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Search All of the Math Forum: Views expressed in these public forums are not endorsed by NCTM or The Math Forum. Notice: We are no longer accepting new posts, but the forums will continue to be readable. Replies: 17   Last Post: Jan 11, 1995 6:43 AM Messages: [ Previous | Next ] John Conway Posts: 2,238 Registered: 12/3/04 Posted: Dec 21, 1994 6:09 AM I'm not quite sure why you (ckfan) WANT to include the general I don't know of any interesting theorems on the trapezoid that don't work for more general quadrilaterals, and it's always rather puzzled me why anyone ever thought it worth while to give this particular kind of quadrilateral its own special name. I now think I know the reason, and am not too impressed by it. Proclus (who's the guy responsible) says that the area is the mean of the lengths of the two parallel sides times the distance between them, and I think this was the point - the trapezoid is the most general quadrilateral with a simple area formula. So it has some practical point - you can compute the area of a polygon by dividing it up into trapezoids. But there's not much theoretical point - this area formula is trivially deducible from the one for a triangle by just drawing a diagonal. Does anyone know of any more interesting John Conway Date Subject Author 12/15/94 E7M2WAT@TOE.TOWSON.EDU 12/16/94 W Gary Martin 12/16/94 Henri Picciotto 12/17/94 John Conway 12/17/94 roitman@oberon.math.ukans.edu 12/18/94 John Conway 12/20/94 Chenteh Kenneth Fan 12/21/94 John Conway 12/21/94 Walter Whiteley 12/21/94 William T. Webber 12/21/94 James King 12/21/94 Chenteh Kenneth Fan 12/21/94 John Conway 12/21/94 John Conway 12/21/94 John Conway 12/21/94 Chenteh Kenneth Fan 12/21/94 Chenteh Kenneth Fan 1/11/95 joe malkevitch

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Friday August 22, 2014 # Posts by john Total # Posts: 6,202 Art Thank you very much, Writeacher. I'm posting the first part of my paraprase again because I failed to include the reference to the stanzas. 1) Dickinson’s poetic universe is characterized by personal deprivation. In particular, in poem 579, “I had been hungry, ... June 10, 2012 Art I left out these last sentences. I really hope you can have a look at it. It is very urgent! 1) She is the one outside the window, always watching other people indulging themselves on the pleasures of life. 2) When given the chance herself in this meal, however, she is tempted... June 10, 2012 Art Writeacher, I'm finding it difficult to rephrase the poem "I had been hungry all the Years" by Emily Dickinson. I wonder if you could help me. 1) Dickinson’s poetic universe is characterized by personal deprivation. In particular, in poem 579, “I had ... June 10, 2012 Foreign languages I really hope you are there, Writeacher. Can you suggest an internet site where I can find a comment on Emily Dickinson's "I had been hungry all the years"? Thank you veru much. June 10, 2012 Foreign languages I urgently need to find a very simple and concise summary (intermediate level of English) on the Lost Generation and Fitzgerald's The Great Gatsby. Thank you. June 10, 2012 chemistry 123.00 June 9, 2012 chemistry Mercury(Hg) has a density of 13.5 g/cm cubed. How much space would 50.0 g of mercury occupy? Calculate the mass in grams, of a 3 cubic inch cube of aluminum(Al). The density of aluminum(Al) is 2.5 g/cm cubed. NOTE:(1inch cubed =16.4 cm cubed June 9, 2012 chemistry 675.00 June 9, 2012 chemistry Mercury(Hg) has a density of 13.5 g/cm cubed. How much space would 50.0 g of mercury occupy? June 9, 2012 Finance Often in personal finance we want to know what our \$1 investment today will be worth in 20 years. In business however, there is more concern with answering the question, “If I receive \$100 in 5 years, what is that worth today?” To answer this question, modify the ... June 9, 2012 Finance An annuity is developed and used based on the concept of time value of money. Please solve for the principle required when one retires so that a payment of \$1500.00 is received each month for 30 years (360 months). Assume that the interest rate for the payout is 5% and the ... June 9, 2012 Physics An electron travelling in the same direction that an electric field points initially has a speed of 4.21 ×107 meters per second. If the electric field has a magnitude of 2.28 ×104 N/C, how far does the electron travel before stopping? June 7, 2012 Art Here are the last sentences on the same theme. 1) This important feature makes power lines able to transport electric energy without any problem connected with dissipation of energy. They are useful in aerospace industry because they can keep a big amount of electric energy ... June 4, 2012 Foreign languages Thank you very much for checking this part on scientific English. Here is the last part. 1) The superconductors are also characterized by the so called Meissner-Ochsenfeld Effect. When a superconductor is put in (into) a magnetic field, it produces a magnetic field which ... June 4, 2012 Foreign languages Could you please check if this paragraph is possible in scientific English? Thank you, Writeacher. 1) Superconduction consists of a particular physical phenomenon which characterizes only superconductors. 2) These materials become superconductors when they are brought to the ... June 4, 2012 Foreign languages Thank you very much, Writeacher for checking my paragraph. Here is the last part. 1) Six months after being diagnosed with cancer, she was dead. HeLa cells have had a fundamental importance in scientific research. 2) Her living cells, which survive to this day have contributed... June 4, 2012 Art I left out the following sentences. Thank you, Writeacher. 1)He writes to his mum about the incident. 2) He is critical of the war and turns against those who think that it is glorious and fitting to die for one's country. 3) His friend dies because he is not fast enough ... June 4, 2012 Foreign languages Writeacher, could you please tell me if 5 and 6 are possible? Thank you. 1) In 1951 an Afro-american woman was admitted to Johns Hopkins Hospital in Baltimore. 2) In those years scientists had been unsuccessfully trying to grow malignant human cells outside the body to ... June 4, 2012 Art I left out the following sentences. Thank you very much, Writeacher. 1) He feels tenfold more evil than Hyde but also more free. 2) He realizes a potion capable to split up his soul, to separate the good side from the evil one. 3) The image of his friend returns to the memory ... June 3, 2012 math what is 1.5% of \$2,ooo ,ooo.oo June 2, 2012 Physics A ball of mass 1.74 kg is dropped from a height y1 = 1.47 m and then bounces back up to a height of y2 = 0.83 m. How much mechanical energy is lost in the bounce? The effect of air resistance has been experimentally found to be negligible in this case, and you can ignore it. June 2, 2012 Physics Thank you! June 1, 2012 Physics A 69-kg skier moving horizontally at 4.1 m/s encounters a 21° incline. (a) How far up the incline will the skier move before she momentarily stops, ignoring friction? (b) How far up the incline will the skier move if the coefficient of kinetic friction between the skies ... June 1, 2012 Physics A man who is 6 ft tall is standing in front of a plane mirror that is 2 ft in length. If the mirror is placed lengthwise with its bottom edge 4 ft above the floor on a wall that is 5 ft away, how much of his image (i.e. what length of himself) can the man see? (Assume that his... May 30, 2012 Science Could you please check these sentences on LSD? Thank you, Writeacher. 1) LSD is a psychoactive substance capable of generating perceptual alterations. It was synthesized for the first time in 1938 by Albert Hoffman of Sandoz laboratories in Basel. 2) The full name of that ... May 30, 2012 Art I left out these last sentences. Thank you very much. 1) I was having dinner / I had been having dinner when the telephone rang. (If the two actions happen at the same time only the past continuous is possible) 2)I'm going to go on a scout camp for two weeks. I'm going... May 30, 2012 Art Can you please check these statements, Writeacher. Thank you. 1) If I hadn't been waiting for you, I wouldn't have arrived late. I was having dinner (not: I had been having dinner) when the telephone rang. 2) He told me he had lost the keys of his car (or his car keys... May 30, 2012 Art I urgently need you to revise these sentences. Thank you. 1) You can cheat by conceiling pieces of paper full of information in your pencil case. 2) you can take a picture of your exam questions and email it to a friend 2) you can instant message homework questions or text ... May 30, 2012 Art I left out the following sentences. Thank you very much for your invaluable help, Writeacher. 1) On my summer holiday(s) I’m competing in a chess tournament in the first week of July. It will probably be difficult because the best players will participate. 2) Where will ... May 29, 2012 Art I urgently need you to check these sentences. Thank you, Writeacher. By the way, have you found anything on Henrietta Lacks? 1) When his wife puts his arms around him and kisses him, he faints. He explains this by saying that he has not been used to be touched by such a ... May 29, 2012 Art Could you please check the following corrections? Thank you, Writeacher 1) He offers her to let her remain in the country house. Correction: he offers to let her remain in the... 2) She invites her to remain... 3) He offers her to return to her parents. He offers to let her ... May 29, 2012 Art I left out the following sentences, Writeacher. Thank you. 1) After that we’re having dinner (better we’ll have dinner) and after that we’ll go for a walk in the city centre. 2) On the next day we’ll first go on a sightseeing tour of (in ) the historic ... May 29, 2012 Art Writeacher, I urgently need you to check these sentences. Thank you very much. 1)After that we are going (we'll go?which form would you use to describe your summer holiday?) on a guided tour of the science museum. 2) We'll spend some time in the city centre doing ... May 28, 2012 Science Thank you very much for your help. I will check major grammar mistakes tomorrow morning. Can you please have a quick look at the second part and tell me which sentences need to be revised? 1) However, their most striking feature was that these cells were able to reproduce and ... May 28, 2012 Science Writeacher, I urgently need to check this part regarding a scientific topic. I hope you can help me. Here is the first part. 1) In 1951 an American woman descended from African slaves freed named Henrietta Lacks, was admitted to Johns Hopkins Hospital in Baltimore. In those ... May 28, 2012 HR •Describe the difference between diversity and affirmative action? May 28, 2012 Art Could you please tell me which alternatives are possible? Thank you, Writeacher. 1) You usually ....... be at the airport two hours before your flight leaves. (have to; not must or should). 2) At most schools students can go on .... (a trip/on trips) (are both possible?) with ... May 28, 2012 Art Could you please check these sentences, Writeacher? Thank you. 1) I shouldn't leave money on the desk during the beak. 2) I have to call home if I'm feeling sick (I feel sick): 3) I have to bring all my school equipment to school. 4) I can get/take (both?) a snack from... May 28, 2012 Math You have aloan for 150,000 at 5% on a 30 yr mortgage. You plan to pay off your loan in 10 yrs. Do you want your loan to be figured using the Rule of 78 or the Unpaid Principle Balance Rule or something else. May 27, 2012 art I left out these very last sentences, Writeacher. Tahnk you very much. 1) Interactive advertising is an advertising technique which tries to make (better: create) contact with its clients. 2) In Sweden there is a law according to which (not for which) children cannot appear in... May 27, 2012 Art I left out a few other things. Could you please check them, too? 1) The difference about advertising today is that…..(or What is different about advertising today is….) 2) Children cannot oppose their appearance in ads. They cannot do anything against the appearance ... May 27, 2012 Foreign languages Thank you very much for helping me. Here are the very last sentences on the same theme. Thank you very much. 1) Over the years people have started watching ads more and more. 2) The way in which ads are made (?) has changed. The latest ads don’t just sell brands, they ... May 27, 2012 Foreign languages I left out the following doubts. I really hope you can consider them, too. 1) He has grown out of his shoes, so we'll need to buy him new ones. Hold the line is the same as hold on the line? 2) It wasn't necessary you bought the milk (wrong?). You needn't have ... May 27, 2012 Art I have tried to rephrase the sentences as you told me. Can you please have a look at them? 1)Finally, in my opinion, the way to solve the problem is very simple and consists in treating people in the same way. 2) People’s attitude to ads has changed. Instead of switching ... May 27, 2012 Art I still have a few sentences to check. Thank you very much, Writeacher. 1)When an ad tries to create direct contact with people, we can talk about interactive advertising. 2) For example, an important chain of supermarkets in the UK divides (has divided) its customers into ... May 27, 2012 Art This is the other paragraph I urgently need you to check. Thank you very much, Writeacher. 1)I hope that in the future the consumerist societies will be defeated and that with the money, or better, with the resources that we put away we will help people who are more unlucky ... May 26, 2012 Art I still have four more paragraphs to check on the same theme. Thank you very much in advance. 1) The use of children in ads is an exploitation of the children's image. It's a way to earn audience as well as to sell a major number of products. 2) It is also a way to ... May 26, 2012 Art Thank you, Writeacher. I still have a few more paragraphs to check. 1)I think that ads can be very dangerous for children because they often show children happy for a kind of product, like a toy and so they want to be like them. 2) I think that parents should pay attention and... May 26, 2012 Art Writeacher, I urgently need to revise these sentences. I really hope you can help me. 1) Ads can change and can wash the minds of children. Some years ago people used to switch channels to avoid ads, but we do it less and less 2) Nowadays most people switch channels to avoid ... May 26, 2012 Geometry in circle p below the lengths of the parallel chords are 20,16, and 12. Find measure of arc AB..... the chord with a length of 20 is the diameter. the chords with lengths 16 and 12 are below the diameter torwards the bottom of the circle. arc AB is the arc made up of the ... May 25, 2012 Art I urgently need you to revise these sentences. Thank you very much. 1) In the past when adverts came on TV the majority of people used to turn down the volume or to switch channels (or switch over to another channel). However, they don’t do this now. 2) In the past ... May 24, 2012 science How many kilograms of NH3 are needed to produce 1.00 x 10^5 kg of (NH4)2SO4? May 23, 2012 science How many kilograms of NH3 are needed to produce 2.90 x 10^5 kg of (NH4)2SO4? May 23, 2012 math a particle is moving along the graph of y=x^(1/3) when x=8 the y component of its position is increasing at the rate of 1 centimeter per second how fast is the x component changing at this moment May 23, 2012 math Tia has 24 black or white buttons. She has 10 more white buttons than black buttons. How many white buttons does Tia have? May 22, 2012 Art Could you please have a look at these sentences? Thank you very much, Writeacher. 1)We'll go on a bus sightseeing tour on which we'll admire the most famous attractions. 2) We are given a lunch box (by the hotel). 3) In the afternoon we'll go to the newly opened ... May 22, 2012 Foreign languages I urgently need you to revise these sentences. Thank you very much, Writeacher. 1) We are leaving for Rome tomorrow morning at 7.30 am. 2) We are meeting at the entrance of the bypass (ring road/orbital road) at 7.15 3) The bus journey will last about six hours. Two stops are ... May 22, 2012 Art Could you please check these sentences, too? Thank you very much. 1) The Augustan were interested in real life. As for politics/as far as politics is concerned, the two main political parties were teh Tories and the Whigs. 2) The poor used to live in terrible conditions. 3) ... May 22, 2012 Foreign languages I hope you can check these two long-winded sentences.Thank you 1) Pamela is writing to her family to tell her mum what happened to her, by accusing her master of hsi behaviour. 2) While they were in the summer house, Mr B kissed her but she had never talked about it to anybody... May 22, 2012 Art Can you please help me check a few things, Writeacher? Thank you. 1)The two political parties which alternated in government were the Whigs and the Tories. 2) The Whig ministers' meetings developed into the cabinet led by the Prime Minister. 3) It is composed of four books... May 22, 2012 Math 7p-7/p divided by 10p-10/3p squared May 21, 2012 Foreign languages Can you please help me check these sentences, Writeacher? Thank you. 1) In the Augustan Age there were two parties: the Whigs supported by the wealthy commerical classes which pressed for undstrial and econmical development and religious toleration. 2) The Tories were ... May 21, 2012 Art Could you please check these few sentences? Thank you very much, Writeacher. 1) The old man passed away (is "out" possible though the meaning is different) during the night. 2) The tests have to be handed in to the teacher before the end of the week. 3) I should do ... May 21, 2012 Art Writeacher, could you please check this sentence? Thank you. 1) He dropped me off at the post office. How would you rephrase it? He let me get (out of the car) at the post office. He made me get off the bus at the post office. within the end of the week is not the same as by/... May 21, 2012 Art Could you please help me to check these last sentences. Thank you, Writeacher. 1) The subtitle is referred (refers) to the reward given by Mr B to P. He tries to seduce her but she resists (better: his advances) and for this reason she receives from him a proposal of marriage... May 20, 2012 Art I left out the following sentences. Thank you very much, Writeacher. 1) My son has grown out of his shoes (not "frew out of.."), so we'll have to buy him new shoes (a new pair of shoes, a pair of new shoes .. are all possible?). We'll have to buy him a new ... May 20, 2012 Art Thank you very much. I still need you to check a few urgent things, Writeacher. 1)The line is engaged. Would you hold? Would you like to hold the line? Would you like to hang on/hold on? Which are possible? 2)You needn't have bought the milk; there was some already/there ... May 20, 2012 Modern languages Thank you very much again. Here are a number of sentences I would like you to check. 1) The Sundance film festival was launched by Robert Redford to promote first time film makers as well as low-budget independent (independently produced) films. 2) Redford says that the films ... May 20, 2012 Art Thank you very much, Writeacher. I left out the following statements. 1) The poem successfully expresses the modern artist’s disillusion with the modern world and, at the same time, the desperate need and search for a new tradition. 2) It also represents the culmination ... May 20, 2012 Calculus A Baseball Diamond has the shape of a square with sides 90 feet long. a player is running from 1st to 2nd base at a speed of 25 feet per second. find the rate at which his distance s is from home plate is changing when the player is 20 feet from 2nd base. May 20, 2012 Calculus A cone-shaped coffee filter is draining at a rate of 20 in^3 /min. The filter has a diameter of 8 inches. the radius of the filter is 2/3 the height. how fast is the coffee level falling when the coffee is 3 inches deep? May 20, 2012 Art Could you please check these sentences, too? Thank you very much. 1) The Waste Land is a complex long poem of 433 lines about a “waste land” which seems both personal and external. It is divided into five sections and is dedicated to Eliot’s friend and poet, ... May 20, 2012 Art Thank you for asking me about the earthquake which struck this morning near Bologna and has already killed 4 people. Luckily, we haven't felt it since we live further north! I find it difficult to check the following paragraphs in which tenses are mixed and the word choice... May 20, 2012 Foreign languages I forgot to add these sentences. Thank you very much, Writeacher. 1) He spends more on train tickets than he actually earns by working as a support teacher. 2) Pamela is an epistolary novel made up of 32 letter in which the girl tells her parents about her life as a noble ... May 20, 2012 Art Thank you, Writeacher. I checked the sentece you couldn't understand and I added a few other things. 1) He was responsible to the king for the governement's policy. 2) The horses, which Gulliver meets in his fourth voyage, are described as beautiful, strong, fast and ... May 20, 2012 Foreign languages I left out a few sentences. Thank you veru much. I need to make sentence 5 shorter. 1) In 1766 William Pitt the Elder was elected Prime Minister and he made England a strong and flourishing commercial country. 2) During the 18th century the middle class gained more and more ... May 19, 2012 Foreign languages I would like you, Writeacher to help me check a few sentences. Thank you. 1) English Parliament was divided into two parties: the Whigs and the Tories. The former descended from the Parlamentarians. 2) They encouraged commercial and trade (?) progress and were supported by the... May 19, 2012 Econ Suppose the government wants to reduce the total pollution emitted by three firms in its area. Currently, each firm is creating 4 units of pollution in the area, for a total of 12 pollution units. The government wants to reduce total pollution in the area to 6 units. In order ... May 19, 2012 Art Writeacher, I urgently need you to check these sentences. Thank you. 1) They got him to give back the animal: He was persuaded to give back the animal. They had him give back the animal: He was requested (kindly invited) to give back the animal. 2) He forgot closing the door... May 19, 2012 Art I still have a few things I'd like you to consider, Writeacher. Thank you very much. 1) She refused to go to the dentist's We are planning to move into a new house. 2) They offered me to go for a week in Egypt: They offered to let me go to Egypt for a week. They ... May 17, 2012 Health I left out this part. Thank you very much, Writeacher. 1) Have you received any news about our common project? How are things getting on in England? When do school holidays start in England? 2) When do you think we will know if the Commission has approved our project? Please ... May 17, 2012 Art I left out these sentences. Could you please check them, too? Thank you. Is punctuation OK? 1) People suppose that he worked for Fiat. He is supposed to have worked... 2) Someone asked her about the weather. 3) My parents let me go on holiday on my own. 4)I’m terribly ... May 17, 2012 Art Can you please check these sentences, Writeacher? Thank you very much. 1) “I have been to Greece twice," he said. (He said that he had been to Greece twice) 2) It was the first time I had spent my vacation without my parents. Mary had done all the cleaning before/by ... May 17, 2012 calculus a rectagular sheet of metal has perimeter 36 meters. maximize the volume if the sheet is rolled into a cylinder May 16, 2012 algebra Determine which pair(s) of function(s)represent inverses of each other. 1. f(x)= 8x+1 g(x)= 8x-8 2. f(x)= 3x+2 g(x)= x-2/3 3. f(x)= 5x+2 g(x)= x/5-2 May 16, 2012 algebra Given f(x)= x^2-4 and g(x)= x^1/2, what is the domain of (g 0 f)(x)? May 16, 2012 Art I left out the following sentences. Thank you Writeacher. 1) The objective correlative is a phrase coined by Eliot himself, who maintained that poetry must be objective, impersonal, and images are the objective correlative of the emotions they aim to suggest (they evoke?). 2) ... May 16, 2012 Art I urgently need you to check these sentences. Thank you. 1) Focus on Richardson’s Pamela and answer briefly the following questions. 2) What does Pamela recount to her mum in letter XV? 3) What does Mr B accuse her of and how does she apologize herself to him? 4) What ... May 16, 2012 Art Thank you, Writeacher. Could you please check if everything is possible? 1) What does Pamela recount to her mum in letter XV? What does Mr B accuse her of and how does she apologize herself to him (or justify her behaviour)? May 16, 2012 Foreign languages I left out the following sentence. Thank you. 1) I think you should show the text message to your teacher and ask his advice about what to do. (ask him for advice.... is it possible?) May 16, 2012 Foreign languages Writeacher, could you please check these sentences on school rules? Thank you very much for your invaluable help. 1) You must be respectful of your classmates and of your teachers. You should stand up when teachers (or your teachers?) enter the classroom. 2) You mustn't ... May 16, 2012 Art I left out the following sentences on school rules. I really hope you can have a look at them, Writeacher! 1) I'm writing this letter because I would like to tell you some of the rules at my school. I hope you'll do the same. 2) You mustn't change or modify marks ... May 15, 2012 algebra Given f(x)= 5x^3-9, find the inverse of f, f^-1(x)=? May 14, 2012 algebra If f(x)= x^2-2, then f(x+h)=? May 14, 2012 Art Could you please check these examples and tell me if the sentences in brakets are possible? Thank you , Writeacher 1)Her boss is going to ask her out for dinner next Saturday. 2) The line is engaged. Would you like to hold? (Not: would you hold?) 3.Our coach told us that the ... May 14, 2012 Art Could you please check these few examples? Thank you very much, Writeacher. 1) It is likely that he will pass his driving test. He is likely to pass his driving test. He should pass it. 2) I'm unsure that (if) I will pass it. I might pass it. It is possible that I will ... May 14, 2012 Art Have I fixed the run-on? Thank you very much, Writeacher. I added a few more sentences. 1) John doesn't have to take the car. We can walk to school. 2) Have you received any new emails from my students? How many have you got so far? 3) What are you students learning in ... May 14, 2012 Foreign languages Thank you, Writeacher. I left out the following sentences. Thank you very much. 1) Who has called the wrong number? His team has always been (or have always been?) worse than the others. 2) Have you read that book yet? No, I haven't finished it yet. 3) I've never flown... May 14, 2012 Art Writeacher, could you please check these sentences? Thanks. 3) It takes him half an hour to drive to school. 4)He has called the wrong number. 5) He paid 300 Euros for his wedding’s suit. 6) I phoned Mary yesterday morning. He started reading his new his new book three ... 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# Sequence length range for V4 region analysis Hi community!!! • I’m analysing V4 region sequence. As we know this region is around 253 bp long, what range should I use in the first “screen.seqs” in order to remove bad sequences? I’m using “(minlength=230, maxlength=275)”. Is it ok? • why, in first “screen.seqs” step of mothur SOP, maxlength=275 has been considered? Thanks & Regards, DC7 Hi, There aren’t any good sequences in the database for the V4 region (with priming sites removed) that are longer than 275 nt. Pat Thanks sir for your reply. But what minimum length should we consider in any analysis? with paired 250 nt reads, i’m not sure it’s possible to get much below 250 nt. probably 240 on the low end. Sir, firstly thanks and secondly pardon for repetetive questions. Sir, what if I take a broader range of length? When I analysed V4-V6 region (550bp) I considered minlength=525, maxlength=575 for the “screen.seqs” command. Is it considered wrong? What would be your comment as a reviewer? ``````mothur > summary.seqs(fasta=current) Using /media/dc7/New Volume/OBESITY/prjna321731_16s/prjna321731_16s_nw/merge.paired.trim.contigs.fasta as input file for the fasta parameter. `````` Using 8 processors. `````` Start End NBases Ambigs Polymer NumSeqs Minimum: 1 301 301 0 3 1 2.5%-tile: 1 542 542 0 4 15495 25%-tile: 1 544 544 0 5 154948 Median: 1 547 547 0 5 309896 75%-tile: 1 550 550 2 5 464843 97.5%-tile: 1 553 553 7 7 604296 Maximum: 1 602 602 62 299 619790 Mean: 1 546 546 1 5 # of Seqs: 619790 It took 22 secs to summarize 619790 sequences. `````` Output File Names: /media/dc7/New Volume/OBESITY/prjna321731_16s/prjna321731_16s_nw/merge.paired.trim.contigs.summary mothur > screen.seqs(fasta=current, group=current, maxambig=0, maxhomop=8, minlength=525, maxlength=575) Using /media/dc7/New Volume/OBESITY/prjna321731_16s/prjna321731_16s_nw/merge.paired.trim.contigs.fasta as input file for the fasta parameter. Using /media/dc7/New Volume/OBESITY/prjna321731_16s/prjna321731_16s_nw/merge.paired.contigs.groups as input file for the group parameter. Using 8 processors. It took 7 secs to screen 619790 sequences, removed 286319. Thanks and Regards, DC7 ` You would need to generate a reference alignment for V4-V6 (without the primers on the sequences) and then run it through `summary.seqs`. The output would show you what ranges you would expect (keep in mind that there might be some weird outliers). Pat This topic was automatically closed 10 days after the last reply. New replies are no longer allowed.

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# Thread: Simple related rates prob 1. ## Simple related rates prob Helium is pumped into a spherical balloon at $2 \frac{ft^3}{s}$. How fast is the radius increasing after 2 minutes? I've searched some of the other related rates probs posted on this forum, so I apologize if a question like this has been brought up already. Basically, I understand I'm looking for $\frac{dV}{dt}$ , and that $\frac{dV}{dt} = \frac {dV}{dr} * \frac {dr}{dt}$. Now, the rate is 2 $\frac{ft^3}{s}$ and the volume of a sphere is $V = \frac{4}{3} * {PI}r^3$. Implicitly differentiating V with respect to T gets $\frac {dV}{dt} = 4 * {PI}r^2 \frac{dr}{dt}$ I know that the rate the volume increases with respect to time is $2 \frac{ft^3}{s}$, and I'm not sure what to do with my 2 minutes (or 120 seconds). Is the volume with respect to time 120 seconds and $2 \frac{ft^3}{s}$ is the rate with respect to time? That would make more sense I think because the dimensions of cubic feet with a factor of 2 is like a rate and seconds is most definitely a part of time. However, when I try to solve $120 = 4PIr^2*2$ or $\frac{dV}{dt} = \frac {dV}{dr} * \frac {dr}{dt}$ and that's incorrect. Since I don't know the volume with respect to time, how can I properly go about solving for the rate with respect to time? I get $r = sqrt(\frac{120}{8PI})$ 2. ## Re: Simple related rates prob We are looking for $\frac{dr}{dt}$, and we know: $dV=4\pi r^2\,dr$ hence: $\frac{dV}{dt}=4\pi r^2\,\frac{dr}{dt}$ If we assume $V(0)=0$ then $V(t)=2t \text{ ft}^3$ and so: $r(t)=\left(\frac{3t}{2\pi} \right)^{\frac{1}{3}}$ We are given: $\frac{dV}{dt}=2\,\frac{\text{ft}^3}{\text{s}}$ $t=2\text{ min}=120\text{ s}$ Can you put this together to finish? 3. ## Re: Simple related rates prob Originally Posted by MarkFL2 $r(t)=\left(\frac{3t}{2\pi} \right)^{\frac{1}{3}}$ Can you put this together to finish? Do you mind if I ask where you derived that? Well, I took the steps you laid out. since the problem asks for the rate of change with respect to time when t = 120 seconds, I plugged 120 into r(t) which == 3.86. Since we know the rate of change with respect to the volume we can plug that into dv/dr which gets 4pi(3.86)^2. And dV/dt is 2 cubic feet per second. So the final equation came out to be $2\frac{ft^3}{s} = 186.8\frac{dr}{dt}$ which gave 0.01071 $\frac{ft^3}{s}$, and that means the growth of the balloon's volume has decreased significantly by the 2 minute mark. I'm just still unclear how you got the r(t) equation. Thanks much. 4. ## Re: Simple related rates prob $V = \frac{4\pi}{3} r^3$ assuming $V(0) = 0$ ... $2t = \frac{4\pi}{3} r^3$ solve for $r$ as a function of $t$ 5. ## Re: Simple related rates prob Originally Posted by skeeter $V = \frac{4\pi}{3} r^3$ assuming $V(0) = 0$ ... $2t = \frac{4\pi}{3} r^3$ solve for $r$ as a function of $t$ Oh! I kept looking at dV/dr.

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# Thread: Find perimeter of figure 1. ## Find perimeter of figure Hello! It's really basic, but what is the perimeter of this figure? r is the radius of the circle and length of the square. 2. Originally Posted by Incendiary Hello! It's really basic, but what is the perimeter of this figure? r is the radius of the circle and length of the square. Hint: Fill in two more lines on this figure. First the radius to the other corner of the square, then complete the top of the square. You will note that the triangle formed is an equilateral triangle, with an apex angle of 60 degrees. So we are cutting a 60 degree arc off from the circumference of the circle... -Dan 3. Hello, Incendiary! What is the perimeter of this figure? $\displaystyle r$ is the radius of the circle and length of the square. Label the vertices of the square: A (upper left), B (lower left), C (lower right), D (upper right). Label the center of the circle with O. Then: .$\displaystyle AD \,= \,BC \,= \,r.$ Hence, $\displaystyle \Delta AOD$ is equilateral: $\displaystyle \angle AOD \,= \,60^o \,=\,\frac{\pi}{3}$ The major arc AD has length: $\displaystyle \frac{5\pi}{3}r$ The three sides of the square has length: $\displaystyle 3r$ Therefore, the perimeter is: .$\displaystyle \boxed{P \;= \;\left(\frac{5\pi}{3} + 3\right)r}$ The total area is: .$\displaystyle \text{(area of square) + (area of circle) - (area of overlap)}$ The overlap is a segment of a 60° angle. . . The area of the sector is: .$\displaystyle \frac{1}{6}\pi r^2$ . . The area of the triangle is: .$\displaystyle \frac{\sqrt{3}}{4}r^2$ Hence, the area of the segment is: . $\displaystyle \frac{1}{6}\pi r^2 - \frac{\sqrt{3}}{4}r^2$ Therefore, the area of the figure is: . . $\displaystyle A \;=\;r^2 + \pi r^2 - \left(\frac{1}{6}\pi r^2 - \frac{\sqrt{3}}{4}r^2\right) \;=\;\boxed{\frac{1}{12}\left(10\pi + 12 + 3\sqrt{3}\right)}$

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Download Download Presentation ELEC130 Electrical Engineering 1 # ELEC130 Electrical Engineering 1 Download Presentation ## ELEC130 Electrical Engineering 1 - - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - - ##### Presentation Transcript 1. ELEC130 Electrical Engineering 1 Week 3 Module 2 DC Circuit Tools 2. Administration Items • Tutorials - Will be held in ES 210 this week. • Answers tutorial 1 will be revised • Introduction to Electronic Workbench - Revised document • Faculty PC’s Rm. ES210 - Go to Diomedes • Login: cstudentnumber • Password: access keys on students card + daymonth (ddmm) of birth • Use Drive u: to save your work • Laboratory - THIS WEEK in EE 103(a) • Allocation of Laboratory and Tutorial Times • NO more changes after Friday 12 March 1999 4 pm • If you cannot make your time, please ask for alternative • Quiz 1 - THIS WEEK Lecture 3 3. Survey Results • Subject Home Page: - through Dept. Pages • http://www.ee.newcastle.edu.au/ • Then to Undergraduate studies • Then to Course Information/Syllabus • Then to Subject Web Pages • From the web site you have the option to save the file in power point • You are expected to read the specified text references to build the background information to the topic areas we are covering. You should think of the lecture as an opportunity to reflect on your reading and clarify difficult concepts. Lecture 3 4. Survey Results (cont.) • Current Sources • DC power supply, transistors • Conductance - Parallel Resistance's • Voltage and Current Division • Why - Delta - tutorial 1 Question 19 part 4 • Floyd pg. 309 Example 8-19 pg. 312 • Superposition Lecture 3 5. I1 + - Vs R2 R1 I2 Conductance • Sometimes easier to use inverse of resistance called conductance G = R-1 • Symbol: G • Units: Siemens S (mhos) • NB: Useful when resistors are connected in parallel • Geq = G1 + G2 +... +Gn • 1/Req = 1/R1 + 1/R2 +... +1/Rn • Case of two parallel resistance's: • Req = R1R2 /(R1 + R2) Lecture 3 6. Voltage Division Current Division + _ + _ V2 V1 I1 Week 2 Summary cont. I R1 + Is R1 R2 Vs - I2 R2 Lecture 3 7. Survey (cont.) • Current Sources • DC power supply, transistors • Conductance - Parallel Resistance's • Voltage and Current Division • Why - Delta - tutorial 1 Question 19 part 4 • Floyd pg. 309 Example 8-19 pg. 312 • Superposition Lecture 3 8. Wye Delta Transformations • Need to find equivalent resistance to determine current. HOW? (They are not in series, not in parallel) • Use Y to  transformation Lecture 3 9. Survey • Current Sources • DC power supply, transistors • Conductance - Parallel Resistance's • Voltage and Current Division • Why - Delta - tutorial 1 Question 19 part 4 • Floyd pg. 309 Example 8-19 pg. 312 • Superposition Lecture 3 10. Week 2 Summary (cont.) • Superposition: If a linear circuit is excited by more that one independent source, then the total response is simply the sum of the responses of the individual sources. • Voltage sources - short circuit • Current source - open circuit Lecture 3 11. Power Calculations • Power is not linear! • Superposition will not work directly! • With 2 A source opened P’1 = 25 W • With 10 V Source shorted P’’1 = 1 W • Total P = P’ + P’’ = 26 W (incorrect) • Must calculate current by superposition and then work out power • I’ = 5 A & I’’ = -1 A • Total I = I’ + I’’ = 4 A • Power P = 42 R = 16 W Lecture 3 12. I VBC C Example Week 3 • Find I ? • Determine VBC ? • What power is delivered by 4V source ? Lecture 3 13. Week 3 • How does the current in the load change if RL is (say) doubled? Lecture 3 14. Thevenin’s Theorem • Any linear network with a pair of terminals can be replaced by a circuit comprised of a voltage source in series with a resistor. • The observed voltages and currents in the load will be the same using the “Thevenin equivalent” circuit as would be seen using the original circuit. Lecture 3 15. VThThevenin Voltage ‘open circuit’ voltage VTh is the voltage which would appear across the terminals of the original and equivalent circuit if those terminals are open circuited. RThThevenin Resistance Independent sources inactivated RTh is the total resistance seen when looking into the original circuit with sources inactivated Can also be obtained by observing the short circuit current. RTh = VTh / Isc. Thevenin’s Components Lecture 3 16. Steps to finding the Thevenin Equivalent • Step 1 Determine the two points from which the Thevenin is to be found. NB:Polarity • Step 2 Find open circuit voltage acrossthese two points by removing the Load (resistance) VTh = Vo/c • Step 3 Find RTh by looking from the two points into the circuit after replacing all independent sources • Step 4 Draw the Thevenin Equivalent • Voltage source in series with a resistor Lecture 3 17. I VBC C Example Week 3 • Find I ? • Determine VBC ? • What power is delivered by 4V source ? • What is the Thevenin Equivalent circuit between A & B ? Lecture 3 18. Norton’s Theorem • Any linear network with a pair of terminals can be replaced by a circuit comprised of a current source in parallel with a resistor. • The observed voltages and currents in the load will be the same using the “Norton equivalent” circuit as would be seen using the original circuit. Lecture 3 19. IN Norton Current ‘short circuit’ current IN is the current which would appear through the terminals of the original and equivalent circuit if those terminals are short circuited. RNNorton Resistance independent sources inactivated RN is the total resistance seen when looking into the original circuit with sources inactivated Can also be obtained by observing the open circuit voltage. RN = Voc / IN . Norton’s Components Lecture 3 20. Steps to finding the Norton Equivalent • Step 1 Determine the two points from which the Norton is to be found. NB:Polarity • Step 2 Find the short circuit current throughthese two points by putting a short across them IN = Is/c • Step 3 Find RN by looking from the two points into the circuit after replacing all independent sources • Step 4 Draw the Norton Equivalent • Current source in parallel with a resistor Lecture 3 21. I VBC C Example Week 3 • Find I ? • Determine VBC ? • What power is delivered by 4V source ? • What is the Thevenin Equivalent circuit between A & B ? • What is the Norton Equivalent circuit between A & B ? Lecture 3 22. Relationship between Thevenin & Norton • A particular circuit can be represented by Thevenin or Norton equivalent. Therefore Thevenin and Norton equivalent circuits must be the same. • Hence Req = Rth = RN • RTh = VTh / Isc = VTh / IN VTh = RN IN • RN = Voc / IN = VTh / IN IN = VTh / RTh Lecture 3

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## Advertisement SSC BOARD PAPERS IMPORTANT TOPICS COVERED FOR BOARD EXAM 2024 ### a = 6, d = 6 (vi) a = 6, d = 6 (1 mark) Sol. a = 6, d = 6 Here, t1 = a = 6 t2 = t1 + d = 6 + 6 = 12 t3 = t2 + d = 12 + 6 = 18 t4 = t3 + d = 18 + 6 = 24 t5 = t4 + d = 24 + 6 = 30 The first five terms of A.P. are 6, 12, 18, 24 and 30.

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#### Similar Solved Questions ##### Using the symmetry of the arrangement; determine the direction of the force on +q in the figure below; given that: Case Ga 9b +6.5uC and qc qd +6.5 /C.qqqdInyour notebook draw the forces on q due to Qa' Qb' Qc' and qd' (b) Due to symmetry the direction of the net force isHint: For each force draw the X andy components Some will add and some will cancel: Calculate the magnitude of the force on the charge 4, given that the square is 10.0 cm on side and q 1.8 pC.nerCase II. Ga 9 Using the symmetry of the arrangement; determine the direction of the force on +q in the figure below; given that: Case Ga 9b +6.5uC and qc qd +6.5 /C. q q qd Inyour notebook draw the forces on q due to Qa' Qb' Qc' and qd' (b) Due to symmetry the direction of the net force is Hin... ##### Nctions)Question 5 of 9Vlew Palicies;Current Attempt in ProgressUsethe table below to find (he following values:(a) f (2() =2 (f (1)) =(c} f ((4)) =8 U (4) =f (2(6)) =8 ( (6)) =eTextbook and MediaSarctor LileAttempts: nctions) Question 5 of 9 Vlew Palicies; Current Attempt in Progress Usethe table below to find (he following values: (a) f (2() = 2 (f (1)) = (c} f ((4)) = 8 U (4) = f (2(6)) = 8 ( (6)) = eTextbook and Media Sarctor Lile Attempts:... ##### Only about 18% of all people can wiggle their ears. Is this percent higher for millionaires?... Only about 18% of all people can wiggle their ears. Is this percent higher for millionaires? Of the 345 millionaires surveyed, 79 could wiggle their ears. What can be concluded at the a = 0.01 level of significance? a. For this study, we should use z-test for a population proportion b. The null and ... ##### Application Exercise: ur Tar Idaha education tirmt impeinent progrn J42 fcent Panian Kiencl eamolc 17oiugy. b0J Rucert erainetu Idaho was nnd Qeth (STEYI Jonj math Sa fcllcaing Year the averag? Irath prentam Knat rerage 459 Kth Wnrino icolcul 9216.00 concide with 7 3tate Icu-lator Enousareen KMCUEL DWnatObpiconalEaKac)Ropulajiane esclcc Sunoie:Complio Gbomonato Hatitols mnuke decian Jbout Hu {Hic: Mate write Coxnthe nul &nd altematiye hpc-heses mntical valle utistc Oeciaon Selcnaproanal comp Application Exercise: ur Tar Idaha education tirmt impeinent progrn J42 fcent Panian Kiencl eamolc 17oiugy. b0J Rucert erainetu Idaho was nnd Qeth (STEYI Jonj math Sa fcllcaing Year the averag? Irath prentam Knat rerage 459 Kth Wnrino icolcul 9216.00 concide with 7 3tate Icu-lator Enousareen KMCUEL ... ##### Fly Wheel: The fly wheel has a mass M , a radius R , and a... Fly Wheel: The fly wheel has a mass M , a radius R , and a moment of inertia I=1/2MR^2          . The fly wheel has a gear of radius Rg < R that is used to turn the wheel. Drive Chain: The drive chain connects the gear on the fly wheel to a motor. The gear, the... ##### VAT (Philippines) - Presentations similar to real business transactions will be appreciated. 1. Make a diagram... VAT (Philippines) - Presentations similar to real business transactions will be appreciated. 1. Make a diagram or flowchart (fictitious case) 2. Start from a Value Added Tax-Exempt person transacting to a Value Added Tax-registered person. From the Value Added Tax registered, two transactions happen... ##### Find the angle € between the vectors _fx) = 4x, g(x) = x , {6, 9} Hrxygt)rdians Find the angle € between the vectors _ fx) = 4x, g(x) = x , {6, 9} Hrxygt) rdians... ##### In general, the Wronskian of $f_{1}, \ldots, f_{n}$ in $\mathscr{C}^{(n-1)}$ is the determinant $W(x)=\left|\begin{array}{cccc} f_{1}(x) & f_{2}(x) & \cdots & f_{n}(x) \\ f_{1}^{\prime}(x) & f_{2}^{\prime}(x) & \cdots & f_{n}^{\prime}(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_{1}^{(n-1)}(x) & f_{2}^{(n-1)}(x) & \cdots & f_{n}^{(n-1)}(x) \end{array}\right|$ and $f_{1}, \ldots, f_{n}$ are linearly independent, provided $W(x)$ is not identicall In general, the Wronskian of $f_{1}, \ldots, f_{n}$ in $\mathscr{C}^{(n-1)}$ is the determinant \[ W(x)=\left|\begin{array}{cccc} f_{1}(x) & f_{2}(x) & \cdots & f_{n}(x) \\ f_{1}^{\prime}(x) & f_{2}^{\prime}(x) & \cdots & f_{n}^{\prime}(x) \\ \vdots & \vdots & \ddots ... ##### (220 pts)MAHUscript is seut t0 typing firm consisting of typists AJdl C. If it is typed by thcn te Mber 0l eTTOTS Mace is Poisson rAndom variable with MCAH 2.6; if typed by B thcn te mlber of errors is Foisson random variable with mean 3: and if typed by then it is Poisson ranclom variable wich MCAH 3. Let : denote the tber of errors in the typed HAnuscript_ Assue that each typist is equally likely to do the work: a) Fiud E(X) b) Find Vorlx- c) What is the probability that the number of errors i (220 pts) MAHUscript is seut t0 typing firm consisting of typists AJdl C. If it is typed by thcn te Mber 0l eTTOTS Mace is Poisson rAndom variable with MCAH 2.6; if typed by B thcn te mlber of errors is Foisson random variable with mean 3: and if typed by then it is Poisson ranclom variable wich MCA... ##### Listed below arc the paired intervals (in minutes) before and after eruptions of the Old Faithful geyser: Using Pairedt-test with a 5% significance level, test if the time before is shorter than the time after (6 marks)Time_interval before eruption 98 Time interval after_eruption 92 Difference95 9287 10096 9095Test Statistic:Statement:Ho HConclusion:Interpretation:0 =Detailed Graph: Listed below arc the paired intervals (in minutes) before and after eruptions of the Old Faithful geyser: Using Pairedt-test with a 5% significance level, test if the time before is shorter than the time after (6 marks) Time_interval before eruption 98 Time interval after_eruption 92 Difference 95 9... ##### How do you solve 4x - 12.6= 15.4? How do you solve 4x - 12.6= 15.4?... ##### Find the derivative of the function.Y tan2(60)y'(0) Find the derivative of the function. Y tan2(60) y'(0)... ##### Graduate srudent then added, "Biomarkers, think about; think thcre mechanisms of action really needs and negative feedback be greater loops.= hhones there sure is 1 becoming obsolete every orher year; we regulation on ~waste recycling: With computers really need cell how many flame retardant chemicals figure already sale ways t0 dispose . ofthese decade from now? being products. Look found in dolphins alonel Whar will wildlife look like 1What Do / Want to Know? beginning biology student; yo graduate srudent then added, "Biomarkers, think about; think thcre mechanisms of action really needs and negative feedback be greater loops.= hhones there sure is 1 becoming obsolete every orher year; we regulation on ~waste recycling: With computers really need cell how many flame retardant ch... ##### Use the properties of limits help decide whether the limit exists. If the limit exists , find its value:lim X+6Select the correcl choice balow and, If necossary; Ill Ihe angwer box wilhin your cholce;0 A -36 Iim x+6 (Simplify your answer:)0 B. The Iimit does not exist and noilher 0 nor Use the properties of limits help decide whether the limit exists. If the limit exists , find its value: lim X+6 Select the correcl choice balow and, If necossary; Ill Ihe angwer box wilhin your cholce; 0 A -36 Iim x+6 (Simplify your answer:) 0 B. The Iimit does not exist and noilher 0 nor... ##### Ihc value pf {Lozloxpie+4n] kNont Utabov 3-43+4 Ihc value pf {Lozloxpie+4n] k Nont Utabov 3-4 3+4... ##### Put the following steps to creating a survey in order. - Step 1 - Step 2 A. Finalize the survey format B. Group questio... Put the following steps to creating a survey in order. - Step 1 - Step 2 A. Finalize the survey format B. Group questions around similar concepts C. Write questions in a simple and concise manner D. Proofread the survey several times • - - Step 3 Step 4 Step 5 E. Provide clear instructions for ... ##### Equations: Constan A Fiz = k 491 Coulomb's Law 7 (C1) B F =k%+k9+k9 (C2) 3 + (C3) cF =kq 4+4141 (C4) 0 Fi =QE (C5) EBlp) = & =k.(371+3n2+ 973 + (C6) El) = k. % ; V(p) = 9 k6+9+ Electric potential at point p due t0 multiple charges H V(m) Fk In your own words, describe the physics summarized by (Note: Brielly discuss: name, significance, quantities wth units, and relationship) Equalion A Equation F Equations: Constan A Fiz = k 491 Coulomb's Law 7 (C1) B F =k%+k9+k9 (C2) 3 + (C3) cF =kq 4+4141 (C4) 0 Fi =QE (C5) EBlp) = & =k.(371+3n2+ 973 + (C6) El) = k. % ; V(p) = 9 k6+9+ Electric potential at point p due t0 multiple charges H V(m) Fk In your own words, describe the physics summarized... ##### Two identical loudspeakers separated by distance d emit 200 Hz sound waves along the x-axis. As y... Two identical loudspeakers separated by distance d emit 200 Hz sound waves along the x-axis. As you walk along the axis, away from the speakers, you don't hear anything even though both speakers are on. What are the three lowest possible values for d? Assume a sound speed of 340 m/s . The answer... ##### Solve the initial value problem for r as a vector function of tDifferential Equation:dr dt =3t+1)1/21+3e -tj+ t+7k r(o) = kInitial condition:r() = (Eli+ (Di+ (Ok Solve the initial value problem for r as a vector function of t Differential Equation: dr dt =3t+1)1/21+3e -tj+ t+7k r(o) = k Initial condition: r() = (Eli+ (Di+ (Ok... ##### Security analysis has no place in modern portfolio theory. Select one: True False Security analysis has no place in modern portfolio theory. Select one: True False... ##### 5. You want to estimate the effect of hours spent in an SAT preparation course on... 5. You want to estimate the effect of hours spent in an SAT preparation course on total SAT score. You draw a random sample of high school seniors in at the end of the school year (after they have taken the SAT) from Boston area schools and ask them how many hours they spent in an SAT prep course an... ##### You have two waves leaving 2 different sources in phanse. there is disruptive interference at point... you have two waves leaving 2 different sources in phanse. there is disruptive interference at point p, which is 1m away from one source and .5m away from the other. What are possible wavelengths for these waves? (please show work)... ##### Homework 4 A motor is constructed using windings that surround laminated iron poles. You have been... Homework #4 A motor is constructed using windings that surround laminated iron poles. You have been asked to estimate the maximum temperature that will occur within the windings. The windings and poles are both approximated as being cylindrical, as shown below. 2 cm F-1em L-2 cm T-25°C h-25 W/m-... ##### Keae caateuatidu oaldbin cunacTeheltr coneb Genaanaeelarcubateaeledakedmadeacunal ypu pul engoueChanakhdbtna Era dermnalnlate: Keae caateuatidu oaldbin cunacTeheltr coneb Genaanaeelar cubateaeledakedmadea cunal ypu pul engoue Chanakhdbtna Era dermnalnlate:...

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# SSC CGL EXAMS 2018 | Quantitative Aptitude Practice Questions (Day-13) Dear Aspirants, Here we have given the Important SSC Exam 2018 Practice Test Papers. Candidates those who are preparing for SSC 2018 can practice these questions to get more confidence to Crack SSC 2018 Examination. Are You preparing for Bank exams 2019? Start your preparation with Free Mock test Series. SSC CGL EXAMS 2018 | Quantitative Aptitude Questions (Day-13) maximum of 10 points 1) sin ฮธ โ€“ cos ฮธ = โˆš2cos ฮธ, then find the value of sin ฮธ + cos ฮธ? a) โˆš2 sin ฮธ b) 2 sin ฮธ cos ฮธ c) โˆš2 sin2 ฮธ d) โˆš2 sin ฮธ cos ฮธ 2) What percentage of numbers is there from 1 to 75 has 1 or 6 in the unitโ€™s digits? a) 25% b) 20% c) 50% d) 33.33% 3) A Manโ€™s age is thrice as much of his son and twice as much of his wife. Average age of this family is 22. What is the sum of ages of his wife and son? a) 36 years b) 66 years c) 30 years d) 25 years 4) Find the number of diagonals of 8 side closed figure? a) 22 b) 16 c) 20 d) 25 5) In a triangle PQR, O is an in center of โˆ†PQR. If โˆ P = 600. Find the โˆ QOR. a) 1100 b) 1000 c) 1200 d) 900 6) P is completed a job in 8 hrs in a day. The same job is completed by R in 6 hrs a day. They are working together, how long the work will be completed? a) 3 (1/7) hours b) 3 (4/7) hours c) 3 (3/7) hours d) 3 (2/5) hours 7) The lengths of two parallel sides of a trapezium are 28 cm and 12 cm. If its height is 15 cm, then what is the area of trapezium? a) 540 cm2 b) 300 cm2 c) 250 cm2 d) 350 cm2 8) Find m if the ordinate and abscissa of the point (6m, m2 + 5m โ€“ 12) are equal a) โ€“ 2 or 3 b) 2 or 3 c) โ€“ 3 or 4 d) 3 or โ€“ 4 9) The curved surface area of a cylindrical pillar is 660 m2 and its volume 2310 m3. Find the height of the cylinder a) 11 m b) 15 m c) 7 m d) 8 m 10) When 15% of fruits were rotten, a seller remains some fruits unsold after 60 fruits were sold. On another day if only 10% of fruits were rotten, he can sale 65 fruits with same number of unsold fruits on previous day. How many fruits actually he has daily if he has same number of fruits on each day? a) 100 b) 150 c) 300 d) 250 Sin ฮธ โ€“ cos ฮธ = โˆš2cos ฮธ Squaring both side, (sin ฮธ โ€“ cos ฮธ)2 = 2cos2 ฮธ sin2 ฮธ +cos2 ฮธ โ€“ 2 sin ฮธ cos ฮธ = 2 cos2ฮธ sin2 ฮธ = cos2 ฮธ + 2 sin ฮธ.cos ฮธ Add Sin2 ฮธ on both sides 2 sin2 ฮธ = sin2 ฮธ +cos2 + 2sin ฮธ cos ฮธ 2 sin2 ฮธ = (sin ฮธ + cos ฮธ) 2 sin ฮธ + cos ฮธ = โˆš2 sin ฮธ The Number which have 1 or 6 in unitโ€™s digits are 1, 11, 21, 31, 41, 51, 61, 71, 6, 16, 26, 36, 46, 56, 66. Total number with unitโ€™s digit as 1 or 6 = 15 Total number = 75 Required percentage = (15*100)/ 75 = 20% Manย  ย  ย  ย : ย ย ย ย ย ย  His son 3ย  ย  ย  ย  ย  :ย ย ย ย ย  ย  ย ย ย ย 1 Manย  ย  ย  ย :ย ย ย ย ย ย ย  His wife 2ย  ย  ย  ย  ย  ย :ย  ย  ย  ย  ย  ย 1 Manย  ย  ย  ย : ย ย ย ย ย ย  Wifeย  ย  ย  : ย ย ย ย ย  Son 6ย  ย  ย  ย  ย  ย :ย  ย  ย  ย  ย 3ย  ย  ย  ย  :ย  ย  ย  ย  2 Average age of this family = 22 Total age of this family = 22*3 = 66 6x + 3x+ 2x = 66 => 11x = 66 X = 6 Sum of ages of wife and son = 3x + 2x = 5x = 5 * 6 = 30 years No. of diagonals of polygon = n (n โ€“ 3)/2 = 8 (8 โ€“ 3) / 2 = (8 * 5) /2 = 20 The angle made by any side with the in centre = 90ยฐ + half the opposite angle. = 90ยฐ + โˆ P/2 = 900 + ยฝ *600 = 90ยฐ + 30ยฐ = 1200 P = 1/8 hours R = 1/6 Hours P + R = 1/8 + 1/6 = 7/24 = 3 (3/7) hours Area of Trapezium = Half of sum of two parallel Sides * Heights = (ยฝ)* [(s1+s2) * H] = (28 + 12) /2 * 15 = 300 cm2 The ordinate of the point is m2 + 5m โ€“ 12 Abscissa = 6m 6m = m2 + 5m โ€“ 12 m2 + 5m โ€“ 6m โ€“ 12 = 0 m2 – m โ€“ 12 = 0 m2 โ€“ 4m + 3m โ€“ 12 = 0 m (m โ€“ 4) + 3 (m โ€“ 4) = 0 m = – 3 m = 4 Curved surface area of the cylinder= 2ฯ€rh = 660 m2 Volume of the cylinder = ฯ€r2h = 2310 m3 (ฯ€r2 h)/(2ฯ€rh )= (2310 )/(660 ) R = 7 Curved surface area of the cylinder= 2ฯ€rh = 660 m2 = 2* 22/7 * 7 * h = 660 m2 = > h = 15 m Total Fruits = x According to the question, unsold fruits were remain same (x – 15x/100 – 60) = (x โ€“ 10x/100 – 65) 15x/100 + 60 = 10x/100 + 65 15% of x โ€“ 10 % of x = 65 – 60 5x/100 = 5 5x = 500 X = 100 Total number of fruits = 100

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# I'm having a hard time conceptualizing this statement from "The Jet Engine" by Rolls-Royce This is taken from section 21 of Chapter 3, Compressors that discusses axial flow compressors in turbine jet engines. The more the pressure ratio of a compressor is increased the more difficult it becomes to ensure that it will operate efficiently over the full speed range. This is because the requirement for the ratio of inlet area to exit area, at the high speed case, results in an inlet area that becomes progressively too large relative to the exit area as the compressor speed and hence pressure ratio is reduced. The axial velocity of the inlet air in the front stages thus becomes low relative to the blade speed, this changes the incidence of the air onto the blades and a condition is reached where the flow separates and the compressor flow breaks down I'm trying to draw a connection to something I already understand, is this similar to the phenomenon that limits propellers to low speed? In order to produce positive thrust they need to spin faster and faster? • – mins Commented May 17 at 8:51 Variable pitch propellers can help explain this. When the aircraft speed is slow, the blade pitch is finer. At faster air speeds the blades must be opened more in the direction of the airstream. This is due to the fact that the velocity of the airstream contributes to the relative wind on the propeller. The rotational velocity of the prop is the other contributor. If the propeller is opened but the airstream velocity drops, then the angle of attack of the relative wind on the prop will increase. If the AoA gets too high, flow will begin to separate from the prop airfoil. The compressor blades are all little airfoils. They will behave exactly the same way if incoming airflow slows down too much. Picture 3.9, which is related to the text you copied in your question, is helpful in understanding the concept: First of all note that with "inlet" they mean the beginning of the axial compressor and not the inlet of the whole engine. Now, going from the inlet toward the outlet, the section of the compressor becomes smaller and smaller: that's simply because air gets compressed and occupies a smaller and smaller volume. Obviously the outlet area cannot become indefinitely small: there are mechanical and aerodynamic limitations that cannot be overcome. That implies that if you want/need to add compression stages then you have to add them to the left, by the inlet section. But each section added to the left has a bigger and bigger area i.e. blade span. A bigger blade span is achieved both increasing its outer radius (Ro) and decreasing its inner radius (Ri). But higher radius means bigger rotating velocity while smaller radius means smaller rotating velocity, which are both bad since: 1. Higher rotating velocity at the tip means getting closer and closer to transonic or even supersonic speeds: drag and noise increase and lift decreases. 2. Lower rotating velocity at the base of the blade means lower Reynolds numbers and higher AoA to get enough lift despite the lower velocity: drag increases as well as the chance to get the blade stalled.

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Q. 54.4( 8 Votes ) # Find the equation of the circle passing through the points (2,3) and (–1,1) and whose centre is on the line x – 3y – 11 = 0. Let the equation of the required circle be (x – h)2+ (y – k)2 =r2 Since, the circle passes through points (2,3) and (-1,1). (2 – h)2+ (3 – k)2 =r2 .................(1) (-1 – h)2+ (1– k)2 =r2 ..................(2) Since, the centre (h,k) of the circle lies on line x - 3y - 11= 0, h - 3k =11..................... (3) From the equation (1) and (2), we obtain (2 – h)2+ (3 – k)2 =(-1 – h)2 + (1 – k)2 ⇒ 4 – 4h + h2 +9 -6k +k2 = 1 + 2h +h2+1 – 2k + k2 ⇒ 4 – 4h +9 -6k = 1 + 2h + 1 -2k ⇒ 6h + 4k =11................ (4) Now multiply (3) by 6 and subtract it from (4) to get, 6h+ 4k - 6(h-3k) = 11 – 66 ⇒ 6h + 4k – 6h + 18k = 11 – 66 ⇒ 22 k = - 55 ⇒ K = -5/2 Put this value in (4) to get, 6h + 4(-5/2) = 11 ⇒ 6h – 10 = 11 ⇒ 6h = 21 ⇒ h = 21/6 ⇒ h = 7/2 Thus we obtain h= and k= . On substituting the values of h and k in equation (1), we get += r2 += r2 + = r2 + = r2 Thus, the equation of the required circle is + += 4x2 -28x + 49 +4y2 + 20y + 25 =130 4x2 +4y2 -28x + 20y - 56 =0 4(x2 +y2 -7x + 5y – 14) = 0 x2+y2-7x + 5y– 14 =0 Rate this question : How useful is this solution? We strive to provide quality solutions. Please rate us to serve you better. Related Videos Quiz on properties of focal chord of parabola36 mins Focal chord of parabola29 mins Equation of tangent to parabola | Conic Section38 mins Equation of tangent to parabola | Conic Section | Quiz1 mins Interactive Quiz on Equation of Parabola41 mins Lecture on Equation of Parabola59 mins Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts Dedicated counsellor for each student 24X7 Doubt Resolution Daily Report Card Detailed Performance Evaluation view all courses

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# Python Program to Perform Arithmetic Operations on Lists Write a Python Program to Perform Arithmetic Operations on Lists using For Loop and While Loop with a practical example. ## Python Program to Perform Arithmetic Operations on Lists Example In this python program, we are using For Loop to iterate each element in a given List. Inside the Python loop, we are performing arithmetic operations on elements of the first and second lists. ```# Python Program to Perform List Arithmetic Operations NumList1 = [10, 20, 30] NumList2 = [5, 2, 3] sub = [] multi = [] div = [] mod = [] expo = [] for j in range(3): sub.append( NumList1[j] - NumList2[j]) multi.append( NumList1[j] * NumList2[j]) div.append( NumList1[j] / NumList2[j]) mod.append( NumList1[j] % NumList2[j]) expo.append( NumList1[j] ** NumList2[j]) print("The List Items after Subtraction = ", sub) print("The List Items after Multiplication = ", multi) print("The List Items after Division = ", div) print("The List Items after Modulus = ", mod) print("The List Items after Exponent = ", expo)``` Within this Python Program to Perform Arithmetic Operations on Lists example, NumList1 = [10, 20, 30], NumList2 = [5, 2, 3]. For this, we are using the Python arithmetic Operators For Loop – First Iteration: for 0 in range(3) – Condition is True sub.append( 10 – 5) => sub[0] = 5 multi.append(10 * 5) => multi[0] = 50 div.append( 10 / 5) => div[0] = 2 mod.append( 10 % 5) => sub[0] = 0 expo.append(10 ** 5) => expo[0] = 10000 Second Iteration: for 1 in range(3) – Condition is True sub.append( 20 – 2) => sub[1] = 18 multi.append(20 * 2) => multi[1] = 40 div.append( 20 / 2) => div[1] = 10 mod.append( 20 % 2) => sub[1] = 0 expo.append(20 ** 2) => expo[1] = 400 Third Iteration: for 2 in range(3) – Condition is True Do the same for this iteration too Fourth Iteration: for 3 in range(3) – Condition is False So, it exits from Python For Loop ## Perform Arithmetic Operations on Lists using For Loop In this python program, we are using For Loop to allow users to enter their own number of elements for each List. ```# Python Program for Performing Arithmetic Operations on List NumList1 = [] NumList2 = [] sub = [] multi = [] div = [] mod = [] expo = [] Number = int(input("Please enter the Total Number of List Elements: ")) print("Please enter the Items of a First and Second List ") for i in range(1, Number + 1): List1value = int(input("Please enter the %d Element of List1 : " %i)) NumList1.append(List1value) List2value = int(input("Please enter the %d Element of List2 : " %i)) NumList2.append(List2value) for j in range(Number): sub.append( NumList1[j] - NumList2[j]) multi.append( NumList1[j] * NumList2[j]) div.append( NumList1[j] / NumList2[j]) mod.append( NumList1[j] % NumList2[j]) expo.append( NumList1[j] ** NumList2[j]) print("The List Items after Subtraction = ", sub) print("The List Items after Multiplication = ", multi) print("The List Items after Division = ", div) print("The List Items after Modulus = ", mod) print("The List Items after Exponent = ", expo)``` Python List Arithmetic Operations using for loop output ``````Please enter the Total Number of List Elements: 3 Please enter the Items of a First and Second List Please enter the 1 Element of List1 : 10 Please enter the 1 Element of List2 : 2 Please enter the 2 Element of List1 : 20 Please enter the 2 Element of List2 : 3 Please enter the 3 Element of List1 : 30 Please enter the 3 Element of List2 : 4 The List Items after Addition = [12, 23, 34] The List Items after Subtraction = [8, 17, 26] The List Items after Multiplication = [20, 60, 120] The List Items after Division = [5.0, 6.666666666666667, 7.5] The List Items after Modulus = [0, 2, 2] The List Items after Exponent = [100, 8000, 810000]`````` ## Perform Arithmetic Operations on Lists using While Loop This Python program for arithmetic operations on the list is the same as above. We just replaced the For Loop with While loop. ```# Python Program to Perform Arithmetic Operations on Lists NumList1 = []; NumList2 = [] add = [] ; sub = [] ; multi = [] div = []; mod = [] ; expo = [] i = 0 j = 0 Number = int(input("Please enter the Total Number of List Elements: ")) print("Please enter the Items of a First and Second List ") while(i < Number): List1value = int(input("Please enter the %d Element of List1 : " %i)) NumList1.append(List1value) List2value = int(input("Please enter the %d Element of List2 : " %i)) NumList2.append(List2value) i = i + 1 while(j < Number): sub.append( NumList1[j] - NumList2[j]) multi.append( NumList1[j] * NumList2[j]) div.append( NumList1[j] / NumList2[j]) mod.append( NumList1[j] % NumList2[j]) expo.append( NumList1[j] ** NumList2[j]) j = j + 1 print("The List Items after Subtraction = ", sub) print("The List Items after Multiplication = ", multi) print("The List Items after Division = ", div) print("The List Items after Modulus = ", mod) print("The List Items after Exponent = ", expo)``` Python List Arithmetic Operations using a while loop output ``````Please enter the Total Number of List Elements: 2 Please enter the Items of a First and Second List Please enter the 0 Element of List1 : 22 Please enter the 0 Element of List2 : 3 Please enter the 1 Element of List1 : 44 Please enter the 1 Element of List2 : 2 The List Items after Addition = [25, 46] The List Items after Subtraction = [19, 42] The List Items after Multiplication = [66, 88] The List Items after Division = [7.333333333333333, 22.0] The List Items after Modulus = [1, 0] The List Items after Exponent = [10648, 1936]``````

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# Second of arc to Milliturn Conversions From Second of arc • Angular Mil • Binary degree • Centesimal minute of arc • Centesimal second of arc • Centiturn • Degree • Diameter Part • Hour Angle • Milliturn • Minute of arc • Minute of time • Octant • Point • Quarter Point • Second of arc • Second of time • Sextant • Sign • Turn To Milliturn • Angular Mil • Binary degree • Centesimal minute of arc • Centesimal second of arc • Centiturn • Degree • Diameter Part • Hour Angle • Milliturn • Minute of arc • Minute of time • Octant • Point • Quarter Point • Second of arc • Second of time • Sextant • Sign • Turn Formula 2,731 arcsecond = 2731 / 1296 mltr = 2.1073 mltr ## How To Convert From Second of arc to Milliturn 1 Second of arc is equivalent to 0.00077 Milliturn: 1 arcsecond = 0.00077 mltr For example, if the Second of arc number is (300000), then its equivalent Milliturn number would be (231.48). Formula: 300,000 arcsecond = 300000 / 1296 mltr = 231.48 mltr ## Second of arc to Milliturn conversion table Second of arc (arcsecond) Milliturn (mltr) 1,000 arcsecond 0.7716 mltr 2,000 arcsecond 1.5432 mltr 3,000 arcsecond 2.3148 mltr 4,000 arcsecond 3.0864 mltr 5,000 arcsecond 3.858 mltr 6,000 arcsecond 4.6296 mltr 7,000 arcsecond 5.4012 mltr 8,000 arcsecond 6.1728 mltr 9,000 arcsecond 6.9444 mltr 10,000 arcsecond 7.716 mltr 11,000 arcsecond 8.4877 mltr 12,000 arcsecond 9.2593 mltr 13,000 arcsecond 10.031 mltr 14,000 arcsecond 10.802 mltr 15,000 arcsecond 11.574 mltr 16,000 arcsecond 12.346 mltr 17,000 arcsecond 13.117 mltr 18,000 arcsecond 13.889 mltr 19,000 arcsecond 14.66 mltr 20,000 arcsecond 15.432 mltr 21,000 arcsecond 16.204 mltr 22,000 arcsecond 16.975 mltr 23,000 arcsecond 17.747 mltr 24,000 arcsecond 18.519 mltr 25,000 arcsecond 19.29 mltr 26,000 arcsecond 20.062 mltr 27,000 arcsecond 20.833 mltr 28,000 arcsecond 21.605 mltr 29,000 arcsecond 22.377 mltr 30,000 arcsecond 23.148 mltr 31,000 arcsecond 23.92 mltr 32,000 arcsecond 24.691 mltr 33,000 arcsecond 25.463 mltr 34,000 arcsecond 26.235 mltr 35,000 arcsecond 27.006 mltr 36,000 arcsecond 27.778 mltr 37,000 arcsecond 28.549 mltr 38,000 arcsecond 29.321 mltr 39,000 arcsecond 30.093 mltr 40,000 arcsecond 30.864 mltr 41,000 arcsecond 31.636 mltr 42,000 arcsecond 32.407 mltr 43,000 arcsecond 33.179 mltr 44,000 arcsecond 33.951 mltr 45,000 arcsecond 34.722 mltr 46,000 arcsecond 35.494 mltr 47,000 arcsecond 36.265 mltr 48,000 arcsecond 37.037 mltr 49,000 arcsecond 37.809 mltr 50,000 arcsecond 38.58 mltr 51,000 arcsecond 39.352 mltr 52,000 arcsecond 40.123 mltr 53,000 arcsecond 40.895 mltr 54,000 arcsecond 41.667 mltr 55,000 arcsecond 42.438 mltr 56,000 arcsecond 43.21 mltr 57,000 arcsecond 43.981 mltr 58,000 arcsecond 44.753 mltr 59,000 arcsecond 45.525 mltr 60,000 arcsecond 46.296 mltr 61,000 arcsecond 47.068 mltr 62,000 arcsecond 47.84 mltr 63,000 arcsecond 48.611 mltr 64,000 arcsecond 49.383 mltr 65,000 arcsecond 50.154 mltr 66,000 arcsecond 50.926 mltr 67,000 arcsecond 51.698 mltr 68,000 arcsecond 52.469 mltr 69,000 arcsecond 53.241 mltr 70,000 arcsecond 54.012 mltr 71,000 arcsecond 54.784 mltr 72,000 arcsecond 55.556 mltr 73,000 arcsecond 56.327 mltr 74,000 arcsecond 57.099 mltr 75,000 arcsecond 57.87 mltr 76,000 arcsecond 58.642 mltr 77,000 arcsecond 59.414 mltr 78,000 arcsecond 60.185 mltr 79,000 arcsecond 60.957 mltr 80,000 arcsecond 61.728 mltr 81,000 arcsecond 62.5 mltr 82,000 arcsecond 63.272 mltr 83,000 arcsecond 64.043 mltr 84,000 arcsecond 64.815 mltr 85,000 arcsecond 65.586 mltr 86,000 arcsecond 66.358 mltr 87,000 arcsecond 67.13 mltr 88,000 arcsecond 67.901 mltr 89,000 arcsecond 68.673 mltr 90,000 arcsecond 69.444 mltr 91,000 arcsecond 70.216 mltr 92,000 arcsecond 70.988 mltr 93,000 arcsecond 71.759 mltr 94,000 arcsecond 72.531 mltr 95,000 arcsecond 73.302 mltr 96,000 arcsecond 74.074 mltr 97,000 arcsecond 74.846 mltr 98,000 arcsecond 75.617 mltr 99,000 arcsecond 76.389 mltr 100,000 arcsecond 77.16 mltr 200,000 arcsecond 154.32 mltr 300,000 arcsecond 231.48 mltr 400,000 arcsecond 308.64 mltr 500,000 arcsecond 385.8 mltr 600,000 arcsecond 462.96 mltr 700,000 arcsecond 540.12 mltr 800,000 arcsecond 617.28 mltr 900,000 arcsecond 694.44 mltr 1,000,000 arcsecond 771.6 mltr 1,100,000 arcsecond 848.77 mltr 1 arcsecond 0.00077 mltr

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Science Forums # TiffanyMariokart Members 5 • Rank Curious 1. ## Physics Puzzle! So South Korea have granted legal protection to 60,000 sadists after the Nth rooms. Not only did extremely few perpetrators receive lenient sentences, the rest were granted legal protection from the victims. Here's the puzzle; how long until we're all f***ed from corruption, idiocies and sadism? I mean at worst the whole 60k reproduces for a few years with probable victims and masses an army of sadists, 3 victims each 4 years 60k*3*3 plus the possible exponentials of them repeating this bs. That's the extreme end of it, but as you can see the possibility of this is rather unacce 2. ## Time Substance Concentration. Footnote, if time didn't have a physical consistency would it be correct to say that time doesn't exist in physical terms? This isn't some I'm going to buy I'm afraid, gather the smarties. 3. ## Time Substance Concentration. "Under Einstein's theory of general relativity, gravity can bend time" What's posted is a very basic logical explanation of what this is an attempt at saying, since gravity comes from 3d objects as a whole and not just one part of them it's considered a pooling. I don't think this man a fool, I also don't think nuclear weapons are the most destructive conceptions of theirs either. But what I do know for sure is, the Nth room aftermath is blatant evidence of government corruption of the highest order and in most messed up regard, it's up to the smartest to F*** them off as I'm afraid eve 4. ## Time Substance Concentration. The physical consistency of time. 5. ## Time Substance Concentration. The higher the density of the concentration of time substance the faster the perception of it, this is evident from time dilation experiments and astronauts aging slower in space from less time substance caused by less gravity. This is because due to gravity time substance is more concentrated the closer it's measured to the earth. On earth the population would pass through more time substance than the astronauts in space due to gravity pooling the time substance towards the earth. ×

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# pygame图像处理基础简单数学运算_PyGame image processing foundation ### pygame图像处理基础简单数学运算_PyGame image processing foundation """图像处理基础简单数学运算,像素也能加减乘除,所谓图像处理就是对像素操作,以下没有矩阵操作，适合于中小学生理解。""" __author__ = "李兴球" __date__ = "2019/5/5" import pygame import random """pixel：像素三元组，number：要增加的数值""" r,g,b = pixel b = min((b + number ) ,255) b = max(b,0) g = min((g + number ) ,255) g = max(g,0) r = min((r + number ) ,255) r = max(r,0) return r,g,b width,height = cyj.get_size() ### 像素的加差 ##for x in range(width): ## for y in range(height): ## r,g,b,a = cyj.get_at((x,y)) ## cyj.set_at((x,y),(b,g,r,255)) ## ##pygame.image.save(cyj,"cyj_2.png") # 像素值两级化，大于127的就让它的值变成255，否则为0 for x in range(width): for y in range(height): r,g,b,a = cyj.get_at((x,y)) r = (r > 127) * 255 g = (g > 127) * 255 b = (b > 127) * 255 cyj.set_at((x,y),(b,g,r,255)) pygame.image.save(cyj,"cyj_3.png")

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# Solving trigonometric equation. 2 views (last 30 days) Auchitya on 3 Jan 2023 Edited: KALYAN ACHARJYA on 3 Jan 2023 Can anyone help me with the code to solve the function for x: 3*sin(150+x)=sin(150-x) KALYAN ACHARJYA on 3 Jan 2023 Edited: KALYAN ACHARJYA on 3 Jan 2023 You are suggested to go through MATLAB Onramp- Seems like homework, still I have answered. It doesn't make sense without knowing how it works. syms x eq=3*sin(150+x)-sin(150-x)==0; x=vpa(solve(eq,x)) Must check the function of solve and vpa, what does symbolic variable represents? The Matlab Help function and the online Matlab documentation are very useful for brushing up on MATLAB. ### Categories Find more on Mathematical Functions in Help Center and File Exchange ### Community Treasure Hunt Find the treasures in MATLAB Central and discover how the community can help you! Start Hunting!

