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Metamath Proof Explorer < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > 2sqlem1 Structured version Visualization version GIF version Theorem 2sqlem1 24942 Description: Lemma for 2sq 24955. (Contributed by Mario Carneiro, 19-Jun-2015.) Hypothesis Ref Expression 2sq.1 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) Assertion Ref Expression 2sqlem1 (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) Distinct variable groups: 𝑥,𝑤 𝑥,𝐴 𝑥,𝑆 Allowed substitution hints: 𝐴(𝑤) 𝑆(𝑤) Proof of Theorem 2sqlem1 StepHypRef Expression 1 2sq.1 . . 3 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) 21eleq2i 2680 . 2 (𝐴𝑆𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))) 3 fveq2 6103 . . . . 5 (𝑤 = 𝑥 → (abs‘𝑤) = (abs‘𝑥)) 43oveq1d 6564 . . . 4 (𝑤 = 𝑥 → ((abs‘𝑤)↑2) = ((abs‘𝑥)↑2)) 54cbvmptv 4678 . . 3 (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) = (𝑥 ∈ ℤ[i] ↦ ((abs‘𝑥)↑2)) 6 ovex 6577 . . 3 ((abs‘𝑥)↑2) ∈ V 75, 6elrnmpti 5297 . 2 (𝐴 ∈ ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2)) ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) 82, 7bitri 263 1 (𝐴𝑆 ↔ ∃𝑥 ∈ ℤ[i] 𝐴 = ((abs‘𝑥)↑2)) Colors of variables: wff setvar class Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 2c2 10947 ↑cexp 12722 abscabs 13822 ℤ[i]cgz 15471 This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-cnv 5046 df-dm 5048 df-rn 5049 df-iota 5768 df-fv 5812 df-ov 6552 This theorem is referenced by: 2sqlem2 24943 mul2sq 24944 2sqlem3 24945 2sqlem9 24952 2sqlem10 24953 Copyright terms: Public domain W3C validator	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://de.metamath.org/mpeuni/2sqlem1.html", "fetch_time": 1726691658000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-38/segments/1725700651941.8/warc/CC-MAIN-20240918201359-20240918231359-00685.warc.gz", "warc_record_offset": 6806357, "warc_record_length": 4394, "token_count": 1205, "char_count": 2124, "score": 3.546875, "int_score": 4, "crawl": "CC-MAIN-2024-38", "snapshot_type": "latest", "language": "en", "language_score": 0.17379130423069, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00013-of-00064.parquet"}
# Resources tagged with: Video Filter by: Content type: Age range: Challenge level: There are 80 NRICH Mathematical resources connected to Video, you may find related items under Physical and Digital Manipulatives. Broad Topics > Physical and Digital Manipulatives > Video ### Subtraction Slip ##### Age 5 to 7Challenge Level Can you spot the mistake in this video? How would you work out the answer to this calculation? ### Tumbling Down ##### Age 7 to 11Challenge Level Watch this animation. What do you see? Can you explain why this happens? ### Eightness of Eight ##### Age 5 to 7Challenge Level What do you see as you watch this video? Can you create a similar video for the number 12? ### Pouring Problem ##### Age 7 to 11Challenge Level What do you think is going to happen in this video clip? Are you surprised? ### Subtraction Surprise ##### Age 7 to 14Challenge Level Try out some calculations. Are you surprised by the results? ### Perimeter Possibilities ##### Age 11 to 14Challenge Level I'm thinking of a rectangle with an area of 24. What could its perimeter be? ### That Number Square! ##### Age 5 to 11Challenge Level Exploring the structure of a number square: how quickly can you put the number tiles in the right place on the grid? ### Seven Squares ##### Age 11 to 14Challenge Level Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100? ### All Change ##### Age 5 to 7Challenge Level There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice? ### Bryony's Triangle ##### Age 7 to 11Challenge Level Watch the video to see how to fold a square of paper to create a flower. What fraction of the piece of paper is the small triangle? ### Dotty Six ##### Age 5 to 11Challenge Level Dotty Six is a simple dice game that you can adapt in many ways. ### How Many? ##### Age 5 to 7Challenge Level This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask? ### Strike it Out ##### Age 5 to 11Challenge Level Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game. ### Twisting and Turning ##### Age 11 to 14Challenge Level Take a look at the video and try to find a sequence of moves that will untangle the ropes. ##### Age 7 to 14Challenge Level Watch our videos of multiplication methods that you may not have met before. Can you make sense of them? ### Whirlyball ##### Age 16 to 18Challenge Level Whirl a conker around in a horizontal circle on a piece of string. What is the smallest angular speed with which it can whirl? ### Painted Cube ##### Age 14 to 16Challenge Level Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces? ### Seven Squares - Group-worthy Task ##### Age 11 to 14Challenge Level Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? ### Amazing Card Trick ##### Age 11 to 14Challenge Level How is it possible to predict the card? ### Take Three from Five ##### Age 11 to 16Challenge Level Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him? ### Counting Cards ##### Age 7 to 11Challenge Level A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work? ### Totality ##### Age 5 to 11Challenge Level This is an adding game for two players. ##### Age 11 to 16Challenge Level The items in the shopping basket add and multiply to give the same amount. What could their prices be? ### Beelines ##### Age 14 to 16Challenge Level Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses? ### Angle Trisection ##### Age 14 to 16Challenge Level It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square. ### Summing Consecutive Numbers ##### Age 11 to 14Challenge Level 15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers? ### Surprising Split ##### Age 7 to 11Challenge Level Does this 'trick' for calculating multiples of 11 always work? Why or why not? ### Unexpected Ordering ##### Age 5 to 7Challenge Level Watch the video of Fran re-ordering these number cards. What do you notice? Try it for yourself. What happens? ### Truth or Lie ##### Age 7 to 11Challenge Level Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic? ### Bundles of Cubes ##### Age 7 to 11Challenge Level Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle? ### Maths for Parents 2017 Resources from Charlie's and Fran's 2017 Madingley course for parents. ### Vanishing Roots ##### Age 14 to 18 If $y=x^2-6x+c$, and we vary $c$, what happens to the roots when $c>9$? ### Introductory Video ##### Age 11 to 16 An introductory video to the Probability and Evidence collection ### The ELISA Test ##### Age 14 to 18Challenge Level In 1% of cases, an HIV test gives a positive result for someone who is HIV negative. How likely is it that someone who tests positive has HIV? ### Probability in Court ##### Age 14 to 18Challenge Level When you're on trial for murder, it can be crucial that the court understands probability... ### Statins and Risk ##### Age 14 to 16Challenge Level "Statins cut the risks of heart attacks and strokes by 40%" Should the Professor take statins? Can you help him decide? ### How Risky Is My Diet? ##### Age 11 to 16Challenge Level Newspapers said that eating a bacon sandwich every day raises the risk of bowel cancer by 20%. Should you be concerned? ### Roll over the Dice ##### Age 7 to 11Challenge Level Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded? ### Sketching Graphs - Transformations ##### Age 16 to 18Challenge Level If you can sketch y=f(x), there are several related functions you can also sketch... ### Areas on a Grid ##### Age 11 to 16 Take a look at the video showing areas of different shapes on dotty grids... ### Rhombuses from Diagonals ##### Age 11 to 16 Take a look at the video showing rhombuses and their diagonals... ### Drawing Rhombuses ##### Age 11 to 16 Take a look at the video showing rhombuses drawn on dotty grids... ### Squares from Diagonals ##### Age 11 to 16 Take a look at the video showing squares and their diagonals... ### Drawing Squares ##### Age 11 to 16 Take a look at the video showing squares drawn on dotty grids... ### Dotty Six for Two ##### Age 5 to 11Challenge Level Dotty Six game for an adult and child. Will you be the first to have three sixes in a straight line? ### Strike it Out for Two ##### Age 5 to 11Challenge Level Strike it Out game for an adult and child. Can you stop your partner from being able to go? ### Up, Down, Flying Around ##### Age 11 to 14Challenge Level Play this game to learn about adding and subtracting positive and negative numbers ### Strange Bank Account ##### Age 11 to 14Challenge Level Imagine a very strange bank account where you are only allowed to do two things...	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://nrich.maths.org/public/topic.php?group_id=22&code=-455", "fetch_time": 1620967396000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2021-21/segments/1620243991737.39/warc/CC-MAIN-20210514025740-20210514055740-00361.warc.gz", "warc_record_offset": 451078626, "warc_record_length": 8594, "token_count": 1844, "char_count": 7650, "score": 3.6875, "int_score": 4, "crawl": "CC-MAIN-2021-21", "snapshot_type": "latest", "language": "en", "language_score": 0.8480259776115417, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00060-of-00064.parquet"}
# Search by Topic #### Resources tagged with Mathematical modelling similar to Pairing Up: Filter by: Content type: Stage: Challenge level: ### There are 62 results Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical modelling ### Tree Tops ##### Stage: 3 Challenge Level: A manager of a forestry company has to decide which trees to plant. What strategy for planting and felling would you recommend to the manager in order to maximise the profit? ### Bell Ringing ##### Stage: 3 Challenge Level: Suppose you are a bellringer. Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once? ### Crossing the Atlantic ##### Stage: 3 Challenge Level: Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey? ##### Stage: 3 Challenge Level: In a league of 5 football teams which play in a round robin tournament show that it is possible for all five teams to be league leaders. ### Königsberg ##### Stage: 3 Challenge Level: Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? ### Buses ##### Stage: 3 Challenge Level: A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end? ### Troublesome Triangles ##### Stage: 2 and 3 Challenge Level: Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . . ### Twenty20 ##### Stage: 2, 3 and 4 Challenge Level: Fancy a game of cricket? Here is a mathematical version you can play indoors without breaking any windows. ### Spot the Card ##### Stage: 4 Challenge Level: It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards? ### Flight of the Flibbins ##### Stage: 3 Challenge Level: Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . . ### Pattern of Islands ##### Stage: 3 Challenge Level: In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island... ### Problem Solving, Using and Applying and Functional Mathematics ##### Stage: 1, 2, 3, 4 and 5 Challenge Level: Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information. ### Learning Mathematics Through Games Series: 4. from Strategy Games ##### Stage: 1, 2 and 3 Basic strategy games are particularly suitable as starting points for investigations. Players instinctively try to discover a winning strategy, and usually the best way to do this is to analyse. . . . ### Chemnrich ##### Stage: 4 and 5 Challenge Level: chemNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of chemistry, designed to help develop the mathematics required to get the most from your study. . . . ### Slippage ##### Stage: 4 Challenge Level: A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . . ### Bionrich ##### Stage: 4 and 5 Challenge Level: bioNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of the biological sciences, designed to help develop the mathematics required to get the most from your. . . . ### Physnrich ##### Stage: 4 and 5 Challenge Level: PhysNRICH is the area of the StemNRICH site devoted to the mathematics underlying the study of physics ### Stringing it Out ##### Stage: 4 Challenge Level: Explore the transformations and comment on what you find. ### Epidemic Modelling ##### Stage: 4 and 5 Challenge Level: Use the computer to model an epidemic. Try out public health policies to control the spread of the epidemic, to minimise the number of sick days and deaths. ### Celtic Knotwork Patterns ##### Stage: 2 and 3 This article for pupils gives an introduction to Celtic knotwork patterns and a feel for how you can draw them. ### What's a Knot? ##### Stage: 2, 3 and 4 Challenge Level: A brief video explaining the idea of a mathematical knot. ### Elastic Maths ##### Stage: 4 and 5 How do you write a computer program that creates the illusion of stretching elastic bands between pegs of a Geoboard? The answer contains some surprising mathematics. ### Rocking Chairs, Railway Games and Rayboxes ##### Stage: 1, 2, 3, 4 and 5 In this article for teachers, Alan Parr looks at ways that mathematics teaching and learning can start from the useful and interesting things can we do with the subject, including. . . . ### Where to Land ##### Stage: 4 Challenge Level: Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible? ### Stemnrich - the Physical World ##### Stage: 3 and 4 Challenge Level: PhysNRICH is the area of the StemNRICH site devoted to the mathematics underlying the study of physics ### Investigating Epidemics ##### Stage: 3 and 4 Challenge Level: Simple models which help us to investigate how epidemics grow and die out. ### Designing Table Mats ##### Stage: 3 and 4 Challenge Level: Formulate and investigate a simple mathematical model for the design of a table mat. ### Witch's Hat ##### Stage: 3 and 4 Challenge Level: What shapes should Elly cut out to make a witch's hat? How can she make a taller hat? ### Straw Scaffold ##### Stage: 3 Challenge Level: Build a scaffold out of drinking-straws to support a cup of water ### Food Web ##### Stage: 3 Challenge Level: Is this eco-system sustainable? ### Guessing the Graph ##### Stage: 4 Challenge Level: Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from? ### Drawing Doodles and Naming Knots ##### Stage: 2, 3, 4 and 5 This article for students introduces the idea of naming knots using numbers. You'll need some paper and something to write with handy! ### Triathlon and Fitness ##### Stage: 3 Challenge Level: The triathlon is a physically gruelling challenge. Can you work out which athlete burnt the most calories? ### Observing the Sun and the Moon ##### Stage: 2 and 3 Challenge Level: How does the time of dawn and dusk vary? What about the Moon, how does that change from night to night? Is the Sun always the same? Gather data to help you explore these questions. ### The Legacy ##### Stage: 4 Challenge Level: Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money? ### Circuit Training ##### Stage: 4 Challenge Level: Mike and Monisha meet at the race track, which is 400m round. Just to make a point, Mike runs anticlockwise whilst Monisha runs clockwise. Where will they meet on their way around and will they ever. . . . ### Konigsberg Plus ##### Stage: 3 Challenge Level: Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges. ### Shaping the Universe III - to Infinity and Beyond ##### Stage: 3 and 4 The third installment in our series on the shape of astronomical systems, this article explores galaxies and the universe beyond our solar system. ### Fixing the Odds ##### Stage: 4 Challenge Level: You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two. . . . ### Rule of Three ##### Stage: 3 Challenge Level: If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall? ### Scratch Cards ##### Stage: 4 Challenge Level: To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize? ### Logic, Truth Tables and Switching Circuits ##### Stage: 3, 4 and 5 Learn about the link between logical arguments and electronic circuits. Investigate the logical connectives by making and testing your own circuits and record your findings in truth tables. ### On Time ##### Stage: 3 Challenge Level: On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands? ### Covering Cups ##### Stage: 3 Challenge Level: What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible. ### Christmas Trees ##### Stage: 3 Challenge Level: Christmas trees are planted in a rectangular array of 10 rows and 12 columns. The farmer chooses the shortest tree in each of the columns... the tallest tree from each of the rows ... Which is. . . . ### Escalator ##### Stage: 4 Challenge Level: At Holborn underground station there is a very long escalator. Two people are in a hurry and so climb the escalator as it is moving upwards, thus adding their speed to that of the moving steps. . . . ### Hands Together ##### Stage: 3 Challenge Level: Sometime during every hour the minute hand lies directly above the hour hand. At what time between 4 and 5 o'clock does this happen? ### Concrete Calculation ##### Stage: 4 Challenge Level: The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to. . . . ### Chocolate 2010 ##### Stage: 4 Challenge Level: First of all, pick the number of times a week that you would like to eat chocolate. 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Atmospheric pressure Essays & Research Papers Best Atmospheric pressure Essays • Atmospheric Pressure - 309 Words Atmospheric pressure is the force per unit area exerted on a surface by the weight of air above that surface in the atmosphere of Earth (or that of another planet). In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the mass of air above the measurement point. Low-pressure areas have less atmospheric mass above their location, whereas high-pressure areas have more atmospheric mass above their location. Likewise, as elevation increases, there... 309 Words \| 1 Page • Breaking a Ruler with Atmospheric Pressure Breaking a Ruler with Atmospheric Pressure Introduction In this experiment, I will try to use air pressure, along with some sheets of newspaper, to attempt to break a ruler. Air pressure is the weight of the atmosphere pressing down on the earth. A device called a barometer measures it in units called millibars. Most barometers use mercury in a glass column, like a thermometer, to measure the change in air pressure. I came up with this idea from when I read in a book about how some kids... 1,783 Words \| 5 Pages • Atmospheric Pressure Test Questions Chapter 3 Pressure Measurement Examples Example 3.3 A special high pressure U tube manometer is constructed to measure pressure differential in air at 13.8 MPa and 20oC. When an oil having a specific gravity of 0.83 is used as the fluid, calculate the differential pressure in N/m2 that would be indicated by a 135 mm reading. p − pa = m g h( gc m − f ) = (0.83)(1000) = 830kg/m 3 = = p 13.8 × 10 6 = f a RT (287)(293) g h( m − f ) p − pa = gc = 9.81(0.135)(830 − 164.11) = 881.51N/m 2... 