distinguish between the different classifiers, these set notations will use superscripts accordingly, e.g., ${\mathfrak{I}}^{LR}$. By combining both types of data points, we aim at identifying elements for explanations that meets the objectives $O_1$-$O_3$. [**$\bullet$ LR Representative Data Points**]{} Consider a test data point $x_0$ predicted to be of class $y_0$. ${\mathfrak{I}}_g$ is constructed by selecting data points that have a (standard Euclidean) distance to the decision boundary i.e., a line, within certain threshold $t_g$. We also consider the proximity between the data points and the test data point $x_0$. That is, neighboring points of $x_0$ will be included, denoted ${\mathfrak{I}}_l$. The distance between a neighbor and $x_0$ should be within the threshold $t_l$. The set ${\mathfrak{I}}_g$ is obtained as follows. First all instances on or close to the decision boundaries are potential elements in ${\mathfrak{I}}_g$. There is a tradeoff between the size of ${\mathfrak{I}}_g$ and the representativeness of ${\mathfrak{I}}_g$. To solve this problem, we find $X_v$ such that $X_v$ has the largest variance among all features (or, the most important feature can be chosen if feature importance is available from the model). First all data points that are close enough (determined by the threshold $t_g$) to the decision boundary are collected. In the context of LR, we have: $$\begin{aligned} {\mathfrak{I}}_{g}^{LR} &= \{x_g ~\mid~ \frac{ \lvert\sum_{i=0}^{n} w_{i}\cdot X_{ig} \rvert }{ \sqrt{\sum_{i=1}^{n} w_{i} } } \le t_g ~\wedge~ y_g=y_0 \}\label{DBLR1}\\ \label{locallr}{\mathfrak{I}}_{l}^{LR} &= \{x_j~\mid~d(x_j, x_0)\le t_l ~\wedge~ y_j=y_0\}\end{aligned}$$ Note that the class labels of $x_j$ must be the same as that of the input $x_0$. Finally, we can obtain the set of representative points by combining the two sets: ${\mathfrak{I}}^{LR} = {\mathfrak{I}}_{g}^{LR} \cup {\mathfrak{I}}_{l}^{LR}$. It also follows that, for an arbitrary data point $x'$, ${\mathfrak{F}_r}(x')=1$ if $x'\in{\mathfrak{I}}^{LR}$, and ${\mathfrak{F}_r}(x')=0$ otherwise. The definition of ${\mathfrak{F}_r}$ in the k-NN case is the same, and is thus omitted. [**$\bullet$ k-NN Representative Data Points**]{}. The local neighbors in k-NN, represented as ${\mathfrak{I}}_{l}^{kNN}$, are computed in the same manner as in . A simple version of ${\mathfrak{I}}_{g}^{kNN}$ is defined as follows: $$\begin{aligned} {\mathfrak{I}}_{g}^{kNN'} = \{x_g\mid x_g\in\text{ convex hull points labelled $y_0$} \}.\label{DB-KNN}\end{aligned}$$ Observe that ${\mathfrak{I}}_{g}^{kNN'}$ might contain, in the worst case scenario and particularly for high dimensional data, exactly all points of class label $y_0$ in the convex hull. Therefore the set of representative data points can be large. Its size can be further reduced by selecting points in the decision boundary. To this end, we consider how these points spread over the decision boundary and aim to sample points that best represent the decision boundary. To achieve this, we consider the feature that has the largest variance, say $X_v$. Data points in ${\mathfrak{I}}_{g}^{kNN'}$ are linearly projected onto this dimension, $X_v$. For each of the $m$ equally-spaced values on $X_v$, say $\langle v_1, \ldots, v_m\rangle$, a random sampling is performed on data points in ${\mathfrak{I}}_{g}^{kNN'}$ that have a value of $X_v$ close to $v_i$. Ultimately, a set of data points $x_m$ for each $v_i$ is selected. This way, the representative data points to the decision boundaries will be spread over the decision boundary. Alternatively, data points can be obtained by iteratively using the features ranked by variance or any other metrics (such as feature importance). This would work like a $k$-dimensional tree until a single data point can be found. In this case, no random sampling is required. Let $t_d$ be a threshold value. The final step is to weigh the points in ${\mathfrak{I}}_g^{kNN}$. The [[rationale]{}]{} is that contour points closer to the test data point $x_0$ are more useful in explaining the prediction of $x_0$. The weighting can be achieved using the distance between the points and $x_0$. $$\begin{aligned} \begin{split} \label{knng} {\mathfrak{I}}_{g}^{kNN} = \bigcup_{j=1}^{m} \{ & \frac{1}{1+d(x_0, x_k)}\cdot x_k ~\mid~ x_k\in{\mathfrak{I}}_{g}^{kNN'}~\wedge~ \\ & \lvert X_v(x_k) - v_j\rvert \le t_d \}, \end{split} \end{aligned}$$ where $X_v(x_k)$ denotes the value projecting $x_k$ on $X_v$. [**$\bullet$ Uniform and Contrastive Explanations**]{}. Computing representative data points depends on the input class label, e.g., $y_0$ in the previous descriptions. We compute not only representative points that are of the same class label $y_0$, but also compute points that are predicted with the different class label. By observation $O_2$, humans need to see more than just *uniform* explanations, i.e., *contrastive* explantions, i.e., why the input is not labeled $y_1$ and alike. For binary classification, we define the class that $x_0$ is labelled to be the positive class and the other is the negative class. The uniform and contrastive explanations are, for simplicity, called positive and negative explanations, respectively. Consequently, the representative data points with respect to $y_0$ are the positive points ${\mathfrak{I}}^{+}$. For negative class, ${\mathfrak{I}}^{-}$ can be computed analogously: we just need to change the labels from $y_0$ to $y_1$ in , , and . The two sets of data points thus serve as positive and negative evidences for explaining why $x_0$ is of class $y_0$. [**(k-NN Representative Data Points)**]{}\ Figure \[fig:progress\] shows the various steps of representative data points discovery. Assume the test data point is predicted to be the positive class and no more than 8 points are to be chosen in each step. Figure \[fig:gp\] first computes some points of positive labels that spread over the convex hull (decision boundary). Note that there are 42 data points in the convex hull, but only 8 points are sampled from the 42 points to approximate the convex hull, based on the spread feature, the age of patients. These 8 points are considered to be the uniform evidences, i.e., they form the set ${\mathfrak{I}}_{g}^{kNN}$ as given in . Figure \[fig:gplp\] then further computes the neighboring points (positive local evidences) that are also in the positive class. These points, denoted by plus signs, form the set ${\mathfrak{I}}_{l}^{kNN}$. After collecting all the positive evidences, Figures \[fig:gplpgn\] and \[fig:gplpgnln\] show the additional data points in the negative class (contrastive information) based on the convex hull and the local neighborhood, which form *negative* extreme (${\mathfrak{I}}_{g}^{kNN-}$) and local (${\mathfrak{I}}_{l}^{kNN-}$) evidences, respectively. Figure \[fig:gplpgnln\] gives a nice visualisation of our representative selection idea: extreme evidences are spread globally, while local evidences gather around the test data points. In addition, negative evidences appear visually more distant from the test point than positive ones. [0.16]{} ![k-NN Representative Data Points. The cross-squared point is the test point.[]{data-label="fig:progress"}](gp.png "fig:"){width="\textwidth"} [0.16]{} ![k-NN Representative Data Points. The cross-squared point is the test point.[]{data-label="fig:progress"}](gplp.png "fig:"){width="\textwidth"} \ [0.16]{} ![k-NN Representative Data Points. The cross-squared point is the test point.[]{data-label="fig:progress"}](gplpgn.png "fig:"){width="\textwidth"} [0.16]{} ![k-NN Representative Data Points. The cross-squared point is the test point.[]{data-label="fig:progress"}](gplpgnln.png "fig:"){width="\textwidth"} Explaining Predictions {#sec:Core} ====================== Definition \[defn:explanation\] provides the basis for constructing informative explanations. To design the explanation algorithm, a few prerequisites need to be elaborated. In particular, the role of a domain knowledge base (or ontology) is indispensable in that semantic abstraction of data points is drawn from the knowledge base. [**$\bullet$ Context:**]{} The explanation algorithm is formulated as $g({\mathbb{M}_{}}, X, x_i, {{\mathcal{O}}{}}, {\mathfrak{F}_s})$, where ${\mathbb{M}_{}}$ is a classifier, $X$ is the set of training data points, $x_i$ is the input data point, ${\mathcal{O}}$ is a [domain ontology]{}, [and ${\mathfrak{F}_s}$ is the semantic uplift function as given in Definition \[defn:explanation\].]{} From ${\mathbb{M}_{}}$, $X$, and $x_i$, [(\[DBLR1\]-\[locallr\]) and (\[DB-KNN\]-\[knng\])]{} provide a way to compute two sets of data points (evidences) based on the decision boundaries of ${\mathbb{M}_{}}$. The positive and negative evidences, denoted ${\mathfrak{I}}^{+}$ and ${\mathfrak{I}}^{-}$ respectively, can then be used to [drive the extraction of relevant information for explanation]{}. [We]{} discuss how an ontology ${\mathcal{O}}$ can be used to abstract the semantics of these data points, which are leveraged to compute explanations. [Our approach]{} can be applied independently to ${\mathfrak{I}}^{+}$ and ${\mathfrak{I}}^{-}$, thus, the general notation ${\mathfrak{I}}$ denotes either set. [**$\bullet$ Notations:**]{} To ease the presentation, ${\mathfrak{I}}$ is given as a matrix-like structure ${\mathcal{M}_{}}$ of size $m\times n$, where $\lvert{\mathfrak{I}}\rvert = m$ and there are $n$ predictors. A single row in ${\mathcal{M}_{}}$ is a data point, represented as a set of feature-value pairs, with a weight given as $\alpha_i$ computed from . $$\alpha_i \cdot x_i:\bigwedge_{i=1\ldots n} f_i=v_i$$ [**$\bullet$ Semantic Uplift of Data Points:**]{} There have been much prior work on deducing concepts from relational-style data. We use the existing work to uplift data semantically. [[For each $f_i=v_i$, we aim at finding its Basic-level Categorization [@DBLP:conf/cikm/WangWWX15], denoted by $\text{blc}(f_i=v_i)$ with respect to a domain ontology ${\mathcal{O}}$. Categorization is achieved in two steps. First, concepts $\text{blc}(f_i=v_i)|_{G}$ are identified in a large knowledge graph $G$ i.e., a graph dominated by instance and *is-a* relationships such as Dbpedia [@DBLP:journals/semweb/LehmannIJJKMHMK15] and Microsoft Concept Graph [@DBLP:conf/sigmod/WuLWZ12] following [@DBLP:conf/ijcai/WangZWMW15]. Then a mapping step from concepts $\text{blc}(f_i=v_i)|_{G}$ to ${\mathbb{C}_{i}}$ in the domain ontology ${\mathcal{O}}$ is required to contextualized categorization in a targeted domain. In other words a concept ${\mathbb{C}_{i}}\in{\mathcal{O}}$ is identified such that: $$\label{eq:mapping} {\mathbb{C}_{i}} \doteq m(\text{blc}(f_i=v_i)|_{G})$$ where $m$ is a mapping function from $G$ to domain ontology $\mathcal{O}$. The concept mapping is achieved following [@ehrig2004qom] where both syntactic and semantic similarities (distance among similar concepts) are considered. Therefore the semantic uplift function ${\mathfrak{F}_s}$ in Definition \[defn:explanation\] is computed through the composition of $m$ and blc i.e., ${\mathfrak{F}_s}\doteq m \circ \text{blc}$. We adopted a [[2-step]{}]{} process to ensure a maximum coverage of $f_i=v_i$ in ${\mathcal{O}}$. Indeed a more direct approach from $f_i=v_i$ in ${\mathcal{O}}$ could result in no mapping, and then no semantic association for $f_i=v_i$. The knowledge graph layer provides a much larger [[input set]{}]{} to be mapped in ${\mathcal{O}}$, and then a better semantic coverage for $f_i=v_i$. ]{}]{} [**$\bullet$ Representation of Data Points:**]{} Since not all $f_i=v_i$ can be automatically matched with concepts in ${\mathcal{O}}$, we differentiate the semantic and non-semantic parts. To this end we assume there is a set $K\subseteq\{1, \ldots, n\}$ such that the final representation of a data point can be defined as two components, which are the projections of points for the feature-value pairs that cannot be matched with ontological concepts (${\mathcal{P}_{f}(x_i)}$) and the semantic counterpart (${\mathcal{P}_{c}(x_i)}$), respectively. $$\begin{aligned} {\label{eq:RDPs}} \begin{split} x_i: &\bigwedge_{k\in K}\alpha_i\cdot\{f_{ik}=v_{ik}\}\wedge\bigwedge_{j\in\{1, \ldots, n\}\backslash K}\alpha_i\{{\mathbb{C}_{ij}}\} \\ & = {\mathcal{P}_{f}(x_i)}\wedge{\mathcal{P}_{c}(x_i)} \end{split}\end{aligned}$$ Projections are applied to each row in ${\mathcal{M}_{}}$, and the concept components of all rows forms a set, ${\mathcal{C}_{in}}$, that serves as the input of our explanation approach (cf. Algorithm \[alg:core\]). $$\label{eq:inputconcepts} {\mathcal{C}_{in}} = \bigcup_{i\in[1, m]}{\mathcal{P}_{c}(x_i)}$$ Note that for duplicate concepts across data points, the weights of these concepts will increase accordingly. For positive data points ${\mathfrak{I}}^{+}$, the set is ${\mathcal{C}_{in}}^{+}$. Similarly, ${\mathcal{C}_{in}}^{-}$ is computed for negative data points ${\mathfrak{I}}^{-}$. [**(Semantic Uplift)**]{}\ [[ The features of Haberman’s survival data set have a natural categorization of values using wikipedia as $G$ in . For instance ]{}]{} $TheSilentGeneration$ denotes people born between $1925$ and $1941$. Patient $16$, [[classified as a patient who survived 5 years or longer,]{}]{} is described as , and as after semantic encoding of the data point. [[Note that the study has been conducted between $1958$ and $1970$ hence a $35$ year-old patient in $1963$.]{}]{} $$\begin{aligned} p_{16} :~ & age = 35 \sqcap yearOp=1963\nonumber\\ & \sqcap numberNodes = 0\label{eq:p21}\\ p_{16} :~ & TheSilentGeneration\sqcap OperationIn1960s\nonumber\\ & \sqcap NoPosAxillaryNode\label{eq:p22}\end{aligned}$$ Semantic representations of human population have been extracted[^3] for appropriate semantic mapping. [**$\bullet$ Explanation Concept Completion**]{} The input concepts given in (\[eq:inputconcepts\]) are not necessarily easy to understand for two reasons: *a*) they tend to be loosely connected to each other as not all feature value pairs can be semantically uplifted. *b*) these concepts may be data specific due to the semantic uplift so that humans may not understand the low-level concepts well. To address these issues, we show how to introduce more human-comprehensible and semantically connected concepts from the ontology. Furthermore, to optimize the completion process and to ensure the final explanation concepts are succinct, the following constraints are stipulated: - Minimize the size of $e$ for succinct explanations. - Maximize the number of matching among input concepts and ontological concepts. - Maximize the total weight of matching. Note that the concepts in the input ${\mathcal{C}_{in}}=\{\alpha_i {\mathbb{C}_{i}}\}$ are weighted. To fully leverage the semantics of concepts, the structure of ${\mathcal{O}}$ is used to find concepts that can abstract input concepts. Our graph-based traversal requires the following notions for defining relationships among any concepts. [**(Distance between Concepts)**]{}[\[defn:DC\]]{}\ Suppose ${\mathcal{O}}$ be an ontology represented as a graph. Given two weighted concepts $\alpha_1{\mathbb{C}_{1}}, \alpha_2{\mathbb{C}_{2}}$ over graph ${\mathcal{O}}$. The distance ${\texttt{dist}({\mathbb{C}_{1}},{\mathbb{C}_{2}})}$ between ${\mathbb{C}_{1}}$ and ${\mathbb{C}_{2}}$ is defined to be the minimum length of the path from ${\mathbb{C}_{1}}$ to ${\mathbb{C}_{2}}$. [**(Concept Matching)**]{}[\[defn:SPBC\]]{}\ Let a mapping be a partial function ${\mathfrak{F}_{io}}:{\mathcal{C}_{in}} \rightarrow {\mathcal{C}_{out}}$, which defines a set of matching between concepts in ${\mathcal{O}}$. A matching of $({\mathbb{C}_{1}}, {\mathbb{C}_{2}})$ is the shortest path [following any labelled edges]{} in ${\mathcal{O}}$ from ${\mathbb{C}_{1}}$ to ${\mathbb{C}_{2}}$. Note that each edge on the path of a matching carries a weight, $\lambda_i$, that denotes the semantic relatedness between concepts. For a matching $({\mathbb{C}_{1}}, {\mathbb{C}_{2}})$ of distance one, an aggregated weight of ${\mathbb{C}_{2}}$ can be computed as follows, e.g., $\gamma_i = \sum_{1}^{k}\beta_k\cdot\lambda_k$ for a concept with matching from $k$ concepts in ${\mathcal{O}}{}$, each of which has a weight $\beta_i$. In the initial case, i.e., concepts in ${\mathcal{C}_{in}}$, the weight $\beta_i = \alpha_i$. Following Definitions \[defn:DC\] and \[defn:SPBC\], we are ready to instantiate the constraints $C_1$-$C_3$. Let ${\texttt{dist}({\mathbb{C}_{1}},{\mathbb{C}_{2}})}$ denote the number of hops between two nodes in ${\mathcal{O}}$ and $\gamma_i$ denote the weight of ${\mathbb{C}_{i}}$. The algorithm aims to find a set of output concepts defined as follows: $$\begin{aligned} \label{optimalConditions} \begin{split} &~~~~~~~~~~~~~~{\mathcal{C}_{O}}=\operatorname*{\arg\!\max}_{{\mathcal{C}_{e}}}(a_1{s_{v}} + a_2{s_{l}}+a_3{s_{d}}),\\ &\text{where $a_{i, i\in\{1, 2, 3\}}$ are weighting factors, and } \\ &{s_{v}} = \frac{1}{\lvert {\mathfrak{F}_{io}}^{-1}({\mathcal{C}_{e}})\rvert}, ~~{s_{l}} = \frac{1}{\lvert{\mathcal{C}_{e}} \rvert}, \\ &{s_{d}} = \sum\frac{\gamma_i}{{\texttt{dist}({\mathbb{C}_{j}},{\mathbb{C}_{i}})}} \text{ for any ${\mathfrak{F}_{io}}({\mathbb{C}_{j}}) = {\mathbb{C}_{i}}$.} \end{split}\end{aligned}$$ For tractable computation, our algorithm uses random hill climbing to find ${\mathcal{C}_{O}}$, as discussed in the next section. Computing Informative Explanations {#sec:ExplanationAlgorithm} ---------------------------------- The main algorithm, Algorithm \[alg:core\], computes the explanation concept completion based on (\[optimalConditions\]). [**$\bullet$ Input:**]{} The input to the algorithm [[includes]{}]{} a set of concepts ${\mathcal{C}_{I}}$, an ontology ${\mathcal{O}}$ represented as a graph, a given integer $k$ to restrict the depth for traversing ${\mathcal{O}}$, and two additional control parameters, $h$ and $mp$. ${\mathcal{C}_{I}} \leftarrow removeDuplicate({\mathcal{C}_{I}})$ $V \leftarrow sort({\mathcal{C}_{I}})$ $s_{p} = 0,\;\;{\mathcal{C}_{O}} = \emptyset$ [**$\bullet$ Algorithm \[alg:core\]:**]{} Line 2 sorts the concepts decreasingly by weights. Lines 3-7 remove concepts that are subsumed by any concepts in $V$ because the purpose of our algorithm is to uplift special concepts into more general ones. The loop in lines 9-19 is a random-restart step to reduce the exponential search space of subsets of $V$ to $h$ restarts. Here, $h$ specifies the number of restarts desired. Line 10 obtains a random subset of $V$, which is then used to find matching successors in ${\mathcal{O}}$ as shown in lines 12-17. Line 13 collects matching concepts that can match at least two different concepts so as to further limit the search space. There is also an implicit condition to ensure correctness: $V_i'$ should collect only concepts that have never been collected before in each restart, due to possible cycles in the ontological graph. Line 14 first sorts the matching concepts and then [[picks]{}]{} the first $mp$ concepts. Here $mp$ specifies the number of concepts to be chosen for next matching. This step is necessary as potentially all concepts in ${\mathcal{O}}$ can be in $V_i'$, so a constant $mp$ can significantly reduce the search space. The output ${\mathcal{C}_{out}}$ is the set of matching concepts that maximizes the weight. [**$\bullet$ Contraction**]{} Applying Algorithm \[alg:core\] to ${\mathcal{C}_{in}}^{+}$ results in ${\mathcal{C}_{out}}^{+}$. For ${\mathcal{C}_{in}} = {\mathcal{C}_{in}}^{+}\cup{\mathcal{C}_{in}}^{-}$, we also apply Algorithm \[alg:core\] to ${\mathcal{C}_{in}}^{-}$ and obtain ${\mathcal{C}_{out}}^{-}$. The two sets of concepts form the basis to provide both uniform and contrastive explanations, each set being a *group* of explanations in that class. Note that in case of multi-classification, there will be many groups of contrastive explanations. To avoid excessive contrastive evidences, it is reasonable to restrict the groups of contrastive explanation to one or two. This can be realized by selecting the next one or two most probable class labels predicted by the classifiers. Now consider the binary classification case. The uniform explanations may contain knowledge already entailed by the contrastive explanations. It makes sense to keep only the essential information in the uniform explanations for succinctness. As an example, assume the uniform explanations have only one concept $\{GraduateStudent\}$ and the contrastive explanations have $\{{PhDStudent}\}$, a more informative uniform explanation would be *“$GraduateStudent$ but not $PhDStudent$"*, which means $\{MasterStudent\}$ w.r.t. a common sense ontology. For this purpose *concept difference* [@dlhb] is used to find out the concepts that entail some positive concept but not any of the negative concepts. Given concepts ${\mathbb{C}_{p}},{\mathbb{C}_{n}}$ from ${\mathcal{C}_{out}}^{+},{\mathcal{C}_{out}}^{-}$, respectively, the difference between two concepts is computed as follows: $$\begin{aligned} {\mathbb{C}_{p}}\backslash{\mathbb{C}_{n}} = \bigcup\{{\mathbb{C}_{d}} ~\mid~ {\mathbb{C}_{d}}\sqsubseteq{\mathbb{C}_{p}}\text{ and } {\mathbb{C}_{d}}\not\sqsubseteq{\mathbb{C}_{n}} \} \label{eq:defdiff}\end{aligned}$$ The subsumption relation $\sqsubseteq$ may introduce many unseen ontological concepts as difference concepts. To select the useful difference concepts, these concepts are ranked according to a weight that indicates how closely a difference concept is semantically related to concepts found in the data. We define the importance of each ${\mathbb{C}_{d}}$ to be: $$\begin{aligned} \label{eq:importance} imp({\mathbb{C}_{d}}) = \frac{1}{ n} \sum_{v=1}^{n} \alpha_v{\texttt{dist}({\mathbb{C}_{d}},{\mathbb{C}_{v}})} \text{ for } {\mathbb{C}_{v}}\in{\mathcal{C}_{out}}^{+},\end{aligned}$$ where $n$ is the size of the input ${\mathcal{C}_{out}}^{+}$. The final set of difference concepts used to replace the grop ${\mathcal{C}_{out}}^{+}$, with a defined threshold $\delta$, is: $$\begin{aligned} \label{eq:diffrep} \begin{split} {\mathcal{C}_{diff}}^{+} = \bigcup\{{\mathbb{C}_{p}}\backslash{\mathbb{C}_{n}}\mid &~{\mathbb{C}_{p}}\in{\mathcal{C}_{out}}^{+}, {\mathbb{C}_{n}}\in{\mathcal{C}_{out}}^{-},\\ & \text{ and } imp({\mathbb{C}_{p}}\backslash{\mathbb{C}_{n}}) \ge\delta \} \end{split}\end{aligned}$$ [**$\bullet$ Ranking**]{} The grouping step generates two groups, ${\mathcal{C}_{diff}}^{+}$ and ${\mathcal{C}_{out}}^{-}$. Recall that humans select explanations *relevant* to them. To allow for human subjectivity, we associate all concepts with a rank order within the same group such that concepts across different groups with the same rank order represent a single rational explanation. We show one possible method to compute the rank orders: 1. A dense rank is computed for concepts in ${\mathcal{C}_{diff}}^{+}$ based on the importance defined in (\[eq:importance\]). 2. For any ${\mathbb{C}_{n}}\in{\mathcal{C}_{out}}^{-}$ used in computing ${\mathcal{C}_{diff}}^{+}$, i.e., (\[eq:diffrep\]), the rank order of ${\mathbb{C}_{n}}$ is the same as that of the majority of the results ${\mathbb{C}_{d}}\in{\mathcal{C}_{diff}}^{+}$. Note a single ${\mathbb{C}_{n}}$ may be used to compute many difference concepts. 3. For any ${\mathbb{C}_{n}}\in{\mathcal{C}_{out}}^{-}$ *not* used in (\[eq:diffrep\]), the rank order of ${\mathbb{C}_{n}}$ is the same as the majority order of the most similar difference concepts. Note that an upper limit threshold, $\sigma$, is set to ensure concepts are sufficiently similar to some difference concepts. Otherwise, next step is required. $$\begin{aligned} \{{\mathbb{C}_{d}}~\mid~&{\mathbb{C}_{d}}\in{\mathcal{C}_{diff}}^{+}, {\texttt{dist}({\mathbb{C}_{n}},{\mathbb{C}_{d}})}<\sigma, \neg\exists {\mathbb{C}_{d'}}\in{\mathcal{C}_{diff}}^{+} \\ & \text{ s.t. } {\texttt{dist}({\mathbb{C}_{n}},{\mathbb{C}_{d'}})} < {\texttt{dist}({\mathbb{C}_{n}},{\mathbb{C}_{d}})}\}.\end{aligned}$$ 4. Otherwise, the rank order of any ${\mathbb{C}_{n}}\in{\mathcal{C}_{out}}^{-}$ is the next order to the lowest order in ${\mathcal{C}_{diff}}^{+}$. After ranking, concepts of the same order from both the uniform and contrastive explanations form an explanation of that rank order. Human users can choose, among these succinct, informative explanations, the ones that they believe to be most relevant using the rank order. [**(Informative Explanations)**]{}\ Consider the test data point given in , representing a $30$ year-old individual with an operation which occurred in $1964$, and a number of nodes equal to $1$. $$\begin{aligned} {\label{eq:exFinal}} p_1: ~& age=30 \sqcap yearOp=1964 \nonumber\\ & \sqcap numberNodes = 1\end{aligned}$$ Although concepts are not drawn from the test point, it is easy to see that $p_1$ is also in “TheSilentGeneration", had “OperationIn1960s," and had “OnePosAxillaryNode." The objective is to understand why such individual is classified to be positive, i.e., survived 5 years or longer. The concepts from the representative data points fall into positive and negative groups. Computing the concept difference in this example discards the concept “NoPosAxillaryNode", though it does not introduce new concepts : [$$\begin{aligned} {\mathcal{C}_{out}}^{+} & = \{0.9TheSilentGeneration, 0.7OperationIn1960s,\nonumber\\ & 0.5NoPosAxillaryNode\}\label{eq:plus}\\ {\mathcal{C}_{out}}^{-}& = \{ 0.6TheGIGeneration, 0.3OperationIn1950s,\nonumber\\ & 0.5OperationIn1960s, 0.5NoPosAxillaryNode\}\label{eq:minus}\\ {\mathcal{C}_{diff}}^{+} & = \{0.85TheSilentGeneration \}\label{eq:expdiff} \end{aligned}$$ ]{} The weights are computed by semantic uplifts and also reflect the proportion of data points that share the concepts cf. , e.g., in the positive data points there are more patients in the silent generation, while the G. I. generation is the majority in the negative data points. The final explanations were ranked within groups:\ ---------- ----------------------------- ---------------------------- ${\mathcal{C}_{diff}}^{+} $ ${\mathcal{C}_{out}}^{-} $ rank$_1$ $TheSilentGeneration$ $TheGIGeneration$ $OperationIn1960s$ $OperationIn1950s$ $NoPosAxillaryNode$ ---------- ----------------------------- ---------------------------- \ The rank orders of concepts in the contrastive explanations ${\mathcal{C}_{out}}^{-} $ are derived from their semantic similarity to the only uniform explanation $TheSilentGeneration$, as given in Steps 3&4 of the ranking process. Conclusions and Future Work =========================== \[sec:ConclusionFutureWork\] Our approach, exploiting the semantics of data points, tackles the problem of explaining the predictions in an informative manner to human users. Semantic reasoning and machine learning have been combined by revisiting decisions boundaries and its representative elements as semantic characteristics of explanations. Such characteristics are then leveraged to derive informative explanations with respect to the domain ontologies. The core contributions of our approach include its ability to capture interesting data points that exhibit extremity (in the form of decision boundaries) and local context of the test data points (in the form of neighborhood) and its manipulation of semantics to enhance *informativeness* for human-centric explanations. To generalize our approach for multi-label classifiers, the key difference is on the computation of ${\mathfrak{I}}^+$ and ${\mathfrak{I}}^-$. For a particular test point, all the predicted labels are considered to be positive classes. Computing ${\mathfrak{I}}^+$ is equivalent to computing the union of ${\mathfrak{I}}^+$ in each positive class. For computing ${\mathfrak{I}}^-$, it can be done for each negative class *separately*. So the contrastive evidences are for all positive labels against each and every of the negative labels. However, we advise choosing only the top few negative classes (by user specification or some predefined popularity metrics of classes). There are several extensions to be considered in our future work. First, explanation relevancy can be improved by considering user profiles, instead of allowing for user choices. Second, we will investigate other types of machine learning models, for instance, random forest classifiers. [^1]: Authors contributed equally to the work. [^2]: http://archive.ics.uci.edu/ml/datasets/ [^3]: https://en.wikipedia.org/wiki/Generation\#List\_of\_generations

--- abstract: 'Classical topological phases derived from point degeneracies in photonic bandstructures show intriguing and unique behaviour. Previously identified exceptional points are based on accidental degeneracies and subject to engineering on a case-by-case basis. Here we show that symmetry induced (deterministic) pseudo Weyl points with non-trivial topology and hyper-conic dispersion exist at the centre of the Brillouin zone of chiral cubic systems. We establish the physical implications by means of a $P2_13$ sphere packing, realised as a nano plasmonic system and a photonic crystal.' author: - Matthias Saba - 'Joachim M. Hamm' - 'Jeremy J. Baumberg' - Ortwin Hess title: A group theoretical route to deterministic Weyl points in chiral photonic lattices --- Current broad interest in topological phases, triggered by the discovery of the quantum Hall effect [@PhysRevLett.45.494] and its theoretical investigation [@PhysRevLett.49.405; @PhysRevB.48.11851; @KOHMOTO1985343], can mainly be attributed to the fact that topological features are, due to their discrete nature, insensitive to system perturbations, and can, for example, give rise to the existence of topologically induced edge states for bulk systems [@RevModPhys.83.1057; @RevModPhys.82.3045]. Plasmonic [@PhysRevB.93.241402] and single electron [@Siroki2016] surface states of Weyl semi-metals, with an isolated exceptional point of non-trivial topology, are stable against perturbations and give rise to peculiar dynamics. Recently, it has been demonstrated that topological quantization occurs in entirely classical systems such as two-dimensional (2D) photonic crystals [@PhysRevLett.93.083901; @PhysRevA.78.033834], sparking a new wave of research on photonic topology [@Lu2014]. In particular, topologically protected Weyl points with hyperconic dispersion have been identified in double gyroid photonic crystals with broken parity-time symmetry [@Lu2013]. Concurrently, group theory provides a tool to predict whether a given spatio-temporal symmetry permits topologically non-trivial exceptional points, or induces them deterministically. This idea has successfully found its way and been applied to classical [@PhysRevB.89.134302] and quantum mechanical [@PhysRevLett.116.186402; @Bradlynaaf5037] systems. Indeed, the existence of deterministic two and three-fold degeneracies at the center of the Brillouin zone (BZ), aka the $\Gamma$ point, for cubic symmetries is well known and documented in the literature [@Bradley_GT]. Recently, it has been shown that some of these degeneracies are topologically non-trivial in electronic systems [@1957JPCS1249K; @Bradlynaaf5037]. [./P2\_13\_combined.png]{} (2,91)[[[()]{}]{}]{} (52,91)[[[()]{}]{}]{} (2,49)[[[()]{}]{}]{} (52,49)[[[()]{}]{}]{} Here we show on the basis of group and perturbation theory that symmetry induced three-fold degenerate *pseudo Weyl points* (PWPs) exist at the $\Gamma$ point in classical (photonic) systems. They split isotropically in first order in the Bloch wave vector ${\ensuremath{\bm{k}}}$ for any chiral cubic space group with time reversal symmetry. The PWPs studied here constitute a deterministic 3D analog to previously studied accidental Dirac points [@Huang2011]. We show and demonstrate that they are of non-trivial topology, leading to protected surface states. In this letter, we first derive a 3D perturbation model that leads to hyperconic dispersion with non-trivial topology, and an intermediate flat band. We then construct a minimalistic geometry, a $P2_13$ sphere packing ([Fig. \[fig:spheres\]]{}), which satisfies the symmetry requirements, and apply it to a quasistatic coupled-dipole model, before discussing topologically protected surface states that emanate from a PWP in a photonic crystal analog. This underscores that the existence of PWPs, including the peculiar transport properties of associated bulk and surface states, only depends on the underlying symmetry irrespective of the particular physical realization. The theory applies to all linear and self-consistent physical systems with time reversal invariance and chiral cubic symmetry, with dynamics described by a Fourier integral over Hilbert states $|v(\omega)\rangle\in\mathcal{H}$ which individually solve a homogeneous (generally non-linear) frequency domain eigenproblem $M(\omega)\,|v(\omega)\rangle = 0$. Symmetry requires that the $\mathcal{H}$ operator $M$ commutes with all elements of the underlying space group $\mathcal{G}$, represented by $\mathcal{H}$ operators $g$. As a consequence, a set of $N$ degenerate eigenvectors $|v_n(\omega)\rangle$ form a *representation* of $\mathcal{G}$, i.e. they span an $N$-dimensional vector space with an associated algebra that is homomorphic to $\mathcal{G}$. *Irreducible representations* impose a lower limit on the dimensionality of the respective vector space, resulting in deterministic degeneracies [@Bradley_GT; @Dresselhaus; @PhysRevB.88.245116]. [Weyl\_dispersion.png]{} (1,38)[[[()]{}]{}]{} (66,38)[[[()]{}]{}]{} First order degenerate perturbation theory and representation theory, the latter of which provides the selection rules for the matrix elements within the former, allows to derive the slopes of the bandstructure at determinstic points of degeneracy (see supplementary material for details). For deterministic three-fold degeneracies at the $\Gamma$ point, this procedure yields a perturbation matrix $W({\ensuremath{\bm{k}}})$ which is valid for small $k\ll 2\pi/a$ (with lattice constant $a$) and time reversal invariance $$W_{\alpha\beta} ({\ensuremath{\bm{k}}}) = \imath d \sum_\gamma \epsilon_{\alpha\beta\gamma}\, k_\gamma\text{ ,} \label{eq:M_pert}$$ with a free parameter $d\in\mathbb{R}$ and $\alpha, \beta, \gamma$ iterating over the three partners of the irreducible representation with ${\ensuremath{\bm{k}}}=0$ that span the degenerate eigenspace. Note the similarity of $W_{\alpha\beta}$ to the Weyl Hamiltonian $\mathcal{W}_{ij}$: the Pauli matrices $\sigma_{ij}^{(\gamma)}$ ($i,j\in\left\{ 1,2 \right\}$) that occur in the latter are here replaced by the 3D Levi-Civita tensor $\epsilon_{\alpha\beta\gamma}$. The first order perturbation eigenvalues corresponding to $W({\ensuremath{\bm{k}}})$ are $k_0^{(1)}:=\omega^{(1)}/c=\{0,\pm d k\}$: they only depend on the absolute value of ${\ensuremath{\bm{k}}}$ and describe isotropic hyperconic dispersion. In the following, we shall define a PWP as the exceptional point $(0,k_0^{(0)})$ at which the two Weyl hypercones $({\ensuremath{\bm{k}}},k_0^{(0)}\pm d k)$ in the four dimensional $({\ensuremath{\bm{k}}},k_0)$ parameter space meet. Although the bandstructure does not support a frequency with vanishing density of states due to the flat band, this definition is justified from a topological perspective: the correlated Chern numbers can be analytically calculated when integrating the Berry curvature over a small sphere in ${\ensuremath{\bm{k}}}$ space for each of the three bands. They evaluate to $C=0$ for the flat band and $C=\pm2$ for the two hyperconic bands, showing a non-trivial topological signature, similar to a genuine Weyl point with Chern numbers $C=\pm 1$. Analysing all 3D space groups [@0792365909], it is straightforward to show that deterministic PWPs at the center of the BZ require chiral cubic symmetry. Interestingly, the trigonal groups $P312$ (149) and $P321$ (150) have two-dimensional representations which split into an anisotropic hypercone if time inversion is present, albeit not at the $\Gamma$ point [@2015arXiv151204681C]. A closely related matter is the non-existence of deterministic Dirac points at the $\Gamma$ point of two-dimensional crystals, including the famous honeycomb lattice [@PhysRevB.89.134302]. To demonstrate the predicted behaviour we construct a chiral cubic sphere packing which is minimalistic in the sense that it generates the lowest dimensional vector space possible in models based on for example tight binding or pair interaction. A periodic sphere packing can be constructed by placing spheres on the Wyckoff point of multiplicity $N$ within a given space group [@0792365909]. Following this procedure for any non-symmorphic chiral cubic space group $\mathcal{G}$ and Wyckoff multiplicities smaller than $12$ yields a sphere packing that has the symmetry of an achiral supergroup $\mathcal{G}_S$. This counter-intuitive behaviour is related to the isotropy of the sphere as seed object that allows to introduce additional spurious (irregular) rotations. These unwanted rotations can be suppressed by the finite translation part in non-symmorphic symmetries. A seed sphere on the $4a$ Wyckoff point $(x,x,x)$ of $\mathcal{G}=P2_13$ thus induces a chiral cubic sphere packing with $\mathcal{G}_S=\mathcal{G}$ and only $4$ spheres in the unit cell. Only if $x = n/4$ ($n\in\mathbb{Z}$), the sphere packing acquires the symmetry of the achiral supergroup $Pa\bar{3}$ (205); the chiral supergroup $P4_332$ (212) is induced for $x=1/8+n/2$, and $P4_132$ (213) for $x=-1/8+n/2$. Note that the introduction of an octahedral isogonal point symmetry instead of the tetrahedral symmetry of $P2_13$ in these cases does not impose a change of bandstructure behaviour close to $\Gamma$. [Fig. \[fig:spheres\]]{} illustrates the sphere packing for $x=3/8\,a$ ($P4_132$ symmetry). To elucidate the physics (ahead of a concrete experimental realization) let us consider an effective plasmonic model consisting of metallic nano-spheres of radius $\rho$ in vacuum (as in [@Han2009; @PhysRevLett.110.106801]). The position ${\ensuremath{\bm{r}}}_i$ of sphere $i$ shall be such that the distance $d_{ij}=|{\ensuremath{\bm{r}}}_i-{\ensuremath{\bm{r}}}_j|\gg \rho$ for any pair of spheres $(i,j)$. In the quasistatic approximation, Maxwell’s equations take the self-consistent form (acting on the dipole moments ${{\ensuremath{\bm{p}}}_i}$) [@Draine:94]: $${\ensuremath{\bm{p}}}_i = \alpha(k_0)\sum_{j\ne i} \mathcal{G}({\ensuremath{\bm{r}}}_i-{\ensuremath{\bm{r}}}_j,k_0){\ensuremath{\bm{p}}}_j\text{ .} \label{eq:dipoles}$$ Here, $\alpha(k_0)=\rho^3(1-3k_0^2/k_p^2)^{-1}$ is the polarizability of a metallic sphere in vacuum, that is modelled by a non-dissipative Drude response with plasma wave number $k_p$; $\mathcal{G}({\ensuremath{\bm{r}}},k_0)$ is the dyadic Green function for the monochromatic Maxwell wave operator. If the spheres are arranged periodically as introduced above ([*cf.*]{}[Fig. \[fig:spheres\]]{}), the index $i$ is conveniently substituted by a multi-index $({\ensuremath{\bm{n}}},\mu)\in\mathbb{Z}^3\times\{1,2,3,4\}$, with ${\ensuremath{\bm{r}}}_{{\ensuremath{\bm{n}}},\mu}={\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}}+{\ensuremath{\bm{r}}}_\mu$ given by the sum of the lattice vector ${\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}}=a\,{\ensuremath{\bm{n}}}$ and the position within the unit cell ${\ensuremath{\bm{r}}}_\mu$. Bloch’s theorem then implies ${\ensuremath{\bm{p}}}_{{\ensuremath{\bm{n}}},\mu} = {\ensuremath{\bm{p}}}_\mu \exp\{ \imath {\ensuremath{\bm{k}}}\cdot{\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}} \}$, so that [Eq. \[eq:dipoles\]]{} reduces to a family of $12$-dimensional non-linear Hermitian eigenproblems: $$\alpha^{-1}(k_0)\, {\ensuremath{\bm{p}}}_\mu = \sum_{\nu} M_{\mu\nu}({\ensuremath{\bm{k}}},k_0)\,{\ensuremath{\bm{p}}}_\nu\text{ .} \label{eq:eigen}$$ Numerical challenges related to the convergence of the lattice sum $M_{\mu\nu}({\ensuremath{\bm{k}}},k_0)$ and their solution are solved in the supplementary material. Since the matrix $M$ generally imposes a small perturbation to the single sphere resonance solution $K_0^{(n)} := \sqrt{3}\,k_0^{(n)}/k_p = 1$ (due to $\rho^{-3}\gg 1$ in [Eq. \[eq:eigen\]]{}), the eigenvalue problem is linearized by approximating $M_{\mu\nu}({\ensuremath{\bm{k}}},k_0)\approx M_{\mu\nu}({\ensuremath{\bm{k}}},k_p/\sqrt{3})=:M_{\mu\nu}({\ensuremath{\bm{k}}})$. This assumption is inadequate close to the Ewald sphere $k_0=|{\ensuremath{\bm{k}}}|$, caused by poles in the diagonal entries of $M({\ensuremath{\bm{k}}}=k_0{\ensuremath{\bm{\hat{k}}}})$, however, only affecting the two modes at the top and the bottom of the bandstructure on either side of the pole, [*cf.*]{}dashed red line in [Fig. \[fig:bandstruct\]]{}[[()]{}]{}. The eigenvalues $\lambda_n({\ensuremath{\bm{k}}})=\alpha^{-1}(k_0)$ ($n=1,2,\dots,N$) of $M({\ensuremath{\bm{k}}})$ can be obtained numerically with low computational cost. They produce the respective dispersion relation $K_0^{(n)}({\ensuremath{\bm{k}}}) = [1-\rho^3\lambda_n({\ensuremath{\bm{k}}})]^{1/2}$, as shown in [Fig. \[fig:bandstruct\]]{} for parameters $x/a=0.175$, $k_0a/(2\pi)=0.1$ and $\rho/a=0.1$. [Fig. \[fig:bandstruct\]]{}[[()]{}]{} shows an example within our model, where the first order perturbation outweighs higher orders even for relatively large Bloch wave number $k\approx\pi/(5a)$, so that an almost perfect hypercone can be observed. We find that, in contrast to this 3D representation, all first order perturbation matrix elements vanish for two-fold degeneracies at the $\Gamma$ point for any space group, cubic or non-cubic ($K_0=0.995$ in [Fig. \[fig:bandstruct\]]{}[[()]{}]{} constitutes an example). These exceptional points are henceforth not lifted in first order and are topologically trivial, with Chern number $C=0$ in both bands. The universality of our results is vividly demonstrated if we replace the small metallic spheres by larger spheres of radius $\rho/a=0.25$ (fill fraction of $\pi/12\approx 26\%$), made of a high refractive index material with $n=4$, thus constructing a photonic crystal analog. The associated bandstructure (calculated with MPB [@Johnson2001:mpb]) close to $k_0a/(2\pi)=0.5$ (supplementary figure $1$) resembles [Fig. \[fig:bandstruct\]]{}. A partial band gap opens in the projected bulk bandstructure with respect to a $[001]$ inclination in [Fig. \[fig:surface\_modes\]]{}[[()]{}]{}: this is the blue area of all $({\ensuremath{\bm{k}}}_\parallel,k_0)$ for which at least one bulk mode exists for arbitrary $k_z\in\mathbb{R}$ [@JoannopoulosJohnsonWinnMeade:2008]. Since the PWP degeneracy as well as the four-fold degeneracy at $R$ (projected onto $A$) is protected by cubic symmetry, this gap can be opened completely by e.g. perturbing the sphere positions (supplementary figure $2$). Topological surface states exist in the band gap at the interface between two enantiomorphic structures (with same bulk bands, but opposite chirality and Chern characteristics): [Fig. \[fig:surface\_modes\]]{}[[()]{}]{} shows the surface mode dispersion of $12$ unit cells of a right handed crystal ($x/a=0.175$) and $12$ unit cells of a left handed crystal ($x/a=-0.175$) stacked in $[001]$ direction in a supercell geometry. The space group of the supercell is monoclinic with $P2_1/c$ (14) symmetry (note, however, that the Bravais lattice is tetragonal). [surface\_bandstruct.png]{} (1,72)[[[()]{}]{}]{} (103,72)[[[()]{}]{}]{} (58,53)[**1**]{} (58,58)[**2**]{} (58,65)[**3**]{} (58,69)[**4**]{} (67.5,53.3)[**1**]{} (67.5,57.5)[**2**]{} (67.5,63.5)[**3**]{} (67.5,68)[**4**]{} The supercell symmetry including time reversal requires all modes along $Z-A-X$ ([Fig. \[fig:surface\_modes\]]{}[[()]{}]{}, left inset) to be two-fold degenerate. However, the space group representations along $\Gamma-Z$ and $X-\Gamma$ (including $\Gamma$ itself) are 1D. The surface states still stick together. This can be understood as follows: consider a single interface surface mode along $\Gamma-Z$. Along this path, ${\ensuremath{\bm{k}}}$ is invariant under the two-fold (screw) rotation $(2)$ in $P2_1/c$ [@0792365909] (note that our $x$ axis corresponds to their $y$ axis). This screw rotation further maps the field profile from one interface to the other, so that a 1D representation has to have intensities of equal magnitude on both interfaces. Since the two interfaces are well separated by a zero field bulk region, Maxwell’s equations are also satisfied for the same frequency by a field that is non-zero at one of the two interfaces only. The mode must thus be two-fold degenerate. Close to the $\Gamma$ point, the decay length becomes larger than $6$ unit cells, so that the argument breaks down and the degeneracy is lifted. For $X-\Gamma$, the same line of thought applies to the glide plane $(4)$. [Fig. \[fig:surface\_modes\]]{}[[()]{}]{} demonstrates that the modes within the bulk gap are indeed localized at the surface, in contrast to the bulk modes within the blue region. There are two degenerate mode pairs at ${\ensuremath{\bm{k}}}=0.2\pi/a\times(1,0,0)^T$ (brown points $2$ and $3$ in [Fig. \[fig:surface\_modes\]]{}[[()]{}]{}, only one states is shown, respectively). The degeneracy splits for ${\ensuremath{\bm{k}}}=0.2\pi/a\times(\cos(\phi),\sin(\phi),0)^T$ with arbitrary $\phi$ (green points for $\phi=0.28\pi$). We have thus shown that surface states exist. But are these also of topological nature? The conventional path $\Gamma{-}Z{-}A{-}X{-}\Gamma$ does not reveal the topological nature of the four surface states emanating from the PWP. To show that these are, indeed, protected, we follow [@Wan2011] and consider the cylinder ${\ensuremath{\bm{k}}}(\varphi,k_z)=(k\cos(\varphi),k\sin(\varphi),k_z)^T$ (with constant $k$ and $-\pi/a<k_z\le\pi/a$, $0<\varphi\le2\pi$). This cylinder constitutes a closed surface (a torus) in ${\ensuremath{\bm{k}}}$-space within which the bandstructure exhibits a band gap, so that a gap Chern number (sum over all bands below the gap) is well defined. The change in gap Chern number $|\Delta C|$ across an interface equals the number of topologically protected surface states that connect the bulk bands below the gap with those above [@PhysRevA.78.033834; @Lu2014]. The gap Chern number for the above torus and a hyperconic band at a PWP is given by $|C|=2$, as shown above (note that only the Chern number of the PWP at the gap frequency needs to be considered). This results in $8$ surface bands for the supercell geometry with two $|\Delta C|=4$ interfaces, four of which are observed along the half cylinder in [Fig. \[fig:surface\_modes\]]{}[[()]{}]{}: each of these bands touches and connects the projected bulk bands above and below the gap and thus is, veritably, protected. In this letter, we have shown that isotropic hyperconic dispersion can be found at the $\Gamma$-point of chiral cubic photonic lattices. The associated exceptional PWP points exhibit the topological characteristics of a double Weyl point. A natural application for the unique dispersion behaviour of these PWPs are zero refractive index materials that have been suggested previously in the context of accidental Dirac points in two-dimensional photonic crystals [@Huang2011]. The localization of associated protected surface states and their flatness yield a gigantic density of states, making PWP systems an ideal starting point to explore topological lasing applications [@Harari:16] in three dimensions. This work was supported by the EPSRC through program grant EP/L027151/1. We would like to thank Dr. Paloma Arroyo Huidobro, Gleb Siroki and Prof. Sir John Pendry for helpful discussions. [31]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1103/PhysRevLett.45.494) [****,  ()](\doibase 10.1103/PhysRevLett.49.405) [****,  ()](\doibase 10.1103/PhysRevB.48.11851) [****,  ()](\doibase http://dx.doi.org/10.1016/0003-4916(85)90148-4) [****,  ()](\doibase 10.1103/RevModPhys.83.1057) [****, ()](\doibase 10.1103/RevModPhys.82.3045) [****,  ()](\doibase 10.1103/PhysRevB.93.241402) [****,  ()](\doibase 10.1038/ncomms12375) [****,  ()](\doibase 10.1103/PhysRevLett.93.083901) [****,  ()](\doibase 10.1103/PhysRevA.78.033834) [****,  ()](\doibase 10.1038/nphoton.2014.248) [****,  ()](\doibase 10.1038/nphoton.2013.42) [****,  ()](\doibase 10.1103/PhysRevB.89.134302) [****,  ()](\doibase 10.1103/PhysRevLett.116.186402) [**** (), 10.1126/science.aaf5037](\doibase 10.1126/science.aaf5037) @noop [**]{} (, ) , ed., @noop [**]{}, Vol.  (, ) [****,  ()](\doibase 10.1016/0022-3697(57)90013-6) [****,  ()](\doibase 10.1038/nmat3030) @noop [**]{} (, ) [****,  ()](\doibase 10.1103/PhysRevB.88.245116) @noop [**]{} (, , ) @noop [ ()]{},  [****,  ()](\doibase 10.1103/physrevlett.102.123904) [****,  ()](\doibase 10.1103/PhysRevLett.110.106801) [****,  ()](\doibase 10.1364/JOSAA.11.001491) [****,  ()](http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173) @noop [**]{} (, , ) [****,  ()](\doibase https://doi.org/10.1016/j.commatsci.2010.05.010) [****,  ()](\doibase 10.1103/physrevb.83.205101) in [**](\doibase 10.1364/CLEO_QELS.2016.FM3A.3) (, ) p.  [****, ()](\doibase 10.1103/PhysRev.52.361) @noop [**]{} (, ) @noop [**]{} (, , ) [****,  ()](\doibase 10.1002/andp.19213690304) [****, ()](\doibase 10.1103/PhysRev.92.569) [****,  ()](\doibase 10.1109/TAP.2004.838793) The canonical representation of the plasmonic sphere packing at $\Gamma$ \[app:canonical\_rep\] =============================================================================================== The existence of three-fold degeneracies at the $\Gamma$ point in chiral tetrahedral and octahedral space groups is enumerated in the literature [@Bradley_GT]. This general results stems from the fact that space group representations at $\Gamma$ can be written as a direct product between the representation of the isogonal point group and the trivial representation of the translation group, [*i.e.*]{}$$\Delta_{{\ensuremath{\bm{k}}}=0,i} (\left\{ p_\alpha|t_\alpha+T \right\}) := D_i ( p_\alpha )\text{ .}$$ It is straightforward to verify that $\Delta_{{\ensuremath{\bm{k}}}=0,i}$ is an irreducible representation of the space group if $D_i$ is an irreducible representation of the isogonal point group. There are therefore 3D irreducible representations ($T$) for any chiral cubic space group associated with ${\ensuremath{\bm{k}}}=0$ (rigorous analysis [@Bradley_GT] shows that there are no others apart from the above): we tabulate the characters for tetrahedral point group in table \[tab:23\_characters\], and refer to [@PhysRevB.88.245116] for the octahedral group. This generic finding is very useful for the application of the perturbation theory below. Additionally, we here derive the canonical representation of the plasmonic sphere packing and thereby the exact split of the underlying $12$ dimensional vector space into irreducible representations. $23$ ${\ensuremath{\mathlarger{\mathbbm{1}}}}$ $3C_2$ $4C_3$ $4C_3'$ FS ------- ------------------------------------------- -------- ------------ ------------ ----- $A$ 1 1 1 1 (a) $E_+$ 1 1 $\omega$ $\omega^2$ (b) $E_-$ 1 1 $\omega^2$ $\omega$ (b) $T$ 3 -1 0 0 (a) : Character table for the $23$ tetrahedral point group, with $\omega:=\exp\left\{ 2\imath\pi/3 \right\}$. The last column lists the Frobenius Schur type of the representation. \[tab:23\_characters\] The representation matrix of the vector space $V$ corresponding to different spheres (within the equivalence class of translations) is given by permutation matrices. The following table lists the spheres $\nu$ onto which a sphere $\mu$ is mapped under the symmetry operation $\alpha$ in $P2_13$. The symmetry indices are adapted from [@0792365909]. ------------------------ ----- --- --- --- --- --- --- --- --- ---- ---- ---- Class $E$ $\mu \setminus \alpha$ 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 1 4 2 3 1 3 4 2 2 2 1 4 3 4 1 3 2 3 1 2 4 3 3 4 1 2 2 3 1 4 4 2 1 3 4 4 3 2 1 3 2 4 1 2 4 3 1 Trace $\chi$ 4 ------------------------ ----- --- --- --- --- --- --- --- --- ---- ---- ---- The space within which the polarisation vector is rotated is the three-dimensional Euclidean space $W$. We only list the relevant trace (or character) of the rotation matrix $\chi(\phi)=1+2\cos(\phi)$ here, where $\phi$ denotes the angle of rotation. The characters of the symmetry operations in $V$ and $W$, and of the canonical representation for the system vector space $V\otimes W$ are summarised below. Class $E$ $3C_2$ $4C_3$ $4C_3'$ --------------------- ----- -------- -------- --------- $\chi^V$ 4 0 1 1 $\chi^W$ 3 -1 0 0 $\chi^{V\otimes W}$ 12 0 0 0 The characters of the irreducible representations of the associated tetrahedral point group are shown in [Tab. \[tab:23\_characters\]]{}. Note that we use the Schoenflies notation $23$ in this section in order to avoid confusion with the symbol $T$ that is conventionally used for three-dimensional irreducible representations. The canonical representation at the $\Gamma$ point reduces into $$A+E_++E_-+3T$$ (where $(+)$ denotes a direct sum operation in this context) according to the reduction formula [@Dresselhaus] $$n_i = \frac{1}{N} \sum_R {\ensuremath{\,\overline{\chi\vphantom{\bar{\chi}}}\,}}(R)\,\chi_i(R)\text{ ,} \label{eq:reduction}$$ where $n_i$ quantifies how often an irreducible representation $i$ is contained in the reducible representation with characters $\chi(R)$ (${\ensuremath{\,\overline{\cdot\vphantom{\bar{\cdot}}}\,}}$ denotes complex conjugation), $N$ is the number of elements $R$ in the group, and $\chi_i(R)$ is the character of the irreducible representation. Note that the representation $E_\pm$ form a two-fold degenerate pair $E$ because the system is also invariant under time inversion, as evident from the Schur type $(b)$ in [Tab. \[tab:23\_characters\]]{} here. For details on time inversion symmetry in addition to space group symmetries see for example Herring [@PhysRev.52.361] and Frobenius [@Frobenius], page 354 ff. The predicted split is thus reproduced in Fig. 2 in the main manuscript: the bandstructure shows one trivial $A$ representation, one 2D $E$ representation, and three 3D $T$ representations, two of which produce genuine hyperconic dispersion, while the third is degenerate in first order and corresponds to the special case $d=0$ below. Perturbation theory and selection rules \[app:pert\] ==================================================== Reduction into the irreducible representations of the respective subgroup of the space group $\mathcal{G}$ as described in [@PhysRevB.88.245116] predicts how and whether a degenerate mode splits when going away from that point. If not perturbed along a high symmetry direction in reciprocal space it always splits into 1D representations as the Bloch representations, characterized by ${\ensuremath{\bm{k}}}$, with respect to the invariant subgroup of translations generate a full star, with the same dimension as the underlying point group. We are here however concerned with the order of splitting: for linear Weyl dispersion, the first order cannot vanish. We here derive an approach that is analogous to, but more general than the well established ${\ensuremath{\bm{k}}}\cdot{\ensuremath{\bm{p}}}$ perturbation theory [@Dresselhaus]. Consider a generic non-linear eigenproblem $$M(\lambda) {\ensuremath{\bm{v}}} = 0\text{ ,} \label{eq:generic_eigen}$$ characterized by an operator $M(\lambda)$ that commutes with all operators of $\mathcal{G}$. The eigenvectors ${\ensuremath{\bm{v}}}$ are partners $({\ensuremath{\bm{k}}}_n,\alpha)$ of the irreducible representations of the space group enumerated by $({\ensuremath{\bm{k}}},i)$. In this context, ${\ensuremath{\bm{k}}}$ is a Bloch wave vector in the irreducible Brillouin zone and $i$ denotes different induced representations from the small representations of the associated little group, whereas ${\ensuremath{\bm{k}}}_n$ iterates over different representatives in the star of ${\ensuremath{\bm{k}}}$ and $\alpha$ over partners of the small representations of the little group. We hence write: $${\ensuremath{\bm{v}}} = {\ensuremath{\bm{v}}}^{(i)}_{\alpha}({\ensuremath{\bm{k}}},{\ensuremath{\bm{k}}}_n)\text{ .}$$ Working in the spatial Fourier domain, we can suppress the explicit ${\ensuremath{\bm{k}}}_n$ dependence in the eigenvector, dividing by the phase factor $\exp\left\{ \imath {\ensuremath{\bm{k}}}_n\cdot{\ensuremath{\bm{r}}} \right\}$ corresponding to a translation group representation. At an arbitrary point within the BZ we define $$\exp\left\{ -\imath {\ensuremath{\bm{k}}}_n\cdot{\ensuremath{\bm{r}}} \right\} {\ensuremath{\bm{v}}}^{(i)}_{\alpha}({\ensuremath{\bm{k}}},{\ensuremath{\bm{k}}}_n) \equiv |{\ensuremath{\bm{k}}}i,n\alpha\rangle\text{ .}$$ Importantly, these are partners $\alpha$ of the irreducible representations of the little group at ${\ensuremath{\bm{k}}}_n$. The eigenvalue equation becomes $$M({\ensuremath{\bm{k}}}_n,\lambda) |{\ensuremath{\bm{k}}}i,n \alpha\rangle = 0\text{ .}$$ For ${\ensuremath{\bm{k}}}_n^{(1)} = {\ensuremath{\bm{k}}}_n^{(0)}+\delta{\ensuremath{\bm{k}}}$, with small $|\delta{\ensuremath{\bm{k}}}| \ll \pi/a$, and hence $\lambda_1=\lambda_0+\delta\lambda$, a first order Taylor expansion of $M$ around $M({\ensuremath{\bm{k}}}_n^{(0)},\lambda_0)$ and an expansion of the eigenfunctions into the aforementioned partners $\alpha$ ([*cf.*]{}representation theorem (iii) in [@PhysRevB.88.245116]) yields: $$\begin{aligned} \Big( M({\ensuremath{\bm{k}}}_n^{(0)},\lambda_0) & \notag \\ + \delta{\ensuremath{\bm{k}}}\cdot\left[\nabla_{{\ensuremath{\bm{k}}}}M({\ensuremath{\bm{k}}},\lambda_0)\right]_{{\ensuremath{\bm{k}}}={\ensuremath{\bm{k}}}_n^{(0)}} & \notag\\ + \delta\lambda\left[\partial_{\lambda}M({\ensuremath{\bm{k}}}_n^{(0)},\lambda)\right]_{\lambda=\lambda_0} \Big) & \notag\\ \sum_\alpha c_\alpha |{\ensuremath{\bm{k}}}i,n\alpha \rangle & = 0\text{ ,} \label{eq:pert}\end{aligned}$$ where we have assumed that $i$ labels the irreducible representation corresponding to the eigenvalue $\lambda_0$ and used the fact that $M({\ensuremath{\bm{k}}}_n^{(0)},\lambda_0)|{\ensuremath{\bm{k}}}j,n\beta\rangle\ne 0$ for any representation $j\ne i$ ([*i.e.*]{} we exclude accidental degeneracies), so that a finite coefficient would contradict [Eq. \[eq:pert\]]{} for arbitrary but small $\delta{\ensuremath{\bm{k}}}$. In the weak form, [Eq. \[eq:pert\]]{} reads: $$\sum_\beta{\ensuremath{\bm{W}}}_{\alpha\beta}\cdot\delta{\ensuremath{\bm{k}}}\,c_\beta = -\sum_\beta \delta\lambda\, E_{\alpha\beta} c_\beta\text{ ,} \label{eq:pert_weak}$$ with $${\ensuremath{\bm{W}}}_{\alpha\beta} := \langle {\ensuremath{\bm{k}}}i,n\alpha| \left[\nabla_{{\ensuremath{\bm{k}}}}M({\ensuremath{\bm{k}}},\lambda_0)\right]_{{\ensuremath{\bm{k}}}={\ensuremath{\bm{k}}}_n^{(0)}}|{\ensuremath{\bm{k}}}i,n\beta\rangle$$ and $$E_{\alpha\beta} := \langle {\ensuremath{\bm{k}}}i,n\alpha| \left[\partial_{\lambda}M({\ensuremath{\bm{k}}}_n^{(0)},\lambda)\right]_{\lambda=\lambda_0} |{\ensuremath{\bm{k}}}i,n\beta\rangle\text{ .}$$ The partial derivative $\partial\lambda$ is trivially invariant under all operations $P$ of the little group at ${\ensuremath{\bm{k}}}_n^{(0)}$. $M$ is invariant under all space group operations, and $P\in\mathcal{G}$, while $P|{\ensuremath{\bm{k}}}i,n\alpha\rangle = \sum_\beta D_{\alpha\beta}(P)|{\ensuremath{\bm{k}}}i,n\beta\rangle$. Hence, the action on $W$ and $E$ is generally given by $$\begin{aligned} P E_{\alpha\beta} &= \sum_{\gamma\delta} {\ensuremath{\,\overline{D\vphantom{\bar{D}}}\,}}_{\alpha\gamma}(P) D_{\beta\delta}(P)\, E_{\gamma\delta} \\ P W^{(n)}_{\alpha\beta} &= \sum_{\gamma\delta,m} {\ensuremath{\,\overline{D\vphantom{\bar{D}}}\,}}_{\alpha\gamma}(P) D_{\beta\delta}(P) R_{nm}(P)\, W^{(m)}_{\gamma\delta}\text{ ,}\end{aligned}$$ where $R(P)$ denotes the Rodrigues’ matrix representation that acts in the standard 3D Euclidean vector space. Since $\delta{\ensuremath{\bm{k}}}$ is arbitrary, according to [Eq. \[eq:pert\_weak\]]{}, ${\ensuremath{\bm{W}}}$ and $E$ are also required to be invariant under all $P$. Introducing the respective direct product space, $E$ and $W$ must hence for all $P$ be in the kernel of $\Delta(P)$ and $\Lambda(P)$, respectively, where $$\begin{aligned} \Delta_{(\alpha\beta),(\gamma\delta)}(P) :=& {\ensuremath{\,\overline{D\vphantom{\bar{D}}}\,}}_{\alpha\gamma}(P) D_{\beta\delta}(P) - \delta_{\alpha\gamma}\delta_{\beta\delta}\label{eq:ker_del}\\ \Lambda_{(\alpha\beta n),(\gamma\delta m)}(P) := &{\ensuremath{\,\overline{D\vphantom{\bar{D}}}\,}}_{\alpha\gamma}(P) D_{\beta\delta}(P) R_{nm}(P)\notag\\ &- \delta_{\alpha\gamma}\delta_{\beta\delta}\delta_{nm}\label{eq:ker_del}\text{ .}\end{aligned}$$ This reveals the group theoretically allowed form of $E$ and $W$, substantially reducing their degrees of freedom and providing selection rules for many of their entries. $E$ and $W$ are respectively non-zero only if $\Delta+{\ensuremath{\mathlarger{\mathbbm{1}}}}$ and $\Lambda+{\ensuremath{\mathlarger{\mathbbm{1}}}}$ contain the trivial representation if decomposed into the irreducible representations of $i$ via [Eq. \[eq:reduction\]]{} (representation theorem (ii) in [@PhysRevB.88.245116]), [*i.e.*]{}$$\begin{aligned} \frac{1}{\sum_P}\sum_P {\ensuremath{\,\overline{\chi_D\vphantom{\bar{\chi_D}}}\,}}(P)\chi_D(P) &\ne 0 \label{eq:char_del}\\ \frac{1}{\sum_P}\sum_P {\ensuremath{\,\overline{\chi_D\vphantom{\bar{\chi_D}}}\,}}(P)\chi_D(P)\text{Tr}\{R(P)\} &\ne 0 \text{ .} \label{eq:char_lam}\end{aligned}$$ More precisely, the integer number on the left side of [Eq. \[eq:char\_del\]]{} and [Eq. \[eq:char\_lam\]]{} is equal to the dimension of the intersection of the nullspaces of $\Delta$ and $\Lambda$, respectively, and hence specifies the degrees of freedom in $E$ and $W$ (but not their form). Perturbation theory at $\Gamma$ \[app:pert\_Gamma\] =================================================== Consider now a perturbation around the centre of the BZ using the equations derived in the previous section. The isogonal little group is here the isogonal point group of the lattice, [*i.e.*]{}for chiral cubic symmetry either tetrahedral ($23$) or octahedral ($432$). We here discuss the tetrahedral symmetry, but note that the main results regarding PWPs are exactly the same for the octahedral point group. Consider first a non-degenerate mode with trivial irreducible representation $A$ (see table \[tab:23\_characters\]). While [Eq. \[eq:char\_del\]]{} is satisfied, [Eq. \[eq:char\_lam\]]{} is not. $W$ is therefore trivial, with all its elements vanishing and [Eq. \[eq:pert\_weak\]]{} yielding $\delta\lambda = 0$. A non-degenerate mode is henceforth flat at the centre of the BZ in a cubic symmetry, even in the absence of inversion or time reversal symmetry. Note, that the $E_\pm$ modes show the same behaviour, so that a deterministic two-fold degenerate mode, as present in the case of time reversal only in tetrahedral symmetries is also predicted to be flat at the centre of the BZ. Further note that the two-fold $E$ representation in the octahedral point group (table 3.33 in [@Dresselhaus]) also yields a flat band. The representation $T$ in table \[tab:23\_characters\] (the Rodrigues’ representation itself) that corresponds to a three-fold degeneracy, however, satisfies both [Eq. \[eq:char\_del\]]{} and [Eq. \[eq:char\_lam\]]{}, with $\text{dim}(\text{ker}(\Delta))=1$ and $\text{dim}(\text{ker}(\Lambda))=2$. We compute the intersection of the kernels of $\Delta(P)$ and $\Lambda(P)$ for all $P$ numerically, and find explicitly for the $T$ degeneracy that $E$ is a scalar matrix $$\begin{aligned} E = c {\ensuremath{\mathlarger{\mathbbm{1}}}}\end{aligned}$$ and that ${\ensuremath{\bm{W}}}$ is given by $${\ensuremath{\bm{W}}} = d_1 \begin{pmatrix} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \\ \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \\ \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \\ \end{pmatrix} + d_2 \begin{pmatrix} \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \\ \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \\ \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \\ \end{pmatrix}\text{ .}$$ In the presence of time reversal symmetry we further require that ${\ensuremath{\,\overline{M\vphantom{\bar{M}}}\,}}(-{\ensuremath{\bm{k}}})=M({\ensuremath{\bm{k}}})$, and hence $c\in\mathbb{R}$ and $W_\gamma$ Hermitian, so that [Eq. \[eq:pert\_weak\]]{} corresponds to the perturbation matrix given in eq. $1$ of the main manuscript: $$\imath d \sum_{\beta\gamma} \varepsilon_{\alpha\beta\gamma}\, \delta k_\gamma\,c_\beta = \delta\lambda\, c_\alpha\text{ ,}$$ where we set [w.l.o.g.]{}$d_1=-\imath d$, $d_2=\imath d$ with $d\in\mathbb{R}$ and $c=-1$. Chern numbers for the hyperconic bands \[app:Chern\] ==================================================== We here calculate the Chern numbers for the three bands that meet at the PWP. We start from the perturbation matrix (Eq. $1$ in the main manuscript, derived above) $$W_{\alpha\beta} ({\ensuremath{\bm{k}}}) = \imath\,d \sum_\gamma \epsilon_{\alpha\beta\gamma}\, k_\gamma\text{ .}$$ To study the topological nature of the PWP, it is convenient to calculate the Chern number over a small sphere in reciprocal space, with the PWP at its centre. We first introduce spherical coordinates with ${\ensuremath{\bm{k}}}:=k(\cos{\phi}\sin{\theta},\sin{\phi}\sin{\theta},\cos{\theta})^T$ and associated basis vectors $\{{\ensuremath{\bm{\hat{k}}}},{\ensuremath{\bm{\hat{\theta}}}},{\ensuremath{\bm{\hat{\phi}}}}\}$. With the eigenvectors expanded as ${\ensuremath{\bm{v}}}=v_k{\ensuremath{\bm{\hat{k}}}}+v_\theta{\ensuremath{\bm{\hat{\theta}}}}+v_\phi{\ensuremath{\bm{\hat{\phi}}}}$, the perturbation equation $$\imath d k\, {\ensuremath{\bm{\hat{k}}}}\times{\ensuremath{\bm{v}}} = \lambda {\ensuremath{\bm{v}}}$$ has the solution set $$\left[ \lambda, \begin{pmatrix} v_k \\ v_\theta \\ v_\phi \end{pmatrix} \right] = \left\{ \left[ 0, \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \right], \left[ \pm dk, \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ 1 \\ \pm\imath \end{pmatrix} \right] \right\}\text{ .}$$ We note that ${\ensuremath{\bm{v}}}$ above is the vector over the coefficients $c_\alpha$ of partners $|\alpha\rangle$ of the irreducible $T$ representation. These are by definition orthonormalized and ${\ensuremath{\bm{k}}}$ independent. Topologically, the discussion of any physical Hilbert space therefore collapses to the three-dimensional vector space of the coefficients. The Gaussian curvature $$K = dt_1\wedge dt_2\,\left[(\partial_2{\ensuremath{\bm{e}}}_1,\partial_1{\ensuremath{\bm{e}}}_2) - (\partial_1{\ensuremath{\bm{e}}}_1,\partial_2{\ensuremath{\bm{e}}}_2)\right]$$ is a measure of intrinsic curvature of the tangent space of a surface spanned by local basis vectors ${\ensuremath{\bm{e}}}_1$ and ${\ensuremath{\bm{e}}}_2$. The Berry curvature is a similar measure concerning a generic complex vector field $|v({\ensuremath{\bm{x}}})\rangle$ that is attached to a manifold $\mathcal{M}$ parametrized by ${\ensuremath{\bm{x}}}$. It can be conveniently written as the two-form ![Bulk bandstructure for the photonic crystal sphere packing with same parameters as in figure $3$ of the main manuscript.[]{data-label="fig:bulk"}](bulk_bandstruct.png){width=".48\textwidth"} $$\mathcal{B} = \imath \text{Tr}\{P dP\wedge dP\}\text{ ,}$$ with the projector $P:=|v\rangle\langle v|$ and its exterior derivative $dP=\sum_i dx_i \partial_{x_i}P$ in the tangent space of $\mathcal{M}$. Note that the wedge product does not vanish as $P$ is an operator, but the Berry curvature is easily shown to be real. In our case, the manifold is 2D, with $x_1=\theta$ and $x_2=\phi$, so that the Berry curvature reduces to $$\mathcal{B} = 2\, d\theta \wedge d\phi \, {\text{Im}\left\{\langle \partial_\phi v | \partial_\theta v \rangle\right\}} \text{ .} \label{eq:Berry}$$ Similar to the Euler number $\chi:=(2\pi)^{-1}\int_\mathcal{M} K$ that approaches integer values for any closed surface and is directly related to its genus, the Chern characteristics $$C = \frac{1}{2\pi}\oint_\mathcal{M} \mathcal{B}$$ is also quantized and constitutes a topological invariant of the vector field. The flat band eigenvector ${\ensuremath{\bm{v}}}={\ensuremath{\bm{\hat{k}}}}$ is real and trivially leads to a vanishing Berry curvature when substituted into [Eq. \[eq:Berry\]]{}. The eigenvectors of the hyperconic bands ${\ensuremath{\bm{v}}}_\pm = ({\ensuremath{\bm{\hat{\theta}}}} \pm \imath {\ensuremath{\bm{\hat{\phi}}}})/\sqrt{2}$ on the other hand yield a Berry curvature of $$\mathcal{B} = \pm\, d\theta \wedge d\phi \, \left( \partial_\theta {\ensuremath{\bm{\hat{\theta}}}}, \partial_\phi {\ensuremath{\bm{\hat{\phi}}}} \right) = \pm \sin(\theta)\, d\theta \wedge d\phi\text{ ,}$$ and hence a Chern characteristics of $C=\pm 2$, in direct analogy to the Euler number $\chi=2$ of the sphere. Additional figures ================== [Fig. \[fig:bulk\]]{} shows the plain bulk bandstructure of the photonic crystal sphere packing close to the PWP of interest around $k_0a/(2\pi)=0.5$. In [Fig. \[fig:pert\]]{} the sphere positions in the unit cell are distorted, thereby destroying cubic symmetry and both degeneracies at the $\Gamma$ and at the $R$ point. We note that, despite the fact that the Bravais lattice remains simple cubic by construction, the path followed in [Fig. \[fig:pert\]]{} is thus in principle inappropriate, but kept for the sake of comparability with [Fig. \[fig:bulk\]]{}. A full bandgap is, however, observed for the particular realization in the whole (triclinic) Brillouin zone. Opening the gap does not change the topological characteristics of the bands above and below the gap. Frequency isolated protected surface states can hence be found for the perturbed structure. ![Bulk bandstructure for a perturbed photonic crystal sphere packing with same parameters as in figure $3$ of the main manuscript, except that the sphere positions are shifted by random vectors $\delta{\ensuremath{\bm{r}}}$, with $r_i$ uniformly distributed between $0$ and $0.1a$.[]{data-label="fig:pert"}](distorted_bulk_bandstruct.png){width=".48\textwidth"} The eigenvalue equation for the sphere packing \[app:spheres\] ============================================================== Since a proper derivation of Eq. 1 of the main manuscript, reproduced here for clarity $${\ensuremath{\bm{p}}}_i = \alpha(k_0)\sum_{j\ne i} \mathcal{G}({\ensuremath{\bm{r}}}_i-{\ensuremath{\bm{r}}}_j,k_0){\ensuremath{\bm{p}}}_j\text{ ,} \label{eq:dipoles}$$ seems to be absent in the literature, we here rigorously show its validity in the assumed limits. We model the dielectric matrix intrinsically via an effective displacement field with the corresponding permittivity $\varepsilon_d$, and the polarisation ${\ensuremath{\bm{P}}}_m$ of the metallic spheres explicitly. The macroscopic Maxwell equations for monochromatic fields can then be transformed into a wave equation for the electric field ${\ensuremath{\bm{E}}}$ and the polarisation field ${\ensuremath{\bm{P}}}_m$, that vanishes in the dielectric matrix: $$\underbrace{-(4\pi k_0^2)^{-1}\, \left( \kappa^2 - \text{curl}^2 \right)}_{:=\mathcal{M}}{\ensuremath{\bm{E}}} = {\ensuremath{\bm{P}}}_m\text{ .} \label{eq:Maxwell}$$ We have here introduced the wave number $\kappa:=\sqrt{\varepsilon_d} k_0$ and the Maxwell operator $\mathcal{M}$ for convenience. [Eq. \[eq:Maxwell\]]{} can be inverted using the dyadic Green function $$\mathcal{G}_\mathcal{M}({\ensuremath{\bm{r}}}) = -4\pi k_0^2\, \left( {\ensuremath{\mathlarger{\mathbbm{1}}}}+ \frac{\nabla\otimes\nabla}{\kappa^2} \right) \mathcal{G}_\mathcal{H}(r) \label{eq:Green}$$ that solves the equation $\mathcal{M}\mathcal{G}_\mathcal{M}(r) = {\ensuremath{\mathlarger{\mathbbm{1}}}}\delta^{(3)}({\ensuremath{\bm{r}}})$, with the (retarded) Helmholtz Green function $\mathcal{G}_\mathcal{H}(r) = -(4\pi r)^{-1}\exp\left\{ \imath\kappa r \right\}$ that solves $$\left( \kappa^2 + \Delta \right)\mathcal{G}_\mathcal{H} = \delta^{(3)}({\ensuremath{\bm{r}}})\text{ .}$$ We assume that the metallic material response $\varepsilon_m(k_0)$ is known and frequency dependent. Therefore, with ${\ensuremath{\bm{E}}} = 4\pi/(\varepsilon_m-\varepsilon_d)\,{\ensuremath{\bm{P}}}_m$, [Eq. \[eq:Maxwell\]]{} transforms into a relation on the metallic subdomain only: $$\frac{4\pi}{\varepsilon_m-\varepsilon_d}\,{\ensuremath{\bm{P}}}_m({\ensuremath{\bm{r}}}) = \int d^3r'\,\mathcal{G}_\mathcal{M}({\ensuremath{\bm{r}}}-{\ensuremath{\bm{r}}}')\,{\ensuremath{\bm{P}}}_m({\ensuremath{\bm{r}}}') \label{eq:inverseMaxwell}$$ The polarisation field inside a single sphere can be considered constant for sufficiently small frequencies, in zero order in $k_0\rho \ll 1$.[^1] We hence make the ansatz $${\ensuremath{\bm{P}}}_m({\ensuremath{\bm{r}}}) = \frac{1}{V_S}\sum_i {\ensuremath{\bm{p}}}_i \chi({\ensuremath{\bm{r}}}-{\ensuremath{\bm{r}}}_i)\;\text{ , with }\chi({\ensuremath{\bm{r}}})= \begin{cases} 1 & \text{if } |{\ensuremath{\bm{r}}}| < \rho\\ 0 & \text{else} \end{cases}$$ and $V_S=(4\pi\rho^3)/3$ the sphere volume. Substitution into [Eq. \[eq:inverseMaxwell\]]{} and testing with the basis functions $\chi({\ensuremath{\bm{r}}}-{\ensuremath{\bm{r}}}_i)$ yields an equation for the dipole moments ${\ensuremath{\bm{p}}}_i$ only: $$\frac{3\rho^3}{\varepsilon_m-\varepsilon_d} {\ensuremath{\bm{p}}}_i = \sum_j I_\rho({\ensuremath{\bm{r}}}_i,{\ensuremath{\bm{r}}}_j)\,{\ensuremath{\bm{p}}}_j\text{ .} \label{eq:invpolMaxwell}$$ Here, $$I_\rho({\ensuremath{\bm{r}}}_i,{\ensuremath{\bm{r}}}_j) = \frac{1}{V_S^2}\int_{\mathcal{B}_\rho({\ensuremath{\bm{r}}}_i)}d^3r\;\int_{\mathcal{B}_\rho({\ensuremath{\bm{r}}}_j)}d^3r'\;\mathcal{G}_\mathcal{M}({\ensuremath{\bm{r}}}-{\ensuremath{\bm{r}}}')$$ is a double integral over balls $\mathcal{B}_\rho({\ensuremath{\bm{r}}})$ with radius $\rho$ at position ${\ensuremath{\bm{r}}}$. For any two spheres $i\ne j$ and in the limit of small spheres $\rho \ll d_{ij}$, $I_\rho({\ensuremath{\bm{r}}}_i,{\ensuremath{\bm{r}}}_j)$ reduces to $V_S\, \mathcal{G}_\mathcal{M}({\ensuremath{\bm{r}}}_i-{\ensuremath{\bm{r}}}_j)$. For $i=j$, we use the following theorem. [**Theorem I:**]{} Let $$I = \int_{\mathcal{B}_\rho}d^3r\;\int_{\mathcal{B}_\rho}d^3r'\; f({\ensuremath{\bm{x}}})$$ be a double integral over balls (at the origin) $\mathcal{B}_\rho$ of a function $f({\ensuremath{\bm{x}}})$ that only depends on the difference in coordinates ${\ensuremath{\bm{x}}}={\ensuremath{\bm{r}}}-{\ensuremath{\bm{r}}}'$. The integral can then be expressed as a single integral over a Ball $\mathcal{B}_{2\rho}$ $$I = \frac{\pi}{12}\int_{\mathcal{B}_{2\rho}}d^3x\;(4\rho+x)(2\rho-x)^2\,f({\ensuremath{\bm{x}}})\text{ .}$$\ [**Proof:**]{} A change of coordinates $({\ensuremath{\bm{r}}},{\ensuremath{\bm{r}}}')\mapsto({\ensuremath{\bm{x}}},{\ensuremath{\bm{R}}}:={\ensuremath{\bm{r}}}+{\ensuremath{\bm{r}}}')$ yields: $$I = \frac{1}{8} \int_{\mathcal{B}_{2\rho}}d^3x\;f({\ensuremath{\bm{x}}})\underbrace{\int_{\mathcal{V}({\ensuremath{\bm{x}}})}d^3R}_{=:V[\mathcal{V}({\ensuremath{\bm{x}}})]}\text{ ,}$$ where, with the binary operator $(+)$ on spatial domains denoting the Minkowski sum, $$\mathcal{V}({\ensuremath{\bm{x}}}) = \left( \mathcal{B}_\rho \cap \mathcal{B}_\rho({\ensuremath{\bm{x}}}) \right) + \left( \mathcal{B}_\rho \cap \mathcal{B}_\rho(-{\ensuremath{\bm{x}}}) \right)\text{.}$$ Using the translation invariance of the volume and a mixed volume expansion [@burago1988geometric], we arrive at: $$\begin{aligned} V[\mathcal{V}({\ensuremath{\bm{x}}})] &= V\left[ \left( \mathcal{B}_\rho \cap \mathcal{B}_\rho({\ensuremath{\bm{x}}}) \right) + \left( \mathcal{B}_\rho \cap \mathcal{B}_\rho(-{\ensuremath{\bm{x}}}) \right) \right] \\ &= V\left[ \left( \mathcal{B}_\rho \cap \mathcal{B}_\rho({\ensuremath{\bm{x}}}) \right) + \left( \mathcal{B}_\rho \cap \mathcal{B}_\rho({\ensuremath{\bm{x}}}) \right) \right] \\ &= 8 V\left[ \mathcal{B}_\rho \cap \mathcal{B}_\rho({\ensuremath{\bm{x}}}) \right] \\ &= \frac{2\pi}{3}\left( 2\rho-x \right)^2 \left( 4\rho+x \right) \text{ .}\end{aligned}$$ In the last line we used that the volume is twice that of a spherical cap with sphere radius $\rho$ and height $\rho-x/2$. The proposition follows from resubstitution. $\square$ Applying theorem I with $f({\ensuremath{\bm{x}}}) = \mathcal{G}_\mathcal{M}({\ensuremath{\bm{x}}})$ and the low frequency condition on the Green function yields the simplified expression $$\mathcal{G}_\mathcal{M}^{(\alpha\beta)}({\ensuremath{\bm{x}}}) = \varepsilon_d^{-1}\partial_\alpha\partial_\beta\frac{1}{x}\text{ ,}$$ with only two independent entries $\mathcal{G}_\mathcal{M}^{(zz)}$ and $\mathcal{G}_\mathcal{M}^{(xz)}$. The integral over the latter entry vanishes as the integrand is antisymmetric in both the azimuth $\phi$ and the polar angle $\theta$. The former integral evaluates to the simple expression $$I^{(zz)}({\ensuremath{\bm{r}}}_i,{\ensuremath{\bm{r}}}_i) = -\rho^{-3}\,\varepsilon_d^{-1}\text{ .}$$ Substitution of these results into [Eq. \[eq:invpolMaxwell\]]{} yields $$\underbrace{\frac{\varepsilon_m+2\varepsilon_d}{\varepsilon_m-\varepsilon_d}\rho^{-3}}_{:=\alpha^{-1}}\, {\ensuremath{\bm{p}}}_i = \varepsilon_d \sum_{j\ne i} \mathcal{G}_\mathcal{M} ({\ensuremath{\bm{r}}}_i-{\ensuremath{\bm{r}}}_j){\ensuremath{\bm{p}}}_j\text{ .}$$ For spheres made of a lossless Drude material $\varepsilon_m(k_0)=1-k_p^2/k_0^2$, embedded in vacuum $\varepsilon_d=1$, this equation is identical to [Eq. \[eq:dipoles\]]{}. We note that this describes an assembly of point dipoles with polarisability $\alpha$, that are interacting via the Maxwell Greens function without self-interaction. This simple picture apparently describes the correct physics in the assumed limits, although it is not possible to derive [Eq. \[eq:dipoles\]]{} within its scope. The lattice sum =============== Substituting the Bloch form ${\ensuremath{\bm{p}}}_{{\ensuremath{\bm{n}}},\mu} = {\ensuremath{\bm{p}}}_\mu \exp\left\{ \imath{\ensuremath{\bm{k}}}\cdot{\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}} \right\}$ into [Eq. \[eq:dipoles\]]{} yields $$\sum_{m,\nu}\mathcal{G}({\ensuremath{\bm{r}}}_\mu-{\ensuremath{\bm{r}}}_\nu+{\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}}-{\ensuremath{\bm{T}}}_m)\,e^{\imath{\ensuremath{\bm{k}}}\cdot({\ensuremath{\bm{T}}}_m-{\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}})}\,{\ensuremath{\bm{p}}}_\nu = \alpha^{-1}\,{\ensuremath{\bm{p}}}_\mu\text{ ,}$$ where we define $\mathcal{G}(0):=0$ to simplify the notation. Eq. 2 from the main manuscript, namely $$\alpha^{-1}(k_0)\, {\ensuremath{\bm{p}}}_\mu = \sum_{\nu} M_{\mu\nu}({\ensuremath{\bm{k}}},k_0)\,{\ensuremath{\bm{p}}}_\nu\text{ ,} \label{eq:eigen}$$ results from a lattice vector shift ${\ensuremath{\bm{T}}}_m\mapsto{\ensuremath{\bm{T}}}_m+{\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}}$, with the matrix $$M_{\mu\nu}({\ensuremath{\bm{k}}}) := \sum_{{\ensuremath{\bm{n}}}} \mathcal{G}_\mathcal{M}({\ensuremath{\bm{r}}}_\mu-{\ensuremath{\bm{r}}}_\nu-{\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}})e^{\imath{\ensuremath{\bm{k}}}\cdot{\ensuremath{\bm{T}}}_{{\ensuremath{\bm{n}}}}}\text{ .} \label{eq:eigen_matrix}$$ This numerical lattice sum that is truncated by for example $|{\ensuremath{\bm{n}}}|<N$ converges very slowly to the infinite series [Eq. \[eq:eigen\_matrix\]]{}.[^2] Convergence is generally guaranteed by the alternating phase factor, but the rate of convergence is governed by the long range $r^{-3}$ interaction from the Green function, and can only be algebraic. This problem is well known and dates back to Ewald [@Ewald_lattice_sum], and also appears in the study of the electronic structure in crystals [@PhysRev.92.569]. In order to circumvent the numerically expensive and inaccurate real space representation in [Eq. \[eq:eigen\_matrix\]]{}, we follow an approach introduced by Silveirinha and Fernandes [@Silve2005] for the Helmholtz Green function $\mathcal{G}_\mathcal{H}({\ensuremath{\bm{r}}})$. Their main idea is to split the lattice sum into real space contribution and a spectral contribution, using a radial weight function $f(r)$ that approaches $1$ at exponential (or even Gaussian) rate for $r\rightarrow\infty$ and vanishes at the origin $r=0$, thereby splitting $M=Mf(r)+M[1-f(r)]$. The first term is then effectively calculated in real space, whereas the second term converges rapidly in the spectral representation, for which a plane wave expansion yields a summation over reciprocal vectors ${\ensuremath{\bm{G}}}$ instead of the real space vectors ${\ensuremath{\bm{T}}}$. This technique can be trivially generalised to the dyadic case exploiting [Eq. \[eq:Green\]]{}. To highlight the importance of this method, we note that the procedure enables us to obtain converged solutions, with relative errors in the eigenvalues at machine precision, for cut-offs as small as $N=5$. For comparison, the naive representation [Eq. \[eq:eigen\_matrix\]]{} yields relative errors as large as $10^{-2}$ for $N=100$. Under these circumstances, numerical summation over almost $10^7$ elements per Bloch wave vector makes band structure calculations slow, while still inaccurate and suffering from ringing artefacts. On the contrary, the eigenvalue problem itself is of dimension $12$ in our example and comes at virtually no computational cost. Hamilton formulation and energy flow \[app:Hamilton\] ===================================================== Consider the coupled harmonic oscillator Hamiltonian $$H = \frac{1}{2} \left[ \sum_n \left( {\ensuremath{\bar{\bm{\Pi}}}}_n^2 + \omega_r^2 {\ensuremath{\bar{\bm{p}}}}_n^2 \right) + \omega_r^2 \sum_{nm} {\ensuremath{\bar{\bm{p}}}}_n^T I_{nm}^2 {\ensuremath{\bar{\bm{p}}}}_m \right] \label{eq:Ham}$$ with the dipole moments ${\ensuremath{\bar{\bm{p}}}}_n$ taking the role of canonical coordinates, with conjugated momenta ${\ensuremath{\bar{\bm{\Pi}}}}_n = \frac{d{\ensuremath{\bar{\bm{p}}}}_n}{dt}$.[^3] The canonical coordinates and momenta are proportional to the centre of charge with respect to the corresponding sphere centre and the average charge flow, respectively (see also [@PhysRevLett.110.106801]). With the single particle resonance $\omega_r:=\omega_p/\sqrt{3}$ and the interaction matrix given by $I_{nm}:=-\rho^3\mathcal{G}({\ensuremath{\bm{r}}}_n-{\ensuremath{\bm{r}}}_m)$ for $n\ne m$ and $I_{nn}=0$, it is straightforward to show that the equations of motion of this Hamiltonian reduce to [Eq. \[eq:dipoles\]]{} in the monochromatic case. The Hamilton formulation has the advantage, that we can easily derive expressions for the average energy flow of the plasmonic modes. The single particle energy $E_i=\frac{1}{2}\left({\ensuremath{\bar{\bm{\Pi}}}}_i^2 + \omega_r^2 {\ensuremath{\bar{\bm{p}}}}_i^2\right)$ has the time derivative $$\dot{E}_i = -\omega_r^2 \sum_n {\ensuremath{\bar{\bm{\Pi}}}}_i^T I_{in}\, {\ensuremath{\bar{\bm{p}}}}_n\text{.}$$ That means that the amount of energy that flows from point $i$ to point $j$ per time is given by $-\omega_r^2 {\ensuremath{\bar{\bm{\Pi}}}}_i^TI_{ij}\,{\ensuremath{\bar{\bm{p}}}}_j$. At the same time, $-\omega_r^2{\ensuremath{\bar{\bm{\Pi}}}}_j^T I_{ji}\,{\ensuremath{\bar{\bm{p}}}}_i=-\omega_r^2{\ensuremath{\bar{\bm{p}}}}_i^T I_{ij}\,{\ensuremath{\bar{\bm{\Pi}}}}_j$ obviously flows from $j$ to $i$. The total power that is transferred from $i$ to $j$ is hence given by the difference of these two contributions (as physically required, it is antisymmetric by construction): $$P_{ij} = \omega_r^2 \left( {\ensuremath{\bar{\bm{p}}}}_i^T I_{ij}\,{\ensuremath{\bar{\bm{\Pi}}}}_j - {\ensuremath{\bar{\bm{\Pi}}}}_i^T I_{ij}\,{\ensuremath{\bar{\bm{p}}}}_j \right)\text{ .}$$ The symmetric energy flow between $i$ and $j$ can now be defined as $${\ensuremath{\bm{S}}}_{ij} = P_{ij}\left( {\ensuremath{\bm{r}}}_j - {\ensuremath{\bm{r}}}_i \right)\text{ .}$$ The total energy flow of a system of spheres, averaged over time for the monochromatic field is then given by $$\langle{\ensuremath{\bm{S}}}\rangle_t = \omega_r^2 \frac{\omega}{4}\sum_{nm} \text{Im}\left\{ {\ensuremath{\bm{p}}}_n^\dagger I_{nm}\, {\ensuremath{\bm{p}}}_m \right\} \left( {\ensuremath{\bm{r}}}_m-{\ensuremath{\bm{r}}}_n \right)\text{ ,} \label{eq:flow}$$ whereas the time average of the total energy is $$\begin{aligned} \langle E \rangle_t = &\frac{1}{4} \big[ \left( \omega^2+\omega_r^2 \right)\sum_n |{\ensuremath{\bm{p}}}_n|^2 \notag \\ & + \omega_r^2\sum_{nm} \text{Re}\left\{ {\ensuremath{\bm{p}}}_n^\dagger I_{nm} {\ensuremath{\bm{p}}}_m \right\} \big]\text{ .} \label{eq:energy}\end{aligned}$$ The mode velocity is then given by $${\ensuremath{\bm{v}}} = \frac{\langle {\ensuremath{\bm{S}}}\rangle_t}{\langle E \rangle_t} \label{eq:velocity}$$ For the periodic system, we can replace $\sum_n\mapsto N\sum_\mu$ and $\sum_{nm}\mapsto N\sum_{\mu\nu}$ in [Eq. \[eq:energy\]]{}, if we also replace $I_{nm}\mapsto I_{\mu\nu}:=-\rho^3 M_{\mu\nu}$ and ${\ensuremath{\bm{p}}}_n\mapsto{\ensuremath{\bm{p}}}_\mu$. The procedure is similar to the one that led to [Eq. \[eq:eigen\_matrix\]]{} above. For the flow, we have $$\begin{aligned} &I_{nm}({\ensuremath{\bm{r}}}_m-{\ensuremath{\bm{r}}}_n)_i\\ \mapsto I_{\mu\nu}^{(i)} &:= -\rho^3\sum_n \left[ {\ensuremath{\bm{r}}}_\mu-{\ensuremath{\bm{r}}}_\nu-{\ensuremath{\bm{T}}}_n \right]_i \mathcal{G}\left( {\ensuremath{\bm{r}}}_\mu-{\ensuremath{\bm{r}}}_\nu-{\ensuremath{\bm{T}}}_n \right)e^{\imath{\ensuremath{\bm{k}}}\cdot{\ensuremath{\bm{T}}}_n}\\ &\phantom{:}= -\rho^3\left( \left[ {\ensuremath{\bm{r}}}_\mu-{\ensuremath{\bm{r}}}_\nu \right]_i M_{\mu\nu} + \imath\partial_{k_i} M_{\mu\nu} \right)\end{aligned}$$ so that, in the lossless Drude case, the average energy velocity reduces to the simple expression for the group velocity, as is well known for photonic crystals [@JoannopoulosJohnsonWinnMeade:2008] $${\ensuremath{\bm{v}}} = \omega_r\,\nabla_{{\ensuremath{\bm{k}}}}K_0({\ensuremath{\bm{k}}}) = \nabla_{{\ensuremath{\bm{k}}}}\omega({\ensuremath{\bm{k}}})\text{ ,} \label{eq:group_velocity}$$ where we have used [Eq. \[eq:eigen\]]{} and the Hellmann-Feynman theorem. Note that for any more sophisticated material model $\alpha(k_0)$, the substitution of [Eq. \[eq:flow\]]{} and [Eq. \[eq:energy\]]{} into [Eq. \[eq:velocity\]]{} yields the general form $${\ensuremath{\bm{v}}} = \frac{-2\text{Im}\left\{ \alpha^{-1} \right\} \frac{\sum_\mu {\ensuremath{\bm{r}}}_\mu|{\ensuremath{\bm{p}}}_\mu|^2}{\sum_\mu |{\ensuremath{\bm{p}}}_\mu|^2}-\nabla_{{\ensuremath{\bm{k}}}}\left( \text{Re}\left\{ \alpha^{-1} \right\} - \text{Im}\left\{ \alpha^{-1} \right\} \right)}{\left( 1+K_0^2 \right) - \text{Re}\left\{ \alpha^{-1} \right\}} k_0\,c\text{ ,}$$ for which we implicitly assume $\alpha^{-1}=\alpha^{-1}(K_0({\ensuremath{\bm{k}}}))$. This expression does not reduce to the group velocity [Eq. \[eq:group\_velocity\]]{}, even for a more general lossless dispersion. [^1]: Note that this condition does not impose a strong constraint on $k_0$ itself, since we assume small spheres $\rho\ll \min(d_{\mu\nu}) < a$. [^2]: Precisely speaking, it does not converge at all in the vicinity of Ewald’s sphere $|{\ensuremath{\bm{k}}}|=k_0$ due to a ringing like phenomenon (spatial and temporal dispersion counteract and prevent the summands from alternating in one direction). [^3]: Note that the dipole moments ${\ensuremath{\bm{p}}}_n$ in [Eq. \[eq:dipoles\]]{} denote the phasors of monochromatic physical dipole moments ${\ensuremath{\bar{\bm{p}}}}_n$. Further note that the above Hamiltonian has the odd dimension $(\text{charge}\times\text{length}/\text{time})^2$. This has no direct implication on the following, particularly on the dimensions of the mode velocity. However, scaling the coordinates with $\sqrt{m}/e$, with the effective mass $m$ and the charge $e$ of the microscopic charges, yields correct physical dimensions.

