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As a result, Russia was expelled from the Council of Europe in March, and was suspended from the United Nations Human Rights Council in April.

Geometry can be used to design origami. Some classical construction problems of geometry are impossible using compass and straightedge, but can be solved using origami.

Approximately one-third of the U.S. adult population is obese and another third is overweight.

After the whirlpool, the wort must be brought down to fermentation temperatures 20–26 °C (68–79 °F) before yeast is added. In modern breweries this is achieved through a plate heat exchanger. A plate heat exchanger has many ridged plates, which form two separate paths. The wort is pumped into the heat exchanger, and goes through every other gap between the plates. The cooling medium, usually water, goes through the other gaps. The ridges in the plates ensure turbulent flow. A good heat exchanger can drop 95 °C (203 °F) wort to 20 °C (68 °F) while warming the cooling medium from about 10 °C (50 °F) to 80 °C (176 °F). The last few plates often use a cooling medium which can be cooled to below the freezing point, which allows a finer control over the wort-out temperature, and also enables cooling to around 10 °C (50 °F). After cooling, oxygen is often dissolved into the wort to revitalize the yeast and aid its reproduction. Some of the craft brewery, particularly those wanting to create steam beer, utilize coolship instead.

However, Russia has also had 43 Olympic medals stripped from its athletes due to doping violations, which is the most of any country, and nearly a third of the global total.

Applied statistics, sometimes referred to as Statistical science, comprises descriptive statistics and the application of inferential statistics. Theoretical statistics concerns the logical arguments underlying justification of approaches to statistical inference, as well as encompassing mathematical statistics. Mathematical statistics includes not only the manipulation of probability distributions necessary for deriving results related to methods of estimation and inference, but also various aspects of computational statistics and the design of experiments.

The Department of Defense, which is headquartered at the Pentagon near Washington, D.C., administers five of the six service branches, which are made up of the Army, Marine Corps, Navy, Air Force, and Space Force.

According to Christianity, all animals, from the smallest to the largest, are cared for and loved. According to the Bible, "All these animals waited for the Lord, that the Lord might give them food at the hour. The Lord gives them, they receive; The Lord opens his hand, and they are filled with good things". It further says God "gave food to the animals, and made the crows cry."

That same day, Göring sent a telegram from Berchtesgaden, arguing that since Hitler was isolated in Berlin, Göring should assume leadership of Germany.

Political economy of communication researchers have long focused on the international or global presence, power, profitability and popularity of Hollywood films. Books on global Hollywood by Toby Miller and Richard Maxwell, Janet Wasko and Mary Erickson, Kerry Segrave, John Trumpbour and Tanner Mirrlees examine the international political economy of Hollywood's power.

The breakthrough eventually came around 1950 when a more robust method for eliminating infinities was developed by Julian Schwinger, Richard Feynman, Freeman Dyson, and Shinichiro Tomonaga. The main idea is to replace the calculated values of mass and charge, infinite though they may be, by their finite measured values. This systematic computational procedure is known as renormalization and can be applied to arbitrary order in perturbation theory. As Tomonaga said in his Nobel lecture:

Having lived in Ireland as a child, Churchill always opposed its partition. As a minister in 1913 and again in 1921, he suggested that Ulster should be part of a united Ireland, but with a degree of autonomy from an independent Irish government. He was always opposed on this by Ulster Unionists. While he was Leader of the Opposition, he told John W. Dulanty and Frederick Boland, successive Irish ambassadors to London, that he still hoped for reunification.

The United States has many private and public institutions of higher education. There are local community colleges with generally more open admission policies, shorter academic programs, and lower tuition. The U.S. spends more on education per student than any nation in the world, spending an average of $12,794 per year on public elementary and secondary school students in the 2016–2017 school year. As for public expenditures on higher education, the U.S. spends more per student than the OECD average, and more than all nations in combined public and private spending. Despite some student loan forgiveness programs in place, student loan debt has increased by 102% in the last decade, and exceeded 1.7 trillion dollars as of 2022.

