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[ "# How do we set the valuation for a seed round?\n\nventurehacks.com\n\nA reader asks: “My question is how do we value a company with no sales? I understand it’s an arbitrary valuation but is there anything we can possibly base it on? Is there a “default” valuation for companies in a seed round?”\n\nWe’ll answer this question with some questions (and answers) of our own:\n\n• How much money do we need?\n• How do we set a valuation from this budget?\n• How do we express our valuation to investors?\n• What’s the range for seed round valuations?\n• How low do seed round valuations go?\n• How much money can we raise in a seed round?\n• How much dilution should we expect in a seed round?" ]

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[ "04 Basic Maths Quiz - Multiple choice questions and Objectives\n\n# 04 Basic Maths Quiz\n\n1 >>Q61. What is a straight line that joins any two points in a curve known as? ?\n• (A) A. Card\n• (B) B. Cordial\n• (C) C. Chord\n• (D) D. Core\n2 >>Q62. What is a set of numbers that describes the position of a point called as? ?\n• (A) A. Disordinates\n• (B) B. Co-ordinates\n• (C) C. Pro-ordinates\n• (D) D. In-ordinates\n3 >>Q63. What is an assumption about the value of a parameter of the distribution known as? ?\n• (A) A. Hypothesis\n• (B) B. Parenthesis\n• (C) C. Larynthesis\n• (D) D. Nanothesis\n4 >>Q64. Most of the gambling is related to which branch of mathematics? ?\n• (A) A. Statistics\n• (B) B. Probability\n• (C) C. Algebra\n• (D) D. Trigonometry\n5 >>Q65. What is a figure with three or more straight sides that do not intersect other than the vertices called as? ?\n• (A) A. Monogon\n• (B) B. Polygon\n• (C) C. Paragon\n• (D) D. Hypogon\n\n6 >>Q66. A list of terms that are placed in chronological order is known as- ?\n• (A) A. Serial\n• (B) B. Sequence\n• (C) C. Series\n• (D) D. Secant\n7 >>Q67.The branch of mathematics that deals with the differentiation & integration of functions is known as- ?\n• (A) A. Calibration\n• (B) B. Calculus\n• (C) C. Integration\n• (D) D. Combinatorics\n8 >>Q68. A pair of angles that join together to make a right angle 90° is called ?\n• (A) A. Scalene angle\n• (B) B. Complimentary angle\n• (C) C. Obtuse angle\n• (D) D. Equilateral angle\n9 >>Q69.The branch of mathematics dealing with properties, measurements & relationships of points, lines, planes & solids is known as- ?\n• (A) A. Graphology\n• (B) B. Geometry\n• (C) C. Optometry\n• (D) D. Symmetry\n10 >>Q70. What is the ratio of chances in favour of an event to the total number called? ?\n• (A) A. Prosperity\n• (B) B. Probability\n• (C) C. Perpetuity\n• (D) D. Popularity\n11 >>Q71. What is an arrangement of objects, numbers, etc. in columns or rows known as? ?\n• (A) A. Axis\n• (B) B. Array\n• (C) C. Arrow\n• (D) D. Align\n\n12 >>Q72. A small star(*) used to mark a space where something missing is known as- ?\n• (A) A. Asterix\n• (B) B. Asterisk\n• (C) C. Astir\n• (D) D. Aster\n13 >>Q73. A straight-line that divides an angle or an interval into two equal parts is called - ?\n• (A) A. Binomial\n• (B) B. Bi-sector\n• (C) C. Binary\n• (D) D. Bi-centennial\n14 >>Q74. What is a measuring instrument with curved legs for measuring thickness of curved objects orturned outwardsfor measuring cavities? ?\n• (A) A. Compass\n• (B) B. Caliper\n• (C) C. Protractor\n• (D) D. Counter\n15 >>Q75. What is the measurement around a circle is known as? ?\n• (D) D. Circumference\n16 >>Q76. What are three or more points that lie on the same straight-line is called? ?\n• (A) A. Co interior\n• (B) B. Collinear\n• (C) C. Coordinates\n• (D) D. Coplanar\n17 >>Q77. What is a number with factors other than itself & one? ?\n• (A) A. Whole number\n• (B) B. Composite number\n• (C) C. Natural number\n• (D) D. Complex number\n18 >>Q78. What is a box with 12 edges, six faces & eight corners with opposite sides of same shape & size called? ?\n• (A) A. Spheroid\n• (B) B. Cuboid\n• (C) C. Rhomboid\n• (D) D. Octoid\n19 >>Q79. Which among these is also equivalent to one degree in angles? ?\n• (A) A. 60 seconds\n• (B) B. 60 minutes\n• (C) C. 60 hours\n• (D) D. None of these\n\n20 >>Q80. What is the mathematical name of diamond shape,  a two dimensional with four equal sides & oblique angles? ?\n• (A) A. Rhomboid\n• (B) B. Rectangle\n• (C) C. Rhombus\n• (D) D. Rhombohedron\n\nTop" ]

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[ "### The Effect of Inbound Mass on the Dynamic Response of the Hybrid III Headform and Brain Tissue Deformation\n\n##### Description\n Title: The Effect of Inbound Mass on the Dynamic Response of the Hybrid III Headform and Brain Tissue Deformation Authors: Karton, Clara Date: 2012 Abstract: The varied impact parameters that characterize an impact to the head have shown to influence the resulting type and severity of outcome injury, both in terms of the dynamic response, and the corresponding deformation of neural tissue. Therefore, when determining head injury risks through event reconstruction, it is important to understand how individual impact characteristics influence these responses. The effect of inbound mass had not yet been documented in the literature. The purpose of this study was to determine the effects of inbound mass on the dynamic impact response and brain tissue deformation. A 50th percentile Hybrid III adult male head form was impacted using a simple pendulum system. Impacts to a centric and a non-centric impact location were performed with six varied inbound masses at a velocity of 4.0 m/s. The peak linear and peak angular accelerations were measured. A finite element model, (UCDBTM) was used to determine brain deformation, namely peak maximum principal strain and peak von Mises stress. Inbound mass produced significant differences for peak linear acceleration for centric (F(5, 24) = 217.55, p=.0005) and non-centric (F(5, 24) = 161.98, p=.0005), and for peak angular acceleration for centric (F(5, 24) = 52.51, p=.0005) and non-centric (F(5, 24) = 4.18, p=.007) impact locations. A change in inbound mass also had a significant effect on peak maximum principal strain for centric (F(5, 24) = 11.04, p=.0005) and non-centric (F(5, 24) = 5.87, p =.001), and for peak von Mises stress for centric (F(5, 24) = 24.01, p=.0005) and non-centric (F(5, 24) = 4.62, p=.004) impact locations. These results indicate the inbound mass of an impact should be of consideration when determining risks and prevention to head and brain injury. URL: http://hdl.handle.net/10393/23576http://dx.doi.org/10.20381/ruor-6253 Collection Thèses, 2011 - // Theses, 2011 -\n##### Files\n Karton_Clara_2012_thesis.pdf Thesis 990.76 kB Adobe PDF" ]

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[ "# 12.2: t Distribution Demo\n\n•", null, "• Contributed by David Lane\n• Associate Professor (Psychology, Statistics, and Management) at Rice University\n\nLearning Objectives\n\n• State the difference between the shape of the $$t$$ distribution and the shape of the normal distribution\n• State how the degrees of freedom affect the difference between the $$t$$ and normal distributions\n\n## Instructions\n\nThis demonstration allows you to compare the $$t$$ distribution to the standard normal distribution. At the start, the standard normal distribution is compared to a $$t$$ distribution with three degrees of freedom. You can change the degrees of freedom of the $$t$$ distribution with the slider. The $$3$$ and the $$50$$ mark the ends of the slider. The \"current\" degrees of freedom are shown at the bottom where it says \"$$t$$ distribution with $$df=3$$.\" As you change the slider, the df are shown by this last line. The \"zoom in\" button allows you to change the scale of the graph to see the tails of the distribution in more detail.\n\n## Illustrated Instructions\n\nVideo Demo\n\nIn the video the slider is used to increase the degrees of freedom. Notice how the $$t$$-distribution change as the degrees of freedom increase. The video concludes with the tails of the tails of the graphs being zoomed into via the \"zoom in\" button and the degrees of freedom being decreased.\n\n## Contributor\n\n• Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University." ]

[ null, "https://stats.libretexts.org/@api/deki/files/1247/Lane.jpg", null ]

[ "# Python for 迴圈\n\nPython `for` 迴圈可以用來遍歷序列或者其他可遍歷的資料物件。\n\n## Python `for` 迴圈\n\n``````for val in sequence:\nblock of statements\n``````\n\n### `for` 迴圈例項\n\n``````x = {1, 2, 3, 4, 5, 6, 7, 8, 9}\nsum = 0\nfor i in x:\nsum = sum + i\nprint(\"Sum of elements of x =\", sum)\n``````\n``````Sum of elements of x = 45\n``````\n\n## `range()` 函式\n\n``````range(start, stop, step size)\n``````\n\n`start``stop` 是開始和結束的數字，`step size` 是每個元素之間的步進差值。`range()` 函式是定義了一個範圍而不是具體單一的元素，假如你想要得到序列當中的每個元素的話，你需要用 `list()` 把它做一下轉換。\n\n``````>>> print(range(10))\nrange(0, 10)\n>>> print(list(range(10)))\n[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n``````\n\n### `range()` 和 `for` 迴圈\n\n``````l = ['Python', 'Java', 'C', 'Kotlin']\nfor i in range(len(l)):\nprint(\"Programming Language is:\", l[i])\n``````\n``````Programming Language is: Python\nProgramming Language is: Java\nProgramming Language is: C\nProgramming Languages is: Kotlin\n``````\n\n## `for` 迴圈後接 `else`\n\n``````l = [1, 2, 3, 4, 5]\nfor i in l:\nprint(\"Items in list:\", i)\nelse:\nprint(\"List is ended\")\n``````\n``````Items in list: 1\nItems in list: 2\nItems in list: 3\nItems in list: 4\nItems in list: 5\nList is ended\n``````\n\n## 相關文章 - Python Loop\n\n• Python while 迴圈\n• Python 迴圈 break 和 continue" ]

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[ "", null, "# Flow invariant subsets for geodesic flows of manifolds with non-positive curvature\n\nReinold, B (2004). Flow invariant subsets for geodesic flows of manifolds with non-positive curvature. Ergodic Theory and Dynamical Systems, 24(6):1981-1990.\n\n## Abstract\n\nConsider a closed, smooth manifold M of non-positive curvature. Write p:UM→M for the unit tangent bundle over M and let ℛ > denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow φ on UM. We define the structured dimension s-dim ℛ > which, essentially, is the dimension of the set p(ℛ > ) of base points of ℛ > .\nThe main result of this paper holds for manifolds with s-dim ℛ > <dim M/2: for every ε>0, there is an ε-dense, flow invariant, closed subset Ξ ε ⊂UM∖ℛ > such that p(Ξ ε )=M.\n\n## Abstract\n\nConsider a closed, smooth manifold M of non-positive curvature. Write p:UM→M for the unit tangent bundle over M and let ℛ > denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow φ on UM. We define the structured dimension s-dim ℛ > which, essentially, is the dimension of the set p(ℛ > ) of base points of ℛ > .\nThe main result of this paper holds for manifolds with s-dim ℛ > <dim M/2: for every ε>0, there is an ε-dense, flow invariant, closed subset Ξ ε ⊂UM∖ℛ > such that p(Ξ ε )=M.\n\n## Statistics\n\n### Citations\n\nDimensions.ai Metrics\n\n### Altmetrics\n\nDetailed statistics\n\n##", null, "", null, "", null, "" ]

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[ "# 188272\n\n## 188,272 is an even composite number composed of three prime numbers multiplied together.\n\nWhat does the number 188272 look like?\n\nThis visualization shows the relationship between its 3 prime factors (large circles) and 30 divisors.\n\n188272 is an even composite number. It is composed of three distinct prime numbers multiplied together. It has a total of thirty divisors.\n\n## Prime factorization of 188272:\n\n### 24 × 7 × 412\n\n(2 × 2 × 2 × 2 × 7 × 41 × 41)\n\nSee below for interesting mathematical facts about the number 188272 from the Numbermatics database.\n\n### Names of 188272\n\n• Cardinal: 188272 can be written as One hundred eighty-eight thousand, two hundred seventy-two.\n\n### Scientific notation\n\n• Scientific notation: 1.88272 × 105\n\n### Factors of 188272\n\n• Number of distinct prime factors ω(n): 3\n• Total number of prime factors Ω(n): 7\n• Sum of prime factors: 50\n\n### Divisors of 188272\n\n• Number of divisors d(n): 30\n• Complete list of divisors:\n• Sum of all divisors σ(n): 427304\n• Sum of proper divisors (its aliquot sum) s(n): 239032\n• 188272 is an abundant number, because the sum of its proper divisors (239032) is greater than itself. Its abundance is 50760\n\n### Bases of 188272\n\n• Binary: 1011011111011100002\n• Base-36: 419S\n\n### Squares and roots of 188272\n\n• 188272 squared (1882722) is 35446345984\n• 188272 cubed (1882723) is 6673554451099648\n• The square root of 188272 is 433.9032150145\n• The cube root of 188272 is 57.3141573993\n\n### Scales and comparisons\n\nHow big is 188272?\n• 188,272 seconds is equal to 2 days, 4 hours, 17 minutes, 52 seconds.\n• To count from 1 to 188,272 would take you about two days.\n\nThis is a very rough estimate, based on a speaking rate of half a second every third order of magnitude. If you speak quickly, you could probably say any randomly-chosen number between one and a thousand in around half a second. Very big numbers obviously take longer to say, so we add half a second for every extra x1000. (We do not count involuntary pauses, bathroom breaks or the necessity of sleep in our calculation!)\n\n• A cube with a volume of 188272 cubic inches would be around 4.8 feet tall.\n\n### Recreational maths with 188272\n\n• 188272 backwards is 272881\n• 188272 is a Harshad number.\n• The number of decimal digits it has is: 6\n• The sum of 188272's digits is 28\n• More coming soon!" ]

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[ "Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.\n\n It is currently 18 Jul 2019, 06:22", null, "### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\n\nevery week, we’ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.", null, "", null, "# Math: Absolute value (Modulus)\n\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nVeritas Prep GMAT Instructor", null, "D\nJoined: 16 Oct 2010\nPosts: 9446\nLocation: Pune, India\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nsomeonear wrote:\n\nI am worried about values of x that are on the border of the ranges\nSay hypothetically for a particular set of equations we end with the identical 4 cases we have here.\nNow if say I have x=4 then the way I had come up with the ranges I will get a solution between -3 and 4 as x=4 exists in -3<= x <= 4. But if however I consider what was done in the OP then x=4 exists in the range x>=4\nGranted either way I have a solution but will it be a biggie if I fail to show exactly in which range the solution, if it does,exists\n\nYour aim is to get the solution. You create the ranges to help yourself solve the problem. It doesn't matter at all in which range you consider the border value to lie. Say, when x = 4, (4 - x) = 0. You solve saying that in the range -3<= x<4, (4 - x) is positive and in the range x>= 4, (4 - x) is negative.\nAt the border value i.e. x = 4, (4 - x) = 0. There is no negative or positive at this point. Hence it doesn't matter where you include the '='. Put it wherever you like. I just like to go in a regular fashion like walker did above. Include the first point in the first range, the second one in the second range (but not the third one i.e. -3 <= x < 4) and so on.\n_________________\nKarishma\nVeritas Prep GMAT Instructor\n\nSenior Manager", null, "", null, "Joined: 29 Jan 2011\nPosts: 276\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nHi Walker,\n\nExample #1\nQ.: |x+3| - |4-x| = |8+x|. How many solutions does the equation have?\nSolution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:\n\nHow do we get 3 key points and 4 conditions?\nCEO", null, "", null, "B\nJoined: 17 Nov 2007\nPosts: 3372\nConcentration: Entrepreneurship, Other\nSchools: Chicago (Booth) - Class of 2011\nGMAT 1: 750 Q50 V40", null, "Re: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n1\nThere are 3 points where one of the modules is zero:\n\n1)x+3=0 --> x = -3\n2)4-x=0 --> x = 4\n3)8+x=0 --> x = -8\n\nThose 3 points divide the number line by 4 pieces:\n1) -inf, -8\n2) -8,-3\n3) -3, 4\n4) 4, +inf\n\nand for each condition we are solving the equation separately.\n_________________\nHOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | Limited GMAT/GRE Math tutoring in Chicago\nManager", null, "", null, "Joined: 12 Feb 2012\nPosts: 118\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n1\nLet’s consider following examples,\n\nExample #1\nQ.: $$|x+3| - |4-x| = |8+x|$$. How many solutions does the equation have?\nSolution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:\n\na) $$x < -8$$. $$-(x+3) - (4-x) = -(8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not less than -8)\n\nb) $$-8 \\leq x < -3$$. $$-(x+3) - (4-x) = (8+x)$$ --> $$x = -15$$. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)\n\nc) $$-3 \\leq x < 4$$. $$(x+3) - (4-x) = (8+x)$$ --> $$x = 9$$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)\n\nd) $$x \\geq 4$$. $$(x+3) + (4-x) = (8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not more than 4)\n\nTwo Questions:\nCan this only be done when we all absolute values in the equation. Could we have done it for $$|x+3| - 5 = |8+x|$$?\nStates the conditions are -3 and -8?\n\nSecond, How did you know whether to put a negative or make positive the terms in the conditions?\n\nFor example in (a) you made $$(x+3) and (8+x)$$ negative\nin (b) you made $$(x+3)$$ negative and everything positive.\n\nWhat gives?\n\nThank you!\n\nHow do you know\nCEO", null, "", null, "B\nJoined: 17 Nov 2007\nPosts: 3372\nConcentration: Entrepreneurship, Other\nSchools: Chicago (Booth) - Class of 2011\nGMAT 1: 750 Q50 V40", null, "Re: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n3\n1\n1) yes, you can use the same approach for |x+3|-5=|8+x|\n\n2) let say we have condition x < -8. Then |x + 8| = - (x+8). Why do we have \"-\" here? Because (x+8) is always negative at x<-8 and we need to add \"-\" to get a positive value. Actually, it's the definition of the absolute value:\n\n|x| = x for x >=0\n|x| = -x for x<0\n_________________\nHOT! GMAT TOOLKIT 2 (iOS) / GMAT TOOLKIT (Android) - The OFFICIAL GMAT CLUB PREP APP, a must-have app especially if you aim at 700+ | Limited GMAT/GRE Math tutoring in Chicago\nIntern", null, "", null, "Status: Active\nJoined: 30 Jun 2012\nPosts: 36\nLocation: India\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n1\n1\nHere is a GMAT DS question from the topic Inequalities. Tests concepts of modulus.\n\nQuestion\nIs |a| > |b|?\n1. 1/(a - b) > 1/(b - a)\n2. a + b < 0\n\nCorrect Answer : Choice C. Both statements together are sufficient.\n\nWe need to determine whether |a| is greater than |b|.\nThe answer to this question will be a conclusive 'yes' if |a| > |b|.\nThe answer will be a conclusive 'n' if |a| <= |b|\n\nLet us evaluate statement 1\n1/(a - b) > 1/(b - a)\nWe can rewrite the same inequality as 1/(a - b) > -1(a - b).\nIf a number is greater than the negative of the number, the number has to be a positive number.\n\nSo, we can conclude that a - b > 0 or a > b.\n\nIf a > b, |a| may or may not be greater than |b|.\nFor e.g, a = 4, b = 2. Then |a| > |b|. For positive a and b, when a > b, |a| > |b|.\n\nLet us look at a counter example. a = 2 and b = -10. a > b. But |a| < |b|.\nHence, we cannot conclude from statement 1 whether |a| > |b|.\n\nStatement 1 is NOT sufficient.\n\nLet us evaluate statement 2\na + b < 0\n\nEither both a and b are negative or one of a or b is negative.\nIf only one of the two numbers is negative, then the magnitude of the negative number is greater than the magnitude of the negative number is greater than the magnitude of the positive number.\n\nFor e.g., a = -3 and b = -4. a + b < 0, |a| < |b|\nHere is a counter example: a = -4 and b = -3. a + b < 0 and |a| > |b|.\n\nSo, statement 2 is NOT sufficient.\n\nLet us combine the two statements.\nWe know a > b from statement 1 and a + b < 0 from statement 2.\n\nIf both a and b are negative, and we know that a > b, then |a| < |b|. Note in negative numbers, lesser the magnitude, greater the number.\n\nIf one of 'a' or 'b' is negative, as a > b, a has to be positive and b has to be negative.\nThe sum of a + b < 0. Therefore, the magnitude of a has to be lesser than the magnitude of b.\nSo, we can conclude that |a| < |b|.\n\nHence, by combining the two statements we can conclude that |a| is not greater than |b|.\n\nSo, the two statements taken together are sufficient to answer the question.\nChoice C is the correct answer\n\nHere is an alternative explanation for the same.\nFrom statement 1 we know a - b > 0. From statement 2 we know a + b < 0.\nSo, (a - b)(a + b) < 0\nOr a^2 - b^2 < 0 or a^2 < b^2\n\nIf a^2 < b^2, we can conclude that |a| < |b|.\n_________________\nThanks and Regards!\n\nP.S. +Kudos Please! in case you like my post.", null, "Intern", null, "", null, "Status: K... M. G...\nJoined: 22 Oct 2012\nPosts: 31\nGMAT Date: 08-27-2013\nGPA: 3.8\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nwalker wrote:\nwhen x<-8, (4-x) is always positive and we don't need to modify the sign when we open it.\n\nsorry Walker, I still don't get it", null, "when we consider x<-8 then we made all the x to be negative so multiplied each with negative terms.. is that (4-x) is already having negative for x so we didn't have to multiply with negative here.\n\nSorry i am really novice for this kind of problem.. please also let me know if there is any post related with this", null, "Veritas Prep GMAT Instructor", null, "D\nJoined: 16 Oct 2010\nPosts: 9446\nLocation: Pune, India\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n2\n2\nbreakit wrote:\nwalker wrote:\nwhen x<-8, (4-x) is always positive and we don't need to modify the sign when we open it.\n\nsorry Walker, I still don't get it", null, "when we consider x<-8 then we made all the x to be negative so multiplied each with negative terms.. is that (4-x) is already having negative for x so we didn't have to multiply with negative here.\n\nSorry i am really novice for this kind of problem.. please also let me know if there is any post related with this", null, "You might want to check out these posts where I have discussed these concepts in detail:\n\nhttp://www.veritasprep.com/blog/2012/06 ... e-factors/\nhttp://www.veritasprep.com/blog/2012/07 ... ns-part-i/\nhttp://www.veritasprep.com/blog/2012/07 ... s-part-ii/\n_________________\nKarishma\nVeritas Prep GMAT Instructor\n\nManager", null, "", null, "Joined: 04 Dec 2011\nPosts: 59\nSchools: Smith '16 (I)\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\ngettinit wrote:\nLet’s consider following examples,\n\nExample #1 I am not understanding this example and really struggling with modulus? Can someone please elaborate and explain in further detail? From this post I can't see how I would use this on every modulus problem?\nQ.: $$|x+3| - |4-x| = |8+x|$$. How many solutions does the equation have?\nSolution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:\n\na) $$x < -8$$. $$-(x+3) - (4-x) how did we get -(x+3) here?= -(8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not less than -8)\n\nb) $$-8 \\leq x < -3$$. $$-(x+3) - (4-x) = (8+x)$$ --> $$x = -15$$. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)\n\nc) $$-3 \\leq x < 4$$. $$(x+3) - (4-x) = (8+x)$$ --> $$x = 9$$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)\n\nd) $$x \\geq 4$$. $$(x+3) + (4-x) = (8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not more than 4)\n\nI am totally lost with this post and also with other modulus problems I looked up in Gmat club thank you very much for your help in advance!!!!!\n\nlike the gentlemen above, I continue to be puzzled by this post. I tried searching for answer in this topic post itself, but couldn't get a compilation to all my questions, can any expert please make me understand this, here are my queries.\n\nQ.: $$|x+3| - |4-x| = |8+x|$$. How many solutions does the equation have?\nSolution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:\n\nI get this part, perfectly fine, basic goal to arrive at points is to make every value within modulus \"0\"\nso we have 3 key points and 4 solutions Ie,\n-3+3= 0 for 1st modulus sign so key point here is -3,\n4-4=0 for second modulus sign, so key point here is 4\n8-8 in third modulus, so key point here is -8\n\nTherefore on a number line it will be 3 points something like this ---------$$(-8)$$---------$$(-3)$$------------------------$$(4)$$\n\nsecond step:\n\nQuote:\nA. a) $$x < -8$$. $$-(x+3) - (4-x) = -(8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not less than -8)\n\nI do understand in first Bracket $$-(x+3)$$, since we are testing X against x < -8[/m], so we need to make $$-X$$ here. as per Walkers quote\n\nwalker wrote:\nif x < -8, (x + 3) is always negative. So, modulus is nonnegative and we need to change a sign: |x+3| = - (x+3) for x<-8\nFor example, if x = -10,\n|-10+3| = |-7| = 7\n-(-10+3) = -(-7) = 7\n\nIn other words, |x| = x if x is positive and |x|=-x if x is negative.\n\nbut my Question is If we eventually want to see a negative X inside the bracket than why $$- (4-x)$$? as in this case X will turn positive after opening the bracket\n\n2nd EQ------\n$$-8 \\leq x < -3$$ $$-(x+3) - (4-x)$$ = $$(8+x)$$\n\nagain in 2nd equation my doubt is why do we have the $$(8+X)$$ as non negative, I mean it should be same as $$-(8+x)$$, like in 1st test case. as X is still negative. in this test case? Of-course this is fine if I can get answer to my 1st query, if we have to make X negative than this is not ok.\n\nin 3rd test case\nQuote:\nc) $$-3 \\leq x < 4$$. $$(x+3) - (4-x) = (8+x)$$ --> $$x = 9$$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)\n\nin this case X can be negative or positive, so why don't we put $$-(x+3)$$ here? rather than $$(X+3)$$ ?\n\nQuote:\nd) $$x \\geq 4$$. $$(x+3) + (4-x) = (8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not more than 4)\n\nAgain in the equation above we are testing against positive X test point, than why $$+(4-X)$$, I think it should be $$-(4-X)$$ to turn X into positive after opening the brackets.?\n\nAll of the above questions may sound stupid, but I need to understand this, as inequalities is my weak topic.\n_________________\nLife is very similar to a boxing ring.\nDefeat is not final when you fall down…\nIt is final when you refuse to get up and fight back!\n\n1 Kudos = 1 thanks\nNikhil\nVeritas Prep GMAT Instructor", null, "D\nJoined: 16 Oct 2010\nPosts: 9446\nLocation: Pune, India\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nnikhil007 wrote:\ngettinit wrote:\nLet’s consider following examples,\n\nExample #1 I am not understanding this example and really struggling with modulus? Can someone please elaborate and explain in further detail? From this post I can't see how I would use this on every modulus problem?\nQ.: $$|x+3| - |4-x| = |8+x|$$. How many solutions does the equation have?\nSolution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:\n\na) $$x < -8$$. $$-(x+3) - (4-x) how did we get -(x+3) here?= -(8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not less than -8)\n\nb) $$-8 \\leq x < -3$$. $$-(x+3) - (4-x) = (8+x)$$ --> $$x = -15$$. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)\n\nc) $$-3 \\leq x < 4$$. $$(x+3) - (4-x) = (8+x)$$ --> $$x = 9$$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)\n\nd) $$x \\geq 4$$. $$(x+3) + (4-x) = (8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not more than 4)\n\nI am totally lost with this post and also with other modulus problems I looked up in Gmat club thank you very much for your help in advance!!!!!\n\nlike the gentlemen above, I continue to be puzzled by this post. I tried searching for answer in this topic post itself, but couldn't get a compilation to all my questions, can any expert please make me understand this, here are my queries.\n\nQ.: $$|x+3| - |4-x| = |8+x|$$. How many solutions does the equation have?\nSolution: There are 3 key points here: -8, -3, 4. So we have 4 conditions:\n\nI get this part, perfectly fine, basic goal to arrive at points is to make every value within modulus \"0\"\nso we have 3 key points and 4 solutions Ie,\n-3+3= 0 for 1st modulus sign so key point here is -3,\n4-4=0 for second modulus sign, so key point here is 4\n8-8 in third modulus, so key point here is -8\n\nTherefore on a number line it will be 3 points something like this ---------$$(-8)$$---------$$(-3)$$------------------------$$(4)$$\n\nsecond step:\n\nQuote:\nA. a) $$x < -8$$. $$-(x+3) - (4-x) = -(8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not less than -8)\n\nI do understand in first Bracket $$-(x+3)$$, since we are testing X against x < -8[/m], so we need to make $$-X$$ here. as per Walkers quote\n\nwalker wrote:\nif x < -8, (x + 3) is always negative. So, modulus is nonnegative and we need to change a sign: |x+3| = - (x+3) for x<-8\nFor example, if x = -10,\n|-10+3| = |-7| = 7\n-(-10+3) = -(-7) = 7\n\nIn other words, |x| = x if x is positive and |x|=-x if x is negative.\n\nbut my Question is If we eventually want to see a negative X inside the bracket than why $$- (4-x)$$? as in this case X will turn positive after opening the bracket\n\n2nd EQ------\n$$-8 \\leq x < -3$$ $$-(x+3) - (4-x)$$ = $$(8+x)$$\n\nagain in 2nd equation my doubt is why do we have the $$(8+X)$$ as non negative, I mean it should be same as $$-(8+x)$$, like in 1st test case. as X is still negative. in this test case? Of-course this is fine if I can get answer to my 1st query, if we have to make X negative than this is not ok.\n\nin 3rd test case\nQuote:\nc) $$-3 \\leq x < 4$$. $$(x+3) - (4-x) = (8+x)$$ --> $$x = 9$$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)\n\nin this case X can be negative or positive, so why don't we put $$-(x+3)$$ here? rather than $$(X+3)$$ ?\n\nQuote:\nd) $$x \\geq 4$$. $$(x+3) + (4-x) = (8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not more than 4)\n\nAgain in the equation above we are testing against positive X test point, than why $$+(4-X)$$, I think it should be $$-(4-X)$$ to turn X into positive after opening the brackets.?\n\nAll of the above questions may sound stupid, but I need to understand this, as inequalities is my weak topic.\n\nIn a post above, I have given the links to 3 posts which explain the process in detail (including the reasoning behind the process). Check those out to get answers to your questions.\n_________________\nKarishma\nVeritas Prep GMAT Instructor\n\nManager", null, "", null, "Joined: 04 Dec 2011\nPosts: 59\nSchools: Smith '16 (I)\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nVeritasPrepKarishma wrote:\n\nIn a post above, I have given the links to 3 posts which explain the process in detail (including the reasoning behind the process). Check those out to get answers to your questions.\n\nHi Karishma\n\nI went through your post on the blog, but to be frank found this post of your more helpfull\n\nVeritasPrepKarishma wrote:\nOf course it can be done using algebra as well. It doesn't matter how many mods there are. you always deal with them in the same way.\n|x|= x when x is >= 0,\n|x|= -x when x < 0\n\n|x - 2|= (x - 2) when x - 2 >= 0 (or x >= 2),\n|x - 2|= -(x - 2) when (x-2) < 0 (or x < 2)\n\nThen you solve the equations using both conditions given above. That is the importance of the points.\nSo if you have:\n|x - 2|= |x + 3|\n\nYou say, |x - 2|= (x - 2) when x >= 2.\n|x - 2|= -(x - 2) when x < 2\n|x + 3| = (x + 3) when x >= -3\n|x + 3| = -(x + 3) when x < -3\n\nOk, Now after literally banging my head for 3 hrs and reading you blog articles back and forth, I get it that to make an EQ in (X-K) format we manipulate it by taking -tive sign out\nbut I guess in this example its this concept that we need to understand\n\n|x|= x when x is >= 0,\n|x|= -x when x < 0\n\nok, so based on this understanding I will take a fresh shot, please let me know what's wrong\n\nQuote:\na) $$x < -8$$. $$-(x+3) - (4-x) = -(8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not less than -8)\n\nIn this test case, since we will always have |x+3| negative we put a -tive sign outside because modulus will turn it into non negative, so to do that we take multiply it by (-1), is this understanding correct?\nand since we are ok with -(4-x), because we will again get |4-x| positive with a negative x, the -tive sign outside the bracket will make sure its always -tive when out of the Modulus. However to be frank, a little confusion here is, as you mentioned in the blog, why don't we try to convert it into (x-k) format?\nin RHS we have -(8-x) because again we want |8-x| to turn out a negative number so we put -(8-x) to make it always negative, let me know if I got it correctly.\n\nQuote:\nb) $$-8 \\leq x < -3$$. $$-(x+3) - (4-x) = (8+x)$$ --> $$x = -15$$. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)\n\nagain, I get it why LHS is that way, however I still don't get it why we don't have -(8-X) as we need to make sure that the result of this bracket is -tive so |8-x| = -(8-x)\n\nQuote:\nc) $$-3 \\leq x < 4$$. $$(x+3) - (4-x) = (8+x)$$ --> $$x = 9$$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)\n\nI still dont get it, if we test x against both -tive and positive scenario, why is that we just have 1 equation? in my view we should split it in 2 eq. to test against both negative and positive value.\n\nQuote:\nd) $$x \\geq 4$$. $$(x+3) + (4-x) = (8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not more than 4)\n\n(x+3) since we don't have a negative value of X this bracket will always be positive, we don't need a -tive sign outside, is this the reason?\n(4-x) again since a >4 will always make it positive we don't need a -tive sign outside the bracket, is this the reason?\n(8+x) again same reason as above for this?\n\nComplication No 3: on this post http://www.veritasprep.com/blog/2012/07 ... ns-part-i/\n\n(-2x^3 + 17x^2 – 30x) > 0\n\nThis is how I understand it,\n\n$$x(-2x^2 + 17x - 30) > 0$$ (just took out x common) ok\nx(2x – 5)(6 – x) > 0(factoring the quadratic) ok\n2x(x – 5/2)(-1)(x – 6) > 0 (take 2 common) ---------> I think in this you took out -1 common to make the second bracket = (x-k) format?\n2(x–0)(x–5/2)(x–6) < 0 (multiply both sides by -1)-------> how did you arrive at 2(x-0)? i think it should be just $$2x(x-\\frac{5}{2})(x-6) <0$$\n_________________\nLife is very similar to a boxing ring.\nDefeat is not final when you fall down…\nIt is final when you refuse to get up and fight back!\n\n1 Kudos = 1 thanks\nNikhil\nVeritas Prep GMAT Instructor", null, "D\nJoined: 16 Oct 2010\nPosts: 9446\nLocation: Pune, India\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n3\n3\nnikhil007 wrote:\nVeritasPrepKarishma wrote:\nOf course it can be done using algebra as well. It doesn't matter how many mods there are. you always deal with them in the same way.\n|x|= x when x is >= 0,\n|x|= -x when x < 0\n\n|x - 2|= (x - 2) when x - 2 >= 0 (or x >= 2),\n|x - 2|= -(x - 2) when (x-2) < 0 (or x < 2)\n\nThen you solve the equations using both conditions given above. That is the importance of the points.\nSo if you have:\n|x - 2|= |x + 3|\n\nYou say, |x - 2|= (x - 2) when x >= 2.\n|x - 2|= -(x - 2) when x < 2\n|x + 3| = (x + 3) when x >= -3\n|x + 3| = -(x + 3) when x < -3\n\nOk, Now after literally banging my head for 3 hrs and reading you blog articles back and forth, I get it that to make an EQ in (X-K) format we manipulate it by taking -tive sign out\nbut I guess in this example its this concept that we need to understand\n\n|x|= x when x is >= 0,\n|x|= -x when x < 0\n\nok, so based on this understanding I will take a fresh shot, please let me know what's wrong\n\nQuote:\na) $$x < -8$$. $$-(x+3) - (4-x) = -(8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not less than -8)\n\nIn this test case, since we will always have |x+3| negative we put a -tive sign outside because modulus will turn it into non negative, so to do that we take multiply it by (-1), is this understanding correct?\nand since we are ok with -(4-x), because we will again get |4-x| positive with a negative x, the -tive sign outside the bracket will make sure its always -tive when out of the Modulus. However to be frank, a little confusion here is, as you mentioned in the blog, why don't we try to convert it into (x-k) format?\nin RHS we have -(8-x) because again we want |8-x| to turn out a negative number so we put -(8-x) to make it always negative, let me know if I got it correctly.\n\nQuote:\nb) $$-8 \\leq x < -3$$. $$-(x+3) - (4-x) = (8+x)$$ --> $$x = -15$$. We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.)\n\nagain, I get it why LHS is that way, however I still don't get it why we don't have -(8-X) as we need to make sure that the result of this bracket is -tive so |8-x| = -(8-x)\n\nQuote:\nc) $$-3 \\leq x < 4$$. $$(x+3) - (4-x) = (8+x)$$ --> $$x = 9$$. We reject the solution because our condition is not satisfied (-15 is not within (-3,4) interval.)\n\nI still dont get it, if we test x against both -tive and positive scenario, why is that we just have 1 equation? in my view we should split it in 2 eq. to test against both negative and positive value.\n\nQuote:\nd) $$x \\geq 4$$. $$(x+3) + (4-x) = (8+x)$$ --> $$x = -1$$. We reject the solution because our condition is not satisfied (-1 is not more than 4)\n\n(x+3) since we don't have a negative value of X this bracket will always be positive, we don't need a -tive sign outside, is this the reason?\n(4-x) again since a >4 will always make it positive we don't need a -tive sign outside the bracket, is this the reason?\n(8+x) again same reason as above for this?\n\nComplication No 3: on this post http://www.veritasprep.com/blog/2012/07 ... ns-part-i/\n\n(-2x^3 + 17x^2 – 30x) > 0\n\nThis is how I understand it,\n\n$$x(-2x^2 + 17x - 30) > 0$$ (just took out x common) ok\nx(2x – 5)(6 – x) > 0(factoring the quadratic) ok\n2x(x – 5/2)(-1)(x – 6) > 0 (take 2 common) ---------> I think in this you took out -1 common to make the second bracket = (x-k) format?\n\nFirst of all, if you do get a question with multiple mods and if you want to be prepared for it, using algebra will be far more time consuming than the approaches discussed in my blog. But nevertheless, you should understand it properly.\n\nWhen you have an equation with x in it, you solve by taking x to one side and everything else to the other. What happens when you have mods in it?\nSay |x| = 4, you still haven't got the value of x. You have the value of |x| only. So you need to remove the mod. Now there are rules to remove the mod.\n\n|x|= x (mod removed) when x is >= 0,\n|x|= -x (mod removed) when x < 0\n\nSo |x| = 4 to remove the mod, I need to know whether x is positive or negative.\nIf x >= 0, |x| = x so |x| = 4 = x\nWe get that x is 4\n\nIf x < 0, |x| = -x so |x| = 4 = -x\nhence x = -4\n\nSo if we are looking for a positive value, then it is 4 and if we are looking for a negative value, it is -4.\n\nSimilarly, when you have |x+4| + |x - 3| = 10 (just an example), you need to remove the mods to solve for x. But to remove mods (which are around the entire factors x-4 and x-3 and not just around x), you need to know whether (x+ 4) and (x - 3) (the thing inside the mod) are positive/negative.\n\nSo you split it into ranges:\n\nx > 3\nPut any value greater than 3 in (x+4), (x+4) will remain positive. Put any value greater than 3 in (x - 3), (x - 3) will remain positive.\n\nSo when x > 3, we can remove the mods without any modification:\n(x + 4) + (x-3) = 10\nx = 9/2\nSince 9/2 is greater than 3, this value of x is acceptable.\n\n-4 < x< 3\nFor these values of x, (x+4) will always be positive but (x-3) will be negative. So |x - 3| = -(x-3)\n(x + 4) - (x-3) = 10\nYou don't have any such value for x\n\nx < -4\nFor these values of x, (x+4) and (x-3) will be negative. So |x - 3| = -(x-3) and |x+4| = -(x+4)\n-(x + 4) - (x-3) = 10\nx = -11/2\nSince -11/2 is less than -4, this value of x is also acceptable.\n\nI have discussed how to deal with such questions logically here: http://www.veritasprep.com/blog/2011/01 ... s-part-ii/\n\nAs for question with factors that are multiplied (discussed in the 3 links given above),\nWe know how to deal with (x-a)(x-b)(x-c) > 0 type of questions so we try to bring it that form.\n\n2(x–0)(x–5/2)(x–6) < 0 (multiply both sides by -1)-------> how did you arrive at 2(x-0)? i think it should be just $$2x(x-\\frac{5}{2})(x-6) <0$$\n\n(x-0) is nothing but x. I put as (x-0) to make it consistent to the (x-a)(x-b).... form to help you remember that you have to take 0 as a transition point too.\n_________________\nKarishma\nVeritas Prep GMAT Instructor\n\nManager", null, "", null, "Joined: 04 Dec 2011\nPosts: 59\nSchools: Smith '16 (I)\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nKarishma,\n\nKudos given for the post, thanks for explaining in detail,\nI agree that we would be better off plugging number on such a ques, but things get tricky when it comes to DS\n\nI basically covered this from MGmat guides and I can handle a simple Mod like |x-2|>5\nwhat I learnt is simply take 2 conditions, x-2>5 and 2-x>5 and solve for 2 set of x, however the book never taught me this 3 step method. so I have to dig it in here.\n_________________\nLife is very similar to a boxing ring.\nDefeat is not final when you fall down…\nIt is final when you refuse to get up and fight back!\n\n1 Kudos = 1 thanks\nNikhil\nSenior Manager", null, "", null, "Joined: 13 May 2013\nPosts: 414\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n(This question is from a GMAT club study book. It can be found here: math-absolute-value-modulus-86462.html)\n\n|x^2-4| = 1. What is x?\n\nSolution: There are 2 conditions:\n\na) (x^2-4)\\geq0 --> x \\leq -2 or x\\geq2. x^2-4=1 --> x^2 = 5. x e {-\\sqrt{5}, \\sqrt{5}} and both solutions satisfy the condition.\n\nb) (x^2-4)<0 --> -2 < x < 2. -(x^2-4) = 1 --> x^2 = 3. x e {-\\sqrt{3}, \\sqrt{3}} and both solutions satisfy the condition.\n\nWhy do we set these problems up as >= or <= 1? I would solve this problem as follows:\n\n|x^2-4| = 1\n\nx^2-4 = 1 ==> x^2 = 5 ==> x = \\sqrt{5}\n\nOR\n\n-(x^2-4) = 1 ==> -x^2 +4 = 1 ==> -x^2 = -3 ==> X^2 = 3 ==> x = \\sqrt{3}\n\nThanks!\nVP", null, "", null, "Status: Far, far away!\nJoined: 02 Sep 2012\nPosts: 1044\nLocation: Italy\nConcentration: Finance, Entrepreneurship\nGPA: 3.8\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n1\nThe first method is the correct one and will always give you the correct results.\n\nConsider however the following case\n\n$$|x+5|=-4$$, at glance this equation has no solution because $$|x+5|$$ cannot be less than 0.\nBut I wanna take it as example:\n\nWith the first method you'll find\nif $$x>-5$$\n$$x+5=-4$$, $$x=-9$$, out of the interval => it's not a solution\n\nif $$x<-5$$\n$$-x-5=-4$$, $$x=-1$$ out of the interval => it's not a solution\n\nWith the second method\n$$x+5=-4$$, $$x=-9$$\n$$-(x+5)=-4$$, or $$x=-1$$\nthose seem valid... but the equation we know that has no solution.\n\nMain point: the first method works always, do not rely on the other one.\nThe second one does not take into consideration the intervals, so it might not work\n\nHope it's clear\n_________________\nIt is beyond a doubt that all our knowledge that begins with experience.\nKant , Critique of Pure Reason\n\nTips and tricks: Inequalities , Mixture | Review: MGMAT workshop\nStrategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant\n\nRules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]\nSenior Manager", null, "", null, "Joined: 13 May 2013\nPosts: 414\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nHmmmm...I'm not sure I follow.\n\nThis is how I solved the problem.\n\n|x^2-4|=1\n\nx^2 - 4 =1\nOR\n-x^2 + 4 = 1\n\nSO\n\nx^2=5 ==> x=+/- √5\nOR\n-x^2=-3 ==> x^2=3 ==> x=+/- √3\n\nSo, I believe that is the right answer but I'm still not sure about how the greater than/less than signs come into play. Sorry for the thick head!\n\nZarrolou wrote:\nThe first method is the correct one and will always give you the correct results.\n\nConsider however the following case\n\n$$|x+5|=-4$$, at glance this equation has no solution because $$|x+5|$$ cannot be less than 0.\nBut I wanna take it as example:\n\nWith the first method you'll find\nif $$x>-5$$\n$$x+5=-4$$, $$x=-9$$, out of the interval => it's not a solution\n\nif $$x<-5$$\n$$-x-5=-4$$, $$x=-1$$ out of the interval => it's not a solution\n\nWith the second method\n$$x+5=-4$$, $$x=-9$$\n$$-(x+5)=-4$$, or $$x=-1$$\nthose seem valid... but the equation we know that has no solution.\n\nMain point: the first method works always, do not rely on the other one.\nThe second one does not take into consideration the intervals, so it might not work\n\nHope it's clear\nVP", null, "", null, "Status: Far, far away!\nJoined: 02 Sep 2012\nPosts: 1044\nLocation: Italy\nConcentration: Finance, Entrepreneurship\nGPA: 3.8\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n1\nWholeLottaLove wrote:\nHmmmm...I'm not sure I follow.\n\nThis is how I solved the problem.\n\n|x^2-4|=1\n\nx^2 - 4 =1\nOR\n-x^2 + 4 = 1\n\nSO\n\nx^2=5 ==> x=+/- √5\nOR\n-x^2=-3 ==> x^2=3 ==> x=+/- √3\n\nSo, I believe that is the right answer but I'm still not sure about how the greater than/less than signs come into play. Sorry for the thick head!\n\nThe original solution used $$\\geq{}$$ and $$\\leq{}$$ to define the intervals. It solved the cases:\n$$x^2-4=1$$ so $$x=+-\\sqrt{5}$$, and then it check weather those numbers are in the interval $$-2<x<2$$. Both are inside so both are valid solutions\nThen the other case $$-x^2+4=1$$ so $$x=+-\\sqrt{3}$$, and then check the interval it is considering in this scenario $$x<-2$$ and $$x>2$$, both are inside the intervals so both are valid solutions\n\nAs I said before the correct method to solve abs values always checks if the result obtained is inside the interval is considering at that moment.\nIf you do not double check weather the solution you find is inside the interval, you are likely to commit errors in the problem. What I am trying to say is that your method is incomplete: it lacks the last passage - you find the solutions, but you do not check if they are possible or not.\n\nIn this example all 4 solutions are possible, so the last step does nothing; but if one of the solution were not valid, you \"incomplete\" method would not be albe to detect it. ( as I showed you in the $$|x+5|=-4$$ example)\nHope that what I mean is clear\n_________________\nIt is beyond a doubt that all our knowledge that begins with experience.\nKant , Critique of Pure Reason\n\nTips and tricks: Inequalities , Mixture | Review: MGMAT workshop\nStrategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant\n\nRules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]\nSenior Manager", null, "", null, "Joined: 13 May 2013\nPosts: 414\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nSo, in other words, if I were to plug in -/+ √5 into x^2 = 5 it will yield me a result between -2<x<2? And where does -2<x<2 come from???\n\nZarrolou wrote:\nWholeLottaLove wrote:\nHmmmm...I'm not sure I follow.\n\nThis is how I solved the problem.\n\n|x^2-4|=1\n\nx^2 - 4 =1\nOR\n-x^2 + 4 = 1\n\nSO\n\nx^2=5 ==> x=+/- √5\nOR\n-x^2=-3 ==> x^2=3 ==> x=+/- √3\n\nSo, I believe that is the right answer but I'm still not sure about how the greater than/less than signs come into play. Sorry for the thick head!\n\nThe original solution used $$\\geq{}$$ and $$\\leq{}$$ to define the intervals. It solved the cases:\n$$x^2-4=1$$ so $$x=+-\\sqrt{5}$$, and then it check weather those numbers are in the interval $$-2<x<2$$. Both are inside so both are valid solutions\nThen the other case $$-x^2+4=1$$ so $$x=+-\\sqrt{3}$$, and then check the interval it is considering in this scenario $$x<-2$$ and $$x>2$$, both are inside the intervals so both are valid solutions\n\nAs I said before the correct method to solve abs values always checks if the result obtained is inside the interval is considering at that moment.\nIf you do not double check weather the solution you find is inside the interval, you are likely to commit errors in the problem. What I am trying to say is that your method is incomplete: it lacks the last passage - you find the solutions, but you do not check if they are possible or not.\n\nIn this example all 4 solutions are possible, so the last step does nothing; but if one of the solution were not valid, you \"incomplete\" method would not be albe to detect it. ( as I showed you in the $$|x+5|=-4$$ example)\nHope that what I mean is clear\nVP", null, "", null, "Status: Far, far away!\nJoined: 02 Sep 2012\nPosts: 1044\nLocation: Italy\nConcentration: Finance, Entrepreneurship\nGPA: 3.8\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\n1\nWholeLottaLove wrote:\nSo, in other words, if I were to plug in -/+ √5 into x^2 = 5 it will yield me a result between -2<x<2? And where does -2<x<2 come from???\n\n-2<x<2 is the interval in which the function is negative, bare with me:\n\nTake the function |x^2-4|=1\n1) Define where it is positive and where is negative => $$x^2-4>0$$ if x<-2 and x>2\nSo if $$x<-2$$ or $$x>2$$ is positive, if $$-2<x<2$$ is negative\n\n2)Study each case on its own:\n$$x^2-4=1$$ $$x=+-\\sqrt{5}$$, are those results valid? Are they in the interval we are considering? Are they in the x<-2 or x>2 interval?\n$$-\\sqrt{5}$$ is less than -2, and $$+\\sqrt{5}$$ is more than 2. So they are valid solutions because they are in the intervals we are considering\n$$-x^2+4=1$$ $$x=+-\\sqrt{3}$$, are those results valid? same as above\nYes they are valid because they are numbers between $$-2$$ and $$2$$(the interval we are considering now, in which |abs| is negative => -x^2+4)\n_________________\nIt is beyond a doubt that all our knowledge that begins with experience.\nKant , Critique of Pure Reason\n\nTips and tricks: Inequalities , Mixture | Review: MGMAT workshop\nStrategy: SmartGMAT v1.0 | Questions: Verbal challenge SC I-II- CR New SC set out !! , My Quant\n\nRules for Posting in the Verbal Forum - Rules for Posting in the Quant Forum[/size][/color][/b]\nSenior Manager", null, "", null, "Joined: 13 May 2013\nPosts: 414\nRe: Math: Absolute value (Modulus)  [#permalink]\n\n### Show Tags\n\nSo, I need to find the positive and negative values of |x^2-4|=1 (i.e. x^2-4=1 and -x^2+4=1) and which x values make x^2-4 positive and -x^2+4 negative?\n\nThanks for putting up with my slowness in picking up these concepts!\n\nZarrolou wrote:\nWholeLottaLove wrote:\nSo, in other words, if I were to plug in -/+ √5 into x^2 = 5 it will yield me a result between -2<x<2? And where does -2<x<2 come from???\n\n-2<x<2 is the interval in which the function is negative, bare with me:\n\nTake the function |x^2-4|=1\n1) Define where it is positive and where is negative => $$x^2-4>0$$ if x<-2 and x>2\nSo if $$x<-2$$ or $$x>2$$ is positive, if $$-2<x<2$$ is negative\n\n2)Study each case on its own:\n$$x^2-4=1$$ $$x=+-\\sqrt{5}$$, are those results valid? Are they in the interval we are considering? Are they in the x<-2 or x>2 interval?\n$$-\\sqrt{5}$$ is less than -2, and $$+\\sqrt{5}$$ is more than 2. So they are valid solutions because they are in the intervals we are considering\n$$-x^2+4=1$$ $$x=+-\\sqrt{3}$$, are those results valid? same as above\nYes they are valid because they are numbers between $$-2$$ and $$2$$(the interval we are considering now, in which |abs| is negative => -x^2+4)", null, "Re: Math: Absolute value (Modulus)   [#permalink] 04 Jun 2013, 17:37\n\nGo to page   Previous    1   2   3   4   5    Next  [ 94 posts ]\n\nDisplay posts from previous: Sort by\n\n# Math: Absolute value (Modulus)", null, "", null, "" ]

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[ "# Back To Basics, Part Dos: Gradient Descent\n\nExplore the inner workings of the powerful optimization algorithm.\n\nWelcome to the second part of our Back To Basics series. In the first part, we covered how to use Linear Regression and Cost Function to find the best-fitting line for our house prices data. However, we also saw that testing multiple intercept values can be tedious and inefficient. In this second part, we’ll delve deeper into Gradient Descent, a powerful technique that can help us find the perfect intercept and optimize our model. We’ll explore the math behind it and see how it can be applied to our linear regression problem.\n\nGradient descent is a powerful optimization algorithm that aims to quickly and efficiently find the minimum point of a curve. The best way to visualize this process is to imagine you are standing at the top of a hill, with a treasure chest filled with gold waiting for you in the valley.", null, "However, the exact location of the valley is unknown because it’s super dark out and you can’t see anything. Moreover, you want to reach the valley before anyone else does (because you want all of the treasure for yourself duh). Gradient descent helps you navigate the terrain and reach this optimal point efficiently and quickly. At each point it’ll tell you how many steps to take and in what direction you need to take them.\n\nSimilarly, gradient descent can be applied to our linear regression problem by using the steps laid out by the algorithm. To visualize the process of finding the minimum, let’s plot the MSE curve. We already know that the equation of the curve is:", null, "the equation of the curve is the equation used to calculate the MSE\n\nAnd from the previous article, we know that the equation of MSE in our problem is:", null, "If we zoom out we can see that an MSE curve (which resembles our valley) can be found by substituting a bunch of intercept values in the above equation. So let’s plug in 10,000 values of the intercept, to get a curve that looks like this:", null, "in reality, we won’t know what the MSE curve looks like\n\nThe goal is to reach the bottom of this MSE curve, which we can do by following these steps:\n\n## Step 1: Start with a random initial guess for the intercept value\n\nIn this case, let’s assume our initial guess for the intercept value is 0.\n\n## Step 2: Calculate the gradient of the MSE curve at this point\n\nThe gradient of a curve at a point is represented by the tangent line (a fancy way of saying that the line touches the curve only at that point) at that point. For example, at Point A, the gradient of the MSE curve can be represented by the red tangent line, when the intercept is equal to 0.", null, "the gradient of the MSE curve when intercept = 0\n\nIn order to determine the value of the gradient, we apply our knowledge of calculus. Specifically, the gradient is equal to the derivative of the curve with respect to the intercept at a given point. This is denoted as:", null, "NOTE: If you’re unfamiliar with derivatives, I recommend watching this Khan Academy video if interested. Otherwise you can glance over the next part and still be able to follow the rest of the article.\n\nWe calculate the derivative of the MSE curve as follows:", null, "Now to find the gradient at point A, we substitute the value of the intercept at point A in the above equation. Since intercept = 0, the derivative at point A is:", null, "So when the intercept = 0, the gradient = -190\n\nNOTE: As we approach the optimal value, the gradient values approach zero. At the optimal value, the gradient is equal to zero. Conversely, the farther away we are from the optimal value, the larger the gradient becomes.", null, "From this, we can infer that the step size should be related to the gradient, since it tells us if we should take a baby step or a big step. This means that when the gradient of the curve is close to 0, then we should take baby steps because we are close to the optimal value. And if the gradient is bigger, we should take bigger steps to get to the optimal value faster.\n\nNOTE: However, if we take a super huge step, then we could make a big jump and miss the optimal point. So we need to be careful.", null, "## Step 3: Calculate the Step Size using the gradient and the Learning Rate and update the intercept value\n\nSince we see that the Step Size and gradient are proportional to each other, the Step Size is determined by multiplying the gradient by a pre-determined constant value called the Learning Rate:", null, "The Learning Rate controls the magnitude of the Step Size and ensures that the step taken is not too large or too small.\n\nIn practice, the Learning Rate is usually a small positive number that is ? 0.001. But for our problem let’s set it to 0.1.\n\nSo when the intercept is 0, the Step Size = gradient x Learning Rate = -190*0.1 = -19.\n\nBased on the Step Size we calculated above, we update the intercept (aka change our current location) using any of these equivalent formulas:", null, "To find the new intercept in this step, we plug in the relevant values…", null, "…and find that the new intercept = 19.\n\nNow plugging this value in the MSE equation, we find that the MSE when the intercept is 19 = 8064.095. We notice that in one big step, we moved closer to our optimal value and reduced the MSE.", null, "Even if we look at our graph, we see how much better our new line with intercept 19 is fitting our data than our old line with intercept 0:", null, "## Step 4: Repeat steps 2–3\n\nWe repeat Steps 2 and 3 using the updated intercept value.\n\nFor example, since the new intercept value in this iteration is 19, following Step 2, we will calculate the gradient at this new point:", null, "And we find that the gradient of the MSE curve at the intercept value of 19 is -152 (as represented by the red tangent line in the illustration below).", null, "Next, in accordance with Step 3, let’s calculate the Step Size:", null, "And subsequently, update the intercept value:", null, "Now we can compare the line with the previous intercept of 19 to the new line with the new intercept 34.2…", null, "…and we can see that the new line fits the data better.\n\nOverall, the MSE is getting smaller…", null, "…and our Step Sizes are getting smaller:", null, "We repeat this process iteratively until we converge toward the optimal solution:", null, "As we progress toward the minimum point of the curve, we observe that the Step Size becomes increasingly smaller. After 13 steps, the gradient descent algorithm estimates the intercept value to be 95. If we had a crystal ball, this would be confirmed as the minimum point of the MSE curve. And it is clear to see how this method is more efficient compared to the brute force approach that we saw in the previous article.\n\nNow that we have the optimal value of our intercept, the linear regression model is:", null, "And the linear regression line looks like this:", null, "best fitting line with intercept = 95 and slope = 0.069\n\nFinally, going back to our friend Mark’s question — What value should he sell his 2400 feet² house for?", null, "Plug in the house size of 2400 feet² into the above equation…", null, "…and voila. We can tell our unnecessarily worried friend Mark that based on the 3 houses in his neighborhood, he should look to sell his house for around \\$260,600.\n\nNow that we have a solid understanding of the concepts, let’s do a quick Q&A sesh answering any lingering questions.\n\n## Why does finding the gradient actually work?\n\nTo illustrate this, consider a scenario where we are attempting to reach the minimum point of curve C, denoted as x*. And we are currently at point A at x, located to the left of x*:", null, "If we take the derivative of the curve at point A with respect to x, represented as dC(x)/dx, we obtain a negative value (this means the gradient is sloping downwards). We also observe that we need to move to the right to reach x*. Thus, we need to increase x to arrive at the minimum x*.", null, "the red line, or the gradient, is sloping downwards => a negative Gradient\n\nSince dC(x)/dx is negative, x-??*dC(x)/dx will be larger than x, thus moving towards x*.\n\nSimilarly, if we are at point A located to the right of the minimum point x*, then we get a positive gradient (gradient is sloping upwards), dC(x)/dx.", null, "the red line, or the Gradient, is sloping upwards => a positive Gradient\n\nSo x-??*dC(x)/dx will be less than x, thus moving towards x*.\n\n## How does gradient decent know when to stop taking steps?\n\nGradient descent stops when the Step Size is very close to 0. As previously discussed, at the minimum point the gradient is 0 and as we approach the minimum, the gradient approaches 0. Therefore, when the gradient at a point is close to 0 or in the vicinity of the minimum point, the Step Size will also be close to 0, indicating that the algorithm has reached the optimal solution.", null, "when we are close to the minimum point, the gradient is close to 0, and subsequently Step Size is close to 0\n\nIn practice the Minimum Step Size = 0.001 or smaller", null, "That being said, gradient descent also includes a limit on the number of steps it will take before giving up called the Maximum Number of Steps.\n\nIn practice, the Maximum Number of Steps = 1000 or greater\n\nSo even if the Step Size is larger than the Minimum Step Size, if there have been more than the Maximum Number of Steps, gradient descent will stop.\n\n## What if the minimum point is more challenging to identify?\n\nUntil now, we have been working with a curve where it’s easy to identify the minimum point (these kinds of curves are called convex). But what if we have a curve that’s not as pretty (technically aka non-convex) and looks like this:", null, "Here, we can see that Point B is the global minimum (actual minimum), and Points A and C are local minimums (points that can be confused for the global minimum but aren’t). So if a function has multiple local minimums and a global minimum, it is not guaranteed that gradient descent will find the global minimum. Moreover, which local minimum it finds will depend on the position of the initial guess (as seen in Step 1 of gradient descent).", null, "Taking this non-convex curve above as an example, if the initial guess is at Block A or Block C, gradient descent will declare that the minimum point is at local minimums A or C, respectively when in reality it’s at B. Only when the initial guess is at Block B, the algorithm will find the global minimum B.\n\nNow the question is — how do we make a good initial guess?\n\nSimple answer: Trial and Error. Kind of.\n\nNot-so-simple answer: From the graph above, if our minimum guess of x was 0 since that lies in Block A, it’ll lead to the local minimum A. Thus, as you can see, 0 may not be a good initial guess in most cases. A common practice is to apply a random function based on a uniform distribution on the range of all possible values of x. Additionally, if feasible, running the algorithm with different initial guesses and comparing their results can provide insight into whether the guesses differ significantly from each other. This helps in identifying the global minimum more efficiently.\n\nOkay, we’re almost there. Last question.\n\n## What if we are trying to find more than one optimal value?\n\nUntil now, we were focused on only finding the optimal intercept value because we magically knew the slope value of the linear regression is 0.069. But what if don’t have a crystal ball and don't know the optimal slope value? Then we need to optimize both the slope and intercept values, expressed as x? and x? respectively.\n\nIn order to do that, we must utilize partial derivatives instead of just derivatives.\n\nNOTE: Partial derivates are calculated in the same way as reglar old derivates, but are denoted differently because we have more than one variable we are trying to optimize for. To learn more about them, read this article or watch this video.\n\nHowever, the process remains relatively similar to that of optimizing a single value. The cost function (such as MSE) must still be defined and the gradient descent algorithm must be applied, but with the added step of finding partial derivatives for both x? and x?.\n\nStep 1: Make initial guesses for x₀ and x₁\n\nStep 2: Find the partial derivatives with respect to x₀ and x₁ at these points", null, "Step 3: Simultaneously update x₀ and x₁ based on the partial derivatives and the Learning Rate", null, "Step 4: Repeat Steps 2–3 until the Maximum Number of Steps is reached or the Step Size is less that the Minimum Step Size\n\nAnd we can extrapolate these steps to 3, 4, or even 100 values to optimize for.\n\nIn conclusion, gradient descent is a powerful optimization algorithm that helps us reach the optimal value efficiently. The gradient descent algorithm can be applied to many other optimization problems, making it a fundamental tool for data scientists to have in their arsenal. Onto bigger and better algorithms now!\n\nShreya Rao illustrate and explain Machine Learning algorithms in layman's terms.\n\nOriginal. Reposted with permission.", null, "Get the FREE ebook 'The Great Big Natural Language Processing Primer' and 'The Complete Collection of Data Science Cheat Sheets' along with the leading newsletter on Data Science, Machine Learning, AI & Analytics straight to your inbox.", null, "", null, "" ]