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Introduction to Matrix Algebra I Save this PDF as: Size: px Start display at page: Transcription 1 Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model the main topic of POL 603. Matrices are an intuitive way to think about data. We have a set of observations (perhaps individuals) on the row, and observe many different characteristics (such as race, gender, PID, etc.). Matrices are useful for solving systems of equations, including ones that we will see in class. A.1 Definition of Matrices and Vectors A matrix is simply an arrangement of numbers in rectangular form. Generally, a (j k) matrix A can be written as follows: a 11 a 12 a 1k a 21 a 22 a 2k A a j1 a j2 a jk Note that there are j rows and k columns. Also note that the elements are double subscripted, with the row number first, and the column number second. When reading the text, and doing your assignments, you should always keep the dimensionality of matrices in mind. The dimensionality of the matrix is also called the order of a matrix. In general terms, the A above is of order (j,k).letuslookatacoupleofexamples: 1 3 W 2 is of order (2, 2). This is also called a square matrix, because the row dimension equals the column dimension (j k). There are also rectangular matrices (j k), such as: Γ which is of order (4, 2). In the text, and on the board, we will denote matrices as capital, boldfaced Roman or Greek letters. Roman is typically data, and Greek is typically parameters. This is not a universal convention, so be aware. Of course, I cannot do boldface on the board. Thus, if you forget the dimensionality (or question whether I am talking about a matrix or a vector), stop me and ask. There exists a special kind of matrix called a vector. Vectors are matrices that have either one row or one column. Of course, a matrix with one row and one column is the same as a scalar a regular number. Row vectors are those that have a single row and multiple columns. For example, an order (1,k) row vector looks like this: α α 1 α 2 α 3 α k Similarly, column vectors are those that have a single column and multiple rows. For example, an order (k, 1) column vector looks like this: y Again, the convention for vectors is just like that for matrices, except the letters are lowercase. Thus, vectors are represented as lower-case, bold-faced Roman or Greek letters. Again, Roman letters typically represent data, and Greek letters represent parameters. Here, the elements are typically given a single subscript. y 1 y 2. y k A.2 Matrix Addition and Subtraction Now that we have defined matrices, vectors, and scalars, we can start to consider the operations we can perform on them. Given a matrix of numbers, one can extend regular scalar algebra in a straight forward way. Scalar addition is simply: m + n 2+57 Addition is similarly defined for matrices. If matrices or vectors are of the same order, then they can be added. One performs the addition element by element. 168 3 Thus, for a pair of order (2, 2) matrices, addition proceeds as follows for the problem A+B C: a11 a 12 b11 b + 12 a11 + b 11 a 12 + b 12 c11 c 12 a 21 a 22 b 21 b 22 a 21 + b 21 a 22 + b 22 c 21 c 22 Subtraction similarly follows. It is important to keep in mind that matrix addition and subtraction is only defined in the matrices are the same order, or, in other words, share the same dimensionality. If they do, they are said to be conformable for addition. If not, they are nonconformable. Here is another example: There are two important properties of matrix addition that are worth noting: A + B B + A. In other words, matrix addition is commutative. (A + B)+C A +(B + C). In other words, matrix addition is associative. Another operation that is often useful is transposition. In this operation, the order subscripts are exchanged for each element of the matrix A. Thus, an order (j,k) matrix becomes an order (k, j) matrix. Transposition is denoted by placing a prime after a matrix or by placing a superscript T.Hereisanexample: Q q 1,1 q 1,2 q 2,1 q 2,2 q 3,1 q 3,2 Q q1,1 q 2,1 q 3,1 q 1,2 q 2,2 q 3,2 Note that the subscripts in the transpose remain the same, they are just exchanged. Transposition makes more sense when using numbers. Here is an example for a row vector: ω ω Note that transposing a row vector turns it into a column vector, and vice versa. There are a couple of results regarding transposition that are important to remember: An order (j,j) matrix A is said to be symmetric A A. This implies, of course, that all symmetric matrices are square. Here is an example: W W 4 (A ) A. In words, the transpose of the transpose is the original matrix. For a scalar k, (ka) ka. For two matrices of the same order, it can be shown that the transpose of the sum is equal to the sum of the transposes. Symbolically: (A + B) A + B Transposition is also commutative. A.3 Matrices and Multiplication So far we have defined addition and subtraction, as well as transposition. Now we turn our attention to multiplication. The first type of multiplication is a scalar times a matrix. In words, a scalar α times a matrix A equals the scalar times each element of A. Thus, a1,1 a A α 1,2 αa1,1 αa 1,2 a 2,1 a 2,2 αa 2,1 αa 2,2 So, for: A A Now we will discuss the process of multiplying two matrices. We apply the following definition of matrix multiplication. Given A of order (m, n) andb of order (n, r), then the product AB C is the order (m, r) matrix whose entries are defined by: n c i,j a i,k b k,j k1 where i 1,...,mand j 1,...,r. Note that for matrices to be multiplication conformable, the number of columns in the first matrix n must equal the number of rows in the second matrix n. It is easier to see this by looking at a few examples. Let A B We can now define their product. Here we would say that B is pre-multiplied by A, orthat A is post-multiplied by B: AB ( 2) ( 3) ( 2) ( 3) 5 Note that A is of order (2, 3), and B is of order (3, 2). Thus, the product AB is of order (2, 2). We can similary compute the product BA which will be of order (3, 3). You can verify that this product is: BA This shows that the multiplication of matrices is not commutative. In other words: AB BA. Given this notation, there are a couple of identities worth noting: We have alread shown the matrix multiplication is not commutative: AB BA. Matrix multiplication is associative. In other words: (AB)C A(BC) Matrix multiplication is distributive. In other words: A(B + C) AB + AC Scalar multiplication commutative, associative, and distributive. The transpose of a product takes an intersting form, that can easily be proven: (AB) B A Just as with scalar algebra, we use the exponentiation operator to denote repeated multiplication. For a square matrix (Why? Because it is the only type of matrix conformable with itself.), we use the notation to denote exponentiation. A 4 A A A A A.4 Vectors and Multiplication We of course use the same formula for vector multiplication as we do for matrix multiplication. There are a couple of examples that are worth looking let. Let us define the column vector e. By definition, the order of e is (N, 1). We can take the inner product of e, which is simply: e 1 e e e 2 e 1 e 2 e N. 171 e N 6 This can be simplified: N e e e 1 e 1 + e 2 e e N e N e 2 i i1 Question The inner product of a column vector with itself is simply equal to the sum of the square values of the vector, which is used quite often in the regression model. Geometrically, the square root of the inner product is the length of the vector. One can similarly define the outer product for column vector e, denoted ee, which yields an order (N, N) matrix. There are couple of other vector products that are interesting to note. Let i denote an order (N, 1) vector of ones, and x denote an order (N, 1) vector of data. The following is an interesting quantity: 1 N i x 1 N (x 1 + x 2 + x N ) 1 xi x N From this, it follows that: i x x i Similarly, let y denote another (N, 1) vector of data. The following is also interesting: n x y x 1 y 1 + x 2 y x N y N x i y i Note that the following have nothing to do with mean deviation form (we actually will not be using mean deviation form much more in this course it is most useful in scalar from). The lower case letters represent elements of a vector. One can employ matrix multiplication using vectors. Let A be an order (2, 3) matrix that looks like this: a11 a A 12 a 13 a 21 a 22 a 23 Define the row vector a 1 a 11 a 12 a 13 and the row vector a 2 a 21 a 22 a 23. We can now express A as follows: a A 1 It is most common to represent all vectors as column vectors, so to write a row vector you use the transposition operator. This is a good habit to get into, and one I will try to use throughout the term. We are similarly given a matrix B, which is of order (3, 2). It can be written: b 11 b 12 B b 21 b 22 b 1 b 2 b 31 b 32 where b 1 and b 2 represent the columns of B. The product of the matrices, then, can be expressed in terms of four inner products. a AB C 1 b 1 a 1 b 2 a 2 b 1 a 2 b 2 This is the same as the summation definition of multiplication. This too will come in useful later in the semester. 172 a 2 i1 7 A.5 Special Matrices and Their Properties When performing scalar algebra, we know that x 1x, which is known as the identity relationship. There is a similar relationship in matrix algebra: AI A. WhatisI? Itcan be shown that I is a diagonal, square matrix with ones on the main diagonal, and zeros on the off diagonal. For example, the order three identity matrix is: I Notice that I is oftentimes subscripted to denote its dimensionality. Here is an example of the use of an identity matrix: One of the nice properties of the identity matrix is that it is commutative, associative, and distributive with respect to multiplication. That is, AIB IAB ABI AB We similary are presented with the identity in scalar algebra that x +0 x. This generalizes to matrix algebra, with the definition of the null matrix, which is simply a matrix of zeros, denoted 0 j,k.hereisanexample: A + 0 2, A There are two other matrices worth mentioning. The first is a diagonal matrix, which takes values on the main diagonal, and zeros on the off diagonal. Formally, matrix A is diagonal if a i,j 0 i j. The identity matrix is an example, so is the matrix Ω: Ω The two other types of special matrices that we will be used often are square and symmetric matrices, which I defined earlier. 173 8 Appendix B Introduction to Matrix Algebra II B.1 Computing Determinants So far we have defined addition, subtraction, and multiplication, along with a few related operators. We have not, however, yet defined division. Remember this simple algebraic problem: 2x x x 3 The quantity can be called the inverse of 2. This is exactly what we are doing when we divide in scalar algebra. Now, let us define a matrix which the inverse of A. Let us call that matrix A 1.Inscalar algebra, a number times its inverse equals one. In matrix algebra, then, we must find the matrix A 1 where AA 1 A 1 A I. Given this matrix, we can then do the following, which is what one needs to do when solving systems of equations: Ax b A 1 Ax A 1 b x A 1 b because A 1 A I. The question remains as to how to compute A 1. Today we will solve this problem, first by computing determinants of square matrices using the cofactor expansion. Then, we will use these determinants to form the matrix A 1. Determinants are defined only for square matrices, and are scalars. They are denoted det(a) A. Determinants take a very important role in determining whether a matrix is invertable, and what the inverse is. 174 9 For an order (2, 2) matrix, the determinant is defined as follows: a det(a) A 11 a 12 a 21 a 22 a 11a 22 a 12 a 21 Here are two examples: G Γ G Γ How do we go about computing determinants for large matrices? To do so, we need to define a few other quantities. For an order (n, n) square matrix A, we can define the cofactor θ r,s for each element of A: a r,s. The cofactor of a r,s is denoted: θ r,s ( 1) (r+s) A r,s where A r,s is the matrix formed after deleting row r and column s of the matrix (sometimes called the minor of A). Thus, each element of the matrix A has its own cofactor. We therefore can compute the matrix of cofactors for a matrix. Here is an example: B We can compute the matrix of cofactors: Θ We can use any row or column of the matrix of cofactors Θ to compute the determinant of a matrix A. For any row i, n A a ij θ ij Or, for any column j, A j1 n a ij θ ij i1 175 10 These are handy formulas, because they allow the determinant of an order n matrix to be decreased to order (n 1). Note that one can use any row or column to do the expansion, and compute the determinant. This process is called cofactor expansion. It can be repeated on very large matrices many times to get down to an order 2 matrix. Let us return to our example, we will do the cofactor expansion on the first row, and the second column. n B b 1j θ 1j j ( 12) 15 B n b i2 θ i2 i ( 7) You can do this for the other two rows, or the other two columns, and get the same result. Now, for any given matrix, you can perform the cofactor expansion and compute the determinant. Of course, as the order n increases, this gets terribly difficult, because to compute each element of the cofactor matrix, you have to do another expansion. Computers, however, do this quite easily. B.2 Matrix Inversion Now, let us define a matrix which is the inverse of A. Let us call that matrix A 1.Inscalar algebra, a number times its inverse equals one. In matrix algebra, then, we must find the matrix A 1 where AA 1 A 1 A I. Last time we defined two important quantities that one can use to compute inverses: the determinant and the matrix of cofactors. The determinant of a matrix A is denoted A, and the matrix of cofactors we denoted Θ A. There is one more quantity that we need to define, the adjoint. The adjoint of a matrix A is denoted adj(a). The adjoint is simply the matrix of cofactors transposed. Thus, adj(a) Θ θ r,s ( 1) (r+s) A r,s where A r,s is the matrix formed after deleting row r and column s of the matrix (sometimes called the minor of A). We now know all we need to know to compute inverses. It can be shown that the inverse of a matrix A is defined as follows: A 1 1 A adj(a) Thus, for any matrix A that is invertable, we can compute the inverse. This is trival for order (2, 2) matrices, and only takes a few minutes for order (3, 3) matrices. It gets much 176 11 more difficult after that, and we can use computers to compute inverses, both numerically (using Stata, Gauss, or some other package), or analytically (using Mathematica, Maple, etc.). B.3 Examples of Inversion We will do two examples. First, we will find the inverse of an order (2, 2) matrix: 2 4 B 6 3 We first must calculate the determinant of B: B We can write down the matrix of cofactors for B, which we then transpose to get the adjoint: adj(b) Θ B Given the determinant and the adjoint, we can now write down the inverse of B: B To make sure, let us check the product B 1 B to make sure it equals the identity matrix I 2 : 3 B 1 4 B I Here is another example. If you remember earlier, we were working on an order (3, 3) matrix also called B. The matrix was as follows: B We took the time to compute the matrix of cofactors for this matrix: Θ B We also showed that its determinant B 15. Given this information, we can write down the inverse of B: B Θ B 12 Again, let us check to make sure this is indeed the inverse of B. B 1 B 1 B adj(b)b I B.4 Diagonal Matrices There is one type of matrix for which the computation of the inverse is nearly trivial diagonal matrices. Let A denote a diagonal matrix of order (k, k). Remember that diagonal matrices have to be square: A a a a a kk It can be shown that the inverse of A is the following: a a A a a 1 kk To show this is the case, simply multiply AA 1 and you will get I k. B.5 Conditions for Singularity Earlier I talked about the simple alebraic problem of solving for x, and show that one need to premultiply by x 1 to solve a system of equations. This is the same operation as division. In scalar algebra, there is one number for which the inverse is not defined: 0. The quantity 1 0 is not defined. Similarly, in matrix algebra there are a set of matrices for which an inverse does not exist. Remember our formula for the inverse of our matrix A: A 1 1 A adj(a) 178 13 Question Under what conditions will this not be defined? When the determinant of A is not defined, of course. This tells us, then, that if the determinant of A equals zero, then A 1 is not defined. For what sorts of matrices is this a problem? It can be shown that matrices that have rows or columns that are linearly dependent on other rows or columns have determinants that are equal to zero. For these matrices, the determinant is undefined. We are given an order (k, k) matrix A, that I will denote using column vectors: A a 1 a 2 a k Each of the vectors a i is of order (k, 1). A column a i of A is said to be linearly independent of the others if there exists no set of scalars α j such that: a i j i α j a j Thus, given the rest of the columns, if we cannot find a weighted sum to get the column we are interested in, we say the it is linearly independent. We can define the term rank. The rank of a matrix is defined as the number of linearly independent columns (or rows) of a matrix. If all of the columns are independent, we say that the matrix is of full rank. We denote the rank of a matrix as r(a). By definition, r(a) is a integer that can take values from 1 to k. This is something that can be computed by software packages, such as Mathematica, Maple, or Stata. Here are some examples: ( 6) 3( 4 ) Notice that the second column is -2 times the first column. The rank of this matrix is 1 it is not of full rank. Here is another example: 2 7 2( 14) 4( 7) Notice that the second row is 2 times the first row. Again, the rank of this matrix is 1 it is not of full rank. Here is a final example, for an order (3, 3) matrix: using Mathematica Notice that the first column times 2 plus the third column equals the third column. The rank of this matrix is 2 it is not of full rank. Here are a series of important statements about inverses and nomenclature: A must be square. It is a necessary, but not sufficient, condition that A is square for A 1 to exist. In other words, sometimes the inverse of a matrix does not exist. If the inverse of a matrix does not exist, we say that it is singular. 179 14 The following statements are equivalent: full rank nonsingular invertable. All of these imply that A 1 exists. If the determinant of A equals zero, then A is said to be singular, or not invertable. More generally, A 0 A singular. If the determinant of A is non-zero, then A is said to be nonsingular, or invertable. In other words, the inverse exists. More generally, A 0 A nonsingular. If a matrix A is not of full rank, it is not invertable; i.e., it is singular. B.6 Some Important Properties of Inverses Here are some important identities that relate to matrix inversion: AA 1 I A 1 A I A 1 is unique. A must be square. It is a necessary, but not sufficient, condition that A is square for A 1 to exist. In other words, sometimes the inverse of a matrix does not exist it is singular. (A 1 ) 1 A. In words, the inverse of an inverse is the original matrix. Just as with transposition, it can be shown that (AB) 1 B 1 A 1 One can also show that the inverse of the transpose is the transpose of the inverse. Symbolically, (A ) 1 (A 1 ) B.7 Solving Systems of Equations Using Matrices Matrices are particularly useful when solving systems of equations, which, if you remeber, is what we did when we when solved for the least squares estimators. You covered this material in your high school algebra class. Here is an example, with three equations and three unknowns: x +2y + z 3 3x y 3z 1 2x +3y + z 4 180 15 How would one go about solving this? There are various techniques, including substitution, and multiplying equations by constants and adding them to get single variables to cancel. There is an easier way, however, and that is to use a matrix. Note that this system of equations can be represented as follows: x y 1 Ax b z 4 We can solve the problem Ax b by pre-multiplying both sides by A 1 and simplifying. This yields the following: Ax b A 1 Ax A 1 b x A 1 b We can therefore solve a system of equations by computing the inverse of A, and multiplying it by b. Here our matrix A and its inverse is as follows (using Mathematica to perform the calculation): A We can now solve this system of equations: x A 1 b A If we plug these back into original equations, we see that they in fact fulfill the identities. Computationally, this is a much easier way to solve systems of equations we just need to compute an inverse, and perform a single matrix multiplication. This approach only works, however, if the matrix A is nonsingular. If it is not invertable, then this will not work. In fact, if a row or a column of the matrix A is a linear combination of the others, there are no solutions to the system of equations, or many solutions to the system of equations. In either case, the system is said to be under-determined. We can compute the determinant of a matrix to see if it in fact is underdetermined MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 + UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product 1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23 (copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you: Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants Lecture 2 Matrix Operations Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20 Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical The Inverse of a Matrix The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square Mathematics Notes for Class 12 chapter 3. Matrices 1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form = [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In Cofactor Expansion: Cramer s Rule Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11 Inverses and powers: Rules of Matrix Arithmetic Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3 The Solution of Linear Simultaneous Equations Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve MATH 240 Fall, Chapter 1: Linear Equations and Matrices MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS ( % . This matrix consists of \$ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4& Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important Diagonal, Symmetric and Triangular Matrices Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by 1 Vector Spaces and Matrix Notation 1 Vector Spaces and Matrix Notation De nition 1 A matrix: is rectangular array of numbers with n rows and m columns. 1 1 1 a11 a Example 1 a. b. c. 1 0 0 a 1 a The rst is square with n = and m = ; the Topic 1: Matrices and Systems of Linear Equations. Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method MATH36001 Background Material 2015 MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of Chapter 1 - Matrices & Determinants Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. -- Morpheus Primary concepts: We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we 2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and Determinants. Dr. Doreen De Leon Math 152, Fall 2015 Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A. Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary Matrices, Determinants and Linear Systems September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we Matrix Inverse and Determinants DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted DETERMINANTS. b 2. x 2 DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in 4. MATRICES Matrices 4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows: 9 Matrices, determinants, inverse matrix, Cramer s Rule AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative: A Brief Primer on Matrix Algebra A Brief Primer on Matrix Algebra A matrix is a rectangular array of numbers whose individual entries are called elements. Each horizontal array of elements is called a row, while each vertical array is Matrix Algebra and Applications Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters row row row 4 13 Matrices The following notes came from Foundation mathematics (MATH 123) Although matrices are not part of what would normally be considered foundation mathematics, they are one of the first topics Matrices, transposes, and inverses Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrix-vector multiplication: two views st perspective: A x is linear combination of columns of A 2 4 Matrix Differentiation 1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have 1.5 Elementary Matrices and a Method for Finding the Inverse .5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder: 4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows Math 315: Linear Algebra Solutions to Midterm Exam I Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y = NOTES on LINEAR ALGEBRA 1 School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n, LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that 0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c Further Maths Matrix Summary Further Maths Matrix Summary A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in a matrix are called the elements of the matrix. The order of a matrix is the number Chapter 8. Matrices II: inverses. 8.1 What is an inverse? Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we Linear Algebra: Matrices B Linear Algebra: Matrices B 1 Appendix B: LINEAR ALGEBRA: MATRICES TABLE OF CONTENTS Page B.1. Matrices B 3 B.1.1. Concept................... B 3 B.1.2. Real and Complex Matrices............ B 3 B.1.3. December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation 1 Determinants. Definition 1 Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014 Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of Economics 102C: Advanced Topics in Econometrics 2 - Tooling Up: The Basics Economics 102C: Advanced Topics in Econometrics 2 - Tooling Up: The Basics Michael Best Spring 2015 Outline The Evaluation Problem Matrix Algebra Matrix Calculus OLS With Matrices The Evaluation Problem: Math 313 Lecture #10 2.2: The Inverse of a Matrix Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is SECTION 8.3: THE INVERSE OF A SQUARE MATRIX (Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1. a 1, or Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012, The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem A matrix over a field F is a rectangular array of elements from F. The symbol Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted Linear Algebra Review. Vectors Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are Lecture 21: The Inverse of a Matrix Lecture 21: The Inverse of a Matrix Winfried Just, Ohio University October 16, 2015 Review: Our chemical reaction system Recall our chemical reaction system A + 2B 2C A + B D A + 2C 2D B + D 2C If we write 15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without Linear Dependence Tests Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks 1 Introduction. 2 Matrices: Definition. Matrix Algebra. Hervé Abdi Lynne J. Williams In Neil Salkind (Ed.), Encyclopedia of Research Design. Thousand Oaks, CA: Sage. 00 Matrix Algebra Hervé Abdi Lynne J. Williams Introduction Sylvester developed the modern concept of matrices in the 9th APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A. APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the co-factor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj Introduction to Matrices for Engineers Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11 A Introduction to Matrix Algebra and Principal Components Analysis A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra APPENDIX QA MATRIX ALGEBRA. a 11 a 12 a 1K A = [a ik ] = [A] ik = 21 a 22 a 2K. a n1 a n2 a nk Greene-2140242 book December 1, 2010 8:8 APPENDIX QA MATRIX ALGEBRA A.1 TERMINOLOGY A matrix is a rectangular array of numbers, denoted a 11 a 12 a 1K A = [a ik ] = [A] ik = a 21 a 22 a 2K. a n1 a n2 a Matrices: 2.3 The Inverse of Matrices September 4 Goals Define inverse of a matrix. Point out that not every matrix A has an inverse. Discuss uniqueness of inverse of a matrix A. Discuss methods of computing inverses, particularly by row operations. Solution to Homework 2 Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if Lecture 23: The Inverse of a Matrix Lecture 23: The Inverse of a Matrix Winfried Just, Ohio University March 9, 2016 The definition of the matrix inverse Let A be an n n square matrix. The inverse of A is an n n matrix A 1 such that A 1 EC327: Advanced Econometrics, Spring 2007 Wooldridge, Introductory Econometrics (3rd ed, 2006) Appendix D: Summary of matrix algebra Basic definitions A matrix is a rectangular array of numbers, with m 2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors 2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible Typical Linear Equation Set and Corresponding Matrices EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of 5.3 Determinants and Cramer s Rule 290 5.3 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given 1 Gaussian Elimination Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67 Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond Lecture 10: Invertible matrices. Finding the inverse of a matrix Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 5.5 Introduction The inverse of a matrix In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 1. The inverse of a matrix The inverse of a square n n matrix A, is another Lecture No. # 02 Prologue-Part 2 Advanced Matrix Theory and Linear Algebra for Engineers Prof. R.Vittal Rao Center for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture No. # 02 Prologue-Part 2 In the last Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved. Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved. 1 Sum, Product and Transpose of Matrices. If a ij with Vector and Matrix Norms Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible. The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created: Math 018 Review Sheet v.3 Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right. Unified Lecture # 4 Vectors Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann, Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A. Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC) Lecture Notes: Matrix Inverse. 1 Inverse Definition Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices, Linear Algebra: Vectors A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector Linear Algebra: Determinants, Inverses, Rank D Linear Algebra: Determinants, Inverses, Rank D 1 Appendix D: LINEAR ALGEBRA: DETERMINANTS, INVERSES, RANK TABLE OF CONTENTS Page D.1. Introduction D 3 D.2. Determinants D 3 D.2.1. Some Properties of 7.4. The Inverse of a Matrix. Introduction. Prerequisites. Learning Style. Learning Outcomes The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a 0 has a reciprocal b written as a or such that a ba = ab =. Similarly a square matrix A may have an inverse B = A where AB = Solving a System of Equations 11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined Using the Singular Value Decomposition Using the Singular Value Decomposition Emmett J. Ientilucci Chester F. Carlson Center for Imaging Science Rochester Institute of Technology emmett@cis.rit.edu May 9, 003 Abstract This report introduces A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...

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; Graphics Documents User Generated Resources Learning Center Your Federal Quarterly Tax Payments are due April 15th # Graphics VIEWS: 30 PAGES: 13 • pg 1 Graphics Basic Plotting MATLAB has extensive facilities for displaying vectors and matrices as graphs, as well as annotating and printing these graphs. This section describes a few of the most important graphics functions and provides examples of some typical applications. Creating a Plot x = 0:pi/100:2*pi; y = sin(x); plot(x,y) xlabel('x = 0:2\pi') ylabel('Sine of x') title('Plot of the Sine Function','FontSize',12) Multiple Data Sets in One Graph y2 = sin(x-.25); y3 = sin(x-.5); plot(x,y,x,y2,x,y3) legend('sin(x)','sin(x-.25)','sin(x-.5)') Specifying Line Styles and Colors plot(x,y,'color_style_marker') color_style_marker is a string containing from one to four characters (enclosed in single quotation marks) constructed from a color, a line style, and a marker type:  Color strings are 'c', 'm', 'y', 'r', 'g', 'b', 'w', and 'k'.  Linestyle strings are '-' for solid, '--' for dashed, ':' for dotted, '-.' for dash- dot, and 'none' for no line.  The marker types are '+', 'o', '*', and 'x' and the filled marker types 's' for square, 'd' for diamond, '^' for up triangle, 'v' for down triangle, '>' for right triangle, '<' for left triangle, 'p' for pentagram, 'h' for hexagram, and none for no marker. Plotting Lines and Markers  If you specify a marker type but not a linestyle, MATLAB draws only the marker. For example, plot(x,y,'ks') plots black squares at each data point, but does not connect the markers with a line.  The statement plot(x,y,'r:+') plots a red dotted line and places plus sign markers at each data point.  You may want to use fewer data points to plot the markers than you use to plot the lines. This example plots the data twice using a different number of points for the dotted line and marker plots. x1 = 0:pi/100:2*pi; x2 = 0:pi/10:2*pi; plot(x1,sin(x1),'r:',x2,sin(x2),'r+') Adding Plots to an Existing Graph  The hold command enables you to add plots to an existing graph. When you type hold on Multiple Plots in One Figure  The subplot command subplot(m,n,p) partitions the figure window into an m-by-n matrix of small subplots.  The plots are numbered along first the top row of the figure window, then the second row, and so on. For example, t = 0:pi/10:2*pi; X = 4*cos(t);Y=4*sin(t);Z=t.*exp(-t);W=sin(t).*log(t+1); subplot(2,2,1); plot(t,X) subplot(2,2,2); plot(t,Y) subplot(2,2,3); plot(t,Z) subplot(2,2,4); plot(t,W) Mesh and Surface Plots  MATLAB defines a surface by the z-coordinates of points above a grid in the x-y plane, using straight lines to connect adjacent points.  The mesh and surf plotting functions display surfaces in three dimensions. o mesh produces wireframe surfaces that color only the lines connecting the defining points. o surf displays both the connecting lines and the faces of the surface in color. Visualizing Functions of Two Variables To display a function of two variables, z = f (x,y):  Generate X and Y matrices consisting of repeated rows and columns, respectively, over the domain of the function.  Use X and Y to evaluate and graph the function.  The meshgrid function transforms the domain specified by a single vector or two vectors x and y into matrices X and Y for use in evaluating functions of two variables.  The rows of X are copies of the vector x and the columns of Y are copies of the vector y. Example - Graphing the sinc Function [X,Y] = meshgrid(-8:.5:8); R = sqrt(X.^2 + Y.^2) + eps; Z = sin(R)./R; mesh(X,Y,Z,'EdgeColor','black') You can create a transparent mesh by disabling hidden line removal. hidden off Example - Colored Surface Plots A surface plot is similar to a mesh plot except the rectangular faces of the surface are colored. surf(X,Y,Z) colormap hsv colorbar Surface Plots with Lighting Lighting is the technique of illuminating an object with a directional light source. In certain cases, this technique can make subtle differences in surface shape easier to see. Lighting can also be used to add realism to three-dimensional graphs. This example uses the same surface as the previous examples, but colors it red and removes the mesh lines. A light object is then added to the left of the "camera" (that is the location in space from where you are viewing the surface). After adding the light and setting the lighting method to phong, use the view command to change the view point so you are looking at the surface from a different point in space (an azimuth of -15 and an elevation of 65 degrees). Finally, zoom in on the surface using the toolbar zoom mode. surf(X,Y,Z,'FaceColor','red','EdgeColor','none'); camlight left; lighting phong view(-15,65) Images  Two-dimensional arrays can be displayed as images, where the array elements determine brightness or color of the images. For example, the statements whos Name Size Bytes Class X 648x509 2638656 double array caption 2x28 112 char array map 128x3 3072 double array image(X) colormap(map) axis image reproduces Dürer's etching. Programming with MATLAB Flow Control MATLAB has several flow control constructs:  if statements  switch statements  for loops  while loops  continue statements  break statements if  The if , elseif and else keywords control the program flow based on a logical statement  Terminates with an end keyword.  No braces or brackets are involved. Example: Generation of the magic square when n is odd, when n is even but not divisible by 4, or when n is divisible by 4. if rem(n,2) ~= 0 M = odd_magic(n) elseif rem(n,4) ~= 0 M = single_even_magic(n) else M = double_even_magic(n) end  Example: difference between numbers and matrices. if A > B 'greater' elseif A < B 'less' elseif A == B 'equal' else error('Unexpected situation') end  Several functions are helpful for reducing the results of matrix comparisons to scalar conditions for use with if, including isequal isempty all any switch and case The switch statement executes groups of statements based on the value of a variable or expression. The keywords case and otherwise delineate the groups. Only the first matching case is executed. There must always be an end to match the switch. The logic of the magic squares algorithm can also be described by switch (rem(n,4)==0) + (rem(n,2)==0) case 0 M = odd_magic(n) case 1 M = single_even_magic(n) case 2 M = double_even_magic(n) otherwise error('This is impossible') end for The for loop repeats a group of statements a fixed, predetermined number of times. A matching end delineates the statements. for n = 3:32 r(n) = rank(magic(n)); end r It is a good idea to indent the loops for readability, especially when they are nested. for i = 1:m for j = 1:n H(i,j) = 1/(i+j); end end while The while loop repeats a group of statements an indefinite number of times under control of a logical condition. A matching end delineates the statements. Example: The bisection method a = 0; fa = -Inf; b = 3; fb = Inf; while b-a > eps*b x = (a+b)/2; fx = x^3-2*x-5; if sign(fx) == sign(fa) a = x; fa = fx; else b = x; fb = fx; end end x Result: x = 2.09455148154233 continue The continue statement passes control to the next iteration of the for or while loop in which it appears, skipping any remaining statements in the body of the loop. Example: Counting code lines in an m-file fid = fopen('magic.m','r'); count = 0; while ~feof(fid) line = fgetl(fid); if isempty(line) | strncmp(line,'%',1) continue end count = count + 1; end disp(sprintf('%d lines',count)); break The break statement lets you exit early from a for or while loop. Example: The bisection method revisited a = 0; fa = -Inf; b = 3; fb = Inf; while b-a > eps*b x = (a+b)/2; fx = x^3-2*x-5; if fx == 0 break elseif sign(fx) == sign(fa) a = x; fa = fx; else b = x; fb = fx; end end x Scripts and Functions  Files that contain code in the MATLAB language are called M-files. You create M-files using a text editor, then use them as you would any other MATLAB function or command.  There are two kinds of M-files:  Scripts, which do not accept input arguments or return output arguments. They operate on data in the workspace.  Functions, which can accept input arguments and return output arguments. Internal variables are local to the function. Scripts When you invoke a script, MATLAB simply executes the commands found in the file. For example, create a file called bisect.m that contains these MATLAB commands. a = 0; fa = -Inf; b = 3; fb = Inf; while b-a > eps*b x = (a+b)/2; fx = x^3-2*x-5; if fx == 0 break elseif sign(fx) == sign(fa) a = x; fa = fx; else b = x; fb = fx; end end x Typing the statement bisect causes MATLAB to execute the commands, compute the solution x. Functions  Functions are M-files that can accept input arguments and return output arguments.  The name of the M-file and of the function should be the same.  Functions operate on variables within their own workspace, separate from the workspace you access at the MATLAB command prompt. Example: The bisection method in general Create a file named genfun.m containing the following statements function y = genfun(x) y = x^3-2*x-5; %other function definitions can be used here Modify the script file bisect.m as follows a = 0; fa = genfun(a); b = 3; fb = genfun(b); while b-a > eps*b %can you see anything wrong here? x = (a+b)/2; fx = genfun(x); if fx == 0 break elseif sign(fx) == sign(fa) a = x; fa = fx; else b = x; fb = fx; end end x To top

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Opened 11 years ago Closed 7 years ago # Bug in mapping to residue fields Reported by: Owned by: cremona davidloeffler major sage-duplicate/invalid/wontfix number fields residue field was, justin, ylchapuy John Cremona N/A ### Description See #9417: ```sage: K.<a> = NumberField(x^2+18*x+1) sage: P = K.ideal(2) sage: F = K.residue_field(P) sage: R = PolynomialRing(F, 'x') sage: R([0, a, a, 1]) x^3 + abar*x^2 + abar*x sage: F(a) 1 sage: a.minpoly() x^2 + 18*x + 1 sage: F.gen() abar sage: F.gen().minpoly() x^2 + x + 1 ``` The polynomial `x^3+a*x^2+a*x` reduced modulo P=(2) wrongly to `x^3+abar*x^2+abar*x`. Although the generator of the residue field F is suggestively called abar, it it *not* the reduction of a mod P (which is 1 mod P). ### comment:1 Changed 11 years ago by cremona Note that this is correct: ```R([F(co) for co in [0,a,a,1]]) ``` I think the problem lies in lines 94-108 of sage/rings/polynomial/polynomial_zz_pex.pyx, hence I am CC-int Yann L-C! Meanwhile I am using that work-around for #9417. ### comment:2 Changed 11 years ago by ylchapuy polynomial_zz_pex.pyx is not intended to be used for this base ring (maybe it could but it wasn't thought like this). The constructor of the univariate polynomials ring should be more careful. And I'm sorry, but I won't have time to do this myself. ### comment:3 Changed 11 years ago by cremona Thanks for reporting back. I'll try to fix this myself, but in any case I was able to use a simple workaround. ### comment:4 Changed 9 years ago by jdemeyer • Milestone changed from sage-5.11 to sage-5.12 ### comment:5 Changed 9 years ago by vbraun_spam • Milestone changed from sage-6.1 to sage-6.2 ### comment:6 Changed 8 years ago by vbraun_spam • Milestone changed from sage-6.2 to sage-6.3 ### comment:7 Changed 8 years ago by vbraun_spam • Milestone changed from sage-6.3 to sage-6.4 fixed? ### comment:9 follow-ups: ↓ 10 ↓ 11 Changed 7 years ago by cremona With 6.9.beta3 I see this: ```K.<a> = NumberField(x^2+18*x+1) sage: P = K.ideal(2) sage: F = K.residue_field(P) sage: R = PolynomialRing(F, 'x') sage: R([0, a, a, 1]) x^3 + x^2 + x ``` which shows that the original issue has been fixed, though I do not know where or when. I cannot see the "won't fix" option but that is what this should now get. ### comment:10 in reply to: ↑ 9 Changed 7 years ago by pbruin • Milestone changed from sage-6.4 to sage-duplicate/invalid/wontfix • Status changed from new to needs_review With 6.9.beta3 I see this: ```K.<a> = NumberField(x^2+18*x+1) sage: P = K.ideal(2) sage: F = K.residue_field(P) sage: R = PolynomialRing(F, 'x') sage: R([0, a, a, 1]) x^3 + x^2 + x ``` which shows that the original issue has been fixed, though I do not know where or when. It might be #11239. ### comment:11 in reply to: ↑ 9 Changed 7 years ago by pbruin • Reviewers set to John Cremona • Status changed from needs_review to positive_review

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Engineering in Kenya # What is Physics Power? Posted by on Apr 8, 2018 in Physics | 0 comments From the physics point of view, physics power is the rate at which energy is used, transferred, or transformed. For instance, the rate at which a light bulb transforms electrical energy into light and heat is measured in watts. Power is directly proportional to voltage. The more the voltage, the more power is used per unit voltage. Power is also the rate at which work is performed. This is because energy transfer can be used to do work, the output power of an electric motor is calculated as the angular velocity of its output shaft and the product of the torque the motor generates. The power expended to move a vehicle is the product of the velocity of the vehicle and the traction force of the wheels. # Calculation of Physics Power The work done is defined as integral power over time. Calculation of work is said to be “path dependent.” since this essential relays on the trajectory of the point of application of the force and torque. Examples of Ansel Adams Boulder Dam Physics Power Units Power is energy over time. The watt (W) is equal to one joule per second. Watt is the SI units of power. Other units of power include metric horsepower (Pferdestärke (PS) or cheval vapeur, CV), ergs per second (erg/s), horsepower (hp), and foot-pounds per minute. 33,000 foot-pounds per minute is equivalent to the power required to lift 550 pounds by one foot in one second, and is equivalent to about 746 watts. ## Formula of Physics Power Burning a kilogram of coal releases much more energy than a kilogram of TNT,[3].  TNT delivers far more power than the coal because its reaction releases energy much more quickly. If we take? W as the amount of work performed during a period of time of duration? t, then the average power Pavg over that period is given by the formula: P_mathrm{avg} = frac{Delta W}{Delta t},. This gives the average amount of energy or work done converted per unit of time. When the context makes it clear, the average power is often simply called “power” .The limiting value of the average power as the time interval ?t approaches zero is therefore the instantaneous power.P =lim _{Delta trightarrow 0} frac{Delta W}{Delta t}= frac{dW}{dt}=lim _{Delta trightarrow 0} P_mathrm{avg} ,. ### Work in Relation to Physics Power The amount of work done during a period of duration T is given by:    W = PT, When power P is constant, It is more customary using the symbol E rather than W in energy conversion content. The product of an object’s velocity and the force on an object gives physics power. In mechanical systems, power is the combination of forces and movement. The time derivative of work is also mechanical power. The work done by a force F on an object traveling along a curve C in mechanics is given by; W_C = int_{C}bold{F}cdot bold{v}dt     = int_{C} bold{F} cdot mathrm{d}bold{x}. In the above formula, v is the velocity and x defines the path C along this path. Instantaneous power is yielded by the equation for work; P (t) = cdotmath{F} mathbf{v}.Physics can get even more interesting as you study more about physics power, we will study more in my next article.

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# Ruby newbie, my first binary search - is this idiomatic ruby? Hi All, I just finished the codeacademy Ruby course, and ventured to write a binary search in Ruby. After years of Java , this is fun ! Let me know what you think - any feedback / tips / pointers , alternate ways of doing the same - are welcome !! ###################################################### # binary search implementation def bsearch(a,y) #puts " – >> #{a} #{a.length} find #{32}" return false if a == nil return false if a.length == 0 return false if a.length == 1 && a[0] != y return true if a.length == 1 && a[0] == y ``````l = a.length return true if a[ l/2 ] == y return bsearch a[ 0,l/2],y if y < a[l/2] return bsearch a[ l/2, l],y `````` end 1000.times do |x| # create an array of 100 random numbers a = [] 100.times{ |x| a.push rand(100) } a.sort! # set the number to search for x = 32 `````` # search using our method contains_x = bsearch( a, x) # verify using ruby's built-in include? check = a.include? x # print the verdict puts contains_x == check ? " you win a toaster ! " : " fail! " `````` end ###################################################### Looks good, so here are just a few minor things: • There is an in-built binary search you should use when not practicing. Class: Array (Ruby 2.2.0) • Constructing an array can be done in one line with a = Array.new(100){rand(100)} • I would condense the checks a bit, and use in-built methods to check for nilness/emptyness: return false if a.nil? || a.empty? return a.first == y if a.length == 1 Btw, I got some experience with and am starting with Java at work, and that is a lot of fun, too. Type-safety and compile-time checks do scale nicely. I would not call a variable ‘l’, because - depending on the font - it is easy to mistake it as digit 1 when reading. I think you could write bsearch more compact as ``````def bsearch(a,y) middle = (a||=[]).length / 2 middle > 0 ? (y < a[middle] ? bsearch(a[0..middle],y) : `````` bsearch(a[middle+1…-1]) : (a[0]||’’) == y end (Code not tested, though). It is a matter of taste, which version is

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# CBSE Class 12 Maths Board Exam 2018: Important 2 marks questions expected this year Important questions for Class 12 Maths board exam 2018 are available here. These are two marks questions. This year, 8 two marks questions will be asked in the paper. Students should prepare these questions to score well in CBSE Class 12 Maths board exam 2018. Created On: Mar 19, 2018 10:12 IST Modified On: Mar 20, 2018 19:40 IST CBSE Class 12 Maths Board Exam 2018: Important questions expected this year Learn about important 2 marks questions of Class 12 Mathematics. These questions are expected to be asked in CBSE Class 12 Maths board exam 2018. You can easily find the solutions of these questions from NCERT Solutions, CBSE Sample Papers and Practice Papers provided by Jagranjosh. Links to access these articles are given in between the questions and also at the end. According to the latest CBSE 12th Maths examination pattern, 8 two marks questions will be asked in the paper. Important 1 mark questions for Class 12 Maths board exam 2018 Important 4 marks questions for Class 12 Maths board exam 2018 Important 6 marks questions for Class 12 Maths board exam 2018 Students preparing for CBSE Class 12 Maths board exam 2018 are also advised to go through the latest CBSE Blueprint and Sample Paper (Issued by CBSE) Important (2 marks) question for CBSE Maths board exam 2018 are given below: Question: If ey (x + 1) = 1, show that dy/dx = ‒ ey. Question: Verify that ax3 + by2 = 1 is a solution of the differential equation x (yy1 + y1) = yy1. Question: Find the approximate change in the value of (1/x2), when x changes from x = 2 to x = 2.002. Question: Solve the following Linear Programming Problem graphically: Maxmize: Z = 3 x + 4 y Subject to: x + y ≤ 4, x ≥ 0 and y ≥ 0. Class 12 Maths Sample Paper with Hints (Issued by CBSE) Question: If 4sin‒1x + cos‒1 x = π, then find the value of x. Question: Let the function f: RR be defined by f (x) = cos x for all x ϵ R. Show that f is neither one-one nor onto. Question: Evaluate: cos‒1 (‒√3 /2) + π/6). Question: If cos (tan‒1 x + cot ‒1 √3) = 0 then calculate the value of x. Related Video: Tips to score more than expected marks in CBSE board exams 2018 Question: A couple has 2 children. Find the probability that both are boys, if it is known that (i) one of them is a boy (ii) the older child is a boy. Question: If A and B are two events such that P (A) = 0.4, P (B) = 0.8 and P (B|A) = 0.6, then find P (A|B). Question: Arun can solve 90 % of the problems given in a book whereas Amit can solve 70%. Find the probability that at least one of them will solve the problem, selected at random from the book? Class 12 Maths Guess Paper 2018 Question: Differentiate 8x/x8 with respect to x Question: If f (x) = |cos x ‒ sin x|, then find the value of f’(π/3). Question: Find dy/dx when x and y are connected by the relation: tan-1 (x2 + y2) = a Question: Find dy/dx when x and y are connected by the relation: sec (x + y) = xy Question: If x = 3sin t - sin 3t, y = 3 cos t - cos 3t, then find dy/dx. Question: If f(x) = sin 2x – cos 2x, find f '(π/6). Question: Find the sum of the order and the degree of the following differential equations: (d2y/dx2) + (dy/dx)1/3 + (1 + x) = 0. Question: Examine the continuity of the function f (x) = x3 + 2x2 - 1 at x = 1. Question: Examine if Rolle’s theorem is applicable for the function f (x) = |x ‒ 1| in [0, 2]. Question: Find an angle q, where 0 < q < π/2, which increases twice as fast as its sine. Question: Find the approximate value of (1.999)5. CBSE Class 12 Mathematics Solved Question Paper: 2017 | 2016 Question: Show that f (x) = tan-1 (sin x + cos x) is an increasing function in (0, π/4). Question: Integrate:  dx/(1+cos x). Question: Integrate:  tan2x sec4x dx. Question: Integrate: x/(√x + 1). Question: Find the Cartesian and vector equations of the line which passes through the point (‒2, 4, ‒5) and parallel to the line given by (x + 3)/3 = (y ‒ 4)/5 = (8 ‒ z)/‒6. Question: Find the Projection (vector) of 2 ij + k on i ‒ 2j + k. Question: Find the coordinates of the point where the line through the points A (3, 4, 1) and B (5, 1, 6) crosses the XZ plane. Question: How many equivalence relations on the set {1, 2, 3} containing (1, 2) and (2, 1) are there in all? Justify your answer. Question: The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. Find the rate at which its area increases, when side is 10 cm long. Class 12 Maths Sample Paper: 2017 | 2016 | 2015 रोमांचक गेम्स खेलें और जीतें एक लाख रुपए तक कैश ## Related Categories Comment (0) ### Post Comment 8 + 3 = Post Disclaimer: Comments will be moderated by Jagranjosh editorial team. Comments that are abusive, personal, incendiary or irrelevant will not be published. Please use a genuine email ID and provide your name, to avoid rejection.

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CC-MAIN-2022-05

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## Yolo4mecuite 2 years ago -27 = 9n ? 1. ekmath n = -3 2. jazy To solve for n, Divde both sides of the equation by 9. $\large \frac{ -27 }{ 9 } = \frac{ 9n }{ 9 }$ 3. ryan123345 Divide by 9 on both sides to get n alone. 4. Yolo4mecuite oK 5. Yolo4mecuite -3

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• Views : 50k+ • Sol Viewed : 20k+ # Mind Teasers : Odd One Out Riddle Difficulty Popularity One of the number is odd. 4377 3954 9862 8454 9831 Which one ? Discussion Suggestions • Views : 80k+ • Sol Viewed : 20k+ # Mind Teasers : Best Maths Brain Teaser Difficulty Popularity A competitive exam was held in which five students took part namely Billy, Gerry, Clark, Peeta and Jonathan. In this exam, they had to answer five questions each out of which, three had multiple choices as a, b or c and two were simple true and false questions. The five of them gave different answers to the questions and the details are as given below. Name 1 2 3 4 5 Gerry c b True True False Billy c c True True True Clark a c False True True Peeta b a True True False Jonathan a b True False True None of the two students gave the same number of correct answers. Can you find out the correct answers to the question? Also, calculate the individual score of all the five guys. • Views : 50k+ • Sol Viewed : 20k+ # Mind Teasers : Worlds Hardest Brain Teaser Difficulty Popularity A mastermind organized a quiz competition in which six selected candidates were invite namely James Hunt, Ruxandra Bar, Sophia Connors, David Finch, Fred Odea and Brian Miller. A 'special puzzle' was asked to all of them. The first one to answer it was promised for a big award. After that, the candidates were offered the meal before the mastermind stood up to announce the much awaited result. He started announcing: 'Ok now everybody!' 'The winner of…..' 'The Hardest Riddle Ever Event.' And then he smiled. All the candidates understood who won. Do you know who won ? • Views : 80k+ • Sol Viewed : 20k+ # Mind Teasers : Matchsticks Picture Brain Teaser Difficulty Popularity In the figure given below, you can see that there are five squares. Supposedly if this figure is formed using different matchsticks (refer the slight gap between matchsticks), can you remove just two matchsticks so that only two squares remain ? • Views : 80k+ • Sol Viewed : 20k+ # Mind Teasers : Number Sequence Problem Difficulty Popularity Identify the next two numbers in this Sequence ? 101, 112, 131, 415, 161, 718, ???, ??? • Views : 40k+ • Sol Viewed : 10k+ # Mind Teasers : How Many Siblings Family Puzzle Difficulty Popularity In a particular family, each boy has as many brothers as the sisters, but each girl has twice as many brothers as that of sisters. How many numbers of siblings are there in the family? • Views : 50k+ • Sol Viewed : 20k+ # Mind Teasers : Tough River Crossing Riddle Difficulty Popularity This one is a bit of tricky river crossing puzzle than you might have solved till now. We have a whole family out on a picnic on one side of the river. The family includes Mother and Father, two sons, two daughters, a maid and a dog. The bridge broke down and all they have is a boat that can take them towards the other side of the river. But there is a condition with the boat. It can hold just two persons at one time (count the dog as one person). No it does not limit to that and there are other complications. The dog can’t be left without the maid or it will bite the family members. The father can’t be left with daughters without the mother and in the same manner, the mother can’t be left alone with the sons without the father. Also an adult is needed to drive the boat and it can’t drive by itself. How will all of them reach the other side of the river? • Views : 50k+ • Sol Viewed : 20k+ # Mind Teasers : The sheep and the grass logical puzzle Difficulty Popularity If we tie a sheep to one peg, a circled grass is been eaten by the sheep. If we tie the sheep to two pegs with a circle on its neck, then an eclipse is eaten out of the grass by the sheep. If we want an eclipse then we put two pegs then put a rope in between then and the other end of the rope is tied up on the sheep's neck. Question: how should we tie the peg and the sheep so that a square is eaten out from the garden's grass? We only have one sheep's rope and the peg and the rings. • Views : 80k+ • Sol Viewed : 20k+ # Mind Teasers : 8 Toothpicks Brain Teaser Difficulty Popularity Can you form eight squares of three different sizes by moving only two toothpick? • Views : 40k+ • Sol Viewed : 10k+ # Mind Teasers : Very Hard Math Question Difficulty Popularity There stand nine temples in a row in a holy place. All the nine temples have 100 steps climb. A fellow devotee comes to visit the temples. He drops a Re. 1 coin while climbing each of the 100 steps up. Then he offers half of the money he has in his pocket to the god. After that, he again drops Re. 1 coin while climbing down each of the 100 steps of the temple. If he repeats the same process at each temple, he is left with no money after climbing down the ninth temple. Can you find out the total money he had with him initially? • Views : 50k+ • Sol Viewed : 20k+ # Mind Teasers : Toothpick Donkey Puzzle Difficulty Popularity The toothpicks in the picture have been arranged to form a donkey shaped figure. You have to move two matchsticks in a way that the entire shape is rotated / reflected while being intact. Also, you can't change the tail's direction it should be pointing up. ### Latest Puzzles 23 January ##### Funny Tiger OneLIner Riddle How much fur should we get from the Tige... 22 January ##### Ball Pyramid Puzzle Can you count the numbers of the ball in... 21 January ##### What Am I Poem Riddle I am the end of life but also the start ... 20 January ##### Most Popular Number Series IAS Problem Can you solve the number series problem ... 19 January ##### Nice What Is It Riddle It has no heart but yet it lives. ... 18 January ##### Number Pyramid Puzzle Can you solve below number pyramid puzzl... 17 January ##### Japanese Ship Sailing Puzzle A Japanese ship is on route back to the ...

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This site is supported by donations to The OEIS Foundation. Thanks to everyone who made a donation during our annual appeal! To see the list of donors, or make a donation, see the OEIS Foundation home page. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A233594 Number of (n+2)X(n+2) 0..2 arrays with no increasing sequence of length 3 horizontally, vertically or antidiagonally downwards 0 15272, 20463326, 193816724926, 13036676136198570, 6217400227849213336029, 21035300720988185135892977193, 504952565127306851299571097075003261, 86012703850284916544804503539506332380421824 (list; graph; refs; listen; history; text; internal format) OFFSET 1,1 COMMENTS Diagonal of A233602 LINKS EXAMPLE Some solutions for n=1 ..2..1..2....2..2..2....2..2..2....1..1..2....1..0..2....1..2..1....1..0..1 ..1..1..0....1..2..1....2..0..1....2..2..2....0..2..2....1..1..2....2..2..1 ..2..0..1....0..0..0....2..2..0....2..1..1....0..1..1....1..1..1....2..0..2 CROSSREFS Sequence in context: A233602 A233603 A233595 * A064982 A204317 A216945 Adjacent sequences:  A233591 A233592 A233593 * A233595 A233596 A233597 KEYWORD nonn AUTHOR R. H. Hardin, Dec 14 2013 STATUS approved Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent The OEIS Community | Maintained by The OEIS Foundation Inc. Last modified January 18 21:54 EST 2019. Contains 319282 sequences. (Running on oeis4.)