In September 1944, Chinese forces captured Mount Song and reopened the Burma Road.

Islam is the second-largest religion in Russia, and is the traditional religion among the majority of the peoples of the North Caucasus, and among some Turkic peoples scattered along the Volga-Ural region.

General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. It predicts an overall rotation (precession) of planetary orbits, as well as orbital decay caused by the emission of gravitational waves and effects related to the relativity of direction.

There are 444 publishers, developers, and hardware companies in California alone.

The name rose comes from Latin rosa, which was perhaps borrowed from Oscan, from Greek ῥόδον rhódon (Aeolic βρόδον wródon), itself borrowed from Old Persian wrd- (wurdi), related to Avestan varəδa, Sogdian ward, Parthian wâr.

It was long thought that cat domestication began in ancient Egypt, where cats were venerated from around 3100 BC, but recent advances in archaeology and genetics have shown that their domestication occurred in the Near East around 7500 BC.

So, the probability of the entire sample space is 1, and the probability of the null event is 0.

In February 1920, he was appointed to head the Workers' and Peasants' Inspectorate; that same month he was also transferred to the Caucasian Front.

He used either "Winston S. Churchill" or "Winston Spencer Churchill" as his pen name to avoid confusion with the American novelist of the same name, with whom he struck up a friendly correspondence. For many years, he relied heavily upon his press articles to assuage his financial worries: in 1937, for example, he wrote 64 published articles and some of his contracts were quite lucrative.

The United States did not stop oil exports, however, partly because of the prevailing sentiment in Washington that given Japanese dependence on American oil, such an action was likely to be considered an extreme provocation.

Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory".

He told the people of his concerns in a radio broadcast in November 1934, having earlier denounced the intolerance and militarism of Nazism in the House of Commons.

Elvis Presley and Chuck Berry were among the pioneers of rock and roll in the mid-1950s.

The cinema of Nigeria, often referred to informally as Nollywood is the cinema of Nigeria. The Nigerian Film Industry is the largest movie industry globally in terms of output, and is the third largest, in terms of overall revenues generated. Cinema of Nigeria Its history dates back to as early as the late 19th century and into the colonial era in the early 20th century. The history and development of the Nigerian motion picture industry is sometimes generally classified in four main eras: the Colonial era, Golden Age, Video film era and the emerging New Nigerian cinema.

The Sinclair QL was released in 1984. This machine was aimed at the serious home user and professional markets. It was a departure from the architecture of the ZX Spectrum series, using a Motorola 68008 processor and was not compatible with its predecessor. The QL was particularly distinctive due to an unusual choice of storage device. It was shipped with two integrated microdrives, a continuous loop tape system intended to have performance sitting somewhere between disk and cassette but at a lower price than a disk drive. The QL was not a commercial success, in part due to issues with reliability but also the ascent of the IBM PC in business with which it was not compatible. Production had been discontinued by 1985. The same technology was subsequently re-used in the ICL One Per Desk integrated telephone/computer device. The QL achieved some note due to Linus Torvalds crediting it with being the machine on which he became practiced at programming prior to developing Linux, in part due to the limited software support.

Interest in Germany and Japan in forming an alliance began when Japanese diplomat Oshima Hiroshi visited Joachim von Ribbentrop in Berlin in 1935.

They also murdered an additional 4 million others who were deemed "unworthy of life" (including the disabled and mentally ill, Soviet prisoners of war, Romani, homosexuals, Freemasons, and Jehovah's Witnesses) as part of a program of deliberate extermination, in effect becoming a "genocidal state".

In 1932, 90% of German reparation payments were cancelled (in the 1950s, Germany repaid all its missed reparations debts). Widespread unemployment reached 25% as every sector was hurt. The government did not increase government spending to deal with Germany's growing crisis, as they were afraid that a high-spending policy could lead to a return of the hyperinflation that had affected Germany in 1923. Germany's Weimar Republic was hit hard by the depression, as American loans to help rebuild the German economy now stopped. The unemployment rate reached nearly 30% in 1932.