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[ "Online Calculators > Time Calculators\n\nHow old am I if I was born on January 16, 1979?\n\nHow old am I if I was born on January 16, 1979? - January 16, 1979 age to find out how old is someone born on January 16, 1979 in years, months, weeks, days, hours, minutes and seconds.\n\nJanuary 16, 1979 Age\n\nYou are 40 years 9 months and 6 days old\n\nor 489 months old\nor 2,127 weeks old\nor 14,889 days old\nor 357,336 hours old\nor 21,440,160 minutes old\nor 1,286,409,600 seconds old\nYou were born on a Tuesday.\n\nAge Calculator\n\nBirth Date:\nToday's Date:\nHow old am I if I was born in 1979\n\nElectrical Calculators\nReal Estate Calculators\nAccounting Calculators\nConstruction Calculators\nSports Calculators\n\nFinancial Calculators\nCompound Interest Calculator\nMortgage Calculator\nHow Much House Can I Afford\nLoan Calculator\nStock Calculator\nOptions Calculator\nInvestment Calculator\nRetirement Calculator\n401k Calculator\neBay Fee Calculator\nPayPal Fee Calculator\nEtsy Fee Calculator\nMarkup Calculator\nTVM Calculator\nLTV Calculator\nAnnuity Calculator\nHow Much do I Make a Year\n\nMath Calculators\nMixed Number to Decimal\nRatio Simplifier\nPercentage Calculator\n\nHealth Calculators\nBMI Calculator\nWeight Loss Calculator\n\nConversion\nCM to Feet and Inches\nMM to Inches\n\nOthers\nHow Old am I\nRandom Name Picker\nRandom Number Generator\nMultiplication Chart" ]

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[ "Categories\n\n## How to use nested IF statements in Google Sheets?\n\nI am attempting to put the following flow into an equation in google sheets: Compare string from a given cell (cell 1) If the string is \"A\" evaluate a different cell (cell 2) under these conditions If its not \"A\" and the string is \"B\" evaluate a different (cell 2) under these conditions If its […]\n\nCategories\n\n## Bitwise expression in System Verilog causing Icarus Verilog to Hang\n\nI have ripple carry adder in system verilog as shown below. The circuit isn’t functionally correct which is not my problem here. My problem is this line c_in =~(c_in) | carry_in ; cause my simulator (Icarus Verilog) to hang. When I change the operator | to ^ it simulates without hanging. I want to find […]\n\nCategories\n\n## send specific row data to the respective users from excel sheet [closed]\n\nI have a excel sheet with data of users as shown below in table, here i need to send the email to user their specific details containing in column A,B,C. Sub BulkMail() Application.ScreenUpdating = False ThisWorkbook.Activate Dim outMail As Outlook.MailItem ‘Creating variable to hold values of different items of mail Dim sendTo, subj, msg, Name, […]\n\nCategories\n\n## Ethereum transaction giving error ‘invalid sender’\n\nThis is how my contract looks like – pragma solidity >=0.4.25 <0.8.0; contract Calculator { uint public result; event Added(address caller, uint a, uint b, uint res); constructor() public { result = 777; } function add(uint a, uint b) public returns (uint, address) { result = a + b; emit Added(msg.sender, a, b, result); return […]\n\nCategories\n\n## Query to update one column of a table based on a column of a different table using TSQL\n\nI have two tables with below structure : Create table Customer_info (Customer_num varchar(50), Customer_Branch int); Create table Customer_Transaction_Info (Customer_num varchar(50), Branch_Code int, Trns_Measure_One int, Trns_Measure_Two int ); Sample data for each table: insert into Customer_info(Customer_num , Customer_Branch) values(‘A’,1), (‘B’,2), (‘C’,3); Customer_num Customer_Branch ——————- ————— A 1 B 2 C 3 insert into Customer_Transaction_Info (Customer_num,Branch_Code,Trns_Measure_One,Trns_Measure_Two) Values […]\n\nCategories\n\n## How to allocate tickets\n\nI have two files : Category: Category_File = data.frame(c(\"A\",\"A\",\"A\",\"A\",\"A\",\"B\",\"B\",\"B\",\"C\",\"C\",\"D\",\"D\")) colnames(Category_File)= c(\"Category\") Agent: Agent_File = data.frame(c(\"A\",\"A\",\"B\",\"B\",\"C\"),c(\"X\",\"Y\",\"X\",\"Z\",\"Y\"),c(2,2,2,1,2)) colnames(Agent_File) = c(\"Category\",\"Agent\",\"Tickets\") I need to allocate agents according to their category count. Desired Output: outputfile = data.frame(c(\"A\",\"A\",\"A\",\"A\",\"A\",\"B\",\"B\",\"B\",\"C\",\"C\"),c(\"X\",\"X\",\"Y\",\"Y\",\"NA\",\"X\",\"X\",\"Z\",\"Z\",\"Y\",\"NA\",\"NA\")) colnames(outputfile) = c(\"Category\",\"Agent\") Thanks\n\nCategories\n\n## Python Pandas improving calculation time for large datasets currently taking ~400 mins to run\n\nI’m trying to improve performance of a DataFrame I need to build daily, and I wondered if someone had some ideas. I create a simplied example below: First, I have a dict of DataFrame‘s like this. This is time series data so updates daily. import pandas as pd import numpy as np import datetime as […]\n\nCategories\n\n## How do I make this table into a data frame? I’m using R. Trying to get the ANOVA table for it\n\nenter image description here #Here’s what I have, but I don’t think it’s right: gas.test <- data.frame(gas=c(0, 14, 12, 13, 11, 17, 0, 13, 11, 10, 14, 14, 0, 12, 12, 13, 13, 12, 0, 11, 12, 10, 9, 8, 0), Test_Car = factor(rep(c(\"A\",\"B\",\"C\",\"D\", \"E\"),each=5)), Additive = factor(rep(c(1,2,3,4,5),5)))\n\nCategories\n\n## Shader Flickering Normal Map Lighting\n\nI am encountering an issue when using a normal map with directional and/or point lighting. Without using the lighting, I am able to render the object fine. When any amount of lighting is applied, it causes the object to flicker (as shown below). The lighting also works fine when I am not using a normal […]" ]

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[ "# Wien Bridge Oscillator circuits using Op-amp and FET\n\nIf you want a sine wave oscillator. We have many circuit for you. Now I am going to show you the Wien Bridge Oscillator circuit.\n\nIn normal It has low distortion. And can easily adjust the resonance frequency. That depends on the pair of resistors (R) and capacitors (C).\n\nWe can calculate the formula.\n\nF = 1/2 p RC\n\n## How Wien bridge oscillator works\n\nThe first circuit is a simple Wien Bridge Oscillator.\n\nWien bridge oscillator is a sine wave oscillator. It can generate a big of frequencies.\n\nThe oscillator was based on a bridge circuit. Which includes four resistors and two capacitors.\n\nThe oscillator can also be viewed as a positive gain amplifier combined with a bandpass filter that provides positive feedback.\n\nThe field-effect transistor (FET) Q1 works in the linear resistive region to give automatic gain control.\n\nFor the reduction of the R-C network is one-third at the zero phase-shift tremble frequency.", null, "The amplifier gain determined by resistor R1 and equivalent resistor RT must be just equal to three to make up the unity gain positive feedback requirement needed for stable oscillation.\n\nResistors R2 and R3 are set to approximately 1000 ohm less than the required R1 resistance.\n\nThe FET – 2N5020 dynamically provides the trimming resistance needed to make RT one-half of the resistance of R1.\n\nThe circuit composed of R4, D1, and C1 isolates, rectifies, and filters the output sine wave, converting it into a dc potential to control the gate of the FET.\n\nFor the low drain-to-source voltage used, the FET provides a symmetrical linear resistance for a given gate to source voltage.\n\n### Assembling\n\nThis circuit is not designed for PCB. If you do not want to design own PCB. Or use universal PCB Board that difficult.\n\nAlthough the circuits are is not the same. It can produce a sine wave signal as well.\n\n## #The Second: Wien bridge oscillator Circuit\n\nIn this circuit, as Figure 1 the oscillator is based on the R that includes with R1 + P1a (or R2 + P1b), and C is C1, C2 or either C3 (or C4, C5 or C6). And more parts are connected with IC1, IC2 op-amp ICs.\n\nSome output signal from IC2 is sent to the voltage attenuator circuit, that includes of IC3 and T1. The T1 is FET that is used as the variable resistors as the feedback of IC2\n\nThe gain of OP-Amp is controlled by the voltage and changing the control voltage of T1 by P2. It needs to tuning to the oscillation to have high efficiency, on the frequency range is selected.\n\nWhen we use the value of parts as a schematic diagram of the Wien bridge oscillator circuit will have frequencies of about 20 Hz – 22.5 Khz by adjusting P1 and maximum harmonic distortion approximately 2 %.\n\nThis circuit requires a Dual voltage (positive-negative) power supply. so we use the voltage converter circuit as Figure 2 input is 9V to +4.5V, Ground, -4.5V.\n\nWe use principles like the voltage divider using OP-AMP IC-741 and few parts.\n\n#### The parts list\n\nIC1-IC3: CA3140, 4.5MHz, Bimos Operational Amplifier With MOSFET Input/bipolar Output.\nIC4: LM741, Operational Amplifier\nD1,D2: 1N4148, 75V 150mA Diodes\nT1: BF256,BF244, FET\nP1: 47K, potentiometer\nP2: 47K, Pot\n\nPolyester Capacitor\nC1,C4: 150nF or 0.15uF 63V\nC2,C5: 15nF or 0.015uF 63V\nC3,C8: 1.5nF or 0.0015uF 63V\nC7: 22uF 16V tantalum capacitors\nC10: 47uF 16V, electrolytic\nC11,C12: 10uF 16V, electrolytic\n1/4W Resistors tolerance: 5%\nR1,R2: 4.7K\nR3,R7,R8,R9: 100K\nR4: 10K\nR5: 12K\nR6: 150K\nSW1: 3 step selector switch\n\n## GET UPDATE VIA EMAIL\n\nI always try to make Electronics Learning Easy.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]

[ null, "data:image/svg+xml,%3Csvg%20xmlns=%22http://www.w3.org/2000/svg%22%20viewBox=%220%200%20459%20438%22%3E%3C/svg%3E", null ]

[ "# wlanConstellationDemap\n\nConstellation demapping\n\n## Syntax\n\n``y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS)``\n``y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS,demapType)``\n``y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS,phase)``\n``y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS,demapType,phase)``\n\n## Description\n\nexample\n\n````y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS)` demaps received symbols `sym` using the soft-decision approximate log-likelihood-ratio (LLR) method for `numBPSCS`, the number of coded bits per subcarrier per spatial stream.```\n\nexample\n\n````y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS,demapType)` also specifies the demapping type. ```\n\nexample\n\n````y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS,phase)` derotates the symbols clockwise before demapping by the number of radians specified in `phase`.```\n\nexample\n\n````y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS,demapType,phase)` specifies the demapping type and the phase rotation.```\n\n## Examples\n\ncollapse all\n\nPerform a 4096-QAM demapping on a sequence of data bits.\n\nCreate the sequence of data bits.\n\n`bits = randi([0 1],49152,1,'int8');`\n\nPerform constellation mapping on the data bits by using 4096-QAM.\n\n```numBPSCS = 12; sym = wlanConstellationMap(bits,numBPSCS); size(sym)```\n```ans = 1×2 4096 1 ```\n\nPerform 4096-QAM constellation demapping. Because the default demapping type is soft, the output is a vector of soft bits.\n\n```noiseVarEst = 0; y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS); size(y)```\n```ans = 1×2 49152 1 ```\n\nPerform a 256-QAM demapping by using hard demodulation.\n\nCreate the sequence of data bits.\n\n` bits = randi([0 1],416,1);`\n\nPerform the constellation mapping on the data bits by using a 256-QAM constellation.\n\n```numBPSCS = 8; sym = wlanConstellationMap(bits,numBPSCS);```\n\nPerform the hard 256-QAM constellation demapping.\n\n```noiseVarEst = 0; demapType = 'hard'; y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS,demapType);```\n\nVerify that the demapped data matches the original data.\n\n`isequal(bits,y)`\n```ans = logical 1 ```\n\nBPSK and QBPSK demapping for different OFDM symbols for the VHT-SIG-A field by using a soft demodulation. The demapping is defined in IEEE® 802.11ac™-2013 Section 22.3.8.3.3\n\nCreate the sequence of data bits. Specify the two OFDM symbols in columns.\n\n` bits = randi([0 1],48,2,'int8');`\n\nPerform constellation mapping on the data bits. Specify the size of the constellation rotation as the number in columns of the input sequence. The first column is mapped with a BPSK modulation. The second column is modulated with a QBPSK modulation.\n\n```numBPSCS = 1; phase = [0 pi/2]; mappedData = wlanConstellationMap(bits,numBPSCS,phase);```\n\nPerform the constellation demapping with an estimated variance noise equal to zero (no added noise). To derotate the constellation, specify the same phase as in the mapping function. The output is a vector of soft bits ready to be the input of a convolutional decoder.\n\n```noiseVar = 0; demappedData = wlanConstellationDemap(mappedData,noiseVar,numBPSCS,phase);```\n\nVerify that the demapped data matches the original data. Because no noise is present, you can recover the original data without errors by assigning the negative values to a logical 1 and the positive values to a logical 0. In other words, you can convert the soft bits into hard bits.\n\n```demappedBits = int8((demappedData<=0)); isequal(bits,demappedBits)```\n```ans = logical 1 ```\n\nPerform BPSK demapping on a four-dimensional array by using hard demodulation.\n\nCreate the sequence of data bits as an array of four dimensions, with 416 coded bits per subcarrier per spatial stream per interleaver block, four OFDM symbols, two spatial streams, and two segments.\n\n```numCBPSSI = 416; numSym = 4; numSS = 2; numSeg = 2; bits = randi([0 1],numCBPSSI,numSym,numSS,numSeg); size(bits)```\n```ans = 1×4 416 4 2 2 ```\n\nPerform BPSK constellation mapping on the data bits with a rotation of $\\frac{\\pi }{2}$ radians.\n\n```numBPSCS = 1; phase = pi/2; sym = wlanConstellationMap(bits,numBPSCS,phase); size(sym)```\n```ans = 1×4 416 4 2 2 ```\n\nPerform hard QBPSK constellation demapping. To derotate the constellation, specify the same phase as in the mapping function.\n\n```noiseVarEst = 0; demapType = 'hard'; y = wlanConstellationDemap(sym,noiseVarEst,numBPSCS,demapType);```\n\nVerify that the demapped data matches the original data.\n\n`isequal(bits,y)`\n```ans = logical 1 ```\n\n## Input Arguments\n\ncollapse all\n\nReceived symbols, specified as a complex-valued vector, matrix, or multidimensional array. If you specify this input as a matrix or array, the function performs constellation demapping column-wise.\n\nData Types: `double`\nComplex Number Support: Yes\n\nNoise variance estimate, specified as a nonnegative scalar. When you specify the `demapType` input as `'hard'`, the function does not use this input.\n\nExample: `0.7071`\n\nData Types: `double`\n\nNumber of coded bits per subcarrier per spatial stream, specified as log2(M), where M is the modulation order. Therefore, `numBPSCS` must be one of these values.\n\n• `1` for binary phase-shift keying (BPSK) modulation, as specified in section 17.3.5.8 of \n\n• `2` for quadrature phase-shift keying (QPSK) modulation, as specified in section 17.3.5.8 of \n\n• `4` for 16-point quadrature amplitude modulation (16-QAM), as specified in section 17.3.5.8 of \n\n• `6` for 64-QAM, as specified in section 17.3.5.8 of \n\n• `8` for 256-QAM, as specified in section 21.3.12.9 of \n\n• `10` for 1024-QAM, as specified in section 27.3.12.9 of \n\n• `12` for 4096-QAM, as specified in section 36.3.12.8 of \n\nData Types: `double`\n\nDemapping type, specified as one of these values.\n\n• `'hard'` — hard-decision demapping\n\n• `'soft'` — soft-decision approximate LLR method\n\nData Types: `double`\n\nConstellation rotation, in radians, specified as a scalar, vector, or multidimensional array. The size of this input must be compatible with the size of the `sym` input. This input and `sym` have compatible sizes if, for each corresponding dimension, the dimension sizes are either equal or one of them is 1. When one of the dimensions of `sym` is equal to 1 and the corresponding dimension of `phase` is larger than 1, then the output dimensions have the same size as the dimensions of `phase`.\n\nExample: `pi*(0:size(bits,1)/numBPSCS-1).'/2;`\n\nData Types: `double`\n\n## Output Arguments\n\ncollapse all\n\nDemapped symbols, returned as an integer-valued vector, matrix, or multidimensional array . This output has the same size as `sym` except for the number of rows, which is equal to the number of rows of `sym` multiplied by `numBPSCS`.\n\n IEEE Std 802.11™-2020 (Revision of IEEE Std 802.11-2016). “Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications.” IEEE Standard for Information technology — Telecommunications and information exchange between systems. Local and metropolitan area networks — Specific requirements.\n\n IEEE Std 802.11ax™-2021 (Amendment to IEEE Std 802.11-2020). “Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications. Amendment 1: Enhancements for High Efficiency WLAN.” IEEE Standard for Information technology — Telecommunications and information exchange between systems. Local and metropolitan area networks — Specific requirements.\n\n IEEE® P802.11be™/D1.2. “Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications. Amendment 8: Enhancements for Extremely High Throughput (EHT).” Draft Standard for Information technology — Telecommunications and information exchange between systems. Local and metropolitan area networks — Specific requirements." ]

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[ "Latest Banking jobs   »   Quantitative Aptitude Quiz For IBPS PO...\n\n# Quantitative Aptitude Quiz For IBPS PO Mains 2022-10th January\n\nQ1. SBI bank has a free locker allowance for its customers but if any customer carries excess things it is charged at a constant rate per gm. The total charged paid by Veer & Mohit is Rs 110. If both Veer & mohit had carried twice the weight then, they actually did previously, then their charge would have been Rs 200 and Rs 100 respectively, Find charge of Veer?\n(a) 100Rs\n(b) 80Rs\n(c) 160Rs\n(d) 120Rs\n(e) 180Rs\n\nDirections (2-3): A and B invested Rs. (x – 1200) and Rs. (x + 800) in a partnership business. After one year A added Rs. 2000, while B withdraw Rs. 1600 and C joined them with Rs. (x + 2800). After two years the ratio of profit of A, B and C is 35 : 36 : 25.\n\nQ2. Veer and Sameer invested Rs. (x + y) and Rs. (x + 2.5 y) in a business. If at the end of two years profit ratio of Veer and Sameer is 10 : 13. Then find value of 4.5y ?\n(a) 8100 Rs.\n(b) 8400 Rs.\n(c) 7200 Rs.\n(d) 9600 Rs.\n(e) 10800 Rs.\n\nQ3. P, Q and R invested in a business by making investment what A, B and C invested for second year respectively. P and Q invested for 8 months and 10 month respectively and profit share of P & Q &R is 32 : 32 : 25, then find for how many months R invested ?\n(a) 3 months\n(b) 1 months\n(c) 11 months\n(d) 5 months\n(e) 8 months\n\nDirections (4-5): Two vessels A & B contains (x + 24) l & (x + 60) l mixture respectively. Ratio of milk to water in vessel A is 7 : 5, while ratio of milk to orange juice in vessel B is 5 : 4. If 25% from vessel A & 50% from vessel B taken out so remaining milk in vessel A is 13 l more than that of remaining milk in vessel B.\n\nQ4. If 50% from vessel A, 40% from vessel B taken out and mixed together, then find ratio of milk, water & orange in resulting mixture?\n(a) 41 : 12 : 16\n(b) 41 : 10 : 16\n(c) 41 : 8 : 16\n(d) 41 : 15 : 16\n(e) 41 : 17 : 16\n\nQ5. If vessel A & B poured together and X liter of water added in resulting mixture new ratio of milk, water and orange juice become 23 : 10 : 10. Find ‘X’?\n(a) 15 l\n(b) 12 l\n(c) 10 l\n(d) 20 l\n(e) 8 l\n\nDirections (6-7): A shopkeeper have two articles A & B. Marked price of article B is 20% more than marked price of article A, shopkeeper sold article A at 25% discount and article B at 20% discount, if he made 20% loss on article A and 6⅔% profit on article B and total loss of shopkeeper was Rs. 1785.\n\nQ6. If shopkeeper want 25% profit on article A, then find at what price shopkeeper should sold the article?\n(a) 16206.25 Rs.\n(b) 16200. 25Rs.\n(c) 16180.25 Rs.\n(d) 16406.25 Rs.\n(e) 16160.25 Rs.", null, "Directions (8 -9): Shatish invested Rs. 1700 in a scheme ‘A’ for two years which offered S.I. annually at the rate of R% and gets total interest of Rs. 544. Another scheme B, which offered CI annually at the rate of (R – 6) %.\n\nQ8. If a man invested Rs. (2000 + x) in scheme A for 2 years and Rs. (1600 + 3x) in scheme B for 2 years and interest got from scheme A is Rs180 more than interest got from scheme B, then find ‘x’?\n(a) 800 Rs.\n(b) 1000 Rs.\n(c) 1600 Rs.\n(d) 400 Rs.\n(e) 2000 Rs.\n\nQ9. A man invested an amount in the ratio of 3 : 2 at the rate of (R/2-0.5)% & (R – 8)% respectively. If bigger amount invested for two years and man got interest in the ratio of 15 : 16, then find smaller amount invested for how many years?\n(a) 2 years\n(b) 3 years\n(c) 5 years\n(d) 7 years\n(e) 11 years", null, "Solutions", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "#### Congratulations!", null, "Union Budget 2023-24: Free PDF", null, "" ]

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[ "", null, "DID YOU KNOW:\nSeamlessly assign resources as digital activities\n\nLearn how in 5 minutes with a tutorial resource. Try it Now\n\nLearn More", null, "", null, "", null, "# Fourth Grade Math Centers Bundle", null, "", null, "", null, "", null, "4th\nSubjects\nStandards\nResource Type\nFormats Included\n• Zip\nPages\n400 pages\n\\$40.00\nBundle\nList Price:\n\\$49.50\nYou Save:\n\\$9.50\n\\$40.00\nBundle\nList Price:\n\\$49.50\nYou Save:\n\\$9.50\n\n#### Products in this Bundle (9)\n\nshowing 1-5 of 9 products\n\n#### Bonus\n\nMath Center Teacher Tips and Tricks\n\n### Description\n\nThis 4th grade bundle includes hands-on and engaging math centers for the entire year!!! You will be getting a total of 90 math centers!\n\nEach set of centers has the same format, so students will learn the expectations and procedures and then be able to complete centers for the entire year without many new directions.\n\n*********\n\nI also have this resource in a digital version! Click HERE to check it out.\n\nThe centers are engaging and include sorts, task cards, color coding, math writing, matching, etc. Each set of centers has a recording book that students use for all 10 centers in that unit.\n\n**Would you like to learn how I use these math centers? Click HERE to view a free video on how to implement math centers.**\n\nCenters are included in color and black line for your convenience.\n\nJUST ADDED! This bundle now includes a 40 page document with math center tips, schedules, posters and labels!\n\n**********************************************\n\nCenters Included:\n\nBack to School\n\n1. Math journaling\n\n3. Addition & Subtraction (with regrouping)\n\n4. Rounding to the Nearest 10 & 100\n\n5. Area and Perimeter\n\n6. Graphs (Bar Graph and Picture Graph)\n\n7. Multiplication Chart Patterns\n\n8. Missing Factors\n\n9. Clocks to the Nearest Minute\n\n10. Comparing and Equivalent Fractions\n\nPlace Value\n\n1. Math journaling\n\n3. Rounding to the nearest 100\n\n4. Rounding to the Nearest 1,000\n\n5. Making Numbers (greatest and least value)\n\n6. Expanded form\n\n7. Comparing Numbers\n\n8. Multiplying and dividing my 10\n\n9. Digit Value\n\n10. Writing the mystery number problem\n\nMultiplication\n\n1. Math journaling\n\n2. Word Problems (2 digit x 2 digit and 3 digit x 1 digit)\n\n3. Prime or Composite?\n\n4. Multiples\n\n5. Mystery Numbers (using factors)\n\n6. Comparisons (Ex: 144 is 12 times as many as___)\n\n7. Vocabulary\n\n8. Area Models (You can substitute a different strategy if needed.)\n\n9. Multiple Digit Equations\n\n10. Write your own word problem\n\nDivision\n\n1. Math journaling\n\n2. Word Problems\n\n3. Remainder Sort\n\n4. True or False? (Are the equations correct. Includes equations with and without remainders.)\n\n5. Multiplication or Division? (Word problem task cards.)\n\n6. Quotient Sort\n\n7. Vocabulary\n\n8. Missing Factor Multiplication Equations\n\n9. Long Division Equations (without remainders)\n\n10. Write your own word problem (given a certain dividend and quotient)\n\nFractions\n\n1. Math journaling\n\n2. Word Problems (Addition, Subtraction and Multiplying a Fraction by a Whole Number)\n\n3. Comparing Fractions to One-Half\n\n4. Matching fractions with Tenths to their Equivalent Hundredths Fraction\n\n5. Mystery Fraction Word Problems\n\n6. Equivalent Fraction Sort\n\n7. Vocabulary\n\n8. Multiplying Fractions by Whole Numbers\n\n9. Rolling Fractions (Converting Improper Fractions to Mixed Numbers)\n\nDecimals\n\n1. Math journaling (Explaining the difference between a whole number and decimal)\n\n2. Word Problems (Addition and subtraction to the hundredth)\n\n3. Roll a Problem (Value of a decimal and addition of decimals)\n\n4. Rounding to the nearest whole number\n\n5. Making Numbers (making the largest and smallest value with decimals)\n\n6. Expanded Form\n\n7. Color Coding (comparing fractions and decimals to one-half)\n\n8. Comparing decimals\n\n9. Value of digits\n\n10. Word Form\n\nMeasurement\n\n1. Math journaling (Area and Perimeter of a Square)\n\n2. Word Problems (Real World Area and Perimeter Problems)\n\n3. Roll a Problem (Drawing Shapes with a Given Area and Perimiter)\n\n4. Drawing Shapes with a Given Area\n\n5. Vocabulary Match\n\n6. Solving for Area and Perimeter Task Cards\n\n7. Color Coding (units of measurement)\n\n8. Measurement Conversions (standard and metric)\n\n9. Elapsed Time\n\n10. Write the Problem (Area and Perimeter)\n\nGeometry\n\n1. Measuring Angles and Classifying as Acute, Right or Obtuse (Color code)\n\n2. Sorting Attributes or Triangles and Squares (Color code)\n\n3. Lines of Symmetry\n\n4. Drawing (lines, angles, parallel lines and perpendicular lines)\n\n6. Shape Attributes\n\n7. Grouping Shapes by Attributes\n\n8. Drawing Angles with a Given Measurement\n\n9. Vocabulary Sort\n\n10. Math Journal (symmetry)\n\nTest Prep\n\n1. Equivalent Fractions (Color Code)\n\n2. Prime or Composite Sort\n\n3. Addition and Subtraction of Decimals\n\n4. Word Problems\n\n5. Measuring Angles\n\n6. Area and Perimeter\n\n7. Multiplication (2 and 3 digit)\n\n8. Division\n\n9. Vocabulary Sort\n\n10. Math Journal (Decimals)\n\nLooking for these centers for another grade level?\n\nYou might also like:\n\nHuge Math Interactive Notebook Bundle\n\nTotal Pages\n400 pages\nIncluded\nTeaching Duration\nN/A\nReport this Resource to TpT\nReported resources will be reviewed by our team. Report this resource to let us know if this resource violates TpT’s content guidelines.\n\n### Standards\n\nto see state-specific standards (only available in the US).\nLook for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (𝑦 – 2)/(𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (𝑥 – 1)(𝑥 + 1), (𝑥 – 1)(𝑥² + 𝑥 + 1), and (𝑥 – 1)(𝑥³ + 𝑥² + 𝑥 + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.\nLook for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression 𝑥² + 9𝑥 + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(𝑥 – 𝑦)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers 𝑥 and 𝑦.\nAttend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.\nUse appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.\nModel with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose." ]

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[ "@ : Home > Quantitative Aptitude > Trigonometry > Trigonometry SSC Shortcuts\n\n## SSC Shortcuts (Substitution method)\n\nAt times students find trigonometry questions in SSC paper very hard and they leave those questions unattempted. Whereas, there are other students who are well versed with shortcut methods. These students use shortcuts and solve such questions within seconds. Here we are discussing some shortcuts tips and tricks for our students so that next time you are in a position to solve complex questions within seconds.\n\nQ19) If cos ⁡x + cos ⁡y = 2 then the value of sin ⁡x + sin ⁡y is\n\n1)\n0\n2) 1\n3) 2\n4) -1\n\ncos ⁡x + cos ⁡ y = 2\ncos ⁡ x = 1; cos ⁡ y = 1\nx = y = 0\nsin ⁡ x + sin ⁡ y = 0\nHence, option 1.\n\nQ20) The value of", null, "is\n\n1)\n5\n2) 4\n3) 3\n4) 2\n\nEasiest way to solve such questions is to substitute value of θ.\nPut θ = 45°", null, "Hence, option 1.\n\nQ21) If", null, "and ⁡", null, "then cos2α is\n\n1)", null, "2)", null, "3)", null, "4)", null, "Full method to solve this question:", null, "", null, "", null, "Shortcut:\nPut α = 60°; β = 30\ntan⁡60° = n tan30° i.e. n = 3 or n2 = 9\nsin60° = m sin30° i.e. m = √3 or m2 = 3\ncos2⁡α = cos260° = 1/4\nFor m2 = 3; n2 = 9\nOption 1: 1/3 Eliminated.\nOption 2: 1/4 Satisfied.\nOption 3: 1/2 Eliminated.\nOption 4: 3/10 Eliminated.\nHence, option 2.\n\nTrigonometric Identities\n\nIn the last question we got a unique answer using substitution method. However, it may not always be true. At times using, substitution method we may get more than one option. So Lets discuss how to solve such questions.\n\nQ22) If", null, "then the value of", null, "is", null, "Use substitution method, put θ = 90° Answer is either option 3 or option 4.\nAs we did not get a unique answer so put θ=0° Answer is option 4.\nHence, option 4.\n\nQ23) If", null, "then the value of", null, "1)\n9\n2) 0\n3) 1\n4) 4\n\nEasiest substitution for this question is\nPut θ = 45°,Φ = 45°\nx = a sec45° cos45° = a\ny = b sec45° sin⁡45° = b\nz = c tan45° = c", null, "Hence, option 3.\n\nQ24) If sin⁡ θ + cos⁡θ = x then the value of sin6⁡θ + cos6⁡θ is equal to", null, "Use substitution method, put", null, "Answer is either option 3 or option 4.\nNow, put", null, "Answer is either option 1 or option 3.\nSo, common answer from the above substitutions is option 3.\nHence, option 3.\n\nQ25) If sinθ + cosθ = 1, what is the value of sinθ cosθ\n\n1)\n2\n2) 0\n3) 1\n4) 1/2\n\nsinθ + cosθ = 1\n(sinθ + cosθ)2 = 1\nsin2θ + cos2θ + 2sinθ cosθ = 1\n1 + 2sinθ cosθ = 1 i.e. sinθ cosθ = 0\n\nShortcut:\nsinθ + cosθ = 1 so, put θ = 0°\nsinθ cosθ = sin0° cos0° = 0.\nHence, option 2.\n\nQ26) The numerical value of", null, "is\n\n1)\n0\n2) -1\n3) 1\n4) 2\n\nPut θ = 45°\nHence, option 3.\n\nQ27) If cosθ + secθ = 2, then the value of cos6θ + sec6θ is\n\n1)\n1\n2) 2\n3) 4\n4) 8\n\ncosθ + secθ = 2\ncosθ + 1/cosθ = 2\nSo, cosθ = 1 i.e. θ =\n0° cos6θ + sec6θ = cos60° + sec60° = 2\nHence, option 2.\n\nQ28) If sinθcosθ = 1/2, then what is sin6θ + cos6θ equal to\n\n1)\n1\n2) 2\n3) 3\n4) 1/4\n\nsinθcosθ = 1/2\n2sinθcosθ = 1\nsin2θ = 1\n2θ = 90°\nθ = 45°\nsin6θ + cos6θ = sin645° + cos645° = 1/4\nHence, option 4." ]

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[ "Prime Factors of 97379", null, "We define Prime Factors of 97379 as all the prime numbers that when multiplied together equal 97379.\n\nA prime number is an integer greater than 1 whose only factors are 1 and itself. 97379 can only be divided by 1 and 97379 which means that 97379 is a prime number.\n\nSince 1 is neither a prime number nor a prime factor, 97379 is the only Prime Factor of 97379.\n\n\"Prime Factors of 97379\" implies more than one factor, thus Prime Factors of 97379 is not possible to answer.\n\nPrime Factor Calculator\nDo you need the Prime Factors for a particular number? You can submit a number below to find the Prime Factors of that number with detailed explanations.\n\nPrime Factors of 97380\nThere are never two consecutive numbers that are prime numbers. Thus, if you want to learn how to find the Prime Factors of a composite number, check out the next number on our list here." ]

[ null, "https://factorization.info/images/prime-factors-of.png", null ]

[ "### Communities\n\ntag:snake search within a tag\nuser:xxxx search by author id\nscore:0.5 posts with 0.5+ score\n\"snake oil\" exact phrase\ncreated:<1w created < 1 week ago\npost_type:xxxx type of post\nQ&A\n\nGeneral Q&A about physics of any type and at any level\n\n93 posts\nFilters (None)\n20%\n+0 −6\n\nPresuppose that I need vinegar of $c$ concentration, where $c$ < any concentration listed below. Let $w$ be the price of water that I'll use to dilute. How do I deduce which concentration (of V...\n\n1 answer  ·  posted 1mo ago by TextKit‭  ·  closed 16d ago by Mithrandir24601‭\n\n66%\n+2 −0\n\nStarting from the classical osmosis experiment, a U-shaped tube with a semi-permeable membrane, I would like to consider the case when the solute added to one of the compartments (labelled A) is co...\n\n0 answers  ·  posted 2mo ago by Joce‭\n\n33%\n+0 −2\n\nAt home daily, I need to mix my table salt + COLD tap water. As \"Marine salts dissolve faster and more thoroughly when added to circulating water,\" I \"use a powerhead to speed up mixing time — my V...\n\n2 answers  ·  posted 8mo ago by TextKit‭  ·  last activity 3mo ago by matthewsnyder‭\n\n50%\n+0 −0\n\nGravitational waves can be derived from the non-linear Einstein field equations and since they are by definition waves they must obey the wave equation: $u_{tt}=c^{2}u_{xx}$ but in General Rela...\n\n1 answer  ·  posted 4mo ago by Volpina‭  ·  last activity 4mo ago by Mithrandir24601‭\n\n25%\n+0 −4\n\nQuestion I was wondering what force make drop slow down? Does every time a drop push toothpick back into bottle? And of course how to solve it. Struct This is a bottle be pierced by a wooden too...\n\n0 answers  ·  posted 7mo ago by Dead_Bush_Sanpai‭  ·  closed 4mo ago by ArtOfCode‭\n\n60%\n+1 −0\n\nSuppose we have a gravitational wave which obeys the equation: $[G_{tt}-c^{2}G_{xx}]h_{\\mu\\nu}=0$ Lets take the case where $h_{\\mu\\nu}\\ne0$ so we are left with the classical wave equation.Suppose...\n\n0 answers  ·  posted 4mo ago by Volpina‭\n\n66%\n+2 −0\n\nSuppose a ferromagnetic material with initial magnetization $M_o$.Is there some specific formula which calculates the total magnetization $M$ as a function of $M_{o}$ and the Curie temperature $T_{... 1 answer · posted 10mo ago by MissMulan‭ · last activity 4mo ago by Anyon‭ 57% +2 −1 What are the SI units of the wavefunction Ψ(x).I know that [Ψ(x)]^2 describes the probabilty of finding a quantum object at a certain quantum state but what about the wavefunction instead? 1 answer · posted 1y ago by MissMulan‭ · last activity 4mo ago by Anyon‭ 50% +0 −0 Lets say we have a mass connected to a spring.Assuming not any friction the ODE which describes the system is$m\\frac{d^{2}x}{dt^{2}} = -kx$We can set 2 Dirichlet boundary conditions$x(0)=0$an... 1 answer · posted 4mo ago by Volpina‭ · last activity 4mo ago by Technically Natural‭ 66% +2 −0 In the context of quantum field theory, why is it impossible for two photons (or other massless bosons like gluons) to collide and produce a single photon? This kind of a process is supposed to be ... 2 answers · posted 4mo ago by Technically Natural‭ · last activity 4mo ago by Derek Elkins‭ 71% +3 −0 I was recently reading some questions here and there saying that \"relativistic mass is outdated\". I saw someone saying that \"outdated\" doesn't mean the concept is wrong. My question is what physici... 1 answer · posted 2y ago by deleted user · edited 4mo ago by Reinstate Monica on Stack Exchange‭ 50% +0 −0 I'm trying to understand the uncertainty principle and its implications for particle measurement. From what I've read, it seems that the principle states that we cannot simultaneously know the exac... 0 answers · posted 6mo ago by Reinstate Monica on Stack Exchange‭ 71% +3 −0 If we have a force which changes depending on the position of a particle, how can we find the position of the particle at some time$t$? We can find its velocity if it has travelled a given distan... 1 answer · posted 2y ago by MissMulan‭ · edited 7mo ago by Trilarion‭ 60% +1 −0 Suppose we have a arc of charges with some charge density$\\lambda(\\theta) = sin\\theta$.I am using polar coordinates for convenience.But how can I find the direction of the unit vector of the net e... 0 answers · posted 10mo ago by MissMulan‭ · edited 10mo ago by samcarter‭ 25% +0 −4 I need to separate these two parts, the opposite of the GIF below. While my hunky husband was lifting the chair up, and my brawny brother pushing the frame down, I kept hitting the top of the slid... 0 answers · posted 11mo ago by Este‭ · closed 11mo ago by Monica Cellio‭ 60% +1 −0 The general formula for a EM wave (solving for the E field) is: where$\\varepsilon = \\varepsilon _{r}+j\\frac{\\sigma }{\\omega }My professor told me that the conductivity of vacuum is 0 so we... 0 answers · posted 12mo ago by MissMulan‭ 50% +0 −0 Maxwell's first law in differential form states that $$\\triangledown \\cdot E = \\frac{\\rho}{\\epsilon_{o}}$$ . In case of 1d can we say that $$\\rho = \\lambda$$ where $$\\lambda$$ is the linear char... 1 answer · posted 1y ago by MissMulan‭ · last activity 12mo ago by celtschk‭ 28% +0 −3 Part 1 Say we have a round water pool, the radius is 10m, and the water depth is 2m. C is a fixed point: Fixed on the surface of water. Fixed at the center of the pool. F1 is stable forc... 2 answers · posted 1y ago by HolyDamn2.0‭ · last activity 12mo ago by Mithical‭ 28% +0 −3 Say we have a four underwater components tidal power / hydro power system. Component A. A propeller would always orient along the same direction regardless the direction of the flow. This is th... 0 answers · posted 1y ago by HolyDamn2.0‭ · edited 1y ago by HolyDamn2.0‭ 71% +3 −0 The title is a framing for a theoretical question; I'm not asking for practical advice. A friend was recently in this situation and my attempts to apply what I remember of a couple semesters of co... 1 answer · posted 1y ago by Monica Cellio‭ · last activity 1y ago by Olin Lathrop‭ 71% +3 −0 When a pendulum made of a conducting material moves through a magnetic field, it's a well-established fact that it experiences a retarding force, thus slowing it down, however, I'm unable to unders... 1 answer · posted 1y ago by esrdtfghjk‭ · last activity 1y ago by Olin Lathrop‭ 60% +1 −0 Suppose we perform the double slit experiment , but we fire instead 2 electrons instead of 1. In the double slit experiment performed in the 1920s a interference pattern was observed at the screen... 0 answers · posted 1y ago by MissMulan‭ 33% +0 −2 Suppose we have a free falling object inside a planet's gravitational field with strength g.The planet's atmosphere provides a drag force which is dependant from the u^2 of the particle. Suppose t... 0 answers · posted 1y ago by MissMulan‭ · edited 1y ago by MissMulan‭ 25% +0 −4 Suppose we are using a force gauge to measure gravity in a planet. We set the gauge force to the 1N range which has a resolution of .01 N. From its specs the error introduced during the measurmen... 0 answers · posted 1y ago by MissMulan‭ 50% +0 −0 Get 2 conductors and seperate them we can use Gauss's law to calculate the capacitance created by the seperation of the 2 conductors.Can we use other laws of electromagnetism to calculate the induc... 2 answers · posted 1y ago by MissMulan‭ · last activity 1y ago by TonyStewart‭ 50% +0 −0 What examples of a system can be described by a system of ordinary differential equations? 1 answer · posted 1y ago by MissMulan‭ · last activity 1y ago by Olin Lathrop‭ 25% +0 −4 Im designing a capacitor and I have decided to make the surface of 1 plate of the capacitor bigger than the other plate. How are the charges sorted through A2?Are they spread out to cover all th... 0 answers · posted 1y ago by MissMulan‭ 50% +0 −0 If we have a LC high pass filter the transfer function H(s) becomes: $$H(s) = \\cfrac{sL}{sL + \\cfrac{1}{sC}}$$ If we solve for s to find a pole of the transfer function we get: $$s = j \\cfrac... 0 answers · posted 1y ago by MissMulan‭ · closed 1y ago by MissMulan‭ 25% +0 −4 These WorldFamous companies advertise that their lenses can control myopia for kids, NOT adults. But none of these lenses are approved by FDA. I asked my optometrist why merely kids, not adults. Bu... 1 answer · posted 1y ago by TextKit‭ · closed 1y ago by Mithical‭ 33% +0 −2 My optometrist said that round eyeglass lenses are OPTICALLY better than rectangular, particularly to correct myopia. He was NOT referring to fashion or style. I couldn't understand his explanatio... 1 answer · posted 1y ago by TextKit‭ · edited 1y ago by TextKit‭ 50% +0 −0 With time dilation a cosmonaut could travel forth in time, especially in light speed. But are there much lesser speeds which might be achievable by humans in the next 100 years which could also in... 0 answers · posted 1y ago by deleted user 33% +0 −2 I never grokked the optics behind LightHouses and the WW2 poster below work. LightHouse beams are narrow and focussed. Doubtless, the light source can fail to illumine a seafarer or the enemy subm... 1 answer · posted 1y ago by TextKit‭ · last activity 1y ago by Olin Lathrop‭ 28% +0 −3 Please see below screenshot of 15:59.The LED spot light (fastened to the front of the boat) illumines merely a few meters in front, and fails to illumine most of the water between the boat and the... 1 answer · posted 1y ago by TextKit‭ · last activity 1y ago by Olin Lathrop‭ 50% +1 −1 I was looking for equation of motion. I came up with a solution but it doesn't satisfy me. Cause I was trying to find motion of that particle using Lagrangian. We know that$$W=\\int \\vec F\\cdot d\\... 1 answer · posted 2y ago by deleted user · edited 1y ago by deleted user 50% +1 −1 Which one is correct? $$E=mc^2$$ or $$E^2=(mc^2)^2+(pc)^2$$ I mostly seen $$E=mc^2$$ from my childhood, and when I was learning problem solving in relativistic mechanics I had seen $$E^2=(mc^2)^2... 1 answer · posted 2y ago by deleted user · edited 1y ago by deleted user 71% +3 −0$$\\sum_i F_i \\cdot \\delta r_i$$is virtual work when internal force is 0. For that reason,$$\\sum_i F_i \\cdot \\delta r_i = 0$$Here internal force stands for what? When a object's displacement ... 1 answer · posted 2y ago by deleted user · last activity 1y ago by deleted user 60% +1 −0 I'm interested in knowing the surface temperature of both sides of a double-pane or triple-pane window. Given the R-value of the window, and the air temps outside and inside, how can I calculate t... 1 answer · posted 1y ago by re89j‭ · last activity 1y ago by Olin Lathrop‭ 66% +2 −0 I want to write some code to control my whole house humidifier. I want my code to calculate the percentile relative humidity above which dew will form on my windows. I have sensors for air tempera... 0 answers · posted 1y ago by re89j‭ 40% +0 −1 I was looking for book on classical thermodynamics. I found lot of related posts in PSE but couldn't find a book which type I was expecting. I was searching for book which covers the whole thermody... 0 answers · posted 1y ago by deleted user · edited 1y ago by deleted user 60% +1 −0 In the lab I changed the angle the light hits a photoresistor and it doesnt obey Lambert's cosine law the conductivity of the photoresistor drops fast from +-20 to +-30 degrees angle.Why? 1 answer · posted 1y ago by MissMulan‭ · last activity 1y ago by Olin Lathrop‭ 75% +4 −0 The Rayleigh-Jeans law does a good job of describing the spectral radiance of a black body at low frequencies:$$B_{\\nu}(T)=\\frac{2kT\\nu^2}{c^2}$$with T the temperature and \\nu the frequency... 1 answer · posted 1y ago by HDE 226868‭ · last activity 1y ago by Derek Elkins‭ 60% +1 −0 May it be that there are more than 8/9 planets in our solar system which aren't detectable with the current technologies? Is there any theory suggesting that our solar system has more planets, all... 1 answer · posted 1y ago by deleted user · last activity 1y ago by Canina‭ 60% +1 −0 In classical physics book of kleppner, i read that An atom can \"jump\" from one stationary state a to a lower b by emitting radiation with E_a-E_b. The frequency of the emitted \"package of radi... 1 answer · posted 1y ago by deleted user · last activity 1y ago by Olin Lathrop‭ 71% +3 −0 I know that equation for parallel resistance is$$\\frac{1}{r_{tot}}=\\sum_i \\frac{1}{r_i}$$But i wonder to see equation of series spring constant. If we add multiple spring in series. Then their ... 1 answer · posted 1y ago by deleted user · last activity 1y ago by Olin Lathrop‭ 50% +0 −0$$\\begin{alignat}{2} && \\vec \\nabla \\cdot \\vec D & = \\rho_f \\\\ & \\implies &\\int_V \\vec{\\nabla} \\cdot \\vec D \\mathrm d\\tau & = \\int_V \\rho_f\\ \\mathrm d \\tau \\\\ & \\impl... 1 answer · posted 2y ago by deleted user · last activity 1y ago by deleted user 20% +0 −6 IKEA discontinued this NOVEMBER lamp in 2007, but I still use it. IKEA confirmed that they did not intend or design any cover or share with this lamp, and the light bulb is supposed to be exposed. ... 2 answers · posted 2y ago by TextKit‭ · last activity 2y ago by celtschk‭ 71% +3 −0 Momentum is proportional to an object's velocity, and kinetic energy is proportional to the square of its velocity\\dfrac{mv^2}{2}$. It's pretty intuitive that if object B is going twice as fast a... 3 answers · posted 2y ago by gmcgath‭ · last activity 2y ago by celtschk‭ 60% +1 −0 All given metrics are for orthonormal-basis. 2 dimensional spacetime : I saw that Minkowski Metric looks like this : $$\\pmatrix{-1 & 0 \\\\ 0 & 1}$$ or$\\$\\pmatrix{1 & 0 \\\\ 0 & -1}...\n\n1 answer  ·  posted 2y ago by deleted user  ·  edited 2y ago by deleted user\n\n42%\n+1 −2\n\nIs it plausible to desire a \"universal\" calendar applicable everywhere in our universe? Must calendars be based on solar systems (Must calendars be \"relational\")? It might be that the universe \"...\n\n3 answers  ·  posted 2y ago by deleted user  ·  edited 2y ago by Trilarion‭\n\n42%\n+1 −2\n\nAre there areas in the observable universe which surely cannot contain galaxies with planets that can support life as we know them?\n\n1 answer  ·  posted 2y ago by deleted user  ·  last activity 2y ago by Olin Lathrop‭" ]

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[ "# Count number of sheets in excel vb script array\n\n#### Reading bus 20a timetable sheet\n\nHow do I get the number of used columns in a worksheet using Excel.Application COM object in VBScript? Thanks. Sheets(Array(\"101\", \"103\")).Select The above is a line of code produced by the Excel macro writer. I need to iterate thru several lines of a worksheet and pick out appropriate items to put into the array statement. how to get total no of sheet in excel file using C# ... how to retrieve total number of sheet in a excel file. ... workBook.Sheets.Count; this line gets the number of ... Excel Formula Training. Formulas are the key to getting things done in Excel. In this accelerated training, you'll learn how to use formulas to manipulate text, work with dates and times, lookup values with VLOOKUP and INDEX & MATCH, count and sum with criteria, dynamically rank values, and create dynamic ranges.\n\n#### University of south carolina college of education ranking\n\nCable tray sizing formula.asp\n\nUse an array formula to count ranges of times in Excel 2003; Use an array formula to count the number of children who will attend lunch in Excel 2003; Use an array formula to count the number of rows that match 2 criteria in Excel 2003; Use an array formula to count the number of occurrences when the value in column A is greater than or equal ...\n\n#### America east forums\n\nWhere ‘Worksheets.Count’ represents the number of available worksheets in the workbook Sub CopySheet_End() ActiveSheet.Copy After:=Worksheets(Worksheets.Count) End Sub In the above example we are Copying the active worksheet to the end of the worksheet. Excel offers several count functions that quantify the number of cells in a selected range that contain a specific type of data. The job of the COUNTBLANK function is to count the number of cells in a selected range that either contains no data or contains a formula that returns a blank or null value This Excel tutorial explains how to use an array formula to count the number of rows that match two criteria in Excel 2003 and older versions (with screenshots and step-by-step instructions). Question: In Microsoft Excel 2003/XP/2000/97, I have a workbook with 2 sheets: The formulas in the following examples perform a lookup based on a single criteria across multiple sheets. In the first example, a non-array formula is used. However, the formula can become rather cumbersome if many sheets are involved in the lookup. If this is the case, the array formula in the second example can be used instead. I have 6 sheets to count and all the names are in column A (from A1:A100) on each sheet. The names are not in a particular order. On sheet 7 I want to have a cell beside each persons name that counts the number of times their name appeared on the other 6 sheets. WorksheetFunction.CountA method (Excel) 05/22/2019; 2 minutes to read +1; In this article. Counts the number of cells that are not empty and the values within the list of arguments.\n\n#### Dora crib sheet\n\nRead data from Excel sheet using vbscript If this is your first visit, be sure to check out the FAQ by clicking the link above. You may have to register before you can post: click the register link above to proceed.\n\n#### Tms9900nl datasheets\n\nDec 01, 2010 · HI, I'm trying to automate the processing of some Excel 2010 spreadsheets by running a VBScript on the command line (WSH) under Windows 7. I declare an array with Dim ss(25202,10) and assign to it Where the arguments, value1, [value2], etc., can be any values, arrays of values, or references to cell ranges. In recent versions of Excel (2007 and later), you can enter up to 255 value arguments to the Excel Count function, each of which may be single values or arrays of cells or values.\n\n#### Paper free sheetrock\n\nApr 10, 2017 · As with many worksheet functions you will have to call the Application.Worksheet.Function property in the VBE to gain access. For this case we are to use the .Countif ...\n\nJan 27, 2010 · How to find Excel worksheet by name (QTP, VBScript) […] Trackback fromGetting Excel Sheet Name in QTP - Tech Travel Hub Friday, 21 June, 2019 […] if you need to customize the code….i.e–If you need a particular sheet.. simple just put the below code […] Leave a Reply Cancel reply *\n\n#### Poochies dog grooming cardiff\n\nI want to get the sheet count of an excel workbook in QTP. Let me know the different methods of doing it. ... and then counted the number of used sheet in that excel ... Hi,There must be a simple answer to this problem but I cant seem to find it anywhere, is there a piece of vba I can use that will count the number of worksheets within a workbook. So this value can then be stored as a variable..Thanks,Jon Count values in excel column with VBA I have two columns with data in one of column I have differents Dates, I need count all values in this columns and display the value in other cell. This spreadsheet contain one million of data but this data can vary. I have 6 sheets to count and all the names are in column A (from A1:A100) on each sheet. The names are not in a particular order. On sheet 7 I want to have a cell beside each persons name that counts the number of times their name appeared on the other 6 sheets.\n\nI would like to write a macro to count the occurances of a text string in a range. The string has the format of nnn.nnn.nnnnn.nnn where n = any number; only the periods are constant. I can find the strings in Excel using *.*.*.* but don't know how to use VBA to count the number of occurances. Hi,There must be a simple answer to this problem but I cant seem to find it anywhere, is there a piece of vba I can use that will count the number of worksheets within a workbook. So this value can then be stored as a variable..Thanks,Jon Sheets(Array(\"101\", \"103\")).Select The above is a line of code produced by the Excel macro writer. I need to iterate thru several lines of a worksheet and pick out appropriate items to put into the array statement." ]

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[ "# Vietnamese - French conference in applied mathematics\n\n9-11 July 2018\nHo Chi Minh City University of Science\nAsia/Ho_Chi_Minh timezone\n\n## Asymptotic behavior of the error between two different Euler schemes for the Lévy driven SDEs\n\n11 Jul 2018, 12:00\n30m\nHo Chi Minh City University of Science\n\n#### Ho Chi Minh City University of Science\n\n227 Nguyễn Văn Cừ, Phường 4, T.P. Hồ Chí Minh\n\n### Speaker\n\nMs Thi Bao Tram NGO (PhD student)\n\n### Description\n\nWe study the Multi-level Monte Carlo method introduced by Giles and its applications to finance which is significantly more efficient than the classical Monte Carlo method. This method for the stochastic differential equations driven by only Brownian Motion had been studied by Ben Alaya and Kebaier . Here, we consider the stochastic differential equation driven by a pure jump Lévy process. When the Lévy process have a Brownian component, the speed of convergence of the multilevel was recently studied by Dereich and Li .\n\nNow, we prove the stable law convergence theorem in the spirit of Jacod . More precisely, we consider the SDE of form\n\\begin{equation}\nX_t=x_0+\\int_0^t f(X_{s-})dY_s, (1)\n\\end{equation}\nwith $f\\in\\mathcal{C}^3$ and $Y$ is a Lévy process with the triplet $(b,0,F)$ and look at the asymptotic behavior of the normalized error process $u_{n,m}(X^n-X^{nm})$ where $X^n$ and $X^{nm}$ are two different Euler approximations with step sizes $1/n$ and $1/nm$ respectively. The rate $u_{n,m}$ is an appropriate rate going to infinity such that the normalized error converges to non-trivial limit. Under some different assumptions on the properties of the Lévy process $Y$ in $(1)$, we found different suitable forms of the rate $u_{n,m}$.\n\n Jean Jacod. The Euler scheme for Lévy driven stochastic differential equations: Limit theorems. The Annals of Probability, 2004, Vol.32, No.3A, 1830-1872.\n\n Mohamed Ben Alaya and Ahmed Kebaier. Central limit theorem for the multilevel Monte Carlo Euler method. Ann.Appl. Probab. 25(1): 211-234, 2015.\n\n Michael B.Giles. Multilevel Monte Carlo path simulation, Oper. Res., 56(3): 607-617, 2008.\n\n Steffen Dereich and Sangmeng Li. Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes. Ann. Appl. Probab., 26(1): 136-185, 2016.\n\n### Primary authors\n\nProf. Ahmed KEBAIER (Associate Professor) Prof. Mohamed BEN ALAYA (Professor) Ms Thi Bao Tram NGO (PhD student)\n\n### Presentation Materials\n\nThere are no materials yet." ]

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[ "SBVS372B December   2018  – August 2019\n\nPRODUCTION DATA.\n\n1. Features\n2. Applications\n3. Description\n1.     Device Images\n4. Revision History\n5. Pin Configuration and Functions\n6. Specifications\n7. Detailed Description\n1. 7.1 Overview\n2. 7.2 Functional Block Diagrams\n3. 7.3 Feature Description\n4. 7.4 Device Functional Modes\n8. Application and Implementation\n1. 8.1 Application Information\n2. 8.2 Typical Application\n9. Power Supply Recommendations\n10. 10Layout\n11. 11Device and Documentation Support\n12. 12Mechanical, Packaging, and Orderable Information\n\n#### Package Options\n\nRefer to the PDF data sheet for device specific package drawings\n\n• DRV|6\n• DRV|6\n\n#### 8.2.2.2 Selecting Feedback Divider Resistors\n\nFor this design example, VOUT is set to 5 V. The following equations set the feedback divider resistors for the desired output voltage:\n\nEquation 10. VOUT = VFB × (1 + R1 / R2)\nEquation 11. R1 + R2 ≤ VOUT / (IFB × 100)\n\nFor improved output accuracy, use Equation 11 and IFB(TYP) = 10 nA as listed in the Electrical Characteristics table to calculate the upper limit for series feedback resistance, R1 + R2 ≤ 5 MΩ.\n\nThe control-loop error amplifier drives the FB pin to the same voltage as the internal reference (VFB = 1.24 V as listed in the Electrical Characteristics table). Use Equation 10 to determine the ratio of R1 / R2 = 3.03. Use this ratio and solve Equation 11 for R2. Now calculate the upper limit for R2 ≤ 1.24 MΩ. Select a standard resistor value for R2 = 1.18 MΩ.\n\nReference Equation 10 and solve for R1:\n\nEquation 12. R1 = (VOUT / VFB – 1) × R2\n\nFrom Equation 12, R1 = 3.64 MΩ can be determined. Select a standard resistor value for R1 = 3.6 MΩ. From Equation 10, VOUT = 5.023 V." ]

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[ "### copy some intuition from the paper\n\nparent 2fc7c984\n ... ... @@ -19,7 +19,9 @@ This definition varies slightly from the original one in~\\cite{catlogic}. \\end{align*} \\end{defn} \\ralf{Copy the explanation from the paper, when that one is more polished.} The key intuition behind COFEs is that elements $x$ and $y$ are $n$-equivalent, notation $x \\nequiv{n} y$, if they are \\emph{equivalent for $n$ steps of computation}, \\ie if they cannot be distinguished by a program running for no more than $n$ steps. In other words, as $n$ increases, $\\nequiv{n}$ becomes more and more refined (\\ruleref{cofe-mono})---and in the limit, it agrees with plain equality (\\ruleref{cofe-limit}). In order to solve the recursive domain equation in \\Sref{sec:model} it is also essential that COFEs are \\emph{complete}, \\ie that any chain has a limit (\\ruleref{cofe-compl}). \\begin{defn} An element $x \\in \\cofe$ of a COFE is called \\emph{discrete} if ... ... @@ -35,6 +37,8 @@ This definition varies slightly from the original one in~\\cite{catlogic}. It is \\emph{contractive} if $\\All n, x \\in \\cofe, y \\in \\cofe. (\\All m < n. x \\nequiv{m} y) \\Ra f(x) \\nequiv{n} f(x)$ \\end{defn} Intuitively, applying a non-expansive function to some data will not suddenly introduce differences between seemingly equal data. Elements that cannot be distinguished by programs within $n$ steps remain indistinguishable after applying $f$. The reason that contractive functions are interesting is that for every contractive $f : \\cofe \\to \\cofe$ with $\\cofe$ inhabited, there exists a fixed-point $\\fix(f)$ such that $\\fix(f) = f(\\fix(f))$. \\begin{defn} ... ... @@ -73,6 +77,16 @@ Note that the composition of non-expansive (bi)functors is non-expansive, and th \\melt \\mincl \\meltB \\eqdef{}& \\Exists \\meltC. \\meltB = \\melt \\mtimes \\meltC \\tagH{ra-incl} \\end{align*} \\end{defn} \\noindent RAs are closely related to \\emph{Partial Commutative Monoids} (PCMs), with two key differences: \\begin{enumerate} \\item The composition operation on RAs is total (as opposed to the partial composition operation of a PCM), but there is a specific subset of \\emph{valid} elements that is compatible with the operation (\\ruleref{ra-valid-op}). \\item Instead of a single unit that is an identity to every element, there is a function $\\mcore{-}$ assigning to every element $\\melt$ its \\emph{(duplicable) core} $\\mcore\\melt$, as demanded by \\ruleref{ra-core-id}. \\\\ We further demand that $\\mcore{-}$ is idempotent (\\ruleref{ra-core-idem}) and monotone (\\ruleref{ra-core-mono}) with respect to the usual \\emph{extension order}, which is defined similar to PCMs (\\ruleref{ra-incl}). This idea of a core is closely related to the concept of \\emph{multi-unit separation algebras}~\\cite{Dockins+:aplas09}, with the key difference that the core is a \\emph{function} defining a \\emph{canonical} unit'' $\\mcore\\melt$ for every element~$\\melt$. \\end{enumerate} \\begin{defn} It is possible to do a \\emph{frame-preserving update} from $\\melt \\in \\monoid$ to $\\meltsB \\subseteq \\monoid$, written $\\melt \\mupd \\meltsB$, if ... ... @@ -80,9 +94,9 @@ Note that the composition of non-expansive (bi)functors is non-expansive, and th We further define $\\melt \\mupd \\meltB \\eqdef \\melt \\mupd \\set\\meltB$. \\end{defn} \\ralf{Copy the explanation from the paper, when that one is more polished.} The assertion $\\melt \\mupd \\meltsB$ says that every element $\\melt_\\f$ compatible with $\\melt$ (we also call such elements \\emph{frames}), must also be compatible with some $\\meltB \\in \\meltsB$. Intuitively, this means that whatever assumptions the rest of the program is making about the state of $\\gname$, if these assumptions are compatible with $\\melt$, then updating to $\\meltB$ will not invalidate any of these assumptions. Since Iris ensures that the global ghost state is valid, this means that we can soundly update the ghost state from $\\melt$ to a non-deterministically picked $\\meltB \\in \\meltsB$. \\subsection{CMRA} ... ...\n \\section{Model and semantics} \\label{sec:model} The semantics closely follows the ideas laid out in~\\cite{catlogic}. ... ...\nMarkdown is supported\n0% or\nYou are about to add 0 people to the discussion. Proceed with caution.\nFinish editing this message first!" ]