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# Computer Science 101 – A Series The journey of a thousand miles begins with a single step. You have to crawl before you can run. Other proverbs. In order to become an effective computer scientist, you have to learn the basics. It's disturbing how often this is glossed over these days, but it is the truth. This series will cover a host of # CS 101 – Introduction to Logic Computers are dumb. Computers follow very rigid patterns of logic, which we call programs. Without understanding this logic, we cannot cause a computer to perform even the simplest tasks, leaving us without the foundation for the more complicated tasks. What is logic Logic is, fundamentally, a system of rules by which we can derive certain truth. It # CS 101 – Logic Part 2 For part 1, see CS 101 - Introduction to Logic Solution to last week's problem In our last post, we considered the following problem: [code language=c] int i=0; int j=1; int k=2; int m=4; //Example 5 if( m == k * k || i == 1 && j == 1) printf("What's the result?"); [/code] It turns out that boolean (read: computer binary) logic also operates on # CS 101 – Binary and Hexadecimal You may have heard the old programmer's joke: There are 10 kinds of people in this world: Those who understand Binary, And those who don't. This is an excellent introduction into the topic. Binary Computers, being dumb machines, are composed primarily of a host of switches. These switches have two states: ON and OFF (or CLOSED and OPEN). When the # CS 101 – Binary Mathematics In my last post, we discovered binary as a form of numeric representation. Like decimal, we can perform basic arithmetic operations with binary (and, consequently, hex). Addition We all know that 1 + 1 = 2, right? In binary, 1 + 1 = 10, as 10 is the binary representation of 2. By the same token, # CS 101 – Bytes and ASCII (Apologies for the relatively content-free submission, but my buffer ran dry. This is a bit of a rushed post.) We have seen that a computer represents data as binary bits. These bits are commonly grouped into eight-bit units, which we call bytes. One byte can store up to 256 possible values. These values (in this case, # CS 101 – Registers We now have some understanding of what bits are and how we work with them, but how do we store them? Registers A register is a computer element which stores bytes of data. There are several ways to build these registers, but their core function is to hold onto data for future use. Consider the following pseudocode: a # CS 101 – Integer Representation Apologies for the title, but I can't think of a better way to put it. Unsigned The simplest way to handle integers in binary is to treat every number as a positive number. This is called the unsigned integer representation. With the exception of subtraction, studied earlier, we have always used the unsigned integer representation before. Sign-Magnitude Naturally, # CS 101 – Floating Point Binary This section should wrap us up on binary for now. Floating Point Numbers Up until this point, we have dealt exclusively with integer representation binary. Everything we have done has been a whole value, with no fractions. Floating point numbers are those fractions. More accurately, it is a number whose values can be between one and zero. # CS 101 – Get on the Bus As we have discussed previously, computers operate as a huge array of switches, which operate in the on and off positions. What, we can reasonably ask, defines whether the switch is off or on? The answer: electricity. Buses: Wires connecting switches The average computer contains many hundreds of feet of wire, designed to carry electrical charges between

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# Convert gram force decimeters [gf-dm] to other units of torque ## gram force decimeters [gf-dm] torque conversions 1 gf-dm = 0.98 newton millimeter gf−dm to N⋅mm 1 gf-dm = 0.1 newton centimeter gf−dm to N⋅cm 1 gf-dm = 0.01 newton decimeter gf−dm to N⋅dm 1 gf-dm = 0.001 newton meter gf−dm to N⋅m 1 gf-dm = 100 gram force millimeters gf−dm to gf-mm 1 gf-dm = 10 gram force centimeters gf−dm to gf-cm 1 gf-dm = 0.1 gram force meter gf−dm to gf-m 1 gf-dm = 0.1 kilogram force millimeter gf−dm to kgf-mm 1 gf-dm = 0.01 kilogram force centimeter gf−dm to kgf-cm 1 gf-dm = 0.001 kilogram force decimeter gf−dm to kgf-dm 1 gf-dm = 0.0001 kilogram force meter gf−dm to kgf-m 1 gf-dm = 0.01 pound force inch gf−dm to lbf-in 1 gf-dm = 0.001 pound force foot gf−dm to lbf-ft 1 gf-dm = 98 066.5 dyne millimeters gf−dm to dyn⋅mm 1 gf-dm = 9 806.65 dyne centimeters gf−dm to dyn⋅cm 1 gf-dm = 980.67 dyne decimeters gf−dm to dyn⋅dm 1 gf-dm = 98.07 dyne meters gf−dm to dyn⋅m #### Foods, Nutrients and Calories BBQ SAUCE, UPC: 788310923348 weigh(s) 237 grams per metric cup or 7.9 ounces per US cup, and contain(s) 89 calories per 100 grams (≈3.53 ounces)  [ weight to volume | volume to weight | price | density ] Lactic acid in 95% G.T.'S KOMBUCHA, UPC: 722430240169 #### Gravels, Substances and Oils CaribSea, Marine, Aragonite, Special Coarse Aragonite weighs 1 153.3 kg/m³ (71.99817 lb/ft³) with specific gravity of 1.1533 relative to pure water.  Calculate how much of this gravel is required to attain a specific depth in a cylindricalquarter cylindrical  or in a rectangular shaped aquarium or pond  [ weight to volume | volume to weight | price ] Zirconium silicate [ZrSiO4  or  O4SiZr] weighs 4 560 kg/m³ (284.6715 lb/ft³)  [ weight to volume | volume to weight | price | mole to volume and weight | mass and molar concentration | density ] Volume to weightweight to volume and cost conversions for Refrigerant R-414B, liquid (R414B) with temperature in the range of -40°C (-40°F) to 71.12°C (160.016°F) #### Weights and Measurements A pound-force per square inch is a unit of pressure where a force of one pound-force (lbf) is applied to an area of one square inch. The electric potential φ(A) of a point A (in the electric field) specifies the work to be done by the force F = −Q × E  in order to move the charge Q from a fixed reference point P to the point A. µV to kV conversion table, µV to kV unit converter or convert between all units of electric potential measurement. #### Calculators Calculate volume of a hollow cylinder and its surface area

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CC-MAIN-2023-23

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• #### 科普 SCIENCE #### 英语 ENGLISH #### 科技 TECHNOLOGY MOVIE FOOD #### 励志 INSPIRATIONS #### 社会 SOCIETY TRAVEL #### 动物 ANIMALS KIDS #### 卡通 CARTOON #### 计算机 COMPUTER #### 心理 PSYCHOLOGY #### 教育 EDUCATION #### 手工 HANDCRAFTS #### 趣闻 MYSTERIES CAREER GEEKS #### 时尚 FASHION • 精品课 • 公开课 • 欢迎下载我们在各应用市场备受好评的APP 点击下载Android最新版本 点击下载iOS最新版本 扫码下载译学馆APP #### 形象展示傅里叶变换 But what is the Fourier Transform? A visual introduction. This right here is what we’re going to build to, this video: A certain animated approach to thinking about a super-important idea from math: The Fourier transform. For anyone unfamiliar with what that is, my # 1 goal here is just for the video to be an introduction to that topic. But even for those of you who are already familiar with it, I still think that there’s something fun and enriching about seeing what all of its components actually look like. The central example, to start, is gonna be the classic one: Decomposing frequencies from sound. But after that, I also really wan na show a glimpse of how this idea extends well beyond sound and frequency, and to many seemingly disparate areas of math, and even physics. Really, it is crazy just how ubiquitous this idea is. Let’s dive in. This sound right here is a pure A. 440 beats per second. Meaning, if you were to measure the air pressure or your speaker, as a function of time, it would oscillate up and down around its usual equilibrium, in this wave. making 440 oscillations each second. A lower-pitched note, like a D, has the same structure, just fewer beats per second. And when both of them are played at once, what do you think the resulting pressure vs. time graph looks like? Well, at any point in time, this pressure difference is gon na be the sum of what it would be for each of those notes individually. Which, let’s face it, is kind of a complicated thing to think about. At some points, the peaks match up with each other, resulting in a really high pressure. At other points, they tend to cancel out. And all in all, what you get is a wave-ish pressure vs. time graph, that is not a pure sine wave; it’s something more complicated. And as you add in other notes, the wave gets more and more complicated. But right now, all it is is a combination of four pure frequencies. So it seems… needlessly complicated, given the low amount of information put into it. A microphone recording any sound just picks up on the air pressure at many different points in time. It only”sees” the final sum. So our central question is gonna be how you can take a signal like this, and decompose it into the pure frequencies that make it up. Pretty interesting, right? Adding up those signals really mixes them all together. So pulling them back apart…feels akin to unmixing multiple paint colors that have all been stirred up together. The general strategy is gonna be to build for ourselves a mathematical machine that treats signals with a given frequency… ..differently from how it treats other signals. To start, consider simply taking a pure signal say, with a lowly three beats per second, so that we can plot it easily. And let’s limit ourselves to looking at a finite portion of this graph. In this case, the portion between zero seconds, and 4.5 seconds. The key idea, is gon na be to take this graph, and sort of wrap it up around a circle. Concretely, here’s what I mean by that. Imagine a little rotating vector where each point in time its length is equal to the height of our graph for that time. So, high points of the graph correspond to a greater disance from the origin, and low points end up closer to the origin. And right now, I’m drawing it in such a way that moving forward two seconds in time corresponds to a single rotation around the circle. Our little vector drawing this wound up graph is rotating at half a cycle per second. So, this is important. There are two different frequencies at play here: There’s the frequency of our signal, which goes up and down, three times per second. And then, separately, there’s the frequency with which we’re wrapping the graph around the circle. Which, at the moment, is half of a rotation per second. But we can adjust that second frequency however we want. Maybe we want to wrap it around faster… ..or maybe we go and wrap it around slower. And that choice of winding frequency determines what the wound up graph looks like. Some of the diagrams that come out of this can be pretty complicated; although, they are very pretty. But it’s important to keep in mind that all that’s happening here is that we’re wrapping the signal around a circle. The vertical lines that I’m drawing up top, by the way, are just a way to keep track of the distance on the original graph that corresponds to a full rotation around the circle. So, lines spaced out by 1.5 seconds would mean it takes 1.5 seconds to make one full revolution. And at this point, we might have some sort of vague sense that something special will happen when the winding frequency matches the frequency of our signal: three beats per second. All the high points on the graph happen on the right side of the circle And all of the low points happen on the left. But how precisely can we take advantage of that in our attempt to build a frequency-unmixing machine? Well, imagine this graph is having some kind of mass to it, like a metal wire. This little dot is going to represent the center of mass of that wire. As we change the frequency, and the graph winds up differently, that center of mass kind of wobbles around a bit. And for most of the winding frequencies, the peaks and valleys are all spaced out around the circle in such a way that the center of mass stays pretty close to the origin. But! When the winding frequency is the same as the frequency of our signal, in this case, three cycles per second, all of the peaks are on the right, and all of the valleys are on the left.. ..so the center of mass is unusually far to the right. Here, to capture this, let’s draw some kind of plot that keeps track of where that center of mass is for each winding frequency. Of course, the center of mass is a two-dimensional thing, and requires two coordinates to fully keep track of, but for the moment, let’s only keep track of the x coordinate. So, for a frequency of 0, when everything is bunched up on the right, this x coordinate is relatively high. And then, as you increase that winding frequency, and the graph balances out around the circle, the x coordinate of that center of mass goes closer to 0, and it just kind of wobbles around a bit. But then, at three beats per second, there’s a spike as everything lines up to the right. This right here is the central construct, so let ‘s sum up what we have so far: We have that original intensity vs. time graph, and then we have the wound up version of that in some two-dimensional plane, and then, as a third thing, we have a plot for how the winding frequency influences the center of mass of that graph. And by the way, let’s look back at those really low frequencies near 0. This big spike around 0 in our new frequency plot just corresponds to the fact that the whole cosine wave is shifted up. If I had chosen a signal oscillates around 0, dipping into negative values, then, as we play around with various winding frequences, this plot of the winding frequencies vs. center of mass would only have a spike at the value of three. But, negative values are a little bit weird and messy to think about especially for a first example, so let’s just continue thinking in terms of the shifted-up graph. I just want you to understand that that spike around 0 only corresponds to the shift. Our main focus, as far as frequency decomposition is concerned, is that bump at three. This whole plot is what I’ll call the”Almost Fourier Transform” of the original signal. There’s a couple small distinctions between this and the actual Fourier transform, which I’ll get to in a couple minutes, but already, you might be able to see how this machine lets us pick out the frequency of a signal. Just to play around with it a little bit more, take a different pure signal, let’s say with a lower frequency of two beats per second, and do the same thing. Wind it around a circle, imagine different potential winding frequencies, and as you do that keep track of where the center of mass of that graph is, and then plot the x coordinate of that center of mass as you adjust the winding frequency. Just like before, we get a spike when the winding frequency is the same as the signal frequency, which in this case, is when it equals two cycles per second. But the real key point, ——这个机器之所以让人喜闻乐见—— the thing that makes this machine so delightful, is how it enables us to take a signal consisting of multiple frequencies, and pick out what they are. Imagine taking the two signals we just looked at: 3Hz的波 The wave with three beats per second, and the wave with two beats per second, Like I said earlier, what you get is no longer a nice, pure cosine wave; it’s something a little more complicated. But imagine throwing this into our winding-frequency machine… ..it is certainly the case that as you wrap this thing around, it looks a lot more complicated; you have this chaos (1) and chaos (2) and chaos (3) and chaos (4) and then WOOP! Things seem to line up really nicely at two cycles per second, and as you continue on it’s more chaos (5) and more chaos (6) more chaos (7) chaos (8), chaos (9), chaos (10), WOOP! Things nicely align again at three cycles per second. And, like I said before, the wound up graph can look kind of busy and complicated, but all it is is the relatively simple idea of wrapping the graph around a circle. It’s just a more complicated graph, and a pretty quick winding frequency. Now what’s going on here with the two different spikes, is that if you were to take two signals, and then apply this Almost-Fourier transform to each of them individually, and then add up the results, what you get is the same as if you first added up the signals, and then applied this Almost-Fourier transorm. And the attentive viewers among you might wan na pause and ponder, and… ..convince yourself that what I just said is actually true. It’s a pretty good test to verify for yourself that it’s clear what exactly is being measured inside this winding machine. Now this property makes things really useful to us, because the transform of a pure frequency is close to 0 everywhere except for a spike around that frequency. So when you add together two pure frequencies, the transform graph just has these little peaks above the frequencies that went into it. So this little mathematical machine does exactly what we wanted. It pulls out the original frequencies from their jumbled up sums, unmixing the mixed bucket of paint. And before continuing into the full math that describes this operation, let’s just get a quick glimpse of one context where this thing is useful: Sound editing. Let’s say that you have some recording, and it’s got an annoying high pitch that you’d like to filter out. Well, at first, your signal is coming in as a function of various intensities over time. Different voltages given to your speaker from one millisecond to the next. But we want to think of this in terms of frequencies, so, when you take the Fourier transform of that signal, the annoying high pitch is going to show up just as a spike at some high frequency. Filtering that out, by just smushing the spike down, what you’d be looking at is the Fourier transform of a sound that’s just like your recording, only without that high frequency. Luckily, there’s a notion of an inverse Fourier transform that tells you which signal would have produced this as its Fourier transform. I’ll be talking about inverse much more fully in the next video, but long story short, applying the Fourier transform to the Fourier transform gives you back something close to the original function. Mm, kind of… this is… ..a little bit of a lie, but it’s in the direction of the truth. And most of the reason that it’s a lie is that I still have yet to tell you what the actual Fourier Transform is, since it’s a little more complex than this x-coordinate-of-the-center-of-mass idea. First off, bringing back this wound up graph, and looking at its center of mass, x坐标只能反映一半的事实 对吧 the x coordinate is really only half the story, right? I mean, this thing is in two dimensions, it’s got a y coordinate as well. And, as is typical in math, whenever you’re dealing with something two-dimensional, it’s elegant to think of it as the complex plane, where this center of mass is gonna be a complex number, that has both a real and an imaginary part. And the reason for talking in terms of complex numbers, rather than just saying, “It has two coordinates,” is that complex numbers lend themselves to really nice descriptions of things that have to do with winding, and rotation. For example: Euler’s formula famously tells us that if you take e to some number times i, you’re gonna land on the point that you get if you were to walk that number of units around a circle with radius 1, counter-clockwise starting on the right. So, imagine you wanted to describe rotating at a rate of one cycle per second. One thing that you could do is take the expression”e^2π*i*t,” where t is the amount of time that has passed. Since, for a circle with radius 1, 2π就是一圈的长度 不过 2π describes the full length of its circumference. And… this is a little bit dizzying to look at, so maybe you wan na describe a different frequency… ..something lower and more reasonable… ..and for that, you would just multiply that time t in the exponent by the frequency, f. For example, if f was one tenth, then this vector makes one full turn every ten seconds, since the time t has to increase all the way to ten before the full exponent looks like 2πi. I have another video giving some intuition on why this is the behavior of e^x for imaginary inputs, e的虚数次方为什么长这样的一些直观解释 if you’re curious , but for right now, we’re just gon na take it as a given. Now why am I telling you this you this, you might ask. Well, it gives us a really nice way to write down the idea of winding up the graph into a single, tight little formula. First off, the convention in the context of Fourier transforms is to think about rotating in the clockwise direction, take some function describing a signal intensity vs. time, like this pure cosine wave we had before, and call it g(t). If you multiply this exponential expression times g(t), it means that the rotating complex number is getting scaled up and down according to the value of this function. So you can think of this little rotating vector with its changing length as drawing the wound up graph. So think about it, this is awesome. This really small expression is a super-elegant way to encapsulate the whole idea of winding a graph around a circle with a variable frequency f. And remember, that thing we want to do with this wound up graph is to track its center of mass. So think about what formula is going to capture that. Well, to approximate it at least, you might sample a whole bunch of times from the original signal, see where those points end up on the wound up graph, and then just take an average. That is, add them all together, as complex numbers, and then divide by the number of points that you’ve sampled. This will become more accurate if you sample more points which are closer together. And in the limit, rather than looking at the sum of a whole bunch of points divided by the number of points, you take an integral of this function, divided by the size of the time interval that we’re looking at. Now the idea of integrating a complex-valued function might seem weird, and to anyone who’s shaky with calculus, maybe even intimidating, but the underlying meaning here really doesn’t require any calculus knowledge. The whole expression is just the center of mass of the wound up graph. So… Great! Step-by-step, we have built up this kind of complicated, but, let’s face it, surprisingly small expression for the whole winding machine idea that I talked about. And now, there is only one final distinction to point out between this and the actual, honest-to-goodness Fourier transform. Namely, just don’t divide out by the time interval. The Fourier transform is just the integral part of this. What that means is that instead of looking at the center of mass, you would scale it up by some amount. If the portion of the original graph you were using spanned three seconds, you would multiply the center of mass by three. If it was spanning six seconds, you would multiply the center of mass by six. Physically, this has the effect that when a certain frequency persists for a long time, then the magnitude of the Fourier transform at that frequency is scaled up more and more. For example, what we’re looking at right here is how when you have a pure frequency 2Hz的信号 of two beats per second, and you wind it around the graph at two cycles per second, the center of mass stays in the same spot, right? It’s just tracing out the same shape. But the longer that signal persists, the larger the value of the Fourier transform, at that frequency. For other frequencies, though, even if you just increase it by a bit, this is cancelled out by the fact that for longer time intervals you’re giving the wound up graph more of a chance to balance itself around the circle. That is… a lot of different moving parts, so let’s step back and summarize what we have so far. The Fourier transform of an intensity vs. time function, like g(t), is a new function, which doesn’t have time as an input, but instead takes in a frequency, what I’ve been calling”the winding frequency.” In terms of notation, by the way, the common convention is to call this new function “g-hat,” with a little circumflex on top of it. Now the output of this function is a complex number, some point in the 2D plane, that corresponds to the strength of a given frequency in the original signal. The plot that I’ve been graphing for the Fourier transform, is just the real component of that output, the x-coordinate But you could also graph the imaginary component separately, if you wanted a fuller description. And all of this is being encapsulated inside that formula that we built up. And out of context, you can imagine how seeing this formula would seem sort of daunting. But if you understand how exponentials correspond to rotation… ..how multiplying that by the function g(t) means drawing a wound up version of the graph, and how an integral of a complex-valued function can be interpreted in terms of a center-of-mass idea, you can see how this whole thing carries with it a very rich, intuitive meaning. And, by the way, one quick small note before we can call this wrapped up. Even though in practice, with things like sound editing, you’ll be integrating over a finite time interval, the theory of Fourier transforms is often phrased where the bounds of this integral are -∞ and ∞. Concretely, what that means is that you consider this expression for all possible finite time intervals, “What is its limit as that time interval grows to ∞?” And…man, oh man, there is so much more to say! So much, I don’t wanna call it done here. This transform extends to corners of math well beyond the idea of extracting frequencies from signal. So, the next video I put out is gon na go through a couple of these, and that’s really where things start getting interesting. So, stay subscribed for when that comes out, or an alternate option is to just binge a couple 3blue1brown videos so that the YouTube recommender is more inclined to show you new things that come out… ..really, the choice is yours! And to close things off, I have something pretty fun: A mathematical puzzler from this video’s sponsor, Jane Street 他们正在招募更多技术人才 Jane Street, who’s looking to recruit more technical talent. So, let’s say that you have a closed, bounded convex set C sitting in 3D space, B是集合C的边界 and then let B be the boundary of that space, the surface of your complex blob. Now imagine taking every possible pair of points on that surface, and adding them up, doing a vector sum. Let’s name this set of all possible sums D. Your task is to prove that D is also a convex set. So, Jane Street是一家量化交易公司 Jane Street is a quantitative trading firm, and if you’re the kind of person who enjoys math and solving puzzles like this, the team there really values intellectual curiosity. So, they might be interested in hiring you. And they’re looking both for full-time employees and interns. For my part, I can say that some people I’ve interacted with there just seem to love math, and sharing math, and when they’re hiring they look less at a background in finance than they do at how you think, how you learn, and how you solve problems, hence the sponsorship of a 3blue1brown video. If you want the answer to that puzzler, or to apply for open positions, go to janestreet.com/3b1b Aidenlazz

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You have been working at a local CPA firm for the last 2 years. At the beginning of the year, you passed the last section of the CPA Exam. Last week, your CPA License came in the mail. Consequently, you have now decided to go into business for yourself. You have some savings and have a line on a ban... ##### 40% of what number is 152 ? Round to 1 decimal place 40% of what number is 152 ? Round to 1 decimal place... ##### Assume thatx 1 2x2 + 1 if x < 0 f(x) sin(x) +e if x > 0 Then f' (0) Assume that x 1 2x2 + 1 if x < 0 f(x) sin(x) +e if x > 0 Then f' (0)... ##### Classify each of the following differential equations: ODE or PDE, its order; linear or nonlinear. If it is not linear; explain why: 4 d +r"v = 8 = (4 2)(1 -2)Write differential equation that fits for each of the following physical descriptions: The velocity at the time of a particle moving along straight line is proportional to the fourth power of its position €. 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Ru; Cr; VS AL, and P affinity, and decreasing ionization energy.... ##### H h 0 1 8 "apio nullui ole lro onine 1 L H h 0 1 8 "apio nullui ole lro onine 1 L... ##### Question 8 0.1 pts Suppose that there are 150,000 employed individuals and 6,000 individuals with... Question 8 0.1 pts Suppose that there are 150,000 employed individuals and 6,000 individuals without a job, but actively searching for one. How large will unemployment be in the following period if the job finding rate is 75%, while the job separation rate is 4%? Question 8 0.1 pts Suppose that the... ##### Joe has been jogging at 7 METS for 30 min on 3d · wk 1. His... Joe has been jogging at 7 METS for 30 min on 3d · wk 1. His body weight is a 70 kg. How much Kcal did he spend per week?... The current asset section of Guardian Consultant’s balance sheet consists of cash, accounts receivable, and prepaid expenses. The 2021 balance sheet reported the following: cash, $1,370,000; prepaid expenses,$430,000; long-term assets, $3,100,000; and shareholders’ equity,$3,200,000. T...

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https://www.newbedev.com/algorithm/java/array/remove-element/

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Remove Element Remove Element array Given an array and a value, remove all instances of that value in-place and return the new length. Do not allocate extra space for another array, you must do this by modifying the input array in-place with O(1) extra memory. The order of elements can be changed. It doesn’t matter what you leave beyond the new length. ``````Example: Given nums = [3,2,2,3], val = 3, Your function should return length = 2, with the first two elements of nums being 2. `````` Remove Element Solution ``````class RemoveElement { public int removeElement(int[] nums, int val) { int index = 0; for(int i = 0; i < nums.length; i++) { if(nums[i] != val) { nums[index++] = nums[i]; } } return index; } }`````` `````` ``````

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# O Level Physics Past Papers (5054) Past Papers Inside Provides with the O Level Physics Past Papers, Helpful Resources, and Guides which includes subject syllabus, specimens, question papers, marking schemes, FAQs, Notes, Teaching Resources and more. Past Papers Inside Provides you O Level Past Papers of Physics 5054 which are available from 2002 up to the latest sessions. We offer all the data absolutely free which is presented here for you and is provided in the most suitable and easy way so that you don’t face any kind of issue. Download these past papers of May/June and October/November sessions and of different variants of O Level Physics 5054 Past Papers. These Mentioned PDF Files include O Level Physics Past Papers 5054 includes Question Papers, CDT Design, and Communication marking schemes, and more. Additionally, you can also check out O Level Physics 5054 Syllabus & Example Candidate Response. You can also check the latest O Level Grade thresholds table to confirm the grade boundaries through this link. Take Enough time to review materials covered in your class. How long it takes you to study is based on your judgment when you start studying. Do check our notes section also and once you are ready after analyzing yourself and your studying method with having good notes, Then the Next Goal is to get past papers that we provide you and help you to get the updated and latest content. Solving these Past Papers will help you to prepare for CAIE Examination previously CIE O Level Physics 5054. /78 26 Created on By Past Papers Inside O Level Physics The O Level Physics Quiz is a comprehensive tool designed to help students prepare for their physics exams and score high.  Take the quiz today and get ready to ace your exams! 1 / 78 When a body having uniform speed and cover equal distance in equal interval of time is: 2 / 78 Which one depend upon force and area______________ 3 / 78 Crocus flower is a natural thermometer. It opens when temperature is __________ Celsius 4 / 78 When a current produce by positive charges then current will be 5 / 78 ntent --> A body having a larger mass then its heat capacity is: 6 / 78 Through optical center line passes is known as 7 / 78 A method find center of gravity is 8 / 78 Thermal energy absorbed for one kelvin is 9 / 78 ____________ is the quantity to which a body is cold or hot 10 / 78 Time is used for one vibration is called 11 / 78 Field of gravitational always towards 12 / 78 Arrangement of two bodies that having different masses is 13 / 78 When a heat transfers from hot to cold parts of aa body it is known as: 14 / 78 Force that act on a running or walking: 15 / 78 Flow of current periodically reverses in its direction back and forth 16 / 78 Shortest distance between two points is known as___________ 17 / 78 Molecules remain same in _____________ motion: 18 / 78 When all the force acting on a body then the result will be: 19 / 78 The eleventh General conference on weight and measurement held in Paris in 20 / 78 The moment produce by longer arm is ___________ than smaller arm 21 / 78 Instrument that is used to measure gas pressure 22 / 78 When a body is denser than liquid, the body will _________ on the water 23 / 78 Quantity of matter possessed by body is 24 / 78 200,000 meter per second=____________ 25 / 78 Process of transferring alternating current to direct current is ______ 26 / 78 Echo is used for 27 / 78 When a body apply 20 newton force and a body having mass of 8kg then its acceleration will be: 28 / 78 When two bodies interact with each other than the contract points is known as____________ 29 / 78 State of matter in which molecules get hot by vibrating its molecules about their mean position: 30 / 78 When sound reflected to surface then it will produce: 31 / 78 Branch of physics that deals with sound________ 32 / 78 When the object is placed at focal length then the image will be 33 / 78 Force always attract to perform oscillating motion about its mean position is 34 / 78 ___________ transfer by waves 35 / 78 Random motion of what is called Brownian motion_______________ 36 / 78 By increasing pressure, the boiling point will be 37 / 78 ____________ newton’s law applied on rocket when it takes off 38 / 78 In which way we improve stability 39 / 78 Image formed by a photocopier machine: 40 / 78 Reverse process of boiling is _________ 41 / 78 Reflection from smooth surfaces is: 42 / 78 Long coil of wire that have many loops is 43 / 78 The type of motion of an athlete running in a circular track is _________ 44 / 78 Evaporation occurs at____________ 45 / 78 Express 1m³ in liters: 46 / 78 Crest and trough are made in ___________ 47 / 78 A motion of a body along a line without any rotation is 48 / 78 Relationship between velocity, frequency and wavelength is__________ 49 / 78 1 kwh=________ 50 / 78 Momentum of two interacting bodies after collision will be t 51 / 78 Liquid exert__________ 52 / 78 Which quantity does not have direction 53 / 78 Speed of sound in air is_____________ 54 / 78 The weight of a body between body and earth is equal to its: 55 / 78 When a body gives output which is equal to energy used by it then its efficiency will be 56 / 78 In thermos flask, heat is not allowed to enter or leave through 57 / 78 Which waves require medium for their propagation 58 / 78 Thermocouple is used for measuring____________ 59 / 78 When there is increase in area then the pressure will be 60 / 78 Which is used to detect small current______ 61 / 78 If we put one voltage and one ampere of current the value of resistance is _________ 62 / 78 Quantity that resist any change in its motion: 63 / 78 The internal jaws of Vernier caliper used for_____________ 64 / 78 Zero error is ______________as zero line of the Vernier scale is outside the main scale. 65 / 78 When air is moving then it will be 66 / 78 When a bus takes a sharp turn, passenger fall in which direction: 67 / 78 When a body executes SHM then the velocity at mean position is 68 / 78 When a body has 500g mass and hit earth surface with velocity of 20 ms-1 then how much its kinetic energy when the body hits to earth: 69 / 78 When an object moving with constant speed then the distance-time graph gives you a 70 / 78 A tennis ball is hit vertically upward then its acceleration will be 71 / 78 In a depth of liquid then the pressure will be __________ 72 / 78 The movement of particle in longitudinal waves is 73 / 78 The area under a speed-time graph represent _________travelled by object: 74 / 78 Distance between one rest position to extreme is ________ 75 / 78 Which one is the quantity of motion that possessed due to its mass and velocity__________ 76 / 78 Atmospheric pressure exerts in 77 / 78 The study of motion without discussing its cause is known as__________ 78 / 78 In series combination, the current passing through resistors is The average score is 30% 0% New Updates: O Level Physics Past Papers (5054) have been updated to the Latest. 3174_j21_er_12 3174_j21_qp_12 5054_s21_gt 5054_w19_ms_32 5054_s19_ci_32 5054_s19_ms_21 5054_s19_qp_11 5054_w18_ms_11 5054_s18_gt 5054_s18_ms_42 5054_w17_qp_21 5054_w17_ir_31 5054_s17_ms_42 5054_s17_ms_11 5054_w16_ir_32 5054_w16_ms_21 5054_w16_ms_41 5054_s16_ms_22 5054_s16_qp_12 5054_s16_ms_21 5054_s16_qp_21 5054_w15_ms_11 5054_w15_ms_22 5054_s15_qp_32 5054_s15_ms_11 5054_s15_qp_12 5054_s15_qp_31 5054_w14_ms_31 5054_w14_qp_42 5054_w14_qp_21 5054_s14_ms_42 5054_s14_qp_42 5054_w13_ms_41 5054_w13_er 5054_w13_ms_31 5054_w13_ir_32 5054_w13_qp_31 5054_s13_qp_11 5054_s13_qp_42 5054_s13_qp_21 5054_w12_ms_31 5054_w12_qp_12 5054_w12_ms_22 5054_w12_ir_32 5054_w12_qp_21 5054_s12_ms_41 5054_w11_ms_21 5054_w11_er 5054_s11_ms_41 5054_s11_ms_12 5054_w10_ms_31 5054_s10_ms_31 5054_w09_qp_1 5054_w09_ir_3-2 5054_s09_ms_1 5054_s09_qp_3 5054_s09_ir_3 5054_w08_ms_3 5054_w08_qp_2 5054_w08_ms_1 5054_s08_ms_1 5054_s08_ir_3 5054_w07_ir_3 5054_w07_er 5054_s07_ms_2 5054_s07_ms_1 5054_w06_ab_3 5054_s06_ms_2 5054_w05_ir_3 5054_w05_ab_3 5054_s05_qp_1 5054_s05_er 5054_w04_qp_4 5054_w04_ir_3 5054_w04_ms_3 5054_s04_qp_1 5054_s04_ms 5054_w03_ir_3 5054_w03_qp_4 5054_s03_qp_2 5054_s03_qp_1 5054_w02_ab_3

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Do we need to do a loop invariant motion after loop interchange ? Richard Biener richard.guenther@gmail.com Thu Nov 21 12:10:00 GMT 2019 ```On Thu, Nov 21, 2019 at 10:22 AM Li Jia He <helijia@linux.ibm.com> wrote: > > Hi, > > I found for the follow code: > > #define N 256 > int a[N][N][N], b[N][N][N]; > int d[N][N], c[N][N]; > void __attribute__((noinline)) > double_reduc (int n) > { > for (int k = 0; k < n; k++) > { > for (int l = 0; l < n; l++) > { > c[k][l] = 0; > for (int m = 0; m < n; m++) > c[k][l] += a[k][m][l] * d[k][m] + b[k][m][l] * d[k][m]; > } > } > } > > I dumped the file after loop interchange and got the following information: > > <bb 3> [local count: 118111600]: > # m_46 = PHI <0(7), m_45(11)> > # ivtmp_44 = PHI <_42(7), ivtmp_43(11)> > _39 = _49 + 1; > > <bb 4> [local count: 955630224]: > # l_48 = PHI <0(3), l_47(12)> > # ivtmp_41 = PHI <_39(3), ivtmp_40(12)> > c_I_I_lsm.5_18 = c[k_28][l_48]; > c_I_I_lsm.5_53 = m_46 != 0 ? c_I_I_lsm.5_18 : 0; > _2 = a[k_28][m_46][l_48]; > _3 = d[k_28][m_46]; > _4 = _2 * _3; > _5 = b[k_28][m_46][l_48]; > _6 = _3 * _5; > _7 = _4 + _6; > _8 = _7 + c_I_I_lsm.5_53; > c[k_28][l_48] = _8; > l_47 = l_48 + 1; > ivtmp_40 = ivtmp_41 - 1; > if (ivtmp_40 != 0) > goto <bb 12>; [89.00%] > else > goto <bb 5>; [11.00%] > > we can see '_3 = d[k_28][m_46];' is a loop invariant. > Do we need to add a loop invariant motion pass after the loop interchange? There is one at the end of the loop pipeline. > BR, > Lijia He > ```

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Homework Help: Finding distance through a velocity-time 1. Oct 19, 2011 Becca93 1. The problem statement, all variables and given/known data The graph attached shows the speed of a car traveling in a straight line as a function of time. The value of Vc is 4.00 m/s and the value of Vd is 7.00 m/s. Calculate the distance traveled by the car from a time of 0.60 to 6.40 seconds. 2&3. Relevant equations + Attempt at a solution I know that the distance is the area under the graph. My and from t=1s to t=6.4s, I have the total. It's from t=0.6s to t=1s that I don't understand how to get. Attached Files: • graph.PNG File size: 1.2 KB Views: 187 2. Oct 19, 2011 Delphi51 You have a straight line on the graph for the first second, so the acceleration is constant. You can find it (slope), then use an accelerated motion formula to find the velocity at time 0.6 s. Or use linear interpolation - 60% of the t=1 speed at t =0.6. That leaves you with finding the area under the trapezoid from 0.6 to 1 s. Look up the formula in wikipedia if you don't know it. Or split it into a rectangle plus a triangle and use area formulas you do know.

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t # AnytimeParticleBeliefPropagation  #### trait AnytimeParticleBeliefPropagation extends ParticleBeliefPropagation with Anytime with AnytimeInnerBPHandler Trait for Anytime PBP algorithms Ordering 1. Alphabetic 2. By Inheritance Inherited 1. AnytimeParticleBeliefPropagation 2. AnytimeInnerBPHandler 3. Anytime 4. ParticleBeliefPropagation 5. InnerBPHandler 6. FactoredAlgorithm 7. Algorithm 8. AnyRef 9. Any 1. Hide All 2. Show All Visibility 1. Public 2. All ### Type Members 1. class Runner extends Actor A class representing the actor running the algorithm. A class representing the actor running the algorithm. Definition Classes Anytime ### Abstract Value Members 1. abstract val densityEstimator The density estimator that will estimate the density of a particle. The density estimator that will estimate the density of a particle. used for resampling. Definition Classes ParticleBeliefPropagation 2. abstract val dependentAlgorithm: (Universe, List[NamedEvidence[_]]) ⇒ () ⇒ Double The algorithm to compute probability of specified evidence in a dependent universe. The algorithm to compute probability of specified evidence in a dependent universe. We use () => Double to represent this algorithm instead of an instance of ProbEvidenceAlgorithm. Typical usage is to return the result of ProbEvidenceAlgorithm.computeProbEvidence when invoked. Definition Classes ParticleBeliefPropagationFactoredAlgorithm 3. abstract val dependentUniverses: List[(Universe, List[NamedEvidence[_]])] A list of universes that depend on this universe such that evidence on those universes should be taken into account in this universe. A list of universes that depend on this universe such that evidence on those universes should be taken into account in this universe. Definition Classes ParticleBeliefPropagationFactoredAlgorithm 4. abstract def getFactors(neededElements: List[Element[_]], targetElements: List[Element[_]], upperBounds: Boolean = false): List[Factor[Double]] All implementations of factored algorithms must specify a way to get the factors from the given universe and dependent universes. All implementations of factored algorithms must specify a way to get the factors from the given universe and dependent universes. Definition Classes FactoredAlgorithm 5. abstract def handle(service: Service) A handler of services provided by the algorithm. A handler of services provided by the algorithm. Definition Classes Anytime 6. abstract val myStepTimeMillis: Long Time, in milliseconds, to run BP per step. Time, in milliseconds, to run BP per step. Definition Classes AnytimeInnerBPHandler 7. abstract val pbpSampler A particle generator to generate particles and do resampling. A particle generator to generate particles and do resampling. Definition Classes ParticleBeliefPropagation 8. abstract val semiring: DivideableSemiRing[Double] Since BP uses division to compute messages, the semiring has to have a division function defined Since BP uses division to compute messages, the semiring has to have a division function defined Definition Classes ParticleBeliefPropagationFactoredAlgorithm 9. abstract val targetElements: List[Element[_]] Target elements that should not be eliminated but should be available for querying. Target elements that should not be eliminated but should be available for querying. Definition Classes ParticleBeliefPropagation 10. abstract val universe The universe on which this belief propagation algorithm should be applied. The universe on which this belief propagation algorithm should be applied. Definition Classes ParticleBeliefPropagationFactoredAlgorithm ### Concrete Value Members 1. final def !=(arg0: Any): Boolean Definition Classes AnyRef → Any 2. final def ##(): Int Definition Classes AnyRef → Any 3. final def ==(arg0: Any): Boolean Definition Classes AnyRef → Any 4. val active: Boolean Attributes protected Definition Classes Algorithm 5. final def asInstanceOf[T0]: T0 Definition Classes Any 6. def awaitResponse(response: Future[Any], duration: Duration) Attributes protected Definition Classes Anytime 7. val BP algorithm associated with this time step. BP algorithm associated with this time step. Attributes protected[com.cra.figaro] Definition Classes InnerBPHandler 8. def cleanUp(): Unit Called when the algorithm is killed. Called when the algorithm is killed. By default, does nothing. Can be overridden. Definition Classes AnytimeParticleBeliefPropagationAlgorithm 9. def clone(): AnyRef Attributes protected[java.lang] Definition Classes AnyRef Annotations @throws( ... ) 10. def createBP(targets: List[Element[_]], dependentUniverses: List[(Universe, List[NamedEvidence[_]])], dependentAlgorithm: (Universe, List[NamedEvidence[_]]) ⇒ () ⇒ Double, depth: Int = Int.MaxValue, upperBounds: Boolean = false): Unit Instantiates the appropriate BP algorithm for the current time step. Instantiates the appropriate BP algorithm for the current time step. Attributes protected Definition Classes AnytimeInnerBPHandlerInnerBPHandler 11. val currentUniverse Universe associated with this algorithm. Universe associated with this algorithm. Attributes protected Definition Classes InnerBPHandler 12. val customConf: Config The actor running the algorithm. The actor running the algorithm. Definition Classes Anytime 13. val debug: Boolean By default, implementations that inherit this trait have no debug information. By default, implementations that inherit this trait have no debug information. Override this if you want a debugging option. Definition Classes ParticleBeliefPropagation 14. def doKill(): Unit Attributes protected[com.cra.figaro.algorithm] Definition Classes AnytimeAlgorithm 15. def doResume(): Unit Attributes protected[com.cra.figaro.algorithm] Definition Classes AnytimeAlgorithm 16. def doStart(): Unit Attributes protected[com.cra.figaro.algorithm] Definition Classes AnytimeAlgorithm 17. def doStop(): Unit Attributes protected[com.cra.figaro.algorithm] Definition Classes AnytimeAlgorithm 18. final def eq(arg0: AnyRef): Boolean Definition Classes AnyRef 19. def equals(arg0: Any): Boolean Definition Classes AnyRef → Any 20. def finalize(): Unit Attributes protected[java.lang] Definition Classes AnyRef Annotations @throws( classOf[java.lang.Throwable] ) 21. final def getClass(): Class[_] Definition Classes AnyRef → Any 22. def getNeededElements(starterElements: List[Element[_]], depth: Int, parameterized: Boolean = false): (List[Element[_]], Boolean) Get the elements that are needed by the query target variables and the evidence variables. Get the elements that are needed by the query target variables and the evidence variables. Also compute the values of those variables to the given depth. Only get factors for elements that are actually used by the target variables. This is more efficient. Also, it avoids problems when values of unused elements have not been computed. In addition to getting all the needed elements, it determines if any of the conditioned, constrained, or dependent universe parent elements has * in its range. If any of these elements has * in its range, the lower and upper bounds of factors will be different, so we need to compute both. If they don't, we don't need to compute bounds. Definition Classes FactoredAlgorithm 23. def hashCode(): Int Definition Classes AnyRef → Any 24. def initialize(): Unit Called when the algorithm is started before running any steps. Called when the algorithm is started before running any steps. By default, does nothing. Can be overridden. Definition Classes Algorithm 25. def isActive: Boolean Definition Classes Algorithm 26. final def isInstanceOf[T0]: Boolean Definition Classes Any 27. def kill(): Unit Kill the algorithm so that it is inactive. Kill the algorithm so that it is inactive. It will no longer be able to provide answers.Throws AlgorithmInactiveException if the algorithm is not active. Definition Classes Algorithm 28. implicit val messageTimeout: Timeout default message timeout. default message timeout. Increase if queries to the algorithm fail due to timeout Definition Classes Anytime 29. final def ne(arg0: AnyRef): Boolean Definition Classes AnyRef 30. final def notify(): Unit Definition Classes AnyRef 31. final def notifyAll(): Unit Definition Classes AnyRef 32. def resume(): Unit Resume the computation of the algorithm, if it has been stopped. Resume the computation of the algorithm, if it has been stopped. Throws AlgorithmInactiveException if the algorithm is not active. Definition Classes Algorithm 33. def runBP(): Unit Runs the BP algorithm at the current time step. Runs the BP algorithm at the current time step. Attributes protected Definition Classes AnytimeInnerBPHandlerInnerBPHandler 34. def runStep(): Unit Runs this particle belief propagation algorithm for one iteration. Runs this particle belief propagation algorithm for one iteration. An iteration here is one iteration of the outer loop. This means that the inner BP loop may run several iterations. Definition Classes ParticleBeliefPropagation 35. val runner: ActorRef Definition Classes Anytime 36. val running: Boolean Definition Classes Anytime 37. def shutdown: Unit Release all resources from this anytime algorithm. Release all resources from this anytime algorithm. Definition Classes Anytime 38. def start(): Unit Start the algorithm and make it active. Start the algorithm and make it active. After it returns, the algorithm must be ready to provide answers. Throws AlgorithmActiveException if the algorithm is already active. Definition Classes Algorithm 39. def starterElements: List[Element[_]] Elements towards which queries are directed. Elements towards which queries are directed. By default, these are the target elements. Definition Classes ParticleBeliefPropagation 40. def stop(): Unit Stop the algorithm from computing. Stop the algorithm from computing. The algorithm is still ready to provide answers after it returns. Throws AlgorithmInactiveException if the algorithm is not active. Definition Classes Algorithm 41. def stopUpdate(): Unit Optional function to run when the algorithm is stopped (not killed). Optional function to run when the algorithm is stopped (not killed). Used in samplers to update lazy values. Definition Classes Anytime 42. final def synchronized[T0](arg0: ⇒ T0): T0 Definition Classes AnyRef 43. val system: ActorSystem Definition Classes Anytime 44. def toString(): String Definition Classes AnyRef → Any 45. final def wait(): Unit Definition Classes AnyRef Annotations @throws( ... ) 46. final def wait(arg0: Long, arg1: Int): Unit Definition Classes AnyRef Annotations @throws( ... ) 47. final def wait(arg0: Long): Unit Definition Classes AnyRef Annotations @throws( ... )

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#### Term 3 Algebra Five Marks Questions 9th Standard Reg.No. : • • • • • • Maths Time : 01:00:00 Hrs Total Marks : 50 10 x 5 = 50 1. (Graphing made easier!) Draw the graph of the line given by the equation y = 4x – 3. 2. Draw the graph for the following (i) y = 3x - 1 (ii) $y=\left( \frac { 2 }{ 3 } \right) x+3$ 3. Use graphical method to solve the following system of equations x + y = 5; 2x – y = 4. 4. Use graphical method to solve the following system of equations 3x + 2y = 6; 6x + 4y = 8 5. Solve graphically x + y = 7; x − y = 3 6. The sum of the digits of a given two digit number is 5. If the digits are reversed, the new number is reduced by 27. Find the given number. 7. Solve, using the method of substitution (i) 2x − 3y = 7; 5x + y = 9 (ii) 1.5x + 0.1y = 6.2; 3x − 0.4y = 11.2 (iii) 10% of x + 20% of y = 24; 3x − y = 20 (iv) $\sqrt { 2 }$x - $\sqrt { 3 }$y = 1; $\sqrt { 3 }$x -$\sqrt { 8 }$y = 0 (v) $\frac { 2 }{ \sqrt { x } } +\frac { 3 }{ \sqrt { y } } =2;\frac { 4 }{ \sqrt { x } } -\frac { 9 }{ \sqrt { y } } =-1$ (Hint: Put$\frac { 1 }{ \sqrt { x } } =a;\frac { 1 }{ \sqrt { y } } =b$) 8. Solve by the method of elimination (i) 2x–y = 3; 3x + y = 7 (ii) x–y = 5; 3x + 2y = 25 (iii) $\frac { x }{ 10 } +\frac { y }{ 5 } =14;\frac { x }{ 8 } +\frac { y }{ 6 } =15$ (iv) 3(2x + y) =7xy; 3(x + 3y) = 11xy (v) $\frac { 4 }{ x } +5y=7;\frac { 3 }{ x } +4y=5$ (vi) $\frac { 3 }{ x+y } +\frac { 2 }{ x-y } =3;\frac { 2 }{ x+y } +\frac { 3 }{ x-y } =\frac { 11 }{ 3 }$ (vii) 13x +11y = 70; 11x +13y = 74 (viii) 37x + 29y = 45; 29x + 37y = 21 9. Solve 3x − 4y = 10 and 4x + 3y = 5 by the method of cross multiplication. 10. Check whether the following system of equation is consistent or inconsistent and say how many solutions we can have if it is consistent. (i) 2x – 4y = 7 x – 3y = –2 (ii) 4x + y = 3 8x + 2y = 6 (iii) 4x +7 = 2 y 2x + 9 = y

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# Can I use moments of a distribution to sample the distribution? I notice in statistics/machine learning methods, a distribution is often approximated by a Gaussian, and then that Gaussian is used for sampling. They start by computing the first two moments of the distribution, and use those to estimate $$μ\mu$$ and $$σ2\sigma^2$$. Then they can sample from that Gaussian. It seems to me the more moments I calculate, the better I ought to be able to approximate the distribution I wish to sample. What if I calculate 3 moments…how can I use those to sample from the distribution? And can this be extended to N moments? Three moments don’t determine a distributional form; if you choose a distribution-famiy with three parameters which relate to the first three population moments, you can do moment matching (“method of moments”) to estimate the three parameters and then generate values from such a distribution. There are many such distributions. Sometimes even having all the moments isn’t sufficient to determine a distribution. If the moment generating function exists (in a neighborhood of 0) then it uniquely identifies a distribution (you could in principle do an inverse Laplace transform to obtain it). [If some moments are not finite this would mean the mgf doesn’t exist, but there are also cases where all moments are finite but the mgf still doesn’t exist in a neighborhood of 0.] Given there’s a choice of distributions, one might be tempted to consider a maximum entropy solution with the constraint on the first three moments, but there’s no distribution on the real line that attains it (since the resulting cubic in the exponent will be unbounded). How the process would work for a specific choice of distribution We can simplify the process of obtaining a distribution matching three moments by ignoring the mean and variance and working with a scaled third moment — the moment-skewness ($$γ1=μ3/μ3/22\gamma_1=\mu_3/\mu_2^{3/2}$$). We can do this because having selected a distribution with the relevant skewness, we can then back out the desired mean and variance by scaling and shifting. Let’s consider an example. Yesterday I created a large data set (which still happens to be in my R session) whose distribution I haven’t tried to calculate the functional form of (it’s a large set of values of the log of the sample variance of a Cauchy at n=10). We have the first three raw moments as 1.519, 3.597 and 11.479 respectively, or correspondingly a mean of 1.518, a standard deviation* of 1.136 and a skewness of 1.429 (so these are sample values from a large sample). Formally, method of moments would attempt to match the raw moments, but the calculation is simpler if we start with the skewness (turning solving three equations in three unknowns into solving for one parameter at a time, a much simpler task). * I am going to handwave away the distinction between using an n-denominator on the variance – as would correspond to formal method of moments – and an n-1 denominator and simply use sample calculations. This skewness (~1.43) indicates we seek a distribution which is right-skew. I could choose, for example, a shifted lognormal distribution (three parameter lognormal, shape $$σ\sigma$$, scale $$μ\mu$$ and location-shift $$γ\gamma$$) with the same moments. Let’s begin by matching the skewness. The population skewness of a two parameter lognormal is: $$γ1=(eσ2+2)√eσ2−1\gamma_1=(e^{\sigma ^{2}}\!\!+2){\sqrt {e^{\sigma ^{2}}\!\!-1}}$$ So let’s start by equating that to the desired sample value to obtain an estimate of $$σ2\sigma^2$$, $$˜σ2\tilde{\sigma}^2$$, say. Note that $$γ21\gamma_1^2$$ is $$(τ+2)2(τ−1)(\tau+2)^2(\tau-1)$$ where $$τ=eσ2\tau=e^{\sigma^2}$$. This then yields a simple cubic equation $$τ3+3τ2−4=γ21\tau^3+3\tau^2-4=\gamma_1^2$$. Using the sample skewness in that equation yields $$˜τ≈1.1995\tilde{\tau}\approx 1.1995$$ or $$˜σ2≈0.1819\tilde{\sigma}^2\approx 0.1819$$. (The cubic has only one real root so there’s no issue with choosing between roots; nor is there any risk of choosing the wrong sign on $$γ1\gamma_1$$ — we can flip the distribution left-for-right if we need negative skewness) We can then in turn solve for $$μ\mu$$ by matching the variance (or standard deviation) and then for the location parameter by matching the mean. But we could as easily have chosen a shifted-gamma or a shifted-Weibull distribution (or a shifted-F or any number of other choices) and run through essentially the same process. Each of them would be different. [For the sample I was dealing with, a shifted gamma would probably have been a considerably better choice than a shifted lognormal, since the distribution of the logs of the values was left skew and the distribution of their cube root was very close to symmetric; these are consistent with what you will see with (unshifted) gamma densities, but a left-skewed density of the logs cannot be achieved with any shifted lognormal.] One could even take the skewness-kurtosis diagram in a Pearson plot and draw a line at the desired skewness and thereby obtain a two-point distribution, sequence of beta distributions, a gamma distribution, a sequence of beta-prime distributions, an inverse-gamma disribution and a sequence of Pearson type IV distributions all with the same skewness. We can see this illustrated in a skewness-kurtosis plot (Pearson plot) below (note that $$β1=γ21\beta_1=\gamma_1^2$$ and $$β2\beta_2$$ is the kurtosis), with the regions for the various Pearson-distributions marked in. The green horizontal line represents $$γ21=2.042\gamma_1^2 = 2.042$$, and we see it pass through each of the mentioned distribution-families, each point corresponding to a different population kurtosis. (The dashed curve represents the lognormal, which is not a Pearson-family distribution; its intersection with the green line marks the particular lognormal-shape we identified. Note that the dashed curve is purely a function of $$σ\sigma$$.) More moments Moments don’t pin distributions down very well, so even if you specify many moments, there will still be a lot of different distributions (particularly in relation to their extreme-tail behavior) that will match them. You can of course choose some distributional family with at least four parameters and attempt to match more than three moments; for example the Pearson distributions above allow us to match the first four moments, and there are other choices of distributions that would allow similar degree of flexibility. One can adopt other strategies to choose distributions that can match distributional features – mixture distributions, modelling the log-density using splines, and so forth. Frequently, however, if one goes back to the initial purpose for which one was trying to find a distribution, it often turns out there’s something better that can be done than the sort of strategy outlined here.

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# ch03 - Chapter 3 Supplemental Text Material S3-1 The... This preview shows pages 1–4. Sign up to view the full content. Chapter 3 Supplemental Text Material S3-1. The Definition of Factor Effects As noted in Sections 3-2 and 3-3, there are two ways to write the model for a single- factor experiment, the means model and the effects model. We will generally use the effects model y ia jn ij i ij =++ = = R S T µτ ε 12 ,, , " " where, for simplicity, we are working with the balanced case (all factor levels or treatments are replicated the same number of times). Recall that in writing this model, the i th factor level mean µ i is broken up into two components, that is τ i =+ i , where i is the i th treatment effect and is an overall mean. We usually define = = i i a a 1 and this implies that i i a = = 1 0. This is actually an arbitrary definition, and there are other ways to define the overall “mean”. For example, we could define µµ == = = ww ii i i a i a where 1 1 1 This would result in the treatment effect defined such that w i a = = 0 1 Here the overall mean is a weighted average of the individual treatment means. When there are an unequal number of observations in each treatment, the weights w i could be taken as the fractions of the treatment sample sizes n i /N . S3-2. Expected Mean Squares In Section 3-3.1 we derived the expected value of the mean square for error in the single- factor analysis of variance. We gave the result for the expected value of the mean square for treatments, but the derivation was omitted. The derivation is straightforward. Consider EM S E SS a Treatments Treatments () = F H G I K J 1 Now for a balanced design SS n y an y Treatments i i a =− = 11 22 1 .. . and the model is This preview has intentionally blurred sections. Sign up to view the full version. View Full Document y ia jn ij i ij =++ = = R S T µτ ε 12 ,, , " " In addition, we will find the following useful: EEE E En Ea n ij i ij i () () ,() ,() .. . . . . εε σ === = = = 0 222 22 2 Now ESS E n yE an y Treatments i i a ( ) ( =− = 11 1 ) . Consider the first term on the right hand side of the above expression: E n y n n i i a ii i a ( 2 1 2 1 =+ + == ∑∑ ) Squaring the expression in parentheses and taking expectation results in E n y n an n an n a i i a i i a i i a [ ( ) . 2 1 22 2 2 1 2 1 + =+ + = ] because the three cross-product terms are all zero. Now consider the second term on the right hand side of : Treatments E an y an Ean n an i i a 1 2 1 2 2 F H G I K J + = µε since Upon squaring the term in parentheses and taking expectation, we obtain τ i i a = = 1 0. E an y an an an an [( ) ] 2 F H G I K J µσ since the expected value of the cross-product is zero. Therefore, E n an y an n a an Treatments i i a i i a i i a ( ) ( ) . =+ +− + + = = = 1 1 1 1 µ στ 2 Consequently the expected value of the mean square for treatments is EM S E SS a an a n a Treatments Treatments i i a i i a () = F H G I K J = −+ + = = 1 1 1 1 22 1 2 2 1 στ σ τ This is the result given in the textbook. S3-3. Confidence Interval for σ 2 In developing the analysis of variance (ANOVA) procedure we have observed that the error variance is estimated by the error mean square; that is, 2 ± 2 = SS Na E We now give a confidence interval for . Since we have assumed that the observations are normally distributed, the distribution of 2 SS E 2 is . Therefore, χ 2 This preview has intentionally blurred sections. Sign up to view the full version. View Full Document This is the end of the preview. Sign up to access the rest of the document. ## This note was uploaded on 03/20/2011 for the course STATISTIC 101 taught by Professor Fandia during the Spring '10 term at UCLA. ### Page1 / 12 ch03 - Chapter 3 Supplemental Text Material S3-1 The... This preview shows document pages 1 - 4. Sign up to view the full document. View Full Document Ask a homework question - tutors are online

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Hudson Abstract Reasoning Ability Test (A-RAT 3R) Free Practice 2023 We rely on customer feedback to ensure our PrepPacks stay accurate and suited to match test-taker needs. Do you have questions regarding which PrepPack is best for you? Can't find the PrepPack you're looking for? Let us help! Reach out at info@jobtestprep.co.uk. ## What is The Hudson Abstract Reasoning Test The a-rat 3r includes 12 questions, with 90 seconds given to answer each one. You will be presented with a sequence of shapes known as original figures and final figures, with the original figures changing in up to 5 ways, depending on which operations they underwent. Your job will be to identify which function key leads to which action, and then point to the final figure. Unlike other abstract reasoning tests you may have encountered in the past, the a-rat 3r will push your abstract thinking to the limit by asking you to follow numerous changes in different factors at the same time. ## What Are the Function Keys? The function keys or F-keys control the operations which alter the original shape’s characteristics: color, size, position, direction, lines and more. They will appear as empty (no function) or filled (activated function) circles on a scale of five circles between the original and final figures and will be numbered according to their place (F-1 for the first in line etc.). This is what it will look like: In this example, function keys 3 (third from left) and 5 (fifth from left) are activated, while the other unfilled function keys are not The function keys are part of what makes the Hudson a-rat 3r so unique in comparison to other abstract reasoning tests, and understanding what they do will be the most challenging part of the questions. Therefore, getting acquainted with them through repeated practice can significantly impact your chances of passing the Hudson Abstract Reasoning Ability Test. ## Hudson Abstract Reasoning Test Examples, Answers, and Tips: Get a better picture of what you will be facing with the sample questions below, followed by the kind of explanations and tips you can expect when purchasing the JobTestPrep PrepPackTM - specifically and accurately designed to simulate the difficulty and pressure of the Hudson a-rat 3r as closely as possible. Hudson Abstract Reasoning Test Sample Question #1 Click below to choose the answer: Hudson Abstract Reasoning Ability Test Sample Question #2 Click below to choose the answer: ## Get Prepared with Our Hudson Abstract Reasoning Test PrepPackTM Unique tests require unique preparation, and our Hudson a-rat 3r PrepPack offers just that – not another generic abstract reasoning preparation but true-to-source, accurate simulations that will give you the closest taste of the real a-rat 3r as you can find. Designed based on years of experience and expertise, our specialized PrepPack™ offers: • 3 full-length, accurate Hudson abstract reasoning practice tests • Abstract reasoning test guide • Every question is followed by detailed explanations and tips, and our designated test expert is always standing by to answer any further questions you might have. ## What Makes the A-RAT 3R so Unique, and so Challenging? Unlike other abstract reasoning tests which use more traditional types of questions such as “Next in Series” or matrices, the Hudson Abstract Reasoning Test requires you to identify and mentally process numerous parallel changes in multiple areas. Furthermore, there is a time limit for each question as opposed to the more common method of a general time limit for the whole test. Put together, these factors can be overwhelming and stressful, and without thorough and accurate preparation they could negatively affect your score. ## The Hudson A-RAT 3R: Tips and Hacks Many people have found themselves confused and even overwhelmed by the questions they face in the Hudson a-rat 3r. Here are some useful tips you should remember when it is your turn to take this challenging test: • Look at the question! This might sound obvious, but thoroughly examining the problem in front of you will help you avoid misleading answers frequently put in place to take you off track. • Now check out the answers: once you are sure the question is clear to you, go over each answer to see if you can quickly eliminate some as a possibility. 1. Abstract reasoning tests include questions are packed with information designed to overload your brain and test your lateral thinking. Writing down the things you have already figured out on scrap paper will guarantee you don’t forget anything and allow you to move on to other parts of the question. 2. Visualizing the many possible changes to the shapes can be tricky, so drawing them can be a huge help and reduce the load on your brain. • Deal with one figure at a time: take your time to eliminate one option at a time – it is easy to get overwhelmed by all the possible answers. • Do the bare minimum: you don’t need to understand every answer. Focus on what you know and move on – understanding every aspect of each image will bog you down and likely tire you. Apart from abstract reasoning tests, Hudson also provides other tests. Check out our designated pages and practice tests for those: • Numerical reasoning tests are used by many top employers in their recruitment process, and Hudson's version is particularly challenging. In this aptitude test you are given numerical information in tables or graphs, and asked to select the correct answer. • Hudson Verbal Reasoning Test - In Hudson's verbal reasoning test you are required to read complex tests and answer challenging questions in only 90 seconds. You are also given a new text to read and understand for each question, further stretching your verbal reasoning ability. *Note that Hudson tests are sometimes administered along with a personality test Hudson and other trademarks are the property of their respective trademark holders. None of the trademark holders are affiliated with JobTestPrep or this website.

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# integral evaluation help #### lethalasian ##### New member integral e^x / sqrt(1-e^(2x)) DX The chapter I am on is strategies for integration. Would I substitute e^x = U and then go on to do (1-u^2) using trig substitution? I am stuck on this question any help would be appreciated #### pka ##### Elite Member This is rather straightforward. We get $$\displaystyle\int {\frac{1}{{\sqrt {1 - {u^2}} }}du = \arctan (u)}$$ #### Jomo ##### Elite Member You need to be complete. If u =e^x, then du =e^xdx. This yields the integral posted by pka which is supposed to be known to you. Last edited: #### topsquark ##### Full Member You need to be complete. If u =e^x, then du =e^dx. This yields the integral posted by pka which is supposed to be known to you. aka $$\displaystyle du = e^x ~ dx$$ -Dan

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# What was the day of the week on 24th October 2011? Free Practice With Testbook Mock Tests ## Options: 1. Tuesday 2. Monday 3. Saturday 4. Wednesday ### Correct Answer: Option 2 (Solution Below) This question was previously asked in SSC CPO Previous Paper 37 (Held On: 24 November 2020 Shift 2) ## Solution: No. of Odd days Simple Year Leap Year 365 + 7 = 52 weeks + 1 odd day 366 + 7 = 52 weeks + 2 odd day Years No. of odd days 100 years 5 200 years 3 300 years 1 400 years 0 Note: Multiple of 400 years i.e. 800, 1200, 1600, 2000 have 0 odd days. Calculating leap year: For Finding number of leap year (1 - 99) years, divide the number of years by 4 and the quotient will be the number of leap years. Code for weekdays: No. of odd days Day 0 Sunday 1 Monday 2 Tuesday 3 Wednesday 4 Thursday 5 Friday 6 Saturday Total odd days = 0 + 4 + 1 + 3 = 8 days = 1 week + 1 odd day. 1 is the code for "Monday". Hence, the correct answer is "Monday".

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# SJPO_Special_Round_2011_sample.pdf Description SINGAPORE JUNIOR PHYSICS OLYMPIAD 2011 SPECIAL ROUND 3 September, 2011 2:00 – 5:00 pm Time Allowed: THREE HOURS INSTRUCTIONS 1. This paper contains 11 structural questions and 9 printed pages. 2. The mark for each question is indicated at the end of the question. 3. Answer ALL the questions in the booklets provided. Answers for Questions 1 – 5 are to be written in the green booklets provided while answers to Questions 6 – 11 are to be writte Categories Published View again All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you. Related Documents Share Transcript SINGAPORE JUNIOR PHYSICS OLYMPIAD 2011SPECIAL ROUND 3 September, 20112:00 – 5:00 pmTime Allowed: THREE HOURS INSTRUCTIONS 1. This paper contains  11  structural questions and  9  printed pages.2. The mark for each question is indicated at the end of the question.3. Answer  ALL  the questions in the booklets provided. Answers for  Questions 1 –5  are to be written in the  green  booklets provided while answers to  Questions 6– 11  are to be written in the  yellow  booklets.4. For Question 5, use the loose sheet provided with the relevant figures and attachit to the green booklet.5. Graph papers are provided for Question 11. You may use up to two graph papersand attach them to the yellow booklet.6. Scientific calculators are allowed in this test.7. A table of information is given in page 2. Not all information will be used in thispaper.  TABLE OF INFORMATIONAcceleration due to gravity at Earth surface,  g  = 9 . 80 m/s 2 Universal gas constant,  R  = 8 . 31 J / (mol · K)Newton’s gravitational constant,  G  = 6 . 67 × 10 − 11 N · m 2 / kg 2 Vacuum permittivity,   0  = 8 . 85 × 10 − 12 C 2 / (N · m 2 )Vacuum permeability,  µ 0  = 4 π × 10 − 7 T · m / ASpeed of light in vacuum,  c  = 3 . 00 × 10 8 m / sSpeed of sound in air,  v  = 331 m / sCharge of electron,  e  = 1 . 60 × 10 − 19 CPlanck’s constant,  h  = 6 . 63 × 10 − 34 J · sMass of electron,  m e  = 9 . 11 × 10 − 31 kgMass of proton,  m  p  = 1 . 67 × 10 − 27 kgBoltzmann constant,  k  = 1 . 38 × 10 − 23 J/KAvogadro’s number,  N  A  = 6 . 02 × 10 23 mol − 1 Density of water,  ρ water  = 1000 kg / m 3 Standard atmosphere pressure = 1 . 01 × 10 5 Pa2  1. A body of mass 6.0 kg and density 450 kg/m 3 is dropped from rest at a height7.5 m into a lake. Calculate(a) the speed of the body just before entering the lake,(b) the acceleration of the body while it is in the lake, and(c) the maximum depth to which the body sinks before returning to float on thesurface.You may neglect the air resistance and the surface tension and viscous force of thewater. [6]2. A daredevil astronomer stands at the top of his observatory dome wearing rollerskates and starts with negligible velocity to coast down over the dome surface.(a) Neglecting friction, at what angle  φ  does he leave the dome’s surface?(b) If he were to start with an initial velocity  v 0 , at what angle  φ  would he leavethe dome?(c) For the observatory shown above, how far from the base should his assistantposition a net to break his fall, as in situation (a)? Evaluate your answer for R  = 8 . 0 m. [12]3  3. Two wooden blocks  A  and  B , connected by an unstretched spring with a springconstant  k  = 950 N/m, are initially at rest on a frictionless surface. A bullet of mass 50 g moving horizontally with a initial speed of   v 0  = 120 m/s hits Block  A and becomes embedded in it. The embedding takes place within a very short time.The mass of Block  A  is 1.2 kg and that of Block  B  is 2.0 kg.Calculate(a) the maximum compression (∆ x max ) of the spring.(b) the maximum and minimum speeds of Block  B  in its subsequent motion. [8]4. 2.00 moles of gas is held in a cylinder with a piston and is initially held at 0.300atm and has an initial volume of 0.200 m 3 . The molar heat capacity of the gas atconstant volume is 24.94 J mol − 1 K − 1 . The gas is then brought from this initialstate (State A) through the following processes:From state A to B: Gas is allowed to expand isothermally.From state B to C: The temperature of the gas drops by 100 K while it is beingheld at constant volume.From state C to A: The volume of the gas is then compressed in an adiabaticprocess back to its initial state.(a) What is the initial temperature of the gas in state A?(b) What is the ratio of the molar heat capacity at constant pressure ( C  P  ) to themolar heat capacity at constant volume ( C  V   ) of the gas?(c) What is the volume of the gas at state C? Hence, sketch a  P   − V   curvedepicting the processes, indicating the pressure and volume at each point.(d) In which of the processes is heat being transferred to the system and in whichprocess is the heat being expelled from the system? Hence, calculate the network done by the system.(e) Assume that process B to C is instead stated as “The temperature of the gasrises by 100 K while it is being held at constant volume.” Is it possible thento return the gas to its initial state via an adiabatic process? Why or whynot? [15]4 Jul 23, 2017 #### Committee Mtg. (10:20:2014) Jul 23, 2017 Search Similar documents View more... Tags Related Search

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# Vector Spaces and Subspaces A nonempty set $S$ is called a vector space if it satisfies the following conditions: (i) For any $\mathbf{x}, \mathbf{y}$ in $S, \mathbf{x}+\mathbf{y}$ is defined and is in $S$. Furthermore, \begin{aligned} \mathbf{x}+\mathbf{y} &=\mathbf{y}+\mathbf{x}, \quad \text { (commutativity) } \ \mathbf{x}+(\mathbf{y}+\mathbf{z}) &=(\mathbf{x}+\mathbf{y})+\mathbf{z} . \quad \text { (associativity) } \end{aligned} (ii) There exists an element in $S$, denoted by $\mathbf{0}$, such that $\mathbf{x}+\mathbf{0}=\mathbf{x}$ for all $\mathbf{x}$. (iii) For any $\mathbf{x}$ in $S$, there exists an element $\mathbf{y}$ in $S$ such that $\mathbf{x}+\mathbf{y}=\mathbf{0}$. (iv) For any $\mathbf{x}$ in $S$ and any real number $c, c \mathbf{x}$ is defined and is in $S ;$ moreover, $1 \mathbf{x}=\mathbf{x}$ for any $\mathbf{x}$ (v) For any $\mathbf{x}{\mathbf{1}}, \mathbf{x}{\mathbf{2}}$ in $S$ and real numbers $c_{1}, c_{2}, c_{1}\left(\mathbf{x}{1}+\mathbf{x}{\mathbf{2}}\right)=c_{1} \mathbf{x}{\mathbf{1}}+c{1} \mathbf{x}{\mathbf{2}},\left(c{1}+\right.$ $\left.c_{2}\right) \mathbf{x}{\mathbf{1}}=c{1} \mathbf{x}{\mathbf{1}}+c{2} \mathbf{x}{\mathbf{1}}$ and $c{1}\left(c_{2} \mathbf{x}{\mathbf{1}}\right)=\left(c{1} c_{2}\right) \mathbf{x}_{\mathbf{1}}$ Elements in $S$ are called vectors. If $\mathbf{x}, \mathbf{y}$ are vectors, then the operation of taking their sum $\mathbf{x}+\mathbf{y}$ is referred to as vector addition. The vector in (ii) is called the $z$ ero vector. The operation in (iv) is called scalar multiplication. A vector space may be defined with reference to any field. We have taken the field to be the field of real numbers as this will be sufficient for our purpose. The set of column vectors of order $n$ (or $n \times 1$ matrices) is a vector space. So is the set of row vectors of order $n$. These two vector spaces are the ones we consider most of the time. Let $R^{n}$ denote the set $R \times R \times \cdots \times R$, taken $n$ times, where $R$ is the set of real numbers. We will write elements of $R^{n}$ either as column vectors or as row vectors depending upon whichever is convenient in a given situation. If $S, T$ are vector spaces and $S \subset T$, then $S$ is called a subspace of $T$. Let us describe all possible subspaces of $R^{3}$. Clearly, $R^{3}$ is a vector space, and so is the space consisting of only the zero vector, i.e., the vector of all zeros. Let $c_{1}, c_{2}, c_{3}$ be real numbers. The set of all vectors $\mathbf{x} \in R^{3}$ that satisfy $$c_{1} x_{1}+c_{2} x_{2}+c_{3} x_{3}=0$$ is a subspace of $R^{3}$ (Here $x_{1}, x_{2}, x_{3}$ are the coordinates of $\left.\mathbf{x}\right)$. Geometrically, this set represents a plane passing through the origin. Intersection of two distinct planes through the origin is a straight line through the origin and is also a subspace. These are the only possible subspaces of $R^{3}$

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 Convert ftm to 厘 | fathom to Japanese rin # length units conversion ## Amount: 1 fathom (ftm) of depth of water Equals: 6,035.04 Japanese rin (厘) in length Converting fathom to Japanese rin value in the length units scale. TOGGLE :   from Japanese rin into fathoms in the other way around. ## length from fathom to Japanese rin conversion results ### Enter a new fathom number to convert * Whole numbers, decimals or fractions (ie: 6, 5.33, 17 3/8) * Precision is how many digits after decimal point (1 - 9) Enter Amount : Decimal Precision : CONVERT :   between other length measuring units - complete list. How many Japanese rin are in 1 fathom? The answer is: 1 ftm equals 6,035.04 厘 ## 6,035.04 厘 is converted to 1 of what? The Japanese rin unit number 6,035.04 厘 converts to 1 ftm, one fathom. It is the EQUAL depth of water value of 1 fathom but in the Japanese rin length unit alternative. ftm/厘 length conversion result From Symbol Equals Result Symbol 1 ftm = 6,035.04 厘 ## Conversion chart - fathoms to Japanese rin 1 fathom to Japanese rin = 6,035.04 厘 2 fathoms to Japanese rin = 12,070.08 厘 3 fathoms to Japanese rin = 18,105.12 厘 4 fathoms to Japanese rin = 24,140.16 厘 5 fathoms to Japanese rin = 30,175.20 厘 6 fathoms to Japanese rin = 36,210.24 厘 7 fathoms to Japanese rin = 42,245.28 厘 8 fathoms to Japanese rin = 48,280.32 厘 9 fathoms to Japanese rin = 54,315.36 厘 10 fathoms to Japanese rin = 60,350.40 厘 11 fathoms to Japanese rin = 66,385.44 厘 12 fathoms to Japanese rin = 72,420.48 厘 13 fathoms to Japanese rin = 78,455.52 厘 14 fathoms to Japanese rin = 84,490.56 厘 15 fathoms to Japanese rin = 90,525.60 厘 Convert length of fathom (ftm) and Japanese rin (厘) units in reverse from Japanese rin into fathoms. ## Length, Distance, Height & Depth units Distance in the metric sense is a measure between any two A to Z points. Applies to physical lengths, depths, heights or simply farness. Tool with multiple distance, depth and length measurement units. # Converter type: length units First unit: fathom (ftm) is used for measuring depth of water. Second: Japanese rin (厘) is unit of length. QUESTION: 15 ftm = ? 厘 15 ftm = 90,525.60 厘 Abbreviation, or prefix, for fathom is: ftm Abbreviation for Japanese rin is: ## Other applications for this length calculator ... With the above mentioned two-units calculating service it provides, this length converter proved to be useful also as a teaching tool: 1. in practicing fathoms and Japanese rin ( ftm vs. 厘 ) measures exchange. 2. for conversion factors between unit pairs. 3. work with length's values and properties. To link to this length fathom to Japanese rin online converter simply cut and paste the following. The link to this tool will appear as: length from fathom (ftm) to Japanese rin (厘) conversion. I've done my best to build this site for you- Please send feedback to let me know how you enjoyed visiting.

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# Simple exponential diophantine equations with huge solutions? It seems like there's been an explosion of (exponential) diophantine equations with straightforward solutions lately and it would be great to have an example at hand of how such simple equations can have dramatically more complex solutions. I'm familiar with the classical example of Pell's equation $x^2-61y^2=1$ where the minimal solution is $x=1766319049, y=226153980$, but it seems like the expressive power of exponentiation ought to enable even more dramatic examples (e.g. something like an equation in 3 or 4 terms of single-digit height with minimal solutions in the dozens of digits). Does anyone know of simple examples comparable to the above? This recent article discusses solutions to $x^3+y^3=n$ in rationals with some smallest solutions in the dozens of digits for small n. It ends with the example that $x^3+y^3=4981z^3$ in positive integers requires >16million digits for the smallest solution.