359 Words \| 2 Pages • Atmospheric Pressure and Possible Answer Choices The following document provides sample items for each knowledge subtest of the ASTB. The sample items are not meant to provide an exhaustive list of the types of questions that will be found in the test. Instead, these questions are meant to familiarize examinees with the format and content of questions found within each section. Item difficulty ratings are provided for each question where applicable, and can be used to gauge how hard a given question is in comparison to similar types of... 2,200 Words \| 9 Pages • All Atmospheric pressure Essays • Pressure - 634 Words According to the essay “Too Much Pressure” by Colleen Wenke, the reason that students cheat on their tests is because they are under too much pressure to get good grades, which is accurately portrayed since cheating is usually seen as one of the only ways to pass tests and that’s what students are being stressed to do. Based on her essay, teachers should be teaching students right from wrong as opposed to pressuring to the extent of making them cheat. This is important to see because it is a... 634 Words \| 2 Pages • ‘Low pressure atmospheric systems have more of a short term impact than high pressure systems.’ Discuss. ‘Low pressure atmospheric systems have more of a short term impact than high pressure systems.’ Discuss. Low pressure atmospheric systems are also known as depressions or cyclones and they form in mid- and high-latitudes. They are formed by the mixing of cold and warm air, the warm air is lighter, so it rises above the denser, cold air and forms a centre of low pressure. High pressure atmospheric systems are also known as anticyclones and have very different characteristics to depressions.... 946 Words \| 3 Pages • Academic Pressure - 361 Words Academic Pressure In the movie The Dead Poet Society, Neil, the protagonist was getting academically pressured by his parents, his dad especially. He was told to get all A’s or he would be punished. He was threatened at the beginning to be forced to leave that field he wanted to study, but he did not listen to his dad. Academic pressure plays a big role in why some people have anxiety, or get really mad at themselves if they do something wrong or even get a bad grade. In my perspective I... 361 Words \| 2 Pages • Air Pressure - 451 Words Air Pressure Air is composed of molecules. Air is matter. It has mass and takes up space. Air is composed of different gases such as nitrogen, oxygen, carbon dioxide, water vapor, and other gases. Air molecules are in constant motion. As they move, they come in contact with surfaces. Air molecules push and press on the surfaces they contact. The amount of force per unit area that air molecules exert on a surface is called air pressure. (What is Air Pressure 6) Air pressure is caused by... 451 Words \| 2 Pages • College Pressures - 402 Words Miguel Casey 3-1-2012 English 110 In his articles , Zinsser takes a negative view of the college pressures he identifies Pressures that an individual feels affect his disposition towards life . The pressure may be taken as positive or negative depending on the weight it brings a person . Most of the time pressures are viewed to bring about negative effect to the person but some just do not realize that it is the pressure felt by an individual which motivates him to finish a goal . For... 402 Words \| 2 Pages • Peer Pressure - 298 Words Peer Pressure “Come on grab it hurry just grab it, it’s easy to steal the video game and I will let you play it first hurry, and grab it.” Peer pressure is basically someone around you who is trying to get you to do something you are not comfortable with, or something that is against your standards. For instance trying to get you to smoke, or drink with them is something you may not be okay with. Although some might say all peer pressure is bad I would argue that because peer pressure can... 298 Words \| 1 Page • pressure gauge - 380 Words A pressure gauge is an instrument that measures the pressure in a vessel, a line, or whatever the pressure gauge is connected to. A Bourdon gauge consists of a C-shaped pipe with one end closed and the other end attached to a chamber whose pressure is being measured. When there is a pressure difference between the inside of the pipe and the outside, there will be a net force acting on the C-shaped pipe which will either try to curl the pipe into a tighter C shape (if the pressure in the pipe is... 380 Words \| 2 Pages • Peer Pressure - 1092 Words Peer Pressure, Cause and Effect Peer pressure has become a big issue in our modern world. Many people experience it every day and a lot of times it leads to bad decision making. Peer pressure can affect you in the long run and knowing how to say no to certain things can sometimes save your life. There are many causes to peer pressure but the more noticeable one would be just pure excitement. Say that your peers are jumping their bikes off ramps and they ask you to try.... 1,092 Words \| 3 Pages • Pressure Hole - 329 Words The crew live and work inside the pressure hull. It must be strong enough to withstand the pressure of the water at the depth the submarine is designed to operate. When the air tanks are full of air the submarine will float - usually submarines are designed to float on the surface quite low in the water (only a little freeboard). When the submarine dives (submerges) water is let into the tanks; to surface again air is blown into them. This air must be stored inside the submarine in compressed... 329 Words \| 1 Page • Peer Pressure - 618 Words Jonathan Bertolotti English 101 Summary Response #2 15 September 2014 Peer PressureShooting an Elephant, by George Orwell, was a very emotional and graphic story that opens the eyes of many people. Beginning the story with some background information, Orwell describes how difficult it is being a white man in Lower Burma, and discusses how much he is hated and made fun of by the people. He is a police officer, which gives people even more of a reason to hate him. After all the background... 618 Words \| 2 Pages • Atmospheric Sciences Assignment - 436 Words Introduction to Atmospheric Sciences (ATOC-210) Assignment # 5, due date March 19, 2009 Explain why, on a sunny day, an aneroid barometer would indicate "stormy" weather when carried to the top of a hill or a mountain? The higher you go up the more the atmospheric pressure decreases. With an aneroid barometer, there are weather related words printed above atmospheric pressure values. If you move up the... 436 Words \| 2 Pages • Pressure Measurement and Calibration - 6982 Words 52 PRESSURE MEASUREMENT AND CALIBRATION (TH2) 53 EQUIPMENT DIAGRAMS 54 55 56 EQUIPMENT DESCRIPTION Refer to the drawing on pages 56, 57 and 58. This equipment is a bench top unit designed to introduce students to pressure, pressure scales and common devices available to measure pressure. The equipment comprises a Dead-weight Pressure Calibrator to generate a number of predetermined pressures, connected to a Bourdon gauge and electronic pressure sensor to allow their... 6,982 Words \| 21 Pages • Pressures of Finding Salvation - 1225 Words Chandler Hoffman Professor Turley Writing 150 Section 5 25 September 2012 The Pressures of Finding Salvation Langston Hughes’ story “Salvation” is one that raises many questions about his life and childhood experiences. Hughes patterns this story to portray the pressures that caused his faith to be lost. Hughes sat on the mourners’ bench waiting for God to save him but, due to these pressures, he chose to stand and pretend that he found his salvation. Pressure is the influences of outside... 1,225 Words \| 3 Pages • Pressure in Youth Sports - 1021 Words If you ever played in competitive sports as a child, then what I am about to tell you about will sound very familiar. Have you ever been pressured by your parents, friends or coaches to do extremely well in any sport or activity you did? Growing up in the late 90’s and into the future, the way parents and coaches act towards their children and players has changed a lot. I have been playing basketball since I was five years old. Luckily for me my parents have never pressured me or pushed me too... 1,021 Words \| 2 Pages • Th2 Pressure Measurement and Calibration PRESSURE MEASUREMENT AND CALIBRATION YEDĐTEPE UNIVERSITY DEPARTMENT OF MECHANICAL ENGINEERING 1 YEDITEPE UNIVERSITY ENGINEERING FACULTY MECHANICAL ENGINEERING LABORATORY Pressure Measurement and Calibration 1. Objective: To convert an arbitrary scale of pressure sensor output into engineering units. To calibrate a semiconductor pressure sensor. 2. Equipment: The equipment comprises a Dead-weight Pressure Calibrator (DPC), Bourdon gauge and diaphragm-type pressure sensor.... 1,573 Words \| 9 Pages • Pressure and the Gas Laws - 396 Words A barometer is a widely used weather instrument that measures atmospheric pressure (also known as air pressure or barometric pressure) - the weight of the air in the atmosphere A barometer is an instrument used to measure atmospheric pressure. It can measure the pressure exerted by the atmosphere by using water, air, or mercury. From the variation of air pressure, one can forecast short-term changes in the weather. There are two main types of barometers – Mercury Barometers newer digital... 396 Words \| 2 Pages • Air Pressure in Footballs - 523 Words What is the relationship between feedback from air pressure of a football to the performance of a athlete? Alex Long Purpose The purpose for my experiment is to work out scientifically whether the air pressure inside a football effects the performance of a sports performer. There are several different types and brands of footballs all with different structures and pressure recommended for the ball. It would be interesting to find out the differences between the... 523 Words \| 4 Pages • How to Handle Peer Pressure How To Handle Peer Pressure By: Kristina Failla Submitted to: Dr. Jaballah M. Hasan Specific Goal: I would like to inform the audience how to handle peer pressure Introduction: 1. What is Peer Pressure? A. Peer Pressure is when one person tries to talk another unwilling person into doing something. B. Peer Pressure can happen anywhere and anytime between people of all ages, but mainly around students in school. C. Many that pressure others are known to be the “popular kids”... 821 Words \| 4 Pages • Effects of Peer Pressure - 462 Words RELATED STUDIES On their study in examining the nature of peer pressure perceive by adolescent, Brown, B.Bradford, et al (1896),states that 373 students in grades 7-12 were asked to indicate, on a 12-item index, the degree and direction of peer pressures they perceived from friends and acquaintances, and to describe their personal attitudes and behavior in areas corresponding to index items. Analyses revealed that peers were seen as encouraging misconduct less than other types of behavior.... 462 Words \| 2 Pages • Positive Peer Pressure - 629 Words Positive Peer Pressure Whenever you hear the word peer pressure every one immediately refers to the negative influences. Have you ever explored the possibilities of positive peer pressure happening in people’s lives today? There are several examples of peer pressure out there that are positive, experienced mainly by teenagers that go unnoticed. The big one that needs to be focused on is the influence of not... 629 Words \| 2 Pages • Parents and Teens, Pressure to Get Good Grades Peer pressure, it has been questioned alot. Students are pressured to do good in school, they are also pressured by their family. So, children are pressured to grow up too fast, by many ways; Such as, pushed by their parents to get good grades, teens get pressured by a tradition to a college that was carried onto the families by generation and the pressures to get a job. Therefore, there are multiple ways teens can get pressured by family to do things. Teens are the future. They have to learn... 383 Words \| 1 Page IACS Guideline for Procedures of Testing Tanks and Tight Boundaries 1. General These test procedures are to ensure the weathertightness of structures/shipboard outfitting, the watertightness of tanks and watertight boundaries and structural adequacy of tanks. Tightness of all tanks and tight boundaries of the ships at the new construction and, when major conversions or repairs* have been made, those relevant to the major conversions/repairs should be confirmed by these test procedures... 2,813 Words \| 11 Pages • Ce 371 Homework 2 CE 371 HOMEWORK 2 1) Find the difference in pressure between tanks A and B if d1=300mm, d2=150mm, d3=460mm, d4=200 mm and S.GHG =13.6 w=9.80 kN/m3 2) Determine the elevation difference, h, between the water levels in the two open tanks shown in the figure. w=9.80 kN/m3 3) An air-filled, hemispherical shell is attached to the ocean floor at a depth of 10 m as shown in the figure. A mercury barometer located inside the shell reads 765 mm Hg, and a mercury U-tube manometer designed to give the... 283 Words \| 3 Pages • Weather and Our Health - 384 Words Weather and our health Throughout history, mankind has always been in awe of the weather. Ancient Civilizations considered natural disasters to be the work of the Gods. The weather still plays a big part in our lives today. It affects many of the things that we do, from the clothes we wear and the food we eat, to where we live and how we travel. As a result, the weather is of great interest to people everywhere, from meteorologists, the scientists who study it. In fact, one of the main... 384 Words \| 1 Page • Assignment Wk3 - 710 Words 1. The pressure announced on last night's television weather broadcast was 29.92. Explain how this was measured and give the units. Would this be considered an unusually large or low pressure value? A pressure announced on the weather forecast of 29.92 is an average measurement. It is measured with a barometer and in the United States the units of measure are inches of mercury, or inHg. This is what meteorologist are referring to in their forecasts. 29.92 inHg is a measurement within the... 710 Words \| 2 Pages • science vs nature - 818 Words Science vs. Nature What is weather? Weather is the heat of the sun that shines through a window. It is the mist on a foggy morning. It can also be a deadly storm that wipes away a house. Weather is not something people can control but that does not mean that they can not protect themselves from it. That is why we have technology. Using weather instruments such as radars, satellites, and computers, forecasters can predict when and where certain storms will occur. Throughout the... 818 Words \| 3 Pages • Molecular Weight of Volatile Liquid Using Dumas Method OBJECTIVE: * To determine the molecular weight of a volatile liquid by using Dumas method. METHOD: MATERIAL \| CHEMICALS \| 125 mL Erlenmeyer flask \| Known liquid (2-propanol) \| Rubber band \| Unknown liquid \| Boiling chips \| \| Watch glass \| \| 100 mL graduated cylinder \| \| Pin \| \| 600 mL beaker \| \| Hot plate \| \| Thermometer \| \| Room temperature water \| \| 6 × 6 and 8 × 8 aluminium foil \| \| PROCEDURE: SAFETY * Lab Coat and Safety Goggles. * Keep the... 1,013 Words \| 4 Pages • Determining the Molar Mass of a Volatile Liquid Determining the Molar Mass of a Volatile Liquid Purpose: The purpose of this lab was to find the molar mass of a volatile liquid. Data Table: Mass of Test Tube and Stopper (g) \| 10.864 g \| Barometric Pressure (mmHg) \| 749.31 mmHg \| Temperature of Boiling Water (C) \| 97.1 C \| Mass of Test Tube, Stopper, and Condensed Liquid (g) \| 10.890 g \| Volume of Flask (mL) \| 9.90 mL \| Calculations: 749.31 mmHG1 atm760 mmHg= .98593 atm 9.90 mL1 L1000 mL= .00990 L 97.1 C+273=370.1 K... 298 Words \| 1 Page • aaaaaaaaaaaa - 2773 Words A N AIR - POLYMER ANALOGY FOR MODELING AIR FLOW THROUGH RUBBER - METAL INTERFACE – T ECHNICAL NOTE by GILAD PAGI , ELI ALTUS T ECH NICAL R EPORT ETR-2007-02 July 2007 ³©¤§ ³ª©¥ ¡¥° ¥±²¢¥ ¢¥©¤¡ ¨¤§ » ¨¢©¤¡ TECHNION — Israel Institute of Technology, Faculty of Mechanical Engineering An air-polymer analogy for modeling air flow through rubber-metal interface – Technical note GILAD PAGI, ELI ALTUS Faculty of Mechanical engineering Technion – Israel Institute of... 2,773 Words \| 13 Pages • Mechanical Properties of Fluids - 445 Words MECHANICAL PROPERTIES OF FLUIDS 1 Torricelli’s barometer used mercury. Pascal duplicated it using French wine of density 984 kg m-3. Determine the height of the wine column for normal atmospheric pressure. 2 A vertical off-shore structure is built to withstand a maximum stress of 109 Pa. Is the structure suitable for putting up on top of an oil well in the ocean? Take the depth of the ocean to be roughly 3 km, and ignore ocean currents. 3 A hydraulic automobile lift is designed to lift cars... 445 Words \| 1 Page • AMU SCIN 137 Wk 3 Prof. Wayne MacKenzie SCIN 137 22 February 2015 1. The pressure announced on last night's television weather broadcast was 29.92. Explain how this was measured and give the units. Would this be considered an unusually large or low pressure value? a. 29.92 is the standard sea level pressure, identified by inches of Mercury in a barometer, and identifies the pressure over an area using the millibar. This would not be considered a large/low pressure measurement. 2. If the earth did not rotate,... 402 Words \| 2 Pages • chapter 13 notes - 2208 Words Conceptual Physics 11th Edition Chapter 14: GASES © 2010 Pearson Education, Inc. This lecture will help you understand: • • • • • • • The Atmosphere Atmospheric Pressure The Barometer Boyle’s Law Buoyancy of Air Bernoulli’s Principle Plasma © 2010 Pearson Education, Inc. The Atmosphere Atmosphere • Ocean of air • Exerts pressure The Magdeburghemispheres demonstration in 1654 by Otto von Guericke showed the large magnitude of atmosphere’s pressure. ©... 2,208 Words \| 18 Pages • Fuid Mechanics Assignment - 315 Words Assignment 1 Problem No 4 is due 1- A body weighs 4800N when exposed to a standard earth gravity g = 9.8066 m/s2. (a) What is its mass in kg? (b) What will the weight of this body be in N ewton if it is exposed to the moon's standard acceleration gmoon = 1.62 m/s2? 2- A rigid pipe of diameter d = 3500mm and 5km long is used to pump water. An obstruction plugs the pipe at an unknown location so that no liquid can flow. A piston is placed in one end of the pipe and slides (without... 315 Words \| 1 Page • Altimeter Aircraft - 1163 Words Chapter – 3 ALTIMETER Principle of operation: A simple altimeter consists of a thin corrugated metal capsule which is partially evacuated, sealed and prevented from collapsing completely by means of a leaf spring. In some cases complete collapsing is prevented by its own rigidity. The capsule is mounted inside a case. The case is fed with static pressure from aircraft static tube/ vent. As the aircraft climbs the static pressure in the case decreases allowing the spring... 