--- abstract: 'We give a pedagogical introduction to quantum discord. We the discuss the problem of separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy as a distance measure of correlations. This allows us to put all correlations on an equal footing. Entanglement and dissonance, whose definition is introduced here, jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We finally show, using relative entropy, how different notions of quantum correlations are related to each other. This gives a single theory that incorporates all correlations, quantum, classical, etc.' author: - Kavan Modi - Vlatko Vedral bibliography: - 'gencorr.bib' title: Unification of quantum and classical correlations and quantumness measures --- Introduction ============ Quantum systems are correlated in ways inaccessible to classical objects. A distinctive quantum feature of correlations is quantum entanglement [@PhysRev.47.777; @Schrodinger:1935eq; @PhysRev.48.696]. Entangled states are nonclassical as they cannot be prepared with the help of local operations and classical communication (LOCC) [@horodecki:865]. However, it is not the only aspect of nonclassicality of correlations due to the nature of operations allowed in the framework of LOCC. To illustrate this, one can compare a classical bit with a quantum bit; in the case of full knowledge about a classical bit, it is completely described by one of two locally distinguishable states, and the only allowed operations on the classical bit are to keep its value or flip it. To the contrary, quantum operations can prepare quantum states that are indistinguishable for a given measurement. Such operations and classical communication can lead to separable states (those which can be prepared via LOCC) which are mixtures of locally indistinguishable states. These states are nonclassical in the sense that they cannot be prepared using classical operations on classical bits. Recent measures of nonclassical correlations are motivated by different notions of classicality and operational means to quantify nonclassicality [@henderson01a; @PhysRevLett.88.017901; @PhysRevLett.89.180402; @PhysRevA.72.032317; @luo:022301; @PhysRevA.80.032319; @PhysRevLett.104.080501]. Quantum discord has received much attention in studies involving thermodynamics and correlations [@zurek; @PhysRevA.71.062307; @Rodriguez07a; @Devi-2008a; @dattashaji; @piani; @mazzola; @PhysRevLett.105.190502; @PhysRevA.83.020101; @cavalcantietal; @pianietal]. These works are concerned with understanding the role of quantumness of correlations in a variety of systems and tasks. In some of the studies, it is also desirable to compare various notions of quantum correlations. It is well known that the different measures of quantum correlation are not identical and conceptually different. For example, the discord does not coincide with entanglement or measurement induced disturbance and a direct comparison of any two of these notions can be rather meaningless. Therefore, an unified classification of correlations is in demand as well as a unification of different notions of quantumness. In this article, using relative entropy, we resolve some of these issues by introducing measures for classical and nonclassical correlations for quantum states under a single theory. Our theory further allows us to connect different notions of quantumness. This will allow us to generalize all measures of quantumness for multipartite systems in symmetric and asymmetric manners. We begin with a pedagogical introduction to quantumness of correlations. Conditional entropy =================== The story of quantumness of correlations beyond-entanglement begins with the non-uniqueness of quantum conditional entropy. In classical probability theory, conditional entropy is defined as $$\begin{gathered} H(b|a)=H(ab)-H(a).\label{clce1}\end{gathered}$$ It is the measure ignorance of $b$ has given some knowledge of state of $a$. Fig. \[condent\]) depicts this relationship in a graphical manner. Another way to express the conditional entropy is as the ignorance of $b$ when the state of $a$ is known to be in the $i$th state, weighted by the probability for $i$th outcome as $$\begin{gathered} H(b|a)=\sum_{i} p^a_i H(b|a=i)\label{clce2},\end{gathered}$$ It is the classical-equivalency of Eqs. \[clce1\] and \[clce2\] give rise to quantumness of correlations and in specific quantum discord [@datta08]. This is due to the fact that these two equations are not the same in quantum theory. While the first simply takes difference in the joint ignorance and the ignorance of $a$, the second formula depends on specific outcomes of $a$, which requires a measurement. Measurements in quantum theory are basis dependent and change the state of the system. In generalizing the classical concepts above to quantum we replace joint classical-probability distributions with density operators and Shannon’s with von Neumann’s entropy. How do we deal with conditional entropy then? Clearly there are two options: Eqs. \[clce1\] and \[clce2\]. Let is deal with Eq. \[clce1\] first and define quantum conditional entropy as $$\begin{gathered} \label{qce1} S^{(1)}(B|A)=S(AB)-S(A).\end{gathered}$$ This is well known quantity in quantum information theory [@Schumacher] and negative of this quantity is known as coherent information. However, this is a troubling quantity as it can be negative for entangled states and for a long time there was no way to interpret the negativity. This is in start contrast with the classical conditional entropy which has a clear interpretation and is always positive. On the other hand, we can give define the quantum version of Eq. \[clce2\] by making measurements on party $A$. To put in the details, the joint state $\rho^{AB}$ is measured by $A$ giving $i$th outcome: $$\begin{gathered} \rho^{AB}\rightarrow \sum_i \Pi^A_i \rho^{AB} \Pi^A_i = \sum_i p_i {| i \rangle}{\langle i |} \otimes \rho^B_i,\end{gathered}$$ where $\Pi_i$ are rank one *positive operator values measures*, ${| i \rangle}$ are classical flags on measuring apparatus indicating the measurement outcome, $p_i=\mbox{Tr}[\Pi^A_i \rho^{AB}]$ is probability of $i$th outcome, $\rho^B_i=\mbox{Tr}_A[\Pi^A_i\rho^{AB}]$. The conditional entropy of $B$ is then clearly defined as $$\begin{gathered} \label{qce2} S^{(2)}(B|A)=S(B|\{\Pi_i\})=\sum_i p_i S(\rho^B_i),\end{gathered}$$ This definition of conditional entropy is always positive. The obvious problem with this definition is that the state $\rho^{AB}$ changes after the measurement. Also note that this quantity is not symmetric under party swap. A different approach to conditional entropy is taken in [@PhysRevLett.79.5194; @PhysRevA.60.893], where a quantum *conditional amplitude* (analogous to classical conditional probability) is defined such that it satisfies Eq. \[qce1\]. We only mean to suggest that the two approaches above are not the only options available. Different approaches give us different distinctions of quantum theory from the classical theory. And in someway different notions of quantumness. Quantumness of correlations =========================== Clearly the two definition of conditional entropies above are different in quantum theory. The first one suffers from negativity and second one is ‘classicalization’ of a quantum state. Let us now derive quantum discord and relate it to the preceding section. We start with the concept of mutual information: $$\begin{gathered} I(a:b)=H(a)+H(b)-H(ab)\end{gathered}$$ and using Eq. \[clce2\] $$\begin{gathered} J(b|a)=H(b)-\sum_{i} p^a_i H(b|a=i).\end{gathered}$$ Clearly the two classical mutual information above are the same, but not in quantum theory. This is precisely what was noted by Ollivier and Zurek, and they called the difference between $I$ and $J$ *quantum discord*: $$\begin{aligned} \delta(B|A)=I(A:B)-J(B|A).\end{aligned}$$ Working out the details one finds that quantum discord is simply, $$\begin{aligned} \delta(B|A)=S^{(2)}(B|A)-S^{(1)}(B|A),\end{aligned}$$ the difference in two definition of conditional entropy. Henderson and Vedral [@henderson01a] had also looked at $J(B|A)$ called it classical correlations. In fact they advocated to that $\max_{\{\Pi_i\}}J(B|A)$ to be the classical correlations. Which meant that quantum discord is best defined as $$\begin{aligned} \delta(B|A)=\min_{\{\Pi_i\}}[I(A:B)-J(B|A)].\end{aligned}$$ Since conditional entropy in Eq. \[qce2\] is asymmetric under party swap, quantum discord is also asymmetric under party swap. A side note should be made at this point. The interpretation of negativity of quantum conditional entropy in Eq. \[qce1\] was given in terms of task known as state merging [@merging], and we will see shortly that a similar task gives quantum discord an operational meaning. While the minimum of Eq. \[qce2\] over all POVM is related to entanglement of formation between $B$ and $C$, a purification of $AB$: $E_F(BC) =\min_{\{\Pi_i\}} S^{(2)}(B|A)$ [@KoashiWinter]. Putting the two together leads to an task dependent operation interpretation of quantum discord [@cavalcantietal]. Barring the details, we can say that quantum discord between $A$ and $B$, as measured by $A$ is equal to the consumption of entanglement in state preparation of $BC$ plus state merging in those two parties. Additionally, state merging and other tasks that involve conditional entropies are asymmetric under party swap and a natural interpretation of asymmetry of quantum discord arises. The minimization over all POVM of quantum is not an easy problem to deal with in general. A similar quantity called *measurement induced disturbance* (MID) was introduced to deal with this difficulty. MID is defined as the difference in the mutual information of a joint state, $\rho^{AB}$ and it’s dephased version $\chi^{AB}$. The dephasing takes place in the marginal basis, leaving the marginal states unchanged: $$\begin{aligned} MID=I(\rho^{AB})-I(\chi^{AB})=S(\rho^{AB})-S(\chi^{AB}).\end{aligned}$$ We will come back to MID later in the article. Unification of correlations =========================== Both of the measures above are defined in terms of mutual information and therefore very difficult to generalize for multipartite case [@arXiv:1006.5784]. Below we will get over that hurdle by examining classical states, states that have no quantum correlations. It is easy verify that a state has zero discord and MID simultaneously. What that means is that such a state has equal value for conditional entropies in Eqs. \[qce1\] and \[qce2\]. Such a state is called a *classical state* and has the form $$\begin{gathered} \chi^{AB}=\sum_i p_i {| i \rangle}{\langle i |}\otimes\rho^B_i\end{gathered}$$ when measurements are made by $A$ and $\chi^{AB}=\sum_j \rho^A_j\otimes p_j {| j \rangle}{\langle j |}$ when measurements are made by $B$. It is then easy to see that a symmetric classical state must have the form $$\begin{gathered} \chi^{AB}=\sum_{ij} p_{ij} {| ij \rangle}{\langle ij |}.\end{gathered}$$ Further the conditional amplitude defined in [@PhysRevLett.79.5194; @PhysRevA.60.893] is not a density operator and may behave strangely. In [@PhysRevA.81.062103] it is shown that the conditional amplitude reduces to a density operator when the state is classical. Based on the definition of classical states we may now introduce a measure of quantum correlations as a distance from a given state to the closest classical state. The distance from a state to a state without the desired property (e.g. entanglement or discord) is a measure of that property. For example, the distance to the closest separable state is a meaningful measure of entanglement. If the distance is measured with relative entropy, the resulting measure of entanglement is the relative entropy of entanglement [@VPRK; @VP]. Using relative entropy we define measures of nonclassical correlations as a distance to the closest classical states [@PhysRevLett.104.080501], though many other distance measures can serve just as well [@PhysRevLett.105.190502]. We call our measure of quantum correlations *relative entropy of discord*. Since all the distances are measured with relative entropy, this provides a consistent way to compare different correlations, such as entanglement, discord, classical correlations, and *quantum dissonance*, a new quantum correlation that may be present in separable states. Dissonance is a similar notion to discord, but it excludes entanglement. Lastly, here we have to make no mention of whether we want a symmetric discord measure or asymmetric, or the number of parties to be involved. We simply have to choose the appropriate classical state and no ambiguity is left. A graphical illustration is given in Fig. \[CORRELATIONS\]. Let us briefly define the types states discussed below. A product state of $N$-partite system, a state with no correlations of any kind, has the form of $\pi=\pi_1 \otimes \dots \otimes \pi_N$, where $\pi_n$ is the reduced state of the $n$th subsystem. The set of product states, $\mathcal{P}$, is not a convex set in the sense a mixture of product states may not be another product state. The set of classical states, $\mathcal{C}$, contains mixtures of locally distinguishable states $\chi = \sum_{k_n} p_{k_1 \dots k_N} {| k_1\dots k_N \rangle}{\langle k_1 \dots k_N |} = \sum_{\vec{k}} p_{\vec{k}} {| \vec{k} \rangle}{\langle \vec{k} |}$, where $p_{\vec k} $ is a joint probability distribution and local states ${| k_n \rangle}$ span an orthonormal basis. The correlations of these states are identified as classical correlations [@henderson01a; @PhysRevLett.88.017901; @PhysRevLett.89.180402; @PhysRevLett.104.080501]. Note that $\mathcal{C}$ is not a convex set; mixing two classical states written in different bases can give rise to a nonclassical state. The set of separable states, $\mathcal{S}$, is convex and contains mixtures of the form $\sigma = \sum_{i} p_i \pi_1^{(i)} \otimes \dots \otimes \pi_N^{(i)}$. These states can be prepared using only local quantum operations and classical communication [@PhysRevA.40.4277] and can possess nonclassical features [@henderson01a; @PhysRevLett.88.017901]. The set of product states is a subset of the set of classical states which in turn is a subset of the set of separable states. Finally, entangled states are all those which do not belong to the set of separable states. The set of entangled states, $\mathcal{E}$, is not a convex set either. The relative entropy between two quantum states $x$ and $y$ is defined as $S(x||y) \equiv - \mbox{tr}(x\log y)-S(x)$, where $S(x) \equiv - \mbox{tr}(x\log x)$ is the von Neumann entropy of $x$. The relative entropy is a non-negative quantity and due to this property it often appears in the context of distance measure though technically it is not a distance, e.g. it is not symmetric. In Fig. \[ALLSTATES\], we present all possible types of correlations present in a quantum state $\rho$. $T_{\rho}$ is the *total mutual information* of $\rho$ given by the distance to the closest product state. If $\rho$ is entangled, its entanglement is measured by the relative entropy of entanglement, $E$, which is the distance to the closest separable state $\sigma$. Having found $\sigma$, one then finds the closest classical state, $\chi_\sigma$, to it. This distance, denoted by $Q$, contains the rest of nonclassical correlations (it is similar to discord [@henderson01a; @PhysRevLett.88.017901] but entanglement is excluded). We call this quantity *quantum dissonance*. Alternatively, if we are interested in relative entropy of discord, $D$, then we find the distance between $\rho$ and closest classical state $\chi_{\rho}$. Summing up, we have the following nonclassical correlations: $$\begin{aligned} E =& \min_{\sigma \in \mathcal{S}} S(\rho || \sigma) \quad \textrm{(entanglement)}, \\\label{dis} D =& \min_{\chi \in \mathcal{C}} S(\rho || \chi) \quad \textrm{(quantum discord)}, \\ Q =& \min_{\chi \in \mathcal{C}} S(\sigma || \chi) \quad \textrm{(quantum dissonance)}.\end{aligned}$$ Next, we compute classical correlations as the minimal distance between a classically correlated state, $\chi$, and a product state, $\pi$: $C = \min_{\pi\in\mathcal{P}} S(\chi||\pi)$. Finally, we compute the quantities labeled $L_\rho$ and $L_\sigma$ in Fig. \[ALLSTATES\], which give us additivity conditions for correlations. (2.2,0.75) (0.55,0.7)[$\sigma$]{} (0.52,.7)[(-1,0)[.50]{}]{} (0.02,.7)[(0,-1)[.54]{}]{} (0.04,0.38)[$T_\sigma$]{} (0.56,.66)[(0,-1)[.2]{}]{} (.49,0.54)[$Q$]{} (1.15,0.7)[$\rho$]{} (1.11,.7)(-.03,0)[16]{}[(1,0)[.02]{}]{} (.65,.7)[(-1,0)[.05]{}]{} (.85,0.72)[$E$]{} (1.16,.66)[(0,-1)[.2]{}]{} (1.09,0.54)[$D$]{} (1.2,0.7)[(1,0)[.49]{}]{} (1.69,.7)[(0,-1)[.54]{}]{} (1.6,0.38)[$T_\rho$]{} (0.55,0.4)[$\chi_\sigma$]{} (0.56,.36)[(0,-1)[.2]{}]{} (0.47,0.24)[$C_\sigma$]{} (1.15,0.4)[$\chi_\rho$]{} (1.16,.36)[(0,-1)[.2]{}]{} (1.07,0.24)[$C_\rho$]{} (0.01,0.1)[$\pi_\sigma$]{} (0.09,0.11)[(1,0)[.44]{}]{} (0.28,0.04)[$L_\sigma$]{} (0.55,0.1)[$\pi_{\chi_\sigma}$]{} (1.15,0.1)[$\pi_{\chi_\rho}$]{} (1.69,0.1)[$\pi_\rho$]{} (1.67,0.11)[(-1,0)[.41]{}]{} (1.44,0.04)[$L_\rho$]{} Skipping the details of the calculations (presented in [@PhysRevLett.104.080501]) we give the final results. The surprisingly simple results and is summarized in Fig. \[ALLSTATES\]. First, we find that the closest classical state is obtained by making rank one POVM measurement on the quantum state $$\begin{gathered} \chi=\sum_{\vec{k}} {| \vec{k} \rangle}{\langle \vec{k} |} \rho {| \vec{k} \rangle}{\langle \vec{k} |},\end{gathered}$$ where ${| \vec{k} \rangle}$ is projection in space at most of dimension $d^2$. To find relative entropy of discord one has to optimize over all rank one POVM. $$\begin{gathered} D= S(\rho||\chi)=\min_{\{\Pi_i\}}S(\chi)-S(\rho).\end{gathered}$$ Therefore finding the closest classical state is still a very difficult problem and has the same challenged as faced in computing original discord. We find that all correlations (except entanglement) in Fig. \[ALLSTATES\] are given by simply taking the difference in entropy of the state at tail of the arrow from the entropy of the state at the tip of the arrow, i.e. $S(x||y)=S(y)-S(x)$ for all solid lines. Which means that a closed loop of (solid lines) yield correlations that are additive, i.e. $D+C_\rho=T_\rho+L_\rho$ or $T_\rho-C_\rho =D-L_\rho$, which is actually the original discord. We show how these two measures are related to each other below. See [@PhysRevLett.104.080501] for details presented in this section. Unifying quantumness measures ============================= Next we note, there are four fundamental elements involved in study of quantumness of correlations. The first of it is the quantum state, $\rho$. Given a quantum state we immediately have its marginals, $\pi_\rho$. The third element is the classical state $\chi$, obtained by dephasing $\rho$ in some basis. And the final element is the marginals of $\chi_\rho$. It then turns out that different measures of quantumness are different because they put different constrain in the relationships these four elements have with each other. We have illustrated this in Fig. \[ALLSTATES2\]. In Fig. \[ALLSTATES2\]a the four fundamental elements are shown. Figs. \[ALLSTATES2\]b-d show how three measures of quantumness are found using the four elements. The original discord maximizes the distance from the classical state and its marginals. This has the meaning that the classical state is least confusing from its marginals. Quantum discord is the defined as the difference in confusion of a quantum state with its marginals and the a classical state obtained from that quantum state and it confusion with its marginals. Similarly MID attempts to minimize the confusion between the classical state and marginals of the original quantum state. This has the effect that the marginals of the quantum state are the same as the marginals of the classical state. Finally relative entropy of discord is defined as the distance between a quantum state and its closest classical state. (2.2,1.2) (.15,1.1)[$\rho$]{} (.16,1.06)[(0,-1)[.2]{}]{} (.18,.94)[$T$]{} (.13,0.8)[$\pi\rho$]{} (.2,1.11)[(1,0)[.37]{}]{} (.37,1.05)[$D$]{} (.6,1.1)[$\chi_\rho$]{} (.63,1.06)[(0,-1)[.2]{}]{} (.56,.94)[$C$]{} (.22,0.81)[(1,0)[.35]{}]{} (.37,.83)[$L$]{} (.6,0.8)[$\pi_{\chi}$]{} (.15,.66)[a. Elements of ]{} (.24,.6)[quantumness ]{} (.15,0.4)[$\rho$]{} (.16,.36)[(0,-1)[.2]{}]{} (.13,0.1)[$\pi\rho=\pi_{\chi}$]{} (.2,0.41)[(1,0)[.37]{}]{} (.6,0.4)[$\chi_\rho$]{} (.62,.36)[(-2,-1)[.4]{}]{} (.44,.22)[minimize]{} (.25,0)[c. MID]{} (1.05,1.1)[$\rho$]{} (1.06,1.06)[(0,-1)[.2]{}]{} (1.03,0.8)[$\pi\rho$]{} (1.1,1.11)[(1,0)[.37]{}]{} (1.5,1.1)[$\chi_\rho$]{} (1.12,0.81)[(1,0)[.35]{}]{} (1.5,0.8)[$\pi_{\chi}$]{} (1.23,0.95)[maximize]{} (1.52,1.06)[(0,-1)[.2]{}]{} (1.12,.66)[b. Original]{} (1.21,.6)[discord]{} (1.05,0.4)[$\rho$]{} (1.06,.36)[(0,-1)[.2]{}]{} (1.03,0.1)[$\pi\rho$]{} (1.15,0.36)[minimize]{} (1.1,0.41)[(1,0)[.37]{}]{} (1.5,0.4)[$\chi_\rho$]{} (1.12,0.11)[(1,0)[.35]{}]{} (1.5,0.1)[$\pi_{\chi}$]{} (1.52,.36)[(0,-1)[.2]{}]{} (1.12,.0)[d. RED]{} We now turn our attention to show that using relative entropy we can describe (and generalize) other quantumness measures such as original *quantum discord* [@PhysRevLett.88.017901], *symmetric quantum discord* [@PhysRevA.80.032319], and *measurement induced disturbance*. [@luo:022301]. Vedral et al. [@PhysRevA.56.4452] show that quantum relative entropy has the operational meaning of being able to confuse two quantum states. The argument goes as the following: suppose you are given either $\rho$ or $\sigma$ and you have to determine which by making $N$ measurements (POVM). The probability of confusing the two states is $$\begin{gathered} P_N=e^{-N S(\rho||\sigma)}.\end{gathered}$$ Now suppose $\rho$ is an entangled state. Then for what separable state $\sigma$ can be confused for $\rho$ the most? The answer is $$\begin{gathered} P_N=e^{-N \min_{\sigma\in\mathcal{S}}S(\rho||\sigma)},\end{gathered}$$ where $\mathcal{S}$ is the set of separable states. This is the meaning of relative entropy of entanglement: $$\begin{gathered} E(\rho)=\min_{\sigma\in\mathcal{S}}S(\rho||\sigma).\end{gathered}$$ In similar manner we can give meaning to relative entropy of discord as $$\begin{gathered} D(\rho)=\min_{\chi\in\mathcal{C}}S(\rho||\chi)\end{gathered}$$ the classical state $\chi$ that imitates $\rho$ the most. The great advantage of looking at these measures in this manner is that, now they are no longer constrained to be bipartite measures. Nor they are constrained to be symmetric or asymmetric under party exchanges. We can now define quantum discord by $n-$partite systems with measurements on $m$ subsystems. Similarly, MID can be defined in such a manner as well. The other advantage is that we know how these elements are related to each other and that there are only finite number of relationships among them that make sense, e.g. maximization of distance between a quantum state and a classical state does not make sense as on may get infinity for the result. Conclusions =========== We have given a pedagogical review of ideas behind quantum correlations beyond entanglement. In doing so we were able to generalize the concepts of quantum discord to multipartite case, with no ambiguity regarding the asymmetry of quantum discord under party exchange. We have shown how three measures of quantumness can be viewed under a single formalism using relative entropy. We acknowledge the financial support by the National Research Foundation and the Ministry of Education of Singapore. We are grateful to the organizers of the *75 years of entanglement* in Kolkata for inviting our participation.

--- abstract: 'In order to understand the structure of the “typical” element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen’s notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper, we generalize the theorems of Dougherty and Mycielski about $S_\infty$ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the *Cofinal Strong Amalgamation Property*. As an application we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set.' address: - | Department of Mathematics, University of Louisville, Louisville, KY 40292, USA\ Ashoka University, Rajiv Gandhi Education City, Kundli, Rai 131029, India - 'Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary and Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, 1117 Budapest, Hungary' - 'Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, 1117 Budapest, Hungary' - 'Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary and Eötvös Loránd University, Institute of Mathematics, Pázmány Péter s. 1/c, 1117 Budapest, Hungary' - 'Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Stra[ß]{}e 25, 1090 Wien, Austria and Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary' author: - 'Udayan B. Darji' - Márton Elekes - Kende Kalina - Viktor Kiss - Zoltán Vidnyánszky bibliography: - 'ran.bib' title: The structure of random automorphisms of countable structures --- [^1] Introduction ============ The study of typical elements of Polish groups is a flourishing field with a large number of applications. The systematic investigation of typical elements of automorphism groups of countable structures was initiated by Truss [@truss1992generic]. He conjectured that the existence of a co-meager conjugacy class can be characterized in model theoretic terms. This question was answered affirmatively by Kechris and Rosendal [@KechrisRosendal]. They, extending the work of Hodges, Hodkinson, Lascar and Shelah [@shelah] also investigated the relation between the existence of co-meager conjugacy classes in every dimension and other group theoretic properties, such as the small index property, uncountable cofinality, automatic continuity and Bergman’s property. The existence and description of typical elements frequently have applications in the theory of dynamical systems as well. For example, it is easy to see that the automorphism group of the countably infinite atomless Boolean algebra is isomorphic to the homeomorphism group of the Cantor set, which is a central object in dynamics. Thus, from their general results Kechris and Rosendal deduced the existence of a co-meager conjugacy class in the homeomorphism group of the Cantor set. A description of an element with such a class was given by Glasner and Weiss [@glasner2003universal] and from a different perspective by Bernardes and the first author [@bernardes]. Thus, it is natural to ask whether there exist measure theoretic analogues of these results. Unfortunately, on non-locally compact groups there is no natural invariant $\sigma$-finite measure. However, a generalization of the ideal of measure zero sets can be defined in every Polish group as follows: \[d:haarnull\] Let $G$ be a Polish group and $B \subset G$ be Borel. We say that $B$ is *Haar null* if there exists a Borel probability measure $\mu$ on $G$ such that for every $g,h \in G$ we have $\mu(gBh)=0$. An arbitrary set $S$ is called Haar null if $S \subset B$ for some Borel Haar null set $B$. It is known that the collection of Haar null sets forms a $\sigma$-ideal in every Polish group (see [@cohen] and [@mycorig]) and it coincides with the ideal of measure zero sets in locally compact groups with respect to every left (or equivalently right) Haar measure. Using this definition, it makes sense to talk about the properties of random elements of a Polish group. A property $P$ of elements of a Polish group $G$ is said to *hold almost surely* or *almost every element of G has property $P$* if the set $\{g \in G: g \text{ has property } P\}$ is co-Haar null. Since we are primarily interested in homeomorphism and automorphism groups, and in such groups conjugate elements can be considered isomorphic, we are only interested in the conjugacy invariant properties of the elements of our Polish groups. Hence, in order to describe the random element, one must give a complete description of the size of the conjugacy classes with respect to the Haar null ideal. The investigation of this question has been started by Dougherty and Mycielski [@DM] in the permutation group of a countably infinite set, $S_\infty$. If $f \in S_\infty$ and $a$ is an element of the underlying set then the set $\{f^{k}(a):k \in \mathbb{Z}\}$ is called the *orbit of $a$ (under $f$)*, while the cardinality of this set is called *orbit length*. Thus, each $f \in S_\infty$ has a collection of orbits (associated to the elements of the underlying set). It is easy to show that two elements of $S_\infty$ are conjugate if and only if they have the same (possibly infinite) number of orbits for each possible orbit length. Almost every element of $S_\infty$ has infinitely many infinite orbits and only finitely many finite ones. Therefore, almost all permutations belong to the union of a countable set of conjugacy classes. \[t:DMold\] All of these countably many conjugacy classes are non-Haar null. Thus, the above theorems give a complete description of the non-Haar null conjugacy classes and the (conjugacy invariant) properties of a random element. The aim of our paper is to initiate a systematic study of the size of the conjugacy classes of automorphism groups of countable structures. Our work is centered around questions of the following type: Let $\mathcal{A}$ be a countable (first order) structure. 1. \[q:darji\] What properties of $\mathcal{A}$ ensure that (an appropriate) generalization of the theorem of Dougherty and Mycielski holds for $\operatorname{Aut}(\mathcal{A})$? 2. \[q:general\] Describe the (conjugacy invariant) properties of almost every element of $\operatorname{Aut}(\mathcal{A})$: Which conjugacy classes of $\operatorname{Aut}(\mathcal{A})$ are non-Haar null? How many non-Haar null conjugacy classes are there? Is almost every element of $\operatorname{Aut}(\mathcal{A})$ contained in a non-Haar null class? In this paper we answer the first question, see Section \[s:results\] and Theorem \[t:gen\]. One can prove that in $S_\infty$ the collection of elements that have no infinite orbits is a co-meager set. This shows that the typical behavior in the sense of Baire category is quite different from the typical behavior in the measure theoretic sense. In particular, $S_\infty$ can be decomposed into the union of a Haar null and a meager set. It is well known that this is possible in every locally compact group, but the situation is not clear in the non-locally compact case. Thus, the below question of the first author arises naturally: Suppose that $G$ is an uncountable Polish group. Can it be written as the union of a meager and a Haar null set? We investigate this question for various automorphism groups, and solve it for a large class, see Corollary \[c:inffixed -&gt; decomp\]. The paper is organized as follows. First, in Section \[s:prel\] we summarize facts and notations used later, then in Section \[s:results\] we give a detailed description of our results. For the sake of the transparency of the topic we also include in this section the results of two upcoming papers [@autqcikk] and [@autrcikk]. Section \[s:genres\] contains our main theorem, while in Section \[s:appl\] we apply the general result to prove a theorem about Haar null-meager decompositions. After this, in Section \[s:poss\] we investigate the possible cardinality of non-Haar null conjugacy classes of (locally compact and non-locally compact) Polish groups. Finally, we conclude with listing a number of open questions in Section \[s:open\]. Preliminaries and notations {#s:prel} =========================== We will follow the notations of [@kechrisbook]. For a detailed introduction to the theory of Polish groups see [@becker1996descriptive Chapter 1], while the model theoretic background can be found in [@hodges Chapter 7]. Nevertheless, we summarize the basic facts which we will use. As usual, a countable structure $\mathcal{A}$ is a first order structure on a countable set with countably many constants, relations and functions. The underlying set will be denoted by $\operatorname{dom}({\mathcal}{A})$. The automorphism group of the structure $\mathcal{A}$ is denoted by $\operatorname{Aut}(\mathcal{A})$ which we consider as a topological (Polish) group with the topology of pointwise convergence. Isomorphisms between topological groups are considered to be group automorphisms that are also homeomorphisms. The structure $\mathcal{A}$ is called *ultrahomogeneous* if every isomorphism between its finitely generated substructures extends to an automorphism of $\mathcal{A}$. The *age* of a structure $\mathcal{A}$ is the collection of the finitely generated substructures of $\mathcal{A}$. An injective homomorphism between structures will be called an *embedding*. A structure is said to be *locally finite* if every finite set of elements generates a finite substructure. A countable set ${\mathcal}{K}$ of finitely generated structures of the same language is called a *Fraïssé class* if it satisfies the hereditary (HP), joint embedding (JEP) and amalgamation properties (AP) (see [@hodges Chapter 7]). We will need the notion of the strong amalgamation property: A Fraïssé class ${\mathcal}{K}$ satisfies the *strong amalgamation property (SAP)* if for every ${\mathcal}{B} \in {\mathcal}{K}$ and every pair of structures ${\mathcal}{C}, {\mathcal}{D} \in {\mathcal}{K}$ and embeddings $\psi: {\mathcal}{B} \to {\mathcal}{C}$ and $\chi:{\mathcal}{B} \to {\mathcal}{D}$ there exist ${\mathcal}{E} \in {\mathcal}{K}$ and embeddings $\psi': {\mathcal}{C} \to {\mathcal}{E}$ and $\chi':{\mathcal}{D} \to {\mathcal}{E}$ such that $$\psi' \circ \psi=\chi'\circ \chi \text{ and } \psi'({\mathcal}{C})\cap \chi'({\mathcal}{D})= (\psi' \circ \psi)({\mathcal}{B})=(\chi'\circ \chi) ({\mathcal}{B}).$$ For a Fraïssé class ${\mathcal}{K}$ the unique countable ultrahomogeneous structure ${\mathcal}{A}$ with $age({\mathcal}{A})={\mathcal}{K}$ is called the *Fraïssé limit of* ${\mathcal}{K}$. If $G$ is the automorphism group of a structure $\mathcal{A}$, we call a bijection $p$ a *partial automorphism* or a *partial permutation* if it is an automorphism between two finitely generated substructures of $\mathcal{A}$ such that $p \subset g$ for some $g \in G$. As mentioned before, $S_\infty$ stands for the permutation group of the countably infinite set $\omega$. It is well known that $S_\infty$ is a Polish group with the pointwise convergence topology. This coincides with the topology generated by the sets of the form $[p]=\{f \in S_\infty: p \subset f\}$, where $p$ is a finite partial permutation. Let $\mathcal{A}$ be a countable structure. By the countability of ${\mathcal}{A}$, every automorphism $f \in \operatorname{Aut}({\mathcal}{A})$ can be regarded as an element of $S_\infty$, and it is not hard to see that in fact $\operatorname{Aut}({\mathcal}{A})$ will be a closed subgroup of $S_\infty$. Moreover, the converse is also true, namely every closed subgroup of $S_\infty$ is isomorphic to the automorphism group of a countable structure. Let $G$ be a closed subgroup of $S_\infty$. The *orbit* of an element $x \in \omega$ (under $G$) is the set $G(x) = \{y \in \omega : \exists g \in G \; (g(x) = y)\}$. For a set $S \subset \omega$ we denote the *pointwise stabiliser* of $S$ by $G_{(S)}$, that is, $G_{(S)} = \{g \in G : \forall s \in S\; (g(s) = s)\}$. In case $S = \{x\}$, we write $G_{(x)}$ instead of $G_{(\{x\})}$. As in the case of $S_\infty$, for a countable structure $\mathcal{A}$, an element $a \in \operatorname{dom}({\mathcal}{A})$ and $f \in \operatorname{Aut}({\mathcal}{A})$ the set $\{f^{k}(a):k \in \mathbb{Z}\}$ is called the *orbit of $a$* and denoted by ${\mathcal}{O}^f(a)$, while the cardinality of this set is called *orbit length*. The *collection of the orbits of $f$*, or *the orbits of $f$* is the set $\{{\mathcal}{O}^f(a):a \in \operatorname{dom}({\mathcal}{A})\}$. If $S \subset \operatorname{dom}({\mathcal}{A})$ we will also use the notation $\mathcal{O}^f(S)$ for the set $\bigcup_{a \in S}{\mathcal}{O}^f(a)$. We will constantly use the following fact. \[f:compactchar\] Let $\mathcal{A}$ be a countable structure. A closed subset $C$ of $\operatorname{Aut}(\mathcal{A})$ is compact if and only if for every $a \in \operatorname{dom}({\mathcal}{A})$ the set $\{f(a),f^{-1}(a):f \in C\}$ is finite. We denote by $\mathcal{B}_\infty$ the countable atomless Boolean algebra, by $(\mathbb{Q},<)$ or $\mathbb{Q}$ the rational numbers as an ordered set. Let us use the notation $\mathcal{R}$ (or $(V,R)$) for the countably infinite random graph, that is, the unique countable graph with the following property: for every pair of finite disjoint sets $A,B \subset V$ there exists $v \in V$ such that $(\forall x \in A)(x R v)$ and $(\forall y \in B)(y {\lnot R}v)$. Let us consider the following notion of largeness: \[d:catcherbiter\] Let $G$ be a Polish topological group. A set $A \subset G$ is called *compact catcher* if for every compact $K \subset G$ there exist $g,h \in G$ so that $gKh \subset A$. $A$ is *compact biter* if for every compact $K \subset G$ there exist an open set $U$ and $g,h \in G$ so that $U \cap K \not = \emptyset$, and $g(U \cap K)h \subset A$. The following easy observation is one of the most useful tools to prove that a certain set is not Haar null. \[f:biter\] If $A$ is compact biter then it is not Haar null. Suppose that this is not the case and let $B \supset A$ be a Borel Haar null set and $\mu$ be a witness measure for $B$. Then, by the regularity of $\mu$, there exists a compact set $K \subset G$ such that $\mu(K)>0$. Subtracting the relatively open $\mu$ measure zero subsets of $K$ we can suppose that for every open set $U$ if $U \cap K \not =\emptyset$ then $\mu(U \cap K)>0$. But, as $A$ is compact biter, so is $B$, thus for some open set $U$ with $\mu(U \cap K)>0$ there exist $g,h \in G$ so that $g(U\cap K)h \subset B$. This shows that $\mu$ cannot witness that $B$ is Haar null, a contradiction. Note that the proof of Theorem \[t:DMold\] by Dougherty and Mycielski actually shows that every non-Haar null conjugacy class is compact biter and the unique non-Haar null conjugacy class which contains elements without finite orbits is compact catcher. It is sometimes useful to consider right and left Haar null sets: a Borel set $B$ is *right (resp. left) Haar null* if there exists a Borel probability measure $\mu$ on $G$ such that for every $g \in G$ we have $\mu(Bg)=0$ (resp. $\mu(gB)=0$). An arbitrary set $S$ is called *right (resp. left) Haar null* if $S \subset B$ for some Borel right (resp. left) Haar null set $B$. The following observation will be used several times. \[l:conjugacyinvariant\] Suppose that $B$ is a Borel set that is invariant under conjugacy. Then $B$ is left Haar null iff it is right Haar null iff it is Haar null. Let $\mu$ be a measure witnessing that $B$ is left Haar null. We check that it also witnesses the Haar nullness of $B$. Indeed, let $g,h \in G$ arbitrary, $\mu(gBh)=\mu(ghh^{-1}Bh)=\mu(ghB)=0$. The proof is analogous when $B$ is right Haar null. Description of the results {#s:results} ========================== We start with defining the crucial notion for the description of the orbits of a random element of an automorphism group. Informally, the following definition says that our structure is free enough: if we want to extend a partial automorphism defined on a finite set, there are only finitely many points for which we have only finitely many options. \[NACdef\] Let $G$ be a closed subgroup of $S_\infty$. We say that *$G$ has the finite algebraic closure property ($FACP$)* if for every finite $S \subset \omega$ the set $\{b:|G_{(S)}(b)|<\infty\}$ is finite. The following model theoretic property of Fraïssé classes turns out to be essentially a reformulation of the $FACP$ for the automorphism groups of the limits. \[CSAPdef\] Let ${\mathcal}{K}$ be a Fraïssé class. We say that *${\mathcal}{K}$ has the cofinal strong amalgamation property (CSAP)* if there exists a subclass of ${\mathcal}{K}$ cofinal under embeddability, which satisfies the strong amalgamation property, or more formally: for every ${\mathcal}{B}_0 \in {\mathcal}{K}$ there exists a ${\mathcal}{B} \in {\mathcal}{K}$ and an embedding $\phi_0:{\mathcal}{B}_0 \to {\mathcal}{B}$ so that the *strong amalgamation property holds over ${\mathcal}{B}$*, that is, for every pair of structures ${\mathcal}{C}, {\mathcal}{D} \in {\mathcal}{K}$ and embeddings $\psi: {\mathcal}{B} \to {\mathcal}{C}$ and $\chi:{\mathcal}{B} \to {\mathcal}{D}$ there exist ${\mathcal}{E} \in {\mathcal}{K}$ and embeddings $\psi': {\mathcal}{C} \to {\mathcal}{E}$ and $\chi':{\mathcal}{D} \to {\mathcal}{E}$ such that $$\psi' \circ \psi=\chi'\circ \chi \text{ and } \psi'({\mathcal}{C})\cap \chi'({\mathcal}{D})= (\psi' \circ \psi)({\mathcal}{B})=(\chi'\circ \chi) ({\mathcal}{B}).$$ A Fraïssé limit ${\mathcal}{A}$ is said to *have the cofinal strong amalgamation property* if $age({\mathcal}{A})$ has the CSAP. Generalizing the results of Dougherty and Mycielski we show that the $FACP$ is equivalent to some properties of the orbit structure of a random element of the group. [Theorem \[t:gen\]]{} Let $\mathcal{A}$ be a locally finite Fraïssé limit. Then the following are equivalent: 1. \[tc:finfin\] almost every element of $\operatorname{Aut}({\mathcal}{A})$ has finitely many finite orbits, 2. \[tc:FACP\] $\operatorname{Aut}({\mathcal}{A})$ has the $FACP$, 3. \[tc:CSAP\] ${\mathcal}{A}$ has the CSAP. Moreover, any of the above conditions implies that almost every element of ${\mathcal}{A}$ has infinitely many infinite orbits. Note that every relational structure and also ${\mathcal}{B}_\infty$ is locally finite, moreover, it is well known that the ages of the structures $\mathcal{R}, (\mathbb{Q},<)$ and ${\mathcal}{B}_\infty$ have the strong amalgamation property which clearly implies the CSAP (it is also easy to directly check the $FACP$ for these groups). Hence we obtain the following corollary. \[c:fininf\] In $\operatorname{Aut}(\mathcal{R}), \operatorname{Aut}(\mathbb{Q}, <)$ and $\operatorname{Aut}(\mathcal{B}_\infty)$ almost every element has finitely many finite and infinitely many infinite orbits. As a corollary of our results, in Section \[s:appl\] we show that a large number of groups can be partitioned in a Haar null and a meager set. [Corollary \[c:inffixed -&gt; decomp\]]{} Let $G$ be a closed subgroup of $S_\infty$ satisfying the $FACP$ and suppose that the set $F = \{g \in G : \text{$\operatorname{Fix}(g)$ is infinite}\}$ is dense in $G$. Then $G$ can be decomposed into the union of an (even conjugacy invariant) Haar null and a meager set. [Corollary \[c:Aut(R), Aut(Q), Aut(B\_inf) decomposes\]]{} $\operatorname{Aut}(\mathcal{R})$, $\operatorname{Aut}(\mathbb{Q}, <)$ and $\operatorname{Aut}({\mathcal}{B}_\infty)$ (and hence ${Homeo}(2^\mathbb{N})$) can be decomposed into the union of an (even conjugacy invariant) Haar null and a meager set. However, these results are typically far from the full description of the behavior of the random elements. We continue with summarizing our results from [@autrcikk] and [@autqcikk] about two special cases, $\operatorname{Aut}(\mathbb{Q}, <)$ and $\operatorname{Aut}(\mathcal{R})$, where we gave a complete description of the Haar positive conjugacy classes. Summary of the random behavior in Aut(Q, &lt;) and Aut(R) --------------------------------------------------------- In order to describe our results about $\operatorname{Aut}({\mathbb{Q}}, <)$ we need the concept of orbitals (defined below, for more details on this topic see [@Glass]). Let $p, q \in {\mathbb{Q}}$. The interval $(p, q)$ will denote the set $\{r \in {\mathbb{Q}}: p < r < q\}$. For an automorphism $f \in \operatorname{Aut}({\mathbb{Q}}, <)$, we denote the set of fixed points of $f$ by $\operatorname{Fix}(f)$. \[d:orbital\] The set of *orbitals* of an automorphism $f \in \operatorname{Aut}({\mathbb{Q}}, <)$, $\mathcal{O}^*_f$, consists of the convex hulls (relative to ${\mathbb{Q}}$) of the orbits of the rational numbers, that is $$\mathcal{O}^*_f = \{ \operatorname{conv}(\{f^n(r) : n \in {\mathbb{Z}}\}) : r \in {\mathbb{Q}}\}.$$ It is easy to see that the orbitals of $f$ form a partition of ${\mathbb{Q}}$, with the fixed points determining one element orbitals, hence “being in the same orbital” is an equivalence relation. Using this fact, we define the relation $<$ on the set of orbitals by letting $O_1 < O_2$ for distinct $O_1, O_2 \in \mathcal{O}^*_f$ if $p_1 < p_2$ for some (and hence for all) $p_1 \in O_1$ and $p_2 \in O_2$. Note that $<$ is a linear order on the set of orbitals. It is also easy to see that if $p, q \in {\mathbb{Q}}$ are in the same orbital of $f$ then $f(p) > p \Leftrightarrow f(q) > q$, $f(p) < p \Leftrightarrow f(q) < q$ and $f(p) = p \Leftrightarrow f(q) = q \Rightarrow p = q$. This observation makes it possible to define the *parity function*, $s_f : \mathcal{O}^*_f \to \{-1, 0, 1\}$. Let $s_f(O) = 0$ if $O$ consists of a fixed point of $f$, $s_f(O) = 1$ if $f(p) > p$ for some (and hence, for all) $p \in O$ and $s_f(O) = -1$ if $f(p) < p$ for some (and hence, for all) $p \in O$. \[t:qintro\] (see [@autqcikk]) For almost every element $f$ of $\operatorname{Aut}(\mathbb{Q}, <)$ 1. for distinct orbitals $O_1, O_2 \in \mathcal{O}^*_f$ (see Definition \[d:orbital\]) with $O_1 < O_2$ such that $s_f(O_1) = s_f(O_2) = 1$ or $s_f(O_1) = s_f(O_2) = -1$, there exists an orbital $O_3 \in \mathcal{O}^*_f$ with $O_1 < O_3 < O_2$ and $s_f(O_3) \neq s_f(O_1)$, 2. (follows from Theorem \[t:gen\]) $f$ has only finitely many fixed points. These properties characterize the non-Haar null conjugacy classes, i. e., a conjugacy class is non-Haar null if and only if one (or equivalently each) of its elements has these properties. Moreover, every non-Haar null conjugacy class is compact biter and those non-Haar null classes in which the elements have no rational fixed points are compact catchers. This yields the following surprising corollary (for the details see [@autqcikk]): \[c:autqcont\] There are continuum many non-Haar null conjugacy classes in $\operatorname{Aut}(\mathbb{Q},<)$, and their union is co-Haar null. Note that it was proved by Solecki [@openlyhaarnull] that in every non-locally compact Polish group that admits a two-sided invariant metric there are continuum many pairwise disjoint non-Haar null Borel sets, thus the above corollary is an extension of his results for $\operatorname{Aut}(\mathbb{Q},<)$ (see also the case of $\operatorname{Aut}({\mathcal}{R})$ below). We would like to point out that in a sharp contrast to this result, in $\operatorname{Homeo^+}([0,1])$ (that is, in the group of order preserving homeomorphisms of the interval) the random behavior is quite different (see [@homeo]), more similar to the case of $S_\infty$: there are only countably many non-Haar null conjugacy classes and their union is co-Haar null. The characterization of non-Haar null conjugacy classes of the automorphism group of the random graph appears to be similar to the characterization of the non-Haar null classes of $\operatorname{Aut}({\mathbb{Q}}, <)$, however their proofs are completely different. \[t:randomintro\] (see [@autrcikk]) For almost every element $f$ of $\operatorname{Aut}({\mathcal}{R})$ 1. for every pair of finite disjoint sets, $A,B \subset V$ there exists $v \in V$ such that $(\forall x \in A)( x R v)$ and $(\forall y \in B)(y {\lnot R}v)$ *and $v \not \in \mathcal{O}^f(A \cup B)$, i. e., the union of orbits of the elements of $A \cup B$*, 2. (from Theorem \[t:gen\]) $f$ has only finitely many finite orbits. These properties characterize the non-Haar null conjugacy classes, i. e., a conjugacy class is non-Haar null if and only if one (or equivalently each) of its elements has these properties. Moreover, every non-Haar null conjugacy class is compact biter and those non-Haar null classes in which the elements have no finite orbits are compact catchers. It is not hard to see that this characterization again yields the following corollary (see [@autrcikk]): \[c:autrcont\] There are continuum many non-Haar null classes in $\operatorname{Aut}({\mathcal}{R})$ and their union is co-Haar null. Various behaviors ----------------- Examining any Polish group we can ask the following questions: \[q:how\] 1. How many non-Haar null conjugacy classes are there? 2. Is the union of the Haar null conjugacy classes Haar null? Note that these are interesting even in compact groups. Table \[tab:1\] summarizes our examples and the open questions as well (the left column indicates the number of non-Haar null conjugacy classes, while C, LC $\setminus$ C and NLC stands for compact, locally compact non-compact and non-locally compact groups, respectively). HNN denotes the well known infinite group, constructed by G. Higmann, B. H. Neumann and H. Neumann [@higman1949embedding], with two conjugacy classes, while $\mathbb{Q}_d$ stands for the rationals with the discrete topology. The action, $\phi$, of $\mathbb{Z}_2$ on $\mathbb{Z}^\omega_3$ and $\mathbb{Q}^\omega_d$ is the map defined by $a \mapsto -a$. ----- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------ ----- C LC $\setminus$ C NLC $0$ **[–]{} & **[–]{} & **[–]{}\ $n$ & $\mathbb{Z}_n$ & HNN & $???$\ $\aleph_0$ & $???$ & $\mathbb{Z}$ & $S_\infty$\ $\mathfrak{c}$ & **[–]{} & **[–]{} & $\operatorname{Aut}(\mathbb{Q}, <)$; $\operatorname{Aut}({\mathcal}{R})$\ &\ & C & LC $\setminus$ C & NLC\ $0$ & $2^\omega$ & $\mathbb{Z} \times 2^\omega$ & $\mathbb{Z}^\omega$\ $n$ & $\mathbb{Z}_n \times(\mathbb{Z}_2 \ltimes_\phi \mathbb{Z}_3^\omega)$ & HNN $\times (\mathbb{Z}_2 \ltimes_\phi \mathbb{Z}^\omega_3)$ & $\mathbb{Z}_n \times (\mathbb{Z}_2 \ltimes_\phi \mathbb{Q}_d^\omega)$\ $\aleph_0$ & $???$ & $\mathbb{Z} \times( \mathbb{Z}_2 \ltimes_\phi \mathbb{Z}_3^\omega)$ & $S_\infty \times (\mathbb{Z}_2\ltimes_\phi \mathbb{Z}_3^\omega)$\ $\mathfrak{c}$ & **[–]{} & **[–]{} & $\operatorname{Aut}(\mathbb{Q}, <) \times (\mathbb{Z}_2 \ltimes_\phi \mathbb{Z}_3^\omega)$\ ************** ----- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------ ----- : Examples of various behaviors[]{data-label="tab:1"} Main results {#s:genres} ============ This section contains our generalization of the result of Dougherty and Mycielski to automorphism groups of countable structures. For the sake of simplicity we will use the following notation. Let $G$ be a closed subgroup of $S_\infty$ and let $S \subset \omega$ be a finite subset. The *group-theoretic algebraic closure* of $S$ is: $$\operatorname{ACL}(S)=\{x \in \omega : \text{the orbit of $x$ under $G_{(S)}$ is finite}\}.$$ Obviously $G$ has the finite algebraic closure property (see Definition \[NACdef\]) if and only if for every finite set $S$ the set $\operatorname{ACL}(S)$ is finite. We start with proving a simple observation about the operator $\operatorname{ACL}$. \[l:idempotentness\] If a group $G$ has the $FACP$ then the corresponding operator $\operatorname{ACL}$ is idempotent. We have to show that for every finite set $S \subset \omega$ the identity $\operatorname{ACL}(\operatorname{ACL}(S)) = \operatorname{ACL}(S)$ holds. Let $S \subset \omega$ be an arbitrary finite set and let $x \in \operatorname{ACL}(\operatorname{ACL}(S))$. We will show that $x$ has a finite orbit under $G_{(S)}$ which implies $x \in \operatorname{ACL}(S)$. It is enough to show that $G_{(S)}(x)$ is finite. Enumerate the elements of $\operatorname{ACL}(S)$ as $\{x_1, x_2 ,\dots, x_k\}$. The group $G_{(S)}$ acts on $\operatorname{ACL}(S)^k$ coordinate-wise. Under this group action the stabiliser of the tuple $(x_1, x_2, \dots, x_k)$ is $G_{(\operatorname{ACL}(S))}$. The Orbit-Stabiliser Theorem states that for any group action the index of the stabiliser of an element in the whole group is the same as the cardinality of its orbit. This yields that the index $[G_{(S)} : G_{(\operatorname{ACL}(S))}]$ is the same as the cardinality of the orbit of $(x_1, x_2, \dots, x_k)$. This orbit is finite because the whole space $\operatorname{ACL}(S)^k$ is finite. So $G_{(\operatorname{ACL}(S))}$ has finite index in $G_{(S)}$. Let $g_1, g_2, \dots, g_n \in G_{(S)}$ be a left transversal for $G_{(\operatorname{ACL}(S))}$ in $G_{(S)}$, then $G_{(S)} = g_1 G_{ACL(S)} \cup \dots \cup g_n G_{ACL(S)}$. Since $G_{(S)}(x) = g_1 G_{(\operatorname{ACL}(S))}(x) \cup g_2 G_{(\operatorname{ACL}(S))}(x) \cup \dots \cup g_n G_{(\operatorname{ACL}(S))}(x)$ is a finite union of finite sets, it must be finite. \[l:ACLtrans\] The operator $\operatorname{ACL}$ is translation invariant in the following sense: if $S \subset \omega$ is a finite set and $g \in G$ then $$\operatorname{ACL}(gS) = g\operatorname{ACL}(S).$$ Let $x\in \omega$ be an arbitrary element, then $$\begin{gathered} x \mbox{ and } y \mbox{ are in the same orbit under } G_{(S)} \Leftrightarrow \\ \exists h \in G_{(S)} : h(y) = x \Leftrightarrow \exists h \in G_{(S)} : gh(y) = g(x) \Leftrightarrow \\ \exists h \in G_{(S)} : ghg^{-1}(g(y)) = g(x) \Leftrightarrow \exists f \in G_{(gS)} : f(g(y)) = g(x) \Leftrightarrow \\ g(x) \mbox{ and } g(y) \mbox{ are in the same orbit under } G_{(gS)}. \end{gathered}$$ So an element $x$ has finite orbit under $G_{(S)}$ if and only if $g(x)$ has finite orbit under $G_{(gS)}$. Now we describe a process to generate a probability measure on $G$, a closed subgroup of $S_\infty$ that has the $FACP$. This probability measure will witness that certain sets are Haar null (see Theorem \[t:DM\]). Our random process will define a permutation $p \in G$ in stages. It depends on integer sequences $(M_{i})_{i \in \omega}$ and $(N_i)_{i \in \omega}$ with $M_i, N_i \ge 1$. We denote the partial permutation completed in stage $i$ by $p_i$. We start with $p_0 = \operatorname{id}_{ACL(\emptyset)}$ and maintain throughout the following Property \[dm:extension\] for every $i \ge 1$, and Properties \[dm:algebraically closed\] and \[dm:partial permutation\] for $i \in \omega$: 1. \[dm:extension\] $p_{i - 1} \subset p_{i}$, 2. \[dm:algebraically closed\] $\operatorname{dom}(p_i)$ and $\operatorname{ran}(p_i)$ are finite sets such that $\operatorname{ACL}(\operatorname{dom}(p_i)) = \operatorname{dom}(p_i)$, $\operatorname{ACL}(\operatorname{ran}(p_i) )= \operatorname{ran}(p_i)$, 3. \[dm:partial permutation\] there is a permutation $g \in G$ that extends $p_i$. Let $O_0, O_1, \ldots \subset \omega$ be a sequence of infinite sets with the property that for every finite set $F \subset \omega$ and every infinite orbit $O$ of $G_{(F)}$, the sequence $(O_i)_{i \in \omega}$ contains $O$ infinitely many times. It is easy to see that such a sequence exists, since there exists only countably many such finite sets $F$, and for each one, there exist only countably many orbits of $G_{(F)}$. At stage $i \ge 1$, we proceed the following way. First suppose that $i$ is even. We now choose a set $S_i \subset \omega$ with $|S_i| = M_i$ such that $S_i \cap \operatorname{ran}(p_{i - 1}) = \emptyset$. If $i \equiv 0 \pmod{4}$ we require that $S_i$ contains the least $M_i$ elements of $\omega \setminus \operatorname{ran}(p_{i - 1})$, and if $i \equiv 2 \pmod{4}$ we require that $S_i$ contains the least $M_i$ elements of $O_{(i - 2) / 4} \setminus \operatorname{ran}(p_{i - 1})$. Now we will extend $p_{i - 1}$ to a partial permutation $p_i$ such that $$\label{e:ran(p_i) = ACL(...)} \operatorname{ran}(p_i)= \operatorname{ACL}(\operatorname{ran}(p_{i - 1})\cup S_i).$$ Let us enumerate the elements of $\operatorname{ACL}(\operatorname{ran}(p_{i - 1})\cup S_i) \setminus \operatorname{ran}(p_{i - 1})$ as $(x_1, \dots, x_j)$ such that if $x_1, \dots, x_{k - 1}$ are already chosen then we choose $x_k$ so that $$\label{e:ordering chosen to be minimal} \text{$\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_k\})$ is minimal with respect to inclusion.}$$ \[c:ordering is nice for ran\] For every $1 \le k \le \ell \le m \le j$, if $$\label{e:x_m in ACL(..)} x_m \in \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{k}\})$$ then $$x_\ell \in \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{k - 1}\} \cup \{x_m\}) \subset \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{k}\}).$$ Thus, letting $\ell=k$ yields $$\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{k - 1}\} \cup \{x_m\}) = \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{k}\}).$$ The last containment holds, using Lemma \[l:idempotentness\] and . If $\ell = m$ then there is nothing to prove. Now suppose towards a contradiction that there exists $\ell < m$ violating the statement of the claim, and suppose that $\ell$ is minimal with $k \le \ell < m$ and $$\label{e:x_ell not in ACL(...)} x_{\ell} \notin \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_{k - 1}\} \cup \{x_m\}).$$ Using the minimality of $\ell$, $\{x_1, \ldots, x_{\ell - 1}\} \subset \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_{k - 1}\} \cup \{x_m\})$, thus an application of Lemma \[l:idempotentness\] and the fact that $k \le \ell$ shows that $\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_{k - 1}\} \cup \{x_m\}) = \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_{\ell - 1}\} \cup \{x_m\})$. By it follows that $x_{\ell} \notin \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_{\ell - 1}\} \cup \{x_m\})$. Using this, the fact that $k \le \ell$ and , $\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_{\ell - 1}\} \cup \{x_m\}) \subsetneqq \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_{\ell}\})$ contradicting , since $x_{\ell}$ was chosen after $\{x_1, \dots, x_{\ell - 1}\}$ to satisfy that $\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \ldots, x_{\ell}\})$ is minimal. We will determine the preimages of $(x_1, x_2, \ldots, x_j)$ in this order. Denote the partial permutations defined in these sub-steps by $p_{i, k}$ so that $\operatorname{ran}(p_{i, k}) = \operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_k\}$ for $k = 0, \dots, j$. If the first $k$ preimages are determined then there are two possibilities for $x_{k + 1}$: - The set of possible preimages of $x_{k + 1}$ under $p_{i, k}$ is finite, that is, the set $\{g^{-1}(x_{k + 1}) : g \in G, g \supset p_{i, k}\}$ is finite. Then choose one from them randomly with uniform distribution. - The set of possible preimages of $x_{k + 1}$ under $p_{i, k}$ is infinite. Then choose one from the smallest $N_i$ many possible values uniformly. We note that the orbit of $x_k$ under the stabiliser $G_{(\operatorname{ran}(p_{i - 1}))}$ is infinite because $x_k \notin \operatorname{ran}(p_{i - 1}) = \operatorname{ACL}(\operatorname{ran}(p_{i - 1}))$ so $$\label{e:possibility (b) for x_1} \text{possibility (b) must occur for at least $x_1$.