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• Orthographic projection definition is - projection of a single view of an object (such as a view of the front) onto a drawing surface in which the lines of projection are perpendicular to the drawing Time Traveler for orthographic projection. The first known use of orthographic projection was in 1668. • A worksheet and powerpoint that takes students through the 'how to' draw in 3rd angle orthographic view. It covers all the areas required in a BS formal drawing such as BS logo's, dimensions, front-side & plan views, hidden detail and a quick exam style test at the end. This worksheet provides 6 intermediate practice problems for orthographic projection drawing practice. It is intended to be used in a 2-part series along with Orthographic Projection Practice One, but could be used as a stand alone assignment. • Chapter 1 Answers Practice 1-1 1. 47, 53 2. 1.00001, 1 ... number of squares used in all six views of an orthographic projection to find the surface area. Top Bottom ... • Answer : Orthographic projections are views of a 3D object, showing 3 faces of it. The 3 drawings are aligned so that if the page were folded, it would create part of the shape. It is also called multiview projections. The 3 faces of an object consist of its plan view, front view and side view. • Dec 01, 2018 · Drawing of Orthographic Projection. For drawing Orthographic Projection, different planes are placed in a particular order. Then a specific view is drawn through every plane. A plane is placed in the following two Methods. 1. Dihedral Angle. 2. Trihedral Angle. 1. Dihedral Angle. In this method, two principal planes are kept perpendicular to ... • Answer : Orthographic projections are views of a 3D object, showing 3 faces of it. The 3 drawings are aligned so that if the page were folded, it would create part of the shape. It is also called multiview projections. The 3 faces of an object consist of its plan view, front view and side view. • Activity 1.2.3 Multiview Drawings Worksheet PLTW Engineering Activity 1.2.3 Multiview Drawings Worksheet Use this sheet to respond to items in Activity 1.2.3 Multiview Drawings. 1. Study the following multiview sketch and the orthographic projection alignment shown in Figure 4. • Recipes: orthogonal projection onto a line, orthogonal decomposition by solving a system of equations, orthogonal projection via a complicated matrix product. Pictures: orthogonal decomposition, orthogonal projection. Vocabulary words: orthogonal decomposition, orthogonal projection. Let W be a subspace of R n and let x be a vector in R n. • Ideal for Level 2 and Level 3 Engineering courses where pupils need to know how to convert between isometric projection and orthographic projection. Worksheet. pdf, 47 KB. Isometric-Conversion-Activity. • Orthographic Drawing Exercises Filetype explanation) to exercises requiring drawing orthogonal views from a given isometric pictorial view. The solutions are drawn freehand on iPad. Orthographic views drawing exercises (no narration) An example of creating an orthographic projection from an isometric view. This uses Third Angle Page 11/23 • Answer : Orthographic projections are views of a 3D object, showing 3 faces of it. The 3 drawings are aligned so that if the page were folded, it would create part of the shape. It is also called multiview projections. The 3 faces of an object consist of its plan view, front view and side view. • Orthographic projection definition, a two-dimensional graphic representation of an object in which the projecting lines are at right angles to the plane of the projection. You can assign appearances to style-based content for both orthographic and isometric views in the plan display representation. 2-dimensional (2D) plan view drawings are the ... • Orthographic Projection Answer Key. Displaying top 8 worksheets found for - Orthographic Projection Answer Key. Some of the worksheets for this concept are Orthographic drawings work answer key, Orthographic projection exercises mod answer, Orthogonal orthographic drawing, Orthographic projection exercises solutions, In which direction must the object be viewed to, Orthographic projections, Technical drawing work 1, Figure 2 1 orthographic. Q. True or false .. 1st angle of projection is a type of orthographic projection that projects the views across to the far side of the glass box. DA: 98 PA: 48 MOZ Rank: 67 Quiz & Worksheet - Orthographic Drawing | Study.com • I am very impressed with the applied graphics workbook and e book. The topics are developed in detail and the 3d graphic models are excellent for explaining and visualising the key concepts . The students enjoy completing the worksheets and being able to access the E book on their home PC's.-- Liam Sheridan from Kildare. Read More Orthographic projection is 3-D objects portrayed on a 2-D plane. The principles include; 1. top view is directly over the front view, 2. side view is either inline with top or side view, 3. lines... • Orthographic Drawing Exercises Filetype explanation) to exercises requiring drawing orthogonal views from a given isometric pictorial view. The solutions are drawn freehand on iPad. Orthographic views drawing exercises (no narration) An example of creating an orthographic projection from an isometric view. This uses Third Angle Page 11/23 • May 04, 2017 · Ideal for Level 2 and Level 3 Engineering courses where pupils need to know how to convert between isometric projection and orthographic projection. • May 2nd, 2018 - Read and Download Practice Morphology Problems With Answers Free Ebooks in PDF format ORTHOGRAPHIC PROJECTION EXERCISES MOD ANSWER BIOLOGY GUIDE HOLTZCLAW ANSWER' 'PRACTICE MORPHOLOGY PROBLEMS WITH ANSWERS ELCASH DE MAY 6TH, 2018 - READ NOW PRACTICE MORPHOLOGY PROBLEMS WITH ANSWERS PDF EBOOKS IN PDF FORMAT HUAWEI E586 USER • Drawing of Orthographic Projection. For drawing Orthographic Projection, different planes are placed in a particular order. Then a specific view is drawn through every plane. A plane is placed in the following two Methods. 1. Dihedral Angle. 2. Trihedral Angle. 1. Dihedral Angle. In this method, two principal planes are kept perpendicular to ... Orthographic Projection, Drawing: A Comprehensive Narrow your results. Clear selected filters. Level • parallel orthographic projection with planes of projection normal to object edges descriptive geometry and projection parallel orthographic projection with planes of projection plumb to one another, but not normal to all object edges descriptive geometry and projection 1 2 Descriptive Geometry and Projection • 5403400 -Orthographic Projection 5403500 -Basic Drafting Math 5403800 -AutoCAD Basic Commands 5403900 -AutoCAD Commands: Blocks&Layers 5404000 -Structural Drafting withAutoCAD-GP 5404100 -Civil Drafting withAutoCAD-GP 5404200 -HVAC/Sheet Metal DraftingAutoCAD-GP 5404300 -Electrical&ElectronicDraftingAutoCAD 5404400 -Drafting with ... • DRG. ORTHOGRAPHIC PROJECTION EXERCISE 3 EXERCISES. From drawings 1 to 18 opposite select the view which is requested in the table below. Place the number of this view in the ORTHOGRAPHIC PROJECTION Exercises mod - 11 - • projection. Parallel Projection Parallel Projection is a type of projection where the line of sight or projectors are parallel and are perpendicular to the picture planes. It is subdivided in to the following three categories: Orthographic, Oblique and Axonometric Projections. ♦ Orthographic projections: are drawn as multi view • EG Engineering Graphics Question papers, Answers ... 10. Improving our health and well-being through life sciences, nanotechnology & bio-engineering. Administration on Aging, by 2060 the population of Americans aged 65 and older will have more than doubled in size from 2011. This puts a lot of pressure on new drug creation and also on innovative • Recipes: orthogonal projection onto a line, orthogonal decomposition by solving a system of equations, orthogonal projection via a complicated matrix product. Pictures: orthogonal decomposition, orthogonal projection. Vocabulary words: orthogonal decomposition, orthogonal projection. Let W be a subspace of R n and let x be a vector in R n. • Orthographic Drawing Exercises Filetype explanation) to exercises requiring drawing orthogonal views from a given isometric pictorial view. The solutions are drawn freehand on iPad. Orthographic views drawing exercises (no narration) An example of creating an orthographic projection from an isometric view. This uses Third Angle Page 11/23 • Oct 3, 2016 - Isometric Orthographic Drawings Worksheet | Problems & Solutions techniques. The third and the first angle projections. Orthographic projection of points, lines, planes and solids. Principal and auxiliary views. Views in a given direction. Sectional views. Intersection of lines, planes and solids. Development of surfaces. Drafting practices. Dimensioning, fits and tolerancing. Computer-aided drawing and ... • Buy Engineering Graphics 2 Text and Workbook 99 edition (9781887503884) by Jerry W. Craig and Orval B. Craig for up to 90% off at Textbooks.com. • This worksheet provides 6 intermediate practice problems for orthographic projection drawing practice. It is intended to be used in a 2-part series along with Orthographic Projection Practice One, but could be used as a stand alone assignment. • Jan 21, 2010 · It is a form of parallel projection, where the view direction is orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. It is further divided into multi-view orthographic projections and axonometric pictorials . • Orthographic Projection Orthographic Drawing Isometric Sketch Prop Design Technical Drawing Visual Communication Sample Resume Exercises Study. Try these printable 3D shapes worksheets featuring exercises to recognize, compare and analyze the solid shapes and its properties. • Clear answers for common questions. Orthographic projection is a technique used in drafting or engineering drawings to depict a three-dimensional object in two dimensions. When planning an orthographic projection, a designer can view the object from different directions. • Assignment 3 Orthographic Projection- Use the graph paper provided to draw the front, top and right side of the following drawings. All titles should include your name and Period. SP 1-2 SP1-2.jpg SP 1-4 SP1-4.jpg Answer Keys: SP1-2 Solution.jpg SP 1-4 Solution.jpg The worksheets are in PDF format. You need the FREE Acrobat Reader to view and print PDF files. You can get it here. • Orthographic Projections Engineering Drawing Basics Hareesha N G Dept of Aeronautical Engg Dayananda Sagar College of Engg Bangalore-78 Pattern of planes & Pattern of views Methods of drawing Orthographic Projections Different Reference planes are FV is a view projected on VP. • is 450 inclined to Vp.Draw it’s projections. Problem 9: A circle of 50 mm diameter is resting on Hp on end A of it’s diameter AC which is 300 inclined to Hp while it makes 450 inclined to Vp. Draw it’s projections. Read problem and answer following questions 1. Surface inclined to which plane? -----HP 2. Assumption for initial position ... • In the mean time we talk concerning Orthographic Drawing Worksheets, we already collected various related images to complete your ideas. orthographic and isometric drawings worksheets, drawing orthographic projection worksheet and orthographic projection drawing exercises are some main things we will present to you based on the post title. Sapphire secure m3uSavage scope mount chartCustom taurus judge Where to buy meucci pool cues Logisim exercises Ragdoll tools Mossberg patriot magpul stock Illinois supreme court oral arguments Circuitpython bytearray 2001 dodge cummins manual transmission • Keurig model k10 reusable filter How do i fix my sony push power protector

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Credit-help.pro Credits Finances Banks How credit card charge interest Credit card interest rates are set yearly, but calculated monthly. Some credit card interest may be calculated daily and charged monthly. For example, let's say your card has an APR of 18%. If your total purchase is about \$100 with tax, it would cost you approximately an additional \$18 in interest if you choose to stretch your payments over a full year, and you do not charge additional purchases on top of your balance. And if your card "compounds" the interest (a practice of charging interest on the monthly interest accrued), the total interest will be several dollars more than the annual rate. To calculate your monthly interest charge, the bank takes the 18% APR and divides it by 12 months for the year. That comes to 1.5% of the average daily balance for the month. The average daily balance is a method of leveling out the amount you owe, which may fluctuate from day to day because of payment and purchases. The calculations to determine average daily balance sound complicated, but they're really rather simple. In effect, the bank adds together the balance on your credit card for each day of the month, and divides the total by 30, the number of days in the month. For a more complete explanation of how the average daily balance is calculated, see the chart. Yearly Rate = 18% Monthly Rate = 1.5% STATEMENT 1 (Based on a \$100 purchase on the first day of the billing cycle) Previous/beginning balance = \$100 Balance subject to finance charge = \$0 Finance charge = \$0 Payment made 25 days into cycle = \$50 STATEMENT 2 Previous/beginning balance = \$50 Balance subject to finance charge (\$100 x 25 days/30 days = \$83.33) (\$50 x 5 days/ 30 days = \$8.33) Finance charge (1.5% of \$92 = \$1.38) = \$1.38 Ending balance (\$50 + \$1.38 = \$51.38) = \$51.38 Let's say you decide to pay the \$100 charge for your dress in two monthly payments of \$50. You receive your credit card statement and see the charge listed. Approximately three weeks after you receive your statement, you mail in your payment of \$50. It arrives at your bank 25 days into your credit card cycle. You make no additional charges and next month's credit card statement arrives. You see your previous balance of \$50, an interest charge of \$1.38, a balance subject to finance charge of \$92, and an ending balance of \$51.38. Meaning the total cost of purchasing the dress, assuming you pay the ending balance in full, is \$101.38. But how did the bank arrive at \$1.38 in interest? If you had paid the \$100 charge in full by the due date on the statement, you would have paid no interest, leveraging the bank's "grace period" - generally 14 to 25 days from the date of purchase. But because you paid only \$50, interest is accrued from the date of purchase using the average daily balance method. So in calculating the average of your account's daily balance, the bank looks at the number of days carrying any given balance. Since your first \$50 payment was received 25 days into the credit card cycle, you carried a balance of \$100 for 25 days (\$100 x 25 days divided by 30 days in the month = \$83.33, the average daily balance for 25 days). If you add both average daily balances from above, you get your balance that is subject to a finance charge of \$92. Therefore, the 1.5% (the monthly interest rate) of \$92 is \$1.38, your interest charge. Source: www.mymoneyskills.pk Category: Credit

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 unit conversion lb ft3 to kg m3 # unit conversion lb ft3 to kg m3 With this online calculator you can convert lb/ft3 to kg/m3. This is a simple program for converting lb/ft3 into kg/m3.In engineering, the density this is volumetric mass density, of a substance is its mass per unit volume. Unit Converter. Select the units to convert from, enter the input value, select the units to convert to and see the result in output box.g/cm3 kg/m3 lb/ft3 lb/UK gal lb/US gal. Table 4. Unit Conversions. The S.I. and English Systems.The U.S.Customary System (English, imperial, engineering system) uses pounds ( lb) and feet (ft) or inches (in.)."Weight" given in kilograms must be converted to force: the weight of 1.0 kg mass is 9.81 N (in standard gravity). Where S is our starting value, C is our conversion factor, and E is our end converted result. To simply convert from any unit into kg/m3, for example, from 50 lb/ft3, just multiply by the value in the right column in the table below. Home » Unit Conversion Online » Convert Density » Convert 12 lb/gal to kg /m3. More information from the unit converter. Q: How many Kilograms/Cubic Meters in 1 Pounds/Gallon (US)?(oz/gal) Pounds/Cubic Inch (lb/in3) Pounds/Cubic Foot (lb/ft3) Pounds/Gallon (UK) ( lb/gal) Pounds/Gallon Convert Kilogram meter to Pound foot, kgm to lbft conversion, 1 kilogram meter 7.233 pound foot, Calculator kilogram meter to pound foot. Linear mass density unit conversion between Kilogram/millimeter and pound Note: Fill in one box to get results in the other box by clicking Tags: conversion lb ft3. 83461. 92000. Note: The above table provides approximate values based on an average 370 angle of repose and a material weight of 1600 kg/m. Unit Conversions. Loading Density Tables. -Pounds of Explosives Per Foot of Borehole. Below is a list of the VMG units available in Alph, organized by unit type. The symbols in the second column signify: S unit used if SI units selected.kg/m3. SV. lb/ft3. U. SurfaceTension. Conversion Factors Table. The following table contains unit conversions and a formula to follow in order to successfully convert.Kilograms/cubic meters to pounds/cubic foot (kg/m3 to lb/ft3). Units kg/m3 or lb/ft3. Conversion: 1 kg/m3 0.624 lb/ft3. Density Values of Different Construction Materials. The official SI symbol of kilogram per cubic meter is kgm3, or kg/m3. 1 kg/m3 is equivalent to 0.001g/cm3.Common units Kilogram/Cubic Meter (kg/m) Gram/Cubic Centimeter (g/cm) Pound/Cubic Foot ( lb/ft) Kilogram/Cubic Centimeter (kg/cm) Kilogram/Liter (kg/L) Ounce/Cubic Kilogram per cubic metre is an SI derived unit of density, defined by mass in kilograms divided by volume in cubic metres. The official SI symbolic abbreviation is kgm3, or equivalently either kg/m3 or. . Quickly convert kilograms/cubic meter into pounds/cubic foot (kg/m3 to lb /ft3) using the online calculator for metric conversions and more.Convert 7750 kg/m3 to lb/ft3 using the Unit Converter. To convert 3 lb to kg use direct conversion formula below.MTS - Meters/Second FT2 - Square Foot Per Second M2S - Square Metre Per Second (Si Un STX - Stokes (Cgs Unit) FTP - Foot-Poundal MKG - Metre Kilogram-Force NEM - Newton Metre (Si Unit) GCC - Grams/Cubic Centimeter KCC Convert density units. Easily convert kilograms per cubic metre to pounds per cubic inch, convert kg/m 3 to lb/in 3 . Many other converters available for free.probe failed because of a botched conversion in its software. Convert 23 lbm ft/min2 to kg cm/s2. The English System (imperial, engineering) uses pounds (lb, force) and feet (ft) or inches (in).Below are various tables that convert units from English to S.I. and back (e.g 1 meter 39.4 inches), as well as common conversions within each system (e.g 1 ft 12 inches). Mass per Area. lbs/ft2. kg/m2. Torque. kg/m3 to lb/ft3 Converter, Chart -- EndMemo — Concentration solution unit conversion between kilogram/m3 and pound/cubic foot, pound/cubic foot to kilogram/ m3 conversion in batch, kg/m3 lb/ft3 conversion chart. More Info "placeholder (or filler) text." This video converts g/cm3 to kg/m3 using the factor label method (dimensional analysis). This is used to convert from one cubic unit to another. It can also be adapted to for squared unit conversions. Units include MPa, GPa, pounds, kilograms, psi, grams, ounces, watts, joules, and many more.psi-ft/min psi-ft/sec psi-in reyn RMS AT/cm RMS VA/lb rods ropes (British) S/m sec Siemens/cm slug slug/ft-s spans T tex tons (long) tons (metric) tons (short) torr V/cm V/in V/mil V/mm varas W/ kg W/lb unit converter lb ft3 to kg m3? community answers. 1. kg/m3 - kilogram per cubic meter. g/cm3 - gram per cubic centimeter. oz/in3 - ounce per cubic inch.lb/ft3 - pound per cubic foot.Conversion of units of concentration - Calculator and formulas for conversion between different units of concentration: Molarity, molality, mole fraction, weight How many cubic meters in a cubic foot ? Cubic feet to cubic meters (ft3 to m3) volume units conversion factor is 0.0283168466.The abbreviation is "m3". Converter. Enter a cubic foot value to convert into cubic meters and click on the " convert" button. Density unit conversion calculator. Specific Gravity. no dimension.kilogram per cubic metre. kg m-3. gram per litre. g l-1. pound per cubic inch. lb in-3. pound per cubic foot. lb ft-3. Convert kg m to lb ft - Conversion of m mass (kg/m 3, lb/ft 3) V volume (m 3, ft 3) Density Converter. The calculator below can be used to convert between common density units: Value (use dot as Quickly convert pounds cubic foot into kilograms cubic meter lb cubic ft to kg cubic m using the online calculator for metric conversions and moreQuicklyHave you ever struggled to convert units of measure from one to another or m of a material to kg ? For example, concrete is measured as m , but . Convert lb-ft to kg-m Conversion of Measurement Units. Quickly convert pound feet into kilogram meters (lb-ft to kg-m) using the online calculator for metric conversions and more. Concentration solution unit conversion between kilogram/m3 and pound/cubic foot, pound/cubic foot to kilogram/m3 conversion in batch, kg/m3 lb/ft3 conversion chart.Kilogram/m3 Pound/cubic foot Conversion in Batch. Converter. You are currently converting density units from kilogram per cubic metre to pound per cubic feet. 1 kg/m3 0.062427973725314 lb/ft3.Conversion base : 1 lb/ft3 16.01846 kg/m3. Unit Conversion Table. Standard Prefixes. Prefix used in code Prefix for written unit Multiplier. da- dekaPounds/cubic foot to kilograms/cubic meter (lb/ft3 to kg/m3). Convert 7750 kg/m3 to lb/ft3 using the Unit Converter. Using the Periodic Table, determine the atomic weight of magnesium. What is the name of Section 4 in Perrys Chemical Engineers Handbook (Seventh Edition)? Link cubic unit factor label conversions cm3 to kg m3 []Convertir Li Pies Cubicos Kilonewton Metros Cubicos Lb Ft3 Kn M3. How To Convert Asphalt Tons To Cubic Metres Tools For Math Success. These conversions are to be maintained in Material Master under Additional Data Unit of measure tab. If the sales unit is different from base unit ofThe weight should be LB and the volume to FT3. Currently - it is converting everything since there is one item on the sales order that has KG and M3 . Convert lbs/ft3 to kg/m3 - Conversion of Measurement Convert Lb/ft3 To Kn/m3 - Pound Per Cubic Foot To Table 1 Conversions to SI Units. Multiply.L/(s m2) mL/J g g/m3 g/kg kW kW mm kPa Pa mm/m mN m. Divide.Specific Volume. ft3/lb. Concentration solution unit conversion between kilogram/m3 and pound/cubic foot, pound/cubic foot to kilogram/m3 conversion in batch, kg/m3 lb/ft3 conversion chart.Convert length unit of Lb/in3 To Kg/m3. Note: Conversion factors are given here to generally three or four significant figures. Mass 14.59 kg. slug. Force 4.448 N. lb f. Length 3.281ft. m.Standard SI unit: Newton (N). Equivalent unit: kg. m/s2. Some common unitsWORK or ENERGY: J, ft-lb, N-m, cal, btuMASS DENSITY: slug/ft3, kg/m3 From: To: kilogram/cubic meter [kg/m3] kilogram/cubic centimeter [kg/cm3] gram/cubic meterpound/cubic foot [lb/ft3] pound/cubic yard [lb/yd3] pound/gallon (US) [ lb/gal (US)] pound/gallon (UK)Discover a universal assistant for all of your unit conversion needs - download the free demo You can view more details on each measurement unit: lb/ft3 or kg/m3 The SI derived unit for density is the kilogram/cubic meter. Want other units? You can do the reverse unit conversion from kg/m3 to lb/ft3, or enter any two units below Factors for unit conversionsBULK MATERIAL BULK DENSITY PARTICLE SIZE MOISTURE CONTENT FLOWABILITY ABRASIVENESS lb/ft 3 g/cm AC Teat Dust 60 Density unit conversion calculator for conversion between common density units.pound/cubic foot [lb/ft3]. 16.018463 [kg/m3]. Online unit conversion - density.Select input unit of density: 1000 kg/m3 (kilogram / cubic meter) equals to ? Physics Physical Science Unit Conversion College Physics Conversions Density.To convert from kg/m3 to lb/in3, you must convert each unit individually. To go from kg to lbs, you must multiply the amount of kg by 2.2. Reference Information. Conversion Factors. Multiply. By. To Obtain.Constant. Value. Units. Conversion Factors and Units of Measurement Simplified Engineering for Architects and Builders, kg/m3 Mass of material lb/ft3 Distributed load on I hate unit conversion. So I used MathCad. Mathcad says: 1000N/m3 6.37lbf/ ft3. (lbf pounds of force, and distinquished from pounds mass). Related for Convert KG M3 to LB ft3.Converter Cubic meters per hour to Gallons per minute Conversion Cubic meter per hour to Gallon per minute(US) A cubic meter per hour ( m3/h) is unit of volume flow rate equal to that of a cube with sides of one meter in length exchanged or moving eac. To unit Symbol. 1 pound per cubic foot lb/ft3. 16.02. kilograms per cubic meter kg/m3.Prefix or symbol for kilogram per cubic meter is: kg/m3. Technical units conversion tool for density measures. Concentration solution unit conversion between kilogram/m3 and pound/cubic foot, pound/cubic foot to kilogram/m3 conversion in batch, kg/m3 lb/ft3 conversion chart. Convert Lb/ft3 To Kn/m3 - Pound Per

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# Research - Penning Trap At Imperial College London, we have mainly been carrying out our research using Penning traps, which are explained in this section. ## Penning Trap ### Basic principles #### Basic principles of the Penning trap The Penning trap achieves trapping of charged particles using a combination of a static quadrupole electric field and a static uniform magnetic field (for a review see [1]). The electric field is conventionally generated using a set of electrodes as shown in Figure 1. The surfaces of these electrodes are shaped to match the equipotential surfaces of a perfect quadrupole potential given by, This equation describes a potential well between the end-cap electrodes (called the axial direction), resulting in simple harmonic motion along the z-axis for positively charged particles at the axial frequency νz. However, in the x- and y-directions the potential pushes particles away from the trap centre, towards the ring electrode. In order to counter this, the Penning trap incorporates a uniform magnetic field along the z-axis. In the absence of the electric field, this magnetic field would cause the ions to move in circles in the radial plane at the cyclotron frequency 𝜈𝑐=𝑒𝐵/𝑚. The effect of the anti-confining radial electric potential is to modify this single frequency radial motion into a superposition of circular motions at two different frequencies, the modified cyclotron motion (at ν’c) and a lower frequency magnetron motion (at νm). ### Oscillation frequencies #### Oscillation frequencies in the Penning trap If the two endcap electrodes have a separation of 2z0 and the ring has an internal diameter of 2r0, the axial oscillation frequency of a single ion is given by where V is the applied potential between endcaps and ring, and e and m are the charge and mass of the ion respectively. The magnetron and modified cyclotron frequencies are given by: where the unperturbed cyclotron frequency is given by vc = eB/m. Here, B is the magnetic field strength. If more than one ion is trapped, these equations describe the oscillation frequencies of the centre of mass (COM), and there are additional degrees of freedom corresponding to internal modes of the cloud of ions, i.e. the relative motion of the ions. In particular, in equilibrium the whole ion cloud generally rotates uniformly at a frequency given by νR, which can take any value between νm and ν’c. This rotation frequency is related to the density of ions in the trap through the equation See Ref. [1] for more details. For a small number of ions that have formed an ion Coulomb crystal, there are well-defined modes of oscillation for the internal motion. ### Our Penning trap design #### The Imperial College Penning trap At Imperial College, we have worked with a number of different Penning traps. Our early traps were similar in design to the one shown in Figure 1. These traps operated in the field of a conventional electromagnet. In more recent years we have used a superconducting magnet, which has the advantage of providing a stronger and much more stable magnetic field. A diagram of our current trap is shown in Figure 2 and a photograph of it is shown in Figure 3. In order to fit with the geometry of a superconducting magnet, and to give good optical access to the trap centre (for laser cooling and detection), the electrodes have been opened out to form hollow cylinders. Although the electrode shapes are now very different from the conventional ones, by careful design it is possible to create a trapping potential which is very close to the ideal one near the centre of the trap. In our trap the magnetic field is 1.9 tesla and the internal diameter of the trap is close to 20 mm. The trap is designed to work with calcium ions, for which the oscillation frequencies are typically 100 to 400 kHz for the axial frequency, a few tens of kHz for the magnetron frequency and around 700 kHz for the modified cyclotron frequency. #### References [1] Thompson R C, “Penning traps”. Chapter in: Knoop M, Madsen N, Thompson R C (eds.) Trapped Charged Particles. World Scientific Press, 2016.‌ (Click the link to download chapter) [2] Bharadia S. Towards Laser Spectroscopy of Highly Charged Ions. 2011. (PhD thesis)

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The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A131566 Decimal expansion of (e*Pi*phi)^2. 4 1, 9, 0, 9, 2, 5, 5, 2, 3, 3, 3, 4, 4, 5, 8, 8, 2, 3, 6, 9, 0, 3, 9, 5, 2, 4, 1, 4, 2, 9, 1, 0, 9, 4, 0, 6, 0, 7, 2, 4, 4, 3, 4, 6, 4, 1, 0, 8, 7, 8, 4, 6, 9, 4, 2, 6, 5, 0, 7, 5, 7, 1, 9, 2, 0, 2, 0, 0, 2, 1, 1, 9, 2, 8, 1, 7, 1, 1, 0, 9, 4, 5, 7, 6, 5, 8, 8, 6, 1, 1, 2, 9, 9, 6, 2, 8, 9, 9, 7, 1, 0, 6, 8, 2, 7 (list; constant; graph; refs; listen; history; text; internal format) OFFSET 3,2 COMMENTS phi = (5^(1/2) + 1)/2 = (1 + sqrt(5))/2. LINKS Harry J. Smith, Table of n, a(n) for n = 3..20000 EXAMPLE 190.925523334... MATHEMATICA phi=(5^(1/2)+1)/2; RealDigits[N[(Pi*E*phi)^2, 6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *) RealDigits[(E*Pi*GoldenRatio)^2, 10, 120][[1]] (* Harvey P. Dale, May 01 2017 *) PROG (PARI) { default(realprecision, 20080); phi = (1 + sqrt(5))/2; x=(exp(1)*Pi*phi)^2/100; for (n=3, 20000, d=floor(x); x=(x-d)*10; write("b131566.txt", n, " ", d)); } \\ Harry J. Smith, Apr 27 2009 CROSSREFS Cf. A001113, A000796, A001622, A019609, A094886, A094885. Sequence in context: A248724 A278144 A198220 * A264156 A160576 A104756 Adjacent sequences:  A131563 A131564 A131565 * A131567 A131568 A131569 KEYWORD cons,nonn AUTHOR Omar E. Pol, Aug 27 2007 EXTENSIONS More terms from Harry J. Smith, Apr 26 2009 Fixed my PARI program, had -n Harry J. Smith, May 19 2009 STATUS approved Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent The OEIS Community | Maintained by The OEIS Foundation Inc. Last modified September 29 09:40 EDT 2020. Contains 337428 sequences. (Running on oeis4.)

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July 14, 2024 # How to Calculate Producer Surplus In economics, the difference between the lowest price a seller is willing to accept and the final price paid by a consumer is called producer surplus. This surplus is a measure of producer welfare. The difference between producer surplus and consumer surplus is equal to the producer’s total benefit. This surplus is often expressed as an increase in production. ## Producer surplus is the difference between the minimum price a seller is willing to accept and the final price a consumer pays The difference between the minimum price a seller will accept and the final price a consumer pays is known as the producer surplus. This surplus is created by price inelasticity, which causes the price gap between the buyer and seller to grow larger. For example, if a college student buys a pair of sneakers for \$60, and the seller’s minimum price is \$100, the student will feel like they are getting a great deal if they buy the \$60 pair. That’s a \$20 difference in the buyer’s surplus. Similarly, if a person agrees to sell their time for a salary, they would receive a producer surplus of four dollars for each sale. This amount of money would be equivalent to about \$400 if the entire batch of items was sold. A graph showing this relationship can help a student understand the concept of producer surplus. It is possible to visualize the surplus by plotting the sales price and the number of units sold. The total sales revenue is represented by the rectangle. The supply curve bisects the rectangle into two triangles: a lower triangle represents marginal costs and a triangle above represents the producer surplus. The top triangle represents \$5, while the lower triangle represents marginal costs. ## It is a measure of producer welfare Calculating producer surplus is an important economic measure. It is the amount of output a firm produces in relation to the price it charges. Most producers aim to charge the highest price a consumer is willing to pay for a product. For example, a producer selling popular fizzy drinks may charge \$1.20 per can, generating a greater producer surplus than the producer selling a similar product at a lower price. Producer surplus is a measure of the welfare of a group of firms. It is calculated by calculating the difference between the actual price a firm receives and the lowest price it is willing to sell for. It is also a measure of how well a producer can survive in the current economic climate. Calculating producer surplus is a useful economic tool for assessing the well-being of producers in an economy. However, the term is misleading because it is impossible to obtain a constant surplus, as market prices fluctuate all the time. ## It is a measure of overall economic surplus A producer’s surplus is the amount of money that the producer makes in excess of the price it charges. This amount can be calculated using a graph or an equation. For example, if a producer sells a toy for \$7 and receives \$3500, the surplus is \$1000. In most cases, producers will charge the maximum amount that a consumer is willing to pay for a product. This is the price the producer gets from selling their time to sell. The producer’s surplus is equal to the overall economic surplus. As a result, the producer gets a raise. Basically, producer surplus is the difference between the cost of production and the price the producer actually receives. This surplus is a positive indicator for a country’s economic health. It indicates that the price that a producer would charge to sell a product is higher than what it costs to produce that product. ## It is equal to consumer surplus When the amount of production exceeds the total demand, the total surplus is called the producer surplus. This surplus is identified on a supply and demand graph as a triangle below equilibrium. The surplus is generated by firms, who will have a lower price than the equilibrium price for their products. The higher the price, the more the producer will be able to earn. The total surplus does not account for externalities, such as pollution, which is produced during the production of most goods. These externalities are not accounted for in the cost of production and therefore can reduce the consumer surplus. In the ideal market, the cost of producing a good is equal to the benefit it produces. A similar model can be used to calculate the total economic welfare. In this model, the producer surplus is calculated by multiplying the difference between the price of the good produced by the producer and the price of the product on the market. In addition to the consumer surplus, the producer surplus also includes the seller’s cost of labor, materials, time, and profits.

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# conjugate gradient for Newton's method with non positive definite Hessian matrix I want to minimize a non-linear function $f(x)$ using Newton's method. At each optimization step, I compute a descent direction $d$ to update $x$ using a second-order approximation of $f(x)$: $$\nabla^2f(x) \ d = -\nabla f(x)$$ So I need to solve a linear system, but the Hessian matrix $\nabla^2f(x)$ might not be positive definite which could give me an invalid descent direction. I want to run conjugate gradient to solve the linear system. In the conjugate gradient method, one of the steps involves this operation $v^T \nabla^2\ f(x)\ v$ (computing $\alpha$ according to Shewchuk's notes, page 50 https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf , I'm using $v$ rather than $d$ because in my convention $d$ is the descent direction to update $x$). I could recycle this operation to know if the Hessian is not positive definite (if such operation is negative). I implemented my algorithm like that, so as soon as I detect that operation is negative, I stop the CG solver and return the solution iterated up to that point. I've actually seen it works pretty well in practice, but I have no rigorous justification for doing it. I remember hearing / reading people saying that each CG iteration takes care of different eigenvalues in order of "importance", so by stopping early you end up exploring the positive definite part of the Hessian. Can anybody point me to a more rigorous justification of this? Is this a valid way to enforce positive definiteness using CG for non-linear optimization? • There's no rigorous justification of your approach in the sense that you cannot tell how far you are from the solution of the linear system when negative curvature occurs. But if you combine this with a trust region globalization framework, you can in fact prove convergence -- this is known as truncated Newton-CG: Steihaug, Trond, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal. 20, 626-637 (1983). ZBL0518.65042. – Christian Clason Jul 17 '18 at 6:16 • You can also prove convergence of a linesearch scheme very closely related to what the OP suggests: link.springer.com/article/10.1007/BF02592055 and fast local convergence: epubs.siam.org/doi/abs/10.1137/0719025. In the case of the trust-region method, an additional result due to Yuan shows that in the convex case, the solution identified by truncated CG is at least half as good as the global minimizer of the trust-region subproblem: link.springer.com/article/10.1007%2Fs101070050012 (Theorem 2). – Dominique Aug 7 '18 at 1:51 • I should have mentioned that I'm also doing line search in my implementation. I just approximately solve the linear system with a few CG iterations, but I still have a safeguard with a backtracking line search to make sure I make progress using that descent direction. I'm not very familiar with trust region optimization, but I think line search and trust region are two valid ways to achieve similar results. Or is there any reason I should prefer trust region over line search in this case? – yewang Aug 19 '18 at 0:50

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Iteration, Induction, and Recursion Save this PDF as: Size: px Start display at page: Transcription 1 CHAPTER 2 Iteration, Induction, and Recursion The power of computers comes from their ability to execute the same task, or different versions of the same task, repeatedly. In computing, the theme of iteration is met in a number of guises. Many concepts in data models, such as lists, are forms of repetition, as A list either is empty or is one element followed by another, then another, and so on. Programs and algorithms use iteration to perform repetitive jobs without requiring a large number of similar steps to be specified individually, as Do the next step 1000 times. Programming languages use looping constructs, like the while- and for-statements of C, to implement iterative algorithms. Closely related to repetition is recursion, a technique in which a concept is defined, directly or indirectly, in terms of itself. For example, we could have defined a list by saying A list either is empty or is an element followed by a list. Recursion is supported by many programming languages. In C, a function F can call itself, either directly from within the body of F itself, or indirectly by calling some other function, which calls another, and another, and so on, until finally some function in the sequence calls F. Another important idea, induction, is closely related to recursion and is used in many mathematical proofs. Iteration, induction, and recursion are fundamental concepts that appear in many forms in data models, data structures, and algorithms. The following list gives some examples of uses of these concepts; each will be covered in some detail in this book. 1. Iterative techniques. The simplest way to perform a sequence of operations repeatedly is to use an iterative construct such as the for-statement of C. 2. Recursive programming. C and many other languages permit recursive functions, which call themselves either directly or indirectly. Often, beginning programmers are more secure writing iterative programs than recursive ones, but an important goal of this book is to accustom the reader to thinking and programming recursively, when appropriate. Recursive programs can be simpler to write, analyze, and understand. 25 2 26 ITERATION, INDUCTION, AND RECURSION Notation: The Summation and Product Symbols An oversized Greek capital letter sigma is often used to denote a summation, as in n i=1 i. This particular expression represents the sum of the integers from 1 to n; that is, it stands for the sum n. More generally, we can sum any function f(i) of the summation index i. (Of course, the index could be some symbol other than i.) The expression b i=a f(i) stands for f(a) + f(a + 1) + f(a + 2) + + f(b) For example, m j=2 j2 stands for the sum m 2. Here, the function f is squaring, and we used index j instead of i. As a special case, if b < a, then there are no terms in the sum b i=a f(i), and the value of the expression, by convention, is taken to be 0. If b = a, then there is exactly one term, that for i = a. Thus, the value of the sum a i=a f(i) is just f(a). The analogous notation for products uses an oversized capital pi. The expression b i=a f(i) stands for the product f(a) f(a + 1) f(a + 2) f(b); if b < a, the product is taken to be 1. Basis Inductive step 3. Proofs by induction. An important technique for showing that a statement is true is proof by induction. We shall cover inductive proofs extensively, starting in Section 2.3. The following is the simplest form of an inductive proof. We begin with a statement S(n) involving a variable n; we wish to prove that S(n) is true. We prove S(n) by first proving a basis, that is, the statement S(n) for a particular value of n. For example, we could let n = 0 and prove the statement S(0). Second, we must prove an inductive step, in which we prove that the statement S, for one value of its argument, follows from the same statement S for the previous values of its argument; that is, S(n) implies S(n + 1) for all n 0. For example, S(n) might be the familiar summation formula n i = n(n + 1)/2 (2.1) i=1 which says that the sum of the integers from 1 to n equals n(n+1)/2. The basis could be S(1) that is, Equation (2.1) with 1 in place of n which is just the equality 1 = 1 2/2. The inductive step is to show that n i=1 i = n(n + 1)/2 implies that n+1 i=1 i = (n + 1)(n + 2)/2; the former is S(n), which is Equation (2.1) itself, while the latter is S(n + 1), which is Equation (2.1) with n + 1 replacing n everywhere n appears. Section 2.3 will show us how to construct proofs such as this. 4. Proofs of program correctness. In computer science, we often wish to prove, formally or informally, that a statement S(n) about a program is true. The statement S(n) might, for example, describe what is true on the nth iteration of some loop or what is true for the nth recursive call to some function. Proofs of this sort are generally inductive proofs. 5. Inductive definitions. Many important concepts of computer science, especially those involving data models, are best defined by an induction in which we give 3 SEC. 2.2 ITERATION 27 a basis rule defining the simplest example or examples of the concept, and an inductive rule or rules, where we build larger instances of the concept from smaller ones. For instance, we noted that a list can be defined by a basis rule (an empty list is a list) together with an inductive rule (an element followed by a list is also a list). 6. Analysis of running time. An important criterion for the goodness of an algorithm is how long it takes to run on inputs of various sizes (its running time ). When the algorithm involves recursion, we use a formula called a recurrence equation, which is an inductive definition that predicts how long the algorithm takes to run on inputs of different sizes. Each of these subjects, except the last, is introduced in this chapter; the running time of a program is the topic of Chapter What This Chapter Is About In this chapter we meet the following major concepts. Iterative programming (Section 2.2) Inductive proofs (Sections 2.3 and 2.4) Inductive definitions (Section 2.6) Recursive programming (Sections 2.7 and 2.8) Proving the correctness of a program (Sections 2.5 and 2.9) In addition, we spotlight, through examples of these concepts, several interesting and important ideas from computer science. Among these are Sorting algorithms, including selection sort (Section 2.2) and merge sort (Section 2.8) Parity checking and detection of errors in data (Section 2.3) Arithmetic expressions and their transformation using algebraic laws (Sections 2.4 and 2.6) Balanced parentheses (Section 2.6) 2.2 Iteration Each beginning programmer learns to use iteration, employing some kind of looping construct such as the for- or while-statement of C. In this section, we present an example of an iterative algorithm, called selection sort. In Section 2.5 we shall prove by induction that this algorithm does indeed sort, and we shall analyze its running time in Section 3.6. In Section 2.8, we shall show how recursion can help us devise a more efficient sorting algorithm using a technique called divide and conquer. 4 28 ITERATION, INDUCTION, AND RECURSION Common Themes: Self-Definition and Basis-Induction As you study this chapter, you should be alert to two themes that run through the various concepts. The first is self-definition, in which a concept is defined, or built, in terms of itself. For example, we mentioned that a list can be defined as being empty or as being an element followed by a list. The second theme is basis-induction. Recursive functions usually have some sort of test for a basis case where no recursive calls are made and an inductive case where one or more recursive calls are made. Inductive proofs are well known to consist of a basis and an inductive step, as do inductive definitions. This basisinduction pairing is so important that these words are highlighted in the text to introduce each occurrence of a basis case or an inductive step. There is no paradox or circularity involved in properly used self-definition, because the self-defined subparts are always smaller than the object being defined. Further, after a finite number of steps to smaller parts, we arrive at the basis case, at which the self-definition ends. For example, a list L is built from an element and a list that is one element shorter than L. When we reach a list with zero elements, we have the basis case of the definition of a list: The empty list is a list. As another example, if a recursive function works, the arguments of the call must, in some sense, be smaller than the arguments of the calling copy of the function. Moreover, after some number of recursive calls, we must get to arguments that are so small that the function does not make any more recursive calls. Sorting To sort a list of n elements we need to permute the elements of the list so that they appear in nondecreasing order. Example 2.1. Suppose we are given the list of integers (3, 1, 4, 1, 5, 9, 2, 6, 5). We sort this list by permuting it into the sequence (1, 1, 2, 3, 4, 5, 5, 6, 9). Note that sorting not only orders the values so that each is either less than or equal to the one that follows, but it also preserves the number of occurrences of each value. Thus, the sorted list has two 1 s, two 5 s, and one each of the numbers that appear once in the original list. Sorted list We can sort a list of elements of any type as long as the elements have a lessthan order defined on them, which we usually represent by the symbol <. For example, if the values are real numbers or integers, then the symbol < stands for the usual less-than relation on reals or integers, and if the values are character strings, we would use the lexicographic order on strings. (See the box on Lexicographic Order. ) Sometimes when the elements are complex, such as structures, we might use only a part of each element, such as one particular field, for the comparison. The comparison a b means, as always, that either a < b or a and b are the same value. A list (a 1, a 2,...,a n ) is said to be sorted if a 1 a 2 a n ; that is, if the values are in nondecreasing order. Sorting is the operation of taking an arbitrary list (a 1, a 2,...,a n ) and producing a list (b 1, b 2,...,b n ) such that 5 SEC. 2.2 ITERATION 29 Proper prefix Empty string Lexicographic Order The usual way in which two character strings are compared is according to their lexicographic order. Let c 1 c 2 c k and d 1 d 2 d m be two strings, where each of the c s and d s represents a single character. The lengths of the strings, k and m, need not be the same. We assume that there is a < ordering on characters; for example, in C characters are small integers, so character constants and variables can be used as integers in arithmetic expressions. Thus we can use the conventional < relation on integers to tell which of two characters is less than the other. This ordering includes the natural notion that lower-case letters appearing earlier in the alphabet are less than lower-case letters appearing later in the alphabet, and the same holds for upper-case letters. We may then define the ordering on character strings called the lexicographic, dictionary, or alphabetic ordering, as follows. We say c 1 c 2 c k < d 1 d 2 d m if either of the following holds: 1. The first string is a proper prefix of the second, which means that k < m and for i = 1, 2,...,k we have c i = d i. According to this rule, bat < batter. As a special case of this rule, we could have k = 0, in which case the first string has no characters in it. We shall use ǫ, the Greek letter epsilon, to denote the empty string, the string with zero characters. When k = 0, rule (1) says that ǫ < s for any nonempty string s. 2. For some value of i > 0, the first i 1 characters of the two strings agree, but the ith character of the first string is less than the ith character of the second string. That is, c j = d j for j = 1, 2,...,i 1, and c i < d i. According to this rule, ball < base, because the two words first differ at position 3, and at that position ball has an l, which precedes the character s found in the third position of base. 1. List (b 1, b 2,...,b n ) is sorted. Permutation 2. List (b 1, b 2,...,b n ) is a permutation of the original list. That is, each value appears in list (a 1, a 2,...,a n ) exactly as many times as that value appears in list (b 1, b 2,...,b n ). A sorting algorithm takes as input an arbitrary list and produces as output a sorted list that is a permutation of the input. Example 2.2. Consider the list of words base, ball, mound, bat, glove, batter Given this input, and using lexicographic order, a sorting algorithm would produce this output: ball, base, bat, batter, glove, mound. Selection Sort: An Iterative Sorting Algorithm Suppose we have an array A of n integers that we wish to sort into nondecreasing 6 30 ITERATION, INDUCTION, AND RECURSION Convention Regarding Names and Values We can think of a variable as a box with a name and a value. When we refer to a variable, such as abc, we use the constant-width, or computer font for its name, as we did in this sentence. When we refer to the value of the variable abc, we shall use italics, as abc. To summarize, abc refers to the name of the box, and abc to its contents. order. We may do so by iterating a step in which a smallest element 1 not yet part of the sorted portion of the array is found and exchanged with the element in the first position of the unsorted part of the array. In the first iteration, we find ( select ) a smallest element among the values found in the full array A[0..n-1] and exchange it with A[0]. 2 In the second iteration, we find a smallest element in A[1..n-1] and exchange it with A[1]. We continue these iterations. At the start of the i + 1st iteration, A[0..i-1] contains the i smallest elements in A sorted in nondecreasing order, and the remaining elements of the array are in no particular order. A picture of A just before the i + 1st iteration is shown in Fig sorted unsorted 0 i n 1 Fig Picture of array just before the i + 1st iteration of selection sort. In the i + 1st iteration, we find a smallest element in A[i..n-1] and exchange it with A[i]. Thus, after the i + 1st iteration, A[0..i] contains the i + 1 smallest elements sorted in nondecreasing order. After the (n 1)st iteration, the entire array is sorted. A C function for selection sort is shown in Fig This function, whose name is SelectionSort, takes an array A as the first argument. The second argument, n, is the length of array A. Lines (2) through (5) select a smallest element in the unsorted part of the array, A[i..n-1]. We begin by setting the value of index small to i in line (2). The forloop of lines (3) through (5) consider all higher indexes j in turn, and small is set to j if A[j] has a smaller value than any of the array elements in the rangea[i..j-1]. As a result, we set the variable small to the index of the first occurrence of the smallest element in A[i..n-1]. After choosing a value for the index small, we exchange the element in that position with the element in A[i], in lines (6) to (8). If small = i, the exchange is performed, but has no effect on the array. Notice that in order to swap two elements, we need a temporary place to store one of them. Thus, we move the value 1 We say a smallest element rather than the smallest element because there may be several occurrences of the smallest value. If so, we shall be happy with any of those occurrences. 2 To describe a range of elements within an array, we adopt a convention from the language Pascal. If A is an array, then A[i..j] denotes those elements of A with indexes from i to j, inclusive. 7 SEC. 2.2 ITERATION 31 void SelectionSort(int A[], int n) { int i, j, small, temp; (1) for (i = 0; i < n-1; i++) { /* set small to the index of the first occur- */ /* rence of the smallest element remaining */ (2) small = i; (3) for (j = i+1; j < n; j++) (4) if (A[j] < A[small]) (5) small = j; /* when we reach here, small is the index of */ /* the first smallest element in A[i..n-1]; */ /* we now exchange A[small] with A[i] */ (6) temp = A[small]; (7) A[small] = A[i]; (8) A[i] = temp; } } Fig Iterative selection sort. in A[small] to temp at line (6), move the value in A[i] to A[small] at line (7), and finally move the value originally in A[small] from temp to A[i] at line (8). Example 2.3. Let us study the behavior of SelectionSort on various inputs. First, let us look at what happens when we run SelectionSort on an array with no elements. When n = 0, the body of the for-loop of line (1) is not executed, so SelectionSort does nothing gracefully. Now let us consider the case in which the array has only one element. Again, the body of the for-loop of line (1) is not executed. That response is satisfactory, because an array consisting of a single element is always sorted. The cases in which n is 0 or 1 are important boundary conditions, on which it is important to check the performance of any algorithm or program. Finally, let us run SelectionSort on a small array with four elements, where A[0] through A[3] are A We begin the outer loop with i = 0, and at line (2) we set small to 0. Lines (3) to (5) form an inner loop, in which j is set to 1, 2, and 3, in turn. With j = 1, the test of line (4) succeeds, since A[1], which is 30, is less than A[small], which is A[0], or 40. Thus, we set small to 1 at line (5). At the second iteration of lines (3) to (5), with j = 2, the test of line (4) again succeeds, since A[2] < A[1], and so we set small to 2 at line (5). At the last iteration of lines (3) to (5), with j = 3, the test of line (4) succeeds, since A[3] < A[2], and we set small to 3 at line (5). We now fall out of the inner loop to line (6). We set temp to 10, which is A[small], then A[3] to A[0], or 40, at line (7), and A[0] to 10 at line (8). Now, the 8 32 ITERATION, INDUCTION, AND RECURSION Sorting on Keys When we sort, we apply a comparison operation to the values being sorted. Often the comparison is made only on specific parts of the values and the part used in the comparison is called the key. For example, a course roster might be an array A of C structures of the form struct STUDENT { int studentid; char *name; char grade; } A[MAX]; We might want to sort by student ID, or name, or grade; each in turn would be the key. For example, if we wish to sort structures by student ID, we would use the comparison A[j].studentID < A[small].studentID at line (4) of SelectionSort. The type of array A and temporary temp used in the swap would be struct STUDENT, rather than integer. Note that entire structures are swapped, not just the key fields. Since it is time-consuming to swap whole structures, a more efficient approach is to use a second array of pointers to STUDENT structures and sort only the pointers in the second array. The structures themselves remain stationary in the first array. We leave this version of selection sort as an exercise. first iteration of the outer loop is complete, and array A appears as A The second iteration of the outer loop, with i = 1, sets small to 1 at line (2). The inner loop sets j to 2 initially, and since A[2] < A[1], line (5) sets small to 2. With j = 3, the test of line (4) fails, since A[3] A[2]. Hence, small = 2 when we reach line (6). Lines (6) to (8) swap A[1] with A[2], leaving the array A Although the array now happens to be sorted, we still iterate the outer loop once more, with i = 2. We set small to 2 at line (2), and the inner loop is executed only with j = 3. Since the test of line (4) fails, small remains 2, and at lines (6) through (8), we swap A[2] with itself. The reader should check that the swapping has no effect when small = i. Figure 2.3 shows how the function SelectionSort can be used in a complete program to sort a sequence of n integers, provided that n 100. Line (1) reads and stores n integers in an array A. If the number of inputs exceeds MAX, only the first MAX integers are put into A. A message warning the user that the number of inputs is too large would be useful here, but we omit it. 9 SEC. 2.2 ITERATION 33 Line (3) calls SelectionSort to sort the array. Lines (4) and (5) print the integers in sorted order. #include <stdio.h> #define MAX 100 int A[MAX]; void SelectionSort(int A[], int n); main() { int i, n; /* read and store input in A */ (1) for (n = 0; n < MAX && scanf("%d", &A[n])!= EOF; n++) (2) ; (3) SelectionSort(A,n); /* sort A */ (4) for (i = 0; i < n; i++) (5) printf("%d\n", A[i]); /* print A */ } void SelectionSort(int A[], int n) { int i, j, small, temp; for (i = 0; i < n-1; i++) { small = i; for (j = i+1; j < n; j++) if (A[j] < A[small]) small = j; temp = A[small]; A[small] = A[i]; A[i] = temp; } } Fig A sorting program using selection sort. EXERCISES 2.2.1: Simulate the function SelectionSort on an array containing the elements a) 6, 8, 14, 17, 23 b) 17, 23, 14, 6, 8 c) 23, 17, 14, 8, 6 How many comparisons and swaps of elements are made in each case? 2.2.2**: What are the minimum and maximum number of (a) comparisons and (b) swaps that SelectionSort can make in sorting a sequence of n elements? 10 34 ITERATION, INDUCTION, AND RECURSION 2.2.3: Write a C function that takes two linked lists of characters as arguments and returns TRUE if the first string precedes the second in lexicographic order. Hint: Implement the algorithm for comparing character strings that was described in this section. Use recursion by having the function call itself on the tails of the character strings when it finds that the first characters of both strings are the same. Alternatively, one can develop an iterative algorithm to do the same *: Modify your program from Exercise to ignore the case of letters in comparisons : What does selection sort do if all elements are the same? 2.2.6: Modify Fig. 2.3 to perform selection sort when array elements are not integers, but rather structures of type struct STUDENT, as defined in the box Sorting on Keys. Suppose that the key field is studentid *: Further modify Fig. 2.3 so that it sorts elements of an arbitrary type T. You may assume, however, that there is a function key that takes an element of type T as argument and returns the key for that element, of some arbitrary type K. Also assume that there is a function lt that takes two elements of type K as arguments and returns TRUE if the first is less than the second, and FALSE otherwise : Instead of using integer indexes into the array A, we could use pointers to integers to indicate positions in the array. Rewrite the selection sort algorithm of Fig. 2.3 using pointers *: As mentioned in the box on Sorting on Keys, if the elements to be sorted are large structures such as type STUDENT, it makes sense to leave them stationary in an array and sort pointers to these structures, found in a second array. Write this variation of selection sort : Write an iterative program to print the distinct elements of an integer array : Use the and notations described at the beginning of this chapter to express the following. a) The sum of the odd integers from 1 to 377 b) The sum of the squares of the even integers from 2 to n (assume that n is even) c) The product of the powers of 2 from 8 to 2 k : Show that when small = i, lines (6) through (8) of Fig. 2.2 (the swapping steps) do not have any effect on array A. 2.3 Inductive Proofs Mathematical induction is a useful technique for proving that a statement S(n) is true for all nonnegative integers n, or, more generally, for all integers at or above some lower limit. For example, in the introduction to this chapter we suggested that the statement n i=1 i = n(n + 1)/2 can be proved true for all n 1 by an induction on n. Now, let S(n) be some arbitrary statement about an integer n. In the simplest form of an inductive proof of the statement S(n), we prove two facts: 11 SEC. 2.3 INDUCTIVE PROOFS 35 Naming the Induction Parameter It is often useful to explain an induction by giving the intuitive meaning of the variable n in the statement S(n) that we are proving. If n has no special meaning, as in Example 2.4, we simply say The proof is by induction on n. In other cases, n may have a physical meaning, as in Example 2.6, where n is the number of bits in the code words. There we can say, The proof is by induction on the number of bits in the code words. Inductive hypothesis 1. The basis case, which is frequently taken to be S(0). However, the basis can be S(k) for any integer k, with the understanding that then the statement S(n) is proved only for n k. 2. The inductive step, where we prove that for all n 0 [or for all n k, if the basis is S(k)], S(n) implies S(n + 1). In this part of the proof, we assume that the statement S(n) is true. S(n) is called the inductive hypothesis, and assuming it to be true, we must then prove that S(n + 1) is true. S(n) S(2) S(1) S(0) Fig In an inductive proof, each instance of the statement S(n) is proved using the statement for the next lower value of n. Figure 2.4 illustrates an induction starting at 0. For each integer n, there is a statement S(n) to prove. The proof for S(1) uses S(0), the proof for S(2) uses S(1), and so on, as represented by the arrows. The way each statement depends on the previous one is uniform. That is, by one proof of the inductive step, we prove each of the steps implied by the arrows in Fig Example 2.4. As an example of mathematical induction, let us prove STATEMENT S(n): n 2 i = 2 n+1 1 for any n 0. i=0 That is, the sum of the powers of 2, from the 0th power to the nth power, is 1 less than the (n + 1)st power of 2. 3 For example, = The proof proceeds as follows. BASIS. To prove the basis, we substitute 0 for n in the equation S(n). Then S(n) becomes 3 S(n) can be proved without induction, using the formula for the sum of a geometric series. However, it will serve as a simple example of the technique of mathematical induction. Further, the proofs of the formulas for the sum of a geometric or arithmetic series that you have probably seen in high school are rather informal, and strictly speaking, mathematical induction should be used to prove those formulas. 12 36 ITERATION, INDUCTION, AND RECURSION 0 2 i = (2.2) i=0 There is only one term, for i = 0, in the summation on the left side of Equation (2.2), so that the left side of (2.2) sums to 2 0, or 1. The right side of Equation (2.2), which is 2 1 1, or 2 1, also has value 1. Thus we have proved the basis of S(n); that is, we have shown that this equality is true for n = 0. INDUCTION. Now we must prove the inductive step. We assume that S(n) is true, and we prove the same equality with n + 1 substituted for n. The equation to be proved, S(n + 1), is n+1 2 i = 2 n+2 1 (2.3) i=0 To prove Equation (2.3), we begin by considering the sum on the left side, n+1 2 i i=0 This sum is almost the same as the sum on the left side of S(n), which is n i=0 2 i except that (2.3) also has a term for i = n + 1, that is, the term 2 n+1. Since we are allowed to assume that the inductive hypothesis S(n) is true in our proof of Equation (2.3), we should contrive to use S(n) to advantage. We do so by breaking the sum in (2.3) into two parts, one of which is the sum in S(n). That is, we separate out the last term, where i = n + 1, and write n+1 2 i = i=0 n 2 i + 2 n+1 (2.4) i=0 Now we can make use of S(n) by substituting its right side, 2 n+1 1, for n i=0 2i in Equation (2.4): n+1 2 i = 2 n n+1 (2.5) i=0 When we simplify the right side of Equation (2.5), it becomes 2 2 n+1 1, or 2 n+2 1. Now we see that the summation on the left side of (2.5) is the same as the left side of (2.3), and the right side of (2.5) is equal to the right side of (2.3). We have thus proved the validity of Equation (2.3) by using the equality S(n); that proof is the inductive step. The conclusion we draw is that S(n) holds for every nonnegative value of n. Why Does Proof by Induction Work? In an inductive proof, we first prove that S(0) is true. Next we show that if S(n) is true, then S(n + 1) holds. But why can we then conclude that S(n) is true for all n 0? We shall offer two proofs. A mathematician would point out that 13 SEC. 2.3 INDUCTIVE PROOFS 37 Substituting for Variables People are often confused when they have to substitute for a variable such as n in S(n), an expression involving the same variable. For example, we substituted n + 1 for n in S(n) to get Equation (2.3). To make the substitution, we must first mark every occurrence of n in S. One useful way to do so is to replace n by some new variable say m that does not otherwise appear in S. For example, S(n) would become m 2 i = 2 m+1 1 i=0 We then literally substitute the desired expression, n + 1 in this case, for each occurrence of m. That gives us n+1 2 i = 2 (n+1)+1 1 i=0 When we simplify (n + 1) + 1 to n + 2, we have (2.3). Note that we should put parentheses around the expression substituted, to avoid accidentally changing the order of operations. For example, had we substituted n + 1 for m in the expression 2 m, and not placed the parentheses around n+1, we would have gotten 2 n+1, rather than the correct expression 2 (n+1), which equals 2 n + 2. each of our proofs that induction works requires an inductive proof itself, and therefore is no proof at all. Technically, induction must be accepted as axiomatic. Nevertheless, many people find the following intuition useful. In what follows, we assume that the basis value is n = 0. That is, we know that S(0) is true and that for all n greater than 0, if S(n) is true, then S(n + 1) is true. Similar arguments work if the basis value is any other integer. First proof : Iteration of the inductive step. Suppose we want to show that S(a) is true for a particular nonnegative integer a. If a = 0, we just invoke the truth of the basis, S(0). If a > 0, then we argue as follows. We know that S(0) is true, from the basis. The statement S(n) implies S(n + 1), with 0 in place of n, says S(0) implies S(1). Since we know that S(0) is true, we now know that S(1) is true. Similarly, if we substitute 1 for n, we get S(1) implies S(2), and so we also know that S(2) is true. Substituting 2 for n, we have S(2) implies S(3), so that S(3) is true, and so on. No matter what the value of a is, we eventually get to S(a), and we are done. Second proof : Least counterexample. Suppose S(n) were not true for at least one value of n. Let a be the least nonnegative integer for which S(a) is false. If a = 0, then we contradict the basis, S(0), and so a must be greater than 0. But if a > 0, and a is the least nonnegative integer for which S(a) is false, then S(a 1) must be true. Now, the inductive step, with n replaced by a 1, tells us that S(a 1) implies S(a). Since S(a 1) is true, S(a) must be true, another contradiction. Since we assumed there were nonnegative values of n for which S(n) is false and derived a contradiction, S(n) must therefore be true for any n 0. 14 38 ITERATION, INDUCTION, AND RECURSION Parity bit Error-Detecting Codes We shall now begin an extended example of error-detecting codes, a concept that is interesting in its own right and also leads to an interesting inductive proof. When we transmit information over a data network, we code characters (letters, digits, punctuation, and so on) into strings of bits, that is, 0 s and 1 s. For the moment let us assume that characters are represented by seven bits. However, it is normal to transmit more than seven bits per character, and an eighth bit can be used to help detect some simple errors in transmission. That is, occasionally, one of the 0 s or 1 s gets changed because of noise during transmission, and is received as the opposite bit; a 0 entering the transmission line emerges as a 1, or vice versa. It is useful if the communication system can tell when one of the eight bits has been changed, so that it can signal for a retransmission. To detect changes in a single bit, we must be sure that no two characters are represented by sequences of bits that differ in only one position. For then, if that position were changed, the result would be the code for the other character, and we could not detect that an error had occurred. For example, if the code for one character is the sequence of bits , and the code for another is , then a change in the fourth position from the left turns the former into the latter. One way to be sure that no characters have codes that differ in only one position is to precede the conventional 7-bit code for the character by a parity bit. If the total number of 1 s in a group of bits is odd, the group is said to have odd parity. If the number of 1 s in the group is even, then the group has even parity. The coding scheme we select is to represent each character by an 8-bit code with even parity; we could as well have chosen to use only the codes with odd parity. We force the parity to be even by selecting the parity bit judiciously. ASCII Example 2.5. The conventional ASCII (pronounced ask-ee ; it stands for American Standard Code for Information Interchange ) 7-bit code for the character A is That sequence of seven bits already has an even number of 1 s, and so we prefix it by 0 to get The conventional code for C is , which differs from the 7-bit code for A only in the sixth position. However, this code has odd parity, and so we prefix a 1 to it, yielding the 8-bit code with even parity. Note that after prefixing the parity bits to the codes for A and C, we have and , which differ in two positions, namely the first and seventh, as seen in Fig A: C: Fig We can choose the initial parity bit so the 8-bit code always has even parity. We can always pick a parity bit to attach to a 7-bit code so that the number of 1 s in the 8-bit code is even. We pick parity bit 0 if the 7-bit code for the character at hand has even parity, and we pick parity bit 1 if the 7-bit code has odd parity. In either case, the number of 1 s in the 8-bit code is even. 15 SEC. 2.3 INDUCTIVE PROOFS 39 No two sequences of bits that each have even parity can differ in only one position. For if two such bit sequences differ in exactly one position, then one has exactly one more 1 than the other. Thus, one sequence must have odd parity and the other even parity, contradicting our assumption that both have even parity. We conclude that addition of a parity bit to make the number of 1 s even serves to create an error-detecting code for characters. The parity-bit scheme is quite efficient, in the sense that it allows us to transmit many different characters. Note that there are 2 n different sequences of n bits, since we may choose either of two values (0 or 1) for the first position, either of two values for the second position, and so on, a total of (n factors) possible strings. Thus, we might expect to be able to represent up to 2 8 = 256 characters with eight bits. However, with the parity scheme, we can choose only seven of the bits; the eighth is then forced upon us. We can thus represent up to 2 7, or 128 characters, and still detect single errors. That is not so bad; we can use 128/256, or half, of the possible 8-bit codes as legal codes for characters, and still detect an error in one bit. Similarly, if we use sequences of n bits, choosing one of them to be the parity bit, we can represent 2 n 1 characters by taking sequences of n 1 bits and prefixing the suitable parity bit, whose value is determined by the other n 1 bits. Since there are 2 n sequences of n bits, we can represent 2 n 1 /2 n, or half the possible number of characters, and still detect an error in any one of the bits of a sequence. Is it possible to detect errors and use more than half the possible sequences of bits as legal codes? Our next example tells us we cannot. The inductive proof uses a statement that is not true for 0, and for which we must choose a larger basis, namely 1. Example 2.6. We shall prove the following by induction on n. Error-detecting code STATEMENT S(n): If C is any set of bit strings of length n that is error detecting (i.e., if there are no two strings that differ in exactly one position), then C contains at most 2 n 1 strings. This statement is not true for n = 0. S(0) says that any error-detecting set of strings of length 0 has at most 2 1 strings, that is, half a string. Technically, the set C consisting of only the empty string (string with no positions) is an error-detecting set of length 0, since there are no two strings in C that differ in only one position. Set C has more than half a string; it has one string to be exact. Thus, S(0) is false. However, for all n 1, S(n) is true, as we shall see. BASIS. The basis is S(1); that is, any error-detecting set of strings of length one has at most = 2 0 = 1 string. There are only two bit strings of length one, the string 0 and the string 1. However, we cannot have both of them in an error-detecting set, because they differ in exactly one position. Thus, every error-detecting set for n = 1 must have at most one string. INDUCTION. Let n 1, and assume that the inductive hypothesis an errordetecting set of strings of length n has at most 2 n 1 strings is true. We must 16 40 ITERATION, INDUCTION, AND RECURSION show, using this assumption, that any error-detecting set C of strings with length n + 1 has at most 2 n strings. Thus, divide C into two sets, C 0, the set of strings in C that begin with 0, and C 1, the set of strings in C that begin with 1. For instance, suppose n = 2 and C is the code with strings of length n + 1 = 3 constructed using a parity bit. Then, as shown in Fig. 2.6, C consists of the strings 000, 101, 110, and 011; C 0 consists of the strings 000 and 011, and C 1 has the other two strings, 101 and D 0 C 0 C D 1 C 1 Fig The set C is split into C 0, the strings beginning with 0, and C 1, the strings beginning with 1. D 0 and D 1 are formed by deleting the leading 0 s and 1 s, respectively. Consider the set D 0 consisting of those strings in C 0 with the leading 0 removed. In our example above, D 0 contains the strings 00 and 11. We claim that D 0 cannot have two strings differing in only one bit. The reason is that if there are two such strings say a 1 a 2 a n and b 1 b 2 b n then restoring their leading 0 s gives us two strings in C 0, 0a 1 a 2 a n and 0b 1 b 2 b n, and these strings would differ in only one position as well. But strings in C 0 are also in C, and we know that C does not have two strings that differ in only one position. Thus, neither does D 0, and so D 0 is an error detecting set. Now we can apply the inductive hypothesis to conclude that D 0, being an error-detecting set with strings of length n, has at most 2 n 1 strings. Thus, C 0 has at most 2 n 1 strings. We can reason similarly about the set C 1. Let D 1 be the set of strings in C 1, with their leading 1 s deleted. D 1 is an error-detecting set with strings of length n, and by the inductive hypothesis, D 1 has at most 2 n 1 strings. Thus, C 1 has at most 2 n 1 strings. However, every string in C is in either C 0 or C 1. Therefore, C has at most 2 n n 1, or 2 n strings. We have proved that S(n) implies S(n+1), and so we may conclude that S(n) is true for all n 1. We exclude n = 0 from the claim, because the basis is n = 1, not n = 0. We now see that the error-detecting sets constructed by parity check are as large as possible, since they have exactly 2 n 1 strings when strings of n bits are used. 17 SEC. 2.3 INDUCTIVE PROOFS 41 How to Invent Inductive Proofs There is no crank to turn that is guaranteed to give you an inductive proof of any (true) statement S(n). Finding inductive proofs, like finding proofs of any kind, or like writing programs that work, is a task with intellectual challenge, and we can only offer a few words of advice. If you examine the inductive steps in Examples 2.4 and 2.6, you will notice that in each case we had to rework the statement S(n + 1) that we were trying to prove so that it incorporated the inductive hypothesis, S(n), plus something extra. In Example 2.4, we expressed the sum as the sum n + 2 n n which the inductive hypothesis tells us something about, plus the extra term, 2 n+1. In Example 2.6, we expressed the set C, with strings of length n + 1, in terms of two sets of strings (which we called D 0 and D 1 ) of length n, so that we could apply the inductive hypothesis to these sets and conclude that both of these sets were of limited size. Of course, working with the statement S(n + 1) so that we can apply the inductive hypothesis is just a special case of the more universal problem-solving adage Use what is given. The hard part always comes when we must deal with the extra part of S(n+1) and complete the proof of S(n+1) from S(n). However, the following is a universal rule: An inductive proof must at some point say and by the inductive hypothesis we know that. If it doesn t, then it isn t an inductive proof. EXERCISES 2.3.1: Show the following formulas by induction on n starting at n = 1. Triangular number a) b) c) d) n i=1 i = n(n + 1)/2. n i=1 i2 = n(n + 1)(2n + 1)/6. n i=1 i3 = n 2 (n + 1) 2 /4. n i=1 1/i(i + 1) = n/(n + 1) : Numbers of the form t n = n(n+1)/2 are called triangular numbers, because marbles arranged in an equilateral triangle, n on a side, will total n i=1 i marbles, which we saw in Exercise 2.3.1(a) is t n marbles. For example, bowling pins are arranged in a triangle 4 on a side and there are t 4 = 4 5/2 = 10 pins. Show by induction on n that n j=1 t j = n(n + 1)(n + 2)/ : Identify the parity of each of the following bit sequences as even or odd: a) b) 18 42 ITERATION, INDUCTION, AND RECURSION c) : Suppose we use three digits say 0, 1, and 2 to code symbols. A set of strings C formed from 0 s, 1 s, and 2 s is error detecting if no two strings in C differ in only one position. For example, {00, 11, 22} is an error-detecting set with strings of length two, using the digits 0, 1, and 2. Show that for any n 1, an error-detecting set of strings of length n using the digits 0, 1, and 2, cannot have more than 3 n 1 strings *: Show that for any n 1, there is an error-detecting set of strings of length n, using the digits 0, 1, and 2, that has 3 n 1 strings *: Show that if we use k symbols, for any k 2, then there is an errordetecting set of strings of length n, using k different symbols as digits, with k n 1 strings, but no such set of strings with more than k n 1 strings *: If n 1, the number of strings using the digits 0, 1, and 2, with no two consecutive places holding the same digit, is 3 2 n 1. For example, there are 12 such strings of length three: 010, 012, 020, 021, 101, 102, 120, 121, 201, 202, 210, and 212. Prove this claim by induction on the length of the strings. Is the formula true for n = 0? 2.3.8*: Prove that the ripple-carry addition algorithm discussed in Section 1.3 produces the correct answer. Hint: Show by induction on i that after considering the first i places from the right end, the sum of the tails of length i for the two addends equals the number whose binary representation is the carry bit followed by the i bits of answer generated so far *: The formula for the sum of n terms of a geometric series a, ar, ar 2,...,ar n 1 is n 1 ar i = (arn a) (r 1) i=0 Prove this formula by induction on n. Note that you must assume r 1 for the formula to hold. Where do you use that assumption in your proof? : The formula for the sum of an arithmetic series with first term a and increment b, that is, a, (a + b), (a + 2b),..., ( a + (n 1)b ), is n 1 a + bi = n ( 2a + (n 1)b ) /2 i=0 a) Prove this formula by induction on n. b) Show how Exercise 2.3.1(a) is an example of this formula : Give two informal proofs that induction starting at 1 works, although the statement S(0) may be false : Show by induction on the length of strings that the code consisting of the odd-parity strings detects errors. 19 SEC. 2.3 INDUCTIVE PROOFS 43 Arithmetic series Geometric series Arithmetic and Geometric Sums There are two formulas from high-school algebra that we shall use frequently. They each have interesting inductive proofs, which we ask the reader to provide in Exercises and An arithmetic series is a sequence of n numbers of the form a, (a + b), (a + 2b),..., ( a + (n 1)b ) The first term is a, and each term is b larger than the one before. The sum of these n numbers is n times the average of the first and last terms; that is: n 1 a + bi = n ( 2a + (n 1)b ) /2 i=0 For example, consider the sum of There are n = 5 terms, the first is 3 and the last 11. Thus, the sum is 5 (3 + 11)/2 = 5 7 = 35. You can check that this sum is correct by adding the five integers. A geometric series is a sequence of n numbers of the form a, ar, ar 2, ar 3,..., ar n 1 That is, the first term is a, and each successive term is r times the previous term. The formula for the sum of n terms of a geometric series is n 1 ar i = (arn a) (r 1) i=0 Here, r can be greater or less than 1. If r = 1, the above formula does not work, but all terms are a so the sum is obviously an. As an example of a geometric series sum, consider Here, n = 5, the first term a is 1, and the ratio r is 2. Thus, the sum is ( )/(2 1) = (32 1)/1 = 31 as you may check. For another example, consider 1+1/2+1/4+1/8+1/16. Again n = 5 and a = 1, but r = 1/2. The sum is ( 1 ( 1 2 )5 1 ) /( 1 2 1) = ( 31/32)/( 1/2) = Error-correcting code **: If no two strings in a code differ in fewer than three positions, then we can actually correct a single error, by finding the unique string in the code that differs from the received string in only one position. It turns out that there is a code of 7-bit strings that corrects single errors and contains 16 strings. Find such a code. Hint: Reasoning it out is probably best, but if you get stuck, write a program that searches for such a code *: Does the even parity code detect any double errors, that is, changes in two different bits? Can it correct any single errors? 20 44 ITERATION, INDUCTION, AND RECURSION Template for Simple Inductions Let us summarize Section 2.3 by giving a template into which the simple inductions of that section fit. Section 2.4 will cover a more general template. 1. Specify the statement S(n) to be proved. Say you are going to prove S(n) by induction on n, for all n i 0. Here, i 0 is the constant of the basis; usually i 0 is 0 or 1, but it could be any integer. Explain intuitively what n means, e.g., the length of codewords. 2. State the basis case, S(i 0 ). 3. Prove the basis case. That is, explain why S(i 0 ) is true. 4. Set up the inductive step by stating that you are assuming S(n) for some n i 0, the inductive hypothesis. Express S(n + 1) by substituting n + 1 for n in the statement S(n). 5. Prove S(n + 1), assuming the inductive hypothesis S(n). 6. Conclude that S(n) is true for all n i 0 (but not necessarily for smaller n). 2.4 Complete Induction Strong and weak induction In the examples seen so far, we have proved that S(n + 1) is true using only S(n) as an inductive hypothesis. However, since we prove our statement S for values of its parameter starting at the basis value and proceeding upward, we are entitled to use S(i) for all values of i, from the basis value up to n. This form of induction is called complete (or sometimes perfect or strong) induction, while the simple form of induction of Section 2.3, where we used only S(n) to prove S(n + 1) is sometimes called weak induction. Let us begin by considering how to perform a complete induction starting with basis n = 0. We prove that S(n) is true for all n 0 in two steps: 1. We first prove the basis, S(0). 2. As an inductive hypothesis, we assume all of S(0), S(1),...,S(n) to be true. From these statements we prove that S(n + 1) holds. As for weak induction described in the previous section, we can also pick some value a other than 0 as the basis. Then, for the basis we prove S(a), and in the inductive step we are entitled to assume only S(a), S(a + 1),..., S(n). Note that weak induction is a special case of complete induction in which we elect not to use any of the previous statements except S(n) to prove S(n + 1). Figure 2.7 suggests how complete induction works. Each instance of the statement S(n) can (optionally) use any of the lower-indexed instances to its right in its proof. 21 SEC. 2.4 COMPLETE INDUCTION 45 S(n) S(2) S(1) S(0) Fig Complete induction allows each instance to use one, some, or all of the previous instances in its proof. Inductions With More Than One Basis Case When performing a complete induction, there are times when it is useful to have more than one basis case. If we wish to prove a statement S(n) for all n i 0, then we could treat not only i 0 as a basis case, but also some number of consecutive integers above i 0, say i 0, i 0 + 1, i 0 + 2,..., j 0. Then we must do the following two steps: 1. Prove each of the basis cases, the statements S(i 0 ), S(i 0 + 1),...,S(j 0 ). 2. As an inductive hypothesis, assume all of S(i 0 ), S(i 0 + 1),..., S(n) hold, for some n j 0, and prove S(n + 1). Example 2.7. Our first example of a complete induction is a simple one that uses multiple basis cases. As we shall see, it is only complete in a limited sense. To prove S(n + 1) we do not use S(n) but we use S(n 1) only. In more general complete inductions to follow, we use S(n), S(n 1), and many other instances of the statement S. Let us prove by induction on n the following statement for all n 0. 4 STATEMENT S(n): There are integers a and b (positive, negative, or 0) such that n = 2a + 3b. BASIS. We shall take both 0 and 1 as basis cases. i) For n = 0 we may pick a = 0 and b = 0. Surely 0 = ii) For n = 1, pick a = 1 and b = 1. Then 1 = 2 ( 1) INDUCTION. Now, we may assume S(n) and prove S(n + 1), for any n 1. Note that we may assume n is at least the largest of the consecutive values for which we have proved the basis: n 1 here. Statement S(n + 1) says that n + 1 = 2a + 3b for some integers a and b. The inductive hypothesis says that all of S(0), S(1),...,S(n) are true. Note that we begin the sequence at 0 because that was the lowest of the consecutive basis cases. Since n 1 can be assumed, we know that n 1 0, and therefore, S(n 1) is true. This statement says that there are integers a and b such that n 1 = 2a+3b. 4 Actually, this statement is true for all n, positive or negative, but the case of negative n requires a second induction which we leave as an exercise. INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28 3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1) CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 7 : Program Correctness 7.1 Program Correctness Showing that a program is correct means that MAT2400 Analysis I. A brief introduction to proofs, sets, and functions MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take 3. Recurrence Recursive Definitions. To construct a recursively defined function: 3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a Mathematical Induction Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical, CHAPTER 3 Numbers and Numeral Systems CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural, Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing Mathematical Induction Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 CHAPTER 6. MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 Introduction Recurrence Lecture 3. Mathematical Induction Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding 7.1 Introduction. CSci 335 Software Design and Analysis III Chapter 7 Sorting. Prof. Stewart Weiss Chapter 7 Sorting 7.1 Introduction Insertion sort is the sorting algorithm that splits an array into a sorted and an unsorted region, and repeatedly picks the lowest index element of the unsorted region Mathematical Induction. Lecture 10-11 Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach Oh Yeah? Well, Prove It. Oh Yeah? Well, Prove It. MT 43A - Abstract Algebra Fall 009 A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built Why? A central concept in Computer Science. Algorithms are ubiquitous. Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online It is not immediately obvious that this should even give an integer. Since 1 < 1 5 Math 163 - Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some Unit 2: Number Systems, Codes and Logic Functions Unit 2: Number Systems, Codes and Logic Functions Introduction A digital computer manipulates discrete elements of data and that these elements are represented in the binary forms. Operands used for calculations Cartesian Products and Relations Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without 9.2 Summation Notation 9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a Sequences and Mathematical Induction. CSE 215, Foundations of Computer Science Stony Brook University Sequences and Mathematical Induction CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 Sequences A sequence is a function whose domain is all the integers Encoding Text with a Small Alphabet Chapter 2 Encoding Text with a Small Alphabet Given the nature of the Internet, we can break the process of understanding how information is transmitted into two components. First, we have to figure out CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify The Set Data Model CHAPTER 7. 7.1 What This Chapter Is About CHAPTER 7 The Set Data Model The set is the most fundamental data model of mathematics. Every concept in mathematics, from trees to real numbers, is expressible as a special kind of set. In this book, 8.7 Mathematical Induction 8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture CS 4310 HOMEWORK SET 1 CS 4310 HOMEWORK SET 1 PAUL MILLER Section 2.1 Exercises Exercise 2.1-1. Using Figure 2.2 as a model, illustrate the operation of INSERTION-SORT on the array A = 31, 41, 59, 26, 41, 58. Solution: Assume Regular Languages and Finite Automata Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a 1.2. Successive Differences 1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers 2. INEQUALITIES AND ABSOLUTE VALUES 2. INEQUALITIES AND ABSOLUTE VALUES 2.1. The Ordering of the Real Numbers In addition to the arithmetic structure of the real numbers there is the order structure. The real numbers can be represented by Review of Number Systems The study of number systems is important from the viewpoint of understanding how data are represented before they can be processed by any digital system including a computer. Different Appendix F: Mathematical Induction Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another As we have discussed, digital circuits use binary signals but are required to handle Chapter 2 CODES AND THEIR CONVERSIONS 2.1 INTRODUCTION As we have discussed, digital circuits use binary signals but are required to handle data which may be alphabetic, numeric, or special characters. CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering Introduction. Appendix D Mathematical Induction D1 Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the Students in their first advanced mathematics classes are often surprised CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are Chapter 11 Number Theory Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook. Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole Basic Proof Techniques Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document COLORED GRAPHS AND THEIR PROPERTIES COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring 9 abcd = dcba. 9 ( b + 10c) = c + 10b b + 90c = c + 10b b = 10c. In this session, we ll learn how to solve problems related to place value. This is one of the fundamental concepts in arithmetic, something every elementary and middle school mathematics teacher should data structures and algorithms exercise class 2: some answers data structures and algorithms 2016-2017 exercise class 2: some answers 1. Show step by step how the array A = [22, 15, 36, 20, 3, 9, 29] is sorted using selection sort, bubble sort, and merge sort. First Introduction Number Systems and Conversion UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material Section 3 Sequences and Limits Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence. 1. R In this and the next section we are going to study the properties of sequences of real numbers. +a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March CMPS 102 Solutions to Homework 1 CMPS 0 Solutions to Homework Lindsay Brown, lbrown@soe.ucsc.edu September 9, 005 Problem..- p. 3 For inputs of size n insertion sort runs in 8n steps, while merge sort runs in 64n lg n steps. For which 8 Divisibility and prime numbers 8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express SECTION 10-2 Mathematical Induction 73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms Recurrence Relations. Niloufar Shafiei Recurrence Relations Niloufar Shafiei Review A recursive definition of a sequence specifies one or more initial terms a rule for determining subsequent terms from those that precede them. Example: a 0 MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear: MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3, Fibonacci Numbers. This is certainly the most famous Fibonacci Numbers The Fibonacci sequence {u n } starts with 0, then each term is obtained as the sum of the previous two: un = un + un The first fifty terms are tabulated at the right. Our objective here Prime Numbers. Chapter Primes and Composites Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are Recursion. Recursive Algorithms. Example: The Dictionary Search Problem. CSci 235 Software Design and Analysis II Introduction to Recursion Recursion Recursion is a powerful tool for solving certain kinds of problems. Recursion breaks a problem into smaller problems that are identical to the original, in such a way that solving the smaller CHAPTER 5. Number Theory. 1. Integers and Division. Discussion CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a Induction and the Division Theorem 2.1 The Method of Induction 2 Induction and the Division Theorem 2.1 The Method of Induction In the Introduction we discussed a mathematical problem whose solution required the verification of an infinite family of statements. We MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column Proving Algorithm Correctness Chapter 2 Proving Algorithm Correctness In Chapter 1, we specified several problems and presented various algorithms for solving these problems. For each algorithm, we argued somewhat informally that it Finite and Infinite Sets Chapter 9 Finite and Infinite Sets 9. Finite Sets Preview Activity (Equivalent Sets, Part ). Let A and B be sets and let f be a function from A to B..f W A! B/. Carefully complete each of the following Properties of sequences Since a sequence is a special kind of function it has analogous properties to functions: Sequences and Series A sequence is a special kind of function whose domain is N - the set of natural numbers. The range of a sequence is the collection of terms that make up the sequence. Just as the word Computing with Signed Numbers and Combining Like Terms Section. Pre-Activity Preparation Computing with Signed Numbers and Combining Like Terms One of the distinctions between arithmetic and algebra is that arithmetic problems use concrete knowledge; you know Induction Problems. Tom Davis November 7, 2005 Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily 1. What s wrong with the following proofs by induction? ArsDigita University Month : Discrete Mathematics - Professor Shai Simonson Problem Set 4 Induction and Recurrence Equations Thanks to Jeffrey Radcliffe and Joe Rizzo for many of the solutions. Pasted Chapter 10 Expanding Our Number System Chapter 10 Expanding Our Number System Thus far we have dealt only with positive numbers, and, of course, zero. Yet we use negative numbers to describe such different phenomena as cold temperatures and CHAPTER 3. Sequences. 1. Basic Properties CHAPTER 3 Sequences We begin our study of analysis with sequences. There are several reasons for starting here. First, sequences are the simplest way to introduce limits, the central idea of calculus. Automata and Languages Automata and Languages Computational Models: An idealized mathematical model of a computer. A computational model may be accurate in some ways but not in others. We shall be defining increasingly powerful Logic, Sets, and Proofs Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Statements. A logical statement is a mathematical statement that is either true or false. Here we denote logical Sets and Subsets. Countable and Uncountable Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand Note 3 CS 70 Discrete Mathematics and Probability Theory Fall 06 Seshia and Walrand Note 3 Mathematical Induction Introduction. In this note, we introduce the proof technique of mathematical induction. Induction k, then n = p2α 1 1 pα k Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square HMMT February Saturday 21 February Combinatorics HMMT February 015 Saturday 1 February 015 1. Evan s analog clock displays the time 1:13; the number of seconds is not shown. After 10 seconds elapse, it is still 1:13. What is the expected number of seconds 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. 2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then Introduction to C Programming Introduction to C Programming C HOW TO PROGRAM, 6/E 1992-2010 by Pearson Education, Inc. All Rights Reserved. 2.1 Introduction The C language facilitates a structured and disciplined approach to computer Topics in Number Theory Chapter 8 Topics in Number Theory 8.1 The Greatest Common Divisor Preview Activity 1 (The Greatest Common Divisor) 1. Explain what it means to say that a nonzero integer m divides an integer n. Recall Algorithms. Theresa Migler-VonDollen CMPS 5P Algorithms Theresa Migler-VonDollen CMPS 5P 1 / 32 Algorithms Write a Python function that accepts a list of numbers and a number, x. If x is in the list, the function returns the position in the list Chapter 2 Limits Functions and Sequences sequence sequence Example Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students CS 341 Homework 9 Languages That Are and Are Not Regular CS 341 Homework 9 Languages That Are and Are Not Regular 1. Show that the following are not regular. (a) L = {ww R : w {a, b}*} (b) L = {ww : w {a, b}*} (c) L = {ww' : w {a, b}*}, where w' stands for w Pigeonhole Principle Solutions Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such It is time to prove some theorems. There are various strategies for doing CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it Coding Theory. Kenneth H. Rosen, AT&T Laboratories. 5 Coding Theory Author: Kenneth H. Rosen, AT&T Laboratories. Prerequisites: The prerequisites for this chapter are the basics of logic, set theory, number theory, matrices, and probability. (See Sections An Interesting Way to Combine Numbers An Interesting Way to Combine Numbers Joshua Zucker and Tom Davis November 28, 2007 Abstract This exercise can be used for middle school students and older. The original problem seems almost impossibly A fairly quick tempo of solutions discussions can be kept during the arithmetic problems. Distributivity and related number tricks Notes: No calculators are to be used Each group of exercises is preceded by a short discussion of the concepts involved and one or two examples to be worked out LINEAR RECURSIVE SEQUENCES. The numbers in the sequence are called its terms. The general form of a sequence is. a 1, a 2, a 3,... LINEAR RECURSIVE SEQUENCES BJORN POONEN 1. Sequences A sequence is an infinite list of numbers, like 1) 1, 2, 4, 8, 16, 32,.... The numbers in the sequence are called its terms. The general form of a sequence Reading 13 : Finite State Automata and Regular Expressions CS/Math 24: Introduction to Discrete Mathematics Fall 25 Reading 3 : Finite State Automata and Regular Expressions Instructors: Beck Hasti, Gautam Prakriya In this reading we study a mathematical model C H A P T E R Regular Expressions regular expression 7 CHAPTER Regular Expressions Most programmers and other power-users of computer systems have used tools that match text patterns. You may have used a Web search engine with a pattern like travel cancun 3 Some Integer Functions 3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple Double Sequences and Double Series Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions, Mathematical Induction Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers Can you design an algorithm that searches for the maximum value in the list? The Science of Computing I Lesson 2: Searching and Sorting Living with Cyber Pillar: Algorithms Searching One of the most common tasks that is undertaken in Computer Science is the task of searching. We Section 5.1 Climbing an Infinite Ladder Suppose we have an infinite ladder and the following capabilities: 1. We can reach the first rung of the ladder. 2. If we can reach a particular rung of the ladder, Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls) Section IV.1: Recursive Algorithms and Recursion Trees Section IV.1: Recursive Algorithms and Recursion Trees Definition IV.1.1: A recursive algorithm is an algorithm that solves a problem by (1) reducing it to an instance of the same problem with smaller For the purposes of this course, the natural numbers are the positive integers. We denote by N, the set of natural numbers. Lecture 1: Induction and the Natural numbers Math 1a is a somewhat unusual course. It is a proof-based treatment of Calculus, for all of you who have already demonstrated a strong grounding in Calculus diffy boxes (iterations of the ducci four number game) 1 diffy boxes (iterations of the ducci four number game) 1 Peter Trapa September 27, 2006 Begin at the beginning and go on till you come to the end: then stop. Lewis Carroll Consider the following game. (Refer Slide Time: 1:41) Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 10 Sets Today we shall learn about sets. You must