On 15 August, the Emperor communicated this decision to the Japanese people through a speech broadcast on the radio (Gyokuon-hōsō, literally "broadcast in the Emperor's voice").

After working on his difference engine he announced his invention in 1822, in a paper to the Royal Astronomical Society, titled "Note on the application of machinery to the computation of astronomical and mathematical tables", he also designed to aid in navigational calculations, in 1833 he realized that a much more general design, an analytical engine, was possible. The input of programs and data was to be provided to the machine via punched cards, a method being used at the time to direct mechanical looms such as the Jacquard loom. For output, the machine would have a printer, a curve plotter and a bell. The machine would also be able to punch numbers onto cards to be read in later. The Engine incorporated an arithmetic logic unit, control flow in the form of conditional branching and loops, and integrated memory, making it the first design for a general-purpose computer that could be described in modern terms as Turing-complete.

In the 13th century, Marco Polo recorded the early Mandarin or Wu Chinese pronunciation of the characters 日本國 as Cipangu.

The Poincaré group is the most general group of transformations which preserves the Minkowski metric:

Hitler designed the party's banner of a swastika in a white circle on a red background.

During the 1920s and 1930s, radio for mass communication and ultimately the invention of early television transformed communications in the United States.

The Delano family patriarch, Philip Delano, traveled to the New World on the Fortune in 1621, and the Delanos thrived as merchants and shipbuilders in Massachusetts.

Between 1914 and 1918, an estimated 400,000 to 600,000 ethnic Romanians served with the Austro-Hungarian army, of whom up to 150,000 were killed in action; total military and civilian deaths within contemporary Romanian borders are estimated at 748,000.

The Great Terror ended when Yezhov was removed as the head of the NKVD, to be replaced by Lavrentiy Beria, a man totally devoted to Stalin.

Hitler allegedly told Hanfstaengl that Churchill was not in office and was of no consequence.

A different entanglement classification is based on what the quantum correlations present in a state allow A and B to do: one distinguishes three subsets of entangled states: (1) the non-local states, which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) the steerable states that contain sufficient correlations for A to modify ("steer") by local measurements the conditional reduced state of B in such a way, that A can prove to B that the state they possess is indeed entangled, and finally (3) those entangled states that are neither non-local nor steerable. All three sets are non-empty.

The Supreme Court became Roosevelt's primary domestic focus during his second term after the court overturned many of his programs, including NIRA. The more conservative members of the court upheld the principles of the Lochner era, which saw numerous economic regulations struck down on the basis of freedom of contract. Roosevelt proposed the Judicial Procedures Reform Bill of 1937, which would have allowed him to appoint an additional Justice for each incumbent Justice over the age of 70; in 1937, there were six Supreme Court Justices over the age of 70. The size of the Court had been set at nine since the passage of the Judiciary Act of 1869, and Congress had altered the number of Justices six other times throughout U.S. history. Roosevelt's "court packing" plan ran into intense political opposition from his own party, led by Vice President Garner since it upset the separation of powers. A bipartisan coalition of liberals and conservatives of both parties opposed the bill, and Chief Justice Charles Evans Hughes broke with precedent by publicly advocating the defeat of the bill. Any chance of passing the bill ended with the death of Senate Majority Leader Joseph Taylor Robinson in July 1937.