}$$ Let $p_i = p_{i, j}$. Properties \[dm:extension\] and \[dm:partial permutation\] obviously hold for $i$. Let $g \in G$ be a permutation with $g \supset p_i$. Now $\operatorname{ran}(p_i) = \operatorname{ACL}(\operatorname{ran}(p_i))$ using and Lemma \[l:idempotentness\]. Then $\operatorname{dom}(p_i) = g^{-1} \operatorname{ran}(p_i)$, hence using Lemma \[l:ACLtrans\], $\operatorname{ACL}(\operatorname{dom}(p_i)) = \operatorname{ACL}(g^{-1} \operatorname{ran}(p_i)) = g^{-1}\operatorname{ACL}(\operatorname{ran}(p_i)) = g^{-1}\operatorname{ran}(p_i) = \operatorname{dom}(p_i)$, showing Property \[dm:algebraically closed\]. This concludes the case where $i$ is even. If $i$ is odd we let $S_i \subset \omega$ be the set of the least $M_i$ elements of $\omega \setminus \operatorname{dom}(p_{i - 1})$, if $i \equiv 1 \pmod{4}$ and the least $M_i$ elements of $O_{(i - 3) / 4} \setminus \operatorname{dom}(p_{i - 1})$, if $i \equiv 3 \pmod{4}$. We extend $p_{i - 1}$ to a partial permutation $p_i$ such that $$\label{e:dom(p_i) = ACL(...)} \operatorname{dom}(p_i)=\operatorname{ACL}(\operatorname{dom}(p_{i - 1}) \cup S_i).$$ Again, we enumerate the elements of $\operatorname{ACL}(\operatorname{dom}(p_{i - 1}) \cup S_i) \setminus \operatorname{dom}(p_{i - 1})$ as $(x_1, \dots, x_j)$ such that if $x_1, \dots, x_{k - 1}$ are already chosen then we choose $x_k$ from the rest so that $\operatorname{ACL}(\operatorname{dom}(p_{i - 1}) \cup \{x_1, \ldots, x_k\})$ is minimal with respect to inclusion. The proof of the following claim is analogous to the proof of Claim \[c:ordering is nice for ran\]. \[c:ordering is nice for dom\] For every $1 \le k \le \ell \le m \le j$, $x_m \in \operatorname{ACL}(\operatorname{dom}(p_{i - 1}) \cup \{x_1, \dots, x_{k}\})$ implies $x_\ell \in \operatorname{ACL}(\operatorname{dom}(p_{i - 1}) \cup \{x_1, \dots, x_{k - 1}\} \cup \{x_m\}) \subset \operatorname{ACL}(\operatorname{dom}(p_{i - 1}) \cup \{x_1, \dots, x_{k}\})$. Thus, letting $\ell=k$ yields $ \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{k - 1}\} \cup \{x_m\}) = \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{k}\}).$ We determine the images of $(x_1, x_2, \ldots, x_j)$ in this order. Denote the partial permutations defined in these sub-steps by $p_{i, k}$ so that $\operatorname{dom}(p_{i, k}) = \operatorname{dom}(p_{i - 1}) \cup \{x_1, \dots, x_k\}$ for $k = 0, \dots, j$. If the first $k$ images are determined then there are two possibilities for $x_{k + 1}$: - The set of possible images of $x_{k + 1}$ under $p_{i, k}$ is finite, that is, the set $\{g(x_{k + 1}) : g \in G, g \supset p_{i, k}\}$ is finite. Then choose one from them randomly with uniform distribution. - The set of possible images of $x_{k + 1}$ under $p_{i, k}$ is infinite. Then choose one from the smallest $N_i$ many possible values uniformly. Again, the orbit of $x_k$ under the stabiliser $G_{(\operatorname{dom}(p_{i - 1}))}$ is infinite because $x_k \notin \operatorname{dom}(p_{i - 1}) = \operatorname{ACL}(\operatorname{dom}(p_{i - 1}))$ for every $k$, so possibility (b) must occur for at least $x_1$. Let $p_i = p_{i, j}$. Again, Properties \[dm:extension\] and \[dm:partial permutation\] hold for $i$. Let $g \in G$ be a permutation with $g \supset p_i$. Now $\operatorname{dom}(p_i) = \operatorname{ACL}(\operatorname{dom}(p_i))$ using and Lemma \[l:idempotentness\]. Then using Lemma \[l:ACLtrans\], $\operatorname{ACL}(\operatorname{ran}(p_i)) = \operatorname{ACL}(g\operatorname{dom}(p_i)) = g\operatorname{ACL}(\operatorname{dom}(p_i)) = g\operatorname{dom}(p_i) = \operatorname{ran}(p_i)$, showing Property \[dm:algebraically closed\]. This concludes the construction for odd $i$. Now let $p = \bigcup_i p_i$. This makes sense using \[dm:extension\]. $p \in G$. First we show that $p \in S_\infty$. Using \[dm:partial permutation\], each $p_i$ is a partial permutation, hence injective. Using \[dm:extension\], $p$ is the union of compatible injective functions, hence $p$ is an injective function. It is clear from the construction that $\{0, 1, \dots, i - 1\} \subset \operatorname{dom}(p_{4i}) \cap \operatorname{ran}(p_{4i})$ for every $i$, hence $p \in S_\infty$. Using \[dm:partial permutation\], we can find an element $g_i \in G$ such that $g_i \supset p_i$. It is clear that $g_i \to p$, and since $G$ is a closed subgroup of $S_\infty$, we conclude that $p \in G$. The following lemma is crucial in proving that almost every element of $G$ has finitely many finite and infinitely many infinite orbits. \[l:badelements\] Suppose that the parameters of the random process $M_1, \dots, M_{i}$ and $N_1, \dots, N_{i - 1}$ are given along with the numbers $K \in \omega$ and $\varepsilon > 0$. Then we can choose $N_i$ so that for every set $S \subset \omega$ with $|S| = K$, the probability that $S \cap (\operatorname{dom}(p_i) \setminus \operatorname{dom}(p_{i - 1})) \neq \emptyset$ if $i$ is even, or that $S \cap (\operatorname{ran}(p_i) \setminus \operatorname{ran}(p_{i - 1})) \neq \emptyset$ if $i$ is odd, is at most $\varepsilon$. We suppose that $i$ is even and prove the lemma only in this case. The proof for the case when $i$ is odd is analogous. One can easily see using induction on $i$ that if $M_1, \dots, M_{i - 1}$ and $N_1, \dots, N_{i - 1}$ are given then the random process can yield only finitely many different $p_{i - 1}$ as a result. Let $p_{i - 1}$ be one of the possible outcomes, and let $(x_1, x_2, \dots, x_{j})$ denote the elements of $\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup S_i) \setminus \operatorname{ran}(p_{i - 1})$ enumerated in the same order as they appear during the construction. Note that this only depends on $p_{i - 1}$ and $M_i$. Let $a_1$ be the index for which $\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1\}) = \operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{a_1}\}$, such an index exists using Claim \[c:ordering is nice for ran\]. Hence, for every $m \le a_1$, $x_m \in \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1\})$, thus using Claim \[c:ordering is nice for ran\] again, it follows that $$\label{e:x_1 in ACL(... x_m)} \text{$x_1 \in \operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_m\})$ for every $1 \le m \le a_1$.}$$ \[c:k\_m exists\] For every such $m$, there is a unique positive integer $k_m$ such that if $q$ is an extension of $p_{i - 1}$ with $\operatorname{ran}(q) = \operatorname{ran}(p_{i - 1}) \cup \{x_m\}$ (such that $q \subset g$ for some $g \in G$) then $|\{g^{-1}(x_1) : g \in G, g \supset q\}| = k_m$. Let $H = G_{(\operatorname{ran}(p_{i - 1}) \cup x_m)}$, then $$\label{e:k def} k = |\{g(x_1) : g \in H\}| = |\{g^{-1}(x_1) : g \in H\}|$$ is finite using and the fact that $H$ is a subgroup. It is enough to show that if $q$ is an extension of $p_{i - 1}$ with $\operatorname{ran}(q) = \operatorname{ran}(p_{i - 1}) \cup \{x_m\}$ then $|\{g^{-1}(x_1) : g \in G, g \supset q\}| = k$. Let $g_1, \dots, g_k \in H$ with $g^{-1}_\ell(x_1) \ne g^{-1}_n(x_1)$ if $\ell \ne n$. If $h \in G$ is a permutation with $h \supset q$ then $g_n h \supset q$ for every $1 \le n \le k$. Then using the identity $(g_n h)^{-1}(x_1) = h^{-1} (g_n^{-1}(x_1))$, $(g_\ell h)^{-1}(x_1) \ne (g_n h)^{-1}(x_1)$ if $\ell \ne n$. This shows that $|\{g^{-1}(x_1) : g \in G, g \supset q\}| \ge k$. To prove the other inequality, suppose towards a contradiction that there exist $g_1, \dots, g_{k + 1}$ with $g_n \supset q$ for every $n \le k + 1$ and $g_\ell^{-1}(x_1) \neq g_n^{-1}(x_1)$ for every $\ell \neq n$. It is easy to see that $g_n g_1^{-1} \in H$ for every $n$, but the values $(g_ng_1^{-1})^{-1}(x_1) = g_1(g_n^{-1}(x_1))$ are pairwise distinct, contradicting . Thus the proof of the claim is complete. Now let $k = \max\{k_2, k_3, \dots, k_{a_1}\}$, if $a_1 \ge 2$, otherwise let $k = 1$. If $N_i > \frac{kKj}{\varepsilon}$ then for every fixed set $S \subset \omega$ with $|S| = K$ we have $\mathbb{P}(p_i^{-1}(x_m) \in S) < \frac{\varepsilon}{j}$ for every $1 \le m \le a_1$. This is immediate for $m = 1$, since $k \ge 1$, and the preimage of $x_1$ is chosen from $N_i$ many elements using . Now let $m > 1$, using Claim \[c:k\_m exists\] and the fact that $k \ge k_m$, it follows that for every $y \in \omega$, $|\{g^{-1}(x_1) : g \in G, g \supset p_{i - 1}, g(y) = x_m\}| \le k$, hence for the set $R = \{g^{-1}(x_1) : g \in G, g \supset p_{i - 1}, g^{-1}(x_m) \in S\}$, $|R| \le kK$. In order to be able to extend $p_{i - 1}$ to $p_i$ with $p_i^{-1}(x_m) \in S$, we need to choose $p_i^{-1}(x_1)$ from $R$. Since during the construction of the random automorphism, $p_i^{-1}(x_1)$ is chosen uniformly from a set of size $N_i > \frac{kKj}{\varepsilon}$, we conclude that $\mathbb{P}(p_i^{-1}(x_m) \in S) \le \mathbb{P}(p_i^{-1}(x_1) \in R) \le \frac{|R|}{N_i} < \frac{\varepsilon}{j}$. For the rest of the proof, we need to repeat the above argument until we reach $j$. If $a_1 < j$, let $a_2$ be the index satisfying $\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{a_1 + 1}\}) = \operatorname{ran}(p_{i - 1}) \cup \{x_1, \dots, x_{a_2}\}$, such an index exists using Claim \[c:ordering is nice for ran\] as before. Again, we can set a lower bound for $N_i$ so that the for every $a_1 < m \le a_2$, $\mathbb{P}(p_i^{-1}(x_m) \in S) < \frac{\varepsilon}{j}$. Repeating the argument, we can choose $N_i$ so that $\mathbb{P}(p_{i}^{-1}(x_m) \in S) < \frac{\varepsilon}{j}$ for every $1 \le m \le j$, thus $\mathbb{P}(p_i^{-1}(\{x_1, \dots, x_j\}) \cap S \neq \emptyset) < \varepsilon$. Completing the proof of the lemma. Now we prove a proposition from which our main result will easily follow. \[p:F and C co-Haar null\] Let $G \leq S_\infty$ be a closed subgroup. If $G$ has the $FACP$ then the sets $$\begin{split} \mathcal{F} = \{g \in G :& \text{$g$ has finitely many finite orbits}\}, \\ \mathcal{C} = \{g \in G :& \forall F \subset \omega \text{ finite } \forall x \in \omega \;(\text{if $G_{(F)}(x)$ is infinite} \\ &\text{then it is not covered by finitely many orbits of $g$})\} \end{split}$$ are co-Haar null. The set $\mathcal{C}$ could seem unnatural for the first sight. However, from the above fact about the set $\mathcal{C}$ not only our main theorem will be deduced, but this fact also plays a crucial role in proving Theorem \[t:randomintro\] (see [@autrcikk]). We first show the following lemma. \[l:conin\] The sets $\mathcal{F}$ and $\mathcal{C}$ are conjugacy invariant Borel sets. The fact that $\mathcal{F}$ is conjugacy invariant follows form the fact that conjugation does not change the orbit structure of a permutation. To show that $\mathcal{C}$ is conjugacy invariant, let $c \in \mathcal{C}$, $h \in G$, we need to show that $h^{-1}ch \in \mathcal{C}$. Let $F \subset \omega$ be finite and $x \in \omega$ so that $|G_{(F)}(x)|=\aleph_0$. Note that $G_{(h(F))}(h(x))=hG_{(F)}h^{-1}(h(x))=hG_{(F)}(x)$, hence the first set is also infinite. By $c \in C$ there exists an infinite set $\{x_n: n \in \omega\} \subset G_{(h(F))}(h(x))$ so that for $n \not = n'$ the points $x_n$ and $x_{n'}$ are in different $c$ orbits. But then the points $\{h^{-1}(x_n): n \in \omega\}\subset G_{(F)}(x)$ are in pairwise distinct $h^{-1}ch$ orbits, as desired. To show that $\mathcal{F}$ is Borel, notice that the set of permutations containing a given finite orbit is open for every finite orbit. Thus for any finite set of finite orbits the set of permutations containing those finite orbits in their orbit decompositions is open: it can be obtained as the intersection of finitely many open sets. Thus for every $n \in \omega$ the set of permutations containing at least $n$ finite orbits is open: it can be obtained as the union of open sets (one open set for each possible set of $n$ orbits). Thus $S_\infty \setminus \mathcal{F}$ is $G_\delta$: it is the intersection of the above open sets. Hence $\mathcal{F}$ is Borel. Now we show that $\mathcal{C}$ is also Borel. It is enough to show that if $H \subset \omega$ is arbitrary then the set $H^* = \{g \in G : \text{finitely many orbits of $g$ cannot cover $H$}\}$ is Borel, since $\mathcal{C}$ can be written as the countable intersection of such sets. And $H^*$ can be easily seen to be Borel for any $H$, since its complement, $\{g \in G : \exists n \ \forall m \in H \ \exists k \ \exists i < n \ (g^k(i) = m)\}$ is $G_{\delta\sigma}$, hence $H^*$ is $F_{\sigma\delta}$. To prove the proposition, we use the above construction to generate a random permutation $p$. We set $M_i = 2^i$ for every $i \ge 1$ and we define $(N_i)_{i \ge 1}$ recursively. If $N_1, \dots, N_{i - 1}$ are already defined, then, as before, the random process can yield only finitely many distinct $p_{i - 1}$. Hence, there is a bound $m_i$ depending only on $N_1, \dots, N_{i - 1}$ such that $|\operatorname{dom}(p_i)| = |\operatorname{ran}(p_i)| \le m_i$, since $|\operatorname{ran}(p_i)| = |\operatorname{ACL}(\operatorname{ran}(p_{i - 1}) \cup S_i)|$ if $i$ is even and $|\operatorname{dom}(p_i)| = |\operatorname{ACL}(\operatorname{dom}(p_{i - 1}) \cup S_i)|$ if $i$ is odd, which is independent of $N_i$. Now we use Lemma \[l:badelements\] to choose $N_i$ so that the conclusion of the lemma is true with $K = m_i$ and $\varepsilon = \frac{1}{2^i}$. Using Lemma \[l:conjugacyinvariant\] and the fact that the sets $\mathcal{F}$ and $\mathcal{C}$ are conjugacy invariant, it is enough to show that $$\label{e:ph-ban veges sok veges} \mathbb{P}(\text{$ph$ has finitely many finite orbits} ) = 1$$ and $$\label{e:ph-ban veges sok ciklus nem fedi O-t} \mathbb{P}(\text{finitely many orbits of $ph$ do not cover $O$} ) = 1$$ for every $h \in G$, every finite $F \subset \omega$ and every infinite orbit $O$ of $G_{(F)}$, since there exist only countably many such orbits. So let us fix $h \in G$ and an infinite orbit $O \subset \omega$ of $G_{(F)}$ for some finite $F \subset \omega$ for the rest of the proof. For a partial permutation $q$, a *partial path* in $q$, is a sequence $(y, q(y), \dots, q^n(y))$ with $n \ge 1$, $q^n(y) \notin \operatorname{dom}(q)$ and $y \notin \operatorname{ran}(q)$. Note that $p_ih$ is considered a partial permutation with $\operatorname{dom}(p_ih) = h^{-1}(\operatorname{dom}(p_i))$ and $\operatorname{ran}(p_ih) = \operatorname{ran}(p_i)$. During the construction of the random permutation, an *event* occurs when the partial permutation is extended to a new element at some stage regardless of whether it happens for possibility (a) or (b). Suppose that during an event, the partial permutation $p'$ is extended to $p'' = p' \cup (x, y)$. We call this event *bad* if the number of partial paths decreases or $h^{-1}(x) = y$. Note that an event is bad if the extension connects two partial paths of $p'h$ or it completes an orbit (possibly a fixed point). \[c:dm:1\] Almost surely, only finitely many bad events happen. Let $i$ be fixed and suppose first that it is even. It is easy to see that a bad event can only happen at stage $i$ if a preimage is chosen from $h(\operatorname{ran}(p_{i - 1}))$, that includes the case when a fixed point is constructed. Note that $|\operatorname{ran}(p_i)| \le m_i$, thus the probability of choosing a preimage from this set is at most $\frac{1}{2^i}$, using Lemma \[l:badelements\]. We proceed similarly if $i$ is odd. Then to connect partial paths or complete orbits, an image has to be chosen from the set $h^{-1}(\operatorname{dom}(p_{i - 1}))$. Since $|\operatorname{dom}(p_i)| \le m_i$, the probability of choosing from this set is at most $\frac{1}{2^i}$. Using the Borel–Cantelli lemma, the number of $i$ such that a bad event happens at stage $i$ is finite almost surely. The fact that only a finite number of bad events can happen at a particular stage completes the proof of the claim. Since a finite orbit can only be created during a bad event, follows immediately from the claim. Thus $\mathcal{F}$ is co-Haar null. Now we prove that $\mathcal{C}$ is also co-Haar null by showing . Let $n_0, n_1, \ldots \in \omega$ be a sequence with $n_0 < n_1 < \dots $ and $O_{n_i} = O$ for every $i \in \omega$. Let $c_i$ be the number of partial paths of $p_{4n_i + 2}h$ intersecting $O$. It is enough to show that the sequence $(c_i)_{i \in \omega}$ is unbounded almost surely, since using Claim \[c:dm:1\], only finitely many of such partial paths can be connected in later stages, hence infinitely many orbits of $ph$ will intersect $O$, almost surely. At stage $4n_i + 2$, $p_{4n_i + 1}$ is extended to $p_{4n_i + 2}$ with $\operatorname{ran}(p_{4n_i + 2}) \setminus \operatorname{ran}(p_{4n_i + 1}) \supset S_{4n_i + 2}$, $|S_{4n_i + 2}| = M_{4n_i + 2} = 2^{4n_i + 2}$ and $S_{4n_i + 2} \subset O_{(4n_i + 2-2) / 4} = O_{n_i} = O$. Hence, it is enough to prove that apart from a finite number of exceptions, the elements of $\operatorname{ran}(p_{4n_i + 2}) \setminus \operatorname{ran}(p_{4n_i + 1})$ are in different partial paths in $p_{4n_i + 2}h$, almost surely. The proof of this fact is similar to the proof of Claim \[c:dm:1\]. An element $y \in O\cap (\operatorname{ran}(p_{4n_i + 2}) \setminus \operatorname{ran}(p_{4n_i + 1}))$ can only be contained in a completed orbit (of $p_{4n_i + 2}h$), if $h^{-1}p_{4n_i + 2}^{-1}(y) \in \operatorname{ran}(p_{4n_i + 2})$, hence $p_{4n_i + 2}^{-1}(y) \in h(\operatorname{ran}(p_{4n_i + 2}))$. Similarly, if $y, y' \in O \cap (\operatorname{ran}(p_{4n_i + 2}) \setminus \operatorname{ran}(p_{4n_i + 1}))$ are in the same partial path (in $p_{4n_i + 2}h$) such that $y$ is the not the first element of this path, then $p_{4n_i + 2}^{-1}(y) \in h(\operatorname{ran}(p_{4n_i + 2}))$. Again using Lemma \[l:badelements\], the probability of this happening at stage $4n_i + 2$ is at most $\frac{1}{2^{4n_i + 2}}$, since $|\operatorname{ran}(p_{4n_i + 2})| \le m_{4n_i + 2}$. As before, the application of the Borel–Cantelli lemma completes the proof of . And thus the proof of the proposition is also complete. \[t:DM\] Let $G \leq S_\infty$ be a closed subgroup. If $G$ has the $FACP$ then the sets $$\begin{split} \mathcal{F} = \{g \in G : \text{$g$ has finitely many finite orbits}\}, \\ \mathcal{I} = \{g \in G : \text{$g$ has infinitely many infinite orbits}\} \end{split}$$ are both co-Haar null. Moreover, if $\mathcal{F}$ is co-Haar null then $G$ has the $FACP$. The fact that $\mathcal{F}$ is co-Haar null follows immediately from Proposition \[p:F and C co-Haar null\]. Let $\mathcal{C}$ denote the set as in Proposition \[p:F and C co-Haar null\]. If $g \in \mathcal{C}$ then $g$ contains infinitely many orbits, since otherwise finitely many orbits of $g$ could cover $\omega$, hence every infinite orbit of $G_{(F)}$ for some finite $F \subset \omega$. It follows that the co-Haar null set $\mathcal{C} \cap \mathcal{F}$ is contained in $\mathcal{I}$, hence $\mathcal{I}$ is also co-Haar null. And thus the proof of the first part of the theorem is complete. Now we prove the second assertion. We have to show that if $G$ does not have the $FACP$ then $\mathcal{F}$ is not co-Haar null. If $G$ does not have the $FACP$ then there is a finite set $S \subset \omega$ such that $\operatorname{ACL}(S)$ is infinite. This means that all of the permutations in $G_{(S)}$ have infinitely many finite orbits, hence $G_{(S)} \cap \mathcal{F} = \emptyset$. The stabiliser $G_{(S)}$ is a non-empty open set, thus it cannot be Haar null. Therefore the proof of the theorem is complete. Now we are ready to prove the main result of this section. \[t:gen\] Let $\mathcal{A}$ be a locally finite Fraïssé limit. Then the following are equivalent: 1. almost every element of $\operatorname{Aut}({\mathcal}{A})$ has finitely many finite orbits, 2. $\operatorname{Aut}({\mathcal}{A})$ has the FACP, 3. ${\mathcal}{A}$ has the CSAP. Moreover, any of the above conditions implies that almost every element of ${\mathcal}{A}$ has infinitely many infinite orbits. The equivalence $\iff$ , and the last statement of the theorem is just the application of Theorem \[t:DM\] to $G=\operatorname{Aut}({\mathcal}{A})$. Thus, it is enough to show that $\iff$ . Let ${\mathcal}{K}=age({\mathcal}{A})$. Since ${\mathcal}{A}$ is the limit of ${\mathcal}{K}$, using that ${\mathcal}{A}$ is ultrahomogeneous it follows that ${\mathcal}{K}$ has the extension property, that is, for every ${\mathcal}{B},{\mathcal}{C} \in {\mathcal}{K}$ and embeddings $\phi:{\mathcal}{B} \to {\mathcal}{C}$ and $\psi:{\mathcal}{B} \to {\mathcal}{A}$ there exists an embedding $\psi':{\mathcal}{C} \to {\mathcal}{A}$ with $\psi' \circ \phi=\psi$. Thus, the embeddings between the structures in ${\mathcal}{K}$ can be considered as partial automorphisms of ${\mathcal}{A}$. ( $\Rightarrow$ ) Take an arbitrary ${\mathcal}{B}_0 \in {\mathcal}{K}$ and fix an isomorphic copy of it inside ${\mathcal}{A}$. Let ${\mathcal}{B}=\operatorname{ACL}(\operatorname{dom}({\mathcal}{B}_0))$ and note that by the fact that $\operatorname{Aut}({\mathcal}{A})$ has the $FACP$ ${\mathcal}{B}$ is a finite substructure of ${\mathcal}{A}$. We will show that over ${\mathcal}{B}$ the strong amalgamation property holds (see Definition \[CSAPdef\]). In order to see this, let ${\mathcal}{C}, {\mathcal}{D} \in {\mathcal}{K}$ and let $\psi: {\mathcal}{B} \to {\mathcal}{C}$ and $\phi:{\mathcal}{B} \to {\mathcal}{D}$ be embeddings. By the extension property we can suppose that ${\mathcal}{B}<{\mathcal}{C} < {\mathcal}{A}$, ${\mathcal}{B} < {\mathcal}{D}<{\mathcal}{A}$ and $\psi=\phi=id_{{\mathcal}{B}}$. By Lemma \[l:idempotentness\] $\operatorname{ACL}(\operatorname{dom}({\mathcal}{B}))={\mathcal}{B}$, hence the $\operatorname{Aut}({\mathcal}{A})_{(\operatorname{dom}({\mathcal}{B}))}$ orbit of every point in $\operatorname{dom}({\mathcal}{C})\setminus \operatorname{dom}({\mathcal}{B})$ is infinite. By M. Neumann’s Lemma [@hodges Corollary 4.2.2.] $\operatorname{dom}({\mathcal}{C})\setminus \operatorname{dom}({\mathcal}{B})$ has infinitely many pairwise disjoint copies under the action of $\operatorname{Aut}({\mathcal}{A})_{(\operatorname{dom}({\mathcal}{B}))}$. In particular, by the pigeonhole principle, there exists an $f \in \operatorname{Aut}({\mathcal}{A})_{(\operatorname{dom}({\mathcal}{B}))}$ such that $f({\mathcal}{C}) \cap {\mathcal}{D}={\mathcal}{B}$. Letting ${\mathcal}{E}$ to be the substructure of ${\mathcal}{A}$ generated by $\operatorname{dom}(f({\mathcal}{C}))\cup \operatorname{dom}({\mathcal}{D})$, $\psi'=f|_{{\mathcal}{C}}$ and $\phi'=\operatorname{id}_{{\mathcal}{D}}$ shows that SAP holds over ${\mathcal}{B}$ and hence CSAP holds as well. ( $\Leftarrow$ ) Let $S \subset \operatorname{dom}({\mathcal}{A})$ be finite. Let ${\mathcal}{B}_0$ be the substructure generated by $S$. Clearly, ${\mathcal}{B}_0 \in {\mathcal}{K}$, hence there exists a ${\mathcal}{B} \in {\mathcal}{K}$ over which the strong amalgamation property holds and which contains an isomorphic copy of ${\mathcal}{B}_0$. By the extension property of ${\mathcal}{A}$ we can suppose that ${\mathcal}{B}$ and all the structures constructed later on in this part of the proof are substructures of ${\mathcal}{A}$ containing ${\mathcal}{B}_0$. We claim that for every $b \in \operatorname{dom}({\mathcal}{A}) \setminus \operatorname{dom}({\mathcal}{B})$ the orbit $\operatorname{Aut}({\mathcal}{A})_{(\operatorname{dom}({\mathcal}{B}))}(b)$ is infinite. Indeed, let ${\mathcal}{C}$ be the substructure generated by $\operatorname{dom}({\mathcal}{B}) \cup \{b\}$. Using the strong amalgamation property repeatedly, first for ${\mathcal}{B},{\mathcal}{C}$ and ${\mathcal}{D}={\mathcal}{C}$ obtaining an ${\mathcal}{E}_1$, then for ${\mathcal}{B},{\mathcal}{C}$ and ${\mathcal}{D}={\mathcal}{E}_1$ obtaining an ${\mathcal}{E}_2$ etc. for every $n$ we can find a substructure ${\mathcal}{E}_n$ of ${\mathcal}{A}$ which contains $n+1$ isomorphic copies of ${\mathcal}{C}$ which intersect only in ${\mathcal}{B}$, and the isomorphisms between these copies fix ${\mathcal}{B}$. Extending the isomorphisms to automorphisms of $\operatorname{Aut}({\mathcal}{A})$ shows that the orbit $\operatorname{Aut}({\mathcal}{A})_{(\operatorname{dom}({\mathcal}{B}))}(b)$ is infinite. It is not hard to construct countable Fraïssé classes to show that CSAP is neither equivalent to SAP, nor to AP. An example showing that CSAP $\not \Rightarrow $ SAP is $age(\mathcal{B}_\infty)$. Indeed, using a result of Schmerl [@SAP-NoAlg] that states that a Fraïssé class has the SAP if and only if its automorphism group has no algebraicity (that is, $\operatorname{ACL}(F) = F$ for every finite $F$), $age(\mathcal{B}_\infty)$ cannot have the SAP. To see that AP $\not \Rightarrow $ CSAP, let $\mathcal{Z}$ be the structure on the set $\mathbb{Z}$ of integers with a relations $R_n$ for each $n \ge 1$, $n \in {\mathbb{N}}$ satisfying that $a R_n b \Leftrightarrow |a - b| = n$ for each $a, b \in \mathbb{Z}$ and $n \ge 1$. It can be easily checked that $age(\mathcal{Z})$ satisfies AP, but $\operatorname{Aut}(\mathcal{Z})$ does not satisfy FACP, since the algebraic closure of any two points is $\mathbb{Z}$. Thus Theorem \[t:gen\] implies that $\mathcal{Z}$ cannot satisfy CSAP. An application to decompositions {#s:appl} ================================ In this section we present an application of our results: we use Theorem \[t:DM\] to show that a large family of automorphism groups of countable structures can be decomposed into the union of a Haar null and a meager set. \[c:inffixed -&gt; decomp\] Let $G$ be a closed subgroup of $S_\infty$ satisfying the $FACP$ and suppose that the set $F = \{g \in G : \text{$\operatorname{Fix}(g)$ is infinite}\}$ is dense in $G$. Then $G$ can be decomposed into the union of an (even conjugacy invariant) Haar null and a meager set. Clearly, $F$ is conjugacy invariant, and since it can be written as $F = \{g \in G : \forall n \in \omega \ \exists m > n \ \left(g(m) = m\right) \}$, $F$ is $G_\delta$. Using the assumptions of this corollary, it is dense $G_\delta$, hence co-meager. Using Theorem \[t:DM\], it is Haar null, hence $F \cup (G \setminus F)$ is an appropriate decomposition of $G$. \[c:Aut(R), Aut(Q), Aut(B\_inf) decomposes\] $\operatorname{Aut}(\mathcal{R})$, $\operatorname{Aut}(\mathbb{Q}, <)$ and $\operatorname{Aut}({\mathcal}{B}_\infty)$ can be decomposed into the union of an (even conjugacy invariant) Haar null and a meager set. In order to show that the set of elements in these groups with infinitely many fixed points is dense, in each case it is enough to show that if $p$ is a finite, partial automorphism then there is another partial automorphism $p'$ extending $p$ such that $p' \supset p \cup (x,x)$ with $x \not \in \operatorname{dom}(p)$. For $\operatorname{Aut}({\mathbb{Q}}, <)$, let $x$ be greater than each element in $\operatorname{dom}(p) \cup \operatorname{ran}(p)$, then it is easy to see that $p \cup (x, x)$ is also a partial automorphism. For $\operatorname{Aut}(\mathcal{R})$, let $x$ be an element different from each of $\operatorname{dom}(p) \cup \operatorname{ran}(p)$ with the property that $x$ is connected to every vertex in $\operatorname{dom}(p) \cup \operatorname{ran}(p)$. Then it is easy to see that $p \cup (x, x)$ is a partial automorphism. For $\operatorname{Aut}(\mathcal{B}_\infty)$, let $a_0 \cup a_1 \cup \dots \cup a_{n-1}$ be a partition of $\mathbf{1}$ with the property that $\operatorname{dom}(p) \cup \operatorname{ran}(p)$ is a subset of the algebra generated by $A = \{a_0, a_1, \dots, a_{n - 1}\}$. Then there is a permutation $\pi$ of $\{0, 1, \dots, n - 1\}$ compatible with $p$, that is, $p(a_i) = a_{\pi(i)}$ for every $i$. Let us write each $a_i$ as a disjoint union $a_i = a_i' \cup a_i''$ of non-zero elements. Again, a partial permutation can be described by a permutation of the elements $\{a_1', \dots, a_n'\} \cup \{ a_1'', \dots, a_n''\}$. Hence, let $p'$ be defined by $p'(a_i') = a_{\pi(i)}'$, $p'(a_i'') = a_{\pi(i)}''$. Then $p'$ is a partial automorphism extending $p$ with a new fixed point $\bigcup_{i < n} a_i'$. Various behaviors {#s:poss} ================= It turns out, that in natural Polish groups we may encounter very different behaviors of conjugacy classes with respect to the ideal of Haar null sets (see [@homeo], [@autrcikk], [@autqcikk]). In this section we address the questions from \[q:how\], namely, given a Polish group, how many non-Haar null conjugacy classes are there and decide whether the union of the Haar null classes is Haar null. Note that these questions make perfect sense in the locally compact case as well. In this section we construct a couple of examples. If $(A,+)$ is an abelian group we will denote by $\phi$ the automorphism of $A$ defined by $a \mapsto -a$. Let $(A,+)$ be an abelian Polish group such that for every $a \in A$ there exists an element $b$ with $2b=a$. Observe that $\phi \in \operatorname{Aut}(A)$, $\phi^2=id_A$ and $({\mathbb{Z}}_2 \ltimes_\phi A,\cdot)$ can be partitioned into $\{0\} \times A$ and $\{1\} \times A$. Moreover, in the group ${\mathbb{Z}}_2 \ltimes_\phi A$ the conjugacy class of every element of $\{0\} \times A$ is of cardinality at most $2$, whereas the set $\{1\} \times A$ is a single conjugacy class. Let $(0,a) \in \{0\} \times A$ and $(i,b) \in {\mathbb{Z}}_2 \ltimes_\phi A$ arbitrary. We claim that the conjugacy class of $(0,a)$ is $\{(0,a),(0,-a)\}$. If $i=0$ then $(0,a)$ and $(i,b)$ commute, so let $i=1$. By definition $$(1,b)^{-1} \cdot (0,a) \cdot (1,b)=(1,b) \cdot (1,b+a)=(0,b-(b+a))=(0,-a),$$ which shows our claim. Now let $(1,a),(1,a') \in {\mathbb{Z}}_2 \ltimes_\phi A$ be arbitrary. Now for an arbitrary element $(1,b)$ we get $$(1,b)^{-1} \cdot (1,a) \cdot (1,b)=(1,b) \cdot (0,-b+a)=(1,b-(-b+a))=(1,2b-a),$$ thus, choosing $b$ so that $2b=a'+a$ we obtain $$(1,b)^{-1} \cdot (1,a) \cdot (1,b)=(1,a').$$ \[c:semidir\] Let $A={\mathbb{Z}}^\omega_3$ or $A=({\mathbb{Q}}_d)^\omega$, (that is, the countable infinite power of the rational numbers with the discrete topology). Then ${\mathbb{Z}}_2 \ltimes_\phi A$ has a non-empty clopen conjugacy class, namely $\{(1,a):a \in A\}$ and every other conjugacy class has cardinality at most $2$. Hence, the union of the Haar null classes $\{(0,a):a \in A\}$ is also non-empty clopen. \[l:prod\] Suppose that $G_1$ and $G_2$ are Polish groups and $A_1 \subset G_1$ is Borel and $U \subset G_2$ is non-empty and open. Then $A_1 \times U$ is Haar null in $G_1 \times G_2$ iff $A_1$ is Haar null. Suppose first that $A_1$ is Haar null witnessed by a measure $\mu_1$. Then, if $\mu'$ is the same measure copied to $G_1 \times \{1\}$, it is easy to see that $\mu'$ witnesses the Haar nullness of $A_1 \times G_2$, in particular, the Haar nullness of $A_1 \times U$. Conversely, suppose that $A_1 \times U$ is Haar null witnessed by the measure $\mu$. Clearly, as countably many translates of $U$ cover $G_2$, countably many translates of $A_1 \times U$ cover $A_1 \times G_2$, hence $A_1 \times G_2$ is Haar null as well, and this is also witnessed by the measure $\mu$. Let $\mu_1=\operatorname{proj}_{G_1*}\mu$, then $\mu_1$ witnesses the Haar nullness of $A_1$. \[p:prod\] If $G$ is a Polish group with $\kappa$ many non-Haar null conjugacy classes then $G \times \left({\mathbb{Z}}_2 \ltimes_\phi {\mathbb{Z}}^\omega_3 \right)$ has $\kappa$ many non-Haar null conjugacy classes and the union of the Haar null conjugacy classes is not Haar null. Clearly, the conjugacy classes of $G \times \left({\mathbb{Z}}_2 \ltimes_\phi {\mathbb{Z}}^\omega_3 \right)$ are of the form $C_1 \times C_2$ where $C_1$ is a conjugacy class in $G$ and $C_2$ is a conjugacy class in ${\mathbb{Z}}_2 \ltimes_\phi {\mathbb{Z}}^\omega_3$. By Corollary \[c:semidir\] we have that every conjugacy class in the latter group is finite with one exception, this exceptional conjugacy class is clopen; let us denote it by $U$. Now, since the finite sets are Haar null in ${\mathbb{Z}}_2 \ltimes_\phi {\mathbb{Z}}^\omega_3$ by Lemma \[l:prod\], the set of non-Haar null conjugacy classes in $G \times \left({\mathbb{Z}}_2 \ltimes_\phi {\mathbb{Z}}^\omega_3\right)$ is equal to $\{C \times U:C \text{ is a non-Haar null conjugacy class in } G\}$, hence the cardinality of the non-Haar null classes is $\kappa$. Moreover, the union of the Haar null conjugacy classes contains $G \times (({\mathbb{Z}}_2 \ltimes_\phi {\mathbb{Z}}^\omega_3) \setminus U)$, which is non-empty and open, consequently it is not Haar null. Finally, we would like to recall the following well known theorem. \[t:hnn\] There exists a countably infinite group with two conjugacy classes. We denote such a group by HNN, and consider it as a discrete Polish group. Combining Proposition \[p:prod\], Corollaries \[c:autqcont\], \[c:autrcont\], \[c:semidir\], Lemma \[l:prod\] and Theorems \[t:DMold\] and \[t:hnn\] we obtain Table \[tab:1\] (see the end of Section \[s:results\]). (Recall that $C$, $LC \setminus C$ and $NLC$ stand for compact, locally compact non-compact, and non-locally compact, respectively.) Open problems {#s:open} ============= We finish with a couple of open questions. In Section \[s:poss\] we produced several groups with various numbers of non-Haar null conjugacy classes. However, our examples are somewhat artificial. Are there natural examples of automorphism groups with given cardinality of non-Haar null conjugacy classes? The following question is maybe the most interesting one from the set theoretic viewpoint. Suppose that a Polish group has uncountably many non-Haar null conjugacy classes. Does it have continuum many non-Haar null conjugacy classes? The answer is of course affirmative under e.g. the Continuum Hypothesis. Since the definition of Haar null sets is complicated (the collection of non-Haar null closed sets can already be $\mathbf{\Sigma}^1_1$-hard and $\mathbf{\Pi}^1_1$-hard [@openlyhaarnull]), it is unlikely that this question can be answered with an absoluteness argument. The characterization result of Section \[s:genres\] and the similarity between Theorems \[t:qintro\] and \[t:randomintro\] suggest that a general theory of the behavior of the random automorphism (similar to the one built by Truss, Kechris and Rosendal) could exist. Formulate necessary and sufficient model theoretic conditions which characterize the measure theoretic behavior of the conjugacy classes. In particular, it would be very interesting to find a unified proof of the description of the non-Haar null classes of $\operatorname{Aut}({\mathbb{Q}}, <)$ and $\operatorname{Aut}({\mathcal}{R})$. **Acknowledgements.** We would like to thank to R. Balka, Z. Gyenis, A. Kechris, C. Rosendal, S. Solecki and P. Wesolek for many valuable remarks and discussions. We are also grateful to the anonymous referee for their comments and suggestions, particularly for pointing out a simplification of the proof of Lemma \[l:conin\]. [^1]: The second, fourth and fifth authors were partially supported by the National Research, Development and Innovation Office – NKFIH, grants no. 113047, no. 104178 and no.  124749. The fifth author was also supported by FWF Grant P29999.

--- abstract: 'Attention is focused on q-deformed quantum algebras with physical importance, i.e. $U_{q}(su_{2})$, $U_{q}(so_{4})$ and q-deformed Lorentz algebra. The main concern of this article is to assemble important ideas about these symmetry algebras in a consistent framework which shall serve as starting point for representation theoretic investigations in physics, especially quantum field theory. In each case considerations start from a realization of symmetry generators within the differential algebra. Formulae for coproducts and antipodes on symmetry generators are listed. The action of symmetry generators in terms of their Hopf structure is taken as q-analog of classical commutators and written out explicitly. Spinor and vector representations of symmetry generators are calculated. A review of the commutation relations between symmetry generators and components of a spinor or vector operator is given. Relations for the corresponding quantum Lie algebras are computed. Their Casimir operators are written down in a form similar to the undeformed case.' author: - | Alexander Schmidt[^1], Hartmut Wachter[^2]\ Max-Planck-Institute\ for Mathematics in the Sciences\ Inselstr. 22, D-04103 Leipzig, Germany\ Arnold-Sommerfeld-Center\ Ludwig-Maximilians-Universität\ Theresienstr. 37, D-80333 München, Germany title: 'q-Deformed quantum Lie algebras' --- Introduction ============ It is an old idea that limiting the precision of position measurements by a fundamental length will lead to a new method for regularizing quantum field theories [@Heis38] . It is also well-known that such a modification of classical spacetime will in general break its Poincaré symmetry [@Sny47]. One way out of this difficulty is to change not only spacetime, but also its underlying symmetry. Quantum groups can be seen as deformations of classical spacetime symmetries, as they describe the symmetry of their comodules, the so-called quantum spaces. From a physical point of view the most realistic examples for quantum groups and quantum spaces arise from q-deformation [@Ku83; @Dri85; @Drin86; @Jim85; @Wor87; @Man88; @RFT90]. In our work we are interested in q-deformed versions of Minkowski space and Euclidean spaces as well as their corresponding symmetries, given by q-deformed Lorentz algebra and algebras of q-deformed angular momentum, respectively [@CSSW90; @Pod90; @SWZ91; @Maj91; @LWW97]. Remarkably, Julius Wess and his coworkers were able to show that q-deformation of spaces and symmetries can indeed lead to discretizations, as they result from the existence of a smallest distance [@Fich97; @CW98]. This observation nourishes the hope that q-deformation might give a new method to regularize quantum field theories [@MajReg; @GKP96; @Oec99; @Blo03]. In our previous work [@WW01; @BW01; @Wac02; @Wac04; @WacTr; @Mik04; @SW04] attention was focused on q-deformed quantum spaces of physical importance, i.e. two-dimensional Manin plane, q-deformed Euclidean space in three or four dimensions and q-deformed Minkowski space. If we want to describe fields on q-deformed quantum spaces we need to consider representations of the corresponding quantum symmetries, given by $U_{q}(su_{2})$, $U_{q}(so_{4})$ and q-deformed Lorentz algebra. The study of such quantum algebras has produced a number of remarkable results during the last two decades. For a review we recommend the reader the presentations in [@KS97; @Maj95; @ChDe96] and references therein. In this article we want to adapt these general ideas to our previous considerations about q-deformed quantum spaces. In doing so, we provide a basis for performing concrete calculations, as they are necessary in formulating and evaluating field theories on quantum spaces. In particular, we intend to proceed as follows. In Sec. \[BasSec\] we cover the ideas our considerations about q-deformed quantum symmetries are based on. In the subsequent sections we first recall for each quantum algebra under consideration how its generators are realized within the corresponding q-deformed differential calculus. Then we are going to present explicit formulae for coproduct and antipode on a set of independent symmetry generators. With this knowledge at hand we should be able to write down explicit formulae for so-called q-commutators between symmetry generators and representation space elements. In addition to this, we are going to consider spinor and vector representations of the independent symmetry generators and give a complete review of the commutation relations between symmetry generators and components of a spinor or vector operator. Furthermore we are going to calculate the adjoint action of the independent symmetry generators on each other. In this manner, we will get relations for quantum Lie algebras. We will close our considerations by writing down q-analogs of Casimir operators. Finally, Sec. \[Concl\] shall serve as a short conclusion. For reference and for the purpose of introducing consistent and convenient notation, we provide a review of key notations and results in Appendix \[AppA\]. We should also mention that most of our results were obtained by applying the computer algebra system Mathematica [@Wol]. We are convinced that in the future this powerful tool will be inevitable in managing the extraordinary complexity of q-deformation. Basic ideas on q-deformed quantum symmetries\[BasSec\] ====================================================== Roughly speaking a quantum space is nothing else than an algebra generated by non-commuting coordinates $X_{1},X_{2},\ldots,X_{n},$ i.e.$$\mathcal{A}_{q}=\mathbb{C}{[}{[}X_{1},\ldots X_{n}{]}{]}/\mathcal{I},$$ where $\mathcal{I}$ denotes the ideal generated by the relations of the non-commuting coordinates. The quantum spaces we are interested in for physical reasons are two-dimensional Manin plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. For their definition we refer the reader to Appendix \[AppA\]. On each of these quantum spaces exist two differential calculi [@WZ91; @CSW91; @Song92] with $$\begin{aligned} \partial^{i}X^{j} & =g^{ij}+k(\hat{R}^{-1})_{kl}^{ij}X^{k}\partial ^{\,l},\qquad k\in\mathbb{R},\\ \hat{\partial}^{i}X^{j} & =g^{ij}+k^{-1}(\hat{R})_{kl}^{ij}X^{k}% \hat{\partial}^{\,l},\nonumber\end{aligned}$$ where $\hat{R}$ and $g^{ij}$ denote respectively the $R$-matrix and the corresponding quantum metric of the underlying quantum symmetry. There is a q-deformed antisymmetriser $P_{A}$ that enables us to define the components of q-deformed orbital angular momentum as antisymmetrized products of coordinates and derivatives [@Fio95]. Specifically, we have$$L^{ij}\sim(P_{A})_{kl}^{ij}\,X^{k}\partial^{\,l}\sim(P_{A})_{kl}^{ij}% \,X^{k}\hat{\partial}^{\,l}.$$ It can be shown that the $L^{ij}$ together with a central generator $U$ being subject to $$U-1\sim\lambda g_{ij}\,X^{i}\hat{\partial}^{\,j},\quad\lambda\equiv q-q^{-1},$$ span the quantum algebras describing the underlying quantum symmetry. This way we will obtain $U_{q}(su_{2})$, $U_{q}(so_{4})$ as well as q-deformed Lorentz algebra in the sequel of this article. In accordance with the classical case, a single particle wave function should be defined as element of the tensor product of a finite vector space $\mathcal{S}$ holding the spin degrees of freedom and the algebra of space functions $\mathcal{M}$, where $\mathcal{S}$ and $\mathcal{M}$ are both modules of the q-deformed symmetry algebra under consideration. In contrast to the classical situation, we now have to distinguish between right and left wave functions, i.e. $$\psi_{R}\equiv\sum\nolimits_{j}e^{j}\otimes\psi^{j},\quad\psi_{L}\equiv \sum\nolimits_{j}\psi^{j}\otimes e^{j},$$ where ${\{}e^{j}{\}}$ is a basis in $\mathcal{S}$ and $\psi^{j}$ are elements of $\mathcal{M}$. The reason for this lies in the fact that due to the braiding between the two tensor factors we are not allowed to interchange them by a trivial flip [@Maj94Kat]. The transformation properties of our wave functions are determined by the coproduct of the symmetry generators, since we have (we write the coproduct in the so-called Sweedler notation, i.e. $\Delta(a)=a_{(1)}\otimes a_{(2)})$ $$\begin{aligned} L^{ij}\rhd\psi_{R} & =\sum\nolimits_{k}(L^{ij})_{(1)}\rhd e^{k}% \otimes(L^{ij})_{(2)}\rhd\psi^{k},\\ L^{ij}\rhd\psi_{L} & =\sum\nolimits_{k}(L^{ij})_{(1)}\rhd\psi^{k}% \otimes(L^{ij})_{(2)}\rhd e^{k}.\nonumber\end{aligned}$$ For scalar wave functions $\mathcal{S}$ has to be a one-dimensional vector space with the corresponding representation on it being the trivial one. In this manner, the symmetry transformation is reduced to the action on space functions. In Ref. [@BW01] we derived explicit formulae for such actions. In the sequel of this article we will restrict ourselves to representations of symmetry generators on a spinorial or vectorial basis. Physical fields of higher spin should then be built up from spinor or vector fields. It should be noted that we can combine a quantum algebra $\mathcal{H}$ with its representation space $\mathcal{A}$ to form a left cross product algebra $\mathcal{A}\rtimes\mathcal{H}$ built on $\mathcal{A}\otimes\mathcal{H}$ with product$$(a\otimes h)(b\otimes g)=a(h_{(1)}\triangleright b)\otimes h_{(2)}g,\quad a,b\in\mathcal{A},\mathcal{\quad}h,g\in\mathcal{H}. \label{LefCrosPro}%$$ There is also a right-handed version of this notion called a right cross product algebra $\mathcal{H}\ltimes\mathcal{A}$ and built on $\mathcal{H}% \otimes\mathcal{A}$ with product$$(h\otimes a)(g\otimes b)=hg_{(1)}\otimes(a\triangleleft g_{(2)})b. \label{RigCrosPro}%$$ The last two identities tell us that the commutation relations between symmetry generators and representation space elements are completely determined by coproduct and action of the symmetry generators. In this article it can be seen that there is a remarkable correspondence between q-deformed symmetry algebras and their classical counterparts. Towards this end we have to introduce the notion of a q-commutator which is nothing other than the action of a symmetry generator $L^{ij}$ on a representation space element $V.$ Expressing that action in terms of the Hopf structure of $L^{ij}$ the q-brackets become$$\begin{aligned} {[}L^{ij},V{]}_{q} & \equiv L^{ij}\triangleright V=L_{(1)}^{ij}% VS(L_{(2)}^{ij}),\label{q-KommAllg}\\ {[}V,L^{ij}{]}_{q} & \equiv V\triangleleft L^{ij}=S^{-1}(L_{(2)}% ^{ij})VL_{(1)}^{ij},\nonumber\end{aligned}$$ where $S$ and $S^{-1}$ denote the antipode and its inverse, respectively. From their very definition it follows that q-commutators obey the q-deformed Jacobi identities $$\begin{aligned} {[}L^{ij},{[}L^{kl},V{]}_{q}{]}_{q} & ={[}{[}L_{(1)}^{ij},L^{kl}{]}_{q}% ,{[}L_{(2)}^{ij},V{]}_{q}{]}_{q},\\ {[}{[}L^{ij},L^{kl}{]}_{q},V{]}_{q} & ={[}L_{(1)}^{ij},{[}L^{kl}% ,{[}S(L_{(2)}^{ij}),V{]}_{q}{]}_{q}{]}_{q}.\nonumber\end{aligned}$$ Now, we are able to introduce the notion of a quantum Lie algebra as it was given in [@Schupp93; @Maj94; @Sud92]. A quantum Lie algebra can be regarded as a subspace of a q-deformed enveloping algebra $U_{q}(g)$ being invariant under the adjoint action of $U_{q}(g)$. The point now is that the $L^{ij}$ are the components of a tensor operator and this is the reason why their adjoint action on each other equals to a linear combination of the $L^{ij}$ [@Bie90], i.e. the $L^{ij}$ span a quantum Lie algebra with $${[}L^{ij},L^{kl}{]}_{q}(=L^{ij}\triangleright L^{kl}=L^{ij}\triangleleft L^{kl})=(C^{ij})_{mn}^{kl}L^{mn}, \label{Liecom}%$$ where the $C^{ij}$ are the so-called quantum structure constants. In the subsequent sections we are going to determine those constants in the case of $U_{q}(su_{2})$, $U_{q}(so_{4})$ and q-deformed Lorentz algebra. Finally, let us mention that the q-brackets give another way to write down commutation relations between symmetry generators and components of a vector or spinor operator. More formally, we have $${[}L^{ij},X^{k}{]}_{q}=(\tau_{L}^{ij})^{k}{}_{l}\,X^{l},\quad{[}X_{k}% ,L^{ij}{]}_{q}=X_{l}\,(\tau_{R}^{ij})^{l}{}_{k}, \label{VerLX}%$$ and $${[}L^{ij},\theta^{\alpha}{]}_{q}=(\sigma_{L}^{ij})^{\alpha}{}_{\beta}% \,\theta^{\beta},\quad\left[ \theta^{\alpha},L^{ij}\right] _{q}% =\theta_{\beta}\,(\sigma_{R}^{ij})^{\beta}{}_{\alpha}, \label{VerLTheta}%$$ where $X^{k}$ and $\theta^{\alpha}$ stand respectively for components of vector and spinor operators. Notice that for $\tau_{L/R}$ and $\sigma_{L/R}$ we have to substitute the representation matrix of $L^{ij}$, as it comes out for the vector and spinor representation, respectively. It should be appreciated that the relations in (\[Liecom\]), (\[VerLX\]), and (\[VerLTheta\]) become part of a q-deformed Super-Euclidean or Super-Poincaré algebra, if such objects exist. Quantum Lie algebra of three-dimensional angular momentum\[QuLie3dim\] ====================================================================== Representation of three-dimensional angular momentum within q-deformed differential calculus -------------------------------------------------------------------------------------------- In the case of three-dimensional $q$-deformed Euclidean space (for its definition see Appendix \[AppA\]) the generators of orbital angular momentum are defined by [@LWW97] $$L^{A}\equiv\Lambda^{1/2}X^{C}\hat{\partial}^{D}\epsilon_{DC}{}^{A},\quad A\in\{+,3,-\}, \label{DrehGen3dim}%$$ where $\varepsilon_{DC}{}^{A}$ denotes a q-analog of the completely antisymmetric tensor of third rank and $\Lambda$ stands for a scaling operator subject to $$\Lambda X^{A}=q^{4}X^{A}\Lambda,\quad\Lambda\hat{\partial}^{A}=q^{-4}% \hat{\partial}^{A}\Lambda,\quad A\in\{+,3,-\}.$$ If not stated otherwise summation over all repeated indices is to be understood. Substituting for $\varepsilon_{DC}{}^{A}$ their explicit form, we obtain from Eq. (\[DrehGen3dim\]) $$\begin{aligned} L^{+} & =-q^{-1}\Lambda^{1/2}X^{+}\hat{\partial}^{3}+q^{-3}\Lambda ^{1/2}X^{3}\hat{\partial}^{+},\\ L^{3} & =-q^{-2}\Lambda^{1/2}X^{+}\hat{\partial}^{-}+q^{-2}\Lambda ^{1/2}X^{-}\hat{\partial}^{+}-q^{-2}\lambda\Lambda^{1/2}X^{3}\hat{\partial }^{3},\nonumber\\ L^{-} & =-q^{-1}\Lambda^{1/2}X^{3}\hat{\partial}^{-}+q^{-3}\Lambda ^{1/2}X^{-}\hat{\partial}^{3}.\nonumber\end{aligned}$$ Using the Leibniz rules for partial derivatives in the form $$X^{A}\hat{\partial}^{B}=g^{AB}+(\hat{R}^{-1})^{AB}{}_{CD}\,\hat{\partial}% ^{C}X^{D},$$ and taking into account the identities$$g^{AB}\epsilon_{BA}{}^{C}=0,\quad(\hat{R}^{-1})^{AB}{}_{CD}\,\epsilon_{BA}% {}^{E}=-q^{4}\epsilon_{DC}{}^{E},$$ the generators in Eq. (\[DrehGen3dim\]) can alternatively be written as $$L^{A}=-q^{4}\Lambda^{1/2}\hat{\partial}^{C}X^{D}\epsilon_{DC}{}^{A}.$$ As already mentioned, there is a second set of derivatives ${\partial}^{A}$ which can be linked to the first one via the relation [@LWW97]$$\hat{\partial}^{A}=\Lambda^{-1}(\partial^{A}+q^{3}\lambda X^{A}\partial ^{B}\partial^{C}g_{BC}).$$ Making use of this identity together with $X^{C}X^{D}\epsilon_{DC}{}^{A}=0$, one can show that we additionally have $$L^{A}=q^{4}\Lambda^{-1/2}X^{C}\partial^{D}\epsilon_{DC}{}^{A}=-\Lambda ^{-1/2}\partial^{C}X^{D}\epsilon_{DC}{}^{A}.$$ Hopf structure of $U_{q}(su_{2})$ and corresponding q-commutators ----------------------------------------------------------------- As it is well-known, the generators $L^{+},$ $L^{3},$ and $L^{-}$ together with a generator $\tau^{1/2}$ can be viewed as elements of the quantum algebra $U_{q}(su_{2})$ [@LWW97]. Its defining relations read $$\begin{gathered} \tau^{1/2}L^{\pm}=q^{\mp2}L^{\pm}\tau^{1/2},\quad\tau^{1/2}L^{3}=L^{3}% \tau^{1/2},\label{UqSu2}\\ L^{-}L^{+}-L^{+}L^{-}=\tau^{-1/2}L^{3},\nonumber\\ L^{\pm}L^{3}-L^{3}L^{\pm}=\mp q^{\pm1}L^{\pm}\tau^{-1/2}.\nonumber\end{gathered}$$ Furthermore, this algebra has a Hopf structure given by $$\begin{aligned} \Delta(L^{\pm}) & =L^{\pm}\otimes\tau^{-1/2}+1\otimes L^{\pm},\\ \Delta(L^{3}) & =L^{3}\otimes\tau^{-1/2}+\tau^{1/2}\otimes{L^{3}}\nonumber\\ & +\;\lambda\tau^{1/2}(qL^{+}\otimes L^{-}+q^{-1}L^{-}\otimes{L^{+}% }),\nonumber\\[0.16in] S(L^{\pm}) & =q^{\mp2}S^{-1}(L^{\pm})=-L^{\pm}\tau^{1/2},\\ S(L^{3}) & =S^{-1}(L^{3})=-q^{-2}L^{3}+\lambda\lambda_{+}\tau^{1/2}% L^{+}L^{-},\nonumber\\[0.16in] \varepsilon(L^{A}) & =0,\quad A\in\{+,3,-\},\end{aligned}$$ where $\lambda_{+}\equiv q+q^{-1}$. In addition to this, let us notice that $\tau^{1/2}$ is a grouplike generator. With this Hopf structure at hand we are in a position to write down expressions for q-commutators. Specifically, we get from Eq. (\[q-KommAllg\]) for left-commutators $$\begin{aligned} {[}L^{\pm},V{]}_{q} & =(L^{\pm}V-VL^{\pm})\tau^{1/2},\label{commeu3l}\\ {[}L^{3},V{]}_{q} & =L^{3}V\tau^{1/2}-q^{-2}\tau^{1/2}VL^{3}\nonumber\\ & -\;\lambda\tau^{1/2}(q^{-1}L^{+}V\tau^{1/2}L^{-}+qL^{-}V\tau^{1/2}% L^{+})\nonumber\\ & +\;\lambda\lambda_{+}\tau^{1/2}V\tau^{1/2}L^{+}L^{-},\nonumber\end{aligned}$$ and likewise for right-commutators,$$\begin{aligned} {[}V,L^{\pm}{]}_{q} & =\tau^{1/2}(VL^{\pm}-L^{\pm}V)\label{comeu3r}\\ {[}V,L^{3}{]}_{q} & =\tau^{1/2}VL^{3}-q^{-2}L^{3}V\tau^{1/2}\nonumber\\ & -\;\lambda\tau^{1/2}(q^{-1}L^{+}V\tau^{1/2}L^{-}+qL^{-}V\tau^{1/2}% L^{+})\nonumber\\ & +\;\lambda\lambda_{+}\tau^{1/2}L^{+}L^{-}V\tau^{1/2},\nonumber\end{aligned}$$ where $V$ denotes an element living in a representation space of $U_{q}% (su_{2}).$ Matrix representations of $U_{q}(su_{2})$ and commutation relations with tensor operators ----------------------------------------------------------------------------------------- Next, we would like to turn our attention to some special representations of the symmetry generators $L^{+},L^{3},$ and $L^{-}$, namely spinor and vector representations [@KS97; @Bie95]. The finite dimensional representations of $U_{q}(su_{2})$ are already well-known (see for example Refs. [@KS97] and [@Bie95]). With our conventions (see also Appendix \[AppA\]) the spinor representations on symmetry generators become$$\begin{aligned} L^{A}\rhd\theta^{\,\alpha} & =(\sigma^{A})^{\alpha}{}_{\beta}\;\theta ^{\beta}, & \tau^{1/2}\rhd\theta^{\,\alpha} & =(\tau^{1/2})^{\alpha}% {}_{\beta}\;\theta^{\beta},\label{RepDreh2dim}\\ \theta_{\alpha}\lhd L^{A} & =\theta_{\beta}\,(\sigma^{A})^{\beta}{}_{\alpha }, & \theta_{\alpha}\lhd\tau^{1/2} & =\theta_{\beta}\,(\tau^{1/2})^{\beta}% {}_{\alpha},\nonumber\end{aligned}$$ where we have introduced as some kind of q-deformed sigma matrices $$\begin{gathered} (\sigma^{+})^{\alpha}{}_{\beta}=-q^{1/2}\lambda_{+}^{-1/2}\left( \begin{array} [c]{cc}% 0 & 0\\ 1 & 0 \end{array} \right) ,\quad(\sigma^{-})^{\alpha}{}_{\beta}=q^{-1/2}\lambda_{+}% ^{-1/2}\left( \begin{array} [c]{cc}% 0 & 1\\ 0 & 0 \end{array} \right) ,\label{SigmMatr3dim}\\ (\sigma^{3})^{\alpha}{}_{\beta}=\lambda_{+}^{-1}\left( \begin{array} [c]{cc}% -q & 0\\ 0 & q^{-1}% \end{array} \right) ,\quad(\tau^{1/2})^{\alpha}{}_{\beta}=\left( \begin{array} [c]{cc}% q & 0\\ 0 & q^{-1}% \end{array} \right) .\nonumber\end{gathered}$$ Here and in what follows, we shall take the convention that **upper and lower matrix indices refer to columns and rows, respectively**. That the above matrices indeed give a representation can easily be checked. Towards this end, we have to substitute in (\[UqSu2\]) the sigma matrices for the algebra generators. Then we can show by usual matrix multiplication that the algebra relations are fulfilled. The above results enable us to write down commutation relations between symmetry generators and components of a spinor operator. Such a spinor operator with components $\theta^{\,\alpha},$ $\alpha=1,2,$ is completely determined by its transformation properties$${[}L^{A},\theta^{\,\alpha}{]}_{q}=L^{A}\rhd\theta^{\,\alpha},\quad{[}% \theta_{\alpha},L^{A}{]}_{q}=\theta_{\alpha}\lhd L^{A}. \label{TransTensOp}%$$ Inserting the results of (\[RepDreh2dim\]) and (\[SigmMatr3dim\]) into the relations of (\[TransTensOp\]) and then rearranging, it follows that $$\begin{aligned} L^{+}\theta^{1} & =\theta^{1}L^{+}-q^{1/2}\lambda_{+}^{-1/2}\theta^{2}% \tau^{-1/2},\\ L^{+}\theta^{2} & =\theta^{2}L^{+},\nonumber\\ L^{3}\theta^{1} & =q\theta^{1}L^{3}-q^{-1/2}\lambda\lambda_{+}^{-1/2}% \theta^{2}L^{-}-q\lambda_{+}^{-1}\theta^{1}\tau^{-1/2},\nonumber\\ L^{3}\theta^{2} & =q^{-1}\theta^{2}L^{3}+q^{1/2}\lambda\lambda_{+}% ^{-1/2}\theta^{1}L^{+}+q^{-1}\lambda_{+}^{-1}\theta^{2}\tau^{-1/2},\nonumber\\ L^{-}\theta^{1} & =\theta^{1}L^{-},\nonumber\\ L^{-}\theta^{2} & =\theta^{2}L^{-}+q^{-1/2}\lambda_{+}^{-1/2}\theta^{1}% \tau^{-1/2},\nonumber\end{aligned}$$ and likewise, $$\begin{aligned} \theta_{1}L^{+} & =L^{+}\theta_{1},\\ \theta_{2}L^{+} & =L^{+}\theta_{2}-q^{1/2}\lambda_{+}^{-1/2}\tau ^{-1/2}\theta_{1},\nonumber\\ \theta_{1}L^{3} & =qL^{3}\theta_{1}+q^{-1/2}\lambda\lambda_{+}^{-1/2}% L^{+}\theta_{2}+q\lambda_{+}^{-1}\tau^{-1/2}\theta_{1},\nonumber\\ \theta_{2}L^{3} & =q^{-1}L^{3}\theta_{2}-q^{1/2}\lambda\lambda_{+}% ^{-1/2}L^{-}\theta_{1}+q^{-1}\lambda_{+}^{-1}\tau^{-1/2}\theta_{2},\nonumber\\ \theta_{1}L^{-} & =L^{-}\theta_{1}+q^{-1/2}\lambda_{+}^{-1/2}\tau ^{-1/2}\theta_{2},\nonumber\\ \theta_{2}L^{-} & =L^{-}\theta_{2}.\nonumber\end{aligned}$$ Next, we turn to the vector representations of $U_{q}(su_{2})$. They take the form $$L^{A}\rhd X^{B}=(\tau^{A})^{B}{}_{C}\;X^{C},\quad X_{B}\lhd L^{A}=X_{C}% \,(\tau^{A})^{C}{}_{B},$$ with $$\begin{gathered} (\tau^{+})^{B}{}_{C}=\left( \begin{array} [c]{ccc}% 0 & -q & 0\\ 0 & 0 & -1\\ 0 & 0 & 0 \end{array} \right) ,\quad(\tau^{-})^{B}{}_{C}=\left( \begin{array} [c]{ccc}% 0 & 0 & 0\\ -1 & 0 & 0\\ 0 & q^{-1} & 0 \end{array} \right) ,\\ (\tau^{3})^{B}{}_{C}=\left( \begin{array} [c]{ccc}% q^{-1} & 0 & 0\\ 0 & -\lambda & 0\\ 0 & 0 & -q \end{array} \right) ,\quad(\tau^{1/2})^{B}{}_{C}=\left( \begin{array} [c]{ccc}% q^{-2} & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & q^{2}% \end{array} \right) ,\nonumber\end{gathered}$$ where rows and columns are arranged in the order $+,$ $3,$ and $-$(from left to right and top to bottom). Their property of being a representation can be verified in very much the same way as was done for the spinor representation. Finally, let us note that the above representation matrices and the q-deformed $\varepsilon$-tensor are linked via $$(\tau^{A})^{B}{}_{C}=q^{2}\varepsilon^{AB}{}_{C},\quad A\in\{+,3,-\}.$$ With the same reasonings already applied to the case of spinor representations we can derive commutation relations between symmetry generators and the components of a vector operator. In complete accordance to the spinor case, their general form is given by$${[}L^{A},X^{B}{]}_{q}=L^{A}\rhd X^{B},\quad{[}X_{B},L^{A}{]}_{q}=X_{B}\lhd L^{A}, \label{VecDar3dim}%$$ from which we find by specifying q-commutator and vector representation$$\begin{aligned} L^{\pm}X^{\pm} & =X^{\pm}L^{\pm},\\ L^{\pm}X^{\mp} & =X^{\mp}L^{\pm}\mp X^{3}\tau^{-1/2},\nonumber\\ L^{\pm}X^{3} & =X^{3}L^{\pm}\mp q^{\pm1}X^{\pm}\tau^{-1/2},\nonumber\\ L^{3}X^{\pm} & =q^{\mp2}X^{\pm}L^{3}\pm q^{\mp1}\lambda X^{3}L^{\pm}\pm q^{\mp1}X^{\pm}\tau^{-1/2},\nonumber\\ L^{3}X^{3} & =X^{3}L^{3}+\lambda(X^{-}L^{+}-X^{+}L^{-})-\lambda X^{3}% \tau^{-1/2},\nonumber\end{aligned}$$ and $$\begin{aligned} X_{\mp}L^{\pm} & =L^{\pm}X_{\mp},\\ X_{\pm}L^{\pm} & =L^{\pm}X_{\pm}\mp q^{\pm1}\tau^{-1/2}X_{3},\nonumber\\ X_{3}L^{\pm} & =L^{\pm}X_{3}\mp\tau^{-1/2}X_{\mp},\nonumber\\ X_{\pm}L^{3} & =q^{\mp2}L^{3}X_{\pm}\mp\lambda L^{\mp}X^{3}\pm q^{\mp1}% \tau^{-1/2}X_{\pm},\nonumber\\ X_{3}L^{3} & =L^{3}X_{3}+\lambda(q^{-1}L^{+}X^{+}-qL^{-}X_{-})-\lambda \tau^{-1/2}X_{3}.