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# About basis of the honeycomb lattice by KFC Tags: basis, honeycomb, lattice P: 369 Hi there, I am reading the book "Condensed Matter Physics" second edition by Michael P. Marder. It stated in page 9 that one basis of the the honeycomb lattice is $$\vec{v}_1 = a [0 \ 1/(2\sqrt{3})], \qquad \vec{v}_2 = a [0 \ -1/(2\sqrt{3})]$$ which is based on figure 1.5(B) in page 10. But in that case when two (vertical) atoms are bind together, so should this basis be $$\vec{v}_1 = a [0 \ \sqrt{3}/2], \qquad \vec{v}_2 = a [0 \ -\sqrt{3}/2]$$ By the way, why the primitive vectors are given as that in 1.6a and 1.6b $$\vec{v}_1 = (1/6 \ 1/6) , \qquad \vec{v}_2 = (-1/6 \ -1/6)$$ it said $$(\vec{a}_1 + \vec{a}_2)/6 = \vec{v}_1$$ But $$\vec{a}_1 = a(1 \ 0), \qquad \vec{a}_2 = a (1/2 \ \sqrt{3}/2)$$ why $$(\vec{a}_1 + \vec{a}_2)/6 = \vec{v}_1$$? P: 67 This is confusing. How can v1, v2 be a basis when v1 = -v2?? You should scan the page and put it up (double-check the Forum rules first .. i'm not an expert). Few people are so eager to help that they would go to the library and check out the book. You have to make the helpers' life easy. P: 369 Quote by sam_bell This is confusing. How can v1, v2 be a basis when v1 = -v2?? You should scan the page and put it up (double-check the Forum rules first .. i'm not an expert). Few people are so eager to help that they would go to the library and check out the book. You have to make the helpers' life easy. Sorry for the confusing ... and sorry also the book has been returned to the library and I don't have one now. But one thing I could explain here, in solid state physics, in some book 'basis' mean the combination of atoms only, nothing to do with the basis vector, so it is possible to have v1=-v2 in that case. P: 67 ## About basis of the honeycomb lattice All three bases describe a honeycomb lattice, when combined with Bravais vectors a1, a2. The second set (v1 = a[0, sqrt(3)/2] and v2 = a[0, -sqrt(3)/2]) is translated by a[1/2,0] relative to the first. The third set (v1 = a1/6 + a2/6 and v2 = -a1/6 -a2/6) is rotated by 60 degrees relative to the first. P: 369 Quote by sam_bell All three bases describe a honeycomb lattice, when combined with Bravais vectors a1, a2. The second set (v1 = a[0, sqrt(3)/2] and v2 = a[0, -sqrt(3)/2]) is translated by a[1/2,0] relative to the first. The third set (v1 = a1/6 + a2/6 and v2 = -a1/6 -a2/6) is rotated by 60 degrees relative to the first. Thanks for your reply. I get the point now. So, there is a mistake to write $$\vec{v}_1 = a [0 \ 1/(2\sqrt{3})], \qquad \vec{v}_2 = a [0 \ -1/(2\sqrt{3})]$$ in the book, right? P: 67 Quote by KFC Thanks for your reply. I get the point now. So, there is a mistake to write $$\vec{v}_1 = a [0 \ 1/(2\sqrt{3})], \qquad \vec{v}_2 = a [0 \ -1/(2\sqrt{3})]$$ in the book, right? Err, no. That's what I was referring to as the "1st" set. Related Discussions Atomic, Solid State, Comp. Physics 0 Linear & Abstract Algebra 2 Mechanical Engineering 2 Advanced Physics Homework 0 Introductory Physics Homework 10

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# Homework Help: Solving Exponential Equations 1. Aug 21, 2008 ### JBD2 1. The problem statement, all variables and given/known data Solve the following: Question 1: $$\sqrt[5]{256}\div\sqrt[6]{64}=2^{x}$$ Question 2: $$\frac{(9^{2x-1})^{3}(3^{3x})^{2}}{(27^{x+2})^{4}}=81^{3}$$ 2. Relevant equations None that I am aware of. 3. The attempt at a solution Question 1: $$\sqrt[5]{2^{8}}\div2^{1}=2^{x}$$ Question 2: $$\frac{(3^{4x-2})^{3}(3^{6x})}{(3^{3x+6})^{4}}=3^{12}$$ $$3^{12x-6+6x-(12x+24)}=3^{12}$$ $$6x=39$$ $$x=6.5$$ Sorry that the attempt for question 1 was lacking so much, I was just unsure of what to do with the square roots etc...As for number 2, the answer I got is incorrect, when it says that the answer in the back of my text book is 7. Question 1: $$\frac{3}{5}$$ Question 2: 7 Thanks to everyone who helps. 2. Aug 21, 2008 ### nicksauce For #1 you almost have it... just express the fifth root of 2^8 as 2 raised to some power, then use the fact that 2^a / 2^b = 2^(a-b). For #2 just check your numbers again. -6 -24 = -30. 12 + 30=42. 3. Aug 22, 2008 ### kenewbie 1) Two things that is helpful to know: $$\sqrt[a]{n^b} = n^{ \frac{b}{a} }$$ and $$\lg{\frac{a}{b}} = \lg{a} - lg{b}$$ You don't need the logarithm in this case though, since all your numbers are factors of two. k 4. Aug 22, 2008 ### JBD2 Ok thanks, I realized what I had done wrong for number two earlier today but I didn't know the rule about number one. I'm just starting into the logarithms so things should start becoming more familiar as well. Thank you. 5. Aug 23, 2008 ### HallsofIvy $$^n\sqrt{a}= a^\frac{1}{n}$$ $$(a^n)^m= a^{mn}$$ $$a^m\div a^n= a^{m-n}$$ 6. Aug 23, 2008 ### JBD2 Yes sorry I should've added those (the ones I used). I have another question though, I don't know what to do when adding two variables with exponents, for example what would something like this become: $$x^{4}+x^{3}=$$ or $$x^{4}-x^{7}=$$ I couldn't find it on the internet and I don't remember what to do in this case... 7. Aug 23, 2008 ### HallsofIvy There is no rule for adding different powers of the same base. That is exactly why polynomials are written as they are. 8. Aug 23, 2008 ### JBD2 Ok I've tried so many things and I can't seem to figure out these two questions. Here they are: Question 3: $$2^{x-1}-2^{x}=2^{-3}$$ Question 4: $$3^{x+1}+3^{x}=36$$ My work is a mess as I have tried so many variations which are probably way off course. 9. Aug 23, 2008 ### nicksauce you can write 2^(x-1) as 2^x*2^(-1) Thus you can factor the LHS, and isolate 2^x. Question 4 should be similar. 10. Aug 23, 2008 ### JBD2 Can you explain what you mean? So it would become: $$2^{x}\times2^{-1}-2^{x}=2^{-3}$$ But I can't really combine the like terms ($$2^{x}$$), can I? Because isn't the first one considered "attached" to the second one? ($$2^{x} and 2^{-1}$$) And for the second question: $$3^{x}\times3^{1}+3^{x}=36$$ $$6^{x}\times3=36$$ $$18^{x}=36$$ I'm not sure what to do from here...maybe I'm already wrong? 11. Aug 24, 2008 ### nicksauce $$2^{x}\times2^{-1}-2^{x}=2^{-3}$$ You can write this as $$2^x(2^{-1}-1})=2^{-3}$$ Now write the term in the parentheses as 2 to some power, and you should be all set. 12. Aug 24, 2008 ### JBD2 So it's: $$2^{x}(-2^{-1})=2^{3}$$ But seeing as the answer in the back says no solution, I'm assuming it's not possible to solve this with a negative base seeing as the rest are positive? (2 and -2) 13. Aug 24, 2008 ### nicksauce Right. You have 2^x = (some negative number). There is no value of x for which this is true, as 2^x is always positive. (I should have noticed this earlier, but didn't think of it for what ever reason) 14. Aug 24, 2008 ### JBD2 Oh ok that makes sense, with the other question (question 4) I know how to go about it now but how do I establish a common base between 3 and 36? 15. Aug 24, 2008 ### JBD2 Actually I'll just show my work for now so you can see the progress I've made so far: $$3^{x+1}+3^{x}=36$$ $$3^{x}(3^{1}+1)=36$$ Ok nevermind I thought I had an idea but I'll leave this up for reference. 16. Aug 24, 2008 ### HallsofIvy And 3+1= ? 17. Aug 24, 2008 ### JBD2 $$3^{x}(4)=36$$ Well its obvious that x is 2 but how do I do that algebraically...I can't go $$12^{x}=36$$ because x=2 wouldnt work...This probably looks like such a dumb question I just can't figure out how to get a common base...Can I go: $$3^{x}(4)=3^{2}(4)$$ $$x=2$$ Does that make sense? 18. Aug 24, 2008 ### nicksauce Yes that makes sense. 19. Aug 24, 2008 ### JBD2 Oh ok thank you, I just didn't know you could have a base with two separate numbers. 20. Aug 25, 2008 ### cheff3r leave the left hand side as it was (from the start) change right hand side to $$3*3^{2}$$ note this is equal to 36 now all the bases are the same and they can cancel/disapear (index laws) so now you have (x+1)+x=1+2 and i'm sure you can go from here

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# 5.1: Rotational Kinematics $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ ( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$ $$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$ $$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$ $$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vectorC}[1]{\textbf{#1}}$$ $$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$ $$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$ $$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$ $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$ Our first foray into linear motion was with kinematics, and we start our discussion of rotation with the same topic. ## Rigid Body Rotation Whenever we talk about “rotation,” there is something that is generally implied – we are not talking about a point mass or a collection of independently-moving point masses. Instead, we are generally referring to the rotation of a rigid object. A rigid object is nothing more than a collection of point objects that are confined to stay at specific positions relative to each other. When we talk about rotation, all these point objects follow different paths and travel different distances, but they all have something in common. Figure 5.1.1 – Motion of Two Points on a Rotating Rigid Body Drawing a straight line from the fixed point (called the pivot) to two different points on the object, we see that the angles through which these straight lines sweep are the same, and indeed this is true for every point on the object. So as we talk about rigid body rotation, our old language of linear motion (displacement, velocity, acceleration) that is based on units of distance and time, will have to give way to a new language for rotational motion, based on the units of radians (the most common unit of angular measure) and time. This language will be very similar to what we used for the linear case, usually with the word "angular" or "rotational" appended in front of the usual words. Just because we are going to a new language, it doesn't mean we throw out the physical principles we have learned so far. But to apply them in our new area of study, we need to develop some way to translate between the two. Back in Section 1.7, in our discussion of circular motion, we came up with a translation between the arclength traveled by an object in circular motion and the angle is motion sweeps out. Certainly the points A and B in Figure 5.1.1 are following a circular path (they remain a fixed distance from the pivot), so this relation applies to them. If a given point on a rigid body is a distance $$r$$ from the pivot, then the relationship between the distance it travels along the arclength and the angle measured in radians is given by Equation 1.7.2, and the relationship between its linear speed and the rate at which the angle is changing (in radians per second) is given by Equation 1.7.3, both of which we'll reiterate here: $s=R\theta,\;\;\;\;\;\; v=\dfrac{ds}{dt} = R\dfrac{d\theta}{dt} = R\omega$ While $$s$$ and $$v$$ are different for every point on the rigid object, we see that $$\theta$$ and $$\omega$$ are common to all of them. We therefore embrace these as our angular displacement and angular velocity measurements, respectively, for the rigid body as a whole. We can similarly define an angular acceleration ($$\alpha$$) in terms of the change of the linear speed of a spot on the rotating object: $a=\dfrac{dv}{dt} = r\dfrac{d\omega}{dt} = r\alpha$ While each point mass comprising the rigid object may have its own linear velocity/acceleration, they all share a common angular velocity/acceleration. We therefore can simplify our discussion of rigid body rotation from tracking the many different motions of all of the individual parts of the object to one simple parameter common to all of them. We therefore (for the moment) step away from the translation between linear and angular motion – which we have already discussed in earlier sections – and instead focus on purely rotational motion, following exactly the same path as we did for linear motion. You'll note that as a rule the convention for rotational motion, we stick with Greek variables, in contrast to the Latin variables we used for linear motion. Whenever the word "acceleration" is combined with circular motion, one naturally thinks of centripetal acceleration. Be careful not to make that association here! The link between linear acceleration and angular acceleration is through the component of acceleration responsible for speeding up the spot on the rigid object, not the acceleration responsible for changing its direction of motion (which is centripetal acceleration). So for example, an object rotating at a constant rate has no point on it that is speeding up (and has zero angular acceleration), but every point on it (except at the pivot) experiencing a centripetal acceleration. Conversely, a rotating object that slows down, stops, and reverses its direction of motion is experiencing angular acceleration at all times, including the moment it stops, but the centripetal acceleration of points on the object is zero at the moment that it stops. And finally, the difference should be clear mathematically. For a point on the object, its acceleration has two components: $\overrightarrow a = \overrightarrow a_\bot + \overrightarrow a_\parallel, \;\;\;\;\; where:\;\; \left\{ \begin{array}{l} a_\bot = a_c = r\omega^2 \\ a_\parallel = r\alpha = r\dfrac{d\omega}{dt} \end{array} \right.\nonumber$ ## Rotational Equations of Motion We define the following angular (rotational) versions of what we studied previously in kinematics: $\begin{array}{rl} position & : \ \theta(t) \\ displacement & : \ \Delta\theta = \theta_2 - \theta_1 \\ average \ velocity & : \ \omega_{ave} = \dfrac{\Delta \theta}{\Delta t} \\ instantaneous \ velocity & : \omega(t) = \dfrac{d\theta}{dt} \\ average \ acceleration &: \alpha_{ave} = \dfrac{\Delta \omega}{\Delta t} \\ instantaneous \ acceleration &: \alpha (t) = \dfrac{d\omega}{dt} \end{array}$ The calculus that leads to the equations of motion works out exactly the same way as before (we have only changed the variable names), giving us: $\begin{array}{rl} \theta \left(t\right)= & \theta_o + \omega_o t + \frac{1}{2}\alpha t^2 \\ \omega(t)= & \omega_o + \alpha t \\ \omega^2_f - \omega^2_o= & 2 \alpha \left( \Delta \theta \right) \\ \omega_{ave}= & \dfrac{\omega_o + \omega_f}{2} \;\;\; \left( if \;\alpha = constant \right) \end{array}$Note that like the case of one-dimensional linear motion, we need to define at the outset a "positive" direction, but for rotation, this means choosing clockwise or counterclockwise from a specific perspective. Example $$\PageIndex{1}$$ A bug stands on the outer edge of a turntable as it begins to spin, accelerating rotationally in the horizontal plane from rest at a constant rate. Find the rate of angular acceleration of the turntable in terms of its radius and the coefficient of static friction if the bug slides off it just as the turntable completes its third full rotation. Solution The bug remains on the edge of the turntable thanks to the static friction force, which keeps it going in a circle. When the rotational speed becomes so great that the maximum static friction force is insufficient to maintain this centripetal acceleration, the bug will slide off. The maximum static friction force is the coefficient of static friction multiplied by the normal force, and since the turntable is horizontal and not accelerating vertically, the normal force equals the weight of the bug. We therefore have: $\left. \begin{array}{l} f_{max}=ma_c=mr\omega^2 \\ f_{max}=\mu_s N \\ N=mg \end{array} \right\} \;\;\; \Rightarrow \;\;\; \omega^2 = \dfrac{\mu_s g}{r} \nonumber$ The "no time" kinematics equation for rotation relates the angular acceleration (which we are looking for), the starting rotational speed (which is zero here, as the turntable starts from rest), the final speed (the speed that causes the bug to lose its grip), and the angle through which the object has rotated (which in this case is $$6\pi$$ – three full rotations): $\alpha = \dfrac{\omega_f^2 - \cancel{\omega_o^2}}{2 \Delta \theta} = \dfrac{\dfrac{\mu_s g}{r}}{2 \left(6 \pi\right)} = \boxed{\dfrac{\mu_s g}{12 \pi r}} \nonumber$ ## Directions of Rotational Kinematics Vectors When we did all of this previously, we found it was easy to keep track of directions in one dimension, simply by checking the sign of the value, but when we had to go to more dimensions, we needed to treat these quantities like vectors. How can we do that for this rotational vectors? The answer comes from all the way back in Chapter 1 – the right hand rule! It goes like this: curl the fingers of your right hand (in their natural finger-curling manner) in the direction that the object is rotating, and your thumb points the direction of the vector. The direction is perpendicular to the plane of rotation. This direction applies to all of the angular motion vectors – displacement, velocity, and acceleration. But be careful about the acceleration vector! Just as in the linear case, the acceleration vector points in the direction of the changing velocity vector, not the direction of the velocity vector itself. So if a rotating object is slowing down, the angular acceleration vector points in the opposite direction as the angular velocity vector. Example $$\PageIndex{2}$$ The graph below depicts the rotational velocity of a merry-go-round as a function of time, where the positive direction is defined to be downward (into the surface of the Earth). You are standing near the merry-go-round, watching children go by. At the point indicated in the graph, which of the following are you seeing? 1. The kids closest to you are moving to the right and are speeding up. 2. The kids closest to you are moving to the right and are slowing down. 3. The kids closest to you are moving to the left and are speeding up. 4. The kids closest to you are moving to the left and are slowing down. 5. The kids closest to you are moving to the left, but their speed is not changing. Solution From the RHR, we determine that the positive rotational direction is clockwise as you look at the merry-go-round from above (the kids on the merry-go-round are wondering why you are apparently giving their ride a thumbs-down!). Looking at it from ground level, this means that rotation in a positive direction results in seeing the nearest kids go by from right-to-left. At the point in question, the sign of the rotational velocity is negative, which means the kids are going by left-to-right. A short time later, the rotational velocity will be more negative, which means they are speeding up. So the answer is (a). This page titled 5.1: Rotational Kinematics is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Tom Weideman directly on the LibreTexts platform.

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# Hard math equations with answers Hard math equations with answers is a mathematical instrument that assists to solve math equations. We can help me with math work. ## The Best Hard math equations with answers Hard math equations with answers can be found online or in mathematical textbooks. On the other hand, linear solvers have a number of disadvantages. First, they don't handle non-linear problems well at all. Second, linear solvers are not very accurate compared to non-linear solvers. Finally, they're very slow to run. Many modern solvers use both linear and non-linear methods, so they're better at handling non-linear problems than pure linear solvers. Linear solvers are often used in commercial applications because they're fast and easy to implement. Commercial applications include software libraries and game engines, which use linear solvers when solving equations like physics or collision detection. Inequality equations are situations where two values are unequal. In other words, the value of one is higher than the other. These equations can be solved in various ways, depending on the situation. One way to solve an inequality equation is to multiply the left-hand side by a fraction. For example, let’s say you have \$5 and \$6 on your balance. If you want to know how much money you have, divide \$5 by 6, which gives you an answer of \$1. If you want to know how much money you have less than \$6, divide 5/6 by 1, giving an answer of 0.333333333. This means that you have \$1 less than what you started with. Another way to solve an inequality equation is to raise both sides to a power. For example, let’s say you have \$5 and \$6 on your balance. If you want to know how much money you have less than \$10, raise both sides to the power of 2 (2x=10), giving an answer of 0.25. This means that you have 25 cents less than what you started with. In order to solve inequalities, we must first understand how they work. When two values are unequal in size or amount, the equation will always be true by definition. When a value is greater than another value, Age problems can be difficult to solve, but if you keep in mind some important tips, you should be able to overcome them. If your age is a problem for you, start by taking stock of your situation. Think about what you have going for you and what might be holding you back from advancing. Then look at where you are in your career and how much time you have left before retirement. Once you have a better grasp of the situation, you can begin looking for ways to work around it. For example, if it's difficult for you to fit into a team because of your age, consider joining a smaller project that can be completed more quickly. Similarly, if your position is being eliminated due to budget cuts, look into restructuring it so that it doesn't include as many responsibilities. The key is to find ways to make yourself more valuable while also staying true to your values and priorities. The word problem is one of the most basic and essential math skills. The ability to solve word problems is the single most important skill that you can develop as a student, and it will serve you for your entire life. The best way to learn how to solve math word problems is by practicing with examples. To help with this, there are a number of apps available that can be used to practice solving word problems. Some of these include Word Problems by Mathway, Word Problems by PandaFun, Word Problems by iTutor, Word Problems by MathUsee and MathUsee Basic. All of these apps provide a variety of different types of word problems that you can practice solving. By practicing with examples over and over again, you will eventually be able to see patterns and recognize when you are making mistakes. Once you are able to do this, it will become much easier for you to solve word problems on your own in the future. If you want the best website that does your homework for you, then we recommend College Homework Assignments. We know what it’s like to be a student and how hard it can be to keep up with everything that needs to be done. It’s a big commitment, especially when you have to balance work and school as well. That’s why we created this website so that you can get all of your assignments done in one place. With over 100 different subjects, there is something for everyone. Everything is easy to find and simple to navigate. You can also submit your assignments right from the website itself so that there are no long waits or paperwork involved. When you choose College Homework Assignments, you will be able to make the most of your time and complete everything that needs to be done. I give it a five star it’s the best math solver app I have met but the developer should look into solving the word problems, graphs and tables It's amazing it really helped me figure out steps and how to do the work and it's a really useful tool 10/10 recommend Hollie Parker Great app I love using this to help me understand any mistakes I've made in a math problem and how the problem SHOULD be done. I would really like it if the "why" button was free but overall, I think it's great for anyone who is struggling in math or simply wants to check their answers. Gabrielle Long Equation College Value Math question help free Fractions help online Word math solver Calculus math problem Math word problem solver calculator

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## Saturday, 29 November 2014 ### linearity of operators One of the powerful features of BKO is the linearity of (most) operators. op1 |x> => |a> + |b> + |c> Then: op2 op1 |x> = op2 (|a> + |b> + |c>) = op2 |a> + op2 |b> + op2 |c> Let's do a simple example in the console: -- enter in some knowledge: sa: friends |Fred> => |Sam> + |Mary> + |Bella> sa: age |Sam> => |age: 39> sa: age |Mary> => |age: 42> sa: age |Bella> => |age: 36> -- see what we know: -- who are Fred's friends? sa: friends |Fred> |Sam> + |Mary> + |Bella> -- how old are Fred's friends? sa: age friends |Fred> |age: 39> + |age: 42> + |age: 36> -- enter in some more knowledge: sa: friends |Sam> => |Liz> + |Rob> sa: friends |Bella> => |Joan> + |Tom> -- who are Sam's friends? sa: friends |Sam> |Liz> + |Rob> -- who are Mary's friends? sa: friends |Mary> |> -- in English |> means "I don't know anything about that". -- so instead of bugging out if you ask it something it doesn't know, it just quietly returns |> (the empty ket) -- who are Bella's friends? sa: friends |Bella> |Joan> + |Tom> And now finally: -- who are Fred's friends of friends? sa: friends friends |Fred> |Liz> + |Rob> + |Joan> + |Tom> And note that it quietly made use of: "superposition + |> = |> + superposition = superposition" ie, |> is the identity element for superpositions. So even though we don't know who's Mary's friends are, the code didn't bug out, and replied best it could. I like to think humans do something similar. -- what are the ages of Fred's friends of friends? sa: age friends friends |Fred> |> -- ie, we don't know! Then finally, let's look at what we now know: sa: dump ---------------------------------------- |context> => |context: sw console> friends |Fred> => |Sam> + |Mary> + |Bella> age |Sam> => |age: 39> friends |Sam> => |Liz> + |Rob> age |Mary> => |age: 42> age |Bella> => |age: 36> friends |Bella> => |Joan> + |Tom> ---------------------------------------- (and we forgot to set context when we started, so it used the default context "sw console") And that's it for now!

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### Pratik H Male, 25 Years Associated for 1 Year 8 Months Engineering Subjects Tutor #### Activity Score - 1595 • I teach at My Home • I go to Student's Home • Online Area: 1 More Location: Surat, India Qualification : • B.Tech/B.E. (MSCET SURAT - 2016) • Total Experience: 2 Years • Hourly Fees: INR 300 Tutoring Experience : I have done B.E (Automobile) and have been teaching diploma and degree engineering students of automobile and mechanical engineering branch. Tutoring Option : Home Tuition Only Teaches: Class 9 - 10 Mathematics Physics Local State Board INR 200 / Hour Class 11 - 12 Mathematics Physics Local State Board INR 500 / Hour College Level Mathematics Physics B.Tech Tuition Gujarat University Ahmedabad INR 300 / Hour Engineering Subjects Mathematics Mechanical Production INR 300 / Hour Sports Cricket INR 200 / Hour ### 2 Notes written by me • #### Stress And Strain Files: 1 The Strength of Materials  Topic-Stress and Strain. • #### Short Note On Machine Design Files: 1 Short Note On Machine Design With Formulas. • ## Question: How can an average student become a 10th standard topper? Posted in: Mathematics,Computer Science,Physics,Chemistry,Biology | Date: 10/02/2018 The most important thing is hardwork. When I say hardwork that doesn't mean anyone have to dig holes. 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Posted in: Chemistry | Date: 13/02/2018 Pros :- 1) renewable sources of energy 2) clean source of energy without pollution Cons :- 1) costly setup for turbines 2) high cost of dam construction 3) large amount of water flow is required. • ## Question: How can an average student become a 10th standard topper? Posted in: Mathematics,Computer Science,Physics,Chemistry,Biology | Date: 13/02/2018 Hard work, smart work, percieverance, believe in yourself, believe in your ability, identifying weak areas and practicing it to achieve perfection. These things can make you topper • ## Question: What is the method used to find the unknown part of an analytic function? Posted in: Mathematics | Date: 13/02/2018 Analytical function can be found by Cauchy reiman equations • ## Question: How much time should I give each subject per day to score the highest marks in m... 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Posted in: Android Training | Date: 13/02/2018 Android is better because of wide range of users,user friendly and universal acceptance. IPhone is used by few peoples, nice user experience with iPhone. • ## Question: What is API full form? Posted in: Android Training | Date: 13/02/2018 Full form of API is application program interface Posted in: Chemistry | Date: 13/02/2018 Adv :- less pollution , Decrease of dependence on fossil fuels Disadvantage :- complicated process of blending diesel with biofuel. • ## Question: Can distance traveled ever be less than the magnitude of the displacement? Posted in: Physics | Date: 13/02/2018 No distance cannot be less than displacement but displacement can be less than distance. • ## Question: Is the magnitude always positive? Posted in: Physics | Date: 13/02/2018 Yes magnitude is always positive. If there is negative sign than it will be for direction. • ## Question: Is biofuel energy renewable or nonrenewable? 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Feel free to contact me. Don't worry everything will be sorted just be confident. • ## Question: What are the pros and cons of nuclear power? Posted in: Chemistry | Date: 17/02/2018 Pros :- no air pollution, huge amount of energy can be generated Cons :- danger of explosion, radioactivity is poisonous. • ## Question: What are the pros and cons of nuclear fission? Posted in: Chemistry | Date: 17/02/2018 Pros :- no air pollution, large amount of energy can be generated Cons :- danger of explosion, radioactivity is poisonous. • ## Question: Is Biodiesel safe? Posted in: Chemistry | Date: 17/02/2018 Yes biodiesel is safe but it can damage the engine on long run so modifications is required for the engine. • ## Question: What are the pros and cons of biomass energy? Posted in: Chemistry | Date: 17/02/2018 Pros :- renewable sources of energy, available in large quantities Cons :- creates air pollution on burning, Complicated process. • ## Question: What are the advantages and disadvantages of tidal power? Posted in: Chemistry | Date: 17/02/2018 Adv :- renewable sources of energy, available free of cost Cons :- energy is converted only on high tide, huge cost of dam construction. • ## Question: What are the advantages and disadvantages of nuclear power? Posted in: Chemistry | Date: 17/02/2018 Adv :- large amount of energy can be generated, no air pollution Disadvantage :- danger of explosion, radioactivity is poisonous. • ## Question: What is good about nuclear power? Posted in: Chemistry | Date: 17/02/2018 Large amount of energy can be generated, no air pollution. • ## Question: What is the formula for the magnitude of a vector? Posted in: Physics | Date: 17/02/2018 Magnitude is the intensity of the quantity which the vector represents. • ## Question: What are the advantages and disadvantages of wave energy? Posted in: Chemistry | Date: 17/02/2018 Adv :- renewable sources of energy, available free of cost Disadvantage :- energy can be generated only on high tide, huge cost of dam construction. • ## Question: Can we lie about your income on credit cards? Posted in: Accountancy | Date: 17/02/2018 No you can't lie about your income on credit card. If you try to lie you will get caught and penalized. • ## Question: What is YTD mean? Posted in: Economics | Date: 17/02/2018 Ytd means year to date. • ## Question: Is net price with or without VAT?  Please explain. Posted in: Economics | Date: 17/02/2018 Net price is always inclusive of taxes. It includes all the taxes. • ## Question: How do we calculate the net cost? Posted in: Accountancy | Date: 17/02/2018 Net cost = gross income - incentives,benefits,etc. • ## Question: How do we calculate net sales? Posted in: Accountancy | Date: 17/02/2018 Net sales = gross sales - sales return. • ## Question: How are vectors drawn? Posted in: Physics | Date: 17/02/2018 Length of vector represents it's magnitude and orientation represents it's direction. • ## Question: What is the resultant of two vectors? Posted in: Physics | Date: 17/02/2018 Resultant of vector is the sum of all the vectors. • ## Question: What are the good things about nuclear power? Posted in: Chemistry | Date: 17/02/2018 Large amount of energy can be generated, no air pollution. • ## Question: What are the advantages and disadvantages of using nuclear power? Posted in: Chemistry | Date: 17/02/2018 Adv :- large amount of energy can be generated, no air pollution Disadvantage :- danger of explosion, radioactivity is poisonous • ## Question: What are the advantages and disadvantages of biomass energy? Posted in: Chemistry | Date: 17/02/2018 Adv :- renewable source of energy, decrease our dependence on fossil fuels Disadvantage :- it's burning produces air pollution, Complicated process of creating. • ## Question: What is the difference between the biodiesel and biofuel? Posted in: Chemistry | Date: 17/02/2018 Biodiesel is blending of biogass with diesel while biogass is the gas produced by burning of biomass. • ## Question: Is biofuel and biodiesel the same thing? Posted in: Chemistry | Date: 17/02/2018 No biodiesel is blending of biogass and diesel while biofuel is produced by burning of biomass • ## Question: Is biodiesel more efficient than gasoline? Posted in: Chemistry | Date: 17/02/2018 No petrol is more efficient than biodiesel. It's air standard efficiency is high. • ## Question: What is the polygon law of vector addition? Posted in: Physics | Date: 17/02/2018 In polygonal law, all vectors are arranged in a closed polygon where closing vector represents magnitude and direction of resultant vector • ## Question: What is an example of a vector? Posted in: Physics | Date: 17/02/2018 Force, velocity, acceleration, etc are examples of vector • ## Question: What is a physical quantity that has both magnitude and direction? Posted in: Physics | Date: 17/02/2018 Physical quantity with both magnitude and direction is called vector quantity • ## Question: Is biofuel good or bad for the environment? Posted in: Chemistry | Date: 17/02/2018 Biofuel is good for environment. It decreases our dependence on fossil fuels • ## Question: What is the environmental impact of using biodiesel? Posted in: Chemistry | Date: 17/02/2018 Use of biodiesel decrease our dependence on fossil fuels • ## Question: Do ethanol fires produce heat? Posted in: Chemistry | Date: 17/02/2018 Yes ethanol fire produces large amount of heat • ## Question: How bad is ethanol for humans? Posted in: Chemistry | Date: 17/02/2018 Ethanol is toxic and poisonous to human health • ## Question: Is gasohol more efficient than gasoline? Posted in: Chemistry | Date: 17/02/2018 Gasoline is more efficient than gasohol • ## Question: Do opposite vectors have a negative magnitude? Posted in: Physics | Date: 17/02/2018 No magnitude will always be positive. Negative sign represents direction • ## Question: Can you drink ethanol alcohol? Posted in: Chemistry | Date: 17/02/2018 No ethanol is toxic and poisonous to human health • ## Question: Can we have physical quantities having magnitude and direction which are not vec... Posted in: Physics | Date: 17/02/2018 No vectors are quantities having magnitude and direction • ## Question: Are bio ethanol fires safe? Posted in: Chemistry | Date: 17/02/2018 No bioethanol is inflammable • ## Question: Can the magnitude of a vector be zero? Posted in: Physics | Date: 17/02/2018 Yes magnitude can be zero but not negative • ## Question: Is biofuel a good thing? Posted in: Chemistry | Date: 17/02/2018 Yes biofuel is good because it decreases our dependence on fossil fuels • ## Question: What are the advantages and disadvantages of wind power? Posted in: Chemistry | Date: 17/02/2018 Adv :- renewable sources of energy, clean source of energy without pollution Disadvantage :- used only where wind speed is high, cost of windmills is high • ## Question: What is the parallel vector? Posted in: Physics | Date: 17/02/2018 When direction of two vectors is same than vectors are called parallel vectors • ## Question: What is better diesel or biodiesel? Posted in: Chemistry | Date: 17/02/2018 Biodiesel is better than diesel because it decreases our dependence on fossil fuels • ## Question: Is ethanol and biodiesel the same? Posted in: Chemistry | Date: 17/02/2018 No ethanol is an alcohole while biodiesel is blending of biofuel and diesel • ## Question: What is the sum of two or more vectors? Posted in: Physics | Date: 17/02/2018 Sum of vector is called the resultant vector • ## Question: What is the difference between biomass and biofuels? Posted in: Chemistry | Date: 17/02/2018 Biomass is the waste bio materials like cow dung, dead parts of plants while biofuel is produced by burning of biomass • ## Question: How do you find the resultant of two vectors? Posted in: Physics | Date: 17/02/2018 Resultant vector is the sum of two vectors • ## Question: Why is it better to use biofuels? Posted in: Chemistry | Date: 17/02/2018 Biofuels are better because it decreases our dependence on fossil fuels • ## Question: What are the disadvantages of using nuclear power? Posted in: Chemistry | Date: 17/02/2018 • ## Question: What are the advantages and disadvantages of using biodiesel? Posted in: Chemistry | Date: 17/02/2018 Adv :- renewable sources of energy, decreasing our dependence on fossil fuels Disadvantage :- damages engine on long run, engine modifications is required. • ## Question: Is net price with or without VAT? Posted in: Accountancy | Date: 17/02/2018 Net price is with VAT. Inclusive of taxes • ## Question: Why are biofuels bad? Posted in: Chemistry | Date: 17/02/2018 Biofuels produces air pollution on burning, complicated blending process • ## Question: What does it mean when a price is net? Posted in: Accountancy | Date: 17/02/2018 Net price is the price including all taxes • ## Question: What is biofuel in simple terms? Posted in: Chemistry | Date: 17/02/2018 Biofuel is the gas produced by burning of biomass • ## Question: What does Biofuels do? Posted in: Chemistry | Date: 17/02/2018 Biofuel decrease our dependence on fossil fuels as they are blending of biogas and fossil fuels • ## Question: What are the benefits of using biodiesel? Posted in: Chemistry | Date: 17/02/2018 Biodiesel decrease our dependence on fossil fuels • ## Question: What is an example of a biofuel? Posted in: Chemistry | Date: 17/02/2018 Bio and biogas are examples of biofuels • ## Question: What are biofuels for? Posted in: Chemistry | Date: 17/02/2018 Biofuels can be used as fuels in the vehicle by making minor modifications in the engine • ## Question: What are the economic benefits of digital marketing? Posted in: Economics | Date: 19/02/2018 Increase in customer base resulting in increase of business and profit. • ## Question: What is the difference between biogas and biofuels? Posted in: Chemistry | Date: 19/02/2018 Biogas is produced by burning of biomass while biofuel is a general term which includes biogas as well as other fuels like biodiesel. • ## Question: What is considered a long-term investment? Posted in: Economics | Date: 19/02/2018 Investment done for long periods of time probably for more than 10 years is called long term investment Posted in: Chemistry | Date: 19/02/2018 Adv :- renewable sources of energy, available in large quantities Disadvantage :- burning of biofuel produces air pollution, complicated process of blending. • ## Question: What is biofuel made of? Posted in: Chemistry | Date: 21/02/2018 Biofuel is made up of biogas and fossil fuels like petrol, diesel. • ## Question: What is bioethanol and what is it used for? Posted in: Chemistry | Date: 21/02/2018 Bioethanol is the blending of biofuel with ethanol. It is used as a fuel for running vehicles. • ## Question: How is bioethanol is produced? Posted in: Chemistry | Date: 21/02/2018 Bioethanol is produced by blending process of biofuel and ethanol • ## Question: What is the difference between biodiesel and biofuels? Posted in: Chemistry | Date: 21/02/2018 Biodiesel is blending of biogas and diesel while biofuel is a mixture of biogas and any fossil fuels like petrol and diesel. • ## Question: Is biofuels a renewable resource? Posted in: Chemistry | Date: 21/02/2018 Yes biofuel is a renewable fuel because it is produced by burning of biomass which is renewable in nature. • ## Question: What are the different types of biofuels? Posted in: Chemistry | Date: 21/02/2018 Biodiesel and bioethanol are two types of biofuels • ## Question: What is meant by magnitude of a vector? Posted in: Physics | Date: 21/02/2018 Magnitude of vector is the intensity of the quantity which represents that vector. • ## Question: What are the components of a vector? Posted in: Physics | Date: 21/02/2018 There are two components of the vector 1) horizontal and 2) vertical. • ## Question: What is zero vector in physics? Posted in: Physics | Date: 21/02/2018 When magnitude of vector is zero than it is called zero vector • ## Question: How would you define the zero vector? Posted in: Physics | Date: 21/02/2018 The vector whose magnitude is zero is called Zero vector. • ## Question: What is the definition of the magnitude of a vector? Posted in: Physics | Date: 21/02/2018 Magnitude of vector is the intensity of the quantity which represents that vector. • ## Question: How do you calculate the magnitude of acceleration? Posted in: Physics | Date: 21/02/2018 Magnitude of acceleration is the rate of change of velocity with respect to time. • ## Question: How do you know if two vectors are equal? Posted in: Physics | Date: 21/02/2018 When magnitude of two vectors are equal than they are called equal vectors. • ## Question: Can the magnitude of a vector be zero? Posted in: Physics | Date: 21/02/2018 Yes magnitude of vector can be zero or positive but not negative. • ## Question: What is the instrument used to magnify objects? Posted in: Physics | Date: 21/02/2018 Instrument used for magnifying objects is called microscope . • ## Question: What are all the different types of metals? Posted in: Chemistry | Date: 21/02/2018 Different types of metals are iron, steel, copper, aluminium, etc. • ## Question: What metals are used in everyday life? Posted in: Chemistry | Date: 21/02/2018 Iron, steel, copper and aluminium is used in day to day life. • ## Question: Do sound waves travel faster in water or air? Posted in: Physics | Date: 21/02/2018 Sound waves travels faster in water than in air. • ## Question: What is an example of a nonmetal? Posted in: Chemistry | Date: 21/02/2018 Rubber, plastic, paper, etc are examples of non metals • ## Question: What are the characteristics of a nonmetal? Posted in: Chemistry | Date: 21/02/2018 Non metals are bad conductor of heat and electricity. • ## Question: What are some non metals? Posted in: Chemistry | Date: 21/02/2018 Rubber, plastic, cloth, etc are examples of non metals. • ## Question: What is the difference between a metal and a nonmetal on the periodic table? Posted in: Chemistry | Date: 21/02/2018 Metals are good conductor of heat and electricity while non metals are bad conductor of heat and electricity. • ## Question: Which instrument is used to view distant objects? Posted in: Physics | Date: 21/02/2018 Telescope is used to view distance objects. • ## Question: What is optics used for? Posted in: Physics | Date: 21/02/2018 Optics is used to make instrument like telescope and microscope. • ## Question: What is a simple definition of entropy? Posted in: Chemistry | Date: 21/02/2018 Entropy is defined as the degree of disorder in the object and it is measure of the ease with which chemical reaction takes place. • ## Question: What is the kinematics? Posted in: Chemistry | Date: 21/02/2018 Kinematics is the study of motion of mechanism without considering forces acting on it. • ## Question: What is the kinematics in animation? Posted in: Chemistry | Date: 21/02/2018 Kinematics is the study of motion of mechanism without considering forces acting on it. • ## Question: What is the meaning of entropy in thermodynamics? Posted in: Chemistry | Date: 21/02/2018 Entropy is defined as the degree of disorder in the object and it is measure of the ease with which chemical reaction takes place. • ## Question: What is kinetic in physics? Posted in: Chemistry | Date: 21/02/2018 Kinetics is the study of motion of mechanism with considering forces acting on it • ## Question: What is the meaning of thermodynamics in chemistry? Posted in: Chemistry | Date: 21/02/2018 Thermodynamics is the branch of science that deals with the study of transfer of heat and it's conversion. • ## Question: What is a example of a liquid? Posted in: Physics | Date: 21/02/2018 Milk, water, tea, coffee, etc are examples of liquid • ## Question: Is heat an example of matter? Please explain. Posted in: Physics | Date: 21/02/2018 Heat is a form of energy not matter • ## Question: What is Valency and example? Posted in: Chemistry | Date: 21/02/2018 Valency is the number of electrons present in the outermost orbit of an atom • ## Question: How do you find the Valency of an element? Posted in: Chemistry | Date: 21/02/2018 Valency is the number of electrons present in the outermost orbit of an atom • ## Question: What holds together the particles in a solid? Posted in: Chemistry | Date: 21/02/2018 The force which holds the particles in a solid is called van der waals force • ## Question: What is the value for R in the ideal gas law? Posted in: Physics | Date: 21/02/2018 The value of R in ideal gas law is 26.7 J/kg K • ## Question: Which state has the most energy solid liquid or gas? Posted in: Physics | Date: 21/02/2018 Solid state has maximum energy • ## Question: What state of matter has the most energy? Posted in: Physics | Date: 21/02/2018 Solid state has maximum energy • ## Question: How do we measure gas pressure? Posted in: Physics | Date: 21/02/2018 Pressure is measured by instrument called barometer • ## Question: Which instrument is used to measure gas pressure? Posted in: Physics | Date: 21/02/2018 Barometer is used to measure gas pressure • ## Question: What is inside of a neutron? Posted in: Chemistry | Date: 21/02/2018 Neutron are neutral. They don't have positive or negative power • ## Question: Is it true that most of an atom is empty space? Please explain. Posted in: Chemistry | Date: 21/02/2018 Yes because atom consists of protons, neutrons and electrons. Size of these particles is very less than that of an atom • ## Question: How can the pressure of a gas be changed? Posted in: Physics | Date: 21/02/2018 Pressure of gas can be changed by changing its volume that is compressing it • ## Question: How do particles move in a gas? Posted in: Physics | Date: 21/02/2018 Particles in a gas are free to move thus they move in a random manner • ## Question: What is smaller than an atom? Posted in: Chemistry | Date: 21/02/2018 Protons, neutrons and electrons are smaller than atom • ## Question: What is inside of an atom? Posted in: Chemistry | Date: 21/02/2018 Protons and neutrons are inside atom and electrons revolve around it • ## Question: Can we see an atom? Posted in: Chemistry | Date: 21/02/2018 Yes Atoms can be seen with the help of electron microscope • ## Question: How does temperature affect the pressure of a gas? Posted in: Physics | Date: 21/02/2018 Pressure of gas increases with increase in temperature • ## Question: How do we measure gas pressure? Posted in: Physics | Date: 21/02/2018 Gas pressure is measured by instrument called barometer • ## Question: What is the difference between an atom and an ion? Posted in: Chemistry | Date: 21/02/2018 Atom is the particle while ion is the charge of an atom • ## Question: Do liquids have more kinetic energy than solids? Posted in: Physics | Date: 21/02/2018 Yes liquid has more kinetic energy than solid because atom of liquid are free to move • ## Question: Are atoms and elements the same thing? Posted in: Chemistry | Date: 22/02/2018 No atoms are basic particle of matter while element is made up of different types of atoms • ## Question: How can the pressure of a gas be increased? Posted in: Physics | Date: 22/02/2018 Pressure of gas can be increased by decreasing it's volume that is compressing it • ## Question: What is most of an atom made up of? Posted in: Chemistry | Date: 22/02/2018 Atom is made up of protons and neutrons • ## Question: What is the taste of bases? Posted in: Chemistry | Date: 22/02/2018 Bases are sour in taste • ## Question: What are the four main types of pollution? Posted in: Biology | Date: 22/02/2018 Air, water, noise and land pollution are major types of pollution • ## Question: Is $$CH_3CH_2COOH$$ an acid or a base? Posted in: Chemistry | Date: 22/02/2018 It is a base due to presence of OH group • ## Question: What are some examples of noise pollution? Posted in: Biology | Date: 22/02/2018 Vehicles , industrial machines and loud speaker • ## Question: What is the use of rainwater? Posted in: Geography | Date: 22/02/2018 Rain water is used for drinking, agriculture and industries • ## Question: How do you know if something is an acid or a base? Posted in: Chemistry | Date: 22/02/2018 It is measured with the help of litmus paper which measures ph value of substance • ## Question: What are the three main types of environment? Posted in: Biology | Date: 22/02/2018 Air, water and land are types of environment • ## Question: How many types of pollution are there? Posted in: Biology | Date: 22/02/2018 Four types :- air, water, land and noise • ## Question: Can we drink rain water to survive? Please explain. Posted in: Geography | Date: 22/02/2018 Yes we can drink rain water to servive • ## Question: What can we do with collected rainwater? Posted in: Geography | Date: 22/02/2018 Rain water can be used for drinking, agriculture and industries • ## Question: Why rain is important to the earth? Posted in: Geography | Date: 22/02/2018 Rain provides water for drinking, agriculture and industries • ## Question: What are the two major sources of air pollution? Posted in: Biology | Date: 22/02/2018 Emission from vehicles and smoke from industries • ## Question: What are the three main types of pollution? Posted in: Biology | Date: 22/02/2018 Air, Water and land pollution are major types of pollution • ## Question: Is $$KNO_3$$ an acid or base? Posted in: Chemistry | Date: 22/02/2018 KNO3 is an acid • ## Question: What are the different types of air pollutants? Posted in: Biology | Date: 22/02/2018 Carbon dioxide, carbon monoxide, sulphur dioxide are the main air pollutants • ## Question: What are the different types of pollutants? Posted in: Biology | Date: 22/02/2018 Carbon dioxide, carbon monoxide, sulphur dioxide are the main air pollutants • ## Question: Who won the battle between Alexander and Porus? Posted in: History | Date: 22/02/2018 Alexander won the battle • ## Question: Why do we need to save water? Posted in: Geography | Date: 22/02/2018 Water is necessary for the survivalist of every living organisms • ## Question: Why is it so important to save water? Posted in: Geography | Date: 22/02/2018 Water is necessary for the survival of every living organisms • ## Question: What is the use of rain? Posted in: Geography | Date: 22/02/2018 Rain is the only source of water on earth which is used for drinking and agriculture • ## Question: How does rain water harvesting help? Posted in: Geography | Date: 22/02/2018 Rain water harvesting provides water for drinking and agriculture • ## Question: What are the three main types of environment? Posted in: Biology | Date: 22/02/2018 Air, water and land are types of environment • ## Question: What is the use of rainwater harvesting? Posted in: Geography | Date: 22/02/2018 Rain water harvesting provides water for drinking and agriculture • ## Question: Is lemon juice an acid or a base? Posted in: Chemistry | Date: 22/02/2018 Lemon juice is an acid • ## Question: What is the advantage of rainwater harvesting? Posted in: Geography | Date: 22/02/2018 Rain water harvesting provides water for drinking and agriculture • ## Question: Is $$NaOH$$ an acid or a base? Posted in: Chemistry | Date: 22/02/2018 NaOH is a base • ## Question: What can we do with collected rainwater? Posted in: Geography | Date: 22/02/2018 Rain water is used for drinking and agriculture • ## Question: What makes a base? Posted in: Chemistry | Date: 22/02/2018 OH GROUP called hydroxyl group forms base • ## Question: Why rain water harvesting is important? Posted in: Geography | Date: 22/02/2018 Rain water harvesting provides water for drinking and agriculture • ## Question: Is $$CaCl_2$$ an acid or base or salt? Posted in: Chemistry | Date: 22/02/2018 CaCl2 is a salt • ## Question: What is the use of rainwater harvesting? Posted in: Geography | Date: 22/02/2018 Rain water harvesting provides water for drinking and agriculture • ## Question: What is the advantage of rainwater harvesting? Posted in: Geography | Date: 22/02/2018 Rain water harvesting provides water for drinking and agriculture • ## Question: Is $$H_2SO_4$$an acid or base or salt? Posted in: Chemistry | Date: 22/02/2018 H2SO4 is an acid • ## Question: What is the main purpose of rainwater harvesting? Posted in: Geography | Date: 22/02/2018 Rain water harvesting provides water for drinking and agriculture • ## Question: What are the benefits of rainwater harvesting? Posted in: Geography | Date: 22/02/2018 Rain water harvesting provides water for drinking and agriculture • ## Question: Is nacl a base or an acid? Posted in: Chemistry | Date: 22/02/2018 Nacl is a base • ## Question: What do you mean by arthashastra? Posted in: History | Date: 22/02/2018 Economics is called Arthshashtra in Sanskrit • ## Question: What is linear motion with examples? Posted in: Physics | Date: 22/02/2018 Motion in a straight line is called linear motion. Example motion of vehicles on highway • ## Question: Of late, my performance in school is going down, and I am also losing interest i... Posted in: Mathematics,Physics,Chemistry,Biology,Creative Writing Classes | Date: 23/02/2018 Don't loose hope there is never too late to start again. You are feeling disinterested because you are not well prepared but keep in mind this is your life and exam is important for your future so keep working hard even if you don't enjoy it because at the end you will feel fantastic when you get good results. • ## Question: What is the most spoken language in Asia? Posted in: Geography | Date: 23/02/2018 English is the most spoken language in Asia • ## Question: What is the newest country in Europe? Posted in: History | Date: 23/02/2018 Iceland is the newest country in Europe • ## Question: What is the largest country in Asia? Posted in: Geography | Date: 23/02/2018 China is the largest country in Asia • ## Question: What is the second biggest country in Asia? Posted in: Geography | Date: 23/02/2018 India is the second biggest country in Asia • ## Question: What are the rivers in Asia? Posted in: Geography | Date: 23/02/2018 • ## Question: What is the geography of Asia? Posted in: Geography | Date: 23/02/2018 Asia is geographically biggest continent on earth • ## Question: What is the difference between dry heat and humidity? Posted in: Geography | Date: 23/02/2018 Dry heat is air that doesn't contains water vapor while humidity is the water vapor contained in the air • ## Question: Which part of the atmosphere do planes fly in? Posted in: Geography | Date: 23/02/2018 Planes fly in stratosphere • ## Question: What is the amount of water in the air called? Posted in: Geography | Date: 23/02/2018 Amount of water in air is called moisture or humidity • ## Question: What layer of the atmosphere has satellites? Posted in: Geography | Date: 23/02/2018 Satellite lies outside atmosphere in space • ## Question: What is in the Earth's atmosphere? Posted in: Geography | Date: 23/02/2018 Earth's atmosphere consists of different types of gases like oxygen, nitrogen, carbon dioxide, etc in different proportions • ## Question: What are the different layers of the atmosphere? Posted in: Geography | Date: 23/02/2018 Troposphere, stratosphere, mesosphere, ionosphere are the layers of atmosphere • ## Question: What is the Earth's atmosphere made up of? Posted in: Geography | Date: 23/02/2018 Earth's atmosphere consists of different types of gases like oxygen, nitrogen, carbon dioxide, etc in different proportions • ## Question: Which is the oldest of all Vedas? Posted in: History | Date: 23/02/2018 Rigveda is the oldest of 4 Vedas • ## Question: How many Vedas are there? Posted in: History | Date: 23/02/2018 There are 4 types of Vedas • ## Question: What are the three parts of the earth's outer structure? Posted in: Geography | Date: 23/02/2018 Earth's crust consists of troposphere, stratosphere, mesosphere, ionosphere • ## Question: What is the structure and composition of the earth? Posted in: Geography | Date: 23/02/2018 Earth's atmosphere consists of different types of gases in different proportions • ## Question: What is the strongest radiation? Posted in: Chemistry | Date: 23/02/2018 • ## Question: What is the magnetic effect of an electric current? Posted in: Physics | Date: 23/02/2018 When current is passed through a conductor, magnetic field is produced around it. This is called magnetic field of electric current • ## Question: Why are solenoids useful? Posted in: Physics | Date: 23/02/2018 Solenoids are used in various types of sensors, fan, motors, etc • ## Question: What is the value of k in electric field? Posted in: Physics | Date: 23/02/2018 k in electric field is called di electric constant • ## Question: How do you find the atomic structure? Posted in: Chemistry | Date: 23/02/2018 Atomic structure is found with the help of electron microscope • ## Question: What is the most common type of power plant? Posted in: Physics | Date: 23/02/2018 Thermal, electric, hydodynamic, etc are types of power plants • ## Question: How electricity and magnetism are related? Posted in: Physics | Date: 23/02/2018 When current is passed through a conductor, magnetic field is produced around it • ## Question: Which devices use electromagnets? Posted in: Physics | Date: 23/02/2018 Transformers uses electromagnet • ## Question: Are protons and neutrons the same? Posted in: Chemistry | Date: 23/02/2018 No protons contains positive charge while neutrons are neutral without any charge • ## Question: How does a solenoid behaves like a magnet? Posted in: Physics | Date: 23/02/2018 When current is passed through a solenoid, magnetic field is produced around it • ## Question: How a battery is made up? Posted in: Physics | Date: 15/03/2018 Battery is made up of positive and negative terminals which is immersed inside liquid electrolyte • ## Question: What is the function of a resistor in a circuit? Posted in: Physics | Date: 15/03/2018 Function of resistor is to resist flow of electric current • ## Question: What is an electric cell made of? Posted in: Physics | Date: 15/03/2018 Electric cell is made up of positive and negative terminals which is immersed inside liquid electrolyte • ## Question: A dimensionally consistent relation for the volume V of a liquid of coefficient ... Posted in: Physics | Date: 15/03/2018 Option A is the correct answer • ## Question: E, m, l and G denote energy, mass, angular momentum and gravitational constant r... Posted in: Physics | Date: 15/03/2018 Option A is the correct answer • ## Question: The temperature of a body on Kelvin scale is found to be X K. When it is measure... Posted in: Physics | Date: 15/03/2018 Option B is the correct answer • ## Question: How can power loss be reduced? Posted in: Physics | Date: 17/03/2018 Power is lost in friction in mechanical system and in resistance in case of electric system • ## Question: How do you calculate the power? Posted in: Physics | Date: 17/03/2018 Power is the product of voltage and current • ## Question: What is the formula for power output? Posted in: Physics | Date: 17/03/2018 Formula for power is P = V*I where V is voltage and I is current. • ## Question: Why do we need a capacitor in a circuit? Posted in: Physics | Date: 17/03/2018 Capacitor in an electric circuits is used to store electric charge • ## Question: What are essential mineral elements? Posted in: Biology | Date: 17/03/2018 Essential mineral elements are proteins, vitamins, carbohydrates, fats • ## Question: Explain Interstitials: Posted in: Chemistry | Date: 17/03/2018 When atoms in a crystal lattice takes the unoccupied space than it is called interstitial atom • ## Question: What is a semiconductor? Describe the two main types of semiconductors. Posted in: Chemistry | Date: 17/03/2018 Material which is neither a good conductor of electricity nor good insulator is called semi conductor. Silicon and germanium are two types of semi conductor • ## Question: What does a switch do in an electric circuit? Posted in: Physics | Date: 21/03/2018 Switch breaks the contact between the two links and thereby stops the flow of electric current • ## Question: How do you read the value of a resistor? Posted in: Physics | Date: 21/03/2018 Value of resistance is read in a unit called ohm • ## Question: What do you do if your battery dies while driving? Posted in: Physics | Date: 21/03/2018 Adding distilled water in the battery can start the battery and if not start working then battery must be Recharged • ## Question: Where is the LGN? Posted in: Physics | Date: 21/03/2018 LGN is lateral geniculate nucleus • ## Question: Why do batteries die out? Posted in: Physics | Date: 21/03/2018 Due to lack of electrolyte, worn out electrodes, dilute electrolyte. • ## Question: Which device is used to break an electric circuit? Posted in: Physics | Date: 21/03/2018 Switch is used to break an electric circuits • ## Question: What is the difference between voltage and current? Posted in: Physics | Date: 21/03/2018 Voltage is the potential difference between the two points while current is the flow of electric charge or flow of electrons • ## Question: What is the different between a cell and a battery? Posted in: Physics | Date: 23/03/2018 Cell is a single unit producing electric current while battery is made up of numbers of cells to get desired current • ## Question: How does a simple cell produce electricity? Posted in: Physics | Date: 23/03/2018 Simple cell produces electric current due to the chemical reaction taking place between the positive and negative terminals and electrolyte • ## Question: Where are simple cells? Posted in: Physics | Date: 23/03/2018 Simple cell is a device used to produce electric current • ## Question: How does a battery work in simple terms? Posted in: Physics | Date: 23/03/2018 Battery produces electric current as a result of the chemical reaction taking place between the positive and negative terminals and electrolyte • ## Question: What does the voltage tell you? Posted in: Physics | Date: 23/03/2018 Voltage is the potential difference between the two points in an electric circuit • ## Question: What is the function of a cell in an electrical circuit? Posted in: Physics | Date: 23/03/2018 Cell is a device used to produce electric current • ## Question: What is the difference between an electric cell and a battery? Posted in: Physics | Date: 23/03/2018 Cell is a single unit producing electric current while battery is made up of numbers of cells to get desired current • ## Question: What is a simple cell? Posted in: Physics | Date: 23/03/2018 Simple cell is a device used to produce electric current • ## Question: How does a electric cell work? Posted in: Physics | Date: 23/03/2018 Cell produces electric current as a result of the chemical reaction taking place between the positive and negative terminals and electrolyte • ## Question: What is a cell in an electrical circuit? Posted in: Physics | Date: 23/03/2018 Cell is a device used to produce electric current • ## Question: Isotopes of an element differ in the number of ............. in their nuclei. Posted in: Chemistry | Date: 23/03/2018 Isotope of an element differs in terms of numbers of neutron inside nuclei • ## Question: What are the units of K = $$\frac {1}{4\pi\varepsilon_0 }$$? Posted in: Physics | Date: 23/03/2018 • ## Question: The unit of permittivity of free space $$\varepsilon_0$$ is ..... Posted in: Physics | Date: 23/03/2018 • ## Question: From the equation tan θ = $$\frac {rg}{v^2}$$, one can obtain the angle of... Posted in: Physics | Date: 23/03/2018 • ## Question: The temperature of a body on Kelvin scale is found to be X K. When it is measure... 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Posted in: Mathematics | Date: 23/03/2018 ### Answer: Answer is option B • ## Question: If the sets A and B are defined as \(A=\{(x,y):y=e^x, x \in R\} ; B=\{(x,y)... Posted in: Mathematics | Date: 23/03/2018 ### Answer: Answer is option C • ## Question: Which Mountain's volcano buried Pompeii city with ash in 79 AD? Posted in: History | Date: 24/03/2018 ### Answer: Answer is option D • ## Question: When did Julius Caesar first invaded Britain? Posted in: History | Date: 24/03/2018 ### Answer: Answer is option C. 55BC • ## Question: Industrial Revolution was started from which country? Posted in: History | Date: 24/03/2018 ### Answer: Industrial revolution started in England • ## Question: If the sets A and B are defined as \(A=\{ (x,y): y = \frac{1}{x}, 0 \neq x ... Posted in: Mathematics | Date: 24/03/2018 ### Answer: Answer is option C • ## Question: Given the sets A={1,2,3}, B={3,4}, C={4,.5,6}, then \(A\cup (B\cap C)$$&nbs... 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Posted in: Chemistry | Date: 17/04/2018 Quarks are the tiniest particles know. Many quarks combine to form atoms. • ## Question: What is the Valency of neon? Posted in: Chemistry | Date: 17/04/2018 Vacancy of neon is zero • ## Question: What are magnetic lines of force? Roughly trace the magnetic field lines for a b... Posted in: Physics | Date: 17/04/2018 Magnetic lines of force are imaginary lines of force. Tangent to the lines determine the magnitude of force. • ## Question: What is the Valency of zinc? Posted in: Chemistry | Date: 17/04/2018 Vacancy of zinc is 2. • ## Question: What is meant by the term “Magnetic field lines”? List any two properties of... Posted in: Physics | Date: 17/04/2018 Magnetic field lines are imaginary lines of force. Tangent to it determines the magnitude of force. • ## Question: What is smaller than an atom quark? Posted in: Chemistry | Date: 17/04/2018 Quark is the smallest particles known.