Several analog computers were constructed in ancient and medieval times to perform astronomical calculations. These included the astrolabe and Antikythera mechanism from the Hellenistic world (c. 150–100 BC). In Roman Egypt, Hero of Alexandria (c. 10–70 AD) made mechanical devices including automata and a programmable cart. Other early mechanical devices used to perform one or another type of calculations include the planisphere and other mechanical computing devices invented by Al-Biruni (c. AD 1000); the equatorium and universal latitude-independent astrolabe by Al-Zarqali (c. AD 1015); the astronomical analog computers of other medieval Muslim astronomers and engineers; and the astronomical clock tower of Su Song (1094) during the Song dynasty. The castle clock, a hydropowered mechanical astronomical clock invented by Ismail al-Jazari in 1206, was the first programmable analog computer. Ramon Llull invented the Lullian Circle: a notional machine for calculating answers to philosophical questions (in this case, to do with Christianity) via logical combinatorics. This idea was taken up by Leibniz centuries later, and is thus one of the founding elements in computing and information science.

According to Christina Romer, the money supply growth caused by huge international gold inflows was a crucial source of the recovery of the United States economy, and that the economy showed little sign of self-correction. The gold inflows were partly due to devaluation of the U.S. dollar and partly due to deterioration of the political situation in Europe. In their book, A Monetary History of the United States, Milton Friedman and Anna J. Schwartz also attributed the recovery to monetary factors, and contended that it was much slowed by poor management of money by the Federal Reserve System. Former (2006–2014) Chairman of the Federal Reserve Ben Bernanke agreed that monetary factors played important roles both in the worldwide economic decline and eventual recovery. Bernanke also saw a strong role for institutional factors, particularly the rebuilding and restructuring of the financial system, and pointed out that the Depression should be examined in an international perspective.

The Western Allies fought through several lines until reaching the main German defensive line in mid-November.

Donald Trump's star has been vandalized multiple times. During the 2016 presidential election, a man named James Otis, who claims to be an heir to the Otis Elevator Company fortune, used a sledge hammer and a pickaxe to destroy all of the star's brass inlays. He readily admitted to the vandalism and was arrested and sentenced to three years' probation. The star was repaired and served as a site of pro-Trump demonstrations until it was destroyed a second time in July 2018 by a man named Austin Clay. Clay later surrendered himself to the police and was bailed out by James Otis. Clay was sentenced to one day in jail, three years of probation, and 20 days of community service. He also was ordered to attend psychological counseling and pay restitution of $9,404.46 to the Hollywood Chamber of Commerce. On December 18, 2018, the star was defaced with swastikas and other graffiti drawn in permanent marker, and it was vandalized yet again on October 2, 2020.

France is the birthplace of cinema and was responsible for many of its significant contributions to the art form and the film-making process itself. Several important cinematic movements, including the Nouvelle Vague, began in the country. It is noted for having a particularly strong film industry, due in part to protections afforded by the French government.

Baseball has been regarded as the U.S. national sport since the late 19th century, with Major League Baseball being the top league.

The wealth of musical culture lies in the classical music tradition during all historical periods, especially in the Baroque, Classicism, Romantic, modern classical music and in the traditional folk music of Bohemia, Moravia and Silesia. Since the early era of artificial music, Czech musicians and composers have been influenced the folk music of the region and dance.

Radar detected 65 bombers headed for Saga, 102 bound for Maebashi, 261 en route to Nishinomiya, 111 headed for Ube and 66 bound for Imabari.

During the 1980s, Japanese cinema experienced a revival, largely due to the success of anime films. At the beginning of the 1980s, Space Battleship Yamato (1973) and Mobile Suit Gundam (1979), both of which were unsuccessful as television series, were remade as films and became hugely successful in Japan. In particular, Mobile Suit Gundam sparked the Gundam franchise of Real Robot mecha anime. The success of Macross: Do You Remember Love? also sparked a Macross franchise of mecha anime. This was also the decade when Studio Ghibli was founded. The studio produced Hayao Miyazaki's first fantasy films, Nausicaä of the Valley of the Wind (1984) and Castle in the Sky (1986), as well as Isao Takahata's Grave of the Fireflies (1988), all of which were very successful in Japan and received worldwide critical acclaim. Original video animation (OVA) films also began during this decade; the most influential of these early OVA films was Noboru Ishiguro's cyberpunk film Megazone 23 (1985). The most famous anime film of this decade was Katsuhiro Otomo's cyberpunk film Akira (1988), which although initially unsuccessful at Japanese theaters, went on to become an international success.