\nonumber\end{aligned}$$ This way, we regain the commutation relations already presented in [@LWW97]. Quantum Lie algebra of $U_{q}(su_{2})$ and its Casimir operator --------------------------------------------------------------- Recalling that $L^{A}$, $A\in\left\{ +,3,-\right\} ,$ is a vector operator, the identities in (\[VecDar3dim\]) also apply to the case with $X^{A}$ being replaced by $L^{A}$. In doing so, we are led to the relations for the quantum Lie algebra of $U_{q}(su_{2})$, i.e. $${[}L^{A},L^{B}{]}_{q}=q^{2}\varepsilon^{AB}{}_{C}\,L^{C},$$ or more concretely,$$\begin{aligned} {[}L^{A},L^{A}{]}_{q} & =0,\qquad A\in\left\{ +,-\right\} ,\\ {[}L^{3},L^{3}{]}_{q} & =-\lambda L^{3},\nonumber\\ {[}L^{\pm},L^{3}{]}_{q} & =\mp q^{\pm1}L^{\pm},\nonumber\\ {[}L^{\pm},L^{\mp}{]}_{q} & =\mp L^{3}.\nonumber\end{aligned}$$ These relations are equivalent to those in (\[UqSu2\]), as can be seen in a straightforward manner by writing out the q-commutators explicitly. Instead of the vector operator $L^{A}$, one can just as well use a complete antisymmetric tensor operator of second rank given by $$M^{AB}\equiv\varepsilon^{AB}{}_{C}\,L^{C}.$$ Its antisymmetry requires the following identities to hold, which can easily be read off from its very definition:$$\begin{gathered} M^{++}=M^{--}=0,\quad M^{\pm3}=-q^{\pm2}M^{3\pm},\\ M^{-+}=-M^{+-},\quad M^{33}=\lambda M^{+-}.\nonumber\end{gathered}$$ Consequently, we have three independent components, for which we can choose$$M^{3\pm}=-q^{\pm1}L^{\pm},\quad M^{+-}=-q^{-2}L^{3}.$$ Last but not least we want to deal with the Casimir operator of our quantum Lie algebra. Let us recall that a Casimir operator $C$ has to be subject to $${[}L^{A},C{]}_{q}={[}C,L^{A}{]}_{q}=0\quad\mbox{for all}\quad A\in\left\{ +,3,-\right\} . \label{CasOp}%$$ By a direct calculation making use of $$\begin{aligned} {[}L^{A},UV{]}_{q} & ={[}L_{(1)}^{A},U{]}_{q}{[}L_{(2)}^{A},V{]}_{q},\\ {[}UV,L^{A}{]}_{q} & ={[}U,L_{(2)}^{A}{]}_{q}{[}V,L_{(1)}^{A}{]}% _{q},\nonumber\end{aligned}$$ it can be seen that a solution to (\[CasOp\]) is given by the expression$$L^{2}\equiv g_{AB}L^{A}L^{B}=-qL^{+}L^{-}+L^{3}L^{3}-q^{-1}L^{-}L^{+}, \label{L2}%$$ which, in fact, is a q-analog for the Casimir operator of classical angular momentum algebra. Equivalently, the Casimir operator for $U_{q}(su_{2})$ can also be written in terms of the $M^{\prime}s$. In this manner, it becomes$$\begin{aligned} M^{2} & \equiv g_{AB}g_{CD}M^{AC}M^{BD}\\ & =q^{-1}M^{+3}M^{3-}-M^{+-}M^{+-}+q^{-3}M^{3-}M^{+3},\nonumber\end{aligned}$$ which agrees with $L^{2}$ up to a constant factor. Before closing this section, we wish to specify our Casimir operator for the different representations addressed so far. Substituting for the symmetry generators their representations we finally obtain 1. (operator representation)$$L^{2}=-(X\circ X)(\partial\circ\partial)+q^{2}(X\circ\partial)(X\circ \partial)+q^{-2}X\circ\partial,$$ with $U\circ V\equiv g_{AB}U^{A}V^{B}$, 2. (spinor representation)$$L^{2}=q^{-2}\lambda_{+}^{-2}{[}{[}3{]}{]}_{q^{2}}% \mbox{1 \kern-.59em {\rm l}}_{2\times2},$$ 3. (vector representation)$$L^{2}=q^{-2}{[}{[}2{]}{]}_{q^{4}}\mbox{1 \kern-.59em {\rm l}}_{3\times3},$$ where the antisymmetric q-numbers are defined by $${[[}n{]]}_{q}\equiv\frac{1-q^{an}}{1-q^{a}},\quad n\in\mathbb{N},\quad a\in\mathbb{C}.$$ Quantum Lie algebra of four-dimensional angular momentum ======================================================== As next example we would like to consider the q-deformed algebra of four-dimensional angular momentum. This case can be treated in very much the same way as the three-dimensional one. Thus, we restrict ourselves to stating the results, only. Representation of four-dimensional angular momentum within q-deformed differential calculus ------------------------------------------------------------------------------------------- First of all, let us start with the defining representation. Within the differential calculus the generators of q-deformed four-dimensional angular momentum are represented by (see for example Ref. [@Oca96]) $$L^{ij}\equiv-q^{-2}\lambda_{+}\Lambda^{1/2}(P_{A})^{ij}{}_{kl}\,X^{k}% \hat{\partial}^{l}=\lambda_{+}\Lambda^{1/2}(P_{A})^{ij}{}_{kl}\,\hat{\partial }^{k}X^{l},$$ or $$L^{ij}\equiv-\lambda_{+}\Lambda^{-1/2}(P_{A})^{ij}{}_{kl}\,X^{k}\partial ^{l}=q^{-2}\lambda_{+}\Lambda^{-1/2}(P_{A})^{ij}{}_{kl}\,\partial^{k}X^{l}, \label{Def4dimAngMom}%$$ where $P_{A}$ is a q-analog of an antisymmetrizer and $\Lambda$ denotes a scaling operator subject to $$\Lambda X^{i}=q^{2}X^{i}\Lambda,\quad\Lambda\partial^{i}=q^{-2}\partial ^{i}\Lambda.$$ Eq. (\[Def4dimAngMom\]) shows us that the $L^{ij}$ are components of an antisymmetric tensor operator. More specifically, they satisfy $$\begin{gathered} L^{ii}=0,\quad i=1,\ldots,4,\\ L^{j1}=-qL^{1j},\quad L^{4j}=-qL^{j4},\quad j=2,3,\nonumber\\ L^{41}=-L^{14},\quad L^{32}=-L^{23}+\lambda L^{14}.\nonumber\end{gathered}$$ Thus, we have in complete analogy to the classical case only six independent generators, for which we can choose the set of $L^{ij}$ with $i<j.$ Taking the explicit form of $P_{A}$ into consideration, we get for them more explicitly $$\begin{aligned} L^{1i} & =-\Lambda^{-1/2}(q^{-1}X^{1}\hat{\partial}^{i}-X^{i}\hat{\partial }^{1}),\quad i=1,2,\\ L^{i4} & =-\Lambda^{-1/2}(q^{-1}X^{i}\hat{\partial}^{4}-X^{4}\hat{\partial }^{i}),\nonumber\\ L^{14} & =2\lambda_{+}^{-1}\Lambda^{-1/2}(X^{1}\hat{\partial}^{4}-X^{4}% \hat{\partial}^{1})\nonumber\\ & -\;\lambda\lambda_{+}^{-1}\Lambda^{-1/2}(X^{2}\hat{\partial}^{3}+X^{3}% \hat{\partial}^{2}),\nonumber\\ L^{23} & =(q^{2}+q^{-2})\lambda_{+}^{-1}\Lambda^{-1/2}X^{2}\hat{\partial }^{3}\nonumber\\ & -\;2\lambda_{+}^{-1}\Lambda^{-1/2}X^{3}\hat{\partial}^{2}+\lambda \lambda_{+}^{-1}\Lambda^{-1/2}(X^{1}\hat{\partial}^{4}-X^{4}\hat{\partial}% ^{1}).\nonumber\end{aligned}$$ Together with two grouplike braiding operators $K_{i},i=1,2$, the generators $L^{kl},$ $k<l$, span the Hopf algebra $U_{q}(so_{4})$. Applying the commutation relations for coordinates, partial derivatives and the two braiding operators (see for example Appendix \[AppA\] and Ref. [@BW01]) leaves us with the nontrivial relations$$\begin{gathered} L^{14}L^{1j}-L^{1j}L^{14}=-K_{i}L^{12},\quad(i,j)\in \{(2,2),(1,3)\},\label{UqSO4Anf}\\ L^{l4}L^{14}-L^{14}L^{l4}=-q^{-2}K_{k}L^{l4},\quad(k,l)\in \{(1,2),(2,3)\},\nonumber\\[0.16in] L^{23}L^{12}-L^{12}L^{23}=-qK_{2}L^{12},\quad L^{23}L^{13}-L^{13}L^{23}% =q^{-1}K_{1}L^{13},\\ L^{24}L^{23}-L^{23}L^{24}=q^{-3}K_{1}L^{24},\quad L^{34}L^{23}-L^{23}% L^{34}=-q^{-1}K_{2}L^{34},\nonumber\\[0.16in] K_{1}L^{13}=q^{-2}L^{13}K_{1},\quad K_{2}L^{12}=q^{-2}L^{12}K_{2}% ,\label{Uqsu4End}\\ K_{1}L^{24}=q^{2}L^{24}K_{1},\quad K_{2}L^{34}=q^{2}L^{34}K_{2}.\nonumber\end{gathered}$$ Hopf structure of $U_{q}(so_{4})$ and corresponding q-commutators ----------------------------------------------------------------- As in the three-dimensional case we need the Hopf structure to proceed further. For the independent generators this reads as (see also Ref. [@Oca96])$$\begin{aligned} \Delta(L^{1j}) & =L^{1j}\otimes K_{i}+1\otimes L^{12},\quad(i,j)\in \{(2,2),(1,3)\},\\ \Delta(L^{l4}) & =L^{l4}\otimes K_{k}+1\otimes L^{l4},\quad(k,l)\in \{(1,2),(2,3)\},\nonumber\\ \Delta(L^{14}) & =q\lambda_{+}^{-1}L^{14}\otimes K_{1}+q^{-1}\lambda _{+}^{-1}L^{14}\otimes K_{2}\nonumber\\ & +\;q\lambda_{+}^{-1}K_{1}^{-1}\otimes L^{14}+q^{-1}\lambda_{+}^{-1}% K_{2}^{-1}\otimes L^{14}\nonumber\\ & -\;\lambda_{+}^{-1}L^{23}\otimes K_{1}+\lambda_{+}^{-1}L^{23}\otimes K_{2}\nonumber\\ & -\;\lambda_{+}^{-1}K_{1}^{-1}\otimes L^{23}+\lambda_{+}^{-1}K_{2}% ^{-1}\otimes L^{23}\nonumber\\ & -\;q\lambda\lambda_{+}^{-1}L^{24}K_{1}^{-1}\otimes L^{13}-q\lambda \lambda_{+}^{-1}K_{1}^{-1}L^{13}\otimes L^{24}\nonumber\\ & -\;q\lambda\lambda_{+}^{-1}L^{34}K_{2}^{-1}\otimes L^{12}-q\lambda \lambda_{+}^{-1}K_{2}^{-1}L^{12}\otimes L^{34},\nonumber\\ \Delta(L^{23}) & =-\lambda_{+}^{-1}K_{1}^{-1}\otimes L^{14}+\lambda_{+}% ^{-1}K_{2}^{-1}\otimes L^{14}\nonumber\\ & -\;\lambda_{+}^{-1}L^{14}\otimes K_{1}+\lambda_{+}^{-1}L^{14}\otimes K_{2}\nonumber\\ & +\;q^{-1}\lambda_{+}^{-1}L^{23}\otimes K_{1}+q\lambda_{+}^{-1}L^{23}\otimes K_{2}\nonumber\\ & +\;q^{-1}\lambda_{+}^{-1}K_{1}^{-1}\otimes L^{23}+q\lambda_{+}^{-1}% K_{2}^{-1}\otimes L^{23}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}L^{24}K_{1}^{-1}\otimes L^{13}+\lambda\lambda _{+}^{-1}K_{1}^{-1}L^{13}\otimes L^{24}\nonumber\\ & -\;q^{2}\lambda\lambda_{+}^{-1}L^{34}K_{2}^{-1}\otimes L^{12}-q^{2}% \lambda\lambda_{+}^{-1}K_{2}^{-1}L^{12}\otimes L^{34},\nonumber\\[0.16in] S(L^{1j}) & =q^{-2}S^{-1}(L^{1j})=-L^{1j}K_{i}^{-1},\quad(i,j)\in \{(2,2),(1,3)\},\\ S(L^{l4}) & =q^{2}S^{-1}(L^{l4})=-L^{l4}K_{k}^{-1},\quad(k,l)\in \{(1,2),(2,3)\},\nonumber\\ S(L^{14}) & =S^{-1}(L^{14})=L^{14}-q^{-1}\lambda^{-1}((K_{1}+K_{2}% )-(K_{1}^{-1}+K_{2}^{-1})),\nonumber\\ S(L^{23}) & =S^{-1}(L^{23})=L^{23}+q^{-1}\lambda^{-1}(q^{-1}(K_{1}% -K_{1}^{-1})-q(K_{2}-K_{2}^{-1})),\nonumber\\[0.16in] \varepsilon(L^{mn}) & =0.\end{aligned}$$ Again, this Hopf structure enables us to introduce q-commutators in complete analogy to the three-dimensional case. Explicitly written out we obtain for the q-commutator with an element living in a representation space of $U_{q}(so_{4})$$$\begin{aligned} {[}L^{1j},V{]}_{q} & =(L^{1j}V-VL^{1j})K_{i}^{-1},\quad(i,j)\in \{(2,2),(1,3)\},\label{commeu4l}\\ {[}L^{l4},V{]}_{q} & =(L^{l4}V-VL^{l4})K_{k}^{-1},\quad(k,l)\in \{(1,2),(2,3)\},\nonumber\\ {[}L^{14},V{]}_{q} & =-q^{-1}\lambda^{-1}(K_{1}^{-1}VK_{1}+K_{2}^{-1}% VK_{2})\nonumber\\ & +\;q^{-1}\lambda^{-1}(K_{1}^{-1}VK_{1}^{-1}+K_{2}^{-1}VK_{2}^{-1}% )\nonumber\\ & -\;\lambda_{+}^{-1}(K_{2}^{-1}VL^{23}+L^{23}VK_{2}^{-1}-K_{1}^{-1}% VL^{23}+L^{23}VK_{1}^{-1})\nonumber\\ & +\;q^{-1}\lambda_{+}^{-1}(K_{2}^{-1}VL^{14}+L^{14}VK_{2}^{-1})\nonumber\\ & +\;q\lambda_{+}^{-1}(K_{1}^{-1}VL^{14}+L^{14}VK_{1}^{-1})\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}(K_{1}^{-1}L^{13}VL^{24}K_{1}^{-1}+K_{2}% ^{-1}L^{12}VL^{34}K_{2}^{-1})\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}(L^{24}K_{1}^{-1}VL^{13}K_{1}^{-1}+L^{34}% K_{2}^{-1}VL^{12}K_{2}^{-1}),\nonumber\\ {[}L^{23},V{]}_{q} & =q\lambda_{+}^{-1}(K_{2}^{-1}VL^{23}+L^{23}VK_{2}% ^{-1})\nonumber\\ & +\;q^{-1}\lambda_{+}^{-1}(K_{1}^{-1}VL^{23}+L^{23}VK_{1}^{-1})\nonumber\\ & +\;\lambda_{+}^{-1}(L^{14}VK_{2}^{-1}+K_{2}^{-1}VL^{14}-L^{14}VK_{1}% ^{-1}-K_{1}^{-1}VL^{14})\nonumber\\ & +\;\lambda^{-1}(K_{2}^{-1}VK_{2}^{-1}-K_{2}^{-1}VK_{2})\nonumber\\ & +\;q^{-2}\lambda^{-1}(K_{1}^{-1}VK_{1}-K_{1}^{-1}VK_{1}^{-1})\nonumber\\ & +\;q^2\lambda\lambda_{+}^{-1}(K_{2}^{-1}L^{12}% VL^{34}K_{2}^{-1}+L^{34}K_{2}^{-1}VL^{12}K_{2}% ^{-1})\nonumber\\ & -\;\lambda\lambda_{+}^{-1}(K_{1}^{-1}L^{13}VL^{24}K_{1}^{-1}-L^{24}% K_{1}^{-1}VL^{13}K_{1}^{-1}),\nonumber\end{aligned}$$ and likewise for their right versions, $$\begin{aligned} {[}V,L^{1j}{]}_{q} & =K_{i}^{-1}(VL^{1j}-L^{1j}V),\quad(i,j)\in \{(2,2),(1,3)\},\label{commeu4r}\\ {[}V,L^{l4}{]}_{q} & =K_{k}^{-1}(VL^{l4}-L^{l4}V),\quad(k,l)\in \{(1,2),(2,3)\},\nonumber\\ {[}V,L^{14}{]}_{q} & =q\lambda_{+}^{-1}(L^{14}VK_{1}^{-1}+K_{1}^{-1}% VL^{14})\nonumber\\ & +\;q^{-1}\lambda_{+}^{-1}(L^{14}VK_{2}^{-1}+K_{2}^{-1}VL^{14})\nonumber\\ & +\;\lambda_{+}^{-1}(L^{23}VK_{2}^{-1}+K_{2}^{-1}VL^{23}-L^{23}VK_{1}% ^{-1}-K_{1}^{-1}VL^{23})\nonumber\\ & +\;q^{-1}\lambda^{-1}(K_{1}^{-1}VK_{1}^{-1}-K_{1}VK_{1}^{-1})\nonumber\\ & +\;q^{-1}\lambda^{-1}(K_{2}^{-1}VK_{2}^{-1}-K_{2}VK_{2}^{-1})\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}(K_{1}^{-1}L^{13}VL^{24}K_{1}^{-1}+K_{1}% ^{-1}L^{24}VK_{1}^{-1}L^{13})\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}(K_{2}^{-1}L^{34}VK_{2}^{-1}L^{12}+K_{2}% ^{-1}L^{12}VL^{34}K_{2}^{-1}),\nonumber\\ {[}V,L^{23}{]}_{q} & =q\lambda_{+}^{-1}(L^{23}VK_{2}^{-1}+K_{2}^{-1}% VL^{23})\nonumber\\ & +\;q^{-1}\lambda_{+}^{-1}(L^{23}VK_{1}^{-1}+K_{1}^{-1}VL^{23})\nonumber\\ & +\;\lambda_{+}^{-1}(L^{14}VK_{2}^{-1}+K_{2}^{-1}VL^{14}-L^{14}VK_{1}% ^{-1}-K_{1}^{-1}VL^{14})\nonumber\\ & +\;\lambda^{-1}(K_{2}^{-1}VK_{2}^{-1}-K_{2}VK_{2}^{-1})\nonumber\\ & +\;q^{-2}\lambda^{-1}(K_{1}VK_{1}^{-1}-K_{1}^{-1}VK_{1}^{-1})\nonumber\\ & +\;\lambda\lambda_{+}^{-1}(q^{2}K_{2}^{-1}L^{34}VK_{2}^{-1}L^{12}% +q^2K_{2}^{-1}L^{12}VL^{34}K_{2}^{-1})\nonumber\\ & -\;\lambda\lambda_{+}^{-1}(K_{1}^{-1}L^{13}VL^{24}K_{1}^{-1}-K_{1}% ^{-1}L^{24}VK_{1}^{-1}L^{13}).\nonumber\end{aligned}$$ Matrix representations of $U_{q}(so_{4})$ and commutation relations with tensor operators ----------------------------------------------------------------------------------------- Let us now go on to the spinor and vector representations of the $L^{ij}$. As in the classical case, we can distinguish two types of spinor representations, i.e. $(1/2,0)$ and $(0,1/2)$ (see for example Ref. [@Oca96]). Explicitly, we have $$\begin{aligned} L^{ij}\rhd\theta^{\,\alpha} & =(\sigma^{ij})^{\alpha}{}_{\beta}% \,\theta^{\beta}, & \quad L^{ij}\rhd\tilde{\theta}^{\,\alpha} & =(\tilde{\sigma}^{ij})^{\alpha}{}_{\beta}\,\tilde{\theta}^{\beta}\\ \theta_{\alpha}\lhd L^{ij} & =\theta_{\beta}(\sigma^{ij})^{\beta}{}_{\alpha }, & \quad\tilde{\theta}_{\alpha}\lhd L^{ij} & =\tilde{\theta}_{\beta }(\tilde{\sigma}^{ij})^{\beta}{}_{\alpha},\nonumber\end{aligned}$$ with $$\begin{gathered} (\sigma^{13})^{\alpha}{}_{\beta}=\left( \begin{array} [c]{cc}% 0 & -q^{-2}\\ 0 & 0 \end{array} \right) ,\qquad(\sigma^{24})^{\alpha}{}_{\beta}=\left( \begin{array} [c]{cc}% 0 & 0\\ -q^{-1} & 0 \end{array} \right) ,\\ (\sigma^{14})^{\alpha}{}_{\beta}=-q(\sigma^{23})^{\alpha}{}_{\beta}% =q^{-1}\lambda_{+}^{-1}\left( \begin{array} [c]{cc}% -q & 0\\ 0 & q^{-1}% \end{array} \right) ,\nonumber\\ (\sigma^{34})^{\alpha}{}_{\beta}=(\sigma^{12})^{\alpha}{}_{\beta}=0.\nonumber\end{gathered}$$ The matrices with tilde one gets most easily from the identities $$\begin{aligned} (\tilde{\sigma}^{13})^{\alpha}{}_{\beta} & =(\sigma^{12})^{\alpha}{}_{\beta }, & (\tilde{\sigma}^{12})^{\alpha}{}_{\beta} & =(\sigma^{13})^{\alpha}% {}_{\beta},\\ (\tilde{\sigma}^{14})^{\alpha}{}_{\beta} & =(\sigma^{14})^{\alpha}{}_{\beta }, & (\tilde{\sigma}^{23})^{\alpha}{}_{\beta} & =-q^{-2}(\sigma ^{23})^{\alpha}{}_{\beta},\nonumber\\ (\tilde{\sigma}^{34})^{\alpha}{}_{\beta} & =(\sigma^{24})^{\alpha}{}_{\beta }, & (\tilde{\sigma}^{24})^{\alpha}{}_{\beta} & =(\sigma^{34})^{\alpha}% {}_{\beta}.\nonumber\end{aligned}$$ Notice that the tilde on the spinor components shall remind us of the fact that they transformed differently. That these matrices determine a representation of $U_{q}(so_{4})$ can again be proven by inserting them together with$$\begin{gathered} (K_{1})^{\alpha}{}_{\beta}=\left( \begin{array} [c]{cc}% q & 0\\ 0 & q^{-1}% \end{array} \right) ,\qquad(\tilde{K}_{2})^{\alpha}{}_{\beta}=\left( \begin{array} [c]{cc}% q^{-1} & 0\\ 0 & q \end{array} \right) ,\\ (\tilde{K}_{1})^{\alpha}{}_{\beta}=(K_{2})^{\alpha}{}_{\beta}% =\mbox{1 \kern-.59em {\rm l}},\nonumber\end{gathered}$$ into relations (\[UqSO4Anf\]) - (\[Uqsu4End\]). From the spinor representation of $U_{q}(so_{4})$ we can - as usual - compute commutation relations between the $L^{ij}$ and the components of a spinor operator. For this to achieve we apply the identities $$\left[ L^{ij},\theta^{\,\alpha}\right] _{q}=(\sigma^{ij})^{\alpha}{}_{\beta }\,\theta^{\beta},\quad\left[ \theta^{\,\alpha},L^{ij}\right] _{q}% =\theta_{\beta}\,(\sigma^{ij})^{\beta}{}_{\alpha}.$$ If we write out the q-commutators and substitute the representation matrices without tilde for the $L^{ij}$ we get the commutation relations $$\begin{aligned} L^{13}\theta^{2} & =\theta^{2}L^{13}-q^{-2}\theta^{1}K_{1},\\ L^{24}\theta^{1} & =\theta^{1}L^{24}-q^{-1}\theta^{2}K_{1},\nonumber\\ L^{14}\theta^{1} & =(q^{2}+q^{-1})\lambda_{+}^{-1}\theta^{1}L^{14}% -(q-1)\lambda_{+}^{-1}\theta^{1}L^{23}\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}\theta^{2}L^{13}-\lambda_{+}^{-1}\theta^{1}% K_{1},\nonumber\\ L^{14}\theta^{2} & =(q^{-1}+1)\lambda_{+}^{-1}\theta^{2}L^{14}% +(q^{-1}-1)\lambda_{+}^{-1}\theta^{2}L^{23}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}\theta^{1}L^{24}+q^{-2}\lambda_{+}^{-1}\theta ^{2}K_{1},\nonumber\\ L^{23}\theta^{1} & =(q+1)\lambda_{+}^{-1}\theta^{1}L^{23}-(q-1)\lambda _{+}^{-1}\theta^{1}L^{14}\nonumber\\ & -\;\lambda\lambda_{+}^{-1}\theta^{2}L^{13}+q^{-1}\lambda_{+}^{-1}\theta ^{1}K_{1},\nonumber\\ L^{23}\theta^{2} & =(q^{-2}+q)\lambda_{+}^{-1}\theta^{2}L^{23}% +(1-q^{-1})\lambda_{+}^{-1}\theta^{2}L^{14}\nonumber\\ & -\;q^{-1}\lambda\lambda_{+}^{-1}\theta^{1}L^{24}-q^{-3}\lambda_{+}% ^{-1}\theta^{2}K_{1},\nonumber\end{aligned}$$ and$$\begin{aligned} \theta_{1}L^{13} & =L^{13}\theta_{1}-q^{-2}K_{1}\theta_{2},\\ \theta_{2}L^{24} & =L^{24}\theta_{2}-q^{-1}K_{1}\theta_{1},\nonumber\\ \theta_{1}L^{14} & =(q^{2}+q^{-1})\lambda_{+}^{-1}L^{14}\theta _{1}-(q-1)\lambda_{+}^{-1}L^{23}\theta_{1}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}L^{24}\theta_{2}-\lambda_{+}^{-1}K_{1}\theta _{1},\nonumber\\ \theta_{2}L^{14} & =(q^{-1}+1)\lambda_{+}^{-1}L^{14}\theta_{2}% +(1-q^{-1})\lambda_{+}^{-1}L^{23}\theta_{2}\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}L^{13}\theta_{1}+q^{-2}\lambda_{+}^{-1}% K_{1}\theta_{2},\nonumber\\ \theta_{1}L^{23} & =(q+1)\lambda_{+}^{-1}L^{23}\theta_{1}-(q-1)\lambda _{+}^{-1}L^{14}\theta_{1}\nonumber\\ & -\;q^{-1}\lambda\lambda_{+}^{-1}L^{24}\theta_{2}+q^{-1}\lambda_{+}% ^{-1}K_{1}\theta_{1},\nonumber\\ \theta_{2}L^{23} & =(q+q^{-2})\lambda_{+}^{-1}L^{23}\theta_{2}% +(1-q^{-1})\lambda_{+}^{-1}L^{14}\theta_{2}\nonumber\\ & -\;\lambda\lambda_{+}^{-1}L^{13}\theta_{1}-q^{-3}\lambda_{+}^{-1}% K_{1}\theta_{2}.\nonumber\end{aligned}$$ The remaining relations are trivial, i.e. these relations take the form $L^{ij}\theta^{k}=\theta^{k}L^{ij}.$ Repeating the same steps as before for spinors (and the representation matrices) with tilde we arrive at the following nontrivial relations: $$\begin{aligned} L^{12}\tilde{\theta}^{2} & =\tilde{\theta}^{2}L^{12}-q^{-2}\tilde{\theta }^{1}K_{2},\\ L^{34}\tilde{\theta}^{1} & =\tilde{\theta}^{1}L^{34}-q^{-1}\tilde{\theta }^{2}K_{2},\nonumber\\ L^{14}\tilde{\theta}^{1} & =(q+1)\lambda_{+}^{-1}\tilde{\theta}^{1}% L^{14}+(q-1)\lambda_{+}^{-1}\tilde{\theta}^{1}L^{23}\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}\tilde{\theta}^{2}L^{12}-\lambda_{+}^{-1}% \tilde{\theta}^{1}K_{2},\nonumber\\ L^{14}\tilde{\theta}^{2} & =(q^{-2}+q)\lambda_{+}^{-1}\tilde{\theta}% ^{2}L^{12}+(q^{-1}-1)\lambda_{+}^{-1}\tilde{\theta}^{2}L^{23}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}\tilde{\theta}^{1}L^{34}+q^{-2}\lambda_{+}% ^{-1}\tilde{\theta}^{2}K_{2},\nonumber\\ L^{23}\tilde{\theta}^{1} & =(q^{-1}+q^{2})\lambda_{+}^{-1}\tilde{\theta}% ^{1}L^{23}+(q-1)\lambda_{+}^{-1}\tilde{\theta}^{1}L^{14}\nonumber\\ & +\;q^{2}\lambda\lambda_{+}^{-1}\tilde{\theta}^{2}L^{12}-q\lambda_{+}% ^{-1}K_{2},\nonumber\\ L^{23}\tilde{\theta}^{2} & =(q^{-1}+1)\lambda_{+}^{-1}\tilde{\theta}% ^{2}L^{23}+(q^{-1}-1)\lambda_{+}^{-1}\tilde{\theta}^{2}L^{14}\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}\tilde{\theta}^{1}L^{34}+q^{-1}\lambda_{+}% ^{-1}\tilde{\theta}^{2}K_{2},\nonumber\end{aligned}$$ and $$\begin{aligned} \tilde{\theta}_{1}L^{12} & =L^{12}\tilde{\theta}_{1}-q^{-2}K_{2}% \tilde{\theta}_{2},\\ \tilde{\theta}_{2}L^{34} & =L^{34}\tilde{\theta}_{2}-q^{-1}K_{2}% \tilde{\theta}_{1},\nonumber\\ \tilde{\theta}_{1}L^{14} & =(q+1)\lambda_{+}^{-1}L^{14}\tilde{\theta}% _{1}+(q-1)\lambda_{+}^{-1}L^{23}\tilde{\theta}_{1}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}L^{34}\tilde{\theta}_{2}-\lambda_{+}^{-1}% K_{2}\tilde{\theta}_{1},\nonumber\\ \tilde{\theta}_{2}L^{14} & =(q+q^{-2})\lambda_{+}^{-1}L^{14}\tilde{\theta }_{2}+(q^{-1}-1)\lambda_{+}^{-1}L^{23}\tilde{\theta}_{2}\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}L^{12}\tilde{\theta}_{1}+q^{-2}\lambda_{+}% ^{-1}K_{2}\tilde{\theta}_{2},\nonumber\\ \tilde{\theta}_{1}L^{23} & =(q^{2}+q^{-1})\lambda_{+}^{-1}L^{23}% \tilde{\theta}_{1}+(q-1)\lambda_{+}^{-1}L^{14}\tilde{\theta}_{1}\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}L^{34}\tilde{\theta}_{2}-q\lambda_{+}^{-1}% K_{2}\tilde{\theta}_{1},\nonumber\\ \tilde{\theta}_{2}L^{23} & =(1+q^{-1})\lambda_{+}^{-1}L^{23}\tilde{\theta }_{2}+(q^{-1}-1)\lambda_{+}^{-1}L^{14}\tilde{\theta}_{2}\nonumber\\ & +\;q^{2}\lambda\lambda_{+}^{-1}L^{12}\tilde{\theta}_{1}+q^{-1}\lambda _{+}^{-1}K_{2}\tilde{\theta}_{2}.\nonumber\end{aligned}$$ These considerations carry over to the vector representations of the independent generators of $U_{q}(so_{4})$. Its right and left versions are given by $$L^{ij}\rhd X^{k}=(\tau^{ij})^{k}{}_{m}\,X^{m},\quad X_{k}\lhd L^{ij}% =X_{m}((\tau^{ij})^{m}{}_{k},$$ with $$\begin{gathered} (\tau^{14})^{k}{}_{m}=q^{-1}\lambda_{+}^{-1}\left( \begin{array} [c]{cccc}% -2q & 0 & 0 & 0\\ 0 & -\lambda & 0 & 0\\ 0 & 0 & -\lambda & 0\\ 0 & 0 & 0 & 2q^{-1}% \end{array} \right) ,\\ (\tau^{12})^{k}{}_{m}=q^{-2}\left( \begin{array} [c]{cccc}% 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array} \right) ,\quad(\tau^{13})^{k}{}_{m}=q^{-2}\left( \begin{array} [c]{cccc}% 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 \end{array} \right) ,\nonumber\\ (\tau^{24})^{k}{}_{m}=q^{-1}\left( \begin{array} [c]{cccc}% 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \end{array} \right) ,\quad(\tau^{34})^{k}{}_{m}=q^{-1}\left( \begin{array} [c]{cccc}% 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{array} \right) ,\nonumber\\ (\tau^{23})^{k}{}_{m}=q^{-1}\lambda_{+}^{-1}\left( \begin{array} [c]{cccc}% -q\lambda & 0 & 0 & 0\\ 0 & -(q^{2}+q^{-2}) & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & q^{-1}\lambda \end{array} \right) .\nonumber\end{gathered}$$ If we want to check that the above matrices give a representation of $U_{q}(so_{4})$ we additionally need the matrices corresponding to the braiding operators, i.e. $$(K_{1})^{k}{}_{m}=\left( \begin{array} [c]{cccc}% q^{-1} & 0 & 0 & 0\\ 0 & q & 0 & 0\\ 0 & 0 & q^{-1} & 0\\ 0 & 0 & 0 & q \end{array} \right) ,\quad(K_{2})^{k}{}_{m}=\left( \begin{array} [c]{cccc}% q^{-1} & 0 & 0 & 0\\ 0 & q^{-1} & 0 & 0\\ 0 & 0 & q & 0\\ 0 & 0 & 0 & q \end{array} \right) .$$ We are again in a position to write down commutation relations between the independent $L^{ij}$ and the components of a vector operator. Starting from$${[}L^{ij},X^{k}{]}_{q}=(\tau^{ij})^{k}{}_{l}\,X^{l}, \label{73}%$$ and proceeding in very much the same way as in the spinor case we obtain the relations $$\begin{aligned} L^{1j}X^{1} & =X^{1}L^{1j},\\ L^{1j}X^{j} & =X^{j}L^{1j},\nonumber\\ L^{1k}X^{l} & =X^{l}L^{1k}-q^{-2}X^{1}K_{i},\nonumber\\ L^{1m}X^{4} & =X^{4}L^{1m}+q^{-2}X^{m}K_{n},\nonumber\\[0.16in] L^{m^{\prime}4}X^{1} & =X^{1}L^{m^{\prime}4}-q^{-1}X^{m^{\prime}}K_{n},\\ L^{j4}X^{j} & =X^{j}L^{j4},\nonumber\\ L^{l4}X^{k} & =X^{k}L^{l4}+q^{-1}X^{4}K_{i},\nonumber\\ L^{j4}X^{4} & =X^{4}L^{j4},\nonumber\end{aligned}$$ [and]{} $$\begin{aligned} L^{14}X^{1} & =qX^{1}L^{14}-\lambda_{+}^{-1}X^{1}(K_{1}+K_{2})\\ & +\;q\lambda\lambda_{+}^{-1}X^{2}% L^{13}+X^{3}L^{12}),\nonumber\\ L^{14}X^{2} & =2\lambda_{+}^{-1}X^{2}L^{14}+\lambda_{+}^{-1}X^{2}(q^{2}% K_{1}-K_{2})+\lambda\lambda_{+}^{-1}X^{2}L^{23}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}(X^{1}L^{24}-qX^{4}L^{12}),\nonumber\\ L^{14}X^{3} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{3}L^{14}+\lambda_{+}% ^{-1}X^{3}(q^{-2}K_{2}-K_{1}% )\nonumber\\ & -\;\lambda\lambda_{+}^{-1}X^{3}L^{23}+\lambda\lambda_{+}^{-1}(X^{1}% L^{34}-qX^{4}L^{13}),\nonumber\\ L^{14}X^{4} & =q^{-1}X^{4}L^{14}+q^{-2}\lambda_{+}^{-1}X^{4}(K_{1}% +K_{2})\nonumber\\ & -\;\lambda\lambda_{+}^{-1}(X^{2}L^{34}+X^{3}L^{24}),\nonumber\\[0.16in] L^{23}X^{1} & =qX^{1}L^{23}+\lambda_{+}^{-1}X^{1}(q^{-1}K_{1}-qK_{2})\\ & -\;\lambda\lambda_{+}^{-1}(X^{2}L^{13}-q^{2}X^{3}L^{12}),\nonumber\\ L^{23}X^{2} & =q^{-2}(q^{4}+1)X^{2}L^{23}-\lambda_{+}^{-1}X^{2}(q^{-3}% K_{1}+qK_{2})\nonumber\\ & -\;q^{2}\lambda_+^{-1} X^{4}L^{12}-\lambda\lambda_{+}^{-1}(q^{-1}X^{1}L^{24}% +X^{2}L^{14}),\nonumber\\ L^{23}X^{3} & =2\lambda_{+}^{-1}X^{3}L^{23}+q^{-1}\lambda_{+}^{-1}% X^{3}(K_{1}+K_{2})\nonumber\\ & -\;\lambda\lambda_{+}^{-1}X^{3}L^{14}+\lambda\lambda_{+}^{-1}(qX^{1}% L^{34}+X^{4}L^{13}),\nonumber\\ L^{23}X^{4} & =q^{-1}X^{4}L^{23}-q^{-1}\lambda_{+}^{-1}X^{4}(q^{-2}% K_{1}-K_{2})\nonumber\\ & -\;\lambda\lambda_{+}^{-1}(qX^{2}L^{34}-q^{-1}X^{3}L^{24}),\nonumber\end{aligned}$$ where$$\begin{gathered} j=2,3,\quad(k,l,i)\in\{(2,3,2),(3,2,1)\},\\ (m,n)\in\{(2,2),(3,1)\},\quad m^{\prime}\equiv5-m^{\prime}.\nonumber\end{gathered}$$ The right version of (\[73\]), i.e. $${[}X_{k},L^{ij}{]}_{q}=X_{l}\,(\tau^{ij})^{l}{}_{k}\,,$$ gives us likewise $$\begin{aligned} X_{1}L^{1m} & =L^{1m}X_{1}-q^{-2}K_{n}X_{3},\\ X_{l^{\prime}}L^{1k} & =L^{1k}X_{l^{\prime}}+q^{-2}K_{i}X_{4},\nonumber\\ X_{j^{\prime}}L^{1j} & =L^{1j}X_{j^{\prime}},\nonumber\\ X_{4}L^{1j} & =L^{1j}X_{4},\nonumber\\[0.16in] X_{1}L^{j4} & =L^{j4}X_{1},\\ X_{k^{\prime}}L^{l4} & =L^{l4}X_{2}-q^{-1}K_{i}X_{1},\nonumber\\ X_{j^{\prime}}L^{j4} & =L^{j4}X_{j^{\prime}},\nonumber\\ X_{4}L^{m^{\prime}4} & =L^{m^{\prime}4}X_{4}+q^{-1}K_{n}X_{m},\nonumber\end{aligned}$$ [and]{} $$\begin{aligned} X_{1}L^{14} & =qL^{14}X_{1}-\lambda_{+}^{-1}(K_{1}+K_{2})X_{1}\\ & +\;\lambda\lambda_{+}^{-1}(L^{24}X_{2}+L^{34}X_{3}),\nonumber\\ X_{2}L^{14} & =2\lambda_{+}^{-1}L^{14}X_2+\lambda _{+}^{-1}(q^{-2}K_{1}-K_{2})X_{2}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}L^{23}X_{2}-\lambda\lambda_{+}^{-1}(L^{34}% X_{4}+qL^{13}X_{1}),\nonumber\\ X_{3}L^{14} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}L^{14}X_{3}+\lambda_{+}% ^{-1}(q^{-2}K_{2}-K_{1})X_{3}\nonumber\\ & -\;\lambda\lambda_{+}^{-1}L^{23}X_{3}+\lambda\lambda_{+}^{-1}(qL^{12}% X_{1}-L^{24}X_{4}),\nonumber\\ X_{4}L^{14} & =q^{-1}L^{14}X_{4}+q^{-2}\lambda_{+}^{-1}(K_{1}+K_{2}% )X_{4}\nonumber\\ & -\;q\lambda\lambda_{+}^{-1}(L^{12}X_{2}+L^{13}X_{3}),\nonumber\\[0.16in] X_{1}L^{23} & =qL^{23}X_{1}+q^{-1}\lambda_{+}^{-1}(K_{1}-q^{2}K_{2})X_{1}\\ & +\;\lambda\lambda_{+}^{-1}(qL^{34}X_{3}-q^{-1}L^{24}X_{2}),\nonumber\\ X_{2}L^{23} & =q^{-2}(q^{4}+1)L^{23}X_{2}-\lambda_{+}^{-1}(q^{-3}% K_{1}+qK_{2})X_{2}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}L^{14}X_{2}-\lambda\lambda_{+}^{-1}(L^{13}% X_{1}+qL^{34}X_{4}),\nonumber\\ X_{3}L^{23} & =2\lambda_{+}^{-1}L^{23}X_3+q^{-1}% \lambda_{+}^{-1}(K_{1}+K_{2})X_{3}\nonumber\\ & -\;\lambda\lambda_{+}^{-1}L^{14}X_{3}+\lambda\lambda_{+}^{-1}(q^{2}% L^{12}X_{1}+q^{-1}L^{24}X_{4}),\nonumber\\ X_{4}L^{23} & =q^{-1}L^{23}X_4+q^{-1}\lambda_{+}% ^{-1}(K_{2}-q^{-2}K_{1})X_{4}\nonumber\\ & +\;\lambda\lambda_{+}^{-1}(L^{13}X_{3}-q^{2}L^{12}X_{2}).\nonumber\end{aligned}$$ Quantum Lie algebra of $U_{q}(so_{4})$ and its Casimir operators ---------------------------------------------------------------- The $L^{ij}$ can act on themselves through the adjoint representation, which can be obtained via the following simple reasoning. We calculate the action of the $L^{ij}$ on a product of antisymmetrised vector coordinates (for their definition see Ref. [@Mik04]), i.e. $$L^{ij}\triangleright\xi^{k}\xi^{l}=\left( (L^{ij})_{(1)}\triangleright\xi ^{k}\right) \left( (L^{ij})_{(2)}\triangleright\xi^{l}\right) =\sum\nolimits_{mn}(a^{ij,kl})_{mn}\,\xi^{m}\xi^{n}. \label{ActLijLkl}%$$ As already mentioned, the $L^{ij}$ are components of an antisymmetric tensor operator. Hence, they have to act on themselves in the same way as they act on a product of antisymmetrised coordinates. In this sense we read off from (\[ActLijLkl\]) for their quantum Lie algebra $${[}L^{ij},L^{kl}{]}_{q}=L^{ij}\triangleright L^{kl}=\sum\nolimits_{mn}% (a^{ij,kl})_{mn}\,L^{mn}.$$ Explicitly, the non vanishing q-commutators for the independent $L^{ij}$ become $$\begin{aligned} {[}L^{12},L^{23}{]}_{q} & =-q^{-2}{[}L^{23},L^{12}{]}_{q}=q^{-1}L^{12},\\ {[}L^{12},L^{34}{]}_{q} & =-{[}L^{34},L^{12}{]}_{q}=-q^{-2}L^{23}% -q^{-3}L^{14},\nonumber\\ {[}L^{13},L^{23}{]}_{q} & =-q^{-2}{[}L^{23},L^{13}{]}_{q}=-q^{-3}% L^{13},\nonumber\\ {[}L^{13},L^{24}{]}_{q} & =-{[}L^{24},L^{13}{]}_{q}=q^{-2}L^{23}% -q^{-1}L^{14},\nonumber\\[0.16in] {[}L^{14},L^{1i}{]}_{q} & =-q^{2}{[}L^{1i},L^{14}{]}_{q}=-L^{1i},\quad i=2,3,\\ {[}L^{14},L^{14}{]}_{q} & =-q^{-1}\lambda L^{14},\nonumber\\ {[}L^{14},L^{23}{]}_{q} & ={[}L^{23},L^{14}{]}_{q}=-q^{-1}\lambda L^{23},\nonumber\\ {[}L^{14},L^{i4}{]}_{q} & =-q^{-2}{[}L^{i4},L^{14}{]}_{q}=q^{-2}L^{i4},\quad i=2,3,\nonumber\\[0.16in] {[}L^{23},L^{23}{]}_{q} & =-q^{-1}\lambda L^{14}-q^{-1}\lambda^{2}L^{23},\\ {[}L^{23},L^{24}{]}_{q} & =-q^{-2}{[}L^{24},L^{23}{]}_{q}=-q^{-3}% L^{24},\nonumber\\ {[}L^{23},L^{34}{]}_{q} & =-q^{-2}{[}L^{34},L^{23}{]}_{q}=q^{-1}% L^{34}.\nonumber\end{aligned}$$ Let us remark that these identities can be checked by means of the spinor and vector representations of the $L^{ij}$. Towards this end, one has to write out the q-commutators using (\[commeu4l\]) or (\[commeu4r\]). Then we replace the generators by their representation matrices and apply usual matrix multiplication. Proceeding in this manner will again show us the validity of the above identities. At last, let us consider the Casimir operators, which can be introduced as in the three-dimensional case. The first one is given by the expression $$\begin{aligned} C_{1} & =g_{ik}g_{jm}L^{ij}L^{km}=2L^{23}L^{23}+\lambda_{+}(L^{12}% L^{34}+L^{13}L^{24})\\ & =+q^{2}\lambda_{+}(L^{24}L^{13}+L^{34}L^{12})+(q^{2}+q^{-2})L^{14}% L^{14}\nonumber\\ & -\;\lambda(L^{14}L^{23}+L^{23}L^{14}),\nonumber\end{aligned}$$ and the second one by $$\begin{aligned} C_{2} & =\varepsilon_{ijkl}L^{ij}L^{kl}=q^{2}\lambda_{+}^{2}(L^{14}% L^{23}+L^{23}L^{14})+q^{2}\lambda_{+}^{2}(L^{12}L^{34}-L^{13}L^{24})\\ & +\;q^{4}\lambda_{+}^{2}(L^{34}L^{12}-L^{24}L^{13})-q^{2}\lambda\lambda _{+}^{2}L^{14}L^{14},\nonumber\end{aligned}$$ where $g_{ik}$ and $\varepsilon_{ijkl}$ denote respectively quantum metric and q-deformed epsilon-tensor (for its definition see Ref. [@Fiore]) corresponding to $U_{q}(so_{4})$. Again, we would like to specify the Casimir operators for the different representations. In doing so, we get the following: 1. (operator representation) $$\begin{aligned} C_{1} & =2q^{-2}(X\circ X)(\partial\circ\partial)+2q^{2}(X\circ \partial)(X\circ\partial)\\ & +\;2q^{-1}\lambda_{+}X\circ\partial,\nonumber\\ C_{2} & =0,\nonumber\end{aligned}$$ with $U\circ V\equiv g_{AB}U^{A}V^{B}$, 2. (spinor representation) $$\begin{aligned} C_{1} & =[[3]]_{q^{-4}}\mbox{1 \kern-.59em {\rm l}}_{2\times2},\\ C_{2} & =q[[3]]_{q^{4}}\lambda_{+}\mbox{1 \kern-.59em {\rm l}}_{2\times 2},\nonumber\end{aligned}$$ 3. (vector representation) $$\begin{aligned} C_{1} & =2[[3]]_{q^{-4}}\mbox{1 \kern-.59em {\rm l}}_{3\times3},\\ C_{2} & =0.\nonumber\end{aligned}$$ Quantum Lie algebra of Lorentz transformations ============================================== In this section we deal with the quantum Lie algebra of Lorentz transformations. Everything so far applies to this case, which from a physical point of view is the most interesting one we consider in this article. Representation of Lorentz generators within q-deformed differential calculus ---------------------------------------------------------------------------- In complete analogy to the previous section we start with a realization of the $V^{\mu\nu}$ given by [@LWW97] $$V^{\mu\nu}\equiv\Lambda^{1/2}(P_{A})^{\mu\nu}{}_{\rho\sigma}\,X^{\rho}% \hat{\partial}^{\sigma}=-q^{-2}\Lambda^{1/2}(P_{A})^{\mu\nu}{}_{\rho\sigma }\,\hat{\partial}^{\rho}X^{\sigma},$$ or equivalently$$V^{\mu\nu}\equiv q^{-2}\Lambda^{-1/2}(P_{A})^{\mu\nu}{}_{\rho\sigma}\,X^{\rho }\partial^{\sigma}=-\Lambda^{-1/2}(P_{A})^{\mu\nu}{}_{\rho\sigma}% \,\partial^{\rho}X^{\sigma},$$ where $\Lambda$ is a scaling operator being subject to $$\Lambda X^{\mu}=q^{-2}X^{\mu}\Lambda,\quad\Lambda\partial^{\mu}=q^{2}% \partial^{\mu}\Lambda,$$ and $P_{A}$ denotes the antisymmetrizer for q-deformed Lorentz symmetry. More explicitly, we have for example $$\begin{aligned} V^{+3} & =2q^{2}\lambda_{+}^{-2}X^{+}\hat{\partial}^{3}-2\lambda_{+}% ^{-2}X^{3}\hat{\partial}^{+}\\ & -\;q\lambda\lambda_{+}^{-2}(X^{+}\hat{\partial}^{0}-X^{0}\hat{\partial}% ^{+}),\nonumber\\ V^{+0} & =2\lambda_{+}^{-2}X^{+}\hat{\partial}^{0}-q^{-2}(q^{4}% +1)\lambda_{+}^{-2}X^{0}\hat{\partial}^{+}\nonumber\\ & +\;\lambda\lambda_{+}^{-2}(qX^{+}\hat{\partial}^{3}-q^{-1}X^{3}% \hat{\partial}^{+}),\nonumber\\ V^{+-} & =2\lambda_{+}^{-2}(\lambda^{2}+1)X^{+}\hat{\partial}^{-}% -2\lambda_{+}^{-2}X^{-}\hat{\partial}^{+}\nonumber\\ & -\;\lambda\lambda_{+}^{-2}(X^{3}\hat{\partial}^{0}+X^{0}\hat{\partial}% ^{3}),\nonumber\\ V^{30} & =2\lambda_{+}^{-2}X^{3}\hat{\partial}^{0}-q^{-2}(q^{4}% +1)\lambda_{+}^{-2}X^{0}\hat{\partial}^{3}\nonumber\\ & +\;\lambda\lambda_{+}^{-2}(\lambda^{2}+1)(X^{+}\hat{\partial}^{-}-X^{-}% \hat{\partial}^{+}),\nonumber\\ V^{3-} & =2q^{2}\lambda_{+}^{-2}X^{3}\hat{\partial}^{-}-2\lambda_{+}% ^{-2}X^{-}\hat{\partial}^{3}\nonumber\\ & -\;q\lambda\lambda_{+}^{-2}(X^{0}\hat{\partial}^{-}+X^{-}\hat{\partial}% ^{0}),\nonumber\\ V^{0-} & =2\lambda_{+}^{-2}X^{0}\hat{\partial}^{-}-q^{-2}(q^{4}% +1)\lambda_{+}^{-2}X^{-}\hat{\partial}^{0}\nonumber\\ & -\;\lambda\lambda_{+}^{-2}(qX^{3}\hat{\partial}^{-}-q^{-1}X^{-}% \hat{\partial}^{3}).\nonumber\end{aligned}$$ The antisymmetry of the $V^{\mu\nu}$ implies that we have six independent components for which we can choose $$V^{+3},V^{+0},V^{+-},V^{30},V^{3-},V^{0-}, \label{inGenLor}%$$ since the remaining ones are related to them by the relations$$\begin{gathered} V^{++}=V^{00}=V^{--}=0,\\ V^{33}=\lambda V^{+-},\quad V^{-+}=-V^{+-},\nonumber\\ V^{\pm3}=-q^{\pm2}V^{3\pm},\nonumber\\ V^{\pm0}=\mp q^{\pm1}\lambda V^{3\pm}-V^{0\pm},\nonumber\\ V^{03}=\lambda V^{+-}-V^{30}.\nonumber\end{gathered}$$ In Ref. [@LWW97] it was shown that the $V^{\mu\nu}$ together with two additional generators, denoted in the following by $U^{1},U^{2}$, span q-deformed Lorentz algebra. However, in what follows it is convenient to introduce another set of generators which are related to the $V^{\mu\nu}$ by $$\begin{aligned} R^{+} & =q^{-1}\lambda_{+}^{-1}(V^{+0}-V^{+3}),\label{RS-Form}\\ R^{3} & =q^{-1}\lambda_{+}^{-1}(V^{30}-qV^{+-}),\nonumber\\ R^{-} & =q^{-1}\lambda_{+}^{-1}(q^{-2}V^{3-}+V^{0-}),\nonumber\\[0.1in] S^{+} & =q^{-1}\lambda_{+}^{-1}(V^{+0}+q^{-2}V^{+3}),\label{RS-Form2}\\ S^{3} & =q^{-1}\lambda_{+}^{-1}(V^{30}+q^{-1}V^{+-}),\nonumber\\ S^{-} & =q^{-1}\lambda_{+}^{-1}(V^{3-}-V^{0-}).\nonumber\end{aligned}$$ In terms of the generators $R^{A}$ and $S^{A}$ the relations of q-deformed Lorentz algebra take the rather compact form $$\begin{aligned} {\epsilon_{CB}}^{A}R^{B}R^{C} & =q^{-4}\lambda_{+}^{-1}U^{1}R^{A}% ,\label{RelRS}\\ {\epsilon_{CB}}^{A}S^{B}S^{C} & =-q^{-4}\lambda_{+}^{-1}U^{2}S^{A}% ,\nonumber\\ R^{A}S^{B} & =q^{2}(\hat{R}_{SO_{q}(3)})_{CD}^{AB}\,S^{C}R^{D},\nonumber\end{aligned}$$ while $U^{1}$ and $U^{2}$ are both central in the algebra. ${\epsilon_{CB}% }^{A}$ in (\[RelRS\]) denotes the epsilon tensor from Sec. \[QuLie3dim\] and $\hat{R}_{SO_{q}(3)}$ stands for the vector representation of the universal R-matrix of $SO_{q}(3).$ For more details we refer the reader to [@LWW97]. Hopf structure of q-deformed Lorentz algebra and corresponding q-commutators ---------------------------------------------------------------------------- To write down q-commutators with Lorentz generators $V^{\mu\nu}$ we need to know their Hopf structure. From Ref. [@Rohr99], however, we know the coproduct for the generators $R^{A}$ and $S^{A},$ $A\in\{+,3,-\}.$ By solving (\[RS-Form\]) and (\[RS-Form2\]) for the independent generators given in (\[inGenLor\]) and making use of the algebra homomorphism property of the coproduct we are able to find expressions for the coproducts of the $V^{\mu \nu}$. For compactness, we introduce $$\rho\equiv q^{2}\lambda\lambda_{+}R^{3}+U^{1},\quad\sigma\equiv q^{2}% \lambda\lambda_{+}S^{3}-U^{2},$$ and the generators of an $U_{q}(su_{2})$-subalgebra, given by [@Rohr99]$$\begin{aligned} L^{A} & \equiv-q^{2}\lambda_{+}(U^{1}S^{A}-U^{2}R^{A}+q^{2}\lambda \lambda_{+}\,\epsilon_{CB}{}^{A}R^{B}S^{C}),\\ \tau^{-1/2} & \equiv U^{1}U^{2}-q^{4}\lambda\lambda_{+}\,g_{AB}\,R^{A}% S^{B}+\lambda L^{3},\nonumber\end{aligned}$$ where $g_{AB}$ is the quantum metric of Sec. \[QuLie3dim\]. In terms of the new quantities the wanted coproducts read as $$\begin{aligned} \Delta(V^{+3}) & =-\;q^{2}\sigma\otimes R^{+}+q^{2}R^{+}\otimes\rho\\ & -\;q^{2}\tau^{1/2}R^{+}\otimes\tau^{1/2}\rho+q^{2}\tau^{1/2}\sigma \otimes\tau^{1/2}R^{+}\nonumber\\ & -\;q^3\lambda_{+}^{-1}\tau^{1/2}\sigma L^{+}\otimes\sigma-q^3\lambda_{+}^{-1}% \tau^{1/2}\sigma\otimes\tau^{1/2}\sigma L^{+}\nonumber\\ & +\;q^{7}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}R^{+}\otimes\tau^{1/2}% S^{-}L^{+}+q^{9}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}R^{+}L^{+}\otimes S^{-},\nonumber\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Delta(V^{+0}) & =\sigma\otimes R^{+}-R^{+}\otimes\rho\nonumber\\ & -\;q^{2}\tau^{1/2}R^{+}\otimes\tau^{1/2}\rho+q^{2}\tau^{1/2}\sigma \otimes\tau^{1/2}R^{+}\nonumber\\ & -\;q^3\lambda_{+}^{-1}\tau^{1/2}\sigma L^{+}\otimes\sigma-q^3\lambda_{+}^{-1}% \tau^{1/2}\sigma\otimes\tau^{1/2}\sigma L^{+}\nonumber\\ & +\;q^{7}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}R^{+}\otimes\tau^{1/2}% S^{-}L^{+}+q^{9}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}R^{+}L^{+}\otimes S^{-},\nonumber\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Delta(V^{+-}) & =qR^{3}\otimes\rho-q\tau^{1/2}\sigma\otimes R^{3}% \nonumber\\ & +\;qS^{3}\otimes\sigma-q\tau^{1/2}\rho\otimes S^{3}\nonumber\\ & -\;q^{4}\lambda\lambda_{+}S^{-}\otimes R^{+}-q^{2}\lambda\lambda _{+}R^{+}\otimes S^{-}\nonumber\\ & +\;q^{4}\lambda\lambda_{+}\tau^{1/2}R^{+}\otimes R^{-}+q^3\lambda\tau ^{1/2}\sigma L^{-}\otimes R^{+}\nonumber\\ & +\;q^{2}\lambda\lambda_{+}\tau^{1/2}S^{-}\otimes S^{+}+q^{5}\lambda \tau^{1/2}\rho L^{+}\otimes S^{-},\nonumber\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Delta(V^{30}) & =-\;R^{3}\otimes\rho+\tau^{1/2}\sigma\otimes R^{3}% \nonumber\\ & +\;q^{2}S^{3}\otimes\sigma-q^{2}\tau^{1/2}\rho\otimes S^{3}\nonumber\\ & +\;q^{3}\lambda\lambda_{+}S^{-}\otimes R^{+}-q^{3}\lambda\lambda _{+}R^{+}\otimes S^{-}\nonumber\\ & -\;q^{3}\lambda\lambda_{+}\tau^{1/2}R^{+}\otimes R^{-}-q^{2}\lambda \tau^{1/2}\sigma L^{-}\otimes R^{+}\nonumber\\ & +\;q^{3}\lambda\lambda_{+}\tau^{1/2}S^{-}\otimes S^{+}+q^{6}\lambda \tau^{1/2}\rho L^{+}\otimes S^{-},\nonumber\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Delta(V^{3-}) & =q^{2}S^{-}\otimes\sigma-q^{2}\rho\otimes S^{-}\nonumber\\ & -\;q^{2}\tau^{1/2}S^{-}\otimes\tau^{1/2}\sigma+q^{2}\tau^{1/2}\rho \otimes\tau^{1/2}S^{-}\nonumber\\ & -\;q^3\lambda_{+}^{-1}\tau^{1/2}\rho L^{-}\otimes\rho-q^3\lambda_{+}^{-1}% \tau^{1/2}\rho\otimes\tau^{1/2}\rho L^{-}\nonumber\\ & +\;q^{5}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}S^{-}L^{-}\otimes R^{+}% +q^{7}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}S^{-}\otimes\tau^{1/2}R^{+}% L^{-},\nonumber\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Delta(V^{0-}) & =-\;S^{-}\otimes\sigma+\rho\otimes S^{-}\nonumber\\ & -\;q^{2}\tau^{1/2}S^{-}\otimes\tau^{1/2}\sigma+q^{2}\tau^{1/2}\rho \otimes\tau^{1/2}S^{-}\nonumber\\ & -\;q^3\lambda_{+}^{-1}\tau^{1/2}\rho L^{-}\otimes\rho-q^3\lambda_{+}^{-1}% \tau^{1/2}\rho\otimes\tau^{1/2}\rho L^{-}\nonumber\\ & +\;q^{5}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}S^{-}L^{-}\otimes R^{+}% +q^{7}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}S^{-}\otimes\tau^{1/2}R^{+}% L^{-}.\nonumber\end{aligned}$$ For writing down q-commutators we also need the following antipodes and their inverses:$$\begin{aligned} S(R^{+}) & =q^{2}S^{-1}(R^{+})=-q^{2}\tau^{1/2}R^{+},\\ S(R^{3}) & =S^{-1}(R^{3})=-q^{-2}\lambda^{-1}\lambda_{+}^{-1}(U^{1}% +\tau^{1/2})\sigma,\nonumber\\ S(R^{-}) & =q^{-2}S^{-1}(R^{-})=-S^{-}-q^{-1}\lambda_{+}^{-1}\tau^{1/2}% L^{-}\sigma,\nonumber\\[0.16in] S(S^{+}) & =q^{2}S^{-1}(S^{+})=-R^{+}-q^{3}\lambda_{+}^{-1}\tau^{1/2}% L^{+}\rho,\\ S(S^{3}) & =S^{-1}(S^{3})=q^{-2}\lambda^{-1}\lambda_{+}^{-1}(U^{2}% -\tau^{1/2}\rho),\nonumber\\ S(S^{-}) & =q^{-2}S^{-1}(S^{-})=-q^{-2}\tau^{1/2}S^{-},\nonumber\\[0.16in] S(U^{i}) & =S^{-1}(U^{i})=U^{i},\quad i=1,2,\\[0.16in] S(\rho) & =S^{-1}(\rho)=-\tau^{1/2}\sigma,\\ S(\sigma) & =S^{-1}(\sigma)=-\tau^{1/2}\rho.\nonumber\end{aligned}$$ After these preparations we are ready to write down q-commutators. For their left versions we have found $$\begin{aligned} {[}V^{+3},V{]}_{q} & =q^3\lambda_{+}^{-1}\tau^{1/2}\sigma(L^{+}V-VL^{+}% )\tau^{1/2}\rho\\ & +\;q^{7}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}R^{+}(VL^{+}-L^{+}V)\tau ^{1/2}S^{-}\nonumber\\ & +\;q^{2}(q^{2}\sigma V\tau^{1/2}R^{+}-\tau^{1/2}\sigma VR^{+})\nonumber\\ & +\;q^{2}(\tau^{1/2}R^{+}V\sigma-R^{+}V\tau^{1/2}\sigma),\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%% {[}V^{+0},V{]}_{q} & =q^3\lambda_{+}^{-1}\tau^{1/2}\sigma(L^{+}V-VL^{+}% )\tau^{1/2}\rho\\ & +\;q^{7}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}R^{+}(VL^{+}-L^{+}V)\tau ^{1/2}S^{-}\nonumber\\ & -\;q^{2}(\sigma V\tau^{1/2}R^{+}+\tau^{1/2}\sigma VR^{+})\nonumber\\ & +\;q^{2}\tau^{1/2}R^{+}V\sigma+R^{+}V\tau^{1/2}\sigma,\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {[}V^{+-},V{]}_{q} & =\;q(R^{3}V\tau^{1/2}\sigma+S^{3}V\tau^{1/2}% \rho)\\ & +\;q^{-1}\lambda^{-1}\lambda_{+}^{-1}\tau^{1/2}(\sigma V(U^{1}+\tau ^{1/2}\sigma)-\rho V(U^{2}-\tau^{1/2}\rho))\nonumber\\ & -\;q^3\lambda\tau^{1/2}(\rho L^{+}V\tau^{1/2}S^{-}+q^{2}S^{-}V\tau^{1/2}% L^{+}\rho)\nonumber\\ & -\;q^3\lambda\tau^{1/2}(q^{2}\sigma L^{-}V\tau^{1/2}R^{+}+R^{+}V\tau ^{1/2}L^{-}\sigma)\nonumber\\ & +\;q^{4}\lambda\lambda_{+}(q^{2}S^{-}V\tau^{1/2}R^{+}-\tau^{1/2}R^{+}% VS^{-})\nonumber\\ & +\;\lambda\lambda_{+}(R^{+}V\tau^{1/2}S^{-}-q^{2}\tau^{1/2}S^{-}% VR^{+}),\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {[}V^{30},V{]}_{q} & =R^{3}V\tau^{1/2}\sigma-q^{2}S^{3}V\tau^{1/2}\rho\\ & -\;q^{-2}\lambda^{-1}\lambda_{+}^{-1}\tau^{1/2}(\sigma V(U^{1}+\tau ^{1/2}\sigma))+q^{2}\rho V(U^{2}-\tau^{1/2}\rho)\nonumber\\ & -\;q^4\lambda\tau^{1/2}(\rho L^{+}V\tau^{1/2}S^{-}+q^{2}S^{-}V\tau^{1/2}% L^{+}\rho)\nonumber\\ & +\;q^{2}\lambda\tau^{1/2}(q^{2}\sigma L^{-}V\tau^{1/2}R^{+}+R^{+}% V\tau^{1/2}L^{-}\sigma)\nonumber\\ & +\;q^{3}\lambda\lambda_{+}(q^{2}S^{-}V\tau^{1/2}R^{+}-\tau^{1/2}R^{+}% VS^{-})\nonumber\\ & +\;q\lambda\lambda_{+}(R^{+}V\tau^{1/2}S^{-}-q^{2}\tau^{1/2}S^{-}% VR^{+}),\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {[}V^{3-},V{]}_{q} & =\rho V\tau^{1/2}S^{-}-q^{2}\tau^{1/2}\rho VS^{-}\\ & +\;q^{2}(\tau^{1/2}S^{-}V\rho-S^{-}V\tau^{1/2}\rho)\nonumber\\ & -q^3\;\lambda_{+}^{-1}\tau^{1/2}\rho(L^{-}V-VL^{-})\tau^{1/2}\sigma\nonumber\\ & \;q^{7}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}S^{-}(VL^{-}-L^{-}V)\tau ^{1/2}R^{+},\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {[}V^{0-},V{]}_{q} & =-\;q^{-2}\rho V\tau^{1/2}S^{-}-q^{2}\tau^{1/2}\rho VS^{-}\\ & +\;q^{2}\tau^{1/2}S^{-}V\rho+S^{-}V\tau^{1/2}\rho\nonumber\\ & -\;q^3\lambda_{+}^{-1}\tau^{1/2}\rho(L^{-}V-VL^{-})\tau^{1/2}\sigma\nonumber\\ & \;q^{7}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}S^{-}(VL^{-}-L^{-}V)\tau ^{1/2}R^{+},\nonumber\end{aligned}$$ and likewise for the corresponding right versions, $$\begin{aligned} {[}V,V^{+3}{]}_{q} & =q^{2}\sigma V\tau^{1/2}R^{+}-q^{2}\sigma\tau ^{1/2}VR^{+}\\ & -\;R^{+}V\tau^{1/2}\sigma+R^{+}\tau^{1/2}V\sigma\nonumber\\ & -q^3\;\lambda_{+}^{-1}(\rho V\tau^{1/2}\sigma L^{+}-L^{+}\rho V\tau ^{1/2}\sigma)\nonumber\\ & +\;q^{9}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}(L^{+}S^{-}V\tau^{1/2}% R^{+}-q^2 S^{-}V\tau^{1/2}R^{+}L^{+}),\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%%%%% {[}V,V^{+0}{]}_{q} & =q^{2}\sigma V\tau^{1/2}R^{+}+\sigma\tau^{1/2}VR^{+}\\ & -\;R^{+}V\tau^{1/2}\sigma-q^{-2}R^{+}\tau^{1/2}V\sigma\nonumber\\ & -q^3\;\lambda_{+}^{-1}(\rho V\tau^{1/2}\sigma L^{+}-L^{+}\rho V\tau ^{1/2}\sigma)\nonumber\\ & +\;q^{9}\lambda^{2}\lambda_{+}^{2}\tau^{1/2}(L^{+}S^{-}V\tau^{1/2}% R^{+}-q^2 S^{-}V\tau^{1/2}R^{+}L^{+}),\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%% {[}V,V^{+-}{]}_{q} & =-\;q(\rho\tau^{1/2}VS^{3}+\sigma\tau^{1/2}VR^{3})\\ & +\;q^{-1}\lambda^{-1}\lambda_{+}^{-1}((-U^{2}+\rho\tau^{1/2})V\tau^{1/2}% \rho+(U^{1}+\sigma\tau^{1/2})V\tau^{1/2}\sigma)\nonumber\\ & -\;q\lambda(L^{+}\rho\tau^{1/2} V\tau^{1/2}S^{-}+q^{6}S^{-}\tau^{1/2}% V\tau^{1/2}\rho L^{+})\nonumber\\ & -\;q\lambda(q^{4}\tau^{1/2}L^{-}\sigma V\tau^{1/2}R^{+}+R^{+}% \tau^{1/2}V\tau^{1/2}\sigma L^{-})\nonumber\\ & +\;\lambda\lambda_{+}(q^{4}S^{-}\tau^{1/2}VR^{+}-R^{+}V\tau^{1/2}% S^{-})\nonumber\\ & +\;q^{2}\lambda\lambda_{+}(R^{+}\tau^{1/2}VS^{-}-q^{4}S^{-}V\tau^{1/2}% R^{+}),\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {[}V,V^{30}{]}_{q} & =\sigma\tau^{1/2}VR^{3}-q^{2}\rho\tau^{1/2}VS^{3}\\ & -\;q^{-2}\lambda^{-1}\lambda_{+}^{-1}((U^{1}+\sigma\tau^{1/2})V\tau ^{1/2}\sigma+q^{2}(U^{2}-\rho\tau^{1/2})V\tau^{1/2}\rho)\nonumber\\ & +\;\lambda(R^{+}\tau^{1/2}V\tau^{1/2}\sigma L^{-}+q^4\tau^{1/2}% L^{-}\sigma V\tau^{1/2}R^{+})\nonumber\\ & -\;q^{2}\lambda(L^{+}\rho\tau^{1/2}V\tau^{1/2}S^{-}-q^{6}S^{-}\tau ^{1/2}V\tau^{1/2}\rho L^{+})\nonumber\\ & -\;q\lambda\lambda_{+}(R^{+}V\tau^{1/2}S^{-}+R^{+}\tau^{1/2}VS^{-}% )\nonumber\\ & +\;q^{5}\lambda\lambda_{+}(S^{-}V\tau^{1/2}R^{+}+S^{-}\tau^{1/2}% VR^{+}),\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {[}V,V^{3-}{]}_{q} & =-\;q^{2}\rho\tau^{1/2}VS^{-}+q^{2}\rho V\tau ^{1/2}S^{-}\\ & -\;q^{4}S^{-}V\tau^{1/2}\rho+q^{4}S^{-}\tau^{1/2}V\rho\nonumber\\ & +\;q^3\lambda_{+}^{-1}(\sigma\tau^{1/2}V\tau^{1/2}\rho L^{-}-\tau^{1/2}% L^{-}\sigma V\tau^{1/2}\rho)\nonumber\\ & +\;q^3\lambda^{2}\lambda_{+}^{2}(q^2\tau^{1/2}L^{-}R^{+}V\tau^{1/2}S^{-}% -R^{+}\tau^{1/2}V\tau^{1/2}S^{-}L^{-}),\nonumber\\[0.1in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {[}V,V^{0-}{]}_{q} & =\rho\tau^{1/2}VS^{-}+q^{2}\rho V\tau^{1/2}S^{-}\\ & -\;q^{4}S^{-}V\tau^{1/2}\rho-q^{2}S^{-}\tau^{1/2}V\rho\nonumber\\ & +\;q^3\lambda_{+}^{-1}(\sigma\tau^{1/2}V\tau^{1/2}\rho L^{-}-\tau^{1/2}% L^{-}\sigma V\tau^{1/2}\rho)\nonumber\\ & -\;q^3\lambda^{2}\lambda_{+}^{2}(q^2\tau^{1/2}L^{-}R^{+}V\tau^{1/2}S^{-}% -R^{+}\tau^{1/2}V\tau^{1/2}S^{-}L^{-}).\nonumber\end{aligned}$$ Matrix representations of q-deformed Lorentz algebra and commutation relations with tensor operators ---------------------------------------------------------------------------------------------------- Let us now consider spin representations of q-deformed Lorentz algebra. From a physical point of view q-deformed analogs of the two spinor representations $(1/2,0)$ and $(0,1/2)$ as well as the vector representation $(1/2,1/2)$ are the most interesting cases [@CSSW90; @SWZ91; @OSWZ92; @LSW94]. For a systematic treatment of finite dimensional representations of q-Lorentz algebra we refer for example to [@Bloh01]. As in the previous sections, we begin with spinor representations, for which we have $$\begin{aligned} V^{\mu\nu}\rhd\theta^{\,\alpha} & =\left( \sigma^{\mu\nu}\right) ^{\alpha }{}_{\beta}\;\theta^{\beta}, & V^{\mu\nu}\rhd\bar{\theta}^{\,\alpha} & =\left( \bar{\sigma}^{\mu\nu}\right) ^{\alpha}{}_{\beta}\;\bar{\theta }^{\beta},\\ \theta_{\alpha}\lhd V^{\mu\nu} & =\theta_{\beta}\,\left( \sigma^{\mu\nu }\right) ^{\beta}{}_{\alpha}\;, & \bar{\theta}_{\alpha}\lhd V^{\mu\nu} & =\bar{\theta}_{\beta}\,\left( \bar{\sigma}^{\mu\nu}\right) ^{\beta}% {}_{\alpha}\;,\nonumber\end{aligned}$$ and the spin matrices are given by $$\begin{aligned} (\sigma^{+3})^{\alpha}{}_{\beta} & =q^{2}(\sigma^{+0})^{\alpha}{}_{\beta }=-q^{1/2}\lambda_{+}^{-3/2}\left( \begin{array} [c]{cc}% 0 & 0\\ 1 & 0 \end{array} \right) ,\\ (\sigma^{+-})^{\alpha}{}_{\beta} & =-q(\sigma^{30})^{\alpha}{}_{\beta }=q^{-1}\lambda_{+}^{-2}\left( \begin{array} [c]{cc}% -q & 0\\ 0 & q^{-1}% \end{array} \right) ,\nonumber\\ (\sigma^{3-})^{\alpha}{}_{\beta} & =(\sigma^{0-})^{\alpha}{}_{\beta }=q^{-1/2}\lambda_{+}^{-3/2}\left( \begin{array} [c]{cc}% 0 & 1\\ 0 & 0 \end{array} \right) ,\nonumber\end{aligned}$$ in connection with $$\begin{aligned} (\bar{\sigma}^{+3})^{\alpha}{}_{\beta} & =(\sigma^{+3})^{\alpha}{}_{\beta}, & (\bar{\sigma}^{+0})^{\alpha}{}_{\beta} & =-q^{2}(\sigma^{+0})^{\alpha}% {}_{\beta},\\ (\bar{\sigma}^{+-})^{\alpha}{}_{\beta} & =(\sigma^{+-})^{\alpha}{}_{\beta}, & (\bar{\sigma}^{30})^{\alpha}{}_{\beta} & =-q^{2}(\sigma^{30})^{\alpha}% {}_{\beta},\nonumber\\ (\bar{\sigma}^{3-})^{\alpha}{}_{\beta} & =(\sigma^{3-})^{\alpha}{}_{\beta}, & (\bar{\sigma}^{0-})^{\alpha}{}_{\beta} & =-q^{2}(\sigma^{0-})^{\alpha}% {}_{\beta}.\nonumber\end{aligned}$$ The spinor representations of the generators $U^{1}$, $U^{2},$ and $\sigma$ are given by diagonal matrices which fulfill the identities$$-q^{2}(q^{2}+q^{-2})^{-1}\lambda_{+}{(U^{1})^{\alpha}}_{\beta}=-{(U^{2}% )^{\alpha}}_{\beta}=({\sigma)^{\alpha}}_{\beta}={\delta^{\alpha}}_{\beta}.$$ Furthermore, for $\rho$ and the generators of the $U_{q}(su_{2})$-subalgebra we have $$\begin{gathered} {(L^{+})^{\alpha}}_{\beta}=-q^{1/2}\lambda_{+}^{-1/2}% \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix} ,\quad{(L^{-})^{\alpha}}_{\beta}=-q^{-1/2}\lambda_{+}^{-1/2}% \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix} ,\\ {(\tau^{-1/2})^{\alpha}}_{\beta}=-({\rho)^{\alpha}}_{\beta}=% \begin{pmatrix} q^{-1} & 0\\ 0 & q \end{pmatrix} .\nonumber\end{gathered}$$ What we have done so far, enables us to write out the commutation relations between Lorentz generators and components of a spinor operator. In general, they are equivalent to$$\begin{aligned} {[}V^{\mu\nu},\theta^{\,\alpha}{]}_{q} & =(\sigma^{\mu\nu})^{\alpha}% {}_{\beta}\,\theta^{\beta},\quad{[}V^{\mu\nu},\bar{\theta}^{\,\alpha}{]}% _{q}=(\bar{\sigma}^{\mu\nu})^{\alpha}{}_{\beta}\,\bar{\theta}^{\beta },\label{Vtheta}\\ {[}\theta_{\alpha},V^{\mu\nu}{]}_{q} & =\theta_{\beta}\,(\sigma^{\mu\nu })^{\beta}{}_{\alpha},\quad{[}\bar{\theta}_{\alpha},V^{\mu\nu}{]}_{q}% =\bar{\theta}_{\beta}\,(\bar{\sigma}^{\mu\nu})^{\beta}{}_{\alpha}.\nonumber\end{aligned}$$ Unfortunately, it is rather difficult to derive from (\[Vtheta\]) the explicit form of the commutation relations between Lorentz generators and spinors. Thus, we proceed differently and apply the identities$$\begin{aligned} V^{\mu\nu}a & =((V^{\mu\nu})_{(1)}\triangleright a)(V^{\mu\nu}% )_{(2)},\label{VerLoGenSp}\\ aV^{\mu\nu} & =(V^{\mu\nu})_{(2)}(a\triangleleft(V^{\mu\nu})_{(1)}% ),\nonumber\end{aligned}$$ which result from Eqs. (\[LefCrosPro\]) and (\[RigCrosPro\]). This way the first relation in (\[VerLoGenSp\]) implies $$\begin{aligned} V^{+3}\theta^{1} & =(q+1)\lambda_{+}^{-1}\theta^{1}V^{+3}+(q^{2}% -q)\lambda_{+}^{-1}\theta^{1}V^{+0}\\ & -\;q^{1/2}\lambda\lambda_{+}^{-1/2}\theta^{2}(V^{+-}+qV^{30})+q^{1/2}% \lambda_{+}^{-3/2}\theta^{2}\rho,\nonumber\\ V^{+3}\theta^{2} & =(q+q^{-2})\lambda_{+}^{-1}\theta^{2}V^{+3}% +(1-q)\lambda_{+}^{-1}\theta^{2}V^{+0},\nonumber\\[0.16in] V^{+0}\theta^{1} & =(q^{2}+q^{-1})\lambda_{+}^{-1}\theta^{1}V^{+0}% +(1-q^{-1})\lambda_{+}^{-1}\theta^{1}V^{+3}\\ & -\;q^{1/2}\lambda\lambda_{+}^{-1/2}\theta^{2}(V^{+-}+qV^{30})-q^{-3/2}% \lambda_{+}^{-3/2}\theta^{2}\rho,\nonumber\\ V^{+0}\theta^{2} & =(1+q^{-1})\lambda_{+}^{-1}\theta^{2}V^{+0}% +(q^{-2}-q^{-1})\lambda_{+}^{-1}\theta^{2}V^{+3},\nonumber\\[0.16in] V^{+-}\theta^{1} & =q^{-1}(2q^{2}+\lambda_{+})\lambda_{+}^{-2}\theta ^{1}V^{+-}+q^{-1}(q-1)^{2}\lambda_{+}^{-2}\theta^{1}V^{30}\\ & +\;q^{1/2}(\lambda_{+}-1)\lambda\lambda_{+}^{-3/2}\theta^{2}V^{0-}% \nonumber\\ & -\;q^{-3/2}(q^{2}\lambda_{+}+1)\lambda\lambda_{+}^{-3/2}\theta^{2}% V^{3-}+\lambda_{+}^{-2}\theta^{1}U^{1},\nonumber\\ V^{+-}\theta^{2} & =q^{-1}(2q^{2}+\lambda_{+})\lambda_{+}^{-2}\theta ^{2}V^{+-}+q^{-1}(q-1)^{2}\lambda_{+}^{-2}\theta^{2}V^{30}\nonumber\\ & +\;q^{-1/2}\lambda\lambda_{+}^{-3/2}\theta^{1}(V^{+0}-V^{+3})-q^{-2}% \lambda_{+}^{-2}\theta^{2}U^{1},\nonumber\\[0.16in] V^{30}\theta^{1} & =q^{-1}(q^{2}\lambda_{+}+2)\lambda_{+}^{-2}\theta ^{1}V^{30}+q^{-1}(q-1)^{2}\lambda_{+}^{-2}\theta^{1}V^{+-}\\ & +\;q^{-1/2}(1+q^{2}\lambda_{+})\lambda\lambda_{+}^{-3/2}\theta ^{2}V^{0-}\nonumber\\ & +\;q^{-5/2}(1-q^{4}\lambda_{+})\lambda\lambda_{+}^{-3/2}\theta ^{2}V^{3-}-q^{-1}\lambda_{+}^{-2}\theta^{1}U^{1},\nonumber\\ V^{30}\theta^{2} & =q^{-1}(q^{2}\lambda_{+}+2)\lambda_{+}^{-2}\theta ^{2}V^{30}+q^{-1}(q-1)^{2}\lambda_{+}^{-2}\theta^{2}V^{+-}\nonumber\\ & +\;q^{-3/2}\lambda\lambda_{+}^{-3/2}\theta^{1}(V^{+0}-V^{+3})+q^{-3}% \lambda_{+}^{-2}\theta^{2}U^{1},\nonumber\\[0.16in] V^{3-}\theta^{1} & =(q^{-1}+1)\lambda_{+}^{-1}\theta^{1}V^{3-}% +(q-1)\lambda_{+}^{-1}\theta^{2}V^{0-},\\ V^{3-}\theta^{2} & =(q^{2}+q^{-1})\lambda_{+}^{-1}\theta^{2}V^{3-}% -(q^{2}-q)\lambda_{+}^{-1}\theta^{1}V^{0-}\nonumber\\ & -\;q^{-1/2}\lambda_{+}^{-3/2}\theta^{1}\rho,\nonumber\\[0.16in] V^{0-}\theta^{1} & =(q+q^{-2})\lambda_{+}^{-1}\theta^{1}V^{0-}% +(q^{-1}-q^{-2})\lambda_{+}^{-1}\theta^{1}V^{3-},\\ V^{0-}\theta^{2} & =(q+1)\lambda_{+}^{-1}\theta^{2}V^{0-}+(q^{-1}% -1)\lambda_{+}^{-1}\theta^{2}V^{3-}\nonumber\\ & -\;q^{-1/2}\lambda_{+}^{-3/2}\theta^{1}\rho.\nonumber\end{aligned}$$ For the second set of spinors we get from (\[VerLoGenSp\]) $$\begin{aligned} V^{+3}\bar{\theta}^{\,1} & =(q+q^{-1})\lambda_{+}^{-1}\bar{\theta}% ^{\,1}V^{+3}+(q-1)\lambda_{+}^{-1}\bar{\theta}^{\,1}V^{+0}\\ & -\;q^{1/2}\lambda_{+}^{-3/2}\bar{\theta}^{\,2}\,\sigma,\nonumber\\ V^{+3}\bar{\theta}^{\,2} & =(q^{2}+q^{-1})\lambda_{+}^{-1}\bar{\theta}% ^{\,2}V^{+3}-(q^{2}-q)\lambda_{+}^{-1}\bar{\theta}^{\,2}V^{+0}% ,\nonumber\\[0.16in] V^{+0}\bar{\theta}^{\,1} & =(q+q^{-2})\lambda_{+}^{-1}\bar{\theta}% ^{\,1}V^{+0}+(q^{-1}-q^{-2})\lambda_{+}^{-1}\bar{\theta}^{\,1}V^{+3}\\ & -\;q^{1/2}\lambda_{+}^{-3/2}\bar{\theta}^{\,2}\,\sigma,\nonumber\\ V^{+0}\bar{\theta}^{\,2} & =(q+1)\lambda_{+}^{-1}\bar{\theta}^{\,2}% V^{+0}-(1-q^{-1})\lambda_{+}^{-1}\bar{\theta}^{\,2}V^{+3},\nonumber\\[0.16in] V^{+-}\bar{\theta}^{\,1} & =q^{-1}(2+q^{2}\lambda_{+})\lambda_{+}^{-2}% \bar{\theta}^{\,1}V^{+-}-q^{-1}(q-1)^{2}\lambda_{+}^{-2}\bar{\theta}% ^{\,1}V^{30}\\ & +\;q^{1/2}\lambda\lambda_{+}^{-3/2}\bar{\theta}^{\,2}(V^{0-}-V^{3-}% )+\lambda_{+}^{-2}\bar{\theta}^{\,1}U^{2},\nonumber\\ %%%%%%%%%%%%%%%%%% V^{+-}\bar{\theta}^{\,2} & =q^{-1}(2+q^{2}\lambda_{+})\lambda_{+}^{-2}% \bar{\theta}^{\,2}V^{+-}+q^{-1}(q-1)^{2}\lambda_{+}^{-2}\bar{\theta}% ^{\,2}V^{30}\nonumber\\ & -\;q^{-1/2}\lambda(\lambda_++1)\lambda_+^{-3/2}\bar{\theta}^1V^{+0}\nonumber\\ & +\;q^{-5/2}\lambda(q^2\lambda_++1)\lambda_+^{-3/2}\bar{\theta}^1V^{+3} -q^{-2}\lambda_{+}^{-2}\bar{\theta}^{\,2}U^{1},\nonumber\\[0.16in] %%%%%%%%%%%% V^{30}\bar{\theta}^{\,1} & = q^{-1}(\lambda_++2q^2)\lambda_+^{-2}\bar{\theta}^1V^{30} -q^{-1}(q-1)^2\lambda_+^{-2}\bar{\theta}^1V^{+-}\\ & +\;q^{3/2}\lambda\lambda_+^{-3/2}\bar{\theta}^2(V^{0-}-V^{3-}) +q\lambda_+^{-2}\bar{\theta}^1U^2,\nonumber\\ %%%%%%%%%%%%%%%%% V^{30}\bar{\theta}^{\,2} & =q^{-1}(\lambda_++2q^2)\lambda_+^{-2}\bar{\theta}^2V^{30} -q^{-1}(q-1)^2\lambda_+^{-2}\bar{\theta}^2V^{+-}\\ & -\;q^{-3/2}\lambda(\lambda_++1)\lambda_+^{-3/2}\bar{\theta}^1V^{+3}\nonumber\\ & +\;q^{-3/2}\lambda(\lambda_++q^2)\lambda_+^{-3/2}\bar{\theta}^1V^{+0} -q^{-1}\lambda_+^{-2}\bar{\theta}^2U^2,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%% V^{3-}\bar{\theta}^{\,1} & =(q+1)\lambda_{+}^{-1}\bar{\theta}% ^{\,1}V^{3-}+(q^2-q)\lambda_{+}^{-1}\bar{\theta}^{\,1}V^{0-},\\ V^{3-}\bar{\theta}^{\,2} & =(q+q^{-2})\lambda_+^{-1}\bar{\theta^2}V^{3-} -(q-1)\lambda_+^{-1}\bar{\theta}^2V^{0-}\nonumber\\ & +\;q^{1/2}\lambda\lambda_+^{-1/2}\bar{\theta}^1(qV^{+-}-V^{30}) +q^{-1/2}\lambda_+^{-3/2}\bar{\theta}^1\sigma,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%%%% V^{0-}\bar{\theta}^{\,1} & =(q^2+q^{-1})\lambda_{+}^{-1}\bar{\theta}% ^{\,1}V^{0-}+(1-q^{-1})\lambda_{+}^{-1}\bar{\theta}^{\,1}V^{3-},\\ V^{0-}\bar{\theta}^{\,2} & =(1+q^{-1})\lambda_+^{-1}\bar{\theta}^2V^{0-} -(q^{-1}-q^{-2})\lambda_+^{-1}\bar{\theta}^2V^{3-}\nonumber\\ & +\;q^{1/2}\lambda\lambda_+^{-1/2}\bar{\theta}^1(q V^{+-}-V^{30}) -q^{-5/2}\lambda_+^{-3/2}\bar{\theta}^1\sigma.