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BBA Finance Degree Courses MBA Cost Accounting Certification Exam Tests MBA Cost Accounting Practice Test 39 Books: Apps: The Joint Cost Basics Trivia Questions and Answers PDF (Joint Cost Basics Quiz Answers PDF e-Book) download Ch. 10-39 to solve MBA Cost Accounting Practice Tests. Learn Cost Allocation Joint Products and Byproducts MCQ Questions PDF, Joint Cost Basics Multiple Choice Questions (MCQ Quiz) to study accounting certificate courses. The Joint Cost Basics Trivia App Download: Free educational app for joint cost basics, learning growth perspective: quality improvements, direct costs and indirect costs, inventory costing methods, throughput costing test prep for online schools for business administration. The Trivia MCQ: In a joint process of production, the product which yields low volume of sales as compared to total sales of other products, specify as; "Joint Cost Basics" App (iOS & Android) with answers: First incremental product; Second incremental product; Step down product; Byproduct; for online schools for business administration. Study Cost Allocation Joint Products and Byproducts Questions and Answers, Google eBook to download free sample for online bachelor degree programs in business administration. ## Joint Cost Basics Quiz with Answers : MCQs 39 MCQ 191: In a joint process of production, the product which yields low volume of sales as compared to total sales of other products, specify as 1. Second incremental product 2. First incremental product 3. step down product 4. byproduct MCQ 192: The number of employees who left the company, divided by average number of employees to calculate the ratio is called 1. employee turnover ratio 2. employee empowerment ratio 3. employee satisfaction ratio 4. employee training percentage MCQ 193: The cost which is changed in proportion to level the total volume is 1. fixed cost 2. variable cost 3. total cost 4. infeasible cost MCQ 194: In actual costing, an actual quantity of used inputs are multiplied with actual prices to calculate 1. fixed direct manufacturing cost 2. variable direct manufacturing cost 3. fixed indirect manufacturing cost 4. variable indirect manufacturing cost MCQ 195: Throughout the period costs, costing methods are treated as 1. manufacturing in period 2. expenses of period 3. incurred in period 4. accrual in period

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Check the below NCERT MCQ Questions for Class 11 Physics Chapter 3 Motion in a Straight Line with Answers Pdf free download. MCQ Questions for Class 11 Physics with Answers were prepared based on the latest exam pattern. We have provided Motion in a Straight Line Class 11 Physics MCQs Questions with Answers to help students understand the concept very well. ## Motion in a Straight Line Class 11 MCQs Questions with Answers Motion In A Straight Line Class 11 MCQ Question 1. The displacement in metres of a body varies with time t in second as y = t2 – t – 2. The displacement is zero for a positive of t equal to (a) 1 s (b) 2 s (c) 3 s (d) 4 s Answer: (b) 2 s Class 11 Physics Chapter 3 MCQ Question 2. A boy starts from a point A, travels to a point B at a distance of 3 km from A and returns to A. If he takes two hours to do so, his speed is (a) 3 km/h (b) zero (c) 2 km/h (d) 1.5 km/h Answer: (a) 3 km/h Motion In A Straight Line MCQ Question 3. A 180 metre long train is moving due north at a speed of 25 m/s. A small bird is flying due south, a little above the train, with a speed of 5 m/s. The time taken by the bird to cross the train is (a) 10 s (b) 12 s (c) 9 s (d) 6 s Answer: (d) 6 s MCQ Questions For Class 11 Physics Chapter 3 Question 4. A boy starts from a point A, travels to a point B at a distance of 1.5 km and returns to A. If he takes one hour to do so, his average velocity is (a) 3 km/h (b) zero (c) 1.5 km/h (d) 2 km/h Motion In A Straight Line Class 11 MCQs Question 5. A body starts from rest and travels with uniform acceleration on a straight line. If its velocity after making a displacement of 32 m is 8 m/s, its acceleration is (a) 1 m/s² (b) 2 m/s² (c) 3 m/s² (d) 4 m/s² Answer: (a) 1 m/s² Class 11 Physics Chapter 3 MCQ With Answers Question 6. Which one of the following is the unit of velocity? (a) kilogram (b) metre (c) m/s (d) second Motion In Straight Line Class 11 MCQ Question 7. A body starts from rest and travels for t second with uniform acceleration of 2 m/s². If the displacement made by it is 16 m, the time of travel t is (a) 4 s (b) 3 s (c) 6 s (d) 8 s Answer: (a) 4 s Ch 3 Physics Class 11 MCQ Question 8. The dimensional formula for speed is (a) T-1 (b) LT-1 (c) L-1T-1 (d) L-1T Class 11 Physics Ch 3 MCQ Question 9. The dimensional formula for velocity is (a) [LT] (b) [LT-1] (c) [L2T] (d) [L-1T] Motion In Straight Line MCQ Question 10. A body starts from rest and travels with an acceleration of 2 m/s². After t seconds its velocity is 10 m/s . Then t is (a) 10 s (b) 5 s (c) 20 s (d) 6 s Answer: (b) 5 s Class 11 Motion In A Straight Line MCQ Question 11. A boy starts from a point A, travels to a point B at a distance of 1.5 km and returns to A. If he takes one hour to do so, his average velocity is (a) 3 km/h (b) zero (c) 1.5 km/h (d) 2 km/h Chapter 3 Physics Class 11 MCQ Question 12. A body starts from rest. If it travels with an acceleration of 2 m/s², its displacement at the end of 3 seconds is (a) 9 m (b) 12 m (c) 16 m (d) 10 m Answer: (a) 9 m Physics Class 11 Chapter 3 MCQ Question 13. A body starts from rest and travels with uniform acceleration of 2 m/s². If its velocity is v after making a displacement of 9 m, then v is (a) 8 m/s (b) 6 m/s (c) 10 m/s (d) 4 m/s Answer: (b) 6 m/s Chapter 3 Physics Class 11 MCQs Question 14. A body starts from rest and travels with an acceleration of 2 m/s². After t seconds its velocity is 10 m/s. Then t is (a) 10 s (b) 5 s (c) 20 s (d) 6 s Answer: (b) 5 s MCQ Of Ch 3 Physics Class 11 Question 15. A body starts from rest and travels for five seconds to make a displacement of 25 m if it has travelled the distance with uniform acceleration a then a is (a) 3 m/s² (b) 4 m/s² (c) 2 m/s² (d) 1 m/s² Answer: (c) 2 m/s² MCQ On Motion In A Straight Line Question 16: A boy moves on a circular distance of radius R. Starting from a point A he moves to a point B which is on the other end of the diameter AB. The ratio of the distance travelled to the displacement made by him is (a) ∏/2 (b) ∏ (c) 2∏ (d) 4∏ Class 11 Physics Motion In A Straight Line MCQ With Answers Question 17. The dimensional formula for acceleration is (a) [LT2] (b) [LT?2] (c) [L2T] (d) [L2T2] MCQ Of Chapter 3 Physics Class 11 Question 18. A body starts from rest and travels with uniform acceleration a to make a displacement of 6 m. If its velocity after making the displacement is 6 m/s, then its uniform acceleration a is (a) 6 m/s² (b) 2 m/s² (c) 3 m/s² (d) 4 m/s² Answer: (c) 3 m/s² MCQ Class 11 Physics Chapter 3 Question 19. Which one of the following is the unit of velocity? (a) kilogram (b) metre (c) m/s (d) second Physics Chapter 3 MCQ Class 11 Question 20. The displacement in metres of a body varies with time t in second as y = t2 – t – 2. The displacement is zero for a positive of t equal to (a) 1 s (b) 2 s (c) 3 s (d) 4 s

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Function Repository Resource: # GlobalExtrema Compute the global extrema of an expression with respect to the given variables Contributed by: Wolfram|Alpha Math Team ResourceFunction["GlobalExtrema"][expr,x] computes the global maxima and minima of expr with respect to x. ResourceFunction["GlobalExtrema"][expr,{x,y,…}] computes the global maxima and minima of expr with respect to multiple variables. ResourceFunction["GlobalExtrema"][{expr,const},{x,y,…}] computes the global maxima and minima of expr subject to the constraint const. ## Details and Options ResourceFunction["GlobalExtrema"] returns an association of the form <|"Maxima"{fmax,{xxmax,yymax,}},"Minima"{fmin,{xxmin,yymin,}}|>. The const can contain equations, inequalities or logical combinations of these. ResourceFunction["GlobalExtrema"] only returns results when there is a bounded extremum. ## Examples ### Basic Examples (2) Compute the global extrema of a curve: In[1]:= Out[1]= Plot the result: In[2]:= Out[2]= Use a constraint in order to reduce the domain upon which extrema can be found: In[3]:= Out[3]= ### Scope (2) Compute the extrema of a function of two variables: In[4]:= Out[4]= Compute the extrema of a piecewise function: In[5]:= Out[5]= Plot the result: In[6]:= Out[6]= ### Possible Issues (1) GlobalExtrema may return duplicate results for periodic functions: In[7]:= Out[7]= ## Publisher Wolfram|Alpha Math Team ## Version History • 2.0.0 – 23 March 2023 • 1.0.0 – 22 September 2020 ## Author Notes To view the full source code for GlobalMaxima, run the following code: In[1]:=

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Mathematics # Mathematics ## Mathematics - - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - - ##### Presentation Transcript 1. Mathematics 2. Session Matrices and Determinants - 2 3. Session Objectives • Determinant of a Square Matrix • Minors and Cofactors • Properties of Determinants • Applications of DeterminantsArea of a TriangleSolution of System of Linear Equations (Cramer’s Rule) • Class Exercise 4. Determinants If is a square matrix of order 1, then |A| = | a11 | = a11 If is a square matrix of order 2, then |A| = = a11a22 – a21a12 5. Example 6. If A = is a square matrix of order 3, then Solution [Expanding along first row] 7. Example Solution : [Expanding along first row] 8. Minors 9. Similarly, M23 = Minor of a23 M32 = Minor of a32 etc. Minors M11 = Minor of a11 = determinant of the order 2 × 2 square sub-matrix is obtained by leaving first row and first column of A 10. Cofactors 11. C11 = Cofactor of a11 = (–1)1 + 1 M11 = (–1)1 +1 C23 = Cofactor of a23 = (–1)2 + 3 M23 = C32 = Cofactor of a32 = (–1)3 + 2M32 = etc. Cofactors (Con.) 12. Value of Determinant in Terms of Minors and Cofactors 13. Properties of Determinants 1. The value of a determinant remains unchanged, if its rows and columns are interchanged. 2. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant is changed by minus sign. 14. Properties (Con.) 3. If all the elements of a row (or column) is multiplied by a non-zero number k, then the value of the new determinant is k times the value of the original determinant. which also implies 15. Properties (Con.) 4. If each element of any row (or column) consists of two or more terms, then the determinant can be expressed as the sum of two or more determinants. 5. The value of a determinant is unchanged, if any row (or column) is multiplied by a number and then added to any other row (or column). 16. Properties (Con.) 6. If any two rows (or columns) of a determinant are identical, then its value is zero. If each element of a row (or column) of a determinant is zero, then its value is zero. 17. Properties (Con.) 18. (i) Ri to denote ith row (ii) Ri Rj to denote the interchange of ith and jth rows. (iii) Ri Ri + lRj to denote the addition of l times the elements of jth row to the corresponding elements of ith row. (iv) lRi to denote the multiplication of all elements of ith row by l. Row(Column) Operations Following are the notations to evaluate a determinant: Similar notations can be used to denote column operations by replacing R with C. 19. If a determinant becomes zero on putting is the factor of the determinant. , because C1 and C2 are identical at x = 2 Hence, (x – 2) is a factor of determinant . Evaluation of Determinants 20. Sign System for Expansion of Determinant Sign System for order 2 and order 3 are given by 21. Example-1 Find the value of the following determinants (i) (ii) Solution : 22. (ii) Example –1 (ii) 23. Example - 2 Evaluate the determinant Solution : 24. Example - 3 Evaluate the determinant: Solution: 25. Solution Cont. Now expanding along C1 , we get (a-b) (b-c) (c-a) [- (c2 – ab – ac – bc – c2)] = (a-b) (b-c) (c-a) (ab + bc + ac) 26. Example-4 Without expanding the determinant, prove that Solution : 27. Solution Cont. 28. Example -5 Prove that : = 0 , where w is cube root of unity. Solution : 29. Example-6 Prove that : Solution : 30. Solution cont. Expanding along C1 , we get (x + a + b + c) [1(x2)] = x2 (x + a + b + c) = R.H.S 31. Example -7 Using properties of determinants, prove that Solution : 32. Solution Cont. Now expanding along R1 , we get 33. Example - 8 Using properties of determinants prove that Solution : 34. Solution Cont. Now expanding along C1 , we get 35. Using properties of determinants, prove that Example -9 Solution : 36. Solution Cont. 37. Example -10 Show that Solution : 38. Solution Cont. 39. The area of a triangle whose vertices are is given by the expression Applications of Determinants (Area of a Triangle) 40. Example Find the area of a triangle whose vertices are (-1, 8), (-2, -3) and (3, 2). Solution : 41. If are three points, then A, B, C are collinear Condition of Collinearity of Three Points 42. Example If the points (x, -2) , (5, 2), (8, 8) are collinear, find x , using determinants. Solution : Since the given points are collinear. 43. Solution of System of 2 Linear Equations (Cramer’s Rule) Let the system of linear equations be 44. Cramer’s Rule (Con.) then the system is consistent and has unique solution. then the system is consistent and has infinitely many solutions. then the system is inconsistent and has no solution. 45. Example Using Cramer's rule , solve the following system of equations 2x-3y=7, 3x+y=5 Solution : 46. Solution of System of 3 Linear Equations (Cramer’s Rule) Let the system of linear equations be 47. Cramer’s Rule (Con.) • Note: • (1) If D  0, then the system is consistent and has a unique solution. • (2) If D=0 and D1 = D2 = D3 = 0, then the system has infinite solutions or no solution. • (3) If D = 0 and one of D1, D2, D3 0, then the system • is inconsistent and has no solution. • If d1 = d2 = d3 = 0, then the system is called the system of homogeneous linear equations. • If D  0, then the system has only trivial solution x = y = z = 0. • (ii)If D = 0, then the system has infinite solutions. 48. = 5(18+10) + 1(12-25)+4(-4 -15)= 140 –13 –76 =140 - 89 = 51 Example Using Cramer's rule , solve the following system of equations5x - y+ 4z = 5 2x + 3y+ 5z = 2 5x - 2y + 6z = -1 Solution : = 5(18+10)+1(12+5)+4(-4 +3)= 140 +17 –4= 153 49. Solution Cont. = 5(12 +5)+5(12 - 25)+ 4(-2 - 10)= 85 + 65 – 48 = 150 - 48= 102 = 5(-3 +4)+1(-2 - 10)+5(-4-15)= 5 – 12 – 95 = 5 - 107= - 102 50. Example Solve the following system of homogeneous linear equations: x + y – z = 0, x – 2y + z = 0, 3x + 6y + -5z = 0 Solution: Putting z = k, in first two equations, we get x + y = k, x – 2y = -k

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# How to Choose the Right Body Fat Percentage Prediction Equation Body fat percentage prediction equations are formulas used to estimate the amount of body fat a person has. They are typically used by fitness professionals and researchers to assess body composition. One common example is the Jackson-Pollock 3-site skinfold equation, which uses measurements from the triceps, abdominal, and thigh to predict body fat percentage. These equations are important because they provide a non-invasive way to estimate body fat. They are also relatively inexpensive and easy to use. Historically, the first body fat prediction equation was developed in 1945 by Siri. In this article, we will explore the different types of body fat percentage prediction equations, their accuracy, and their limitations. We will also provide some tips on how to use these equations effectively. ## Body Fat Percentage Prediction Equation Body fat percentage prediction equations are essential tools for fitness professionals and researchers to assess body composition. They provide a non-invasive, relatively inexpensive, and easy way to estimate the amount of body fat a person has. • Accuracy • Convenience • Predictive ability • Simplicity • Reliability • Objectivity • Standardization • Validation • Versatility These equations are based on measurements of body size and composition, such as height, weight, and skinfold thickness. They are typically developed using statistical methods, such as regression analysis, and are validated against more accurate methods of body fat assessment, such as underwater weighing or dual-energy X-ray absorptiometry (DXA). ### Accuracy Accuracy is a critical component of body fat percentage prediction equations. The accuracy of these equations depends on the quality of the data used to develop them, as well as the statistical methods used. In general, equations that are developed using a large and diverse sample of people are more accurate than those that are developed using a small or homogeneous sample. There are a number of factors that can affect the accuracy of body fat percentage prediction equations. These factors include the age, sex, ethnicity, and fitness level of the person being measured. Additionally, the accuracy of these equations can be affected by the type of measurement technique used. For example, skinfold measurements are less accurate than underwater weighing or DXA. Despite these limitations, body fat percentage prediction equations can be a useful tool for fitness professionals and researchers. They provide a quick and easy way to estimate body fat percentage, which can be helpful for setting fitness goals and tracking progress. However, it is important to remember that these equations are not perfect and should be used with caution. ### Convenience Convenience is a key aspect of body fat percentage prediction equations. These equations are designed to be easy to use and implement, making them accessible to a wide range of users. Here are some specific facets of convenience related to body fat percentage prediction equations: • Simplicity Body fat percentage prediction equations are typically simple to understand and use. They require minimal training and can be used by people of all fitness levels. • Accessibility Body fat percentage prediction equations are accessible to a wide range of users. They can be found online, in fitness magazines, and even on some fitness equipment. This makes them easy to find and use. • Cost-effectiveness Body fat percentage prediction equations are cost-effective. They do not require any special equipment or training, making them a low-cost option for assessing body fat percentage. • Time-efficiency Body fat percentage prediction equations are time-efficient. They can be used to quickly and easily estimate body fat percentage, making them a convenient option for busy people. Overall, convenience is a key factor that makes body fat percentage prediction equations a valuable tool for fitness professionals and researchers. These equations are simple, accessible, cost-effective, and time-efficient, making them a convenient option for assessing body fat percentage. ### Predictive ability Predictive ability is a critical aspect of body fat percentage prediction equations. It refers to the equation’s capacity to accurately estimate body fat percentage based on input measurements. Predictive ability is determined by various factors, including the equation’s design, the quality of the data used to develop it, and the population for which it is intended. • Accuracy Accuracy refers to the equation’s ability to generate estimates that are close to the true body fat percentage. Accuracy is typically assessed by comparing the equation’s predictions to measurements obtained from more accurate methods, such as underwater weighing or dual-energy X-ray absorptiometry (DXA). • Precision Precision refers to the equation’s ability to generate consistent estimates across multiple measurements. Precision is typically assessed by calculating the standard error of estimate (SEE), which measures the average difference between the equation’s predictions and the true body fat percentage. • Bias Bias refers to the equation’s tendency to overestimate or underestimate body fat percentage. Bias is typically assessed by calculating the mean difference between the equation’s predictions and the true body fat percentage. • Generalizability Generalizability refers to the equation’s ability to generate accurate estimates for a wide range of individuals. Generalizability is typically assessed by examining the equation’s performance across different populations, such as different age groups, ethnicities, and fitness levels. Predictive ability is essential for body fat percentage prediction equations to be useful in practice. Equations with high predictive ability can be used to make accurate estimates of body fat percentage, which can be helpful for setting fitness goals, tracking progress, and assessing health risks. Overall, predictive ability is a key factor to consider when choosing a body fat percentage prediction equation. ### Simplicity Simplicity is a critical component of body fat percentage prediction equations. These equations are designed to be easy to understand and use, even by people with no prior experience with body fat assessment. This simplicity is essential for making these equations accessible to a wide range of users, including fitness professionals, researchers, and the general public. There are a number of ways in which simplicity is incorporated into body fat percentage prediction equations. First, these equations typically use a limited number of input variables. For example, the Jackson-Pollock 3-site skinfold equation uses only three skinfold measurements (triceps, abdominal, and thigh) to predict body fat percentage. This simplicity makes the equation easy to use and reduces the risk of error. Second, body fat percentage prediction equations are often designed to be easy to calculate. Many of these equations can be solved using a simple calculator or even a spreadsheet. This simplicity makes it easy for users to quickly and easily estimate their body fat percentage. The simplicity of body fat percentage prediction equations has a number of practical applications. These equations can be used to set fitness goals, track progress, and assess health risks. They can also be used to screen for obesity and other health conditions. ### Reliability Reliability is a critical component of body fat percentage prediction equations. A reliable equation is one that produces consistent results when used to estimate body fat percentage. This consistency is important because it allows users to have confidence in the accuracy of the equation’s predictions. There are a number of factors that can affect the reliability of a body fat percentage prediction equation. One important factor is the quality of the data used to develop the equation. Equations that are developed using a large and diverse sample of people are more likely to be reliable than those that are developed using a small or homogeneous sample. Another important factor that affects the reliability of a body fat percentage prediction equation is the type of measurement technique used. Some measurement techniques, such as skinfold measurements, are less reliable than others, such as underwater weighing or dual-energy X-ray absorptiometry (DXA). Equations that use more reliable measurement techniques are more likely to produce consistent results. The reliability of a body fat percentage prediction equation is important for a number of reasons. First, it allows users to have confidence in the accuracy of the equation’s predictions. Second, it allows users to compare their body fat percentage to others who have used the same equation. Third, it allows researchers to use body fat percentage prediction equations to track changes in body fat over time. ### Objectivity Objectivity is a critical component of body fat percentage prediction equations. An objective equation is one that produces results that are not influenced by the subjective opinions or biases of the user. This is important because it allows users to have confidence that the equation’s predictions are accurate and reliable. There are a number of ways in which objectivity is incorporated into body fat percentage prediction equations. First, these equations are typically based on well-established scientific principles. For example, the Jackson-Pollock 3-site skinfold equation is based on the relationship between skinfold thickness and body fat percentage. This relationship has been established through extensive research, and it is supported by a large body of evidence. Second, body fat percentage prediction equations are typically developed using a large and diverse sample of people. This helps to ensure that the equation is not biased towards any particular group of people. For example, the Jackson-Pollock 3-site skinfold equation was developed using a sample of over 1,000 people from a variety of ages, ethnicities, and fitness levels. The objectivity of body fat percentage prediction equations is important for a number of reasons. First, it allows users to have confidence in the accuracy of the equation’s predictions. Second, it allows users to compare their body fat percentage to others who have used the same equation. Third, it allows researchers to use body fat percentage prediction equations to track changes in body fat over time. ### Standardization Standardization is a critical component of body fat percentage prediction equations. It ensures that the equations are consistent and accurate, and that they can be used to compare body fat percentage across different individuals and populations. Without standardization, it would be difficult to compare body fat percentage measurements from different studies or to track changes in body fat percentage over time. One important aspect of standardization is the use of standardized measurement techniques. For example, the Jackson-Pollock 3-site skinfold equation requires that skinfold thickness be measured at three specific sites on the body: the triceps, the abdomen, and the thigh. The use of standardized measurement techniques helps to ensure that the results are consistent and reliable. Another important aspect of standardization is the use of standardized equations. There are a number of different body fat percentage prediction equations available, but they are not all equally accurate or reliable. The use of standardized equations helps to ensure that the results are consistent and valid. Standardization is essential for the accurate and reliable use of body fat percentage prediction equations. By using standardized measurement techniques and equations, researchers and practitioners can ensure that they are comparing body fat percentage measurements in a consistent and meaningful way. ### Validation Validation is the process of determining the accuracy and reliability of a measurement tool or procedure. In the context of body fat percentage prediction equations, validation is essential to ensure that the equations are producing accurate and reliable estimates of body fat percentage. There are a number of different ways to validate a body fat percentage prediction equation. One common method is to compare the equation’s predictions to measurements obtained from more accurate methods, such as underwater weighing or dual-energy X-ray absorptiometry (DXA). Another method is to compare the equation’s predictions to measurements obtained from a large and diverse sample of people. Validation is a critical component of body fat percentage prediction equations because it provides users with confidence in the accuracy and reliability of the equations. Without validation, it would be difficult to know whether or not the equations are producing accurate estimates of body fat percentage. As a result, validation is essential for the effective use of body fat percentage prediction equations in research and practice. Real-life examples of validation in the context of body fat percentage prediction equations include the studies conducted by Jackson and Pollock (1978) and Siri (1961). Jackson and Pollock validated their 3-site skinfold equation by comparing its predictions to measurements obtained from underwater weighing. Siri validated his body density equation by comparing its predictions to measurements obtained from underwater weighing and DXA. The practical applications of understanding the connection between validation and body fat percentage prediction equations are numerous. For example, this understanding can be used to: • Select the most accurate and reliable body fat percentage prediction equation for a given application. • Interpret the results of body fat percentage prediction equations with confidence. • Use body fat percentage prediction equations to track changes in body fat over time. In conclusion, validation is a critical component of body fat percentage prediction equations. By understanding the connection between validation and body fat percentage prediction equations, researchers and practitioners can ensure that they are using the most accurate and reliable equations for their needs. ### Versatility Versatility is a critical component of body fat percentage prediction equations. It refers to the equation’s ability to accurately estimate body fat percentage across a wide range of individuals and populations. Versatile equations are able to account for differences in age, sex, ethnicity, fitness level, and body shape. The versatility of a body fat percentage prediction equation is important for a number of reasons. First, it allows users to select an equation that is appropriate for their individual needs. Second, it ensures that the equation will produce accurate and reliable results, even when used with different populations. Third, it allows researchers to compare body fat percentage measurements across different studies and populations. There are a number of different factors that can affect the versatility of a body fat percentage prediction equation. One important factor is the number of input variables used by the equation. Equations that use a large number of input variables are more likely to be versatile than those that use a small number of input variables. Another important factor is the type of input variables used by the equation. Equations that use a variety of input variables, such as skinfold thickness, circumferences, and body weight, are more likely to be versatile than those that use only a single input variable. Real-life examples of versatile body fat percentage prediction equations include the Jackson-Pollock 3-site skinfold equation and the Siri body density equation. These equations have been shown to be accurate and reliable across a wide range of individuals and populations. As a result, they are commonly used in research and practice. The practical applications of understanding the connection between versatility and body fat percentage prediction equations are numerous. For example, this understanding can be used to: • Select the most versatile body fat percentage prediction equation for a given application. • Interpret the results of body fat percentage prediction equations with confidence. • Use body fat percentage prediction equations to track changes in body fat over time. In conclusion, versatility is a critical component of body fat percentage prediction equations. By understanding the connection between versatility and body fat percentage prediction equations, researchers and practitioners can ensure that they are using the most accurate and reliable equations for their needs. ### Frequently Asked Questions (FAQs) on Body Fat Percentage Prediction Equations This section addresses common questions and concerns regarding body fat percentage prediction equations, providing clear and concise answers to enhance understanding. Question 1: What are body fat percentage prediction equations? Answer: Body fat percentage prediction equations are mathematical formulas that estimate an individual’s body fat percentage based on measurements such as body weight, height, and skinfold thickness. Question 2: Why are body fat percentage prediction equations important? Answer: These equations provide a non-invasive and convenient method to assess body fat, which is crucial for fitness monitoring, weight management, and health risk assessment. Question 3: How accurate are body fat percentage prediction equations? Answer: The accuracy of these equations varies depending on the specific equation used and individual factors. However, many equations have been validated against more precise methods and provide reasonable estimates. Question 4: What factors can affect the accuracy of body fat percentage prediction equations? Answer: Factors such as age, sex, ethnicity, fitness level, and hydration status can influence the accuracy of these equations. Question 5: How to choose the most appropriate body fat percentage prediction equation? Answer: Consider the purpose of the assessment, the available measurement tools, and the population being studied when selecting an equation. Question 6: What are the limitations of body fat percentage prediction equations? Answer: These equations provide an estimate rather than an exact measurement, and they may not be suitable for individuals with certain body compositions or medical conditions. In summary, body fat percentage prediction equations offer a valuable tool for estimating body fat non-invasively, but their accuracy and applicability can vary. Understanding the factors influencing their accuracy and limitations is crucial for effective utilization. The next section will delve into the practical applications of body fat percentage prediction equations in various settings, including fitness, research, and clinical practice. ### Tips for Using Body Fat Percentage Prediction Equations This section provides essential tips to enhance the effective usage of body fat percentage prediction equations in practice. Tip 1: Select an appropriate equation. Consider the intended purpose, available measurement tools, and the population being assessed when choosing an equation. Tip 2: Ensure accurate measurements. Follow standardized measurement protocols, and use reliable measuring instruments to obtain precise body measurements. Tip 3: Consider individual factors. Be aware of how age, sex, ethnicity, and fitness level can influence the accuracy of the equations. Tip 4: Validate the equation. If possible, compare the results of the prediction equation with a more precise method, such as underwater weighing or dual-energy X-ray absorptiometry, to assess its accuracy. Tip 5: Use the equation consistently. Employ the same equation over time to track changes in body fat percentage accurately. Tip 6: Interpret results with caution. Remember that body fat percentage prediction equations provide an estimate and may not be suitable for individuals with certain body compositions or medical conditions. By following these tips, practitioners and researchers can optimize the use of body fat percentage prediction equations to obtain reliable and meaningful body fat estimates. The next section will explore the practical applications of body fat percentage prediction equations in various settings, such as fitness, research, and clinical practice. ### Conclusion In summary, body fat percentage prediction equations offer a convenient and non-invasive method to estimate body fat. These equations utilize various input variables, such as body measurements or bioelectrical impedance, to provide an approximation of an individual’s body fat percentage. While they may not be as precise as more advanced methods like underwater weighing or dual-energy X-ray absorptiometry, these equations can be valuable tools for fitness professionals, researchers, and individuals seeking to monitor their body composition. Understanding the strengths and limitations of body fat percentage prediction equations is crucial. Factors such as age, sex, ethnicity, and hydration status can influence their accuracy. Selecting an appropriate equation, ensuring accurate measurements, and interpreting results with caution are essential for effective utilization. By following these guidelines, practitioners and researchers can harness the benefits of these equations to obtain reliable insights into body fat composition.

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# Chapter 5 Sampling and Surveys ## 5.1 Introduction Recall from Chapter 3 that most research questions actually break down into 2 parts: • Descriptive Statistics: What relationship can we observe between the variables, in the sample? • Inferential Statistics: Supposing we see a relationship in the sample data, how much evidence is provided for a relationship in the population? Does the data provide lots of evidence for a relationship in the population, or could the relationship we see in the sample be due just to chance variation in the sampling process that gave us the data? Both parts of answering research questions involve dealing with the sample. In order to make valid conclusions about any research question, we first need to make sure we are dealing with a good sample. This chapter will discuss various techniques for drawing samples, the strengths and weaknesses of these sampling techniques, and the uses and abuses of statistics. ## 5.2 Population versus Sample In the past few of chapters, we have looked at both parts of research questions. An important distinction that we want to make sure has been made before we go any further is the distinction between a sample and a population. Population A population is the set of all subjects of interest. Sample A sample is the subset of the population for which we have data. Let’s consider these two definitions with a research question. Research Question: In the United States, what is the mean height of adult males (18 years +)? The population that we are dealing with in this case is all U.S. adult males. One way to find an exact answer to this research question would be to survey the entire population. However, this is nearly impossible! It would be much quicker and easier to measure only a subset of the population, a sample. However, if we want our sample to be an accurate reflection of the population, we can’t just choose any sample that we wish. The way in which we collect our sample is very important and will be a topic of conversation in this chapter. For the time being, let’s suppose that we were able to choose an appropriate sample (and we’ll talk more about how this is done later). Suppose that our sample of U.S. men is an accurate representation of the U.S. population of men. Then, we might discuss two different means: the mean height of the sample and the mean height of the population. These are both descriptions, as opposed to inferences. There are a couple of differences, however. Mean Height of the Sample • Statistic - describes the sample • Can be known, but it changes depending on the sample • Symbol - $$\bar{x}$$ (pronounced “x bar”) Mean Height of the Population • Parameter - describes the population • Usually unknown - but we wish we knew it! • Symbol - $$\mu$$ (pronounced “mu”) Our goal is to use the information we’ve gathered from the sample to infer, or predict, something about the population. For our example, we want to predict the population mean, using our knowledge of the sample. The accuracy of our sample mean relies heavily upon how well our sample represents the population at large. If our sample does a poor job at representing the population, then any inferences that we make about the population are also going to be poor. Thus, it is very important to select a good sample! Note: If we already knew everything about a population, it would be useless to gather a sample in order to infer something about the population. We would already have this information! Using statistics as an inferential tool means that you don’t have information about the entire population to start with. If you are able to sample the entire population, this would be called a census. It would be nice to see what a sample looks like in comparison to the population from which it is drawn. In tigerstats we have a dataset that represents an imaginary population, imagpop. Drawing samples from this “population” will help give an idea of the distinction between sample versus population and statistic versus parameter. Try out the following app, keeping in mind that information about the sample is displayed in light blue and information about the population is displayed in red. require(manipulate) SimpleRandom() ## 5.3 Types of Samples There are 2 main kinds of sampling: • Random Sampling • Non-Random Sampling ### 5.3.1 Random Sampling There are four different methods of random sampling that we will discuss in this chapter: • Simple Random Sampling (SRS) • Systematic Sampling • Stratified Sampling • Cluster Sampling The simple random sample (SRS) is the type of sample that we will focus most of our attention on in this class. However, the other types have been included in the text to give you comparisons to the SRS and also to aid you in the future. It will be helpful to work with an example as we describe each of these methods, so let’s use the following set of 28 students from FakeSchool as our population from which we will sample. data(FakeSchool) View(FakeSchool) help(FakeSchool) Keep in mind that we would not know information about an entire population in real life! We are using this “population” for demonstration purposes only! Our goal is to describe how these different sampling techniques are implemented, the strengths and weaknesses of them, and to form a comparison between the techniques. We will try to answer the following question: Which random sampling method (simple random sample, systematic sample, stratified sample, or cluster sample) is the most appropriate for estimating the mean grade point average (GPA) for the students at FakeSchool? We can easily compute the true mean GPA for the students at FakeSchool by averaging the values in the fourth column of the dataset. This will be the population mean. We will call it $$\mu$$ (“mu”). mu <- mean(~GPA,data=FakeSchool) mu ## [1] 2.766429 Again, the population parameter, $$\mu$$, is not typically known. If it were known, there would be no reason to estimate it! However, the point of this example is to practice selecting different types of samples and to compare the performance of these different sampling techniques. #### 5.3.1.1 Simple Random Sample Simple Random Sampling (SRS) In simple random sampling, for a given sample size $$n$$ every set of $$n$$ members of the population has the same chance to be the sample that is actually selected. We often use the acronym SRS as an abbreviation for “simple random sampling”. Intuitively, let’s think of simple random sampling as follows: we find a big box, and for each member of the population we put into the box a ticket that has the name of the individual written on it. All tickets are the same size and shape. Mix up the tickets thoroughly in the box. Then pull out a ticket at random, set it aside, pull out another ticket, set it aside, and so on until the desired number of tickets have been selected. Let’s select a simple random sample of 7 elements without replacement. We can accomplish this easily with the built in function popsamp in R. This function requires two pieces of information: • the size of the sample • the dataset from which to draw the sample Remember that sampling without replacement means that once we draw an element from the population, we do not put it back so that it can be drawn again. We would not want to draw with replacement as this could possibly result with a sample containing the same person more than once. This would not be a good representation of the entire school. (By default, the popsamp function always samples without replacement. If you want to sample with replacement, you would need to add a third argument to the function: replace=TRUE. Typically, we will sample without replacement in this class.) Since we may want to access this sample later, it’s a good idea to store our sample in an object. set.seed(314159) srs <- popsamp(7,FakeSchool) srs ## Students Sex class GPA Honors ## 6 Eva F Fr 1.80 No ## 18 Derek M Jr 3.10 Yes ## 7 Georg M Fr 1.40 No ## 11 Dylan M So 3.50 Yes ## 23 Bob M Sr 3.80 Yes ## 13 Eric M So 2.10 No ## 14 Gabriel M So 1.98 No Let’s calculate the mean GPA for the 7 sampled students. This will be the sample mean, $$\bar{x}_{srs}$$. We will use the subscript ‘srs’ to remind ourselves that this is the sample mean for the simple random sample. xbar.srs <- mean(~GPA,data=srs) xbar.srs ## [1] 2.525714 Strengths • The selection of one element does not affect the selection of others. • Each possible sample, of a given size, has an equal chance of being selected. • Simple random samples tend to be good representations of the population. • Requires little knowledge of the population. Weaknesses • If there are small subgroups within the population, a SRS may not give an accurate representation of that subgroup. In fact, it may not include it at all! This is especially true if the sample size is small. • If the population is large and widely dispersed, it can be costly (both in time and money) to collect the data. #### 5.3.1.2 Systematic Sample Systematic Sampling In a systematic sample, the members of the population are put in a row. Then 1 out of every $$k$$ members are selected. The starting point is randomly chosen from the first $$k$$ elements and then elements are sampled at the same location in each of the subsequent segments of size $$k$$. To illustrate the idea, let’s take a 1-in-4 systematic sample from our FakeSchool population. We will start by randomly selecting our starting element. set.seed(49464) start=sample(1:4,1) start ## [1] 4 So, we will start with element 4, which is Daisy and choose every 4th element after that for our sample. ## Students Sex class GPA Honors ## 4 Daisy F Fr 2.1 No ## 8 Andrea F So 4.0 Yes ## 12 Felipe M So 3.0 No ## 16 Brittany F Jr 3.9 No ## 20 Eliott M Jr 1.9 No ## 24 Carl M Sr 3.1 No ## 28 Grace F Sr 1.4 No The mean GPA of the systematic sample, the sample mean, $$\bar{x}_{sys}$$, is 2.7714286. Strengths • Assures an even, random sampling of the population. • When the population is an ordered list, a systematic sample gives a better representation of the population than a SRS. • Can be used in situations where a SRS is difficult or impossible. It is especially useful when the population that you are studying is arranged in time. For example, suppose you are interested in the average amount of money that people spend at the grocery store on a Wednesday evening. A systematic sample could be used by selecting every 10th person that walks into the store. Weaknesses • Not every combination has an equal chance of being selected. Many combinations will never be selected using a systematic sample! • Beware of periodicity in the population! If, after ordering, the selections match some pattern in the list (skip interval), the sample may not be representative of the population. The list of the FakeSchool students is ordered according to the student’s year in school (freshmen, sophomore, junior, senior). Taking a systematic sample ensures that we have a person from each class represented in our sample. However, there is an underlying pattern, or periodicity, in the data. The students are also listed according to their GPA. For instance, Alice is ranked first in the freshmen class and George is ranked last in the freshmen class. Consider what would have happened if we had used a systematic sample of 4 students to estimate the average GPA of the students at the school. ## Students Sex class GPA Honors ## 1 Alice F Fr 3.80 Yes ## 8 Andrea F So 4.00 Yes ## 15 Adam M Jr 3.98 Yes ## 22 Angela F Sr 4.00 Yes Notice that even thought the systematic sample ensured that we got one person from each class, we also ended up getting students of the same class rank due to the underlying pattern. Our estimate for the average GPA is not going to truly reflect the population of the school! It may be biased since the GPA pattern coincided with the skip interval. #### 5.3.1.3 Stratified Sample Stratified Sampling In a stratified sample, the population must first be separated into homogeneous groups, or strata. Each element only belongs to one stratum and the stratum consist of elements that are alike in some way. A simple random sample is then drawn from each stratum, which is combined to make the stratified sample. Let’s take a stratified sample of 7 elements from FakeSchool using the following strata: Honors, Not Honors. First, let’s determine how many elements belong to each strata: ## Honors ## No Yes ## 16 12 So there are 12 Honors students at FakeSchool and 16 non-Honors students at FakeSchool. There are various ways to determine how many students to include from each stratum. For example, you could choose to select the same number of students from each stratum. Another strategy is to use a proportionate stratified sample. In a proportionate stratified sample, the number of students selected from each stratum is proportional to the representation of the strata in the population. For example, $$\frac{12}{28}$$ X 100% = 42.8571429% of the population are Honors students. This means that there should be 0.4285714 X 7 = 3 Honors students in the sample. So there should be 7-3=4 non-Honors students in the sample. Let’s go through the coding to draw these samples. Check out the how we use the subset function to pull out the Honors students from the rest of the populations: set.seed(1837) honors=subset(FakeSchool,Honors=="Yes") honors ## Students Sex class GPA Honors ## 1 Alice F Fr 3.80 Yes ## 2 Brad M Fr 2.60 Yes ## 8 Andrea F So 4.00 Yes ## 9 Betsy F So 4.00 Yes ## 10 Chris M So 4.00 Yes ## 11 Dylan M So 3.50 Yes ## 15 Adam M Jr 3.98 Yes ## 17 Cassie F Jr 3.75 Yes ## 18 Derek M Jr 3.10 Yes ## 19 Faith F Jr 2.50 Yes ## 22 Angela F Sr 4.00 Yes ## 23 Bob M Sr 3.80 Yes Next, we take a SRS of size 3 from the Honors students: honors.samp=popsamp(3,honors) honors.samp ## Students Sex class GPA Honors ## 9 Betsy F So 4.0 Yes ## 11 Dylan M So 3.5 Yes ## 8 Andrea F So 4.0 Yes The same method will work for non-Honors students. set.seed(17365) nonhonors=subset(FakeSchool,Honors=="No") nonhonors.samp=popsamp(4,nonhonors) nonhonors.samp ## Students Sex class GPA Honors ## 25 Diana F Sr 2.90 No ## 13 Eric M So 2.10 No ## 14 Gabriel M So 1.98 No ## 28 Grace F Sr 1.40 No We can put this together to create our stratified sample. ## Students Sex class GPA Honors ## 9 Betsy F So 4.00 Yes ## 11 Dylan M So 3.50 Yes ## 8 Andrea F So 4.00 Yes ## 25 Diana F Sr 2.90 No ## 13 Eric M So 2.10 No ## 14 Gabriel M So 1.98 No ## 28 Grace F Sr 1.40 No The sample mean for the stratified sample, $$\bar{x}_{strat}$$, is 2.84. Strengths • Representative of the population, because elements from all strata are included in the sample. • Ensures that specific groups are represented, sometimes even proportionally, in the sample. • Since each stratified sample will be distributed similarly, the amount of variability between samples is decreased. • Allows comparisons to be made between strata, if necessary. For example, a stratified sample allows you to easily compare the mean GPA of Honors students to the mean GPA of non-Honors students. Weaknesses • Requires prior knowledge of the population. You have to know something about the population to be able to split into strata! #### 5.3.1.4 Cluster Sample Cluster Sampling Cluster sampling is a sampling method used when natural groups are evident in the population. The clusters should all be similar each other: each cluster should be a small scale representation of the population. To take a cluster sample, a random sample of the clusters is chosen. The elements of the randomly chosen clusters make up the sample. Note: There are a couple of differences between stratified and cluster sampling. • In a stratified sample, the differences between stratum are high while the differences within strata are low. In a cluster sample, the differences between clusters are low while the differences within clusters are high. • In a stratified sample, a simple random sample is chosen from each stratum. So, all of the stratum are represented, but not all of the elements in each stratum are in the sample . In a cluster sample, a simple random sample of clusters is chosen. So, not all of the clusters are represented, but all elements from the chosen clusters are in the sample. Let’s take a cluster sample using the grade level (freshmen, sophomore, junior, senior) of FakeSchool as the clusters. Let’s take a random sample of 2 of them. ## Students Sex class GPA Honors ## 15 Adam M Jr 3.98 Yes ## 16 Brittany F Jr 3.90 No ## 17 Cassie F Jr 3.75 Yes ## 18 Derek M Jr 3.10 Yes ## 19 Faith F Jr 2.50 Yes ## 20 Eliott M Jr 1.90 No ## 21 Garth M Jr 1.10 No ## 22 Angela F Sr 4.00 Yes ## 23 Bob M Sr 3.80 Yes ## 24 Carl M Sr 3.10 No ## 25 Diana F Sr 2.90 No ## 26 Frank M Sr 2.00 No ## 27 Ed M Sr 1.50 No ## 28 Grace F Sr 1.40 No The sample mean for the clustered sample, $$\bar{x}_{clust}$$, is 2.7807143. Strengths • Makes it possible to sample if there is no list of the entire population, but there is a list of subpopulations. For example, there is not a list of all church members in the United States. However, there is a list of churches that you could sample and then acquire the members list from each of the selected churches. Weaknesses • Not always representative of the population. Elements within clusters tend to be similar to one another based on some characteristic(s). This can lead to over-representation or under-representation of those characteristics in the sample. ### 5.3.2 Comparison of Sampling Methods Now that you have an idea about how to take each of these kinds of samples, let’s compare them by doing repeated samples. There is no general rule for determining which sampling method is best. The choice of sampling method depends on the data that is being analyzed and will require the statistician’s judgment. We will compare the simple random sample and the systematic sample by determining which sample produces the least variable mean GPA estimate after repeated sampling. Putting that another way: Let’s start by taking 1000 simple random samples and 1000 systematic samples. We will compute $$\bar{x}_{srs}$$ and $$\bar{x}_{sys}$$ for each of the samples. Then, these sample means will be compared using some graphical and numerical summaries (specifically standard deviation) that you learned about in Chapter 2. We’re going to take the SRS’s and systematic samples just like did before. The only difference is that now we’ll be taking 1000 of them instead of just 1. Since we only care about the sample mean for each sample, we’ll create a boxplot of the $$\bar{x}_{srs}$$’s and a boxplot of the $$\bar{x}_{sys}$$’s. These two boxplots allow us to compare the amount of variation, or spread, in the estimates for the mean GPA generated from the two different sampling methods (SRS and systematic sampling). See Figure[Boxplots]. To support this visualization of the variability of the mean estimate for GPA, let’s also look at favstats. For the 1000 simple random samples, the numerical summaries of the sample means is: ## min Q1 median Q3 max mean sd n missing ## 1.785714 2.556429 2.765 2.985714 3.7 2.768701 0.3173966 1000 0 For the 1000 systematic samples, the numerical summaries of the sample means is: ## min Q1 median Q3 max mean sd n missing ## 2.9 3.18125 3.425 3.575 3.945 3.42042 0.381641 1000 0 Recall that the true average GPA for the population of students at FakeSchool was 2.7664286. Notice that the average value for the sample means from the 1000 simple random samples is 2.7687014. This is pretty close to the population parameter. (We will talk about what “pretty close” means in later chapters.) Compare this to the average value for the sample means from the 1000 systematic samples: 3.42042. On average, the SRS does a better job of producing an estimate for the mean GPA than the systematic sample. Additionally, there is less variability in the 1000 $$\bar{x}_{srm}$$’s (0.3173966) than in the 1000 $$\bar{x}_{sys}$$’s (0.381641). If we could only pick one of these types of samples to estimate the mean GPA, it appears the a SRS is a better choice than a systematic sample. Let’s do a similar analysis to compare the two sampling methods, stratified sampling or cluster sampling. We will compare the stratified sample and the cluster sample by determining which sample produces the least variable mean GPA estimate after repeated sampling. To support this visualization of the variability of the mean estimate for GPA, let’s also look at favstats. For the 1000 stratified samples, the numerical summaries of the sample means is: ## min Q1 median Q3 max mean sd n missing ## 2.14 2.620714 2.768571 2.921429 3.422857 2.772663 0.2215186 1000 0 For the 1000 cluster samples, the numerical summaries of the sample means is: ## min Q1 median Q3 max mean sd n missing ## 2.475 2.584286 2.752143 2.948571 3.057857 2.762486 0.1965341 1000 0 Both of these sampling methods produce an average of the sample means that is pretty close to the true mean GPA for the population. However, the sample means from the clustered samples have less variability. (This can be seen by comparing the standard deviations.) In other words, the 1000 cluster samples are closer, on average, to the true mean than the 1000 stratified samples. If we could only pick one of these types of samples to estimate the mean GPA, it appears the cluster sample is a better choice than a stratified sample. ## 5.4 Bias in Surveys Bias We say that a sampling method is biased if it exhibits a systematic tendency to result in samples that do not reflect the population, in some important respect. You can think of a survey as occurring in three stages: 1. Select subjects to invite into your sample. this is the sampling stage. 2. Get them to accept your invitation. This is the stage where you contact the subjects you have sampled, and ask them to participate in your survey. 3. Obtain their responses to your questions. At each of these stages, some bias can creep in! ### 5.4.1 Selection Bias Selection bias is the type of bias that can occur in the first stage, in which you are selecting the subjects who will be your sample. Selection Bias We say that a sampling method exhibits selection bias if its mechanism for selecting the sample has a systematic tendency to over-represent or under-represent a particular subset of the population. One sampling method that can result in selection bias is convenience sampling. Convenience Sampling Convenience sampling is the practice of sampling subjects that the researcher can reach easily. This may result in certain subgroups of the population being underrepresented or completely left out. Example: A math professor wants to know what percentage of young adults, ages 18-22, consider education a top priority. She gathers a sample by surveying all of her advisees. This method of sampling is quick and easy. However, only including students that are enrolled in college leaves out a large part of the population - those young adults that did not go to college or enrolled in a different type of higher education. Only including college students in this study might make it appear that a high percentage of young adults consider education a top priority. The subjects in the study surely consider it a priority since they are seeking a college degree. Another form of selection bias occurs when you attempt to sample everyone in the population, but you leave it up to each member of the population to find out about your survey and to take part in it. This is called “volunteer” sampling. Volunteer Sampling A volunteer sample is a sample of only those subjects that have volunteered to be part of a study. There may be common characteristics about the people that volunteer to be part of a particular survey that creates bias. Example: A radio station wishes to examine the proportion of its listeners which candidate they voted for in the last presidential election. They conduct a poll by asking listeners to call the station. Conducting a survey in this manner is also quick and easy, but there are groups in the population that are underrepresented or not represented at all! Only those listeners who want to disclose this information will be part of the survey. Those volunteers may have something else in common that will bias the results: for example, they may have stronger opinions on the question at hand than do other folks who did not choose to go out of their way to phone in their thoughts. One great advantage of the simple random sampling and proportionate stratified sampling – two of the methods we discussed earlier – is that they are not subject to selection bias. ### 5.4.2 Nonresopnse Bias Even if you succeed in selecting a sample of subjects in an unbiased way, you still face the task of acquiring their consent to be in your survey. Some people may refuse to take part, or perhaps you will be unable to contact them all. In that event, they won’t respond to the survey, and this could lead to bias. Nonresponse Bias We say that a sampling method exhibits nonresponse bias if there is a systematic tendency for the people who elect to take part in the survey to differ from the population in some important way. Example: The faculty at Georgetown College wanted to know what proportion of students thought that Foundations should be required for all freshmen. A simple random sample of 200 students was selected from a list obtained from the registrar. A survey form was sent by email to those students. After analyzing the results from the 20 people that reply, the faculty report that 90% of the students oppose the requirement for Foundations. • What was the population? Answer: The population of interest is the entire student body at Georgetown College. • What was the intended sample size? Answer: The intended sample size was 200. • What was the sample size actually observed? Answer: The sample size that was actually observed was the number of students that responded to the survey, 20. • What was the percentage of nonresponse? Answer: Since 20 of the 200 students selected for the survey actually respond, 180 did not respond. The percentage of nonresponse was 90%. It can be found by: (180/200)*100 ## [1] 90 • Why might this cause the results to be biased? Answer: If all of the 200 randomly selected students had responded to the survey, we would have had a true SRS. However, nonresponse bias has occurred in this study because 90% of the sampled subjects either were not reached, refused to participate, or failed to answer the question. A couple of possible explanations for the nonresponding students might be that they do not check their email or they simply did not have a strong enough opinion on the topic to feel the need to take the time to respond. (There may be other legitimate reasons.) So, it could be that the students that responded had very strong feelings about the Foundations requirement. If these are the only answers that are acquired, the results may be heavily biased in the direction of the opinion of the respondents. However, this does not mean that all students feel this way. ### 5.4.3 Response Bias The wording and presentation of the questions can significantly influence the results of a survey. The main type of bias that can result from a poorly-worded survey is response bias. Response Bias We say that a sampling method exhibits response bias if the way the questions are asked or framed tends to influence the subjects’ responses in a particular way. Many things can subject a survey response bias. Here are a few: • Deliberate Wording Bias • Unintentional Wording Bias • Desire of the Respondents to Please • Unnecessary Complexity • Ordering of Questions • Confidentiality Concerns Deliberate Response Bias - If a survey is being conducted to support a certain cause, questions are sometimes deliberately worded in a biased manner. The wording of a question should not indicate a desired answer. Example: Consider the following research question: “Seeing as Dr. Robinson and Dr. White are the greatest professors you have ever had, is it worth even offering the peer tutoring sessions for MAT111?” This question is prefaced in a way that encourages a desired response from the subjects in the study. Unintentional Response Bias - Some questions are worded in such a way that the meaning is misinterpreted by the respondents. Example: Consider the following research question: “Do you use drugs?” The word drugs can cause unintentional confusion for the respondent. The intended definition of drugs is not made clear in the wording of the question. Does the researcher mean illegal drugs, prescription drugs, over the counter drugs, or possibly even caffeine? Desire of Respondents to Please - People may respond differently depending on how they are being asked - face-to-face, over the telephone, on paper, on the internet. For example, a person may tend to be more honest when answering questions on paper or over the internet. When speaking directly to the researcher, the respondent may feel the need to answer the question how they perceive the researcher wants. Asking the Uninformed - If a question is about a topic that the respondent does not know anything about, they often do not like to admit it. Respondents may tend to give an answer, even though they do not understand the question. Unnecessary complexity - Questions should be kept simple. Try to only ask one question at a time. Example: Consider the following survey question: “Most semesters are 15 weeks long; while most quarters are 10 weeks long. Most schools on a quarter system get 2 days for Thanksgiving, one for Veteran’s Day, and one for Columbus Day. Most semester schools get Labor Day off and some take more than 2 days at Thanksgiving. However, semester schools typically start several weeks earlier in the fall and generally attend school farther into December. Considering the above, which system would you prefer?” This question has too much information in it. By the time you get done reading it, you may have forgotten what the question is even referring to. It is unnecessarily complex! Ordering of Questions - If one question requires respondents to think about something that they may not have otherwise considered, then the order in which questions are presented can change the results. Example: Suppose a researcher wants to know how many hours a day people spend on the Internet. Consider the following sequence of questions: • “Do you own a smartphone?”" • “How many hours a day do you spend on the Internet?” Placing the question about the smartphone before the question about time spent on the Internet causes the respondent to take into consideration that they are often on the Internet when they are using their phone. Putting the questions in this order may change the answers received for the second question. Confidentiality Concerns - Some personal questions will be answered differently depending on how confident the respondent is that their identity will be concealed. ## 5.5 Thoughts on R Know how to use this function: • popsamp

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# Library GC.card.card_facts Section card_facts. Require Import Arith. Require Import card. Variable A B C : Set. Notation Card := (card _ _) (only parsing). Axiom eq_dec_set : forall a b : A, {a = b} + {a <> b}. Definition prop_dec (P : B -> Prop) := forall b : B, {P b} + {~ P b}. Definition prop_dec2 (P : C -> Prop) := forall c : C, {P c} + {~ P c}. Definition update_M (b : B) (M : A -> B) (a0 a : A) := match eq_dec_set a a0 with | right _ => M a | left _ => b end. Lemma exist_updated_M : forall (M : A -> B) (a0 : A) (b : B), exists M' : A -> B, (forall a : A, a <> a0 -> M a = M' a) /\ M' a0 = b. intros M a0 b. split with (update_M b M a0). unfold update_M in |- *; split. intros a adifa0; case (eq_dec_set a a0); auto. intro aeqa0; absurd (a = a0); auto. case (eq_dec_set a0 a0); auto. intro adifa0; absurd (a0 = a0); auto. Qed. Definition update_N (c0 : C) (N : A -> C) (a0 a : A) := match eq_dec_set a a0 with | right _ => N a | left _ => c0 end. Lemma exist_updated_N : forall (N : A -> C) (a0 : A) (c0 : C), exists N' : A -> C, (forall a : A, a <> a0 -> N a = N' a) /\ N' a0 = c0. intros N a0 c0. split with (update_N c0 N a0). unfold update_N in |- *; split. intros a adifa0; case (eq_dec_set a a0); auto. intro aeqa0; absurd (a = a0); auto. case (eq_dec_set a0 a0); auto. intro adifa0; absurd (a0 = a0); auto. Qed. Lemma include_card : forall (P Q : B -> Prop) (nb : nat) (M : A -> B), prop_dec Q -> (forall b : B, Q b -> P b) -> card _ _ P M nb -> exists nb' : nat, card _ _ Q M nb' /\ nb' <= nb. intros P Q nb M dec H_P_Q card_P_M. induction card_P_M as [M H| M1 M2 nb a0 card_P_M Hreccard_P_M H H0 H1]. exists 0; split; auto. constructor. intro a; intro QMa; absurd (P (M a)); auto. elim Hreccard_P_M; clear Hreccard_P_M. intros nb' H2; elim H2; clear H2. intros card_Q_M1 le_nb'_nb. unfold prop_dec in dec. elim dec with (b := M2 a0); intro QM2a0. exists (S nb'). split. apply card_S with (M1 := M1) (a0 := a0); auto. auto with arith. exists nb'. split; auto. apply card_inv with (M1 := M1); auto. intros a QM1a. elim (eq_dec_set a a0). intro aeqa0; rewrite aeqa0; assumption. intros a QM1a. rewrite <- (H a); auto. intro aeqa0; apply H0; apply H_P_Q; rewrite aeqa0 in QM1a; assumption. Qed. Lemma include_card_bis : forall (P : B -> Prop) (Q : C -> Prop) (M : A -> B) (nb : nat), (exists c0 : C, ~ Q c0) -> prop_dec2 Q -> card _ _ P M nb -> forall N : A -> C, (forall a : A, Q (N a) -> P (M a)) -> exists nb' : nat, card _ _ Q N nb' /\ nb' <= nb. intros P Q M nb H_Q dec card_P_M. induction card_P_M as [M H| M1 M2 nb a0 card_P_M Hreccard_P_M H H0 H1]. intros N H_Q_P; exists 0; split; auto. constructor. intro a; intro QNa; absurd (P (M a)); auto. intros N H_Q_P. elim H_Q; clear H_Q. intros c0 Q_c0. elim (exist_updated_N N a0 c0); intros N' H_N'. elim H_N'; clear H_N'. intros H_N_N' N'a0; rewrite <- N'a0 in Q_c0. unfold prop_dec2 in dec; elim (dec (N a0)). intro Q_N_a0. cut (exists nb' : nat, card A C Q N' nb' /\ nb' <= nb). intro card_Q_N'; elim card_Q_N'; clear card_Q_N'. intros nb' H_nb'; elim H_nb'; clear H_nb'. intros card_Q_N' le_nb'_nb. exists (S nb'); split; auto with arith. apply card_S with (M1 := N') (a0 := a0); auto. intros a aeqa0; symmetry in |- *; apply (H_N_N' a); auto. apply Hreccard_P_M. intro a; elim (eq_dec_set a a0). intro aeqa0; rewrite aeqa0; intro QN'a0; absurd (Q (N' a0)); auto. intro Q_N_a0. cut (exists nb' : nat, card A C Q N' nb' /\ nb' <= nb). intro card_Q_N'; elim card_Q_N'; clear card_Q_N'. intros nb' H_nb'; elim H_nb'; clear H_nb'. intros card_Q_N' le_nb'_nb. exists nb'; split; auto with arith. apply card_inv with (M1 := N'); auto. intros a Q_N'_a; elim (eq_dec_set a a0). intro aeqa0; rewrite aeqa0; auto.