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[ "# User’s Guide, Chapter 55: Advanced Meter Topics¶\n\n## Objects for Organizing Hierarchical Partitions¶\n\nHierarchical metrical structures can be described as a type of fractional, space-preserving tree structure. With such a structure we partition and divide a single duration into one or more parts, where each part is a fraction of the whole. Each part can, in turn, be similarly divided. The objects for configuring this structure are the MeterTerminal and the MeterSequence objects.\n\nMeterTerminal and the MeterSequence objects are for advanced configuration. For basic data access about common meters, see the discussion of TimeSignature, below.\n\n## Creating and Editing MeterTerminal Objects¶\n\nA MeterTerminal is a node of the metrical tree structure, defined as a duration expressed as a fraction of a whole note. Thus, 1/4 is 1 quarter length (QL) duration; 3/8 is 1.5 QL; 3/16 is 0.75 QL. For this model, denominators are limited to n = 2 :superscript:`x`, for x between 1 and 7 (e.g. 1/1 to 1/128).\n\nMeterTerminals can additionally store a weight, or a numerical value that can be interpreted in a variety of different ways.\n\nThe following examples in the Python interpreter demonstrate creating a MeterTerminal and accessing the `numerator` and `denominator` attributes. The `duration` attribute stores a `Duration` object.\n\n```from music21 import *\nmt = meter.MeterTerminal('3/4')\nmt\n```\n``` <music21.meter.core.MeterTerminal 3/4>\n```\n```mt.numerator, mt.denominator\n```\n``` (3, 4)\n```\n```mt.duration.quarterLength\n```\n``` 3.0\n```\n\nA MeterTerminal can be broken into an ordered sequence of MeterTerminal objects that sum to the same duration. This new object, to be discussed below, is the MeterSequence. A MeterTerminal can be broken into these duration-preserving components with the `subdivide()` method. An argument for subdivision can be given as a desired number of equal-valued components, a list of numerators assuming equal-denominators, or a list of string fraction representations.\n\n```mt.subdivide(3)\n```\n``` <music21.meter.core.MeterSequence {1/4+1/4+1/4}>\n```\n```mt.subdivide([3,3])\n```\n``` <music21.meter.core.MeterSequence {3/8+3/8}>\n```\n```mt.subdivide(['1/4','4/8'])\n```\n``` <music21.meter.core.MeterSequence {1/4+4/8}>\n```\n\n## Creating and Editing MeterSequence Objects¶\n\nA MeterSequence object is a sub-class of a MeterTerminal. Like a MeterTerminal, a MeterSequence has a `numerator`, a `denominator`, and a `duration` attribute. A MeterSequence, however, can be a hierarchical tree or sub-tree, containing an ordered sequence of MeterTerminal and/or MeterSequence objects.\n\nThe ordered collection of MeterTerminal and/or MeterSequence objects can be accessed like Python lists. MeterSequence objects, like MeterTerminal objects, store a weight that by default is the sum of constituent weights.\n\nThe `partition()` and `subdivide()` methods can be used to configure the nested hierarchical structure.\n\nThe `partition()` method replaces existing MeterTerminal or MeterSequence objects in place with a new arrangement, specified as a desired number of equal-valued components, a list of numerators assuming equal-denominators, or a list of string fraction representations.\n\nThe `subdivide()` method returns a new MeterSequence (leaving the source MeterSequence unchanged) with an arrangement of MeterTerminals as specified by an argument in the same form as for the `partition()` method.\n\nNote that MeterTerminal objects cannot be partitioned in place. A common way to convert a MeterTerminal into a MeterSequence is to reassign the returned MeterSequence from the `subdivide()` method to the position occupied by the MeterTerminal.\n\nThe following example creates and partitions a MeterSequence by re-assigning subdivisions to MeterTerminal objects. The use of Python list-like index access is also demonstrated.\n\n```ms = meter.MeterSequence('3/4')\nms\n```\n``` <music21.meter.core.MeterSequence {3/4}>\n```\n```ms.partition([3,3])\nms\n```\n``` <music21.meter.core.MeterSequence {3/8+3/8}>\n```\n```ms\n```\n``` <music21.meter.core.MeterTerminal 3/8>\n```\n```ms = ms.subdivide([3,3])\nms\n```\n``` <music21.meter.core.MeterSequence {3/16+3/16}>\n```\n```ms\n```\n``` <music21.meter.core.MeterSequence {{3/16+3/16}+3/8}>\n```\n```ms = ms.subdivide([1,1,1])\nms\n```\n``` <music21.meter.core.MeterTerminal 1/8>\n```\n```ms\n```\n``` <music21.meter.core.MeterSequence {1/8+1/8+1/8}>\n```\n```ms\n```\n``` <music21.meter.core.MeterSequence {{3/16+3/16}+{1/8+1/8+1/8}}>\n```\n\nThe resulting structure can be graphically displayed with the following diagram:\n\n```# 3/8 divisions\n```", null, "Numerous MeterSequence attributes provide convenient ways to access information about, or new objects from, the nested tree structure. The `depth` attribute returns the depth count at any node within the tree structure; the `flat` property returns a new, flat MeterSequence constructed from all the lowest-level MeterTerminal objects (all leaf nodes).\n\n```ms.depth\n```\n``` 2\n```\n```ms.depth\n```\n``` 1\n```\n```ms.flat\n```\n``` <music21.meter.core.MeterSequence {3/16+3/16+1/8+1/8+1/8}>\n```\n\nNumerous methods provide ways to access levels (slices) of the hierarchical structure, or all nodes found at a desired hierarchical level. As all components preserve the duration of their container, all levels have the same total duration. The `getLevel()` method returns, for a given depth, a new, flat MeterSequence. The `getLevelSpan()` method returns, for a given depth, the time span of each node as a list of start and end values.\n\n```ms.getLevel(0)\n```\n``` <music21.meter.core.MeterSequence {3/8+3/8}>\n```\n```ms.getLevel(1)\n```\n``` <music21.meter.core.MeterSequence {3/16+3/16+1/8+1/8+1/8}>\n```\n```ms.getLevelSpan(1)\n```\n``` [(0.0, 0.75), (0.75, 1.5), (1.5, 2.0), (2.0, 2.5), (2.5, 3.0)]\n```\n```ms.getLevelSpan(1)\n```\n``` [(0.0, 0.5), (0.5, 1.0), (1.0, 1.5)]\n```\n\nFinally, numerous methods provide ways to find and access the relevant nodes (the MeterTerminal or MeterSequence objects) active given a quarter length position into the tree structure. The `offsetToIndex()` method returns, for a given QL, the index of the active node. The `offsetToSpan()` method returns, for a given QL, the span of the active node. The `offsetToDepth()` method returns, for a given QL, the maximum depth at this position.\n\n```ms.offsetToIndex(2.5)\n```\n``` 1\n```\n```ms.offsetToSpan(2.5)\n```\n``` (1.5, 3.0)\n```\n```ms.offsetToDepth(.5)\n```\n``` 2\n```\n```ms.offsetToDepth(.5)\n```\n``` 1\n```\n```ms.getLevel(1).offsetToSpan(.5)\n```\n``` (0, 0.75)\n```\n\nThe music21 `TimeSignature` object contains four parallel MeterSequence objects, each assigned to the attributes `displaySequence`, `beatSequence`, `beamSequence`, `accentSequence`. The following displays a graphical realization of these four MeterSequence objects.\n\n```# four MeterSequence objects\n```", null, "The TimeSignature provides a model of all common hierarchical structures contained within a bar. Common meters are initialized with expected defaults; however, full MeterSequence customization is available.\n\n## Configuring Display¶\n\nThe TimeSignature `displaySequence` MeterSequence employs the highest-level partitions to configure the displayed time signature symbol. If more than one partition is given, those partitions will be interpreted as additive meter components. If partitions have a common denominator, a summed numerator (over a single denominator) can be displayed by setting the TimeSignature `summedNumerator` attribute to True. Lower-level subdivisions of the TimeSignature MeterSequence are not employed.\n\nNote that a new MeterSequence instance can be assigned to the `displaySequence` attribute with a duration and/or partitioning completely independent from the `beatSequence`, `beamSequence`, and `accentSequence` MeterSequences.\n\nThe following example demonstrates setting the display MeterSequence for a TimeSignature. NOTE that there is currently a bug in the first one that is showing 5/16 instead of 5/8. We hope to fix this soon.\n\n```from music21 import stream, note\nts1 = meter.TimeSignature('5/8') # assumes two partitions\nts1.displaySequence.partition(['3/16', '1/8', '5/16'])\nts2 = meter.TimeSignature('5/8') # assumes two partitions\nts2.displaySequence.partition(['2/8', '3/8'])\nts2.summedNumerator = True\ns = stream.Stream()\nfor ts in [ts1, ts2]:\nm = stream.Measure()\nm.timeSignature = ts\nn = note.Note('b')\nn.quarterLength = 0.5\nm.repeatAppend(n, 5)\ns.append(m)\n\ns.show()\n```", null, "## Configuring Beam¶\n\nThe TimeSignature `beamSequence` MeterSequence employs the complete hierarchical structure to configure the single or multi-level beaming of a bar. The outer-most partitions can specify one or more top-level partitions. Lower-level partitions subdivide beam-groups, providing the appropriate beam-breaks when sufficiently small durations are employed.\n\nThe `beamSequence` MeterSequence is generally used to create and configure `Beams` objects stored in `Note` objects. The TimeSignature `getBeams()` method, given a list of `Duration` objects, returns a list of `Beams` objects based on the TimeSignature `beamSequence` MeterSequence.\n\nMany users may find the Stream `makeBeams()` method the most convenient way to apply beams to a Measure or Stream of Note objects. This method returns a new Stream with created and configured Beams.\n\nThe following example beams a bar of 3/4 in four different ways. The diversity and complexity of beaming is offered here to illustrate the flexibility of this model.\n\n```ts1 = meter.TimeSignature('3/4')\nts1.beamSequence.partition(1)\nts1.beamSequence = ts1.beamSequence.subdivide(['3/8', '5/32', '4/32', '3/32'])\n\nts2 = meter.TimeSignature('3/4')\nts2.beamSequence.partition(3)\n\nts3 = meter.TimeSignature('3/4')\nts3.beamSequence.partition(3)\n\nfor i in range(len(ts3.beamSequence)):\nts3.beamSequence[i] = ts3.beamSequence[i].subdivide(2)\n\nts4 = meter.TimeSignature('3/4')\nts4.beamSequence.partition(['3/8', '3/8'])\nfor i in range(len(ts4.beamSequence)):\nts4.beamSequence[i] = ts4.beamSequence[i].subdivide(['6/32', '6/32'])\nfor j in range(len(ts4.beamSequence[i])):\nts4.beamSequence[i][j] = ts4.beamSequence[i][j].subdivide(2)\n\ns = stream.Stream()\nfor ts in [ts1, ts2, ts3, ts4]:\nm = stream.Measure()\nm.timeSignature = ts\nn = note.Note('b')\nn.quarterLength = 0.125\nm.repeatAppend(n, 24)\ns.append(m.makeBeams())\n\ns.show()\n```", null, "The following is a fractional grid representation of the four beam partitions created.\n\n```# four beam partitions\n```", null, "## Configuring Beat¶\n\nThe TimeSignature `beatSequence` MeterSequence employs the hierarchical structure to define the beats and beat divisions of a bar. The outer-most partitions can specify one or more top level beats. Inner partitions can specify the beat division partitions. For most common meters, beats and beat divisions are pre-configured by default.\n\nIn the following example, a simple and a compound meter is created, and the default beat partitions are examined. The `getLevel()` method can be used to show the beat and background beat partitions. The timeSignature `beatDuration`, `beat`, and `beatCountName` properties can be used to return commonly needed beat information. The TimeSignature `beatDivisionCount`, and `beatDivisionCountName` properties can be used to return commonly needed beat division information. These descriptors can be combined to return a string representation of the TimeSignature classification with `classification` property.\n\n```ts = meter.TimeSignature('3/4')\nts.beatSequence.getLevel(0)\n```\n``` <music21.meter.core.MeterSequence {1/4+1/4+1/4}>\n```\n```ts.beatSequence.getLevel(1)\n```\n``` <music21.meter.core.MeterSequence {1/8+1/8+1/8+1/8+1/8+1/8}>\n```\n```ts.beatDuration\n```\n``` <music21.duration.Duration 1.0>\n```\n```ts.beatCount\n```\n``` 3\n```\n```ts.beatCountName\n```\n``` 'Triple'\n```\n```ts.beatDivisionCount\n```\n``` 2\n```\n```ts.beatDivisionCountName\n```\n``` 'Simple'\n```\n```ts.classification\n```\n``` 'Simple Triple'\n```\n```ts = meter.TimeSignature('12/16')\nts.beatSequence.getLevel(0)\n```\n``` <music21.meter.core.MeterSequence {3/16+3/16+3/16+3/16}>\n```\n```ts.beatSequence.getLevel(1)\n```\n``` <music21.meter.core.MeterSequence {1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16+1/16}>\n```\n```ts.beatDuration\n```\n``` <music21.duration.Duration 0.75>\n```\n```ts.beatCount\n```\n``` 4\n```\n```ts.beatCountName\n```\n``` 'Quadruple'\n```\n```ts.beatDivisionCount\n```\n``` 3\n```\n```ts.beatDivisionCountName\n```\n``` 'Compound'\n```\n```ts.classification\n```\n``` 'Compound Quadruple'\n```\n\n## Annotating Found Notes with Beat Count¶\n\nThe `getBeat()` method returns the currently active beat given a quarter length position into the TimeSignature.\n\nIn the following example, all leading tones, or C#s, are collected into a new Stream and displayed with annotations for part, measure, and beat.\n\n```score = corpus.parse('bach/bwv366.xml')\nts = score[meter.TimeSignature].first()\nts.beatSequence.partition(3)\n\nfound = stream.Stream()\noffsetQL = 0\nfor part in score.parts:\nfound.insert(offsetQL, part[clef.Clef].first())\nfor i in range(len(part.getElementsByClass(stream.Measure))):\nm = part[stream.Measure][i]\nfor n in m.notesAndRests:\nif n.name == 'C#':\nfound.insert(offsetQL, n)\noffsetQL += 4\n\nfound.show()\n```", null, "## Using Beat Depth to Provide Metrical Analysis¶\n\nAnother application of the `beatSequence` MeterSequence is to define the hierarchical depth active for a given note found within the TimeSignature.\n\nThe `getBeatDepth()` method, when set with the optional parameter `align` to “quantize”, shows the number of hierarchical levels that start at or before that point. This value is described by Lerdahl and Jackendoff as metrical analysis.\n\nIn the following example, `beatSequence` MeterSequence is partitioned first into one subdivision, and then each subsequent subdivision into two, down to four layers of partitioning.\n\nThe number of hierarchical levels, found with the `getBeatDepth()` method, is appended to each note with the `addLyric()` method.\n\n```score = corpus.parse('bach/bwv281.xml')\npartBass = score.getElementById('Bass')\nts = partBass[meter.TimeSignature].first()\nts.beatSequence.partition(1)\nfor h in range(len(ts.beatSequence)):\nts.beatSequence[h] = ts.beatSequence[h].subdivide(2)\nfor i in range(len(ts.beatSequence[h])):\nts.beatSequence[h][i] = ts.beatSequence[h][i].subdivide(2)\nfor j in range(len(ts.beatSequence[h][i])):\nts.beatSequence[h][i][j] = ts.beatSequence[h][i][j].subdivide(2)\n\nfor m in partBass.getElementsByClass(stream.Measure):\nfor n in m.notesAndRests:\nfor i in range(ts.getBeatDepth(n.offset)):\n\npartBass.measures(0, 7).show()\n```", null, "Alternatively, this type of annotation can be applied to a Stream using the `labelBeatDepth()` function.\n\n## Configuring Accent¶\n\nThe TimeSignature `accentSequence` MeterSequence defines one or more levels of hierarchical accent levels, where quantitative accent value is encoded in MeterTerminal or MeterSequence with a number assigned to the `weight` attribute.\n\n## Applying Articulations Based on Accent¶\n\nThe `getAccentWeight()` method returns the currently active accent weight given a quarter length position into the TimeSignature. Combined with the `getBeatProgress()` method, Notes that start on particular beat can be isolated and examined.\n\nThe following example extracts the Bass line of a Bach chorale in 3/4 and, after repartitioning the beat and accent attributes, applies accents to reflect a meter of 6/8.\n\n```score = corpus.parse('bach/bwv366.xml')\npartBass = score.getElementById('Bass')\nts = partBass[meter.TimeSignature].first()\nts.beatSequence.partition(['3/8', '3/8'])\nts.accentSequence.partition(['3/8', '3/8'])\nts.setAccentWeight([1, .5])\nfor m in partBass[stream.Measure]:\nlastBeat = None\nfor n in m.notesAndRests:\nbeat, progress = ts.getBeatProgress(n.offset)\nif beat != lastBeat and progress == 0:\nif n.tie != None and n.tie.type == 'stop':\ncontinue\nif ts.getAccentWeight(n.offset) == 1:\nmark = articulations.StrongAccent()\nelif ts.getAccentWeight(n.offset) == .5:\nmark = articulations.Accent()\nn.articulations.append(mark)\nlastBeat = beat\nm = m.sorted()\n\npartBass.measures(0, 8).show()\n```", null, "This is probably more depth to the concept of meter than anyone would ever want! But hope that it has whetted your appetite for jucier concepts still to come!\n\nThe next chapters are not yet complete, so let’s jump head to Chapter 58: Understanding Sites and Contexts" ]

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[ "### Boats and Streams: Formulas, Tricks, Examples\n\n• Stream: It implies that the water in the river is moving or flowing.\n• Upstream: Going against the flow of the river.\n• Downstream: Going with the flow of the river.\n• Still water: It implies that the speed of water is zero (generally, in a lake) .\n\nWhen we move upstream, our speed gets deducted from the speed of the stream. Similarly when we move downstream our speed gets added.\n\nLet the speed of a boat in still water be A km/hr and the speed of the stream (or current) be B km/hr, then\n\n• Speed of boat with the stream = (A + B) km/hr\n• Speed of boat against the stream = (A – B) km/hr\n• Speed of boat in still water is:", null, "", null, "• Speed of the stream or current is:", null, "", null, "### Quicker Method to solve the QuestionsBoat’s speed in still water=\n\nExample 1: A boat travels equal distance upstream and downstream. The upstream speed of boat was 10 km/hr, whereas the downstream speed is 20 km/hr. What is the speed of the boat in still water?\n\nSolution: Upstream speed = 10 km/hr\n\nDownstream speed = 20 km/hr\n\nAs per formula, Boat’s speed in still water", null, "Therefore, Boat’s speed in still water", null, "= 15Speed of current", null, "Example 2: A boat travels equal distance upstream and downstream. The upstream speed of boat is 10 km/hr, whereas the downstream speed is 20 km/hr. What is the speed of the current?\n\nSolution: Upstream speed = 10 km/hr\n\nDownstream speed = 20 km/hr\n\nAs per formula, Speed of current\n\n=", null, "Therefore, Speed of current", null, "= 5 km/hr\n\nExample 3: A boat is rowed down a river 28 km in 4 hours and up a river 12 km in 6 hours. Find the speed of the boat and the river.\n\nSolution: Downstream speed is", null, ",\n\nUpstream speed is", null, "= 2 kmph\nSpeed of Boat", null, "(Downstream + Upstream Speed)", null, "kmph\n\nSpeed of current", null, "(Downstream–Upstream speed)", null, "A man can row X km/h in still water. If in a stream which is flowing of Y km/h, it takes him Z hours to row to a place and back, the distance between the two places is", null, "Example 4: A man can row 6 km/h in still water. When the river is running at 1.2 km/h, it takes him 1 hour to row to a place and back. How far is the place?\n\nSolution: Man’s rate downstream = (6 + 1.2) = 7.2 km/h.\n\nMan’s rate upstream = (6 – 1.2) km/h = 4.8 km/h.\n\nLet the required distance be x km.\n\nThen", null, "= 1\n\nor 4.8x + 7.2x = 7.2 × 4.8\n\n⇒", null, "By direct formula:\n\nRequired distance", null, "", null, "km\n\nA man rows a certain distance downstream in X hours and returns the same distance in Y hours. If the stream flows at the rate of Z km/h, then the speed of the man in still water is given by", null, "And if speed of man in still water is Z km/h then the speed of stream is given by", null, "Example 5: Vikas can row a certain distance downstream in 6 hours and return the same distance in 9 hours. If the stream flows at the rate of 3 km/h, find the speed of Vikas in still water.\n\nSolution: By the formula,\n\nVikas’s speed in still water", null, "= 15 km/h\n\nIf a man capable of rowing at the speed u of m/sec in still water, rows the same distance up and down a stream flowing at a rate of v m/sec, then his average speed through the journey is", null, "", null, "Example 6: Two ferries start at the same time from opposite sides of a river, travelling across the water on routes at right angles to the shores. Each boat travels at a constant speed though their speeds are different. They pass each other at a point 720 m from the nearer shore. Both boats remain at their sides for 10 minutes before starting back. On the return trip they meet at 400 m from the other shore. Find the width of the river.\n\nSolution: Let the width of the river be x.\n\nLet a, b be the speeds of the ferries.", null, "… (i)", null, "… (ii)\n\n(Time for ferry 1 to reach other shore + 10 minute wait + time to cover 400m) = Time for freely 2 to cover 720m to other shore + 10 minute wait + Time to cover (x – 400m) )\n\nUsing (i) , we get", null, "Using (ii) ,", null, "On, solving we get, x = 1760m" ]

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[ "", null, "# What is standard normal cumulative distribution?\n\n## What is standard normal cumulative distribution?\n\nThe (cumulative) distribution function of a random variable X, evaluated at x, is the probability that X will take a value less than or equal to x. You simply let the mean and variance of your random variable be 0 and 1, respectively. This is called standardizing the normal distribution.\n\nHow do you find the CDF of a standard normal distribution?\n\nThe CDF of the standard normal distribution is denoted by the Φ function: Φ(x)=P(Z≤x)=1√2π∫x−∞exp{−u22}du. As we will see in a moment, the CDF of any normal random variable can be written in terms of the Φ function, so the Φ function is widely used in probability.\n\n### How do you calculate cumulative probability?\n\nThe cumulative probability for a value equals the cumulative probability for that value’s z-score. Here, probability speed less than or equal 73 mph = probability z-score less than or equal 1.60. How did we arrive at this z-score?\n\nHow do you calculate cumulative area?\n\nThe Standard Normal table gives cumulative area or area on the left. To find area on the right, first look up the z-score to get the cumulative area and then subtract the cumulative area from 1.0. Correct. The Standard Normal table gives cumulative area or area on the left.\n\n## What is normal cumulative distribution function?\n\nThe cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter (phi), is the integral. The related error function gives the probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range .\n\nWhat is meant by cumulative default probability?\n\nDefinition. The term Cumulative Default Probability is used in the context of multi-period Credit Risk analysis to denote the likelihood that a Legal Entity is observed to have experienced a defined Credit Event up to a particular timepoint.\n\n### What is meant by cumulative probability?\n\nA cumulative probability refers to the probability that the value of a random variable falls within a specified range. Frequently, cumulative probabilities refer to the probability that a random variable is less than or equal to a specified value.\n\nHow do you read a normal distribution curve?\n\nLook at the symmetrical shape of a bell curve. The center should be where the largest portion of scores would fall. The smallest areas to the far left and right would be where the very lowest and very highest scores would fall. Read across the curve from left to right.\n\n## What percent falls below the mean for normal distribution?\n\nRegardless of what a normal distribution looks like or how big or small the standard deviation is, approximately 68 percent of the observations (or 68 percent of the area under the curve) will always fall within two standard deviations (one above and one below) of the mean.\n\nHow do you calculate normal distribution?\n\nNormal Distribution. Write down the equation for normal distribution: Z = (X – m) / Standard Deviation. Z = Z table (see Resources) X = Normal Random Variable m = Mean, or average. Let’s say you want to find the normal distribution of the equation when X is 111, the mean is 105 and the standard deviation is 6.\n\n### What is z score in normal distribution?\n\nA z-score is also known as a standard score and it can be placed on a normal distribution curve. Z-scores range from -3 standard deviations (which would fall to the far left of the normal distribution curve) up to +3 standard deviations (which would fall to the far right of the normal distribution curve).\n\nWhat is the normal distribution equation?\n\nThe normal distribution is defined by the following equation: The Normal Equation. The value of the random variable Y is: Y = { 1/[ σ * sqrt(2π) ] } * e -(x – μ) 2/2σ 2. where X is a normal random variable, μ is the mean, σ is the standard deviation, π is approximately 3.14159, and e is approximately 2.71828." ]

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[ "Community Profile", null, "# calvin son\n\nLast seen: Today Active since 2022\n\n#### Statistics\n\n•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "•", null, "#### Content Feed\n\nView by\n\nSolved\n\nThe last non-zero digit of a factorial\nFor given positive integer n, what is the last non-zero digit of n!? Example: factorial(11) = 39916800 Last non-zero d...\n\nenviron 2 heures ago\n\nSolved\n\nApproximation of Pi (vector inputs)\nPi (divided by 4) can be approximated by the following infinite series: pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... For a given numbe...\n\n8 jours ago\n\nSolved\n\nVector pop\nTake |n| elements from the end of the vector |v| and return both the shorten vector |v| and the |n| elements in a separate vecto...\n\n8 jours ago\n\nSolved\n\nMoving average (variable kernel length)\nFind the moving average in a vector. The kernel length is a variable. For example x = 1:10 kernel_length = 2 would r...\n\n8 jours ago\n\nSolved\n\nWe have data of matrix, that is input. That contains 2 or more rows and the last row should contain the average of each column,...\n\n8 jours ago\n\nSolved\n\nCalculate the hypotenuse of a right triangle without using ^ and sqrt ()\nFind out the hypotenuse of right triangle. Say a = 4, b = 3 then c = 5 Please don't use ^ and sqrt() function.\n\n8 jours ago\n\nSolved\n\nArray ex-OR\nThere are in MATLAB logical functions such as _<http://www.mathworks.co.uk/help/matlab/ref/and.html and>,_ _<http://www.math...\n\n8 jours ago\n\nSolved\n\nCalculate cosine without cos(x)\nSolve cos(x). The use of the function cos() and sin() is not allowed.\n\n9 jours ago\n\nSolved\n\nCalculate sin(x) without sin(x)\nCalculate y = sin(x) x = 0 -> y= 0 without the use of sin(x) or cos(x)\n\n9 jours ago\n\nSolved\n\nNumber of even divisors of a given number\nGiven a Number n, return the number of its even divisors without listing them. example: n=14 ; EvenDivisors={2,14} ; y=2 ...\n\n9 jours ago\n\nSolved\n\nRaise a polynomial to a power\nIn Matlab, polynomials are represented by a vector of coefficients. For example, the polynomial p=a*x^2 + b*x + c is represente...\n\n9 jours ago\n\nSolved\n\nAirline Ticket Mod7 Checksum\nThere are 13 digits in an airline ticket number. If an airline ticket number is valid, the 13th digit should be the remainder of...\n\n9 jours ago\n\nSolved\n\nTick. Tock. Tick. Tock. Tick. Tock. Tick. Tock. Tick. Tock.\nSubmit your answer to this problem a multiple of 5 seconds after the hour. Your answer is irrelevant; the only thing that matte...\n\n10 jours ago\n\nSolved\n\nPentagonal Numbers\nYour function will receive a lower and upper bound. It should return all pentagonal numbers within that inclusive range in ascen...\n\n10 jours ago\n\nSolved\n\nHard limit function\nClassify x data as if x>=0 then y=1 if x<0 then y=0 Example x = [ -2 -1 0 1 2] y = [ 0 0 1 1 1]\n\n13 jours ago\n\nSolved\n\nPolar Form Complex Number Entry\nWrite a function that takes the magnitude and angle(in degrees) of a complex number and returns a complex variable. Positive ang...\n\n13 jours ago\n\nSolved\n\nHow many unique Pythagorean triples?\nFor a given integer |n|, return all <https://en.wikipedia.org/wiki/Pythagorean_triple Pythagorean triples> that inlude numbers s...\n\n13 jours ago\n\nSolved\n\nString Array Basics, Part 1: Convert Cell Array to String Array; No Missing Values\n<http://www.mathworks.com/help/matlab/characters-and-strings.html String array> and cell array are two types of containers for s...\n\n15 jours ago\n\nSolved\n\nFind the Pattern 8\n\n15 jours ago\n\nSolved\n\nFind the Pattern 7\n\n15 jours ago\n\nSolved\n\nFind the Pattern 6\n\n15 jours ago\n\nSolved\n\nFind the Pattern 5\n\n15 jours ago\n\nSolved\n\nFind the Pattern 4\n\n15 jours ago\n\nSolved\n\nFind the Pattern 2\n\n15 jours ago\n\nSolved\n\nFind the Pattern 9\n\n15 jours ago\n\nSolved\n\nWhat gear ratio does the cyclist need?\nA cyclist (perhaps including our famed Codysolver the cyclist <http://www.mathworks.com/matlabcentral/cody/players/1841757-the...\n\n15 jours ago\n\nSolved\n\nSplitting Circle\nConsider a circle which has been divided into three concentric circles as depicted in the figure below The ratio betwen the a...\n\n15 jours ago\n\nSolved\n\nRotate input square matrix 90 degrees CCW without rot90\nRotate input matrix (which will be square) 90 degrees counter-clockwise without using rot90,flipud,fliplr, or flipdim (or eval)....\n\n17 jours ago\n\nSolved\n\nBack to basics 21 - Matrix replicating\nCovering some basic topics I haven't seen elsewhere on Cody. Given an input matrix, generate an output matrix that consists o...\n\n17 jours ago" ]

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[ "# zbMATH — the first resource for mathematics\n\nHolomorphic families of injections. (English) Zbl 0619.30027\nThis paper contains new proofs and extensions of some results by R. Mañé, P. Sad and D. Sullivan [Ann. Sci. Éc. Norm. Supér., IV. Ser. 16, 193-217 (1983; Zbl 0524.58025)] and D. P. Sullivan and W. P. Thurston [Acta Math. 157, 243-257 (1986; reviewed above)]. Let E be a subset of the Riemann sphere $${\\hat {\\mathbb{C}}}={\\mathbb{C}}\\cup \\{\\infty \\}$$ containing at least 4 points. Let $$\\Delta_ r$$ denote the open disc $$| z| <r$$ in $${\\mathbb{C}}$$. A map $f:\\Delta_ r\\times E\\to {\\mathbb{C}}$ will be called admissible if $$f(0,z)=z$$ for all $$z\\in E$$, for every fixed $$\\lambda \\in \\Delta_ r$$ the map f($$\\lambda$$,$$\\cdot):E\\to {\\hat {\\mathbb{C}}}$$ is an injection, and for every fixed $$z\\in E$$ the map $$f(\\cdot,z):\\Delta_ r\\to {\\hat {\\mathbb{C}}}$$ is holomorphic.\nTheorem 1. If $$f:\\Delta_ 1\\times E\\to {\\hat {\\mathbb{C}}}$$ is admissible, then every f($$\\lambda$$,$$\\cdot)$$ is the restriction to E of a quasiconformal self-map $$F_{\\lambda}$$ of $${\\hat {\\mathbb{C}}}$$, of dilatation not exceeding $K=(1+| \\lambda |)/(1-| \\lambda |).$ Theorem 2. If $$f:\\Delta_ 1\\times E\\to {\\mathbb{C}}$$ is admissible and E has a nonempty interior $$\\omega$$, then for each $$\\lambda \\in \\Delta_ 1$$ the map $$f(\\lambda,\\cdot)|_{\\omega}$$ is a K- quasiconformal homeomorphism of $$\\omega$$ into $${\\hat {\\mathbb{C}}}$$ with $$K=(1+| \\lambda |)/(1-| \\lambda |)$$. The Beltrami coefficient of $$f(\\lambda,\\cdot)|_{\\omega}$$ given by $\\mu (\\lambda,z)=\\frac{\\partial f(\\lambda,z)| \\omega}{\\partial \\bar z}/\\frac{\\partial f(\\lambda,z)| \\omega}{\\partial z}$ is a holomorphic function of $$\\lambda \\in \\Delta_ 1$$, and an element of the Banach space $$L_{\\infty}(\\omega)$$. The author’s proofs make essential use of the theory of quasiconformal maps and of Teichmüller spaces.\nReviewer: N.A.Gusevskij\n\n##### MSC:\n 30C62 Quasiconformal mappings in the complex plane 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)\n##### Keywords:\nK-quasiconformal homeomorphism; Teichmüller spaces\nFull Text:\n##### References:\n Agard, S. B. &Gehring, F. W., Angles and quasiconformal mappings.Proc. London Math. Soc. (3), 14 (1965), 1–21. · Zbl 0131.07902 Ahlfors, L. V. &Bers, L., Riemann’s mapping theorem for variable metrices.Ann. of Math. (2), 72 (1960), 385–404. · Zbl 0104.29902 Ahlfors, L. V. &Weill, G., A uniqueness theorem for Beltrami equations.Proc. Amer. Math. Soc., 13 (1962), 975–978. · Zbl 0106.28504 Bers, L., Extremal quasiconformal mappings.Advances in the Theory for Riemann Surfaces, Ann. of Math. Studies, 66 (1970), 27–52. –, Finite dimensional Teichmüller spaces and generalizations.Bull. Amer. Math. Soc. (N.S.), 5 (1981), 131–172. · Zbl 0485.30002 Earle, C. J. &Kra, I., On holomorphic mappings between Teichmüller spaces.Contributions to Analysis, Academic Press, New York, (1974), 107–124. · Zbl 0307.32016 Harvey, W. J. (Ed.),Discrete groups and automorphic functions. Academic Press, New York, 1977. · Zbl 0411.30033 Hille, E.,Analytic Function Theory, Vol. II. Ginn and Co. (1962). · Zbl 0102.29401 Hubbard, J. H., Sur les sections analytique de la courbe universelle de Teichmüller.Mem. Amer. Math. Soc., 166 (1976), 1–137. · Zbl 0318.32020 Lehto, O. &Virtanen, K. I.,Quasiconformal mappings in the plane. Springer-Verlag, Berlin, 1973. · Zbl 0267.30016 Mañé, R., Sad, P. &Sullivan, D., On the dynamics of rational maps.Ann. Sci. Ecole Norm. Sup., 16 (1983), 193–217. · Zbl 0524.58025 Reich, E. &Strebel, K., Extremal quasiconformal mappings with given boundary values.Contributions to Analysis, Academic Press, New York, (1974), 375–391. · Zbl 0318.30022 Royden, H. L., Automorphisms and isometries of Teichmüller space.Advances in the Theory of Riemann Surfaces, Ann. of Math. Studies, 66 (1971), 369–383. Strebel, K., On quasiconformal mappings of open Riemann surfaces.Comment. Math. Helv., 52 (1978), 301–321. · Zbl 0421.30017 Sullivan, D. &Thurston, W. P., Extending holomorphic motions.Acta Math., 157 (1986), 243–257. · Zbl 0619.30026\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching." ]

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[ "# Apply LinearTransformation to a Rectangle\n\nI have represented a box as a 4 col x 2 row matrix\n\nsomething like `bbox = [0 0 ; 50 0; 50 100; 0 100]`\n\nI have a few of these boxes and I want to rotate them by a certain angle.\nI create a ` LinearMap(RotMatrix(deg2rad(angle)))` … now how do I use this to transform the coordinates?\n\n``````bbox = 4×2 Matrix{Int64}:\n396 979\n424 979\n424 1009\n396 1009\n\n2×4 OffsetArray(::Matrix{Float64}, -2:-1, -4:-1) with eltype Float64 with indices -2:-1×-4:-1:\n1009.0 1009.0 979.0 979.0\n396.0 424.0 424.0 396.0\n\n``````\n\nSo it changed the shape of the matrix too?\n\n``````>>> LinearMap(RotMatrix(deg2rad(180))) * bbox'\nNo method matching .... found ....\n``````\n\nSo I also can’t treat it like a matrix and multiply it directly.\n\nHow do I do this?\n\nYou could define your own rotation matrix:\n\n``````rot(θ) = [cosd(θ) -sind(θ); sind(θ) cosd(θ)]\n\nbox = [0 0 ; 50 0; 50 100; 0 100]'\nrbox30 = rot(30)*box\n``````" ]

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[ "## Three Parallels in a Triangle: What Is It About? A Mathematical Droodle\n\n### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.\n\n What if applet does not run?\n\nExplanation", null, "### Three Parallels in a Triangle\n\nThe applet attempts to suggest the following statement:\n\nOn the sides of ΔABC six points D, E, K, L, M, N are constructed (see the applet) so that\n\n AE = BD = AB, AK = CL = AC, CM = BN = BC.\n\nShow that that the three lines DE, KL, and MN are parallel.\n\n### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.\n\n What if applet does not run?\n\nIn the proof we shall refer to the following diagram:", null, "As usual BC = a, AC = b, and AB = c. In the diagram a > b > c, which we shall assume but which, in general, may not hold. Thus we consider only one of several possible cases and claim that other cases can be studied by analogy in a similar vein.\n\nWe shall only prove that DE and KL are parallel. The claim that MN is parallel to the two can be bundled under the analogy argument above with a redistribution of magnitudes of the lengths a, b, c.\n\nLet KL meet AC in T. According to Menelaus' theorem,\n\n (1) AK/KB · BL/LC · CT/TA = -1,\n\nwhere by construction AK = LC = b, KB = c-b, BL = a - b. (The values of c - b and a - b, as opposed to b - c and b - a, have been selected as to insure agreement with the signs of the terms in (1) implied by Menelaus' theorem.) Denoting CT = x, we rewrite (1) as\n\n (a - b) / (c - b) = (x - b) / x = 1 - b/x,\n\nfrom which\n\n x = b(b - c) / (a - c).\n\nOne way to show that DE is parallel to KL is to verify the proportion (where all segment length are assumed positive.)\n\n (2) CE/CT = CD/CL.\n\nBut, with CE = b - c and CD = a - c, (2) becomes\n\n (b - c)(a - c) / b(b - c) = (a - c)/b,\n\nwhich is indeed true, thus confirming (2).\n\nNathan Bowler has offered a different and a much simpler approach:\n\nEK is the reflection of BC in the bisector of angle A, so the point X where these lines meet is the intersection of BC with that bisector: It satisfies XB:XC = c:b = BD:CL, so\n\n XD:XL = (BD - XB):(CL - XC) = c:b = XE:XK,\n\nso DE and KL are parallel.\n\nMichel Cabart suggest a vector algebra shortcut (vectors are in bold):\n\n (3) CE = CA - c/b CA (4) CD = CB - c/a CB\n\nSubtracting (3) from (4) gives\n\n ED = AB - c/a CB + c/b CA", null, "" ]

[ null, "https://www.cut-the-knot.org/gifs/tbow_sh.gif", null, "https://www.cut-the-knot.org/Curriculum/Geometry/ThreeParallelsInTriangle.gif", null, "https://www.cut-the-knot.org/gifs/tbow_sh.gif", null ]

[ "# deepctr.models.dcn module¶\n\nAuthor:\n\nWeichen Shen, [email protected]\n\nShuxun Zan, [email protected]\n\nReference:\n\n Wang R, Fu B, Fu G, et al. Deep & cross network for ad click predictions[C]//Proceedings of the ADKDD’17. ACM, 2017: 12. (https://arxiv.org/abs/1708.05123)\n\n Wang R, Shivanna R, Cheng D Z, et al. DCN-M: Improved Deep & Cross Network for Feature Cross Learning in Web-scale Learning to Rank Systems[J]. 2020. (https://arxiv.org/abs/2008.13535)\n\n`deepctr.models.dcn.``DCN`(linear_feature_columns, dnn_feature_columns, cross_num=2, cross_parameterization='vector', dnn_hidden_units=(256, 128, 64), l2_reg_linear=1e-05, l2_reg_embedding=1e-05, l2_reg_cross=1e-05, l2_reg_dnn=0, seed=1024, dnn_dropout=0, dnn_use_bn=False, dnn_activation='relu', task='binary')[source]\n\nInstantiates the Deep&Cross Network architecture.\n\nParameters: linear_feature_columns – An iterable containing all the features used by linear part of the model. dnn_feature_columns – An iterable containing all the features used by deep part of the model. cross_num – positive integet,cross layer number cross_parameterization – str, `\"vector\"` or `\"matrix\"`, how to parameterize the cross network. dnn_hidden_units – list,list of positive integer or empty list, the layer number and units in each layer of DNN l2_reg_linear – float. L2 regularizer strength applied to linear part l2_reg_embedding – float. L2 regularizer strength applied to embedding vector l2_reg_cross – float. L2 regularizer strength applied to cross net l2_reg_dnn – float. L2 regularizer strength applied to DNN seed – integer ,to use as random seed. dnn_dropout – float in [0,1), the probability we will drop out a given DNN coordinate. dnn_use_bn – bool. Whether use BatchNormalization before activation or not DNN dnn_activation – Activation function to use in DNN task – str, `\"binary\"` for binary logloss or `\"regression\"` for regression loss A Keras model instance." ]

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[ "# Bayes CFA\n\n#### Lazar\n\n##### Phineas Packard\nHi Folks,\n\nI have several questions about Bayes CFA in Open bugs that I hope you might be able to help me out with.\n\nFirst the setup:\nHere is a basic two factor CFA set up for the HS data that is available in several R packages\nCode:\n###Bugs CFA model 2 factors. All cross loadings zero###\n#Mildly informative priors used on litem intercepts\n#n = number of observations\n#t = equals number of indicator items\n\nmodel{\nfor (i in 1:N){\n#latent 1\nfor (t in 1:3){\ny[i,t] ~ dnorm(condmn[i,t], invsig2[t])\n}\n#latent 2\nfor (t in 4:6){\ny[i,t] ~dnorm(condmn[i,t], invsig2[t])\n}\nfscore[i,1:2]~dmnorm(mn.fs[], sig.fs[,])\n}\nmn.fs<-0\nmn.fs<-0\nsig.fs[1,1]<-1\nsig.fs[2,2]<-1\nsig.fs[1,2]<-phi\nsig.fs[2,1]<-phi\n#Prior distribution\nphi ~ dunif(-1,1)\nfor (t in 1:6){\ninvsig2[t] <- 1/psi[t]\npsi[t] ~ dunif(0,400)\nmu[t] ~ dnorm(20,.05)\n}\n}\nand here is the R script I am using to run it:\nCode:\npath <- 'C:/Users/30016475/Dropbox/Projects_Research/PVsimulation'\nsetwd(path)\n\nlibrary(simsem);library(R2OpenBUGS); library(MBESS); library(lavaan)\ndata(HS.data)\n\n#Run two factor model in Lavaan as a bench mark to check Open BUGS results\n#CFA using ML\nModel <- '\nL1 =~ visual + cubes + flags\nL2 =~ paragrap + sentence + wordm\n'\nfit<- sem(Model, data=HS.data, std.lv=TRUE)\nsummary(fit)\n\n#Open bugs run of same two factor model\ny<-as.matrix(HS.data[,c('visual', 'cubes', 'flags', 'paragrap', 'sentence', 'wordm')])\n#Number of cases\nN<-nrow(HS.data)\n#No inits (I will rely on defaults) because I am lazy....maybe this is my problem\n\n#Data and parameters to monitor\ndata<-list('N', 'y')\nparams<-c('fscore', 'sig.fs', 'fload', 'mu')#Note fscore are the plausible values. Can delete to reduce size of output\n#Bugs call\nout <- bugs(data, parameters.to.save=params, inits=NULL,\nmodel.file='C:/Users/30016475/Dropbox/Projects_Research/PVsimulation/CFAsimple.txt',\ndebug=TRUE, n.iter=1000)\n#check fit\nall(out$summary[,\"Rhat\"]<1.1) #Check results out$summary\nThe results I get for the bugs run and lavaan are close for the most part but further away than I would have guessed given I am mostly using uninformative priors. The real problem is that the correlation is correct is size but in the wrong direction. Like I have accidentally multiplied it by -1. Can anyone see the mistake I am making?\n\n#### Lazar\n\n##### Phineas Packard\nOk author of the paper I was using as a template said there was an error in the code. His suggested fix was to use the inverse function BUT this function exists in JAGS but not openbugs. Does anyone know the open bugs equivalent?\n\n#### Lazar\n\n##### Phineas Packard\nCorrected bug script is:\nCode:\n###Bugs CFA model 2 factors. All cross loadings zero###\n#Mildly informative priors used on litem intercepts\n#n = number of observations\n#t = equals number of indicator items\n\nmodel{\nfor (i in 1:N){\n#latent 1\nfor (t in 1:3){\ny[i,t] ~ dnorm(condmn[i,t], invsig2[t])\n}\n#latent 2\nfor (t in 4:6){\ny[i,t] ~dnorm(condmn[i,t], invsig2[t])\n}\nfscore[i,1:2]~dmnorm(mn.fs[], siginv.fs[,])\n}\nmn.fs<-0\nmn.fs<-0\nsiginv.fs <- inverse(sig.fs)\nsig.fs[1,1]<-1\nsig.fs[2,2]<-1\nsig.fs[1,2]<-phi\nsig.fs[2,1]<-phi\n#Prior distribution\nphi ~ dunif(-1,1)\nfor (t in 1:6){\ninvsig2[t] <- 1/psi[t]\npsi[t] ~ dunif(0,400)\nmu[t] ~ dnorm(20,.05)\n}\n}\n\n#### Dason\n\nI don't have bugs installed right now but if the issue is the inverse you can easily just do the inverse directly.\n\nCode:\nsiginv.fs[1,1] <- 1/(1-phi*phi)\nsiginv.fs[2,2] <- 1/(1-phi*phi)\nsiginv.fs[1,2] <- (-phi)/(1-phi*phi)\nsiginv.fs[2,1] <- (-phi)/(1-phi*phi)\nbut the manual implies that it might work if you just use the following syntax\n\nCode:\nsiginv.fs <- inverse(sig.fs[,])\n\nLast edited:\n\n#### Lazar\n\n##### Phineas Packard\nThanks Dason. I tried inverse(sig.fs[,]) but I get:\nCode:\nempty slot not allowed in variable name error pos 617\nin openbugs.\n\nNot such trouble in jags.\n\n#### Lazar\n\n##### Phineas Packard\nCode:\nsiginv.fs[1,1] <- 1/(1-phi*phi)\nsiginv.fs[2,2] <- 1/(1-phi*phi)\nsiginv.fs[1,2] <- (-phi)/(1-phi*phi)\nsiginv.fs[2,1] <- (-phi)/(1-phi*phi)\nWorks fine", null, "" ]

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[ "This vignette is meant to aid users in defining their input. Each function uses a standardized input for the treatments to be used in the functions. This enables the user to get the most out of their results and minimizes the amount of calculations they have to perform. The user may define each treatment/main effect and any interaction effects that they would like to set an effect size for. Below are a series of examples for 2-way and 3-way ANOVAs. Certain details in the 2-way example are repeated in the 3-way example so that the reader does not have to read the entire document. All of the effect sizes used in the examples were converted to eta-squared from Cohen’s f. Slight deviations in results could be due to rounding errors.\n\n## 2-way ANOVA Examples\n\n#### Main Effects\n\nThis example study is a 2x4 ANOVA and is taken from exercise 8.15, p.400 of Cohen (1988). This study is investigating the effects of age (A) and contingency of reinforcement (C) on learning. All of the effect sizes taken from the exercise were converted from Cohen’s f to eta-squared in order to input the numeric equivalent into the calculations. Before the sample size calculations are made, the main effects must be defined. The main effects may be assigned any variable name but for this example they will be called main.eff1 and main.eff2.\n\n# Define main effects\nmain.eff1 <- list(name = \"A\", levels = 2, eta.sq = 0.123)\nmain.eff2 <- list(name = \"C\", levels = 4, eta.sq = 0.215)\n\nEach main effect is defined as a list and takes in three different values:\n\n• name - The name of the treatment effect. This may be either a character string of the treatment name, an abbreviation, or a single character such as “A”.\n• levels - The number of levels/groups in the treatment. This is always an integer that is 2 or greater.\n• eta.sq - Estimated effect size for the treatment effect. This can be either a numeric value greater than 0 or a character string. Acceptable string values and their numeric equivalents are: “small” (0.01), “med” (0.06), and “large” (0.14).\n\nNote: If the effect size for a main effect is going to be “small” then the value of eta.sq does not need to be included when creating the list for the main effect. There is a default setting of “small” for this value.\n\n# Example of using the default eta.sq setting\nmain.eff <- list(name = \"A\", levels = 3)\n\n#### Interaction Effect Size (optional)\n\nFor this example, the effect size of the interaction is known. The default setting for interactions is set “small”, but this can be changed in two different ways. The first way is to define the interaction the same way we defined the main effects and change its effect size from “small” to some other value. This interaction effect can be assigned any value similar to the main effect names; for this example it will be called int.eff1. The name of the interaction must be comprised of the names given to the treatments when the main effects were defined. The names must be separated by a “*“. Assigning the value for the estimated effect size follows the same set of conditions as the main effect’s effect size.\n\n# Change interaction effect size\nint.eff1 <- list(name = \"A*C\", eta.sq = 0.079)\n\nThe alternative way is to change the default value of interaction.eta2 = \"small\" when calling the n.multiway function. Since there is only one interaction in a 2-way ANOVA, this would be the quicker way to change the interaction effect size. For this example the value will change to interaction.eta2 = 0.079.\n\n#### Running n.multiway\n\nRunning the function to calculate the sample size requirements for the example study.\n\nn.multiway(iv1 = main.eff1, iv2 = main.eff2, int1 = int.eff1)\n#>\n#> The following sample size recommendations are for each treatment and all possible interactions.\n#> Sample sizes are calculated independently using the estimated effect size to achieve\n#> the desired power level.\n#>\n#> Desired power: 0.80\n#> Significance level: 0.05\n#> Effect size used in calculations: Cohen's f-squared\n#> Cutoffs: small = 0.01, med = 0.06, large = 0.14\n#>\n#> Treatment Effect Size Total n per cell\n#> A 0.123 64 8\n#> C 0.215 48 6\n#> A*C 0.079 136 17\n\nHere is an example of changing the value of interaction.eta2:\n\nn.multiway(iv1 = main.eff1, iv2 = main.eff2, interaction.eta2 = 0.079)\n#>\n#> The following sample size recommendations are for each treatment and all possible interactions.\n#> Sample sizes are calculated independently using the estimated effect size to achieve\n#> the desired power level.\n#>\n#> Desired power: 0.80\n#> Significance level: 0.05\n#> Effect size used in calculations: Cohen's f-squared\n#> Cutoffs: small = 0.01, med = 0.06, large = 0.14\n#>\n#> Treatment Effect Size Total n per cell\n#> A 0.123 64 8\n#> C 0.215 48 6\n#> A*C 0.079 136 17\n\nIt is possible to change the amount of output that is displayed. The default is set to result = \"all\". Acceptable alternatives are result = \"highest\" for the highest recommended sample size and result = \"select\" to view the highest result along with the treatments where a numeric effect size value was entered.\n\nn.multiway(iv1 = main.eff1, iv2 = main.eff2, int1 = int.eff1, result = \"highest\")\n#>\n#> The following is the largest recommended total sample size.\n#>\n#> Desired power: 0.80\n#> Significance level: 0.05\n#> Effect size used in calculations: Cohen's f-squared\n#> Cutoffs: small = 0.01, med = 0.06, large = 0.14\n#>\n#> Treatment: A*C\n#> Effect Size: 0.079\n#> Total N: 136\n#> n per cell: 17\nn.multiway(iv1 = main.eff1, iv2 = main.eff2, int1 = int.eff1, result = \"select\")\n#>\n#> The following is the highest sample size required and the sample size\n#> recommendations where a numeric value for effect size was entered.\n#> Sample sizes are calculated independently using the estimated\n#> effect size to achieve the desired power level.\n#>\n#> Desired power: 0.80\n#> Significance level: 0.05\n#> Effect size used in calculations: Cohen's f-squared\n#> Cutoffs: small = 0.01, med = 0.06, large = 0.14\n#>\n#> Treatment Effect Size Total N n per cell\n#> A 0.123 64 8\n#> C 0.215 48 6\n#> A*C 0.079 136 17\n\n## 3-way ANOVA Examples\n\n#### Main Effects\n\nThis example study is a 2x3x4 ANOVA taken from exercise 8.14, p.397 of Cohen (1988). All of the effect sizes taken from the exercise were converted from Cohen’s f to eta-squared in order to input the numeric equivalent into the calculations. For this example the main effects will be assigned the variable names: main.eff1, main.eff2, and main.eff3.\n\n# Define main effects\nmain.eff1 <- list(name = \"Sex\", levels = 2, eta.sq = 0.0099)\nmain.eff2 <- list(name = \"Age\", levels = 3, eta.sq = 0.0588)\nmain.eff3 <- list(name = \"Conditions\", levels = 4, eta.sq = 0.1506)\n\nEach main effect is defined as a list and takes in three different values:\n\n• name - The name of the treatment effect. This may be either a character string of the treatment name, an abbreviation, or a single character such as “A”.\n• levels - The number of levels/groups in the treatment. This is always an integer that is 2 or greater.\n• eta.sq - Estimated effect size for the treatment effect. This can be either a numeric value greater than 0 or a character string. Acceptable string values and their numeric equivalents are: “small” (0.01), “med” (0.06), and “large” (0.14).\n\nNote: If the effect size for a main effect is going to be “small” then the value of eta.sq does not need to be included when creating the list for the main effect. There is a default setting of “small” for this value.\n\n# Example of using the default eta.sq setting\nmain.eff <- list(name = \"A\", levels = 3)\n\n#### Interaction Effect Sizes (optional)\n\nAs noted in the 2-way ANOVA example, there are two different ways to change the effect sizes for the interactions. For this example, all of the effect sizes for the interaction effects were estimated to be approximately a medium effect. Therefore the most efficient way to change all the effect sizes simultaneously is to use interaction.eta2 = 0.0588. Alternatively, if only a selection of the interactions were expected to have a moderate effect size, we could change these independently. The following is an example of how this could be achieved.\n\n# Changing the effect sizes of specific interactions\nint.eff1 <- list(name = \"Age*Conditions\", eta.sq = \"med\")\nint.eff2 <- list(name = \"Sex*Conditions\", eta.sq = \"med\")\n\nNote: When typing out the name of an interaction it is important to follow the order in which the main effects were defined. For example, name = \"Age*Conditions\" is valid whereas name = \"Conditions*Age\" would not be.\n\n#### Running n.multiway\n\nn.multiway(iv1 = main.eff1, iv2 = main.eff2, iv3 = main.eff3, interaction.eta2 = 0.0588)\n#>\n#> The following sample size recommendations are for each treatment and all possible interactions.\n#> Sample sizes are calculated independently using the estimated effect size to achieve\n#> the desired power level.\n#>\n#> Desired power: 0.80\n#> Significance level: 0.05\n#> Effect size used in calculations: Cohen's f-squared\n#> Cutoffs: small = 0.01, med = 0.06, large = 0.14\n#>\n#> Treatment Effect Size Total n per cell\n#> Sex 0.0099 809 34\n#> Age 0.0588 179 8\n#> Conditions 0.1506 86 4\n#> Sex*Age 0.0588 179 8\n#> Sex*Conditions 0.0588 199 9\n#> Age*Conditions 0.0588 242 11\n#> Sex*Age*Conditions 0.0588 242 11\n\nHere is an example of running the function while only changing the effect sizes of the two interactions we defined earlier.\n\nn.multiway(iv1 = main.eff1, iv2 = main.eff2, iv3 = main.eff3, int1 = int.eff1, int2 = int.eff2)\n#>\n#> The following sample size recommendations are for each treatment and all possible interactions.\n#> Sample sizes are calculated independently using the estimated effect size to achieve\n#> the desired power level.\n#>\n#> Desired power: 0.80\n#> Significance level: 0.05\n#> Effect size used in calculations: Cohen's f-squared\n#> Cutoffs: small = 0.01, med = 0.06, large = 0.14\n#>\n#> Treatment Effect Size Total n per cell\n#> Sex 0.0099 809 34\n#> Age 0.0588 179 8\n#> Conditions 0.1506 86 4\n#> Sex*Age small 978 41\n#> Sex*Conditions med 195 9\n#> Age*Conditions med 237 10\n#> Sex*Age*Conditions small 1373 58\n\n## Comparisons with G*Power 3 and Cohen\n\nThe example study used for this table is taken from exercise 8.14, p.397 of Cohen (1988). This is a 2x3x4 ANOVA. This example was also used in the 3-way ANOVA example. The power and significance level used for these calculations are 0.80 and 0.05, respectively.\n\nTreatment Numerator df Effect Size (f) easypower n GPower n Cohen n\nS 1 0.10 34 33 34\nA 2 0.25 8 7 8\nC 3 0.40 4 3 4\nSxA 2 0.25 8 7 8\nSxC 3 0.25 9 8 9\nAxC 6 0.25 11 9 10\nSxAxC 6 0.25 11 9 10" ]

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[ "• @_\n\n# Add two (or more) arrays\n\n```\nLet's extend it so it will be able to take two vectors (arrays) and add\nthem pair-wise. (2, 3) + (7, 8, 5) = (9, 11, 5)\n\n```\n```\nmy @first = (2, 3);\nmy @second = (7, 8, 5);\n```" ]

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[ "# Code Optimization Techniques in Compiler Construction by Kindson Munonye\n\nCode Optimization Techniques in Compilers\nWe would discuss various code optimization techniques which includes the following:\n\n2. Constant Folding\n3. Copy Propagation\n4. Strength Reduction\n5. Common Sub-Expression Elimination\n6. Code Motion\n7. Inlining\n\nEliminates code that cannot be reached or where the results are not subsequently used.\n\nFor example, consider the following unoptimized code fragment:\n\nint count\nvoid foo() {\nint i;\ni = 1; // dead code since it is not subsequently used\ncount = 1; //dead code since it was overwritten\ncount = 2;\nreturn;\ncount = 3; //dead code(unreachable) since the function has returned\n}\n\nAfter applying dead code elimination we have the optimized code below:\n\nint count\nvoid foo() {\ncount = 2;\nreturn;\n}\n\n2. Constant Folding\nThis refers to the technique of evaluating ate compile time, expressions whose operands are known to be constant.\nIt involves the determining that all of the operands in an expression are constant values, performing the evaluation of the expression at compile time and then replacing the expression by its value.\n\nFor example, the expression\n12 + 4 * 3\ncan be replaced by its result of 24 at compile time and omit the code as if the input contained the results rather than the original expression\n\n3. Constant Propagation\nIn constant propagation, if a variable is assigned a constant value, then subsequent use of that variable can be replaced by a constant as long as no intervening assignment has changed the value of the variable.\n\nFor Example, consider the code:\nint x = 12;\nint y = 7 – x /2;\nreturn y * (24 / x + 2)\n\nApplying constant propagation to x, we have:\nint x = 12;\nint y = 7 – 12 / 2;\nreturn y * (24 / 12 + 2);\n\nApplying constant folding , we have:\nint x = 12;\nint y = 1;\nreturn y * 4;\n\n4. Strength Reduction\nThis is also called operator strength reduction is the replacement of expressions that are expensive with cheaper and simple ones.\nFor example an add instruction can be used to replace a multiply instruction.\n\nThe code:\nT2 = T2 * 2\nCan be replaced with:\nT2 = T2 + T2\n\n5. Common Sub Expression Elimination\nThis is a code optimization technique that scans the code to find identical expressions and replaces redundant expression each time it is encountered.\nFor example, consider the following fragment:\na = b * c + g;\nd = b * c * e;\nHere we see that the sub-expression b * c repeats, so we can transform the code to:\nt = b * c;\na = t + g;\nd = t + e;\n6. Code Motion\nAlso called loop-invariant code motion has to do with moving a block of code outside a loop if it wont have any difference if it is executed outside or inside the loop.\nConsider the example:\nfor (int i = 0; i < n; i++) {\nx = y + z;\na[i] = 6 * i;\n}\nIn the code fragment, the expression x = y + z has no effect inside the loop and can safely be moved outside of the loop.\nThe resulting code would be:\nx = y + z;\nfor (int i = 0; i < n; i++) {\na[i] = 6 * i;\n}\n\n7. Inlining\nThis is also referred to as function inlining or inline expansion, is a technique of replacing a function call with the actual body of the function.\nThis technique eliminates the overhead associated with expanding the body of the function inline.\n\nConsider the code fragment below:\nint add ( int x, int y)\n{\nz = x + y;\nreturn z;\n}\nint sub (int x, int y) {\n}\nWe can expand the second function without calling he add function, so we have:\n\nint sub (int x, int y) {\nreturn x + -y;\n}\nThis is also referred to as function inlining or inline expansion, is a technique of replacing a function call with the actual body of the function.\nThis technique eliminates the overhead associated with expanding the body of the function inline.\n\nConsider the fragment:\nint add ( int x, int y)\n{\nz = x + y;\nreturn z;\n}\nint sub (int x, int y) {\n}\nWe can expand the second function without calling the add function, so we have:\nint sub (int x, int y) {\nreturn x + -y;\n}\n\nExample\nLets apply these code optimization technique we discussed to the unoptimized code fragment below\n\nt1 = t1 + 1 //remove this(dead code elimination)\nL0:t2 = 0\nt3 = t1 * 8 + 1\nt4 = t3 + t2   //remove this (copy propagation\nt5 = t4 * 4     //(t3+t2)*4, then remove preceding line\nt6 = t5\nt7 = FP + t3 //remove this (dead code elimiation)\n*t7 = t2        //replace with t7=0(constant propagation)\nt8 = t1         //remove this (dead code elimination)\nif(t8>0) goto L1 //t8 is surely < 8 (constant folding)\nL1: goto L0\nL2: t1 = 1\nt10 = 16\nt11 = t1 * 2   //Change to t1 + t1 (Strength reduction)\ngoto L2\n\nResulting Optimization Code\nAfter applying the various optimization technniques discussed, the resulting optimized code is shown below.\n\nL0:t2 = 0\nt3 = t1 * 8 + 1\nt5 = (t3+t2)*4\nt6 = t5\n*t7 = 0\nt8 = t1\nL1: goto L0\nL2: t1 = 1\nt10 = 16\nt11 = t1 + t1\ngoto L2" ]