\nonumber\end{aligned}$$ The right versions of the above relations read $$\begin{aligned} \theta_{1}V^{+3} & =(q+1)\lambda_{+}^{-1}V^{+3}\theta_{1}+(q^{2}% -q)\lambda_{+}^{-1}V^{+0}\theta_{1}\;,\\ \theta_{2}V^{+3} & =(q+q^{-2})\lambda_{+}^{-1}V^{+3}\theta_{2}% -(q-1)\lambda_{+}^{-1}V^{+0}\theta_{2}\nonumber\\ & -\;q^{1/2}\lambda\lambda_{+}^{-1/2}(V^{+-}+qV^{30})\theta_{1}% +q^{1/2}\lambda_{+}^{-3/2}\rho\,\theta_{1}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%% \theta_{1}V^{+0} & =(q^2+q)\lambda_{+}^{-1}V^{+0}\theta_{1}% +(1-q^{-1})\lambda_{+}^{-1}V^{+3}\theta_{1}\;,\\ \theta_{2}V^{+0} & =(1+q^{-1})\lambda_{+}^{-1}V^{+0}\theta_{2}% -(q^{-1}-q^{-2})\lambda_{+}^{-1}V^{+3}\theta_{2}\nonumber\\ & -\;q^{1/2}\lambda\lambda_{+}^{-1/2}(V^{+-}+qV^{30})\theta_{1}% -q^{-3/2}\lambda_{+}^{-3/2}\rho\,\theta_{1}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \theta_{1}V^{+-} & =q^{-1}(q^{4}+1+\lambda_{+})\lambda_{+}^{-2}V^{+-}% \theta_{1}\\ & -\;q^{-2}(q^{4}+1-q^{2}\lambda_{+})\lambda_{+}^{-2}V^{30}\theta _{1}\nonumber\\ & +\;q^{-1/2}\lambda\lambda_{+}^{-3/2}(V^{+3}-V^{+0})\theta_{2}\nonumber\\ & +\;q^{-1}\lambda\lambda_{+}^{-2}\rho\,\theta_{1}+q^{-2}\lambda_{+}% ^{-2}U^{1}\theta_{1},\;\nonumber\\ \theta_{2}V^{+-} & =q^{-1}(q^{4}+1+\lambda_{+})\lambda_{+}^{-2}V^{+-}% \theta_{2}\nonumber\\ & -\;q^{-2}(q^{4}+1-q^{2}\lambda_{+})\lambda_{+}^{-2}V^{30}\theta _{2}\nonumber\\ & -\;q^{-3/2}(q^{2}\lambda_{+}+1)\lambda\lambda_{+}^{-3/2}V^{3-}\theta _{1}\nonumber\\ & -\;q^{1/2}(\lambda_{+}-1)\lambda\lambda_{+}^{-3/2}V^{0-}\theta _{1}\nonumber\\ & +\;q^{-1}\lambda\lambda_{+}^{-2}\rho\,\theta_{2}-\lambda_{+}^{-2}% U^{1}\theta_{2},\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \theta_{1}V^{30} & =q^{-3}(q^{4}+1+q^{4}\lambda_{+})\lambda_{+}^{-2}% V^{30}\theta_{1}\\ & -\;q^{-2}(q^{4}+1-q^{2}\lambda_{+})\lambda_{+}^{-2}V^{+-}\theta _{1}\nonumber\\ & +\;q^{-3/2}\lambda\lambda_{+}^{-3/2}(V^{+0}-V^{+3})\theta_{2}\nonumber\\ & -\;q^{-2}\lambda\lambda_{+}^{-2}\rho\,\theta_{1}-q^{-3}\lambda_{+}% ^{-2}U^{1}\theta_{1}\;,\nonumber\\ \theta_{2}V^{30} & =q^{-3}(q^{4}+1+q^{4}\lambda_{+})\lambda_{+}^{-2}% V^{30}\theta_{2}\nonumber\\ & -\;q^{-2}(q^{4}+1-q^{2}\lambda_{+})\lambda_{+}^{-2}V^{+-}\theta _{2}\nonumber\\ & +\;q^{-1/2}(q^{2}\lambda_{+}+1)\lambda\lambda_{+}^{-3/2}V^{0-}\theta _{1}\nonumber\\ & +\;q^{-5/2}(1-q^{4}\lambda_{+})\lambda\lambda_{+}^{-3/2}V^{3-}\theta _{1}\nonumber\\ & -\;q^{-2}\lambda\lambda_{+}^{-2}\rho\,\theta_{2}+q^{-1}\lambda_{+}% ^{-2}U^{1}\theta_{2}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%% \theta_{1}V^{3-} & =(1+q^{-1})\lambda_{+}^{-1}V^{3-}\theta_{1}% +(q-1)\lambda_{+}^{-1}V^{0-}\theta_{1}\\ & -\;q^{-1/2}\lambda_{+}^{-3/2}\rho\,\theta_{2}\;,\nonumber\\ \theta_{2}V^{3-} & =q^{-1}(q^{3}+1)\lambda_{+}^{-1}V^{3-}\theta_{2}% -(q^{2}-q)\lambda_{+}^{-1}V^{0-}\theta_{2}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%% \theta_{1}V^{0-} & =q^{-2}(q^{3}+1)\lambda_{+}^{-1}V^{0-}\theta_{1}% +q^{-2}(q-1)\lambda_{+}^{-1}V^{3-}\theta_{1}\\ & -\;q^{-1/2}\lambda_{+}^{-3/2}\rho\,\theta_{2},\;\nonumber\\ \theta_{2}V^{0-} & =(q+1)\lambda_{+}^{-1}V^{0-}\theta_{2}-(1-q^{-1}% )\lambda_+^{-1}V^{3-}\theta_{2}\;,\nonumber\end{aligned}$$ and similar for spinors with bar,$$\begin{aligned} \bar{\theta}_{1}V^{+3} & =(1+q^{-1})\lambda_{+}^{-1}V^{+3}\bar{\theta}% _{1}+(q-1)\lambda_{+}^{-1}V^{+0}\bar{\theta}_{1}\;,\\ \bar{\theta}_{2}V^{+3} & =q^{-1}(q^{3}+1)\lambda_{+}^{-1}V^{+3}\bar{\theta }_{2}-(q^2-q)\lambda_{+}^{-1}V^{+0}\bar{\theta}_{2}\nonumber\\ & -\;q^{1/2}\lambda_{+}^{-3/2}\sigma\,\bar{\theta}_{1}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bar{\theta}_{1}V^{+0} & =q^{-2}(q^{3}+1)\lambda_{+}^{-1}V^{+0}\bar{\theta }_{1}+(q^{-1}-q^{-2})\lambda_{+}^{-1}V^{+3}\bar{\theta}_{1}\;,\\ \bar{\theta}_{2}V^{+0} & =(q^{-1}-1)\lambda_{+}^{-1}V^{+3}% \bar{\theta}_{2}+(q+1)\lambda_{+}^{-1}% V^{+0}\bar{\theta}_{2}\nonumber\\ & -\;q^{1/2}\lambda_{+}^{-3/2}\sigma\,\bar{\theta}_{1}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bar{\theta}_{1}V^{+-} & =q^{-3}(q^{4}+1+q^{4}\lambda_{+})\lambda_{+}% ^{-2}V^{+-}\bar{\theta}_{1}\\ & +\;q^{-1}(q-1)^{2}(\lambda_{+}+1)\lambda_{+}^{-2}V^{30}\bar{\theta}% _{1}\nonumber\\ & +\;(q^{2}\lambda_{+}+1)q^{-5/2}\lambda\lambda_{+}^{-3/2}V^{+3}\bar{\theta }_{2}\nonumber\\ & +\;q^{-1/2}(1-\lambda_{+})\lambda\lambda_{+}^{-3/2}V^{+0}\bar{\theta}% _{2}\nonumber\\ & -\;q^{-1}\lambda\lambda_{+}^{-2}\sigma\,\bar{\theta}_{1}+q^{-2}\lambda _{+}^{-2}U^{2}\bar{\theta}_{1}\;,\nonumber\\ \bar{\theta}_{2}V^{+-} & =q^{-3}(q^{4}+1+q^{4}\lambda_{+})\lambda_{+}% ^{-2}V^{+-}\bar{\theta}_{2}\nonumber\\ & +\;q^{-1}(q-1)^{2}(\lambda_{+}+1)\lambda_{+}^{-2}V^{30}\bar{\theta}% _{2}\nonumber\\ & +\;q^{1/2}\lambda\lambda_{+}^{-3/2}(V^{0-}-V^{3-})\bar{\theta}% _{1}\nonumber\\ & -\;q^{-1}\lambda\lambda_{+}^{-2}\sigma\,\bar{\theta}_{2}-\lambda_{+}% ^{-2}U^{2}\bar{\theta}_{2}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bar{\theta}_{1}V^{30} & =q^{-1}(q^{4}+1+\lambda_{+})\lambda_{+}% ^{-2}V^{30}\bar{\theta}_{1}\\ & +\;q^{-1}(q-1)^{2}(\lambda_{+}+1)\lambda_{+}^{-2}V^{+-}\bar{\theta}% _{1}\nonumber\\ & +\;q^{-3/2}(1-\lambda_{+})\lambda\lambda_{+}^{-3/2}V^{+3}\bar{\theta}% _{2}\nonumber\\ & +\;q^{-3/2}(q^{4}+\lambda_{+})\lambda\lambda_{+}^{-3/2}V^{+0}\bar{\theta }_{2}\nonumber\\ & -\;\lambda\lambda_{+}^{-2}\sigma\,\bar{\theta}_{1}+q^{-1}\lambda_{+}% ^{-2}U^{2}\bar{\theta}_{1}\;,\nonumber\\ \bar{\theta}_{2}V^{30} & =q^{-1}(q^{4}+1+\lambda_{+})\lambda_{+}% ^{-2}V^{30}\bar{\theta}_{2}\nonumber\\ & +\;q^{-1}(q-1)^{2}(\lambda_{+}+1)\lambda_{+}^{-2}V^{+-}\bar{\theta}% _{2}\nonumber\\ & +\;q^{3/2}\lambda\lambda_{+}^{-3/2}(V^{0-}-V^{3-})\bar{\theta}% _{1}\nonumber\\ & -\;\lambda\lambda_{+}^{-2}\sigma\,\bar{\theta}_{2}-q\lambda_{+}^{-2}% U^{2}\bar{\theta}_{2}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bar{\theta}_{1}V^{3-} & =(q+1)\lambda_{+}^{-1}V^{3-}\bar{\theta}_{1}% +(q^{2}-q)\lambda_{+}^{-1}V^{0-}\bar{\theta}_{1}\\ & +\;q^{1/2}\lambda\lambda_{+}^{-1/2}(qV^{+-}-V^{30})\bar{\theta}% _{2}+q^{-1/2}\lambda_{+}^{-3/2}\sigma\,\bar{\theta}_{2}\;,\nonumber\\ \bar{\theta}_{2}V^{3-} & =q^{-2}(q^{3}+1)\lambda_{+}^{-1}V^{3-}\bar{\theta }_{2}-(q-1)\lambda_{+}^{-1}V^{0-}\bar{\theta}_{2}\;,\nonumber\\[0.16in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bar{\theta}_{1}V^{0-} & =q^{-1}(q^{3}+1)\lambda_{+}^{-1}V^{0-}\bar{\theta }_{1}+(1-q^{-1})\lambda_{+}^{-1}V^{3-}\bar{\theta}_{1}\\ & +\;q^{1/2}\lambda\lambda_{+}^{-1/2}(qV^{+-}-V^{30})\bar{\theta}% _{2}-q^{-5/2}\lambda_{+}^{-3/2}\sigma\,\bar{\theta}_{2}\;,\nonumber\\ \bar{\theta}_{2}V^{0-} & =(q+q^{-1})\lambda_{+}^{-1}V^{0-}\bar{\theta}% _{2}-(q^{-1}-q^{-2})\lambda_{+}^{-1}V^{3-}\bar{\theta}_{2}\;.\nonumber\end{aligned}$$ Next, we turn to the vector representations for the $V^{\mu\nu},$ which are seen to be $$V^{\mu\nu}\rhd X^{\rho}=(\tau^{\mu\nu})^{\rho}{}_{\sigma}\;X^{\sigma},\quad X_{\rho}\lhd V^{\mu\nu}=X_{\sigma}(\tau^{\mu\nu})^{\sigma}{}_{\rho},$$ with $$\begin{aligned} (\tau^{+3})^{\rho}{}_{\sigma} & =\lambda_{+}^{-2}\left( \begin{array} [c]{cccc}% 0 & -2q & -\lambda & 0\\ 0 & 0 & 0 & -2\\ 0 & 0 & 0 & -q\lambda\\ 0 & 0 & 0 & 0 \end{array} \right) ,\\ (\tau^{+0})^{\rho}{}_{\sigma} & =\lambda_{+}^{-2}\left( \begin{array} [c]{cccc}% 0 & -\lambda & 2q^{-1} & 0\\ 0 & 0 & 0 & -q^{-1}\lambda\\ 0 & 0 & 0 & -(q^{2}+q^{-2})\\ 0 & 0 & 0 & 0 \end{array} \right) ,\nonumber\\ (\tau^{+-})^{\rho}{}_{\sigma} & =\lambda_{+}^{-2}\left( \begin{array} [c]{cccc}% 2q^{-2} & 0 & 0 & 0\\ 0 & -2q^{-1}\lambda & -q^{-1}\lambda & 0\\ 0 & q^{-1}\lambda & 0 & 0\\ 0 & 0 & 0 & -2 \end{array} \right) ,\nonumber\\ (\tau^{30})^{\rho}{}_{\sigma} & =\lambda_{+}^{-2}\left( \begin{array} [c]{cccc}% \lambda q^{-2} & 0 & 0 & 0\\ 0 & -q^{-1}\lambda^{2} & 2q & 0\\ 0 & q^{-1}(q^{2}+q^{-2}) & 0 & 0\\ 0 & 0 & 0 & -\lambda \end{array} \right) ,\nonumber\\ (\tau^{3-})^{\rho}{}_{\sigma} & =\lambda_{+}^{-2}\left( \begin{array} [c]{cccc}% 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0\\ -q^{-1}\lambda & 0 & 0 & 0\\ 0 & 2q^{-1} & -\lambda & 0 \end{array} \right) ,\nonumber\\ (\tau^{0-})^{\rho}{}_{\sigma} & =\lambda_{+}^{-2}\left( \begin{array} [c]{cccc}% 0 & 0 & 0 & 0\\ q^{-1}\lambda & 0 & 0 & 0\\ 2q^{-2} & 0 & 0 & 0\\ 0 & q^{-2}\lambda & -q^{-1}(q^{2}+q^{-2}) & 0 \end{array} \right) .\nonumber\end{aligned}$$ Notice that rows and columns are labeled in the order $+,3,0,-$. For $U^{1}$, $U^{2}$, $\sigma$ and $\rho$ we obtain the matrices$$\begin{aligned} {(U^{1})^{\rho}}_{\sigma} & ={(U^{1})^{\rho}}_{\sigma}=-(q^{2}% +q^{-2})\lambda_{+}^{-1}{\delta^{\rho}}_{\sigma},\\ {(\rho)^{\rho}}_{\sigma} & =\lambda_{+}^{-1}% \begin{pmatrix} -q\lambda_{+} & 0 & 0 & 0\\ 0 & -2 & q\lambda & 0\\ 0 & q^{-1}\lambda & -(q^{2}+q^{-2}) & 0\\ 0 & 0 & 0 & -q^{-1}\lambda_{+}% \end{pmatrix} ,\nonumber\\ {(\sigma)^{\rho}}_{\sigma} & =\lambda_{+}^{-1}% \begin{pmatrix} q\lambda_{+} & 0 & 0 & 0\\ 0 & 2 & q^{-1}\lambda & 0\\ 0 & q\lambda & (q^{2}+q^{-2}) & 0\\ 0 & 0 & 0 & q^{-1}\lambda_{+}% \end{pmatrix} ,\nonumber\end{aligned}$$ and likewise for the generators of the $U_{q}(su_{2})$-subalgebra,$$\begin{gathered} {(L}^{+}{)^{\rho}}_{\sigma}=% \begin{pmatrix} 0 & -q & 0 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{pmatrix} ,\quad{(L}^{-}{)^{\rho}}_{\sigma}=% \begin{pmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & q^{-1} & 0 & 0 \end{pmatrix} ,\\ {(\tau}^{-1/2}{)^{\rho}}_{\sigma}=% \begin{pmatrix} q^{2} & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & q^{-2}% \end{pmatrix} .\nonumber\end{gathered}$$ We can now proceed to write down commutation relations between the $V^{\mu\nu }$ and the components of a vector operator. Let us recall that we mean by a vector operator a set of objects $X^{\rho}$ which transform under Lorentz transformations according to$${[}V^{\mu\nu},X^{\rho}{]}_{q}=(\tau^{\mu\nu})^{\rho}{}_{\sigma}\,X^{\sigma },\quad{[}X_{\sigma},V^{\mu\nu}{]}_{q}=X_{\sigma}\,(\tau^{\mu\nu})^{\sigma}% {}_{\rho}.$$ These relations are equivalent to those in (\[VerLoGenSp\]) if we substitute the vector components for the general element $a$. By taking the coproduct of the ${V}^{\mu\nu}$ together with the vector representations of the Lorentz generators we can again compute from (\[VerLoGenSp\]) the explicit form for the commutation relations between the ${V}^{\mu\nu}$ and the $X^{\rho}.$ Unfortunately, the results are rather lengthy: $$\begin{aligned} V^{+3}X^{+} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{+}V^{+3}-q\lambda \lambda_{+}^{-1}X^{+}V^{+0},\\ V^{+3}X^{3} & =2\lambda_{+}^{-1}X^{3}V^{+3}-q\lambda X^{+}V^{+-}% +q\lambda\lambda_{+}^{-1}X^{0}V^{+3}\nonumber\\ & -q\lambda\lambda_{+}^{-1}X^{+}V^{30}+q\lambda_{+}^{-2}X^{+}(U^{1}% +U^{2}),\nonumber\\ V^{+3}X^{0} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{0}V^{+3}+q^{-1}\lambda \lambda_{+}^{-1}X^{3}V^{+3}\nonumber\\ & -q^{2}\lambda\lambda_{+}^{-1}X^{+}V^{+-}+\lambda_{+}^{-2}X^{+}% (qU^{1}-q^{-1}U^{2}),\nonumber\\ V^{+3}X^{-} & =2\lambda_{+}^{-1}X^{-}V^{+3}-2q\lambda\lambda_{+}^{-1}% X^{3}V^{+-}+q\lambda\lambda_{+}^{-1}X^{0}V^{+-}\nonumber\\ & +\;q\lambda\lambda_{+}^{-1}X^{-}V^{+0}-q^{2}\lambda\lambda_{+}^{-1}% X^{3}V^{30}\nonumber\\ & -\;q^{2}\lambda^{2}\lambda_{+}^{-1}X^{+}V^{0-}+q^2\lambda^2\lambda_+^{-1}X^+V^{3-}\nonumber\\ & +\lambda_{+}^{-2}X^{3}(U^{1}+U^{2})-\lambda_{+}^{-2}X^{0}(U^{1}-q^{2}% U^{2}),\nonumber\\[0.2in] %%%%%%%%%%%%%%%%%%%%%%% V^{+0}X^{+} & =2\lambda_{+}^{-1}X^{+}V^{+0}-q^{-1}\lambda\lambda_{+}^{-1}% X^{+}V^{+3},\\ V^{+0}X^{3} & =2\lambda_{+}^{-1}X^{3}V^{+0}-\lambda\lambda_{+}^{-1}% X^{+}V^{+3}-2q\lambda\lambda_{+}^{-1}X^{+}V^{30}\nonumber\\ & +q\lambda\lambda_{+}^{-1}X^{0}V^{+0}-\lambda_{+}^{-2}X^{+}(q^{-1}U^{1}% -qU^{2}),\nonumber\\ V^{+0}X^{0} & =q^{-2}\lambda_{+}^{-1}(q^{4}+1)X^{0}V^{+0}-q\lambda \lambda_{+}^{-1}X^{+}V^{30}\nonumber\\ & +q^{-1}\lambda\lambda_{+}^{-1}X^{3}V^{+0}-q^{-1}\lambda_{+}^{-2}X^{+}% (U^{1}+U^{2}),\nonumber\\ V^{+0}X^{-} & =q^{-2}\lambda_{+}^{-1}(q^{4}+1)X^{-}V^{+0}-q\lambda X^{3}V^{30}+\lambda\lambda_{+}^{-1}X^{0}V^{30}\nonumber\\ & +q^{-1}\lambda\lambda_{+}^{-1}X^{-}V^{+3}-q\lambda\lambda_{+}^{-1}% X^{3}V^{+-}-q^{2}\lambda^{2}\lambda_{+}^{-1}X^{+}V^{0-}\nonumber\\ & +q^{2}\lambda^{2}\lambda_{+}^{-1}X^{+}V^{3-}+\lambda_{+}^{-2}X^{0}% (q^{-2}U^{1}+q^{2}U^{2})\nonumber\\ & -\lambda_{+}^{-2}X^{3}(q^{-2}U^{1}-U^{2}),\nonumber\\[0.2in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V^{+-}X^{+} & =2\lambda_{+}^{-1}X^{+}V^{+-}+q^{-1}\lambda X^{3}% V^{+3}-\lambda\lambda_{+}^{-1}X^{3}V^{+0}\\ & -q^{-2}\lambda\lambda_{+}^{-1}X^{0}V^{+3}-q^{-2}\lambda_{+}^{-2}X^{+}% (U^{1}+U^{2}),\nonumber\\ V^{+-}X^{3} & =(2-\lambda^{2})\lambda_{+}^{-1}X^{3}V^{+-}-\lambda X^{+}V^{3-}+q\lambda\lambda_{+}^{-1}X^{+}V^{0-}\nonumber\\ & +2q^{-1}\lambda\lambda_{+}^{-1}X^{-}V^{+3}-q^{-1}\lambda\lambda_{+}% ^{-1}X^{-}V^{+0}+\lambda^{2}\lambda_{+}^{-1}X^{0}V^{+-}\nonumber\\ & +q^{-2}\lambda_{+}^{-2}X^{0}U^{1}% -U^{2})+q^{-1}\lambda\lambda_{+}^{-2}X^{3}(U^{1}% +U^{2}),\nonumber\\ V^{+-}X^{0} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{0}V^{+-}-\lambda^{2}% \lambda_{+}^{-1}X^{3}V^{+-}+q^{-1}\lambda\lambda_{+}^{-1}X^{-}V^{+3}% \nonumber\\ & -q^{-1}\lambda\lambda_{+}^{-1}X^{+}V^{3-}+\;\lambda_{+}^{-2}X^{3}% (U^{1}-q^{-2}U^{2}),\nonumber\\ V^{+-}X^{-} & =2\lambda_{+}^{-1}X^{-}V^{+-}-2\lambda\lambda_{+}^{-1}% X^{3}V^{3-}+\lambda\lambda_{+}^{-1}X^{3}V^{0-}\nonumber\\ & +\lambda\lambda_{+}^{-1}X^{0}V^{3-}+\lambda_{+}^{-2}X^{-}(U^{1}% +U^{2}),\nonumber\\[0.2in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V^{30}X^{+} & =2\lambda_{+}^{-1}X^{+}V^{30}-q^{-1}\lambda\lambda_{+}% ^{-1}X^{0}V^{+0}-q^{-1}\lambda\lambda_{+}^{-1}X^{3}V^{+3}\\ & +\;2q^{-1}\lambda\lambda_{+}^{-1}X^{3}V^{+0}+q^{-1}\lambda_{+}^{-2}% X^{+}(q^{-2}U^{1}-U^{2}),\nonumber\\ V^{30}X^{3} & =(2-\lambda^{2})\lambda_{+}^{-1}X^{3}V^{30}+q\lambda X^{+}V^{0-}+q^{-1}\lambda X^{-}V^{+0}\nonumber\\ & +\;q^{-2}\lambda\lambda_{+}^{-1}(1-q^{3}\lambda_{+})X^{+}V^{3-}% -q^{-2}\lambda\lambda_{+}^{-1}X^{-}V^{+3}\nonumber\\ & +\lambda^{2}\lambda_{+}^{-1}X^{0}V^{30}-\lambda\lambda_{+}^{-2}X^{3}% (q^{-2}U^{1}-U^{2})\nonumber\\ & -\lambda_{+}^{-2}X^{0}(q^{-3}U^{1}+qU^{2}),\nonumber\\ V^{30}X^{0} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{0}V^{30}+\lambda\lambda _{+}^{-1}X^{+}V^{0-}+\lambda\lambda_{+}^{-1}X^{-}V^{+0}\nonumber\\ & -\;q^{-1}\lambda^{2}\lambda_{+}^{-1}X^{+}V^{3-}-\lambda^{2}\lambda_{+}% ^{-1}X^{3}V^{30}-q^{-1}\lambda_{+}^{-2}X^{3}(U^{1}+U^{2}),\nonumber\\ V^{30}X^{-} & =2\lambda_{+}^{-1}X^{-}V^{30}-\lambda\lambda_{+}% ^{-1}(q+1)X^{3}V^{3-}-q\lambda\lambda_{+}^{-1}X^{0}V^{0-}\nonumber\\ & +2q\lambda\lambda_{+}^{-1}X^{3}V^{0-}+\lambda^{2}\lambda_{+}^{-1}% X^{0}V^{3-}-\lambda_{+}^{-2}X^{-}(q^{-1}U^{1}-qU^{2}),\nonumber\\[0.2in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V^{3-}X^{+} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{+}V^{3-}-q\lambda \lambda_{+}^{-1}X^{0}V^{+-}\\ & -\;q\lambda\lambda_{+}^{-1}X^{+}V^{0-}+2q\lambda\lambda_{+}^{-1}X^{3}% V^{+-}+\lambda^{2}\lambda_{+}^{-1}X^{-}(V^{+3}-V^{+0})\nonumber\\ & -\lambda\lambda_+^{-1}X^3V^{30}-\lambda_{+}^{-2}X^{3}(U^{1}+U^{2})-\lambda_{+}^{-2}X^{0}(q^{-2}U^{1}% -U^{2}),\nonumber\\ V^{3-}X^{3} & =2\lambda_{+}^{-1}X^{3}V^{3-}+q\lambda X^{-}V^{+-}% -q^{-1}\lambda\lambda_{+}^{-1}X^{0}V^{3-}\nonumber\\ & -q\lambda\lambda_{+}^{-1}X^{-}V^{30}-q^{-1}\lambda_{+}^{-2}X^{-}% (U^{1}+U^{2}),\nonumber\\ V^{3-}X^{0} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{0}V^{3-}-q\lambda \lambda_{+}^{-1}X^{3}V^{3-}\nonumber\\ & +\lambda\lambda_{+}^{-1}X^{-}V^{+-}+\lambda_{+}^{-2}X^{-}(qU^{1}% -q^{-1}U^{2}),\nonumber\\ V^{3-}X^{-} & =2\lambda_{+}^{-1}X^{-}V^{3-}+q\lambda\lambda_{+}^{-1}% X^{-}V^{0-},\nonumber\\[0.2in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V^{0-}X^{+} & =2\lambda_{+}^{-1}X^{+}V^{0-}+\lambda\lambda_{+}% ^{-1}(q+1)X^{3}V^{+-}-2\lambda\lambda_{+}^{-1}X^{3}V^{30}\\ & -q^{-1}\lambda\lambda_{+}^{-1}X^{+}V^{3-}+\lambda\lambda_{+}^{-1}% X^{0}V^{30}\nonumber\\ & +\lambda^{2}\lambda_{+}^{-1}X^{-}(V^{+3}-V^{+0})-\lambda^{2}\lambda _{+}^{-1}X^{0}V^{+-}\nonumber\\ & -\lambda_{+}^{-2}X^{3}(U^{1}-q^{-2}U^{2})-q^{-2}\lambda_{+}^{-2}X^{0}% (U^{1}+U^{2}),\nonumber\\ V^{0-}X^{3} & =2\lambda_{+}^{-1}X^{3}V^{0-}-\lambda X^{-}V^{30}% -q^{-2}\lambda\lambda_{+}^{-1}X^{-}(1-q^3\lambda_{+})V^{+-}\nonumber\\ & -q^{-1}\lambda\lambda_{+}^{-1}X^{0}V^{0-}-q^{-1}\lambda_{+}^{-2}X^{-}% (U^{1}-q^{-2}U^{2}),\nonumber\\ V^{0-}X^{0} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{0}V^{0-}-q\lambda \lambda_{+}^{-1}X^{3}V^{0-}\nonumber\\ & -q^{-1}\lambda\lambda_{+}^{-1}X^{-}V^{30}+q^{-1}\lambda^{2}\lambda_{+}% ^{-1}X^{-}V^{+-}\nonumber\\ & +\lambda_{+}^{-2}X^{-}(qU^{1}+q^{-3}U^{2}),\nonumber\\ V^{0-}X^{-} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}X^{-}V^{0-}+q^{-1}% \lambda\lambda_{+}^{-1}X^{-}V^{3-}.\nonumber\end{aligned}$$ If we want to commute the $V^{\mu\nu}$ to the left of a vector operator, we have to apply instead $$\begin{aligned} X_{+}V^{+3} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}V^{+3}X_{+}-q\lambda V^{+-}X_{3}\\ & -q\lambda\lambda_{+}^{-1}V^{30}X_{3}-q\lambda\lambda_{+}^{-1}V^{+0}% X_{+}-q^{2}\lambda\lambda_{+}^{-1}V^{+-}X_{0}\nonumber\\ & +q^{2}\lambda^{2}\lambda_{+}^{-1}(V^{3-}-V^{0-})X_{-}+q\lambda_{+}% ^{-2}(U^{1}+U^{2})X_{3}\nonumber\\ & +\lambda_{+}^{-2}(qU^{1}-q^{-1}U^{2})X_{0}\;,\nonumber\\ X_{3}V^{+3} & =2\lambda_{+}^{-1}V^{+3}X_{3}-2q\lambda\lambda_{+}^{-1}% V^{+-}X_{-}+q^{-1}\lambda\lambda_{+}^{-1}V^{+3}X_{0}\nonumber\\ & -q^{2}\lambda\lambda_{+}^{-1}V^{30}X_{-}+\lambda_{+}^{-2}(U^{1}+U^{2}% )X_{-}\;,\nonumber\\ X_{0}V^{+3} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}V^{+3}X_{0}+q\lambda \lambda_{+}^{-1}V^{+3}X_{3}\nonumber\\ & +q\lambda\lambda_{+}^{-1}V^{+-}X_{-}-\lambda_{+}^{-2}(U^{1}-q^{2}% U^{2})X_{-}\;,\nonumber\\ X_{-}V^{+3} & =2\lambda_{+}^{-1}V^{+3}X_{-}+q\lambda\lambda_{+}^{-1}% V^{+0}X_{-}\;,\nonumber\\[0.2in] %%%%%%%%%%%%%%%%%%%%%% X_{+}V^{+0} & =2\lambda_{+}^{-1}V^{+0}X_{+}-\lambda\lambda_{+}^{-1}% V^{+-}X_{3}-q^{-1}\lambda\lambda_{+}^{-1}V^{+3}X_{+}\\ & -2q\lambda\lambda_{+}^{-1}V^{30}X_{3}-q\lambda\lambda_{+}^{-1}V^{30}% X_{0}+q^{2}\lambda^{2}\lambda_{+}^{-2}(V^{3-}-V^{0-})X_{-}\nonumber\\ & -\lambda_{+}^{-2}(q^{-1}U^{1}-qU^{2})X_{3}-q^{-1}\lambda_{+}^{-2}% (U^{1}+U^{2})X_{0}\;,\nonumber\\ X_{3}V^{+0} & =2\lambda_{+}^{-1}V^{+0}X_{3}-q\lambda V^{30}X_{-}% +q^{-1}\lambda\lambda_{+}^{-1}V^{+0}X_{0}\nonumber\\ & -q\lambda\lambda_{+}^{-1}V^{+-}X_{-}-\lambda_{+}^{-2}(q^{-2}U^{1}% -U^{2})X_{-}\;,\nonumber\\ X_{0}V^{+0} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}V^{+0}X_{0}+\lambda\lambda _{+}^{-1}V^{30}X_{-}\nonumber\\ & +q\lambda\lambda_{+}^{-1}V^{+0}X_{3}+\lambda_{+}^{-2}(q^{-2}U^{1}% +q^{2}U^{2})X_{-}\;,\nonumber\\ X_{-}V^{+0} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}% V^{+0}X_-+q^{-1}\lambda\lambda_{+}^{-1}V^{+3}X_{-}\;,\nonumber\\[0.2in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X_{+}V^{+-} & =2\lambda_{+}^{-1}V^{+-}X_{+}-\lambda V^{3-}X_{3}% -q^{-1}\lambda\lambda_{+}^{-1}V^{3-}X_{0}\\ & +q\lambda\lambda_{+}^{-1}V^{0-}X_{3}-q^{-2}\lambda_{+}^{-2}(U^{1}% +U^{2})X_{+}\;,\nonumber\\ X_{3}V^{+-} & =(2-\lambda^{2})\lambda_{+}^{-1}V^{+-}X_{3}+q^{-1}\lambda V^{+3}X_{+}-\lambda\lambda_{+}^{-1}V^{+0}X_{+}\nonumber\\ & +\lambda\lambda_{+}^{-1}V^{0-}X_{-}-2\lambda\lambda_{+}^{-1}V^{3-}% X_{-}-\lambda^{2}\lambda_{+}^{-1}V^{+-}X_{0}\nonumber\\ & +q^{-1}\lambda\lambda_{+}^{-1}(U^{1}+U^{2})X_{3}+\lambda_{+}^{-2}% (U^{1}-q^{-2}U^{2})X_{0},\nonumber\\ X_{0}V^{+-} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}V^{+-}X_{0}-q^{-2}% \lambda\lambda_{+}^{-1}V^{+3}X_{+}\nonumber\\ & +\lambda\lambda_{+}^{-1}V^{3-}X_{-}+\lambda^{2}\lambda_{+}^{-1}V^{+-}% X_{3}+\lambda_{+}^{-2}(q^{-2}U^{1}-U^{2})X_{3}\;,\nonumber\\ X_{-}V^{+-} & =2\lambda_{+}^{-1}V^{+-}X_{-}+2q^{-1}\lambda\lambda_{+}% ^{-1}V^{+3}X_{3}+q^{-1}\lambda\lambda_{+}^{-1}V^{+3}X_{0}\nonumber\\ & -q^{-1}\lambda\lambda_{+}^{-1}V^{+0}X_{3}+\lambda_{+}^{-2}(U^{1}% +U^{2})X_{-}\;,\nonumber\\[0.2in] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X_{+}V^{30} & =2\lambda_{+}^{-1}V^{30}X_{+}+q^{-2}\lambda\lambda_{+}% ^{-1}(1-q^{3}\lambda_{+})V^{3-}X_{3}\\ & +q\lambda V^{0-}X_{3}+\lambda\lambda_{+}^{-1}V^{0-}X_{0}-q^{-1}\lambda ^{2}\lambda_{+}^{-1}V^{3-}X_{0}\nonumber\\ & +q^{-1}\lambda_{+}^{-2}(q^{-2}U^{1}-U^{2})X_{+}\;,\nonumber\\ X_{3}V^{30} & =(2-\lambda^{2})\lambda_{+}^{-1}V^{30}X_{3}-(q+\lambda )\lambda\lambda_{+}^{-1}V^{3-}X_{-}\nonumber\\ & -q^{-1}\lambda\lambda_{+}^{-1}V^{+3}X_{+}+2\lambda\lambda_{+}^{-1}% (q^{-1}V^{+0}+qV^{0-})X_{+}\nonumber\\ & -\lambda^{2}\lambda_{+}V^{30}X_{0}-\lambda\lambda_{+}^{-1}(q^{-2}% U^{1}-U^{2})X_{3}\nonumber\\ & +q^{-1}\lambda_{+}^{-2}(U^{1}-U^{2})X_{0}\;,\nonumber\\ X_{0}V^{30} & =q^{-2}(q^{4}+1)V^{30}X_{0}-q^{-1}\lambda\lambda_{+}% ^{-1}V^{+0}X_{+}-q\lambda\lambda_{+}^{-1}V^{0-}X_{-}\nonumber\\ & +\lambda^{2}\lambda_{+}^{-1}V^{30}X_{3}+\lambda^{2}\lambda_{+}^{-1}% V^{3-}X_{-}-\lambda_{+}^{-2}(q^{-3}U^{1}+qU^{2})X_{3}\;,\nonumber\\ X_{-}V^{30} & =2\lambda_{+}^{-1}V^{30}X_-+q^{-1}\lambda V^{+0}X_{0}-q^{-2}\lambda\lambda_{+}^{-1}V^{+3}X_{3}\nonumber\\ & +\lambda\lambda_{+}^{-1}V^{+0}X_{0}-\lambda_{+}^{-2}(q^{-1}U^{1}% -qU^{2})X_{-}\;,\nonumber\\[0.2in] X_{+}V^{3-} & =q^{-2}(q^{4}+1)V^{3-}X_{+}-q\lambda\lambda_{+}^{-1}% V^{0-}X_{+}\;,\\ X_{3}V^{3-} & =2\lambda_{+}^{-1}V^{3-}X_{3}+2q\lambda\lambda_{+}^{-1}% V^{+-}X_{+}-\lambda\lambda_{+}^{-1}V^{30}X_{+}\nonumber\\ & -q\lambda\lambda_{+}^{-1}V^{3-}X_{0}-\lambda_{+}^{-2}(U^{1}+U^{2}% )X_{+}\;,\nonumber\\ X_{0}V^{3-} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}V^{3-}X_{0}-q\lambda \lambda_{+}^{-1}V^{+-}X_{+}\nonumber\\ & -q^{1}\lambda\lambda_{+}^{-1}V^{3-}X_{3}-\lambda_{+}^{-2}(q^{-2}U^{1}% -U^{2})X_{+}\;,\nonumber\\ X_{-}V^{3-} & =2\lambda_{+}^{-1}V^{3-}X_{-}+q\lambda V^{+-}X_{3}% +\lambda\lambda_{+}^{-1}V^{+-}X_{0}\nonumber\\ & +\lambda^{2}\lambda_{+}^{-1}(V^{+3}-V^{+0})X_{+}+q\lambda\lambda_{+}% ^{-1}V^{0-}X_{-}-q\lambda\lambda_{+}^{-1}V^{30}X_{3}\nonumber\\ & +\lambda_{+}^{-2}(qU^{1}-q^{-1}U^{2})X_{0}-q^{-1}\lambda_{+}^{-2}% (U^{1}+U^{2})X_{3}\;,\nonumber\\[0.2in] X_{+}V^{0-} & =2\lambda_{+}^{-1}V^{0-}X_{+}-q^{-1}\lambda\lambda_{+}% ^{-1}V^{3-}X_{+}\;,\\ X_{3}V^{0-} & =2\lambda_{+}^{-1}V^{0-}X_{3}+\lambda(q+\lambda)\lambda _{+}^{-1}V^{+-}X_{+}-2\lambda\lambda_{+}^{-1}V^{30}X_{+}\nonumber\\ & -q\lambda\lambda_{+}^{-1}V^{0-}X_{0}-\lambda_{+}^{-2}(U^{1}-q^{-2}% U^{2})X_{+}\;,\nonumber\\ X_{0}V^{0-} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}V^{0-}X_{0}+\lambda\lambda _{+}^{-1}V^{30}X_{+}\nonumber\\ & -q^{-1}\lambda\lambda_{+}^{-1}V^{0-}X_{3}-\lambda^{2}\lambda_{+}^{-1}% V^{+-}X_{+}-q^{-2}\lambda_{+}^{-2}(U^{1}+U^{2})X_{+}\;,\nonumber\\ X_{-}V^{0-} & =q^{-2}(q^{4}+1)\lambda_{+}^{-1}V^{0-}X_{-}-\lambda V^{30}X_{3}+q^{-1}\lambda\lambda_{+}^{-1}V^{3-}X_{-}\nonumber\\ & -q^{-1}\lambda\lambda_{+}^{-1}V^{30}X_{0}+q^{-3}\lambda\lambda_{+}% ^{-1}(-q+q^{-2}\lambda_{+})V^{+-}X_{3}\nonumber\\ & -q^{-1}\lambda_{+}^{-2}(U^{1}-q^{-1}U^{2})X_{3}+\lambda_{+}^{-2}% (qU^{1}+q^{-2}U^{2})X_{0}.\nonumber\end{aligned}$$ Quantum Lie algebra of q-deformed Lorentz algebra and its Casimir operators --------------------------------------------------------------------------- Last but not least we would like to present the quantum Lie algebra of q-deformed Lorentz algebra. For this to achieve, we calculate the adjoint action of the independent $V^{\mu\nu}$ on themselves in a fashion as was done in the previous section and set the results equal to the q-commutators of the corresponding Lorentz generators. In the cases where the adjoint actions do not vanish, we obtain $$\begin{aligned} {[}V^{+3},V^{+-}{]}_{q} & =-q^{2}{[}V^{+-},V^{+3}{]}_{q}=-\lambda_{+}% ^{-1}V^{+3},\\ {[}V^{+3},V^{30}{]}_{q} & =-q^{2}{[}V^{30},V^{+3}{]}_{q}=-q\lambda_{+}% ^{-1}V^{+0},\nonumber\\ {[}V^{+3},V^{3-}{]}_{q} & =-{[}V^{3-},V^{+3}{]}_{q}=-q\lambda_{+}^{-1}% V^{+-},\nonumber\\ {[}V^{+3},V^{0-}{]}_{q} & =-{[}V^{0-},V^{+3}{]}_{q}=\lambda_{+}^{-1}% (V^{30}-\lambda V^{+-}),\nonumber\\[0.16in] {[}V^{+0},V^{+-}{]}_{q} & =-q^{2}{[}V^{+-},V^{+0}{]}_{q}=-\lambda_{+}% ^{-1}V^{+0},\\ {[}V^{+0},V^{30}{]}_{q} & =-q^{2}{[}V^{30},V^{+0}{]}_{q}=-\lambda_{+}% ^{-1}(q^{-1}V^{+3}+\lambda V^{+0}),\nonumber\\ {[}V^{+0},V^{3-}{]}_{q} & =-{[}V^{3-},V^{+0}{]}_{q}=-\lambda_{+}^{-1}% V^{30},\nonumber\\ {[}V^{+0},V^{0-}{]}_{q} & =-{[}V^{0-},V^{+-}{]}_{q}=q^{-1}\lambda_{+}% ^{-1}V^{+-},\nonumber\\[0.16in] {[}V^{+-},V^{+-}{]}_{q} & =-q^{-1}\lambda\lambda_{+}^{-1}V^{+-},\\ {[}V^{+-},V^{30}{]}_{q} & ={[}V^{30},V^{+-}{]}_{q}=-q^{-1}\lambda\lambda _{+}^{-1}V^{30},\nonumber\\ {[}V^{+-},V^{3-}{]}_{q} & =-q^{2}{[}V^{3-},V^{+-}{]}_{q}=-\lambda_{+}% ^{-1}V^{3-},\nonumber\\ {[}V^{+-},V^{0-}{]}_{q} & =-q^{2}{[}V^{0-},V^{+-}{]}_{q}=-\lambda_{+}% ^{-1}V^{0-},\nonumber\\[0.16in] {[}V^{30},V^{30}{]}_{q} & =-q^{-1}\lambda\lambda_{+}^{-1}(V^{+-}+\lambda V^{30}),\\ {[}V^{30},V^{3-}{]}_{q} & =-q^{2}{[}V^{3-},V^{30}{]}_{q}=\lambda_{+}^{-1}(qV^{0-}-\lambda V^{3-}% ),\nonumber\\ {[}V^{30},V^{0-}{]}_{q} & =-q^{2}{[}V^{0-},V^{30}{]}_{q}=q^{-1}\lambda _{+}^{-1}V^{3-}.\nonumber\end{aligned}$$ Of course, the quantum Lie algebra relations are consistent with spinor and vector representation of q-deformed Lorentz algebra. This can again be checked in a familiar way by writing out q-commutators and substituting the representation matrices for the Lorentz generators. As Casimirs of this quantum Lie algebra we have found the two operators $$\begin{aligned} C^{1} & =\eta_{\mu\nu}\eta_{\rho\sigma}V^{\mu\rho}V^{\nu\sigma}% =2V^{30}V^{30}+(q^{2}+q^{-2})V^{+-}V^{+-}\\ & +\;2(qV^{+0}V^{0-}-q^{-1}V^{+3}V^{3-}-q^{-3}V^{3-}V^{3+})\nonumber\\ & -\;\lambda(V^{+3}V^{0-}+V^{+0}V^{3-}+V^{+-}V^{30}+V^{30}V^{+-})\nonumber\\ & -\;q^{-2}\lambda(V^{3-}V^{+0}+V^{0-}V^{+3}),\nonumber\\[0.16in] C^{2} & =\varepsilon_{\mu\nu\rho\sigma}V^{\mu\nu}V^{\rho\sigma}% =(3+q^{-4}-2q^{-3}\lambda)(V^{+3}V^{0-}-V^{+0}V^{3-})\\ & +\;(3+q^{-4}-2q^{-3})(V^{+-}V^{30}+V^{30}V^{+-})\nonumber\\ & +\;q^{-6}(3q^{4}+1-2q\lambda)(V^{0-}V^{+3}-V^{3-}V^{+0})\nonumber\\ & -\;q^{-2}\lambda(2q^{2}+2q\lambda+\lambda\lambda_{+})V^{+-}V^{+-},\nonumber\end{aligned}$$ with $\eta_{\mu\nu}$ and $\varepsilon_{\mu\nu\rho\sigma}$ being q-analogs of Minkowski metric and corresponding epsilon tensor (see also Appendix \[AppA\]). Specifying the two Casimirs for the different representations finally yields the results: 1. (operator representation) $$\begin{aligned} C^{1} & =2q\lambda_{+}^{-2}(X\circ X)(\partial\circ\partial)+2q^{-2}% \lambda_{+}^{-2}(X\circ\partial)(X\circ\partial)\\ & +\;2q\lambda_{+}^{-1}X\circ\partial,\nonumber\\ C^{2} & =0,\nonumber\end{aligned}$$ 2. (spinor representation) $$\begin{aligned} C^{1} & =[[3]]_{q^{-4}}\lambda_{+}^{-2}\mbox{1 \kern-.59em {\rm l}}_{2\times 2},\\ C^{2} & =[[3]]_{q^{4}}(3+q^{4}+2q^{3}\lambda )\mbox{1 \kern-.59em {\rm l}}_{2\times2},\nonumber\end{aligned}$$ 3. (vector representation) $$\begin{aligned} C^{1} & =2[[3]]_{q^{-4}}\lambda_{+}^{-2}% \mbox{1 \kern-.59em {\rm l}}_{3\times3},\\ C^{2} & =0.\nonumber\end{aligned}$$ Conclusion\[Concl\] =================== Let us end with a few comments on our results. We dealt with quantum algebras describing q-deformed versions of physical symmetries. This way we considered $U_{q}(su_{2}),$ $U_{q}(so_{4}),$ and q-deformed Lorentz algebra. It was our aim to provide a consistent framework of basic definitions and relations which allow for representation theoretic investigations in physics. In each case the starting point of our reasonings was the realization of symmetry generators within q-deformed differential calculus. We listed the relations of the corresponding symmetry algebras and presented expressions for coproducts and antipodes on symmetry generators. We realized that the Hopf structure of the symmetry generators allows us to define q-analogs of classical commutators. Furthermore, we concerned ourselves with q-deformed versions of such finite dimensional representations that play an important role in physics, i.e. spinor and vector representation. With the help of these representations we were able to write down q-deformed commutation relations between symmetry generators and components of a spinor or vector operator. Moreover, we calculated the adjoint action of the symmetry generators on each other and obtained relations for quantum Lie algebras this way. Finally, we gave expressions for the corresponding Casimir operators and specified them for the different representations. Our reasonings were in complete analogy to the classical situation, but compared with the undeformed case our results are modified by terms proportional to $\lambda=q-q^{-1}$. Hence, we regain in the classical limit as $q\rightarrow1$ the familiar expressions. This observation nourishes the hope that a field theory based on quantum group symmetries can be developed along the same lines as its undeformed counterpart. In this sense we also hope that implementing our results on a computer algebra system like Mathematica will prove useful in a systematical search for new q-identities. Furthermore, such an undertaking can be helpful to make it obvious that everything presented in this article works fine. **Acknowledgements**First of all we are very grateful to Eberhard Zeidler for his invitation to the MPI Leipzig, his very interesting and useful discussions, his special interest in our work and his financial support. Also we want to express our gratitude to Julius Wess for his efforts and his steady support. Furthermore we would like to thank Fabian Bachmaier for teaching us Mathematica. Finally, we thank Dieter Lüst for kind hospitality. q-Deformed quantum spaces\[AppA\] ================================= The aim of this appendix is the following. For quantum spaces of physical importance we list their defining commutation relations as well as the nonvanishing elements of their quantum metric and q-deformed epsilon tensor. The coordinates $\theta^{\,\alpha},$ $\alpha=1,2,$ of two-dimensional antisymmetrised quantum plane fulfill the relation [@Man88; @SS90] $$\theta^{1}\theta^{2}=-q^{-1}\theta^{2}\theta^{1},\quad q\in\mathbb{R},~q>1,$$ whereas the spinor metric is given by a matrix $\varepsilon^{\alpha\beta}$ with nonvanishing elements $$\varepsilon^{12}=q^{-1/2},\quad\varepsilon^{21}=-q^{1/2}.$$ Furthermore, we can raise and lower indices as usual, i.e. $$\theta^{\,\alpha}=\varepsilon^{\alpha\beta}\theta_{\beta},\quad\theta_{\alpha }=\varepsilon_{\alpha\beta}\theta^{\beta},$$ where $\varepsilon_{\alpha\beta}$ denotes the inverse of $\varepsilon ^{\alpha\beta}$. In the case of three-dimensional q-deformed Euclidean space the commutation relations between its coordinates $X^{A},$ $A\in\{+,3,-\},$ read $$\begin{aligned} X^{3}X^{\pm} & =q^{\pm2}X^{\pm}X^{3},\\ X^{-}X^{+} & =X^{+}X^{-}+\lambda X^{3}X^{3}.\nonumber\end{aligned}$$ The nonvanishing elements of the quantum metric are $$g^{+-}=-q,\quad g^{33}=1,\quad g^{-+}=-q^{-1}.$$ Now, the covariant coordinates are given by $$X_{A}=g_{AB}X^{B},$$ with $g_{AB}$ being the inverse of $g^{AB}$. The nonvanishing components of the three-dimensional q-deformed epsilon tensor take the form$$\begin{aligned} \varepsilon_{-3+} & =1, & \varepsilon_{3-+} & =-q^{-2},\\ \varepsilon_{-+3} & =-q^{-2}, & \varepsilon_{+-3} & =q^{-2},\nonumber\\ \varepsilon_{3+-} & =q^{-2}, & \varepsilon_{+3-} & =-q^{-4},\nonumber\\ \varepsilon_{333} & =-q^{-2}\lambda. & & \nonumber\end{aligned}$$ In Sec. \[QuLie3dim\] we need especially$$\epsilon_{AB}{}^{C}\equiv g^{CD}\varepsilon_{ABD},\quad\epsilon^{AB}{}% _{C}\equiv g^{AD}g^{BE}\varepsilon_{DEC}.$$ Next we come to four-dimensional Euclidean space. For its coordinates $X^{i},$ $i=1,\ldots,4,$ we have the relations $$\begin{aligned} X^{1}X^{j} & =qX^{j}X^{1},\quad j=1,2,\\ X^{j}X^{4} & =qX^{4}X^{j},\nonumber\\ X^{2}X^{3} & =X^{3}X^{2},\nonumber\\ X^{4}X^{1} & =X^{1}X^{4}+\lambda X^{2}X^{3}.\nonumber\end{aligned}$$ The metric has the nonvanishing components $$g^{14}=q^{-1},\quad g^{23}=g^{32}=1,\quad g^{41}=q.$$ Its inverse denoted by $g_{ij}$ can again be used to introduce covariant coordinates, i.e. $$X_{i}=g_{ij}X^{j}.$$ For the epsilon tensor of q-deformed Euclidean space with four dimensions one can find as components$$\begin{aligned} \varepsilon_{1234} & =q^{4}, & \varepsilon_{1432} & =-q^{2}, & \varepsilon_{2413} & =-q^{2},\\ \varepsilon_{2134} & =-q^{3}, & \varepsilon_{4132} & =q^{2}, & \varepsilon_{4213} & =q,\nonumber\\ \varepsilon_{1324} & =-q^{4}, & \varepsilon_{3412} & =q^{2}, & \varepsilon_{2341} & =-q^{2},\nonumber\\ \varepsilon_{3124} & =q^{3}, & \varepsilon_{4312} & =-q, & \varepsilon _{3241} & =q^{2},\nonumber\\ \varepsilon_{2314} & =q^{2}, & \varepsilon_{1243} & =-q^{3}, & \varepsilon_{2431} & =q,\nonumber\\ \varepsilon_{3214} & =-q^{2}, & \varepsilon_{2143} & =q^{2}, & \varepsilon_{4231} & =-1,\nonumber\\ \varepsilon_{1342} & =q^{3}, & \varepsilon_{1423} & =q^{2}, & \varepsilon_{3421} & =-q,\nonumber\\ \varepsilon_{3142} & =-q^{2}, & \varepsilon_{4123} & =-q^{2}, & \varepsilon_{4321} & =1.\nonumber\end{aligned}$$ In addition to this there are the non-classical components$$\varepsilon_{3232}=-\varepsilon_{2323}=q^{2}\lambda.$$ Now, we come to q-deformed Minkowski space [@SWZ91; @OSWZ92; @Maj91] (for other deformations of spacetime and their related symmetries we refer to [@Lu92; @Cas93; @Dov94; @ChDe95; @ChKu04; @Koch04]). Its coordinates are subjected to the relations $$\begin{gathered} X^{\mu}X^{0}=X^{0}X^{\mu},\quad\mu\in{\{}0,+,-,3{\},}\\ X^{3}X^{\pm}-q^{\pm2}X^{\pm}X^{3}=-q\lambda X^{0}X^{\pm},\nonumber\\ X^{-}X^{+}-X^{+}X^{-}=\lambda(X^{3}X^{3}-X^{0}X^{3}),\nonumber\end{gathered}$$ and its metric is given by$$\eta^{00}=-1,\quad\eta^{33}=1,\quad\eta^{+-}=-q,\quad\eta^{-+}=-q^{-1}.$$ As usual, the metric can be used to raise and lower indices. In analogy to the classical case the q-deformed epsilon tensor has as nonvanishing components$$\begin{aligned} \varepsilon_{+30-} & =-q^{-4}, & \varepsilon_{+-03} & =q^{-2}, & \varepsilon_{3-+0} & =q^{-2},\\ \varepsilon_{3+0-} & =q^{-2}, & \varepsilon_{-+03} & =-q^{-2}, & \varepsilon_{-3+0} & =-1,\nonumber\\ \varepsilon_{+03-} & =q^{-4}, & \varepsilon_{0-+3} & =-q^{-2}, & \varepsilon_{30-+} & =q^{-2},\nonumber\\ \varepsilon_{0+3-} & =-q^{-4}, & \varepsilon_{-0+3} & =q^{-2}, & \varepsilon_{03-+} & =-q^{-2},\nonumber\\ \varepsilon_{30+-} & =-q^{-2}, & \varepsilon_{+3-0} & =q^{-4}, & \varepsilon_{3-0+} & =-q^{-2},\nonumber\\ \varepsilon_{03+-} & =q^{-2}, & \varepsilon_{3+-0} & =-q^{-2}, & \varepsilon_{-30+} & =1,\nonumber\\ \varepsilon_{+0-3} & =-q^{-2}, & \varepsilon_{+-30} & =-q^{-2}, & \varepsilon_{0-3+} & =1,\nonumber\\ \varepsilon_{0+-3} & =q^{-2}, & \varepsilon_{-+30} & =q^{-2}, & \varepsilon_{-03+} & =-1,\nonumber\end{aligned}$$ and$$\begin{aligned} \varepsilon_{0-0+} & =q^{-1}\lambda, & \varepsilon_{-0+0} & =-q^{-1}% \lambda,\\ \varepsilon_{0333} & =-q^{-2}\lambda, & \varepsilon_{3330} & =q^{-2}\lambda,\nonumber\\ \varepsilon_{3033} & =+q^{-2}\lambda, & \varepsilon_{3030} & =-q^{-2}\lambda,\nonumber\\ \varepsilon_{3303} & =-q^{-2}\lambda, & \varepsilon_{+0-0} & =-q^{-3}\lambda,\nonumber\\ \varepsilon_{0303} & =q^{-2}\lambda, & \varepsilon_{0+0-} & =q^{-3}% \lambda.\nonumber\end{aligned}$$ [99]{} W. 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--- abstract: 'Large scale filaments, with lengths that can reach tens of Mpc, are the most prominent features in the cosmic web. These filaments have only been observed indirectly through the positions of galaxies in large galaxy surveys or through absorption features in the spectra of high-redshift sources. In this study, we propose to go one step further and directly detect intergalactic medium filaments through their emission in the HI 21 cm line. We make use of high-resolution cosmological simulations to estimate the intensity of this emission in low-redshift filaments and use it to make predictions for the direct detectability of specific filaments previously inferred from galaxy surveys, in particular the Sloan Digital Sky Survey. Given the expected signal of these filaments, our study shows that HI emission from large filaments can be observed by current and next-generation radio telescopes. We estimate that gas in filaments of length $l \gtrsim$ 15 $h^{-1}$Mpc with relatively small inclinations to the line of sight ($\lesssim 10^\circ$) can be observed in $\sim100$ h with telescopes such as Giant Metrewave Telescope or Expanded Very Large Array, potentially providing large improvements over our knowledge of the astrophysical properties of these filaments. Due to their large field of view and sufficiently long integration times, upcoming HI surveys with the Apertif and Australian Square Kilometre Array Pathfinder instruments will be able to detect the brightest independently of their orientation and curvature. Furthermore, our estimates indicate that a more powerful future radio telescope like Square Kilometre Array can be used detect even the faintest of these filaments with integration times of $\sim10-40$ h.' author: - | Robin Kooistra,$^{1}$ Marta B. Silva$^{1}$ and Saleem Zaroubi$^{1,2}$\ $^1$Kapteyn Astronomical Institute, University of Groningen, Landleven 12, 9747 AD Groningen, the Netherlands\ $^2$Department of Natural Sciences, Open University of Israel, 1 University Road, PO Box 808, Ra’anana 4353701, Israel bibliography: - 'HI.bib' title: 'Filament Hunting: Integrated HI 21 cm Emission From Filaments Inferred by Galaxy Surveys' --- intergalactic medium – cosmology: theory – diffuse radiation – large-scale structure of Universe Introduction ============ Observations of the local Universe have revealed that most galaxies reside in a complex network of filamentary structures known as the cosmic web [@art:web]. Within the ${\Lambda}$ cold dark matter framework, these structures are an expected result from nonlinear gravitational evolution. According to this picture, the dark matter haloes within which galaxies reside are connected to each other through a patchwork of filaments and sheets that constitute the structure of the intergalactic medium [IGM; e.g., @art:cen99]. Whereas galaxies reside in high density regions, the diffuse gaseous component, if observed, gives an alternative view into filaments and their baryonic component. Furthermore, it potentially gives a more faithful tracer to the spatial distribution of the IGM than galaxies by themselves, which in turn allows for the study of the connection between galaxies and the filaments of gas.\ Outside galaxies most baryons are expected to be in the relatively dense circumgalactic medium, which is mostly ionized and warm to hot. The remaining gas is in a filamentary structure with gas heated and ionized due to gravitational collapse and the UV and X-ray backgrounds. However, shielding from ionizing radiation due to recombinations in the denser pockets of gas still allows some of this gas to have a higher neutral fraction and emit potentially observable HI 21 cm radiation. Generally, the neutral fraction in these regions is well below a percent of the total gas content [@art:popping; @art:takeuchi]. Nonetheless, its existence has been proved observationally through Lyman alpha forest absorption in the spectra of high redshift quasars [@art:GP]. However, meaningful Lyman alpha absorption requires neutral gas above a given density threshold. Moreover, the technique is limited by the lines of sight to the available quasars and the matter content inferred from it depends on several assumptions about the gas conditions that are deduced from comparisons with simulations [e.g., @art:borde].\ Thus far, large scale filaments have mostly been detected through tracing the spatial distribution of galaxies in large galaxy surveys, such as the Sloan Digital Sky Survey [SDSS, @art:sdss1; @art:sdss8], the 2-degree Field (2dF) Galaxy Redshift Survey [@art:2df] and the Two Micron All-Sky Survey [2MASS, @art:2mass]. In such surveys the larger structures of the cosmic web can be easily identified. Indeed, some effort has been made to catalogue filaments traced by SDSS galaxies and many filaments with lengths ranging from a few to tens of $h^{-1}$Mpc and diameters of $\sim$0.5-2 $h^{-1}$Mpc have been found [e.g., @art:tempel; @art:sousbie08; @art:jasche10; @art:smith12]. In recent years, some effort have been devoted to tracing the large scale structure of the cosmic web through the gas distribution. In a ground braking work, @art:chang10 used HI 21 cm intensity maps obtained from the Green Bank Telescope (GBT), to constrain the large scale structure at a redshift of z $\sim0.8$ by cross-correlating with data from galaxy surveys. The same group followed this work up with a more significant detection using the WiggleZ galaxy survey data together with GBT 21cm data [@art:masui13]. At higher redshifts, evidence for filamentary structures was found in the spectra of background sources due to scattering of Lyman alpha photons by the neutral IGM along the line of sight [e.g., @art:abs1; @art:abs2; @art:abs3; @art:finley14]. Recently, a cosmic web filament at z $\sim$ 2-3 illuminated by a bright quasar was detected due to its Lyman alpha fluorescent emission [@art:fildet]. At lower redshifts, however, the gas in the IGM is on average more ionized and the Lyman alpha line is observed in the UV, and so it is much more challenging to detect it in emission.\ @art:takeuchi explored the possibility of using the HI 21 cm line to directly observe IGM filaments. By estimating the integrated HI 21 cm line intensity from simulated filaments they found that filaments with a length of $\sim100\, {\rm Mpc}$ can be detected in a 100 hours of integration by reasonably sensitive telescopes, such as the Giant Metrewave Radio Telescope (GMRT) or the Five hundred meter Aperture Spherical Telescope (FAST). They also found that the signal from filaments aligned along the line of sight can be more easily detected. Obviously, detection of such a signal would allow for the study of both the baryonic content of the filament and the ionization state of the gas. This would in turn help constrain the UV background, since this radiation is the main agent responsible for the thermal and ionization state of the gas.\ Recently, @art:horii17 performed a similar exercise, but then for the warm hot intergalactic medium (WHIM). Their simulations include strong feedback, resulting in very high temperature filaments. This in turn leads to a large ionization fraction and an HI 21 cm signal that is more difficult to observe.\ In this work we go one step further by directly linking the properties of cosmic web filaments in our simulations to those that have previously been identified from observational galaxy catalogues. Specifically, we use the filament catalogue by @art:tempel, obtained from SDSS data, to find real filaments that are aligned along the line of sight and extract similar filaments from the simulation. With this identification we then employ the simulations to calculate the neutral gas fraction in these specific filaments and predict their observability by current and future radio telescopes. To do this, we use a high resolution cosmological n-body simulation and apply a semi-analytical prescription to the density field to estimate the temperature and ionization state of the IGM, from which we determine the HI 21 cm signal from these filaments. The content of the paper is organized as follows. We begin by describing the model that is used to determine the thermal and ionization states of the IGM in Section \[secmodel\]. The method to calculate the 21cm brightness temperature signal is shown in Section \[sec:sig\]. The simulation details are presented in Section \[sec:sim\]. We then describe how we selected the observed filaments and how we estimated their integrated HI 21 cm signal in Section \[sec:fils\]. In the same section we introduce the instruments that are considered for possible observations and compare their sensitivity to the signals of the filaments. We explore the advantages of survey instruments with large field of view in more detail in Section \[sec:surveys\] and finally the removal of contamination due to emission from galaxies in the HI 21 cm line is discussed in Section \[sec:contam\].\ Throughout this work, we assume the @art:planck15para cosmological parameters ($\Omega_{\mathrm{m}}$ = 0.3089, $\Omega_{\mathrm{\Lambda}}$ = 0.6911, $\Omega_{\mathrm{b}}h^{2}$ = 0.02230, $H_{\rm{0}}$ = 67.74 $\mathrm{km\,s^{-1}Mpc^{-1}}$ and $Y_{\rm{P}}$ = 0.249). Model: Ionization and thermal state in the IGM {#secmodel} ============================================== The conditions in the IGM drive the observed HI signal. This depends on the complex interplay between the different ionization, recombination, heating and cooling processes in the gas. The thermal and ionization state of the cold IGM is mainly set by the strength of the UV/X-ray background and can be estimated by assuming both thermal and ionization equilibrium. The equilibrium assumptions are a good approximation for most of the gas in filaments, given that the relevant timescales for ionization and recombination are relatively short. However, they break down in very low density regions, where the gas cannot efficiently cool through recombination and collisional emission, and in the vicinity of galaxies or active galactic nuclei, where the ionizing radiation is much stronger and therefore the gas is highly ionized. Here we describe the full details of the model used to determine the ionization and thermal states of the IGM. The ionization state of the intergalactic medium is governed by the balance between ionization and recombination processes. Even at low redshift, the gas in filaments far from local sources is expected to be very metal poor and so the cooling and heating processes in this medium are dominated by reactions involving only hydrogen and helium. The fractions of the different states of hydrogen and helium can be found by solving the following set of balance equations [@art:fukkaw]. {width="48.00000%"}{width="48.00000%"} $$\begin{aligned} \frac{\mathrm{d}x_{\rm{HII}}}{\mathrm{d}t} & = \Gamma_{\rm{HI}}x_{\rm{HI}} + \beta_{\rm{HI}}n_{\mathrm{e}}x_{\rm{HI}} - \alpha_{\rm{HII}}n_{\mathrm{e}}x_{\rm{HII}}\label{rate:HII},\\ \frac{\mathrm{d}x_{\rm{HeII}}}{\mathrm{d}t} & = \Gamma_{\rm{HeI}}x_{\rm{HeI}} + \beta_{\rm{HeI}}n_{\mathrm{e}}x_{\rm{HeI}}\notag\\ &\quad-\left(\alpha_{\rm{HeII}}+\xi_{\rm{HeII}}\right)n_{\mathrm{e}}x_{\rm{HeII}}\notag\\ &\quad- \beta_{\rm{HeII}}n_{\mathrm{e}}x_{\rm{HeII}} - \Gamma_{\rm{HeII}}x_{\rm{HeII}}\notag\\ &\quad+ \alpha_{\rm{HeIII}}n_{\mathrm{e}}x_{\rm{HeIII}},\label{rate:HeII}\\ \frac{\mathrm{d}x_{\rm{HeIII}}}{\mathrm{d}t} & = \Gamma_{\rm{HeII}}x_{\rm{HeII}} + \beta_{\rm{HeII}}n_{\mathrm{e}}x_{\rm{HeII}}\notag\\ &\quad-\alpha_{\rm{HeIII}}n_{\mathrm{e}}x_{\rm{HeIII}}.\label{rate:HeIII}\end{aligned}$$ Here $n_{\mathrm{e}}$ is the electron number density, whereas $x_{\rm{HI}}$, $x_{\rm{HII}}$, $x_{\rm{HeI}}$, $x_{\rm{HeII}}$ and $x_{\rm{HeIII}}$ denote the fractions of HI, HII, HeI, HeII and HeIII, respectively. $\Gamma_i$ is the photoionization rate, $\beta_i$ the collisional excitation rate and $\alpha_i$ the recombination rate of species $i$. $\xi_{\mathrm{HeII}}$ is the dielectronic recombination rate of HeII. For the photoionization rates we interpolate the tables of the ionizing background from @art:hm12. The recombination and collisional rates were determined using known temperature dependent parameterisations (see Appendix \[app:recion\]). From energy conservation of the IGM in an expanding universe, the gas temperature $T_{\mathrm{g}}$ follows $$\frac{dT_{\mathrm{g}}}{dt} = -2H(z)T_{\mathrm{g}} + \frac{2}{3}\frac{\left(\mathcal{H} - \Lambda\right)}{nk_{\mathrm{B}}}, \label{eq:tg}$$ where $H$(z) denotes the Hubble parameter, $\mathcal{H}$ the heating rate, $\Lambda$ the cooling function and $n$ the baryon number density defined as $n\equiv n_H + 4n_{He}$. The first term on the right-hand side accounts for the adiabatic cooling due to the Hubble expansion of the Universe. For the heating function we adopt the values corresponding to the @art:hm12 ionizing background. Our cooling function includes collisional ionization/excitation, (dielectronic) recombination, free-free emission and Compton scattering of the CMB photons. A detailed description of the adopted cooling rates can be found in Appendix \[app:cool\]. As mentioned above, we solve these equations assuming ionization and thermal equilibrium, namely, that the LHS of the equations is zero. Due to the dependence of Equations \[rate:HII\], \[rate:HeII\] and \[rate:HeIII\] on the gas temperature, they need to be solved together with Equation \[eq:tg\], which we do iteratively. Our code solves for the ionization fractions and the gas temperature for a given hydrogen density and redshift. The redshift evolution of the ionization fractions of hydrogen and helium for different densities, relevant for filaments ($\Delta_\mathrm{b} \equiv \rho_{\rm b}/\left<\rho_{\rm b}\right>$), are shown in Figure \[fig:fractions\] and the evolution of the gas temperature in Figure \[fig:gastemp\]. The gas temperature will thus be $\sim 1-3\times 10^4$ K for most of the gas in a filament. At redshift z = 3.5 our temperature-density distribution matches the median equilibrium solution in the hydrodynamical simulations with radiative transfer by @art:puchwein. These simulations also use the evolving @art:hm12 UV background, and were shown to reproduce the IGM temperature as predicted by Lyman alpha forest observations. The computed neutral hydrogen fraction and gas temperature are stable for small variations of the ionizing background. For overdensities of 100 and higher, the HI fraction would be high enough to make a significant contribution to the cosmological mass density of neutral hydrogen $\Omega_{\mathrm{HI}}$ [e.g., Figure 12 in @art:crighton15]. ![Redshift evolution of the gas temperature. The width of the lines denotes the overdensity.[]{data-label="fig:gastemp"}](./images/gastemp_2.pdf){width="48.00000%"} HI 21 cm Brightness temperature signal {#sec:sig} ====================================== Instead of directly measuring the intensity, radio telescopes measure the contrast between the brightness of the observed object and that of the CMB. This signal is expressed as the differential brightness temperature and is given by $$\begin{aligned} \delta T_{\mathrm{b}}^X (z) = &T_{\mathrm{b}}^X(z) - T_{\gamma}(z)\notag\\ = &\frac{\left[T_{\mathrm{s}}^X(z) - T_{\gamma}(z)\right]\cdot\left(1-e^{-\tau_X(z)}\right)}{1+z},\end{aligned}$$ where $T_\gamma$ is the CMB temperature and $T_{\rm s}^X$ and $\tau_X$ are the spin temperature and the optical depth of species X (i.e. HI), respectively. So if the spin temperature is lower than that of the CMB, the signal will appear in absorption and if it is higher, it will be in emission. In general, the optical depth is given by [@art:furlanetto06] $$\tau_X(z) = \frac{g_1}{g_0 + g_1}\frac{c^3A_{10}}{8\pi\nu_{10}^3}\frac{h_\mathrm{p}\nu_{10}}{k_{\mathrm{B}}T_{\mathrm{s}}}\frac{n_X(z)}{1+z}\frac{1}{dv_\parallel/dr_\parallel},$$ where $h_{\mathrm{p}}$ is the Planck constant, $A_{10}$ is the transition probability, $k_\mathrm{B}$ the Boltzmann constant, $g_0$ and $g_1$ are the statistical weights of the ground and excited states, $\nu_{10}$ is the frequency at which the hyperfine transition occurs (for neutral hydrogen $A_{10} = 2.867\times10^{-15} \mathrm{s}^{-1}$, $g_1/g_0 = 3/1$ and $\nu_{10} = 1420.4\,\mathrm{MHz}$), $n_X(z)$ is the physical number density of species X and $dv_\parallel/dr_\parallel$ is the comoving radial velocity gradient along the line of sight. When including peculiar velocities, the latter is given by $dv_\parallel/dr_\parallel = 1/(1+z)\left[H(z) + dv_r/dr\right]$, where $dv_r/dr$ is the comoving gradient of the line of sight component of the comoving velocity. In the optically thin limit, the differential brightness temperature becomes $$\begin{aligned} \delta T_{\mathrm{b}}^X (z) =& \frac{g_1}{g_0 + g_1}\frac{c^3h_{\mathrm{p}} A_{10}}{8\pi k_{\mathrm{B}}\nu_{10}^2}\frac{n_X(z)}{\left(1+z\right)H(z)}\notag\\ &\left(1-\frac{T_\gamma(z)}{T_s}\right)\left[1 + H(z)^{-1}dv_r/dr\right]^{-1}.\label{eq:dtb}\end{aligned}$$ So the signal depends on the general properties of the line, the density of the medium and the cosmology, but it is then modified by the peculiar velocity term of order unity and by the spin temperature term. The latter becomes important when the spin temperature is close to the CMB temperature. In general, in studies of galaxies, the spin temperature is much higher than $T_{\gamma}$ and is therefore safe to ignore. However, for the lower density IGM that is not always the case and so it needs to be properly estimated. Spin temperature ---------------- The spin temperature $T_s$ determines the relative abundance of the exited state versus the ground state through the Boltzmann equation as $$\frac{n_1}{n_0} = \frac{g_1}{g_0}\exp\left(-\frac{h_{\mathrm{p}}\nu_{10}}{k_{\mathrm{B}} T_{\mathrm{s}}}\right),$$ where $n_1$ and $n_0$ are the number of particles in the excited and ground state, respectively. The spin temperature is governed by absorption of CMB photons, collisions with hydrogen atoms, free electrons and protons and by scattering of UV photons (Wouthuysen-Field effect, @art:WFw [@art:field58]), which couple the spin temperature to the CMB photons and to the gas [@art:furlanetto06]. Therefore, the spin temperature can be written as [@art:field58] $$T_{\mathrm{s}}^{-1} = \frac{T_\gamma^{-1} + x_{\mathrm{c}} T_{\mathrm{k}}^{-1} + x_\alpha T_\alpha^{-1}}{1 + x_{\mathrm{c}} + x_\alpha},$$ with $T_{\mathrm{k}}$ the kinetic temperature of the gas and $T_\alpha$ the colour temperature. The collisional coupling factor $x_{\mathrm{c}}$ and the Wouthuysen-Field coupling factor $x_\alpha$ are given by $$x_{\mathrm{c}} = \frac{C_{10}}{A_{10}}\frac{T_*}{T_\gamma},\qquad x_\alpha = \frac{P_{10}}{A_{10}}\frac{T_*}{T_\gamma},$$ where $A_{10}$ is the spontaneous decay rate from state 1 to state 0, which for neutral hydrogen has a value of $2.867\times10^{-15} \mathrm{s}^{-1}$, $C_{10}$ is the collisional de-excitation rate and $P_{10}$ is the de-excitation rate due to absorption of a Lyman alpha photon. The equivalent temperature $T_*$ is defined as $T_*\equiv h_{\mathrm{p}}\nu_{10}/k_{\mathrm{B}}$. When calculating the spin temperature, we assume that the kinetic and colour temperatures follow the gas temperature: $T_\alpha \sim T_{\mathrm{k}} \sim T_{\mathrm{g}}$. The collisional de-excitation rate for neutral hydrogen can be expressed as a sum over the collisional processes with electrons, protons and other neutral hydrogen atoms. $$C_{10}^{\mathrm{HI}} = \kappa_{10}^{\mathrm{HH}}(T_{\mathrm{k}})n_{\mathrm{H}} + \kappa_{10}^{\mathrm{eH}}(T_{\mathrm{k}})n_{\mathrm{e}} + \kappa_{10}^{\mathrm{pH}}(T_{\mathrm{k}})n_{\mathrm{p}}$$ Here $\kappa_{10}^{\mathrm{HH}}$, $\kappa_{10}^{\mathrm{eH}}$ and $\kappa_{10}^{\mathrm{pH}}$ denote the collision rates for each process. Expressions for these can be found in @art:zygelman, @art:sigfurl and @art:furlfurl [@art:furlfurl2]\ The de-excitation rate for the Wouthuysen-Field effect equals $$P_{10}^{\mathrm{HI}} = \frac{16\pi e^2 f_\alpha^{\mathrm{HI}}}{27m_{\mathrm{e}}c}J_{\mathrm{Ly\alpha,HI}},$$ where $e$ is the electron charge, $f_\alpha$ the oscillator length of the Lyman alpha transition ($f_\alpha = 0.4162$ for neutral hydrogen), $m_{\mathrm{e}}$ the electron mass and $c$ the speed of light. For the Lyman alpha photon angle-averaged specific intensity $J_{\mathrm{Ly\alpha,HI}}$ a model of the Lyman alpha emission that interacts locally with the IGM is required. We assume three sources of Lyman alpha photons: collisional excitations, recombinations and high energy background photons that redshift into the the Lyman alpha line. The full details of our calculation can be found in Appendix \[app:lya\] and the resulting evolution of the spin temperature is shown in Figure \[fig:tspin\], where we also show the evolution of the gas temperature and the CMB temperature. For the high densities the spin temperature quickly couples to the kinetic temperature, but at low densities and low redshift the spin temperature approaches the CMB temperature, therefore suppressing the brightness temperature signal. ![Spin temperature evolution of hydrogen compared to the CMB temperature and kinetic temperature. The linewidth denotes the density.[]{data-label="fig:tspin"}](./images/spintemp_2.pdf){width="48.00000%"} Test: slab model ---------------- To test the effect of the spin temperature correction term on the resulting signal of a filament, we assume a simple constant density slab model. The signal from such a slab is calculated through a small adjustment of Equation \[eq:dtb\], giving [@art:takeuchi] $$\delta T_{\mathrm{b}}^X (z) = \frac{g_1}{g_0 + g_1}\frac{c^3h_{\mathrm{p}}A_{10}}{8\pi k_{\mathrm{B}}\nu_{10}^2}\frac{n_X(z)}{\left(1+z\right)^2}\left(1-\frac{T_\gamma(z)}{T_{\mathrm{s}}}\right)\frac{\Delta r}{\Delta v}$$ with the width of the slab $\Delta r$ = 1 Mpc $h^{-1}$ and its proper line of sight velocity $\Delta v$ = 300 km $\rm{s^{-1}}$. The resulting differential brightness temperature evolution of HI is given in Figure \[fig:dtbslab\]. The red dashed lines show the signal in the saturated limit $\left(\left[1-T_\gamma/T_\mathrm{s}\right] \sim 1\right)$ and the blue solid lines show the corrected signal. As can be seen, the spin temperature correction becomes significant with a factor of a few for the lower density filaments at low redshifts and therefore needs to be included in these calculations. ![Differential brightness temperature evolution for the slab model. The density is denoted by the width of the lines and the dashed lines show the signal in the saturated limit $\left(\left[1-T_{\gamma}/T_{\rm s}\right] \sim 1\right)$.[]{data-label="fig:dtbslab"}](./images/brighttemp_2.pdf){width="48.00000%"} {width="49.00000%"} {width="50.00000%"} Cosmological simulations {#sec:sim} ======================== In order to predict the signal from more realistic filaments we make use of high resolution simulations which make it possible to resolve different filament morphologies and allow for differences in the gas properties along a filament. We start by running the parallel Tree-Particle Mesh code Gadget 2 [@art:gadget2] with $1024^3$ particles and a volume of $\left(200\,h^{-1}\mathrm{Mpc}\right)^3$. This corresponds to a mass resolution of $6.514\times10^8\, \mathrm{M_\odot} h^{-1}$. We then use cloud-in-cell interpolation to divide the simulation particles into a grid of N = $600^3$ cells, corresponding to a spatial resolution of $\sim0.33\, {h^{-1}\, {\rm Mpc}}$. This is enough to properly resolve the filament morphology. Then we determine the density contrast $\delta = \rho/<\rho>-1$ for each cell of the simulation and, assuming that the spatial distribution of the baryons follows that of the dark matter, we compute the gas temperature, ionization state and differential brightness temperature following the prescriptions described in Sections \[secmodel\] and \[sec:sig\]. Because we are interested in the emission of the IGM and not from galaxies, we mask the cells that are above the virialized limit determined through the scaling relations by @art:collapse. For z = 0.1 this critical overdensity is $\Delta_c \approx 297$. Figure \[fig:boxes\] shows a slice of the simulation at z = 0.1, with a thickness of 0.33 $h^{-1}$Mpc, where several large filaments can be visually identified. Additionally, we constructed a catalogue with the positions and masses of dark matter haloes using the Amiga halo finder [@art:halofind] on the initial particle catalogue. At low redshift, the most massive haloes are expected to have more than one galaxy. However most of the luminosity of a halo usually originates in a single bright galaxy. This is also the only galaxy that a modest galaxy survey would probably detect, so it is reasonable to assume that each of these haloes would correspond to a single galaxy when comparing with SDSS detected galaxies. 