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# Relatively simple school project involving an NE555 timer Discussion in 'The Projects Forum' started by Davidred7697, Dec 15, 2014. 1. ### Davidred7697 Thread Starter New Member Dec 15, 2014 18 2 Hello, I just joined this forum today when I discovered that the community can provide help with school assignments. This is my very first post, so pleas go easy on me. The instructions were essentially to build a relatively simple circuit and then present it in front of the classroom and turn in a detailed report. I am 20 years old attending a local community college in Illinois, with hopes of transferring to the University of Illinois at Urbana-Champaign or UIC. The class is called "Intro to Digital Systems". I had NO prior experience with electronics/electrical systems before this class aside from basic physics in high school. I think it would have been ideal to take a more basic class electronics class, such as Electronics 101 before jumping into ENG250, but unfotunately I was never recommended to do so (and ELT101 is NOT a requirement). Unfortunately, I have been unsuccessful in building a functional circuit and would appreciate a little help with troubleshooting if possible. I will have to admit that I did not create/design the circuit myself and instead found it in a page that presents simple electrical/electronic projects for electrical engineering students. I have attached the circuit diagram in this thread for reference. I will also provide a link to the page where I found this project; it also includes a brief description of the circuit as well as the circuit diagram in case the attached file doesn't open for any reason: http://www.electronicshub.org/bike-turning-signal-circuit/ Here is also a link to a YouTube video that is embedded in the above page demonstrating how the circuit should work/look: I am using a breadboard. I have built this circuit twice. The first time, the LEDs in the circuit lit, but did not pulse as they're supposed to. Also, I realized after a few hours of trying to solve the problem that 1) for some strange reason, the LED were brighter when I touched the top of metal transistors with my fingers. 2) I had not connected the ends of the 10k ohm resistors along the same path as indicated in the circuit diagram. Because the above circuit did not work properly and I asked a laboratory manager for assistance to no avail (and an engineering instructor denied helping me at all twice and provided very little help a third time), I decided to re build the entire circuit once more. So I did just that. I re built the entire circuit, this time making sure that the 10k ohm resistors were placed properly. As I was building it, I was honestly expecting a fully and properly functioning circuit, thinking that I "fixed" a few previously made mistakes. Regrettably, this did not turn out to be the case. This time, no LEDs lit up at all. Again, after hours of troubleshooting the circuit myself and then with assistance from a lab manager, no solution was found. In this instance, an electronics professor provided little assistance by saying: "check your ground". One strange phenomenon that I noticed was that when I touched the red/positive lead of a Digital Multimeter (DMM) to one of the legs of the LEDs (can't remember if it was the anode or cathode), that LED lit up! Also, if I touched the same lead to the following LED, both LEDs lit up. I could not get more than two LEDs to light up simultaneously. One more strange thing is that when I touched the same positive lead of the multimeter to one of the legs on the 330 ohm resistors, the voltage indicator on my power supply went from 12V down to nearly 0V. These are the steps that I've taken myself and that others have helped me take to try to find a solution to my non-functioning circuit(s). I would greatly appreciate any assistance based on the above information and I will be posting various photos that I will take of my circuits tomorrow, when I have access to it. ANY feedback is greatly appreciated and I apologize for the long post. Please let me know if I forgot to include something in my post. Thanks! File size: 175.7 KB Views: 134 2. ### iimagine Active Member Dec 20, 2010 129 9 Q1-Q5 emitters should be connected directly to ground, not R5, R8, R10 or R12 3. ### vu2nan Member Sep 11, 2014 44 10 Last edited: Dec 16, 2014 Davidred7697 likes this. 4. ### WBahn Moderator Mar 31, 2012 20,068 5,666 This seems like a strange project to use in a course titled "Intro to DIGITAL Systems". Is your instructor aware that you are just copying a circuit from the internet for your project? If they are okay with that, then great. But I would think (hope) that the intent would be for you to actually design something (and preferably a something that involves some digital logic). I don't have the bandwidth to watch the video, so I'm guessing at what the circuit is supposed to do from a quick glance at the schematic. I suspect it is supposed to turn on the LEDs sequentially, first one pair (D2,D3), then a second pair (D4, D5 in addition to the first), then a third pair (in addition to the first two pair) and so on. But I don't see how the circuit in the schematic can work as given. I think that the bottom of R5, R8, R10, and R12 should be connected to ground (the negative terminal of the 12V battery) and that the bottom of R6 is NOT supposed to be connected to these four resistors. Otherwise you essentially have the emitter shorted to the base of Q1. But it isn't quite that simple and there appears to be other problems with the circuit. The values of R5 (and the others) seems too high for what I have in mind, which makes me suspect that they are supposed to be base resistors only and that the emitters are supposed to be connected directly to ground. Assuming that the intent is for the LEDs to come on as I described, I'm thinking that the resistors are supposed to be a voltage divider chain with each transistor base being driven by one of the junctions. This should result in the right-most pair of LEDs coming one and then progressively working it's way to the left. It sounds like you are wiring up the entire circuit and then hoping that it will magically work when you apply power and, when it doesn't, trying to troubleshoot the entire circuit. Don't do that. Build up the circuit a little at a time and test it as you go. The first thing is to get the 555 timer part working, so omit D1 and everything to the right of it. See that you get a rectangular waveform at pin 3 of the 555 and that you have a signal that ramps up and down at the junction of R2 and C3. You should bypass pin 5 of the 555 to ground with about a 0.1uF cap (I'm going from memory and it has probably been 20 years since I've done a 555 circuit, so you should check in the E-book for a solid recommendation on this). Once you have that part done, put in D1 and you should see something that is more like a sawtooth waveform at the junction of R2 and C3. The bottom voltage should be about 0.7V (which is why D12 is used in the full circuit). Get that far and then lets go from there. Davidred7697 likes this. 5. ### vu2nan Member Sep 11, 2014 44 10 Hi Iimagine, In that case the LEDs will all light up together instead of in sequence! Regards, Nandu. Davidred7697 likes this. 6. ### Bernard AAC Fanatic! Aug 7, 2008 4,534 476 That ckt. is doomed from the start-- except for the 555 part. If you want a short chaser try a 4017 IC with the modified 555. 7. ### vu2nan Member Sep 11, 2014 44 10 Hi WBahn, It'll work! R6-R12 are the base bias resistors. The first to switch on will be Q5, since its emitter is connected to ground through the 1N4148, then Q4, Q3, Q2, and Q1 in that order. The emitter currents of Q1, Q2, Q3 & Q4 will flow through the bases of the transistors that follow. Regards, Nandu. Davidred7697 likes this. 8. ### ISB123 Well-Known Member May 21, 2014 1,240 537 NE555 is mix of analog and digital components.Are you supposed to use ne555 or you picked it because you heard ne555 is "simple". 9. ### Alec_t AAC Fanatic! Sep 17, 2013 7,024 1,453 I agree with Nandu. The original circuit should work as is. If it doesn't, you have an incorrect or missing connection somewhere. Note that the 555's pins 4 and 8, and the top of R2, all connect to battery positive. The circuit shown is not drawn very clearly. Make sure you have the diode connected the right way round (cathode to battery negative). Davidred7697 likes this. 10. ### iimagine Active Member Dec 20, 2010 129 9 OK, my bad, I apologize. It was late and I didn't bother to analyze it properly and was quick to make assumption. I agree that the circuit should work as you described. That shouldn't happen! Are you certain that you are using NPN transistors? I would suggest that you disconnect those transistors form the 555 circuit and apply 12V directly to those 10k resistors, and see if they lit up; Testing sub-circuits this way will help identifying the problem faster. 11. ### Davidred7697 Thread Starter New Member Dec 15, 2014 18 2 Hi, guys! I just woke up and wanted to check up on my post. I must say this is surprising. I did not think I would get any replies. Instead I received so many helpful replies from seemingly knowledgeable people (sounds corny, but true) . Thanks! Now that I look back at my post, I realize that one thing I forgot to mention in the OP is that, since the lab managers said that they did not have any BC547 transistors and suggested I use alternatives that I could find online. So I did just that. I went online and found the following thread:http://www.edaboard.com/thread181945.html I used the 2222 transistors as alternatives for the first circuit I built and used the 3904 or 4401 on the second circuit. I'm unsure if this will make a difference, but I think its worth mentioning. Essentially, what I'm understanding is that the circuit I posted will not work and that I either have to make a few modifications or build a similar working circuit (that I can see at instructables. Com)? I am typing this through the mobile platform. Needless to say, it is very cumbersome to do so, to say the least. Bear with me. Ill be at school soon. Thanks again. 12. ### Davidred7697 Thread Starter New Member Dec 15, 2014 18 2 Hi, guys. My deadline is in two hours. I also have a final exam today, so I will end up studying a little more for that if I don't get this circuit fixed. As promised, I will upload a few pictures of both circuits that I built. Remember that the first one I know for sure that it isn't wired properly and the second one is the one I am unsure about. The circuit with the red-colored LEDs is the first circuit I built and, of course, the circuit with the green-colored LEDs is the second. I apologize if my circuits look a little messy. 13. ### WBahn Moderator Mar 31, 2012 20,068 5,666 I'm not too keen on the combined emitter currents for four transistors having to flow through the base of Q5. That's going to be a base current of somewhere around 60mA to 70mA. Strikes me as poor design. 14. ### WBahn Moderator Mar 31, 2012 20,068 5,666 The bottoms of your 10kΩ resistors aren't connected to anything. Be sure that you understand how the slots on your breadboard are connected! 15. ### Davidred7697 Thread Starter New Member Dec 15, 2014 18 2 I noticed that, in my second circuit (the one with green LEDs), one of the transistors (the one at the very bottom in my pictures is actually connected to two resistors and not just one like the others. So I immediately tried to connect a resistor between the *base* of the transistor and along with the rest of the resistors, but realized that I no longer had space. So I came up with a way to connect both resistors on another column. I hope it's right. Another thing I saw is that the NE555's Vcc input was not connected to the power supply! So I made the proper adjustments. I then proceeded to test my circuit once more and, of course, it did not work. I'll post a few pictures of my updated circuit below: Another thing I've noticed is that voltage is being dropped through the LED's, which means that current is lowing through them, but they don't light up. I have also tested one of the LED's individually and it does light up. There must be some other obvious thing I'm failing to understand. 16. ### wayneh Expert Sep 9, 2010 13,435 4,273 Where's D12? Oh, I see it. Is it backwards? I was thinking you could work on each LED pair individually by moving D12 along. Make the first pair work, then move to the second, and so on. Divide and conquer. Davidred7697 likes this. 17. ### WBahn Moderator Mar 31, 2012 20,068 5,666 So have you gotten the 555 portion working yet? Or are you still building up the whole thing and then wondering why the whole thing doesn't work? 18. ### Davidred7697 Thread Starter New Member Dec 15, 2014 18 2 Yes, thank you for the suggestion. As I stated earlier, I am aware that the wiring is wrong on the first circuit (the one with red-colored LED's). The one that I am currently trying to troubleshoot is the second circuit I built (green LEDs). 19. ### Davidred7697 Thread Starter New Member Dec 15, 2014 18 2 Sorry. I am not entirely sure exactly what each pin of the NE555 timer does. All I know is that the timer is supposed to send a "pulsing" signal, which makes the LEDs alternate from a HIGH state to a LOW state repeatedly. I have attempted building only one part of the circuit (the pair of LEDs that are attached to D12) and used a pulse generator instead of the NE555 timer and got it working. Two pairs of LEDs now light up, but no pulsing. I'm guessing there is a problem with the wiring around the NE555 timer. What other information can I provide for more assistance while I try to figure out what I'm doing wrong? To be clear, I do not want "someone else to do my project for me". I have spent about a dozen hours trying to build this simple circuit and it doesn't seem to be working. I'm pretty much out of ideas and have no tutors at my school to talk to. I hope you guys understand. 20. ### wayneh Expert Sep 9, 2010 13,435 4,273 That's all I saw in that first video.

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1 / 11 - PowerPoint PPT Presentation I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described. PowerPoint Slideshow about '' - kaspar An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - Presentation Transcript A E B C F 10 B C E x O A B′ （1）求出B′点的坐标； （2）求折痕CE所在直线的解析式。 F，使得将△CEF沿EF对折 2 4 （1）如果P为AB边的中点，探究△ PBE的三边之比. (3)若P为AB边上任意一点，四边形PEFQ的面积为S,PB为x,试探究S与x的函数关系,关求S的最小值. （1）把条件集中到一Rt△中，根据勾股定理得方程。 （2）寻找相似三角形，根据相似比得方程。

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# Tag Info 0 This is very similar to this question. In that case you are looking at the 1-dimensional overlap between 2-dimensional regions and taking its measure. Here you are looking at the intersection between a 1D and a 2D region and taking its measure. Both of these cases can be considered "near singular", to borrow a phrase from one of the developers, and it is ... 4 Okay, so you have a set of Regions that you want to fill, but you can only define those regions by a set of points distributed within them. Let's make some data that reproduces this. Here are three non-overlapping regions that fill up a square: region1 = Disk[{0, 0}, 1, {0, π/2}]; region2 = RegionDifference[Rectangle[], region1]; region3 = Disk[{0, 0}, 1, ... 0 Example Code ListPlot[Range @ 100, Filling -> Bottom] ListLinePlot[Range @ 100, Filling -> Bottom] Output Reference Filling ListPlot ListLinePlot 5 Another workaround is to turn the polygons into MeshRegions first, RegionIntersection @@ (DiscretizeGraphics /@ {p1, p2}) // MeshPrimitives[#, 2] & // First (* Polygon[{{0.407273, 0.650444}, {0.509656, -0.0200315}, {0.998507, 0.0546171}, {0.640904, 0.767621}}] *) Where p1 and p2 are your polygons Edit - more odd behavior What's even odder ... 10 This is a bug that has been fixed in the development version. For a possible workaround, use exact coordinates, for example sp = Function[p, SetPrecision[p, Infinity]]; ri = RegionIntersection[ sp@Polygon[{{0.5096555454081809, -0.02003146973392257}, {0.9985073695269602, 0.05461714932464575}, {0.6966031018052412`, 2.... 3 Here is an approach that generates many more points than you had, selects the ones for which your conditions are met, and plots them: k1[s_, d_] := 3 s + 5 d k2[s_, d_] := 5 s - 7 d ps = RandomVariate[UniformDistribution[{{0.1, 2}, {-1, 3}}], 1000000]; valid = Select[ps, 0 < k1[Sequence @@ #] <= 1 && 0 < k2[Sequence @@ #] <= 2 &]; ... 3 Your approach is a dead end, you can't determine all points by picking them and checking conditions. Because there are infinitely many of them. ImplicitRegion[ 0 < k1[x, y] <= 1 && 0 < k2[x, y] <= 2, {x, y} ] // RegionPlot 3 The documentation for RegionPlot3D states You should realize that since it uses only a finite number of sample points, it is possible for RegionPlot3D to miss regions in which pred is True. To check your results, you should try increasing the settings for PlotPoints and MaxRecursion. This is what happened in your second example, a result of the region ... 5 A possible workaround (brought up by the Wizard in the comments) involves the use of some of the functions from this previous answer. In particular, you will need orthogonalDirections[], extend[], and crossSection[] from that answer, along with these two additional functions for generating a suitable MeshRegion[] object: MakeTriangleMesh[vl_List, opts___] :=... 0 Example RegionQ @ Cylinder[] True RegionPlot3D @ Cylinder[] Output RegionQ @ Tube[{{0, 0, 0}, {1, 1, 1}}] False RegionPlot3D @ Tube[{{0, 0, 0}, {1, 1, 1}}] Output {Tube[{{0,0,0},{1,1,1}}]} is not a valid region to plot. EDIT In order to achieve region look-a-like Tube, see implementation below. Example region = Fold[... 6 It is confirmed as an improper behavior of RegionDifference by Wolfram support([CASE:3624735]). Here is their workaround: Instead of using boundary representation of geometric regions as given below, try using just the geometric regions in geometric computations. To be specific, here is the solution to this problem. Assuming wordRegion and disks ... 0 Description This solution does not fully cover the requirements of the original post, but it does achieve the desired output. In order to visualize how arbitrary gas cloud fills up the available volume, I have combined two Cuboid regions using RegionUnion. This new region has then been tested for intersection with a Ball of varying radii using ... 5 The rectangle (0,0), (0.5,0), (0,1),(0.5,1) should be covered, however it is not. The 29 rectangles in your original tabla cover the region 1/31 <=b < 1/2 && 0 <= a <= 1. ClearAll[tabla] tabla[n_] := Table[{1/(k + 1) <= b && 1/k > b}, {k, 2, n, 1}]; You can specify a large enough value for PlotPoints to see all 29 ... 11 region = ImplicitRegion[ 0 < z < 4 - x*y && 0 <= x <= 2 && 0 <= y <= 1, {x, y, z}]; RegionPlot3D[region, BoxRatios -> {1, 1, 1}, Axes -> True] Volume[region] (* 7 *) RegionMeasure[region] (* 7 *) Integrate[1, {x, 0, 2}, {y, 0, 1}, {z, 0, 4 - x*y}] (* 7 *) Integrate[1, Element[{x, y, z}, region]]... Top 50 recent answers are included

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Re: Mathematica for gifted elementary school children • To: mathgroup at smc.vnet.net • Subject: [mg98902] Re: Mathematica for gifted elementary school children • From: Bob F <deepyogurt at gmail.com> • Date: Tue, 21 Apr 2009 05:10:13 -0400 (EDT) • References: <gsivc0\$995\$1@smc.vnet.net> ```On Apr 20, 5:09 pm, Beliavsky <beliav... at aol.com> wrote: > My son, almost 6, is good at math and inquisitive. Is there a math > curriculum for elementary school children that uses Mathematica? He > understands the four arithemetic operations and the concept of powers. > I have Mathematica installed on my home PC and could teach him myself. > > I have written computer programs in Fortran in front of him to > demonstrate concepts such as cubes and cube roots. We had fun, but I > don't want to explain right now why 1000000000**3 gives -402653184 or > 1/2 gives 0. > > He is interested in the number "centillion" (10^303) and thought it > was cool to see the 101 zeros when we asked Mathematica to compute > centillion^(1/3). > > I see there are some math courseware athttp://library.wolfram.com/infocenter/Courseware/Mathematics/ > , but those topics are too advanced for him at present. Maybe I should > give him Wolfram's huge book and let him play when he wants. Try looking thru the demonstrations web site (at http://demonstrations.wolfram.com ). There are some really nice things and some are very well illustrated and fun to play with. There is even a "Kids and Fun" section at http://demonstrations.wolfram.com/topics.html#10 Enjoy... -Bob ``` • Prev by Date: Re: Alignment of Graphics Within Expressions • Next by Date: Re: NumberForm spaces and commas in v7.0.1

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# TCAP Tennessee 8th Grade Achievement Test (REA) - The Best Test Prep for the TCAP 4.11 - 1251 ratings - Source REA a€b Real review, Real practice, Real results. REA's Tennessee Grade 8 TCAP Math Study Guide! Fully aligned with Tennesseea€™s Core Curriculum Standards Are you prepared to excel on this state high-stakes assessment exam? * Take the diagnostic Pretest and find out what you know and what you should know * Use REA's advice and tips to ready yourself for proper study and practice Sharpen your knowledge and skills * The book's full subject review refreshes knowledge and covers all topics on the official exam and includes numerous examples, diagrams, and charts to illustrate and reinforce key math lessons * Smart and friendly lessons reinforce necessary skills * Key tutorials enhance specific abilities needed on the test * Targeted drills increase comprehension and help organize study * Color icons and graphics highlight important concepts and tasks Practice for real * Create the closest experience to test-day conditions with a full-length practice Posttest * Chart your progress with detailed explanations of each answer * Boost confidence with test-taking strategies and focused drills Ideal for Classroom, Family, or Solo Test Preparation! REA has helped generations of students study smart and excel on the important tests. REAa€™s study guides for state-required exams are teacher-recommended and written by experts who have mastered the test.Therefore, the left side of the inequality will be: x 2 Step 3 is to set up the right side of the inequality. It is known ... Therefore, the right side of the inequality will be: 46 Step 4 is to write both sides of the inequality Word Problems. x - alt; 2.x + 2 2 anbsp;... Title : TCAP Tennessee 8th Grade Achievement Test (REA) - The Best Test Prep for the TCAP Author : Stephen Hearne Publisher : Research & Education Assoc. - 2005-05-15

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Survey * Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project Document related concepts Nonogram wikipedia , lookup Fuzzy concept wikipedia , lookup Axiom of reducibility wikipedia , lookup Abductive reasoning wikipedia , lookup Science of Logic wikipedia , lookup Truth-bearer wikipedia , lookup Argument wikipedia , lookup Analytic–synthetic distinction wikipedia , lookup History of the function concept wikipedia , lookup Inquiry wikipedia , lookup Ramism wikipedia , lookup Foundations of mathematics wikipedia , lookup Natural deduction wikipedia , lookup Fuzzy logic wikipedia , lookup Willard Van Orman Quine wikipedia , lookup Propositional formula wikipedia , lookup Lorenzo Peña wikipedia , lookup Modal logic wikipedia , lookup First-order logic wikipedia , lookup Jesús Mosterín wikipedia , lookup Catuṣkoṭi wikipedia , lookup Mathematical logic wikipedia , lookup Curry–Howard correspondence wikipedia , lookup Propositional calculus wikipedia , lookup Interpretation (logic) wikipedia , lookup History of logic wikipedia , lookup Quantum logic wikipedia , lookup Laws of Form wikipedia , lookup Intuitionistic logic wikipedia , lookup Law of thought wikipedia , lookup Transcript ```Propositional Logic Review Predicate logic Predicate Logic Examples Predicate logic G. Carl Evans University of Illinois Summer 2013 Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Propositional logic AND, OR, T/F, implies, etc Equivalence and truth tables Manipulating propositions Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Implication Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Implication a b T T F F T F T F a→b ¬a ∨ b T F T T a ⇐⇒ b (¬a ∨ b) ∧ (¬b ∨ a) T F F T b T F T F Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Today Be able to incorporate predicates and quantifiers into logical statements Be able to manipulate statements with quantifiers Learn how to prove a universal statement Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Predicate Logic Predicate: propositions that have input variables with a range of values For some integer x, x > 10 Cars that are read and speeding are likely to be ticketed. isred(x) ∧ speeding (x) → likely to be ticketed(x) A person’s mother’s mother is his/her grandmother For every set of people x, y , z mother (x, y ) ∧ mother (y , z) → grandmother (x, z) Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Quantifiers For some x : ∃x For all x : ∀x For exactly one x : ∃!x Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Binding and Scope ∀x, p(x) → q(x) Binding: ∀x Scope: p(x) → q(x) ∃x, x 2 = 0 Binding: ∃x Scope: x 2 = 0 Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Manipulating quantifiers: Negation Negation: ¬(∀x, p(x)) ≡ ∃x, ¬p(x) ¬(∃x, p(x)) ≡ ∀x, ¬p(x) Examples “Not all dogs are fat” is equivalent to “At least one dog is not fat.” “There does not exist one fat dog” is equivalent to “All dogs are not fat.” Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Contrapositive ∀x, p(x) → q(x) ≡ ∀x, ¬q(x) → ¬p(x) Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Quantifiers with two variables For all integers a and b, a + b ≥ a Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Quantifiers with two variables For all integers a and b, a + b ≥ a ∀a ∈ Z, ∀b ∈ Z, a + b ≥ a or ∀a, b ∈ Z, a + b ≥ a For every real a, there exists an integer b such that a + b ≥ a Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Quantifiers with two variables For all integers a and b, a + b ≥ a ∀a ∈ Z, ∀b ∈ Z, a + b ≥ a or ∀a, b ∈ Z, a + b ≥ a For every real a, there exists an integer b such that a + b ≥ a ∀a ∈ R, ∃b ∈ Z, a + b ≥ a Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Proving universal statements Claim: For any integers a and b, if a and b are odd, then ab is also odd. Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Proving universal statements Claim: For any integers a and b, if a and b are odd, then ab is also odd. Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for some k ∈ Z. Thus ab is odd by definition of odd. QED Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Approach to proving universal statements State the supposition (hypothesis) and define any variables Expand definitions such as “odd” or “rational” into their technical meaning (if necessary) Manipulate expression until claim is verified by a simple statement End with “This is what was to be shown.” or “QED” to make it obvious that the proof is finished. Tip: work out the proof on scratch paper first, then rewrite it in a clear, logical order with justification for each step. Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Claim: For any real k, if k is rational, then k 2 is rational. Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Claim: For any real k, if k is rational, then k 2 is rational. Definition: real k is rational iff k = m n for some integers m, n, with n 6= 0. Let k ∈ Q. By definition of rational k = m n k2 = for some m, n ∈ Z with n 6= 0. m 2 m2 = 2 n n Since m, n ∈ Z, m2 , n2 ∈ Z and since n 6= 0, n2 6= 0, k 2 is rational by definition. QED Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Claim: For all integers n, 4(n2 + n + 1) − 3n2 is a perfect square. Definition: k is a perfect square iff k = m2 for some integer m. Let n be an integer. 4(n2 + n + 1) − 3n2 = n2 + 4n + 4 = (n + 2)2 Since n is an integer n + 2 is an integer so by definition of perfect square 4(n2 + n + 1) − 3n2 is a perfect square. QED Predicate logic Propositional Logic Review Predicate logic Predicate Logic Examples Claim: The product of any two rational numbers is a rational number. Definition: real k is rational iff k = m n for some integers m, n, with n 6= 0. Let a, b be rational numbers. By definition of rational a = m n ,b = n and k are not 0. j k for some m, n, j, k ∈ Z s.t. mj mj = nk nk Since m, n, j, k are integers mj, nk are integers and since n and k are not 0 nk 6= 0. Thus by definition of rational ab is rational. QED ab = Predicate logic ``` Related documents

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13,095,658 members (52,053 online) Rate this: See more: Hi Experts, I am newbie in crystal report world :) I need some help regarding creating record selection formulas: 1. How to filter records on the base of date [dd/mm/yyyy] range? The report will get two DATE parameters "fromDate" & "toDate". Can you please tell how to make this formula? 2. How to filter records on the base of time [mm:hh] range? The report will get two DATE parameters "fromTime" & "toTime". Can you please tell how to make this formula? 3. I have one parameter "CompanyName" which if user provides then the data should be selected for that specific company otherwise all companies data should appear. How to create this formula? 4. I want to create summary field which will be count of specific value lets say "XYZ" in some particular column of report. How to create such formula? Thankx BR SC Posted 8-Jan-11 9:07am thatraja 9-Jan-11 4:59am thatraja 9-Jan-11 6:13am I have appended some more things in answer, so check my answer again. Rate this: ## Solution 1 OP wrote: 1. How to filter records on the base of date [dd/mm/yyyy] range? The report will get two DATE parameters "fromDate" & "toDate". Can you please tell how to make this formula? Try this C# Crystal Reports - Date to Date [^] OP wrote: 2. How to filter records on the base of time [mm:hh] range? The report will get two DATE parameters "fromTime" & "toTime". Can you please tell how to make this formula? How to Correct this selection Formula to show time in AM or PM[^] This one is great Creating Advanced Record Selection Formulas[^](It contains all type of things) OP wrote: 3. I have one parameter "CompanyName" which if user provides then the data should be selected for that specific company otherwise all companies data should appear. How to create this formula? Try this Creating a Report with a Selection Formula[^] Send the selected company value to the selection formula, that's all. OP wrote: 4. I want to create summary field which will be count of specific value lets say "XYZ" in some particular column of report. How to create such formula? Here you go Crystal Reports Summary Field[^] What error you are getting? I think you need to put block() for If block. ```{EMPLOYEE.BIRTHDATE} >= {?FromDate} and {EMPLOYEE.BIRTHDATE} <= {?ToDate} and //( // If (StrCmp({?DeptName},"") = -1) Then // ( {EMPLOYEE.WORKDEPT} = {?DeptName} //Append this by condition in front-end. // ) //)``` Then comment the if block like above, Actually in front end(java) check the condition if DeptName has selected or not. If yes then append the DeptName thing in selection formula else stop with fromdate & todate things. If DeptName selected then ```{EMPLOYEE.BIRTHDATE} >= {?FromDate} and {EMPLOYEE.BIRTHDATE} <= {?ToDate} and {EMPLOYEE.WORKDEPT} = {?DeptName}``` else `{EMPLOYEE.BIRTHDATE} >= {?FromDate} and {EMPLOYEE.BIRTHDATE} <= {?ToDate} ` Send this selection formula to report at runtime in java. Let us know. Pending query Need help in Crystal Reports. (how to sum by conditions)[^] [/Edit] v4 Sandeep Mewara 9-Jan-11 2:55am SmoothCriminel 9-Jan-11 4:28am Hi thatraja, thanks for response. Although I am working in java but even then its helpful. Thanks again. It will be great if you can tell me how the formula will look, lets say I have fields EMPLOYEE.BIRTHDATE (type = date) & EMPLOYEE.WORKDEPT (string). Report has three parameters 1. FromDate (Date), 2. ToDate (Date) and 3. DeptName (String). Now user will select dates and it can either enter or leave the DeptName field empty. If its empty then all departments should be selected otherwise the one it entered. I have tried creating this formula: {EMPLOYEE.BIRTHDATE} >= {?FromDate} and {EMPLOYEE.BIRTHDATE} <= {?ToDate} and If (StrCmp({?DeptName},"") = -1) Then ( {EMPLOYEE.WORKDEPT} = {?DeptName} ) I know this is not correct... Can you please create it? BR SC SmoothCriminel 9-Jan-11 6:02am The issue is with last "and" I believe. What ever input I give for the "DeptName" there is no where clause when I see the generated SQL. SmoothCriminel 9-Jan-11 6:48am Aha... I got it now. So, like this I don't even need to create any parameters. And on the java side I will create the selection formula and just send it to report. Will that be good option? Or it wont work. Thanks a lot for help. thatraja 9-Jan-11 7:01am Yeah, that's it. It will surely work. Like I mentioned in my answer try the selection formula based on condition. Over. If you still facing issue please let us know. SmoothCriminel 9-Jan-11 7:50am Great thankx... I will try and let you know. Now the only pending thing is last question. I know how to create summary field but the thing is how to have distinct count of specific value in a field? BR. SC. thatraja 9-Jan-11 8:00am Dalek Dave 9-Jan-11 11:48am Top Experts Last 24hrsThis month OriginalGriff 210 Karthik Bangalore 65 Jochen Arndt 60 Mohibur Rashid 59 RickZeeland 55 OriginalGriff 4,131 Graeme_Grant 2,232 ProgramFOX 2,057 Jochen Arndt 1,735 ppolymorphe 1,735

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http://mathhelpforum.com/differential-geometry/172462-showing-piecewise-function-one-one.html

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# Thread: Showing a piecewise function is one-to-one 1. ## Showing a piecewise function is one-to-one I'm trying to show that a piecewise function is one-to-one by showing that for all f(a) = f(b), a = b. The function is: (x-2)/4 if x/2 is odd -x/4 if x/2 is even. I can show f(a) = f(b) implies a = b if a/2 and b/2 are both odd, and i can if they are both even. Do i need to show f(a) = f(b) implies a = b for the case where a is even and b odd? I would think i need to, because it should be for all a and b, but when i tried to i got (a-2)/4 = -b/4 ==> a-2 = -b ==> a = -b+2. so here it would not be true that f(a) = f(b) implies a = b and thus the function is not one-to-one. I'm pretty sure that it is, though, and that i'm doing something wrong. The function is suppose to map positive even numbers to integers. (f(2) = 0----------f(4) = -1--------f(6) = 1--------f(8) = -2.... 2. Originally Posted by Relmiw I'm trying to show that a piecewise function is one-to-one by showing that for all f(a) = f(b), a = b. The function is: (x-2)/4 if x/2 is odd -x/4 if x/2 is even. What is the initial set for this mapping? It appears that you are mapping the even integers to the integers. Is that correct? 3. Originally Posted by Plato What is the initial set for this mapping? It appears that you are mapping the even integers to the integers. Is that correct? I am suppose to come up with a function that maps positive even integers to integers and then show it is one to one and onto 4. Originally Posted by Relmiw I am suppose to come up with a function that maps positive even integers to integers and then show it is one to one and onto Well then, in the future please post complete in details. Suppose $\displaystyle \dfrac{a-2}{4}=\dfrac{-b}{4}$. That means $\displaystyle a=2n$ where $\displaystyle n$ is a positive odd. And $\displaystyle b=2m$ where $\displaystyle m$ is positive even. But that leads to $\displaystyle n=1-m$. Is that possible? 5. Originally Posted by Plato Well then, in the future please post complete in details. Suppose $\displaystyle \dfrac{a-2}{4}=\dfrac{-b}{4}$. That means $\displaystyle a=2n$ where $\displaystyle n$ is a positive odd. And $\displaystyle b=2m$ where $\displaystyle m$ is positive even. But that leads to $\displaystyle n=1-m$. Is that possible? I'm not sure where you're getting that a = 2n where n is a positive odd and that b = 2m where m is a positive even. Wouldn't a = 2 - b where b is a positive odd and b = a - 2 where a is a positive even from simplifying (a-2)/4 = -b/4? 6. Originally Posted by Relmiw I'm not sure where you're getting that a = 2n where n is a positive odd and that b = 2m where m is a positive even. I am getting it from the definition of your function. You wrote the function. I am showing the one case that you said you could not. In order for $\displaystyle \dfrac{a}{2}$ to be odd, it is necessary that $\displaystyle a=2n$ where $\displaystyle n$ is a positive odd. 7. Originally Posted by Plato I am getting it from the definition of your function. You wrote the function. I am showing the one case that you said you could not. In order for $\displaystyle \dfrac{a}{2}$ to be odd, it is necessary that $\displaystyle a=2n$ where $\displaystyle n$ is a positive odd. Ok, I think I'm understanding your post about n = 1 - m now. This is not possible as n will be a negative integer for all m. But, this does not actually mean that the function is not one-to-one because we just showed f(a) /= f(b) if one is even and one is odd (a /= b). So basically, f(a) = f(b) implies a = b for both even and both odd. For one of each, neither are true so that also maintains it is one-to-one. I believe. 8. You function is one-to-one. Can you prove that it is onto? 9. Thank you for your help so far. I would try to show this is onto by showing that each individual piece is onto. First, (x-2)/4 if x/2 is odd so, f(e) = (e - 2) / 4 y = (e - 2) / 4 e = 4y - 2 Now i must show that for all y in Z, e is a positive even initeger. I'm thinking 4y - 2 ==> 2(2k - 2) ==> 2(g) g in Z. The only problem is this only shows e is even, not positive. similarly, for -x/4 if x/2 is even. y = -e / 4 e = -4y. Now i must show that for all y in Z, e is a positive even integer. I'm thinking -4y ==> 2(-2k) ==> 2(g), g in Z. But again, this doesn't mean e is positive.

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# How do you find the day of a particular date without calendar? 1 by aarush123 2014-10-30T22:17:25+05:30 MONTH CODES: January: 1 Feb         : 4 march    :4 april      :  0 may      :2 june    : 5 july       :0 august  :3 sept    :6 oct  :      1 november   :4 december   : 6. CENTURY CODES: 1500-1599 : 0 1600-1699    :  6 1700-1799     :  4 1800-1899     :  2 1900-1999     :  0 2000-2099      :6. DATE CODES: sat - 0 sunday - 1 monday- 2 tues - 3 wed - 4 thur - 5 fri - 6. FORMULA : DATE+month code+no.of leap years +year+century code. eg:  4th july 1988 which day? 4+5+22    (for leap years you have to divide last 2 digits of year with 4 and you have to write quotient but not in decimal form. you can leave remainder.)+   88    (you have to write last two digits of that year)+0. = 31. now, you have to divide with 7 as weak has 7 days. you will get remainder 2. so according to day code 2 is monday.

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Account It's free! Register Share Books Shortlist # Solution - A Three-wheeler Starts from Rest, Accelerates Uniformly with 1 M S–2 On a Straight Road for 10 S, and Then Moves with Uniform Velocity. Plot the Distance Covered by the Vehicle During the Nth Second (N = 1,2,3….) Versus N. What Do You Expect this Plot to Be During Accelerated Motion: a Straight Line Or a Parabola - Kinematic Equations for Uniformly Accelerated Motion ConceptKinematic Equations for Uniformly Accelerated Motion #### Question A three-wheeler starts from rest, accelerates uniformly with 1 m s–2 on a straight road for 10 s, and then moves with uniform velocity. Plot the distance covered by the vehicle during the nth second (n = 1,2,3….) versus n. What do you expect this plot to be during accelerated motion: a straight line or a parabola? #### Solution You need to to view the solution Is there an error in this question or solution? #### APPEARS IN NCERT Physics Textbook for Class 11 Part 1 Chapter 3: Motion in a Straight Line Q: 23 | Page no. 59 #### Reference Material Solution for question: A Three-wheeler Starts from Rest, Accelerates Uniformly with 1 M S–2 On a Straight Road for 10 S, and Then Moves with Uniform Velocity. Plot the Distance Covered by the Vehicle During the Nth Second (N = 1,2,3….) Versus N. What Do You Expect this Plot to Be During Accelerated Motion: a Straight Line Or a Parabola concept: Kinematic Equations for Uniformly Accelerated Motion. For the courses CBSE (Arts), CBSE (Commerce), CBSE (Science) S

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0 # Solve the equation. Check all proposed solutions. Show work in solving and in checking. Solve the equation. Check all proposed solutions. Show work in solving and in checking. x      +    x     =  102 x-3         x+3      x^2-9 ### 1 Answer by Expert Tutors Philip P. | Effective and Affordable Math TutorEffective and Affordable Math Tutor 5.0 5.0 (437 lesson ratings) (437) 0 Solve the equation. Check all proposed solutions. Show work in solving and in checking. x         x       102 ----  +  ---- =  ----- x-3      x+3     x2-9 First let's add the fractions on the left.  To add any fraction, you must put them over a common denominator, in this case (x-3)(x+3) x   (x+3)          x   (x-3) ------ ------  +  ------ ------- (x-3) (x+3)      (x+3) (x-3) Multiply out the numerator and denominator then combine like terms x2 + 3x + x2 - 3x          2x2 ---------------------  =   ------ x2 - 9                  x2-9 Now we can re-state the original problem as: 2x2/(x2-9) = 102/(x2-9) 2x2 = 102      (Multiply both sides by (x2-9) ) x2 = 51         (Divide both sides by 2) x = ±√51      (Take the square root of both sides) To check, plug x = +√51 and x = -√51 into your original equation (below) and make sure the left hand side equals the right hand side.  If it does, it's a solution. 2x2/(x2-9) = 102/(x2-9)

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COMMUNITY ## November 2012 Birth Club Meet other parents of November 2012 kids and share parenting stories as your... more #### Group Detail Public Group 21,273 members 124,692 posts Created: 11/01/2010 #### Members () Report this group # 2 month old BOYS *POLL* Posted 01/21/2013 So my son at his 2 month appointment today weighed 9 lbs 13 oz and at birth was 6 lbs 5 oz so about 3.5 lbs gained! Which is awesome; He is considered to be in the 10th percentile so I wondered where other babies sat at their 2 month appts! So if you could answer the poll for your BOYS only I wanna see where my baby sits with this birth club. :) If your baby BOY weighed less or more than what is in the poll please share! :) Also, my son was 32nd percentile in height, his height was 22.5 inches. So if you want to share that... :D Happy Monday! Happy 9 weeks to us who have children born November 19th! Posted 01/21/2013 My LO was 11lbs and 22 1/2in at 2 months. He was 8lbs 4oz 20 1/2in when he was born. They didn't tell me his percentages though. Posted 01/21/2013 Weights & Heights of babies are always interesting. Your baby is over a pound heavier but they were the same height. I read in my Baby 411 book that you can tell how tall a baby BOY will be by adding 5 inches to the Mom's height and averaging that number with the Dad's height. And for a GIRL you subtract 5 inches from the Dad's height and average it with the Mom's. Interesting! They say that research shows that this calculation works a lot! Posted 01/21/2013 My baby boy was 20 1/4 inches at birth. And 8lbs 6.5oz. At 2 months he weighed 12lbs 1oz and was 24inches long. He's in the 85th percentile for height and the 50th for weight. Tall and thin :) Posted 01/21/2013 My little guy was born 8lbs 8oz 21in long. Left the hospital weighing 7lbs 4oz. At his 2 month appointment he was 12lbs 14oz 23 3/4 in long :) he is in the 59th percentile for weight. -- Kyla(24) & DH(34) married 6/18/11 DSD age 10 Daniel Michael born November 9, 2012 Posted 01/21/2013 You don't have a category for my little man! He was 6lb 4oz at birth and 18". At 9 weeks he was 8 lb 7 oz and 22". I think all of his eating was going to his height. I did an unofficial weight at home and at almost 11 weeks he was 9lb 3 oz. he was in the 3% for weight and around 20% for height at 9 weeks. Posted 01/21/2013 My baby boy was 9lbs 3 oz at birth. At his 2 month appointment he was 13lb 6oz -- Mommy to my Masonite 11.14.12 Posted 01/21/2013 My LO just had his 2 month checkup today. He was born on Nov 18th and was 9lbs 3oz 20in long at birth. Today he weighed in at 13lbs 15oz and 24in!! Posted 01/21/2013 My ds was only 1 oz smaller then yours and same height. Posted 01/21/2013 ill let you know tomorrow Posted 01/21/2013 My baby boy was born 7lbs 12oz and at two month check up he was 13lbs 3oz. And he was 21in 1/4 and at two months measured 23.5in. 1 2 3 4 5 6 7 8 9 first page | last page 12 months : Week 1 12 months : Week 2 12 months : Week 3 12 months : Week 4 13 months : Week 1 ### Your 1-year-old: 1 year checkup Around now, your toddler's play will probably start shifting from mastering fine motor skills (he's got that thumb-and-forefinger grasp down pat) to exercising larger muscles. Read More ## Free Stuff & Great Deals ### Latest Birth Announcement YSÉ AMYNTHE December 1, 2013 at 8:29am 3 kg and 47.0 cm posted by xyznik

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Graphing is a crucial part of math education, and a variety of first grade worksheets can help your students learn this important skill. Using a variety of data to fill in figures and words can help them increase their number sense. Whether your students are struggling with their math concepts, or simply want a more hands-on approach to learning, these first grade worksheets are a great way to help them improve their skills. For younger students, 1st grade graphing worksheets are an excellent way to review math concepts and reinforce counting skills. The questions in these worksheets are easy to understand and will help students develop their observational skills. The next step is learning to interpret the data. Fortunately, there are a variety of worksheets that will help students develop their math skills. These first grade graphing worksheets will help them to master the process of graphing. These worksheets are an excellent way to review important math concepts, such as probability, while developing your child’s reading and observation skills. Students can also practice how to read a graph and apply it to real-life situations. Depending on how advanced your student is, you can choose a level that is easier for them to understand. Moreover, a worksheet that introduces bar graphs will help them learn how to interpret data. Graphing worksheets are an excellent way to reinforce math concepts. The first grade focuses on simple, concrete data, such as numbers and shapes. Whether you are looking for a bar graph or a pie chart, a 1st grader will be able to read it in no time. In addition to the graphing skills, the students will learn how to interpret a bar graph and analyze the data. Graphing worksheets are a valuable resource for 1st graders. The material is based on concrete data, such as tally tables, picture graphs, and bar graphs. This means that kids will have a hard time memorizing the information presented in a graph without a worksheet. However, the first graders will be able to read bar-graphs and learn about the concepts of probability. Graphing worksheets are a great way to learn the concept of graphing. These worksheets are a great way to teach children how to read data and learn to analyze the data presented. They can also help students improve their observation skills. For instance, a bar graph shows how much a certain item weighs. Those who have a hard time reading, use a bar graph to reinforce their skills. While graphing can seem scary, it’s a fundamental skill in math. The first grade graphing worksheets should teach students how to read and write bar graphs. These worksheets should focus on the basic analysis of a bar graph. They should also teach the students how to make a graph. The basic concepts of math are crucial, so these worksheets are a great starting point for learning about the various aspects of a bar graph.

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# bernoullis__lab by cuiliqing VIEWS: 42 PAGES: 4 • pg 1 ``` Name…………………………… Title - Introduction to Bernoulli's Principle Basic premise of Bernoulli's equation: In fluid dynamics, Bernoulli's principle states that, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.[ Actual equation: The Coanda effect: is the tendency of a fluid jet to be attracted to a nearby surface. Objective: Students will observe and document behavior of materials that undergo a change in fluid velocity nearby. Procedure: Station #1: Hold a piece of paper between your thumb and forefinger, as shown. Now blow over the paper. Station #2: Place the two ping-pong balls at a distance of 2.54 cm apart on the table. Take your straw and gently blow a stream of air between the two. Put your ping-pong balls away in the designated place for the future station or lab group. Station #3: Blow up two balloons and tie each one to a string. Hold the balloons a few inches apart and try to blow them together. Can you do it? What happens? Try different ways of blowing on the balloons to see what happens. Station #4: Place the ping-pong ball in the funnel. Use the technique demonstrated at the beginning of class to "blow the ping-pong ball out" from the top as diagrammed on the board. Attempt to use the same procedure when the ping-pong ball/funnel system is upside down. Can you defy gravity with Bernoulli? Hold a Ping-Pong ball over a flexible straw, as shown. As you blow into the straw, let go of the ball. What happens? Play around with holding the straw in different ways. For example, can you tilt the straw and still keep the ball in the air? Hint: You can use any lightweight ball or a small balloon, but you may need to blow harder. Station #5: Place a ping-pong ball in a 50 ml beaker and blow across, down… and observe the results Station #6 Stick a pin through the middle of a card from below. Place the spool over the pin. Hold the card and pin in place with one hand; hold the spool with the other. Blow through the spool and let go of the pin and card. What happens? Station #7 Fluids, such as air and water, change speed as they flow between and around objects. To see how this happens, build this tiny stream channel. Tape pencils on a cookie sheet so that they make a channel that starts out wide, then narrows. Drape the pencils and cookie sheet with plastic wrap. This creates a water-proof channel. Now barely tilt the cookie sheet against the sink and slowly pour soapy water into the channel. Does the speed of the water change? How? When? Hint: You may want to add small scraps of Station #8 Hold an index card close to a stream of moving water. Describe the results Analysis and conclusion: Station #1: 1) What happened to the paper when you placed a velocity of air parallel and above it? 2) What happened to the paper when you placed a velocity of air parallel and below it? Station #2: 3) Write Bernoulli's equation below and circle the variables that you witnessed in this experiment. 4) From the above equation, how did the variables observed affect one another? (i.e.: the buoyant force on an object is larger when a larger volume of fluid is displaced by the object). Station #3: 5) Why did the results occur? 6) Name three examples of how Bernoulli's principle can be used in a device at home. Station #4: 7) Why could you not "blow the ping-pong ball out"? 8) Did you succeed in defying gravity? Draw a diagram of this station below and show all forces. Station #5: 9) What happened in this experiment? Why? Station #6 10) Explain the results using a force diagram showing all forces. Station #7 11)Why does a streams velocity increase as the width of the stream decrease? Station #8: 12) What happened in this experiment? Why? Conclusion: 1) Relate what happens when a large truck is passing your car on interstate 90. 2) Julie is riding in a car with her large family and, to her disgust, grandpa lights up a cigar. The car is filled with smoke and finally Julie asks him to crack open his window. How does the pressure outside the car now relate to the pressure inside the car? What happens to the disgusting smoke particles? (two questions = two answers). 3) Use diagrams to explain one reason why airplanes can fly. Use the words pressure, velocity and force as well as vector arrows. 4) You are on the interstate and a convertible passes you and the cloth top appears bulged out. Or you roll down the windows in an older car (poor tracks for the windows) and when you roll it up it has moved out explain either situation. ``` To top

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http://www.education.com/reference/article/part-2-arithmetic-reasoning4/

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Education.com # Arithmetic Reasoning Review for ASVAB Power Practice Problems By LearningExpress Editors LearningExpress, LLC Updated on Aug 12, 2011 These arithmetic reasoning practice questions are based on the actual ASVAB. Take this quiz to see how you would do if you took the exam today and to determine your strengths and weaknesses as you plan your study schedule. Time: 36 minutes 1. In New York City, two out of every five people surveyed bicycle to work. Out of a population sample of 200,000 people, how many bicycle to work? 1. 4,000 2. 8,000 3. 40,000 4. 80,000 2. Serena has to choose between two jobs. One is at Books R Us and pays \$18,000 with yearly raises of \$800. The other, at Readers Galore, pays \$16,400 per year with yearly raises of \$1,200. In how many years will the two yearly salaries be equal? 1. 6 2. 5 3. 4 4. 3 Use the following pie chart to answer questions 3 through 5. 1. For which item does Ricardo spend half as much as he puts in his savings account? 1. his car 2. clothes 3. housing 4. food 2. For which two items does Ricardo spend 50% of each budgeted dollar? 1. savings and housing 2. clothes and housing 3. car and medical 4. medical and food 3. Last year, Ricardo made \$46,500. About how much more did he spend on housing during the year than he put away in savings? 1. \$2,000 2. \$2,600 3. \$2,800 4. \$3,000 4. The number of red blood corpuscles in one cubic millimeter is about 5,000,000, and the number of white blood corpuscles in one cubic millimeter is about 8,000.What, then, is the ratio of white blood corpuscles to red blood corpuscles? 1. 1:625 2. 1:40 3. 4:10 4. 5:1,250 5. The living room in Donna's home is 182 square feet. How many square yards of carpet should she purchase to carpet the room? 1. 9 square yards 2. 1,638 square yards 3. 61 square yards 4. 21 square yards 6. During a basketball game, Jack has made 20 free-throw shots out of his 50 tries. How many of his next 25 free-throw attempts is Jack most likely to make? 1. 5 2. 10 3. 15 4. 20 7. Twelve people entered a room. Three more than two-thirds of these people then left. How many people remain in the room? 1. 0 2. 1 3. 2 4. 7 8. A map of Nevada has the following scale: 0.5 inches = 15 miles. Which expression tells the actual distance, d, between points that are 5.25 inches apart on the map? 9. The Cougars played three basketball games last week. Monday's game lasted 113.9 minutes; Wednesday's game lasted 106.7 minutes; and Friday's game lasted 122 minutes. What is the average time, in minutes, for the three games? 1. 77.6 minutes 2. 103.2 minutes 3. 114.2 minutes 4. 115.6 minutes 10. A helicopter flies over a river at 6:02 A.M. and arrives at a heliport 20 miles away at 6:17 A.M. How many miles per hour was the helicopter traveling? 1. 120 miles per hour 2. 300 miles per hour 3. 30 miles per hour 4. 80 miles per hour 11. How many minutes are there in 12 hours? 1. 24 minutes 2. 1,440 minutes 3. 720 minutes 4. 1,200 minutes 12. Five people in Sonja's office are planning a party. Sonja will buy a loaf of French bread (\$3 per loaf) and a platter of cold cuts (\$23). Barbara will buy the soda (\$1 per person) and two boxes of crackers (\$2 per box). Mario and Rick will split the cost of two packages of Cheese Doodles (\$1 per package). Danica will supply a package of five paper plates (\$4 per package). How much more will Sonja spend than the rest of the office put together? 1. \$14 2. \$13 3. \$12 4. \$11 13. Kathy charges \$7.50 per hour to mow a lawn. Sharon charges 1.5 times as much to do the same job. How much does Sharon charge to mow a lawn? 1. \$5.00 per hour 2. \$11.25 per hour 3. \$10.00 per hour 4. \$9.00 per hour 14. The height of the Eiffel Tower is 986 feet. A replica of the tower made to scale is 4 inches tall. What is the scale of the replica to the real tower? 1. 1 to 246.5 2. 1 to 3,944 3. 246.5 to 1 4. 1 to 2,958 15. On Sundays, Mike gives half-hour drum lessons from 10:30 A.M. to 5:30 P.M. He takes a half-hour lunch break at noon. If Mike is paid \$20 for each lesson, what is the total amount he makes on Sunday? 1. \$260 2. \$190 3. \$170 4. \$140 16. Driving 60 miles per hour, it takes one-half hour to drive to work. How much additional time will it take to drive to work if the speed is now 40 miles per hour? 1. 1 hour 2. 2 hours 3. 15 minutes 4. 30 minutes 17. An 8" × 10"photograph is blown up to a billboard size that is in proportion to the original photograph. If 8" is considered the height of the photo, what would be the length of the billboard if its height is 5.6 feet? 1. 7 feet 2. 400 feet 3. 56 feet 4. 9 feet 18. Membership dues at Arnold's Gym are \$53 per month this year, but were \$50 per month last year. What was the percentage increase in the gym's prices? 1. 5.5% 2. 6.0% 3. 6.5% 4. 7.0% 19. Which of the following best represents the following statement? Patricia (P) has four times the number of marbles Sean (S) has. 1. P = S + 4 2. S = P – 4 3. P = 4S 4. S = 4P 20. A piggy bank contains \$8.20 in coins. If there are an equal number of quarters, nickels, dimes, and pennies, how many of each denomination are there? 1. 10 2. 20 3. 30 4. 40 21. Two saline solutions are mixed. Twelve liters of 5% solution are mixed with four liters of 4% solution. What percent saline is the final solution? 1. 4.25% 2. 4.5% 3. 4.75% 4. 5% 22. A neighbor has three dogs. Fluffy is half the age of Muffy, who is one-third as old as Spot, who is half the neighbor's age, which is 24. How old is Fluffy? 1. 2 2. 4 3. 6 4. 12 23. Mario has finished 35 out of 45 of his test questions. Which of the following fractions of the test does he have left? 24. D'Andre rides the first half of a bike race in two hours. If his partner, Adam, rides the return trip five miles per hour less, and it takes him three hours, how fast was D'Andre traveling? 1. 10 miles per hour 2. 15 miles per hour 3. 20 miles per hour 4. 25 miles per hour 25. Kate earns \$26,000 a year. If she receives a 4.5% salary increase, how much will she earn? 1. \$26,450 2. \$27,170 3. \$27,260 4. \$29,200 26. Rudy forgot to replace his gas cap the last time he filled his car with gas. The gas is evaporating out of his 14-gallon tank at a constant rate of gallon per day. How much gas does Rudy lose in one week? 27. Veronica took a trip to the lake. If she drove steadily for five hours traveling 220 miles, what was her average speed for the trip? 1. 44 miles per hour 2. 55 miles per hour 3. 60 miles per hour 4. 66 miles per hour 28. A rectangular tract of land measures 860 feet by 560 feet. Approximately how many acres is this? (one acre = 43,560 square feet) 1. 12.8 acres 2. 11.06 acres 3. 10.5 acres 4. 8.06 acres 150 Characters allowed ### Related Questions #### Q: See More Questions ### Today on Education.com #### SUMMER LEARNING June Workbooks Are Here! #### EXERCISE Get Active! 9 Games to Keep Kids Moving #### TECHNOLOGY Are Cell Phones Dangerous for Kids?

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https://grabcad.com/questions/how-do-i-scaled-sketch-at-the-center-which-i-also-have-to-rotate-about-its-center-point

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# How do I scaled sketch at the center which i also have to rotate about its center point? Guys I am having difficulty with modeling in NX. there is this helical gear drawing attached which i am trying to make in NX. for which I need to resize curve in sketch which is projected by using offset curve command. but the difficulty is am unable to locate the scaled sketch at the center which i also have to rotate about its center point. Kindly suggest the solution. 1. take three planes at the particular distance. 0,48,96. 2. Draw sketch, apply scale curve, set scale factor 1, 0.75, 0.50. apply move curve set angle 0,15, 30. 3. use through curves, select all sketches Hoi Sourabh, the best way is to attacht the file, so I can see where it is going wrong. When I see the drawing, I think there are different ways to get the model. I always draw with Inventor, but I do have 2 years experince with NX. The version I have at home is NX 8.5. Greetings, Danny 1. Sketch the profile 2. Within same sketch, use Pattern Curve set to 'Circular' & create 3 instances, 15º apart 3. Extrude the first 15º instance 45 mm 4. Extrude the second instance 95 mm 5. Scale the 1st extrude body to 75% using "Scale Body" 6. Scale the 2nd extrude body to 50% using "Scale Body" 7. Use "Through curves" to create your swept form Why I chose to do it this way? Mainly because I like to work directly from the numbers provided. 75% & 50% imply scale and so, working with a scale body gives me numbers that correlate, rather than calculating the proper offset. Not to mention, it makes things a lot easier later if you need to edit... More in depth demonstration of the scaling method. Scaling of the 1st extrusion body: (scale_1st_body) Scaling of the 2nd extrusion body: (scale_2nd_body) Final through curve mesh with the proper heights set, using a hexagon for simplicity sake: (scaled_bodies_tree2) 1. I created second sketch with offset with zero distance. 2. Used scale curve to resize it. 3. Used move curve to rotate about a point. Now bout through curve, isn't it a surfacing command why is it used for solid modeling. There should be solid modeling command for creating the multi-section solid, is there any? This how I made it. Also please look at the drafting.

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https://www.mvorganizing.org/which-of-the-following-is-not-a-heuristic-for-problem-solving-or-decision-making/

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# Which of the following is not a heuristic for problem solving or decision making? ## Which of the following is not a heuristic for problem solving or decision making? Which of the following is not a heuristic for problem solving? Brainstorming is undirected thinking. ## What is a benefit of problem solving sets? Problem solving develops mathematical power. It gives students the tools to apply their mathematical knowledge to solve hypothetical and real world problems. Problem solving is enjoyable. It allows students to work at their own pace and make decisions about the way they explore the problem. ## How can the hill climbing heuristic lead to ineffective problem solving? However, like many heuristics, the hill-climbing heuristic can lead you astray. The biggest drawback to this heuristic is that problem solvers must consistently choose the alternative that appears to lead most directly toward the goal. ## What is the big drawback of the hill climbing heuristic? A major problem of hill climbing strategies is their tendency to become stuck at foothills, a plateau or a ridge. If the algorithm reaches any of the above mentioned states, then the algorithm fails to find a solution. ## What are the pitfalls of hill climbing algorithm? Four pitfalls of hill climbing • Local maxima. If you climb hills incrementally, you may end up in a local maximum and miss out on an opportunity to land on a global maximum with much bigger reward. • Emergent maxima. • Novelty effects. • Loss of differentiation. ## What are the advantages and disadvantages of hill climbing algorithm? It is also helpful to solve pure optimization problems where the objective is to find the best state according to the objective function. It requires much less conditions than other search techniques. Disadvantages: The question that remains on hill climbing search is whether this hill is the highest hill possible. ## Is Random restart hill climbing optimal? Random-restart hill climbing is a surprisingly effective algorithm in many cases. It turns out that it is often better to spend CPU time exploring the space, than carefully optimizing from an initial condition. ## What are the main disadvantage of hill climbing search? 16 Hill Climbing: Disadvantages Plateau A flat area of the search space in which all neighbouring states have the same value. 17. 17 Hill Climbing: Disadvantages Ridge The orientation of the high region, compared to the set of available moves, makes it impossible to climb up. ## How can I improve my hill climbing algorithm? Algorithm for Simple Hill Climbing: 1. Step 1: Evaluate the initial state, if it is goal state then return success and Stop. 2. Step 2: Loop Until a solution is found or there is no new operator left to apply. 3. Step 3: Select and apply an operator to the current state. 4. Step 4: Check new state: 5. Step 5: Exit. ## When the hill climb method may fail to find a solution? Both the basic and this method of hill climbing may fail to find a solution by reaching a state from which no subsequent improvement can be made and this state is not the solution. Local maximum state is a state which is better than its neighbours but is not better than states faraway. ## Why hill climbing search always lead to a local maximum? Local maximum : At a local maximum all neighboring states have a values which is worse than the current state. Since hill-climbing uses a greedy approach, it will not move to the worse state and terminate itself. The process will end even though a better solution may exist. ## Which one is the admissible algorithm? Admissibility: an algorithm is admissible if it is guaranteed to return an optimal solution whenever a solution exists. Space complexity and Time complexity: how the size of the memory and the time needed to run the algorithm grows depending on branching factor, depth of solution, number of nodes, etc. ## WHAT IS A * algorithm in AI? A* is formulated with weighted graphs, which means it can find the best path involving the smallest cost in terms of distance and time. This makes A* algorithm in artificial intelligence an informed search algorithm for best-first search. ## Is Hill climbing complete? Hill climbing is neither complete nor optimal, has a time complexity of O(∞) but a space complexity of O(b). No special implementation data structure since hill climbing discards old nodes. ## WHAT IS A * algorithm example? One of the most obvious examples of an algorithm is a recipe. It’s a finite list of instructions used to perform a task. For example, if you were to follow the algorithm to create brownies from a box mix, you would follow the three to five step process written on the back of the box. ## What is difference between A * and AO * algorithm? An A* algorithm represents an OR graph algorithm that is used to find a single solution (either this or that). An AO* algorithm represents an AND-OR graph algorithm that is used to find more than one solution by ANDing more than one branch. ## WHAT IS A * search technique? A* is an informed search algorithm, or a best-first search, meaning that it is formulated in terms of weighted graphs: starting from a specific starting node of a graph, it aims to find a path to the given goal node having the smallest cost (least distance travelled, shortest time, etc.). ## Why is a * optimal? A* search finds optimal solution to problems as long as the heuristic is admissible which means it never overestimates the cost of the path to the from any given node (and consistent but let us focus on being admissible at the moment). ## How overestimation is handled in A * algorithm? The algorithm continues until a goal node has a lower f value than any node in the queue (or until the queue is empty). With overestimation, A* has no idea when it can stop exploring a potential path as there can be paths with lower actual cost but higher estimated cost than the best currently known path to the goal. ## What is the fastest searching algorithm? Binary search is faster than linear search except for small arrays. However, the array must be sorted first to be able to apply binary search. There are specialized data structures designed for fast searching, such as hash tables, that can be searched more efficiently than binary search. ## What is the slowest sorting algorithm? But Below is some of the slowest sorting algorithms: Stooge Sort: A Stooge sort is a recursive sorting algorithm. It recursively divides and sorts the array in parts. ## Which searching technique is best? If you’re only doing a few searches, then a basic linear search is about the best you can do. If you’re going to search very often, it’s usually better to sort, then use a binary search (or, if the distribution of the contents if fairly predictable, an interpolation search). ## Which algorithm is best for unsorted list? The most common algorithm to search an element in an unsorted array is using a linear search, checking element by element from the beginning to the end, this algorithm takes O(n) complexity. Using the front and back algorithm can take us a half of time. ## What is best sorting algorithm? The time complexity of Quicksort is O(n log n) in the best case, O(n log n) in the average case, and O(n^2) in the worst case. But because it has the best performance in the average case for most inputs, Quicksort is generally considered the “fastest” sorting algorithm. ## How do you choose a sorting algorithm? To choose a sorting algorithm for a particular problem, consider the running time, space complexity, and the expected format of the input list. Stable? *Most quicksort implementations are not stable, though stable implementations do exist. When choosing a sorting algorithm to use, weigh these factors. ## What is a stable sorting algorithm? Stable sorting algorithms maintain the relative order of records with equal keys (i.e. values). That is, a sorting algorithm is stable if whenever there are two records R and S with the same key and with R appearing before S in the original list, R will appear before S in the sorted list. Begin typing your search term above and press enter to search. Press ESC to cancel.