1,163 Words \| 4 Pages Breuna Welch 1.Have you ever heard of the Tri-State tornado before? Do you live anywhere along the path of the tornado? No I haven’t heard of a tri state before . Yes i do live along the path of tornado.2. What quadrant of a storm does the article say a tornado is most likely to be found in ? Northwest quadrant is were the article say a tornado is most likely to be found . 3. The “Barograph charts” in figure 3 and 4 are traces of barometric pressure. What explains the low point on each... 731 Words \| 2 Pages • Assignment 1 Sem 1 20152016_Section 2 BAA 2713 FLUID MECHANICS ASSIGNMENT 1 PART 1 (5%) [CO2 : Relationship between pressure and elevation ; Pressure Measurement] 1. The pressure is an unknown fluid at a depth of 1.22m is measured to be 12.55 kPa(gage). Compute the specific gravity of the fluid. 2. The pressure at the bottom of a tank of propyl alcohol at 250C must be maintained at 52.75kPa(gage). What depth of alcohol should be maintained? 3. Question 3.48 and 3.49 from Applied Fluid Mechanics by Mott and Untener (7th Edition)... 501 Words \| 4 Pages • El Nini - 401 Words El Niño El Niño is a disruption of the ocean-atmosphere system in the Tropical Pacific that directly affects weather and climate around the globe. El Niño is a climate pattern that occurs in the tropical Pacific Ocean and brings warm air to areas that have colder air. El Niño involves warmer-than-usual sea temperatures, great amounts of rainfall (in the northern hemisphere) and low atmospheric pressure. El Niño is normally a climate pattern that lasts for a long time. The average life of an... 401 Words \| 1 Page • Physical Geography Elements of Latin American Essay1 discuss some of the connections between the following physical geography elements of Latin American temperature, precipitation, topography and altitudinal life zones Latin America three-quarters of range in tropical range, on all the continents of the world, its conditions is the best. The temperature, compared to other states, its characteristics is warm; it is not like Asia so cold, also not like Africa so hot. The whole continent annual precipitation average up to 342 mm,... 1,020 Words \| 3 Pages • Atmosphere and Volatile Liquids - 577 Words SECTION - A 1. Name the technique to separate [1] (a) Salt from Sea – water (b) Butter from curd 2. Define velocity. [1] 3. What do you mean by free fall? [1] 4. Mention any 2 advantages of using Italian bee variety in honey production. OR (a) Identify soluble and solvent in the following solutions: [3] (i) Aerated drinks (ii) Tincture of iodine (iii) Lemon water (b) State the principle of each if the following methods of separation of... 577 Words \| 3 Pages • Thermo Fluids - 1712 Words Solutions (Week-01) Chapter-01 1-12 A plastic tank is filled with water. The weight of the combined system is to be determined. Assumptions The density of water is constant throughout. Properties The density of water is given to be  = 1000 kg/m3. Analysis The mass of the water in the tank and the total mass are mw =V =(1000 kg/m3)(0.2 m3) = 200 kg mtotal = mw + mtank = 200 + 3 = 203 kg Thus, 1-14 The variation of gravitational acceleration above the sea level is given... 1,712 Words \| 7 Pages • Hello - 5773 Words ASTB Personal Study Guide MATH SKILLS: • 16 ounces = 1 pound • 8 pints = 1 gallon • 4 quarts = 1 gallon • 2000 lbs = 1 ton • Shapes • (# of sides - 2)*180 = Total # of degrees • Supplemental angle - add up to 180 degrees • Complementary angle - add up to 90 degrees • Words to Equations: • If 2 times r exceeds one-half of t by 5, which of the following represents the relationship between r and t. • 2r - .5t = 5 • or.. 4r - t = 10 • If 3 times x exceeds 1/3 of y by 9, which of the following is the... 5,773 Words \| 20 Pages • Science Paper - 3475 Words EVERYDAY SCIENCE : PHYSICS -NO.6 Time: 30 minutes Marks: 100 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. Which of the following is the best conductor for electricity? A. Distilled water B. Salt water C. Tap water D. Rain water A sprayer works on the principle expounded by A. Newton B. Archimedes C. Boyle D. Pascal Cloudy nights are warmer because A. clouds prevent radiation of heat from the ground into the air B. of low atmospheric pressure C. of the compact... 3,475 Words \| 9 Pages • case study chf afib Case Study #1: Chronic Heart Failure Case study #1 (see attached) tells about a patient, Maria, who has a history of hypertension and high cholesterol and 30 pack a year smoking.i Although normally under control with medication, upon traveling by plane she experienced what she thought was new condition with new symptoms which would ultimately be diagnosed as chronic heart failure. Maria's recent history showed that she was experiencing difficulty breathing on exertion, fatigue, abdominal... 534 Words \| 2 Pages • Unit 1 Lab Precipitation Precipitation In this experiment, you will monitor both precipitation and barometric pressure. You will craft a hypothesis about the impact that barometric pressure has on the level of precipitation. Remember that a hypothesis should be specific and testable. Place a collection vessel of some type (e.g., an empty coffee can or a pot) in an unobstructed area outside your home. If your vessel is light, you may want to weigh it down by placing a stone in it so that it doesn’t blow away. Use... 320 Words \| 1 Page • Yunus Çengel Thermodynamics - 10599 Words 1-1 Chapter 1 INTRODUCTION AND BASIC CONCEPTS Thermodynamics 1-1C Classical thermodynamics is based on experimental observations whereas statistical thermodynamics is based on the average behavior of large groups of particles. 1-2C On a downhill road the potential energy of the bicyclist is being converted to kinetic energy, and thus the bicyclist picks up speed. There is no creation of energy, and thus no violation of the conservation of energy principle. 1-3C There is no truth to his... 10,599 Words \| 40 Pages • Salvation - 402 Words Salvation The young Langston Houghes was pressured into believing in Jesus by the church who is responsible for his loss of faith. Langston was in his aunt’s church were a revival was being held “to bring the young lambs to fold…” Langston along with the other “young lambs” were all placed on the mourners’ bench on the front row. Each child one by one accepted Jesus until Langston was last. Langston eventually stood and claimed to have seen Jesus. Langston’s church was responsible for... 402 Words \| 1 Page • Hot Air Balloon and Wood Block Float Part 1 – Buoyancy http://phet.colorado.edu/en/simulation/buoyancy 1. Open the web browser and enter the link above into the address bar. 2. When the simulation page opens click “Run Now!” 3. Click the “Buoyancy Playground” tab at the top of the window. 4. Look in the yellow data box in the top left corner of the window. Record the mass (m), volume (V), and density (ρ) of the wood block. 5. In the middle of the screen there is a container of water. Record the volume of the water. 6.... 1,249 Words \| 5 Pages • Kaymito Leaves Decoction as Antiseptic Mouthwash San Sebastian College – Recoletos High School Department Manila Principle of Gravitational Force An Investigatory Project Submitted to Mrs. Elizabeth S. Nicolas In Partial Fulfilment of the Requirements in Science IV (PHYSICS) Name: Mark Joseph Delfin Patricia Blessy Gonzales Reymon Musa John Michael Ortiz Date: March 4, 2013 i ABSTRACT Delfin M.J. (2013) The Principle of The Gravitational Force... 1,525 Words \| 8 Pages • ‘Storm Catchers’ Essay - 1121 Words The novel ‘Storm Catchers’ is about a kidnapping of the youngest daughter of the Parnell family ‘Ella’. The term tension means intense emotion either good or bad e.g. when opening a gift it creates tension which is exciting. This is appropriate to the novel, because it’s about kidnapping which creates tension and it’s effective throughout the novel. Atmosphere is the mood of your surroundings e.g. when you enter a dark room it creates a scary atmosphere. This essay will consider of how tension... 1,121 Words \| 3 Pages • Mechanic - 275 Words Form: 16 Version 1.4 1 September 2003 STANDARD OPERATING PROCEDURE TASK: Use of Compressed Air SOP No: VA12 ..................... Version: 1........................... Date: ..................... Dept/Div/School: Visual and Performing Arts Supervisor/Manager: Other Contacts: HAZARDS: High pressure air in storage cylinder. Eye and hearing damage. Air bubbles in bloodstream. PROTECTIVE EQUIPMENT AND EMERGENCY EQUIPMENT Eye protection ie: goggles, visor. Hearing... 275 Words \| 3 Pages • Aircraft Instruments System Reviewer ANEROID – sensitive component in an altimeter or barometer that measures absolute pressure of the air. -Sealed, flat capsule made of thin corrugated disks of metal soldered together and evacuated by pumping all of the air out of it. PRESSURE – amount of force acting on a given unit of area and all pressure must be measured from some known references. BAROMETRICSCALE/KOLLSMAN WINDOW- small window in the dial of a sensitive altimeter in which the pilot sets the barometric pressure level from... 1,449 Words \| 8 Pages • Bourdon Gage - 448 Words "fluid mechanics lab" Experiment number "1" (bourdon gauge apparatus) Report date: 20/2/2011 Delivered date:27/2/2011 Submitted by: Eyyass Bassam Hamdan Major: mechanical engineering Student number: 30815150521 To Doctor : Abd-el hadi Lecture time: Sunday, 8-11 Summary: In this experiment we are going to measure the pressure of gage and absolute by using a dead weight device which it use bourdon tube to measure the pressure. Introduction: We are... 448 Words \| 3 Pages • Questions and Answers on Fluid Mechanics BPH – Tut # 2 Fluid Mechanics – University of Technology, Sydney – Engineering & IT Tutorial # 2 – Questions and Solutions Q.1 For the situation shown, calculate the water level d inside the inverted container, given the manometer reading h = 0.7 m, and depth L = 12 m. Q.2 (from Street, Watters, and Vennard) The weight density γ = ρg of water in the ocean may be calculated from the empirical relation γ = γo + K(h1/2), in which h is the depth below the ocean surface, K a constant,... 520 Words \| 4 Pages • Physics - 1470 Words Physics pg194-195 2.why does barometric pressure drop when there is a storm? -Barometric pressure drops because there are powerful winds in the higher atmosphere that blows in or suck out in any direction and change at random times. 3.Our ears pop when we ride in an airplane or when we drive to the mountains,why? - Our ears pop when we ride in an airplane or when we drive to the mountains because the air high above the surface of the earth is less dense than air near the surface.As... 1,470 Words \| 4 Pages • pneumatic controls - 28537 Words INTRODUCTION 1 INTRODUCTION A fluid power system is one that transmits and controls energy through the use of pressurised liquid or gas. In Pneumatics, this power is air. This of course comes from the atmosphere and is reduced in volume by compression thus increasing its pressure. Compressed air is mainly used to do work by acting on a piston or vane. While this energy can be used in many facets of industry, the field of Industrial Pneumatics is considered here. The correct use of... 28,537 Words \| 253 Pages • Salvation rhetorical analysis - 636 Words AP Portfolio Entry #3 Alayna Baudry “Salvation” Hughes, Langston. "Salvation." [The Big Sea, 1940.] The McGraw-Hill Reader: Issues across the Disciplines. Ed. Gilbert H. Muller. 11th ed. Boston: McGraw-Hill, 2011. 642-643. Print. Question #18: What is the author’s purpose? Does s/he achieve this purpose? What three or four elements most significantly contribute to the success or failure of the passage? Hughes’ purpose in writing salvation was to display that the pressure of... 636 Words \| 2 Pages • Fluid Statistics - 1824 Words Chapter 3 Fluid Statics: Definitions Statics: ∑F = 0. In statics we have only pressure as surface force and weight as body force. Thus, when fluids are still, the pressure is balanced by the fluid weight. No relative motion between adjacent fluid layers. Shear stress is zero Only _______ can be acting on fluid surfaces Gravity force acts on the fluid (____ force) Applications: Pressure variation within a reservoir Forces on submerged surfaces Buoyant forces 9/4/2013 1 Pressure Pressure is... 1,824 Words \| 15 Pages • EXERCISE 9 Solid And Fluid EXERCISE 9 - Mechanical properties of solid and fluid mechanics 1. A stainless-steel wire of length 3.1 cm and a diameter of 0.22 mm. If it is stretched by 0.10 mm, find the tension of the wire. The Young’s modulus for stainless steel is 18 × 1010 Pa. 22 N 2. Determine the elongation of the rod i Figure 1 if it is under a tension of 5.8 × 103 N. Young’s Modulus: Copper, 11.0 x 1010 Nm-2, Aluminium 7.0 x 1010 Nm-2 1.9 cm 3. Air is trapped above liquid ethyl alcohol... 592 Words \| 2 Pages • What If Exams Were Abolished? Exams are tests held for students to show their progress and knowledge in different subjects. These 'assessments' are kept at regular periods of time every academic year. But should exams be abolished? What are the advantages and disadvantages of exams? This topic is an argumentative one. Let's see what would happen if tests and examinations were abolished by looking at the advantages and the disadvantages. Disadvantages of exams: 1.) Students are stressed due to the pressure of exams.... 331 Words \| 2 Pages • 60720891 ATPL 500 Meteorology Questions 050 – METEOROLOGY 050-01 THE ATMOSPHERE 050-01-01 Composition, extent, vertical division 8814. The troposphere is the: A – part of the atmosphere above the stratosphere B – part of the atmosphere below the tropopause C – boundary between the mesosphere and thermosphere D – boundary between the stratosphere and the mesosphere Ref: all Ans: B 8817. What is the boundary layer between troposphere and stratosphere called: A – Tropopause B – Ionosphere C – Stratosphere D – Atmosphere Ref: all Ans:... 66,316 Words \| 332 Pages • I need help - 397 Words Lab Report Format for Meteorology Lab Hypothesis: Using the relationships from weather data write a hypothesis about how weather may be forecasted. Remember this is your hypothesis. Make sure it is reasonable and done before you plot the weather station models and create your graphs. Weather Data - Location: Boston, Logan International Airport Date: August 7th, 2014 Weather Data - Location: Date: Time of Day Temperature (F°) Air Pressure Humidity Cloud... 397 Words \| 2 Pages • Fast Food - 1483 Words Hoo Sze Yen www.physicsrox.com Physics SPM 2011 CHAPTER 3: FORCES AND PRESSURE 3.1 Pressure Units of pressure Unit Note Pa SI unit N m-2 Equivalent to Pa N cm-2 cm Hg m water atm 1 atm = atmospheric pressure at sea level bar 1 bar = 1 atm Pressure is the force which acts normal per unit area of contact. P= F A where P = pressure [Pa] F = force [N] A = area [m2] For atmospheric pressure only 3.2 Pressure in Liquids Pressure in liquids are not dependent on the size or... 1,483 Words \| 9 Pages • Automatic Accident Avoiding Punching Press INTRODUCTION The designer should make the machine as reliable and as reasonable as possible to minimize the maintenance requirement and allow for long intervals between routine maintenance tasks. It is also important to design the machine and its control system so that the maintenance can be carried out safely. For example, hold- to- run controls can be installed that allow a machine to be run at a reduced speed, or a removable tool holder can be used so that sharp tools can be replaced on a... 1,983 Words \| 7 Pages • Lab Earth Science lab 1.06 Name: Chase Carter Date: march 2nd 2015 Graded Assignment Lab Report Answer the questions below. When you are finished, submit this assignment to your teacher by the due date for full credit. (4 points) 1. Complete the following table based on the observations you made during the lab. Experiment Observations Cold water After I had opened the bottle, condensation rose to the top of the water bottle where the air was. Cold water plus match With water plus the match, the smoke from the match... 307 Words \| 2 Pages • Contrast Brach vs Mountains Do you get tired from long hours of work? Do you really need some time off to rest? 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### Home > GC > Chapter 8 > Lesson 8.2.2 > Problem8-81 8-81. Multiple Choice: What is the solution to the system of equations at right? 1. $(2, 0)$ 1. $(16, 4)$ 1. $(−2, −5)$ 1. $(4, −2)$ 1. None of these $\begin{array}{c} y = \frac { 1 } { 2 }x - 4 \\ x - 4y = 12 \end{array}$ First solve the bottom equation for $y$. $4y=x-12$, so $y=\frac{1}{4}x-3$ Now set the two equations equal to each other. $\frac{1}{2}x-4=\frac{1}{4}x-3$ Now solve for $x$. $\frac{1}{4}x=1$ $x=4$, so D	{"found_math": true, "script_math_tex": 12, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://homework.cpm.org/category/CON_FOUND/textbook/gc/chapter/8/lesson/8.2.2/problem/8-81", "fetch_time": 1721801212000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-30/segments/1720763518157.20/warc/CC-MAIN-20240724045402-20240724075402-00021.warc.gz", "warc_record_offset": 267858086, "warc_record_length": 15492, "token_count": 210, "char_count": 493, "score": 4.21875, "int_score": 4, "crawl": "CC-MAIN-2024-30", "snapshot_type": "latest", "language": "en", "language_score": 0.5998222231864929, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00019-of-00064.parquet"}
MagneticFieldduetoCurrent <!--Induction_and_Inductance --> ## General Physics (calculus based) Class Notes Dr. Rakesh Kapoor, M.Sc., Ph.D. Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294 Magnetic Fields Due to Currents Objectives In this chapter we will explore the relationship between an electric current and the magnetic field it generates in the space around it. For problems with low symmetry we will use the law of Biot-Savart law in combination with the principle of superposition. For problems with high symmetry we will introduce Ampere’s law. Both approaches will be used to explore the magnetic field generated by currents in a variety of geometries (straight wire, wire loop, solenoid coil, toroid coil). We will also determine the force between two parallel, current-carrying conductors. We will then use this force to define the SI unit for electric current (the ampere). Magnetic Field Due to a current A moving charge produce magnetic field and its magnitude and direction are given by "Biot-Savart law" (pronounced bee-oh sah-VAR) is the magnetic field at a position due to a charge q, moving with velocity . Current in a wire is an example of moving charge, therefore it should produce a magnetic field. Consider a small segment ds of a wire carrying current i. If is the drift velocity then time t taken by all the conduction electrons, in the segment ds, to cross the line A is given as Total charge q moving in time t through this segment is The magnetic field due to this segment of wire at point P will be given as Since direction of and is same therefore we can re write above equation by substituting the value of q. Now , therefore magnitude of can be written as Since the magnetic field is proportional to the cross product (×) of vector and , therefore the direction of will always be perpendicular to and . Tips to find direction of magnetic field due to a wire carrying current: Point the thumb of right hand in the direction of current, curled fingers point to the direction of magnetic field . Draw a circle on a plane (page) perpendicular to the direction of current with wire at the center. If current is going in the plane (page), magnetic field goes in clockwise direction. If current is coming out of the plane (page), magnetic field is in anti-clockwise direction. Magnetic field at any point P on the circle will point in the direction of the tangent and will always be perpendicular to the radial vector joining the point P and the wire at the center. Magnetic Field of a Current carrying Long Straight Wire Consider a very long (infinitely long ) wire. How can we compute magnetic field due to this wire at a point P? We know how to calculate magnetic field at a point due to a small segment of a wire carrying current. We can divide the wire in several small segments of length ds, and can compute magnetic field due to each of these segments at point P. We know Magnetic field is a vector quantity and net field at any point can be computed by vector sum of the magnetic fields due to all these segments (law of superposition). Net magnetic field at point P due to all the segments will be If we apply right hand rule, the direction of at point P due to each element will point in the page. You can see that it is perpendicular to both and . Therefore magnitude of net magnetic field B can be obtained by adding magnitude of magnetic field due to each segment If the segment ds is very small, the above equation can be written as an integral In the above figure With the substitution of θ and r, the integral can be re written as Solving this integral gives us the value By substituting the value of the integral we get In the above example point P was at the mid of the wire. What will be the magnetic field if point P is located at one edge of the wire? In this case integration limits will be from 0 to ∞, instead of from -∞ to ∞. Therefore the magnetic field at the edge of the wire will be half of the magnetic field at the middle of the wire. Magnetic Field of a Current carrying Circular arc of Wire Let us calculate magnetic field at the center of a circular arc of wire carrying current i. Magnitude of , of each element at the center of the arc is given as Since magnetic field due to all the segments point out of the page, the net field at the center can be computed by adding the sum of magnitude due to all the segments. If the segment ds is very small then The above summation can be written as an integral When arc is a complete circle then φ=2π, therefore field at the center of a circular wire will be Magnetic Field on the axis of a Current carrying Circular loop Let us calculate magnetic field at the center of a circular arc of wire carrying current i. In the figure cross section of the loop is shown. Here angle between and segment is 90°, therefore magnitude of , of each element at point P on the axis of the circular loop is given as As we have discussed earlier, the direction of is perpendicular to the position vector and segment . Let us decompose into two components, One parallel to z-axes and other perpendicular to z-axes. Due to symmetry, due to the segment on left will be opposite to the of symmetric segment on the right. Therefore the sum of all components is equal to zero. Thus only components contributing to the total magnetic field are of all the segments. Net magnetic field will be sum of component of all the segments. Value of cos α and value of r at a distance z from the center is given as If ds is very small summation can be written as an integral and magnetic field at a distance z from the center is Or When observation point is very far from the current loop or Rz, then the magnetic field at that point is given as Here , area of the current loop. In last chapter we have seen that the magnitude of magnetic moment μ of a dipole is In present case number of loops N is one therefore Since direction of and is same therefore in vector form we can write Checkpoint 1 The figure here shows four arrangements of circular loops of radius r or 2r, centered on vertical axes (perpendicular to the loops) and carrying identical currents in the directions indicated. Rank the arrangements according to the magnitude of the net magnetic field at the dot, midway between the loops on the central axis, greatest first. Hint : Magnetic field is proportional to area of the loop. Direction is given by right hand rule. Force Between Two Parallel Currents. Interactive Checkpoint - 1 (Force Between two Parallel Currents) Two current carrying wires are placed parallel to each other. We can say a current carrying wire is placed in the magnetic field produced by another current carrying wire. (a) Will the wires attract or repel if the currents are parallel? (b) Will the wires attract or repel if the currents are anti parallel? Let us consider two parallel wires of length L each with current and . How to calculate magnetic force between two current-carrying wires? First find the magnetic field due to second (b) wire at the position of first (a) wire. Then calculate the force on first (a) wire due to the field of second (b) wire. Magnetic force on first (a) wire placed in a magnetic field of second (b) wire is given as Here is length vector that has magnitude L and is directed along the direction of current in the wire. Magnitude of magnetic field due to second (b) wire at a distance d is Where is the current through second wire. Now the force on first wire is By right hand rule we can find the direction of force and it follows the rule Parallel currents attract each other, and anti parallel currents repel each other. Checkpoint 2 The figure here shows three long, straight, parallel, equally spaced wires with identical currents either into or out of the page. Rank the wires according to the magnitude of the force on each due to the currents in the other two wires, greatest first. Hint : Parallel currents attract and anti parallel repel. Magnitude of field reduces with distance. Ampere's Law Gauss law is used to compute electric field in certain symmetric charge distribution, similarly if the current distribution is considerably symmetric, Ampere's law can be used to find the magnetic field with considerably less effort. According to Amperes law The loop on the integral sign means that the scalar (dot) product is to be integrated around a closed loop, called an Amperian loop. The current is the net current encircled by that closed loop. How to compute integral ? Divide the closed path into n segments , ,.......,. Compute the sum Here is the magnetic field at the location of ith segment. In the limiting case the summation can be replaced by an integral For calculating , choose any arbitrary direction as direction of Amperian loop Curl the fingers of right hand in the direction of Amperian loop and note the direction of thumb. All the currents inside the loop parallel to the thumb are counted as positive. All the currents inside the loop anti parallel to the thumb are counted as negative. All the currents outside the loop are not counted. For the above example Checkpoint 3 The figure here shows three equal currents i (two parallel and one anti parallel) and four Amperian loops. Rank the loops according to the magnitude of along each, greatest first. Hint : Only count currents inside the loop. Application of Ampere' s Law Example - 1 (Magnetic Field Outside a Long Straight Wire with Current) Let us calculate magnetic field due to a long straight wire that carries current i straight through the page. Draw an Amperian loop as a circle of radius r around the wire with its center at the wire center. Since all the points on the circle are equidistant from the wire, the magnitude of will be same at any point on the circle. At any point the angle between and is , therefore According to Ampere's law, the direction of current is parallel to thumb so it will be taken as positive and Or This is the relation we have derived using long calculus method. Ampere's law holds true for any closed path. We choose the path that makes the calculation of as easy as possible. Example - 2 (Magnetic Field inside a Long Straight Wire with Current) Let us calculate magnetic field due to a long straight wire of radius R, that carries current i straight through the page. Assume that the distribution of current with in the cross-section is uniform, or current density in the wire is a constant. Draw an Amperian loop as a circle of radius r<R, around the wire with its center at the wire center. Since all the points on the circle are equidistant from the wire, the magnitude of will be same at any point on the circle. At any point the angle between and is , therefore is the fraction of the total current i which is passing through the area of the circle r. is given as According to Ampere's law, the direction of current parallel to thumb should be taken as positive and Or Let us plot magnetic field inside and outside a wire as function of the distance r from the center of wire. Solenoid and Toroid Magnetic field of a solenoid A solenoid is a coil of conducting wire as shown below Following figure shows a vertical cross-section through the central axis of a stretched out solenoid. Magnetic field lines of a stretched out solenoid. Magnetic field lines of a real solenoid. The field is strong and uniform at the interior point but relatively weak at external points such as . In an ideal solenoid we take field at any external point as zero and uniform at any point inside the solenoid. Let us now apply Ampere's law to a real solenoid. We can consider a rectangular Amperian loop abcda. Since is uniform inside the solenoid and zero out side we can write as sum of four integrals The first integral on the right is Bh, where B is the magnitude of the field and h is the length of segment ab. Second and fourth integral are zero because B is perpendicular to the direction inside the solenoid and B is zero out side the solenoid. Third integral is also zero as B is zero outside the solenoid. total value of integral will be If n is the number of wire turns per unit length then total number of turns enclosed by the Amperian loop will nh. If i is the current through each turn, total enclosed current will be According to Ampere' s law, magnitude of B inside the solenoid will be given as Or Although it is derived for an infinite solenoid but it holds true for actual solenoid if we measure at a point inside the solenoid away from the edges. Magnetic field inside a solenoid depends only on the number of turns per unit length and current, it is independent of the area (radius of solenoid) of the loops Magnetic Field of a Toroid A toroid is a ring shaped solenoid as shown below Following figure shows a horizontal cross-section of the toroid. What is the magnetic field inside a toroid? We can find out by applying Ampere' s law. From symmetry we see that B forms concentric circles inside toroid. Consider a circular Amperian loop of radius r inside the toroid . If N is the total number of turns, the enclosed current . According to Ampere's law The magnetic field B inside the toroid at a distance r from the center of the toroid ring, will be given as	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://rakeshkapoor.us/ClassNotes/classnotes.php?notes=MagneticFieldduetoCurrent&index=2", "fetch_time": 1696366691000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2023-40/segments/1695233511220.71/warc/CC-MAIN-20231003192425-20231003222425-00429.warc.gz", "warc_record_offset": 35973114, "warc_record_length": 8956, "token_count": 2897, "char_count": 13388, "score": 4.0625, "int_score": 4, "crawl": "CC-MAIN-2023-40", "snapshot_type": "latest", "language": "en", "language_score": 0.8977326154708862, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00004-of-00064.parquet"}
# geometry If m∠BCD = 60 and m∠DEC = 80, what is the measure of ∠DEF 1. 👍 0 2. 👎 0 3. 👁 223 1. not having a diagram to refer to, it's impossible to say. Provide some description of the relative positions of B,C,D,E,F. I can think of various arrangements that provide different answers. 1. 👍 0 2. 👎 0 ## Similar Questions 1. ### Geometry Help Triangles DEF and D'E'F' are shown on the coordinate plane below: Triangle DEF and triangle D prime E prime F prime with ordered pairs at D negative 1, 6, at E 1, 3, at F 6, 3, at D prime 6, 1, at E prime 3, negative 1, at F prime 2. ### math Triangle ABC undergoes a series of transformations to result in triangle DEF. Is triangle DEF congruent to triangle ABC ? Is it Congruent or Not congruent? 1.)) Triangle ABC is translated 3 units up and 5 units right, and then 3. ### Accounting On March 1, 2003, a company paid a \$16,200 premium on a 36-month insurance policy for coverage beginning on that date. Refer to that policy and fill in the blanks in the following table: Check 2005 insurance expense: Accrual, 4. ### geometry triangle abc is similar to def, the lengths of the sides of triangle abc is 5,8,11. what is the length of the shortest side of triangle def if its perimeter is 60? 1. ### Precalculus In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles ABC and DEF such that AB = DE, BC = EF, and angle A = angle D, then we cannot deduce that ABC and DEF are congruent. 2. ### math (need help badly) the vertices of triangle DEF are D(5,12)and E(2,7) and F(8,4) Triangle DEF undergoes an enlargement with the centre ,O, and scale factors k.Its image is D`E`F` where D(5,12)arrow D`(7.5,18) a)How do i detremine the value of k 3. ### Math In angle DEF, d=13.5 cm, e=18.2 cm and F = 60 degrees Determine the measure of f to the nearest tenth of a centimetre. I know the answer is 16.4 cm but how do I get it 4. ### algebra Triangles ABC and DEF are similar. Find the perimeter of triangle DEF. Round your answer to the nearest tenth. 1. ### Geometry 1. If line JK is perpendicular to line XY at its midpoint M, which statement is true? a) JX = KY b) JX = KX c) JM = KM d) JX = JY Is it c? 2. What information is needed to conclude that line EF is the bisector of ∠DEG? a) 2. ### Math Given: measure of angle D=~ measure of angle F ; line GE bisects angle DEF Prove: line DG=~ line FG =~ is the congruent sign 3. ### geometry In the accompanying diagram, triangle ABCis similar to triangle DEF, AC = 6, AB = BC = 12, and DF = 8. Find the perimeter of triangle DEF. 4. ### Geometry Given: ABCD is a parallelogram;	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.jiskha.com/questions/742086/if-m-bcd-60-and-m-dec-80-what-is-the-measure-of-def", "fetch_time": 1603443539000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2020-45/segments/1603107880878.30/warc/CC-MAIN-20201023073305-20201023103305-00496.warc.gz", "warc_record_offset": 793885652, "warc_record_length": 4955, "token_count": 778, "char_count": 2654, "score": 3.890625, "int_score": 4, "crawl": "CC-MAIN-2020-45", "snapshot_type": "latest", "language": "en", "language_score": 0.8918665647506714, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00018-of-00064.parquet"}
# Calculating Percentage Increase And Decrease ## Learn About Calculating Percent Increase And Decrease With The Following Examples And Interactive Exercises. Example 1: Ann works in a supermarket for \$10.00 per hour. If her pay is increased to \$12.00, then what is her percent increase in pay? Analysis: When finding the percent increase, we take the absolute value of the difference and divide it by the original value. The resulting decimal is then converted to a percent. Solution: Answer: The percent increase in Ann's pay is 20%. Let's look at an example of percent decrease. Example 2: The staff at a company went from 40 to 29 employees. What is the percent decrease in staff? Analysis: When finding the percent decrease, we take the absolute value of the difference and divide it by the original value. The resulting decimal is then converted to a percent. Solution: Answer: There was a 27.5% decrease in staff. Percent increase and percent decrease are measures of percent change, which is the extent to which something gains or loses value. Percent changes are useful to help people understand changes in a value over time. Let's look at some more examples of percent increase and decrease. In Example 1, we divided by 10, which was the lower number. In Example 2, we divided by 40, which was the higher number. Students often get confused by this. Remember that the procedure above asked us to divide by the original value. Another way to remember the procedure is to subtract the old value from the new value and then divide by the old value. Convert the resulting decimal to a percent. The formula is shown below. Example 3: At a supermarket, a certain item has increased from 75 cents per pound to 81 cents per pound. What is the percent increase in the cost of the item? Solution: Answer: There was an 8% increase in the cost of the item. Example 4: Four feet are cut from a 12-foot board. What is the percent decrease in length? Solution: Answer: There was a 33.3% decrease in length. Summary: Percent increase and percent decrease are measures of percent change, which is the extent to which something gains or loses value. Percent change is useful to help people understand changes in a value over time. The formula for finding percent change is: ### Exercises Directions: Each problem below involves percent change. Enter your answer for each exercise without the percent symbol. Round your answer to the nearest tenth of a percent when necessary. For each exercise below, click once in the ANSWER BOX, type in your answer and then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. 