21cm emission from SDSS Filaments {#sec:fils} ================================= In this section we select gas filaments indirectly detected through the SDSS galaxies and estimate their integrated HI intensity. Given that galaxy surveys, such as the SDSS, do not probe the gas in filaments, we estimate the properties of the filamentary gas from similar length filaments found in our cosmological N-body simulations, which contain a similar number of galaxies. We start with a description of the catalogue and the criteria that were used to select the filaments and follow with the estimation of the HI 21 cm line intensity. The estimates are then used to determine the detectability of these filaments by current and future telescopes. Filament catalogue ------------------ Since we are interested in targeting filaments inferred from galaxy catalogues, a complete sample of galaxies, covering a relatively large volume and down to a relatively low magnitude threshold is required to identify large scale filaments, the largest of which, currently available was obtained by SDSS. The sample includes 499340 galaxies and goes up to a redshift of $z$ = 0.155 with a lower limit of $z$ = 0.009. The lower magnitude limit of the sample of galaxies is set to $m_{\mathrm{r}}$ = 17.77, imposed by the limits of the spectroscopic sample [@art:ssdssspecsamp]. We therefore use the filament catalogue by @art:tempel. This catalogue is obtained by statistically inferring the filamentary pattern in the SDSS data release 8 sample through the Bisous model. The model assumes a fixed maximum distance to which galaxies can be separated from the filament spine and still belong to it, which in this case was chosen to be 1 $h^{-1}$Mpc. In Figure \[fig:fildist\] we show the length distribution of the filaments from the catalogue. Most filaments are short (less than 10 $h^{-1}$Mpc) and would therefore result in a low signal-to-noise in most observations. However, a sample of $\sim$ 4000 long filaments, reaching lengths of $\sim$ 10 $h^{-1}$Mpc to $\sim$ 50 $h^{-1}$Mpc is also found. Within this sample of longer filaments we choose the most suitable candidates to be targeted by observations, taking into account not only their lengths, but also their galaxy densities. It should be noted here that, given the spatial density of galaxies in the SDSS catalogue, the @art:tempel method tends to fragment the filaments and assign smaller lengths to them than in reality. ![Distribution of the length of the filaments found in the @art:tempel catalogue.[]{data-label="fig:fildist"}](./images/fildist_2.pdf){width="45.00000%"} The signal from a filament is highest when it can be observed directly along the line of sight, because then a larger part of the filament can be integrated over with a single pointing of the telescope. Therefore, we select the larger filaments with the smaller alignment angles ($\theta$) with the line of sight. For simplicity we define the inclination by assuming a straight line between the endpoints of the filaments. This method might exclude long filaments whose endpoints are not aligned along the line of sight, but that in between do have large parts that are aligned. However, we still find a number of useful filaments. In Table \[tab:filselect\] we give the properties of the seven selected filaments. ID z $d_{\mathrm{com}}$ ($h^{-1}$Mpc) $l$ ($h^{-1}$Mpc) $\theta\, (^\circ)$ $N_{\mathrm{gal}}$ ---- ------ ---------------------------------- ------------------- --------------------- -------------------- 1 0.05 175 23.5 5.02 44 2 0.04 130 13.9 5.70 32 3 0.07 227 16.3 4.27 40 4 0.11 333 16.8 1.02 44 5 0.12 356 14.1 2.05 55 6 0.10 282 16.6 2.21 42 7 0.06 180 19.2 5.42 21 : Properties of SDSS filaments with small alignment angles. The parameter $l$ is the length of the filament, $\theta$ is the alignment angle and $N_{\mathrm{gal}}$ is the number of galaxies associated with the filament. \[tab:filselect\] Filaments 4, 5 and 6 are the most aligned along the line of sight. They have lengths of $l\sim$ 15 $h^{-1}$Mpc, are all relatively straight spatially and are therefore ideal candidates for observations. Signal estimation ----------------- Ideally one would like to directly link current observables (galaxies) to the gas density in the filaments. Some work has been done to find a relation between the luminosity density of galaxies and the warm hot intergalactic medium (WHIM) using simulations [@art:nevalainen]. However, there are no similar relations between galaxy luminosity and the properties of the cold gas in filaments. Here we do a similar exercise in combining observations with simulations by looking for similar filaments to the ones found by SDSS. From the simulated box we visually identify a number of long filaments which, after a careful selection, can be used to estimate the expected HI signal of the SDSS filaments. Although most of the selected filaments, in the @art:tempel catalogue, have lengths of $\sim$ 15 $h^{-1}$Mpc, the real gas filaments can be more extended, or the catalogue model can identify smaller parts of a larger filament as separate ones. Consequently, the observed signal can be higher than what would be expected by assuming the length inferred from the galaxy distribution. Therefore, in the simulations we search for filaments that are longer ($\sim$ 50 $h^{-1}$Mpc, similar to the maximum length found in the filament catalogue) and, by comparing to the haloes in the simulation, determine what part of the filament would be detected by the SDSS. Although, our simulation box is at a redshift of $z$ = 0.1, the relevant properties of the gas in the filaments is not expected to vary by much for the redshift range covered by the SDSS selected filaments. Figure \[fig:filsel\] shows the extracted filaments from the simulation box. They each have a length of $\sim$ 50 $h^{-1}$Mpc and a radius of $\sim$ 1 $h^{-1}$Mpc and are thus comparable to the filaments that can be detected by SDSS. We now analyse in more detail three filaments that cover the diversity of the selected filaments and the expected range in their intensities. Filament 2 is relatively straight along the line of sight, compared to the other two and is therefore the optimistic case. Filament 3, on the other hand, is relatively faint at the top and middle sections and therefore most of the signal arises from the bottom part of the filament as shown in Figure \[fig:filsel\]. This is our pessimistic case. Filament 1 is somewhere in between the other two in terms of their expected signal strengths. The signal of the simulated filaments is determined by taking a cylindrical skewer centered on the filament, which represents the beam of the telescope and then the signal of the filament is calculated as follows $$\delta T_{\rm b}^{\rm fil} = \frac{\sum_{i,j}\delta T_{\rm b}^{i,j}}{\Delta R\cdot\pi r_{\mathrm{S}}^2},$$ where $\delta T_{\rm b}^{i,j}$ is the signal per cell within the skewer, $\Delta R$ is the length of the skewer, which depends on the observed bandwidth, and $r_\mathrm{S}$ is the radius of the skewer, which is determined by the angular resolution. We set $\Delta R$ = 50 $h^{-1}$Mpc (corresponding to a frequency bandwidth of $\sim$ 15 MHz) and the radius to $r_\mathrm{S}$ = 1 $h^{-1}$Mpc (or an angular resolution of 10 arcmin). The blue lines in Figure \[fig:filsel\] indicate the observational skewer. By comparing to the simulated halo map previously obtained with the Amiga halo finder, we find that most of the massive haloes are located in the region between the magenta lines and a galaxy survey like SDSS would therefore most likely observe the regions between these lines. ![The density fields of the selection of filaments from the simulation. The blue lines denote the observational skewer, whereas the magenta line shows what part would be preferentially detected by SDSS. The colourbar denotes the mean overdensity over the slices in the z-direction.[]{data-label="fig:filsel"}](./images/fildens.pdf){width="48.00000%"} Observability {#sec:observability} ------------- Telescope $D_{\mathrm{dish}}$ (m) $N_{\mathrm{dish}}$ $A_{\mathrm{tot}}$ ($\mathrm{m}^2$) $\epsilon_{\mathrm{ap}}$ $T_{\rm sys}$(K) Spectral range (GHz) $D_{\mathrm{max}}$ (km) $\theta_{\mathrm{res}}$ (’) $\nu_{\mathrm{res}}$ (kHz) FoV ($\mathrm{deg^2}$) ---------------- ------------------------- --------------------- ------------------------------------- -------------------------- ------------------ ---------------------- ------------------------- ----------------------------- ---------------------------- ------------------------ Arecibo 205 - 32,750 0.7 30 0.047 - 10 0.3 3.24 12.2 0.17 FAST 300 - 70,700 0.55 25 0.070 - 3 0.5 2.9 $\lesssim$ 0.5 0.36 Apertif (WSRT) 25 12 5,890 0.75 55 1.13 - 1.75 2.7 0.36 12.2 8 EVLA 25 27 13,300 0.45 26 1 - 50 1 - 36 0.97 - 0.03 31 0.42 GMRT 45 30 47,720 0.4 75 0.05 - 1.5 25 0.04 31 0.13 ASKAP 12 36 4,072 0.8 50 0.7 - 1.8 6 0.5 18.3 30 MeerKAT 13.5 64 9,160 0.8 20 0.580 - 14.5 20 0.05 $\lesssim$ 18 1.44 SKA-2 15 1,500 300,000 0.8 30 0.070 - 10 5 (core) 0.19 $\lesssim$ 18 1.17 {width="\textwidth" height="300"} In order to check the observability of these filaments, we compare the signal we obtain with the sensitivity of multiple instruments for the same conditions. In general, the noise in the measurement of a radio telescope can be written as [@art:furlanetto06] $$\delta T_{\rm N} = \frac{c^2(1+z)^2}{\nu_{0}^2\Delta\theta^2\epsilon_{\rm ap}A_{\rm dish}}\frac{T_{\rm sys}}{\sqrt{2\Delta\nu t_{\rm obs}}},$$ where $\epsilon_{\mathrm{ap}}$ is the aperture efficiency, $A_{\mathrm{dish}}$ the total (illuminated) surface area of a single dish of the array, $\Delta\theta$ the size of the beam, $\Delta\nu$ the frequency bandwidth and $t_{\mathrm{obs}}$ the observation time. The factor 2 in the last term follows from observing two polarizations simultaneously and integrating them together. The system temperature $T_{\mathrm{sys}}$ of a radio telescope has two components, one due to the sky that dominates at low frequencies and another due to the receiver, dominant at high frequencies. The brightness temperature uncertainty (sensitivity) $\delta T_{\mathrm{N}}$ is thus also a combination of the two contributions, where in the case of an interferometer the noise drops by the square root of the number of baselines ($N_{\mathrm{B}}=N_{\mathrm{dish}}(N_{\mathrm{dish}}-1)/2$), giving $$\delta T_{\mathrm{N}} = \left(\delta T_{\mathrm{N}}^{\mathrm{sky}} + \delta T_{\mathrm{N}}^{\mathrm{rec}}\right) \times \begin{cases} 1 & (\mathrm{single\, dish})\\ 1/\sqrt{N_{\mathrm{B}}} & (\mathrm{interferometer}) \end{cases} .$$ We note that this calculation assumes that the filaments contain structure on all scales for which the interferometers have baselines and therefore do not suffer from spatial filtering. This will be discussed in more detail in Kooistra et al. in prep.\ Ideal instruments would be those that have both a large field of view and good sensitivity in order to be able to probe the extended low-surface brightness HI emission. We consider both single dish telescopes and interferometers. The single dish telescopes are Arecibo and FAST. For interferometers we consider Apertif on the Westerbork Synthesis Radio Telescope (WSRT), the Expanded Very Large Array (EVLA), GMRT, the Australian Square Kilometre Array Pathfinder (ASKAP), the Karoo Array Telescope (MeerKAT) and the second phase of the Square Kilometre Array (SKA-2). The relevant parameters for each of them are given in Table \[tab:tels\]. In Figure \[fig:filsig\] we show the brightness temperature uncertainties for each of the instruments as a function of the observation time. Each plot gives a comparison with the signal of one of the filaments, previously shown in Figure \[fig:filsel\], where the blue shaded region gives the signal of the full simulated filament and the white striated region denotes the signal from the filamentary region that we expect to be detected from SDSS data. The minima and maxima of these regions denote the minimum and maximum signal that arises when rotating the observational skewer from -5 to +5 degrees with respect to the filament spine. This uncertainty accounts for a possible misalignment of the telescope beam and the filament spine, which we need to account for, given that this spine has to be defined without information on the gas content of the filament. Depending on the morphology of the filament, the small rotation of $\pm$ 5 degrees can change the signal of the filament by more than an order of magnitude. Since the regions corresponding to the SDSS filaments also contain the brightest parts of the filaments, the difference between the signal from either just that region or the full filaments is usually small. As can be clearly seen in Figure \[fig:filsig\], a number of instruments should be able to observe the HI signal within $\sim 100$ hours. The SKA will be able to detect the signal in all cases. Furthermore, single dish telescopes are the best alternatives, where FAST can detect the signal in all but the worst case scenario. The signal is also detectable for most instruments in the most optimistic case, whereas for the lower signals from Filaments 1 and 3, the signal would still be within reach of FAST, Arecibo and the SKA. Apertif and ASKAP can only make a low signal-to-noise detection of the strongest filament signal considered here.\ There is some uncertainty in the amount of heating that goes into the IGM. Since we do not include a prescription for shock heating in this medium, we might be underestimating the amount of heating and as a consequence also underestimate the ionization fractions. The amount of shock heating predicted by simulations differs a lot, depending on the assumptions on the feedback and on how it is implemented. According to the EAGLE simulation [@art:eagle] this is, however, not a crucial effect in the gas we target and so most of the gas in the EAGLE simulation follows the same temperature density relation we use in our simulation [@art:crain17]. Nevertheless, we account for an extra source of heating by manually increasing the heating and photoionization rates of the @art:hm12 background by a factor of 4. The resulting signal from the filaments reduces by a factor of $\sim$5, as shown by the orange lines denoting the maximum signal under these conditions in Figure \[fig:filsig\]. Even in this pessimistic case the maximum signal from the filaments would be detectable by the SKA. For strong filaments, such as filament 2, it would be possible to get a good detection with some of the other instruments as well. The Apertif and ASKAP HI surveys {#sec:surveys} ================================ In the previous sections we studied the detectability of the HI gas in IGM filaments for the case where the filaments are aligned along the line of sight and so their signal can be integrated over a single pointing, even by instruments with a small FOV. However, some of the considered instruments (i.e. Apertif, ASKAP and SKA) have a very large field of view and so they can, with a single pointing, also detect filaments with different alignments or even curved filaments. Furthermore, Apertif and ASKAP have a number of planned deep HI surveys in the near future. For Apertif there will be a medium deep survey[^1] covering 450 $\mathrm{deg}^2$ with 84 hours integration for each pointing. ASKAP will have the Deep Investigation of Neutral Gas Origins (DINGO) survey[^2]. DINGO will cover two fields on the southern hemisphere, smaller but with longer integration times than the full ASKAP field: the DINGO-Ultradeep field of 60 $\mathrm{deg}^2$ with pointings of 2500 hours and the less deep, but larger DINGO-Deep field of 150 $\mathrm{deg}^2$ and 500 hour pointings. The areas covered by these surveys are larger than the ones in our simulation and so they are likely to contain filaments at least as bright in HI as the ones considered here. In this section we present the feasibility of detecting curved filaments by such surveys. We assume that the spatial location of the filament can be inferred through the positions of the bright galaxies previously observed by the same instrument or by other instruments. In this case it is possible to follow the spine of the filament for the integration. We extract a curved filament from our simulation, as can be seen in Figure \[fig:filselcurv\] and estimate its detectability. ![The density field of the selected curved filament from the simulation. The magenta lines show the observational skewer over which the integration was performed with an angular resolution of 10 arcmin and a frequency bandwidth of 0.6 MHz, corresponding to a filament radius of $\sim$ 1 $h^{-1}\mathrm{Mpc}$. The colourbar denotes the mean overdensity over the filament in the z-direction.[]{data-label="fig:filselcurv"}](./images/fil4_dens.pdf){width="49.00000%"} Filament 4 is similar in density and length as the previously discussed aligned filaments. In order to calculate a curved filament signal, we consider that each small section of the filament is aligned perpendicular to the line of sight and that the instantaneous field of view of the survey encompasses the entire filament. This is a reasonable assumption for the case of both Apertif and ASKAP observations at $z$ = 0.1. Then we integrate along the filament by again assuming an angular resolution element of 10 arcmin, but now the frequency bandwidth will only be 0.6 MHz for the part along the line of sight (in the z-direction in Figure \[fig:filselcurv\]). The section over which we integrate the filament is shown by the magenta lines in Figure \[fig:filselcurv\].\ In this case the noise of the telescope for the integration of a single segment of the filament will be higher than for the case of the aligned filaments discussed in Section \[sec:sim\]. Fortunately, the integration is done over multiple of these 10 arcmin resolution segments and the noise drops as the square root of this number. The resulting signal to noise of the four filaments for both telescopes is shown in Figure \[fig:filsigsurv\]. Filament 2 is the brightest and is thus the only one that can be detected by both Apertif and ASKAP in 100 h. The other filaments fall below the detection threshold and we note that when the heating and photoionization rates are increased, the signal will become hard to detect with these surveys. In all cases, the signal-to-noise is almost the same as when the filament was aligned along the line of sight, showing that the orientation of a filament matters little if the FoV of the survey is large enough to cover the entire spine. ![Expected signal to noise of the simulated filaments with the HI survey instruments Apertif and ASKAP. We assume an angular resolution of 10 arcmin and a frequency bandwith of 0.6 MHz. The colour of the lines denotes the instrument and the linestyle shows for which filament it is.[]{data-label="fig:filsigsurv"}](./images/snr_curved_recal.pdf){width="48.00000%"} With their wide fields of view, survey instruments like Apertif and ASKAP might thus provide excellent tools for detecting HI emission from IGM filaments, provided that the HI gas in these filaments correlates well with the positions of the strongest galaxies in the same field. The cross-correlation of the HI signal with the galaxies could also possibly be used to confirm wether the signal really corresponds to a filament. Furthermore, since the noise from both quantities is not correlated, to first order, the noise terms would cancel in the cross-correlation, which could render the cross-correlation itself another useful probe of the large scale filaments to be used for the understanding of galaxy formation and evolution mechanisms. This will be explored further in a future publication. Contamination from galaxies {#sec:contam} =========================== Besides the IGM, another source of HI 21 cm emission in filaments are the galaxies. These are brighter than the IGM itself and will thus contaminate the integrated signal. Fortunately, the HI signal from galaxies is largely dominated by a few bright galaxies, which can be directly observed by current instruments, such as Arecibo or WSRT, and certainly by SKA-2. Also, these galaxies are relatively small compared to the width of the filament, given that typical sizes of HI disks are of the order of $\sim$30-60 kpc [e.g., @art:h1sizegals]. In the above sections we have calculated the signals by masking all cells in our simulation box with a density above the critical density for collapse. This corresponds to 0.4$\%$, 0.6$\%$, 0.1$\%$ and 0.4$\%$ of the pixels in the skewer for the proposed pixel size of $(0.33\,h^{-1}\mathrm{Mpc})^3$ for filament 1, 2, 3 and 4, respectively. We note that the resolution available from the proposed instruments is usually even higher. This means that, assuming that all dark matter haloes contain galaxies, the emission from all galaxies in the filament could be masked, without losing much of the IGM filamentary emission. In reality it is only necessary to know the positions of the brightest galaxies and mask the corresponding pixels in the observation. This requires deep galaxy surveys and instruments that have high resolution, both spatially and in frequency. The instruments that were considered in Section \[sec:observability\] all have the necessary resolution to be able to mask the most luminous foreground galaxies, without erasing the target signal. In most cases, these surveys can also detect the position and the brightness of these galaxies. In the case of the filaments detected by SDSS, the positions of the galaxies are already known from the survey itself and they only need to be masked. To check the level of contamination to the signal by galaxies, we make a conservative upper estimate of how much HI would be contained by them and compare it to the HI content of the IGM in the filaments in the simulation. The halo finder provides the masses of the dark matter haloes. We can convert these masses to HI masses by using the fitting function to the HI mass-halo mass relation determined through abundance matching by @art:haloh1: $$\begin{aligned} M_{\mathrm{HI}} = & 1.978\times10^{-2}M_{\mathrm{h}}\times\notag\\ &\left[\left(\frac{M_{\mathrm{h}}}{4.58\times10^{11}\,\mathrm{M_\odot}}\right)^{-0.90}+\left(\frac{M_{\mathrm{h}}}{4.58\times10^{11}\,\mathrm{M_\odot}}\right)^{0.74}\right]^{-1}\label{HIHM}.\end{aligned}$$ $M_{\mathrm{h}}$ denotes the mass of the halo and $M_{\mathrm{HI}}$ gives the mass of the HI expected to be inside the halo. The total HI mass in galaxies in the simulated filaments are $2.8\times10^{11}\,\mathrm{M_\odot}$, $5.5\times10^{11}\,\mathrm{M_\odot}$, $2.1\times10^{11}\,\mathrm{M_\odot}$ and $4.0\times10^{11}\,\mathrm{M_\odot}$ for filament 1, 2, 3 and 4, respectively. Their total HI masses contained by the IGM are, respectively, $3.5\times10^{12}\, \mathrm{M_{\odot}}$, $5.2\times10^{12}\, \mathrm{M_{\odot}}$, $1.7\times10^{12}\, \mathrm{M_{\odot}}$ and $4.7\times10^{12}\, \mathrm{M_{\odot}}$. Galaxy surveys can only find the locations of the brightest galaxies. The flux limit of the survey constrains which fraction of the contaminating galaxies can be masked from observations. In order to get an estimate of the amount of HI mass that could be inferred from SDSS data, we convert the flux limit of SDSS ($m_\mathrm{r}^{\mathrm{lim}}$ = 17.77) to a minimum galaxy HI mass ($M_{\mathrm{HI}}^{\mathrm{min}}$) in three steps. We connect the SDSS magnitude limit to the typical colour of galaxies using the colour-magnitude diagram of @art:mrtocolor. In this case the highest limit (thus the least effective in removing the contamination by galaxies) comes from assuming a blue galaxy, giving a colour of $(u-r)_{0.1} \approx$ 1.85. Using this colour we then find the appropriate mass-to-light ratio for these galaxies from @art:colortomasslight, corresponding to $\log\left(M_*/L\right) \approx$ 0.33, which, for the luminosity limit of SDSS, corresponds to a stellar mass of $\log M_* \approx$ 9.62. Finally, we convert the stellar mass into an HI mass by applying the stellar mass to gas mass relation found by @art:stellartogasmass, which results in a limiting HI mass for SDSS of $M_{\mathrm{HI}}^{\mathrm{lim}} \approx 2\times10^{9} \mathrm{M_{\odot}}$. By masking all the haloes in our simulated filaments with HI masses above this threshold, we then estimate the remaining contamination. Filament $M_{\mathrm{HI}}^{\mathrm{IGM}}$ $M_{\mathrm{HI}}^{\mathrm{gal}}$ $M_{\mathrm{HI}}^{\mathrm{cut}}$ ---------- ---------------------------------- ---------------------------------- ---------------------------------- 1 $3.5\times10^{12}$ $2.8\times10^{11}$ $4.5\times10^{10}$ 2 $5.2\times10^{12}$ $5.5\times10^{11}$ $5.9\times10^{10}$ 3 $1.7\times10^{12}$ $2.1\times10^{11}$ $4.3\times10^{10}$ 4 $4.7\times10^{12}$ $4.0\times10^{11}$ $5.1\times10^{10}$ : HI masses of the simulated filaments. $M_{\mathrm{HI}}^{\mathrm{IGM}}$ is the HI mass in the IGM of the filament, $M_{\mathrm{HI}}^{\mathrm{gal}}$ gives the total HI mass contained by the haloes in the filaments and $M_{\mathrm{HI}}^{\mathrm{cut}}$ gives the remaining HI contamination after removing the haloes with masses above the SDSS flux limit. All masses are given in $\mathrm{M_{\odot}}$. \[tab:HIcont\] Table \[tab:HIcont\] gives a summary of the results for each of the four filaments. As can be seen, the remaining contamination due to galaxies is two orders of magnitude smaller than the HI content in the IGM of the filaments. A survey like SDSS will thus be sufficient for dealing with the most severe contamination due to galaxies. Conclusions {#sec:concl} =========== In this work we explored the possibility of observing the integrated HI 21 cm line emission from large scale filaments in the IGM. Directly mapping this emission is a potential new avenue to probe the spatial distribution of the filaments, and their gas content, ionization state and temperature. The properties of the filaments can be useful to construct more realistic models of galaxy formation and evolution, given the constant exchange of gas between galaxies and their surrounding medium, which is usually a large scale IGM filament. Moreover, since the thermal and ionization state of the gas far from strong local sources is dominated by the UV background, these properties can be used to constrain this radiation field. Our study has mainly focused on filaments at z=0.1, primarily because that is the average redshift at which good galaxy catalogues are available. The new generation of surveys will, however, also provide good galaxy catalogues at higher redshifts, and the proposed SKA-2 survey can be used to probe their emission. This would make it possible to constrain the evolution of the UV background even further. In this study, we took observed filaments inferred from SDSS data and estimate their integrated HI 21 cm signal and its detectability by current and upcoming surveys. By selecting the largest filaments with the smallest inclination to the line of sight, we determined the intensity of the easier to observe filaments to be of the order of $10^{-6}-10^{-5}$K. We check the observability of these signals for an integration time of up to 100 hours and found that a number of the radio surveys considered could detect some of these filaments in less than 50 hours. In particular, FAST and the SKA are good candidates to detect the signal. The upcoming HI surveys for instruments with large fields of view, such as Apertif and ASKAP, could furthermore remove the need for observing spatially straight filaments aligned along the line of sight. We find that the integration times of the planned surveys for these two instruments will be sufficient to make the detection of the IGM gas in the strongest filaments feasible and opening up an interesting avenue to explore with these instruments. Contamination to the signal from galaxies has to be taken into account, given that the bulk luminosity at the 21cm line emanates from galaxies and not from the IGM gas we are targeting. We used the observational luminosities of the galaxies obtained by SDSS and compared them to simulated luminosities. This allowed us to safely conclude that this emission is dominated by a small number of sources, whose position can be determined by galaxy surveys and masked from observations before the signal is integrated. We note that these filaments inferred from SDSS data are biased towards the most luminous galaxies and should therefore have considerably more galaxy contamination than most IGM filaments. Our estimates also show that this masking procedure would bring the contamination down to a negligible level compared to the total signal of the filament, without erasing the target signal, even when attributing more HI mass to the remaining galaxies than what they are expected to have. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Marc Verheijen and Adi Nusser for interesting discussions on the subject presented here. We also thank the Netherlands Foundation for Scientific Research support through the VICI grant 639.043.00. Recombination and collisional ionization rates {#app:recion} ============================================== The ionization states of hydrogen and helium depend on the detailed balance between recombinations, ionizations and excitations. Below we list all the rates we adopted following @art:fukkaw. - Collisional ionization rates: - H I $\rightarrow$ H II:$$\beta_{\mathrm{HI}} = 5.85\times10^{-11}T^{1/2}\left(1+T_5^{1/2}\right)^{-1}\exp\left(-1.578/T_5\right)\quad \mathrm{cm^3\cdot s^{-1}}$$ - He I $\rightarrow$ He II:$$\beta_{\mathrm{HeI}} = 2.38\times10^{-11}T^{1/2}\left(1+T_5^{1/2}\right)^{-1}\exp\left(-2.853/T_5\right)\quad \mathrm{cm^3\cdot s^{-1}}$$ - He II $\rightarrow$ He III:$$\beta_{\mathrm{HeII}} = 5.68\times10^{-12}T^{1/2}\left(1+T_5^{1/2}\right)^{-1}\exp\left(-6.315/T_5\right)\quad \mathrm{cm^3\cdot s^{-1}}$$ - Recombination rates: - H II $\rightarrow$ H I:$$\alpha_{\mathrm{HII}} = 3.96\times10^{-13}T_4^{-0.7}\left(1+T_6^{0.7}\right)^{-1}\quad \mathrm{cm^3\cdot s^{-1}}$$ - He II $\rightarrow$ He I:$$\alpha_{\mathrm{HeII}} = 1.50\times10^{-10}T^{-0.6353}\quad \mathrm{cm^3\cdot s^{-1}}$$ - He III $\rightarrow$ He II:$$\alpha_{\mathrm{HeIII}} = 2.12\times10^{-12}T_4^{-0.7}\left(1+0.379T_6^{0.7}\right)^{-1}\quad \mathrm{cm^3\cdot s^{-1}}$$ - Dielectric recombination rate: - He II $\rightarrow$ He I:$$\xi_{\mathrm{HeII}} = 6.0\times10^{-10}T_5^{-1.5}\exp\left(-4.7/T_5\right)\left[1+0.3\exp\left(-0.94/T_5\right)\right]\quad \mathrm{cm^3\cdot s^{-1}}$$ Cooling rates {#app:cool} ============= In this section we list the cooling rates that were included in our model. - Collisional ionization cooling: - H I:$$\zeta_{\mathrm{HI}} = 1.27\times10^{-21}T^{1/2}\left(1+T_5^{1/2}\right)^{-1}\exp\left(-1.58/T_5\right)\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - He I:$$\zeta_{\mathrm{HeI}} = 9.38\times10^{-22}T^{1/2}\left(1+T_5^{1/2}\right)^{-1}\exp\left(-2.85/T_5\right)\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - He II ($2^3$S):$$\zeta_{\mathrm{HeI,2^3S}} = 5.01\times10^{-27}T^{-0.1687}\left(1+T_5^{1/2}\right)^{-1}\exp\left(-5.53/T_4\right)n_{\mathrm{e}}n_{\mathrm{HeII}}/n_{\mathrm{HeI}}\quad \mathrm{cm^3\cdot s^{-1}}$$ - He II:$$\zeta_{\mathrm{HeII}} = 4.95\times10^{-22}T^{1/2}\left(1+T_5^{1/2}\right)^{-1}\exp\left(-6.31/T_5\right)\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - Collisional excitation cooling: - H I:$$\psi_{\mathrm{HI}} = 7.5\times10^{-19}\left(1+T_5^{1/2}\right)^{-1}\exp\left(-1.18/T_5\right)\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - He I:$$\psi_{\mathrm{HeI}} = 9.10\times10^{-27}T^{-0.1687} \left(1+T_5^{1/2}\right)^{-1} \exp\left(-1.31/T_4\right) n_{\mathrm{e}}n_{\mathrm{HeII}}/n_{\mathrm{HeI}}\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - He II:$$\psi_{\mathrm{HeII}} = 5.54\times10^{-17} T^{-0.397}\left(1+T_5^{1/2}\right)^{-1} \exp\left(-4.73/T_5\right)\quad \mathrm{cm^3\cdot s^{-1}}$$ - Recombination cooling: - H II:$$\eta_{\mathrm{HII}} = 2.82\times10^{-26}T_3^{0.3}\left(1+3.54T_6\right)^{-1}\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - He II:$$\eta_{\mathrm{HeII}} = 1.55\times10^{-26}T^{0.3647}\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - He III:$$\eta_{\mathrm{HeIII}} = 1.49\times10^{-25}T^{0.3}\left(1+0.855T_6\right)^{-1}\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - Dielectric recombination cooling: - He II:$$\omega_{\mathrm{HeII}} = 1.24\times10^{-13}T_5^{-1.5} \left(1+0.3\exp\left(-9.4/T_4\right)\right)^{-1}\exp\left(-4.7/T_5\right)\quad \mathrm{erg\cdot cm^3\cdot s^{-1}}$$ - Free-free cooling:$$\theta_{\mathrm{ff}} = 1.42\times10^{-27}g_{\mathrm{ff}}T^{1/2}$$ With $g_{ff}$ = 1.1 - Compton cooling:$$\lambda_{\mathrm{c}} = 4k_{\mathrm{B}}\left(T-T_{\gamma}\right)\frac{\pi^2}{15}\left(\frac{k_{\mathrm{B}}T_\gamma}{\hbar c}\right)^3\left(\frac{k_{\mathrm{B}}T_\gamma}{m_{\mathrm{e}}c^2}\right)n_{\mathrm{e}}\sigma_{\mathrm{T}}c$$ Where $T_\gamma$ is the temperature of the cosmic microwave background ($T_\gamma$ = 2.736(1+z) K). Lyman alpha {#app:lya} =========== For the Wouthuysen-Field coupling it is necessary to assume a model of the Lyman alpha emission. We take into account three sources of Lyman alpha: collional excitations, recombinations and high energy photons from the X-ray/UV-background that redshift into the Lyman alpha line and then interact with the IGM. For the latter we adopt the @art:hm12 model.\ The Lyman alpha photon angle-averaged specific intensity (in units of $\mathrm{s}^{-1}\mathrm{cm}^{-2}\mathrm{Hz}^{-1}\mathrm{sr}^{-1}$) is calculated as follows $$J_{\mathrm{Ly\alpha,x}} = \frac{N_{\mathrm{Ly\alpha,x}}(z)D_\mathrm{A}^2}{4\pi D_\mathrm{L}^2}\frac{dr}{d\nu} = \frac{N_{\mathrm{Ly\alpha,x}}(z)}{4\pi}\frac{\lambda_{\mathrm{Ly\alpha,0}}}{H(z)}$$ where the $D_{\mathrm{A}}$ and $D_{\mathrm{L}}$ are the angular and luminosity distances, respectively, and $\lambda_{\mathrm{Ly\alpha,0}} = 1215.76 \mbox{\AA}$ is the rest wavelength of the Lyman alpha transition. $N_{\mathrm{Ly\alpha,x}}$ is the number of Ly$\alpha$ photons that interact with the IGM per unit volume per unit time. For recombinations this number follows directly from the recombination rate $$N_{\mathrm{Ly\alpha,rec}}(z) = f_{\mathrm{Ly\alpha}}\alpha_{\mathrm{HII}}\left(T_{\mathrm{k}},z\right)n_{\mathrm{e}}n_\mathrm{HII}\quad \mathrm{cm^{-3}s^{-1}}$$ where $f_{\mathrm{Ly\alpha}}$ is the fraction of recombinations that result in a Lyman alpha photon. This fraction depends on the temperature. We use the fitting function obtained by @art:cantalupo $$f_{\mathrm{Ly\alpha}}^{\mathrm{HI}} = 0.686 - 0.106\log T_4 - 0.009T_4^{-0.44},\label{eqrecfrac}$$ which is accurate within 0.1$\%$ at temperatures 100 K $<$ T $<$ $10^5$ K. The recombination rate is given by $$\alpha_{\mathrm{HII}} = 3.96\times10^{-13}T_4^{-0.7}\left(1+T_6^{0.7}\right)^{-1}(1+z)^3\quad \mathrm{cm^{3}s^{-1}}. \label{eqrecrate}$$ Equations \[eqrecfrac\] and \[eqrecrate\] are the rates for case B recombination. The case A recombination rate is higher, but the corresponding probability of emitting a Lyman alpha photon per recombination is lower, effectively negating the difference [@art:dijkstra]. Similarly, the density of Lyman alpha photons due to collisions that interact with the IGM follows from the collisional excitation coefficient $Q_{col}$ as $$N_{\mathrm{Ly\alpha,col}}(z) = Q_{\mathrm{col}}\left(T_k,z\right)n_{\mathrm{e}} n_\mathrm{HI}\quad \mathrm{cm^{-3}s^{-1}}$$ The comoving collisional excitation coefficient for transitions from the ground state to level $n$ is given by [@art:cantalupo] $$q_{1,n}^{\mathrm{HI}} = 8.629\times10^{-6}T^{-0.5}\frac{\Omega_n(T)}{\omega_{\mathrm{1}}}e^{E_n/k_{\mathrm{B}}T}(1+z)^3\quad \mathrm{cm^{3}s^{-1}},$$ where $E_n$ is the energy corresponding to the transition, $\omega_{\rm{1}}$ is the statistical weight of the ground state and the function $\Omega_n$ is the effective collision strength, given by [@art:effcolstren] $$\Omega_n(T) = \begin{cases} 3.44\times10^{-1} &+ 1.293\times10^{-5} T\\ &+ 5.124\times10^{-12} T^2\\ &+ 4.473\times10^{-17} T^3,\, n = 1\\ 5.462\times10^{-2} &- 1.099\times10^{-6} T\\ &+ 2.457\times10^{-11} T^2\\ &- 1.528\times10^{-16} T^3,\, n = 2\\ 4.838\times10^{-2} &+ 8.56\times10^{-7} T\\ &- 2.544\times10^{-12} T^2\\ &+ 5.093\times10^{-18} T^3,\, n = 3. \end{cases}$$ The total collisional excitation coefficient is then the sum over all the collisional excitation coefficients, where we only consider transitions up to n = 3 $$Q_{\mathrm{col}}^{\mathrm{HI}} = \sum^{3}_{n=1} q^{\mathrm{HI}}_{1,n}.$$\ Finally, for the background Lyman alpha emission there are two main contributions. At high redshift, quasars are dominant, whereas at low redshift the main contribution of Lyman alpha photons comes from the galaxies. For the quasars, the comoving emissivity at 1 ryd is [@art:hm12] $$\epsilon_{912}(z) = \left(10^{24.6} \mathrm{erg\, s^{-1} Mpc^{-3} Hz^{-1}}\right)\times(1+z)^{4.68}\frac{\exp(-0.28z)}{26.3 + \exp(1.77z)},$$ which is a fit to the @art:hopkins results. This is then integrated over frequency to get the quasar Lyman alpha photon density $$N_{\mathrm{Ly\alpha,qso}} = \int_{\nu(z_{\mathrm{Ly\alpha}})}^{\nu(z_{\mathrm{max}})}\frac{\epsilon_{912}(z(\nu))}{h\nu}\left(\frac{\nu}{\nu_{912}}\right)^{-1.57}\mathrm{d}\nu,$$ where $z_{Ly\alpha}$ is the redshift at which the emission couples to the local IGM as Lyman alpha, $\nu_{912}$ is the frequency corresponding to 912 $\mbox{\AA}$, $\nu(z) = \nu_{Ly\alpha}(1+z)/(1+z_{\mathrm{Ly\alpha}})$ and the exponent -1.57 comes from the quasar UV SED for wavelengths below 1300 $\mbox{\AA}$. The galactic contribution to the background Lyman alpha photons follows from a fit to the star formation rate density (SFRD) by @art:hm12 $$SFRD(z) = \frac{6.9\times10^{-3}+0.14(z/2.2)^{1.5}}{1 + (z/2.7)^{4.1}}\mathrm{M_\odot yr^{-1}\mathrm{Mpc}^{-3}}$$ To convert the observed luminosity densities $\rho_{1500\mbox{\AA}}$ to ongoing star formation rate densities, @art:hm12 adopted a conversion factor $\kappa = 1.05\times10^{-28}$ $$SFRD(t) = \kappa\times\rho_{1500\mbox{\AA}}(t),$$ where $\rho_{1500\mbox{\AA}}$ is in units of erg $\mathrm{s}^{-1}\mathrm{Mpc}^{-3}\mathrm{Hz}^{-1}$. We adopt the same conversion factor to go from the fitted star formation rate density, back to a luminosity density. The galactic Lyman alpha photon density is then the integral over the SFRD $$N_{\mathrm{Ly\alpha,gal}} = \int_{\nu(z_{\rm Ly\alpha})}^{\nu(z_{\rm{max}})}\frac{SFRD(z(\nu))/\kappa}{h\nu}\frac{B_\nu(T_\mathrm{gal})}{B_{\mathrm{1500}}(T_\mathrm{gal})}\mathrm{d}\nu$$ where $B_\nu$ is the Planck function at frequency $\nu$ and $B_\mathrm{1500}$ is the same at 1500 $\mbox{\AA}$. The Planck functions depend on the temperature of the IGM in the galaxies, for which we adopt $T_{\mathrm{gal}}$ = 6000 K.\ One more effect that needs to be taken into account is that photons can scatter multiple times before they redshift out of the Lyman alpha line. Each additional scattering adds to the Wouthuysen-Field coupling and so the Wouthuysen-Field coupling factor $x_{\rm c}$ needs to be multiplied by the number of scatterings[@art:field59]. This is given by the Gunn-Peterson optical depth [@art:GP] $$\tau_{\mathrm{GP}} = \frac{n_\mathrm{x}\lambda_\mathrm{Ly\alpha}}{H(z)}\frac{f_\alpha\pi e^2}{m_\mathrm{e}c}$$ with $n_{\rm{x}}$ either the HI or $\rm{^3HeII}$ density. Because Lyman alpha photons created through recombinations originate close to the line center, their Lyman alpha specific intensity gets an additional factor of 1.5 [@art:field59].\ The evolution of the Lyman alpha intensity is shown in Figure \[fig:jlya\]. At low redshift, for the higher densities recombination and collisions are the dominant source of Lyman alpha, whereas for the lower densities it is the background emission. ![Lyman alpha photon angle-averaged specific intensity for hydrogen for the different mechanisms. The linewidth denotes the density.[]{data-label="fig:jlya"}](./images/Jlya_3.pdf){width="52.00000%"} [^1]: http://www.astron.nl/radio-observatory/apertif-surveys [^2]: http://askap.org/dingo

--- abstract: | Abundances of Fe, Si, Ni, Ti, Na, Mg, Cu, Zn, Mn, Cr and Ca in the atmosphere of the K-dwarf HD 77338 are determined and discussed. HD 77338 hosts a hot Uranus-like planet and is currently the most metal-rich single star to host any planet. Determination of abundances was carried out in the framework of a self-consistent approach developed by Pavlenko et al. (2012). Abundances were computed iteratively by the program ABEL8, and the process converged after 4 iterations. We find that most elements follow the iron abundance, however some of the iron peak elements are found to be over-abundant in this star.\ [**Key words:**]{} stars: abundances, stars: atmospheres, stars: individual (HD 77338), line: profiles. author: - '*I.O.Kushniruk$^{1}$, Ya.V.Pavlenko$^{2,3}$, J.S.Jenkins$^{4,2}$, H.R.A.Jones$^2$*' title: | Abundances in the atmosphere of the metal-rich planet-host star\ HD 77338 --- [$^1$Taras Shevchenko National University of Kyiv, Glushkova ave., 4, 03127, Kyiv, Ukraine\ $^2$Main Astronomical Observatory of the NAS of Ukraine, 27 Akademika Zabolotnoho St., 03680 Kyiv, Ukraine\ $^3$Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield, Hertfordshire AL10 9AB, UK\ $^4$Departamento de Astronomía, Universidad de Chile, Camino el Observatorio 1515, Las Condes, Santiago, Chile\ [nondanone@gmail.com]{}]{} Introduction ============ Determining the chemical composition of stars is one of the primary goals of astrophysics. Such investigations help us to better understand the chemical enrichment of the Galaxy and to make some assumptions about the mechanisms involved in element evolution in the interstellar medium, and in stellar atmospheres in particular [@cu]. While studying the Sun, the problem of the abundances of certain atoms necessitated a model to explain this. It was finally explained with the introduction of the pp- and CNO- cycles in the interior of the Sun. But this was not enough to explain the presence of large amounts of helium. The next step in studying the evolution of elements was the introduction of nucleosynthesis theory. Modern scientific understanding is that chemical elements were formed as a result of the processes occurring in stars, leading to evolutionary changes of their physical conditions. Therefore, the problem of nuclide formation is also closely related to the issue of the evolution of stars and planetary system. Recently Jenkins et al. [@uranus] announced the discovery of a low-mass planet orbiting the super HD77338 as part of our ongoing Calan-Hertfordshire Extrasolar Planet Search [@jenkins2013a]. The best-fit planet solution has an orbital period of 5.7361 $\pm$ 0.0015 days and with a radial velocity semi-amplitude of only 5.96$\pm$1.74 [ms$^{-1}$]{}, giving a minimum mass of $^{+4.7}_{-5.3}$ [M$_{\rm{\oplus}}$]{}. The best-fit eccentricity from this solution is 0.09$^{+0.25}_{-0.09}$, and is in the agreement with results of a Bayesian analysis and a periodogram analysis. According to modern theory, the formation of the nucleus of chemical elements from carbon to iron is the result of thermonuclear reactions involving He, C, O, Ne and Si in stars. After the depletion of hydrogen reserves, a star’s core starts running a 3$\alpha $ reaction, where it produces a number of elements as a result of the following transformations: $\mathrm{3\,{}^4He\,\longrightarrow\,{}^{12}C}$, $\mathrm{{}^{12}C \,+\, {}^4He \,\longrightarrow\, {}^{16}O \,+\, \gamma}$, $\mathrm{{}^{16}O \,+\, {}^4He \,\longrightarrow\, {}^{20}Ne \,+\, \gamma}$.\ After reaching a specific threshold temperature, carbon begins fusing with the formation of Ne, Na and Mg: $\mathrm{{}^{12}C \,+\, {}^{12}C \,\longrightarrow\, {}^{20}Ne \,+\, {}^{4}He \,+\, 4,62 MeV }$,\ $\mathrm{{}^{12}C \,+\, {}^{12}C \,\longrightarrow\, {}^{23}Na \,+\, p \,+\, 2,24 MeV }$,\ $\mathrm{{}^{12}C \,+\, {}^{12}C \,\longrightarrow\, {}^{24}Mg \,+\,\gamma \,-\, 2,60 MeV }$.\ Aluminum can then be produced by: $\mathrm{{}^{24}Mg \,+\, p \,\longrightarrow\, {}^{25}Al \,+\, \gamma}$.\ The combustion reaction of oxygen is a dual-channel process and causes the presence of Al, S, P, Si and Mg. One of the channels is:\ $\mathrm{{}^{16}O \,+\, {}^{16}O \,\longrightarrow\, {}^{30}Si \,+\,{}^{1}H \,+\, {}^{1}H \, +\, 0,39 MeV }$,\ $\mathrm{{}^{16}O \,+\, {}^{16}O \,\longrightarrow\, {}^{24}Mg \,+\, {}^{4}He \,+\,{}^{4}He \,- \, 0,39 MeV }$,\ $\mathrm{{}^{16}O \,+\, {}^{16}O \,\longrightarrow\, {}^{27}Al \,+\, {}^{4}He \,+\, {}^{1}H \, -\, 1,99 MeV }$,\ With continuous temperature growth, silicon burning is initiated. This process is described by a number of reactions. As a result we can receive, for example, Ar, Ni, S, etc. $\mathrm{{}^{56}Ni}$, after two $ \beta $ decays, turns into $\mathrm{{}^{56}Fe}$. It is the final stage of the fusion of nuclides in massive stars, which forms the nucleus of the iron group.\ The production of heavy elements is provided by other mechanisms. They are called s- and r- processes. - s-process or slow neutron capture: formation of heavier nuclei by lighter nuclei through successive neutron capture. The original element in the s-process is $\mathrm{{}^{56}Fe}$. The reaction chain ends with $\mathrm{{}^{209}Bi}$. It is thought that s-processes occur mostly in stars on the asymptotic giant branch. For the s-process to run, an important condition is the ability to produce neutrons. The main neutron source reactions are: $\mathrm{{}^{13}C \,+\, {}^{4}He \,\longrightarrow\, {}^{16}O \,+\, n }$, $\mathrm{{}^{22}Ne \,+\, {}^{4}He \,\longrightarrow\, {}^{25}Mg \,+\, n }$. Example of s-process reactions are: $$^{56}\textrm{Fe} + n \longrightarrow {}^{57}\textrm{Fe} + n \longrightarrow {}^{58}\textrm{Fe} + n \longrightarrow {}^{59}\textrm{Fe} \stackrel{\beta^-}{\longrightarrow} {}^{59}\textrm{Co} + n \longrightarrow {}^{60}\textrm{Co} \stackrel{\beta^-}{\longrightarrow} {}^{60}\textrm{Ni} + n {\longrightarrow}~~\textrm{...}$$ Elements heavier than H and He are usually called metals in astrophysics. Their concentration is significantly less, relative to hydrogen and helium, but they are the source of thousands of spectral lines originating from a star’s atmosphere. The abundance of iron depends on a stars age and on its position in the galaxy [@book]. Metal-rich stars are also known to be rich in orbiting giant exoplanets. High metallicity appears to be a major ingredient in the formation of planets through core accretion [@uranus].\ HD 77338 is one of the most metal-rich stars in the sample of [@2008] and in the local Solar neighborhood in general. Its spectral type is given as K0IV in the Hipparcos Main Catalogue [@per]. However, is not a subgiant, as labeled in Hipparcos [@uranus], its mass and radius are smaller than the Sun’s: M = 0.93 $ \pm $ 0.05 M$_\odot$, R = 0.88 $ \pm $ 0.04 R$_\odot$. A parallax of 24.54 $ \pm $ 1.06 mas for HD 77338 means the star is located at a distance of 40.75 $\pm$ 1.76 pc. Its effective temperature and surface gravity were found $T_{eff}$ = 5370 $ \pm $ 80 K, log$ g$ = 4.52 $ \pm $ 0.06 [@uranus]. More stellar parameters for HD 77338 and detailed information about its planetary system are in [@uranus]. Using the Simbad database one can find information on the previous assessments of abundances in the atmosphere of HD 77338 (see Table \[tab1\]). In most cases the authors only determine the metallicity of the star, i.e. the iron abundance. In turn, we recomputed the abundances of many elements which show significant absorption lines in the observed spectrum of . Teff $log g$ \[Fe/H\] CompStar Reference ------ --------- ---------- ---------- ----------- -- 5300 4.30 0.36 Sun [@prug] 5290 4.90 0.22 Sun [@felt] 5290 4.60 0.30 Sun [@th] : Simbad’s list of previous assessments of abundances in the atmosphere of HD 77338 []{data-label="tab1"} The observations ================ The observations of were carried out as part of the Calan-Hertfordshire Extrasolar Planet Search (CHEPS) program [@2009]. The main aim of the program is monitoring a sample of metal-rich stars in the southern hemisphere to search for short period planets that have a high probability to transit their host stars, along with improving the existing statistics for planets orbiting solar-type and metal-rich stars. The high-S/N ($ > 50$) and high-resolution ($R = 100\, 000$) spectrum of , observed with the HARPS spectrograph [@mayor], was reduced using the standard automated HARPS pipeline and analyzed in this work in order to determine the chemical abundances and other physical parameters of the stellar atmosphere. The procedure ============= Firstly, we selected “good” absorption lines for all elements of interest that are present in spectra of the Sun and HD 77338. These lines should be not be blended (see [@2008]) and be intense enough in both spectra. We selected lists of lines of each element that were to be used for the abundance investigations. We used line list data, which was taken from database of atomic absorption spectra VALD [@_kupka99], to compute synthetic spectra of the Sun for a plane-parallel model atmosphere with parameters //\[Fe/H\] = 5777/4.44/0.0 [@SAM12]. The model atmosphere was used to compute the synthetic spectra using WITA6 [@97], building a grid of models with different microturbulent velocities = 0 – 3 km/s with a step size of 0.25 km/s. The shape of the line absorption profiles were constructed as Voigt function profiles $H(a,v)$, and a classical approach was used to compute the damping effects [@_unsold56]. To compute the rotational profile we followed the procedure described in Gray [@_gray76]. All abundance determinations were performed by the ABEL8 program [@abel8]. Details of the full procedure we used is described in [@mnras], see [@2008] also for more details on the line selection and fitting procedure. Results ======= The Sun ------- The solar spectrum is well-studied and abundances for the Sun are known to very high accuracy, therefore it represents a very good template. Fig. \[fig1\] and Fig. \[fig2\] illustrate the presence of spectral lines of Cr in the observed spectrum of the Sun as a star [@_kurucz84]. Arrows on the plot show the spectral range which was selected to compute profiles of two Cr I lines to be used later by ABEL8 [@abel8] in the determination of the abundance of chromium. We employ a similar selection in the solar spectrum Sun and the spectrum of HD 77338 for lines of Fe I, Si I, Ni I, Ti I, Na I, Mg I, Cu I, Zn I, Mn I, Cr I, Al I, and Ca I. We verified whether our input data are good enough to reproduce the abundances in the atmosphere of the Sun. We computed abundances for the Sun using the fits of the theoretical spectra to the profiles of the selected lines. In that way we can test our method and estimate the accuracy of our abundance determination. Then we investigated the dependence $E_a = \partial a/\partial E''$, where $a$ and $E''$ are the iron abundance and excitation potential of the correspondent radiative transition forming the absorption line. Best fits of the selected lines of Fe I in the computed spectra, when compared to their observed profiles in the solar spectrum, provides the min $E_a$ of =0.75 km/s. The abundances of iron and other elements were then obtained using this adopted value for the microturbulence, the results are shown in Table \[tab2\]. It is worth noting that our abundances agree with the reference values within an accuracy of $\pm$0.1 dex. HD 77338 -------- The model atmosphere for was computed using the parameters determined by Jenkins et al. [@uranus] using the SAM12 program [@SAM12]. Again, as the first step of our analysis we determined the microturbulent velocity in the atmosphere of . The minimum of the slope of $E_a$ provides = 0.75 km/s for $log$ N(Fe) = $-$ 4.120 $ \pm $ 0.07 or \[Fe/H\] = 0.281 (iteration 1). For other elements we used the same value ($V_{t}$ = 0.75 km/s). We carried out 4 iterations to determine all abundances. In each next step the abundances from the previous determination were used to recompute the model atmosphere by SAM12 [@SAM12] and the synthetic spectra. Each time we are approaching self-consistency by computing the model atmosphere that relates to the final metallicity of the star. Iron $log$ N(X) $log$ N(X)$_\odot$ ABEL8 $log$ N(X)$_\odot$ \[X/H\] \[X/Fe\] $v \sin i$ ($ km/s $) N$_l$ ------ ----------------------- -------------------------- -------------------- ----------- ----------- ----------------------- ------- Al I $-$ 5.403 $\pm$ 0.000 $-$ 5.767 $\pm$ 0.033 $-$ 5.551 $+$ 0.148 $-$ 0.095 2.33 $\pm$ 0.44 3 Ca I $-$ 5.376 $\pm$ 0.029 $-$ 5.588 $\pm$ 0.023 $-$ 5.661 $+$ 0.285 $+$ 0.042 1.25 $\pm$ 0.13 14 Cr I $-$ 6.085 $\pm$ 0.032 $-$ 6.345 $\pm$ 0.026 $-$ 6.441 $+$ 0.356 $+$ 0.113 1.91 $\pm$ 0.14 22 Cu I $-$ 7.670 $\pm$ 0.058 $-$ 8.133 $\pm$ 0.067 $-$ 7.941 $+$ 0.271 $-$ 0.028 2.17 $\pm$ 0.44 3 Fe I $-$ 4.158 $\pm$ 0.038 $-$ 4.439 $\pm$ 0.023 $-$ 4.401 $+$ 0.243 $+$ 0.000 2.06 $\pm$ 0.11 27 Mg I $-$ 4.228 $\pm$ 0.058 $-$ 4.367 $\pm$ 0.095 $-$ 4.441 $+$ 0.213 $-$ 0.030 1.50 $\pm$ 0.50 3 Mn I $-$ 5.957 $\pm$ 0.097 $-$ 6.600 $\pm$ 0.046 $-$ 6.641 $+$ 0.684 $+$ 0.441 2.69 $\pm$ 0.21 8 Na I $-$ 5.387 $\pm$ 0.048 $-$ 5.789 $\pm$ 0.054 $-$ 5.721 $+$ 0.334 $+$ 0.091 1.94 $\pm$ 0.31 9 Ni I $-$ 5.367 $\pm$ 0.033 $-$ 5.756 $\pm$ 0.027 $-$ 5.821 $+$ 0.454 $+$ 0.211 2.29 $\pm$ 0.15 17 Si I $-$ 4.111 $\pm$ 0.054 $-$ 4.469 $\pm$ 0.058 $-$ 4.401 $+$ 0.290 $+$ 0.047 2.39 $\pm$ 0.13 23 Ti I $-$ 6.897 $\pm$ 0.040 $-$ 7.064 $\pm$ 0.028 $-$ 6.981 $+$ 0.084 $-$ 0.159 1.88 $\pm$ 0.13 24 Zn I $-$ 7.028 $\pm$ 0.065 $-$ 7.375 $\pm$ 0.048 $-$ 7.441 $+$ 0.413 $+$ 0.170 2.25 $\pm$ 0.25 4 : Abundances in the atmosphere of HD 77338, iteration 4[]{data-label="tab2"} In Table \[tab2\] we present our results for 12 different ionic species. We compare our abundances with the solar values, obtained using a model atmosphere of 5777/4.44/0.00. They are in good agreement with each other. Fig. \[fig2\] and Fig. \[fig3\] show the line profiles of Cr and Mg calculated using a $V_{t}$ = 0.75 km/s. In Fig. \[fig4\] we show the dependence of \[X/H\] on atomic number of each element for every iteration. The presence of errors can be explained by the presence of noise in the selected spectral lines, along with only having a small number of lines to work with for some elements. In Fig. \[fig5\] we present the dependence of \[X/Fe\] for the final iteration, where different elements are shown using different plotting shapes depending on their mechanism of formation. Discussion ========== We determined abundances for 12 ionic species in the atmosphere of the Sun and the metal-rich exoplanet host star HD 77338. Our values for the solar abundances are in good agreement with results from previous authors, proving the validity of our method. We used the solar spectrum as a reference to select the proper list of absorption lines to be used later in the analysis of the spectrum. Our \[X/H\] correlates well with the condensation temperature of the ions ($T_{cond}$), see discussion in [@_melendez]. This may indicate the presence of a common shell (in the past) and can be an additional criterion for the existence of a planetary system around metal-rich stars. We also computed $ v \sin i $ for both stars. It is worth noting that we determined all parameters in the framework of a fully self-consistent approach (see [@SAM12] for more details). In general, lines of Mn, Cu can not be used to obtain $ v \sin i $ because these lines usually have several close components, but in our case the parameter $v \sin i$ was used to adjust theoretical profiles to get the proper fits to the observed special features. We believe that fits to Fe I lines provide reasonable measures of the rotational velocity. Our results show that the abundances of most elements in the atmosphere can be described well by the overall metallicity. However, we found an overabundance of some of the iron peak elements (e.g. Mn, Cu). Interestingly, Cu is an element formed through the s-process and its abundance follows that of Fe, whereas Zn and elements formed through the p-process, e.g. Ni (see http://www.mao.kiev.ua/staff/yp/TXT/prs.png), show a noticeable overabundance compared to iron. It would be interesting to compare these results for with other metal rich stars to see if this is a common trend for super metal-rich stars. We plan to investigate this issue in a following paper (Ivanyuk et al. 2013, in preparation). acknowledgement {#acknowledgement .unnumbered} =============== JSJ acknowledges the support of the Basal-CATA grant. YP’s work has been supported by an FP7 POSTAGBinGALAXIES grant (No. 269193; International Research Staff Exchange Scheme). Authors thank the compilers of the international databases used in our study: SIMBAD (France, Strasbourg), VALD (Austria, Vienna), and the authors of the atlas of the spectrum of the Sun as a star. We thank anonymous Referee for some reasonable remarks and helpful comments. [3]{} Feltzing S., Gustafsson B., 1998, A&AS, 129, 237-266. Gray D.F. The Onservation and Analysis of Stellar Photospheres, A. Wiley-Interscience Publ., 1976, 1. Jenkins J.S., Jones H.R.A., Pavlenko Y., Pinfield  D.J., Barnes J.R., Lyubchik Y., 2008, A&A, 485, 571-584. Jenkins J.S., Jones H.R.A., Gozdziewski K., Migaszewski C., Barnes J.R., Jones M.I., Rojo P., Pinfield D.J., Day-Jones A.C., Hoyer S., 2009, MNRAS, 398, 2, 911-917. Jenkins J.S., Jones H.R.A., Tuomi M., Murgas F., Hoyer S. Jones M.I., Barnes J.R., Pavlenko Y.V., Ivanyuk O., Rogo P., Jordan A. 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**Nonlinear branching processes with immigration** Pei-Sen Li School of Mathematical Sciences, Beijing Normal University, Beijing 100875 E-mail: peisenli@mail.bnu.edu.cn The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established. *Key words and phrases.