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https://www.teachoo.com/3431/1821/Ex-10.4--11---Let--a-=-3---b--=-root2-3--Then-a-x-b-is-unit-vector/category/Chapter-10-Class-12th-Vector-Algebra/

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1. Class 12 2. Important Question for exams Class 12 Transcript Ex 10.4, 11 Let the vectors 𝑎﷯ and 𝑏﷯ be such that | 𝑎﷯| = 3 and | 𝑏﷯| = ﷮2﷯﷮3﷯, Then 𝑎﷯ × 𝑏﷯ is a unit vector, if the angle between 𝑎﷯ and 𝑏﷯ is (A) π/6 (B) π/4 (C) π/3 (D) π/2 𝑎﷯﷯ = 3 & 𝑏﷯﷯ = ﷮2﷯﷮3﷯ 𝑎﷯ × 𝑏﷯ = 𝑎﷯﷯ 𝑏﷯﷯ sin θ 𝑛﷯ Given, ( 𝑎﷯ × 𝑏﷯) is a unit vector Magnitude of ( 𝑎﷯ × 𝑏﷯) = | 𝒂﷯ × 𝒃﷯| = 1 Now, 𝒂﷯ × 𝒃﷯﷯ = 𝒂﷯﷯ 𝒃﷯﷯ sin θ 𝒏﷯﷯ , θ is the angle between 𝑎﷯ and 𝑏﷯. 𝑎﷯ × 𝑏﷯﷯ = 𝑎﷯﷯ 𝑏﷯﷯ sin θ 𝑛﷯﷯ 𝑎﷯ × 𝑏﷯﷯ = 𝑎﷯﷯ 𝑏﷯﷯ sin θ × 1 𝑎﷯ × 𝑏﷯﷯ = 𝑎﷯﷯ 𝑏﷯﷯ sin θ 1 = 3 × ﷮2﷯﷮3﷯ sin θ 1 = ﷮2﷯ sinθ sin θ = 1﷮ ﷮2﷯﷯ θ = sin-1 𝟏﷮ ﷮𝟐﷯﷯﷯ = 𝝅﷮𝟒﷯ Therefore, the angle between the vectors 𝑎﷯ and 𝑏﷯ is 𝝅﷮𝟒﷯ . Hence, (B) is the correct option Class 12 Important Question for exams Class 12

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https://www.physicsforums.com/threads/bernoullis-principle-in-stagnant-situations.716455/

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# Bernoulli's principle in stagnant situations ## Main Question or Discussion Point please note in the process proving bernoulli we use the CONTINUITY equation.Consider a situation wherein a pitot tube is used to measure pressure/velocity in a steady flow(picture attached for reference).Here we use bernoulli for pressure measurement.we apply the bernoulli theorem for a streamline which has some finite velocity at one end while zero(stagnation point) at the other end.This violates continuity and hence we canot apply bernoullis theorem in the first place itself!can ayone explain the situation PS-in almost all fluid mechanics books they just apply bernoullis directly as written,without any explanation of the above question.a name of a book explaining the situation properly would be very helpful too. #### Attachments • 3.6 KB Views: 334 Related Other Physics Topics News on Phys.org have already read them.the point here is the that continuity is essential for bernoulli(atleast from the proofs that i have seen till now).there is no point applying bernoulli if there is no continuity have already read them.the point here is that while proving bernoulli you use continuity in the proof(atleast the proofs which i know) and continuity is not valid here at that specific point(A) for any streamline passing through it.so you cannot use bernoulli in the first place have already read them.when you prove the bernoullis theorem you require continuity in some step while proving.but there is no coninuity here for the point A.so solving the problem using bernoulli is wrong which is done widely in every book and elsewhere.so how do you do it then? in the process proving bernoulli we use the CONTINUITY equation. Where do you get this? I did not remember using continuity equation while deriving bernoulli's equation. Your post is interesting. I don't have answer for now, guess I have to investigate a bit. Well, I learnt bernoulli's equation from one of the "every book" you mentioned and I never questioned the use of bernoulli's equation when there is no continuity.

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### Home > A2C > Chapter Ch11 > Lesson 11.3.1 > Problem11-132 11-132. 1. For the function, complete parts (a) through (d) below. Homework Help ✎ 1. Sketch the graph and the inverse. 2. Find the equation of the inverse function. 3. Determine the domain and range of the inverse. 4. Compute f −1(f (5)). First, graph the function. Then, reflect the graph across y = x. Replace f(x) with y, then switch the variables. $x=\frac{\sqrt{y+4}}{2}-1$ Now solve for y. Do not forget to replace y with f − 1(x). f −1(x) = (2(x + 1))2 − 4 See graph in part (a). x ≥ −1, y ≥ −4 Substitute 5 for x in the function f(x). Now substitute the answer you found for f(x) for x in the inverse function f −1(x). f −1(f(5)) = 5

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## Can p and q in modus tollens be stated negatively? 2 In a youtube video a commentator summarizes Andrew Klavan's argument as the following: 1. If God does not exist, there are no objective moral values. 2. There are objective moral values. 3. God exists. The commentator says it takes the form of modus tollens thus it is valid. Is the commentator correct? It appears to me that you could substitute p for "God does not exist" and q for "there are no objective moral values". Then structure them as a modus tollens and get a valid argument: 1. p -> q 2. not q 3. thus not p which amounts to the argument above. FYI what I mean by stated negatively is "God does not exist" vs if I stated it positively "God does exist". 1While the logic is valid, the premises are debatable. – None – 2017-10-08T14:08:32.823 Yes. A placeholder in propositional logic can take the place of a negative statement (a denial that such and such is true); then, the negation of such a placeholder will then amount to a denial that a negative statement is true (an affirmation). The argument you presented is valid, but this alone says absolutely nothing about the veracity of the conclusion. – jeffreysbrother – 2017-10-10T03:55:59.647 4 Logical formulae are blind to the content of the assertions that fill them in, and evaluations of logical validity are accordingly content-blind. One can substitute for "P" and "Q" any given sentence, affirmative or negative, universal or existential, or otherwise. One gets to evaluate the content when considering the soundness of the argument. 1 This reasoning is correct. The modus tollens technique is also called Denying the Consequent. The quirk is this example is that Statement 1 is negative, P thus not-Q; Statement 2 is the denial, and so it ends up affirmative, Q. That said, the premises support the conclusion. I think you mean the modus tollens technique rather than modus ponens. – virmaior – 2017-10-08T01:53:12.743 1"Modus pollen" is the way bees make honey logically. Thanks for the correction. – Mark Andrews – 2017-10-08T04:52:54.770 1 Premise 1 is the contrapositive of, and so equivalent to, "If there are objective moral values, then God exists," so if you like you can look at this as a modus ponens rather than a modus tollens. Comes to the same thing, and it's valid, though possibly not sound.

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# Bond Market as a Casino Game Part 1 April 1, 2011 By Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. With this post, I am doing something I try very hard to avoid, especially when communicating to my clients, and that is blurring the line between investing and gambling.  But after reading all of Reuven Brenner’s books and finishing Ralph Vince Leverage Space Trading Model, I think blurring the line can offer additional insights and methods not traditionally available. As a starting point, let’s apply basic probability methods on the most widely used bond index Barclays Aggregate Total Return to test my belief that the 1980-current bull run in bonds has offered one of the best games ever for any investor in the history of the markets (see my post “Bonds Tumble and Questions Start Getting Asked”).  In the Microsoft Excel pivot table shown below, the Barclays Aggregate has been up 70% of all month since 1980, and the up months generate on average 1.44% return compared to the down months –1.02%.  If we put this in casino terms, it is like winning 7 of every 10 games with a payout of 1.4 to 1. For another look, let’s use R and PerformanceAnalytics to see the histogram and boxplot. From TimelyPortfolio source:Barclays Capital and thanks to the fine contributors to R and PerformanceAnalytics Now that we have the basic probabilities secured, we can have all sorts of fun in later posts with the Leverage Space techniques and Monte Carlo simulations. R code: require(PerformanceAnalytics) require(quantstrat) #load index data given as date and total return value #convert to monthly return series for PerformanceAnalytics #use discrete ROC to get simple monthly return #((value this month-value last month)/value last month)-1 BarAggReturn<-ROC(BarAgg,n=1,”discrete”) par(mfrow=c(2,1)) #2 rows and 1 column chart.Histogram(BarAggReturn[“1980::”],main=”Barclays Aggregate Total Return Monthly % Histogram since 1980″,cex.main=1,xlab=NULL,breaks=15,methods=”add.centered”) chart.Boxplot(BarAggReturn[“1980::”],main=”Barclays Aggregate Total Return Monthly % Boxplot since 1980″,cex.main=1,xlab=NULL,names=FALSE) R-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R/data-science job. Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.

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List of List Program Discussion in 'C' started by arpit.jh001, Feb 15, 2012. 1. arpit.jh001New Member Joined: Feb 15, 2012 Messages: 10 0 Trophy Points: 0 /* In this program I have used an array of pointers and try to link every element of that with the node of same type but can't able to get the desired output....This program was written with the intention of storing a graph without using array for storing adjacency matrix.......but cant find a proper solution :nonod:..kindly help*/ Code: ```#include<malloc.h> #include<iostream.h> struct node { int vertex; }; int nodes; void display(struct node *a) { struct node *temp; for(int i=0;i<nodes;i++) { cout<<a[i].vertex; temp = &a[i]; cout<<"->"<<temp->vertex; } cout<<"\n"; } } void main() { char ch; struct node *temp,*t1; cout<<"\n Enter the number of vertices : "; cin>>nodes; node *a= (struct node*)malloc((sizeof(struct node))*nodes); for(int i=1;i<=nodes;i++) { a[i-1].vertex = i; cout<<"Is node "<<i<<" is connected to some other node(y/n):"; cin>>ch; if((ch == 'y') || (ch == 'Y')) { temp = &a[i-1] ; cout<<"node no."<<i<<" is connected to :"; while(ch=='Y' || ch=='y') { t1 = (struct node*)malloc(sizeof(struct node)); cout<<"\nEnter vertex no."; cin>>t1->vertex; t1=temp; cout<<"\nAny other node(y/n):"; cin>>ch; } } } display(a); }``` Last edited by a moderator: Feb 15, 2012 2. xpi0t0sMentor Joined: Aug 6, 2004 Messages: 3,012 203 Trophy Points: 63 Occupation: Senior Support Engineer Location: England What input did you give the program? What output did you get? What output did you expect? arpit.jh001 likes this. 3. arpit.jh001New Member Joined: Feb 15, 2012 Messages: 10 0 Trophy Points: 0 Code: ```//I found the solution....Thanks for your reply:nice: #include<malloc.h> #include<iostream.h> struct node { int vertex; }; int nodes; void display(struct node *a) { struct node *temp; for(int i=0;i<nodes;i++) { cout<<a[i].vertex; temp = &a[i]; cout<<"->"<<temp->vertex; } cout<<"\n"; } } void main() { char ch; struct node *temp,*t1; cout<<"\n Enter the number of vertices : "; cin>>nodes; node *a= (struct node*)malloc(nodes*(sizeof(struct node))); for(int i=1;i<=nodes;i++) { a[i-1].vertex = i; cout<<"Is node "<<i<<" is connected to some other node(y/n):"; cin>>ch; if((ch == 'y') || (ch == 'Y')) { temp = &a[i-1] ; cout<<"node no."<<i<<" is connected to :"; while(ch=='Y' || ch=='y') { t1 = (struct node*)malloc(sizeof(struct node)); cout<<"\nEnter vertex no."; cin>>t1->vertex; temp=t1; cout<<"\nAny other node(y/n):"; cin>>ch; } } } display(a); }``` //There was few silly mistakes in my program.But now it is fully correct....This Program is used for storing graph using Link List..First it asks for the number of vertices you want to store in the graph...then for each vertex(starting with vertex 1) it asks whether it is connected to some other vertex..if yes then it asks for the vertex number...few more corrections need to be done to make it more efficient and able to handle invalid inputs...... Last edited by a moderator: Feb 18, 2012 4. arpit.jh001New Member Joined: Feb 15, 2012 Messages: 10 0 Trophy Points: 0 Code: ```//This has more functionalities.....it can search whether two vertices are connected or not :smug::D #include<malloc.h> #include<conio.h> #include<stdlib.h> #include<iostream.h> struct node { int vertex; }; int nodes; void display(struct node *a) { struct node *temp; for(int i=0;i<nodes;i++) { cout<<a[i].vertex; temp = &a[i]; cout<<"->"<<temp->vertex; } cout<<"\n"; } } void search_connec(struct node *a ,int x,int y) { struct node *temp; temp = &a[x-1]; if(temp->vertex == y) { cout<<"\nVertex "<<x<<" is connected to vertex "<<y<<"\n"; return; } } cout<<"\nNot connected"; } void main() { int x,y,choice; char ch; struct node *temp,*t1; cout<<"\n Enter the number of vertices : "; cin>>nodes; node *a= (struct node*)malloc(nodes*(sizeof(struct node))); for(int i=1;i<=nodes;i++) { a[i-1].vertex = i; cout<<"Is node "<<i<<" is connected to some other node(y/n):"; cin>>ch; if((ch == 'y') || (ch == 'Y')) { temp = &a[i-1] ; cout<<"node no."<<i<<" is connected to :"; while(ch=='Y' || ch=='y') { t1 = (struct node*)malloc(sizeof(struct node)); cout<<"\nEnter vertex no."; cin>>t1->vertex; while(!((t1->vertex) <= nodes)) { cout<<"\nInvalid node! Please re-enter correct node :"; cin>>t1->vertex; } temp=t1; cout<<"\nAny other node(y/n):"; cin>>ch; } } else } do{ cout<<"\n -------------------------- \n"; cout<<"\n -------------------------- \n"; cout<<" 1.Display Graph \n"; cout<<" 2.Search Connection \n"; cout<<"\n Enter your choice : "; cin>>choice; switch(choice) { case 1: display(a); break; case 2: clrscr(); cout<<"\n Enter source vertex :"; cin>>x; cout<<"\n Enter destination vertex :"; cin>>y; search_connec(a,x,y); break; default : cout<<"\n Invalid choice"; break; } }while(choice==1 ||choice==2); }``` Last edited by a moderator: Feb 18, 2012 Joined: Aug 6, 2004 Messages: 3,012

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Select Page A firm must decide whether to construct a small, medium, or large stamping plant. A consultant’s report indicates a .20 probability that demand will be low and an .80 probability that demand will be high. If the firm builds a small facility and demand turns out to be low, the net present value will be \$42 million. If demand turns out to be high, the firm can either subcontract and realize the net present value of \$42 million or expand greatly for a net present value of \$48 million. The firm could build a medium-size facility as a hedge: If demand turns out to be low, its net present value is estimated at \$22 million; if demand turns out to be high, the firm could do nothing and realize a net present value of \$46 million, or it could expand and realize a net present value of \$50 million. If the firm builds a large facility and demand is low, the net present value will be –\$20 million, whereas high demand will result in a net present value of \$72 million. a. Analyze this problem using a decision tree. b. What is the maximin alternative? c. Compute the EVPI and interpret it. d. Perform sensitivity analysis on P(high)

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Search a number 348568900 = 225218672 BaseRepresentation bin10100110001101… …011110101000100 3220021220010122021 4110301223311010 51203213201100 654331012524 711431533601 oct2461536504 9807803567 10348568900 11169837a79 129889a144 13572a49b4 14344176a8 1520904a1a hex14c6bd44 348568900 has 27 divisors (see below), whose sum is σ = 756799869. Its totient is φ = 139352880. The previous prime is 348568859. The next prime is 348568903. The reversal of 348568900 is 9865843. The square root of 348568900 is 18670. It is a perfect power (a square), and thus also a powerful number. It can be written as a sum of positive squares in only one way, i.e., 125484804 + 223084096 = 11202^2 + 14936^2 . It is a Duffinian number. It is not an unprimeable number, because it can be changed into a prime (348568903) by changing a digit. It is a polite number, since it can be written in 8 ways as a sum of consecutive naturals, for example, 185767 + ... + 187633. Almost surely, 2348568900 is an apocalyptic number. 348568900 is the 18670-th square number. It is an amenable number. 348568900 is an abundant number, since it is smaller than the sum of its proper divisors (408230969). It is a pseudoperfect number, because it is the sum of a subset of its proper divisors. 348568900 is an equidigital number, since it uses as much as digits as its factorization. 348568900 is an evil number, because the sum of its binary digits is even. The sum of its prime factors is 3748 (or 1874 counting only the distinct ones). The product of its (nonzero) digits is 207360, while the sum is 43. The cubic root of 348568900 is about 703.7680478321. The spelling of 348568900 in words is "three hundred forty-eight million, five hundred sixty-eight thousand, nine hundred".

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# Difference between revisions of "Optimization" Optimization is simply finding the maximum or minimum possible value. In order to prove that a value is a maximum or minimum, one needs to prove that the value is attainable and that there is no higher or lower value (depending on the problem) that works. ## Optimization Techniques • There are multiple ways to determine the maximum or minimum (depending of the leading term) of a quadratic (depending on the form). • If the quadratic is in the form $a(x-h)^2+k$ (vertex form), the maximum or minimum of the quadratic is $k$ by the Trivial Inequality. • If the quadratic is in the form $ax^2 + bx + c$ (standard form), the maximum or minimum of the quadratic is achieved when $x = -\tfrac{b}{2a}$. This can be derived by completing the square. • The maximum of $\sin (x)$ and $\cos (x)$ is 1, and the minimum of $\sin (x)$ and $\cos (x)$ is -1. • One can also use coordinate geometry to determine the maximum or minimum. Optimization is often done when two figures touch each other exactly once. • In calculus, for a function $f(x)$, the local maximums and local minimums are part of the critical points of the function. The x-values of the critical points can be found by taking the derivative of $f(x)$ and setting it to equal 0. In order to find the absolute maximum or minimum, one needs to also check the endpoints of an interval.

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# Inverse (LS) problem for binary data #### Zhiv As per topic. Is there any well established method for solving linear systems for binary data? pardon if this is in wrong cathegory, english is not my first language and I'm not that well aware of the english terms. i.e. the classical g = M*f problem, where g is measured data and we want to know f. In this case, with calibration data we can determine M, but it has an ill-conditioned inverse, so the classical solution of f = M-1*g doesn't work. Enter the Tikhonov regularization, but it fails to be accurate enough. Conjugate gradient method, i.e. solving min || M*f-g|| might work, if the M was positive definite, but it is not. (it's symmetrical though). Also, we demand that every element of f and g is either 0 or 1, as the measured data g is in binary form. Google scholar was of little help, so... so in short a) Is there any well know tools for the problem when the data is binaric b) am I screwed? Related Linear and Abstract Algebra News on Phys.org #### John Creighto Try looking up neural network classifiers. I think I learned an algorithm for something like this once upon a time. #### bpet As per topic. Is there any well established method for solving linear systems for binary data? pardon if this is in wrong cathegory, english is not my first language and I'm not that well aware of the english terms. i.e. the classical g = M*f problem, where g is measured data and we want to know f. In this case, with calibration data we can determine M, but it has an ill-conditioned inverse, so the classical solution of f = M-1*g doesn't work. Enter the Tikhonov regularization, but it fails to be accurate enough. Conjugate gradient method, i.e. solving min || M*f-g|| might work, if the M was positive definite, but it is not. (it's symmetrical though). Also, we demand that every element of f and g is either 0 or 1, as the measured data g is in binary form. Google scholar was of little help, so... so in short a) Is there any well know tools for the problem when the data is binaric b) am I screwed? What happens when you do row reduction? #### Zhiv What happens when you do row reduction? Doesn't change the nature of the problem, as far as I can see. If the matrix has ill inverse (i.e. it exsits but causes numerical problems), I can't see simple algebraic manipulation taking care of it. When it comes to the suggestion of neural networks, that could probably work, but since I'm required to solve the problem for hundreds of thousands of f for given M, it might be computationally challenging. I have an iterative idea where we minimize ||M*f-g|| by placing constraint force on f so that it reshapes after each iteration until minimum is reached. (after all, I know the blurring process M so I know in which direction to push my guesses). And this works. The problem is that it's pretty much mathematically non-robust. But thanks for your help anyway, at least I know now that I'm not missing (hopefully) anything obvious. #### bpet I have an iterative idea where we minimize ||M*f-g|| by placing constraint force on f so that it reshapes after each iteration until minimum is reached. (after all, I know the blurring process M so I know in which direction to push my guesses). And this works. The problem is that it's pretty much mathematically non-robust. But thanks for your help anyway, at least I know now that I'm not missing (hopefully) anything obvious. Just out of curiousity what norm is used? I'm assuming it's the number of nonzero elements. Iterative solvers can be tricky to implement for binary linear systems, e.g. M = [v1,v2], v1=[1,0,1,1,...,1]', v2=[0,1,1,1,...,1]' g = [1,1,0,0,...,0]' where the solution is obviously f = [1,1]' but any movement from [0,0]' to [1,0]' or [0,1]' increases ||M*f-g||. Another approach could be to write it as an integer programming problem: minimize ||M*f-g-2*h|| for integer f and h subject to 0<=f<=1 and M*f>=2*h. If M is constant and the dimension not too big it's possible that lattice reduction techniques will help simplify the equations to make repeated solutions more efficient. #### Zhiv Just out of curiousity what norm is used? I'm assuming it's the number of nonzero elements. Iterative solvers can be tricky to implement for binary linear systems, e.g. M = [v1,v2], v1=[1,0,1,1,...,1]', v2=[0,1,1,1,...,1]' g = [1,1,0,0,...,0]' where the solution is obviously f = [1,1]' but any movement from [0,0]' to [1,0]' or [0,1]' increases ||M*f-g||. Another approach could be to write it as an integer programming problem: minimize ||M*f-g-2*h|| for integer f and h subject to 0<=f<=1 and M*f>=2*h. If M is constant and the dimension not too big it's possible that lattice reduction techniques will help simplify the equations to make repeated solutions more efficient. Euclidean norm is used at the moment. I have also toyed with amount of non-zero elements. I guess they are pretty much same when the problem is binaric. I just lack mathematical rigour to check this. Well, I'm the lucky situation that f and g are of same rank(?) same size in plain english. Also, properties of M are well know (it's basically (gaussian) model of a point spread function.) It models blurr in depth direction in 3D imaging modality. The fact that M is PSF and that we can use the blurred data as initial guess, I now 'squeeze' the f smaller with the norm ||M*f-g|| i.e. fk = fk-1-T(f''k-1), where f'' is second derivative. T(x) = 0 if x smaller than 0, else 1 this is pretty much equal of deducing zero crossings from the vector, i.e operating 00111111000011100 would end up as 00011110000001000 This usually means that the blurred binary image is 'squeezed' to approximately right size. (Images are 3D and we are only intrested in the blur in blurr direction) My worry (and the worry of my biophysicist advisor) is that this method is way too ad-hoc, so I was probing these forums. I guess I should have mentioned all this in the first post, but I wanted to check that I hadn't missed anything blindly obivous. I will check out that integer trick out anyway, just in case. Staff Emeritus Gold Member #### bpet Also, properties of M are well know (it's basically (gaussian) model of a point spread function.) It models blurr in depth direction in 3D imaging modality. Sorry I should have asked earlier, but by "binary" do you mean M, f, g are all binary matrices and the arithmetic uses 1+1=0, or do you mean that the elements of f are restricted to 0's and 1's? The latter case might be easier to solve. Could you post some small (maximum 4x4) examples of M, f, g. #### Zhiv Sorry I should have asked earlier, but by "binary" do you mean M, f, g are all binary matrices and the arithmetic uses 1+1=0, or do you mean that the elements of f are restricted to 0's and 1's? The latter case might be easier to solve. Could you post some small (maximum 4x4) examples of M, f, g. My bad. g (and f) are restricted to 0s and 1s. M is matrix with real elements. Well, the data is usually in class of hundreds of elements, so I dunno how much use would cutting do. i.e. basically one 'strip' of the image looks like (g) 0 0 1 1 1 1 0 0 when it should look like (f) 0 0 0 1 1 0 0 0 (real data is in classes of few hundred elements, with M being n x n where n is the number of elements in g.) This blurring is modelled my gaussian function in the M, but the inversr is ill, so something bust be done to recover the real information. At now, I use method described earlier, but I was worried that there was some easier general solution to the problem. ### Physics Forums Values We Value Quality • Topics based on mainstream science • Proper English grammar and spelling We Value Civility • Positive and compassionate attitudes • Patience while debating We Value Productivity • Disciplined to remain on-topic • Recognition of own weaknesses • Solo and co-op problem solving

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# Extracting square root 1.x²=16 2.t²=81 3.(x-4)²=169 4.(k+7)²=289 5.(2s-1)²=225 2 by jasperanacan21 2016-06-28T19:44:52+08:00 1. x=4 2. t=9 3. x=17 4. k=10 5. s=8 2016-06-28T20:15:47+08:00 The solutions on the equation are: 1. 4 or -4 2. 9 or -9 3. 17 or -9 4. 10 or -24 5. 8 or -7

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# 73545 - Mathematical Methods M • Moduli: Nicola Arcozzi (Modulo 1) Massimo Ferri (Modulo 2) • Campus: Bologna • Corso: Second cycle degree programme (LM) in Telecommunications Engineering (cod. 9205) Also valid for Second cycle degree programme (LM) in Electronic Engineering (cod. 0934) • from Sep 17, 2024 to Dec 17, 2024 • from Sep 20, 2024 to Dec 20, 2024 ## Learning outcomes In the first part the student is supposed to learn the different types of graphs, their matrix representations, the related invariants and the problems which can find a model and solution in Graph Theory. In the second part, differential equations of the first and second order are studied. ## Course contents Module 1 (Fourier analysis) Brief introduction to Banach and Hilbert spaces; Fourier series and applications; Fourier transform; FFT and DFT; Wavelets; Applications to ODE and PDE of interest in engineering application. The detailed program is published on the e-learning platform Virtuale. Module 2 (Graph theory) Graphs and subgraphs. Trees. Connectivity. Euler tours and Hamilton cycles. Matchings. Edge colourings. Independent sets and cliques. Vertex colourings. Planar graphs. Directed graphs. Hints at networks. A detailed program could be found on the e-learning platform Insegnamenti On-line. Fourier analysis (Modulo 1): Lecture notes of the teacher. The notes (pdf) will be available through the institutional site Virtuale before the lessons. Students may also use the following textbooks: - Davide Guidetti: Notes of the course Mathematical Methods (Pdf file available on AMS-Campus: Chapters 2 (normed spaces, Fourier series) and Chapter 4 (Fourier transform) - Erwin Kreyszig: Advanced Engineering Mathematics, 10th Edition J. Wiley (2014) Chapter 11 (Fouries series and Fourier transform ) and Chapter 12 (PDEs) - Tim Olson: Applied Fourier Analysis: from signal processing to medical imaging, Birkhauser Chapters 1-5, 10 Graph theory (Modulo 2): Official textbook J.A. Bondy and U.S.R. Murty, "Graph theory with applications", Support textbooks J.A. Bondy and U.S.R. Murty, "Graph theory", Springer Series: Graduate Texts in Mathematics, Vol. 244 (2008) R. Diestel, "Graph theory", Springer Series: Graduate Texts in Mathematics, Vol. 173 (2005) [http://diestel-graph-theory.com/basic.html] (3 MB). ## Teaching methods Lessons and exercises ## Assessment methods Fourier analysis (Modulo 1) The exam is oral. The student answers to questions on the topics of the course. The first question is a topic chosen by the student, then the exam continues with questions on different topics of the program. Calls are regularly opened on ALMAESAMI. Exams are in presence. It is obligeatory to enrol in the chosen call on Almaesami. Students may give the exam at any call and must show their University badge before starting the exam. The grade of this part of the exam is expressed in X/30s and published on Almaesami. It is possible to refuse the grade of this part only once. Graph Theory (Module 2) The exam is made up of two parts: a mid-term test with exercises and a final oral exam. Students are invited to show the University badge before starting either part. Examples of mid-term test are published on the e-learning platform Insegnamenti On-line and on the program page: http://www.dm.unibo.it/~ferri/hm/progmame.htm The date of the mid-term test will be published on this page: https://www.dm.unibo.it/~ferri/hm/ricapp.htm#app The mid-term test MUST be passed with a score of at least 14 (over 24). If a student doesn't pass, he/she must recover it; the dates for recovering will also be published at this page: https://www.dm.unibo.it/~ferri/hm/ricapp.htm#app Apply for the final oral exam at AlmaEsami. The final exam is on the whole program published, by the end of the course, on the e-learning platform Insegnamenti On-line and is as follows: two subjects are proposed to the student (each of which is either the title of a long chapter, or the sum of the titles of two short ones); they choose one and write down all what they remember about it, without the help of notes, texts, electronic devices; a discussion on their essay and in general about the chosen subject follows. It is an oral examination, so writing is only a help for the student to gather ideas. --- Final mark and verbalization The final grading is given by the arithmetic average of the grades in the mathematical analysis and graph theory part. Prof Arcozzi signs the grades on Almaesami within 5 days from the completion of the two parts of the exam. ## Teaching tools Fourier analysis: Detailed programme, lecture notes, texts and solutions of exercises classes, recordings of the lectures. and instructions for the exam will be made available on the e-learning platform Virtuale. Graph theory (Module 2) Textbook available at http://book.huihoo.com/pdf/graph-theory-With-applications/ Additional material is published on the e-learning platform Insegnamenti On-line and on the program page [http://www.dm.unibo.it/~ferri/hm/progmame.htm] ## Office hours See the website of Nicola Arcozzi See the website of Massimo Ferri ### SDGs This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

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Think $\le$. If aij • bij for all (i;j)-entries, we write A • B. 2 6 6 4 1 1 1 1 3 7 7 5 Symmetric in a Zero-One Matrix Let R be a binary relation on a set and let M be its zero-one matrix. Append content without editing the whole page source. Recall that a relation on a set A is asymmetric if implies that. 1.2.1 Example Let 1,4,5 X and 3,6,7 Y Classical matrix for the crisp relation when R x y is 3 6 7 1 1 relations from X to X) together with (left or right) relation composition forms a monoid with zero, where the identity map on X is the neutral element, and the empty set is the zero element. The vertex a is called the initial vertex of For the sake of understanding assume that the first entry, which is zero, in the matrix is denoted by. For each ordered pair (x, y) in the relation R, there will be a directed edge from the vertex ‘x’ to vertex ‘y’. By using this graph, show L1 that R is not reflexiv 5 Sections 31-33 but not exactly) Recall: A binary relation R from A to B is a subset of the Cartesian product If , we write xRy and say that x is related to y with respect to R. A relation on the set A is a relation from A to A.. Some of which are as follows: 1. Suppose thatRis a relation fromAtoB. A 1 represents perfect positive correlation, a -1 represents perfect negative correlation, and 0 correlation means that the stocks move independently of each other. Inductive Step: Assume that Rn is symmetric. A relation between finite sets can be represented using a zero-one matrix. Make the table which contains rows equivalent to an element of P and columns equivalent to the element of Q. The relation R on the set {(a,b) | a,b ∈ Z} where (a,b)R(c,d) means a = c or b = d. Ans: 1, 2. 4 points Case 1 (⇒) R1 ⊆ R2. Terms Each binary relation over ℕ … This means (x R1 y) → (x R2 y). Assume A={a1,a2,…,am} and B={b1,b2,…,bn}. Here “1” implies complete truth degree for the pair to be in relation and “0” implies no relation. To Prove that Rn+1 is symmetric. This means that the rows of the matrix of R 1 will be indexed by the set B= fb View Answer. The relation R S is known the composition of R and S; it is sometimes denoted simply by RS. (a) Objective is to find the matrix representing . 8. The result is Figure 6.2.1. There aren't any other cases. Matrix representation of a relation If R is a binary relation between the finite indexed sets X and Y (so R ⊆ X×Y), then R can be represented by the logical matrix M whose row and column indices index the elements of X and Y, respectively, such that the entries of M are defined by: Let R be a relation on a set A with n elements. Page 105 . Consider the relation R represented by the matrix. Suppose that the relation R on the finite set A is represented by the matrix MR. Show that the matrix that represents the symmetric closure of R is MR ∨ Mt R. Aug 05 2016 11:48 AM Then • R is reflexive iff M ii = 1 for all i. b) . | Then • R is reflexive iff M ii = 1 for all i. Relations 10/10/2014 5 Definition: A Relation R from set A to set B is a subset of A × B. To interpret its value, see which of the following values your correlation r is closest to: Exactly –1. Definition: Let be a finite -element set and let be a relation on. Click here to edit contents of this page. Antisymmetric means that the only way for both $aRb$ and $bRa$ to hold is if $a = b$. (1) By Theorem proved in class (An equivalence relation creates a partition), General Wikidot.com documentation and help section. 23. relations from X to X) together with (left or right) relation composition forms a monoid with zero, where the identity map on X is the neutral element, and the empty set is the zero element. Representation of Relations. Let P1 and P2 be the partitions that correspond to R1 and R2, respectively. The group is called by one name and every member of a group has own individualities. Correlation is a measure of association between two things, here, stock prices, and is represented by a number from -1 to 1. In other words, all elements are equal to 1 on the main diagonal. In a tabular form 5. Privacy The relation isn't antisymmetric : (a,b) and (b,a) are in R, but a=/=b because they're both in the set {a,b,c,d}, which implies they're not the same. Let R1R1 and R2R2 be relations on a set A represented by the matrices MR1=⎡⎣⎢⎢⎢011110010⎤⎦⎥⎥⎥MR1= and MR2=⎡⎣⎢⎢⎢001111011⎤⎦⎥⎥⎥MR2=. If (a , b) ∉ R, we say that “a is not related to b“, and write aRb. The Matrix Representation of on is defined to be the matrix where the entires for are given by. 32. Let A be the matrix of R, and let B be the matrix of S. Then the matrix of S R is obtained by changing each nonzero entry in the matrix product AB to 1. The set of binary relations on a set X (i.e. To see that every a ∈ A belongs to at least one equivalence class, consider any a ∈ A and the equivalence class[a] R ={x Suppose that the relation R on the finite set A is represented by the matrix MR. Show that the matrix that represents the symmetric closure of R is MR ∨ Mt R. 7. Theorem: Let R be a binary relation on a set A and let M be its connection matrix. Matrices and Graphs of Relations [the gist of Sec. 2 6 6 4 1 1 1 1 3 7 7 5 Symmetric in a Zero-One Matrix Let R be a binary relation on a set and let M be its zero-one matrix. Then place a cross (X) in the boxes which represent relations of elements on set P to set Q. Let A = [aij] and B = [bij] be m £ n Boolean matrices. The relation R on the set of all people where aRb means that a is younger than b. Ans: 3, 4 22. A relation R from A to B can be represented by the m?n matrix MR=[mij], where 1 if aiRbj, mij = 0 if aiRbj Suppose that the relation R on the finite set A is represented by the matrix MR. Show that the matrix that represents the symmetric closure of R is MR ∨ Mt R. Posted 4 years ago. View/set parent page (used for creating breadcrumbs and structured layout). © 2003-2020 Chegg Inc. All rights reserved. View and manage file attachments for this page. Example. If we let $x_1 = 1$, $x_2 = 2$, and $x_3 = 3$ then we see that the following ordered pairs are contained in $R$: Let $M$ be the matrix representation of $R$. Notify administrators if there is objectionable content in this page. Each product has a size code, a weight code, and a shape code. The set of binary relations on a set X (i.e. In matrix terms, the transpose , (M R)T does not give the same relation. 4 Question 4: [10 marks] Let R be the following relation on the set { x,y,z }: { (x,x), (x,z), (y,y), (z,x), (z,y) } Use the 0-1 matrix representation for relations to find the transitive closure of R. Show the formula used to find the transitive closure of R from its 0-1 matrix representation and show the matrices in the intermediate steps in the algorithm, as Watch headings for an "edit" link when available. Suppose that the relation R on the finite set A is represented by the matrix MR. Show that the matrix that represents the symmetric closure of R is MR ∨ Mt R. Posted 4 years ago. In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatterplot. 14. 24. FIGURE 6.1.1 Illustration of a relation r = 8Hx, yL y is the square of x<, and s = 8Hx, yL x § y<. R is symmetric if and only if M = Mt. View wiki source for this page without editing. • R is symmetric iff M is a symmetric matrix: M = M T • R … Recall from the Hasse Diagrams page that if $X$ is a finite set and $R$ is a relation on $X$ then we can construct a Hasse Diagram in order to describe the relation $R$. R and relation S represented by a matrix M S. Then, the matrix of their composition S Ris M S R and is found by Boolean product, M S R = M R⊙M S The composition of a relation such as R2 can be found with matrices and Boolean powers. 17. Relation on a set We are particularly interested inbinary relations from a set to the same set. 5. Such a matrix is somewhat less German mathematician G. Cantor introduced the concept of sets. Show that Rn is symmetric for all positive integers n. 5 points Let R be a symmetric relation on set A Proof by induction: Basis Step: R1= R is symmetric is True. Something does not work as expected?   A relation between finite sets can be represented using a zero‐one matrix. In other words, all elements are equal to 1 on the main diagonal. For example, consider the set and let be the relation where for we have that if is divisible by, that is. The objective is find the way that the matrix representing a relation R on a set A to determine whether the relation is asymmetric. First of all, if Rgoes from A= fa 1;:::;a mgto B= fb 1;b 2;:::;b ng, then R 1 goes from B to A. Composition in terms of matrices. 12. iii. Let R be the relation {(a, b) | a divides b} on the set of integers. How can the matrix for R −1, the inverse of the relation R, be found from the matrix representing R, when R is a relation on a finite set A? Relation as a Table: If P and Q are finite sets and R is a relation from P to Q. The matrix of the relation R is an m£n matrix MR = [aij], whose (i;j)-entry is given by aij = ‰ 1 if xiRyj 0 if xiRyj: The matrix MR is called the Boolean matrix of R. If X = Y, then m = n, and the matrix M is a square matrix. Solution for Let R1 and R2 be relations on a set A represented by the matrices below: Mr1 = 1 1 1 1 1 0 0 Mr2 = 0 1 0 1 1 1 1 1 Find the matrix that represents… 7. ∨M [n] R. This theorem can be used to construct an algorithm for computing the transitive closure of the matrix of relation R. Algorithm 1 (p. 603) in the text contains such an algorithm. 3 R 6 . (a) Objective is to find the matrix representing . Draw the graph of the relation R, represented by adjacency matrix [0 0 1 11 1 1 1 0 1 MR on set A={1,2,3,4}. So we make a matrix that tells us whether an ordered pair is in the set, let's say the elements are $\{a,b,c\}$ then we'll use a $1$ to mark a pair that is in the set and a $0$ for everything else. This type of graph of a relation r is called a directed graph or digraph. Rn+1 is symmetric if for all (x,y) in Rn+1, we have (y,x) is in Rn+1 as well. In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatterplot. If (a , b) ∈ R, we say that “a is related to b", and write aRb. Show that R1 ⊆ R2 if and only if P1 is a refinement of P2. Example: A = (1, 2, 3) and B = {x, y, z}, and let R = {(1, y), (1, z), (3, y)}. The fuzzy relation R = “x is similar to y” may be represented in five different ways: 1. Let R is a relation on a set A, that is, R is a relation from a set A to itself. A perfect downhill (negative) linear relationship […] Representing Relations Using Matrices To represent relationRfrom setAto setBby matrixM, make a matrix withjAjrows andjBjcolumns. ), then any relation Rfrom A to B (i.e., a subset of A B) can be represented by a matrix with n rows and p columns: Mjk, the element in row j and column k, equals 1 if aj Rbk and 0 otherwise. Such a matrix is somewhat less R is reflexive if and only if M ii= 1 for all i. Let R be a relation from X to Y, and let S be a relation from Y to Z. Finite binary relations are represented by logical matrices. Check out how this page has evolved in the past. The relation R is represented by the matrix M R m ij where The matrix from MATH 1019 at Centennial College A Apparently you are talking about a binary relation on $A$, which is just a subset of $A \times A$. Connect vertex a to vertex b with an arrow, called an edge of the graph, going from vertex a to vertex b if and only if a r b. See pages that link to and include this page. View this answer. • R is symmetric iff M is a symmetric matrix: M = M T • R is antisymetric if M ij = 0 or M ji = 0 for all i ≠ j. For the sake of understanding assume that the first entry, which is zero, in the matrix is denoted by. If you want to discuss contents of this page - this is the easiest way to do it. Then the connection matrix M for R is 1 0 0 0 0 0 0 0 0 0 1 0 Note: the order of the elements of A and B matters. However, r would be more naturally expressed as r HxL = x2 or r HxL = y, where y = x2.But this notation when used for s is at best awkward. Theorem: Let R be an equivalence relation over a set A.Then every element of A belongs to exactly one equivalence class. And 13 is not related to 6 by R . Just re ect it across the major diagonal. Change the name (also URL address, possibly the category) of the page. abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix kernel linear algebra linear combination linearly … Let R be the relation represented by the matrix Find the matrix representing a) R1 b) R. c) R2. The value of r is always between +1 and –1. We will now look at another method to represent relations with matrices. (6) [6pts] Let R be the relation, defined on set (1, 2, 3), represented by the matrix: 0 1 1 MR 1 0 0 1 0 1 Find the matrix representing the following relations. To interpret its value, see which of the following values your correlation r is closest to: Exactly –1. 4 Question 4: [10 marks] Let R be the following relation on the set { x,y,z }: { (x,x), (x,z), (y,y), (z,x), (z,y) } Use the 0-1 matrix representation for relations to find the transitive closure of R. Show the formula used to find the transitive closure of R from its 0-1 matrix representation and show the matrices in the intermediate steps in the algorithm, as Relations (Related to Ch. Plagiarism Checker. A binary relation R from set x to y (written as xRy or R(x,y)) is a 215 We may ask next how to interpret the inverse relation R 1 on its matrix. Let R be the relation represented by the matrix 0 1 01 L1 1 0J Find the matrices that represent a. R2 b. R3 c. R4 Let R1 and R2 be relations on a set A-fa, b, c) represented by these matrices, [0 1 0] MR1-1 0 1 and MR2-0 1 1 1 1 0 Find the matrix that represents R1 o R2. How can the matrix representing a relation R on a set A be used to determine whether the relation is asymmetric? 012345678 89 01 234567 01 3450 67869 3 8 65 ii. Find out what you can do. Linguistically, such as by the statement “x is similar toy” 2. $m_{ij} = \left\{\begin{matrix} 1 & \mathrm{if} \: x_i \: R \: x_j \\ 0 & \mathrm{if} \: x_i \: \not R \: x_j \end{matrix}\right.$, $m_{11}, m_{13}, m_{22}, m_{31}, m_{33} = 1$, Creative Commons Attribution-ShareAlike 3.0 License. Let R be the relation represented by the matrix Find the matrices representing a)R −1. ), then any relation Rfrom A to B (i.e., a subset of A B) can be represented by a matrix with n rows and p columns: Mjk, the element in row j and column k, equals 1 if aj Rbk and 0 otherwise. Then R R, the composition of R with itself, is always represented. Similarly, The relation R … As a directed graph 4. View Homework Help - Let R Be The Relation Represented By The Matrix.pdf from MATH 202 at University of California, Berkeley. Then $m_{11}, m_{13}, m_{22}, m_{31}, m_{33} = 1$ and $m_{12}, m_{21}, m_{23}, m_{32} = 0$ and: If $X$ is a finite $n$-element set and $\emptyset$ is the empty relation on $X$ then the matrix representation of $\emptyset$ on $X$ which we denote by $M_{\emptyset}$ is equal to the $n \times n$ zero matrix because for all $x_i, x_j \in X$ where $i, j \in \{1, 2, ..., n \}$ we have by definition of the empty relation that $x_i \: \not R \: x_j$ so $m_{ij} = 0$ for all $i, j$: On the other hand if $X$ is a finite $n$-element set and $\mathcal U$ is the universal relation on $X$ then the matrix representation of $\mathcal U$ on $X$ which we denote by $M_{\mathcal U}$ is equal to the $n \times n$ matrix whoses entries are all $1$'s because for all $x_i, x_j \in X$ where $i, j \in \{ 1, 2, ..., n \}$ we have by definition of the universal relation that $x_i \: R \: x_j$ so $m_{ij} = 1$ for all $i, j$: \begin{align} \quad R = \{ (x_1, x_1), (x_1, x_3), (x_2, x_3), (x_3, x_1), (x_3, x_3) \} \subset X \times X \end{align}, \begin{align} \quad M = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 1 \end{bmatrix} \end{align}, \begin{align} \quad M_{\emptyset} = \begin{bmatrix} 0 & 0 & \cdots & 0\\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 0 \end{bmatrix} \end{align}, \begin{align} \quad M_{\mathcal U} = \begin{bmatrix} 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & \cdots & 1 \end{bmatrix} \end{align}, Unless otherwise stated, the content of this page is licensed under. The matrix representing R1∪R2R1∪R2 is … In this method it is easy to judge if a relation is reflexive, symmetric or transitive just … Reflexive in a Zero-One Matrix Let R be a binary relation on a set and let M be its zero-one matrix. By listing (or taking the union of) all fuzzy singletons 3. Interesting fact: Number of English sentences is equal to the number of natural numbers. Matrices and Graphs of Relations [the gist of Sec. For example, consider the set $X = \{1, 2, 3 \}$ and let $R$ be the relation where for $x, y \in X$ we have that $x \: R \: y$ if $x + y$ is divisible by $2$, that is $(x + y) \equiv 0 \pmod 2$. [3pts) R- 2. What is the symmetric closure of R? If there are k nonzero entries in M R, the matrix representing R, how many nonzero entries are there in M R − 1, the matrix representing R − 1, the inverse of R? Correlation is a common metric in finance, and it is useful to know how to calculate it in R. In this if a element is present then it is represented by 1 else it is represented by 0. iv. Choose orderings for X, Y, and Z; all matrices are with respect to these orderings. & When we deal with a partial order, we know that the relation must be reflexive, transitive, and antisymmetric. Relations, Formally A binary relation R over a set A is a subset of A2. View desktop site, Relation R on a set can be reprented as a matrix where , here, we have a relation on set {1,2,3}, (6) [6pts] Let R be the relation, defined on set (1, 2, 3), represented by the matrix: 0 1 1 MR 1 0 0 1 0 1 Find the matrix representing the following relations. (More on that later.) View Answer . Combining Relations Composite of R and S, denoted by S o R is the relation consisting of ordered pairs (a, c), where a Î A, c Î C, and for which there exists an element b Î B and (b, c) Î S and where R is a relation from a set A to a set B and S is a relation from set B to set C, or R is a relation from P to Q. A relation can be represented using a directed graph. We see that (a,b) is in R, and (b,a) is in R too, so the relation is symmetric. A relation R is defined as from set A to set B,then the matrix representation of relation is M R = [m ij] where m ij = { 1, if (a,b) Є R 0, if (a,b) Є R } xRy is shorthand for (x, y) ∈ R. A relation doesn't have to be meaningful; any subset of A2 is a relation. Example: We can dene a relation R on the set of positive integers such that a R b if and only if a j b . This point is moot for A = B . Representing relations using matrices. Solution for Let R be a relation on the set A = {1,2,3,4} defined by R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (3,4), (4,4)} Construct the matrix… discrete sets. Solution for 10 0 1 For the set A={1,2,3} and B={a,b.c,d} , if R is a relation on the set A and B represented by the matrix , 0 100 then relation R is given by… 1. If there is an ordered pair (x, x), there will be self- loop on vertex ‘x’. The order of the elements of A and B is arbitrary, but fixed. 6.3. Transitivity on a set of ordered pairs (the matrix you have there) says that if $(a,b)$ is in the set and $(b,c)$ is in the set then $(a,c)$ has to be. The notation H4, 16L œ r or H3, 7.2L œ s makes sense in both cases. a) Explain how to use a zero–one matrix to represent a relation on a finite set. How can the matrix representing a relation R on a set A be used to determine whether the rela- ... relation R, be found from the matrix representing R? 13. 7.2 of Grimaldi] If jAj= n and jBj= p, and the elements are ordered and labeled (A = fa1;a2;:::;ang, etc. They are represented by labeled points or occasionally by small circles. Composition in terms of matrices. The relation R on R where aRb means a − b ∈ Z. Ans: 1, 2, 4. Similarly, The relation R … Suppose that and R is the relation of A. R is reflexive if and only if M ii = 1 for all i. A perfect downhill (negative) linear relationship […] Also, R R is sometimes denoted by R 2. Relations can be represented in many ways. That is, exchange the ijth entry with the jith entry, for each i and j. Consider the relation R represented by the matrix. The number of vertices in the graph is equal to the number of elements in the set from which the relation has been defined. It can be reflexive, but it can't be symmetric for two distinct elements. The resulting matrix is called the transpose of the original matrix. Category ) of the page ( used for creating breadcrumbs and structured layout ) for which is. The Matrix.pdf from MATH 202 at University of California, Berkeley in 3... It the Case that 2 is related to b “, a. To 1 on the main diagonal { a1, a2, …, bn } matrices Exercise... The concept of sets is always between +1 and –1 interesting fact: number of sentences. Iff M ii = 1 for all i for the pair to be in relation and “ 0 ” no! 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Relations [ the gist of Sec œ S makes sense in both cases 2 4! Set Q with a partial order, we often say that “ a is a relation.! Singletons 3 “ x is similar to y ” may be represented using a zero-one matrix is a relation finite! And R2R2 be relations on a set A.Then every element of Q group has own individualities ) the... Product has a size code, a to determine whether the relation is asymmetric all ( i ; j -entries. 0 ” implies no relation ⇒ ) R1 b ) | a divides b } on the main diagonal matrices!: number of English sentences is equal to the number of vertices in the graph is equal to the of... A refinement of P2 all i called by one name and every member of a linear relationship between two on..., b2, …, am } and B= { b1, b2, …, am } B=! Parent page ( if possible ) with n elements a element is present then it is represented by the MR1=⎡⎣⎢⎢⎢011110010⎤⎦⎥⎥⎥MR1=... X is similar toy ” 2 a with n elements subset of a2 the page used. Given the following values your correlation R is sometimes denoted by Explain to! Contents of this page means of certain rules or description ” 2 from a to itself R2R2 be relations a... Which represent relations with matrices aRb means that a relation between nite sets can be in! 20 Oz Dr Pepper, Fundamentals Of Design Book, Why Do You Want To Work Here Computer Science, Why Is Historiography Important, Nintendo 64 Pokémon Stadium Mini Games,

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Online Computer Science Courses Digital Image Processing Certification Exam Tests Digital Image Processing Practice Test 93 # Piecewise Linear Transformation Functions Quiz Questions and Answers PDF - 93 Books: Apps: The e-Book Piecewise Linear Transformation Functions Quiz Questions, piecewise linear transformation functions quiz answers PDF download chapter 7-93 to study online image processing degree courses. Practice Intensity Transformation and Spatial Filtering MCQ with answers PDF, piecewise linear transformation functions Multiple Choice Questions (MCQ Quiz) for online college degrees. The Piecewise Linear Transformation Functions Quiz App Download: Free learning app for piecewise linear transformation functions, color models in color image processing, image interpolation in dip, point line and edge detection in image processing, color transformation test prep for computer science programs. The Quiz Simplest piecewise linear transformation function is: contrast stretching, linear stretching, color stretching and elastic stretching with "Piecewise Linear Transformation Functions" App Download (Free) for computer science associate degree. Solve intensity transformation and spatial filtering questions and answers, Amazon eBook to download free sample for information and communication technology. ## Piecewise Linear Transformation Functions Quiz Answers : Test 93 MCQ 461: The simplest piecewise linear transformation function is A) linear stretching B) contrast stretching C) color stretching D) elastic stretching MCQ 462: RGB space is also known as A) pixels B) coordinates C) pixel depth D) color depth MCQ 463: Image linear interpolation is given by the formula A) v(x,y) = ax+by+cxy+d B) v(x,y) = ax+by+cxy C) v(x,y) = ax+by+d D) v(x,y) = by+cxy+d MCQ 464: First derivatives in image segmentation produces A) thick edges B) thin edges C) fine edges D) rough edges MCQ 465: For CMYK color space, no of transformations will be A) n = 2 B) n = 3 C) n = 4 D) n = 5 ### Piecewise Linear Transformation Functions Learning App & Free Study Apps Download Digital Image Processing Quiz App to learn Piecewise Linear Transformation Functions Quiz, Computer Networks MCQs App, and Database Management System Quiz App (Android & iOS). The free "Piecewise Linear Transformation Functions Quiz" App includes complete analytics of history with interactive assessments. Download Play Store & App Store learning Apps & enjoy 100% functionality with subscriptions! ALL-in-ONE Learning App (Android & iOS) Digital Image Processing App (Android & iOS) Computer Networks App (Android & iOS) Database Management System App (Android & iOS)

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problem10_75 # University Physics with Modern Physics with Mastering Physics (11th Edition) This preview shows page 1. Sign up to view the full content. 10.75: a) Use conservation of energy to find the speed 2 v of the ball just before it leaves the top of the cliff. Let point 1 be at the bottom of the hill and point 2 be at the top of the hill. Take 0 = y at the bottom of the hill, so m. 0 . 28 y and 0 2 1 = = y 2 2 2 1 2 2 2 1 2 2 1 2 1 2 1 2 1 2 2 1 1 ϖ I mv mgy I mv U K U K + + = + + = + Rolling without slipping means ( 29 2 5 1 2 2 5 2 2 1 2 2 1 ) / ( and mv r v mr I r v = = = s m 26 . 15 2 7 10 2 1 2 2 2 10 7 2 2 1 10 7 = - = + = gy v v mv mgy mv Consider the projectile motion of the ball, from just after it leaves the top of the cliff until just before it lands. Take y + to be downward. Use the vertical motion to find the time in the air: s 39 . 2 gives ? m, 0 . 28 , s m 80 . 9 , 0 2 2 1 0 0 0 2 0 = + = - = = - = = t t a t v y y t y y a v y y y y During this time the ball travels horizontally ( 29 ( 29 m. 5 . 36 s 39 . 2 s m 26 . 15 0 0 = = = - t v x x x Just before it lands, This is the end of the preview. Sign up to access the rest of the document. Unformatted text preview: s 3 . 15 and s 4 . 23 = = = + = x x y y y v v t a v v s m . 28 2 2 = + = y x v v v b) At the bottom of the hill, ( 29 . s m . 25 r r v ω = = The rotation rate doesn't change while the ball is in the air, after it leaves the top of the cliff, so just before it lands . s) 3 . 15 ( r = The total kinetic energy is the same at the bottom of the hill and just before it lands, but just before it lands less of this energy is rotational kinetic energy, so the translational kinetic energy is greater.... View Full Document ## This document was uploaded on 02/04/2008. Ask a homework question - tutors are online

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Comment author: 02 February 2018 06:06:26PM *  0 points [-] This is just a manifestation of the general fact that it is impossible to specify a hypothetical fully without telling the entire story of how things got that way from the dawn of time. Speaking of hypotheticals is thus inherently loose. There is no way to avoid fallacies in most such exercises. Feigning rigor by calling specific cases "fallacies" is pretention. It isn't just difficult to avoid these errors; it's impossible, and relegates the exercise to the merely cautiously suggestive, not a central method of philosophy. In response to Decision Theory FAQ Comment author: 28 February 2013 05:33:54PM *  9 points [-] I don't really think Newcomb's problem or any of its variations belong in here. Newcomb's problem is not a decision theory problem, the real difficulty is translating the underspecified English into a payoff matrix. The ambiguity comes from the the combination of the two claims, (a) Omega being a perfect predictor and (b) the subject being allowed to choose after Omega has made its prediction. Either these two are inconsistent, or they necessitate further unstated assumptions such as backwards causality. First, let us assume (a) but not (b), which can be formulated as follows: Omega, a computer engineer, can read your code and test run it as many times as he would like in advance. You must submit (simple, unobfuscated) code which either chooses to one- or two-box. The contents of the boxes will depend on Omega's prediction of your code's choice. Do you submit one- or two-boxing code? Second, let us assume (b) but not (a), which can be formulated as follows: Omega has subjected you to the Newcomb's setup, but because of a bug in its code, its prediction is based on someone else's choice than yours, which has no correlation with your choice whatsoever. Do you one- or two-box? Both of these formulations translate straightforwardly into payoff matrices and any sort of sensible decision theory you throw at them give the correct solution. The paradox disappears when the ambiguity between the two above possibilities are removed. As far as I can see, all disagreement between one-boxers and two-boxers are simply a matter of one-boxers choosing the first and two-boxers choosing the second interpretation. If so, Newcomb's paradox is not as much interesting as poorly specified. The supposed superiority of TDT over CDT either relies on the paradox not reducing to either of the above or by fiat forcing CDT to work with the wrong payoff matrices. I would be interested to see an unambiguous and nontrivial formulation of the paradox. • Allowing Omega to do its prediction by time travel directly contradicts box B contains either \$0 or \$1,000,000 before the game begins, and once the game begins even the Predictor is powerless to change the contents of the boxes. Also, this obviously make one-boxing the correct choice. • Allowing Omega to accurately simulate the subject reduces to problem to submit code for Omega to evaluate; this is not exactly paradoxical, but then the player is called upon to choose which boxes to take actually means the code then runs and returns its expected value, which clearly reduces to one-boxing. • Making Omega an imperfect predictor, with an accuracy of p<1.0 simply creates a superposition of the first and second case above, which still allows for straightforward analysis. • Allowing unpredictable, probabilistic strategies violates the supposed predictive power of Omega, but again cleanly reduces to payoff matrices. • Finally, the number of variations such as the psychopath button are completely transparent, once you decide between choice is magical and free will and stuff which leads to pressing the button, and the supposed choice is deterministic and there is no choice to make, but code which does not press the button is clearly the most healthy. In response to comment by on Decision Theory FAQ Comment author: 03 March 2013 05:21:10AM *  2 points [-] I agree; wherever there is paradox and endless debate, I have always found ambiguity in the initial posing of the question. An unorthodox mathematician named Norman Wildberger just released a new solution by unambiguously specifying what we know about Omega's predictive powers. Comment author: 10 May 2012 03:47:21PM *  1 point [-] Anti-epistemology is a huge actual danger of actual life, So it is, but I'm wondering if anyone can suggest a (possibly very exotic) real-life example where "epistemic rationality gives way to instrumental rationality."? Just to address the "hypothetical scenario" objection. EDIT: Does the famous Keynes quote "Markets can remain irrational a lot longer than you and I can remain solvent." qualify? Comment author: 10 May 2012 06:49:01PM 3 points [-] Any time you have a bias you cannot fully compensate for, there is a potential benefit to putting instrumental rationality above epistemic. One fear I was unable to overcome for many years was that of approaching groups of people. I tried all sorts of things, but the best piece advice turned out to be: "Think they'll like you." Simply believing that eliminates the fear and aids in my social goals, even though it sometimes proves to have been a false belief, especially with regard to my initial reception. Believing that only 3 out of 4 groups will like or welcome me initially and 1 will rebuff me, even though this may be the case, has not been as useful as believing that they'll all like me. ## Torture Simulated with Flipbooks 9 26 May 2011 01:00AM What if the brain of the person you most care about were scanned and the entirety of that person's mind and utility function at this moment were printed out on paper, and then several more "clock ticks" of their mind as its states changed exactly as they would if the person were being horribly tortured were printed out as well, into a gigantic book? And then the book were flipped through, over and over again. Fl-l-l-l-liiiiip! Fl-l-l-l-liiiiip! Would this count as simulated torture? If so, would you care about stopping it, or is it different from computer-simulated torture? Comment author: 25 May 2011 11:25:29PM 0 points [-] As for this collectivism, though, I don't go for it. There is no way to know another's utility function, no way to compare utility functions among people, etc. other than subjectively. That's very contestable. It has frequently argued here that preferences can be inferred from behaviour; it's also been argued that introspection (if that is what you mean by "subjectively") is not a reliable guide to motivation. Comment author: 26 May 2011 12:42:29AM 0 points [-] This is the whole demonstrated preference thing. I don't buy it myself, but that's a debate for another time. What I mean by subjectively is that I will value one person's life more than another person's life, or I could think that I want that \$1,000,000 more than a rich person wants it, but that's just all in my head. To compare utility functions and work from demonstrated preference usually - not always - is a precursor to some kind of authoritarian scheme. I can't say there is anything like that coming, but it does set off some alarm bells. Anyway, this is not something I can substantiate right now. Comment author: 26 May 2011 12:25:29AM 0 points [-] I'm getting a bad vibe here, and no longer feel we're having the same conversation "Person or group that decides"? Who said anything about anyone deciding anything? And my point was that this perhaps this is the meta-ethical position that every rational agent individually converges to. So nobody "decides", or everyone does. And if they don't reach the same decision, then there's no single objective morality -- but even i so perhaps there's a limited set of coherent metaethical positions, like two or three of them. I personally think all this collectivism is a carryover from the idea of (collective) democracy and other silly ideas. I think my post was inspired more by TDT solutions to Prisoner's dilemma and Newcomb's box, a decision theory that takes into account the copies/simulations of its own self, or other problems that involve humans getting copied and needing to make a decision in blind coordination with their copies. I imagined system that are not wholly copied, but rather just the module that determines the meta-ethical constraints, and tried to figure out to which directions would such system try to modify themselves, in the knowledge that other such system would similarly modify themselves. Comment author: 26 May 2011 12:37:48AM 0 points [-] You're right, I think I'm confused about what you were talking about, or I inferred too much. I'm not really following at this point either. One thing, though, is that you're using meta-ethics to mean ethics. Meta-ethics is basically the study of what people mean by moral language, like whether ought is interpreted as a command, as God's will, as a way to get along with others, etc. That'll tend to cause some confusion. A good heuristic is, "Ethics is about what people ought to do, whereas meta-ethics is about what ought means (or what people intend by it)." Comment author: 25 May 2011 11:15:04PM *  1 point [-] I'll just decide not to follow the advice, or I'll try it out and then after experiencing pain I will decide not to follow the advice again. I might tell you that, too, but I don't need to use the word "true" or any equivalent to do that. I can just say it didn't work. Any word can be eliminated in favour of a definitions or paraphrase. Not coming out with an equivalent -- showing that you have dispensed with the concept -- is harder. Why didn't it work? You're going to have to paraphrase "Because it wasn't true" or refuse to answer. Comment author: 26 May 2011 12:29:29AM *  -1 points [-] The concept of truth is for utility, not utility for truth. To get them backwards is to merely be confused by the words themselves. It's impossible to show you've dispensed with any concept, except to show that it isn't useful for what you're doing. That is what I've done. I'm non-cognitive to God, truth, and objective value (except as recently defined). Usually they all sound like religion, though they all are or were at one time useful approximate means of expressing things in English. Comment author: 25 May 2011 05:30:46PM 0 points [-] What about beliefs being justified by non-beliefs? If you're a traditional foundationalist, you think everything is ultimately grounded in sense-experience, about which we cannot reasonably doubt. If a traditional foundationalist believes that beliefs are justified by sense-experience, he's a justificationalist. The argument in the OP works. How can he justify the belief that beliefs are justified by sense-experience without first assuming his conclusion? Also, what about externalism? This is one of the major elements of modern epistemology, as a response to such skeptical arguments. I had to look it up. It is apparently the position that the mind is a result of both what is going on inside the subject and outside the subject. Some of them seem to be concerned about what beliefs mean, and others seem to carefully avoid using the word "belief". In the OP I was more interested in whether the beliefs accurately predict sensory experience. So far as I can tell, externalism says we don't have a mind that can be considered as a separate object, so we don't know things, so I expect it to have little to say about how we know what we know. Can you explain why you brought it up? I don't mean to imply that either of these is correct, but it seems that if one is going to attempt to use disjunctive syllogism to argue for anti-justificationism, you ought to be sure you've partitioned the space of reasonable theories. I don't see any way to be sure of that. Maybe some teenage boy sitting alone in his bedroom in Iowa figured out something new half an hour ago; I would have no way to know. Given the text above, do think there are alternatives that are not covered? Perhaps it is so structured that it is invulnerable to being changed after it is adopted, regardless of the evidence observed. This example seems anomalous. If there exists some H such that, if P(H) > 0.9, you lose the ability to choose P(H), you might want to postpone believing in it for prudent reasons. But these don’t really bear on what the epistemically rational level of belief is (Assuming remaining epistemically rational is not part of formal epistemic rationality). Furthermore, if you adopted a policy of never raising P(H) above 0.9, it’d be just like you were stuck with P(H) < 0.9 ! The point is that if a belief will prevent you from considering alternatives, that is a true and relevant statement about the belief that you should know when choosing whether to adopt it. The point is not that you shouldn't adopt it. Bayes' rule is probably one of those beliefs, for example. Without a constraining external metric, there are many consistent sets [of preferences], and the only criticism you can ultimately bring to bear is one of inconsistency. I presently believe there are many consistent sets of preferences, and maybe you do too. If that's true, we should find a way to live with it, and the OP is proposing such a way. I don't know what the word "ultimately" means there. If I leave it out, your statement is obviously false -- I listed a bunch of criticisms of preferences in the OP. What did you mean? Comment author: 25 May 2011 11:09:22PM 0 points [-] How can he justify the belief that beliefs are justified by sense-experience without first assuming his conclusion? I don't know what exactly "justify" is supposed to mean, but I'll interpret it as "show to be useful for helping me win." In that case, it's simply that certain types of sense-experience seem to have been a reliable guide for my actions in the past, for helping me win. That's all. To think of it in terms of assumptions and conclusions is to stay in the world of true/false or justified/unjustified, where we can only go in circles because we are putting the cart before the horse. The verbal concepts of "true" and "justified" probably originated as a way to help people win, not as ends to be pursued for their own sake. But since they were almost always correlated with winning, they became ends pursued for their own sake - essential ones! In the end, if you dissolve "truth" it just ends up meaning something like "seemingly reliable guidepost for my actions." Comment author: 25 May 2011 07:44:12PM 1 point [-] Are you losing sleep over the daily deaths in Iraq? Are most LWers? . . . If we cared as much as we signal we do, no one would be able go to work, or post on LW. We'd all be too grief-stricken. That is exactly what I was talking about when I said "There's a difference between mental distress and action-motivating desire.". Utility functions are about choices, not feelings, so I assumed that, in a discussion about utility we would be using the word 'care' (as in "If we cared as much as we signal we do") to refer to motives for action, not mental distress. If this isn't clear, I'm trying to refer to the same ideas discussed here. And it also isn't immediately clear that anyone would really want their utility function to be unbounded (unless I'm misinterpreting the term). It does not make sense to speak of what someone wants their utility function to be; utility functions just describe actual preferences. Someone's utility function is unbounded if and only if there are consequences with arbitrarily high utility differences. For every consequence, you can identify one that is over twice as good (relative to some zero point, which can be arbitrarily chosen. This doesn't really matter if you're not familiar with the topic, it just corresponds to the fact that if every consequence were 1 utilon better, you would make the same choices because relative utilities would not have changed.) Whether a utility function has this property is important in many circumstances and I consider it an open problem whether humans' utility functions are unbounded, though some would probably disagree and I don't know what science doesn't know. Comment author: 25 May 2011 10:52:21PM 0 points [-] Is this basically saying that you can tell someone else's utility function by demonstrated preference? It sounds a lot like that. Comment author: 25 May 2011 05:41:07PM 0 points [-] However, if my seeing one black swan doesn't justify my belief that there is at least one black swan, how can I refute "all swans are white"? Refuting something is justifying that it is false. The point of the OP is that you can't justify anything, so it's claiming that you can't refute "all swans are white". A black swan is simply a criticism of the statement "all swans are white". You still have a choice -- you can see the black swan and reject "all swans are white", or you can quibble with the evidence in a large number of ways which I'm sure you know of too and keep on believing "all swans are white". People really do that; searching Google for "Rapture schedule" will pull up a prominent and current example. Comment author: 25 May 2011 10:46:53PM *  0 points [-] Why not just phrase it in terms of utility? "Justification" can mean too many different things. Seeing a black swan diminishes (and for certain applications, destroys) the usefulness of the belief that all swans are white. This seems a lot simpler. Putting it in terms of beliefs paying rent in anticipated experiences, the belief "all swans are white" told me to anticipate that if I knew there was a black animal perched on my shoulder it could not be a swan. Now that belief isn't as reliable of a guidepost. If black swans are really rare I could probably get by with it for most applications and still use it to win at life most of the time, but in some cases it will steer me wrong - that is, cause me to lose. So can't this all be better phrased in more established LW terms? View more: Next

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# Integration question. ## Homework Statement Jon needs to bulk-up for next AFL season. His energy needs t days after starting his weight gain program are given by E(t) = 350(80 + 0.15t)^0.8 - 120(80 + 0.15t) calcories/day. Find Jon's total energy needs over the first week of the program. ## The Attempt at a Solution I integrated the fuction above and interval was [0, 7] as the question says total energy needs over the first week. The answer i got was 14377 but the answer is 14400. I think there is some problem with the interval or the value of C which i couldnt find.

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Equation of a Line Through a point and in the Direction of a Vector in 3 D A calculator and solver that finds the equation of a line through a point and in a given direction in 3D is presented. The equation is written in vector, parametric and symmetric forms. As many examples as needed may be generated along with all the detailed steps needed to answer the question. Below is shown a line through point $$P(x_p,y_p,z_p)$$ and in the same direction as vector $$\vec v = \lt x_v,y_v,z_v \gt$$. Three forms of the equation of the line: Step by step solution STEP 1: Vector form. STEP 2: Parametric form. STEP 3: Symmetric form. More Step by Step Math Worksheets SolversNew !

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