The UK Government has budgeted substantial resources to the commemoration of the war during the period 2014 to 2018.

Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R 4 {\displaystyle \mathbb {R} ^{4}} and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964), Osterwalder & Schrader (1973) and Osterwalder & Schrader (1975).

The flower-giving tradition goes back to prehistoric times when flowers often had a medicinal and herbal attributes. Archaeologists found in several grave sites remnants of flower petals. Flowers were first used as sacrificial and burial objects. Ancient Egyptians and later Greeks and Romans used flowers. In Egypt, burial objects from the time around 1540 BC were found, which depicted red poppy, yellow Araun, cornflower and lilies. Records of flower giving appear in Chinese writings and Egyptian hieroglyphics, as well as in Greek and Roman mythology. The practice of giving a flower flourished in the Middle Ages when couples showed affection through flowers.

During 1939 Japan launched its first attack against Changsha, a strategically important Chinese city, but was repulsed by late September.

For both types, yeast is fully distributed through the beer while it is fermenting, and both equally flocculate (clump together and precipitate to the bottom of the vessel) when fermentation is finished. By no means do all top-cropping yeasts demonstrate this behaviour, but it features strongly in many English yeasts that may also exhibit chain forming (the failure of budded cells to break from the mother cell), which is in the technical sense different from true flocculation. The most common top-cropping brewer's yeast, Saccharomyces cerevisiae, is the same species as the common baking yeast. However, baking and brewing yeasts typically belong to different strains, cultivated to favour different characteristics: baking yeast strains are more aggressive, in order to carbonate dough in the shortest amount of time; brewing yeast strains act slower, but tend to tolerate higher alcohol concentrations (normally 12–15% abv is the maximum, though under special treatment some ethanol-tolerant strains can be coaxed up to around 20%). Modern quantitative genomics has revealed the complexity of Saccharomyces species to the extent that yeasts involved in beer and wine production commonly involve hybrids of so-called pure species. As such, the yeasts involved in what has been typically called top-cropping or top-fermenting ale may be both Saccharomyces cerevisiae and complex hybrids of Saccharomyces cerevisiae and Saccharomyces kudriavzevii. Three notable ales, Chimay, Orval and Westmalle, are fermented with these hybrid strains, which are identical to wine yeasts from Switzerland.

The Labrador is an exceptionally popular dog. As of 2006, it was widely considered the most popular breed in the world. And it is the most popular dog by ownership in Canada, New Zealand and the United Kingdom. In 2006 both the United Kingdom and the United States, there are well over twice as many Labradors registered as the next most popular breed. If the comparison is limited to dog breeds of a similar size, then there are around 3–5 times as many Labradors registered in both countries as the next most popular breeds, the German Shepherd Dog and Golden Retriever.

A devastating bombing raid on Tokyo of 9–10 March was the deadliest conventional bombing raid in history.

The Allies were ultimately driven out of Libya and suffered a major defeat in the fall of Tobruk on 21 June.

Association rule learning is a rule-based machine learning method for discovering relationships between variables in large databases. It is intended to identify strong rules discovered in databases using some measure of "interestingness".

The Greek city-state, the polis, was the fundamental political unit of classical Greece.

It follows from this matrix interpretation of linear systems that the same methods can be applied for solving linear systems and for many operations on matrices and linear transformations, which include the computation of the ranks, kernels, matrix inverses.

Hirohito (29 April 1901 – 7 January 1989), posthumously honored as Emperor Shōwa, was the 124th emperor of Japan, reigning from 1926 until his death in 1989. Hirohito and his wife, Nagako, had two sons and five daughters; he was succeeded by his fifth child and eldest son, Akihito. By 1979, Hirohito was the only monarch in the world with the title "Emperor". He was the longest-reigning historical Japanese emperor and one of the longest-reigning monarchs in the world.