* Nonlinear branching process, immigration, stochastic integral equation, regularity, recurrence, ergodicity, strong ergodicity. Introduction ============ Markov branching processes are models for the evolution of populations of particles. Those processes constitute one of the most important subclasses of continuous-time Markov chains. Standard references on those processes are [@Harris] and [@Athreya72]. The basic property of an ordinary linear branching process is that different particles act independently when giving birth or death. In most realistic situations, however, this property is unlikely to be appropriate. In particular, when the number of particles becomes large or the particles move with high speed, the particles may interact and, as a result, the birth and death rates can either increase or decrease. Those considerations have motivated the study of nonlinear branching processes. On the other hand, a branching process describes a population evolving randomly in an isolated environment. A useful and realistic modification of the model is the addition of new particles from outside sources. This consideration has provided the stimulation for the study of branching models with immigration and/or resurrection. Let $\{r_i: i\ge 0\}$ be a sequence of nonnegative constants with $r_0=0$ and $\{b_i: i\ge 0\}$ a discrete probability distribution [on $\mbb{N}:=\{0,1,\ldots\}$ ]{}with $b_1=0$. A continuous-time Markov chain is called a nonlinear branching process if it has density matrix $R=(r_{ij})$ given by \[1.4\] r\_[ij]{}={ [lcl]{} r\_i b\_[j-i+1]{} & & [ji+1, i1,]{}-r\_i & & [j=i1,]{}r\_i b\_0 & & [j=i-1, i1,]{}0 & & . A typical special case is where $r_i=\alpha i^\theta$ for $\alpha\ge 0$ and $\theta>0$, which reduces to the ordinary linear branching process when $r_i=\alpha i$. Let $\gamma\ge 0$ and let $\{a_i: i\ge 0\}$ be another discrete probability distribution on $\mbb{N}$ satisfying $a_0=0$. A continuous-time Markov chain is called a nonlinear branching process with resurrection if its density matrix is given by \[1.1\] \_[ij]{}={ [lcl]{} r\_i b\_[j-i+1]{} & & [ji+1, i1,]{}-r\_i & & [j=i1,]{}r\_i b\_0 & & [j=i-1, i1,]{}a\_j & & [j&gt; i=0,]{}-& & [j = i=0,]{}0 & & . Here the resurrection means that at each time when the process gets extinct, some immigrants come into the population at rate $\gamma$ according to the distribution $\{a_i\}$. By a nonlinear branching process with immigration we mean a Markov chain with density matrix $Q=(q_{ij})$ given by \[1.2\] q\_[ij]{}={ [lcl]{} r\_i b\_[j-i+1]{}+a\_[j-i]{} & & [ji+1, i0,]{}-r\_i-& & [j=i0,]{}r\_i b\_0 & & [j=i-1, i1,]{}0 & & . In this model, the immigrants come at rate $\gamma$ according to the distribution $\{a_i\}$ independently of the inner population. The purpose of this paper is to investigate the construction and basic properties of the nonlinear branching process with immigration defined by (\[1.2\]). Let m = \_[j=0]{}\^ja\_j, M = \_[j=0]{}\^jb\_j, which represent the birth mean and immigration mean of the process, respectively. Moreover, we introduce the functions F(s)=\_[i=0]{}\^a\_is\^i,   A(s)=(1-F(s)),   G(s)=\_[i=0]{}\^b\_is\^i,  B(s)= G(s)-s, s. Let $q$ be the smaller root of the equation $G(s)=s$ in $[0,1]$. We sometimes denote $r_i$ by $r(i)$ for notational convenience. Suppose that $(\Omega,\mcr{F},\mcr{F}_t,P)$ is a probability space satisfying the usual hypotheses. [ Denote $m(i)=b_i$ and $n(i)=a_i$ for each $i\in \mbb{N}$.]{} Let $\{p(t)\}$ and $\{q(t)\}$ be $(\mcr{F}_t)$-Poisson point processes with characteristic measures $dum(dz)$ and $\gamma n(dz)$, respectively. We assume $\{p(t)\}$ and $\{q(t)\}$ are independent of each other. Let $N_p(ds,du,dz)$ and $N_q(ds,dz)$ be the Poisson random measures associated with $\{p(t)\}$ and $\{q(t)\}$, respectively. Given an $\mbb{N}$-valued $\mcr{F}_0$-measurable random variable $X_0,$ let us consider the stochastic integral equation \[e0.1\] X\_t= X\_0+\_0\^t \_0\^[r([X\_[s-]{}]{})]{}\_(z-1) N\_p(ds,du,dz)+ \_0\^t \_ z N\_q(ds,dz). Let $\zeta=\lim_{k\rightarrow\infty} \tau_k,$ where $\tau_k=\inf\{t\geq0: X_t\geq k\}.$ The above equation only makes sense for $0\leq t< \zeta.$ We call $\zeta$ the explosion time of $\{X_t\}$ and make the convention $X_t=\infty$ for $t\geq \zeta.$ We say the solution is non-explosive if $\zeta= \infty.$ As a special case of (\[e0.1\]) we also consider the equation \[0.2\] X\_t= X\_0+\_0\^t \_0\^[r([X\_[s-]{}]{})]{}\_(z-1) N\_p(ds,du,dz). We now state the main results of the paper. \[t0.2a\] There exists a pathwise unique solution to [(\[e0.1\])]{}. Moreover, if the solution to [(\[0.2\])]{} is non-explosive, then so is the solution to [(\[e0.1\])]{}. \[t0.4\] Let $\{X_t\}$ be the solution to [(\[e0.1\])]{} and let $Q_{ij}(t) = P(X_t=j|X_0=i).$ Then $Q_{ij}(t)$ solves the Kolmogorov forward equation of $Q$. \[t0.2aa\] The solution to [(\[e0.1\])]{} is the minimal process of $Q$ and the solution to [(\[0.2\])]{} is the minimal process of $R$. \[t0.5\] The density matrix $R$ is regular if and only if $Q$ is regular. \[t1.4\] [(1)]{} If $M\leq 1,$ then $Q$ is regular. [(2)]{} Suppose that $\sum^\infty_{i=1}r_i^{-1}<\infty$. Then $Q$ is regular if and only if $M\leq 1.$ [(3)]{} Suppose that $1<M\leq \infty$ and $r_i=\alpha i^{\theta}$ for $\alpha>0$ and $\theta>0$. Then $Q$ is regular if and only if for some $\varepsilon\in (q,1)$, we have $$\int^1_\varepsilon\frac{1}{B(s)}\bigg(\ln \frac{1}{s}\bigg)^{\theta-1}ds=-\infty.$$ In the following three theorems, we assume $\gamma r_i b_0>0$ for every $i\ge 1$, so the matrix $Q$ is irreducible. \[t3.1\] [(1)]{} Suppose that $m< \infty$, $M< 1$ and $\lim_{i\rightarrow\infty}r_i=\infty$. Then the nonlinear branching process with immigration is recurrence. [(2)]{} Suppose that $r_i$ is increasing and there exist constants $\alpha>0$ and $N>0$ such that $r_i/i\geq \alpha$ holds for each $i> N$. Then the nonlinear branching process with immigration is recurrent if $M\leq 1$ and J := \_0\^1 dy=. [(3)]{} Suppose that $M>1$. Then the nonlinear branching process with immigration is transient. [(4)]{} Suppose that $r_i$ is increasing and there exist constants $\alpha>0$ and $N>0$ such that $r_i/i\leq \alpha$ holds for each $i> N$. Then the nonlinear branching process with immigration is transient if $M\leq 1$ and J := \_0\^1 dy&lt;. \[t4.1\] [(1)]{} If $m< \infty$, $M\leq 1$, $r_i$ is increasing and $\sum_{i=1}^\infty r_i^{-1}< \infty$, then the nonlinear branching process with immigration is ergodic. [(2)]{} Suppose that $r_i=\alpha i^\theta$ for $\alpha>0$ and $\theta\ge 1$. Then the recurrent nonlinear branching process with immigration is ergodic if and only if \[4.8\] \_0\^1 ()\^[-1]{} ds &lt; . If $m< \infty, M<1$ and $\lim\inf_{i\rightarrow\infty} r_i/i>0$, then the nonlinear branching process with immigration is exponentially ergodic. \[t5.1\] [(1)]{} If $m< \infty$, $M< 1$, $r_i$ is increasing and $\sum_{i=1}^\infty r_i^{-1}< \infty$, then the process is strongly ergodic. [(2)]{} Suppose that $r_i=\alpha i^\theta$ for $\alpha>0$ and $\theta> 1$. Then the nonlinear branching process with immigration is strongly ergodic if and only if \[5.1\] \^1\_0 ()\^[-1]{}ds &lt; . [(3)]{} If $\sum_{i=1}^\infty r_i^{-1}= \infty$, then the nonlinear branching process with immigration is not strongly ergodic. The nonlinear branching process with resurrection defined above was introduced by [@ChenRR97], who studied the problems of uniqueness, recurrence and ergodicity of the process. The model has attracted the attention of a number of authors. In particular, [@Zhang01] gave criteria for strong ergodicity of the process. [@Chen05] and [@Pakes] established some criteria for their regularity and uniqueness. [@Chen02] studied some interesting differential-integral equations associated with a special class of nonlinear branching processes and gave some characterizations of their mean extinction times. [@Chen06] established a Harris regularity criterion for such processes. The existence and uniqueness of linear branching processes with instantaneous resurrection were studied in [@Chen90]. However, most of the study of models with immigration have been focused on linear branching structures. The branching process with immigration was studied in [@Karlin], who gave a characterization of the one-dimensional marginal distributions of the process starting from zero. An ergodicity criterion for the process was given in [@Yang]. [@LJP06] established some recurrence criteria for linear branching processes with immigration and resurrection. The first three theorems above give constructions of nonlinear branching processes with and without immigration. These provide convenient formulations of the processes. In particular, the result of Theorem \[t0.5\] is derived as an immediate consequence of (\[e0.1\]) and (\[0.2\]). We hope the equations can also be useful in some other similar situations. The proof of Theorem \[t1.4\] is based on Theorem \[t0.5\] and the results of [@ChenRR97] and [@Chen06]. The study of recurrence of the immigration model is more delicate since the problem cannot be reduced to the extinction problem of the original nonlinear branching process as in the case of a resurrection model. Theorem \[t3.1\] was proved by using the results of the minimal nonnegative solutions as developed in [@ChenMF04] and comparing the process with some linear branching processes which was studied by [@LJP06]. The proofs of the ergodicities in Theorems \[t4.1\] and \[t5.1\] are based on comparisons of the process with some suitably designed birth-death process and estimates of the mean extinction time. Stochastic integral equations ============================= Stochastic integral equations with jumps have been playing increasingly important roles in the study of Markov processes. In this section, we give a construction of the solution to (\[e0.1\]) and prove the solution is a minimal nonlinear branching process with immigration. This result is then used to study the regularity of the density matrix $Q$. We refer to [@Ikeda89] for the general theory of stochastic equations with jumps. \[t0.0\] The pathwise uniqueness of solutions holds for the equation [(\[e0.1\])]{}. Let $\{X_t\}$ and $\{X'_t\}$ be any two solutions of equation (\[e0.1\]) with $X_0=X'_0$. By passing to the conditional probability $P(\cdot|\mcr{F}_0),$ we may and do assume $X_0= X'_0$ is deterministic. Let $\tau_m=\inf\Big\{t\geq 0: X_t \geq m\Big\},$ $\tau'_m=\inf\Big\{t\geq 0: X'_t \geq m\Big\}$ and $\sigma_m=\tau_m\wedge \tau'_m.$ It is sufficient to show that $\tau_m=\tau'_m=\sigma_m$ and $X_t=X'_t$ for all $t\leq \sigma_m~(m=1,2,\ldots).$ Then X\_[t\_m]{} - X’\_[t\_m]{} = \^[t\_m]{}\_0\^\_0\_[\_[m+1]{}]{} (z-1)\[1\_[{0 &lt; ur(X\_[s-]{})}]{} - 1\_[{0&lt;ur(X’([s-]{}))}]{}\]N\_p(ds,du,dz), where $\mbb{N}_m=\{0,1,2,\ldots,m\}.$ Taking the expectation, we get E\[|X\_[t\_m]{} - X’\_[t\_m]{}|\] E{\^[t\_m]{}\_0\^\_0\_[\_[m+1]{}]{} |(z-1)\[1\_[{0 &lt; ur(X\_[s-]{})}]{} - 1\_[{0 &lt; ur(X’\_[s-]{})}]{}\]|N\_p(ds,du,dz)} E{\^[t\_m]{}\_0\^\_0\_[\_[m+1]{}]{} (z+1)|1\_[{0 &lt; ur(X\_[s-]{})}]{} - 1\_[{0 &lt; ur(X’\_[s-]{})}]{}|dsdum(dz)} (M\_[m+1]{}+1)E{\^[t\_m]{}\_0 |r(X\_[s-]{})-r(X’\_[s-]{})|ds} (M\_[m+1]{}+1)\^[t]{}\_0 E\[|r(X\_[s\_m-]{})-r(X’\_[s\_m-]{})|\]ds, where $M_m:=\int_{\mbb{N}_m}z m(dz).$ By taking $m\geq X_0,$ we have $X_{s-}\vee X'_{s-} \leq m$ for $0<s\leq \sigma_m.$ Denote $d_m=\sup\{|(r(i)-r(j))/(i-j)|:i\neq j,\quad 0\leq i,j\leq m\}.$ Then we have \[0.3\] E\[|X\_[t\_m]{} - X’\_[t\_m]{}|\] (M\_[m+1]{}+1)d\_m\^[t]{}\_0 E\[|X\_[s\_m-]{}-X’\_[s\_m-]{}|\]ds. Since $X_{s\wedge\sigma_m}$ and $X'_{s\wedge\sigma_m}$ only have countably many discontinuous points, we can also use $X_{s\wedge\sigma_m}$ and $X'_{s\wedge\sigma_m}$ instead of $X_{s\wedge\sigma_m-}$ and $X'_{s\wedge\sigma_m-}$ in the right hand side of (\[0.3\]). Using Gronwall’s inequality we have $E[|X_{s\wedge\sigma_m}-X'_{s\wedge\sigma_m}|]=0$. Thus we can conclude that $X_t=X'_t$ for all $t\in [0,\sigma_m)$ a.s. This clearly implies that $\tau_m=\tau'_m= \sigma_m$ a.s. and the pathwise uniqueness of solutions of (\[e0.1\]) is proven. \[t0.2\] For any $\mbb{N}$-valued $\mcr{F}_0$-measurable random variable $X_0,$ there is a pathwise unique solution to [(\[0.2\])]{}. Without loss of generality, we assume $X_0$ is deterministic. Let $D_1=\{s: p(s)\in (0,r(X_0)]\times \mathbb{N}\}.$ Since $$E[N_p((0,t]\times(0,r(X_0)]\times\mbb{N})]= \int^t_0ds\int^{r(X_0)}_0du\int_{\mathbb{N}}m(dz)= tr(X_0)<\infty,$$ the set $D_1$ is discrete in $(0,\infty)$. Let $\sigma_1$ be the minimal element in $D_1$ and $p(\sigma_1)=(u_1,z_1)$. Then set X\_t= { [lcl]{} X\_0, & & t\[0,\_1)X\_0+ (z\_1-1), & & t=\_1. . The process $\{X_t: 0< t\leq \sigma_1\}$ is clearly the solution of (\[0.2\]). Set $D_2=\{s: p(s+\sigma_1)\in [0,r(X(\sigma_1))]\times \mathbb{N}\},$ $\sigma_2$ be the minimal element in $D_2$ and $p(\sigma_1+\sigma_2)=(u_2,z_2)$. Define $\{X_t: \sigma_1< t\leq \sigma_1+\sigma_2\}$ by X\_t= { [lcl]{} x(\_1), & & t(\_1,\_1+\_2)x(\_1)+ (z\_2-1), & & t=\_1+\_2. . It is easy to see that $\{X_t: 0< t\leq \sigma_1+\sigma_2\}$ is the unique solution of (\[0.2\]). Continuing this process successively, we get a process $\{X_t: 0\leq t< \tau\}$, where $\tau=\sum^\infty_{i=1}\sigma_i.$ Next, we show $\tau=\zeta:=\lim_{k\rightarrow\infty}\tau_k,$ where $\tau_k=\inf\{t\geq 0: X_t\geq k\}.$ Clearly, for each $n\geq 0$ we have $X_t< \infty$ for $t\in [0, \sum^n_{i=0} \sigma_i].$ Then $\sum^n_{i=0} \sigma_i< \zeta$ holds for each $n\geq 0,$ and so $\tau\leq \zeta.$ On the other hand, since $$E\bigg[\int^{t\wedge\tau_m}_0\int^{r(X_{s-})}_0\int_{\mbb{N}}N_p(ds, du, dz)\bigg]\leq t\max_{0\leq k\leq m}r(k)<\infty,$$ the process $\{X_t\}$ has finitely many jumps before $t\wedge\tau_m$, therefore $t\wedge\tau_m< \tau,$ since $t\geq 0$ and $m\geq 1$ can be arbitrary, we get $\zeta\leq \tau.$ Then we have $\tau=\zeta.$ Hence $X_t$ is determined in the time interval $[0, \zeta);$ the uniqueness is clear from Proposition \[t0.0\]. *Proof of Theorem [\[t0.2a\]]{}.*  Let $\{X^0_t\}$ denote the solution to (\[0.2\]). Let $\{v_k: k=1,2,\ldots\}$ be the set of jump times of the Poisson process $$t\longmapsto \int^t_0\int_{\mbb{N}} N_q(ds,dz).$$ We have clearly $v_k\longrightarrow\infty$ as $k\longrightarrow\infty.$ For $0\leq t<v_1$ set $X_t=X^0_t.$ Suppose that $X_t$ has been defined for $0\leq t< v_k$ and let $$\xi= X_{v_k-}+\int_{\{v_k\}}\int_{\mbb{N}} z N_q(ds,dz).$$ Here and in the sequel we make the convention $\infty+\cdots= \infty.$ By the assumption there is also a solution $\{X_t^k\}$ to $$X_t=\xi+\int^t_0\int^{r(X_{s-})}_0\int_{\mbb{N}} (z-1)N_p(v_k+ds,du,dz).$$ Let $\eta_k$ be the explosion time of $\{X_t^k\}$. If $v_k+\eta_k> v_{k+1}$, we define $X_t=X_{t-v_k}^k$ for $v_k\leq t<v_{k+1}.$ If $v_k+\eta_k\leq v_{k+1}$, we set $X_t=X_{t-v_k}^k$ for $v_k\leq t<v_k+\eta_k$ and $X_t=\infty$ for $v_k+\eta_k\leq t< v_{k+1}$. By induction that defines a process $\{X_t\}$, which is clearly the pathwise unique solution to (\[e0.1\]). Obviously, if the solution of (\[0.2\]) is non-explosive for each deterministic initial state $X_0=i\in \mbb{N}$, we have $\eta_k= \infty$ for all $k\in\mbb{N}$, and so $\{X_t\}$ is non-explosive. *Proof of Theorem [\[t0.4\]]{}.*  Let $\tilde{N}_p(ds, du, dz) = N_p(ds, du, dz)-dsdum(dz)$ and $\tilde{N}_q(ds, dz)=\tilde{N}_q(ds, du, dz)-dsdun(dz).$ For any bounded function $f$ on $\mbb{N}$ we have, \[0.5\] f(X\_[t\_m]{}) =f(X\_0)+ \_0\^[t\_m]{} \_0\^[r(X\_[s-]{})]{}\_\[f(X\_[s-]{}+z-1)-f(X\_[s-]{})\] N\_p(ds,du,dz) + \_0\^[t\_m]{} \_ \[f(X\_[s-]{}+z)-f(X\_[s-]{})\] N\_q(ds, dz) =f(X\_0)+ \_0\^[t\_m]{} \_0\^[r(X\_[s-]{})]{}\_\[f(X\_[s-]{}+z-1)-f(X\_[s-]{})\] ds du m(dz) + \_0\^[t\_m]{} \_ \[f(X\_[s-]{}+z)-f(X\_[s-]{})\] ds n (dz) + M\_t(f), where M\_t(f) :=\_0\^[t\_m]{} \_0\^[r(X\_[s-]{})]{}\_\[f(X\_[s-]{}+z-1)-f(X\_[s-]{})\]\_p(ds,du,dz) + \_0\^[t\_m]{} \_ \[f(X\_[s-]{}+z)-f(X\_[s-]{})\] \_q(ds, dz) is a martingale. Since $X_s\neq X_{s-}$ for at most countably many $s\geq 0$, we can also use $X_s$ instead of $X_{s-}$ in the right hand side of (\[0.5\]). In particular, for $f=1_{\{j\}}$ we have 1\_[{X\_[t\_m]{}=j}]{}=1\_[{X\_0=j}]{}+\_[k=0]{}\^b\_k \_0\^[t\_m]{} r(X\_s)\[1\_[{X\_s+k-1=j}]{}-1\_[{X\_s=j}]{}\] ds +\_[k=1]{}\^a\_k \_0\^[t\_m]{} \[1\_[{X\_s+k=j} ]{}-1\_[{X\_s=j}]{}\]ds +M\_t(1\_[{j}]{}). Write $E_i = E(\cdot|X_0=i)$ for $i\in \mbb{N}$. Taking the expectation in both sides of the above equation and letting $m\longrightarrow \infty$ we get E\_i(1\_[{X\_[t]{}=j}]{})=E\_i(1\_[{X\_0=j}]{}) + \_[k=0]{}\^b\_k E\_i(\_0\^[t]{} r(X\_s)\[1\_[{X\_s+k-1=j}]{}-1\_[{X\_s=j}]{}\] ds) + \_[k=1]{}\^a\_k E\_i(\_0\^[t]{} \[1\_[{X\_s+k=j} ]{}-1\_[{X\_s=j}]{}\]ds). Obviously, here we can remove the truncation “$\wedge\zeta$” and obtain Q\_[ij]{}(t)=\_[ij]{}+\_[k=0]{}\^[j]{} b\_k \^t\_0 \[r\_[j-k+1]{} Q\_[i,j-k+1]{}(s)-r\_jQ\_[ij]{}(s)\]ds +\_[k=1]{}\^j \_0\^t a\_k \[Q\_[i,j-k]{}(s)-Q\_[ij]{}(s)\]ds =\_[ij]{}+ \^t\_0 (\_[k=1]{}\^[j+1]{} Q\_[ik]{}(s)r\_[k]{}b\_[j-k+1]{} -Q\_[ij]{}(s)r\_j)ds +\_0\^t ( \_[k=0]{}\^[j-1]{} Q\_[ik]{}(s) a\_[j-k]{}- Q\_[ij]{}(s))ds. Differentiating both sides we get Q\_[ij]{}’(t) =\_[k=1]{}\^[j+1]{} Q\_[ik]{}(t)r\_[k]{}b\_[j-k+1]{} - Q\_[ij]{}(t)r\_j +\_[k=0]{}\^[j-1]{} Q\_[ik]{}(t) a\_[j-k]{}- Q\_[ij]{}(t) = \_[k=0]{}\^ Q\_[ik]{}(t)q\_[kj]{}. This is just the Kolmogorov forward equation of $Q$. *Proof of Theorem [\[t0.2aa\]]{}.*  By Theorem \[t0.2a\], the solution $\{X_t\}$ to (\[e0.1\]) is a time homogeneous Markov process with state space $\bar{\mbb{N}} := \{0,1,2,\dots,\infty\}$. Suppose that $\sigma_1$ and $z_1$ are given in the proof of Theorem \[t0.2\]. Let $q(v_1)=y_1$. By the properties of Poisson point process, we can see that $P(\sigma_1>t)=e^{-r(X_0)t},$ $P(z_1=i)=m(\{i\})=b_i,$ $P(v_1>t)=e^{-\gamma t},$ $P(y_1=i)= n(\{i\})= a_i $ and $\sigma_1$, $z_1$, $v_1,$ $y_1$ are mutually independent. Write $P_i(\cdot) = P(\cdot|X_0=i)$ for $i\in \mbb{N}$. Let $\xi_t=\max\{n+m: \sum^n_{i=0}\sigma_i, \sum^m_{i=0} v_i \leq t \}$. Obviously we have $P_i[X_t=j, \xi_t=0]=\delta_{ij}.$ By the Markov property of $\{X_t\},$ P\_i{X\_t=j, \_t=m+1} = P\_i{1\_[{\_1v\_1&lt;t}]{}P\_[X\_[\_1v\_1]{}]{}} = P\_i{1\_[{\_1&lt; t}]{}1\_[{v\_1\_1}]{}P\_[X\_[\_1]{}]{}} +P\_i{1\_[{v\_1&lt;t}]{}1\_[{v\_1&lt; \_1}]{}P\_[X\_[v\_1]{}]{}} = P\_i{\^t\_0r\_ie\^[-r\_i(t-s)]{}e\^[-(t-s)]{}P\_[X\_[t-s]{}]{}\[X\_s=j, \_s=m\]ds} +P\_i{\^t\_0e\^[-(t-s)]{}e\^[-r\_i (t-s)]{}P\_[X\_[t-s]{}]{}\[X\_s=j, \_s=m\]ds} =P\_i{\^t\_0r\_ie\^[-(r\_i+)(t-s)]{}\^\_[k=i-1]{}P(z\_1=k-i+1)P\_k\[X\_s=j, \_s=m\]ds} +P\_i{\^t\_0e\^[-(r\_i+)(t-s)]{}\^\_[k=i+1]{}P(y\_1=k-i)P\_k\[X\_s=j, \_s=m\]ds} = \_[ki]{}\^t\_0e\^[-(r\_i+)(t-s)]{}q\_[ik]{}P\_k\[X\_s=j, \_s=m\]ds. Notice that $$P_i[X_t=j]=\sum^\infty_{m=0}P_i[X_t=j, \xi_t=m].$$ From the theory of Markov chains we know $P_{ij}(t):=P_i[X_t=j]$ is the minimal solution to the Kolmogorov equation of the density matrix $Q$, see Chen (2004, p.78). Then $\{X_t\}$ is the minimal process of the density matrix $Q.$ *Proof of Theorem [\[t0.5\]]{}.*  Suppose that $R$ is regular. Then the minimal solution of its Kolmogorov backward equation is honest i.e. the minimal process of $R$ is non-explosive. Applying Theorems \[t0.2a\] and \[t0.2aa\] we know the minimal process of $Q$ is non-explosive. Thus $Q$ is regular. Conversely, suppose that $R$ is not regular. Then by Theorem 2.7(3) in [@Anderson91] there exists a non-trivial solution $(u^*_i) $ to u\_i\_[ki]{}u\_k,0u\_i1. Since $r_{ik}\leq q_{ik},$ we see $(u^*_i)$ is also a solution to $$u_i\leq\sum_{k\neq i}\frac{ q_{ik}}{\gamma+q_i}u_k.$$ Using Theorem 2.7(3) in [@Anderson91] again, we see $Q$ is not regular. *Proof of Theorem [\[t1.4\]]{}.*  By Theorem \[t0.5\] we derive the results from Theorem 1.2 of [@ChenRR97] and Theorem 2.3 of [@Chen06]. Recurrence ========== *Proof of Theorem [\[t3.1\]]{}.*  (1) Under the assumption, there exists a constant $N\ge 1$ such that $r_i\geq \gamma m/(1-M)$ holds for each $i\ge N$. Take $x_i=i$ for $i\geq 0$. For $i\geq N$ we have \^\_[j=0]{} q\_[ij]{}x\_j =r\_ib\_0(i-1) + \^\_[j=1]{} (r\_ib\_[j+1]{}+a\_j)(i+j) =(r\_i+)i + r\_i(M-1)+m(r\_i+)i = -q\_[ii]{}x\_i. Let $(\pi_{ij})$ be the embedded chain of $(q_{ij})$. The above calculations imply that $(x_i)$ is a finite solution of \^\_[j=0]{} \_[ij]{}x\_jx\_i, iN. Then $Q$ is recurrent by Theorem 4.24 in [@ChenMF04]. \(2) Suppose that $M\le 1$ and $J=\infty$. We shall prove the process is recurrent by comparison arguments. Let $\bar{Q} = (\bar{q}_{ij})$ be the density matrix defined by |[q]{}\_[ij]{}= { [lcl]{} ib\_[j-i+1]{} + a\_[j-i]{} & & [j i+1]{}\ - i - & & [j= i]{}\ ib\_0 & & [j= i-1]{}\ 0 & & , . which corresponds to a linear branching process with immigration. It was proved in [@LJP06] that this process is recurrent. Next, we define the density matrix $Q^*= (q_{ij}^*)$ by q\_[ij]{}\^\*={ [lcl]{} r\_i b\_[j-i+1]{} + a\_[j-i]{}r\_i/i & & [j i+1]{}\ -r\_i - r\_i/i & & [j= i]{}\ r\_ib\_0 & & [j= i-1]{}\ q\_[ij]{}& & [i&lt; N]{}\ 0 & & . . Let $(\bar{\pi}_{ij})$ and $(\pi_{ij}^*)$ denote the embedded chains of $(\bar{q}_{ij})$ and $(q_{ij}^*)$, respectively. It is easy to see that $\bar{\pi}_{ij}= \pi_{ij}^*$ for $i\ge N$ and $j\ge 0$. Then $Q^*$ is also recurrent. For $l\geq i> N$ we have \_[j=i]{}\^q\_[ij]{} = -r\_i b\_0\_[j=i]{}\^q\_[lj]{}\^\*. Moreover, we have \_[j=k]{}\^q\_[ij]{} = \_[j=k]{}\^q\_[lj]{}\^\*=0, ki-1 and \_[j=k]{}\^q\_[ij]{}\_[j=k]{}\^q\_[ij]{}\^\* \_[j=k]{}\^q\_[lj]{}\^\*, kl+1. Then $Q$ and $Q^*$ are stochastically comparable, so we can construct a $Q$-process $(X_t)$ and a $Q^*$-process $(X^*_t)$ on some probability space in such a way that $X_0=X^*_0$ and $X_t\le X^*_t$ for all $t\ge 0$; see Example 5.51 in [@ChenMF04]. Now the recurrence of $(X_t)$ follows from that of $(X^*_t)$. \(3) Since $M>1$, there exists a $s\in (0,1)$ such that $B(s)<0, \quad i.e. \quad \sum^\infty_{i=0} b_i s^{i-1}<1$. Take $H=\{0\}$ and $x_i=1-s^i.$ For $i\geq 1$ we have \^\_[k=0]{} \_[ik]{}x\_k=\_[i, i-1]{}x\_[i-1]{}+\^\_[k=1]{}\_[i, i+k]{}x\_[i+k]{}=x\_[i-1]{}+\^\_[k=1]{}x\_[i+k]{}==1-1-s\^i=x\_i. Then the process is transient by Theorem 8.0.2 in [@Meyn09]. \(4) Since the proof is similar to that of (2), we omit it. Mean extinction time ==================== In this section, we assume $r_i=\alpha i^\theta$ for $\alpha>0$ and $\theta\ge 1$. Let $(X_t)$ be a realization of the nonlinear branching process with immigration. Its jump times are given successively by $\tau_0=0$ and $\tau_n=\inf\{t:t>\tau_{n-1}, X_t\neq X_{\tau_{n-1}}\}.$ We also define $\sigma_k=\inf\{t\geq \tau_1: X_t=k\}$. In order to prove the criterion for the ergodicity of $(X_t)$, let us consider the absorbing process $\tilde{X}_t := X_{t\wedge \sigma_0}$. The density matrix of this process is given by: \_[ij]{}={ [lcl]{} q\_[ij]{} & & [i0]{}\ 0 & & [i= 0]{}. . For this process, we define $\tilde{\tau}_0=0$, $\tilde{\tau}_n=\inf\{t:t>\tilde{\tau}_{n-1}, \tilde{X}(t)\neq \tilde{X}(\tilde{\tau}_{n-1})\}$ and $\tilde{\sigma}_k=\inf\{t\geq \tau_1: \tilde{X}_t=k\}$. It is easy to see that \[4.2\]E\_i \_0=E\_i \_0. Let $(\tilde{p}_{ij}(t))$ and $(\tilde{\phi}_{ij}(\lambda))$ denote the transition function and the resolvent of $(\tilde{X}_t)$, respectively. \[l4.3\] For any $i\geq 0$ and $s\in [0,1)$, we have \[4.4\] \_[j=0]{}\^\_[ij]{}’(t)s\^j = B(s)\_[j=1]{}\^\_[ij]{}(t)j\^s\^[j-1]{} - A(s)\_[j=1]{}\^\_[ij]{}(t)s\^j, t0, and \[4.10\] \_[j=0]{}\^\_[ij]{}()s\^j-s\^i = B(s)\_[j=1]{}\^\_[ij]{}() j\^s\^[j-1]{} - A(s)\_[j=1]{}\^\_[ij]{}()s\^j, &gt;0. From the Kolmogorov forward equation of the transition function we obtain that \_[ij]{}’(t)=\_[k=1]{}\^[j-1]{} \_[ik]{}(t)(r\_k b\_[j-k+1]{}+a\_[j-k]{}) - \_[ij]{}(t)(r\_j+) + \_[i,j+1]{}(t) r\_[j+1]{} b\_0. Multiplying $s^j$ on both sides of the above equality and then summing over $j$, we have \_[j=0]{}\^\_[ij]{}’(t)s\^j =\_[j=0]{}\^\_[k=1]{}\^[j-1]{}\_[ik]{}(t)r\_k b\_[j-k+1]{}s\^j + \_[j=0]{}\^\_[k=1]{}\^[j-1]{}\_[ik]{}(t) a\_[j-k]{} s\^j + \_[j=0]{}\^\_[i, j+1]{}(t)r\_[j+1]{} s\^j b\_0 - \_[j=1]{}\^\_[ij]{}(t) r\_j s\^j - \_[j=0]{}\^\_[ij]{}(t) s\^j, Then we can interchange the order of summation to see $$\sum_{j=0}^\infty\sum_{k=0}^{j-1}\tilde{p}_{ik}(t)r_k b_{j-k+1}s^j=\sum_{k\neq l} \tilde{p}_{ik}(t)r_k s^{k-1} \sum_{j=k+1}^\infty b_{j-k+1} s^{j-k+1}$$and $$\gamma\sum_{j=0}^\infty \sum_{k=0}^{j-1}\tilde{p}_{ik}(t) a_{j-k} s^j= \gamma \sum_{k\neq l} \tilde{p}_{ik}(t)s^k\sum_{j=k+1}^\infty a_{j-k}s^{j-k}.$$ It follows that $$\sum_{j=0}^\infty \tilde{p}_{ij}'(t)s^j= \sum_{j=1}^\infty \tilde{p}_{ij}(t)r_j s^{j-1}\alpha B(s)-\sum_{j=0}^\infty \tilde{p}_{ij}(t)s^j A(s).$$ That proves (\[4.4\]) and (\[4.10\]) is just the Laplace transform of (\[4.4\]). \[l4.4\] For any $i, k\geq 1,$ we have $\int_0^\infty \tilde{p}_{ik}(t)dt <\infty$ and $\lim_{t\to\infty} \tilde{p}_{ik}(t)= 0$. Furthermore, for $i\geq 1$ and $s\in [0,1)$, we have \[4.5\] \_[k=1]{}\^(\_0\^\_[ik]{}(t)dt) s\^k &lt; . Fixing an $i\geq 1$, we can use the Kolmogorov forward equation to see \_[i0]{}(t)=b\_0\_0\^t \_[i1]{}(u)du, which means that \_0\^\_[i1]{}(t)dtb\_0\^[-1]{}\^[-1]{} &lt;. Suppose that $\int_0^\infty \tilde{p}_{ik}(t)dt <\infty$ for $k\leq j$. By the Kolmogorov forward equations we can see for $j\geq 1$, \_[ij]{}(t)-\_[ij]{} =\_[k=1]{}\^[j-1]{} (k\^b\_[j-k+1]{}+a\_[j-k]{})\_0\^t \_[ik]{}(u)du - (j\^+) \_0\^t \_[ij]{}(u)du + (j+1)\^b\_0 \_0\^t \_[ij+1]{}(u)du. Letting $t\to\infty$, we have \_0\^\_[ij+1]{}(t)dt &lt;. Then $\int_0^\infty \tilde{p}_{ik}(t)dt <\infty$ by induction. Since the limit $\lim_{t\to\infty} \tilde{p}_{ik}(t)$ always exists, we see $\lim_{t\to\infty} \tilde{p}_{ik}(t)= 0$ immediately. We next tend to prove (\[4.5\]). Since $M\leq 1$, we have $B(s)>0$ for a fixed $s\in[0,1)$. Then there exists a $k\ge 1$ so that $k\alpha B(s)-sA(s)> 0$. Using (\[4.4\]), we have \[4.9\] \_[j=0]{}\^’\_[ij]{}(u)s\^j =B(s)\_[j=1]{}\^\_[ij]{}(u)j\^s\^[j-1]{}- A(s)\_[j=1]{}\^\_[ij]{}(u)s\^j B(s)\_[j=k+1]{}\^\_[ij]{}(u)j\^s\^[j-1]{} - A(s)\_[j=1]{}\^\_[ij]{}(u)s\^j \_[j=k+1]{}\^\_[ij]{}(u) s\^[j-1]{} - A(s) \_[j=1]{}\^k \_[ij]{}(u)s\^j. Let $\|A\| = \max_{s\in[0,1]}|A(s)|$ and $\|B\| = \max_{s\in[0,1]}|\alpha B(s)|$. Then for each $s\in[0,1)$, \_0\^t \_[j=0]{}\^|’\_[ij]{}(u)s\^j|du B \_0\^t\_[j=1]{}\^\_[ij]{}(u)j\^s\^[j-1]{}du+A\_0\^t \_[j=1]{}\^\_[ij]{}(u) s\^j du tB\_[j=1]{}\^j\^s\^[j-1]{} + tA\_[j=1]{}\^s\^j&lt;. Then we use Fubini’s theorem to see \_0\^t\_[j=0]{}\^’\_[ij]{}(u)s\^j du=\_[j=0]{}\^\_0\^t ’\_[ij]{}(u)s\^jdu. Integrating both sides of (\[4.9\]), \_[j=0]{}\^\_[ij]{}(t)s\^j-s\^i \_[j=k+1]{}\^(\_0\^t \_[ij]{}(u)du)s\^[j-1]{} - A(s)\_[j=1]{}\^k (\_0\^t \_[ij]{}(u)du)s\^j . Letting $t\to\infty$ and using the fact that $\int_0^\infty \tilde{p}_{ik}(t)dt <\infty,$ we have \_[j=k+1]{}\^(\_0\^\_[ij]{}(u)du)s\^[j-1]{}&lt;, which implies (\[4.5\]). \[l4.5\] Suppose that the nonlinear branching process with immigration is recurrent and [(\[4.8\])]{} holds. Then for $i\ge 1$ we have \[4.15\] E\_i(\_0) \^1\_0 ()\^[-1]{}dy. and \[4.16\] E\_i(\_0)\^1\_0 ()\^[-1]{}dy. Multiplying (\[4.10\]) by $(\ln(s/y))^{\theta-1}$, dividing by $\alpha B(s)$ and integrating both sides we have $$\int^s_0 \sum^\infty_{j=1} \tilde{\phi}_{ij}j^\theta y^{j-1} (\ln \frac{s}{y})^{\theta-1}dy=\int_0^s \frac{(\lambda+A(y))\sum_{j=1}^\infty \tilde{\phi}_{ij}(\lambda)y^j-y^i+ \lambda\tilde{\phi}_{i0}(\lambda)}{\alpha B(y)}\bigg(\ln\frac{s}{y}\bigg)^{\theta-1}dy.$$ Letting $y=se^{-\frac{x}{j}}$ in the left hand side of the above equation we get $$\int^s_0 \sum^\infty_{j=1} \tilde{\phi}_{ij}j^\theta y^{j-1} (\ln \frac{s}{y})^{\theta-1}dy = \int^\infty_0 \sum_{j=1}^\infty \tilde{\phi}_{ij}(\lambda) s^j x^{\theta-1} e^{-x} dx= \Gamma(\theta) \sum^\infty_{j=1} \tilde{\phi}_{ij}(\lambda) s^j.$$ Using the above two equations we obtain \[4.12\] \_[j=1]{}\^\_[ij]{}()s\^j= \_0\^s ()\^[-1]{}dy. For $i\geq 1$, $\lambda>0$ and $s\in[0,1]$ let \_i(,s)=\_[j=1]{}\^\_[ij]{}()s\^j. Note that \_[i0]{}()= \^\_0 e\^[-t]{} p\_[i0]{}()dt \^\_0 e\^[-t]{}dt = 1. Then, by (\[4.12\]), \[4.17\] \_i(,s) \_0\^1 ()\^[-1]{}dy + \_0\^s ()\^[-1]{}dy. By Lemma \[l4.4\], \_[0]{}\_[j=1]{}\^\_[ij]{}() =\_[0]{} \_[j=1]{}\^\^\_0e\^[-t]{}\_[ij]{}()dt =0. It follows that, for $s\in[0, 1]$, \[4.6\] \_[0]{} \_i(,s)\_[0]{}\_[j=1]{}\^\_[ij]{}() =0. Denote C\_i := \_0\^1 ()\^[-1]{}dy\ \_0\^1 ()\^[-1]{}dy\ \_0\^1 ()\^[-1]{}dy&lt;. By (\[4.5\]) we have \_i(0,s)=\_[k=1]{}\^(\_0\^p\_[ik]{}(t)dt) s\^k &lt; for each $0\leq s< 1$. Letting $\lambda\to0$ in (\[4.17\]), we have \_i(0,s)C\_i + \_0\^s ()\^[-1]{}dy. Using the Gronwall’s inequality, we have \[4.13\] \_i(0,s)C\_i . Letting $s\uparrow 1$ we see \_[s1]{}\_i(0,s) = \_[s1]{}\_[j=1]{}\^\^\_0 \_[ij]{}(t)s\^jdt = \^\_0 (1-\_[i0]{}(t)) dt = E\_i (\_0). Hence (\[4.15\]) follows from (\[4.2\]) and (\[4.13\]). Similarly, by (\[4.12\]) we have $$\psi_i(\lambda,s)\geq \frac{1}{\Gamma(\theta)}\int^s_0 \frac{\lambda\tilde{\phi}_{i0}(\lambda)-y^i}{\alpha B(y)}\big(\ln\frac{s}{y}\big)^{\theta-1}dy.$$Letting $\lambda\rightarrow 0$ and then letting $s\rightarrow 1$, we obtain (\[4.16\]). Ergodicity and strong ergodicity ================================ One of the main steps to prove Theorems \[t4.1\] and \[t5.1\] is to compare our nonlinear branching process with immigration with a suitably designed birth-death process, which we now introduce. A similar birth-death process was used by [@ChenRR97] in her study of the regularity of the nonlinear branching process with resurrection. Let L = M+b\_0-1 = \_[k=1]{}\^k b\_[k+1]{} and let $(\hat{X}_t)$ be a birth-death process with birth rate $d_i = r_iL+ \gamma m$ and death rate $c_i = r_ib_0$. We denote the density matrix of $(\hat{X}_t)$ by $(\hat{q}_{ij})$. Let $T_0:= \inf\{t\geq 0: \hat{X}_t= 0\}$. \[l5.2\] [(1)]{} Suppose that $m<\infty$, $M< 1$, $r_i$ is increasing and $\sum_{i=1}^\infty r_i^{-1} <\infty$. Then the birth-death process $(\hat{X}_t)$ is strongly ergodic. [(2)]{} Suppose that $m<\infty$, $M\leq 1$, $r_i$ is non-decreasing and $\sum_{i=1}^\infty r_i^{-1} <\infty$. Then the birth-death process $(\hat{X}_t)$ is ergodic. \(1) It is easy to check that the birth-death process is regular. Fix an $\varepsilon> 0$ satisfying $L + \varepsilon< b_0$. Then there exists an $N$ such that $d_i \leq \gamma _i(L+\varepsilon)$ for each $i> N$. Let \[5.1+\] S = \_[n=1]{}\^(+ \_[k=1]{}\^n ). It is obvious that $\sum_{n=1}^\infty c_{n+1}^{-1}< \infty$. Notice that for each $n> N$ we have \_[k=1]{}\^n \_[1kN]{} N + \_[k=1]{}\^[n-N]{}N \^[n-N]{}\_[1kN]{} + \_[k= 1]{}\^[n-N]{}, where $\rho = b_0^{-1}(L +\varepsilon)<1$. Then $S< \infty$. By Corollary 2.4 of [@Zhang01], we conclude that $(\hat{X}_t)$ is strongly ergodic. \(2) Since $L \leq b_0$, we have R := \_[n=1]{}\^ d\_0\_[n=1]{}\^. Taking logarithm on the right-hand side we get $$\ln \bigg(\frac{(c_1+\gamma m)(c_2+\gamma m)\cdots (c_{n-1}+\gamma m)}{c_1c_2\cdots c_n}\bigg) = \sum_{i=1}^{n-1}\ln \bigg(1+\frac{\gamma m}{r_i b_0}\bigg) + \ln \bigg(\frac{1}{c_n}\bigg).$$ Since $\lim_{i\rightarrow \infty} r_i=\infty$, we have $\ln{\big(1+\frac{\gamma m}{r_i b_0}\big)} \sim \frac{\gamma m}{r_i b_0}$ as $i\rightarrow\infty$. Then there exists a constant $C\ge 0$ such that for sufficiently large $n$, $$\ln \bigg(\frac{(c_1+\gamma m)(c_2+\gamma m)\cdots (c_{n-1}+\gamma m)}{c_1c_2\cdots c_n}\bigg) \leq C\sum_{i=1}^{\infty}\frac{1}{r_i} + \ln \bigg(\frac{1}{c_n}\bigg),$$ and hence $$\frac{(c_1+\gamma m)(c_2+\gamma m)\cdots (c_{n-1}+\gamma m)}{c_1c_2\cdots c_n}\leq \frac{T}{c_n}$$ for another constant $T\ge 0$. That implies $R<\infty$. By Theorem 4.55 in [@ChenMF04] the birth-death process is ergodic. \[l4.2\] If the nonlinear branching process with immigration has a stationary distribution $\mu=(\mu_j)$, then the generating function $f(s):=\sum_{j=0}^\infty \mu_j s^j$ satisfies the following equation: \[n4.1\] ()f(s) = ()\_0+\_0\^s ()\^[-1]{}f(y)dy,s. The stationary distribution $(\mu_j)$ satisfies $\mu Q = 0$. In view of (\[1.2\]), we have \[4.1+\] \_j(+ j\^)=\_[i=0]{}\^[j-1]{} \_ia\_[j-i]{}+\_[i=0]{}\^[j+1]{} \_ii\^b\_[j-i+1]{}. Multiplying $s^j$ on both sides of the above equality and then summing over $j$, we have \[4.7\]\_[j=1]{}\^\_j s\^j + s \_[j=1]{}\^\_j j\^s\^[j-1]{} =\_[j=1]{}\^\_[i=0]{}\^[j-1]{} \_i a \_[j-i]{} s\^j +\^\_[j=1]{} \_[i=1]{}\^[j+1]{} \_i i\^b\_[j-i+1]{} s\^j. Interchanging the order of summation, (\[4.7\])=\_[i=0]{}\^\_i s\^i \_[j=i+1]{}\^a\_[j-i]{} s\^[j-i]{}- \_1b\_0 +\_[i=1]{}\^\_i i\^s\^[i-1]{} \_[j=i-1]{}\^b\_[j-i+1]{} s\^[j-i+1]{} =\_[i=0]{}\^\_i s\^i F(s)- \_1b\_0+\_[i=1]{}\^\_i i\^s\^[i-1]{}G(s). Letting $j=0$ in (\[4.1+\]), we see $\mu_0\gamma=\alpha\mu_1b_0$. Therefore, from (\[4.7\]) it follows that \_[j=1]{}\^\_j j\^s\^[j-1]{} = . Multiplying the above equation by $(\ln\frac{s}{y})^{\theta-1}$ and integrating the both sides, we have \[3.3\] \_0\^s \_[j=1]{}\^\_j j\^y\^[j-1]{} ()\^[-1]{} dy = \_0\^s ()\^[-1]{}f(y) dy. Letting $y=se^{-\frac{x}{j}}$ we get (\[3.3\]) =- \^\_0 \_[j=1]{}\^\_j j\^(s e\^[-]{})\^[j-1]{}()\^[-1]{} (-e\^[-]{}) dx = \_0\^\_[j=1]{}\^\_j s\^j x\^[-1]{} e\^[-x]{} dx = () \[f(s)-\_0\]. Then $f(s)$ is a solution to the differential equation (\[n4.1\]). *Proof of Theorem [\[t4.1\]]{}.*  (1) By Lemma \[l5.2\] the birth-death process $(\hat{X}_t)$ is ergodic. Thus by Theorem 4.45 in [@ChenMF04], the equation \[4.11\] u\_0=0,   d\_i(u\_[i+1]{}-u\_i)+c\_i(u\_[i-1]{}-u\_i)+1=0, i0 has a finite nonnegative solution $(u_i)$. By Remark 2.5 of [@Zhang01], we have \[5.2\] u\_0=0,   u\_i= \_[k=0]{}\^[i-1]{}(+\_[j=k+1]{}\^). It is apparent that $u_i\leq u_{i+1}$. Moreover, we have u\_[i+1]{}-u\_i= + \_[j=i+1]{}\^, u\_i- u\_[i-1]{}= +\_[j=i]{}\^. Since $d_{i+1}/c_{i+1}< d_i/c_i$ and $1/c_{i+1}< 1/c_i$, it is not hard to show that $u_{i+1}- u_i$ is non-increasing in $i\geq 0$. Coming back to the matrix $Q$, for $i\ge 1$, \[4.18\] \_[j=0]{}\^q\_[ij]{}u\_j =\_[j=0]{}\^q\_[ij]{}(u\_j-u\_i) =c\_i(u\_[i-1]{}- u\_i)+ r\_i\_[k=1]{}\^b\_[k+1]{} \_[l=1]{}\^k (u\_[i+l]{}- u\_[i+l-1]{}) + \_[k=1]{}\^a\_k \_[l=1]{}\^k (u\_[i+l]{}- u\_[i+l-1]{}) c\_i(u\_[i-1]{}-u\_i)+ r\_i\_[k=1]{}\^k b\_[k+1]{} (u\_[i+1]{}- u\_[i]{}) + \_[k=1]{}\^k a\_k(u\_[i+1]{}-u\_i) =c\_i(u\_[i-1]{}-u\_i)+ (r\_iL + m)(u\_[i+1]{}-u\_i) =c\_i(u\_[i-1]{}-u\_i)+ d\_i(u\_[i+1]{}-u\_i) = - 1. and \[5.3\] \_[j=1]{}\^ q\_[0j]{}u\_j=\_[j=1]{}\^ q\_[0j]{}(u\_j-u\_1)=\_[j=1]{}\^q\_[0j]{} \_[i=1]{}\^j(u\_i-u\_[i-1]{})&lt; \_[j=1]{}\^q\_[0j]{}j u\_1m u\_1&lt; . Then $(u_i)$ is a nonnegative bounded solution to the following equation \_[j=1]{}\^q\_[0j]{}u\_j&lt; ,   \_[j=0]{}\^q\_[ij]{}u\_j-1, i1. By Theorem 4.45 in [@ChenMF04] we know the process is positive recurrent. \(2) Suppose that the process is ergodic. Then letting $s=1$ in (\[n4.1\]) we get &gt; () \_[j=1]{}\^\_j \_0\^1 ()\^[-1]{}dy \_0\^1 ()\^[-1]{}dy. Since $\mu_0 > 0$, we have (\[4.8\]). Conversely, suppose that (\[4.8\]) holds. By the strong Markov property, we have E\_0(\_0) = E\_0(\_1) + E\_0\[E\_[X\_[\_1]{}]{}(\_0)\] = + \_[i=1]{}\^E\_i(\_0) = + \_[i=1]{}\^a\_iE\_i(\_0). Using (\[4.15\]) we have E\_0(\_0) + . By (\[4.8\]), the right-hand side is finite. Thus the process is ergodic. \(3) By the assumption, there exists $C> 0$ such that $r_i\geq \frac{C}{b_0-\Gamma}i$ for large enough $i$. Therefore \_[j=0]{}\^j q\_[ij]{}=\_[k=1]{}\^ (i+k)r\_i b\_[k+1]{} + \_[k=1]{}\^ (i+k)a\_k +(i-1)r\_ib\_0-(+r\_i)im-r\_i (b\_0-) m-Ci. Applying Corollary 4.49 in [@ChenMF04], we know the process is exponentially ergodic. *Proof of Theorem [\[t5.1\]]{}.*  (1) Using Lemma \[l5.2\], we see the birth-death process $(\hat{X}_t)$ is strongly ergodic. Let $u_i:= E_i(T_0)$ for $i\geq 0$. Applying Theorem 4.44 and Lemma 4.48 in [@ChenMF04], we find that $(u_i)$ is a bounded non-negative solution to equation (\[4.11\]). By (\[4.18\]) and (\[5.3\]), $(u_i)$ is also a non-negative bounded solution to the following equation \_[j=1]{}\^q\_[0j]{}u\_j&lt; ,   \_[j=0]{}\^q\_[ij]{}u\_j-1, i1. By Theorem 4.45 in [@ChenMF04], we know the process is strongly ergodic. \(2) Suppose that (\[5.1\]) holds. Then $$\int_0^1 \frac{A(y)}{\alpha B(y)} \bigg(\ln\frac{1}{y}\bigg)^{\theta-1} dy \leq \gamma\int^1_0 \frac{1}{\alpha B(y)}\bigg(\ln\frac{1}{y}\bigg)^{\theta-1}dy<\infty.$$ Letting $i\rightarrow\infty$ in (\[4.15\]), we get $$\sup_i E_i (\sigma_0)\leq \frac{1}{\Gamma(\theta)}\bigg[ \int^1_0 \frac{1}{\alpha B(y)}\bigg(\ln\frac{1}{y}\bigg)^{\theta-1}dy\bigg]\cdot \exp\bigg[\frac{1}{\Gamma(\theta)}\int_0^1 \frac{A(y)}{\alpha B(y)} \bigg(\ln\frac{1}{y}\bigg)^{\theta-1} dy\bigg]<\infty.$$ Then by Theorem 4.44 in [@ChenMF04] the process is strongly ergodic. Conversely, suppose that $X_t$ is strongly ergodic. By Theorem 4.44 in [@ChenMF04] and (\[4.16\]), we know (\[5.1\]) holds. \(3) By the strong Markov property, for $i\geq 1$ we have $E_i\sigma_0=\sum_{k=1}^i E_k \sigma_{k-1}.$ Notice that $$E_k \sigma_{k-1}\geq E_k [\mbox{time spent at $k$ until the next jump}]=\frac{1}{r_k+\gamma}.$$ Thus $E_i\sigma_0 \geq\sum_{k=1}^i (r_k+\gamma)^{-1}$. By the assumption $\sum_{i=1}^\infty r_i^{-1}=\infty$, we have $\sup_iE_i\sigma_0=\infty$. Applying Theorem 4.44 in [@ChenMF04], we know the process is not strongly ergodic. *Abstract.*The author would like to thank Professors Mu-Fa Chen, Yong-Hua Mao and Yu-Hui Zhang for their advice and encouragement. I am grateful to the two referees for pointing out a number of typos in the first version of the paper. [99]{} Anderson, W.J.: Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York, 1991. Athreya, K.B. and Ney, P.E.: Branching Processes. Springer, Berlin, 1972. Chen, A.Y.: Ergodicity and stability of generalised Markov branching processes with resurrection. *J. Appl. Probab.* **39** (2002), 786–803. Chen, A.Y., Li, J.P. and Ramesh, N.I.: Uniqueness and extinction of weighted Markov branching processes. *Methodol. Comput. Appl. Prob.* **7** (2005), 489–516. Chen, A.Y., Li, J.P. and Ramesh, N.I.: General Harris regularity criterion for non-linear Markov branching process. *Statist. Probab. Letters.* **76** (2006), 446–452. Chen, A.Y. and Renshaw, E.: Markov branching processes with instantaneous immigration. *Probab. Theory Relat. Fields* **87** (1990), 209–240. Chen, M.F.: From Markov Chains to Non-Equilibrium Particle Systems. Second edition. World Scientific, Singapore, 2004. Chen, R.R.: An extended class of time-continuous branching processes. *J. Appl. Probab.* **34** (1997), 14–23. Harris, T.E.: The Theory of Branching Processes. Springer, Berlin, 1963. Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. Second edition, North-Holland/Kodasha, Amsterdam/Tokyo, 1989. Karlin, S. and Taylor, H.M.: A First Course in Stochastic Processes. Second edition. Academic Press, New York, 1975. Li, J.P. and Chen, A.Y.: Markov branching processes with immigration and resurrection. *Markov Processes Relat. Fields* **12** (2006), 139–168. Meyn, S.P. and Tweedie, R.L.: Markov Chains and Stochastic Stability. Second edition. Cambridge Univ. Press, 2009. Pakes, A.G.: Extinction and explosion of nonlinear Markov branching processes. *J. Austr. Math. Soc.* **82** (2007), 403–428. Yang, Y.S.: On branching processes allowing immigration. *J. Appl. Probab.* **9** (1972), 24–31. Zhang, Y.H.: Strong-ergodicity for single-birth processes. *J. Appl. Probab.* **38** (2001), 270–277.

--- abstract: 'In this letter, a novel run-length limited (RLL) code is reported. In addition to providing a strict DC-balance and other relevant features for visible light communication (VLC) applications, the proposed 5B10B code presents enhanced error correction capabilities. Theoretical and simulation results show that the proposed code outperforms its standard enforced counterparts in terms of bit error ratio while preserving low-complexity requirements.' author: - 'Vitalio Alfonso Reguera,  [^1]' bibliography: - 'VLC\_paper.bib' title: | New RLL Code with Improved Error Performance\ for Visible Light Communication --- =1 Run-length limited (RLL) codes, error correction, visible light communication (VLC). Introduction ============ n recent years, there has been a growing interest in exploiting the inherent capabilities of lighting systems to transmit information. The rapid development of LED technologies has made it feasible to implement Visible Light Communication (VLC) systems to satisfy a wide range of applications [@Khan2017] (e.g. indoor positioning systems, Li-Fi, vehicle to vehicle communication, underwater communication, etc.). Combining lighting and communication imposes additional challenges in the design of VLC systems such as brightness control and flicker mitigation. Consequently, line codes for data transmission must be carefully chosen to avoid temporal light artefacts that could potentially harm human health. Particularly, the IEEE standard 802.15.7 [@Standard2018] enforces the use of Manchester, 4B6B and 8B10B run-length limited (RLL) codes with basic modulations such as on-off key (OOK) modulation and variable pulse-position modulation (VPPM). These RLL codes provide an appropriate DC-balance to the transmitted signal at the cost of lowering the resulting data rate. Also, traditional RLL codes have weak error correction capability. Often the code rate is further affected by channel coding stages in order to improve the bit error ratio (BER). In this context, it seems convenient to join forward error correction (FEC) and RLL codes in an effort to improve error performance with a suitable computational complexity. This has been addressed in several previous works using different strategies and coding techniques [@Wang2015; @Lu2016; @Wang2016; @Fang2017; @Babar2017; @Lu2018; @Wang2019]. The works in [@Wang2015] and [@Wang2016] developed a soft decoding strategy for the concatenation of an outer Reed-Solomon (RS) code with an inner 4B6B or 8B10B code. In [@Wang2015] the RLL code is soft decoded and a hard output of one or more possible codeword candidates are passed to the RS decoder. An enhanced RLL decoding with soft output was presented in [@Wang2016]; improving BER when compared to its predecessors. Subsequently, the outer RS encoder was replaced by polar codes in [@Wang2019]. The performance of polar code is boosted using pre-determined frozen bit indexes. As a result, the proposal in [@Wang2019] significantly improves BER performance. These works have in common that they focus on the decoding process, but no new RLL codes were specified besides the classics 4B6B and 8B10B codes. In [@Fang2017; @Babar2017] new RLL encoders are considered. In [@Fang2017] the RLL encoder is replaced by the insertion of compensation symbols within the data frame. Unfortunately, this scheme can cause long runs of equal bits as compared to traditional RLL codes. With the purpose of increasing the capacity of the VLC systems in [@Babar2017], the Unity-Rate Code (URC) was promoted to replace conventional RLL codes. This improvement in the code rate comes at the cost of relaxing the strict DC-balance at the output of the RLL encoder. For both, [@Fang2017] and [@Babar2017], instability in the DC balance could cause flicker if brightness fluctuations exceed the maximum flickering time period (MFTP) for practical systems [@Rajagopal2012; @Std2015; @Standard2018]. A joint FEC-RLL coding mechanism was first introduced in [@Lu2016]. It combines convolutional codes and Miller codes to simultaneously achieve FEC and RLL control. A major drawback of this mechanism is the disappointing BER of Miller codes. To circumvent this problem, a new class of Miller codes, termed eMiller, was recently proposed in [@Lu2018]. The eMiller code match the BER of Manchester code, improving the overall performance when decoded taking advantage of their optimal trellis structure. In this letter a novel RLL code, hereafter named 5B10B, is reported; which, in addition to preserving the desirable characteristics of traditional RLL codes for VLC, allows for enhanced error correction capabilities. Analytical and simulation results show that, for OOK-modulated transmissions over channels with moderate to high signal-to-noise ratio (SNR), the proposed 5B10B code outperforms the standard encouraged transmission techniques in terms of BER performance. Proposed 5B10B code =================== $i$ Data Code $i$ Data Code ----- ------- ------------ ----- ------- ------------ 0 00000 1100110001 16 10000 0111010001 1 00001 1110001001 17 10001 0101111000 2 00010 1110010010 18 10010 0101100011 3 00011 0100011011 19 10011 0110101010 4 00100 1101000101 20 10100 0110110100 5 00101 1100011100 21 10101 0100101101 6 00110 1100100110 22 10110 0101010110 7 00111 1101001010 23 10111 0111001100 8 01000 1001010011 24 11000 1001101001 9 01001 1011011000 25 11001 0010111001 10 01010 1010100011 26 11010 0011110010 11 01011 1000111010 27 11011 0011001011 12 01100 1001110100 28 11100 0011100101 13 01101 1010010101 29 11101 0001011101 14 01110 1011000110 30 11110 0001101110 15 01111 1010101100 31 11111 0010011110 : 5B10B Code[]{data-label="table_1"} The proposed code is listed in Table \[table\_1\]. As suggested by its designation, 5B10B, maps 5-bit datawords to 10-bit codewords. Therefore, the resulting code rate is $\frac{1}{2}$; matching the code rate of Manchester code. A close inspection of Table \[table\_1\] reveals two main features of the proposed code: (*i*) all codewords have the same *weight*, $w=5$, and (*ii*) its *minimum Hamming distance* is $d_m = 4$. Thus, it can be referred to as a constant weight code with error detection and correction capabilities. The existence of such codeword subset was anticipated in [@Brouwer1990], but no explicit construction was given. Here, one of these subsets was found by using computer search techniques. It is worth noting that multiple variations of the code are realizable (preserving their main features) by simply permuting the bit columns and/or by computing the ones’ complement of the codewords. Moreover, according to [@Brouwer1990], within the code space of constant weight, $w = 5$ and codeword length, $n = 10$, there are at least 36 codewords that satisfy $d_m = 4$. In addition to the codewords selected, four other codewords can be included in this subset: 0000110111, 0110000111, 1000001111 and 1111100000. The reason why these codewords were not chosen to encode data bits is because they have larger runs. However, in a practical implementation they could be considered as control symbols (e.g. *comma symbols* to synchronize data frames). Since all codewords have the same weight (with equal proportion of zeros and ones) the code guarantees a strict DC-balance. Making it similar to Manchester and 4B6B codes in this regard, and different from 8B10B which suffers from *running disparity* that adds some extra complexity in the coding/decoding process. The run-length of 5B10B is six; very close to the one offered by 8B10B (five). More specifically, runs of six consecutive zeros appear when the codewords 0001011101 or 0001101110 are preceded by the 0101111000 or 1011011000. Most of the 5B10B properties commented so far do have aspects in common with the standard enforced RLL codes, except for the minimum distance. The proposed code doubles the minimum distance of traditional RLL codes (please, note that for Manchester, 4B6B and 8B10B codes, $d_m = 2$). A detailed analysis of the impact of this increase in minimum distance will be presented in the next subsection. It is worth emphasizing that 5B10B belongs to the group of constant weight codes. Also, other constant weight codes with different codeword length and weight could be take into account in order to increase the minimum distance. In this sense, our proposal aims to offer a balance between code rate, error correction capabilities and computational complexity. For instance, due to its relative small number of valid codewords, coding and decoding can be efficiently implemented by using a lookup table (LUT). Error performance ----------------- In order to evaluate the error performance of 5B10B, in what follows we consider that VLC data is transmitted over the line-of-sight path, for which the received signal can be modeled as: $$\label{eq:Channelmodel} \mathbf{y} = \alpha \mathbf{x}_{i}+\mathbf{z},$$ where $\mathbf{x}_{i}$ represents the $i$-th OOK-modulated codeword $(i=0,2,\dots,31)$, $\alpha$ is a constant that accounts for all transmission losses and electro-optical conversion factors, and $\mathbf{z}$ is an independent and identically distributed vector of Gaussian components with zero mean and power spectral density $\frac{N_0}{2}$, representing both thermal noise and background shot noise. Assuming that signal symbols are equiprobable, the *maximum likelihood* (ML) detector for this channel model is given by: $$\label{eq:ML} \hat{i} = \underset{0 \leq i \leq 31}{\text{arg min}} \Vert \mathbf{y} - \mathbf{x}_i \Vert,$$ and the resulting symbol error probability is upper bounded by the union bound as: $$\label{eq:Pe1} P_e \leq \frac{1}{32} \sum_{r=1}^{5} \sum_{i=0}^{31} A_{i,r} Q\left( \sqrt{\frac{r \Delta^2}{2 N_0}} \right),$$ where $A_{i,r}$ is the number of codewords that are at Hamming distance $2r$ from the $i$-th codeword (please note that since it is a constant weight code, the Hamming distance between any two codewords is a multiple of 2), $Q(\cdot)$ is the complementary cumulative distribution function of the standard normal distribution, and $\Delta$ is the Euclidean distance between two OOK-modulated signal symbol that differ in the position of just one ON pulse. Defining $\mathcal{E}$ as the energy of the difference between individually received ON and OFF pulses, we have: $$\label{eq:D} \Delta = \sqrt{2\mathcal{E}}$$ $r=1$ $r=2$ $r=3$ $r=4$ $r=5$ ------- ------- ------- ------- ------- 0 17.69 8.81 4.50 0 : Average number of codewords at Hamming distance $2r$ over the entire codeword set: $\frac{1}{32}\sum_{i=0}^{31} A_{i,r}$[]{data-label="table_2"} Table \[table\_2\] shows the average number of codewords per symbol at Hamming distance $2r$. Since 5B10B has $d_m = 4$, $A_{i,1} = 0, \forall i$. It is also easy to verify from Table \[table\_1\] that, for any codeword, its one’s complement does not produce a valid codeword. Therefore, $A_{i,5} = 0, \forall i$. Using Equation (\[eq:D\]) and the results in Table \[table\_2\], Equation (\[eq:Pe1\]) can be rewritten as: $$\label{eq:Pe2} P_e \leq 17.69 \cdot Q\left( \sqrt{\frac{2 \mathcal{E}}{N_0}} \right) + 8.81 \cdot Q\left( \sqrt{\frac{3 \mathcal{E}}{N_0}} \right) + 4.5 \cdot Q\left( \sqrt{\frac{4 \mathcal{E}}{N_0}} \right)$$ For the proposed code, it’s not hard to see that the bit error probability, $P_b$, will be bounded as: $$\label{eq:Pb_Bound} \frac{P_e}{5} < P_b < P_e.$$ In order to bring the actual $P_b$ as close as possible to its theoretical lower bound, the *binary switching algorithm* (BSA) [@Zeger1990] was used to efficiently map datawords to codewords (as presented in Table \[table\_1\]). The algorithm is started with an arbitrary assignment of binary labels to codewords. A cost function, $\Phi(i)$, was used to weigh the contribution of each pair-wise assignment to the bit error probability as follows: $$\label{eq:C} \Phi(i) = \frac{1}{32} \sum_{\forall j \neq i} d_{i,j} Q\left(\sqrt{\frac{\Delta_{i,j}^{2}}{2 N_0}} \right),$$ where $d_{i,j}$ is the Hamming distance between the datawords assigned to the $i$-th and $j$-th codewords $(i,j = \{0,1,2\dots,31\})$, and $\Delta_{i,j}$ is the Euclidean distance between the same pair $(i,j)$ of OOK-modulated codewords. The algorithm minimizes the overall cost function iteratively by swapping labels between codewords in a pair-wise fashion. The swapping operation start with those codewords that present a greater contribution to $P_b$. The procedure continues until the label swapping operation does not allow any further reduction of the overall cost function. For any given mapping, the overall cost function is computed as $\sum_{i=0}^{31} \Phi(i)$. For large SNR, this function provides a good estimate of the average number of bits received with errors per transmitted symbol. From where, $P_b$ can be approximated as: $$\label{eq:Pb} P_b \sim \frac{1}{5} \sum_{i=0}^{31} \Phi(i).$$ It is important to point out that, depending on the initial binary assignment, the BSA strategy can lead to different final mappings with very similar expected overall error performance. Thus, alternative mappings can be considered to meet specific application demands. Also, other cost functions that reflect a different system model can be easily adopted if required. Simulation Results ================== ![Block diagram of the VLC system model[]{data-label="fig:System"}](Fig1){width="0.85\columnwidth"} For the sake of validating the proposed code, in this section, the results of extensive simulations are presented. The system model used in the simulations is depicted in Fig. \[fig:System\]. The RLL encoder maps $k$-bit datawords to $n$-bit codewords that are subsequently modulated and transmitted over the optical channel. The optical signal is captured by a photo-detector (PD) at the receiver and the correspondent decoding process is performed. Code 5B10B was contrasted to physical layer RLL codes specified for PHY I and PHY II in [@Standard2018]. As recommended, the 4B6B code was modulated using VPPM; with a 50% duty cycle. While Manchester and 8B10B RLL codes, as well as the proposed 5B10B code, were OOK-modulated. The optical channel was modeled as described by Equation (\[eq:Channelmodel\]). In the experiments, the transmission of $10^8$ codewords carrying a random pattern of data bits was considered in calculating the *power spectral density* (PSD), the *symbol error ratio* (SER) and the BER. ![Power Spectral Density estimates, comparison between standardize RLL codes and the proposed 5B10B code.[]{data-label="fig:PSD"}](Fig2){width="0.95\columnwidth"} Fig \[fig:PSD\] shows the PSD for the analysed RLL codes. The frequency scale is normalized by the inverse of the pulse period ($T^{-1}$). As expected, the spectral content of 5B10B is very close to that produced by the 8B10B encoder. The slight increase in the run-length, makes the proposed code to shift a small portion of the spectral components towards low frequencies, when compared to 8B10B. This marginal increase in low frequency spectral components can be perceived in more detail in the zoomed box on the upper right side of Fig \[fig:PSD\]. Except for the DC component, there is a sharp decrease in spectral components below $0.05 \times T^{-1}$Hz. Taking as reference the minimum clock rate specified in [@Standard2018], noticeable oscillations will appear at frequencies greater than $0.05 \times 200\text{KHz}=10\text{KHz}$. This guarantees that the MFTP stays well below the recommended values to minimize health risks associated with low-frequency modulation of lighting sources [@Std2015; @Standard2018]. The error performance of the 5B10B code is shown in Fig \[fig:SER\]. The upper bound of $P_e$, calculated from Equation (\[eq:Pe2\]), is depicted in Fig. \[fig:SER\] with a dark solid line. As the SNR increases, the union bound becomes a tight bound of the symbol error probability. Moreover, for values of the energy per bit to noise power spectral density ratio, $\frac{E_b}{N_0}$, above $9$dB, Equation (\[eq:Pe2\]) offers a good estimate of the actual SER. For $\frac{E_b}{N_0}$ greater than $6.79$dB, the proposed code outperforms Manchester, 4B6B and 8B10B encoders. Particularly, compared to Manchester coding, 5B10B requires approximately $2$dB less of signal energy to offer a SER $=10^{-6}$. It should be noted that in all cases the error performance of the system can be improved by using additional channel coding techniques (as suggested in Fig. \[fig:System\]) at the cost of lowering the data rate. For instance, the output of a RS encoder can be coupled to the input of the RLL encoder. This follows trivially from the well-known result on concatenated codes, [@Forney1965], that the concatenation of an outer code with 5B10B has the potential to double the error correction capabilities compared to its standardized counterparts. However, to achieve this potential improvement, proper selection of the algorithms and coding parameters is required. Also, decoding algorithms such as those proposed in [@Wang2015] and [@Wang2016] can be implemented together with 5B10B in other to improve the overall BER. ![SER curves for different RLL codes, including the observed and theoretically estimated values for the proposed 5B10B code.[]{data-label="fig:SER"}](Fig3){width="0.95\columnwidth"} Table \[table\_3\] summarizes the main characteristics of the proposed RLL code and establishes a comparison with its counterparts. As discussed above, it resembles in some aspects the RLL codes commonly used for VLC. The data rate that can be achieved with 5B10B is equal to that of the Manchester code, while its run-length is very close to that of the 8B10B code. However, differently from standardized RLL codes, it has the ability not only to detect but also to correct errors. Last column of Table \[table\_3\] shows the $\frac{E_b}{N_0}$ required by the RLL codes under study at BER = $10^{-5}$. Since the error performance for an OOK-modulated Manchester encoded transmission is the same as that of uncoded OOK or VPPM (with 50% duty cycle) [@Abshire1984], its energy requirement can be used as benchmark to estimate the coding gain. Hence, it follows that at BER = $10^{-5}$ the coding gain offered by Manchester, 4B6B, 8B10B and 5B10B is: 0dB, 0.43dB, 0.86dB and 2.17dB, respectively. It is worth noting that the achieved coding gain is comparable to those reported in [@Wang2015] and [@Lu2018]. However, in the cited works this output comes at the cost of a significant increase in computational complexity when compared to the current proposal. ------------ ------ -------- ---------------------- ------------------- Run- Spectral $E_b/N_0$ length efficiency (BER = $10^{-5}$) Manchester OOK 2 0.5 bit/s/Hz 12.59 dB 4B6B VPPM 4 $0.\bar{3}$ bit/s/Hz 12.16 dB 8B10B OOK 5 0.8 bit/s/Hz 11.73 dB 5B10B OOK 6 0.5 bit/s/Hz 10.42 dB ------------ ------ -------- ---------------------- ------------------- : Performance comparison between 5B10B and standard enforced RLL codes for VLC[]{data-label="table_3"} Conclusion ========== The proposed 5B10B code is analogous in many aspects to the RLL codes commonly used in VLC applications. Particularly, it ensures strict DC-balance, run-length limited to six consecutive equal bits and $\frac{1}{2}$ code rate. However, different from its standardized counterparts, it offers enhanced error correction capabilities. The main advantage of the proposed 5B10B RLL code is that it provides an extra coding gain (2.17dB at BER = $10^{-5}$, based on simulation results) without incurring an increase in computational complexity, making it a promising candidate for VLC applications. [^1]: V.A. Reguera, is with the Electrical Engineering Graduate Program, Federal University of Santa Maria, RS, Brazil e-mail: (vitalio.reguera@ufsm.br).