1 At Furnace Woods School, enrollment increased from 320 students in 2006 to 349 students in 2007. What is the percent increase in enrollment? ANSWER BOX: % RESULTS BOX: 2 Stock in Company XYZ decreased from \$14 a share to \$9 a share. What is the percent decrease in stock price? ANSWER BOX: % RESULTS BOX: 3 The tuition at a college increased from 50,000 in 2006 to to 59,000 in 2007. What is the percent increase in tuition? ANSWER BOX: % RESULTS BOX: 4 The price of oil decreased from \$54 per barrel to \$50 per barrel. What is the percent decrease in oil prices? ANSWER BOX: % RESULTS BOX: 5 In a small town, the population increased from 25,000 people in 1990 to 32,000 people in 2000. What is the percent increase in population? ANSWER BOX: % RESULTS BOX:	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.mathgoodies.com/lessons/percent/change", "fetch_time": 1652994117000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2022-21/segments/1652662530066.45/warc/CC-MAIN-20220519204127-20220519234127-00159.warc.gz", "warc_record_offset": 1007121742, "warc_record_length": 19125, "token_count": 816, "char_count": 3519, "score": 4.78125, "int_score": 5, "crawl": "CC-MAIN-2022-21", "snapshot_type": "longest", "language": "en", "language_score": 0.9311457872390747, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00008-of-00064.parquet"}
# Homework Help: Euler's Method and Free Fall 1. Feb 24, 2016 ### njo 1. The problem statement, all variables and given/known data Write a C program to simulate a falling object. The program should ask for the initial height of the object, in feet. The output of the program should be the time for the object to fall to the ground, and the impact velocity, in ft/s and miles/hour. Your program should use Euler’s method to numerically solve the differential equations describing the motion of a falling object. In Euler’s method, the state of the object at some small future time is calculated using the values at the present time. This small future timestep is typically called delta time, or dt. Thus the position (p) and speed (v) of the object in the next timestep t + dt is written as a simple function of the position and speed at the current timestep t (g is the acceleration due to gravity): v(t+dt) = v(t) + g * dt p(t+dt) = p(t) + v(t+dt) * dt You should actually start out with the velocity as zero, and the position at the initial height of the object. Then your position (above the ground) would be: p(t+dt) = p(t) - v(t+dt) * dt And you would integrate until the position becomes less than or equal to zero. I'm having difficulty understanding what to do here. I don't really understand what the point of using these equations when t=sqrt(2d/g) gives us time of free fall. Thanks for the help. 2. Relevant equations v(t+dt) = v(t) + g * dt p(t+dt) = p(t) + v(t+dt) * dt p(t+dt) = p(t) - v(t+dt) * dt 3. The attempt at a solution p(t+dt) = p(t) - v(t+dt) * dt 2. Feb 24, 2016 ### Staff: Mentor Welcome to the PF. Yes, clearly without any air resistance issues, you can solve for this algebraically. But the point of the programming exercise is to give you an easy problem to solve numerically. You should compare the answers you get from your numerical solution with the equation you mention, to see what the accuracy is. You could even vary the dt values to see how they affect the accuracy of the numerical solution. 3. Feb 24, 2016 ### brainpushups I give a similar exercise to my students. The point is to give you some practice with numerical integration. The fact that you know the analytic solution allows you to check your answer. The importance of the exercise is that for many (most?) 'real' problems (systems modeled by differential equations) an analytic solution may not exist, or may be very difficult to find. Numerical integration allows for a relatively easy way to analyze the system. Edit: Sorry for the redundancy - berkeman's post must have posted as soon as I started typing! 4. Feb 24, 2016 ### njo I'm still unsure what v(t+dt) and p(t+dt) represent. 5. Feb 24, 2016 ### Staff: Mentor It's probably best to start with an Excel spreadsheet to get a feel for how these sequential calculations go. The first column is time, starting at zero, and incrementing by dt for each row after that. The second column is y(t), the third is velocity v(t). All are zero in the first row. Then put in formulas into the second row, second and third columns to calculate the new position and velocity based on the previous values in the previous row. That's what v(t+dt) and p(t+dt) represent. Then just do a click-drag down for the y(t) and v(t) cells and do a Fill-Down to paste the same formulas (with adjusted row numbers) in the cells below. You should see how the position and velocity are changing with time when you do this. Makes sense? Then you just need to code up the same thing in C... 6. Feb 24, 2016 ### brainpushups v(t+dt) is the velocity after a short time interval. Even for non-constant acceleration the velocity will be approximately linear for very short intervals of time - that's why the method is so simple. p(t+dt) is the position after a short time interval. Similar to the example above the change in position is approximately linear for short time intervals. You use the velocity you calculated to calculate each subsequent change in position. As @berkeman said, it would be worth your time to set this up in a spreadsheet - that is actually what I instruct my students to do. 7. Feb 24, 2016 ### njo How would I solve for time with these equations though? if g=-32.2 and assume p=100? 8. Feb 24, 2016 ### brainpushups You don't solve for time in the algebraic sense - you iterate the equations until the desired position is reached. Time is a parameter that everything else depends on and you pick the time step, dt. For example, suppose we pick a time step of 1 second (a poor choice if we want to be close to approximate for the position). The first few columns for various rows in a spreadsheet would look like: t a v p 0 10 0 0 1 10 10 0 2 10 20 10 3 10 30 30 4 10 40 60 5 10 50 100 Edit: hopefully you can understand what I've written here; I just quickly typed it in nicely spaced so I didn't need to spend time making a chart, but the spacing didn't show up right... (a is acceleration, v is velocity, p is position and I used an acceleration of 10 to make the numbers look nice.) The nth entry in the velocity column is calculated by using the n-1 acceleration term and adding the n-1 velocity. Similarly, the nth position entry is calculated by using the n-1 velocity and adding the n-1 position. Suppose the question is to find the time for the object to fall 100 meters. The time would be 5 seconds according to this spreadsheet. Does that help? 9. Feb 24, 2016 ### njo It's making a bit more sense. What is your work to calculate those values? If we assume dt=1 and g=10 and p=10, does that mean p(0)=100? p(t+1) = p(t) + v(t) + 10 is this true? I really appreciate the help 10. Feb 24, 2016 ### brainpushups Let me start with the velocity explicitly. Let v0 be the initial velocity. The next velocity (after time dt) will be v1 = a0 * dt + v0 where a0 is the initial acceleration (note that for free fall the acceleration is always constant, but I'm going to give instructions as if it was changing so everything is more general, and hopefully more clear). Then, the velocity after the second time interval is v2 = a1 * dt + v1 And the third velocity is v3 = a2 * dt + v2 And so on. In the line for t = 4 seconds in post 8 above the velocity was calculated as v4 = (10) (1) + 30 = 40 The position works the same way except you use the velocity in place of the acceleration: p1 = v0 * dt + p0 p2 = v1 * dt +p1 p3 = v2 dt +p2 So p4 from the example in post 8 is p4 = 30 1 + 30 = 60 And p5 is p5 = 40 * 1 + 60 = 100 11. Feb 24, 2016 ### njo Alright it makes sense now. Thanks so much for the help.	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.physicsforums.com/threads/eulers-method-and-free-fall.859154/", "fetch_time": 1527268378000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2018-22/segments/1526794867140.87/warc/CC-MAIN-20180525160652-20180525180652-00218.warc.gz", "warc_record_offset": 798958739, "warc_record_length": 19579, "token_count": 1722, "char_count": 6622, "score": 3.96875, "int_score": 4, "crawl": "CC-MAIN-2018-22", "snapshot_type": "longest", "language": "en", "language_score": 0.9047573804855347, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00029-of-00064.parquet"}
# Non-chromatic paths in Hamiltonian graphs What is an example of a Hamiltonian graph $G=(V,E)$ such that there is one path visiting all vertices that is not chromatic (definition see below)? Let $G= (V,E)$ be a simple undirected graph on $n\geq 1$ vertices, and let $b:[n]\to V$ be a bijection. We assign to $b$ the greedy coloring $c_b$ constructed by traversing the graph in the order $b$. Formally, with recursive definition of $c_b:[n] \to [n]$: • $c_b(1) = 1$; • if $k\in[n]$ and $k>1$ let $$c_b(k) = \min\:\big(\mathbb{N}\setminus\{c_b(j): j \in [k-1]\land \{b(j),b(k)\}\in E\}\big).$$ We call $b$ chromatic if $\text{im}(c_b) = [\chi(G)]$. For every graph there is a chromatic bijection (see here). A chromatic path is a chromatic bijection that is also a path. • by "one path" you mean "at least one path"? Feb 24, 2017 at 8:41 Counterexample. Let $G$ be the graph with vertices $v_1,v_2,v_3,v_4,v_5,v_6$ and edges $v_1v_2,v_2v_3,v_3v_4,v_4v_5,v_5v_6,v_6v_1,v_1v_5,v_4v_6.$ The graph $G$ is Hamiltonian, since $v_1,v_2,v_3,v_4,v_5,v_6,v_1$ is a Hamiltonian cycle. The graph $G$ is $3$-chromatic; for a proper coloring, we may color $v_1$ and $v_4$ red, $v_2$ and $v_5$ white, $v_3$ and $v_6$ blue. The path $v_1,v_2,v_3,v_4,v_5,v_6$ is not chromatic.	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://mathoverflow.net/questions/262989/non-chromatic-paths-in-hamiltonian-graphs", "fetch_time": 1656529810000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2022-27/segments/1656103642979.38/warc/CC-MAIN-20220629180939-20220629210939-00209.warc.gz", "warc_record_offset": 441962987, "warc_record_length": 24694, "token_count": 497, "char_count": 1265, "score": 3.53125, "int_score": 4, "crawl": "CC-MAIN-2022-27", "snapshot_type": "latest", "language": "en", "language_score": 0.8202470541000366, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00041-of-00064.parquet"}
### 1: Number #### 1.A: Number Concepts 1.A.1: Demonstrate a knowledge of the interrelationship of the sets of numbers within the real number system. 1.A.1.1: Compare and order integers. 1.A.1.2: Find and be able to model an understanding of common multiples, common factors, lowest common multiple and greatest common factor as they apply to whole numbers. 1.A.1.4: Distinguish between exact values and decimal approximations of square roots and cube roots. 1.A.2: Develop a number sense of powers with integral exponents and rational bases. 1.A.2.7: Explain and apply the exponent laws for powers with integral exponents. #### 1.B: Number Operations 1.B.1: Use a scientific calculator to solve problems involving real numbers. 1.B.1.8: Document and explain the calculator keying sequences used to perform: 1.B.1.8.a: square roots, cube roots 1.B.1.8.d: sine, cosine, tangent 1.B.2: Decide which arithmetic operations can be used to solve a problem and then solve the problem. 1.B.2.9: Perform arithmetic operations with integers concretely, pictorially and symbolically. 1.B.2.10: Illustrate and explain the order of operations. 1.B.2.11: Add, subtract, multiply and divide fractions concretely, pictorially and symbolically. 1.B.2.13: Estimate and calculate operations on rational numbers. 1.B.2.15: Use a variety of methods to solve problems, such as drawing a diagram, making a table, guessing and testing, using objects to model, making it simpler, looking for a pattern, using logical reasoning and working backward. 1.B.3: Illustrate and apply the concepts of rates, ratios, percentages and proportions to solve problems. 1.B.3.16: Understand the meaning of rate, ratio, percentage and proportion; and apply these concepts to solve problems. 1.B.3.17: Express rates and ratios in equivalent forms. 1.B.4: Apply exponent laws to solve problems. 1.B.4.18: Use exponent laws to evaluate expressions with numerical bases. ### 2: Patterns and Relations #### 2.A: Patterns 2.A.1: Generalize, design and justify mathematical procedures, using appropriate patterns and technology. 2.A.1.2: Given a first-degree equation, substitute numbers for variables and graph and analyze the relation. 2.A.1.3: Translate between an oral or written expression and an equivalent algebraic expression. 2.A.1.4: Write equivalent forms of algebraic expressions, or equations, with integral coefficients. #### 2.B: Variables and Equations 2.B.1: Generalize arithmetic operations from the set of rational numbers to the set of polynomials. 2.B.1.5: Identify constant terms, coefficients and variables in polynomial expressions. 2.B.1.8: Perform the operations of addition and subtraction on polynomial expressions. 2.B.1.9: Represent multiplication, division and factoring of monomials, binomials and trinomials of the form x² + bx + c, using concrete materials and diagrams. 2.B.1.10: Find the product of: 2.B.1.10.a: two monomials 2.B.1.10.c: two binomials. 2.B.1.11: Determine equivalent forms of algebraic expressions by identifying common factors. 2.B.1.12: Factor trinomials of the form ax² + bx + c, where a = 1, or of the form ax² + abx + ac. 2.B.1.13: Factor polynomials of the form A² ? B², where A and B are both monomial expressions. 2.B.1.14: Find the quotient when a polynomial is divided by a monomial. 2.B.2: Solve and verify linear equations and inequalities in one variable. 2.B.2.15: Illustrate the solution process for a one-step, single variable, first-degree equation, using concrete materials or diagrams. 2.B.2.15.a: x + a = b 2.B.2.15.b: x ? a = b 2.B.2.15.c: ax = b 2.B.2.15.d: x/a = b 2.B.2.16: Solve and verify, using a variety of techniques, one-step linear equations of the form: 2.B.2.16.a: x + a = b 2.B.2.16.b: x/a = b 2.B.2.16.c: ax = b 2.B.1.17: Illustrate the solution process for a two-step, single variable, first-degree equation, using concrete materials or diagrams. 2.B.2.18: Solve and verify one- and two-step first-degree equations of the form: 2.B.2.18.a: x/a + b = c 2.B.2.18.b: ax + b = c 2.B.2.18.c: where a, b and c are integers. 2.B.2.19: Solve and verify first-degree, single variable equations of the form: 2.B.2.19.a: ax = b + cx 2.B.2.19.b: a(x + b) = c 2.B.2.19.d: a(bx + c) = d(ex + f) ### 3: Shape and Space #### 3.A: Measurement 3.A.1: Solve problems involving perimeter, area, surface area and volume. 3.A.1.1: Estimate, measure and calculate the surface area and volume of any right prism, cylinder, cone or pyramid. 3.A.1.2: Demonstrate concretely, pictorially and symbolically that many rectangles are possible for a given perimeter or a given area. 3.A.2: Solve problems, using right triangles. 3.A.2.3: Use the Pythagorean relationship to calculate the measure of the third side of a right triangle, given the other two sides, in 2-D applications. 3.A.2.5: Explain the meaning of sine, cosine and tangent ratios in right triangles. 3.A.2.6: Calculate an unknown side or an unknown angle in a right triangle, using trigonometric ratios. 3.A.3: Specify conditions under which triangles may be similar, and use these conditions to solve problems. 3.A.3.8: Recognize when, and explain why, two triangles are similar; and use the properties of similar triangles to solve problems. #### 3.B: Transformations 3.B.1: Create and analyze patterns and designs, using symmetry, translation, rotation and reflection. 3.B.1.9: Draw designs, using ordered pairs, in all four quadrants of the coordinate grid. 3.B.1.10: Draw and interpret scale diagrams, including: 3.B.1.10.a: enlargements 3.B.1.10.b: reductions. ### 4: Statistics and Probability #### 4.A: Data Analysis 4.A.1: Develop and implement a plan for the display and analysis of data. 4.A.1.1: Read and interpret graphs that are provided. 4.A.2: Analyze experimental results expressed in two variables. 4.A.2.2: Create scatterplots for discrete and continuous variables. 4.A.2.3: Interpret a scatterplot to determine if there is an apparent relationship. 4.A.2.4: Determine, by inspection, the line of best fit from a scatterplot for an apparent linear relationship. 4.A.2.5: Draw and justify conclusions from the line of best fit, by: 4.A.2.5.a: interpolation 4.A.2.5.b: extrapolation. Correlation last revised: 9/16/2020 This correlation lists the recommended Gizmos for this province's curriculum standards. Click any Gizmo title below for more information.	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.explorelearning.com/index.cfm?method=cResource.dspStandardCorrelation&id=904", "fetch_time": 1627411880000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2021-31/segments/1627046153474.19/warc/CC-MAIN-20210727170836-20210727200836-00423.warc.gz", "warc_record_offset": 789078736, "warc_record_length": 16035, "token_count": 1693, "char_count": 6440, "score": 4.125, "int_score": 4, "crawl": "CC-MAIN-2021-31", "snapshot_type": "latest", "language": "en", "language_score": 0.8554005026817322, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00005-of-00064.parquet"}
.adslot_3 { width: 479px; height: 600px; } @media (min-width:480px) { .adslot_3 { width: 300px; height: 600px; } } @media (max-width:479px) { .adslot_3 { width: 300px; height: 600px; } } For some reason, people always get confused with Carat Weights. They have a tough time with fractions and decimals and points. What the heck are Carats and Weights and Points? Can’t they make this stuff simpler to understand? Well the good thing is, you don’t need to learn math to understand a Diamond’s Carat Weights. If you can count money, then you can count Carats! Let’s begin… One Carat is written as 1.00. It should be familiar looking since it actually looks like a 1 Dollar Bill minus the \$ symbol. And if you think about it as Dollars, then you’re already on the right track. (There are no Cents… But there are Points!) 1.00 = 100 Points One Carat or 1.00, is made up of 100 Points. Once again, 100 Points = One Carat (1.00) Are you still with me? Think about Points as Pennies. 100 Pennies equals One Dollar right? So 100 Points equals One Carat! Makes sense eh? 100 Points = 1 Carat. If you think about relating Carat Weights as money, it’ll be so much easier to comprehend. More Weights • A Half Carat or .50 is like 50 Cents or a Half Dollar (1/2) • 25 Points = 1/4 Carat. (or .25 Cents) • 10 Points = .10 Cents or a Tenth of a Carat, .1/10. Wow, is it really that simple? Yep! • 75 Points = 3/4 Carat or .75 Cents. See how easy it is? When a Jeweler says that a Diamond Solitaire is 58 Points, you’ll know that it’s a little bit bigger than a Half Carat (50/100), but smaller than a 3/4 Carat. (or 75 Points) A 95 Point Diamond is just 5 Points, or .05 Carats shy of a Full Carat. (Remember 1 Carat is 1.00 Points) It really doesn’t get any simpler! As long as you compare Carats and Points to Dollars and Pennies, you’ll quickly grasp all there is to understand about a Diamond’s Weight and Carats. Now that you grasp Weights and Carats and Diamond Points…	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://www.jewelry-secrets.com/Blog/understanding-carat-weights-and-points/", "fetch_time": 1455239815000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2016-07/segments/1454701162938.42/warc/CC-MAIN-20160205193922-00236-ip-10-236-182-209.ec2.internal.warc.gz", "warc_record_offset": 486705428, "warc_record_length": 10169, "token_count": 576, "char_count": 1980, "score": 3.796875, "int_score": 4, "crawl": "CC-MAIN-2016-07", "snapshot_type": "longest", "language": "en", "language_score": 0.8756274580955505, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00055-of-00064.parquet"}