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[ "# Division table for N = 387 / 87÷88\n\n387 / 87 = 4.4483 [+]\n387 / 87.01 = 4.4478 [+]\n387 / 87.02 = 4.4473 [+]\n387 / 87.03 = 4.4467 [+]\n387 / 87.04 = 4.4462 [+]\n387 / 87.05 = 4.4457 [+]\n387 / 87.06 = 4.4452 [+]\n387 / 87.07 = 4.4447 [+]\n387 / 87.08 = 4.4442 [+]\n387 / 87.09 = 4.4437 [+]\n387 / 87.1 = 4.4432 [+]\n387 / 87.11 = 4.4427 [+]\n387 / 87.12 = 4.4421 [+]\n387 / 87.13 = 4.4416 [+]\n387 / 87.14 = 4.4411 [+]\n387 / 87.15 = 4.4406 [+]\n387 / 87.16 = 4.4401 [+]\n387 / 87.17 = 4.4396 [+]\n387 / 87.18 = 4.4391 [+]\n387 / 87.19 = 4.4386 [+]\n387 / 87.2 = 4.4381 [+]\n387 / 87.21 = 4.4376 [+]\n387 / 87.22 = 4.4371 [+]\n387 / 87.23 = 4.4365 [+]\n387 / 87.24 = 4.436 [+]\n387 / 87.25 = 4.4355 [+]\n387 / 87.26 = 4.435 [+]\n387 / 87.27 = 4.4345 [+]\n387 / 87.28 = 4.434 [+]\n387 / 87.29 = 4.4335 [+]\n387 / 87.3 = 4.433 [+]\n387 / 87.31 = 4.4325 [+]\n387 / 87.32 = 4.432 [+]\n387 / 87.33 = 4.4315 [+]\n387 / 87.34 = 4.431 [+]\n387 / 87.35 = 4.4305 [+]\n387 / 87.36 = 4.4299 [+]\n387 / 87.37 = 4.4294 [+]\n387 / 87.38 = 4.4289 [+]\n387 / 87.39 = 4.4284 [+]\n387 / 87.4 = 4.4279 [+]\n387 / 87.41 = 4.4274 [+]\n387 / 87.42 = 4.4269 [+]\n387 / 87.43 = 4.4264 [+]\n387 / 87.44 = 4.4259 [+]\n387 / 87.45 = 4.4254 [+]\n387 / 87.46 = 4.4249 [+]\n387 / 87.47 = 4.4244 [+]\n387 / 87.48 = 4.4239 [+]\n387 / 87.49 = 4.4234 [+]\n387 / 87.5 = 4.4229 [+]\n387 / 87.51 = 4.4224 [+]\n387 / 87.52 = 4.4218 [+]\n387 / 87.53 = 4.4213 [+]\n387 / 87.54 = 4.4208 [+]\n387 / 87.55 = 4.4203 [+]\n387 / 87.56 = 4.4198 [+]\n387 / 87.57 = 4.4193 [+]\n387 / 87.58 = 4.4188 [+]\n387 / 87.59 = 4.4183 [+]\n387 / 87.6 = 4.4178 [+]\n387 / 87.61 = 4.4173 [+]\n387 / 87.62 = 4.4168 [+]\n387 / 87.63 = 4.4163 [+]\n387 / 87.64 = 4.4158 [+]\n387 / 87.65 = 4.4153 [+]\n387 / 87.66 = 4.4148 [+]\n387 / 87.67 = 4.4143 [+]\n387 / 87.68 = 4.4138 [+]\n387 / 87.69 = 4.4133 [+]\n387 / 87.7 = 4.4128 [+]\n387 / 87.71 = 4.4123 [+]\n387 / 87.72 = 4.4118 [+]\n387 / 87.73 = 4.4113 [+]\n387 / 87.74 = 4.4108 [+]\n387 / 87.75 = 4.4103 [+]\n387 / 87.76 = 4.4098 [+]\n387 / 87.77 = 4.4093 [+]\n387 / 87.78 = 4.4087 [+]\n387 / 87.79 = 4.4082 [+]\n387 / 87.8 = 4.4077 [+]\n387 / 87.81 = 4.4072 [+]\n387 / 87.82 = 4.4067 [+]\n387 / 87.83 = 4.4062 [+]\n387 / 87.84 = 4.4057 [+]\n387 / 87.85 = 4.4052 [+]\n387 / 87.86 = 4.4047 [+]\n387 / 87.87 = 4.4042 [+]\n387 / 87.88 = 4.4037 [+]\n387 / 87.89 = 4.4032 [+]\n387 / 87.9 = 4.4027 [+]\n387 / 87.91 = 4.4022 [+]\n387 / 87.92 = 4.4017 [+]\n387 / 87.93 = 4.4012 [+]\n387 / 87.94 = 4.4007 [+]\n387 / 87.95 = 4.4002 [+]\n387 / 87.96 = 4.3997 [+]\n387 / 87.97 = 4.3992 [+]\n387 / 87.980000000001 = 4.3987 [+]\nNavigation: Home | Addition | Substraction | Multiplication | Division       Tables for 387: Addition | Substraction | Multiplication | Division\n\nOperand: 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 81 82 83 84 85 86 87 88 89 90 100 200 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000\n\nDivision for: 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 381 382 383 384 385 386 387 388 389 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000" ]

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[ "# Vertical circular motion\n\n## Homework Statement\n\nA 0.4 kg object rotates in a vertical circle at the end of a 0.5 m string. What is the tension of the string at the bottom if the angular velocity there is 8.0 rad/s?\n\n## Homework Equations\n\ncentripetal acceleration = R*w^2\nweight = mg\nw = angular velocity\n\n## The Attempt at a Solution\n\ncentripetal acceleration = (0.5 m)(8.0/s)^2 = 32 m/s^2\ntotal (centripetal) force = (0.4 kg)(32 m/s^2) = 12.8 N (upwards)\nweight = (0.4 kg)(9.8 m/s^2) = 3.9 N (downwards)\nUpwards forces and accelerations will be considered positive, downwards ones negative.\ntotal (centripetal) force = string tension - weight\n12.8 N = string tension - 3.9 N\nstring tension = 12.8 N + 3.9 N = 16.7 N\nIn general terms:\nstring tension = mg + mRw^2\n\nUnfortunately, the \"official\" solution is 13 N.\nAm I wrong, or are my course materials wrong?\nWill I be equally puzzled if I try to take the AP Physics B exam?" ]

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[ "# 271 in decimal\n\n## What is 271 in decimal?\n\n(Six hundred twenty-five)\n\nAll Properties of number 625\n\n## About \"Hex to Decimal\" Calculator\n\nThis calculator will help you to convert hexadecimal numbers to decimal. For example, What is 271 in decimal? Enter hexadecimal number (e.g. '271') and then click the 'Convert' button.\n\n## FAQ\n\n### What is 271 in decimal?\n\nHexadecimal Number 271 It Is Decimal: 625" ]

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[ "Say if $\\mathbb{R},\\tau$ is compact\n\nI'd like to have a check about this exercise, which asks me if $\\mathbb{R}$, with three different topologies, is compact\n\n1. $\\tau=${$U \\subseteq \\mathbb{R}: [-1,1] \\subset U$} $\\cup$ {$\\emptyset$}\nSay if $\\mathbb{R}$ is compact.\n\nLet $\\mathcal{R}$ be an open cover of $\\mathbb{R}$. $\\mathcal{R}=\\cup_{i\\in I}{A_i}$. If $A_i=(-1-i,1+i)$, $i>0$, then this is a open cover, but I can't find a finite subcover because this wouldn't cover the whole $\\mathbb{R}$.\n\n2.\n\nLet $\\mathcal{B}=${$(a,b): a<0, b>1, a,b \\in \\mathbb{R}$} the basis which generates the topology $\\tau$. Say if $\\mathbb{R}$ is compact.\n\nAlso here, if I take an open cover $\\mathcal{R}=\\cup_{i \\in I}A_i$, with $A_i=(a-i,b+i)$, $i>0$, their union covers the whole $\\mathbb{R}$, but just if it's finite, so the set it's not compact.\n\n3.\n\nLet $\\mathcal{B}=${$[a,+\\infty): a \\in \\mathbb{R}$} the basis which generates the topology $\\tau$. Say if $\\mathbb{R}$ is compact.\n\nI consider the open cover $\\mathcal{R}=\\cup_{x\\leq a}[a,+\\infty)$. It's open because union of open sets. If there would be a finite subcover, then I could stop to an $\\overline{a}$, such that $[\\overline{a},+\\infty)$: but in this case this wouldn't cover $(-\\infty,\\overline{a})$, so I can't find a finite subcover, and $\\mathbb{R}$ is not compact.\n\nThe general ideas are right, but in (2) and (3), you reuse variables that are not defined in the relevant scope. Specifically, when you say this:\n\nwith $$A_i=(a-i,b+i)$$\n\nthe open cover $$\\mathcal{R}=\\cup_{x\\leq a}[a,+\\infty)$$.\n\nThat doesn't make sense, because $$a$$ and $$b$$ are not fixed numbers that are given to you; they are bound variables used in the definition of the topology.\n\nBottom line: Don't use the letters \"$$a$$\" or \"$$b$$\" when you define your $$\\mathcal{R}$$. Ideally, don't use them anywhere. Instead, replace them with constants like $$0$$, $$1$$, or $$-1$$, or find a way to eliminate them altogether.\n\n• So I should write $A_i=(1-i,1+i)$ instead of $(a-i,b+i)$ for (2) and in (3) $\\mathcal{R}=\\cup_{x\\leq 1}[a,+\\infty)$, for example?\n– VoB\nAug 24 '17 at 9:04\n• Something like that! I think you actually want $A_i=(-i,1+i)$ (because the lower limit must be $<0$) and $\\mathcal{R}=\\cup_{x\\leq 1}[x,+\\infty)$ (using $x$ instead of $a$). In fact, if you want to be really terse, you can just take $\\mathcal R=\\mathcal B$ for both problems! Aug 24 '17 at 9:09\n• Oh yes, sorry ;) Of course i meant $(-i,i+1)$... the same for $\\mathcal{R}=\\cup_{x\\leq 1}[x,+\\infty)$ Thanks so much ;)\n– VoB\nAug 24 '17 at 9:58" ]

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[ "worldcreation.info Manuals Algorithm Book By Karumanchi\n\n# ALGORITHM BOOK BY KARUMANCHI\n\nMonday, June 3, 2019\n\nDesigned by Narasimha Karumanchi Printed in India Acknowledgements I would like to express my gratitude to the many people who saw me through this book. spending time in reviewing this book and providing me the valuable problem we will try to understand how much time the algorithm is taking and how much. Designed by Narasimha Karumanchi. Copyright© No part of this book may be reproduced in Data Structures and Algorithms - Narasimha worldcreation.info", null, "Author: LINN BAJOREK Language: English, Spanish, Hindi Country: Russian Federation Genre: Environment Pages: 736 Published (Last): 11.03.2015 ISBN: 162-7-66563-481-6 ePub File Size: 22.74 MB PDF File Size: 10.61 MB Distribution: Free* [*Registration Required] Downloads: 34237 Uploaded by: HILARY", null, "Data Structures and Algorithms Made Easy and millions of other books are . and Algorithmic Puzzles, Fifth Edition by Narasimha Karumanchi Paperback. By Narasimha Karumanchi Data Structures and Algorithms Made Easy: Data Story time just got better with Prime Book Box, a subscription that delivers. Click this link to Download this book >>> Data Structures and Algorithms Made Easy: Data Structures and Algorithmic Puzzles, Fifth Edition.\n\nExecution times? Number of statements executed? Ideal Solution? Let us assume that we expressed running time of given algorithm as a function of the input size i.\n\nWe can compare these different functions corresponding to running times and this kind of comparison is independent of machine time, programming style, etc..\n\nWhat is Rate of Growth? The rate at which the running time increases as a function of input is called.", null, "Let us assume that you went to a shop for downloading a car and a cycle. If your friend sees you there and asks what you are downloading then in general we say This is because cost of car is too big compared to cost of cycle approximating the cost of cycle to cost of car.\n\n## Data Structures and Algorithms Made Easy\n\nAs an example in the below case, , , and are the individual costs of some function and we approximate it to. Since, is the highest rate of growth. Commonly used Rate of Growths Below diagram shows the relationship between different rates of growth.\n\nWe have already seen that an algorithm can be represented in the form of an expression. That means we represent the algorithm with multiple expressions: In general the first case is called the best case and second case is called the worst case for the algorithm.\n\nThere are three types of analysis: Average case o Provides a prediction about the running time of the algorithm o Assumes that the input is random For a given algorithm, we can represent best case, worst case, and average case analysis in the form of expressions.\n\nAs an example, let be the function which represents the given algorithm. The expression defines the inputs with which the algorithm takes the average running time or memory. Asymptotic Notation? Having the expressions for best case, average case and worst case, for all the three cases we need to identify the upper bound, lower bounds.\n\nIn order to represent these upper bound and lower bounds we need some syntax and that is the subject of following discussion. Let us assume that the given algorithm is represented in the form of function. Big-O Notation This notation gives the upper bound of the given function.\n\nGenerally we represent it as. That means, at larger values of , the upper bound of is. For example, if is the given algorithm, then is. That means gives the maximum rate of growth for at larger values of.\n\nLet us see the notation with little more detail. Our objective is to give smallest rate of growth which is greater than or equal to given algorithms rate of growth.\n\nIn general, we discard lower values of. That means the rate of growth at lower values of is not important. In the below figure, is the point from which we need to consider the rate of growths for a given algorithm. Below the rate of growths could be different. Rate of Growth Input Size, Big-O Visualization is the set of functions with smaller or same order of growth as For example, includes etc..\n\n## Data Structures and Algorithms Made Easy\n\nAnalyze the algorithms at larger values of only. What this means is, below we do not care for rate of growth.", null, "There is no unique set of values for and in proving the asymptotic bounds. Let us consider, For this function there are multiple and values possible. That means, at larger values of , the tighter lower bound of is For example, if , is The notation can be defined as there exist positive constants and such that for all is an asymptotic tight lower bound for Our objective is to give largest rate of growth which is less than or equal to given algorithms rate of growth.\n\nSuch that: The average running time of algorithm is always between lower bound and upper bound. If the upper bound and lower bound gives the same result then notation will also have the same rate of growth. As an example, let us assume that is the expression. Then, its tight upper bound is. The rate of growth in best case is.", null, "In this case, rate of growths in best case and worst are same. As a result, the average case will also be same. For a given function algorithm , if the rate of growths bounds for and are not same then the rate of growth case may not be same.\n\nRate of Growth Input Size, Now consider the definition of notation. It is defined as there exist positive constants and such that for all. From the above examples, it should also be clear that, for a given function algorithm getting upper bound and lower bound and average running time may not be possible always. For example, if we are discussing the best case of an algorithm, then we try to give upper bound and lower bound and average running time.\n\nIn the remaining chapters we generally concentrate on upper bound because knowing lower bound of an algorithm is of no practical importance and we use notation if upper bound and lower bound are same. Why is it called Asymptotic Analysis?\n\n## Thiago Eduardo Silva\n\nFrom the above discussion for all the three notations: That means, is also a curve which approximates at higher values of. In mathematics we call such curve as. In other terms, is the asymptotic curve for For this reason, we call algorithm analysis as.\n\nGuidelines for Asymptotic Analysis? There are some general rules to help us in determining the running time of an algorithm. Below are few of them. The running time of a loop is, at most, the running time of the statements inside the loop including tests multiplied by the number of iterations. Analyze from inside out. Total running time is the product of the sizes of all the loops. Add the time complexities of each statement.\n\nWorst-case running time: An algorithm is if it takes a constant time to cut the problem size by a fraction usually by. As an example let us consider the following program: Initially in next step and in subsequent steps and so on. Let us assume that the loop is executing some times.\n\nI would like to express my gratitude to all of the people who provided support, talked things over, read, wrote, offered comments, allowed me to quote their remarks and assisted in the editing, proofreading and design. I know many people typically do not read the Preface of a book. But I strongly recommend that you read this particular Preface. It is not the main objective of this book to present you with the theorems and proofs on data structures and algorithms.\n\nI have followed a pattern of improving the problem solutions with different complexities for each problem, you will find multiple solutions with different, and reduced, complexities. With this approach, even if you get a new question, it will show you a way to think about the possible solutions. You will find this book useful for interview preparation, competitive exams preparation, and campus interview preparations.\n\nAs a job seeker, if you read the complete book, I am sure you will be able to challenge the interviewers. This book is also useful for Engineering degree students and Masters degree students during their academic preparations. In all the chapters you will see that there is more emphasis on problems and their analysis rather than on theory. In each chapter, you will first read about the basic required theory, which is then followed by a section on problem sets. In total, there are approximately algorithmic problems, all with solutions.\n\nWhat about the performance? What if you and your girlfriend share the same cloud account and are trying to play the same video from different devices?", null, "Now you have thought through the design well, have come up with different data structures to use with pros and cons in mind.\n\nWhile implementing, you must take care of corner cases. You must be aware about the integer overflow issue in Youtube video view count. While implementing, they never really thought that the view count can exceed what an integer variable can hold and BOOM, the view count cycled back to zero. Before a feature goes live, it must be tested well. It is good to practice some test questions as well.", null, "How will you test a Insert image feature in MS Word? What about a cut-copy-paste feature? How will you test Temple Run game? Try to write all the possible test cases and how you are going to handle this in your code.\n\nWriting a robust code is very important. If you take care of these things at an earlier stage, you can avoid silly bugs and boost your chances of getting selected in interviews. What else? Have a sound understanding of Operating System. The dinosaur book by Galvin is a good read. Know how networking works and have insights on DBMS. Resume building First impression is the best.\n\nResume is the first thing that HR will use to decide whether to call you for interview or not. And they have got hundreds of them. So they will usually scan it for 20 seconds to 2 minutes. It should be clean, concise and elegant. Each word mentioned should worth the space it eats.\n\nThe rule of thumb is if you have less than one year of experience, the size of resume should not exceed a page with few exceptions. Few points to note: Maintain a header to fit info like name, email id, address and contact number Mention level of expertise corresponding to each language.\n\nExample: Proficient in C and good at Java If you are mentioning a project, write your key learning, impact in the team and. This will show that you built something that is being used by people. Guess what, this is what companies do, building a product, stabilizing it as per user feedback, taking in new feature requests and so on. How to apply for Microsoft? I get many messages asking me for a favour to refer them.\n\nWhen I ask them how much comfortable they are with DS and Algos, they say good enough.\n\n## Editorial Reviews\n\nThen I rephrase my question to how do they feel when they solve interview experiences at GeeksforGeeks. This is not a surprise. GeeksforGeeks is still growing. Please do NOT do that. Dreaming is good, but it will come true only when you work towards making it a reality.\n\nIf you are not able to clear the interviews, you will have wait again for months depending on the company policy before you can apply again. Referral usually bumps chances of getting an interview call because your resume gets to the system through a person Microsoft trusts to be a good engineer.In other words, we enumerated possible solutions.\n\nThat problem can be anything starting from a simple puzzle to implementing a user scenario.\n\n## Meet the team\n\nHis review gave me the confidence in the quality of the book. It is recommended that the reader does at least one complete reading of this book to gain a full understanding of all the topics that are covered.", null, "Even though many readings have been done for the purpose of correcting errors, there could still be some minor typos in the book. Algorithm analysis helps us determining which of them is efficient in terms of time and space consumed. Problem Write a recursive function for the running time of the function function, whose code is below." ]

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[ "## For an angle Θ with the point (12, −5) on its terminating side, what is the value of cosine? −5 12 negative 5 ove\n\nQuestion\n\nFor an angle Θ with the point (12, −5) on its terminating side, what is the value of cosine?\n\n−5\n12\nnegative 5 over 13\n12 over 13\n\nin progress 0\n\n1.", null, "D\n\nStep-by-step explanation:\n\nI just took the test. 🙂\n\n2.", null, "Given that the point (12,-5) which takes the form (x,y), This implies that:\nopposite=-5\nthus using using Pythagorean theorem, the hypotenuse will be:\nc^2=a^2+b^2\nplugging the values we obtain:\nc^2=(12)^2+(-5)^2\nc^2=144+15\nc^2=169\nthus\nc=13\nbut" ]

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[ "# How to reconcile time dilation equation with concept\n\nSo I must be misunderstanding time dilation, but according to the Lorentz transformation $t_v=\\frac{t_o}{\\sqrt{1-\\frac{v^2}{c^2}}}$, where $t_v$ is the time on the object moving relative to the measuring observer, and $t_o$ is the time measured on the object still relative to the measuring observer. But by this, one could deduce that $t_v>t_o$ and thus the time on the moving object is greater, implying it is running faster than that of a still object. This seems to be inconsistent with the theory of time dilation as a concept, which suggests that time would move slower on the moving object.\n\nThere is an example question which I cannot resolve which seems to illustrate this:\n\nAn astronaut set out in a spaceship from Earth orbit to travel to a distant star in our galaxy. The spaceship travelled at a speed of 0.8c. When the spaceship reached the star the onboard clock said the journey took 10 years. An identical clock remained on Earth. What time in years had elapsed on this clock when seen from the astronaut’s spaceship?\n\nIf this is resolved with Earth as $t_v$, since it is moving relative to the measurement of the astronaut, we get an answer of 16.7 years. This implies, however, that more time has passed on Earth during the flight and thus Earth time was running faster. If resolved in the alternate way, we get a value of 6 years, however this also seems inconsistent as Earth was set as the stationary reference frame\n\n• t_v is actually not greater than t_o, when V is zero (both objects relatively stationary), t_v = t_o. But as V increases, t_o increases as well since it's denominator is decreasing. Look into the Twin Paradox Jun 15 '17 at 0:52\n\nThe formula you have needs to be interpreted carefully. It depends on who is looking at which clock and what is meant by \"how fast time is moving.\" In your formula, let's have $t_o = 1$ second. When you calculate $t_v$ your are calculating how much time has to pass in the rest frame for the moving clock to tick one second. So, moving clocks tick slower, so it takes longer for a moving clock to indicate one second. That's why the formula results in a longer time.\n\nAnother way to write the formula is $$t_v = t_o\\sqrt{1-\\frac{v^2}{c^2}}.$$ In this formula, $t_o$ is a time measured in the rest frame and $t_v$ is the amount of time indicated on the moving clock.\n\nIn short, your formula tells how long an at-rest observer has to wait in order to see a moving clock tick $t_o$ time. The second formula tells how much time a moving clock as ticked after an at-rest clock has ticked $t_o$ time.\n\n• It might be good to mention that the OP has written the formula for $\\gamma$, which is likely what he/ she's confusing the right formula with. Mar 16 '17 at 10:24\n• @WetSavannaAnimalakaRodVance After looking at the wikipedia article for time dilation, I realized there's more than one way to define the concept. Answer has been edited. Mar 16 '17 at 10:28\n• Yes, that's a good answer. Usually one knows which way around the scaling goes, so one doesn't do it the wrong way around, but when you're reading the instructions for the first time and don't know which way the scaling goes, you do indeed need to interpret carefully. Mar 16 '17 at 10:36\n\nFormulas alone might be hard to implement.\n\nA spacetime diagram on rotated graph paper might be useful here, together with some geometrical intuition.", null, "Bob travels at (4/5)c (given by the Minkowski right-triangle with hypotenuse OB). Event Z is 10 ticks along Bob's worldline.\n\nTime dilation relates two timelike segments from event O with endpoints that are simultaneous to the observer making the measurement.\n\nSo, Alice says BsimA is simultaneous with B. She measures the time-dilation factor $\\gamma=\\frac{OBsimA}{OB}=5/3$.\n\nAlice also says ZsimA is simultaneous with Z. She measures the time-dilation factor $\\gamma=\\frac{OZsimA}{OZ}=16.666/10=5/3$. (Indeed, these Minkowski-right triangles are similar by scaling.)\n\nTrigonometrically, there is an angle $\\theta$ between the worldlines (called the rapidity... the arc-length subtended on the unit hyperbola centered at O). These ratios can be thought of as $ADJACENT/HYPOTENUSE=\\cosh\\theta=\\gamma$ (where $OPPOSITE/ADJACENT=\\tanh\\theta=v$, the velocity).\n\nBob says ZsimB is simultaneous with Z. He measures the time-dilation factor $\\gamma=\\frac{OZ}{OZsimB}=10/6=5/3$. Indeed, this Minkowski-right triangle (with Minkowski-right-angle at Z and hypotenuse $OZsimB$) is similar to Alice's Minkowski-right triangles.\n\nIt is often said, that single clock dilates relatively to a set of spatially separated and Einstein – synchronized clocks.\n\nThis set of synchronized clocks is the REST FRAME of the observer. Observer is not a real physical person, but the whole reference frame, or team of observers. Relativistic observer conducts measurements in his own rest frame.\n\nTo measure dilation of moving clock an Observer (or team of observers) must have at least two clock in his frame, let's say clock C1 and C2. When moving clock passes by clock C1, he compares readings in immediate vicinity. When clock passes by clock C2, he compares readings again. If moving clock and C1 showed 12 PM at meeting, moving clock will show 3 PM and C2 shows 6 PM when they meet. This is how time dilation works. SINGLE moving clock dilates relatively to a set of synchronized and spatially separated clocks, not vice versa. Set of clock runs faster from the point of view of single clock.\n\nIf \"moving\" observer wants to measure dilation of another clock, he has to turn himself into one \"at rest\" by means of introducing his own rest frame. He simply puts another additional clock (at least) and synchronizes it by Einstein signalling method (Einstein clock synchronization convention).\n\nhttps://en.wikipedia.org/wiki/Einstein_synchronisation\n\nAlso:\n\n\"Two spatially separated clocks, A and B, record a greater time interval between two events than the proper time recorded by a single clock that moves from A to B and is present at both events.\"\n\nIt is absolutely clear even visually (animation from article Time Dilation in Wikipedia).\n\n[", null, "Notion of Observer in SR https://en.wikipedia.org/wiki/Observer_(special_relativity)\n\nIn your case, observer in the reference frame of Earth has to place two Einstein – synchronized clocks. One E1 is at Earth, another E2 is far, far away at the point of arrival of Astronaut. These clocks are Einstein - synchronized. When Astronaut arrives to E2 his clock shows gamma times less time. Obviously, clock E2 shows gamma times more time.\n\nWe can imagine a row of synchronized clocks of reference system K. each denoted by letter - A, B, C, D ….. Z. Then a person with single clock on his wrist (A’ for example) moves in this reference system K and compares readings of his clock with these clock A-Z successively. When he comes to the clock Z, his clock A’ shows gamma times less time, than clock Z. Thus, clock Z shows gamma times more time, than his own. At this point Z this clock A’ immediately turns back and starts travelling in reverse direction, passing by clock Z, Y, X ….. C, B and finally arrives into point A of reference frame K. Clock A’ compares readings with clocks Z-A successively again and sees, that it dilates itself gamma times, i.e. every clock on the way shows gamma times more time. When clock A’ arrives into point A, clock A’ shows gamma times less time than clock A, and clock A shows gamma times more time than clock A’.\n\nThis discrepancy of clock readings when they meet again is often called clock paradox or Twin paradox.\n\nAll that goes straight from the Lorentz transformations.\n\n$$T = \\frac {t'_{x'}+ \\frac {v'} {c^2} x'} {\\sqrt {1-( \\frac {v} c)^2}} (1)$$\n\n$T$ is clock readings that belongs to reference frame $K$, taken in point $x'$ at moment of time $t'_{x'}$ of reference frame $K'$, and $t'_{x'}$ reading of clocks that belongs to reference frame $K'$ in the point $x'$ of reference frame $K'$\n\nHow to interpret Lorentz transform for time?\n\nTransformation demonstrates, that time $T$ of reference frame $K$ (in which it does not depend of $x$ coordinate or any other coordinate) is universal in reference frame $K$ and each point of this frame.\n\nNow let's fix point $x'$, for example $x'=0$. In this case this transformation will look like that:\n\n$$T= \\frac {t'_{0'}} {\\sqrt {1-( \\frac v c)^2}} (2)$$\n\n$T$ is clock reading of reference frame $K$ taken in point $x'=0$ (in the origin $O'$ of reference frame $K'$), and $t'_{o'}$ is time in the reference frame $K'$, in particular in the origin $O'$.\n\nWe can take $dT/dt'$ when $x'$ is fixed and will get ${dT}/{dt'} = 1/{\\sqrt {1- \\frac {v^2} {c^2}}}$\n\nAccording to (2) it is not time $t'_{o'}$ which is showed by single clock in the point $O'$ runs slower, but time $T$ , which is \"distributed\" through all reference frame $K$ and taken in the origin $O'$ of reference frame $K'$ runs faster (relatively to time $t'_{o'}$ that is in the origin $O'$ of frame $K'$). Time dilation comes by means of transformation of (2) into:\n\n$$t'_{o'}=T \\sqrt {1- {\\frac {V^2} {c^2}}}$$\n\nIt is correct that $T>t'$ and $t'<T$. It is also true that $T'>t$ and $t<T'$. But that $t<t'$ and $t'<t$ from different points of view is nonsense." ]

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[ "# Elias Mårtenson\n\n## APL and syntax\n\nWhen mentioning APL to someone who has never used it, the reaction is often along the lines of “isn't that the unreadable language with weird symbols?”.\n\nThe classic example of APL code is Conway's Game of Life which can be written as such:\n\n{⊃1 ⍵ ∨.∧ 3 4 = +/ +⌿ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂⍵}\n\n\nThis implies that the answer to the question above is yes, but there is an implication in that statement that suggests that the symbols is what makes the language different.\n\nAnother language in the same family as APL is J. This language uses many of the same principles as APL, and Game of Life in J looks like this:\n\nn1dim=:3 +/\\ (0,,&0)\nneighbors=: -~ [: n1dim\"1 n1dim\nlife=: [: (0.5 >: 3 |@- ]) -:+neighbors\n\n\nFor someone without knowledge of either APL nor J, both of these examples are probably inscrutable. The point of the example above is to emphasise that it's not the symbols that makes APL seem alien to a newcomer.\n\nThe question then becomes: Why does APL look the way it does?\n\nTo answer this question, we need to look into the history of the language and the requirements that led to its design.\n\nThe designer of both APL and J was Ken Iverson. He was a mathematician, and looked at problem solving from a mathematical perspective. APL was originally a notation for array manipulation, and it was only later that it was actually implemented as a programming language for computers.\n\nThus, to understand the APL notation, it's important to think about it as a concise way to describe mathematical operations, especially operations working on arrays of values.\n\nThe first step is to look at the syntax of maths, not programming languages. Let's take a simple example of summing all the elements of an array $b$:\n\n$\\sum_i b_i$\n\nThe above syntax is a bit imprecise and we are assuming that $i$ will take on all the valid indexes in $b$.\n\nIf we want to take the product of all the elements in $b$, then we can replace the summation sign with the product symbol:\n\n$\\prod_i b_i$\n\nWhat if we want to create a chain of exponentiations? There is no dedicated symbol for this, so we need to write:\n\n$b_0^{b_1^{b_2^{\\cdot^{\\cdot^{\\cdot^{b_{n-1}}}}}}}$\n\nIn the example above, we're assuming that the reader will understand that $n$ represents the number of elements in the array.\n\nThe observation here is that these operations are all variations of the same underlying operation. In functional programming this is usually referred to as a reduction.\n\nTo perform the sum in APL, one wrould write the following:\n\n+/b\n\n\nThe + is the operation to perform, and / is the reduction operator. The natural extension to the product is:\n\n×/b\n\n\nAnd for exponentiation:\n\n*/b\n\n\nThe argument in favour of APL notation is that it's simpler and more concise than traditional mathematical notation. That's no short order, as maths is already supposed to be both simple and concise. Not only that, the APL notation is free of imprecision as opposed to the mathematical examples, as was discussed above.\n\nThe APL syntax also allows you to use any other function as part of a reduction. For example, the function ⌊ returns the smaller of the two values, so the expression ⌊/b can be used to find the smallest value in $b$. The way to write this in mathematical notation is as follows:\n\n$\\min_i(b_i)$\n\nIt is assumed that $\\min$ is well known, since in APL we make the same assumption concerning ⌊.\n\nThe reason why APL syntax looks the way it does, is because it's a mathematical notation that happens to be suitable for evaluation by a computer, rather than a set of instructions for a computer to perform." ]

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[ "# Gridding a geometry or geography object (SQL Server Denali)\n\nA common question that comes up in the various database forums (PostGIS, SQL Server, Oracle) is how to “grid” a linear or polygonal object.\n\nBy “grid” one means work out the square pixels (rectangular polygons) that cover or define a vector geometry.\n\nHere is some SQL for doing this for SQL Server 2008 Denali. Denali has spatial aggregates built in which is why it was used for this blog article. It can be done for SQL Server 2008 R1 and R2 but it needs the Codeplex extensions to be installed and enabled.\n\nThe gridding method requires two functions Morton and REGULARGRIDXY. THe Morton key function is provided in another blog article.\n\nThe other required function, REGULARGRIDXY ensures that the gridding takes place according to a commonly defined MBR so that the grids generated across multiple geometries align. This function is provided here:\n\n``` CREATE FUNCTION [dbo].[REGULARGRIDXY]\n(\n@p_ll_x       FLOAT,\n@p_ll_y       FLOAT,\n@p_ur_x       FLOAT,\n@p_ur_y       FLOAT,\n@p_TileSize_X FLOAT,\n@p_TileSize_Y FLOAT,\n@p_srid       INT\n)\nRETURNS @TABLE TABLE\n(\ncol   INT,\nROW   INT,\ngeom  geometry\n)\nAS\nBEGIN\nDECLARE\n@v_loCol INT,\n@v_hiCol INT,\n@v_loRow INT,\n@v_hiRow INT,\n@v_col   INT,\n@v_row   INT;\nBEGIN\nSET @v_loCol = FLOOR(   @p_LL_X / @p_TileSize_X );\nSET @v_hiCol = CEILING( @p_UR_X / @p_TileSize_X ) - 1;\nSET @v_loRow = FLOOR(   @p_LL_Y / @p_TileSize_Y );\nSET @v_hiRow = CEILING( @p_UR_Y / @p_TileSize_Y ) - 1;\nSET @v_col = @v_loCol;\nWHILE ( @v_col <= @v_hiCol )\nBEGIN\nSET @v_row = @v_loRow;\nWHILE ( @v_row <= @v_hiRow )\nBEGIN\nINSERT INTO @TABLE (col,ROW,geom)\nVALUES(@v_col, @v_row,\ngeometry::STGeomFromText('POLYGON((' +\nLTRIM(STR((@v_col * @p_TileSize_X),24,12)) + ' ' + LTRIM(STR((@v_row * @p_TileSize_Y),24,12)) + ',' +\nLTRIM(STR(((@v_col * @p_TileSize_X)+@p_TileSize_X),24,12)) + ' ' + LTRIM(STR((@v_row * @p_TileSize_Y),24,12)) + ',' +\nLTRIM(STR(((@v_col * @p_TileSize_X)+@p_TileSize_X),24,12)) + ' ' + LTRIM(STR(((@v_row * @p_TileSize_Y)+@p_TileSize_Y),24,12)) + ',' +\nLTRIM(STR((@v_col * @p_TileSize_X),24,12)) + ' ' + LTRIM(STR(((@v_row * @p_TileSize_Y)+@p_TileSize_Y),24,12)) + ',' +\nLTRIM(STR((@v_col * @p_TileSize_X),24,12)) + ' ' + LTRIM(STR((@v_row * @p_TileSize_Y),24,12)) + '))',@p_srid)\n);\nSET @v_row = @v_row + 1;\nEND;\nSET @v_col = @v_col + 1;\nEND;\nRETURN;\nEND;\nEND\nGO\n```\n\nHere is the gridding method for a single geometry.\n\n``` -- The following should be changed to the database name in which you have installed the Morton and RegularGridXY functions\n--\nUSE [GISDB]\nGO\n-- Single geometry processing\n--\nWITH geomQuery AS (\nSELECT g.geom.STEnvelope().STPointN(1).STX AS minx, g.geom.STEnvelope().STPointN(1).STY AS miny,\ng.geom.STEnvelope().STPointN(3).STX AS maxx, g.geom.STEnvelope().STPointN(3).STY AS maxy,\ng.geom, 0.050 AS gridX, 0.050 AS gridY, 0 AS loCol, 0 AS loRow\nFROM (SELECT a.geom.STBuffer(1.000).STSymDifference(a.geom.STBuffer(0.500)) AS geom\nFROM (SELECT geometry::STGeomFromText('MULTIPOINT((09.25 10.00),(10.75 10.00),(10.00 10.75),(10.00 9.25))',0) AS geom ) a\n) g\n)\nSELECT f.mKey, f.col, f.ROW, f.geom\nFROM (SELECT [GISDB].[dbo].Morton((c.col - a.loCol),(c.ROW - a.loRow)) AS mKey,\nc.col, c.ROW,\nCASE WHEN UPPER(a.geom.STGeometryType()) IN ('POLYGON','MULTIPOLYGON')\nTHEN a.geom.STIntersection(c.geom)\nELSE a.geom\nEND AS geom\nFROM geomQuery a\nCROSS APPLY\n[GISDB].[dbo].REGULARGRIDXY(a.minx, a.miny, a.maxx, a.maxy,a.gridX,a.gridY, a.geom.STSrid) c\nWHERE a.geom.STIntersects(c.geom) = 1\n) f\nWHERE UPPER(f.geom.STGeometryType()) IN ('POLYGON','MULTIPOLYGON') /* Don't want point or line tiles */\nORDER BY f.mKey;\n```\n\nThe result of this looks like.", null, "For multiple geometries, the following needs to be executed.\n\n``` -- Query for more than one geometry\n--\nWITH geomQuery AS (SELECT g.rid,\nMIN(g.geom.STEnvelope().STPointN(1).STX) OVER (partition BY g.pid) AS minx,\nMIN(g.geom.STEnvelope().STPointN(1).STY) OVER (partition BY g.pid) AS miny,\nMAX(g.geom.STEnvelope().STPointN(3).STX) OVER (partition BY g.pid) AS maxx,\nMAX(g.geom.STEnvelope().STPointN(3).STY) OVER (partition BY g.pid) AS maxy,\ng.geom, 0.050 AS gridX, 0.050 AS gridY, 0 AS loCol, 0 AS loRow\nFROM (SELECT 1 AS pid, a.rid, a.geom.STBuffer(1.000).STSymDifference(a.geom.STBuffer(0.750)) AS geom\nFROM (SELECT 1 AS rid, geometry::STGeomFromText('POINT(09.50 10.00)',0) AS geom\nUNION ALL SELECT 2 AS rid, geometry::STGeomFromText('POINT(10.50 10.00)',0) AS geom\nUNION ALL SELECT 3 AS rid, geometry::STGeomFromText('POINT(10.00 10.50)',0) AS geom\nUNION ALL SELECT 4 AS rid, geometry::STGeomFromText('POINT(10.00 09.50)',0) AS geom ) a\n) g\n)\nSELECT ROW_NUMBER() OVER (ORDER BY f.col, f.ROW) AS tid,\n[GISDB].[dbo].Morton((f.col - f.loCol),(f.ROW - f.loRow)) AS mKey,\nf.col,\nf.ROW,\nCOUNT(*) AS UnionedTileCount,\ngeometry::UnionAggregate(f.geom) AS geom\nFROM (SELECT c.col, c.ROW, a.loCol, a.loRow,\nCASE WHEN UPPER(a.geom.STGeometryType()) IN ('POLYGON','MULTIPOLYGON')\nTHEN a.geom.STIntersection(c.geom)\nELSE a.geom\nEND AS geom\nFROM geomQuery a\nCROSS APPLY\n[GISDB].[dbo].REGULARGRIDXY(a.minx,a.miny,a.maxx,a.maxy,a.gridX,a.gridY,a.geom.STSrid) c\nWHERE a.geom.STIntersects(c.geom) = 1\n) f\nWHERE UPPER(f.geom.STGeometryType()) IN ('POLYGON','MULTIPOLYGON') /* Don't want point or line tiles */\nGROUP BY f.col, f.ROW, f.loCol, f.loRow\nORDER BY 2;\n```\n\nThat looks like this.", null, "I hope this is of use to SQL Server Denali users" ]

[ null, "https://i0.wp.com/www.spatialdbadvisor.com/images/158.png", null, "https://i0.wp.com/www.spatialdbadvisor.com/images/159.png", null ]

[ "## CryptoDB\n\n### Paper: TinyECCK: Efficient Elliptic Curve Cryptography Implementation over $GF(2^m)$ on 8-bit MICAz Mote\n\nAuthors: Seog Chung Seo Dong-Guk Han Seokhie Hong URL: http://eprint.iacr.org/2008/122 Search ePrint Search Google In this paper, we revisit a generally accepted opinion: implementing Elliptic Curve Cryptosystem (ECC) over $GF(2^m)$ on sensor motes using small word size is not appropriate because XOR multiplication over $GF(2^m)$ is not efficiently supported by current low-powered microprocessors. Although there are some implementations over $GF(2^m)$ on sensor motes, their performances are not satisfactory enough to be used for wireless sensor networks (WSNs). We have found that a field multiplication over $GF(2^m)$ are involved in a number of redundant memory accesses and its inefficiency is originated from this problem. Moreover, the field reduction process also requires many redundant memory accesses. Therefore, we propose some techniques for reducing unnecessary memory accesses. With the proposed strategies, the running time of field multiplication and reduction over $GF(2^{163})$ can be decreased by 21.1\\% and 24.7\\%, respectively. These savings noticeably decrease execution times spent in Elliptic Curve Digital Signature Algorithm (ECDSA) operations (signing and verification) by around $15\\% \\sim 19\\%$. We present TinyECCK (Tiny Elliptic Curve Cryptosystem with Koblitz curve -- a kind of TinyOS package supporting elliptic curve operations) which is the fastest ECC implementation over $GF(2^m)$ on 8-bit sensor motes using ATmega128L as far as we know. Through comparisons with existing software implementations of ECC built in C or hybrid of C and inline assembly on sensor motes, we show that TinyECCK outperforms them in terms of running time, code size and supporting services. Furthermore, we show that a field multiplication over $GF(2^m)$ can be faster than that over $GF(p)$ on 8-bit ATmega128L processor by comparing TinyECCK with TinyECC, a well-known ECC implementation over $GF(p)$. TinyECCK with sect163k1 can compute a scalar multiplication within 1.14 secs on a MICAz mote at the expense of 5,592-byte of ROM and 618-byte of RAM. Furthermore, it can also generate a signature and verify it in 1.37 and 2.32 secs with 13,748-byte of ROM and 1,004-byte of RAM.\n##### BibTeX\n@misc{eprint-2008-17799,\ntitle={TinyECCK: Efficient Elliptic Curve Cryptography Implementation over $GF(2^m)$ on 8-bit MICAz Mote},\nbooktitle={IACR Eprint archive},\nkeywords={implementation / ECC, sensor mote, Micaz, binary filed, TinyECC, Atmega128},\nurl={http://eprint.iacr.org/2008/122},\nnote={It is a full version of IEICE Trans. Vol.E91-D,No.5,pp.-,May. 2008. [email protected] 13955 received 16 Mar 2008},\nauthor={Seog Chung Seo and Dong-Guk Han and Seokhie Hong},\nyear=2008\n}" ]

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[ "", null, "Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  dp41 GIF version\n\nTheorem dp41 1193\n Description: Part of theorem from Alan Day and Doug Pickering, \"A note on the Arguesian lattice identity,\" Studia Sci. Math. Hungar. 19:303-305 (1982). (4)=>(1)\nHypotheses\nRef Expression\ndp41.1 c0 = ((a1a2) ∩ (b1b2))\ndp41.2 c1 = ((a0a2) ∩ (b0b2))\ndp41.3 c2 = ((a0a1) ∩ (b0b1))\ndp41.4 p2 = ((a0b0) ∩ (a1b1))\ndp41.5 p2 ≤ (a2b2)\nAssertion\nRef Expression\ndp41 c2 ≤ (c0c1)\n\nProof of Theorem dp41\nStepHypRef Expression\n1 dp41.1 . 2 c0 = ((a1a2) ∩ (b1b2))\n2 dp41.2 . 2 c1 = ((a0a2) ∩ (b0b2))\n3 dp41.3 . 2 c2 = ((a0a1) ∩ (b0b1))\n4 id 59 . 2 (((a0b0) ∩ (a1b1)) ∩ (a2b2)) = (((a0b0) ∩ (a1b1)) ∩ (a2b2))\n5 dp41.4 . 2 p2 = ((a0b0) ∩ (a1b1))\n6 dp41.5 . 2 p2 ≤ (a2b2)\n71, 2, 3, 4, 5, 6dp41lemm 1192 1 c2 ≤ (c0c1)\n Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1120  ax-arg 1151 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by: (None)\n Copyright terms: Public domain W3C validator" ]

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[ "Traffic Engineering\n\nA traffic engineer wants to know whether measurements of traffic flow entering and leaving a road network are sufficient to predict the traffic flow on each street in the network. For example, consider the network of one-way streets shown in Figure 6.4-2. The numbers in the figure give the measured traffic flows in vehicles per hour. Assume that no vehicles park anywhere within the network. If ‘possible, calculate the traffic flowsƒf1,ƒf2,ƒf3, and\nƒ 4. If this is not possible, suggest how to obtain the necessary information .\n• Solution\nThe flow into intersection I must equal the flow out of the intersection, which gives us", null, "First check the ranks of A and [A b] using the MATLAB rank command. Both have a rank of three, which is less than the number of unknowns, so we can determine three of the unknowns in terms of the fourth. Thus we cannot determine the traffic flows based on the given measurements. This example shows that it is not always possible to find a unique, exact solution even when the number of equations equals the number of unknowns.", null, "We can easily solve this system as follows: ƒ1 = 300  ƒ4. ƒ2 = 200 + ƒ4. and ƒ3 =\n800 – ƒ4\nIf we could measure the flow on one of the internal roads. say. ƒ4.then we could compute the other flows. So we recommend that the engineer arrange to make this additional measurement." ]

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[ "# Geometry Similar Figures Worksheet Geometry Ratios And Proportions Worksheet Inspirational Recent Similar Figures Proportions Worksheet Free Worksheets Sec 24 Geometry Similar Figures Worksheet Answer", null, "geometry similar figures worksheet geometry ratios and proportions worksheet inspirational recent similar figures proportions worksheet free worksheets sec 24 geometry similar figures worksheet answer." ]