--- abstract: 'With a main tool is signed graphs, we give a full description of the characteristic quasi-polynomials of ideals of classical root systems ($ABCD$) with respect to the integer and root lattices. As a result, we obtain a full description of the characteristic polynomials of the toric arrangements defined by these ideals. As an application, we provide a combinatorial verification to the fact that the characteristic polynomial of every ideal subarrangement factors over the dual partition of the ideal in the classical cases.' address: 'Tan Nhat Tran, Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo 060-0810, Japan.' author: - Tan Nhat Tran bibliography: - 'references.bib' date: - - title: 'Characteristic quasi-polynomials of ideals and signed graphs of classical root systems' --- Introduction ============ In recent years, the “finite field method" for studying hyperplane arrangements have been developed, extended and put into practice. Roughly speaking, suppose that the real hyperplane arrangement ${\mathcal{A}}({\mathbb{R}})$ associated to a list ${\mathcal{A}}$ of elements in ${\mathbb{Z}}^\ell$ is given, we can take coefficients modulo a positive integer $q$ and get an arrangement ${\mathcal{A}}({\mathbb{Z}}/q{\mathbb{Z}})$ of subgroups in $({\mathbb{Z}}/q{\mathbb{Z}})^\ell$. The central theorem in the theory asserts that when $q$ is a sufficiently large prime, the arrangement ${\mathcal{A}}({\mathbb{Z}}/q{\mathbb{Z}})$ now is defined over the finite field ${\mathbb{F}}_q$, and the cardinality of its complement $\#{\mathcal{M}}({\mathcal{A}}; {\mathbb{Z}}^\ell, {\mathbb{Z}}/q{\mathbb{Z}})$ coincides with $\chi_{{\mathcal{A}}({\mathbb{R}})}(q)$, the evaluation of the characteristic polynomial $\chi_{{\mathcal{A}}({\mathbb{R}})}(t)$ of ${\mathcal{A}}({\mathbb{R}})$ at $q$ (e.g., [@A96 Theorem 2.2]). Later on, Kamiya-Takemura-Terao showed that $\#{\mathcal{M}}({\mathcal{A}}; {\mathbb{Z}}^\ell, {\mathbb{Z}}/q{\mathbb{Z}})$ is actually a quasi-polynomial in $q$ [@KTT08], and left the task of understanding the constituents of this quasi-polynomial to be an interesting problem. A number of attempts have been made in order to tackle the problem (e.g., [@KTT08], [@LTY17], [@DFM17]), especially, two interpretations for every constituent via subspace and toric viewpoints have been found [@TY18]. The mentioning establishments open a new direction for studying the combinatorics and topology of hyperplane and toric arrangements in one single quasi-polynomial. Much of the motivation for the study of the hyperplane and toric arrangements comes from the arrangements that are defined by irreducible root systems. Apart from the theoretical aspects, the “finite field method" and its toric analogue proved to have efficient applications to compute the characteristic (quasi-)polynomials of several arrangements arising from these vector configurations (e.g., [@A96], [@BS98], [@KTT07], [@ACH15], [@Y18L]). More concrete computational results have also been derived to assist the observation of interesting coincidences, which we choose to mention some important examples in our study. The surprising connection between independent calculations on the Ehrhart quasi-polynomials [@Su98] and the characteristic quasi-polynomials [@KTT07] produced a main flavor to the analysis on the deformations of root system arrangements in [@Y18W]. Combining the computation on the arithmetic Tutte polynomials of classical root systems [@ACH15] with the previously mentioned calculations provided the authors in [@LTY17] with the key observation of the identification between the last constituent of the characteristic quasi-polynomial and the corresponding toric arrangement. Passing from global to local, one may wish to compute the characteristic quasi-polynomials of subsets of a given root system. A particularly well-behaved class of the subsets is that of ideals with the associated ideal subarrangements are proved to be free in the sense of Terao [@ABCHT16]. As a consequence, the characteristic polynomial of every ideal subarrangement factors over the integers with the roots are described combinatorially by the dual partition of the ideal. However, some combinatorial explanations for the factorization may have been hidden because of the freeness. The main goal of this paper is to compute the characteristic quasi-polynomials of the ideals of classical root systems with respect to two different choices of lattices. We were inspired and motivated by the ideas and techniques used in [@KTT07] that will greatly help us in doing so. In addition, we wish to provide more combinatorial insights to the understanding of the constituents in connection with the signed graphs. The remainder of the paper is organized as follows. In Section \[sec:background\], we recall definitions and basic facts of the characteristic quasi-polynomials, irreducible root systems and their ideals. We also recall the constructions of the classical root systems together with the properties of the associated signed graphs. In Section \[sec:ideal\], with a combinatorial ingredient is signed graphs, we compute the characteristic quasi-polynomial of every ideal of a given classical root system with respect to the integer and root lattices. As a result, we obtain a full description of the characteristic polynomials of the toric arrangements defined by the ideals. We will also provide a direct verification to the factorization of the characteristic polynomial of every ideal subarrangement in the classical cases without using the freeness (Theorem \[thm:verify\]). Preliminaries {#sec:background} ============= Characteristic quasi-polynomials -------------------------------- Let $\Gamma:={\mathbb{Z}}^\ell$. Let ${\mathcal{A}}$ be a finite list (multiset) of elements in $\Gamma$. Let $q\in{\mathbb{Z}}_{>0}$. For each $\alpha=(a_1,\ldots,a_\ell) \in {\mathcal{A}}$, define the subgroup $H_{\alpha, {\mathbb{Z}}/q{\mathbb{Z}}}$ of $({\mathbb{Z}}/q{\mathbb{Z}})^\ell$ by $$H_{\alpha, {\mathbb{Z}}/q{\mathbb{Z}}}:=\left\{\textbf{z} = (\overline{z_1}, \ldots, \overline{z_\ell})\in ({\mathbb{Z}}/q{\mathbb{Z}})^\ell \middle| \sum_{i=1}^\ell a_i \overline{z_i}\equiv\overline{0}\right\}.$$ Then the list ${\mathcal{A}}$ determines the *$q$-reduction* arrangement in $({\mathbb{Z}}/q{\mathbb{Z}})^\ell$ $${\mathcal{A}}(\Bbb {\mathbb{Z}}/q{\mathbb{Z}}):=\{H_{\alpha, \Bbb {\mathbb{Z}}/q{\mathbb{Z}}} \mid \alpha \in {\mathcal{A}}\}.$$ The *complement* of ${\mathcal{A}}(\Bbb {\mathbb{Z}}/q{\mathbb{Z}})$ is defined by $${\mathcal{M}}({\mathcal{A}}; {\mathbb{Z}}^\ell,{\mathbb{Z}}/q{\mathbb{Z}}):= ({\mathbb{Z}}/q{\mathbb{Z}})^\ell \smallsetminus\bigcup_{\alpha\in{\mathcal{A}}}H_{\alpha,{\mathbb{Z}}/q{\mathbb{Z}}}.$$ For each ${\mathcal{S}}\subseteq {\mathcal{A}}$, write $\Gamma/\langle{\mathcal{S}}\rangle\simeq\bigoplus_{i=1}^{n_{{\mathcal{S}}}}{\mathbb{Z}}/d_{{\mathcal{S}}, i}{\mathbb{Z}}\oplus {\mathbb{Z}}^{r_{\Gamma}-r_{{\mathcal{S}}}}$ where $n_{{\mathcal{S}}}\geq 0$ and $1<d_{{\mathcal{S}}, i}|d_{{\mathcal{S}}, i+1}$. The *LCM-period* $\rho_{{\mathcal{A}}}$ of ${\mathcal{A}}$ is defined by $$\label{eq:LCM-period} \rho_{\mathcal{A}}:={\operatorname{lcm}}(d_{{\mathcal{S}}, n_{{\mathcal{S}}}}\mid{\mathcal{S}}\subseteq {\mathcal{A}}).$$ It is proved in [@KTT08 Theorem 2.4] that $\#{\mathcal{M}}({\mathcal{A}}; {\mathbb{Z}}^\ell, {\mathbb{Z}}/q{\mathbb{Z}})$ is a monic quasi-polynomial in $q$ for which $\rho_{{\mathcal{A}}}$ is a period. The quasi-polynomial is called the *characteristic quasi-polynomial* of ${\mathcal{A}}$ (or of ${\mathcal{A}}({\mathbb{Z}}/q{\mathbb{Z}})$), and denoted by $\chi^{{\operatorname{quasi}}}_{{\mathcal{A}}}(q)$. More precisely, there exist monic polynomials $f_{{\mathcal{A}}}^k(t)\in{\mathbb{Z}}[t]$ ($1 \le k \le \rho_{\mathcal{A}}$) such that for any $q\in{\mathbb{Z}}_{>0}$ with $q\equiv k\bmod \rho_{\mathcal{A}}$, $$\chi^{{\operatorname{quasi}}}_{{\mathcal{A}}}(q) =f^k_{\mathcal{A}}(q).$$ The polynomial $f^k_{\mathcal{A}}(t)$ is called the *$k$-constituent* of $\chi^{{\operatorname{quasi}}}_{{\mathcal{A}}}(q)$. It is known that (e.g., [@A96], [@KTT08]) the $1$-constituent $f^1_{\mathcal{A}}(t)$ coincides with $\chi_{{\mathcal{A}}({\mathbb{R}})}(t)$ the characteristic polynomial (e.g., [@OT92 Definition 2.52]) of the real hyperplane arrangement (or ${\mathbb{R}}$-plexification in the sense of [@LTY17]) ${\mathcal{A}}({\mathbb{R}})=\{H_{\alpha, \Bbb {\mathbb{R}}} \mid \alpha \in {\mathcal{A}}\}$ with $H_{\alpha, {\mathbb{R}}}:=\left\{\textbf{x} \in {\mathbb{R}}^\ell \middle| \sum_{i=1}^\ell a_ix_i=0\right\}$. Root systems and signed graphs {#subsec:Root-systems} ------------------------------ Our standard reference for root systems is [@B68]. Let $V$ be an $\ell$-dimensional Euclidean space with the standard inner product $(\cdot,\cdot)$. Let $\Phi$ be an irreducible (crystallographic) root system in $V$. Fix a positive system $\Phi^+ \subseteq \Phi$ and the associated set of simple roots (base) $\Delta := \{\alpha_1,\ldots,\alpha_\ell \} \subseteq\Phi^+$. For $\beta = \sum_{i=1}^\ell n_i\alpha_i \in \Phi^+$, the *height* of $\beta$ is defined by $ {\rm ht}(\beta) := \sum_{i=1}^\ell n_i $. *Notation*: For simplicity of notation, we use the same symbol $M$ for the realization of the matrix $M$ of size $\ell \times m$ as the finite list of elements in $\Gamma={\mathbb{Z}}^\ell$ whose elements are the columns of $M$. For each $\Psi\subseteq\Phi^+$, we assume that an $\ell \times \#\Psi$ integral matrix $S_\Psi = [S_{ij}]$ satisfies $$\Psi = \left\{\sum_{i=1}^\ell S_{ij}\alpha_i \,\middle|\, 1 \le j \le \#\Psi \right\}.$$ In other words, $S_\Psi$ is the coefficient matrix of $\Psi$ with respect to the base $\Delta$. Denote $({\mathbb{Z}}/q{\mathbb{Z}})^\times:={\mathbb{Z}}/q{\mathbb{Z}}\smallsetminus \{\overline{0}\}$. We then call $\chi^{{\operatorname{quasi}}}_{S_\Psi}(\Phi, q)$ the characteristic quasi-polynomial of $\Psi$ with respect to the *root lattice*, and interpret it by the formula $$\chi^{{\operatorname{quasi}}}_{S_\Psi}(\Phi, q)=\#\{\textbf{z}\in ({\mathbb{Z}}/q{\mathbb{Z}})^\ell \mid\textbf{z}\cdot S_\Psi \in (({\mathbb{Z}}/q{\mathbb{Z}})^\times)^{\#\Psi}\}.$$ We define ${{\mathcal{H}}}_{\Psi}:= \{H_\alpha \mid \alpha\in\Psi\}$, where $H_\alpha=\{x\in V \mid (\alpha,x)=0\}$ is the hyperplane orthogonal to $\alpha$. It is not hard to see that ${{\mathcal{H}}}_{\Psi}$ is the ${\mathbb{R}}$-plexification of $S_\Psi$ i.e., ${{\mathcal{H}}}_{\Psi}=S_\Psi({\mathbb{R}})$. Note also that ${{\mathcal{H}}}_{\Phi^+}$ is called the *Weyl arrangement* of $\Phi^+$, and ${{\mathcal{H}}}_{\Psi}$ is a Weyl subarrangement. In the remainder of the paper, we are mainly interested in the root system $\Phi$ of classical type ($ABCD$). Let us recall briefly the constructions of these root systems[^1] following [@B68 Chapter VI, $\S$4]. Let $\{\epsilon_1, \ldots, \epsilon_{\ell}\}$ be an orthonormal basis for $V$. If $\ell \ge 2$ then $$\Phi(B_\ell) = \{\pm\epsilon_i \,(1 \le i \le \ell),\pm(\epsilon_i \pm \epsilon_j) \,(1 \le i < j \le \ell )\},$$ with $\#\Phi(B_\ell) =2\ell^2$ is an irreducible root system in $V$ of type $B_{\ell}$. We may choose a positive system $$\Phi^+(B_\ell) = \{\epsilon_i \,(1 \le i \le \ell), \epsilon_i \pm \epsilon_j \,(1 \le i < j \le \ell )\}.$$ Define $\alpha_i := \epsilon_i - \epsilon_{i+1}$, for $1 \le i \le \ell-1$, and $\alpha_{\ell} := \epsilon_{\ell}$. Then $\Delta(B_\ell) = \{\alpha_1,\ldots, \alpha_{\ell}\}$ is the base associated with $\Phi^+(B_\ell) $. We may express $$\begin{aligned} \Phi^+(B_\ell) & = \{ \epsilon_i=\sum_{i\le k \le \ell }\alpha_k \,(1 \le i \le \ell), \epsilon_i-\epsilon_j=\sum_{i\le k < j }\alpha_k \,(1 \le i < j \le \ell), \\ & \epsilon_i+\epsilon_j=\sum_{i\le k<j }\alpha_k+2 \sum_{j\le k \le \ell }\alpha_k\,(1 \le i < j \le \ell) \}.\end{aligned}$$ For any $\Psi\subseteq\Phi^+(B_\ell)$, we write $T_\Psi = [T_{ij}]$ for the coefficient matrix of $\Psi$ with respect to the orthonormal basis. We then call $\chi^{{\operatorname{quasi}}}_{T_\Psi}(\Phi, q)$ the characteristic quasi-polynomial of $\Psi$ with respect to the *integer lattice*. The matrices $T_\Psi$ and $S_\Psi$ are related by $T_\Psi=P(B_\ell)\cdot S_\Psi$, where $P(B_\ell) $ is an unimodular matrix of size $\ell \times \ell$ given by $$P(B_\ell) =\begin{bmatrix} 1 & & & & \\ -1 & 1 & & & \\ & -1 & & & \\ & & \ddots & & \\ & & & 1 & \\ & & & -1 & 1 \end{bmatrix}.$$ Similarly, let $\ell \ge 2$, an irreducible root system of type $C_{\ell}$ is given by $$\begin{aligned} \Phi(C_\ell) &= \{\pm2\epsilon_i \,(1 \le i \le \ell),\pm(\epsilon_i \pm \epsilon_j) \,(1 \le i < j \le \ell )\}, \\ \Phi^+(C_\ell) &= \{2\epsilon_i \,(1 \le i \le \ell), \epsilon_i \pm \epsilon_j \,(1 \le i < j \le \ell )\}, \\ \Delta(C_\ell) &=\{\alpha_i = \epsilon_i - \epsilon_{i+1}\,(1 \le i \le \ell-1),\, \alpha_{\ell} =2\epsilon_{\ell} \} ,\\ \Phi^+(C_\ell) &= \{ 2\epsilon_i=2\sum_{i\le k< \ell }\alpha_k+\alpha_\ell \,(1 \le i \le \ell), \epsilon_i-\epsilon_j=\sum_{i\le k<j }\alpha_k \,(1 \le i < j \le \ell), \\ & \epsilon_i+\epsilon_j=\sum_{i\le k<j }\alpha_k+2 \sum_{j\le k <\ell }\alpha_k+\alpha_\ell\,(1 \le i < j \le \ell) \}.\end{aligned}$$ Finally, let $\ell \ge 3$, an irreducible root system of type $D_{\ell}$ is given by $$\begin{aligned} \Phi(D_\ell) &= \{\pm(\epsilon_i \pm \epsilon_j) \,(1 \le i < j \le \ell )\}, \\ \Phi^+(D_\ell) &= \{ \epsilon_i \pm \epsilon_j \,(1 \le i < j \le \ell )\}, \\ \Delta(D_\ell) &=\{\alpha_i = \epsilon_i - \epsilon_{i+1}\,(1 \le i \le \ell-1),\, \alpha_{\ell} =\epsilon_{\ell-1}+\epsilon_{\ell} \},\\ \Phi^+(D_\ell) &=\{ \epsilon_i + \epsilon_{\ell}=\sum_{i\le k \le \ell-2 }\alpha_k+\alpha_\ell \,(1 \le i< \ell), \\ & \epsilon_i - \epsilon_{j}=\sum_{i< k < j }\alpha_k \,(1 \le i < j \le \ell), \\ & \epsilon_i + \epsilon_{j}=\sum_{i\le k<j }\alpha_k+2 \sum_{j\le k <\ell-1 }\alpha_k+\alpha_{\ell-1}+\alpha_\ell\,(1 \le i < j< \ell) \}.\end{aligned}$$ From the constructions above, we obtain the comparison of the height placements of positive roots in $\Phi(B_\ell)$, $\Phi(C_\ell)$ and $\Phi(D_\ell)$ as in Table \[tab:placement\]. In the language of *signed graphs* following [@Z81 §5], we can associate to each subset $\Psi\subseteq\Phi^+(B_\ell)$ a signed graph $G:=G(\Psi)=(V_G, E_{G^+},E_{G^-},L_G)$ on the vertex set $$V_G:=\{v_i,v_j \mid \mbox{$\epsilon_i \in \Psi$ or $\epsilon_i-\epsilon_{j} \in \Psi$ or $\epsilon_i+\epsilon_{j} \in \Psi$} \},$$ with the set of positive edges $E_{G^+}:=\{e_{ij}^+ \mid \epsilon_i + \epsilon_{j} \in \Psi\}$, the set of negative edges $E_{G^-}:=\{e_{ij}^- \mid \epsilon_i - \epsilon_{j} \in \Psi\}$, and the set of loops $L_G:=\{\ell_i \mid \epsilon_i \in \Psi\}$. Alternatively, if $\Psi\subseteq\Phi^+(C_\ell)$, we can define $L_G:=\{\ell_i \mid 2\epsilon_i \in \Psi\}$. To extract information from $\Psi$ by using $G(\Psi)$, we associate to it an unordered sequence of nonnegative integers, denoted ${{\mathcal{SG}}}(\Psi):=(p_1,\ldots,p_\ell)$ with for each $i$ ($1 \le i \le \ell$) $$\label{eq:di-signed-graphs} p_i :=\# \{e_{ij}^+ \mid e_{ij}^+ \in E_{G^+}\}+ \# \{e_{ij}^- \mid e_{ij}^- \in E_{G^-}\} + \# \{\ell_{i} \mid \ell_{i} \in L_{G}\}.$$ Ideals {#subsec:ideals} ------ Define the partial order $\succeq$ on $\Phi^+$ such that $\beta_1 \succeq \beta_2$ if and only if $\beta_1-\beta_2 = \sum_{i=1}^\ell n_i\alpha_i$ with all $n_i \in {\mathbb{Z}}_{\ge 0}$. A subset $I$ of $ \Phi^+$ is called an *ideal* if, for $\beta_1,\beta_2 \in \Phi^+$, $\beta_1 \succeq \beta_2, \beta_ 1 \in I$ then $\beta_2 \in I$. Let us recall the recent advance towards the study of the ideals. Let $\Theta^{(k)} \subseteq \Phi^+$ be the set consisting of positive roots of height $k$. Let $I$ be an ideal of $ \Phi^+$ and set $M:=\max\{{\rm ht}(\beta)\mid \beta \in I\}$. The *height distribution* of $I$ is defined as a sequence of positive integers: $$(i_1, \ldots , i_k, \ldots , i_{M}),$$ where $i_k := \#\Theta^{(k)}$ for $1 \le k \le M$. The *dual partition* ${{\mathcal{DP}}}(I)$ of (the height distribution of) $I$ is given by a sequence of nonnegative integers: $${{\mathcal{DP}}}(I) := \left( (0)^{\ell-i_1},(1)^{i_1-i_2},\ldots ,(M-1)^{i_{M-1}-i_{M}},(M)^{i_{M}}\right),$$ where notation $(a)^b$ means the integer $a$ appears exactly $b$ times. Although the definition of the dual partition seems to esteem the (increasing) order of components in the sequence, this requirement is not important in this paper. Two dual partitions of an ideal are conventionally identical if the partitions differ only by a re-ordering of the components. \[thm:dual\] Any ideal subarrangement ${{\mathcal{H}}}_I$ is free (in the sense of Terao) with the set of exponents coincides with ${{\mathcal{DP}}}(I)$. \[cor:ideal-factorization\] For any ideal $I\subseteq \Phi^+$, the characteristic polynomial $\chi_{{{\mathcal{H}}}_I}(\Phi, t)$ factors as follows: $$\label{eq:deal-factorization} \chi_{{{\mathcal{H}}}_I}(\Phi,t)= \prod_{i=1}^\ell (t-d_i),$$ where ${{\mathcal{DP}}}(I)=(d_1,\ldots,d_\ell)$. When $\Phi$ is of type $B_\ell$ or $C_\ell$, ${{\mathcal{DP}}}(I)={{\mathcal{SG}}}(I)$, while ${{\mathcal{DP}}}(I)\ne{{\mathcal{SG}}}(I)$ if $\Phi$ is of type $D_\ell$. Also, ${{\mathcal{SG}}}(I)$ does not determine $I$, for instance, $I_1=\{\epsilon_4 - \epsilon_5\}$ and $I_2=\{\epsilon_4 + \epsilon_5\}$ are distinct ideals of $\Phi^+(D_5)$, but ${{\mathcal{SG}}}(I_1)={{\mathcal{SG}}}(I_2)=(0,0,0,1,0)$. Computation on ideals {#sec:ideal} ===================== In the remainder of the paper, we assume that $\Phi$ is of classical type. We summarize some easy cases that the computation of the characteristic quasi-polynomials is manageable thanks to Corollary \[cor:ideal-factorization\]. The minimum period coincides with the LCM-period [@KTT10 Remark 3.3]. So the minimum period of $\chi^{{\operatorname{quasi}}}_{S_I}(A_\ell, q)$ is $1$ for every $I\subseteq\Phi^+$; hence $\chi^{{\operatorname{quasi}}}_{S_I}(A_\ell, q)=\chi_{{{\mathcal{H}}}_I}(A_\ell,q)$. For other cases, the minimum period of $\chi^{{\operatorname{quasi}}}_{S_I}(\Phi,q)$ is at most $2$; hence we know the $1$-constituents: $f^1_{S_I}(\Phi,t)=\chi_{{{\mathcal{H}}}_I}(\Phi,t)$. We are left with the task of determining $f^2_{S_I}(\Phi, t)$, or equivalently, $\chi^{{\operatorname{quasi}}}_{S_I}(\Phi, q)$ when $q$ is even, and $\Phi$ is of type $B$, $C$ or $D$. Turning the problem around, we would like to verify Corollary \[cor:ideal-factorization\] by using the information of ideals via signed graphs without relying on the freeness, which we will do in Theorem \[thm:verify\]. Type $B$ root systems --------------------- By [@KTT07 Theorem 4.1][^2], if $\Psi\subseteq\Phi^+(B_\ell)$, $$\label{eq:B-TS} \chi^{{\operatorname{quasi}}}_{S_\Psi}(B_\ell, q)=\chi^{{\operatorname{quasi}}}_{T_\Psi}(B_\ell, q).$$ Let $I$ be an ideal of $\Phi^+(B_\ell)$. Assume that ${{\mathcal{DP}}}(I)={{\mathcal{SG}}}(I)=(d_1,\ldots,d_\ell)$. For each $k$ $(1 \le k \le \ell)$, write $d_k=d_k^{(+)}+d_k^{(-)}+d_k^{(0)}$ for a partition of $d_k$ with $$\label{eq:partition-dk} \begin{aligned} d_k^{(0)} & := \begin{cases} 0 \quad\mbox{ if $\epsilon_k \notin I$}, \\ 1 \quad\mbox{ if $\epsilon_k \in I$}. \end{cases} \\ d_k^{(\pm)} & :=\#\{\epsilon_k \pm\epsilon_{j} \mid \epsilon_k \pm\epsilon_{j} \in I \}. \end{aligned}$$ The partitions give a partition of $I$ which we call it the *$B$-partition*, as follows: $I =I^{0} \sqcup I^{-} \sqcup I^{+},$ where $$\label{eq:B-partition} \begin{aligned} I^{0} & := \{\epsilon_i \mid \epsilon_i \in I\}\\ I^{\pm} & := \{\epsilon_i\pm\epsilon_j \mid \epsilon_i \pm\epsilon_{j} \in I \}. \end{aligned}$$ If $\epsilon_i + \epsilon_{j} \notin I$ for all $i, j$ (type $A$), then for all $q \in {\mathbb{Z}}_{>0}$, $$\label{eq:type-A} \chi^{{\operatorname{quasi}}}_{T_I}(B_\ell, q) = \prod_{i=1}^\ell (q-d_i).$$ Now assume that some $\epsilon_i + \epsilon_{j} \in I$ with $1 \le i < j \le \ell$. In particular, $\epsilon_k \in I$ for all $i \le k \le \ell$. Set $s:=\min \{ 1 \le k \le \ell \mid \epsilon_k \in I\}$. Denote $R:=I \smallsetminus \{\epsilon_i - \epsilon_{j} \in I \mid 1\le i<s, i <j \le \ell\}$. Thus $R$ is an ideal of the root subsystem of $\Phi(B_\ell)$ of type $B_{\ell-s+1}$ with a base given by $\Delta(B_{\ell-s+1})=\{\alpha_s,\ldots,\alpha_\ell\}$. Furthermore, for all $q \in {\mathbb{Z}}_{>0}$, we have $$\label{eq:inductive-B} \chi^{{\operatorname{quasi}}}_{T_I}(B_\ell, q) =\chi^{{\operatorname{quasi}}}_{T_R}(B_{\ell-s+1}, q)\cdot \prod_{i=1}^{s-1}(q-d_i).$$ Then it suffices to consider $s=1$ i.e., $\epsilon_1 \in I$. For such ideals, $d_k^{(-)} = \ell-k$, $d_k^{(0)}=1$ for $1 \le k \le \ell$. \[lem:B-D\] Let $I$ be an ideal of $\Phi^+(B_\ell)$ with $\epsilon_1 \in I$. Set $J:=I \smallsetminus I^{0}$. (a) $J$ is an ideal of $\Phi^+(D_\ell)$. (b) ${{\mathcal{DP}}}(J)=(p_1,\ldots,p_\ell)$ with $p_k=d_k^{(-)}+d_{k-1}^{(+)}$ for all $1 \le k \le \ell$. Here we agree that $d_{0}^{(2)} \equiv 0$. The proof of (a) is straightforward by the definition of ideals. The proof of (b) follows from the height placements in Table \[tab:placement\]. \[thm:B-and-D\] Under the Lemma \[lem:B-D\]’s assumptions, if $q \in {\mathbb{Z}}_{>0}$ is even, $$\label{eq:B-D} \chi^{{\operatorname{quasi}}}_{T_{I}}(B_\ell, q)=\chi^{{\operatorname{quasi}}}_{T_J}(D_\ell, q-1)= \prod_{i=1}^\ell (q-p_i-1).$$ The proof of the first equality is similar to (but more general than) that of [@KTT07 Lemma 4.4(11)]. $$\begin{aligned} \chi^{{\operatorname{quasi}}}_{T_I}(B_\ell, q) &=\# \left\{ \textbf{z} \in ({\mathbb{Z}}/q{\mathbb{Z}})^\ell \middle| \begin{array}{c} \overline{z_i} \ne \overline{z_j}\,(1 \le i <j \le \ell), \\ \overline{z_i} + \overline{z_j}\ne \overline0\,(\epsilon_i+\epsilon_{j} \in I), \\ \overline{z_i}\ne \overline0\,(1 \le i \le \ell) \end{array} \right\} \\ &= \#\left\{(z_1,\ldots,z_\ell) \in {\mathbb{Z}}^\ell \middle| \begin{array}{c} z_i \ne z_j \,(1 \le i <j \le \ell), \\ z_i +z_j \ne q\,(\epsilon_i+\epsilon_{j} \in I), \\ 1 \le z_i \le q-1\,(1 \le i \le \ell) \end{array} \right\} \\ &= \#\left\{(v_1,\ldots,v_\ell) \in {\mathbb{Z}}^\ell \middle| \begin{array}{c} v_i \ne v_j \,(1 \le i <j \le \ell), \\ v_i +v_j \ne 0\,(\epsilon_i+\epsilon_{j} \in I), \\ -\frac{q-2}2 \le v_i \le \frac{q-2}2\,(1 \le i \le \ell) \end{array} \right\} \\ &= \#\left\{(t_1,\ldots,t_\ell) \in {\mathbb{Z}}^\ell \middle| \begin{array}{c} t_i \ne t_j\,(1 \le i <j \le \ell), \\ t_i +t_j \ne q-1\,(\epsilon_i+\epsilon_{j} \in I), \\ 0 \le t_i \le q-2\,(1 \le i \le \ell) \end{array} \right\} \\ &=\# \left\{ \textbf{u} \in ({\mathbb{Z}}/(q-1){\mathbb{Z}})^\ell \middle| \begin{array}{c} \overline{u_i} \ne \overline{u_j}\,(1 \le i <j \le \ell), \\ \overline{u_i} + \overline{u_j}\ne \overline0\,(\epsilon_i+\epsilon_{j} \in J) \end{array} \right\}\\ &= \chi^{{\operatorname{quasi}}}_{T_J}(D_\ell, q-1).\end{aligned}$$ We have used the following changes of variables $$v_i =z_i- \frac{q}2 \quad \mbox{and}\quad t_i = \begin{cases} v_i \mbox{ if } v_i \ge 0, \\ v_i+q-1 \mbox{ if } v_i< 0. \end{cases}$$ The second equality follows from Lemma \[lem:B-D\] and Corollary \[cor:ideal-factorization\]. \[eg:B-parition\] Table \[tab:B5-Bpartition\] shows the $B$-partition of an ideal $I=\{\alpha \in \Phi^+(B_5) \mid {\rm ht}(\alpha) \le 7\}$ (in colored region), with $I^{0}, I^{-}, I^{+}$ are colored in red, yellow, blue, respectively. Table \[tab:D5-partition-J\] shows the corresponding partition of the ideal $J=I \smallsetminus I^{0}$ in $\Phi^+(D_5)$. In this case, ${{\mathcal{DP}}}(I)=(7,7,5,3,1)$ and ${{\mathcal{DP}}}(J)=(4,5,5,3,1)$. Hence for even $q \in {\mathbb{Z}}_{>0}$, we have $$\chi^{{\operatorname{quasi}}}_{T_{I}}(B_\ell, q)=(q-2)(q-4)(q-5)(q-6)^2.$$ \[rem:B-recover\] When $I=\Phi^+(B_\ell)$, ${{\mathcal{DP}}}(I)=(2\ell-1,2\ell-3,\ldots,3,1)$ and ${{\mathcal{DP}}}(J)=(\ell-1,2\ell-3,\ldots,3,1)$. Thus for even $q \in {\mathbb{Z}}_{>0}$, we have $$\chi^{{\operatorname{quasi}}}_{S_{\Phi^+}}(B_\ell, q)=\chi^{{\operatorname{quasi}}}_{T_{\Phi^+}}(B_\ell, q)=(q-2)(q-4)\ldots(q-(2\ell-2))(q-\ell),$$ which recovers the result of [@KTT07 Theorem 4.8] for type $B$ root systems. Type $C$ root systems --------------------- By [@KTT07 Theorem 4.1], for any $\Psi\subseteq\Phi^+(C_\ell)$ and for even $q \in {\mathbb{Z}}_{>0}$, we have $$\label{eq:C-TS} \chi^{{\operatorname{quasi}}}_{S_\Psi}(C_\ell, q)=\frac12\left( \chi^{{\operatorname{quasi}}}_{T_\Psi}(C_\ell, q)+F_\Psi(C_\ell, q)\right),$$ $$\begin{aligned} F_\Psi(C_\ell, q) & :=\# \{ \textbf{z}\in ({\mathbb{Z}}/q{\mathbb{Z}})^\ell \mid \textbf{z}\cdot T_\Psi+\textbf{g}\cdot S_\Psi\in (({\mathbb{Z}}/q{\mathbb{Z}})^\times)^{\#\Psi}\}, \\ \textbf{g} & := (\overline0,\overline0,\ldots,\overline1) \in ({\mathbb{Z}}/q{\mathbb{Z}})^\ell .\end{aligned}$$ Let $I$ be an ideal of $\Phi^+(C_\ell)$ with ${{\mathcal{DP}}}(I)={{\mathcal{SG}}}(I)=(d_1,\ldots,d_\ell)$. We need only consider $\epsilon_i + \epsilon_{j} \in I$ for some $1 \le i < j \le \ell$. In particular, $2\epsilon_k \in I$ for all $j \le k \le \ell$. Set $s:=\min \{ 1 \le k \le \ell \mid 2\epsilon_k \in I\}$. Define $R:=I \smallsetminus \{ \epsilon_i \pm \epsilon_{j} \in I \mid 1\le i<s, i <j \le \ell\}$. Thus $R$ itself is the positive system of a root system of type $C_{\ell-s+1}$ with a base given by $\Delta(C_{\ell-s+1})=\{\alpha_s,\ldots,\alpha_\ell\}$. Furthermore, for all $q \in {\mathbb{Z}}_{>0}$, we have $$\label{eq:inductive-C} \begin{aligned} \chi^{{\operatorname{quasi}}}_{T_I}(C_\ell, q) & =\chi^{{\operatorname{quasi}}}_{T_R}(C_{\ell-s+1}, q)\cdot \prod_{i=1}^{s-1}(q-d_i),\\ F_I(C_\ell, q) &= F_R(C_{\ell-s+1}, q)\cdot \prod_{i=1}^{s-1}(q-d_i). \end{aligned}$$ Then it suffices to consider $s=1$ or equivalently, $I=\Phi^+(C_\ell)$. The computations of $\chi^{{\operatorname{quasi}}}_{T_{\Phi^+}}(C_\ell, q)$, $F_{\Phi^+}(C_\ell, q)$ and $\chi^{{\operatorname{quasi}}}_{S_{\Phi^+}}(C_\ell, q)$ were already done in [@KTT07 Theorem 4.7 and §4.3]. More direct computations are also obtainable. For instance, when $q$ is even, we have $$\begin{aligned} \chi^{{\operatorname{quasi}}}_{T_{\Phi^+}}(C_\ell, q) & =\# \{ \textbf{z} \in ({\mathbb{Z}}/q{\mathbb{Z}})^\ell \mid \overline{z_i} \notin \{ \overline0, \overline{q/2},\pm\overline{z_j}\}, 1 \le i <j \le \ell \}\\ &= \prod_{i=1}^{\ell}(q-(d_i+1)).\end{aligned}$$ \[eg:C-ideal\] Table \[tab:C5-ideal\] shows an example of an ideal $I\subsetneq \Phi^+(C_5)$ (in enclosed region). In this case, ${{\mathcal{DP}}}(I)=(4,6,5,3,1)$. Hence for even $q \in {\mathbb{Z}}_{>0}$, we have $$\begin{aligned} \chi^{{\operatorname{quasi}}}_{T_{I}}(C_\ell, q) & =(q-6)^2(q-4)^2(q-2),\\ F_{\Phi^+}(C_\ell, q) &= (q-6)(q-4)^2(q-2)q,\\ \chi^{{\operatorname{quasi}}}_{S_{I}}(C_\ell, q) & = (q-6)(q-4)^2(q-3)(q-2).\end{aligned}$$ Type $D$ root systems --------------------- The computation on this type requires a bit more effort. By [@KTT07 Theorem 4.1], if $\Psi\subseteq\Phi^+(D_\ell)$ and $q$ is even, $$\label{eq:D-TS} \chi^{{\operatorname{quasi}}}_{S_\Psi}(D_\ell, q)=\frac12\left( \chi^{{\operatorname{quasi}}}_{T_\Psi}(D_\ell, q)+F_\Psi(D_\ell, q)\right),$$ $$\begin{aligned} F_\Psi(D_\ell, q) & :=\# \{ \textbf{z}\in ({\mathbb{Z}}/q{\mathbb{Z}})^\ell \mid \textbf{z}\cdot T_\Psi+\textbf{g}\cdot S_\Psi\in (({\mathbb{Z}}/q{\mathbb{Z}})^\times)^{\#\Psi}\}, \\ \textbf{g} & := (\overline0,\overline0,\ldots,\overline1) \in ({\mathbb{Z}}/q{\mathbb{Z}})^\ell .\end{aligned}$$ Let $I$ be an ideal of $\Phi^+(D_\ell)$. We need only consider $\epsilon_i + \epsilon_{j} \in I$ for some $1 \le i < j \le \ell$. In particular, $\epsilon_{\ell-1}+\epsilon_\ell \in I$. Define $$\label{eq:s-D} s :=\min \{ 2 \le k \le \ell \mid \epsilon_{k-1}+\epsilon_k \in I\}.$$ If $\epsilon_{\ell-1}-\epsilon_{\ell} \notin I$, we must have $s=\ell$. Then the computation can be reduced (up to a bijection) to that on the type $A$ root systems, which can be done easily. Suppose henceforth that $\epsilon_{\ell-1}-\epsilon_{\ell} \in I$, and set $$\label{eq:r-D} r :=\min \{ 1 \le k \le \ell \mid \epsilon_{k}+\epsilon_{\ell} \in I \,\,\mbox{and}\,\, \epsilon_{k}-\epsilon_{\ell} \in I\}.$$ Obviously, $r \le s-1$. Assume that ${{\mathcal{SG}}}(I)=(p_1,\ldots,p_\ell)$ with for each $i$ $$\label{eq:di-signed-graphs-D} p_i = \# \{ \epsilon_i - \mu_{i,j}\epsilon_{j} \in I \mid \mu_{i,j} \in \{\pm1\}\}.$$ Define $R:=I \smallsetminus \{ \epsilon_i \pm\epsilon_{j} \in I \mid 1\le i<r, i <j \le \ell\}$. Thus $R$ is an ideal of the root subsystem of $\Phi(D_\ell)$ of type $D_{\ell-r+1}$ with a base given by $\Delta(D_{\ell-r+1})=\{\alpha_r,\ldots,\alpha_\ell\}$. Furthermore, for all $q \in {\mathbb{Z}}_{>0}$, we have $$\label{eq:inductive-D} \begin{aligned} \chi^{{\operatorname{quasi}}}_{T_I}(D_\ell, q) & =\chi^{{\operatorname{quasi}}}_{T_R}(D_{\ell-r+1}, q)\cdot \prod_{i=1}^{r-1}(q-p_i),\\ F_I(D_\ell, q) &= F_R(D_{\ell-r+1}, q)\cdot \prod_{i=1}^{r-1}(q-p_i). \end{aligned}$$ Then it suffices to consider $r=1$ i.e., $\epsilon_1\pm\epsilon_{\ell} \in I$. For such ideals, $p_{i}^{(-)} = p_{i+1}^{(-)}+1=\ell-i$ for $1 \le i \le \ell-1$. Moreover, the subset $\{ \epsilon_i \pm \epsilon_{j} \mid s-1 \le i<j \le \ell\}\subseteq I$ is the positive system of a root subsystem of $\Phi(D_\ell)$ of type $D_{\ell-s+2}$. Thus $p_i \le p_{i+1}+1$, $p_{i}^{(+)} \le p_{i+1}^{(+)}$ for all $1 \le i \le s-3$, and $p_i+2=p_{i-1}$ for $s \le i \le \ell$. We will need the following lemma. \[lem:needed-B\] Let $\Psi$ be a subset of $\Phi^+(B_\ell)$ such that $\{ \epsilon_i \pm \epsilon_{j} \mid s-1 \le i<j \le \ell\}\subseteq \Psi$ for some $2 \le s \le \ell$. Assume that ${{\mathcal{SG}}}(\Psi)=(p_1,\ldots,p_\ell)$ with $p_i \le p_{i+1}+1$ for all $1 \le i \le s-3$. Then $\Psi$ is an ideal of $\Phi^+(B_{\ell})$. For $\beta_1,\beta_2 \in \Phi^+(B_\ell)$, $\beta_1 \succeq \beta_2$, $\beta_ 1 \in \Psi$, we will prove that $\beta_2 \in \Psi$. Note that for each $\beta\in \Phi^+(B_\ell)$, we have $\# \{ \gamma \in \Phi^+(B_\ell) \mid \beta - \gamma \in \Delta(B_\ell)\} \le 2$. Since $\beta_1 \succeq \beta_2$, there exists a path in the Hasse diagram of $\Phi^+(B_\ell)$ connecting $\beta_1$ and $\beta_2$.[^3] It follows that this path must lie entirely within $\Psi$, yielding $\beta_2 \in \Psi$. Let $\Pi$ be an irreducible root system of type $B_{\ell-1}$ with a base given by $\Delta=\{\alpha_i = \epsilon_i - \epsilon_{i+1}\,(1 \le i \le \ell-2),\, \alpha_{\ell-1} =\epsilon_{\ell-1} \}$. We define a sequence of subsets $\{U_k\}_{k=1}^\ell$ (depending on $I$) of $\Pi^+(B_{\ell-1})$ classified into two types as follows: (i) Type I, $$\label{eq:1st-Uk} {{\mathcal{SG}}}(U_k)=(p_1,\ldots,p_{k-1}, \widehat{p_{k}}, p_{k+1}+1,\ldots,p_{\ell}+1),$$ for $1 \le k \le s-2$. Here $\widehat{p_{k}}$ means omission. (ii) Type II, $$\label{eq:2nd-Uk} {{\mathcal{SG}}}(U_k)=(p_1-\tau_{1,k},\ldots,p_{s-2}-\tau_{s-2,k}, p_{s-1}-1,\ldots,p_{\ell-1}-1),$$ for $s-1\le k \le \ell$, $1 \le n \le s-2$, with $$\tau_{n,k}:= \begin{cases} 0 \quad\mbox{ if $\epsilon_n+\epsilon_k \notin I$} \\ 1 \quad\mbox{ if $\epsilon_n+\epsilon_k \in I$.} \end{cases}$$ It is easily seen that $\tau_{n,k} \le \tau_{n+1,k}$ (as well as $\tau_{n,k} \le \tau_{n,k+1}$), hence $p_{n}-\tau_{n,k} \le p_{n+1}-\tau_{n+1,k}+1$ for all $1 \le n \le s-3$. By Lemma \[lem:needed-B\], the subsets $\{U_k\}_{k=1}^\ell$ are indeed ideals of $\Pi^+(B_{\ell-1})$. We also define $$\label{eq:D-2nd-summand} K:=I \sqcup \{\epsilon_k \mid 1 \le k \le \ell\}.$$ Then again by Lemma \[lem:needed-B\], $K$ is an ideal of $\Phi^+(B_{\ell})$ with $$\label{eq:DP(K)} {{\mathcal{SG}}}(K)=(p_1+1, p_{2}+1,\ldots,p_{\ell}+1).$$ The following result is a generalization of [@KTT07 Lemma 4.4(12)]. \[lem:D-to-B\] Let $I$ be an ideal of $\Phi^+(D_\ell)$ so that $\epsilon_1\pm\epsilon_{\ell} \in I$. For all $q \in {\mathbb{Z}}_{>0}$, we have $$\label{eq:D-to-B} \chi^{{\operatorname{quasi}}}_{T_{I}}(D_\ell, q)= \sum_{k=1}^{\ell}\chi^{{\operatorname{quasi}}}_{T_{U_k}}(B_{\ell-1}, q)+\chi^{{\operatorname{quasi}}}_{T_K}(B_{\ell}, q),$$ where $U_k$ and $K$are defined in , , . With the notion of contraction lists (e.g., [@Tan18 Section 2]), we can write $\chi^{{\operatorname{quasi}}}_{T_{U_k}}(B_{\ell-1}, q)=\chi^{{\operatorname{quasi}}}_{{\mathcal{A}}_k}(B_\ell, q)$ with ${\mathcal{A}}_k :=T_{I\cup\{\epsilon_\ell,\ldots,\epsilon_{k}\}}/T_{\{\epsilon_k\}}$ for $1 \le k \le \ell$. For all $q \in {\mathbb{Z}}_{>0}$, by applying the Deletion-Contraction formula [@Tan18 Theorem 3.5] recursively, we get $$\begin{aligned} \chi^{{\operatorname{quasi}}}_{T_{I}}(D_\ell, q) & = \chi^{{\operatorname{quasi}}}_{T_{U_\ell}}(B_{\ell-1}, q) + \chi^{{\operatorname{quasi}}}_{T_{I\cup\{\epsilon_\ell\}}}(B_\ell, q)\\ & = \chi^{{\operatorname{quasi}}}_{T_{U_\ell}}(B_{\ell-1}, q) + \chi^{{\operatorname{quasi}}}_{T_{U_{\ell-1}}}(B_{\ell-1}, q) +\chi^{{\operatorname{quasi}}}_{T_{I\cup\{\epsilon_\ell, \epsilon_{\ell-1}\}}}(B_\ell, q)\\ & =\ldots\\ & =\sum_{k=1}^{\ell}\chi^{{\operatorname{quasi}}}_{T_{U_k}}(B_{\ell-1}, q)+\chi^{{\operatorname{quasi}}}_{T_K}(B_{\ell},q).\end{aligned}$$ In Lemma \[lem:D-to-B-F\] and Theorem \[thm:D-to-B-L\] below, we use the same assumption and notation as in Lemma \[lem:D-to-B\]. \[lem:D-to-B-F\] For even $q \in {\mathbb{Z}}_{>0}$, we have $$\label{eq:D-to-B-F} F_I(D_\ell, q) =F_{I\cup\{2\epsilon_s,\ldots,2\epsilon_{\ell}\}}(C_\ell, q)=\prod_{i=1}^{\ell}(q-p_i).$$ This follows from the height placements in Table \[tab:placement\]. \[thm:D-to-B-L\] For even $q \in {\mathbb{Z}}_{>0}$, we have $$\label{eq:D-to-B-L} \chi^{{\operatorname{quasi}}}_{S_{I}}(D_\ell, q)= \frac12\left(\sum_{k=1}^{\ell}\chi^{{\operatorname{quasi}}}_{T_{U_k}}(B_{\ell-1}, q)+\chi^{{\operatorname{quasi}}}_{T_K}(B_{\ell}, q) +\prod_{i=1}^{\ell}(q-p_i) \right).$$ This follows from formula , and Lemmas \[lem:D-to-B\], \[lem:D-to-B-F\]. \[eg:D-ideal\] Table \[tab:D5-ideal-eg\] shows an example of the ideal $I=\{\alpha \in \Phi^+(D_5) \mid {\rm ht}(\alpha) \le 6\}$ (in colored region), with positive roots contributing to $p_1$, $p_2$, $p_3$, $p_4$ are colored in green, yellow, blue, red, respectively. In this case, $s=2$ since $\epsilon_2+\epsilon_3 \in I$, but $\epsilon_1+\epsilon_2 \notin I$. We have ${{\mathcal{SG}}}(I)=(7,6,4,2,0)$, and the computation on the ideals $K$ and $U_k$ for even $q \in {\mathbb{Z}}_{>0}$ is given in Table \[tab:data-D\]. By Theorem \[thm:D-to-B-L\], for even $q \in {\mathbb{Z}}_{>0}$, we have $$\chi^{{\operatorname{quasi}}}_{S_{I}}(D_\ell, q)=(q-2)(q-4)(q^3-13q^2+51q-51).$$ \[rem:D-recover\] When $I=\Phi^+(D_\ell)$, ${{\mathcal{SG}}}(I)=(2\ell-2,2\ell-4,\ldots,2,0)$, ${{\mathcal{DP}}}(I)=(\ell-1, 2\ell-3,\ldots,3,1)$, ${{\mathcal{DP}}}(K)=(2\ell-1,2\ell-3,\ldots,3,1)$, and ${{\mathcal{DP}}}(U_k)=(2\ell-3,\ldots,3,1)$ for all $1 \le k \le \ell$. Note that $s=2$, so there is no ideal $U_k$ of type I. Then by Lemma \[lem:D-to-B\], for odd $q \in {\mathbb{Z}}_{>0}$ $$\chi^{{\operatorname{quasi}}}_{S_{\Phi^+}}(D_\ell, q)=\chi^{{\operatorname{quasi}}}_{T_{\Phi^+}}(D_\ell, q)=(q-1)(q-3)\ldots(q-(2\ell-3))(q-(\ell-1)),$$ which agrees with Corollary \[cor:ideal-factorization\]. Moreover, for even $q \in {\mathbb{Z}}_{>0}$ $$\chi^{{\operatorname{quasi}}}_{S_{\Phi^+}}(D_\ell, q)=(q-2)(q-4)\ldots(q-(2\ell-4))\left(q^2-2(\ell-1)q+\frac{\ell(\ell-1)}2\right),$$ which recovers the result of [@KTT07 Theorem 4.8] for type $D$ root systems. With a recent study on characteristic quasi-polynomials and toric arrangements [@LTY17 Corollary 5.6], our computation gives a full description of the characteristic polynomials of the toric arrangements defined by the ideals. We complete this section by giving a direct verification of Corollary \[cor:ideal-factorization\] when $\Phi$ is any classical root system. We restrict the discussion to type $D$ root systems as the other cases are easy. For any ideal $I\subseteq\Phi^+(D_\ell)$ with ${{\mathcal{SG}}}(I)=(p_1,\ldots,p_\ell)$ defined in , we write $$p_i=p_i^{(+)}+p_i^{(-)}, \mbox{where, } p_i^{(\pm)}:=\#\{ \epsilon_i \pm\epsilon_{j} \mid \epsilon_i \pm\epsilon_{j} \in I \},$$ for each $1 \le i \le \ell$. It is easily seen that ${{\mathcal{DP}}}(I)=(d_1,\ldots,d_\ell)$ with $$d_i=p_i^{(-)}+p_{i-1}^{(+)}.$$ Here we agree that $p_{0}^{(+)} = 0$. \[thm:verify\] Let $I$ be an ideal of $\Phi^+(D_\ell)$. For odd $q \in {\mathbb{Z}}_{>0}$, we have $$\label{eq:verify} \chi^{{\operatorname{quasi}}}_{T_{I}}(D_\ell, q)= \prod_{i=1}^{\ell}\left(q- d_i \right).$$ It suffices to prove Theorem \[thm:verify\] when $\epsilon_1\pm\epsilon_{\ell} \in I$, as the other cases are straightforward. For such ideals, $d_1=\ell-1$, $d_i=p_i^{(-)}+p_{i-1}^{(+)}=p_{i-1}-1$ for all $2 \le i \le \ell$. We recall the notation of the parameter $s$ defined in that $s =\min \{ 2 \le k \le \ell \mid \epsilon_{k-1}+\epsilon_k \in I\}$. It follows from Lemma \[lem:D-to-B\] and Remark \[rem:D-recover\] that both sides of are divisible by $\prod_{i=s}^{\ell}\left(q- p_{i-1}+1 \right)$. Hence we need only prove the following: $$\label{eq:verify-need} A+B+C=(q-\ell+1)\prod_{i=2}^{s-1}\left(q- p_{i-1}+1\right),$$ where $$\begin{aligned} A & := \prod_{i=1}^{s-1}\left(q- p_i-1 \right), \\ B & := \sum_{k=1}^{s-2}(q-p_1)\ldots(q-p_{k-1})(q-p_{k+1}-1)\ldots(q-p_{s-1}-1), \\ C & = \sum_{k=s-1}^{\ell}C_k, \mbox{ with } C_k:= \prod_{n=1}^{s-2}\left(q- p_n+\tau_{n,k} \right),\end{aligned}$$ and $\tau_{n,k}$ is defined in . Since $\tau_{n,s-1}=0$ for all $1 \le n \le s-2$, $C_{s-1}=\prod_{i=1}^{s-2}\left(q- p_i\right)$. It is routine to check that $$\label{eq:verify-routine} A+B+C_{s-1}=\prod_{i=1}^{s-1}\left(q- p_i \right).$$ Write $M_\tau=[\tau_{n,k}]$ for a matrix of size $(s-2) \times (\ell-s+1)$ whose entries are the $\tau_{n,k}$’s (the columns indexed by the set $\{s,\ldots,\ell\}$). Then $$M_\tau = \begin{bmatrix} 0 & \cdots & 0 & \cdots &\cdots & 0 & 1 & \cdots & 1 \\ 0 & \cdots & 0 & \cdots &1 & \cdots & 1 & \cdots & 1 \\ \vdots & & \vdots & & \vdots & & \vdots & & \vdots \\ 0 & \cdots & 0 & 1 & \cdots & \cdots & 1 & \cdots & 1 \\ \end{bmatrix},$$ with the number of $1$’s on the $n$-th row is exactly $p_{n}^{(+)}$, and the entries on the $k$-th column contribute to the evaluation of $C_k$. Thus $$\label{eq:verify-Ck} \sum_{k=s}^{\ell}C_k = \sum_{n=0}^{s-2}\left(p_{n+1}^{(+)}-p_{n}^{(+)} \right)\prod_{i=1}^{n}\left(q- p_i \right)\prod_{i=n+1}^{s-2}\left(q- p_i+1 \right).$$ Now combining and with a rigorous check, we obtain . **Acknowledgements:** The author is greatly indebted to Professor Masahiko Yoshinaga for drawing the author’s attention to the characteristic quasi-polynomials of the ideals and for many helpful suggestions during the preparation of the paper. The author wishes to thank Professor Michele Torielli for helpful comments concerning the signed graphs and thank Ye Liu for stimulating conversations. He also gratefully acknowledges the support of the scholarship program of the Japanese Ministry of Education, Culture, Sports, Science, and Technology (MEXT) under grant number 142506. [^1]: We decided to omit the construction of type $A$ root systems as the calculation on this type follows from those on the other types (e.g., see formula ). [^2]: Note that the number $b(q)$ in [@KTT07 Theorem 4.1] should be read as $b(q)=\prod_{i=1}^m\gcd\{q,d_i\}$, where $d_i$’s are invariant factors of the matrix $P$. In particular, if $\det(P)$ takes the value in $\{1,2\}$, then $b(q)=\gcd\{q,\det(P)\}$. Hence [@KTT07 Theorem 4.1] is valid if $\Phi$ is a classical root system. [^3]: This fact is true for any root system, which is a consequence of, e.g., [@S05 Lemma 3.2].

--- abstract: 'We numerically study the quantum Hall effect (QHE) in bilayer graphene based on tight-binding model in the presence of disorder. Two distinct QHE regimes are identified in the full energy band separated by a critical region with non-quantized Hall Effect. The Hall conductivity around the band center (Dirac point) shows an anomalous quantization proportional to the valley degeneracy, but the $\nu=0$ plateau is markedly absent, which is in agreement with experimental observation. In the presence of disorder, the Hall plateaus can be destroyed through the float-up of extended levels toward the band center and higher plateaus disappear first. The central two plateaus around the band center are most robust against disorder scattering, which is separated by a small critical region in between near the Dirac point. The longitudinal conductance around the Dirac point is shown to be nearly a constant in a range of disorder strength, till the last two QHE plateaus completely collapse.' address: | $^1$Department of Physics and Astronomy, California State University, Northridge, California 91330, USA\ $^2$Department of Physics, Southeast University, Nanjing 210096, China\ $^3$National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China author: - 'R. Ma$^{1,2}$, L. Sheng$^{3}$, R. Shen$^{1,3}$, M. Liu$^{2}$ and D. N. Sheng$^1$' title: ' Quantum Hall Effect in Bilayer Graphene: Disorder Effect and Quantum Phase Transition' --- I. Introduction =============== Since the experimental discovery of an unusual half-integer quantum Hall effect (QHE) [@K.; @S.; @Novoselov1; @Y.; @Zhang] in monolayer graphene, the electronic transport properties of graphene related materials have been extensively studied [@J.Nilsson1; @E.; @McCann; @J.; @Nilsson2; @K.; @S.; @Novoselov2; @J.; @G.; @Checkelsky; @Y.; @Hasegawa; @D.; @A.; @Abanin; @E.; @V.; @Gorbar; @H.; @Min; @E.; @V.; @Castro; @Sheng]. Recently, bilayer graphene is found to show an anomalous behavior in its spectral and transport properties, which has attracted much experimental and theoretical interest. Theoretical studies [@E.; @McCann; @J.; @Nilsson2] show that interlayer coupling modifies the intralayer relativistic spectrum to yield a quasiparticle spectrum with a parabolic energy dispersion, which implies that the quasiparticles in bilayer graphene cannot be treated as massless but have a finite mass. Experiments have shown that bilayer graphene exhibits an unconventional integer QHE [@K.; @S.; @Novoselov2]. The Landau level (LL) quantization results in plateaus of Hall conductivity at integer positions proportional to the valley degeneracy, but the plateau at zero energy is markedly absent. The unconventional QHE behavior derives from the coupling between the two graphene layers. The quasiparticles in bilayer graphene are chiral and carry a Berry phase 2$\pi$, which strongly affects their quantum dynamics. However, a detailed theoretical understanding of the unconventional properties of the QHE in bilayer graphene taking into account of the full band structure and disorder effect is still lacking. As established for a single layer graphene [@Sheng] and conventional quantum Hall systems [@Sheng0], the QHE phase diagram in such a system is crucially depending on the topological properties of the full energy band, and thus can be naturally determined in the band model calculations. In this work, we carry out a numerical study of the QHE in bilayer graphene in the presence of disorder based upon a tight-binding model. We reveal that the experimentally observed unconventional QHE plateaus emerge near the band center, while the conventional QHE plateaus appear near the band edges. The unconventional ones are found to be much more stable to disorder scattering than the conventional ones near the band edges. We further investigate the quantum phase transition and obtain the phase boundaries $W_{c}$ for different QHE states to insulator transition by calculating the Thouless number [@J.T.Edwards]. Our results show that the unconventional QHE plateaus can be destroyed at strong disorder (or weak magnetic field) through the float-up of extended levels toward the band center and higher plateaus always disappear first. While the $\nu =\pm 2$ QHE states are most stable, the Dirac point at the band center separating these two QHE states remains critical with a nearly constant longitudinal conductance. This paper is organized as follows. In Sec. II, we introduce the model Hamiltonian. In Sec. III, numerical results based on exact diagonalization and transport calculations are presented. The final section contains a summary. II. The tight-binding model of bilayer graphene =============================================== {width="2.0in"} We consider the bilayer graphene composed of two coupled hexagonal lattice including inequivalent sublattices $A$, $B$ on the bottom layer and $\widetilde{A}$, $\widetilde{B}$ on the top layer. The two layers are arranged in the AB (Bernal) stacking [@S.; @B.; @Trickey; @K.; @Yoshizawa], as shown in Fig. 1, where $B$ atoms are located directly below $\widetilde{A}$ atoms, and $A$ atoms are the centers of the hexagons in the other layer. The unit cell contains four atoms $A$, $B$, $\widetilde{A}$, and $\widetilde{B}$, and the Brillouin zone is identical with that of monolayer graphene. Here, the in-plane nearest-neighbor hopping integral between $A$ and $B$ atoms or between $\widetilde{A}$ and $\widetilde{B}$ atoms is denoted by $\gamma_{AB} =\gamma_{\widetilde{A}\widetilde{B}}=\gamma_{0}$. For the interlayer coupling, we take into account the largest hopping integral between $B$ and $\widetilde{A}$ atoms $\gamma_{\widetilde{A}B}=\gamma_{1}$, and the smaller hopping integral between $A$ and $\widetilde{B}$ atoms $\gamma_{A\widetilde{B}}=\gamma_{3}$. The values of these hopping integrals are estimated to be $\gamma_{0}=3.16$ eV [@W.; @W.; @Toy], $\gamma_{1}=0.39$ eV [@A.; @Misu], and $\gamma_{3}=0.315$ eV [@R.; @E.; @Doezema]. We assume that each monolayer graphene has totally $L_{y}$ zigzag chains with $L_{x}$ atomic sites on each chain [@Sheng]. The size of the sample will be denoted as $N=L_{x}\times L_{y}\times L_{z}$, where $L_{z}=2$ is the number of monolayer graphene planes along the $z$ direction. In the presence of an applied magnetic field perpendicular to the plane of the bilayer graphene, the lattice model in real space can be written in the tight-binding form: $$\begin{aligned} H&=&-\gamma_{0}\sum\limits_{\langle ij\rangle}e^{ia_{ij}}(c_{i}^{\dagger }c_{j}+ \widetilde{c}_{i}^{\dagger }\widetilde{c}_{j} )+ (-\gamma_{1}\sum\limits_{\langle ij\rangle_1}e^{ia_{ij}}c_{jB}^{\dagger }\widetilde{c}_{i\widetilde{A}}\nonumber\\ &-&\gamma_{3}\sum\limits_{\langle ij\rangle_3}e^{ia_{ij}}c_{iA}^{\dagger }\widetilde{c}_{j\widetilde{B}} +h.c.)+\sum\limits_{i}w_{i}(c_{i}^{\dagger }c_{i}+\widetilde{c}_{i}^{\dagger }\widetilde{c}_{i}),\end{aligned}$$ where $c_{i}^{\dagger}$($c_{iA}^{\dagger}$), $c_{j}^{\dagger}$($c_{jB}^{\dagger}$) are creating operators on $A$ and $B$ sublattices in the bottom layer, and $\widetilde{c}_{i}^{\dagger}$($\widetilde{c}_{i\widetilde{A}}^{\dagger}$), $\widetilde{c}_{j}^{\dagger}$($\widetilde{c}_{j\widetilde{B}}^{\dagger}$) are creating operators on $\widetilde{A}$ and $\widetilde{B}$ sublattices in the top layer. The sum $\sum_{\langle ij\rangle}$ denotes the intralayer nearest-neighbor hopping in both layers, $\sum_{\langle ij\rangle_1}$ stands for interlayer hopping between the $B$ sublattice in the bottom layer and the $\widetilde{A}$ sublattice in the top layer, and $\sum_{\langle ij\rangle_3}$ stands for the interlayer hopping between the $A$ sublattice in the bottom layer and the $\widetilde{B}$ sublattice in the top layer, as described above. $w_{i}$ is a random disorder potential uniformly distributed in the interval $w_{i}\in \lbrack -W/2,W/2]\gamma_0$. The magnetic flux per hexagon $\phi =\sum_{{\small {\mbox{\hexagon}}}}a_{ij}=% \frac{2\pi }{M}$, with $M$ an integer. The total flux through the sample is $N\frac {\phi}{2\pi}$, where $N=L_{x}L_{y}/M$ is taken to be an integer. When $M$ is commensurate with $L_x$ or $L_y$, the magnetic periodic boundary conditions are reduced to the ordinary periodic boundary conditions. III. Results and Discussion =========================== The eigenstates $\vert\alpha\rangle$ and eigenenergies $\epsilon_\alpha$ of the system are obtained through exact diagonalization of the Hamiltonian Eq. (1), and the Hall conductivity $\sigma _{xy}$ is calculated by using the Kubo formula $$\sigma _{xy}= \frac{ie^{2}\hbar}{S}\sum_{\alpha, \beta}\frac{\langle \alpha\mid V_x\mid\beta\rangle\langle\beta\mid V_y\mid\alpha \rangle-h.c.}{(\epsilon_\alpha-\epsilon_\beta)^2},$$ where $S$ is the area of the sample, $V_{x}$ and $V_{y}$ are the velocity operators. In Fig. 2a, the Hall conductivity $\sigma _{xy} $ and electron density of states are plotted as functions of electron Fermi energy $E_{f}$ for a clean sample ($W=0$) at system size $N=96\times 24\times 2$ with magnetic flux $\phi =\frac{2\pi }{48}$, which illustrates the overall picture of the QHE in the full energy band. From the electron density of states, we can see the discrete LLs. We will call central LL at $E_f=0$ the $n=0$ LL, the one just above (below) it the $n=1$ ($n=-1$) LL, and so on. According to the behavior of $\sigma _{xy}$, the energy band is naturally divided into three different regimes. Around the band center, the Hall conductivity is quantized as $\sigma _{xy}=\nu \frac{e^{2}}{h}$, where $\nu =kg_{s}$ with $k$ an integer and $g_{s}=2$ for each LL due to double-valley degeneracy [@E.; @McCann; @Sheng] (the spin degeneracy will contribute an additional factor $2$, which is omitted here). With each additional LL being occupied, the total Hall conductivity is increased by $g_{s}\frac{e^{2}}{h}$. This is an invariant as long as the states between the $n$-th and $(n-1)$-th LL are localized. $\sigma _{xy}=0$ at the particle-hole symmetric point $E_{f}=0$, which corresponds to the half-filling of the central LL. However, there is no $\sigma_{xy}=0$ quantized Hall plateau. These anomalously quantized Hall plateaus agree with the results observed experimentally in bilayer graphene [@K.; @S.; @Novoselov2]. The Hall conductivity near the band edges, however, is quantized as $\sigma _{xy}=k \frac{e^{2}}{h}$ with $k$ an integer, as in the conventional QHE systems. Remarkably, around $E_{f}=\pm \gamma_0 $ (within a narrow energy region $\Delta E\sim 0.4\gamma_0$), there are two critical regions which separate the unconventional and conventional QHE states, where the Hall conductance quantization is lost. These crossover regions also correspond to a novel transport regime, where the Hall resistance changes sign and the longitudinal conductivity exhibits metallic behavior. The singular behavior of the Hall conductivity in the crossover regions is likely to originate from the Van Hove singularity in the electron density of states at $B=0$ limit. In Fig. 2b, the quantization rule of the Hall conductivity in this unconventional region for three different strengths of magnetic flux is shown. With decreasing magnetic flux from $\protect\phi =\frac{2\protect\pi }{12}$ to $\frac{2\protect\pi }{48}$, more quantized Hall plateaus emerge following the same quantization rule as the gap between the LLs is reduced. {width="3.3in"} {width="3.0in"} Now we study the effect of random disorder on the unconventional QHE in bilayer graphene. In Fig. 3, the Hall conductivity around the band center is shown as a function of $E_{f}$ for four different disorder strengths at system size $N=96\times 24\times 2$ with magnetic flux $\phi =\frac{2\pi }{48}$. We can see that the plateaus with $\nu =\pm 10,\pm 6$ and $\pm 2$ remain well quantized at $W=0.5$. We mention that the $\nu=\pm 4,\pm 8$ plateaus are unclear at this relatively weak disorder strength because of very small plateau widths and relatively large localization lengths (the critical $W_c$ for each plateau will be obtained based on our larger size calculations of the Thouless number as presented later). With increasing $W$, higher Hall plateaus (with larger $|\nu |$) are destroyed first. At $W=2.0$, only the $\nu =\pm 2$ QHE remain robust. The last two plateaus $\nu =\pm 2$ eventually disappear around $W\sim 3.2$. For comparison, the QHE near the lower band edge is shown in the inset, where all plateaus disappear at a much weaker disorder strength $W\geq 1.0$. This clearly indicates that under the same conditions, the unconventional QHE around the band center is much more stable than the conventional QHE near the band edges. Clearly, after the destruction of the conventional QHE states near the band edge, these states become localized. Then the topological Chern numbers initially carried by these states will move towards band center in a similar manner to the single-layer graphene case [@Sheng]. Thus we observe that the destruction of the unconventional QHE states near the band center is due to the float-up of extended levels. {width="3.3in"} To study the fate of the IQHE at weak magnetic field limit, we reduce the strength of magnetic field. In Fig. 4, the Hall conductivities around the band center with weaker magnetic flux $\phi =\frac{2\pi }{96}$, $\frac{2\protect\pi }{192}$ and $\frac{2\protect\pi }{288}$ are shown for different disorder strengths and system size $N=96\times 24\times 2$. In Fig. 4a, a lot more well quantized Hall plateaus emerge for a clean sample($W=0$), if we compare them with the results in Fig. 2b. In Fig. 4b, 4c and 4d, we can see that with the increasing of the disorder strength $W$, Hall plateaus are destroyed faster for the system with weaker magnetic flux $\phi$. At $W=2.0$, the most robust Hall plateaus at $\nu =\pm 2$ remain well quantized for magnetic flux $\phi =\frac{2\pi }{96}$ and $\frac{2\protect\pi }{192}$, however, they already disappear for weaker magnetic flux $\phi =\frac{2\pi }{288}$. Our flux $2\pi/M$ in each hexagon the magnetic field is $B \sim 1.3\times 10^5/M$ Tesla[@weakb]. Thus the weakest $B$ we used is about $451$ Tesla. This is a very large magnetic field comparing to the experimental ones around $B\sim 40 $ Tesla. However, the topology of the QHE and how they disappear with the increase of the disorder strength $W$ remain to be the same as the stronger $B$ cases as demonstrated in Fig. 4a-4d. Thus, we establish that the obtained behavior of QHE for bilayer graphene will survive at weak $B$ limit. We further investigate the quantum phase transition of the bilayer graphene electron system. In order to determine the critical disorder strength $W_{c}$ for the different QHE states, the Thouless number $g$ is calculated by using the following formula [@J.T.Edwards], $$g= \frac{\Delta E}{dE/dN}\ . %%%= \frac{2\hbar}{e^{2}}G\ .$$ Here, $\Delta E$ is the geometric mean of the shift in the energy levels of the system caused by replacing periodic by antiperiodic boundary conditions, and $dE/dN$ is the mean spacing of the energy levels. The Thouless number $g$ is proportional to the longitudinal conductance $G$. In Fig. 5, we show some examples of calculated Thouless number for a relatively weak flux $\phi =\frac{2\pi }{48}$ and some different disorder strengths to explain how quantum phase transitions and the related phase boundaries $W_c$ are determined. In Fig. 5a, the calculated Thouless number $g$ and Hall conductivity $\sigma _{xy}$ as a function of $E_{f}$ at a weak disorder strength $W=0.2$ are plotted. Clearly, each valley in Thouless number corresponds to a Hall plateau and each peak corresponds to a critical point between two neighboring Hall plateaus. We can also call the first valley just above (below) $E_f=0$ the $\nu=-2$ ($\nu=2$) QHE state, the second one the $\nu=-4$ ($\nu=4$) state, and so on, as same as the Hall plateaus. In Fig. 5b-5d, we see that with increasing $W$, higher QHE states (valleys) are destroyed first. At $W=W_c=1.0$ (see Fig. 5b), the valleys with $\nu=\pm 12$ disappear, which correspond to the destruction of the $\nu =\pm 12$ Hall plateau states. Therefore, $W_c=1.0$ is the critical disorder strength, at which the $\nu =\pm 12$ plateau states change to an insulating phase. At $W=W_c=1.3$ (see Fig. 5c), the valleys with $\nu =\pm 8$ disappear, which indicates the destruction of the $\nu =\pm 8$ QHE states and their transition into the insulating phase. When $W=W_{c}=3.2$ (see Fig.5d), the most stable QHE states with $\nu=\pm 2$ eventually disappear, which indicates all QHE phases are destroyed by disorder. All the phase boundaries $W_{c}$ between the different QHE states are determined in the same manner and tabulated in Table 1. {width="3.3in"} We now focus on the region around $E_{f}=0$. In Fig. 6, we show the Thouless number for some different disorder strengths at system size $N=96\times 24\times 2$ and magnetic flux $\phi =\frac{2\pi }{48}$. We can see that the Thouless number shows a central peak at $E_{f}=0$. With increasing the disorder strength, the width of the peak increases and its height remains nearly unchanged. This behavior may suggest an interesting effect that the extended states originally sited at the critical point $E_f=0$ splits in the presence of disorder. However, the splitting is too small to induce two separated peaks in the Thouless number for the present sample sizes we can approach. Instead, it leads to a widened peak of unreduced height. This behavior also indicates that the critical longitudinal conductance in a small finite region near $E_f=0$ is almost constant about $2e^2/h$ according to the proportionality of Thouless number to longitudinal conductance. We have also confirmed this conclusion by direct Kubo formula calculation, in which the system size that can be approached is however much smaller. Hall plateaus index critical point $W_c$ --------------------- ---------------------- $\nu=\pm 12$ 1.0 $\nu=\pm 10$ 1.2$\pm 0.1$ $\nu=\pm 8$ 1.3$\pm 0.1$ $\nu=\pm 6$ 1.6$\pm 0.1$ $\nu=\pm 4$ 1.7$\pm 0.1$ $\nu=\pm 2$ 3.2 : The phase boundaries $W_{c}$ for the different Hall plateaus. {width="3.4in"} IV. Summary =========== In summary, we have numerically investigated the QHE in bilayer graphene based on tight-binding model in the presence of disorder. The experimentally observed unconventional QHE is reproduced near the band center. The unconventional QHE plateaus around the band center are found to be much more stable than the conventional ones near the band edges. Our results of quantum phase transition indicate that with increasing disorder strength, the Hall plateaus can be destroyed through the float-up of extended levels toward the band center and higher plateaus always disappear first. At $W=W_{c}=3.2$, the most stable QHE states with $\nu=\pm 2$ eventually disappear, which indicates transition of all QHE phases into the insulating phase. A small critical region is observed between the $\nu=\pm 2$ plateaus, where the longitudinal conductance remains almost constant about $2e^2/h$ in the presence of moderate disorder, possibly due to the splitting of the critical point originally sited at $E_f=0$. We mention that in our numerical calculations, the magnetic field is much stronger than the ones one can realize in the experimental situation, as limited by current computational ability. However, the phase diagram we obtained is robust and applicable to weak field limit since it is determined by the topological property of the energy band as clearly established for single layer graphene [@Sheng] and conventional quantum Hall systems [@Sheng0]. We further point out that the continuum model can not be used to address the fate of the quantum Hall effect in strong disorder or weak magnetic field limit. Because in such a model, both the band bottom and band edge are pushed to infinite energy limit, and thus one will not be able to see the important physics of opposite Chern numbers annihilating each other to destroy the IQHE[@Sheng]. **Acknowledgment:** This work is supported by the US DOE grant DE-FG02-06ER46305 (RS, DNS) and the NSF grant DMR-0605696 (RM, DNS). We thank the KITP for partial support through the NSF grant PHY05-51164. 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CP3-08-01 [**Higher-order QCD effects\ .3cm in the Higgs to $ZZ$ search channel at the LHC**]{} [**Rikkert Frederix**]{}${}^{(a)}$ and [**Massimiliano Grazzini**]{}${}^{(b)}$\ ${}^{(a)}$Center for Particle Physics and Phenomenology (CP3),\ Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium ${}^{(b)}$INFN, Sezione di Firenze\ I-50019 Sesto Fiorentino, Florence, Italy\ [**Abstract**]{} > 10000 We present a consistent analysis of the signal as well as the irreducible background for the search of the SM Higgs boson in the $ZZ$ decay channel at the LHC. Soft-gluons effects are resummed up to next-to-leading logarithmic accuracy, and the results are compared to those obtained with fixed order calculations and the MC@NLO event generator. The soft-gluon effects are typically modest but should be taken into account when precise predictions are demanded. Our results show that the signal over background ratio can be significantly enhanced with a cut on the transverse momentum $\ptZZ$ of the $ZZ$ pair. We also introduce a fully transverse angular variable that could give information about the CP nature of the Higgs boson. January 2008 Introduction {#sec:intro} ============ The elucidation of the mechanism of electroweak symmetry breaking is one of the main goals of the LHC physics program. In the Standard Model (SM) and several popular extensions such as SUSY, mass generation is triggered by the Higgs mechanism, which predicts the existence of (at least) one scalar state, the Higgs boson. The search for the Higgs at collider experiments has now being on-going for two decades. The present direct lower limit of the Higgs mass in the SM is 114.4 GeV (at 95% CL) [@Barate:2003sz], while precision measurements point to a rather light Higgs, $m_h \lesssim 200$ GeV [@Alcaraz:2007ri]. At the LHC, the main production mechanism will be $gg \to H$, and if $m_h>180$ GeV, the Higgs decay into two Z bosons, $h\to ZZ$, will provide one of the cleanest signatures at hadron colliders, [*i.e.*]{}, four leptons. Such a final state will allow a very accurate mass reconstruction and the best of all possible discovery modes, a sharp peak over a rather flat background. At this stage, accurate predictions from theory will be helpful to design the best analysis but are not essential to claim a discovery as data alone will provide all the necessary information. However, to answer the key questions on the nature of the discovered particle, such as its spin, CP nature and couplings, accurate predictions for both signal and backgrounds will be required. As far as the Higgs signal is concerned, QCD corrections at the next-to-leading order (NLO) have been known for some time [@Dawson:1991zj; @Spira:1995rr]: their effect increases the LO cross section by about 80–100%. In recent years, even next-to-next-to-leading order (NNLO) corrections have been computed, first for the total cross section [@NNLOtotal], and more recently implemented in fully exclusive calculations [@Anastasiou:2004xq; @Catani:2007vq]. Note, however, that all the NNLO results use the large-$m_{top}$ approximation, $m_{top}$ being the mass of the top quark. As far as $ZZ$ production is concerned, NLO corrections have been known for some time [@Ohnemus:1990za; @Mele:1990bq; @Ohnemus:1994ff]. More recent NLO calculations exist that, using the one-loop helicity amplitudes of Ref. [@Dixon:1998py], fully take into account spin correlations in the $Z$ boson decay [@Dixon:1999di; @Campbell:1999ah]. The fixed-order calculations provide a reliable estimate of signal and background cross sections and distributions as long as the scales involved in the process are all of the same order. When the total transverse momentum of the $ZZ$ pair is much smaller than its invariant mass the validity of the fixed-order expansion may be spoiled since the coefficients of the perturbative expansion can be enhanced by powers of the large logarithmic terms, $\ln^n M_{ZZ}/\ptZZ$. In the case of the Higgs signal, the resummation of such contributions has been performed up to next-to-next-to-leading-logarithmic (NNLL) accuracy [@Bozzi:2003jy; @Bozzi:2005wk; @Bozzi:2007pn]. The purpose of the present paper is twofold. We first consider transverse momentum resummation for $ZZ$ production at the LHC. The resummation of such logarithmic contributions was first considered in Ref. [@Balazs:1998bm]. Here we use the resummation formalism of Refs. [@Bozzi:2003jy; @Bozzi:2005wk] together with the helicity amplitudes of Ref. [@Dixon:1998py] (including finite width effects from the Z bosons, but neglecting single-resonant contributions). Contrary to Ref. [@Balazs:1998bm] we fully include the decay of the $Z$ bosons, keeping track of their polarization in the leptonic decay. In the large $\ptZZ$ region we use LO perturbation theory ($\ZZ$+1 parton); in the region $\ptZZ\ll M_{ZZ}$ the large logarithmic contributions are resummed to NLL accuracy. The present study parallels the one performed in Ref. [@Grazzini:2005vw] in the case of $WW$ production. By using these results, we perform a detailed comparison of signal and background cross sections and distributions. The paper is organized as follows. In Sect. \[sec:zz\] we analyze the impact of transverse momentum resummation for $ZZ$ production. In Sect. \[sec:sb\] we compare signal and background cross sections and distributions for the search of a Higgs boson of mass $m_h=200$ GeV. In Sect. \[summa\] we conclude with a summary of our results. Transverse-momentum resummation for $ZZ$ production {#sec:zz} =================================================== In this Section we discuss the effect of transverse-momentum resummation for $ZZ$ production at the LHC, and present a comparison to fixed order NLO results obtained with MCFM [@Campbell:1999ah] and to results obtained with MC@NLO [@MCatNLO]. We consider the process $pp\to ZZ+X\to e^+e^-\mu^+\mu^-+X$ and perform the all-order resummation of the logarithmically enhanced contributions at small $\ptZZ$. The implementation is completely analogous to the case of $WW$ pair production discussed in Ref. [@Grazzini:2005vw] and is based on the formalism of Refs. [@Bozzi:2003jy; @Bozzi:2005wk]. We refer the reader to the above papers for the technical details. The large logarithmic contributions at small transverse momenta of the $ZZ$ pair are resummed up to NLL accuracy. The result is then matched to the fixed order LO calculation valid at large $\ptZZ$, to achieve NLL+LO accuracy. We recall that the formalism of Refs. [@Bozzi:2003jy; @Bozzi:2005wk] enforces a unitarity constraint such that resummation effects vanish when total cross sections are considered. As a consequence, at NLL+LO accuracy the integral of our resummed spectra coincides with the total NLO cross section if no cuts are applied. To compute the $ZZ$ cross section we use MRST2002 NLO parton densities [@Martin:2002aw] and ${\alpha_{\mathrm{S}}}$ evaluated at two-loop order. Our resummed predictions depend on renormalization, factorization and resummation scales. The resummation scale parametrizes the arbitrariness in the resummation procedure, and is set equal to the invariant mass $M_{ZZ}$ of the $ZZ$ pair. Variations around this central value can give an idea of the size of yet uncalculated higher-order logarithmic contributions. Renormalization and factorization scales are set to $2M_Z$. The latter choice allows us to exploit our unitarity constraint and to exactly recover the total NLO cross section when no cuts are applied. At NLO we consistently use $\mu_F=\mu_R=2M_Z$ as default choice, whereas in MC@NLO $\mu_F$ and $\mu_R$ are set to the default choice, the average transverse mass of the $Z$ bosons. The predictions of resummation are implemented in a partonic Monte Carlo program which generates the full 5-body final state ($e^+e^-\mu^+\mu^-$ + 1 parton). Nonetheless, since the resummed cross section we use is inclusive over rapidity, we are not able to apply the usual rapidity cuts on the leptons. To the purpose of the present work, we do not expect this limitation to be essential. We start by considering the inclusive cross sections. Our NLL+LO result is 33.76 fb, and agrees with the NLO one (33.99 fb) to about $1\%$. With MC@NLO we obtain 34.60 fb. As expected, the MC@NLO cross section is slightly larger because $ZZ$ production is calculated in the narrow width approximation, while in the NLO and NLL+LO calculations, finite width effects are included. In Fig. \[fig:ptzz\] we show the corresponding $p^{ZZ}_T$ distribution, computed at NLL+LO (solid), with MC@NLO (dashed) and at NLO (dots). As is well known, the NLO result diverges to $+\infty$ as $\ptZZ\to 0$, and this divergence is cancelled by the (negative) weight of the first bin, due to the virtual contribution. On the contrary, the NLL+LO and MC@NLO results are well behaved as $\ptZZ\to 0$ and are very close to each other, showing a peak around $\ptZZ\sim 5$ GeV. In order to study the perturbative uncertainties affecting our resummed calculation, we have varied the renormalization and factorization scale by a factor 2 around the central value. We find that the effect of $\mu_R$ and $\mu_F$ variations is rather small, of the order of $\pm 1\%$, and comparable with the estimated accuracy of our numerical code. Similar effects are found at NLO. The dependence of our NLL+LO results on the resummation scale $Q$ is instead stronger. In Fig. \[fig:ptzz\_Q\] we show the NLL+LO prediction for different choices of the resummation scale $Q$. We see that varying the resummation scale the effect on the $\ptZZ$ spectrum is visible and amounts to about $\pm 10\%$ at the peak. For lower (higher) values of $Q$ the effect of resummation is confined to smaller (larger) values of $\ptZZ$. Thanks to our unitarity constraint, the total rate is instead insensitive to resummation scale variations, within the numerical accuracy of our code. As in the case of Higgs [@Bozzi:2005wk] and $WW$ [@Grazzini:2005vw] production, we find that the choice $Q=2M_{ZZ}$ gives (slightly) negative cross sections at very large $\ptZZ$. In order to define a range of variations of $Q$, we prefer to avoid values that give a bad behaviour at large $\ptZZ$. For this reason, in the following, we will consider resummation scale variations in the range $M_{ZZ}/4\leq Q\leq M_{ZZ}$. We now consider the selection cuts designed for the search of a Higgs boson of mass $m_h=200$ GeV in the $e^+e^-\mu^+\mu^-$ channel [@cmsnote]. The final-state leptons, ordered according to decreasing $p_T$, should fulfil the following thresholds: $$p_{T1}>22~{\rm GeV}~~~p_{T2}>20~{\rm GeV}~~~p_{T3}>15~{\rm GeV}~~~p_{T4}>7~{\rm GeV},$$ the invariant mass of the $e^+e^-$ and the $\mu^+\mu^-$ pairs should be between $$60~{\rm GeV}<M_{e^+e^-\!,\,\mu^+\mu^-}<105~{\rm GeV}$$ and the invariant mass of the $ZZ$ pair should fulfil $$190~{\rm GeV}<M_{ZZ}<210~{\rm GeV}.$$ With these cuts the NLL+LO result is 5.42 fb, which is about 2% smaller than the NLO one (5.51 fb). The cross section from MC@NLO is about 11% larger (6.01 fb). This is mainly due to the fact that MC@NLO calculates the cross section in the narrow width approximation, and therefore the cuts on the invariant masses of the $e^+e^-$ and $\mu^+\mu^-$ pairs are always fulfilled. As in the inclusive case, the effect of scale variations on the rate is very small, of the order of $\pm 1\%$. We point out that single-resonant contributions are neglected in our calculation. We have used MadGraph/MadEvent [@madgraph] to check that these contributions are indeed small and found that at LO the effects are smaller than the permille level. Effects from off-shell photons $pp\to Z\gamma^*\to e^+e^-\mu^+\mu^-$ are larger. At NLO they decrease the cross section by about 1% with the cuts described above, due to negative interference between the $Z$ boson and the photon. The shapes of the distributions are, however, not significantly changed. Hence, it is safe to neglect these two contributions in the NLL+LO approximation with the cuts described above. For selection cuts used in Higgs searches where its mass is smaller than the $ZZ$ threshold, the effects from off-shell photons cannot be neglected and have to be included. In Fig. \[fig:ptz\_cuts\] we show the $p_T$ distribution of one of the $Z$ bosons, computed at NLL+LO, NLO and with MC@NLO. Contrary to the $\ptZZ$ spectrum, this distribution is well behaved at NLO but the effect of resummation is still visible on its shape. This is evident from the lower part of the plot, showing the NLO and MC@NLO result normalized to NLL+LO. In Fig. \[fig:ptleptons\_cuts\] we show the $p_T$ distributions of the charged leptons, ordered according to decreasing $p_T$. Here the NLO prediction is in good agreement with the NLL+LO one. MC@NLO, however, predicts slightly softer leptons. The effect of scale variations is still very small for the above distributions. Only in the high-$p_T$ tail of the $p_T^Z$ distribution resummation scale variations give a visible effect, being of about $\pm 10\%$ at $p_T^Z\sim M_Z$. In Fig. \[fig:dphi\_cuts\] we consider the distribution in the variable $\Delta\phi_T$ defined as follows. We consider the separation between the $e^-$ and the $\mu^-$ where their momenta are taken in the rest frame of their parent $Z$ boson, by neglecting all the longitudinal components. In this way the $\Delta\phi_T$ is manifestly longitudinally invariant. As will be illustrated later, see Sect. \[sec:sb\], this angle is sensitive to the CP nature of a Higgs boson resonance. Due to the fully transverse nature of this angle, it can potentially be reconstructed also if only three leptons are detected together with missing $E_T$. We see that the shapes of the NLO and NLL+LO distributions are qualitatively similar. Both decrease with increasing separation angle, the NLL+LO prediction slightly more at small angles before it flattens out, while the NLO prediction has a more constant slope. The differences are, however, small. The effect of scale variations on the NLO and NLL+LO results is again of the order or smaller than $1\%$. We also plotted the prediction for this angle by MC@NLO, although we remind the reader that MC@NLO does not include spin correlations in the $Z$ decay. Despite this fact, the shape of this distribution is not too different from those obtained at NLO and NLL+LO. Signal and background {#sec:sb} ===================== In this Section we perform a consistent comparison of signal and background cross sections for the Higgs search in the $gg\to h\to ZZ\to e^+e^-\mu^+\mu^-$ channel at the LHC. We consider a Higgs boson with mass $m_h=200$ GeV and use the numerical program of Refs. [@Bozzi:2003jy; @Bozzi:2005wk] to compute its transverse momentum spectrum. To be consistent with the background[^1], we work at NLL+LO accuracy and we generate a set of events containing a Higgs boson which is then let decay using the MadGraph package [@madgraph]. We use the same cuts as in Sect. \[sec:zz\]. The signal cross section is 7.74 fb. Comparing with the background we get $S/B=1.43$. In Fig. \[fig:ptleptonssb\] we plot the $p_T$ spectra of the leptons for signal and background. We see that the spectrum of the leading lepton tends to be slightly harder for the signal, compared to the background, whereas the opposite happens for the lepton with the minimum $p_T$. In Fig. \[fig:dphisb\] we plot the $\Delta\phi_T$ distribution defined in Sect. \[sec:zz\] for signal and background. Since this distribution is expected to be sensitive to the CP nature of the Higgs, we also consider the case of a pseudo-scalar Higgs boson. As in the case of the scalar, the events are generated starting from the transverse momentum spectrum at NLL+LO and then letting the Higgs boson decay using the MadGraph package [@madgraph]. The computation of the spectrum for the pseudoscalar has been done by using a modified version of the numerical program of Refs. [@Bozzi:2003jy; @Bozzi:2005wk], using the results of Ref. [@Kauffman:1993nv][^2]. From Fig. \[fig:dphisb\] we see that the shape of the distribution shows remarkable differences in the three cases. As shown in Fig. \[fig:dphi\_cuts\], for the background the distribution is rather flat. On the contrary, for the pseudoscalar, the distribution is peaked at central values of $\Delta\phi_T$, whereas for the scalar the distribution has a minimum in this region. We conclude that this angular variable has a good discriminating potential to assess the CP nature of the Higgs boson. We finally consider the possibility to apply an additional cut on the total transverse momentum of the four leptons. This idea is inspired by a comparison of the transverse momentum spectra of the Higgs boson and of the $ZZ$ pair in Fig. \[fig:ptzz\_cuts\]. We see from Fig. \[fig:ptzz\_cuts\] that the Higgs signal is definitely harder than the $ZZ$ background, being peaked at $p_T\sim 17$ GeV. The $ZZ$ background is instead peaked at $p_T\sim 5$ GeV. As such, a cut on the total transverse momentum of the leptons may increase the statistical significance. Starting with the standard set of cuts used in the rest of the paper, we compute the efficiency of the additional cut by defining $$\epsilon(\ptcut)=\sigma_{p_{T}>\ptcut}/\sigma\, .$$ In Fig. \[fig:eff\] we plot the efficiency as a function of $\ptcut$ for the signal and the background. We see that the efficiency of this additional cut decreases more rapidly for the background than for the signal. As should be expected, the resummation effect is crucial in this case. The fixed order NLO efficiencies, not shown in Fig. \[fig:eff\], become unphysically larger than unity for small values of $\ptcut$. Due to the fact that the efficiency of the background decreases more rapidly compared to the signal, the signal over background ratio increases with increasing $\ptcut$, as can also be seen from the lower left plot of Fig. \[fig:eff\]. The lower right plot shows the statistical significance for an integrated luminosity of 10 fb$^{-1}$, We observe that the statistical significance is maximum when $\ptcut\sim 15$ GeV. The latter point, however, requires a word of caution. The predictions presented in the present paper are based on resummed calculations obtained in a purely perturbative framework. Intrinsic-$p_T$ effects are known (see e.g. Ref. [@Collins:va] and references therein) to affect transverse-momentum distributions, particularly at small transverse momenta. These effects are not taken into account in our calculation. As noted in Ref. [@Bozzi:2005wk], these non-perturbative effects have the same qualitative impact as the inclusion of higher-order logarithmic contributions, [*i.e.*]{}, they tend to make the resummed $p_T$ distribution harder. The quantitative results shown in Fig. \[fig:eff\] will certainly depend on these effects, although the qualitative picture should not change dramatically. Summary {#summa} ======= Higgs boson production by gluon-gluon fusion, followed by the decay mode $h\to ZZ\to 4$ leptons, provides the best discovery channel at the LHC for Higgs masses above 180 GeV. For a precise determination of the properties of the Higgs resonance, such as its mass and CP nature, detailed theoretical predictions for the signal and backgrounds are necessary. In this work we considered a Higgs boson with mass $m_h=200$ GeV and performed the resummation of multiple soft-gluon emission for the $ZZ$ background. We then compared the results with those for the signal in the case of a (pseudo-)scalar Higgs boson. The effects from the resummation of soft gluons are modest for observables like the transverse momentum of the final state leptons. However, the transverse momentum spectra of the $Z$ bosons and of the $ZZ$ pair are sensitive to these effects. An angle $\Delta\phi_T$ that is sensitive to the CP nature of the Higgs signal is also introduced. This angle is defined in a fully transverse way, such that it is longitudinally boost invariant. This angle can potentially be reconstructed also if only three leptons are detected together with missing $E_T$. We also argued that an additional cut on the transverse momentum of the $ZZ$ pair may significantly increase the signal over background ratio and the statistical significance. The impact of resummation is of course crucial in this case. 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