# Math 202C: Lecture 19 Author: Qihao Ye Andrew Dennehy have already introduced the concept of the discrete Fourier transform (DFT) to us in Lecture 18, but I would like to retake the path with more details, because there are some other concepts (which help for fully understanding) I would like to talk about. We mainly talk about how fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) work, we sightly talk about the calculation error. We will see how FFT and IFFT work from both the perspective of math and computer, along with their applications and some specific problems. Some problem may involve the number theoretic transform (NTT), which is recognized as the integer DFT over finite field. We use Python3 for code examples, since we would not use version related feature, the specific version does not matter. (For instance, Python3.5.2 and Python3.8.1 are both ok.) We would call Python instead of Python3 in the following content. (If you do not install Python, there are many online interpreters that can run Python codes, for instance, Try It Online.) Recommend to view this page with computer for getting the best experience. Recommend to try the codes and solve some problems by yourself, which definitely would be challenging and interesting. There are only 2 exercises in this lecture, other problems are interest based algorithm problems (no need to do as homework), which might be difficult to solve if you are not very familiar with the FFT and NTT in this form, but I will give hints to guide you. Now let us start with an easy example. Example 1: We consider two polynomials Multiply them together (using distributive property), we would get Definition 1 (coefficient representation): For a polynomial $P(x) = a_{0} + a_{1} x + \cdots + a_{n} x^{n}$, the list $[a_{0}, a_{1}, \ldots, a_{n}]$ is its coefficient representation. We denote $P[k]$ as the coefficient of $x^{k}$ in $P(x)$. Use definition 1, we can write $P_{1}(x)$ as $[1, 2, 3, 4]$, $P_{2}(x)$ as $[2, 3, 4, 5]$ and $P_{1}(x) \cdot P_{2}(x) = Q(x)$ as $[2, 7, 16, 30, 34, 31, 20]$. The naïve polynomial multiplication function is not hard to get: def naive_poly_mul(P1, P2): Q = [0] * (len(P1) + len(P2) - 1) for i in range(len(P1)): for j in range(len(P2)): Q[i + j] += P1[i] * P2[j] return Q In the general case, i.e., it is easy to see that the complexity of the naïve polynomial multiplication function is $\mathcal{O}(m n)$. Note that $\text{len}(P_{1}) = n + 1$ (count from $0$ to $n$). If we consider the specific condition $m = n$, then the complexity becomes $\mathcal{O}(n^{2})$. Definition 2 (degree of polynomial): The degree of a polynomial is the highest of the degrees of the polynomial’s monomials (individual terms) with non-zero coefficients. In Example 1, $P_{1}, P_{2}$ are both $3$-degree polynomials. Definition 3 (value representation): Except representing a $n-$th degree polynomial with $n + 1$ coefficients, we can also represent a $n-$th degree polynomial with proper (see below Exercise) $n + 1$ points on the polynomial, which is called the value representation. Exercise 1 : Prove that $n + 1$ points $\{ (x_{i}, y_{i}) \}_{i = 1}^{n + 1}$ with distinct $\{ x_{i} \}$ determine a unique polynomial of degree $n$. Hint: use the fact that a Vandermonde matrix is nonsingular without proof. Back to Example 1, to get $Q$, we can first get $\{ (x_{i}, P_{1}(x_{i}), P_{2}(x_{i})) \}_{i = 1}^{7}$ from $P_{1}, P_{2}$ with distinct $\{ x_{i} \}$, then the $7$ points $\{ (x_{i}, P_{1}(x_{i}) P_{2}(x_{i})) \}_{i = 1}^{7}$ just represent $Q$. When the degree of $P_{1}, P_{2}$ are both $n$, we can see that the multiplication here just needs complexity $\mathcal{O}(n)$. However, at most time, we only need the coefficient representation, because it is more suitable to calculate the values in its domain. It is not hard to figure out we can do the multiplication in this way: If we only consider the situation that we aim to multiply two $n-$th degree polynomials, the multiplication part only costs $\mathcal{O}(n)$, so the bottleneck of the complexity lies in the algorithm changing the coefficient representation into the value representation and the algorithm changing it back. The naïve way to get the value representation of a coefficient represented polynomial cost at least $\mathcal{O}(n^{2})$. (To get each value we need $\mathcal{O}(n)$, and we totally need $\mathcal{O}(n)$ values.) The essential idea is selecting specific points to reduce the calculation cost. The straight thought would be looking at the parity of a function. Because for any odd function $P_{\text{odd}}$, we have $P_{\text{odd}}(-x) = - P_{\text{odd}}(x)$ while for any even function $P_{\text{even}}$, we have $P_{\text{even}}(-x) = P_{\text{even}}(x)$. Actually, we can divide any polynomial into one odd function plus one even function. Take a look back to $Q(x)$ in Example 1, we can write Notice that $Q_{\text{odd}}(x^{2})$ is actually an even function, but write in this form allows us to take $x^{2}$ as the variable. It follows that Note that $Q_{\text{even}}(x^{2})$ has degree $3$ and $Q_{\text{odd}}(x^{2})$ has degree $2$ while $Q$ has degree $6$. Once we get $Q_{\text{even}}(x^{2})$ and $Q_{\text{odd}}(x^{2})$, we would immediately get $Q(x)$ and $Q(-x)$. What we only need to do is recursive process $Q_{\text{even}}(x^{2})$ and $Q_{\text{odd}}(x^{2})$. It would lead to an $\mathcal{O}(n \log n)$ recursive algorithm. However, we would encounter the problem that the symmetric property could not be maintained further (we can pair $x$ and $-x$, but how to pair $x^{2}$). As we already see in the previous Lecture, we can use the roots of unity to solve this problem. Denote $\omega_{n} = \exp(2 \pi i / n)$. Note that $\omega_{n}^{n} = \omega_{n}^{0} = 1$. To make it easier to understand, we now only consider $n = 2^{k}$ for some positive integer $k$, we would evaluate polynomial at $[\omega_{n}^{0}, \omega_{n}^{1}, \ldots, \omega_{n}^{n - 1}]$, then $[\omega_{n}^{0}, \omega_{n}^{2}, \ldots, \omega_{n}^{n - 2}]$, next $[\omega_{n}^{0}, \omega_{n}^{4}, \ldots, \omega_{n}^{n - 4}]$, etc. Just as the figure below. We pair each $\omega_{n}^{j}$ with $\omega_{n}^{-j} = \omega_{n}^{j + n / 2}$. For any polynomial $P$, we can get $[P(\omega_{n}^{0}), P(\omega_{n}^{1}), \ldots, P(\omega_{n}^{n - 1})]$ fast by evaluating $[P_{\text{even}}(\omega_{n}^{0}), P_{\text{even}}(\omega_{n}^{2}), \ldots, P_{\text{even}}(\omega_{n}^{n - 2})]$ and $[P_{\text{odd}}(\omega_{n}^{0}), P_{\text{odd}}(\omega_{n}^{2}), \ldots, P_{\text{odd}}(\omega_{n}^{n - 2})]$ recursively. Recall that we divide $P(x)$ as $P_{\text{even}}(x^{2}) + x P_{\text{odd}}(x^{2})$. The corresponding formula is This leads to the naive FFT code: from math import pi from math import sin from math import cos def get_omega(n): theta = 2 * pi / n omega = cos(theta) + sin(theta) * 1j return omega def naive_FFT(P): # P is the coefficient representation # P = [a_{0}, a_{1}, ..., a_{n-1}] # current we assume n = 2^{k} n = len(P) half_n = n // 2 if n == 1: return P # constant function omega = get_omega(n) P_even, P_odd = P[::2], P[1::2] V_even, V_odd = naive_FFT(P_even), naive_FFT(P_odd) V = [0] * n # the value representation for j in range(half_n): V[j] = V_even[j] + omega ** j * V_odd[j] V[j + half_n] = V_even[j] - omega ** j * V_odd[j] return V Feed $P_{1}$ in example 1 as input, we would get Now we can apply Fourier transform to a coefficient representation to get the corresponding value representation, and the multiplication in the value representation form is easy to implement, what remains to solve is the inverse Fourier transform. In the matrix-vector form for $P(x) = a_{0} + a_{1} x + \cdots + a_{n - 1} x^{n - 1}$, we have Note that the character table of $C_{n}$ is just as the form of the DFT matrix: To get the inverse Fourier transform, we can just use the inverse of the matrix: Exercise 2: Check that the inverse DFT matrix is the inverse of the DFT matrix. The difference between DFT and FFT is just the way of calculating these matrix-vector forms, where DFT uses the direct matrix-vector multiplication way in $\mathcal{O}(n^{2})$ and FFT uses the tricky recursive way to achieve $\mathcal{O}(n \log n)$. The inverse DFT matrix leads to the naive IFFT code: def naive_IFFT(V, is_outest_layer=False): # V is the value representation # w means omega_{n} # V = [P(w^{0}), P(w^{1}), ..., P(w^{n-1})] # current we assume n = 2^{k} n = len(V) half_n = n // 2 if n == 1: return V # constant function omega = 1.0 / get_omega(n) # omega_{n}^{-1} V_even, V_odd = V[::2], V[1::2] P_even, P_odd = naive_IFFT(V_even), naive_IFFT(V_odd) P = [0] * n # the value representation for j in range(half_n): P[j] = P_even[j] + omega ** j * P_odd[j] P[j + half_n] = P_even[j] - omega ** j * P_odd[j] if is_outest_layer: for j in range(n): P[j] /= n return P Use $P_{1}$ in example 1, we would get If we ignore the little error, this is just the coefficient representation of $P_{1}$. The following materials might not be that clear and might not be easy to understand, but I will try my best. (Some material cannot be expanded too much, otherwise that would cost too much space and might confuse the main part.) The lowest layer of Diagram 1 is just the bit-reversal permutation index and there is a neat code to generate: def get_BRI(length): # Bit-Reversal Index n = 1 k = -1 while n < length: n <<= 1 k += 1 BRI = [0] * n for i in range(n): BRI[i] = (BRI[i >> 1] >> 1) \| ((i & 1) << k) return BRI It is more easy to see in a tabular (an example of $8$-length BRI) Use this we can implement the Cooley–Tukey FFT algorithm, which is the most common FFT algorithm. Further, with proper manner of coding, we can devise an in-place algorithm that overwrites its input with its output data using only $\mathcal{O}(1)$ auxiliary storage, which is called the iterative radix-2 FFT algorithm. Moreover, since the form of FFT and IFFT are actually very similar, we can integrate them together. def FFT(X, length, is_inverse=False): # X : input, either coefficient representation # or value representation # length : how much values need to evaluate # is_inverse : indicate whether is FFT or IFFT inverse_mul = [1, -1][is_inverse] BRI = get_BRI(length) n = len(BRI) X += [0] * (n - len(X)) for index in range(n): if index < BRI[index]: # only change once X[index], X[BRI[index]] = X[BRI[index]], X[index] bits = 1 while bits < n: omega_base = cos(pi/bits) + inverse_mul * sin(pi/bits) * 1j j = 0 while j < n: omega = 1 for k in range(bits): even_part = X[j + k] odd_part = X[j + k + bits] * omega X[j + k] = even_part + odd_part X[j + k + bits] = even_part - odd_part omega *= omega_base j += bits << 1 bits <<= 1 if is_inverse: for index in range(length): X[index] = X[index].real / n # only the real part is needed return X[:length] Note that we could ignore the return part, since $X$ is already changed. This algorithm would extend the input length to its closest larger bit number (of form $2^{k}$), but under most condition, we would take the length as $2^{k}$ before we use this algorithm (adding $0$‘s). Because we use the complex number to implement the FFT algorithm, we can see that the error is hard to eliminate. Even though the initial polynomial is integer based, apply FFT to it, then apply IFFT, we would get a decimal list with some calculation error. If we do the FFT in the field $\mathbb{Z} / P \mathbb{Z}$, where $P = Q \cdot 2^{k} + 1$, denote $g$ as a primitive root modulo $P$, then $\{ 1, g^{Q}, g^{2 Q}, \ldots \}$ is a cyclic group of order $2^{k}$, we can replace $\omega_{n}$ with $g$ to do the FFT. This method is called NTT, since only integers are involved, the errors are not possible to appear. For arbitrary modulo $m$, the aiming NTT length $n$, we can take a set of distinct NTT modulo $\{ p_{i} \}_{i = 1}^{r}$ satisfies do NTT respectively on all $p_{i}$, then use the Chinese remainder theorem to combine them together getting the final result modulo $m$. Note that during the NTT algorithm, the maximum intermediate value would not exceed $n (m - 1)^{2}$. We may say the FFT algorithm solves the convolution in the form of in time $\mathcal{O}(n \log n)$. Back in Example 1, we have (Not mandatory) Problem 1: Give the formula of Stirling numbers of the second kind: Use the NTT with some modulo of the form $P = Q \cdot 2^{k} + 1$ to calculate all $S(n, k)$ for $0 \leq k \leq n$ in time complexity $\mathcal{O}(n \log n)$. (Not mandatory) Problem 2: PE 537. Hint: If denote $A$ as the list of the value of $\pi(x)$, i.e., $A[n] = \sum_{x} [\pi(x) = n]$, then the convolution of $A$ and $A$, named $B$, is the list of the value of $\pi(x) + \pi(y)$, i.e., $B[n] = \sum_{x, y} [\pi(x) + \pi(y) = n]$, then the convolution of $B$ and $B$, named $C$, is the list of the value of $\pi(x) + \pi(y) + \pi(z) + \pi(w)$, i.e., $C[n] = \sum_{x, y, z, w} [\pi(x) + \pi(y) + \pi(z) + \pi(w) = n]$, etc. You can consider $A, B, C$ as the generating function. You might need to learn how to sieve prime numbers and use fast exponentiation. There are bunch of similar algorithms, for example, fast Walsh–Hadamard transform (FWHT) and fast wavelet transform (FWT). FWHT can solve the general convolution in time complexity $\mathcal{O}(n \log n)$, where $\star$ is some binary operation, usually $\star$ is bitwise OR, bitwise AND, and bitwise XOR. Using FFT, we can do a lot of things for polynomials fast, for instance, for a polynomial $P(x)$, we can find a polynomial $Q(x)$, such that $P(x) Q(x) \equiv 1 \mod{x^{n}}$, this $Q(x)$ is called the inverse of $P(x)$ under modulo $x^{n}$. The basic idea is to find the inverse polynomial under modulo $x^{n / 2}$, then $x^{n / 4}$, etc. Because the inverse polynomial under modulo $x^{1}$ is trivial, we can solve this recursively. Similar idea may be apply to the Newton’s method under modulo $x^{n}$, specifically, we can find the square root of a polynomial under modulo $x^{n}$. (Not mandatory) Problem 3: PE 258. Hint: Consider the Cayley–Hamilton theorem ($x^{2000} - x - 1 = 0$) and use the polynomial inverse to do polynomial quotient on $x^{10^{18}}$. Or consider solving the homogeneous linear recurrence with constant coefficients by the Berlekamp–Massey algorithm, which involves polynomial multiplication. Note that $20092010 = 8590 \times 2339$. FFT is also used to transform the time domain to the frequency domain in the signal area, while IFFT is used to transform reverse. In the artical [Machine Learning from a Continuous Viewpoint I] by Weinan E, the Fourier representation can be considered as a two-layer neural network model with activation function $\sigma$. In the above figure, $\sigma(\boldsymbol{\omega}, \boldsymbol{x})$ calculates each hidden layer value using the input layer $\boldsymbol{x}$ with weight $\boldsymbol{\omega}$, $\int_{\mathbb{R}^{d}} a(\boldsymbol{\omega}) \sigma(\boldsymbol{\omega}, \boldsymbol{x}) d \boldsymbol{\omega}$ sums all the hidden layer with weight $a(\boldsymbol{\omega})$. Note this is an integral formula, we work on a continuous condition, which means that the hidden layer has infinite width (the hidden layer is considered to have infinite nodes). 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# Sort an array of 0s, 1s and 2s If you want to practice data structure and algorithm programs, you can go through Java coding interview questions. In this post, we will see how to sort an array of 0s, 1s and 2s.We have already seen a post on sort 0s and 1s in an array. ## Problem Given an array containing zeroes, ones and twos only. Write a function to sort the given array in `O(n)` time complexity. Input : [1, 2, 2, 0, 0, 1, 2, 2, 1] Output : [0, 0, 1, 1, 1, 2, 2, 2, 2] ## Solution APPROACH – I : A very basic approach to solve this problem can be keeping the count of number of zeroes, ones and twos in the given array and then manipulate the given array in accordance with the frequency of every number. This approach is a bit inspired by counting sort. No matter what the initial value of that particular index is, we first put all the zeroes we have in the array starting from index zero, then put all the ones and after that put all the twos. Steps: 1.) Traverse the given array once and keep incrementing the count of the number encountered. 2.) Now Traverse the array again starting from index zero and keep changing the value of the element on current index first exhaust all the zeroes then ones and finally all the twos. This way we have a sorted array where all the zeroes are in starting followed by all the ones and then in last section we have all the twos in a time complexity of `O(n)`. • But the major drawback of this approach is, we have to traverse the given array twice once for counting the number of zeroes, ones and twos and second one for manipulating the array to make it sorted, which can be done only in a single pass. APPROACH – II : This algorithm is called as `Dutch national flag algorithm `or` Three way partitioning` in which elements of similar type are grouped together and their collective groups are also sorted in a the correct order. Now we have three types of elements to be sorted, therefore, we divide the given array in four sections out of which 3 sections are designated to `zeroes`, `Ones` and `twos` respectively and one section is `unknown` or the section which is left to be explored. Now for traversing in these sections we need 3 pointers as well which will virtually divide the given array in four segments. Let us name these pointers as low, mid and high. Now we can tell the starting and ending points of these segments. • Segment-1 : zeroes This will be a known section containing only `zeroes` with a range of `[0, low-1]`. • Segment-2: Ones This will also be a know section containing only ones with a range of `[low, mid-1]`. • Segment-3 : Unexplored This will be an unknown section as the elements in this sections are yet to be explored and hence it can contain all types of element that is, zeroes, ones and twos. Range of this segment will be `[mid, high]` • Segment-4 : Twos This will be the last and known area containing only twos having the range of `[high+1, N]` where N is the length of the given array or basically the last valid index of the given array. Steps used in this Algorithm to sort the given array in a single pass : (i) Initialize the low, mid and high pointers to, `low = 0`, `mid = 0`, `high = N` (ii) Now, run a loop and do the following until the `mid` pointer finally meets `high` pointer.As the `mid` pointer moves forward we keep putting the element at `mid` pointer to its right position by swapping that element with the element at pointers of respective sections. (iii) CASE – I : If the element at `mid`, that is, `A[mid] == 0`, this means the correct position of this element is in the range `[0, low-1]`, therefore, we swap `A[mid]` with `A[low] `and increment low making sure that element with index lesser than low is a Zero. (iv) CASE – II : If the element at `mid`, that is, `A[mid] == 2`, this means the correct position of this element is in the range `[hi+1, N]`, therefore, we swap `A[mid]` with `A[hi]` and decrement high making sure that element with index greater than high is a two. (v) CASE – III : If the element at mid, that is, `A[mid]=1`, this means that the element is already in its correct segment because `[low, mid-1]` is the range where it needs to be. Therefore, we do nothing and simply increment the mid pointer. So, there are total three cases, let us take a moment and emphasise on the fact that mid pointer gets only incremented only when the element `A[mid] == 1`. Let us discuss every case individually, For case – I : In this case we increment `mid` as well along with increment `low` pointer, as we are sure that element at low pointer before swapping can surely only be one as had it been a two, it would have already got swapped with `high` pointer when `mid` pointer explored it as the only reason that mid pointer left it because it was a one. For case – II : Now, In this case we swap the element at `mid` and `high`, but unlike case – I, in this case we are not sure about the element which will come at `mid` index after swapping as the element at `high` index before swapping can be any of zero, one or two, therefore, we need to explore this swapped element and hence we do not increment `mid` pointer in this case. For case – III : There is no confusion regarding incrementing `mid` in this case as already discussed, as we know the element at `mid` is one therefore we definitely need to increment mid here. Time complexity of this algorithm is also O(n) but it sorts the array in just a single pass and without any extra space unlike previous approach. That’s about sort an array of 0s, 1s and 2s. import_contacts import_contacts ## Related Posts • 18 June ### Maximum Number of Vowels in a Substring of Given Length Table of ContentsApproach – 1 Generate All Substrings Using substring() MethodApproach – 2 Using Sliding Window Method (Linear Time Solution) In this article, we will look at an interesting problem related to the Strings and [Sliding-Window Algorithm](https://java2blog.com/sliding-window-maximum-java/ “Sliding-Window Algorithm”). The problem is : "Given a String we have to Find the Maximum Number of Vowel […] • 04 June ### Search for a range Leetcode – Find first and last position of element in sorted array Table of ContentsApproach 1 (Using Linear Search)Approach 2 (Using Modified Binary Search-Optimal) In this article, we will look into an interesting problem asked in Coding Interviews related to Searching Algorithms. The problem is: Given a Sorted Array, we need to find the first and last position of an element in Sorted array. This problem is […] • 30 April ### Convert Postfix to Infix in Java Learn about how to convert Postfix to Infix in java. • 30 April ### Convert Prefix to Postfix in Java Learn about how to convert Prefix to Postfix in java. • 16 April	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://java2blog.com/sort-array-of-0s-1s-and-2s/?