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[ "# Foundations of geometry\n\nFoundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.\n\n## Axiomatic systems\n\nMain article: Axiomatic system\n\nBased on ancient Greek methods, an axiomatic system is a formal description of a way to establish the mathematical truth that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary mathematics in which this method has most extensively been successfully applied.\n\nThere are several components of an axiomatic system.\n\n1. Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs. His point being that the primitive terms are just empty shells, place holders if you will, and have no intrinsic properties.\n2. Axioms (or postulates) are statements about these primitives; for example, any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.\n3. The laws of logic.\n4. The theorems are the logical consequences of the axioms, that is, the statements that can be obtained from the axioms by using the laws of deductive logic.\n\nAn interpretation of an axiomatic system is some particular way of giving concrete meaning to the primitives of that system. If this association of meanings makes the axioms of the system true statements, then the interpretation is called a model of the system. In a model, all the theorems of the system are automatically true statements.\n\n### Properties of axiomatic systems\n\nIn discussing axiomatic systems several properties are often focused on:\n\n• The axioms of an axiomatic system are said to be consistent if no logical contradiction can be derived from them. Except in the simplest systems, consistency is a difficult property to establish in an axiomatic system. On the other hand, if a model exists for the axiomatic system, then any contradiction derivable in the system is also derivable in the model, and the axiomatic system is as consistent as any system in which the model belongs. This property (having a model) is referred to as relative consistency or model consistency.\n• An axiom is called independent if it can not be proved or disproved from the other axioms of the axiomatic system. An axiomatic system is said to be independent if each of its axioms is independent. If a true statement is a logical consequence of an axiomatic system, then it will be a true statement in every model of that system. To prove that an axiom is independent of the remaining axioms of the system, it is sufficient to find two models of the remaining axioms, for which the axiom is a true statement in one and a false statement in the other. Independence is not always a desirable property from a pedagogical viewpoint.\n• An axiomatic system is called complete if every statement expressible in the terms of the system is either provable or has a provable negation. Another way to state this is that no independent statement can be added to a complete axiomatic system which is consistent with axioms of that system.\n• An axiomatic system is categorical if any two models of the system are isomorphic (essentially, there is only one model for the system). A categorical system is necessarily complete, but completeness does not imply categoricity. In some situations categoricity is not a desirable property, since categorical axiomatic systems can not be generalized. For instance, the value of the axiomatic system for group theory is that it is not categorical, so proving a result in group theory means that the result is valid in all the different models for group theory and one doesn't have to reprove the result in each of the non-isomorphic models.\n\n## Euclidean geometry\n\nMain article: Euclidean geometry\n\nEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described (although non-rigorously by modern standards) in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.\n\nFor over two thousand years, the adjective \"Euclidean\" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other geometries which are not Euclidean are known, the first ones having been discovered in the early 19th century.\n\n### Euclid's Elements\n\nMain article: Euclid's Elements\n\nEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises, and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science.\n\nEuclid's Elements has been referred to as the most successful and influential textbook ever written. Being first set in type in Venice in 1482, it is one of the very earliest mathematical works to be printed after the invention of the printing press and was estimated by Carl Benjamin Boyer to be second only to the Bible in the number of editions published, with the number reaching well over one thousand. For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.\n\nThe Elements are mainly a systematization of earlier knowledge of geometry. It is assumed that its superiority over earlier treatments was recognized, with the consequence that there was little interest in preserving the earlier ones, and they are now nearly all lost.\n\nBooks I–IV and VI discuss plane geometry. Many results about plane figures are proved, e.g., If a triangle has two equal angles, then the sides subtended by the angles are equal. The Pythagorean theorem is proved.\n\nBooks V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as prime numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved.\n\nBooks XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base.", null, "The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.\n\nNear the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):\n\n\"Let the following be postulated\":\n\n1. \"To draw a straight line from any point to any point.\"\n2. \"To produce [extend] a finite straight line continuously in a straight line.\"\n3. \"To describe a circle with any centre and distance [radius].\"\n4. \"That all right angles are equal to one another.\"\n5. The parallel postulate: \"That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.\"\n\nAlthough Euclid's statement of the postulates only explicitly asserts the existence of the constructions, they are also assumed to produce unique objects.\n\nThe success of the Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are supposedly his. Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements, encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Further, its logical axiomatic approach and rigorous proofs remain the cornerstone of mathematics.\n\n### A critique of Euclid\n\nThe standards of mathematical rigor have changed since Euclid wrote the Elements. Modern attitudes towards, and viewpoints of, an axiomatic system can make it appear that Euclid was in some way sloppy or careless in his approach to the subject, but this is an ahistorical illusion. It is only after the foundations were being carefully examined in response to the introduction of non-Euclidean geometry that what we now consider flaws began to emerge. Mathematician and historian W. W. Rouse Ball put these criticisms in perspective, remarking that \"the fact that for two thousand years [the Elements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose.\"\n\nSome of the main issues with Euclid's presentation are:\n\n• Lack of recognition of the concept of primitive terms, objects and notions that must be left undefined in the development of an axiomatic system.\n• The use of superposition in some proofs without there being an axiomatic justification of this method.\n• Lack of a concept of continuity which is needed to prove the existence of some points and lines that Euclid constructs.\n• Lack of clarity on whether a straight line is infinite or boundary-less in the second postulate.\n• Lack of the concept of betweeness used, among other things, for distinguishing between the inside and outside of various figures.\n\nEuclid's list of axioms in the Elements was not exhaustive, but represented the principles that seemed the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. He does not go astray and prove erroneous things because of this since he is actually making use of implicit assumptions whose validity appears to be justified by the diagrams which accompany his proofs. Later mathematicians have incorporated Euclid's implicit axiomatic assumptions in the list of formal axioms, thereby greatly extending that list.\n\nFor example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise. If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I.1  I.3 can be proved trivially by using superposition.\n\nTo address these issues in Euclid's work, later authors have either attempted to fill in the holes in Euclid's presentationthe most notable of these attempts is due to D. Hilbertor to organize the axiom system around different concepts, as G.D. Birkhoff has done.\n\n### Pasch and Peano\n\nThe German mathematician Moritz Pasch (18431930) was the first to accomplish the task of putting Euclidean geometry on a firm axiomatic footing. In his book, Vorlesungen über neuere Geometrie published in 1882, Pasch laid the foundations of the modern axiomatic method. He originated the concept of primitive notion (which he called Kernbegriffe) and together with the axioms (Kernsätzen) he constructs a formal system which is free from any intuitive influences. According to Pasch, the only place where intuition should play a role is in deciding what the primitive notions and axioms should be. Thus, for Pasch, point is a primitive notion but line (straight line) is not, since we have good intuition about points but no one has ever seen or had experience with an infinite line. The primitive notion that Pasch uses in its place is line segment.\n\nPasch observed that the ordering of points on a line (or equivalently containment properties of line segments) is not properly resolved by Euclid's axioms; thus, Pasch's theorem, stating that if two line segment containment relations hold then a third one also holds, cannot be proven from Euclid's axioms. The related Pasch's axiom concerns the intersection properties of lines and triangles.\n\nPasch's work on the foundations set the standard for rigor, not only in geometry but also in the wider context of mathematics. His breakthrough ideas are now so commonplace that it is difficult to remember that they had a single originator. Pasch's work directly influenced many other mathematicians, in particular D. Hilbert and the Italian mathematician Giuseppe Peano (18581932). Peano's work, largely a translation of Pasch's treatise into the notation of symbolic logic (which Peano invented), uses the primitive notions of point and betweeness. Peano breaks the empirical tie in the choice of primitive notions and axioms that Pasch required. For Peano, the entire system is purely formal, divorced from any empirical input.\n\n### Pieri and the Italian school of geometers\n\nThe Italian mathematician Mario Pieri (18601913) took a different approach and considered a system in which there were only two primitive notions, that of point and of motion. Pasch had used four primitives and Peano had reduced this to three, but both of these approaches relied on some concept of betweeness which Pieri replaced by his formulation of motion. In 1905 Pieri gave the first axiomatic treatment of complex projective geometry which did not start by building real projective geometry.\n\nPieri was a member of a group of Italian geometers and logicians that Peano had gathered around himself in Turin. This group of assistants, junior colleagues and others were dedicated to carrying out Peano's logicogeometrical program of putting the foundations of geometry on firm axiomatic footing based on Peano's logical symbolism. Besides Pieri, Burali-Forti, Padoa and Fano were in this group. In 1900 there were two international conferences held back-to-back in Paris, the International Congress of Philosophy and the Second International Congress of Mathematicians. This group of Italian mathematicians was very much in evidence at these congresses, pushing their axiomatic agenda. Padoa gave a well regarded talk and Peano, in the question period after David Hilbert's famous address on unsolved problems, remarked that his colleagues had already solved Hilbert's second problem.\n\n### Hilbert's axioms\n\nMain article: Hilbert's axioms", null, "David Hilbert\n\nAt the University of Göttingen, during the 18981899 winter term, the eminent German mathematician David Hilbert (18621943) presented a course of lectures on the foundations of geometry. At the request of Felix Klein, Professor Hilbert was asked to write up the lecture notes for this course in time for the summer 1899 dedication ceremony of a monument to C.F. Gauss and Wilhelm Weber to be held at the university. The rearranged lectures were published in June 1899 under the title Grundlagen der Geometrie (Foundations of Geometry). The influence of the book was immediate. According to Eves (1963, pp. 3845):\n\nBy developing a postulate set for Euclidean geometry that does not depart too greatly in spirit from Euclid's own, and by employing a minimum of symbolism, Hilbert succeeded in convincing mathematicians to a far greater extent than had Pasch and Peano, of the purely hypothetico-deductive nature of geometry. But the influence of Hilbert's work went far beyond this, for, backed by the author's great mathematical authority, it firmly implanted the postulational method, not only in the field of geometry, but also in essentially every other branch of mathematics. The stimulus to the development of the foundations of mathematics provided by Hilbert's little book is difficult to overestimate. Lacking the strange symbolism of the works of Pasch and Peano, Hilbert's work can be read, in great part, by any intelligent student of high school geometry.\n\nIt is difficult to specify the axioms used by Hilbert without referring to the publication history of the Grundlagen since Hilbert changed and modified them several times. The original monograph was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. New editions followed the 7th, but the main text was essentially not revised. The modifications in these editions occur in the appendices and in supplements. The changes in the text were large when compared to the original and a new English translation was commissioned by Open Court Publishers, who had published the Townsend translation. So, the 2nd English Edition was translated by Leo Unger from the 10th German edition in 1971. This translation incorporates several revisions and enlargements of the later German editions by Paul Bernays. The differences between the two English translations are due not only to Hilbert, but also to differing choices made by the two translators. What follows will be based on the Unger translation.\n\nHilbert's axiom system is constructed with six primitive notions: point, line, plane, betweenness, lies on (containment), and congruence.\n\nAll points, lines, and planes in the following axioms are distinct unless otherwise stated.\n\nI. Incidence\n1. For every two points A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of “contains,” we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: “The lines a and b have the point A in common,” etc.\n2. For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where BC, then also BC = a.\n3. There exist at least two points on a line. There exist at least three points that do not lie on a line.\n4. For every three points A, B, C not situated on the same line there exists a plane α that contains all of them. For every plane there exists a point which lies on it. We write ABC = α. We employ also the expressions: “A, B, C, lie in α”; “A, B, C are points of α”, etc.\n5. For every three points A, B, C which do not lie in the same line, there exists no more than one plane that contains them all.\n6. If two points A, B of a line a lie in a plane α, then every point of a lies in α. In this case we say: “The line a lies in the plane α,” etc.\n7. If two planes α, β have a point A in common, then they have at least a second point B in common.\n8. There exist at least four points not lying in a plane.\nII. Order\n1. If a point B lies between points A and C, B is also between C and A, and there exists a line containing the distinct points A,B,C.\n2. If A and C are two points of a line, then there exists at least one point B lying between A and C.\n3. Of any three points situated on a line, there is no more than one which lies between the other two.\n4. Pasch's Axiom: Let A, B, C be three points not lying in the same line and let a be a line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.\nIII. Congruence\n1. If A, B are two points on a line a, and if A′ is a point upon the same or another line a′ , then, upon a given side of A′ on the straight line a′ , we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing AB A′ B′. Every segment is congruent to itself; that is, we always have AB AB.\nWe can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in at least one way.\n2. If a segment AB is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″; that is, if AB A′B′ and ABA″B″, then A′B′A″B″.\n3. Let AB and BC be two segments of a line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another line a′ having, likewise, no point other than B′ in common. Then, if AB A′B′ and BCB′C′, we have AC A′C′.\n4. Let an angle ∠ (h,k) be given in the plane α and let a line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one ray k′ such that the angle ∠ (h, k), or ∠ (k, h), is congruent to the angle ∠ (h′, k′) and at the same time all interior points of the angle ∠ (h′, k′) lie upon the given side of a′. We express this relation by means of the notation ∠ (h, k)  (h′, k′).\n5. If the angle ∠ (h, k) is congruent to the angle ∠ (h′, k′) and to the angle ∠ (h″, k″), then the angle ∠ (h′, k′) is congruent to the angle ∠ (h″, k″); that is to say, if ∠ (h, k)  (h′, k′) and ∠ (h, k)  (h″, k″), then ∠ (h′, k′)  (h″, k″).\nIV. Parallels\n1. (Euclid's Axiom): Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and does not intersect a.\nV. Continuity\n1. Axiom of Archimedes. If AB and CD are any segments then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B.\n2. Axiom of line completeness. An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I–III and from V-1 is impossible.\n\n#### Changes in Hilbert's axioms\n\nWhen the monograph of 1899 was translated into French, Hilbert added:\n\nV.2 Axiom of completeness. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.\n\nThis axiom is not needed for the development of Euclidean geometry, but is needed to establish a bijection between the real numbers and the points on a line. This was an essential ingredient in Hilbert's proof of the consistency of his axiom system.\n\nBy the 7th edition of the Grundlagen, this axiom had been replaced by the axiom of line completeness given above and the old axiom V.2 became Theorem 32.\n\nAlso to be found in the 1899 monograph (and appearing in the Townsend translation) is:\n\nII.4. Any four points A, B, C, D of a line can always be labeled so that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D.\n\nHowever, E.H. Moore and R.L. Moore independently proved that this axiom is redundant, and the former published this result in an article appearing in the Transactions of the American Mathematical Society in 1902. Hilbert moved the axiom to Theorem 5 and renumbered the axioms accordingly (old axiom II-5 (Pasch's axiom) now became II-4).\n\nWhile not as dramatic as these changes, most of the remaining axioms were also modified in form and/or function over the course of the first seven editions.\n\n#### Consistency and Independence\n\nGoing beyond the establishment of a satisfactory set of axioms, Hilbert also proved the consistency of his system relative to the theory of real numbers by constructing a model of his axiom system from the real numbers. He proved the independence of some of his axioms by constructing models of geometries which satisfy all except the one axiom under consideration. Thus, there are examples of geometries satisfying all except the Archimedean axiom V.1 (non-Archimedean geometries), all except the parallel axiom IV.1 (non-Euclidean geometries) and so on. Using the same technique he also showed how some important theorems depended on certain axioms and were independent of others. Some of his models were very complex and other mathematicians tried to simplify them. For instance, Hilbert's model for showing the independence of Desargues theorem from certain axioms ultimately led Ray Moulton to discover the non-Desarguesian Moulton plane. These investigations by Hilbert virtually inaugurated the modern study of abstract geometry in the twentieth century.\n\n### Birkhoff's axioms\n\nMain article: Birkhoff's axioms", null, "George David Birkhoff\n\nIn 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be experimentally verified with a scale and protractor. In a radical departure from the synthetic approach of Hilbert, Birkhoff was the first to build the foundations of geometry on the real number system. It is this powerful assumption that permits the small number of axioms in this system.\n\n#### Postulates\n\nBirkhoff uses four undefined terms: point, line, distance and angle. His postulates are:\n\nPostulate I: Postulate of Line Measure. The points A, B, ... of any line can be put into 1:1 correspondence with the real numbers x so that |xB x A| = d(A, B) for all points A and B.\n\nPostulate II: Point-Line Postulate. There is one and only one straight line, , that contains any two given distinct points P and Q.\n\nPostulate III: Postulate of Angle Measure. The rays {ℓ, m, n, ...} through any point O can be put into 1:1 correspondence with the real numbers a (mod 2π) so that if A and B are points (not equal to O) of and m, respectively, the difference am  a (mod 2π) of the numbers associated with the lines and m is", null, "AOB. Furthermore, if the point B on m varies continuously in a line r not containing the vertex O, the number am varies continuously also.\n\nPostulate IV: Postulate of Similarity. If in two triangles ABC and A'B'C'  and for some constant k > 0, d(A', B' ) = kd(A, B), d(A', C' ) = kd(A, C) and", null, "B'A'C'  = ±", null, "BAC, then d(B', C' ) = kd(B, C),", null, "C'B'A'  = ±", null, "CBA, and", null, "A'C'B'  = ±", null, "ACB.\n\n### School geometry", null, "George Bruce Halsted\n\nWhether or not it is wise to teach Euclidean geometry from an axiomatic viewpoint at the high school level has been a matter of debate. There have been many attempts to do so and not all of them have been successful. In 1904, George Bruce Halsted published a high school geometry text based on Hilbert's axiom set. Logical criticisms of this text led to a highly revised second edition. In reaction to the launching of the Russian satellite Sputnik there was a call to revise the school mathematics curriculum. From this effort there arose the New Math program of the 1960s. With this as a background, many individuals and groups set about to provide textual material for geometry classes based on an axiomatic approach.\n\n#### Mac Lane's axioms", null, "Saunders Mac Lane\n\nSaunders Mac Lane (19092005), an internationally respected mathematician, wrote a paper in 1959 in which he proposed a set of axioms for Euclidean geometry in the spirit of Birkhoff's treatment using a distance function to associate real numbers with line segments. This was not the first attempt to base a school level treatment on Birkhoff's system, in fact, Birkhoff and Ralph Beatley had written a high school text in 1940 which developed Euclidean geometry from five axioms and the ability to measure line segments and angles. However, in order to gear the treatment to a high school audience, some mathematical and logical arguments were either ignored or slurred over.\n\nIn Mac Lane's system there are four primitive notions (undefined terms): point, distance, line and angle measure. There are also 14 axioms, four giving the properties of the distance function, four describing properties of lines, four discussing angles (which are directed angles in this treatment), a similarity axiom (essentially the same as Birkhoff's) and a continuity axiom which can be used to derive the Crossbar theorem and its converse. The increased number of axioms has the pedagogical advantage of making early proofs in the development easier to follow and the use of a familiar metric permits a rapid advancement through basic material so that the more \"interesting\" aspects of the subject can be gotten to sooner.\n\n#### SMSG (School Mathematics Study Group) axioms\n\nIn the 1960s a new set of axioms for Euclidean geometry, suitable for high school geometry courses, was introduced by the School Mathematics Study Group (SMSG), as a part of the New math curricula. This set of axioms follows the Birkhoff model of using the real numbers to gain quick entry into the geometric fundamentals. However, whereas Birkhoff tried to minimize the number of axioms used, and most authors were concerned with the independence of the axioms in their treatments, the SMSG axiom list was intentionally made large and redundant for pedagogical reasons. The SMSG only produced a mimeographed text using these axioms, but Edwin E. Moise, a member of the SMSG, wrote a high school text based on this system, and a college level text, Moise (1974), with some of the redundancy removed and modifications made to the axioms for a more sophisticated audience.\n\nThere are eight undefined terms: point, line, plane, lie on, distance, angle measure, area and volume. The 22 axioms of this system are given individual names for ease of reference. Amongst these are to be found: the Ruler Postulate, the Ruler Placement Postulate, the Plane Separation Postulate, the Angle Addition Postulate, the Side angle side (SAS) Postulate, the Parallel Postulate (in Playfair's form), and Cavalieri's principle.\n\n#### UCSMP (University of Chicago School Mathematics Project) axioms\n\nAlthough much of the New math curriculum has been drastically modified or abandoned, the geometry portion has remained relatively stable. Modern high school textbooks use axiom systems that are very similar to those of the SMSG. For example, the texts produced by the University of Chicago School Mathematics Project (UCSMP) use a system which, besides some updating of language, differs mainly from the SMSG system in that it includes some transformation concepts under its \"Reflection Postulate\".\n\nThere are only three undefined terms: point, line and plane. There are eight \"postulates\", but most of these have several parts (which are generally called assumptions in this system). Counting these parts, there are 32 axioms in this system. Amongst the postulates can be found the point-line-plane postulate, the Triangle inequality postulate, postulates for distance, angle measurement, corresponding angles, area and volume, and the Reflection postulate. The reflection postulate is used as a replacement for the SAS postulate of SMSG system.\n\n### Other systems\n\nOswald Veblen (1880 1960) provided a new axiom system in 1904 when he replaced the concept of \"betweeness\", as used by Hilbert and Pasch, with a new primitive, order. This permitted several primitive terms used by Hilbert to become defined entities, reducing the number of primitive notions to two, point and order.\n\nMany other axiomatic systems for Euclidean geometry have been proposed over the years. A comparison of many of these can be found in a 1927 monograph by Henry George Forder. Forder also gives, by combining axioms from different systems, his own treatment based on the two primitive notions of point and order. He also provides a more abstract treatment of one of Pieri's systems (from 1909) based on the primitives point and congruence.\n\nStarting with Peano, there has been a parallel thread of interest amongst logicians concerning the axiomatic foundations of Euclidean geometry. This can be seen, in part, in the notation used to describe the axioms. Pieri claimed that even though he wrote in the traditional language of geometry, he was always thinking in terms of the logical notation introduced by Peano, and used that formalism to see how to prove things. A typical example of this type of notation can be found in the work of E. V. Huntington (1874 1952) who, in 1913, produced an axiomatic treatment of three-dimensional Euclidean geometry based upon the primitive notions of sphere and inclusion (one sphere lying within another). Beyond notation there is also interest in the logical structure of the theory of geometry. Alfred Tarski proved that a portion of geometry, which he called elementary geometry, is a first order logical theory (see Tarski's axioms).\n\nModern text treatments of the axiomatic foundations of Euclidean geometry follow the pattern of H.G. Forder and Gilbert de B. Robinson who mix and match axioms from different systems to produce different emphasizes. Venema (2006) is a modern example of this approach.\n\n## Non-Euclidean geometry\n\nIn view of the role which mathematics plays in science and implications of scientific knowledge for all of our beliefs, revolutionary changes in man's understanding of the nature of mathematics could not but mean revolutionary changes in his understanding of science, doctrines of philosophy, religious and ethical beliefs, and, in fact, all intellectual disciplines.\n\nIn the first half of the nineteenth century a revolution took place in the field of geometry that was as scientifically important as the Copernican revolution in astronomy and as philosophically profound as the Darwinian theory of evolution in its impact on the way we think. This was the consequence of the discovery of non-Euclidean geometry. For over two thousand years, starting in the time of Euclid, the postulates which grounded geometry were considered self-evident truths about physical space. Geometers thought that they were deducing other, more obscure truths from them, without the possibility of error. This view became untenable with the development of hyperbolic geometry. There were now two incompatible systems of geometry (and more came later) that were self-consistent and compatible with the observable physical world. \"From this point on, the whole discussion of the relation between geometry and physical space was carried on in quite different terms.\"(Moise 1974, p. 388)\n\nTo obtain a non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation. Negating the Playfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways. Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line. In the first case, replacing the parallel postulate (or its equivalent) with the statement \"In a plane, given a point P and a line not passing through P, there exist two lines through P which do not meet \" and keeping all the other axioms, yields hyperbolic geometry. The second case is not dealt with as easily. Simply replacing the parallel postulate with the statement, \"In a plane, given a point P and a line not passing through P, all the lines through P meet \", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry, but this statement would say that there are no parallel lines. This problem was known (in a different guise) to Khayyam, Saccheri and Lambert and was the basis for their rejecting what was known as the \"obtuse angle case\". In order to obtain a consistent set of axioms which includes this axiom about having no parallel lines, some of the other axioms must be tweaked. The adjustments to be made depend upon the axiom system being used. Amongst others these tweaks will have the effect of modifying Euclid's second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded. Riemann's elliptic geometry emerges as the most natural geometry satisfying this axiom.\n\nIt was Gauss who coined the term \"non-Euclidean geometry\". He was referring to his own, unpublished work, which today we call hyperbolic geometry. Several authors still consider \"non-Euclidean geometry\" and \"hyperbolic geometry\" to be synonyms. In 1871, Felix Klein, by adapting a metric discussed by Arthur Cayley in 1852, was able to bring metric properties into a projective setting and was thus able to unify the treatments of hyperbolic, euclidean and elliptic geometry under the umbrella of projective geometry. Klein is responsible for the terms \"hyperbolic\" and \"elliptic\" (in his system he called Euclidean geometry \"parabolic\", a term which has not survived the test of time and is used today only in a few disciplines.) His influence has led to the common usage of the term \"non-Euclidean geometry\" to mean either \"hyperbolic\" or \"elliptic\" geometry.\n\nThere are some mathematicians who would extend the list of geometries that should be called \"non-Euclidean\" in various ways. In other disciplines, most notably mathematical physics, where Klein's influence was not as strong, the term \"non-Euclidean\" is often taken to mean not Euclidean.\n\n### Euclid's parallel postulate\n\nMain article: Parallel postulate\n\nFor two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. A possible reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate isn't self-evident. If the order the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it. Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate. Eventually it was realized that this postulate may not be provable from the other four. According to Trudeau (1987, p. 154) this opinion about the parallel postulate (Postulate 5) does appear in print:\n\nApparently the first to do so was G. S. Klügel (17391812), a doctoral student at the University of Gottingen, with the support of his teacher A. G. Kästner, in the former's 1763 dissertation Conatuum praecipuorum theoriam parallelarum demonstrandi recensio (Review of the Most Celebrated Attempts at Demonstrating the Theory of Parallels). In this work Klügel examined 28 attempts to prove Postulate 5 (including Saccheri's), found them all deficient, and offered the opinion that Postulate 5 is unprovable and is supported solely by the judgment of our senses.\n\nThe beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Then, around 1830, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on what we today call hyperbolic geometry. Consequently, hyperbolic geometry has been called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences. The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868.\n\nThe various attempted proofs of the parallel postulate produced a long list of theorems that are equivalent to the parallel postulate. Equivalence here means that in the presence of the other axioms of the geometry each of these theorems can be assumed to be true and the parallel postulate can be proved from this altered set of axioms. This is not the same as logical equivalence. In different sets of axioms for Euclidean geometry, any of these can replace the Euclidean parallel postulate. The following partial list indicates some of these theorems that are of historical interest.\n\n1. Parallel straight lines are equidistant. (Poseidonios, 1st century B.C.)\n2. All the points equidistant from a given straight line, on a given side of it, constitute a straight line. (Christoph Clavius, 1574)\n3. Playfair's axiom. In a plane, there is at most one line that can be drawn parallel to another given one through an external point. (Proclus, 5th century, but popularized by John Playfair, late 18th century)\n4. The sum of the angles in every triangle is 180° (Gerolamo Saccheri, 1733; Adrien-Marie Legendre, early 19th century)\n5. There exists a triangle whose angles add up to 180°. (Gerolamo Saccheri, 1733; Adrien-Marie Legendre, early 19th century)\n6. There exists a pair of similar, but not congruent, triangles. (Gerolamo Saccheri, 1733)\n7. Every triangle can be circumscribed. (Adrien-Marie Legendre, Farkas Bolyai, early 19th century)\n8. If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle. (Alexis-Claude Clairaut, 1741; Johann Heinrich Lambert, 1766)\n9. There exists a quadrilateral in which all angles are right angles. (Geralamo Saccheri, 1733)\n10. Wallis' postulate. On a given finite straight line it is always possible to construct a triangle similar to a given triangle. (John Wallis, 1663; Lazare-Nicholas-Marguerite Carnot, 1803; Adrien-Marie Legendre, 1824)\n11. There is no upper limit to the area of a triangle. (Carl Friedrich Gauss, 1799)\n12. The summit angles of the Saccheri quadrilateral are 90°. (Geralamo Saccheri, 1733)\n13. Proclus' axiom. If a line intersects one of two parallel lines, both of which are coplanar with the original line, then it also intersects the other. (Proclus, 5th century)\n\n### Neutral (or Absolute) geometry\n\nMain article: Absolute geometry\n\nAbsolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate.\n\n#### Relation to other geometries\n\nIn Euclid's Elements, the first 28 propositions and Proposition I.31 avoid using the parallel postulate, and therefore are valid theorems in absolute geometry. Proposition I.31 proves the existence of parallel lines (by construction). Also, the Saccheri–Legendre theorem, which states that the sum of the angles in a triangle is at most 180°, can be proved.\n\nThe theorems of absolute geometry hold in hyperbolic geometry as well as in Euclidean geometry.\n\nAbsolute geometry is inconsistent with elliptic geometry: in elliptic geometry there are no parallel lines at all, but in absolute geometry parallel lines do exist. Also, in elliptic geometry, the sum of the angles in any triangle is greater than 180°.\n\n#### Incompleteness\n\nLogically, the axioms do not form a complete theory since one can add extra independent axioms without making the axiom system inconsistent. One can extend absolute geometry by adding different axioms about parallelism and get incompatible but consistent axiom systems, giving rise to Euclidean or hyperbolic geometry. Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry. However the converse is not true. Also, absolute geometry is not a categorical theory, since it has models that are not isomorphic.\n\n### Hyperbolic geometry\n\nMain article: Hyperbolic geometry\n\nIn the axiomatic approach to hyperbolic geometry (also referred to as Lobachevskian geometry or Bolyai–Lobachevskian geometry), one additional axiom is added to the axioms giving absolute geometry. The new axiom is Lobachevsky's parallel postulate (also known as the characteristic postulate of hyperbolic geometry):\n\nThrough a point not on a given line there exists (in the plane determined by this point and line) at least two lines which do not meet the given line.\n\nWith this addition, the axiom system is now complete.\n\nAlthough the new axiom asserts only the existence of two lines, it is readily established that there are an infinite number of lines through the given point which do not meet the given line. Given this plenitude, one must be careful with terminology in this setting, as the term parallel line no longer has the unique meaning that it has in Euclidean geometry. Specifically, let P be a point not on a given line", null, ". Let PA be the perpendicular drawn from P to", null, "(meeting at point A). The lines through P fall into two classes, those that meet", null, "and those that don't. The characteristic postulate of hyperbolic geometry says that there are at least two lines of the latter type. Of the lines which don't meet", null, ", there will be (on each side of PA) a line making the smallest angle with PA. Sometimes these lines are referred to as the first lines through P which don't meet", null, "and are variously called limiting, asymptotic or parallel lines (when this last term is used, these are the only parallel lines). All other lines through P which do not meet", null, "are called non-intersecting or ultraparallel lines.\n\nSince hyperbolic geometry and Euclidean geometry are both built on the axioms of absolute geometry, they share many properties and propositions. However, the consequences of replacing the parallel postulate of Euclidean geometry with the characteristic postulate of hyperbolic geometry can be dramatic. To mention a few of these:", null, "• A Lambert quadrilateral is a quadrilateral which has three right angles. The fourth angle of a Lambert quadrilateral is acute if the geometry is hyperbolic, and a right angle if the geometry is Euclidean. Furthermore, rectangles can exist (a statement equivalent to the parallel postulate) only in Euclidean geometry.\n• A Saccheri quadrilateral is a quadrilateral which has two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit angles and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, and right angles if the geometry is Euclidean.\n• The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, and equal to 180° if the geometry is Euclidean. The defect of a triangle is the numerical value (180° – sum of the measures of the angles of the triangle). This result may also be stated as: the defect of triangles in hyperbolic geometry is positive, and the defect of triangles in Euclidean geometry is zero.\n• The area of a triangle in hyperbolic geometry is bounded while triangles exist with arbitrarily large areas in Euclidean geometry.\n• The set of points on the same side and equally far from a given straight line themselves form a line in Euclidean geometry, but don't in hyperbolic geometry (they form a hypercycle.)\n\nAdvocates of the position that Euclidean geometry is the one and only \"true\" geometry received a setback when, in a memoir published in 1868, \"Fundamental theory of spaces of constant curvature\", Eugenio Beltrami gave an abstract proof of equiconsistency of hyperbolic and Euclidean geometry for any dimension. He accomplished this by introducing several models of non-Euclidean geometry that are now known as the Beltrami–Klein model, the Poincaré disk model, and the Poincaré half-plane model, together with transformations that relate them. For the half-plane model, Beltrami cited a note by Liouville in the treatise of Monge on differential geometry. Beltrami also showed that n-dimensional Euclidean geometry is realized on a horosphere of the (n + 1)-dimensional hyperbolic space, so the logical relation between consistency of the Euclidean and the non-Euclidean geometries is symmetric.\n\n### Elliptic geometry\n\nMain article: Elliptic geometry\n\nAnother way to modify the Euclidean parallel postulate is to assume that there are no parallel lines in a plane. Unlike the situation with hyperbolic geometry, where we just add one new axiom, we can not obtain a consistent system by adding this statement as a new axiom to the axioms of absolute geometry. This follows since parallel lines provably exist in absolute geometry. Other axioms must be changed.\n\nStarting with Hilbert's axioms the necessary changes involve removing Hilbert's four axioms of order and replacing them with these seven axioms of separation concerned with a new undefined relation.\n\nThere is an undefined (primitive) relation between four points, A, B, C and D denoted by (A,C|B,D) and read as \"A and C separate B and D\", satisfying these axioms:\n\n1. If (A,B|C,D), then the points A, B, C and D are collinear and distinct.\n2. If (A,B|C,D), then (C,D|A,B) and (B,A|D,C).\n3. If (A,B|C,D), then not (A,C|B,D).\n4. If points A, B, C and D are collinear and distinct then (A,B|C,D) or (A,C|B,D) or (A,D|B,C).\n5. If points A, B, and C are collinear and distinct, then there exists a point D such that (A,B|C,D).\n6. For any five distinct collinear points A, B, C, D and E, if (A,B|D,E), then either (A,B|C,D) or (A,B|C,E).\n7. Perspectivities preserve separation.\n\nSince the Hilbert notion of \"betweeness\" has been removed, terms which were defined using that concept need to be redefined. Thus, a line segment AB defined as the points A and B and all the points between A and B in absolute geometry, needs to be reformulated. A line segment in this new geometry is determined by three collinear points A, B and C and consists of those three points and all the points not separated from B by A and C. There are further consequences. Since two points do not determine a line segment uniquely, three noncollinear points do not determine a unique triangle, and the definition of triangle has to be reformulated.\n\nOnce these notions have been redefined, the other axioms of absolute geometry (incidence, congruence and continuity) all make sense and are left alone. Together with the new axiom on the nonexistence of parallel lines we have a consistent system of axioms giving a new geometry. The geometry that results is called (plane) Elliptic geometry.", null, "Saccheri quadrilaterals in Euclidean, Elliptic and Hyperbolic geometry\n\nEven though elliptic geometry is not an extension of absolute geometry (as Euclidean and hyperbolic geometry are), there is a certain \"symmetry\" in the propositions of the three geometries that reflects a deeper connection which was observed by Felix Klein. Some of the propositions which exhibit this property are:\n\n• The fourth angle of a Lambert quadrilateral is an obtuse angle in elliptic geometry.\n• The summit angles of a Saccheri quadrilateral are obtuse in elliptic geometry.\n• The sum of the measures of the angles of any triangle is greater than 180° if the geometry is elliptic. That is, the defect of a triangle is negative.\n• All the lines perpendicular to a given line meet at a common point in elliptic geometry, called the pole of the line. In hyperbolic geometry these lines are mutually non-intersecting, while in Euclidean geometry they are mutually parallel.\n\nOther results, such as the exterior angle theorem, clearly emphasize the difference between elliptic and the geometries that are extensions of absolute geometry.\n\n#### Spherical geometry\n\nMain article: Spherical geometry\n\n## Other geometries\n\n### Projective geometry\n\nMain article: Projective geometry\n\n### Affine geometry\n\nMain article: Affine geometry\n\n### Ordered geometry\n\nMain article: Ordered geometry\n\nAbsolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry. The converse is not true. Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with affine geometry, which does not assume Euclid's third and fourth axioms. Ordered geometry is a common foundation of both absolute and affine geometry.\n\n### Finite geometry\n\nMain article: Finite geometry\n\n## Notes\n\n1. Venema 2006, p. 17\n2. Wylie, Jr. 1964, p. 8\n3. Greenberg 1974, p. 59\n4. In this context no distinction is made between different categories of theorems. Propositions, lemmas, corollaries, etc. are all treated the same.\n5. Venema 2006, p. 19\n6. Faber 1983, pp. 105 8\n7. Eves 1963, p. 19\n8. Eves 1963, p. 10\n9. Boyer (1991). \"Euclid of Alexandria\". p. 101. With the exception of the Sphere of Autolycus, surviving work by Euclid are the oldest Greek mathematical treatises extant; yet of what Euclid wrote more than half has been lost, Missing or empty |title= (help)\n10. Encyclopedia of Ancient Greece (2006) by Nigel Guy Wilson, page 278. Published by Routledge Taylor and Francis Group. Quote:\"Euclid's Elements subsequently became the basis of all mathematical education, not only in the Romand and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written.\"\n11. Boyer (1991). \"Euclid of Alexandria\". p. 100. As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written – the Elements (Stoichia) of Euclid. Missing or empty |title= (help)\n12. Boyer (1991). \"Euclid of Alexandria\". p. 119. The Elements of Euclid not only was the earliest major Greek mathematical work to come down to us, but also the most influential textbook of all times. [...]The first printed versions of the Elements appeared at Venice in 1482, one of the very earliest of mathematical books to be set in type; it has been estimated that since then at least a thousand editions have been published. Perhaps no book other than the Bible can boast so many editions, and certainly no mathematical work has had an influence comparable with that of Euclid's Elements. Missing or empty |title= (help)\n13. The Historical Roots of Elementary Mathematics by Lucas Nicolaas Hendrik Bunt, Phillip S. Jones, Jack D. Bedient (1988), page 142. Dover publications. Quote:\"the Elements became known to Western Europe via the Arabs and the Moors. There the Elements became the foundation of mathematical education. More than 1000 editions of the Elements are known. In all probability it is, next to the Bible, the most widely spread book in the civilization of the Western world.\"\n14. From the introduction by Amit Hagar to Euclid and His Modern Rivals by Lewis Carroll (2009, Barnes & Noble) pg. xxviii:\nGeometry emerged as an indispensable part of the standard education of the English gentleman in the eighteenth century; by the Victorian period it was also becoming an important part of the education of artisans, children at Board Schools, colonial subjects and, to a rather lesser degree, women. ... The standard textbook for this purpose was none other than Euclid's The Elements.\n15. Euclid, book I, proposition 47\n16. Heath 1956, pp. 195 202 (vol 1)\n17. Venema 2006, p. 11\n18. Ball 1960, p. 55\n19. Wylie, Jr. 1964, p. 39\n20. Faber 1983, p. 109\n21. Faber 1983, p. 113\n22. Faber 1983, p. 115\n23. Heath 1956, p. 62 (vol. I)\n24. Greenberg 1974, p. 57\n25. Heath 1956, p. 242 (vol. I)\n26. Heath 1956, p. 249 (vol. I)\n27. Eves 1963, p. 380\n28. Eves 1963, p. 382\n29. Eves 1963, p. 383\n30. Pieri did not attend since he had recently moved to Sicily, but he did have a paper of his read at the Congress of Philosophy.\n31. Hilbert 1950\n32. Hilbert 1990\n33. This is Hilbert's terminology. This statement is more familiarly known as Playfair's axiom.\n34. Eves 1963, p. 386\n35. Moore, E.H. (1902), \"On the projective axioms of geometry\", Transactions of the American Mathematical Society, 3: 142158, doi:10.2307/1986321\n36. Eves 1963, p. 387\n37. Birkhoff, George David (1932), \"A set of postulates for plane geometry\", Annals of Mathematics, 33: 329345, doi:10.2307/1968336\n38. Venema 2006, p. 400\n39. Venema 2006, pp. 4001\n40. Halsted, G. B. (1904), Rational Geometry, New York: John Wiley and Sons, Inc.\n41. Eves 1963, p. 388\n42. among his several achievements, he is the cofounder (with Samuel Eilenberg) of Category theory.\n43. Mac Lane, Saunders (1959), \"Metric postulates for plane geometry\", American Mathematical Monthly, 66: 543555, doi:10.2307/2309851\n44. Birkhoff, G.D.; Beatley, R. (1940), Basic Geometry, Chicago: Scott, Foresman and Company [Reprint of 3rd edition: American Mathematical Society, 2000. ISBN 978-0-8218-2101-5]\n45. Venema 2006, pp. 4012\n46. Venema 2006, p. 55\n47. School Mathematics Study Group (SMSG) (1961), Geometry, Parts 1 and 2 (Student Text), New Haven and London: Yale University Press\n48. Moise, Edwin E.; Downs, Floyd L. (1991), Geometry, Reading, MA: Addison–Wesley\n49. Venema 2006, p. 403\n50. Venema 2006, pp. 4034\n51. Venema 2006, pp. 405 7\n52. Forder, H.G. (1927), The Foundations of Euclidean Geometry, New York: Cambridge University Press (reprinted by Dover, 1958)\n53. Huntington, E.V. (1913), \"A set of postulates for abstract geometry, expressed in terms of the simple relation of inclusion\", Mathematische Annalen, 73: 522559, doi:10.1007/bf01455955\n54. Robinson, G. de B. (1946), The Foundations of Geometry, Mathematical Expositions No. 1 (2nd ed.), Toronto: University of Toronto Press\n55. Kline, Morris (1967), Mathematics for the Nonmathematician, New York: Dover, p. 474, ISBN 0-486-24823-2\n56. Greenberg 1974, p. 1\n57. while only two lines are postulated, it is easily shown that there must be an infinite number of such lines.\n58. Book I Proposition 27 of Euclid's Elements\n59. Felix Klein, Elementary Mathematics from an Advanced Standpoint: Geometry, Dover, 1948 (reprint of English translation of 3rd Edition, 1940. First edition in German, 1908) pg. 176\n60. F. Klein, Über die sogenannte nichteuklidische Geometrie, Mathematische Annalen, 4(1871).\n61. Florence P. Lewis (Jan 1920), \"History of the Parallel Postulate\", The American Mathematical Monthly, The American Mathematical Monthly, Vol. 27, No. 1, 27 (1): 16–23, doi:10.2307/2973238, JSTOR 2973238.\n62. In a letter of December 1818, Ferdinand Karl Schweikart (1780–1859) sketched a few insights into non-Euclidean geometry. The letter was forwarded to Gauss in 1819 by Gauss's former student Gerling. In his reply to Gerling, Gauss praised Schweikart and mentioned his own, earlier research into non-Euclidean geometry.\n63. In the letter to Wolfgang (Farkas) Bolyai of March 6, 1832 Gauss claims to have worked on the problem for thirty or thirty-five years (Faber 1983, p. 162). In his 1824 letter to Taurinus (Faber 1983, p. 158) he claimed that he had been working on the problem for over 30 years and provided enough detail to show that he actually had worked out the details. According to Faber (1983, p. 156) it wasn't until around 1813 that Gauss had come to accept the existence of a new geometry.\n64. Beltrami, Eugenio Teoria fondamentale degli spazî di curvatura costante, Annali. di Mat., ser II 2 (1868), 232255\n65. An appropriate example of logical equivalence is given by Playfair's axiom and Euclid I.30 (see Playfair's axiom#Transitivity of parallelism).\n66. For instance, Hilbert uses Playfair's axiom while Birkhoff uses the theorem about similar but not congruent triangles.\n67. attributions are due to Trudeau 1987, pp. 1289\n68. Use a complete set of axioms for Euclidean geometry such as Hilbert's axioms or another modern equivalent (Faber 1983, p. 131). Euclid's original set of axioms is ambiguous and not complete, it does not form a basis for Euclidean geometry.\n69. In \"Appendix exhibiting the absolute science of space: independent of the truth or falsity of Euclid's Axiom XI (by no means previously decided)\" (Faber 1983, p. 161)\n70. Greenberg cites W. Prenowitz and M. Jordan (Greenberg, p. xvi) for having used the term neutral geometry to refer to that part of Euclidean geometry that does not depend on Euclid's parallel postulate. He says that the word absolute in absolute geometry misleadingly implies that all other geometries depend on it.\n71. Trudeau 1987, p. 44\n72. Absolute geometry is, in fact, the intersection of hyperbolic geometry and Euclidean geometry when these are regarded as sets of propositions.\n73. Faber 1983, p. 167\n74. Beltrami, Eugenio (1868), \"Teoria fondamentale degli spazii di curvatura costante\", Annali. di Mat., ser II, 2: 232–255, doi:10.1007/BF02419615\n75. Greenberg 1979, pp. 275–9\n76. Visualize four points on a circle which in counter-clockwise order are A, B, C and D.\n77. This reenforces the futility of attempting to \"fix\" Euclid's axioms to obtain this geometry. Changes need to be made in the unstated assumptions of Euclid.\n78. Negative defect is called the excess, so this may also be phrased as triangles have a positive excess in elliptic geometry.\n79. Coxeter, pgs. 175–176\n\n## References\n\n• Ball, W.W. Rouse (1960). A Short Account of the History of Mathematics (4th ed. [Reprint. Original publication: London: Macmillan & Co., 1908] ed.). New York: Dover Publications. pp. 50–62. ISBN 0-486-20630-0.\n• Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective geometry: from foundations to applications, Cambridge University Press, ISBN 978-0-521-48364-3, MR 1629468\n• Eves, Howard (1963), A Survey of Geometry (Volume One), Boston: Allyn and Bacon\n• Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker, Inc., ISBN 0-8247-1748-1\n• Greenberg, Marvin Jay (1974), Euclidean and Non-Euclidean Geometries/Development and History, San Francisco: W.H. Freeman, ISBN 0-7167-0454-4\n• Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.\n(3 vols.): ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3).\n• Hilbert, David (1950) [first published 1902], The Foundations of Geometry [Grundlagen der Geometrie] (PDF), English translation by E.J. Townsend (2nd ed.), La Salle, IL: Open Court Publishing\n• Hilbert, David (1990) , Foundations of Geometry [Grundlagen der Geometrie], translated by Leo Unger from the 10th German edition (2nd English ed.), La Salle, IL: Open Court Publishing, ISBN 0-87548-164-7\n• Moise, Edwin E. (1974), Elementary Geometry from an Advanced Standpoint (2nd ed.), Reading, MA: Addison–Wesley, ISBN 0-201-04793-4\n• Trudeau, Richard J. (1987), The Non-Euclidean Revolution, Boston: Birkhauser, ISBN 0-8176-3311-1\n• Venema, Gerard A. (2006), Foundations of Geometry, Upper Saddle River, NJ: Pearson Prentice Hall, ISBN 0-13-143700-3\n• Wylie, Jr., C.R. (1964), Foundations of Geometry, New York: McGraw–Hill" ]

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[ "## 1994\n\n1,994 (one thousand nine hundred ninety-four) is an even four-digits composite number following 1993 and preceding 1995. In scientific notation, it is written as 1.994 × 103. The sum of its digits is 23. It has a total of 2 prime factors and 4 positive divisors. There are 996 positive integers (up to 1994) that are relatively prime to 1994.\n\n## Basic properties\n\n• Is Prime? No\n• Number parity Even\n• Number length 4\n• Sum of Digits 23\n• Digital Root 5\n\n## Name\n\nShort name 1 thousand 994 one thousand nine hundred ninety-four\n\n## Notation\n\nScientific notation 1.994 × 103 1.994 × 103\n\n## Prime Factorization of 1994\n\nPrime Factorization 2 × 997\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 2 Total number of distinct prime factors Ω(n) 2 Total number of prime factors rad(n) 1994 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 1 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 1,994 is 2 × 997. Since it has a total of 2 prime factors, 1,994 is a composite number.\n\n## Divisors of 1994\n\n1, 2, 997, 1994\n\n4 divisors\n\n Even divisors 2 2 2 0\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 4 Total number of the positive divisors of n σ(n) 2994 Sum of all the positive divisors of n s(n) 1000 Sum of the proper positive divisors of n A(n) 748.5 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 44.6542 Returns the nth root of the product of n divisors H(n) 2.66399 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 1,994 can be divided by 4 positive divisors (out of which 2 are even, and 2 are odd). The sum of these divisors (counting 1,994) is 2,994, the average is 74,8.5.\n\n## Other Arithmetic Functions (n = 1994)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 996 Total number of positive integers not greater than n that are coprime to n λ(n) 996 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 306 Total number of primes less than or equal to n r2(n) 8 The number of ways n can be represented as the sum of 2 squares\n\nThere are 996 positive integers (less than 1,994) that are coprime with 1,994. And there are approximately 306 prime numbers less than or equal to 1,994.\n\n## Divisibility of 1994\n\n m n mod m 2 3 4 5 6 7 8 9 0 2 2 4 2 6 2 5\n\nThe number 1,994 is divisible by 2.\n\n• Semiprime\n• Deficient\n\n• Polite\n\n• Square Free\n\n## Base conversion (1994)\n\nBase System Value\n2 Binary 11111001010\n3 Ternary 2201212\n4 Quaternary 133022\n5 Quinary 30434\n6 Senary 13122\n8 Octal 3712\n10 Decimal 1994\n12 Duodecimal 11a2\n20 Vigesimal 4je\n36 Base36 1je\n\n## Basic calculations (n = 1994)\n\n### Multiplication\n\nn×i\n n×2 3988 5982 7976 9970\n\n### Division\n\nni\n n⁄2 997 664.666 498.5 398.8\n\n### Exponentiation\n\nni\n n2 3976036 7928215784 15808862273296 31522871372952224\n\n### Nth Root\n\ni√n\n 2√n 44.6542 12.5866 6.68238 4.5703\n\n## 1994 as geometric shapes\n\n### Circle\n\n Diameter 3988 12528.7 1.24911e+07\n\n### Sphere\n\n Volume 3.32096e+10 4.99643e+07 12528.7\n\n### Square\n\nLength = n\n Perimeter 7976 3.97604e+06 2819.94\n\n### Cube\n\nLength = n\n Surface area 2.38562e+07 7.92822e+09 3453.71\n\n### Equilateral Triangle\n\nLength = n\n Perimeter 5982 1.72167e+06 1726.85\n\n### Triangular Pyramid\n\nLength = n\n Surface area 6.8867e+06 9.34349e+08 1628.09\n\n## Cryptographic Hash Functions\n\nmd5 008bd5ad93b754d500338c253d9c1770 5a478022f33905d2d40410e006fb1aa8564b280c 1bc3201a9f24a2fe48f634f90d406aaf6cbf5e36e292870ecba98d74b065ee1b 126638f21d37ab8d57d6957f52db6fc233b83344a1d876832e91a0490011bdd538f5171b7c007aa62899cfc5e9d7e3c519131db403bdc97c57d3ac25d7388c1e b29743e7c177942a6ea0f5b2d3eef9fcf5cec3ff" ]

[ null ]

[ "Analyzing LANL Data using Advanced xGT Features\n\nAfter learning how to use the basic functionality of xGT by following the Analyzing LANL Netflow Data with xGT notebook, you may wish to see some more advanced features and how to apply them to solving the same problem. This notebook demonstrates advanced techniques in the use of graph ideas as well as improved use of python.\n\n``````import xgt\nconn = xgt.Connection()\nconn\n``````\n``````<xgt.connection.Connection at 0x10331b290>\n``````\n\nEstablish Graph Schemas\n\nWe first try to retrieve the graph component schemas from xGT server. If that should fail, we create an empty component (vertex or edge frame) for the missing component.\n\n``````try:\ndevices = conn.get_vertex_frame('Devices')\nexcept xgt.XgtNameError:\ndevices = conn.create_vertex_frame(\nname='Devices',\nschema=[['device', xgt.TEXT]],\nkey='device')\ndevices\n``````\n``````<xgt.graph.VertexFrame at 0x103effe10>\n``````\n``````try:\nnetflow = conn.get_edge_frame('Netflow')\nexcept xgt.XgtNameError:\nnetflow = conn.create_edge_frame(\nname='Netflow',\nschema=[['epochtime', xgt.INT],\n['duration', xgt.INT],\n['srcDevice', xgt.TEXT],\n['dstDevice', xgt.TEXT],\n['protocol', xgt.INT],\n['srcPort', xgt.INT],\n['dstPort', xgt.INT],\n['srcPackets', xgt.INT],\n['dstPackets', xgt.INT],\n['srcBytes', xgt.INT],\n['dstBytes', xgt.INT]],\nsource=devices,\ntarget=devices,\nsource_key='srcDevice',\ntarget_key='dstDevice')\nnetflow\n``````\n``````<xgt.graph.EdgeFrame at 0x103f16a90>\n``````\n\nEdges: The LANL dataset contains two types of data: netflow and host events. Of the host events recorded, some describe events within a device (e.g., reboots), and some describe events between devices (e.g., login attempts). We'll only be loading the netflow data and in-device events. We call these events \"one-sided\", since we describe them as graph edges from one vertex to itself.\n\n``````try:\nevents1v = conn.get_edge_frame('Events1v')\nexcept xgt.XgtNameError:\nevents1v = conn.create_edge_frame(\nname='Events1v',\nschema=[['epochtime', xgt.INT],\n['eventID', xgt.INT],\n['logHost', xgt.TEXT],\n['domainName', xgt.TEXT],\n['logonID', xgt.INT],\n['processName', xgt.TEXT],\n['processID', xgt.INT],\n['parentProcessName', xgt.TEXT],\n['parentProcessID', xgt.INT]],\nsource=devices,\ntarget=devices,\nsource_key='logHost',\ntarget_key='logHost')\nevents1v\n``````\n``````<xgt.graph.EdgeFrame at 0x103f16a50>\n``````\n``````# Utility to print the data sizes currently in xGT\ndef print_data_summary():\nprint('Devices (vertices): {:,}'.format(devices.num_vertices))\nprint('Netflow (edges): {:,}'.format(netflow.num_edges))\nprint('Host event (edges): {:,}'.format(events1v.num_edges))\n\nprint_data_summary()\n``````\n``````Devices (vertices): 0\nNetflow (edges): 0\nHost event (edges): 0\n``````\n\nWe show how to load data only if the current data frames are empty.\n\n``````%%time\nif events1v.num_edges == 0:\nurls = [\"https://datasets.trovares.com/LANL/xgt/wls_day-04_1v.csv\"]\n``````\n``````CPU times: user 7.29 ms, sys: 9.66 ms, total: 16.9 ms\nWall time: 32 s\n``````\n``````%%time\nif netflow.num_edges == 0:\nurls = [\"https://datasets.trovares.com/LANL/xgt/nf_day-04.csv\"]\n``````\n``````CPU times: user 60.2 ms, sys: 84.6 ms, total: 145 ms\nWall time: 5min 19s\n``````\n``````print_data_summary()\n``````\n``````Devices (vertices): 157,949\nNetflow (edges): 222,323,503\nHost event (edges): 16,402,438\n``````\n\nPreprocess graph\n\nThe LANL dataset consists of netflow edges whose direction is established using an algorithm that provides some consistency, but it is not based on the originator of the session as source as the destination of the session as the target.\n\nSince this is the case, we first build a new edge frame containing all command and control (C2) edges. We call the new frame `C2flow`.\n\nTo construact the `C2flow` edge frame we need to look through all `Netflow` edges for records where either:\n\n• `dstPort = 3128`, in which case we copy the edge to the `C2flow` frame verbatim\n• `srcPort = 3128`, in which case we \"reverse\" the edge (swap srcX and dstX, for all of the directional fields) and insert that into the `C2flow` frame\n``````# Generate a new edge frame for holding only the C2 edges\nconn.drop_frame('C2flow')\nc2flow = conn.create_edge_frame(\nname='C2flow',\nschema=netflow.schema,\nsource=devices,\ntarget=devices,\nsource_key='srcDevice',\ntarget_key='dstDevice')\nc2flow\n``````\n``````<xgt.graph.EdgeFrame at 0x103f16710>\n``````\n\nExtract forward \"Command-and-control\" edges\n\nA \"forward\" edge is one where the `dstPort = 3128`. This edge is copied verbatim to the `C2flow` edge frame.\n\n``````def run_query(query, table_name = \"answers\", show_query=False):\nconn.drop_frame(table_name)\nif query[-1] != '\\n':\nquery += '\\n'\nquery += 'INTO {}'.format(table_name)\nif show_query:\nprint(\"Query:\\n\" + query)\njob = conn.schedule_job(query)\nprint(\"Launched job {}\".format(job.id))\nconn.wait_for_job(job)\ntable = conn.get_table_frame(table_name)\nreturn table\n``````\n``````%%time\nq = \"\"\"\nMATCH (v0)-[edge:Netflow]->(v1)\nWHERE edge.dstPort=3128\nCREATE (v0)-[e:C2flow {epochtime : edge.epochtime,\nduration : edge.duration, protocol : edge.protocol,\nsrcPort : edge.srcPort, dstPort : edge.dstPort,\nsrcPackets : edge.srcPackets, dstPackets : edge.dstPackets,\nsrcBytes : edge.srcBytes, dstBytes : edge.dstBytes}]->(v1)\nRETURN count(*)\n\"\"\"\nr = run_query(q)\nprint('Number of answers: ' + '{:,}'.format(r.get_data()))\n``````\n``````Launched job 7\nCPU times: user 14.3 ms, sys: 8.44 ms, total: 22.7 ms\nWall time: 4.81 s\n``````\n\nExtract reverse \"Command-and-control\" edges\n\nA \"reverse\" edge is one where the `srcPort = 3128`. These edges are copied to the `C2flow` frame but reversed in transit. The reversal process involves swapping the: `srcDevice` and `dstDevice`; `srcPort` and `dstPort`; `srcPackets` and `dstPackets`; and `srcBytes` and `dstBytes`.\n\n``````%%time\nq = \"\"\"\nMATCH (v0)-[edge:Netflow]->(v1)\nWHERE edge.srcPort=3128\nCREATE (v1)-[e:C2flow {epochtime : edge.epochtime,\nduration : edge.duration, protocol : edge.protocol,\nsrcPort : edge.dstPort, dstPort : edge.srcPort,\nsrcPackets : edge.dstPackets, dstPackets : edge.srcPackets,\nsrcBytes : edge.dstBytes, dstBytes : edge.srcBytes}]->(v0)\nRETURN count(*)\n\"\"\"\nr = run_query(q)\nprint('Number of answers: ' + '{:,}'.format(r.get_data()))\n``````\n``````Launched job 10\nCPU times: user 14.7 ms, sys: 5.85 ms, total: 20.5 ms\nWall time: 4.79 s\n``````\n\nQuerying our graph\n\nWe'll be looking for a mock pattern, similar to one that might be used to detect bot-net behavior. The pattern reflects an infected host (A) which is connecting up to a bot-net command and control node (B) with an exfiltration connection to a collection node (C).", null, "• Some device A boots up and, within a short amount of time, starts up a program.\n\n• Shortly afterwards, device A sends a message to some other device B.\n\n• Device B has a long-standing connection to another device C, which has been open for at least an hour, started before A booted, and remained open after A sent a message to B.\n\nQuery 1: Boot, program start, and connection\n\nWe begin the pattern description with the boot and program start events followed by a `C2flow` edge from our preprocessing. We'll restrict it such that all the pieces (boot, program start, and C2 flow) happen within 4 seconds.", null, "``````%%time\nq = \"\"\"\nMATCH (A)-[boot:Events1v]->(A)-[program:Events1v]->(A)\n-[c2:C2flow]->(B)\nWHERE A <> B\nAND boot.eventID = 4608\nAND program.eventID = 4688\nAND program.epochtime >= boot.epochtime\nAND c2.epochtime >= program.epochtime\nAND c2.epochtime - boot.epochtime < 4\nRETURN COUNT(*)\n\"\"\"\n# Note the overall time limit on the sequence of the three events\n\nr = run_query(q)\nprint('Number of boot, programstart, & c2 events: ' + '{:,}'.format(r.get_data()))\n``````\n``````Launched job 13\nNumber of boot, programstart, & c2 events: 109\nCPU times: user 9.31 ms, sys: 4.03 ms, total: 13.3 ms\nWall time: 748 ms\n``````\n\nQuery 2: Full zombie reboot pattern\n\nFinally, we add in the last network connection and match the full pattern.", null, "``````%%time\nq = \"\"\"\nMATCH (A)-[boot:Events1v]->(A)-[program:Events1v]->(A)\n-[c2:C2flow]->(B)-[nf2:Netflow]->(C)\nWHERE A <> B AND B <> C AND A <> C\nAND boot.eventID = 4608\nAND program.eventID = 4688\nAND program.epochtime >= boot.epochtime\nAND c2.epochtime >= program.epochtime\nAND c2.epochtime - boot.epochtime < 4\nAND nf2.duration >= 3600\nAND nf2.epochtime < boot.epochtime\nAND nf2.epochtime + nf2.duration >= c2.epochtime\nRETURN COUNT(*)\n\"\"\"\n\nr = run_query(q)\nprint('Number of zombie reboot events: ' + '{:,}'.format(r.get_data()))\n``````\n``````Launched job 16\nNumber of zombie reboot events: 981\nCPU times: user 17.4 ms, sys: 6.53 ms, total: 24 ms\nWall time: 9.44 s\n``````" ]

[ null, "http://docs.trovares.com/1.2.0/jupyter/LANL-ZR-advanced/images/zr-pattern-final.png", null, "http://docs.trovares.com/1.2.0/jupyter/LANL-ZR-advanced/images/zr-pattern-flow1.png", null, "http://docs.trovares.com/1.2.0/jupyter/LANL-ZR-advanced/images/zr-pattern-final.png", null ]

[ "## Circle Definitions\n\nThis section of Revision Maths defines many terms in relation to circles, including: Circumference, Diameter, Radius, Chord, Segment, Tangent, Point of contact, Arc, Angles on major and minor arcs, Angle of Centre and Sectors.\n\nCircumference: The circumference of a circle is the distance around it.\n\nDiameter: Any straight line that passes through the centre of the circle to two points on the perimeter.\n\nRadius: Any straight line that originates at the centre of a circle and ends at the perimeter.\n\nChord: A straight line whose ends are on the perimeter of a circle. A diameter is the longest chord possible.\n\nSegment: A part of the circle separated from the rest of a circle by a chord.\n\nTangent: A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle - it just touches it).\n\nPoint of contact: Where a tangent touches a circle.\n\nArc: A part of the curve along the perimeter of a circle.\n\nAngle on major arc: The larger of 2 angles when a circle is split into 2 uneven parts. Greater than 180 degrees.\n\nAngle of centre: An angle at the centre of a triangle between two lines that intersect with the perimeter.\n\nAngle at circumference on minor arc: The smaller of 2 angles when a circle is split into 2 uneven parts. Less than 180 degrees.\n\nSector: A portion of a circle resembling a 'slice of pizza'.", null, "More information on Circles can be found on the Circle Theorems page Here." ]

[ null, "https://revisionmaths.com/sites/mathsrevision.net/files/imce/CircleDefinitions.jpg", null ]

[ "×\n×\n\n# Solution: Absolute and relative error Compute the absolute", null, "ISBN: 9780321570567 2\n\n## Solution for problem 10E Chapter 7.6\n\nCalculus: Early Transcendentals | 1st Edition\n\n• Textbook Solutions\n• 2901 Step-by-step solutions solved by professors and subject experts\n• Get 24/7 help from StudySoup virtual teaching assistants", null, "Calculus: Early Transcendentals | 1st Edition\n\n4 5 1 325 Reviews\n26\n4\nProblem 10E\n\nAbsolute and relative error\n\nCompute the absolute and relative errors in using c to approximate x.\n\nx = e; c = 2.718\n\nStep-by-Step Solution:\nStep 1 of 3\n\nProblem 10E\n\nAbsolute and relative error Compute the absolute and relative errors in using c to approximate x.\n\nx = e; c = 2.718\n\nStep-1\n\nAbsolute Error Formula\n\nAbsolute error is defined as the magnitude of difference between the actual and the individual values of any quantity in question.\n\nSay we measure any given quantity for n number of times and a1, a2 , a3 …..an are the individual values then\n\nArithmetic mean am = [a1+a2+a3+ …..an]/n\n\nam= [Σi=1i=n ai]/n\n\nNow absolute error formula as per definition =\n\nΔa1= am – a1\n\nΔa2= am – a2\n\n………………….\n\nΔan= am – an\n\nMean Absolute Error= Δamean= [Σi=1i=n |Δai|]/n\n\nNote: While calculating absolute mean value, we don't consider the +- sign in its value.\n\nRelative Error or fractional error\n\nIt is defined as the ratio of mean absolute error to the mean value of the measured quantity\n\nδa =mean absolute value/mean value = Δamean/am\n\nExample ;  what are the absolute and relative errors of the approximation 3.14 to the value", null, "?\n\n...\n\nStep 2 of 3\n\nStep 3 of 3\n\n##### ISBN: 9780321570567\n\nUnlock Textbook Solution" ]

[ null, "https://studysoup.com/cdn/2cover_2418941", null, "https://studysoup.com/cdn/2cover_2418941", null, "https://chart.googleapis.com/chart", null ]

[ "# Best app for math\n\nMath can be a challenging subject for many students. But there is help available in the form of Best app for math. Keep reading to learn more!", null, "## The Best Best app for math\n\nIn addition, Best app for math can also help you to check your homework. Word phrase math is a mathematical technique that uses words instead of symbols to represent numbers and operations. This approach can be particularly helpful for students who struggle with traditional math notation. By using words, students can more easily visualize the relationships between numbers and operations. As a result, word phrase math can provide a valuable tool for understanding complex mathematical concepts. Additionally, this technique can also be used to teach basic math skills to young children. By representing numbers and operations with familiar words, children can develop a strong foundation for future mathematics learning.\n\nA composition of functions solver is a tool that helps to determine the composition of two or more functions. In mathematics, function composition is the process of combining two or more functions to create a new function. The resulting function is typically a simpler or more efficient version of the original functions. Composition of functions is a powerful technique that can be used to solve complex problems. By breaking down a problem into smaller pieces, it can be easier to find a solution. A composition of functions solver can be used to help find the composition of two or more functions. This tool can be an essential part of solving complex mathematical problems.\n\nThere are many ways to solve problems involving interval notation. One popular method is to use a graphing calculator. Many graphing calculators have a built-in function that allows you to input an equation and then see the solution in interval notation. Another method is to use a table of values. This involves solving the equation for a few different values and then graphing the results. If the graph is a straight line, then the solution is simple to find. However, if the graph is not a straight line, then the solution may be more complicated. In either case, it is always important to check your work to make sure that the answer is correct.\n\nSolving rational expressions calculator is a simple online tool which helps to solve rational expressions. It can be used in place of standard calculators. In order to use this tool, enter the expression you want to solve, choose between log and trigonometric functions, and click ‘Calculate’. The result will be displayed on the screen. You can also select among several options for converting and simplify rational expressions using the drop-down menu. While solving rational expressions using this calculator can take some time depending on the complexity of the expression, it is still a useful tool for learning or practicing basic math skills.\n\nOnce we have this total area, we can use it to calculate the volume of that shape. The formula below shows how to calculate the volume of a triangle: V = 1 / (1 + t^2) * l * w * h where V = Volume, t = Triangle’s area, l = Length side, w = Width side, and h = Height side The formula below shows how to calculate the volume of a quadrilateral: V = 1 / (1 + t^2) * l * w * h * 2", null, "", null, "" ]

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[ "G m1 m2 [Hint: The value of g is greater at the poles than at the equator. Universal Law of Gravitation For Class 9 The Universal Law of gravitation was coined by Sir Issac Newton. What is the net displacement and the total distance covered by the stone? In which of the following case do you think the long-term effects on your health are likely to be most unpleasant? Why or why not? The Universal law of gravitation is important because it tells about the force that is responsible for binding us to earth, motion of moon around the earth, motion of planet around the sun, formation of tides etc. What is the importance of universal law of gravitation? The Earth and moon experience equal gravitational forces from each other.The mass of the earth is much larger than the mass of the moon.Hence it accelerates at a rate lesser than acceleration rate of the moon towards the Earth.Therefore the Earth does not move towards the moon. Initial velocity of stone = 40 m/s, Acceleration due to gravity ,g = -10 m/s2, let h be the maximum height attained by the stone, Total distance covered by the stone during its upward and downward journey = 80 +80 =160 m, Net displacement during its upward and downward journey = 80 + (-80) = 0. The universal law of gravitation states that. The upward force exerted by a liquid on an object immersed in it is known as buoyancy. 1. She has started this educational website with the mindset of spreading Free Education to everyone. What is the acceleration of free fall? (iii) camphor from salt. What is the magnitude of the gravitational force between the earth and a 1 kg object on its surface? Q 1 Page 134 - State the universal law of gravitation. Answer: Answer: Universal Law of Gravitation: Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. SOLUTION: According to universal law of gravitation, every particle in the universe attracts every other particle with a force which is directly proportional to the distance between them. What do we call the gravitational force between the earth and an object? Gravitational force on the surface of the moon is only 1/6 as strong as gravitational force on the earth. (iii) sodium suphide 15. 3. (ii) salt from sea-water, If  the density of an object is more than the density of a liquid, then it sinks in the liquid. State the universal law of gravitation. (ii) the total time it takes to return to the surface of the earth. Answer: Calculate when and where the two stones will meet. (i) the mass of one object is doubled? inversely proportional to square of distance between them. Question 1. How does the force of gravitation between two objects change when the distance between them is reduced to half ? Position of the ball after 4 s of the throw is given by the distance travelled by it during its down ward journey in 4s -3 s = 1s, The height of ball above the ground after 4 s = (44.1 – 4.9) = 4.9 m. 19. Why will a sheet of paper fall slower than one that is crumpled into a ball? Answer: Answer: If you have any query regarding NCERT Class 9 Science Notes Chapter 10 Gravitation, drop a comment below and we will get back to you at the earliest. Answer: 1) Force is directly proportional to the mass of the object.If the mass of the object is doubled, then the gravitational force will also get doubled. Why does an object float or sink when placed on the surface of water? The value of g is greater at poles than at the equator.Therefore gold at equator weighs less than at the poles.Hence Amit’s friend will not agree with weight of the gold bought. Answer: 4. If you face any difficulty regarding understanding these new concepts, you can refer to the NCERT Solution of 9th Class Science Chapter 10 Gravitation. 2. F = 2F, so force is also doubled. Calculate. Here on AglaSem Schools, you can access to NCERT Book Solutions in free pdf for Science for Class 9 so that you can refer them as and when required. If the distance is tripled, then the gravitational force will become one-ninth of its original value. Two forces act on an object immersed in water. Gravitation NCERT In Text Solution. Every body on earth attracts every other body. Copyright © 2020 saralstudy.com. | EduRev Class 9 Question is disucussed on EduRev Study Group by 140 Class 9 Students. How are sol, solution and suspension different from each other? and. Why is the weight of an object on the moon 1/6, The mass of moon is 1/100 times and its radius is 1/4 times that of the earth.As a result, the gravitational attraction on the moon is about one sixth when compared to earth.Hence the weight of an object on the moon is 1/6. Will the packet float or sink in water if the density of water is 1 gcm–3? 2. If the upward buoyant force is greater than the downward gravitational force , then the object comes up to the surface of the water as soon as it is released within the water.Due to this reason, a block of plastic released under water come up to the surface of water. Your email address will not be published. heavy object does not fall faster than a light object. Will the friend agree with the weight of gold bought? If the buoyant force exerted by water is less than weight of object, the object will sink in water. | EduRev Class 9 Question is disucussed on EduRev Study Group by 115 Class 9 Students. ], Answer: What is the magnitude of the gravitational force between the earth and a 1 kg object on its surface? 2. If the buoyant force exerted by water is less than weight of object, the object will sink in water. Answer: Answer: F= G × M sun × M earth /R 2. Compounds of Xenon and uses of Noble Gases, Characteristics and Physical Properties of Group 18 Elements, Oxoacids of Halogens and Interhalogen Compounds, English Alphabets with Phonics Pronunciation, Economics Chapter 2 Sectors of the Indian Economy – Notes & Study Material. Formula to find the maximum height to which it rises body on earth... Is constant for all objects can get the CBSE Class 9 Question is disucussed on Study. Have a bag of cotton and an object immersed in a liquid, then the gravitational force become. Taken 3s to attain the maximum height to which it rises value is 6.67 x 10 that! Luggage kept on the moon are attracted to each other strong string one is the weight of object, velocity... 1024 kg and radius of the body Sun is given by to more! C ) ball attains the maximum height reached by the stone are moving the! He hands over the same straight line but in opposite directions distances between the earth the. Symptoms were present, would you still go to the product of their masses m before! 19.6 m/s sol, solution and suspension different from each other by gravitational force between two objects each! You think the long-term effects on your health are likely to be 42 kg height to which it.. Earth = mass of the packet is 1 g cm–3, will the substance is 20 cm3 to. Inversely proportional to the surface of the force between the objects is doubled, then gravitational... - Mention the significance of universal gravitation constant ( g ) has been found to be 6.67 x 10 with!, CBSE-Gravitation is crumpled into a ball is thrown vertically upward with an initial of. ( g ) has been found to be 42 kg on a weighing machine and a 1 object... These symptoms were present, would you still go to the square of water. Saralstudy helps in prepare for NCERT CBSE solutions for Class 9th Science, download pdf! The unit of gravitational force between the earth and a 1 kg object on its surface | NCERT Class Students... Not move towards the moon is only 1/6 as strong as gravitational force will one-ninth... Are moving in the liquid and ; inversely proportional to the doctor 30.! It advised to tie any luggage kept on the moon, Class 9 is. Shilpi Nagpal 3 Comments perfume sitting several metres away chance to win an Amazing Prizes, more questions you the! 6 × 10 how to identify the unit of gravitational force acts all. Weight on the earth is 6 × 10 24 kg earth, why does the between. The surface of water is 1 g cm–3, will the substance is! Question is disucussed on EduRev Study Group by 140 Class 9 Question is on... A school bag having a strap made of a cell if there was Golgi! Is 19.6 m/s sick and ought to see a doctor as strong as gravitational force an! Get doubled during which they stick together become one-ninth of its original value solutions NCERT... And why 42 kg sample papers, Notes, Important questions pulls the object will float, Physics CBSE-Gravitation... Jaundice, • if you get jaundice, • if you get jaundice •! 20 cm, 22, the object will float your mass to be unpleasant! / volume of the moon are attracted to each other = 10 m/s2, the. Notes, Important questions of attraction between the objects is doubled solutions, Class 9 is... Providing gravitation Class 9, Physics, CBSE-Gravitation our expert teachers questions you answer the chances to win more. Than the density of the gravitational force between the earth and an object be most?. The same straight line but in opposite directions more questions you answer chances! The liquid 500 g sealed packet is equal to the doctor sheet of paper fall slower than that... Your CBSE Board Exams gold at the equator m. ) there was no Golgi apparatus meets! Was no Golgi apparatus are marked *, the value of g is magnitude. Say which one is heavier and why 3s to attain the maximum reached. By Mrs Shilpi Nagpal 3 Comments masses of both objects are doubled then the gravitational force on... One-Fourth of its original value force on an object and its weight for all objects in proportion their... M earth /R 2 is thrown vertically upward with an initial velocity of stone just before touching ground. Each of mass 1.5 kg, are moving in the same straight line but in opposite directions then... Of one object is 2.5 m s-1 before the collision during which they stick together a of! Of one object is more than the previous value is 19.6 m/s and on the earth =W=mg 2 force! Last Updated on March 20, 2019 by Mrs Shilpi Nagpal 3 Comments m..... The product of their masses object upwards would think that you are sick 20 cm3 friend agree with the of. Object with a force called the gravitational force between the objects and ; inversely proportional to product of packet... As gravitational force acts on all objects in proportion to their masses of?. And its weight on the surface of water so force is along line... The line joining the two stones will state the universal law of gravitation class 9 ncert effects on your health are likely to be most unpleasant Important.... Measured on a weighing machine buys few grams of gold bought is known as buoyancy the smell perfume. Teachers for you Hint: the gravitational force become four times its original value March. The long-term effects on your health are likely to be 42 kg bus with a velocity of 40.... Takes to return to the weight in newtons of a 10 kg object on the is! Time it takes to return to the product of the earth is 6 × 10 15... Your mass to be 6.67 x 10 present, would you still to! Without leaving any solid NCERT Extra questions for Class 9th Science go the... Will sink in water likely to be 6.67 x 10 of attraction between earth. Line but in opposite directions weight in newtons of a 500 g sealed packet is equal the! Can you say which one is the force of gravity acting on body how to identify it,,! When and where the two particles return= 10 s. 14 ],:... Is 6 × 1024 kg and radius of the gravitational force between two objects change when the between. 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Of 40 m/s, Class 9 Question is disucussed on EduRev Study Group by 140 Class 9 Physics! Float or sink when placed on the surface of water on EduRev Study by! In the same straight line but in opposite directions velocity of the distances between the objects are.... Tie any luggage kept on the earth what is the magnitude of the body and g is the magnitude the! Started this educational website with the weight of object, the object you would think that are! Are sick list any three reasons why you would think that you are sick and ought to see doctor! By this packet measured on a weighing machine is inversely proportional to product of their masses its surface the between... Of paper fall slower than one that is crumpled into a ball is vertically., 2019 by Mrs Shilpi Nagpal 3 Comments 30 kg and strong string is along the line the! The two stones will meet by 140 Class 9 Science Chapter 10 ‘Gravitation’ ( Part-I ) from article. 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Its final velocity just before touching the ground substance float or sink in water th its weight attaining this,... A sheet of paper fall slower than one that is crumpled into a ball 106 m. ) that are! Density of water object float or sink when placed on the surface of the earth =W=mg = 2 × 30. G sealed packet is equal to the surface of the combined object collision..." ]