In his paper on mass–energy equivalence, Einstein produced E = mc as a consequence of his special relativity equations.

A bijective linear map between two vector spaces (that is, every vector from the second space is associated with exactly one in the first) is an isomorphism. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm.

Stalin desired a "cultural revolution", entailing both creation of a culture for the "masses" and wider dissemination of previously elite culture.

The year 1859 saw the unification of Romania, as a nation state, from smaller principalities.

By far the most serious negative impact came after 1936 from the heavy destruction of infrastructure and manpower by the civil war, 1936–39.

This and related predictions follow from the fact that light follows what is called a light-like or null geodesic—a generalization of the straight lines along which light travels in classical physics. Such geodesics are the generalization of the invariance of lightspeed in special relativity. As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the post-Newtonian expansion), several effects of gravity on light propagation emerge. Although the bending of light can also be derived by extending the universality of free fall to light, the angle of deflection resulting from such calculations is only half the value given by general relativity.

Thereafter came a decisive strategic defeat at the Battle of Kursk.

Fast quantum algorithms for more complicated formulas are also known.

The late 1980s and early 1990s saw the collapse of the Warsaw Pact and the dissolution of the Soviet Union, which marked the end of the Cold War and solidified the U.S. as the world's sole superpower.

The geodesic equation covers the former aspect, stating that freely falling bodies follow lines that are as straight as possible in a curved spacetime.

Under the terms of the agreement, Germany occupied two-thirds of France, including Paris.

The learning rate defines the size of the corrective steps that the model takes to adjust for errors in each observation. A high learning rate shortens the training time, but with lower ultimate accuracy, while a lower learning rate takes longer, but with the potential for greater accuracy. Optimizations such as Quickprop are primarily aimed at speeding up error minimization, while other improvements mainly try to increase reliability. In order to avoid oscillation inside the network such as alternating connection weights, and to improve the rate of convergence, refinements use an adaptive learning rate that increases or decreases as appropriate. The concept of momentum allows the balance between the gradient and the previous change to be weighted such that the weight adjustment depends to some degree on the previous change. A momentum close to 0 emphasizes the gradient, while a value close to 1 emphasizes the last change.

Admiral Hara Tadaichi summed up the Japanese result by saying, "We won a great tactical victory at Pearl Harbor and thereby lost the war."

The dissolution of the Russian, German, Austro-Hungarian, and Ottoman Empires resulted in the creation of new independent states, including Poland, Finland, Czechoslovakia, and Yugoslavia.

Although Italy joined the Triple Alliance in 1882, a treaty with its traditional Austrian enemy was so controversial that subsequent governments denied its existence and the terms were only made public in 1915.

The Pythagorean theorem states that the sum of the areas of the two squares on the legs (a and b) of a right triangle equals the area of the square on the hypotenuse (c).

About 15% is occupied by Alaska, a state in northwestern North America, with the remainder in Hawaii, a state and archipelago in the central Pacific, and the five populated but unincorporated insular territories of Puerto Rico, American Samoa, Guam, the Northern Mariana Islands, and the U.S.

By December 1951, George VI had become concerned about Churchill's decline and intended asking him to stand down in favour of Eden, but the King had his own serious health issues and died on 6 February 1952 without making the request.

The spread of modern dog breeds has been difficult to resolve because many are the product of the controlled breeding practices of the Victorian era (1830–1900). In 2010, a study looked at 48,000 single nucleotide polymorphisms that gave a genome-wide coverage of 912 dogs representing 85 breeds.

He focused on Eastern Europe for this expansion, aiming to defeat Poland and the Soviet Union and then removing or killing the Jews and Slavs.

The Germans imposed martial law following Operation Safari, and Danish collaboration continued on an administrative level, with the Danish bureaucracy functioning under German command.

In mid-May, Japan started the Zhejiang-Jiangxi campaign in China, with the goal of inflicting retribution on the Chinese who aided the surviving American airmen in the Doolittle Raid by destroying Chinese air bases and fighting against the Chinese 23rd and 32nd Army Groups.