_page=4", "fetch_time": 1638054370000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2021-49/segments/1637964358323.91/warc/CC-MAIN-20211127223710-20211128013710-00336.warc.gz", "warc_record_offset": 408633944, "warc_record_length": 32872, "token_count": 1649, "char_count": 6778, "score": 4.125, "int_score": 4, "crawl": "CC-MAIN-2021-49", "snapshot_type": "latest", "language": "en", "language_score": 0.892507016658783, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00036-of-00064.parquet"}
# Proposition 16 If two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another. Let A and B be two numbers, and let A multiplied by B make C, and B multiplied by A make D. I say that C equals D. VII.Def.15 Since A multiplied by B makes C, therefore B measures C according to the units in A. But the unit E also measures the number A according to the units in it, therefore the unit E measures A the same number of times that B measures C. VII.15 Therefore, alternately, the unit E measures the number B the same number of times that A measures C. Again, since B multiplied by A makes D, therefore A measures D according to the units in B. But the unit E also measures B according to the units in it, therefore the unit E measures the number B the same number of times that A measures D. But the unit E measures the number B the same number of times that A measures C, therefore A measures each of the numbers C and D the same number of times. Therefore C equals D. Therefore, if two numbers multiplied by one another make certain numbers, then the numbers so produced equal one another. Q.E.D. ## Guide This proposition states the commutativity of multiplication of formal numbers, ab = ba. #### Outline of the proof Let a = nu and b = mu. Then by the definition of multiplication of numbers, ab = nb, and ba = ma. By the preceding proposition, n(mu) = m(nu). Therefore, ab = ba. #### Use of Proposition 16 This proposition is used in VII.18 and a few others in Book VII.	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://aleph0.clarku.edu/~djoyce/java/elements/bookVII/propVII16.html", "fetch_time": 1723101991000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-33/segments/1722640723918.41/warc/CC-MAIN-20240808062406-20240808092406-00573.warc.gz", "warc_record_offset": 1408020, "warc_record_length": 2025, "token_count": 368, "char_count": 1538, "score": 4.0625, "int_score": 4, "crawl": "CC-MAIN-2024-33", "snapshot_type": "latest", "language": "en", "language_score": 0.8676698803901672, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00032-of-00064.parquet"}
20分钟搞懂神经网络BP算法网络结构和样本数据 BP算法的目标就是优化神经网络的权重使得学习到的模型能够将输入值正确地映射到实际的输出值（也就是，希望模型能够模型真实数据产生的机制。在统计学中就是，我们要学习一个统计模型（统计分布函数），使得真实数据分布与统计模型产生的样本分布尽可能一致）。前向传播过程 $$\sigma(x) = \frac{1}{1+e^{-x}}$$ $$net_{h1} = 0.15 * 0.05 + 0.2 * 0.1 + 0.35 * 1 = 0.3775$$ $$out_{h1} = \frac{1}{1+e^{-net_{h1}}} = \frac{1}{1+e^{-0.3775}} = 0.593269992$$ $$out_{h2} = 0.596884378$$ $$net_{o1} = w_5 * out_{h1} + w_6 * out_{h2} + b_2 * 1$$ $$net_{o1} = 0.4 * 0.593269992 + 0.45 * 0.596884378 + 0.6 * 1 = 1.105905967$$ $$out_{o1} = \frac{1}{1+e^{-net_{o1}}} = \frac{1}{1+e^{-1.105905967}} = 0.75136507$$ $$out_{o2} = 0.772928465$$ 计算模型总误差 $$E_{total} = \sum \frac{1}{2}(target - output)^{2}$$ $$E_{o1} = \frac{1}{2}(target_{o1} - out_{o1})^{2} = \frac{1}{2}(0.01 - 0.75136507)^{2} = 0.274811083$$ $$E_{o2} = 0.023560026$$ $$E_{total} = E_{o1} + E_{o2} = 0.274811083 + 0.023560026 = 0.298371109$$ 后向传播过程输出层（output layer) $$\frac{\partial E_{total}}{\partial w_{5}} = \frac{\partial E_{total}}{\partial out_{o1}} * \frac{\partial out_{o1}}{\partial net_{o1}} * \frac{\partial net_{o1}}{\partial w_{5}}$$ $$E_{total} = \frac{1}{2}(target_{o1} - out_{o1})^{2} + \frac{1}{2}(target_{o2} - out_{o2})^{2}$$ $$\frac{\partial E_{total}}{\partial out_{o1}} = 2 * \frac{1}{2}(target_{o1} - out_{o1})^{2 - 1} * -1 + 0$$ $$\frac{\partial E_{total}}{\partial out_{o1}} = -(target_{o1} - out_{o1}) = -(0.01 - 0.75136507) = 0.74136507$$ $$out_{o1} = \frac{1}{1+e^{-net_{o1}}}$$ $$\frac{\partial out_{o1}}{\partial net_{o1}} = out_{o1}(1 - out_{o1}) = 0.75136507(1 - 0.75136507) = 0.186815602$$ logistic函数对自变量求导，可参考：https://en.wikipedia.org/wiki/Logistic_function#Derivative $$net_{o1} = w_5 * out_{h1} + w_6 * out_{h2} + b_2 * 1$$ $$\frac{\partial net_{o1}}{\partial w_{5}} = 1 * out_{h1} * w_5^{(1 - 1)} + 0 + 0 = out_{h1} = 0.593269992$$ $$\frac{\partial E_{total}}{\partial w_{5}} = \frac{\partial E_{total}}{\partial out_{o1}} * \frac{\partial out_{o1}}{\partial net_{o1}} * \frac{\partial net_{o1}}{\partial w_{5}}$$ $$\frac{\partial E_{total}}{\partial w_{5}} = 0.74136507 * 0.186815602 * 0.593269992 = 0.082167041$$ $$w_5^{+} = w_5 - \eta * \frac{\partial E_{total}}{\partial w_{5}} = 0.4 - 0.5 * 0.082167041 = 0.35891648$$ $w_6^{+} = 0.408666186$ $w_7^{+} = 0.511301270$ $w_8^{+} = 0.561370121$ 隐藏层（hidden layer） $$\frac{\partial E_{total}}{\partial w_{1}} = \frac{\partial E_{total}}{\partial out_{h1}} * \frac{\partial out_{h1}}{\partial net_{h1}} * \frac{\partial net_{h1}}{\partial w_{1}}$$ $$\frac{\partial E_{total}}{\partial out_{h1}} = \frac{\partial E_{o1}}{\partial out_{h1}} + \frac{\partial E_{o2}}{\partial out_{h1}}$$ $$\frac{\partial E_{o1}}{\partial out_{h1}} = \frac{\partial E_{o1}}{\partial net_{o1}} * \frac{\partial net_{o1}}{\partial out_{h1}}$$ $$\frac{\partial E_{o1}}{\partial net_{o1}} = \frac{\partial E_{o1}}{\partial out_{o1}} * \frac{\partial out_{o1}}{\partial net_{o1}} = 0.74136507 * 0.186815602 = 0.138498562$$ $$net_{o1} = w_5 * out_{h1} + w_6 * out_{h2} + b_2 * 1$$ $$\frac{\partial net_{o1}}{\partial out_{h1}} = w_5 = 0.40$$ $$\frac{\partial E_{o1}}{\partial out_{h1}} = \frac{\partial E_{o1}}{\partial net_{o1}} * \frac{\partial net_{o1}}{\partial out_{h1}} = 0.138498562 * 0.40 = 0.055399425$$ $$\frac{\partial E_{o2}}{\partial out_{h1}} = -0.019049119$$ $$\frac{\partial E_{total}}{\partial out_{h1}} = \frac{\partial E_{o1}}{\partial out_{h1}} + \frac{\partial E_{o2}}{\partial out_{h1}} = 0.055399425 + -0.019049119 = 0.036350306$$ $$out_{h1} = \frac{1}{1+e^{-net_{h1}}}$$ $$\frac{\partial out_{h1}}{\partial net_{h1}} = out_{h1}(1 - out_{h1}) = 0.59326999(1 - 0.59326999 ) = 0.241300709$$ $$net_{h1} = w_1 * i_1 + w_3 * i_2 + b_1 * 1$$ $$\frac{\partial net_{h1}}{\partial w_1} = i_1 = 0.05$$ $$\frac{\partial E_{total}}{\partial w_{1}} = \frac{\partial E_{total}}{\partial out_{h1}} * \frac{\partial out_{h1}}{\partial net_{h1}} * \frac{\partial net_{h1}}{\partial w_{1}}$$ $$\frac{\partial E_{total}}{\partial w_{1}} = 0.036350306 * 0.241300709 * 0.05 = 0.000438568$$ $w_1$的更新值为： $$w_1^{+} = w_1 - \eta * \frac{\partial E_{total}}{\partial w_{1}} = 0.15 - 0.5 * 0.000438568 = 0.149780716$$ $$w_2^{+} = 0.19956143$$ $$w_3^{+} = 0.24975114$$ $$w_4^{+} = 0.29950229$$ \| 5天前 \| 【MATLAB】BiGRU神经网络回归预测算法【MATLAB】BiGRU神经网络回归预测算法 45 0 \| 21小时前 \| 【MATLAB】CEEMDAN_ MFE_SVM_LSTM 神经网络时序预测算法【MATLAB】CEEMDAN_ MFE_SVM_LSTM 神经网络时序预测算法 10 4 \| 1天前 \| 【MATLAB】CEEMD_ MFE_SVM_LSTM 神经网络时序预测算法【MATLAB】CEEMD_ MFE_SVM_LSTM 神经网络时序预测算法 7 0 \| 1天前 \| 8 1 \| 2天前 \| 【MATLAB】EEMD_ MFE_SVM_LSTM 神经网络时序预测算法【MATLAB】EEMD_ MFE_SVM_LSTM 神经网络时序预测算法 10 1 \| 3天前 \| 【MATLAB】EMD_MFE_SVM_LSTM神经网络时序预测算法【MATLAB】EMD_MFE_SVM_LSTM神经网络时序预测算法 18 1 \| 4天前 \| 9 0 \| 3天前 \| m基于码率兼容打孔LDPC码nms最小和译码算法的LDPC编译码matlab误码率仿真 m基于码率兼容打孔LDPC码nms最小和译码算法的LDPC编译码matlab误码率仿真 9 0 \| 5天前 \| 11 0 \| 5天前 \| 【MATLAB 】 EEMD-ARIMA联合时序预测算法，科研创新优选算法【MATLAB 】 EEMD-ARIMA联合时序预测算法，科研创新优选算法 18 0 • 机器翻译 • 工业大脑更多更多更多	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://developer.aliyun.com/article/277312", "fetch_time": 1708810897000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-10/segments/1707947474569.64/warc/CC-MAIN-20240224212113-20240225002113-00767.warc.gz", "warc_record_offset": 208047311, "warc_record_length": 20854, "token_count": 2461, "char_count": 4912, "score": 3.5, "int_score": 4, "crawl": "CC-MAIN-2024-10", "snapshot_type": "latest", "language": "en", "language_score": 0.3502867817878723, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00043-of-00064.parquet"}
# Z-Test ## What is a Z-Test : A Z-Test is a statistical test used to determine if there is a significant difference between the mean of a sample and a known population mean. It is used to test the hypothesis that the sample mean is different from the population mean. Example 1: A company wants to determine if the average salary of their employees is different from the national average salary for their industry. They take a sample of 50 employees and calculate the mean salary of the sample to be \$50,000. The national average salary for the industry is \$48,000. The company wants to determine if the difference between the sample mean and the population mean is significant. To conduct a Z-Test, the company would first need to calculate the standard deviation of the sample. The standard deviation is a measure of how spread out the data is. If the standard deviation is small, it means that the data points are close to the mean, while a large standard deviation indicates that the data points are more spread out. Next, the company would need to determine the Z-score, which is the number of standard deviations that the sample mean is from the population mean. To calculate the Z-score, the company would subtract the population mean from the sample mean and divide the result by the standard deviation of the sample. In this example, the Z-score would be calculated as follows: Z-score = (50,000 – 48,000) / (standard deviation of the sample) The company would then use a Z-table to determine the probability of getting a result this extreme if the sample mean and the population mean are the same. If the probability is low, it indicates that the difference between the sample mean and the population mean is statistically significant. Example 2: A high school teacher wants to determine if the average test scores of her students are significantly different from the average test scores of students in the district. She takes a sample of 20 students from her class and calculates the mean test score to be 75. The district average test score is 80. The teacher wants to determine if the difference between the sample mean and the population mean is significant. To conduct a Z-Test, the teacher would first need to calculate the standard deviation of the sample. She would then calculate the Z-score as follows: Z-score = (75 – 80) / (standard deviation of the sample) The teacher would then use a Z-table to determine the probability of getting a result this extreme if the sample mean and the population mean are the same. If the probability is low, it indicates that the difference between the sample mean and the population mean is statistically significant. In conclusion, a Z-Test is a statistical test used to determine if there is a significant difference between the mean of a sample and a known population mean. It is used to test the hypothesis that the sample mean is different from the population mean. To conduct a Z-Test, the standard deviation of the sample and the Z-score must be calculated, and the probability of getting a result this extreme if the sample mean and the population mean are the same must be determined using a Z-table.	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://datasciencewiki.net/z-test/", "fetch_time": 1717076203000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-22/segments/1715971667627.93/warc/CC-MAIN-20240530114606-20240530144606-00428.warc.gz", "warc_record_offset": 156441620, "warc_record_length": 10066, "token_count": 640, "char_count": 3170, "score": 4.4375, "int_score": 4, "crawl": "CC-MAIN-2024-22", "snapshot_type": "latest", "language": "en", "language_score": 0.9508697986602783, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00030-of-00064.parquet"}
math. A theater purchases \$500 worth of Sticky Bears and chocolate bombs. Each bag of Sticky Bears costs \$1.50 and each bag of Chocolate Bombs costs \$1.00. If a total of 400 bags of candy were purchased, how many bags of Chocolate Bombs did the theater buy? 1. 👍 0 2. 👎 0 3. 👁 156 1. Let S=number of bags of sticky bears, then 400-S=number of bags of chocolate bombs. From total cost = \$500, we get 1.5S+1.0(400-S)=500 Solve for S 0.5S = 500-400 S=200 Quick way: each bag of sticky bears costs 1.50 each bag of chocolate costs 1.00 Average cost = \$500/400=1.25 which is smack in between 1.00 and 1.50. So equal number of bags of each, namely 200. 1. 👍 0 2. 👎 0 posted by MathMate Similar Questions 1. theory of evolution If you could help me with this I would appreciate it In Europe, three lineages of bears evolved from a common ancestor. Two of those lineages led to present-day black bears and brown bears. The third lineage led to cave bears that asked by anonymous on May 18, 2015 A shop sells candies by weight. Monisha bought 2 full bags of chocolates and 3 full bags of gummy bears for \$23.50. If 3/4 of a bag of chocolate cost as much as 5/6 of a bag of gummy bears, find the cost of 1 full bag of asked by Ashley on March 16, 2013 3. science 1. which of the following is not part of theory of evolution. a. organisms end to produce more offspring than can survive to reproductive age. b. organisms can acquire changes during their lifetime.*** c. organisms that do not asked by kayla on October 5, 2015 4. Math(Algebra) Estimate the size of the bear population 50 bears, 1 year later 100 bears, 2 with taggs what is the estimated size of the bears can you Please wirite me the stepes so I cam understand what you did. Thank You so much foer your asked by Anonymous on September 26, 2014 5. English I'm sorry but after asking for help and looking up several websites on how to write a really good thesis statement, I don't think I can make one. So I am asking can you please help me by writing it. I tried but I came up with asked by Anonymous on September 28, 2014 6. math april sells specialty teddy bears at various summer festivals. her profit for a week,P, in dollars, can be modelled by P= -0.1n^2 + 30n - 1200, where n is the umber of teddy bearsshe sells during the week. a.) According to this asked by geekgirl95 on June 20, 2011 7. English Yeot is a traditional Korean sweet like taffy. It is made from sweet potatoes and grains. Because it is sticky, Koreans like to give it as a present to students. They hope students to pass their exams. As yeot is sticky, when you asked by John on May 4, 2009 8. Algebra A biologist studied the populations of black bear and brown bears over a 10-year period. The biologist modeled the populations, in thousands, with the following polynomials where x is time, in years. black bears: 2.3x^2 - 5.6x + asked by Anonymous on April 13, 2017 9. Algebra Mai has 3 times as many teddy bears as tony. Altogether they have 24 teddy bears. If let tony has x teddy bears, make an equation to find the number of bears that Mai has? asked by Alana on November 13, 2015 10. Math Wildlife biologists catch, tag, and release 32 bears at a game reserve. Later,10 bears are caught and 4 of them have tags. Estimate the total number of bears at the game reserve. A.320 B.316 C.128 D.80 Is the answer D? asked by Beau on April 29, 2014 More Similar Questions	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.jiskha.com/questions/574937/a-theater-purchases-500-worth-of-sticky-bears-and-chocolate-bombs-each-bag-of-sticky", "fetch_time": 1582252200000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2020-10/segments/1581875145438.12/warc/CC-MAIN-20200221014826-20200221044826-00388.warc.gz", "warc_record_offset": 780199821, "warc_record_length": 5788, "token_count": 975, "char_count": 3431, "score": 3.859375, "int_score": 4, "crawl": "CC-MAIN-2020-10", "snapshot_type": "latest", "language": "en", "language_score": 0.96526038646698, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00044-of-00064.parquet"}