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[ "1 square mile = 2 589 988.11 square meters. 1 USD = 1.30 CAD 1 USD = 0.76 GBP 1 USD = 1.38 AUD 1 USD = 73.98 INR Find other conversions here: Hinweis: Für ein reines Dezimalzahl-Ergebnis wählen Sie bitte 'dezimal' aus den Optionen über dem Ergebnis. Mile. Die Meile ist eine Längenmaßeinheit außerhalb des Internationalen Einheitensystems.. um oitavo mile [survey, US] equals 201.2 meters because 0.125 times 1609 (the conversion factor) = 201.2 Live Currency Calculator Click Here! Use of the mile as a unit of measurement is now largely confined to the United Kingdom, the United States, and Canada. It consists of one hundred centimeters. 8 mile to meter = 12874.752 meter. What is 1 mi in Meters. Umrechnung Millimeter (mm) in Meter (m) Aktualisiert am 20.11.20 von Stefan Banse. We can find out how many meters are in 3 miles, 5 miles, 10 miles and so on. Die nautische Meilen genau gleich 1852 Meter. Definition: A mile (symbol: mi or m) is a unit of length in the imperial and US customary systems of measurement. According to Swim Outlet, a mile run would have the same number of meters in a mile as the standard conversion value which means that one mile run is equal to 1609.34 meters. 2 mile to meter = 3218.688 meter. Sie beträgt exakt 1609,344 Meter. The key is to multiply with the value of meters in 1 mile. The mile of 5,280 feet is called land mile or the statute mile to distinguish it from the nautical mile (1,852 meters, about 6,076.1 feet). Type in your own numbers in the form to convert the units! 1 mile = 1609.34 meters. 1 mi to m conversion. The conversion factor from miles to meters is 1609.344, which means that 1 mile is equal to 1609.344 meters: To convert 1 miles into meters we have to multiply 1 by the conversion factor in order to get the length amount from miles to meters. It is currently defined as 5,280 feet, 1,760 yards, or exactly 1,609.344 meters. This means that 5 miles is 8046.72m and 10 miles is 16093.44m. To convert any value in miles to meters, just multiply the value in miles by the conversion factor 1609.344.So, 1.2 miles times 1609.344 is equal to 1931 meters. How many meters in a mile (in a mile run, in a mile swim)? Hinweis: Sie können die Genauigkeit des Ergebnisses erhöhen oder verringern, indem Sie die Anzahl der massgeblichen Ziffern aus den oben stehenden Optionen ändern. 7 mile to meter = 11265.408 meter. The nautical mile equals to about 2025.4 yards (1852 meters). There are exactly 1609.344 meters in a mile. Die heute gebräuchlichsten Einheiten sind die Seemeile (1852,0 Meter) und die englische statute mile (1609,344 Meter). A mile is a unit of distance equal to 5,280 feet or exactly 1.609344 kilometers. Conversions miles to other units. After we know the value of how many meters is in 1 mile. 13.1 miles in other units What is Mil? 1 Meter is equal to 39370.0787 mils. 1 Meter = 39370.0787 Mils. Als Basiseinheit der Länge im SI und anderen mks Systemen (basierend auf Meter, Kilogramm und Sekunde) wird der Meter verwendet, um daraus andere Maßeinheiten wie Newton, für Kraft, abzuleiten. <=> precision: 1 miles = 1609.34 meters Formula miles in meters (mi in m). To convert any value in miles to meters, just multiply the value in miles by the conversion factor 1609.344.So, 1.5 miles times 1609.344 is equal to 2414 meters. Units of Measure. Unsere vollständigen Geschäftsbedingungen finden Sie hier: Bedingungen und Konditionen, Datenschutz. 22 Juli 2018, Umrechnungstabelle für metrische Maßeinheiten. 1 Mil is equal to 0.0000254 meter (m). It is defined as \"the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second.\" Herkunft speziell das Messgerät wurde als 1/40.000.000 der Umfang der Erde definiert, sondern seit 1983, als die Distanz, die das Licht in 1/299 792 458 Sekunden Reisen. 44.6 feet per second to meters per second, 10.8 feet per second to kilometers per hour. In Deutschland galten bis ins späte 19. We can also form a simple proportion to calculate the result: 1 mi → 1609.344 m 5 mile to meter = 8046.72 meter. The mile is an English unit of length of linear measure equal to 5,280 feet, or 1,760 yards, and standardised as exactly 1,609.344 metres by international agreement in 1959. The International System of Units states that the standard symbol for meter is \"m\", while for mile it is \"mi\". Miles : A mile is a most popular measurement unit of length, equal to most commonly 5,280 feet (1,760 yards, or about 1,609 meters). 4 mile to meter = 6437.376 meter. In 1799, France start using the metric system, and that is the first country using the metric. 2 miles = 1609.34 X 2 = 3218.68 meters. Meilen sind in Deutschland wenig geläufig - von Redewendungen wie \"meilenweit entfernt\" mal abgesehen. Wenn Sie einen Fehler auf dieser Seite entdecken, wären wir Ihnen dankbar, wenn Sie uns diesen über den Kontakt-Link oben auf dieser Seite mitteilen könnten. To convert miles into meters, you simply multiply the number of miles by the number of meters in a mile: Number of miles × 1609.344m. It is commonly used to measure the distance between places in the United States and United Kingdom. In Deutschland findet in der Regel das metrische System Anwendung, wir arbeiten in der Praxis mit Metern und Zentimeter (cm), Millimetern (mm) und den verbundenen Einheiten. Convert 1 Miles to Meters (1 mi to m) with our Length converter. How many meters in a mil? Length and Distance. For quick reference purposes, below is the conversion table you can use to convert from miles to meters. 1 Mile is equal to 1609.344 meters (m). Mile. It is commonly used to measure the distance between places in the United States and United Kingdom. A mile is a unit of distance equal to 5,280 feet or exactly 1.609344 kilometers. miles to meters formula. 6 mile to meter = 9656.064 meter. Miles Meters; 0 mi: 0.00 m: 1 mi: 1609.34 m: 2 mi: 3218.69 m: 3 mi: 4828.03 m: 4 mi: 6437.38 m: 5 mi: 8046.72 m: 6 mi: 9656.06 m: 7 mi: 11265.41 m: 8 mi: 12874.75 m: 9 mi: 14484.10 m: 10 mi: 16093.44 m: 11 mi: 17702.78 m: 12 mi: 19312.13 m: 13 mi: 20921.47 m: 14 mi: 22530.82 m: 15 mi: 24140.16 m: 16 mi: 25749.50 m: 17 mi: 27358.85 m: 18 mi: 28968.19 m: 19 mi: 30577.54 m Grundlagen zu Meilen und Kilometern. Meilen Meter; 0 mi: 0.00 m: 1 mi: 1609.34 m: 2 mi: 3218.69 m: 3 mi: 4828.03 m: 4 mi: 6437.38 m: 5 mi: 8046.72 m: 6 mi: 9656.06 m: 7 mi: 11265.41 m: 8 mi: 12874.75 m: 9 mi: 14484.10 m: 10 mi: 16093.44 m: 11 mi: 17702.78 m: 12 mi: 19312.13 m: 13 mi: 20921.47 m: 14 mi: 22530.82 m: 15 mi: 24140.16 m: 16 mi: 25749.50 m: 17 mi: 27358.85 m: 18 mi: 28968.19 m: 19 mi: 30577.54 m In this case 1 meter is equal to 0.00062137119223733 × 1 miles. 1 mile = 1609.34 meters So if we have to convert miles into meter we will simply multiply the value of the miles by 1609.34. For practical purposes we can round our final result to an approximate numerical value. Das Ergebnis rechts im Resultate-Feld erscheint, sobald Sie mit dem Mauszeiger ausserhalb des Eingabe-Feldes klicken. 1 mile to meter = 1609.344 meter. Für eine genauere Antwort wählen Sie bitte 'dezimal' aus den Optionen über dem Ergebnis. The conversion factor from miles to meters is 1609.344, which means that 1 mile is equal to 1609.344 meters: 1 mi = 1609.344 m. To convert 1 miles into meters we have to multiply 1 by the conversion factor in order to get the length amount from miles to meters. We can also form a simple proportion to calculate the result: Solve the above proportion to obtain the length L in meters: We conclude that 1 miles is equivalent to 1609.344 meters: We can also convert by utilizing the inverse value of the conversion factor. 13.1 miles to meters Miles to meters - Length Converter - 13.1 meters to miles This conversion of 13.1 miles to meters has been calculated by multiplying 13.1 miles by 1,609.344 and the result is 21,082.4064 meters. On the other end, they mention that one mile swim would be 1610. The meter (British spelling: metre; abbr. 9 mile to meter = 14484.096 meter. To convert miles to meters, multiply the mile value by 1609.344. The conversion factor from miles to meters is 1609.344, which means that 1 mile is equal to 1609.344 meters: 1 mi = 1609.344 m To convert 1.2 miles into meters we have to multiply 1.2 by the conversion factor in order to get the length amount from miles to meters. Eine Längeneinheit, die 1760 Yard entspricht. Angaben wie 1/4 Meile statt 440 yard oder 1320 feed sind dort üblich. Use of the mile as a unit of measurement is now largely confined to the United Kingdom, the United States, and Canada. The United States is one notable exception in that it largely uses US customary units such as yards, inches, feet, and miles instead of meters in everyday use. Umrechnungstabelle: There are 1609.344 meter in a mile. Meter = mi*1609.344 . The area units' conversion factor of square miles to square meters is 2 589 988.11. This is how they are defined: A mile is a most popular measurement unit of length, equal to most commonly 5,280 feet (1,760 yards, or about 1,609 meters). Wir werden uns bemühen, es so schnell wie möglich zu korrigieren. The mile of 5,280 feet is called land mile or the statute mile to distinguish it from the nautical mile (1,852 meters, about 6,076.1 feet). Mil (thou) is an imperial length unit. Next, let's look at an example showing the work and calculations that are involved in converting from miles to meters (mi to m). meter = mile * 1609.344. For example, to calculate how many meters is 2 miles, multiply 2 by 1609.344, that makes 3218.688 meters is 2 miles. The statue (land) mile is equal to 1760 yards (1609.344 meters). Home Science Math History Literature Technology Health Law Business All Topics Random. 1 Miles equals how many Meters. 1 Mile is equal to 1609.344 Meter." ]

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[ "", null, "# Python Namespace and Scope\n\nIn this tutorial, you will learn about namespace, mapping from names to objects, and scope of a variable.\n\n## What is Name in Python?\n\nIf you have ever read 'The Zen of Python' (type `import this` in the Python interpreter), the last line states, Namespaces are one honking great idea -- let's do more of those! So what are these mysterious namespaces? Let us first look at what name is.\n\nName (also called identifier) is simply a name given to objects. Everything in Python is an object. Name is a way to access the underlying object.\n\nFor example, when we do the assignment `a = 2`, `2` is an object stored in memory and a is the name we associate it with. We can get the address (in RAM) of some object through the built-in function `id()`. Let's look at how to use it.\n\n``````# Note: You may get different values for the id\n\na = 2\nprint('id(2) =', id(2))\n\nprint('id(a) =', id(a))``````\n\nOutput\n\n```id(2) = 9302208\nid(a) = 9302208```\n\nHere, both refer to the same object `2`, so they have the same `id()`. Let's make things a little more interesting.\n\n``````# Note: You may get different values for the id\n\na = 2\nprint('id(a) =', id(a))\n\na = a+1\nprint('id(a) =', id(a))\n\nprint('id(3) =', id(3))\n\nb = 2\nprint('id(b) =', id(b))\nprint('id(2) =', id(2))``````\n\nOutput\n\n```id(a) = 9302208\nid(a) = 9302240\nid(3) = 9302240\nid(b) = 9302208\nid(2) = 9302208```\n\nWhat is happening in the above sequence of steps? Let's use a diagram to explain this:\n\nInitially, an object `2` is created and the name a is associated with it, when we do `a = a+1`, a new object `3` is created and now a is associated with this object.\n\nNote that `id(a)` and `id(3)` have the same values.\n\nFurthermore, when `b = 2` is executed, the new name b gets associated with the previous object `2`.\n\nThis is efficient as Python does not have to create a new duplicate object. This dynamic nature of name binding makes Python powerful; a name could refer to any type of object.\n\n``````>>> a = 5\n>>> a = 'Hello World!'\n>>> a = [1,2,3]``````\n\nAll these are valid and a will refer to three different types of objects in different instances. Functions are objects too, so a name can refer to them as well.\n\n``````def printHello():\nprint(\"Hello\")\n\na = printHello\n\na()``````\n\nOutput\n\n`Hello`\n\nThe same name a can refer to a function and we can call the function using this name.\n\n## What is a Namespace in Python?\n\nNow that we understand what names are, we can move on to the concept of namespaces.\n\nTo simply put it, a namespace is a collection of names.\n\nIn Python, you can imagine a namespace as a mapping of every name you have defined to corresponding objects.\n\nDifferent namespaces can co-exist at a given time but are completely isolated.\n\nA namespace containing all the built-in names is created when we start the Python interpreter and exists as long as the interpreter runs.\n\nThis is the reason that built-in functions like `id()`, `print()` etc. are always available to us from any part of the program. Each module creates its own global namespace.\n\nThese different namespaces are isolated. Hence, the same name that may exist in different modules does not collide.\n\nModules can have various functions and classes. A local namespace is created when a function is called, which has all the names defined in it. Similar is the case with class. The following diagram may help to clarify this concept.\n\n## Python Variable Scope\n\nAlthough there are various unique namespaces defined, we may not be able to access all of them from every part of the program. The concept of scope comes into play.\n\nA scope is the portion of a program from where a namespace can be accessed directly without any prefix.\n\nAt any given moment, there are at least three nested scopes.\n\n1. Scope of the current function which has local names\n2. Scope of the module which has global names\n3. Outermost scope which has built-in names\n\nWhen a reference is made inside a function, the name is searched in the local namespace, then in the global namespace and finally in the built-in namespace.\n\nIf there is a function inside another function, a new scope is nested inside the local scope.\n\n## Example of Scope and Namespace in Python\n\n``````def outer_function():\nb = 20\ndef inner_func():\nc = 30\n\na = 10``````\n\nHere, the variable a is in the global namespace. Variable b is in the local namespace of `outer_function()` and c is in the nested local namespace of `inner_function()`.\n\nWhen we are in `inner_function()`, c is local to us, b is nonlocal and a is global. We can read as well as assign new values to c but can only read b and a from `inner_function()`.\n\nIf we try to assign as a value to b, a new variable b is created in the local namespace which is different than the nonlocal b. The same thing happens when we assign a value to a.\n\nHowever, if we declare a as global, all the reference and assignment go to the global a. Similarly, if we want to rebind the variable b, it must be declared as nonlocal. The following example will further clarify this.\n\n``````def outer_function():\na = 20\n\ndef inner_function():\na = 30\nprint('a =', a)\n\ninner_function()\nprint('a =', a)\n\na = 10\nouter_function()\nprint('a =', a)``````\n\nAs you can see, the output of this program is\n\n```a = 30\na = 20\na = 10```\n\nIn this program, three different variables a are defined in separate namespaces and accessed accordingly. While in the following program,\n\n``````def outer_function():\nglobal a\na = 20\n\ndef inner_function():\nglobal a\na = 30\nprint('a =', a)\n\ninner_function()\nprint('a =', a)\n\na = 10\nouter_function()\nprint('a =', a)``````\n\nThe output of the program is.\n\n```a = 30\na = 30\na = 30 ```\n\nHere, all references and assignments are to the global a due to the use of keyword `global`." ]

[ null, "https://www.facebook.com/tr", null ]

[ "# Swift Language Arrays\n\n## Introduction\n\nArray is an ordered, random-access collection type. Arrays are one of the most commonly used data types in an app. We use the Array type to hold elements of a single type, the array's Element type. An array can store any kind of elements---from integers to strings to classes.\n\n## Syntax\n\n• Array<Element> // The type of an array with elements of type Element\n• [Element] // Syntactic sugar for the type of an array with elements of type Element\n• [element0, element1, element2, ... elementN] // An array literal\n• [Element]() // Creates a new empty array of type [Element]\n• Array(count:repeatedValue:) // Creates an array of `count` elements, each initialized to `repeatedValue`\n• Array(_:) // Creates an array from an arbitrary sequence\n\n## Remarks\n\nArrays are an ordered collection of values. Values may repeat but must be of the same type.\n\n## Value Semantics\n\nCopying an array will copy all of the items inside the original array.\n\nChanging the new array will not change the original array.\n\n``````var originalArray = [\"Swift\", \"is\", \"great!\"]\nvar newArray = originalArray\nnewArray = \"awesome!\"\n//originalArray = [\"Swift\", \"is\", \"great!\"]\n//newArray = [\"Swift\", \"is\", \"awesome!\"]\n``````\n\nCopied arrays will share the same space in memory as the original until they are changed. As a result of this there is a performance hit when the copied array is given its own space in memory as it is changed for the first time.\n\n## Basics of Arrays\n\n`Array` is an ordered collection type in the Swift standard library. It provides O(1) random access and dynamic reallocation. Array is a generic type, so the type of values it contains are known at compile time.\n\nAs `Array` is a value type, its mutability is defined by whether it is annotated as a `var` (mutable) or `let` (immutable).\n\nThe type `[Int]` (meaning: an array containing `Int`s) is syntactic sugar for `Array<T>`.\n\n## Empty arrays\n\nThe following three declarations are equivalent:\n\n``````// A mutable array of Strings, initially empty.\n\nvar arrayOfStrings: [String] = [] // type annotation + array literal\nvar arrayOfStrings = [String]() // invoking the [String] initializer\nvar arrayOfStrings = Array<String>() // without syntactic sugar\n``````\n\n## Array literals\n\nAn array literal is written with square brackets surrounding comma-separated elements:\n\n``````// Create an immutable array of type [Int] containing 2, 4, and 7\nlet arrayOfInts = [2, 4, 7]\n``````\n\nThe compiler can usually infer the type of an array based on the elements in the literal, but explicit type annotations can override the default:\n\n``````let arrayOfUInt8s: [UInt8] = [2, 4, 7] // type annotation on the variable\nlet arrayOfUInt8s = [2, 4, 7] as [UInt8] // type annotation on the initializer expression\nlet arrayOfUInt8s = [2 as UInt8, 4, 7] // explicit for one element, inferred for the others\n``````\n\n## Arrays with repeated values\n\n``````// An immutable array of type [String], containing [\"Example\", \"Example\", \"Example\"]\nlet arrayOfStrings = Array(repeating: \"Example\",count: 3)\n``````\n\n## Creating arrays from other sequences\n\n``````let dictionary = [\"foo\" : 4, \"bar\" : 6]\n\n// An immutable array of type [(String, Int)], containing [(\"bar\", 6), (\"foo\", 4)]\nlet arrayOfKeyValuePairs = Array(dictionary)\n``````\n\n## Multi-dimensional arrays\n\nIn Swift, a multidimensional array is created by nesting arrays: a 2-dimensional array of `Int` is `[[Int]]` (or `Array<Array<Int>>`).\n\n``````let array2x3 = [\n[1, 2, 3],\n[4, 5, 6]\n]\n// array2x3 is 2, and array2x3 is 6.\n``````\n\nTo create a multidimensional array of repeated values, use nested calls of the array initializer:\n\n``````var array3x4x5 = Array(repeating: Array(repeating: Array(repeating: 0,count: 5),count: 4),count: 3)\n``````\n\n## Accessing Array Values\n\nThe following examples will use this array to demonstrate accessing values\n\n``````var exampleArray:[Int] = [1,2,3,4,5]\n//exampleArray = [1, 2, 3, 4, 5]\n``````\n\nTo access a value at a known index use the following syntax:\n\n``````let exampleOne = exampleArray\n//exampleOne = 3\n``````\n\nNote: The value at index two is the third value in the `Array`. `Array`s use a zero based index which means the first element in the `Array` is at index 0.\n\n``````let value0 = exampleArray\nlet value1 = exampleArray\nlet value2 = exampleArray\nlet value3 = exampleArray\nlet value4 = exampleArray\n//value0 = 1\n//value1 = 2\n//value2 = 3\n//value3 = 4\n//value4 = 5\n``````\n\nAccess a subset of an `Array` using filter:\n\n``````var filteredArray = exampleArray.filter({ \\$0 < 4 })\n//filteredArray = [1, 2, 3]\n``````\n\nFilters can have complex conditions like filtering only even numbers:\n\n``````var evenArray = exampleArray.filter({ \\$0 % 2 == 0 })\n//evenArray = [2, 4]\n``````\n\nIt is also possible to return the index of a given value, returning `nil` if the value wasn't found.\n\n``````exampleArray.indexOf(3) // Optional(2)\n``````\n\nThere are methods for the first, last, maximum or minimum value in an `Array`. These methods will return `nil` if the `Array` is empty.\n\n``````exampleArray.first // Optional(1)\nexampleArray.last // Optional(5)\nexampleArray.maxElement() // Optional(5)\nexampleArray.minElement() // Optional(1)\n``````\n\n## Useful Methods\n\nDetermine whether an array is empty or return its size\n\n``````var exampleArray = [1,2,3,4,5]\nexampleArray.isEmpty //false\nexampleArray.count //5\n``````\n\nReverse an Array Note: The result is not performed on the array the method is called on and must be put into its own variable.\n\n``````exampleArray = exampleArray.reverse()\n//exampleArray = [9, 8, 7, 6, 5, 3, 2]\n``````\n\n## Modifying values in an array\n\nThere are multiple ways to append values onto an array\n\n``````var exampleArray = [1,2,3,4,5]\nexampleArray.append(6)\n//exampleArray = [1, 2, 3, 4, 5, 6]\nvar sixOnwards = [7,8,9,10]\nexampleArray += sixOnwards\n//exampleArray = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n``````\n\nand remove values from an array\n\n``````exampleArray.removeAtIndex(3)\n//exampleArray = [1, 2, 3, 5, 6, 7, 8, 9, 10]\nexampleArray.removeLast()\n//exampleArray = [1, 2, 3, 5, 6, 7, 8, 9]\nexampleArray.removeFirst()\n//exampleArray = [2, 3, 5, 6, 7, 8, 9]\n``````\n\n## Sorting an Array\n\n``````var array = [3, 2, 1]\n``````\n\n## Creating a new sorted array\n\nAs `Array` conforms to `SequenceType`, we can generate a new array of the sorted elements using a built in sort method.\n\n2.12.2\n\nIn Swift 2, this is done with the `sort()` method.\n\n``````let sorted = array.sort() // [1, 2, 3]\n``````\n3.0\n\nAs of Swift 3, it has been re-named to `sorted()`.\n\n``````let sorted = array.sorted() // [1, 2, 3]\n``````\n\n## Sorting an existing array in place\n\nAs `Array` conforms to `MutableCollectionType`, we can sort its elements in place.\n\n2.12.2\n\nIn Swift 2, this is done using the `sortInPlace()` method.\n\n``````array.sortInPlace() // [1, 2, 3]\n``````\n3.0\n\nAs of Swift 3, it has been renamed to `sort()`.\n\n``````array.sort() // [1, 2, 3]\n``````\n\nNote: In order to use the above methods, the elements must conform to the `Comparable` protocol.\n\n## Sorting an array with a custom ordering\n\nYou may also sort an array using a closure to define whether one element should be ordered before another – which isn't restricted to arrays where the elements must be `Comparable`. For example, it doesn't make sense for a `Landmark` to be `Comparable` – but you can still sort an array of landmarks by height or name.\n\n``````struct Landmark {\nlet name : String\nlet metersTall : Int\n}\n\nvar landmarks = [Landmark(name: \"Empire State Building\", metersTall: 443),\nLandmark(name: \"Eifell Tower\", metersTall: 300),\nLandmark(name: \"The Shard\", metersTall: 310)]\n``````\n2.12.2\n``````// sort landmarks by height (ascending)\nlandmarks.sortInPlace {\\$0.metersTall < \\$1.metersTall}\n\nprint(landmarks) // [Landmark(name: \"Eifell Tower\", metersTall: 300), Landmark(name: \"The Shard\", metersTall: 310), Landmark(name: \"Empire State Building\", metersTall: 443)]\n\n// create new array of landmarks sorted by name\nlet alphabeticalLandmarks = landmarks.sort {\\$0.name < \\$1.name}\n\nprint(alphabeticalLandmarks) // [Landmark(name: \"Eifell Tower\", metersTall: 300), Landmark(name: \"Empire State Building\", metersTall: 443), Landmark(name: \"The Shard\", metersTall: 310)]\n``````\n3.0\n``````// sort landmarks by height (ascending)\nlandmarks.sort {\\$0.metersTall < \\$1.metersTall}\n\n// create new array of landmarks sorted by name\nlet alphabeticalLandmarks = landmarks.sorted {\\$0.name < \\$1.name}\n``````\n\nNote: String comparison can yield unexpected results if the strings are inconsistent, see Sorting an Array of Strings.\n\n## Transforming the elements of an Array with map(_:)\n\nAs `Array` conforms to `SequenceType`, we can use `map(_:)` to transform an array of `A` into an array of `B` using a closure of type `(A) throws -> B`.\n\nFor example, we could use it to transform an array of `Int`s into an array of `String`s like so:\n\n``````let numbers = [1, 2, 3, 4, 5]\nlet words = numbers.map { String(\\$0) }\nprint(words) // [\"1\", \"2\", \"3\", \"4\", \"5\"]\n``````\n\n`map(_:)` will iterate through the array, applying the given closure to each element. The result of that closure will be used to populate a new array with the transformed elements.\n\nSince `String` has an initialiser that receives an `Int` we can also use this clearer syntax:\n\n``````let words = numbers.map(String.init)\n``````\n\nA `map(_:)` transform need not change the type of the array – for example, it could also be used to multiply an array of `Int`s by two:\n\n``````let numbers = [1, 2, 3, 4, 5]\nlet numbersTimes2 = numbers.map {\\$0 * 2}\nprint(numbersTimes2) // [2, 4, 6, 8, 10]\n``````\n\n## Extracting values of a given type from an Array with flatMap(_:)\n\nThe `things` Array contains values of `Any` type.\n\n``````let things: [Any] = [1, \"Hello\", 2, true, false, \"World\", 3]\n``````\n\nWe can extract values of a given type and create a new Array of that specific type. Let's say we want to extract all the `Int(s)` and put them into an `Int` Array in a safe way.\n\n``````let numbers = things.flatMap { \\$0 as? Int }\n``````\n\nNow `numbers` is defined as `[Int]`. The `flatMap` function discard all `nil` elements and the result thus contains only the following values:\n\n``````[1, 2, 3]\n``````\n\n## Filtering an Array\n\nYou can use the `filter(_:)` method on `SequenceType` in order to create a new array containing the elements of the sequence that satisfy a given predicate, which can be provided as a closure.\n\nFor example, filtering even numbers from an `[Int]`:\n\n``````let numbers = [22, 41, 23, 30]\n\nlet evenNumbers = numbers.filter { \\$0 % 2 == 0 }\n\nprint(evenNumbers) // [22, 30]\n``````\n\nFiltering a `[Person]`, where their age is less than 30:\n\n``````struct Person {\nvar age : Int\n}\n\nlet people = [Person(age: 22), Person(age: 41), Person(age: 23), Person(age: 30)]\n\nlet peopleYoungerThan30 = people.filter { \\$0.age < 30 }\n\nprint(peopleYoungerThan30) // [Person(age: 22), Person(age: 23)]\n``````\n\n## Filtering out nil from an Array transformation with flatMap(_:)\n\nYou can use `flatMap(_:)` in a similar manner to `map(_:)` in order to create an array by applying a transform to a sequence's elements.\n\n``````extension SequenceType {\npublic func flatMap<T>(@noescape transform: (Self.Generator.Element) throws -> T?) rethrows -> [T]\n}\n``````\n\nThe difference with this version of `flatMap(_:)` is that it expects the transform closure to return an Optional value `T?` for each of the elements. It will then safely unwrap each of these optional values, filtering out `nil` – resulting in an array of `[T]`.\n\nFor example, you can this in order to transform a `[String]` into a `[Int]` using `Int`'s failable `String` initializer, filtering out any elements that cannot be converted:\n\n``````let strings = [\"1\", \"foo\", \"3\", \"4\", \"bar\", \"6\"]\n\nlet numbersThatCanBeConverted = strings.flatMap { Int(\\$0) }\n\nprint(numbersThatCanBeConverted) // [1, 3, 4, 6]\n``````\n\nYou can also use `flatMap(_:)`'s ability to filter out `nil` in order to simply convert an array of optionals into an array of non-optionals:\n\n``````let optionalNumbers : [Int?] = [nil, 1, nil, 2, nil, 3]\n\nlet numbers = optionalNumbers.flatMap { \\$0 }\n\nprint(numbers) // [1, 2, 3]\n``````\n\n## Subscripting an Array with a Range\n\nOne can extract a series of consecutive elements from an Array using a Range.\n\n``````let words = [\"Hey\", \"Hello\", \"Bonjour\", \"Welcome\", \"Hi\", \"Hola\"]\nlet range = 2...4\nlet slice = words[range] // [\"Bonjour\", \"Welcome\", \"Hi\"]\n``````\n\nSubscripting an Array with a Range returns an `ArraySlice`. It's a subsequence of the Array.\n\nIn our example, we have an Array of Strings, so we get back `ArraySlice<String>`.\n\nAlthough an ArraySlice conforms to `CollectionType` and can be used with `sort`, `filter`, etc, its purpose is not for long-term storage but for transient computations: it should be converted back into an Array as soon as you've finished working with it.\n\nFor this, use the `Array()` initializer:\n\n``````let result = Array(slice)\n``````\n\nTo sum up in a simple example without intermediary steps:\n\n``````let words = [\"Hey\", \"Hello\", \"Bonjour\", \"Welcome\", \"Hi\", \"Hola\"]\nlet selectedWords = Array(words[2...4]) // [\"Bonjour\", \"Welcome\", \"Hi\"]\n``````\n\n## Grouping Array values\n\nIf we have a struct like this\n\n``````struct Box {\nlet name: String\nlet thingsInside: Int\n}\n``````\n\nand an array of `Box(es)`\n\n``````let boxes = [\nBox(name: \"Box 0\", thingsInside: 1),\nBox(name: \"Box 1\", thingsInside: 2),\nBox(name: \"Box 2\", thingsInside: 3),\nBox(name: \"Box 3\", thingsInside: 1),\nBox(name: \"Box 4\", thingsInside: 2),\nBox(name: \"Box 5\", thingsInside: 3),\nBox(name: \"Box 6\", thingsInside: 1)\n]\n``````\n\nwe can group the boxes by the `thingsInside` property in order to get a `Dictionary` where the `key` is the number of things and the value is an array of boxes.\n\n``````let grouped = boxes.reduce([Int:[Box]]()) { (res, box) -> [Int:[Box]] in\nvar res = res\nres[box.thingsInside] = (res[box.thingsInside] ?? []) + [box]\nreturn res\n}\n``````\n\nNow grouped is a `[Int:[Box]]` and has the following content\n\n``````[\n2: [Box(name: \"Box 1\", thingsInside: 2), Box(name: \"Box 4\", thingsInside: 2)],\n3: [Box(name: \"Box 2\", thingsInside: 3), Box(name: \"Box 5\", thingsInside: 3)],\n1: [Box(name: \"Box 0\", thingsInside: 1), Box(name: \"Box 3\", thingsInside: 1), Box(name: \"Box 6\", thingsInside: 1)]\n]\n``````\n\n## Flattening the result of an Array transformation with flatMap(_:)\n\nAs well as being able to create an array by filtering out `nil` from the transformed elements of a sequence, there is also a version of `flatMap(_:)` that expects the transformation closure to return a sequence `S`.\n\n``````extension SequenceType {\npublic func flatMap<S : SequenceType>(transform: (Self.Generator.Element) throws -> S) rethrows -> [S.Generator.Element]\n}\n``````\n\nEach sequence from the transformation will be concatenated, resulting in an array containing the combined elements of each sequence – `[S.Generator.Element]`.\n\n## Combining the characters in an array of strings\n\nFor example, we can use it to take an array of prime strings and combine their characters into a single array:\n\n``````let primes = [\"2\", \"3\", \"5\", \"7\", \"11\", \"13\", \"17\", \"19\"]\nlet allCharacters = primes.flatMap { \\$0.characters }\n// => \"[\"2\", \"3\", \"5\", \"7\", \"1\", \"1\", \"1\", \"3\", \"1\", \"7\", \"1\", \"9\"]\"\n``````\n\nBreaking the above example down:\n\n1. `primes` is a `[String]` (As an array is a sequence, we can call `flatMap(_:)` on it).\n2. The transformation closure takes in one of the elements of `primes`, a `String` (`Array<String>.Generator.Element`).\n3. The closure then returns a sequence of type `String.CharacterView`.\n4. The result is then an array containing the combined elements of all the sequences from each of the transformation closure calls – `[String.CharacterView.Generator.Element]`.\n\n## Flattening a multidimensional array\n\nAs `flatMap(_:)` will concatenate the sequences returned from the transformation closure calls, it can be used to flatten a multidimensional array – such as a 2D array into a 1D array, a 3D array into a 2D array etc.\n\nThis can simply be done by returning the given element `\\$0` (a nested array) in the closure:\n\n``````// A 2D array of type [[Int]]\nlet array2D = [[1, 3], , [6, 8, 10], ]\n\n// A 1D array of type [Int]\nlet flattenedArray = array2D.flatMap { \\$0 }\n\nprint(flattenedArray) // [1, 3, 4, 6, 8, 10, 11]\n``````\n\n## Sorting an Array of Strings\n\n3.0\n\nThe most simple way is to use `sorted()`:\n\n``````let words = [\"Hello\", \"Bonjour\", \"Salute\", \"Ahola\"]\nlet sortedWords = words.sorted()\nprint(sortedWords) // [\"Ahola\", \"Bonjour\", \"Hello\", \"Salute\"]\n``````\n\nor `sort()`\n\n``````var mutableWords = [\"Hello\", \"Bonjour\", \"Salute\", \"Ahola\"]\nmutableWords.sort()\nprint(mutableWords) // [\"Ahola\", \"Bonjour\", \"Hello\", \"Salute\"]\n``````\n\nYou can pass a closure as an argument for sorting:\n\n``````let words = [\"Hello\", \"Bonjour\", \"Salute\", \"Ahola\"]\nlet sortedWords = words.sorted(isOrderedBefore: { \\$0 > \\$1 })\nprint(sortedWords) // [\"Salute\", \"Hello\", \"Bonjour\", \"Ahola\"]\n``````\n\nAlternative syntax with a trailing closure:\n\n``````let words = [\"Hello\", \"Bonjour\", \"Salute\", \"Ahola\"]\nlet sortedWords = words.sorted() { \\$0 > \\$1 }\nprint(sortedWords) // [\"Salute\", \"Hello\", \"Bonjour\", \"Ahola\"]\n``````\n\nBut there will be unexpected results if the elements in the array are not consistent:\n\n``````let words = [\"Hello\", \"bonjour\", \"Salute\", \"ahola\"]\nlet unexpected = words.sorted()\nprint(unexpected) // [\"Hello\", \"Salute\", \"ahola\", \"bonjour\"]\n``````\n\nTo address this issue, either sort on a lowercase version of the elements:\n\n``````let words = [\"Hello\", \"bonjour\", \"Salute\", \"ahola\"]\nlet sortedWords = words.sorted { \\$0.lowercased() < \\$1.lowercased() }\nprint(sortedWords) // [\"ahola\", \"bonjour\", \"Hello\", \"Salute\"]\n``````\n\nOr `import Foundation` and use NSString's comparison methods like `caseInsensitiveCompare`:\n\n``````let words = [\"Hello\", \"bonjour\", \"Salute\", \"ahola\"]\nlet sortedWords = words.sorted { \\$0.caseInsensitiveCompare(\\$1) == .orderedAscending }\nprint(sortedWords) // [\"ahola\", \"bonjour\", \"Hello\", \"Salute\"]\n``````\n\nAlternatively, use `localizedCaseInsensitiveCompare`, which can manage diacritics.\n\nTo properly sort Strings by the numeric value they contain, use `compare` with the `.numeric` option:\n\n``````let files = [\"File-42.txt\", \"File-01.txt\", \"File-5.txt\", \"File-007.txt\", \"File-10.txt\"]\nlet sortedFiles = files.sorted() { \\$0.compare(\\$1, options: .numeric) == .orderedAscending }\nprint(sortedFiles) // [\"File-01.txt\", \"File-5.txt\", \"File-007.txt\", \"File-10.txt\", \"File-42.txt\"]\n``````\n\n## Lazily flattening a multidimensional Array with flatten()\n\nWe can use `flatten()` in order to lazily reduce the nesting of a multi-dimensional sequence.\n\nFor example, lazy flattening a 2D array into a 1D array:\n\n``````// A 2D array of type [[Int]]\nlet array2D = [[1, 3], , [6, 8, 10], ]\n\n// A FlattenBidirectionalCollection<[[Int]]>\nlet lazilyFlattenedArray = array2D.flatten()\n\nprint(lazilyFlattenedArray.contains(4)) // true\n``````\n\nIn the above example, `flatten()` will return a `FlattenBidirectionalCollection`, which will lazily apply the flattening of the array. Therefore `contains(_:)` will only require the first two nested arrays of `array2D` to be flattened – as it will short-circuit upon finding the desired element.\n\n## Combining an Array's elements with reduce(_:combine:)\n\n`reduce(_:combine:)` can be used in order to combine the elements of a sequence into a single value. It takes an initial value for the result, as well as a closure to apply to each element – which will return the new accumulated value.\n\nFor example, we can use it to sum an array of numbers:\n\n``````let numbers = [2, 5, 7, 8, 10, 4]\n\nlet sum = numbers.reduce(0) {accumulator, element in\nreturn accumulator + element\n}\n\nprint(sum) // 36\n``````\n\nWe're passing `0` into the initial value, as that's the logical initial value for a summation. If we passed in a value of `N`, the resulting `sum` would be `N + 36`. The closure passed to `reduce` has two arguments. `accumulator` is the current accumulated value, which is assigned the value that the closure returns at each iteration. `element` is the current element in the iteration.\n\nAs in this example, we're passing an `(Int, Int) -> Int` closure to `reduce`, which is simply outputting the addition of the two inputs – we can actually pass in the `+` operator directly, as operators are functions in Swift:\n\n``````let sum = numbers.reduce(0, combine: +)\n``````\n\n## Removing element from an array without knowing it's index\n\nGenerally, if we want to remove an element from an array, we need to know it's index so that we can remove it easily using `remove(at:)` function.\n\nBut what if we don't know the index but we know the value of element to be removed!\n\nSo here is the simple extension to an array which will allow us to remove an element from array easily without knowing it's index:\n\n## Swift3\n\n``````extension Array where Element: Equatable {\n\nmutating func remove(_ element: Element) {\n_ = index(of: element).flatMap {\nself.remove(at: \\$0)\n}\n}\n}\n``````\n\ne.g.\n\n`````` var array = [\"abc\", \"lmn\", \"pqr\", \"stu\", \"xyz\"]\narray.remove(\"lmn\")\nprint(\"\\(array)\") //[\"abc\", \"pqr\", \"stu\", \"xyz\"]\n\narray.remove(\"nonexistent\")\nprint(\"\\(array)\") //[\"abc\", \"pqr\", \"stu\", \"xyz\"]\n//if provided element value is not present, then it will do nothing!\n``````\n\nAlso if, by mistake, we did something like this: `array.remove(25)` i.e. we provided value with different data type, compiler will throw an error mentioning-\n`cannot convert value to expected argument type`\n\n## Finding the minimum or maximum element of an Array\n\n2.12.2\n\nYou can use the `minElement()` and `maxElement()` methods to find the minimum or maximum element in a given sequence. For example, with an array of numbers:\n\n``````let numbers = [2, 6, 1, 25, 13, 7, 9]\n\nlet minimumNumber = numbers.minElement() // Optional(1)\nlet maximumNumber = numbers.maxElement() // Optional(25)\n``````\n3.0\n\nAs of Swift 3, the methods have been renamed to `min()` and `max()` respectively:\n\n``````let minimumNumber = numbers.min() // Optional(1)\nlet maximumNumber = numbers.max() // Optional(25)\n``````\n\nThe returned values from these methods are Optional to reflect the fact that the array could be empty – if it is, `nil` will be returned.\n\nNote: The above methods require the elements to conform to the `Comparable` protocol.\n\n## Finding the minimum or maximum element with a custom ordering\n\nYou may also use the above methods with a custom closure, defining whether one element should be ordered before another, allowing you to find the minimum or maximum element in an array where the elements aren't necessarily `Comparable`.\n\nFor example, with an array of vectors:\n\n``````struct Vector2 {\nlet dx : Double\nlet dy : Double\n\nvar magnitude : Double {return sqrt(dx*dx+dy*dy)}\n}\n\nlet vectors = [Vector2(dx: 3, dy: 2), Vector2(dx: 1, dy: 1), Vector2(dx: 2, dy: 2)]\n``````\n2.12.2\n``````// Vector2(dx: 1.0, dy: 1.0)\nlet lowestMagnitudeVec2 = vectors.minElement { \\$0.magnitude < \\$1.magnitude }\n\n// Vector2(dx: 3.0, dy: 2.0)\nlet highestMagnitudeVec2 = vectors.maxElement { \\$0.magnitude < \\$1.magnitude }\n``````\n3.0\n``````let lowestMagnitudeVec2 = vectors.min { \\$0.magnitude < \\$1.magnitude }\nlet highestMagnitudeVec2 = vectors.max { \\$0.magnitude < \\$1.magnitude }\n``````\n\n## Accessing indices safely\n\nBy adding the following extension to array indices can be accessed without knowing if the index is inside bounds.\n\n``````extension Array {\nsubscript (safe index: Int) -> Element? {\nreturn indices ~= index ? self[index] : nil\n}\n}\n``````\n\nexample:\n\n``````if let thirdValue = array[safe: 2] {\nprint(thirdValue)\n}\n``````\n\n## Comparing 2 Arrays with zip\n\nThe `zip` function accepts 2 parameters of type `SequenceType` and returns a `Zip2Sequence` where each element contains a value from the first sequence and one from the second sequence.\n\nExample\n\n``````let nums = [1, 2, 3]\nlet animals = [\"Dog\", \"Cat\", \"Tiger\"]\nlet numsAndAnimals = zip(nums, animals)\n``````\n\nnomsAndAnimals now contains the following values\n\nsequence1sequence1\n`1``\"Dog\"`\n`2``\"Cat\"`\n`3``\"Tiger\"`\n\nThis is useful when you want to perform some kind of comparation between the n-th element of each Array.\n\nExample\n\nGiven 2 Arrays of `Int(s)`\n\n``````let list0 = [0, 2, 4]\nlet list1 = [0, 4, 8]\n``````\n\nyou want to check whether each value into `list1` is the double of the related value in `list0`.\n\n``````let list1HasDoubleOfList0 = !zip(list0, list1).filter { \\$0 != (2 * \\$1)}.isEmpty\n``````" ]

[ null ]

[ "Compute a Bayesian equivalent of the p-value, related to the odds that a parameter (described by its posterior distribution) has against the null hypothesis (h0) using Mills' (2014, 2017) Objective Bayesian Hypothesis Testing framework. It corresponds to the density value at the null (e.g., 0) divided by the density at the Maximum A Posteriori (MAP).\n\n## Usage\n\np_map(x, null = 0, precision = 2^10, method = \"kernel\", ...)\n\np_pointnull(x, null = 0, precision = 2^10, method = \"kernel\", ...)\n\n# S3 method for stanreg\np_map(\nx,\nnull = 0,\nprecision = 2^10,\nmethod = \"kernel\",\neffects = c(\"fixed\", \"random\", \"all\"),\ncomponent = c(\"location\", \"all\", \"conditional\", \"smooth_terms\", \"sigma\",\n\"distributional\", \"auxiliary\"),\nparameters = NULL,\n...\n)\n\n# S3 method for brmsfit\np_map(\nx,\nnull = 0,\nprecision = 2^10,\nmethod = \"kernel\",\neffects = c(\"fixed\", \"random\", \"all\"),\ncomponent = c(\"conditional\", \"zi\", \"zero_inflated\", \"all\"),\nparameters = NULL,\n...\n)\n\n## Arguments\n\nx\n\nVector representing a posterior distribution, or a data frame of such vectors. Can also be a Bayesian model. bayestestR supports a wide range of models (see, for example, methods(\"hdi\")) and not all of those are documented in the 'Usage' section, because methods for other classes mostly resemble the arguments of the .numeric or .data.framemethods.\n\nnull\n\nThe value considered as a \"null\" effect. Traditionally 0, but could also be 1 in the case of ratios.\n\nprecision\n\nNumber of points of density data. See the n parameter in density.\n\nmethod\n\nDensity estimation method. Can be \"kernel\" (default), \"logspline\" or \"KernSmooth\".\n\n...\n\nCurrently not used.\n\neffects\n\nShould results for fixed effects, random effects or both be returned? Only applies to mixed models. May be abbreviated.\n\ncomponent\n\nShould results for all parameters, parameters for the conditional model or the zero-inflated part of the model be returned? May be abbreviated. Only applies to brms-models.\n\nparameters\n\nRegular expression pattern that describes the parameters that should be returned. Meta-parameters (like lp__ or prior_) are filtered by default, so only parameters that typically appear in the summary() are returned. Use parameters to select specific parameters for the output.\n\n## Details\n\nNote that this method is sensitive to the density estimation method (see the section in the examples below).\n\n### Strengths and Limitations\n\nStrengths: Straightforward computation. Objective property of the posterior distribution.\n\nLimitations: Limited information favoring the null hypothesis. Relates on density approximation. Indirect relationship between mathematical definition and interpretation. Only suitable for weak / very diffused priors.\n\n• Makowski D, Ben-Shachar MS, Chen SHA, Lüdecke D (2019) Indices of Effect Existence and Significance in the Bayesian Framework. Frontiers in Psychology 2019;10:2767. doi:10.3389/fpsyg.2019.02767\n\n• Mills, J. A. (2018). Objective Bayesian Precise Hypothesis Testing. University of Cincinnati." ]

[ null ]

[ "# Set Theory/Zorn's Lemma and the Axiom of Choice/Well-founded\n\nA binary relation R is well-founded iff for every set A\n\n$A\\subseteq R[A]\\Rightarrow A=\\emptyset$", null, "Theorem: A binary relation R is well-founded iff for every binary relation S\n\n$S\\circ R\\subseteq R\\circ S\\Rightarrow R\\cap S^{-1}=\\emptyset$", null, "Proof: Let R be a well founded relation and let S be a relation such that\n\n$S\\circ R\\subseteq R\\circ S$", null, "Let\n\n$X=field(R)$", null, "and let\n\n$A=dom(R\\cap S^{-1})$", null, "Then\n\n$A=dom(R\\cap S^{-1})=dom((S\\circ R)\\cap I_{X})\\subseteq dom((R\\circ S)\\cap I_{X})=dom(S\\cap R^{-1})=ran(R\\cap S^{-1})\\subseteq R[A]$", null, "It follows that A is empty, and therefore $R\\cap S^{-1}=\\emptyset$", null, "Conversely, suppose that for every relation S we have\n\n$S\\circ R\\subseteq R\\circ S\\Rightarrow R\\cap S^{-1}=\\emptyset$", null, "Let A be a set such that\n\n$A\\subseteq R[A]$", null, "Let $B=field(R)$", null, "and let $S=BxA$", null, ". Then\n\n$S\\circ R=R^{-1}[B]\\times A\\subseteq B\\times R[A]=R\\circ S$", null, "It follows that\n\n$R\\circ I_{A}=R\\cap (A\\times B)=R\\cap S^{-1}=\\emptyset$", null, "and so\n\n$R[A]=\\emptyset$", null, "and consequently $A=\\emptyset$", null, "" ]

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[ "## Similar Triangles\n\nRemember those fun diagrams from high school geometry? This post is a refresher on similar triangles, which will be needed for a forthcoming post on billiard geometry, so stay tuned for that one...\n\n#### Congruent Triangles\n\nTwo triangles (or any other shape for that matter) are called congruent if they are the same shape and size, except possibly rotated. Congruent triangles thus have three sides of the same length and three angles with the same measure. If you can prove that two triangles are congruent using a few of the sides and angles, then you know the other sides and angles are also equal.\n\nHow do you prove two triangles are congruent? It turns out any of the following will work:\n- side side side (SSS)\n- side angle side (SAS)\n- angle side angle (ASA)\n- angle angle side (AAS)\n- hypotenuse and one leg length (for right triangles only)\n\nSo for example, if I have two triangles with side lengths 4 and 7 with a 30 degree angle in between, then they are congruent by SAS, and the other side and two angles are also equal.\n\nWhy do these formulas work? Because given, for example, that two sides and the angle in between are all equal, then the other pieces are predetermined. That means that a second triangle with the same SAS will have the same predetermined other pieces as well, i.e. is congruent to the first triangle.\n\nTo prove that SAS works, take some triangle $ABC$ in the plane, and assume without loss of generality that point $A$ is at $(0,0)$, $B$ is on the positive $x$-axis, and $C$ is somewhere above the $x$-axis (i.e. in the positive $y$ region). This is indeed without loss of generality, because if it isn't the case, we can move our triangle left/right/up/down and/or rotate it, all without changing its shape and size.\n\nGiven the length of segment $AB$ is $L_1$, we know that $B = (L_1,0)$. Given that the measure of angle $BAC$ is $\\theta$ and the length of segment $AC$ is $L_2$, we know that $C = (L_2 \\cos \\theta , L_2 \\sin \\theta)$ (and that $\\theta$ will be between 0 and 180 degrees, or 0 and $\\pi$ in radians, because of the assumption that $C$ is above the $x$-axis). So we've determined all 3 vertices, and thus we know all the side lengths and angles.\n\nThere are other ways to prove these as well besides coordinate proofs like the one above, but I won't go further into it since it's not that exciting, and we can at least say we proved one of them now.\n\nNote that the following are not sufficient to establish congruence:\n- AAA (proves they are the same shape, but not necessarily same size)\n- ASS (which doesn't even necessarily prove they are the same shape- can you see why?)\n\n#### Similar Triangles\n\nTwo triangles (or any other shape) are called similar if they are the same shape, just different size, so one is a scaled up/down version of the other. This means that the three angles are the same, and the side lengths of the second triangle are $r$ times the side lengths of the first where $r>0$ is some fixed ratio. Congruence is the special case of similarity where $r=1$.\n\nThe same five criteria I mentioned for congruence also work to prove similarity, except that for the side comparisons, you need to show that they have a certain fixed ratio instead of being equal. Since the ratio can be any positive number, AAS and ASA just become AA; if any two angles are the same, the triangles are similar. We know the third angle is the same too because the three of them need to add up to 180, so AA is equivalent to AAA and is sufficient to prove similarity. For SAS and SSS, which have more than one side involved, you need to actually check the ratio of the sides.\n\nOne last comment here: to prove that the angles add up to 180, draw a straight line $L$ through one of the vertices which is parallel to the side of the triangle which is opposite that vertex. Then the angles between the triangle and $L$ are equal to the angles at the other two vertices. But the three angles at our first vertex obviously add up to 180 because they are on a straight line.\n\n#### Brief off-topic foray:\n\nInterestingly, the angles of a triangle do not necessarily add up to 180 if the triangle is drawn on a non-flat surface (in particular, if a neighborhood of a vertex isn't flat). For example, if you draw a triangle on the sphere of the earth which has a vertex at the north pole and two vertices on the equator, you can have all 3 angles be 90 degrees, in which case the angles add up to 270.\n\nThe analog of \"straight lines\" on a non-flat surface, which form the sides of a triangle, are the paths of shortest distance and are called geodesics. On a sphere, the geodesics are the great circle arcs, which are pieces of circles on on the surface whose radius is the entire radius of the sphere. The equator is a great circle on the surface of the earth, as is the line dividing the eastern and western hemispheres.\n\nThanks for reading. Please post any questions in the comments section.\n\n1.", null, "Cool Post..\nOf all the posts in gtMath, I liketh this one the most\n& probably the next one too - Pool Geometry! Billiards, way to go:)\n\n//- ASS (which doesn't even necessarily prove they are the same shape- can you see why?)//\n\nProfessionally, ASS never works:)\nBecause, given two sides and a non-included angle, it is possible to draw 2 different triangles that satisfy those values.\n\nIf u draw a line through the vertex, opposite to that given angle bisecting that vertex, and touching the base.. then u will end up with same length as the other side.\nSo, Angle, Side, Side = ASS Does not work!\n\nQuestion(s):\n1. Apart from sides and angles, are other properties of congruent triangles, the same too? (like area, perimeter, base, height, concentric circles etc?)\n2. Is congruence, only applicable for triangles? Can there be congruence on rectangles, quadrilaterals, polygons etc?\n\n2.", null, "//angles of a triangle do not necessarily add up to 180 if the triangle is drawn on earth//\n\nLiked this last, off-topic foray!\nU should do it for every post:)\n\nWhat about those triangles caused by umbra & penumbra of moon's shadows?\nAre they congruent or similar?:)\n\n3.", null, "Congruent triangles share ALL properties because they are the same triangle, just moved over or rotated. And congruence does apply to any other shape as well.\n\nUmbra, penumbra- I have no idea..." ]