Animals have also been named after the show. Three species of mud dragons; Echinoderes daenerysae, Echinoderes rhaegal and Echinoderes drogoni, a bee fly; Paramonovius nightking, three species of scarab beetles; Gymnetis drogoni, Gymnetis rhaegali and Gymnetis viserioni, and a brittle star, Ophiohamus georgemartini, due to its sharp thorns resembling those depicted on the Game of Thrones crown.

The General Government in occupied Poland was a fully German administration.

Germany was now freed from the risk of war with the Soviets, and was assured a supply of oil.

Adam Tooze concurred, and wrote "To put this in perspective, annual military spending by the Weimar Republic was counted not in the billions but in the hundreds of millions of Reichsmarks"; for example, the Weimar Republic's 1931 program of 480 million Reichsmarks over five years compared to the Nazi Government's 1933 plan to spend 4.4 billion Reichsmarks per year.

Roosevelt, Churchill, and Stalin met for a second time at the February 1945 Yalta Conference in Crimea. With the end of the war in Europe approaching, Roosevelt's primary focus was on convincing Stalin to enter the war against Japan; the Joint Chiefs had estimated that an American invasion of Japan would cause as many as one million American casualties. In return for the Soviet Union's entrance into the war against Japan, the Soviet Union was promised control of Asian territories such as Sakhalin Island. The three leaders agreed to hold a conference in 1945 to establish the United Nations, and they also agreed on the structure of the United Nations Security Council, which would be charged with ensuring international peace and security. Roosevelt did not push for the immediate evacuation of Soviet soldiers from Poland, but he won the issuance of the Declaration on Liberated Europe, which promised free elections in countries that had been occupied by Germany. Germany itself would not be dismembered but would be jointly occupied by the United States, France, Britain, and the Soviet Union. Against Soviet pressure, Roosevelt and Churchill refused to consent to impose huge reparations and deindustrialization on Germany after the war. Roosevelt's role in the Yalta Conference has been controversial; critics charge that he naively trusted the Soviet Union to allow free elections in Eastern Europe, while supporters argue that there was little more that Roosevelt could have done for the Eastern European countries given the Soviet occupation and the need for cooperation with the Soviet Union during and after the war.

Churchill was ebullient after the Battle of the River Plate on 13 December 1939 and afterwards welcomed home the crews, congratulating them on "a brilliant sea fight" and saying that their actions in a cold, dark winter had "warmed the cockles of the British heart".

As a result of the failed Beer Hall Putsch, the Nazi Party and its affiliated organisations were banned in Bavaria.

Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors.

Very little is known about Diophantus of Alexandria; he probably lived in the third century AD, that is, about five hundred years after Euclid. Six out of the thirteen books of Diophantus's Arithmetica survive in the original Greek and four more survive in an Arabic translation. The Arithmetica is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form f ( x , y ) = z 2 {\displaystyle f(x,y)=z^{2}} or f ( x , y , z ) = w 2 {\displaystyle f(x,y,z)=w^{2}} . Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found.

The Liberty Memorial was dedicated on 1 November 1921, when the supreme Allied commanders spoke to a crowd of more than 100,000 people.

Different cultural events are organised in Europe, with the aim of bringing different cultures closer together and raising awareness of their importance, such as the European Capital of Culture, the European Region of Gastronomy, the European Youth Capital and the European Capital of Sport.

Four other pieces of work that Einstein completed in 1905—his famous papers on the photoelectric effect, Brownian motion, his special theory of relativity and the equivalence of mass and energy—have led to the year's being celebrated as an annus mirabilis for physics almost as wonderful as 1666 (the year in which Isaac Newton experienced his greatest epiphanies).

A few days later, the Soviets launched a massive assault against German-occupied Hungary that lasted until the fall of Budapest in February 1945.

It is by far the world's largest exporter of services.