Education.com # Simplifying Expressions and Solving Equation Word Problems Study Guide based on 1 rating ## Introduction to Simplifying Expressions and Solving Equation Word Problems Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting. —ERNST MACH (1838–1916) This lesson reviews the key words and phrases for the basic operations and provides examples and tips on simplifying algebraic expressions and the equation solving steps. Equation word problems are modeled with explanations to help your understanding in this type of question. ### Key Words and Phrases Translating expressions from words into mathematical symbols was covered in Lesson 1. The following chart below summarizes the key words and phrases studied in that lesson for the four basic operations and the equal sign. Refer to this chart when you are changing sentences in words to math equations. #### Tip: In algebra, the number in front of the letter is called the coefficient and the letter is called the variable. In the expression 8x, 8 is the coefficient and x is the variable. ## Simplifying Expressions - Combining Like Terms and The Distributive Property Two important processes to know when you are simplifying expressions are combining like terms and the distributive property. ### Combining Like Terms Terms, in mathematics, are numbers and symbols that are separated by addition and subtraction. The expressions 3, 5x, and 7xy are each one term. The expressions 2x + 3, and x – 7 each have two terms. The expression 7x + 5y – 9 has three terms. Like terms are terms with the same variable and exponent. Like terms can be combined by addition and subtraction. To do this, add or subtract the coefficients and keep the variable the same. For example 3x + 5x = 8x, and 6y2 – 4y2 = 2y2. #### Tip: Be sure to combine only like terms. Terms without the exact same variable and exponent cannot be combined: 5x2 and 6x cannot be combined because the exponents are not the same. ### Distributive Property The distributive property is used when a value needs to be multiplied, or distributed, to more than one term. For example, in the expression 3(x + 10), the number 3 needs to be multiplied by the term x and the term 10. The use of arrows can help in this process, as shown in the following figure. The result becomes 3 × x + 3 × 10, which simplifies to 3x + 30. ### Solving Equations When you are solving equations, the goal is to get the letter, or variable, by itself. This is called isolating the variable. Each of the following examples goes through the process of isolating the variable for different types of equations. #### Tip: One of the most important rules in equation solving is to do the same thing on both sides of the equation. For example, if you divide on one side to get the variable alone, divide the other side by the same number. This keeps the equation balanced and will lead to the correct solution. ### Ask a Question 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? Welcome!	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://www.education.com/study-help/article/simplifying-expressions-solving-equation-word/", "fetch_time": 1371635631000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2013-20/segments/1368708664942/warc/CC-MAIN-20130516125104-00070-ip-10-60-113-184.ec2.internal.warc.gz", "warc_record_offset": 417224415, "warc_record_length": 28886, "token_count": 736, "char_count": 3308, "score": 4.96875, "int_score": 5, "crawl": "CC-MAIN-2013-20", "snapshot_type": "longest", "language": "en", "language_score": 0.9309056997299194, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00040-of-00064.parquet"}
## Tuesday, November 29, 2011 ### MGRE's solution to last week's Math Beast Challenge Remember last week's MGRE Math Beast Challenge problem? The answer is indeed B, so I would have gotten it right, but according to the answer write-up, the amount of \$173.40 is correct-- it's not \$173.13. Personally, I'm not so sure, even after having read MGRE's explanation. But you decide: here's what they had to say. RECAP OF THE PROBLEM An online bank verifies customers’ ownership of external bank accounts by making both a small deposit and a small debit from each customer’s external account, and asking the customer to verify the amounts. In 70% of these exchanges, the deposit and debit are within two cents of one another (for example, a deposit of \$0.18 and a debit of \$0.16, or a deposit of \$0.37 and a debit of \$0.38), and the deposit and debit are always within five cents of one another. During one week, the online bank attempts to verify 6,000 accounts in this manner, but 0.5% of the transactions do not go through, and thus no money is transferred. What is the maximum amount, in dollars, that the account verification system could have cost the bank that week? (A) \$165.30 (B) \$173.40 (C) \$174 (D) \$256.71 (E) \$258 EXPLANATION This is just a very lengthy problem that requires careful reading and note taking. Of 6,000 accounts, 70% have deposits and debits 2 cents apart, and the other 30% have deposits “within 5 cents” (but not within 2 cents), and thus are 3-5 cents apart. So: 4,200 are 1-2 cents apart 1,800 are 3-5 cents apart 0.5% (that’s one-half of one percent) of 6,000 attempts do not go through, so: 30 do not go through 5,970 do go through We are not told how many of the 30 failed attempts were in the 1-2 cents apart category and how many were in the 3-5 cents apart category. However, we are trying to MAXIMIZE the bank’s cost, so we’re going to finish this problem by presuming the worst possible scenarios for the bank. To maximize the loss, presume that: • All the (2-5 cent) differences are in the customer’s favor • All the costs are as large as possible (so the 1-2 cent ones are all 2 cents, and the 3-5 cent ones are all 5 cents) • The 30 accounts that did not go through were the two-cent ones (that way we can maximize the 5-cent losses) Thus, we WOULD have had: 4,200 2 cent losses 1,800 5 cent losses ...except for the 30 exchanges that didn’t go through. Again, to maximize the bank’s loss, let’s assume that the 30 that didn’t go through were 2-cent losses. Therefore: 4,170 2 cent losses = \$83.40 1,800 5 cent losses = \$90 \$83.40 + \$90 = \$173.40 The correct answer is B. While the explanation sounds plausible, I don't agree with the idea that the failed transactions should be counted as losses. I don't see this implied anywhere in the problem, which to my mind makes MGRE's assumption unwarranted. As a practical matter, though, that's just a quibble because my own reckoning puts me in the proper ballpark. I agree with the MGRE gurus that (B) is the best answer of the bunch. _ 1. "While the explanation sounds plausible, I don't agree with the idea that the failed transactions should be counted as losses." I'm not sure what you mean by this. The failed transactions are not, in fact, counted as losses. Maybe I'm misunderstanding this sentence? Where we differed from MGRE was in our assumptions about the distribution of the failed transactions. There were 30 total failed transactions. If the distribution of these transactions were even, then 21 (70%) would be 2-cent losses and 9 (30%) would be 5-cent losses: (4200 - 21) * 0.02 = 83.58 (1800 - 9) * 0.05 = 89.55 Add these two figures together and you get our original result, 173.13. What MGRE is saying, though, is that if we are looking for the most money the bank could have lost, we should assume that the failed transactions are all of the less expensive variety, as opposed to being equally distributed. This makes sense to me. I just failed to take that little fact into account. 2. Well, the MGRE explanation says "let’s assume that the 30 that didn’t go through were 2-cent losses." I assumed, in my calculations, that the failed transactions were neither profits nor losses: transactions had been attempted in those 30 cases, but no transactions had actually occurred. How, then, assume that those transactions represent a loss? They're neither a loss nor a gain. Anyway, once I had dismissed those 30 transactions from my calculations, the rest followed. 3. Hmm. I see where you're coming from. I guess the confusion boils down the the phrase "go through," which is frankly rather vague. What exactly does this mean? The way I saw it (after reading the explanation), these 30 failed transactions would have been 2-cent transactions had they gone through. What exactly that means, I don't know. All in all, I think this is a rather retarded problem that derives its difficulty from its obfuscatory nature. READ THIS BEFORE COMMENTING! All comments are subject to approval before they are published, so they will not appear immediately. Comments should be civil, relevant, and substantive. Anonymous comments are not allowed and will be unceremoniously deleted. For more on my comments policy, please see this entry on my other blog. AND A NEW RULE (per this post): comments critical of Trump's lying must include criticism of Biden's lying on a one-for-one basis! Failure to be balanced means your comment will not be published.	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://bighominid.blogspot.com/2011/11/mgres-solution-to-last-weeks-math-beast.html", "fetch_time": 1723553577000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-33/segments/1722641076695.81/warc/CC-MAIN-20240813110333-20240813140333-00493.warc.gz", "warc_record_offset": 94734562, "warc_record_length": 19278, "token_count": 1344, "char_count": 5493, "score": 3.984375, "int_score": 4, "crawl": "CC-MAIN-2024-33", "snapshot_type": "latest", "language": "en", "language_score": 0.9506396055221558, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00032-of-00064.parquet"}
How many soap cakes can be placed in a box of size 56 cm × 0.4 m × 0.25 m, Question: How many soap cakes can be placed in a box of size 56 cm × 0.4 m × 0.25 m, if the size of a soap cake is 7 cm × 5 cm × 2.5 cm? Solution: Dimension of a soap cake $=7 \mathrm{~cm} \times 5 \mathrm{~cm} \times 2.5 \mathrm{~cm}$ Its volum $e=$ length $\times$ breadth $\times$ height $=(7 \times 5 \times 2.5) \mathrm{cm}^{3}=87.5 \mathrm{~cm}^{3}$ Also, the dimension of the box that contains the soap cakes is $56 \mathrm{~cm} \times 0.4 \mathrm{~m} \times 0.25 \mathrm{~m}$, i.e., $56 \mathrm{~cm} \times 40 \mathrm{~cm} \times 25 \mathrm{~cm}$$(\because 1 \mathrm{~m}=100 \mathrm{~cm})$. Volume of the box $=$ length $\times$ breadth $\times$ height $=(56 \times 40 \times 25) \mathrm{cm}^{3}=56000 \mathrm{~cm}^{3}$ $\therefore$ The number of soap cakes that can be placed inside the box $=\frac{\text { volume of the box }}{\text { volume of a soap cake }}=\frac{56000 \mathrm{~cm}^{3}}{87.5 \mathrm{~cm}^{3}}=640$	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.esaral.com/q/how-many-soap-cakes-can-be-placed-in-a-box-of-size-56-cm-0-4-m-0-25-m-10007", "fetch_time": 1725905550000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-38/segments/1725700651133.92/warc/CC-MAIN-20240909170505-20240909200505-00657.warc.gz", "warc_record_offset": 718639369, "warc_record_length": 11596, "token_count": 379, "char_count": 1010, "score": 4.375, "int_score": 4, "crawl": "CC-MAIN-2024-38", "snapshot_type": "latest", "language": "en", "language_score": 0.5815998911857605, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00058-of-00064.parquet"}
How to: Make Investment Decisions using Mixed Integer Linear Programming # How to: Make Investment Decisions using Mixed Integer Linear Programming Solver Foundation 3.0 You can make investment decisions by modeling the problem as a mixed integer linear program. In this example, given an initial capital expenditure of \$20 million and five different projects, you must decide which projects to invest in to maximize the total profit. This example is based on problem 12.1.3 in Hillier and Lieberman’s book Introduction to Operations Research. The following table shows the estimated profits and capital requirements for each project. Project Estimated profits Capital requirements 0 \$1 million \$6 million 1 \$1.8 million \$12 million 2 \$1.6 million \$10 million 3 \$0.8 million \$4 million 4 \$1.4 million \$8 million The following steps show how to use Solver Foundation to create and solve the investment decision model using the simplex solver. The total profit is represented as a row to be maximized, and the total capital expenditure cannot be greater than the initial capital expenditure. ### To make investment decisions by using a mixed integer linear program 1. Create a console application named InvestmentDecisions. 2. Add a reference to Microsoft Solver Foundation on the .NET tab of the Add Reference dialog box. 3. Add the following Imports or using statements to the top of the Program code file. ``` using Microsoft.SolverFoundation.Common; using Microsoft.SolverFoundation.Solvers; ``` 4. In the Main method, add a solver by typing the following code. ``` SimplexSolver solver = new SimplexSolver(); ``` 5. Create variables to store the data about the estimated profits, capital expenditures for each project, the initial capital expenditure, and the decision whether to invest in a project. ``` double[] estimatedProfitOfProjectX = new double[] { 1, 1.8, 1.6, 0.8, 1.4 }; double[] capitalRequiredForProjectX = new double[] { 6, 12, 10, 4, 8 }; double availableCapital = 20; int[] chooseProjectX = new int[5]; ``` 6. Create decision variables for the profit and capital expenditure, and then add row identifiers for both of these variables. ``` int profit; int expenditure; solver.SetBounds(expenditure, 0, availableCapital); ``` 7. Add the project names to the solver, and then add coefficients to the constraint rows by using the SetCoefficient method. Set the bounds and choices for the investment decision by using the SetBounds and SetIntegrality methods. ``` for (int i = 0; i < 5; i++) { out chooseProjectX[i]); solver.SetBounds(chooseProjectX[i], 0, 1); solver.SetIntegrality(chooseProjectX[i], true); solver.SetCoefficient(profit, chooseProjectX[i], estimatedProfitOfProjectX[i]); solver.SetCoefficient(expenditure, chooseProjectX[i], capitalRequiredForProjectX[i]); } ``` 8. Configure the solver parameters to generate cuts, and then solve the model. ``` SimplexSolverParams param = new SimplexSolverParams(); param.MixedIntegerGenerateCuts = true; solver.Solve(param); ``` 9. Show whether the solve process is optimal, and print the results of the solve process. ``` Console.WriteLine(solver.MipResult); for (int i = 0; i < 5; i++) { Console.WriteLine("Project {0} is {1} selected.", i, solver.GetValue(chooseProjectX[i]) == 1 ? "" : "not "); } Console.WriteLine("The estimated total profit is: \${0} million.", (double)solver.GetValue(profit).ToDouble()); Console.WriteLine("The total expenditure is: \${0} million.", solver.GetValue(expenditure).ToDouble()); ``` 10. Press F5 to build and run the code. The command window shows the following results. Optimal Project 0 is selected. Project 1 is not selected. Project 2 is selected. Project 3 is selected. Project 4 is not selected. The estimated total profit is: \$3.4. The total expenditure is: \$20.	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://msdn.microsoft.com/en-us/library/ff852840(v=vs.93).aspx", "fetch_time": 1448924610000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2015-48/segments/1448398464253.80/warc/CC-MAIN-20151124205424-00039-ip-10-71-132-137.ec2.internal.warc.gz", "warc_record_offset": 833663096, "warc_record_length": 13396, "token_count": 887, "char_count": 3821, "score": 3.515625, "int_score": 4, "crawl": "CC-MAIN-2015-48", "snapshot_type": "longest", "language": "en", "language_score": 0.8236554861068726, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00012-of-00064.parquet"}
# Physics Homework Physics Homework I need all answers worked out showing how i got the answer 1- Two forces, one equal to 15 N and another equal to 40 N, act on a 50-kg crate resting on a horizontal surface as shown in the accompa- nying figure. (a) What is the net horizontal force on the crate? (b) What is its horizontal acceleration? (c) If the crate starts from rest, what is its horizontal speed after 5 s? (d) How far has the crate traveled along the surface in this time? 2- As a horse and wagon are accelerating from rest, the horse exerts a force of 400 N on the wagon ( ● Figure 2.53). Illustrating Newton’s third law, the wagon exerts an equal and opposite force of 400 N. Because the two forces are in opposite directions, why don’t they cancel each other and produce zero acceleration (i.e., no motion)? 3- Perhaps you’ve noticed that the rockets used to put satellites and spacecraft into orbit are usually launched from pads near the equa- tor. Why is this so? Is the fact that rockets are usually launched to the east also important? Why? 4-A 200-kg communications satellite is placed into a circular orbit around Earth with a radius of 4.23  107 m (26,300 miles) (see ● Figure 2.54). (a) Find the gravitational force on the satellite. (There is some useful information in Section 2.8.) (b) Use the equation for centripetal force to compute the speed of the satellite. (c) Show that the period of the satellite—the time it takes to com- plete one orbit—is 1 day. (The distance it travels during one orbit is 2 p, or 6.28, times the radius.) This is a geosynchronous orbit: the satellite stays above a fixed point on Earth’s equator	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://topnursingassignments.com/physics-homework/", "fetch_time": 1717106082000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2024-22/segments/1715971684053.99/warc/CC-MAIN-20240530210614-20240531000614-00050.warc.gz", "warc_record_offset": 484838361, "warc_record_length": 11365, "token_count": 414, "char_count": 1680, "score": 3.515625, "int_score": 4, "crawl": "CC-MAIN-2024-22", "snapshot_type": "latest", "language": "en", "language_score": 0.910459041595459, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00056-of-00064.parquet"}
# Cauchy's Convergence Criterion/Real Numbers/Necessary Condition/Proof 1 Jump to navigation Jump to search ## Theorem Let $\sequence {x_n}$ be a sequence in $\R$. Let $\sequence {x_n}$ be convergent. Then $\sequence {x_n}$ is a Cauchy sequence. ## Proof Let $\sequence {x_n}$ be convergent. Let $\struct {\R, d}$ be the metric space formed from $\R$ and the usual (Euclidean) metric: $\map d {x_1, x_2} = \size {x_1 - x_2}$ where $\size x$ is the absolute value of $x$. This is proven to be a metric space in Real Number Line is Metric Space. From Convergent Sequence in Metric Space is Cauchy Sequence, we have that every convergent sequence in a metric space is a Cauchy sequence. Hence $\sequence {x_n}$ is a Cauchy sequence. $\blacksquare$ ## Also known as Cauchy's Convergence Criterion is also known as the Cauchy convergence condition.	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://proofwiki.org/wiki/Cauchy%27s_Convergence_Criterion/Real_Numbers/Necessary_Condition/Proof_1", "fetch_time": 1686423030000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2023-23/segments/1685224657735.85/warc/CC-MAIN-20230610164417-20230610194417-00151.warc.gz", "warc_record_offset": 544786078, "warc_record_length": 12233, "token_count": 241, "char_count": 859, "score": 3.5625, "int_score": 4, "crawl": "CC-MAIN-2023-23", "snapshot_type": "latest", "language": "en", "language_score": 0.8624988794326782, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00005-of-00064.parquet"}

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This dataset is associated with the paper Reasoning to Learn from Latent Thoughts. It contains data used for pretraining language models with a focus on improving data efficiency by modeling and inferring latent thoughts underlying the text generation process, such as on reasoning-intensive math corpus. An expectation-maximization algorithm is developed for models to self-improve their self-generated thoughts and data efficiency.

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