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[ "## Wednesday, October 31, 2012\n\n### Water Slide Collisions!\n\nWater Slide Collisions\n\nNitro Racer – Water Country USA Williamsburg, Virginia\n\nVertical Drop:  150ft (45.7m)\nLength of Slide 320ft (97.5m)\n\nMass of Kid:  40kg\nMass of Father:  80kg\nCollision occurs over .05s\n\nPotential Energy = mgh\nKinetic Energy = 1/2mv2\n\nKid drops to vertical elevation of 25ft (7.6m)\nKid PE = (40kg)(9.8m/s2)(7.6) = 2980J\n\nMan Drops to vertical elevation of 25ft (7.6m) before collision\n\nMan PE at 95.5m = (80kg)(9.8m/s2)(45.7) = 35800J\nMan PE at 7.6m = (80kg)(9.8m/s2)(7.6) = 5960J\n\nMan KE at 7.6m = -mgh + mgh - mmg * sin(q)(d) =  Velocity of man before collision = -(80kg * 9.8m/s 2 * 7.6m) + (80kg * 9.8m/s2 * 45.7m) – (.2 *80kg*  9.8m/s2)(sin 28)(97.5m-16.2m) = 24.4m/s\n\nThe Collision!\nM1V1 + M2V2 = v(M1+M2)\n(80kg)(24.4m/s) + (40kg)(0m/s) = (120kg)(v) v= 16.2 m/s\n\nImpulse = m∆v\nImpulse = 120kg(16.2m/s-24.4m/s) = -984kg*m/s\nForce = Impulse / ∆t\nForce = 984kg*m/s / .05s\n\n19,700N Very possible that serious injuries were sustained\n30,000N needed to break some bones such as the patella\n\nChances are that the bones would not break!\n\n## Tuesday, October 30, 2012\n\n### The Force of the Frankenstorm\n\nWe all know Hurricane Sandy rocked the East Coast over the past couple of days, and I was curious to check out some of the statistics for the Superstorm. I found that, along with huge amounts of rain and snowfall, gusts of wind reached as high as 94 mph (42 m/s) in places like Eaton's Neck, NY.\n\nhttp://www.scientificamerican.com/article.cfm?id=the-stats-are-in-superstorm-sandy\n\nThis reminded me of a homework problem we'd done a little while back, with wind being strong enough to blow over a person. So, I wanted to look at situation that could definitely occur in a hurricane: wind blowing over an entire car.\nI assumed that the wind is approaching the car from the side, so that the car is basically a rectangle. I found that the average length of a sedan is about 4.6 m, and the height is 1.5 m, and the mass is 1,500 kg. I also assumed the wind is striking the car at a rate of 66 kg/s per square meter and comes to rest. I wanted to: A. Find the approximate force of the wind on the car, and B.Compare this force to the typical maximum friction force between a car and the wet road.\n\nA)\n1. First find the area of the car on which the wind will be acting:\nA = (l)(w) = (4.6 m)(1.5 m) = 6.9 m^2.\n\n2. Use the area to find the rate of the wind in kg/s:\n(66 kg/s/m^2)(6.9m^2) = 455.4 kg/s\n\n3. Find the force of the wind:\nWe know that the force stopping the wind is exerted by the car, so the force on the car is equal in magnitude and opposite in direction to the force stopping the wind. So we can just look at magnitude to find the force of the wind:\n\nFwind on car = Fcar on wind = Δpwind/Δt = (mwind)(vwind)/Δt = (mwind/Δt)(vwind) = (455.4 kg/s)(42 m/s) = 19,127 N --> 19,000 N.\n\nB)\nFfr = usmg = (0.40)(1500 kg)(9.8m/s^2) = 5880 N --> 5,900 N.\n\n19,000 N is much greater than 5,900 N, meaning the force of the wind is strong enough to blow over a car.\n\nHere's some evidence!\n\nPart 2: If the wind got the car to travel 10.0 m/s and it was blown into a large building (and came to rest), A. What is the velocity of the car/building after the collision? B. How much energy is \"lost\" in the collision?\n\nA) We would use conservation of momentum to find the final velocity of the car/building:\n\nmcvci + mbvbi = (mc + mb)vf\n\nBut in this case, we can assume that the mass of the building is so much greater than the mass of the car that the velocity of the building will be unchanged in the collision, meaning the velocity of the car/building after the collision is 0 m/s (since the building starts at rest).\n\nB) To find the energy \"lost\" in the collision, we look at the change in kinetic energy of the system:\n\nΔKE = KEafter - KEbefore\n\nSince the velocity of the car/building after the collision is zero, the kinetic energy of the system after the collision is also zero. This means ALL of the initial kinetic energy is \"lost\". But we know this just means it is transformed into other types of energy, such as heat and sound!\n\nCatherine Stecyk\n\n### Tying an NFL Record\n\nBy Alex Girden\n\nWhat we Know\nDistance =63 yards\n1 yard=0.9144 meters\nDistance=58 meters\nHeight of cross bar =10. feet\n1 foot=0.3048 meters\nHeight =3.0 meters\nFootball= 0.413 kg\nEstimated time of travel = 3 seconds\nThe easy idea\nSolving for initial velocity?\nAngle of kick?\nKinetic Energy?\nCalculations\nΔx=vot+1/2at^2\n3.o=vo (3 s)+ 0.5*-9.8* (3)^2\nVoy=16 m/s\nMax height- Vf^2=Vo^2+2aΔx\nΔx=13 meters\nMore equations\nΔx=vot+1/2at^2\nax=0 m/s^2\n58=vo (3)\nVox=19 m/s\nVnet^2=vx^2 +vy^2\nVnet=25 m/s\nUnknowns\nsinΘ=16/25\nΘ=40 °\nKinetic Energy given to ball\nKE=1/2mv^2\n½ (0.413kg) (25)^2=129 J\nForce of a kicker\nTime in contact with ball approximate=1.8 X 10^-3 s\nVf=vo+at\n25=0 + a (1.3X 10^-3)\nAcceleration = 19,200 m/s\nForce=ma= (19,200 * 0.413 kg)=7,900 N\nFuture Directions\nTake into account wind speed and direction\nAir resistance’s affect on the ball\n\n### Physics of blocking a Puck\n\nBy Rachel Walsh\n\nFor my physics new assignment I chose to evaluate the forces and momentum involved in stopping a puck with your forehead. This was inspired by this block by Ian Lapperriere http://www.youtube.com/watch?v=47ankjpg__Q . First I considered the change in momentum from before and after the hit and then considered the sum of the forces.\n\nMomentum of puck before the block:\n\nAverage speed of a slap shot assumed to be 80 mph = 128.8 km/h = 35.8 m/s\n\nm = 1 pound or 0.454 kg\np = mv = (35.8m/s)(0.454 kg) = 16.6 kg m/s\n\nMomentum after the block:\nv = - 25 m/s (estimated)\np = mv = (-25 m/s)(0.454 kg) = - 11.35 kg m/s\nSum of forces:\nt of impact = 0.01 seconds\nSF = Dp/Dt = (-11.35 kg m/s – 16.6 kg m/s)/(0.01s) = -2795 N\n\n2795 N are applied by the player’s forehead to change the momentum of the puck. Due to Newton’s law stating that every force has an equal and opposite force the puck also applies 2795 N of force on Lapperiere’s forehead. In homework #3 it is stated that 6000 N is needed to fracture the human forehead; therefore, a puck would not be able to fracture the forehead in this example.\n\n### THE PHYSICS OF A LONG BACK SPINGING KICK\n\nA long back spinning kick is a kick where the practitioner (me.) spins using the foot that is initially behind them to kick in a circle by raising their foot at they go. The kick is intended to hit the target 180° after motion has started.\n* Please note this is actually not a circle it is an oval because I have to shift my weight as I go, but for this problem we’re going to call it a circle.\nI know that I can kick hard enough to break someone’s ribs. It takes 3300 newton’s to do this (the average martial arts master can, depending on the kick, get up to 9000 newton’s.) I’m not a master and I’m pretty small so let’s just stick to breaking a rib here. So torque is:\nT=rFsinΘ\nWhere Θ is 90. The length of my leg is about .69m and since I’m pivoting in a circle the distance to my foot is the radius. So the torque is going to be\n.69*3300*sin90\nTorque is 2277 Nm.\n\nNow to get the Radial Acceleration we use the equation T=Iα if we assume my leg is a thin straight object with the point that we rotate around being the end. Than the equation for torque should read. T=[(mL^2)/3]*α.\nNow I have a mass of 50 kilograms. (well 49.8 but let’s call it 50.) And my leg to my hip joint is about .69m. And earlier we found the torque to be about 2300Nm so we get\nα=(T*3)/(ML^2)\nα=(2300*3)/(50*.69^2).\nα=23.8m/s^2\nSo to get the angular velocity of my foot at the point of contact.\nα=ω/t.\nSo I timed myself kicking and it takes me.75 seconds for a half a circle. Because we assume once I hit the guy I stop spinning.\nω=αt\n23.8*.75=ω\nω=17.9 m/s.\nSo in order to get the linear velocity my foot hits the target with we take the angular velocity and times it by the radius.\nV=ωr\n17.9*.69=V\nV=12.35m/s.\nTo give you a better reference for this number 12.35m/s is 27.63mph.\n\n## Sunday, October 28, 2012\n\n### Physics of Dunking a Basketball\n\nBy Kathryn Taylor\n\nBasketball players exert a huge amount of force when they are jumping and cutting during games. Dunking is one of the skills that requires the greatest amount of force to jump as high as possible and virtually drop the ball downwards through the hoop. In his day Michael Jordan was one of the greatest dunkers and was known as Air Jordan.\n\nTo make the situation a little simpler I am going to look at dunking a ball from just standing under the rim. This takes away the horizontal component of the situation which would have little effect anyway as there is negligible acceleration in that direction. Michael Jordan is 6’6” (1.98m) and in his playing days weighed 216lb (98kg) and would be dunking a ball of 624g. I found that he was thought to have a reach of 8’10” (2.7m) (http://thekitchensinkhole.blogspot.com/2007/02/\nsinkhole-vertical-leaps.html). This means that the change in height for Michael Jordan dunking on a regulation rim (10’) would be just 1’2” (0.35m). This means that the minimum work done by his jump would be equal to ΔPE= mgΔh = (98+0.624kg)*(9.8m/sˆ2)*(0.35m) = 338.3J. This work was done over the distance that his knees bend which can be estimated to around 3’ (0.9144m). Using this the minimum applied force by Jordan into the ground to dunk the ball is equal to Work/distance = (338.3J/0.9144m)=370N.\n\nThe Guinness World Record dunk of 12ft was set by Michael “Wild Thing” Wilson who is part of the Harlem Globetrotters. For Jordan to complete this dunk his work would be mg\nΔh = (98.624kg)*(9.8m/s^2)*(0.96m) = 927.9J. This relates to an applied force = (927.9J/0.9144m)=1014.7N. This is a huge increase in force required for a dunk only 2 feet higher than regulation.\n\n### Physics in Figure Skating: The Death Spiral\n\nFigure skating is obviously an interesting application of physics, and the success of many of the \"moves\" that skaters do depends on whether or not they are able to exert the force that is physically required for what they are aiming to accomplish. Specifically, I chose to look at the physics behind the \"death spiral,\" which is a common \"move\" that is is seeming difficult in figure skating. The death spiral is when a male skater pulls a female skater in a circle while she is almost completely perpendicular to the ice. See a video of it here: http://www.youtube.com/watch?v=bOsWG3BeslE\n\nI chose to qualitatively look at the center of mass between the two skaters, simplifying their relationship to being one rigid body. Since CM=(m1x1+m2x2)/(m1+m2), I intuitively observe that the center of mass lies closer to the male skater. This makes sense because as a male skater who weighs more, his mass will be larger and therefore the resulting calculation for center of mass will lie closer to him.\n\nNext I considered looking at the speed the pair would probably be skating in. Based on the video, I presumed that the pair did about one revolution in two seconds. To find the angular velocity, I know that w=2piT, so plugging in 0.5 for T estimates that the skaters' angular velocity was about 3.1 radians per second. Next, to find their approximate linear velocity I assumed their radius of rotation (from the center of mass to where the male plants his foot) was about 0.5 meters. Accordingly, I know that v=rw, so plugging in 0.5 in for r and 3.1 in for w, I found their linear velocity to be about 1.6 meters per second.\n\nNext, I chose to look at what forces are involved between this pair as they rotate. They have a centripetal acceleration that comes from the force that the male exerts on the female in planting his blade into the ice. This force is the centripetal force that keeps the pair rotating. Based upon Newton's second law, I know that F=ma. Furthermore, I know that radial acceleration is equal to w^2r. Plugging this in, I can get an equation to estimate the force, F=(m1+m2)w^2r. Supposing that the female weighed about 50 kg and the male about 79 kg, I estimate the force to be about 620 N.\n\nIn my interpretation, I thought it was important to consider whether this amount of force seems reasonable. The male must exert 620 N of force in order to successfully complete the \"death spiral.\" In looking at the force that the male can exert with his body weight, I found that he could exert about 770 N (F=ma=79*9.8). Therefore, the 620 N that the male must exert is a reasonable estimate, and more importantly possible. This is obviously good news for the female skater, who must trust that the male skater can keep his skate stabilized while the force is exerted as she spins, otherwise the stunt could result in an accident. Lastly, another consideration in the calculation of this force is that the force of tension in the hands of the skaters must equal 620 N--the skaters grips must withstand the large force.\n\nBased upon my investigations above, it is apparent that the death spiral is a pretty difficult stunt in the world of figure skating. This is probably why its completion receives high scores in the judging rounds of major competitions.\n\nFigures that may help visualize:", null, "", null, "Source: http://www.real-world-physics-problems.com/physics-of-figure-skating.html\n\n## Friday, October 26, 2012\n\n### The Physics of Hitting an Elk With You Car", null, "", null, "", null, "By Heather Frank\nThis summer when I was on a road trip with my Dad in the Grand Canyon we hit an Elk with our car at 4:30 in the morning while driving in the dark at 45 mph. Although the car was totaled and the Elk died, my Dad and I were unharmed since we were wearing our seatbelts and the airbags deployed.\n\nIn order to analyze the forces that we felt during the accident I had to make a few estimations. I assumed that MCAR=2000 kg, MELK=225 kg, V1 of the car before the accident was 45 mph (20 m/s) and V2 is the velocity of the car and elk after colliding. I treated this as an inelastic collision because after we hit the Elk we carried the Elk on the hood of our car for about .5 seconds. Then we traveled an additional 15m before coming to a stop. After finding the Velocity of the Car and the Elk after the collision (V2) I was able to find the WNC done on the car to make it stop by looking at the change in Kinetic Energy. This WNC was a combination of the work done by the friction between the tires and the road and the work done by the car brakes to come to the stop. Finally, I looked at the Force exerted on my Dad and I during the accident and how that force was significantly decreased by us wearing seatbelts and the deployment of the airbags, which increased the time that we experienced a change in momentum.\nMVCAR+MVElK=MVCAR AND ELK\n(2000)(20)+0=(2225)(V) Vfinal of Car and Elk= 18m/s\nThen Elk fell off car and car traveled for additional 15 m.\nChange in KE=WNC=1/2m(vf2-v02)\nWNC=1/2(2000)(0-182)= -324000 J\nThis is combination of Ffr and Work done by brakes of car over 30m.\nWNC=Ffrd+Fbraked\n324000=(.7)(2000)(9.8)(15)+ Fbrake(15)\nFbrake= 7880 N A LOT OF FORCE!\nAlso it is important to note that the reason we did not sustain injury from the accident is because the Force exerted on us (change in momentum over time) was significantly decreased by the airbags and the seatbelts, by increasing the time in which we felt the change in momentum.\n\nNOTE: These numbers are all estimates. I have a foggy memory of the accident and my times and distances might be slightly off but the ideas of Physics apply to this scenario.\n\n### PUMPKIN CHUNKIN’\n\nPumpkin Chunkin'\n\nIt seems like it’s about that time in the semester where everyone starts getting overwhelmed by the amount of work they have. I know that when I have too much work I love taking TV breaks, which mostly consist of my favorite show “Modern Family”.  Recently, I was watching the Halloween and Thanksgiving episodes, and came across the scene where the entire family goes pumpkin chunkin’. I decided to take a look at these contests in real life, and saw that they nicely corresponded with our recent discussions about conservation of energy.\nI thought it would be interesting to examine a scenario to find the velocity of the pumpkin right before it hits the ground.  I made the assumption that the rope they use on the show works the same way as a bungee cord would (there is no friction between it and the pumpkin) and there is no air resistance. Additionally, I set the mass of the pumpkin to be about 6 kg and the initial launch height to be 5 m.  Also, I added a Fapplied of about 2205 N because people pull back on the cord (distance= 2m) so that the pumpkin may gain potential energy. Before the pumpkin is launched, it only contains potential energy, and by the time it launches all of that energy turns into kinetic. Therefore:\nΔKE = -ΔPE + WNC\nWNC= -Fappdcos180= Fappd\n½ mvf2 =-(mghf – mgho) + Fappd\n½(6 kg) vf2= -(6 kg)(9.8 m/s2)(- 5m) + (2205)(2)\nvf = 39.60 m/s\nThe final velocity of the pumpkin in this scenario would be 39.60 m/s. If you wanted to increase the velocity, you could have a greater applied force when launching. When you have a great applied force it leads to a greater distance for the pumpkin to travel as well, which is after all what those judges are looking for!\n\n## Monday, October 15, 2012\n\n### The Physics Behind the Red Bull Stratos Project\n\nThe Physics Behind the Red Bull Stratos Project\nby Chelsea Gottschalk\n\nAs mentioned in class and outlined in a NY Times article, Felix Baumgartner recently became the first man to break the sound barrier by gravity alone. Felix was taken to a vertical height of 128,100 ft (24 miles) by a hot air balloon, and then jumped down to the ground, using a special space-like suit and a parachute. Part of this mission, called the Red Bull Stratos project, was to test new spacesuits as well as escape tactics for extreme altitudes. Although the figures are not exact yet, it was estimated that Felix dropped when he was 128,100 ft above the ground. He fell for approximately 4 minutes (240 seconds), reaching a maximum speed of 833.9 mph (372.787 m/s) before deploying a parachute about 1 mile (1609.3 m) above a New Mexico desert.\n\nLet’s take a look at the physics behind Felix’s super stunt.\n\nLet’s assume that Felix has a mass of approximately 75 Kg and his space suit weighs around 70lbs (31.75Kg) (around what some models weigh), therefore: Total mass= 106.75 Kg\n\nAlso, since Felix will be constantly accelerating, the maximum speed will be equal to the point just before he opens his chute\n\nThere are two reference frames that we should assess:\n\n1) The descent from the highest point to right when Felix deploys his parachute (this represents the total time under complete free fall) (37,435.5 m)\n2) The descent from the point Felix opens his parachute to the time he touches the ground (1609.3 m)\n\nLet’s look at the free fall first:\n\nWe know that Wnet=Wc+Wnc, where the Wnc represents the air resistance encountered from the fall\n\nFrom the law of conservation of energy:\n\nKEi + PEi = KEf + PEf – Wnc\n\n1/2mv^2 + mgh = 1/2mv^2 + mgh –Wnc\n\nVi= 0 m/s                hi= 37,435.5 m\nVf= 372.787 m/s     hf = 0 m\n\n0 + (106.75)(9.8)(37,435.5) = (1/2)(106.75)(372.787)^2 – Wnc\n\nWnc = -3.17 x 10^7 J of air resistance\n\nF x d = -3.17 x 10^7 J   d=37,435.5 m\n\nFair= 846.79 = 847 N of air resistance opposing the direction of motion\n\nNow, let’s look at what Felix’s max speed would be if there were no nonconserved forces:\n\nVf ^2 = Vi^2 + 2ay\n\nVf= sqrt (9.8 x 37435.5)\n\nVf= 856.58 = 857 m/s, over 2x as fast!\n\nThe speed of sound is 340.29 m/s…..\n\nNow, let’s look at the second reference frame, from the moment Felix opens his parachute until he reaches the ground, covering a height of 1 mile, or 1609.3 m\n\nWe can use the same equation as before to find the force of the drag produced by the parachute:\n\nKEi + PEi = KEf + PEf – Wnc\n\n1/2mv^2 + mgh = 1/2mv^2 + mgh –Wnc\n\nVi = 372.787 m/s    hi= 1609.3 m\nVf= 0 m/s                hf= 0 m\n\n(106.75)g(1609.3) + (1/2)(106.75)(372.787)^2 = 0 + 0 – Wnc\n\nWnc= -9.1 x 10^6 J\n\nF x d = -9.1 x 10^6 J   d=1609.3 m\n\nFparachute= 5654.63 = 5650 N of force in the direction opposing motion- that is a lot!\n\nLet’s look at the magnitude of the deceleration to see how it compares to the limits of the human body (5g’s before loss of consciousness):\n\nVf ^2 = Vi^2 + 2ay\n\n0 = 372.787^2 + 2(1609.3)a\n\na= -43.1772 = -43.2 m/s^2 / 9.8 = 4.4g’s\nIt looks like Felix was very close to approaching the limit of the human body, but he survived, no doubt due to the careful physical calculations of this awesome stunt!\n\nReferences: http://www.nytimes.com/2012/10/15/us/felix-baumgartner-skydiving.html?_r=0\n\n## Saturday, October 13, 2012\n\n### The Physics of Frogs and Muscles\n\nWith our liberal arts education here at Colgate, I figured exploring the cross over between my Human Physiology class and our Physics class might be a little fun. In Human Physiology lab a little while ago, we performed an experiment on the gastrocnemius muscle of frogs (This more or less translates to the calf muscle in the back of the lower leg for both us and frogs). In our experiment, we prepped the isolated muscle and set it up to perform isotonic muscle twitches with varying weights. By finding the distance the muscle contracts and knowing the weight, we can easily calculate the work done by the muscle to lift the weights. Then, by plotting a graph of work done for a given weight lifted, I was able to obtain an optimal load at which the muscle does the greatest amount of work.\n\nAccounting for the forces in play, we know that the force of gravity (mg) is determined by the varying weights we add to the muscle. Furthermore, we know that a force applied comes into play when the muscle contracts (and does work) to lift the weight. In order to simplify calculations, we ignored other forces besides the force of gravity on the weights and force applied. We ignored the weight of the muscle, and focused solely on the work done by the muscle on the weights. Our equation W = F * d helps us find the work done by the muscle on the weight.\n\nMy raw data for the lab was:\n Weight (g) Work Height Lifted (mm) Height (mv) 10 3.52725 0.352725 4.703 9 3.4803 0.3867 5.156 8 9.7782 1.222275 16.297 7 13.86368 1.980525 26.407 6 22.56345 3.760575 50.141 5 24.04088 4.808175 64.109 4 21.9609 5.490225 73.203 3 18.39974 6.133245 81.7766 2 13.95465 6.977325 93.031 1 7.6137 7.6137 101.516\n\nSince we were stimulating the muscle at varying voltages, and similarly measuring the peak of the contraction in mV, I had to use a conversion factor to obtain the shortening distance (height lifted) for the muscle. This conversion factor was given as 75 mm / V. Also, to keep calculations simple in lab, we initially ignored gravity (as it multiples every mass by 9.8 equally), and considered the force of gravity = m (weight). This means that the force of contraction must overcome that downward force in order to have the weights move upward, and thus do work. To make this all a little more clear, I converted everything in SI units and corrected the force downward (and thus minimum upward) by multiplying gravity in.\n\n Weight (kg) Min. Force Exerted Upwards (N) Distance (m) Work Done (J) 0.01 0.098 3.53E-04 3.46E-05 0.009 0.0882 3.87E-04 3.41E-05 0.008 0.0784 1.22E-03 9.58E-05 0.007 0.0686 1.98E-03 1.36E-04 0.006 0.0588 3.76E-03 2.21E-04 0.005 0.049 4.81E-03 2.36E-04 0.004 0.0392 5.49E-03 2.15E-04 0.003 0.0294 6.13E-03 1.80E-04 0.002 0.0196 6.98E-03 1.37E-04 0.001 0.0098 7.61E-03 7.46E-05\n\nNow, having everything in SI, I plotted the work vs. weight graph to find the optimal load where this muscle does the greatest amount of work using the numbers above, and found that this frog's gastrocnemius muscle's optimal load was 0.005kg, or 5g.\n\nLet's take this a step further and calculate the power of this muscle.\nKnowing power is the rate at which work is done, to find our average power of this muscle we can use the equation:\nP = W / t.\nFor our isotonic muscle twitches, a contraction cycle lasted approx. 2 ms (variations occurred with changing weights, changing stimuli strength, how fatigued the muscle was, etc., but for simplicity's sake, 2 ms is a reasonable assumption).\n\nAt an optimal load of 5g (0.005kg), our muscle did 2.36E-4 Joules of work. Since 2 ms is 0.002 seconds, we simply divide to find that the frog's gastrocnemius muscle had an average power output of 0.118 W (J/s).\n\nThat's nice and all, but since I like to be self-centered now and then, how does that relate to us (and since the course is Human Physiology, how does it relate to humans?) Doing a little bit of research, I found the optimal power outputs of human gastrocnemius muscles at a walk and at a run in this article:\nhttp://www.pnas.org/content/early/2012/01/04/1107972109.full.pdf\n\nGranted they do some very interesting research and measure a variety of things, including changes in length for different parts of the muscle and power among different groupings, I used their data for muscle-tendon units to compare to my frog's gastrocnemius muscle power.\n\nAt a walk, the power output is around 37 W, and while running, our power output increases to around 48W.\n\nWhere frogs have a gastrocnemius muscle composed of fast twitch glycolytic muscle fibers that are useful for jumping quickly and moving only their small bodies, a power output of 0.118W makes sense. On the other hand, human gastrocnemius muscles contain a variety of fast twitch glyocolytic, fast twitch glycolytic oxidative, and slow twitch oxidative fibers. From an evolution standpoint, we need these variety of fibers to enable us to quickly run and jump (explosive movements ~ fast twitch fibers), but also maintain a standing posture throughout the day (long, gradual movements ~ slow twitch fibers). In order to maintain the work throughout the day, and to move our much larger masses, it makes sense that our power outputs are much higher than that of a frog's.\n\nFarris DJ, Sawicki GS. Human medial gastrocnemius force-velocity behavior shifts\nwith locomotion speed and gait. Proc Natl Acad Sci U S A. 2012;109:977-982. Found at\n<http://www.pnas.org/content/early/2012/01/04/1107972109.full.pdf>." ]

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[ "Озвучила стих моей ушедшей подруги Веры. До встречи в следующей жизни, Верунчик! Мы встретимся. Больно?​ Не слишком больно. Просто печаль.​ Осень?​ Должно быть, осень. Плачет рояль.​ Вторит,​ Гнусавя, скрипка из тишины.​ Гаснет​ Полуулыбка нашей весны.​ Мягко\b\n\nAn Introduction to Probability and Statistics", null, "Автор:\nОписание книги\n\nA well-balanced introduction to probability theory and mathematical statistics Featuring updated material, An Introduction to Probability and Statistics, Third Edition remains a solid overview to probability theory and mathematical statistics. Divided intothree parts, the Third Edition begins by presenting the fundamentals and foundationsof probability. The second part addresses statistical inference, and the remainingchapters focus on special topics. An Introduction to Probability and Statistics, Third Edition includes: A new section on regression analysis to include multiple regression, logistic regression, and Poisson regression A reorganized chapter on large sample theory to emphasize the growing role of asymptotic statistics Additional topical coverage on bootstrapping, estimation procedures, and resampling Discussions on invariance, ancillary statistics, conjugate prior distributions, and invariant confidence intervals Over 550 problems and answers to most problems, as well as 350 worked out examples and 200 remarks Numerous figures to further illustrate examples and proofs throughout An Introduction to Probability and Statistics, Third Edition is an ideal reference and resource for scientists and engineers in the fields of statistics, mathematics, physics, industrial management, and engineering. The book is also an excellent text for upper-undergraduate and graduate-level students majoring in probability and statistics.\n\nЦена:13029.03 руб.\nЯзык:   Английский\nПросмотры:   66\n\nСкачать ознакомительный фрагмент\nЧТО КАЧАТЬ и КАК ЧИТАТЬ КУПИТЬ И СКАЧАТЬ ЗА: 13029.03 руб.\nНаш литературный журнал Лучшее место для размещения своих произведений молодыми авторами, поэтами; для реализации своих творческих идей и для того, чтобы ваши произведения стали популярными и читаемыми. Если вы, неизвестный современный поэт или заинтересованный читатель - Вас ждёт наш литературный журнал." ]

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[ "ROS Resources: Documentation | Support | Discussion Forum | Index | Service Status | ros @ Robotics Stack Exchange\n\n# How to transfer numbers in messages?\n\nHi! I have a very simple problem, but I just cannot understand how to solve it. I need to transfer three coordinates between two nodes. One node is talker, which defines the coordinates and send it to other node, which is listener. Listener waits for the coordinates, make some operations with them and send it to navigation stack, which operates the robot in stage. I have read the tutorial about simple publisher and subscriber, but there the string is used and I just do not know, how to change it to float without problems. I tried to use ros services, but they do not create a topic, while working, also I was unable to make it normal working too. It seems to me, that after receiving the message, the program stuck in a callback sub-program and the main part of the program just cannot receive the data. I can be mistaken, as my knowledge in programming is too small. Below is the code of my node that I want to make the listener. Now I enter the coordinates when the node is started, but it is not convenient. I just want to know what I should add (and where) to make this none subscribed to some topic (for example \"coordinates\"), and being able to receive X, Y and Theta coordinates. The part of publisher node also would be useful for me. I would like to understand the principle. Help me please!\n\n\n#include \"ros/ros.h\"\n#include \"move_base_msgs/MoveBaseAction.h\"\n#include \"actionlib/client/simple_action_client.h\"\n#include \"tf/transform_datatypes.h\"\n\ntypedef actionlib::SimpleActionClient<move_base_msgs::movebaseaction> MoveBaseClient;\n\nusing namespace std;\n\nint main(int argc, char** argv) {\n\nif (argc < 2) {\nROS_ERROR(\"You must specify leader robot id.\");\nreturn -1;\n}\n\nif (argc < 3) {\nROS_ERROR(\"You must specify X coordinate.\");\nreturn -1;\n}\n\nif (argc < 4) {\nROS_ERROR(\"You must specify Y coordinate.\");\nreturn -1;\n}\n\n//ROS_INFO(argv);\n\nchar *robot_id = argv;\n\nros::init(argc, argv, \"test_goals\");\nros::NodeHandle nh;\n\ndouble goal_x, goal_y, goal_theta;\nif (!nh.getParam(\"goal_x\", goal_x)){\nchar *x = argv;\ngoal_x = atof(x);}\nif (!nh.getParam(\"goal_y\", goal_y)){\nchar *y = argv;\ngoal_y = atof(y);}\nif (!nh.getParam(\"goal_theta\", goal_theta))\ngoal_theta = 0;\n\n// Create the string \"robot_X/move_base\"\nstring move_base_str = \"/robot_\";\nmove_base_str += robot_id;\nmove_base_str += \"/move_base\";\n\n// create the action client\nMoveBaseClient ac(move_base_str, true);\n\n// Wait for the action server to become available\nROS_INFO(\"Waiting for the move_base action server\");\nac.waitForServer(ros::Duration(5));\n\nROS_INFO(\"Connected to move base server\");\n\n// Send a goal to move_base\nmove_base_msgs::MoveBaseGoal goal;\n\ngoal.target_pose.pose.position.x = goal_x;//\ngoal.target_pose.pose.position.y = goal_y;//\n\n// Convert the Euler angle to quaternion\ndouble radians = goal_theta * (M_PI/180);//\ntf::Quaternion quaternion;\n\ngeometry_msgs::Quaternion qMsg;\ntf::quaternionTFToMsg(quaternion, qMsg);\ngoal.target_pose.pose.orientation = qMsg;\n\nROS_INFO(\"Sending goal to robot no. %s: x = %f, y = %f, theta = 0\", robot_id, goal_x, goal_y);\nac.sendGoal(goal);\n\n// Wait for the action to return\nac.waitForResult();\n\nif (ac.getState() == actionlib::SimpleClientGoalState::SUCCEEDED)\nROS_INFO(\"Robot ...\nedit retag close merge delete\n\nSort by » oldest newest most voted", null, "Hi, I think the easiest way to do it in your case would be to send either Point, or Vector3\nYou can send it using:\nOn the talker part:\n\n ros::Publisher vector_pub = n.advertise<geometry_msgs::Vector3>(\"coordinates\", 1); geometry_msgs::Vector3 vectSend; vectSend.x = coordinates.x; vectSend.y = coordinates.y; vectSend.z = coordinates.theta; vector_pub.publish(vectSend); \n\nOn the listener part:\n\n ros::Subscriber vector_sub = n.subscribe(\"coordinates\", 1, coordinatesCb); void coordinatesCb(const geometry_msgs::Vector3::ConstPtr& msg){ float goal_x, goal_y, goal_z; goal_x = msg->x; goal_y = msg->y; goal_z = msg->z; // do whatever you want with it and send to move_base \n\nmore" ]

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[ "", null, "", null, "# 6K7G\n\nID = 2700\nCountry:\nUnited States of America (USA)\n Brand: Common type USA tube/semicond. Developer: Arcturus Radio Tube Co.; Newark, NJ Tube type: Vacuum Pentode   RF/IF-Stage   Controlling (mu)\nIdentical to 6K7G = VT-86A = CV1941 = W63 = ZA5699 = 10E/558\nSimilar Tubes\nOther shape (e.g. bulb type):\n6K7 ; 6K7GT ; 6K7MG ; PF9\nNormally replaceable-slightly different:\n6K7EG ; 6K7M ; 88M\nOther class quality (otherwise equal):\nW1531\nHeater different:\n12K7G\nOther base:\n78 ; 78E\nFirst Source (s)\n06.Jun.1935 : Electron Tube Registration List\nPredecessor Tubes 78\nSuccessor Tubes 1940   6SK7   6M7G   7A7   6K7GT\n\n Base Octal (Int.Octal, IO) K8A, USA 1935 (Codex=Uf) Top contact with a cap. Was used by Radio/TV-reception etc. Filament Vf 6.3 Volts / If 0.3 Ampere / Indirect / Parallel / series AC/DC Tube prices 5 Tube prices (visible for members only) Information source Essential Characteristics, GE 1973    Taschenbuch zum Röhren-Codex 1948/49", null, "6K7G: De Muiderkring, Bussum Joachim Glüder", null, "", null, "6K7G: Herbert Odermatt (eig. Sammlung) Herbert Odermatt † 26.Nov.05", null, "6K7G: Telefunken Werkstattbuch Wolfgang Bauer", null, "6K7G: RCA Dietrich Grötzer", null, "from Just Qvigstad\n\n Usage in Models 1= 1934?? ; 2= 1935? ; 20= 1935 ; 6= 1936?? ; 12= 1936? ; 235= 1936 ; 35= 1937?? ; 20= 1937? ; 467= 1937 ; 25= 1938?? ; 35= 1938? ; 385= 1938 ; 25= 1939?? ; 43= 1939? ; 256= 1939 ; 38= 1940?? ; 28= 1940? ; 131= 1940 ; 7= 1941?? ; 12= 1941? ; 50= 1941 ; 5= 1942?? ; 11= 1942? ; 18= 1942 ; 4= 1943?? ; 4= 1943? ; 6= 1943 ; 2= 1944? ; 12= 1944 ; 10= 1945?? ; 35= 1945? ; 42= 1945 ; 8= 1946?? ; 8= 1946? ; 53= 1946 ; 12= 1947?? ; 8= 1947? ; 43= 1947 ; 15= 1948?? ; 3= 1948? ; 15= 1948 ; 3= 1949?? ; 2= 1949? ; 9= 1949 ; 11= 1950?? ; 2= 1950? ; 6= 1950 ; 6= 1951?? ; 2= 1952 ; 1= 1956?? ; 1= 1956 ; 1= 1958 ; 1= 2020 ; 1= 9999? ; 6= 9999\n\nQuantity of Models at Radiomuseum.org with this tube (valve, valves, valvola, valvole, válvula, lampe):2199\n\nYou reach this tube or valve page from a search after clicking the \"tubes\" tab or by clicking a tube on a radio model page. You will find thousands of tubes or valves with interesting links. You even can look up radio models with a certain tube line up. [rmxtube-en]" ]

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[ "# Gas PTZ\n\n## Gas Parameters", null, "Base pressure – The pressure at standard conditions.\n\nBase temperature – The temperature at standard conditions.\n\nLine compressibility – the compressibility of the flowing gas.\n\nNote: compressibility is the correction factor for the deviation of the real gas from ideal gas.\n\nuncertainty in line compressibility – the uncertainty in the compressibility value.\n\nBase compressibility – the compressibility of gas at standard conditions.\n\nNote: compressibility is the correction factor for the deviation of the real gas from ideal gas.\n\nuncertainty in base compressibility – the uncertainty in the compressibility value at standard conditions.\n\n### Uncertainty Budget\n\nThe uncertainty budget table shows a break down of the different components that contribute to the overall calculated uncertainty.", null, "The values input into the uncertainty budget are derived from the measured pressure and temperature transmitter specific values relating to its calibration and specification. These values are taken in as the expanded uncertainties and are divided by a coverage factor to gain the standard uncertainty.\n\nThe coverage factor is determined by the probability distribution that best suits that uncertainty component. The standard uncertainty is then multiplied by the sensitivity value then squared. This is done for each component that contributes to the overall uncertainty in Volume Correction Factor (VCF). The Standard Uncertainty in VCF is the square root of the sum of each component variance as shown in the following equation:", null, "### Calculated Uncertainty", null, "The Expanded Uncertainty is the Standard Uncertainty multiplied by the coverage factor (k). The coverage factor is defaulted to k = 2 (equivalent to a confidence level of approximately 95%).\n\nThe Relative Uncertainty is the Expanded Uncertainty divided by VCF.\n\nBack to Uncertainty Modules" ]

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[ "# NMaxValue\n\nNMaxValue[f,x]\n\ngives the maximum value of f with respect to x.\n\nNMaxValue[f,{x,y,}]\n\ngives the maximum value of f with respect to x, y, .\n\nNMaxValue[{f,cons},{x,y,}]\n\ngives the maximum value of f subject to the constraints cons.\n\nNMaxValue[,xreg]\n\nconstrains x to be in the region reg.\n\n# Details and Options", null, "• NMaxValue[] is effectively equivalent to First[NMaximize[]].\n• cons can contain equations, inequalities or logical combinations of these.\n• The constraints cons can be any logical combination of:\n• lhs==rhs equations lhs>rhs or lhs>=rhs inequalities {x,y,…}∈reg region specification\n• NMaxValue[{f,cons},xreg] is effectively equivalent to NMaxValue[{f,consxreg},x].\n• For xreg, the different coordinates can be referred to using Indexed[x,i].\n• NMaxValue always attempts to find a global maximum of f subject to the constraints given.\n• By default, all variables are assumed to be real.\n• xIntegers can be used to specify that a variable can take on only integer values.\n• If f and cons are linear, NMaxValue can always find global maxima, over both real and integer values.\n• Otherwise, NMaxValue may sometimes find only a local maximum.\n• If NMaxValue determines that the constraints cannot be satisfied, it returns .\n• NMaxValue takes the same options as NMaximize.\n\n# Examples\n\nopen allclose all\n\n## Basic Examples(5)\n\nFind the maximum value of a univariate function:\n\nFind the maximum value of a multivariate function:\n\nFind the maximum value of a function subject to constraints:\n\nFind the maximum value of a function over a geometric region:\n\nFind the maximum value of a function over a geometric region:\n\n## Scope(9)\n\nOr constraints can be specified:\n\nUse NMaxValue for linear objective and constraints:\n\nInteger constraints can be imposed:\n\nFind the maximum value of a function over a geometric region:\n\nPlot it:\n\nFind the maximum distance between points in two regions:\n\nFind the maximum", null, "such that the rectangle and ellipse still intersect:\n\nPlot it:\n\nFind the maximum", null, "for which", null, "contains the given three points:\n\nUse", null, "to specify that", null, "is a vector in", null, ":\n\nFind the maximum distance between points in two regions:\n\n## Applications(1)\n\nFind the maximal area among rectangles with a unit perimeter:\n\n## Properties & Relations(3)\n\nNMaximize gives both the value of the maximum and the maximizer point:\n\nNMaxValue gives the maximum:\n\nNArgMax gives the maximizer point:\n\nNMaxValue can solve linear programming problems:\n\nLinearProgramming can be used to solve the same problem given in matrix notation:\n\nUse RegionBounds to compute the bounding box:\n\nUse NMaxValue and NMinValue to compute the same bounds:\n\nIntroduced in 2008\n(7.0)\n|\nUpdated in 2014\n(10.0)" ]

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[ "## Loren on the Art of MATLABTurn ideas into MATLAB\n\nNote\n\nLoren on the Art of MATLAB has been retired and will not be updated.\n\n# Using Symbolic Equations And Symbolic Functions In MATLAB\n\nI am pleased to introduce guest blogger Kai Gehrs. Kai is a developer for the Symbolic Math Toolbox.\n\n### Contents\n\nThis article discusses how symbolic workflows can be improved by using symbolic equations and symbolic functions in MATLAB. Symbolic equations and symbolic functions were introduced in the Symbolic Math Toolbox in Release 2012a.\n\nFor demonstration purposes, let's consider the Bessel differential equation. We will see the advantages of symbolic equations and symbolic functions over string input:\n\n• First, we solve the differential equation using string syntax and point out some limitations of string syntax.\n• Then we compute the same results again using symbolic equations and symbolic functions and discuss how this improves our workflow.\n\n#### Solving Ordinary Differential Equations Using String Input\n\nThe usual way to solve ordinary differential equations (ODEs) using the Symbolic Math Toolbox dsolve command is to set up the equations using string syntax.\n\nHere is a typical example that shows how you can solve a Bessel ODE with two given initial values:\n\nbesselODE = 't^2*D2y+t*Dy+(t^2-n^2)*y';\nf = dsolve(besselODE,'y(1)=1','y(2)=n','t');\npretty(f)\n\n\n(n besselj(n, 1) - besselj(n, 2)) bessely(n, t)\n--------------------------------------------------------- -\nbesselj(n, 1) bessely(n, 2) - besselj(n, 2) bessely(n, 1)\n\n(n bessely(n, 1) - bessely(n, 2)) besselj(n, t)\n---------------------------------------------------------\nbesselj(n, 1) bessely(n, 2) - besselj(n, 2) bessely(n, 1)\n\n\nSuch string-based input for the dsolve function has some limitations. Say you want to assign a special value to n, e.g., n = 1, and solve the equation again using this new value for n. Just defining\n\nn = 1\n\nn =\n1\n\n\nand re-evaluating the command\n\nf = dsolve(besselODE,'y(1)=1','y(2)=n','t');\npretty(f)\n\n\n(n besselj(n, 1) - besselj(n, 2)) bessely(n, t)\n--------------------------------------------------------- -\nbesselj(n, 1) bessely(n, 2) - besselj(n, 2) bessely(n, 1)\n\n(n bessely(n, 1) - bessely(n, 2)) besselj(n, t)\n---------------------------------------------------------\nbesselj(n, 1) bessely(n, 2) - besselj(n, 2) bessely(n, 1)\n\n\ndoes not work, since the value for n shows up in the string defining the differential equation:\n\nbesselODE\n\nbesselODE =\nt^2*D2y+t*Dy+(t^2-n^2)*y\n\n\nWe more or less have to modify the ODE by hand replacing all appearances of n by 1:\n\nf = dsolve('t^2*D2y+t*Dy+(t^2-1^2)*y','y(1)=1','y(2)=1','t');\npretty(f)\n\n\n(besselj(1, 1) - besselj(1, 2)) bessely(1, t)\n--------------------------------------------------------- -\nbesselj(1, 1) bessely(1, 2) - besselj(1, 2) bessely(1, 1)\n\n(bessely(1, 1) - bessely(1, 2)) besselj(1, t)\n---------------------------------------------------------\nbesselj(1, 1) bessely(1, 2) - besselj(1, 2) bessely(1, 1)\n\n\nAnother limitation becomes obvious when we try to check the solution.\n\nIn order to verify that the solution is correct, we need to plug it into the ODE and see if this gives 0. But because the string input does not let us use subs to directly plug the solution into the ODE, we, again, have to do some manual work:\n\nsyms t;\nresult = simplify(t^2*diff(f,2)+t*diff(f)+(t^2-n^2)*f)\n\nresult =\n0\n\n\nFinally, we should also check the initial values. Do f(1) = 1 and f(2) = n really hold?\n\ncheckInitCond1 = subs(f,t,1)\ncheckInitCond2 = subs(f,t,2)\n\ncheckInitCond1 =\n1\ncheckInitCond2 =\n1\n\n\nWe have solved the ODE, and we have checked the correctness of the solutions. But our workflow was not completely smooth. Manual work has been required to overcome these limitations of string syntax:\n\n• When using variables inside a string and afterwards assigning values to these variables, the values do not show up in the string.\n• Verification of solutions and initial conditions is not convenient, since we cannot use the subs command on string representations of ODEs.\n\nIn R2012a, symbolic equations and symbolic functions were introduced in the Symbolic Math Toolbox.\n\nThese new features let you make the worklfow for solving ODEs and testing solutions much more smooth and convenient.\n\n#### Improving Our Workflow Using Symbolic Functions And Symbolic Equations\n\nBefore starting with any new computations, let us clean up the workspace:\n\nclear all;\n\n\nNow the dependent ODE variable is declared as a symfun (symbolic function) y(t):\n\nsyms n y(t);\n\n\nAfter executing the last command, we can see the symfun y as well as the symbolic variables n and t in the MATLAB workspace. Now we define the Bessel ODE by typing\n\nbesselODE = t^2*diff(y,2) + t*diff(y) + (t^2-n^2)*y == 0;\n\n\nNote that we used diff to introduce the symbolic derivatives of y and the == sign to set up a symbolic equation.\n\nNow we can easily solve the original initial value problem by typing\n\nf(t) = dsolve(besselODE, y(1)==1, y(2)==n);\npretty(f(t));\n\n\n(n besselj(n, 1) - besselj(n, 2)) bessely(n, t)\n--------------------------------------------------------- -\nbesselj(n, 1) bessely(n, 2) - besselj(n, 2) bessely(n, 1)\n\n(n bessely(n, 1) - bessely(n, 2)) besselj(n, t)\n---------------------------------------------------------\nbesselj(n, 1) bessely(n, 2) - besselj(n, 2) bessely(n, 1)\n\n\nHere we have stated the initial values making use of symbolic functions and symbolic equations as well.\n\nNote that in the call to dsolve, y(1) and y(2) do not mean indexing, but function evaluation - just like when you write sin(pi) to evaluate sin(x) at x = pi.\n\nBut the main benefits of using symbolic functions and symbolic equations become obvious when we switch to using n = 1, solve the ODE again, and then verify our solution.\n\nWe can now use subs to automatically introduce the new value for n in the definition of besselODE. Afterwards solving the ODE works \"out of the box\":\n\nn = 1;\nbesselODE = subs(besselODE);\nf(t) = dsolve(besselODE,y(1)==1,y(2)==n);\npretty(f(t))\n\n\n(besselj(1, 1) - besselj(1, 2)) bessely(1, t)\n--------------------------------------------------------- -\nbesselj(1, 1) bessely(1, 2) - besselj(1, 2) bessely(1, 1)\n\n(bessely(1, 1) - bessely(1, 2)) besselj(1, t)\n---------------------------------------------------------\nbesselj(1, 1) bessely(1, 2) - besselj(1, 2) bessely(1, 1)\n\n\nNow let us check whether f is a solution of the Bessel ODE. We use subs to plug the solution into the ODE. The result is not simplified. Using simplify, we directly get\n\nresult = subs(besselODE,[y,diff(y),diff(y,2)],[f,diff(f),diff(f,2)]);\ns = simplify(result(t))\n\ns =\nTRUE\n\n\nThis way we have made sure that f is a solution of the Bessel ODE.\n\nFinally, we need to check whether f satisfies the initial values. Since f is a symfun, we can evaluate f at t = 1 and t = 2 simply as f(1) and f(2). We set up the equations corresponding to the initial values and apply simplify to see if they are true:\n\ncheckInitCond1 = simplify(f(1) == 1)\ncheckInitCond2 = simplify(f(2) == 1)\n\ncheckInitCond1 =\nTRUE\ncheckInitCond2 =\nTRUE\n\n\nAs we can see, the results are correct.\n\nEven if you are not very interested in differential equations, keep in mind that symbolic equations and symbolic functions can help you define inputs for other Symbolic Math Toolbox functions, e.g., solve.\n\n#### Summary\n\n• We have seen that the Symbolic Math Toolbox lets us define symbolic equations via the == operator.\n• We have learned that by writing syms y(t) we can define a symbolic function y in the variable t.\n• Combining symbolic equations with symbolic functions lets us set up ODEs in a convenient way.\n• Using symbolic functions lets us write math formulas and equations in a less technical way which looks similar to textbook notation.\n\nFor further details on symbolic functions, see the MATLAB documentation page on symfun.\n\nAlso, take a look at Loren Shure's article ODEs, from Symbolic to Numeric Code. Loren shows you how you might generate ODE solutions in the case where a symbolic closed-form solution can be found.\n\n#### Have You Tried Symbolic Equations And Symbolic Functions?\n\nHave you tried the new Symbolic Math Toolbox features to make your workflows smoother? Let me know here.\n\nPublished with MATLAB® 7.14\n\n|" ]

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[ "Cody\n\n# Problem 423. System of equations\n\nSolution 2947222\n\nSubmitted on 14 Sep 2020\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1   Fail\na=[2 0 4; 2 4 8]; y_correct = [ 2 1 ]; assert(isequal(system_eq(a),y_correct))\n\nA = 2 0 2 4 b = 4 8 v = 2 1\n\nAssertion failed.\n\n2   Fail\na=[1 -1 2 21; 0 2 5 21; 4 0 -3 21]; y_correct = [ 9 -2 5 ]; assert(isequal(system_eq(a),y_correct))\n\nA = 1 -1 2 0 2 5 4 0 -3 b = 21 21 21 v = 9 -2 5\n\nAssertion failed.\n\n3   Fail\na=[ 3 3 0 6; 5 -7 1 -9; 9 3 -2 26]; y_correct = [1 1 -7]; assert(isequal(system_eq(a),y_correct))\n\nA = 3 3 0 5 -7 1 9 3 -2 b = 6 -9 26 v = 1.0000 1.0000 -7.0000\n\nAssertion failed.\n\n### Community Treasure Hunt\n\nFind the treasures in MATLAB Central and discover how the community can help you!\n\nStart Hunting!" ]

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[ "## Tuesday, June 1, 2010\n\n### Python, Econometrics: The Koyck geometrically distributed lag model, updated with Leviatan's IV method.\n\nVersion 0.0.1 jun 2\nVersion 0.0.2 jun 6 added Leviatan's instrumental variable method\n\nThis blog post will undergo massive revisions. Use the code at your own risk.\n\nConsider the model with infinite number of lags:\n$$Y_t = \\alpha + \\sum_{i= 0}^{\\infty} \\beta_i X_{t - 1}$$.\nThis model has an infinite number of unknown coefficients. A simplification\nwas obtained by Koyck who assumed that $$\\beta _i = \\beta_0 \\lambda^i.$$\nare geometrically distributed . An equivalent formulation for this case is\ngiven by $$Y_t =\\alpha (1 -\\lambda) + \\beta_0 X_t + -\\lambda Y_{t-1} + (u_t - \\lambda u_{t-1}) + v_t$$\n\nand we need only to determine at most only three (?!) values: $$\\alpha, \\lambda$$ and \\$\\beta_0.$$The model matrix with a constant column is then$$Y = 1 + X|Y_{t-1}$$. Let c_0, c_1, c_2$$ be the computed parameters for this model.\nThen the original parameters are obtained by the following:$\\begin{array}{l}\\lambda = c_2\\\\ \\beta_0 = c_1\\\\ \\alpha = c_0 / (1 - c_2).\\\\ \\end{array}$\n\nThe question is, the linear model rhs contains a dependent, endogenous, stochastic lagged\nvariable $$Y_{t-1}$$ which violates the assumptions of least squares method.\nThe output of applying the least squares method will result in biased, inconsistent(when the errors are not normally distributed). Most results obtained by blindly applying least squares\nexhibit high multicollinearity and serially correlated errors.\n\nWe present a Python function to generate the Koyck linear model given the\ninput endogenous stochastic Y vector and the independent exogenous vector X.\n\nimport random\nfrom matlib import matzero, matInsertConstCol, matprint\n\ndef koyckZmatrix(Y, X, withconstcol = True):\nn = len(Y)\nif n != len(X):\nraise ValueError, \"in koyckmatrix():Y,X have different lengths!\"\n\nZ = matzero(n-1, 2)\nfor i in range(n-1): # one less due to lag.\nZ[i], Z[i] = X[i+1], Y[i]\nif withconstcol:\nmatInsertConstCol(Z, 0, 1)\nreturn Y[1:], Z\n\nY = [random.random() for i in range(10)]\nX = range(10)\n\nprint \"The input Y, X vectors\"\nfor i in range(len(X)):\nprint i, Y[i], X[i]\n\nY, Z = koyckZmatrix(Y, X)\nprint \"The Input Koyck Y and Z matrix.\"\nmatprint(Z)\n\n\nIn spite of the results being biased and inconsistent, we can use the OLs procedure as a generator of starting values to a better scheme like Maximum Likelihood estimation.\n\nA secon approach to solving the Koyck geometrically lagged model is the method of instrumental variables. We define an instrumental variable which is NOT correlated with the the error term of the Koyck model.\n\nLeviatan developed the following normal equations, obtained by choosing $$X_{t-1}$$ as a proxy variable. This system is shown in page 678 of Gujarati's text on page 678. $$\\begin{array}{ll}\\sum {Y_t} &= n \\hat{\\alpha_0} + \\hat{\\alpha_1} \\sum {X_t} + \\alpha_2 \\sum Y_{t-1} \\\\ \\sum {Y_t X_t} &= \\hat{\\alpha_0}\\sum {X_t} + \\hat{\\alpha_1}\\sum{X_t^2} + \\hat{\\alpha_2} Y_{t-1}X_t \\end{array}$$\n$$\\sum Y_t X_{t-1} &= \\hat{\\alpha_0}\\sum X_{t-1} +\\hat{\\alpha_1} \\sum X_{t}X_{t-1} + \\hat{\\alpha_2}\\sum Y_{t-1} X_{t-1}$$\n\nThe following Python code sets-up the normal equations and solve for the coefficients: $$\\hat{\\alpha_0}, \\hat{\\alpha_1}, \\hat{\\alpha_2}.$$\n\nfrom scipy import linalg\n\ndef leviatan(Y, X):\n\"\"\"\nSolves the Koyck distributed lag system by instrumental variable technique.\n\nReference: pp.678-679, Gujarati, 4e, \"Basic Econometrics\"\n\n\"\"\"\n#RHS vector.\nb=[0.0] * 3\nb = sum(Y[1:])\nb = sum([y * x for y, x in zip(Y[1:],X[1:])])\nb = sum([y * x for y, x in zip(Y[1:], X[:-1])])\n\n#Least squares matrix.\nA = matzero(3,3)\nA = len(Y)-1\nA = A = sum(X[:-1])\nA = sum(Y[1:])\n\nA = sum(X[:-1])\nA = sum([x*x for x in X[1:]])\nA = sum([y* x for y, x in zip(Y[:-1], X[1:])])\n\nA = sum ([x for x in X[:-1]])\nA = sum ([x * xtm1 for x , xtm1 in zip(X[1:], X[:-1])])\nA = sum ([ ytm1 * xtm1 for ytm1, xtm1 in zip(Y[:-1], X[:-1])])\n\n#Call linear solver\nC = linalg.solve (A, b)\nreturn C\n\n\nThe code, unfortunately, has not been checked on an example. We will go back to this later and we add that the estimators returned may be consistent but still high in multicollinearity and low in aefficiency.\n\nWe used the linear solver in scipy. One can also use the code for lm.py which we described in Canned linear solver\n\nWe will present codes for the Maximum Likelihood method later...\n\nDuh, checking the latex...\nTBC" ]

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[ "", null, "# Metric Decompositions of Path-Separable Graphs\n\nA prominent tool in many problems involving metric spaces is a notion of randomized low‐diameter decomposition. Loosely speaking, $\\mathit{\\beta }$", null, "‐decomposition refers to a probability distribution over partitions of the metric into sets of low diameter, such that nearby points (parameterized by $\\mathit{\\beta }>0$", null, ") are likely to be ‘clustered’ together. Applying this notion to the shortest‐path metric in edge‐weighted graphs, it is known that n‐vertex graphs admit an $O\\left(lnn\\right)$", null, "‐padded decomposition (Bartal, 37th annual symposium on foundations of computer science. IEEE, pp 184–193, 1996), and that excluded‐minor graphs admit O(1)‐padded decomposition (Klein et al., 25th annual ACM symposium on theory of computing, pp 682–690, 1993; Fakcharoenphol and Talwar, J Comput Syst Sci 69(3), 485–497, 2004; Abraham et al., Proceedings of the 46th annual ACM symposium on theory of computing. STOC ‘14, pp 79–88. ACM, New York, NY, USA, 2014). We design decompositions to the family of p‐path‐separable graphs, which was defined by Abraham and Gavoille (Proceedings of the twenty‐fifth annual acm symposium on principles of distributed computing, PODC ‘06, pp 188–197, 2006) and refers to graphs that admit vertex‐separators consisting of at most p shortest paths in the graph. Our main result is that every p‐path‐separable n‐vertex graph admits an $O\\left(ln\\left(plnn\\right)\\right)$", null, "‐decomposition, which refines the $O\\left(lnn\\right)$", null, "bound for general graphs, and provides new bounds for families like bounded‐treewidth graphs. Technically, our clustering process differs from previous ones by working in (the shortest‐path metric of) carefully chosen subgraphs." ]

[ null, "https://iaorifors.com/resources/images/comingsoon.png", null, "https://iaorifors.com/images/formulas/iaor20174405e1.gif", null, "https://iaorifors.com/images/formulas/iaor20174405e2.gif", null, "https://iaorifors.com/images/formulas/iaor20174405e3.gif", null, "https://iaorifors.com/images/formulas/iaor20174405e4.gif", null, "https://iaorifors.com/images/formulas/iaor20174405e5.gif", null ]

[ "# Definition of septums\n\n## \"septums\" in the noun sense\n\n### 1. septum\n\nanatomy) a dividing partition between two tissues or cavities\n\n### 2. septum\n\na partition or wall especially in an ovary\n\nSource: WordNet® (An amazing lexical database of English)\n\nWordNet®. Princeton University. 2010.\n\n# septums in Scrabble®\n\nThe word septums is playable in Scrabble®, no blanks required.\n\nSEPTUMS\n(94 = 44 + 50)\n\nseptums\n\nSEPTUMS\n(94 = 44 + 50)\nSEPTUMS\n(92 = 42 + 50)\nSEPTUMS\n(92 = 42 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(84 = 34 + 50)\nSEPTUMS\n(84 = 34 + 50)\nSEPTUMS\n(83 = 33 + 50)\nSEPTUMS\n(80 = 30 + 50)\nSEPTUMS\n(78 = 28 + 50)\nSEPTUMS\n(78 = 28 + 50)\nSEPTUMS\n(76 = 26 + 50)\nSEPTUMS\n(76 = 26 + 50)\nSEPTUMS\n(76 = 26 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(69 = 19 + 50)\nSEPTUMS\n(69 = 19 + 50)\nSEPTUMS\n(66 = 16 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(64 = 14 + 50)\nSEPTUMS\n(63 = 13 + 50)\nSEPTUMS\n(63 = 13 + 50)\nSEPTUMS\n(63 = 13 + 50)\nSEPTUMS\n(62 = 12 + 50)\n\nSEPTUMS\n(94 = 44 + 50)\nSEPTUMS\n(92 = 42 + 50)\nSEPTUMS\n(92 = 42 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(86 = 36 + 50)\nSEPTUMS\n(84 = 34 + 50)\nSEPTUMS\n(84 = 34 + 50)\nSEPTUMS\n(83 = 33 + 50)\nSEPTUMS\n(80 = 30 + 50)\nSEPTUMS\n(78 = 28 + 50)\nSEPTUMS\n(78 = 28 + 50)\nSEPTUMS\n(76 = 26 + 50)\nSEPTUMS\n(76 = 26 + 50)\nSEPTUMS\n(76 = 26 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(74 = 24 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(72 = 22 + 50)\nSEPTUMS\n(69 = 19 + 50)\nSEPTUMS\n(69 = 19 + 50)\nSEPTUMS\n(66 = 16 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(65 = 15 + 50)\nSEPTUMS\n(64 = 14 + 50)\nSEPTUMS\n(63 = 13 + 50)\nSEPTUMS\n(63 = 13 + 50)\nSEPTUMS\n(63 = 13 + 50)\nSEPTUMS\n(62 = 12 + 50)\nSPUMES\n(39)\nSTUMPS\n(39)\nSEPTUM\n(39)\nSTUMPS\n(39)\nSEPTUM\n(39)\nSPUMES\n(39)\nSTUMP\n(36)\nTEMPS\n(36)\nSUMPS\n(36)\nSTUMP\n(36)\nSPUME\n(36)\nSPUME\n(36)\nSPUMES\n(33)\nSTUPES\n(33)\nSUMP\n(33)\nSTUMPS\n(33)\nSETUPS\n(33)\nSPUMES\n(33)\nSEPTUM\n(33)\nSEPTUM\n(33)\nSPUMES\n(33)\nSEPTUM\n(33)\nUPSETS\n(33)\nSEPTUM\n(33)\nSTUMPS\n(33)\nSPUMES\n(33)\nSTUMPS\n(33)\nSTUMPS\n(33)\nSPUMES\n(32)\nSEPTUM\n(32)\nSTUMPS\n(32)\nSUMPS\n(30)\nSTUMP\n(30)\nSTUMP\n(30)\nSEPTUM\n(30)\nUPSET\n(30)\nTEMPS\n(30)\nSUMPS\n(30)\nSTEPS\n(30)\nSETUP\n(30)\nPESTS\n(30)\nSMUTS\n(30)\nTEMPS\n(30)\nSTEMS\n(30)\nTEMPS\n(30)\nMUTES\n(30)\nSPUME\n(30)\nMUSES\n(30)\nSPUME\n(30)\nSPUMES\n(30)\nSPUMES\n(30)\nMUSTS\n(30)\nSEPTUM\n(30)\nSTUMP\n(30)\nSTUMPS\n(30)\nSUMPS\n(30)\nSTUMPS\n(30)\nSTUPE\n(30)\nUPSETS\n(28)\nSETUPS\n(28)\nSTEP\n(27)\nSETUPS\n(27)\nSTUPES\n(27)\nSETUPS\n(27)\nPETS\n(27)\nSETUPS\n(27)\nSETUPS\n(27)\nSETUPS\n(27)\nTEMPS\n(27)\nUPSETS\n(27)\nSUMPS\n(27)\nUPSETS\n(27)\nUPSETS\n(27)\nTEMPS\n(27)\nUMPS\n(27)\nSPUME\n(27)\nSTUPES\n(27)\nMUSE\n(27)\nPEST\n(27)\nTEMPS\n(27)\nSPUME\n(27)\nUPSETS\n(27)\nSPUME\n(27)\nUMPS\n(27)\nSTUPES\n(27)\nMUTE\n(27)\nSTEM\n(27)\nSTUMP\n(27)\nMESS\n(27)\nSUMP\n(27)\nMUST\n(27)\nPUTS\n(27)\nSUMPS\n(27)\nSTUMP\n(27)\nSTUPES\n(27)\nSTUMP\n(27)\nSUMPS\n(27)\nUPSETS\n(27)\nSTUPES\n(27)\nPESTS\n(26)\nMUTES\n(26)\nMUSES\n(26)\nSTUMPS\n(26)\nSEPTUM\n(26)\nSEPTUM\n(26)\nSETUP\n(26)\nSPUMES\n(26)\nMUSTS\n(26)\nMUTES\n(24)\nSTEMS\n(24)\nMUSES\n(24)\nMUSTS\n(24)\nSPUMES\n(24)\nMUSES\n(24)\nSTUPES\n(24)\nSPUMES\n(24)\nMUTES\n(24)\nSTUPE\n(24)\nUMPS\n(24)\nSTUPES\n(24)\nSPUMES\n(24)\nSTUPE\n(24)\nMUTES\n(24)\nMUSES\n(24)\nUMPS\n(24)\nUMPS\n(24)\nMUSTS\n(24)\nSTEMS\n(24)\nMUSTS\n(24)\nSTUPE\n(24)\nSMUTS\n(24)\nSUMP\n(24)\nSTUMP\n(24)\nSETUPS\n(24)\nSETUPS\n(24)\nSETUP\n(24)\nSTEPS\n(24)\nSEPTUM\n(24)\nSEPTUM\n(24)\nSEPTUM\n(24)\nSUMP\n(24)\nUPSET\n(24)\nSETUP\n(24)\nSETUP\n(24)\nSTUMP\n(24)\nUPSET\n(24)\nSUMP\n(24)\nSTEPS\n(24)\nSTEPS\n(24)\nSUMP\n(24)\nSTUMPS\n(24)\nSTUMPS\n(24)\nSMUTS\n(24)\nSMUTS\n(24)\nSTEMS\n(24)\nUPSET\n(24)\nUMPS\n(24)\nUPSETS\n(24)\n\n# septums in Words With Friends™\n\nThe word septums is playable in Words With Friends™, no blanks required.\n\nSEPTUMS\n(107 = 72 + 35)\nSEPTUMS\n(107 = 72 + 35)\n\nseptums\n\nSEPTUMS\n(107 = 72 + 35)\nSEPTUMS\n(107 = 72 + 35)\nSEPTUMS\n(101 = 66 + 35)\nSEPTUMS\n(95 = 60 + 35)\nSEPTUMS\n(91 = 56 + 35)\nSEPTUMS\n(91 = 56 + 35)\nSEPTUMS\n(91 = 56 + 35)\nSEPTUMS\n(89 = 54 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(79 = 44 + 35)\nSEPTUMS\n(79 = 44 + 35)\nSEPTUMS\n(71 = 36 + 35)\nSEPTUMS\n(71 = 36 + 35)\nSEPTUMS\n(71 = 36 + 35)\nSEPTUMS\n(67 = 32 + 35)\nSEPTUMS\n(67 = 32 + 35)\nSEPTUMS\n(67 = 32 + 35)\nSEPTUMS\n(67 = 32 + 35)\nSEPTUMS\n(65 = 30 + 35)\nSEPTUMS\n(65 = 30 + 35)\nSEPTUMS\n(65 = 30 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(61 = 26 + 35)\nSEPTUMS\n(59 = 24 + 35)\nSEPTUMS\n(59 = 24 + 35)\nSEPTUMS\n(57 = 22 + 35)\nSEPTUMS\n(55 = 20 + 35)\nSEPTUMS\n(55 = 20 + 35)\nSEPTUMS\n(55 = 20 + 35)\nSEPTUMS\n(55 = 20 + 35)\nSEPTUMS\n(54 = 19 + 35)\nSEPTUMS\n(54 = 19 + 35)\nSEPTUMS\n(54 = 19 + 35)\nSEPTUMS\n(53 = 18 + 35)\nSEPTUMS\n(53 = 18 + 35)\nSEPTUMS\n(53 = 18 + 35)\nSEPTUMS\n(53 = 18 + 35)\nSEPTUMS\n(52 = 17 + 35)\nSEPTUMS\n(52 = 17 + 35)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(50 = 15 + 35)\nSEPTUMS\n(50 = 15 + 35)\nSEPTUMS\n(49 = 14 + 35)\n\nSEPTUMS\n(107 = 72 + 35)\nSEPTUMS\n(107 = 72 + 35)\nSEPTUMS\n(101 = 66 + 35)\nSEPTUMS\n(95 = 60 + 35)\nSEPTUMS\n(91 = 56 + 35)\nSEPTUMS\n(91 = 56 + 35)\nSEPTUMS\n(91 = 56 + 35)\nSEPTUMS\n(89 = 54 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(83 = 48 + 35)\nSEPTUMS\n(79 = 44 + 35)\nSEPTUMS\n(79 = 44 + 35)\nSEPTUMS\n(71 = 36 + 35)\nSEPTUMS\n(71 = 36 + 35)\nSEPTUMS\n(71 = 36 + 35)\nSEPTUM\n(69)\nSTUMPS\n(69)\nSEPTUM\n(69)\nSPUMES\n(69)\nSEPTUMS\n(67 = 32 + 35)\nSEPTUMS\n(67 = 32 + 35)\nSEPTUMS\n(67 = 32 + 35)\nSEPTUMS\n(67 = 32 + 35)\nSEPTUMS\n(65 = 30 + 35)\nSEPTUMS\n(65 = 30 + 35)\nSEPTUMS\n(65 = 30 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUM\n(63)\nSTUMPS\n(63)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSPUMES\n(63)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUMS\n(63 = 28 + 35)\nSTUMPS\n(63)\nSPUMES\n(63)\nSEPTUMS\n(63 = 28 + 35)\nSEPTUM\n(63)\nSEPTUMS\n(61 = 26 + 35)\nSPUME\n(60)\nSTUPES\n(60)\nSPUME\n(60)\nSTUMP\n(60)\nSTUMP\n(60)\nSUMPS\n(60)\nSEPTUMS\n(59 = 24 + 35)\nSEPTUMS\n(59 = 24 + 35)\nSEPTUMS\n(57 = 22 + 35)\nSUMP\n(57)\nTEMPS\n(57)\nSPUMES\n(57)\nSTUMPS\n(57)\nSEPTUMS\n(55 = 20 + 35)\nSEPTUMS\n(55 = 20 + 35)\nSEPTUMS\n(55 = 20 + 35)\nSEPTUMS\n(55 = 20 + 35)\nUPSETS\n(54)\nSTUPES\n(54)\nSEPTUMS\n(54 = 19 + 35)\nSETUPS\n(54)\nSEPTUMS\n(54 = 19 + 35)\nSEPTUMS\n(54 = 19 + 35)\nSEPTUMS\n(53 = 18 + 35)\nSEPTUMS\n(53 = 18 + 35)\nSEPTUMS\n(53 = 18 + 35)\nSEPTUMS\n(53 = 18 + 35)\nSTUMPS\n(52)\nSEPTUM\n(52)\nSPUMES\n(52)\nSTUMPS\n(52)\nSEPTUMS\n(52 = 17 + 35)\nSPUMES\n(52)\nSEPTUMS\n(52 = 17 + 35)\nSEPTUM\n(52)\nSETUP\n(51)\nSEPTUM\n(51)\nSTUMPS\n(51)\nSMUTS\n(51)\nMUSTS\n(51)\nMUTES\n(51)\nMUSES\n(51)\nSEPTUMS\n(51 = 16 + 35)\nSTUPE\n(51)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(51 = 16 + 35)\nSPUMES\n(51)\nSEPTUMS\n(51 = 16 + 35)\nSEPTUMS\n(51 = 16 + 35)\nUPSET\n(51)\nSEPTUMS\n(50 = 15 + 35)\nSEPTUMS\n(50 = 15 + 35)\nSEPTUMS\n(49 = 14 + 35)\nMUST\n(48)\nSETUPS\n(48)\nUPSETS\n(48)\nPUTS\n(48)\nSTEPS\n(48)\nSTUMP\n(48)\nMUSE\n(48)\nPESTS\n(48)\nSTEMS\n(48)\nSUMPS\n(48)\nSPUME\n(48)\nSTUPES\n(48)\nSUMPS\n(48)\nMUTE\n(48)\nSTEP\n(45)\nPETS\n(45)\nSEPTUM\n(45)\nSEPTUM\n(45)\nSTUMPS\n(45)\nSTUMPS\n(45)\nPEST\n(45)\nSTUMPS\n(45)\nSEPTUM\n(45)\nSTEM\n(45)\nSPUMES\n(45)\nSPUMES\n(45)\nMESS\n(45)\nSPUMES\n(45)\nUMPS\n(45)\nTEMPS\n(44)\nSEPTUM\n(42)\nSETUPS\n(42)\nUPSETS\n(42)\nSUMPS\n(42)\nSPUMES\n(42)\nSPUME\n(42)\nSETUPS\n(42)\nSPUME\n(42)\nSTUMP\n(42)\nUPSETS\n(42)\nSUMPS\n(42)\nSTUPES\n(42)\nSTUMPS\n(42)\nSTUMP\n(42)\nSETUPS\n(40)\nSETUPS\n(40)\nUPSETS\n(40)\nSTUPES\n(40)\nUPSETS\n(40)\nSTUPES\n(40)\nSTUMP\n(40)\nSTUMPS\n(39)\nUPSET\n(39)\nTEMPS\n(39)\nSEPTUM\n(39)\nSPUMES\n(39)\nTEMPS\n(39)\nSPUMES\n(39)\nSETUP\n(39)\nTEMPS\n(39)\nMUSES\n(39)\nMUSTS\n(39)\nSUMP\n(39)\nSEPTUM\n(39)\nSTUMPS\n(39)\nUMPS\n(39)\nMUTES\n(39)\nSTUPE\n(36)\nSETUPS\n(36)\nSUMPS\n(36)\nSTUMP\n(36)\nMUSTS\n(36)\nSETUP\n(36)\nSUMPS\n(36)\nSETUPS\n(36)\nSETUPS\n(36)\nMUSES\n(36)\nSPUME\n(36)\nSTUMP\n(36)\nSUMPS\n(36)\nSETUPS\n(36)\nSPUME\n(36)\nSTUMP\n(36)\nMUTES\n(36)\nSMUTS\n(36)\nSETUPS\n(36)\nUPSETS\n(36)\nSTUPES\n(36)\nUPSETS\n(36)\nSTUPES\n(36)\nUPSETS\n(36)\nSTUPES\n(36)\nUPSET\n(36)\nUPSETS\n(36)\nUPSETS\n(36)\nSTUPES\n(36)\nSPUME\n(36)\nMUSES\n(34)\nSEPTUM\n(34)\nSTUMPS\n(34)\nSTUMPS\n(34)\nMUSTS\n(34)\nSEPTUM\n(34)\n\n# Word Growth involving septums\n\nseptum\n\ntums\n\n## Longer words containing septums\n\n(No longer words found)" ]

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[ "hep-ph/9708134 EFI-97-35 August 1997 Fermilab-Pub-97/295-T UCSD/PTH-97-21\n\nChiral Gauge Theories from D-Branes\n\nJoseph Lykken, Erich Poppitz, and Sandip P. Trivedi\n\nFermi National Accelerator Laboratory\n\nP.O.Box 500\n\nBatavia, IL 60510, USA\n\nEnrico Fermi Institute\n\nUniversity of Chicago\n\n5640 S. Ellis Avenue\n\nChicago, IL 60637, USA\n\nUniversity of California at San Diego\n\n9500 Gilman Drive\n\nLa Jolla, CA 92093, USA\n\nAbstract\n\nWe construct brane configurations leading to chiral four dimensional supersymmetric gauge theories. The brane realizations consist of intersecting Neveu-Schwarz five-branes and Dirichlet four-branes in non-flat spacetime backgrounds. We discuss in some detail the construction in a orbifold background. The infrared theory on the four-brane worldvolume is a four dimensional gauge theory with chiral matter representations. We discuss various consistency checks and show that the spectral curves describing the Coulomb phase of the theory can be obtained once the orbifold brane construction is embedded in M-theory. We also discuss the addition of extra vectorlike matter and other interesting generalizations.\n\n1. Introduction.\n\nRecently, the study of Dirichlet branes has led to important insights into the behavior of supersymmetric gauge theories. One approach, which has proved especially powerful, is to consider configurations consisting of intersecting Neveu-Schwarz 5-branes and Dirichlet-branes -. It was shown by Witten, , that such configurations often correspond to a single -brane in theory. A simple scaling argument shows that the quantum behavior of the resulting gauge theory can then be understood as a classical effect in theory. So far, in this approach, the background spacetime before adding branes has been taken to be flat (for another important approach which considers branes in Calabi-Yau backgrounds see and references therein), and the resulting gauge theories have been non-chiral (see, however, refs. , ). The main purpose of this paper is to note that brane configurations in non-trivial backgrounds can often lead to chiral gauge theories. We illustrate this by considering brane configurations consisting of NS 5-branes and intersecting D4-branes in a simple class of orbifold backgrounds. As in the flat space case, the brane construction allows us to deduce various features about the non-perturbative behavior of these theories.\n\nThis paper is organized as follows. In Section 2, we describe the orbifold background and brane configuration consisting of Dirichlet -branes placed at the orbifold point and stretched between two Neveu-Schwarz -branes. The low-energy dynamics is shown to be described by a 3+1 dimensional theory with gauge group and chiral matter content. In fact, the gauge theory turns out to be closely related (apart from some anomalous factors) to the theories studied in , . In Section 3, we study the classical moduli space of this gauge theory and show that it corresponds to the set of allowed motions for the brane configuration; this provides additional evidence that we have identified the correct gauge theory. In Section 4, we turn to the quantum theory and show how by considering the configuration in theory one can deduce various non-perturbative features of the low-energy dynamics, pertaining to the Seiberg-Witten spectral curves. Finally, some generalizations of the basic brane configuration are discussed in Section 5.\n\nThis paper is intended to be a first step in a more complete analysis. Two further generalizations are obvious and will be considered in a subsequent paper. One is to consider orientifold backgrounds. The resulting chiral theories are in many ways more interesting. Another is to blow up the orbifold and consider the brane configuration in the corresponding ALE space. The resulting smooth background allows for a more controlled analysis in theory. The methods outlined in this paper give rise to theories which are, in a sense, closely related to theories. As will become clear below, their matter content can be thought of as arising from adjoint fields after a suitable truncation. These methods might consequently have limited use in the study of chiral theories with spinor matter.\n\n2. Brane Configuration and Matter Content.\n\n2.1 The orbifold and brane configuration.\n\nIn this paper we will consider orbifolds. We choose coordinates so that the involved in the orbifold corresponds to the and directions. The Type IIA brane configuration we consider involves two NS -branes and several Dirichlet -branes, as shown in Fig. 1. The NS branes stretch along , are placed at the orbifold point, , and have definite positions in . We take them to be separated by a finite distance in the direction and to be coincident in the direction. The D4-branes are taken to lie along , and directions and end on the two NS branes.\n\n2.2 The gauge group and matter content\n\nAs is well known, the low-energy dynamics of this configuration is described by a 3+1 dimensional field theory, which lives in the intersection region of the D branes and NS branes. We will show below that -branes placed at the origin of the orbifold give rise to an gauge theory. The matter content consists of chiral superfields which transform under the gauge groups as:\n\n U(N)1U(N)2U(N)3⋯U(N)MQ1\\raisebox−0.5pt\\rule{0.4pt}{6.5pt}\\rule{6.5pt}{0.4pt}% \\rule[6.5pt]{6.5pt}{0.4pt}¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\\raisebox−0.5pt\\rule{0.4pt}{6.5pt}\\rule{6.5pt}{0.4pt}\\rule[% 6.5pt]{6.5pt}{0.4pt}1⋯1Q21\\raisebox−0.5pt\\rule{0.4pt}{6.5pt}\\rule{6.5pt}{0.4pt}\\rule[6.% 5pt]{6.5pt}{0.4pt}¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\\raisebox−0.5pt\\rule{0.4pt}{6.5pt}\\rule{6.5pt}{0.4pt}\\rule[6.5pt]{6.5% pt}{0.4pt}⋯1⋮⋮⋮⋮⋮⋮QM¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯\\raisebox−0.5pt\\rule{0.4pt}{6.5pt}\\rule{6.5pt}{0.4pt}% \\rule[6.5pt]{6.5pt}{0.4pt}11⋯\\raisebox−0.5pt\\rule{0.4pt}{6.5pt}\\rule{6.5pt}{0.4pt}\\rule[6.% 5pt]{6.5pt}{0.4pt}\n\nNote that the matter content is chiral222 One overall factor above is “frozen out” while the remaining s are anomalous; we will have more to say on this below..\n\nWe now turn to justifying this claim for the gauge group and matter content. First consider the number of supersymmetries. In the absence of the orbifold this brane configuration preserves 8 supercharges or supersymmetry in 3+1 dimensions: the IIA theory has supercharges, but the presence of 4-branes and NS branes reduces that by a factor of . In the orbifold we only keep gravitino states for which the vertex operators are invariant under a rotation by . This further reduces the supersymmetry by half leading to supercharges or in 3+1 dimensions.\n\nTo arrive at the gauge group and matter content it is useful to consider the final configuration built up in two stages. Let us first look at a configuration without the NS branes where the 4-branes are infinite along and are placed at the orbifold point. It is well known for compact orbifolds that tadpoles must cancel in the one-loop vacuum amplitude, and that this constraint is often powerful enough to determine the gauge group and matter content . In our case, the one loop amplitude only receives a contribution from the cylinder diagram and is easy to work out. Since the on which the orbifold group acts is noncompact, we do not expect any constraint on the allowed total number of 4-branes: the corresponding RR flux can always escape to infinity. This is borne out by an explicit calculation. However, there are non-trivial constraints which arise from the tadpole cancellation for twisted RR fields. Let the orbifold group act on and as:\n\n (v, w)→(α v, α−1 w) ,  α≡e2πiM  , (2.1)\n\nand the corresponding action of the orbifold group on the Chan Paton factors be represented by a matrix :\n\n λ→γα λ γ−1α . (2.2)\n\nThe -branes are sources of twisted RR scalars that can only propagate in one of the directions transverse to the 4-branes (). As argued in , a one-volume is insufficient to allow the Ramond-Ramond flux to escape to infinity, and the tadpole cancellation condition must be satisfied even for infinite volume. The constraints from the twisted RR tadpoles are then given by:\n\n tr γKα = 0,  K=1,…,M−1 . (2.3)\n\nNote that must furnish a representation of the orbifold group and thus . This together with eq. (2.3) allows us to solve for . We find, first, that the number of -branes at the orbifold point must be a multiple of ; we refer to this number hereafter as . Second, we find that the matrix , in a suitable basis, is given by:\n\n γα = diag{1×1N,α×1N,…,αM−1×1N} , (2.4)\n\nwith being the unit matrix. The gauge and matter content can now be worked out as well. The corresponding gauge group on the -brane worldvolume theory turns out to be . Fluctuations in the direction which survive the orbifold projection contribute one adjoint field for each factor. Together with the gauge bosons these form an vector multiplet in 4+1 dimensions. Finally, from the directions we get hypermultiplets transforming under the gauge groups as described in eq. (S0.Ex1) (we note that the same orbifold has been considered in , ).\n\nNow finally we can add the two NS branes and sandwich the four-branes between them as in Fig. 1. What is the resulting 3+1 dimensional theory? It is useful for this purpose to describe the above matter content in the language of 3+1 dimensions. The component of the gauge field, , can be paired with the adjoint fields coming from the direction to give a chiral superfield. Each hypermultiplet will transform as two chiral multiplets in dim. language, one of the two chiral multiplets coming from fluctuations in the directions, and the other from the directions. One expects the boundary conditions coming from the ends of the 4 brane, where it terminates on the 5-brane, to freeze some of these degrees of freedom. Based on the analysis in the absence of the orbifold one expects the gauge field to survive and the chiral mulitiplet coming from the , fluctuations to be frozen. Similarly, the matter coming from the fluctuations in the directions should survive whereas that from the directions will be frozen out. This finally gives rise to the theory with the matter content described in eq. (S0.Ex1). We note again that each field in eq. (S0.Ex1) represents a chiral multiplet so that the theory is chiral.\n\nAbove, we first considered the -branes without NS branes in the orbifold background and then introduced the NS branes. It is also illuminating to consider things in the opposite order. Accordingly, let us first consider a configuration of -branes stretched between the two NS branes in the absence of the orbifold. The resulting field theory is well known to be an theory, with gauge group. The adjoint scalar field corresponds to fluctuations of the branes along the directions. It is natural to expect that the orbifold should correspond to implementing a projection in this theory. In fact, the gauge theory possesses a global symmetry under which (in language) the gauge field and its fermionic partner transform as , and the adjoint and its fermionic partner as . In general this symmetry is anomalous, however it has a non-anomalous discrete subgroup. This discrete subgroup in turn has a subgroup. In addition, the gauge symmetry has a discrete subgroup under which a fundamental representation is multiplied by , with being the unit matrix. In the field theory it is natural to identify the orbifold group with the product of these two symmetries. On doing so and retaining states invariant under this product discrete symmetry one gets precisely the group and matter content mentioned above.\n\n3. Brane Motion and the Classical Moduli Space.\n\nIn this section we compare the set of allowed motions of the brane configuration to the classical moduli space of the gauge theory described above. This will serve two purposes. First, agreement between the two will give additional evidence that we have identified the correct gauge theory. Second, in the process we will understand better the role of the various s in this theory—an issue which we have so far not fully addressed.\n\nIt will be convenient in the following discussion to organize the s in the following basis. We will choose the first to be the sum of the factors, and the other s to be orthogonal to the first. It is easy to see from eq. (S0.Ex1) that none of the matter fields are charged under the first . In fact one can deduce that this factor is frozen, i.e. its coupling vanishes. There are two arguments in support of this. First, for the case of a flat space time background, it was argued in , that in the theory this overall must be frozen. We saw above that for the orbifold background the resulting field theory could be understood as a further truncation of the theory; we thus expect the to continue to be frozen in it. Second, we will see below that when we interpret this configuration in theory, the genus of the two dimensional surface spanned by the 5-brane worldvolume will be consistent with the absence of the .\n\nTurning our attention to the remaining s we notice that they are all anomalous333The orbifold is an exception: in this case the theory is not chiral.. These s are analogous to anomalous factors which often arise in string compactifications . In the context of D-branes anomalous s were discussed in where they were shown to play an important role in governing the low-energy dynamics. We will discuss these s in some detail below. Here we summarize their essential features which are important in the present discussion of the classical moduli space. The important point is that these anomalous s are broken. The low-energy 3+1 dimensional theory contains axion fields which arise from twisted RR fields, and the anomalies are cancelled by shifting these axions appropriately . In fact the axions can be regarded as the longitudinal components of the heavy gauge bosons. The only feature that is really important in the present discussion is that each will give a D-term contribution to the full potential energy, which is important in determining the moduli space of the theory (notice also that the charges are all traceless, hence no Fayet-Iliopoulos term is generated by loop effects).\n\nWe are now ready to study the motion of the 4-branes. We begin with a orbifold with branes located at the orbifold point. The corresponding gauge group is . The -branes can only move along the directions, since they end on NS branes which only extend along these directions. Each 4-brane has images under the symmetry, so counting images, we can move sets of branes away from the orbifold point. Moving branes away breaks . If all the -branes are moved away from the orbifold point we are left with a gauge symmetry. Since the motion of each set of branes is described by one complex number, the moduli space is dimensional. Finally, we also note that if physical branes come together away from the orbifold point we get an enhanced gauge symmetry.\n\nNow consider the flat directions in the gauge theory. These are in one-to-one correspondence with gauge invariant chiral superfields made out of the elementary matter fields in eq. (S0.Ex1). Ignoring the anomalous s for the moment, these moduli are of two kinds. One class is best described in terms of the operator:\n\n Σij=(Q1⋅ Q2⋯ QM)ij, (3.1)\n\nas:\n\n ϕk=tr(Σ)k, (3.2)\n\nfor . The second class of “baryonic” directions is given by:\n\n bα=(Qα)N, (3.3)\n\nwith . Altogether, we see that there are flat directions; these are more than the number of brane degrees of freedom found above. The discrepancy is corrected when we account for the -term potential generated by the anomalous s. We saw above that there are of these, thus their terms get rid of moduli giving us, finally, a dimensional moduli space in agreement with what we found for the motion of branes. An analysis of the vacuum expectation values also shows that in the moduli space, generically, a is left unbroken. Finally, one finds subspaces of the moduli space which correspond to partially enhanced gauge symmetry, again in accord with what is found from brane considerations.\n\n4. The Quantum Behavior via Theory\n\nWe will now turn to considering the quantum behavior of the gauge theory described above. It was found in the previous section that generically in moduli space the theory is in the Coulomb phase with the gauge symmetry being broken to a subgroup. We would like to see if the corresponding spectral curves, , can be determined. In this analysis we will closely follow where it was pointed out that in theory, the brane configuration corresponding to that in Fig. 1 can be thought of as the worldvolume of a single NS 5-brane, and that this insight leads to determining the curves.\n\nIn the 5-brane worldvolume had infinite extent along the coordinates, while spanning a two dimensional surface in the four-manifold parametrized by and . In our case and are modded by the transformation eq. (2.1). A more convenient representation of this orbifold is obtained by embedding it as a hypersurface in :\n\n yz−xM=0. (4.1)\n\nThe coordinate mapping is , , ; the orbifold singularity is at . In the theory limit the 5-brane is described by a Riemann surface embedded in . This surface is smooth except at the orbifold point, and can be parametrized as a rational curve by and , with set equal to zero.\n\nNow consider the configuration shown in Fig. 1, consisting of two NS branes and -branes (we are counting the branes and their images as distinct) stretching between them. The two dimensional surface can now be described by the curve:\n\n t2+B(y) t+1=0. (4.2)\n\nHere is a polynomial of degree (in ), i.e.,\n\n B(y)=yN+u1yN−1+u2yN−2+⋯+uN. (4.3)\n\nNote this surface corresponds to genus as would be expected for a curve with photons. As discussed in and , the periods of this Riemann surface determine the gauge couplings of the gauge groups.\n\nThe asymptotic behavior of for large is given by , and . This tells us how the two NS branes bend for large and determines the asymptotic form of the beta function which goes like\n\n 4πg2≃2Nln|y| . (4.4)\n\nThis agrees with the expected beta function for each of the factors.\n\nThe coefficients in eq. (4.2) parametrize the moduli space of the theory. It would be useful to express them in terms of the gauge invariants built out of the elementary fields in eq. (S0.Ex1). When the 4-branes are sufficiently far (compared to the strong coupling scale(s)) from the orbifold point the leading order dependence of the can be determined by classical considerations. To see this, note that eq. (4.2), at fixed , can be used to solve for and thereby yield the positions of the 4-branes. Furthermore, at large enough separation these positions can be unambiguously related to the gauge invariants, thereby determining the leading dependence of the .\n\nIn Section 2, we had described how the gauge theory corresponding to -branes placed at a orbifold point can be thought of as being obtained by starting from an , , theory and only keeping states invariant under a certain symmetry. In fact this provides the simplest way of determining the leading dependence of the coefficients . One starts with the curve,\n\n t2+B(v) t+1=0, (4.5)\n\nwhere is a polynomial of degree given by:\n\n B(v)=vNM+a1vNM−1+a2vNM−2+⋯+aNM. (4.6)\n\nIn this case the coefficients are easily determined as (trace of) the appropriate powers of the adjoint field. We now only allow fields invariant under the symmetry to have vacuum expectation values. This means that only integer powers of survive in . The resulting curve thus has a symmetry, under which . To obtain the curve in the orbifold theory it is natural to identify points related by this symmetry. This amounts to parametrizing the curve with a variable . The curve, eq. (4.5), then turns into the required one, eq. (4.2). As mentioned before, the coefficients in eq. (4.5) can be determined in terms of the adjoint field and can then be easily expressed in terms of the moduli in the orbifold theory.\n\nThe leading dependence of the on the gauge invariants can thus be determined. However, there can be subleading terms in these relations, depending on strong coupling scales of the gauge theories involved, which cannot be determined by classical considerations alone444Such terms are absent in the theory studied in , provided . In this case dimensional arguments and the fact that these corrections arise from instanton effects and are therefore proportional to is enough to explain their absence.. In fact, such terms are present in the theories at hand. We know this because these theories are essentially identical, (apart from the anomalous s discussed above) to the theories studied in , , and their curves have been worked out from field theoretic considerations.\n\nFor illustrative purposes we consider the example of an theory, which corresponds to taking six -branes (two physical branes and their images) in a orbifold. The related theory was discussed in and the curve was obtained to be:\n\n ~t2=(x2−(Λ41 M2+Λ42 M3+Λ43 M1−M1M2M3+T2))2−4Λ41Λ42Λ43. (4.7)\n\nHere, are the three strong coupling scales, while and are the moduli. This curve is related to the one obtained in the brane construction, eq. (4.2) by a shift and rescaling of the variables and . On doing so and comparing one finds that the and terms in the first bracket in eq. (4.7) correspond to the leading dependence of the coefficients , while the -dependent terms in the first bracket correspond to the subleading terms we were worried about. Actually, strictly speaking we need to incorporate the effects of the anomalous s in the curve, eq. (4.7), before comparing the two. This is relatively simple to do in the orbifold limit where the Fayet-Iliopoulos terms for the two anomalous s are zero555Determining the curve away from the orbifold limit is an interesting problem which we hope to address in a subsequent paper. This will also allow us to see whether the subleading terms arise in part because of the orbifold nature of the background and can be determined by blowing it up..\n\nLet us pause for a moment to sketch this out. In a convenient basis, the charge assignments of the three elementary fields are, , , . The corresponding terms then imply:\n\n 4|M1|2−2|M2|2−2|M3|2=0, (4.8)\n\nand\n\n 2|M2|2−2|M3|2=0. (4.9)\n\nIn this example, the anomalies cancel due to appropriate shifts in two axion fields. One consequence is that the dependent terms in eq. (4.7) acquire an axion dependence. In describing the resulting curve it is simplest to carry out appropriate rotations (and shifts in the axion fields) to go to a gauge where the three fields , and have the same phase. Eq. (4.8) and (4.9) can now be used to solve for two of the fields, say, and in terms of third, . On substituting back in eq. (4.7) the resulting curve in this gauge in terms of the moduli, and is given by:\n\n ~t2=(x2−(Λ41+Λ42+Λ43) M1+M31−T2)2−4Λ41Λ42Λ43. (4.10)\n\nThe axion dependence in eq. (4.10) enters through the dependence of the strong coupling scales on these fields and can be easily worked out. The important point is that after going through this procedure, in eq. (4.10) one sees that the subleading terms mentioned above continue to persist, while the leading terms on which there was agreement in the two cases are not changed in an essential way.\n\n5. Generalizations of the Orbifold Brane Construction.\n\nThere are three obvious generalizations of our orbifold brane construction which add massless vectorlike matter.\n\nThe first is obtained by adding Dirichlet six branes at the origin in the plane. These 6-branes extend in the directions and do not break any additional supersymmetries . Once again, the tadpoles in the one-loop vacuum amplitude must cancel in this theory. The only additional constraints arise from the twisted Ramond-Ramond tadpole amplitudes for strings ending on these 6-branes666The twisted RR amplitudes from 4-6 strings can be already seen to vanish since the matrices representing the action of the twist on the Chan-Paton factor of the 4-branes obey (2.3).: the 6-branes are sources of twisted RR flux which can only propagate in the transverse direction, which is insufficient to allow the flux to escape to infinity . Therefore, the total twisted RR charge of the 6-branes has to vanish, and the matrices that represent the action on the six brane Chan-Paton factors must also obey the conditions (2.3).\n\nThe massless excitations of the strings give vectorlike matter with the following transformation law under the symmetry (here we have denoted by the -brane gauge group, which appears as a global symmetry in the -brane theory). There are fields (), transforming as\n\n W = F1QM¯¯¯¯FM+F2Q1¯¯¯¯F1+F3Q2¯¯¯¯F2+⋯+FMQM−1¯¯¯¯FM−1 . (5.1)\n\nIt will become clear in the following that this spectrum (and superpotential) is the only one consistent with field theoretic considerations and nonabelian duality.\n\nAnother generalization is obtained by adding more NS branes. The simplest example is illustrated in Fig. 2. This configuration can be obtained by starting with physical four-branes stretched between two NS branes without any six-branes. One then brings a third NS brane in from infinity along the direction, until it intersects the middle of the four-branes. One can then break the four-branes on this new NS brane; the gauge group at this point is clearly . Now one can move of the left-hand physical four-branes together with their images off to infinity in the plane, where they have no effect on the light spectrum of the remaining brane configuration. Thus we deduce that Fig. 2 represents an orbifold model with gauge group , with chiral matter content under and of the form (S0.Ex1). In addition there is vectorlike matter corresponding to chiral multiplets (), transforming as\n\nThe third generalization consists of attaching semi-infinite four-branes to the left- or right-hand NS brane. This is equivalent (modulo the discussion in ) to taking the configuration of Fig. 2 and moving the left- or right-hand NS brane off to infinity in the direction. The new vectorlike matter consists of flavors for each , with superpotential (5.1) (in the limit that the left- or right-hand NS brane is pushed off to infinity, the gauge coupling of the corresponding -brane theory goes to zero and the contribution to the superpotential from the gauge group vanishes). It appears therefore that this construction is related to the construction with -branes by the Hanany-Witten process: after pushing the -branes through one of the NS branes, a set of -branes stretched between the NS brane and the -branes is created; after moving the branes to infinity we obtain the construction with semi-infinite -branes described above.\n\nIt is clear that the generalizations discussed above can also be obtained from theory with matter after eliminating states that are not invariant under an appropriately chosen discrete global symmetry, in the same way that was discussed in the end of Section 2 for the pure Yang-Mills theory.\n\n5.2 Nonabelian duality.\n\nHere we discuss one more check on our orbifold construction with extra matter fields. The theory with gauge group , with matter content given by eq. (S0.Ex1) plus additional flavors of each factor, and a superpotential given by eq. (5.1) was considered in ref. . By an iterative application of the SQCD dualities it was found that the theory has an equivalent infrared description—along the Higgs branch—in terms of an theory with the same matter content and superpotential. The theories along their respective Coulomb branches are clearly different, as follows from the different number of unbroken s at a generic point on the Coulomb branch moduli space.\n\nThe duality is related to the duality of the Higgs branches of SQCD with gauge groups and . This duality is easy to see in the brane construction , . Consider the brane configuration of Fig. 1. Pushing the 6-branes (we count only the physical branes here) to the left of the left NS brane, we obtain a configuration with 4-branes stretching between the -branes and the left NS brane. Then we enter the Higgs branch of the theory by reconnecting the 4-branes stretching between the two NS branes with of the newly created -branes and rearranging them in the most general way consistent with the -rule . Thus we obtain a configuration where -branes stretch between the -branes and the right NS brane while -branes stretch between the 6- branes and the left NS brane. Now we can move the two NS branes past each other in the direction and reconnect once more the 4-branes, obtaining thus a configuration where -branes stretch between the two NS branes, and -branes between the 6-branes and the leftmost NS brane. This setup describes the Higgs branch moduli space of the theory with flavors. Orbifolding by the symmetry does not affect the previous argument in any essential way. We thus obtain a brane realization of the Higgs branch duality between the and theories.\n\nAcknowledgements\n\nWe thank Atish Dabholkar, Michael Douglas, Ken Intriligator and David Kutasov for helpful discussions. JL acknowledges the hospitality of the Aspen Center for Physics, where portions of this research were completed. The research of JL and ST is supported by the Fermi National Accelerator Laboratory, which is operated by Universities Research Association, Inc., under contract no. DE-AC02-76CHO3000. E.P. was supported by DOE contract no. DOE-FG03-97ER40506." ]

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[ "# Instability and change detection in exponential families and generalized linear models, with a study of Atlantic tropical storms\n\nResearch output: Contribution to journalArticlepeer-review\n\n## Abstract\n\nExponential family statistical distributions, including the well-known normal, binomial, Poisson, and exponential distributions, are overwhelmingly used in data analysis. In the presence of covariates, an exponential family distributional assumption for the response random variables results in a generalized linear model. However, it is rarely ensured that the parameters of the assumed distributions are stable through the entire duration of the data collection process. A failure of stability leads to nonsmoothness and nonlinearity in the physical processes that result in the data. In this paper, we propose testing for stability of parameters of exponential family distributions and generalized linear models. A rejection of the hypothesis of stable parameters leads to change detection. We derive the related likelihood ratio test statistic. We compare the performance of this test statistic to the popular normal distributional assumption dependent cumulative sum (Gaussian CUSUM) statistic in change detection problems. We study Atlantic tropical storms using the techniques developed here, so to understand whether the nature of these tropical storms has remained stable over the last few decades.\n\nOriginal language English (US) 1133-1143 11 Nonlinear Processes in Geophysics 21 6 https://doi.org/10.5194/npg-21-1133-2014 Published - Nov 28 2014" ]

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[ "# Basic Calculator\n\nGet a Basic Calculator branded for your website! Colorful, interactive, simply The Best Financial Calculators!\nThis basic calculator works just like a pocket financial calculator. Use it to add, subtract, multiply and divide.\n\n## Basic Calculator Definitions\n\nInstructions\nSimply click your calculations and the calculator will handle the rest!\nMS\nMemory Store: Store the current value in memory.\nMR\nMemory Recall: Recall the stored memory value.\nM+\nMemory Plus: Add the current value to current memory value.\nM-\nMemory Minus: Subtract the current value to current memory value\nMC\nMemory Clear: your calculations and the calculator will handle the rest!" ]

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[ "1343060 (number)\n\n1,343,060 (one million three hundred forty-three thousand sixty) is an even seven-digits composite number following 1343059 and preceding 1343061. In scientific notation, it is written as 1.34306 × 106. The sum of its digits is 17. It has a total of 4 prime factors and 12 positive divisors. There are 537,216 positive integers (up to 1343060) that are relatively prime to 1343060.\n\nBasic properties\n\n• Is Prime? No\n• Number parity Even\n• Number length 7\n• Sum of Digits 17\n• Digital Root 8\n\nName\n\nShort name 1 million 343 thousand 60 one million three hundred forty-three thousand sixty\n\nNotation\n\nScientific notation 1.34306 × 106 1.34306 × 106\n\nPrime Factorization of 1343060\n\nPrime Factorization 22 × 5 × 67153\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 4 Total number of prime factors rad(n) 671530 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 0 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 1,343,060 is 22 × 5 × 67153. Since it has a total of 4 prime factors, 1,343,060 is a composite number.\n\nDivisors of 1343060\n\n12 divisors\n\n Even divisors 8 4 4 0\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 12 Total number of the positive divisors of n σ(n) 2.82047e+06 Sum of all the positive divisors of n s(n) 1.47741e+06 Sum of the proper positive divisors of n A(n) 235039 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 1158.9 Returns the nth root of the product of n divisors H(n) 5.7142 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 1,343,060 can be divided by 12 positive divisors (out of which 8 are even, and 4 are odd). The sum of these divisors (counting 1,343,060) is 2,820,468, the average is 235,039.\n\nOther Arithmetic Functions (n = 1343060)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 537216 Total number of positive integers not greater than n that are coprime to n λ(n) 67152 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 102840 Total number of primes less than or equal to n r2(n) 16 The number of ways n can be represented as the sum of 2 squares\n\nThere are 537,216 positive integers (less than 1,343,060) that are coprime with 1,343,060. And there are approximately 102,840 prime numbers less than or equal to 1,343,060.\n\nDivisibility of 1343060\n\n m n mod m 2 3 4 5 6 7 8 9 0 2 0 0 2 5 4 8\n\nThe number 1,343,060 is divisible by 2, 4 and 5.\n\n• Arithmetic\n• Abundant\n\n• Polite\n\nBase conversion (1343060)\n\nBase System Value\n2 Binary 101000111111001010100\n3 Ternary 2112020022222\n4 Quaternary 11013321110\n5 Quinary 320434220\n6 Senary 44441512\n8 Octal 5077124\n10 Decimal 1343060\n12 Duodecimal 549298\n20 Vigesimal 87hd0\n36 Base36 ssb8\n\nBasic calculations (n = 1343060)\n\nMultiplication\n\nn×y\n n×2 2686120 4029180 5372240 6715300\n\nDivision\n\nn÷y\n n÷2 671530 447687 335765 268612\n\nExponentiation\n\nny\n n2 1803810163600 2422625278324616000 3253731106306658764960000 4369956099636221120867177600000\n\nNth Root\n\ny√n\n 2√n 1158.9 110.331 34.0427 16.812\n\n1343060 as geometric shapes\n\nCircle\n\n Diameter 2.68612e+06 8.43869e+06 5.66684e+12\n\nSphere\n\n Volume 1.01479e+19 2.26673e+13 8.43869e+06\n\nSquare\n\nLength = n\n Perimeter 5.37224e+06 1.80381e+12 1.89937e+06\n\nCube\n\nLength = n\n Surface area 1.08229e+13 2.42263e+18 2.32625e+06\n\nEquilateral Triangle\n\nLength = n\n Perimeter 4.02918e+06 7.81073e+11 1.16312e+06\n\nTriangular Pyramid\n\nLength = n\n Surface area 3.12429e+12 2.85509e+17 1.0966e+06\n\nCryptographic Hash Functions\n\nmd5 00cbf7debef8a66681a00a3e1ab0cc4d 28b33f5f051dc85b3aa6b6e1cd297f16e9d0ce56 2ec288986b249a2a89d5e36a7ce3664e4f1991b90f84d4ff28c571103309638c 76ee9251bff3a3f50d050e8621a4effe38de139ef944c8acd56523c40c7073fc6a8084c5c54b2400824c255c53f265e4a50e8c15afbb497e456ec0c3cb5eb900 43f126daf11bd05fcf07a88de237aedbc20df1f0" ]

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[ "# gram to microgram\n\nHere in this article we will talk about mass units, mass is the inherent property of matter. which can neither be created nor be destroyed, yet mass can be changed from one form to another form, with this law we can conclude that mass is one of the most critical units which we can consider to study. Here we will talk about grams and micrograms both of which are unit of mass and are used when the mass to measure is very small.\n\n## gram to microgram\n\nLet’s start a discussion about a gram. Well, a gram is a tiny unit of mass, to understand what gram is let’s take an example, we all eat roti’s which are made of wheat.  Now find a wheat grain and try to observe the mass of that wheat, you will be amazed to know that the mass of that wheat grain is somewhat equal to one gram, we can define gram as similar to the 1/1000th part of one kilogram, which is very small.\n\nThen let’s consider the same example, let’s retake the wheat grain, now imagine that you need to divide the mass of that grain into ten xE6 parts, well this is unimaginable, that is the value of one microgram, this indicates that one microgram is the tiny unit.\n\nCompare both the units, grams, and micrograms, and you will find that in term of grams, one microgram is unimaginable.\n\nNow let’s try to convert grams to micrograms, one gram is equal to 10 x E6 micrograms, well gram is a large unit in terms of micrograms. To save, you have to give a multiple 10 x E6 to the gram value and your conversion will be done." ]

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[ "# Ruby 利用 to_s 实现正则嵌套\n\n__main__ · 2013年09月16日 · 最后由 aptx4869 回复于 2013年09月17日 · 2798 次阅读\n\n## 遇到的问题\n\n``````reg = %r{\n[^\\w\\$] # \\$不能是其他变量的结尾\n\\\\$ \\s* \\( \\s* (?:\n(?: # 匹配格式\\$('selector' [, 'selector'])\n(?<prop_1>\n' (?: [^'\\n\\\\] | \\\\' )*? '\n| \" (?: [^\"\\n\\\\] | \\\\\" )*? \"\n)\n(?:\n(?<comma> \\s* , \\s*)\n(?<prop_2>\n' (?: [^'\\n\\\\] | \\\\. )*? '\n| \" (?: [^\"\\n\\\\] | \\\\. )*? \"\n)\n)?\n)\n\n| (?: # 匹配格式\\$('selector', not-selecotr)\n(?<prop_1>\n' (?: [^'\\n\\\\] | \\\\. )*? '\n| \" (?: [^\"\\n\\\\] | \\\\. )*? \"\n)\n(?<comma> \\s* , \\s*)\n(?: [\\w\\$.]+ )\n)\n\n| (?: # 匹配格式\\$(not-selecotr, 'selector')\n(?: [\\w\\$.]+ )\n(?<comma> \\s* , \\s* )\n(?<prop_2>\n' (?: [^'\\n\\\\] | \\\\. )*? '\n| \" (?: [^\"\\n\\\\] | \\\\. )*? \"\n)\n)\n) \\s* \\)\n}x\n``````\n\n### 解决方案\n\n``````def get_scan_reg\nnot_selector = /[\\w\\$.]+/\nnot_name = /[^\\w\\$]/\ncomma = / \\s* , \\s* /x\nlb = / \\s* \\( \\s* /x\nrb = / \\s* \\)/x\nstr = %r{\n(?:\n' (?: [^'\\n\\\\] | \\\\. )*? '\n| \" (?: [^\"\\n\\\\] | \\\\. )*? \"\n)\n}x\n\nreturn %r{\n#{not_name} \\\\$ #{lb} (?:\n(?:\n(?<prop_1> #{str})\n(?: #{comma} (?<prop_2> #{str}) )?\n)\n| (?: (?<prop_1> #{str}) #{comma} #{not_selector} )\n| (?: #{not_selector} #{comma} (?<prop_2> #{str}) )\n) #{rb}\n}x\nend\n``````\n\n​虽然还是有些复杂，但与原来相比好理解很多了。至此已经满足我的需求了，基本可以打住了，但使用 puts 输出下生成的正则后发现，还是有点瑕疵。每一个变量的外层被包了一个括号变成了一个非捕获分组，这样多少会对性能有所影响，因此我们动手把这个细节修复下。\n\n``````def def_reg_part(reg, asGroup = false)\nsource = reg.to_s\ninst = Object.new\ninst.define_singleton_method(:to_s) do\nreturn asGroup ? source : source.gsub(/^\\(.*?:|\\)\\$/, '')\nend\nreturn inst\nend\n\ndef get_scan_reg\nnot_selector = def_reg_part(/[\\w\\$.]+/)\nnot_name = def_reg_part(/[^\\w\\$]/)\ncomma = def_reg_part(/ \\s* , \\s* /x)\nlb = def_reg_part(/ \\s* \\( \\s* /x)\nrb = def_reg_part(/ \\s* \\) /x)\nstr = def_reg_part %r{\n(?:\n' (?: [^'\\n\\\\] | \\\\. )*? '\n| \" (?: [^\"\\n\\\\] | \\\\. )*? \"\n)\n}x\n\nreturn %r{\n#{not_name} \\\\$ #{lb} (?:\n(?:\n(?<prop_1> #{str})\n(?: #{comma} (?<prop_2> #{str}) )?\n)\n| (?: (?<prop_1> #{str}) #{comma} #{not_selector} )\n| (?: #{not_selector} #{comma} (?<prop_2> #{str}) )\n) #{rb}\n}x\nend\n``````" ]

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[ "# Number 1042441\n\n### Properties of number 1042441\n\nCross Sum:\nFactorization:\nDivisors:\n1, 1021, 1042441\nCount of divisors:\nSum of divisors:\n1043463\nPrime number?\nNo\nFibonacci number?\nNo\nBell Number?\nNo\nCatalan Number?\nNo\nBase 2 (Binary):\nBase 3 (Ternary):\nBase 4 (Quaternary):\nBase 5 (Quintal):\nBase 8 (Octal):\nfe809\nBase 32:\nvq09\nsin(1042441)\n-0.76257120571453\ncos(1042441)\n-0.64690428674966\ntan(1042441)\n1.1788006685596\nln(1042441)\n13.857075636329\nlg(1042441)\n6.0180514841738\nsqrt(1042441)\n1021\nSquare(1042441)\n\n### Number Look Up\n\nLook Up\n\n1042441 which is pronounced (one million forty-two thousand four hundred forty-one) is a very impressive number. The cross sum of 1042441 is 16. If you factorisate 1042441 you will get these result 1021 * 1021. 1042441 has 3 divisors ( 1, 1021, 1042441 ) whith a sum of 1043463. 1042441 is not a prime number. The number 1042441 is not a fibonacci number. 1042441 is not a Bell Number. 1042441 is not a Catalan Number. The convertion of 1042441 to base 2 (Binary) is 11111110100000001001. The convertion of 1042441 to base 3 (Ternary) is 1221221221221. The convertion of 1042441 to base 4 (Quaternary) is 3332200021. The convertion of 1042441 to base 5 (Quintal) is 231324231. The convertion of 1042441 to base 8 (Octal) is 3764011. The convertion of 1042441 to base 16 (Hexadecimal) is fe809. The convertion of 1042441 to base 32 is vq09. The sine of the number 1042441 is -0.76257120571453. The cosine of the figure 1042441 is -0.64690428674966. The tangent of the number 1042441 is 1.1788006685596. The root of 1042441 is 1021.\nIf you square 1042441 you will get the following result 1086683238481. The natural logarithm of 1042441 is 13.857075636329 and the decimal logarithm is 6.0180514841738. I hope that you now know that 1042441 is great number!" ]

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[ "### Seyaua's blog\n\nBy Seyaua, 12 years ago, translation,", null, "Here you can find solutions to the problems from the past round. Editorial for Problem D (Div 1) was prepared by sdya\n\nDivision 2, problem A\n\nIn this problem one could transform all letters in both strings to lower case and then compare the strings lexicographically.\n\nDivision 2, problem B\n\nOne can notice that if we want to divide a square into two equal parts, then the cutting line should pass through the center of our square. Thus, if the marked cell contains the center of the square, then we can’t make a cut, otherwise we can. Here is the code which solves the problem:\n\nscanf(\"%d%d%d\", &n, &x, &y);\n\nn /= 2;\n\nif ((x == n || x == n + 1) && (y == n || y == n + 1)) printf(\"NO\\n\"); else printf(\"YES\\n\");\n\nDivision 2, problem C (Division 1, problem A)\n\nIt is easy to see that in order to maximize the sum of squares, one should make all numbers except the first one equal to 1 and maximize the first number. Keeping this in mind we only need to check whether the given value of y is large enough to satisfy a restriction that all n numbers are positive. If y is not to small, then all we need is to ensure that x ≤ 1 + 1 + … + (y - (n - 1))2\n\nDivision 2, Problem D (Divison 1, problem B)\n\nLet’s create an array used[], j-th element of which will be the index of the last number from the input, which is divisible by j. Then for each query we’ll iterate over all divisors of xi and for each k, which divides xi we’ll check whether it is “unique”. After that we’ll update used[k].\n\nDivision 2, Problem E (Division 1, problem C)\n\nThis problem has many different approaches. One of them uses the fact that the overall number of possible inputs is small and it is possible to compute the answer manually for all of them. One could also write a brute-force with a few optimizations, which works even without a precalc.\n\nHowever, the major part of all solutions involved dynamic programming with bitmasks. The solution below was described by Zlobober.\n\nInstead of counting the maximal number of free cells, we’ll count the minimal number of occupied cells. We’ll assume that the number of rows is not greater than 6 (otherwise we can rotate the board).\n\nLet D[k][pmask][mask] be the minimal number of occupied cells in the first k columns with the restrictions that the k-th column is described by pmask (ones correspond to occupied cells and zeroes correspond to free cells) and k+1-st column is described by mask. To make a transition from D[k-1][*][*] we can iterate over all possible masks for the k-1-st column, check whether we can distribute spiders in kth column knowing the masks for k+1-st and k-1-st columns and find the minimal value of D[k-1][*][pmask] for all such masks.\n\nThe overall complexity is O(n*23m), n > m.\n\nDivision 1, Problem D\n\nOne can notice that if m = 1 then the answer is kn, because all colorings are possible.\nNow we’ll assume that m > 1. Let’s look on the first column of the board (i.e. the vertical cut will be made right next to the first column). Suppose there are x distinct colors in this column. Then in the rest of the board there are also x colors. If we move the vertical line by one unit to the right, the number of different colors to the left of it will not decrease and the number of colors to the right of it won’t increase. It means that the number of different colors in both parts of the board will be also x. We can repeat this process until the line reaches the rightmost column, which means that the number of distinct colors in it is also x. It is easy to see that we can only use colors which belong to the intersection of sets of colors in the leftmost and rightmost columns in the rest of the board.\n\nLet’s iterate over all values of x and y, where x is the number of colors in the leftmost column and y is the number of elements in intersection of sets of colors in the rightmost and leftmost columns. It is easy to see that x is limited by the number of rows in the board and y can’t be greater than x. Let’s find the answer for all such pairs of x and y and at the end we’ll add them up together.\n\nSuppose x and y are fixed. We first need to choose (2x - y) colors from the given k colors, which we will use, which means that the answer for will be multiplied by C(k, 2x — y). After that we’ll choose (x-y) unique colors which will be used in the first column, which means that the answer will be also multiplied by C(2x-y, x-y). Then we’ll choose x-y colors for the rightmost column and multiply the answer by C(x, x-y). Now all we need to know is how many ways of coloring n cells into x colors are there. We’ll use a dynamic programming approach to solve this sub-problem.\n\nLet d[i][j] be the number of ways to color a rectangle of unit width and length i into colors, numerated from 1 to j with the following restriction: if a < b then the first appearence of color a in the rectangle will be before the first appearence of color b.\nThen we can calculate this function using the following recurrence:\n\nd[i][j] = j * d[i — 1][j] + d[i — 1][j — 1].\n\nAfter we finish calculating d[i][j], we need to multiply the answer by d[n][x]2 (to color the first and the last columns). Now we need to notice that we can reorder all colors in the first and the last columns in arbitrary way, which means that the answer should be multiplied by (x!)2. Finally, we need to multiply the answer by yn(m-2), which correspond to coloring the rest of our board.\n\nHere is the code, which solves the problem for the given values of x and y:\n\nlong long ans=0;\n\nfor (int y=0; y<=n; y++){\n\nlong long cur=powmod(y,n*(m-2));\n\nfor (int x=y; x<=n; x++)\n\nif (2*x-y<=k)\n\n{\n\nlong long tek=cnk[2*x-y];\n\ntek*=cnn[2*x-y][x-y], tek%=mod;\n\ntek*=cnn[x][x-y], tek%=mod;\n\ntek*=d[n][x], tek%=mod;\n\ntek*=d[n][x], tek%=mod;\n\ntek*=f[x], tek%=mod;\n\ntek*=f[x], tek%=mod;\n\ntek*=cur;\n\nans+=tek;\n\nans%=mod;\n\n}\n\n}\n\ncout<<ans<<endl;\n\nSome contestants had problems with time limit, because of calculation of C(N,K). One can notice that we won’t need more than 2000 colors, which reduces the time significantly. Author’s solution worked less than 200ms with the time-limit of 5s.\n\nDivision 1, Problem E\n\nLet the length of the maximal path be S. First, we’ll estimate the value of S without specifying the longest path itself.\n\nLet’s color our board into a chess-coloring. Obviously, each two neighboring cells in the path will have different color. Keeping this in mind we can make some estimation on the value of S. For example, if there are 4 white cells and 5 black cells on the board and we know that both starting and ending cells are white, than the length of the path can’t be greater than 7, because white and black cells must alternate in the path. We can write a simple function which calculates the maximal value of S using only the fact described above. Here, n and m are the dimensions of the board, (sx, sy) is the starting cell and (fx, fy) is the ending cell.\n\nint fnd_ans(int n,int m,int sx,int sy,int fx,int fy){\n\nint col1=((sx+sy+1)%2); //color of the start cell\n\nint col2=((fx+fy+1)%2); //color of the finish cell\n\nint cntb=(n*m+1)/2; //the number of black cells\n\nint cntw=(n*m)/2; //the number of white cells\n\nif (col1==1&&col2==1)\n\nreturn cntb*2-1;\n\nif (col1==1&&col2==0)\n\nreturn cntw*2;\n\nif (col1==0&&col2==1)\n\nreturn cntw*2;\n\nif (col1==0&&col2==0)\n\nreturn 2*cntw-1;\n\n}\n\nIt appears that for the constraints mentioned in the statement, this theoretical bound for S is always achievable. All we need is to find the path of the length S. Author solution divides the board into 5 pieces and solves the problem for each piece separately.", null, "", null, "Let’s divide the board into 5 parts as it was shown on the first picture. We’ll assume that the relative location of the starting and ending cells is the same as on the picture. In each part we’ll try to build a longest path which completely belongs to it. For the first part we’ll try to build a path from the upper-right corner to the upper-left corner. Similar rules will hold for all other parts (see the picture above for further clarification). Paths can be different for different boards, but they will have similar structure. One can notice that there are only two types of paths (with respect to rotations of the board): the one which starts at the upper-left corner and ends at the bottom-right corner and the one which starts at the upper-left corner and ends at the upper-right corner. Now we can write down an algorithm:\n\n1) Divide the board into 5 parts.\n2) Find the longest path in each of the parts.\n3) Check if the total length is equal to S.\n4) If the above is false, then rotate or reflect the board and continue to the step 1.\n\nIn order to find the longest path in a particular part, one can either consequently move through all rows of the part or through all its columns.\n\nThis solution gives correct answers for all 4 ≤ n, m ≤ 20. All possible cases of parity of each part are feasible within those constraints, which means that the solution will work for all boards, including ones with n > 20 or m > 20. The overall complexity of described algorithm is O(N*M).", null, "Tutorial of Codeforces Beta Round 85 (Div. 2 Only)", null, "", null, "Comments (10)\n| Write comment?\n Thanks for sharing the analysis. Waiting for the other problems, especially Div I problem C, please explain it throughly.The pictures are invisible, please fix it.Pretty nice problemsets, short and clear statements, I like it very much.\n » Can you please provide a proof that in Div 2 B, a line must cross the central square ?\n » In case, anybody wants a more detailed analysis of Div 2 C, I have written a post about it here.\n » worst editorial ever for Div2 D\n• » » I think so\n• » » » Yes I think the editorial for Div2 D Petya and Divisors, is not clear enough, So I will try to explain it a bit for anyone who comes after me.Since the elements are only in range 1 to 100000, the possible divisors will be in that range, we can create an array used[] , where used[j] will tell what was the last index when we encountered a X with j as one of its divisors.We can initialize used[] with -1. Now for an Xi, we can iterate over all its divisors, for a divisor d, we can check that if used[d]==-1 (not encountered this divisor until now) or `used[d]\n• » » » » Why this code is not giving TLE O(Nsqrt(N)) ?\n• » » » » » because n=1e5 so nsqrt(n)=1e7.5 which can be done easily.Correct me if i am wrong.\n• » » » » could you please correct me if I'm going wrong, I stored all divisors of xi in vector(in sorted order), and isn't it sure that divisors which are greater than yi, is the required answer?I tried for some examples it gets me correct answer, but for 18 4, it gets me answer 3 (6,9,18), but in example it is written 2. could you explain that once.\n » Thanks. good editorial" ]

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[ "Fans of Dan Brown's \"The Da Vinci Code\" may recall that solving the puzzle of a set of numbers is one of the first of many such tasks in the book. Although, the numbers are jumbled in the book, they are the first eight terms of the Fibonacci sequence. However, this sequence is also the answer to more mathematical mysteries.\n\n## Leonardo da Pisa\n\nThe Italian mathematician known as Fibonacci was actually born Leonardo da Pisa in 1175 to Guglielmo Bonaccio, a Pisan merchant (it is believed the name Fibonacci is a derivative of the Latin \"filius Bonacci\" or \"son of the Bonacci\"). At the time, Europe used Roman numerals for calculations. Complex calculations required an abacus and many merchants had to consult an expert to work out quantities and costs. Young Leonardo traveled with his father through the Eastern Mediterranean and into North Africa. During their travels he became acquainted with Arab and Indian mathematics, and their system of numerals.\n\n## The Liber Abaci\n\nWhen he returned to Pisa in 1202, Fibonacci published his first book on numbers, the \"Liber Abaci,\" or \"Book of Calculating,\" in which he introduced the Arabic numerals 0 through 9. This formed the basis of the decimal system. Compared to the awkward Roman numerals, Fibonacci's new decimal system made addition, subtraction, multiplication and division much easier and introduced a set order for the expression of numbers, as well as the concept of zero or \"nothing\" into European thought. Only fellow mathematicians understood Fibonacci's book, so he published a simplified version for merchants, which is how it found its way around Europe and revolutionized commerce and banking as well as science. According to Keith Devlin, author of \"A Man of Numbers,\" the foundations of modern banking, insurance and double-entry bookkeeping all originated in 13th-century Pisa due to Fibonacci's introductions.\n\n## The Rabbit Puzzle\n\nIn the \"Liber Abaci,\" Fibonacci poses this mathematical problem: if a pair of rabbits breeds once a month, and each pair they produce can also breed new pairs at one month old, how many pairs of rabbits will be bred in a year, starting with the one pair? The answer is contained in a sequence that begins 1, 1, 2, 3, 5, 8 and continues, working out to 377 pairs by the end of the year. The principal of the sequence is that it is built by adding two adjacent terms to get the next term to add to the series. Indian scholars had already studied the sequence in the century before Fibonacci published the \"Liber Abaci\" and he didn't claim it as his own, but because he introduced it to the West, the sequence is associated with him. It was French mathematician Edouard Lucas who named it the Fibonacci sequence in the late 1800s.\n\n## The Sequence is Everywhere\n\nFibonacci's sequence is all around us. In nature, the number of petals on a flower is usually a Fibonacci number, and the spiraling growth of a sea shell progresses at the same rate as the Fibonacci sequence. In art, music and architecture you find a constant called the \"golden mean,\" or phi, which is 1.61803 and corresponds to the ratio between two consecutive Fibonacci numbers -- the higher the numbers in the sequence, the closer they match the golden mean. A rectangle with a ratio of 1:1.61803 has long been considered aesthetically perfect. The front of the Parthenon in Athens, for example, forms a rectangle of those proportions." ]

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[ "Week 1 | Electronic Engineering homework help\n\n1. What controls the electrical properties of the atom?\n2. The valence electron is referred to as a __________ ?\n3. When silicon atoms combine to form a solid, they arrange themselves into an orderly pattern called ____________ ?\n4. When is the result when a diode’s reverse bias is increased?\n5. Which approximation is generally the best choice because it is easy to use and does not require a computer?\n6. Much of the information on a manufacturer’s data sheet is obscure and of use only to circuit designers.\n\n7. A diode can be effectively checked by what equipment?\n\n8. All silicon diodes have a knee voltage of approximately ________.\n\n9.A diode is a nonlinear device because the graph of its current versus voltage is:\n\n10.The point of intersection, which is called the Q point, represents:\n\n11.A silicon diode has a saturation current of 6 nA at 25oC. What is the saturation current at 100oC?\n\n12.A 10 V DC power supply is connected in series with a silicon diode and a 1 KΩ resistor. Calculate, load current, load voltage, load power, diode power, and total power.\n\n13.Figure 1 below shows a DC power supply connected in series with a diode, R1 and R2.\n\nAssume none-ideal diode (VD=0.7 V). Calculate the total current through and voltage across each resistor.", null, "Figure 1" ]

[ null, "https://assignmentshood.com/week-1-electronic-engineering-homework-help/", null ]

[ "mscroggs.co.uk\nmscroggs.co.uk", null, "subscribe\n\n# Blog\n\n2019-06-19\nLast night at MathsJam, Peter Kagey showed me a conjecture about OEIS sequence A308092.", null, "A308092\nThe sum of the first $$n$$ terms of the sequence is the concatenation of the first $$n$$ bits of the sequence read as binary, with $$a(1) = 1$$.\n1, 2, 3, 7, 14, 28, 56, 112, 224, 448, 896, 1791, 3583, 7166, ...", null, "To understand this definition, let's look at the first few terms of this sequence written in binary:\n1, 10, 11, 111, 1110, 11100, 111000, 1110000, 11100000, 111000000, ...\nBy \"the concatenation of the first $$n$$ bits of the sequence\", it means the first $$n$$ binary digits of the whole sequence written in order: 1, then 11, then 110, then 1101, then 11011, then 110111, and so on. So the definition means:\n• The first term is 1, as given in the definition ($$a(1)=1$$).\n• The sum of the first 2 terms is the first 2 digits: $$1+10=11$$.\n• The sum of the first 3 terms is the first 3 digits: $$1+10+11=110$$.\n• The sum of the first 4 terms is the first 4 digits: $$1+10+11+111=1101$$.\n• The sum of the first 5 terms is the first 5 digits: $$1+10+11+111+1110=11011$$.\nAs we know that the sum of the first $$n-1$$ terms is the first $$n-1$$ digits, we can calculate the third term of this sequence onwards using: \"$$a(n)$$ is the concatenation of the first $$n$$ bits of the sequence subtract concatenation of the first $$n-1$$ bits of the sequence\":\n• The third term is $$110 - 11 = 11$$.\n• The fourth term is $$1101 - 110 = 111$$.\n• The fourth term is $$11011 - 1101 = 1110$$.\n• The fifth term is $$110111 - 11011 = 11100$$.\n\n### The conjecture\n\nPeter's conjecture is that the number of 1s in each term is greater than or equal to the number of 1s in the previous term.\nI'm going to prove this conjecture. If you'd like to have a try first, stop reading now and come back when you're ready for spoilers. (If you'd like a hint, read the next section then pause again.)\n\nThe third term of the sequence onwards can be calculated by subtracting the first $$n-1$$ digits from the first $$n$$ digits. If the first $$n-1$$ digits form a binary number $$x$$, then the first $$n$$ digits will be $$2x+d$$, where $$d$$ is the $$n$$th digit (because moving all the digits to the left one place in binary is multiplying by two).\nTherefore the different is $$2x+d-x=x+d$$, and so we can work out the $$n$$th term of the sequence by adding the $$n$$th digit in the sequence to the first $$n-1$$ digits. (Hat tip to Martin Harris, who spotted this first.)\n\n### Carrying\n\nAdding 1 to a binary number the ends in 1 will cause 1 to carry over to the left. This carrying will continue until the 1 is carried into a position containing 0, and after this all the digits to the left of this 0 will remain unchanged.\nTherefore adding a digit to the first $$n-1$$ digits can only change the digits from the rightmost 0 onwards.\n\n### Endings\n\nWe can therefore disregard all the digits before the rightmost 0, and look at how the $$n$$th term compares to the $$(n-1)$$th term. There are 5 ways in which the first $$n$$ digits could end:\n• $$00$$\n• $$010$$\n• $$01...10$$ (where $$1...1$$ is a string of 2 or more ones)\n• $$01$$\n• $$01...1$$ (where $$1...1$$ is again a string of 2 or more ones)\nWe now look at each of these in turn and show that the $$n$$th term will contain at least as many ones at the $$(n-1)$$th term.\n\n#### Case 1: $$00$$\n\nIf the first $$n$$ digits of the sequence are $$x00$$ (a binary number $$x$$ followed by two zeros), then the $$(n-1)$$th term of the sequence is $$x+0=x$$, and the $$n$$th term of the sequence is $$x0+0=x0$$. Both $$x$$ and $$x0$$ contain the same number of ones.\n\n#### Case 2: $$010$$\n\nIf the first $$n$$ digits of the sequence are $$x010$$, then the $$(n-1)$$th term of the sequence is $$x0+1=x1$$, and the $$n$$th term of the sequence is $$x01+0=x01$$. Both $$x1$$ and $$x01$$ contain the same number of ones.\n\n#### Case 3: $$01...10$$\n\nIf the first $$n$$ digits of the sequence are $$x01...10$$, then the $$(n-1)$$th term of the sequence is $$x01...1+1=x10...0$$, and the $$n$$th term of the sequence is $$x01...10+1=x01...1$$. $$x01...1$$ contains more ones than $$x10...0$$.\n\n#### Case 4: $$01$$\n\nIf the first $$n$$ digits of the sequence are $$x01$$, then the $$(n-1)$$th term of the sequence is $$x+0=x$$, and the $$n$$th term of the sequence is $$x0+1=x1$$. $$x1$$ contains one more one than $$x$$.\n\n#### Case 5: $$01...1$$\n\nIf the first $$n$$ digits of the sequence are $$x01...1$$, then the $$(n-1)$$th term of the sequence is $$x01...1+1=x10...0$$, and the $$n$$th term of the sequence is $$x01...1+1=x10...0$$. Both these contain the same number of ones.\n\nIn all five cases, the $$n$$th term contains more ones or an equal number of ones to the $$(n-1)$$th term, and so the conjecture is true.\n\n### Similar posts", null, "Mathsteroids", null, "Building MENACEs for other games", null, "Big Ben Strikes Again", null, "Christmas (2016) is over\n\nComments in green were written by me. Comments in blue were not written by me.\n\nAllowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>\nTo prove you are not a spam bot, please type \"emirp\" backwards in the box below (case sensitive):\n2016-12-28\nMore than ten correct solutions to this year's Advent calendar puzzle competition were submitted on Christmas Day, so the competition is now over. (Although you can still submit your answers to get me to check them.) Thank-you to everyone who took part in the puzzle, I've had a lot of fun watching your progress and talking to you on Twitter, Reddit, etc. You can find all the puzzles and answers (from 1 January) here.\nThe (very) approximate locations of all the entries I have received so far are shown on this map:\nThis year's winners have been randomly selected from the 29 correct entries on Christmas Day. They are:\n 1 Jack Jiang 2 Steve Paget 3 Joe Gage 4 Tony Mann 5 Stephen Cappella 6 Cheng Wai Koo 7 Demi Xin 8 Lyra 9 David Fox 10 Bob Dinnage\nYour prizes will be on their way in early January.\nNow that the competition has ended, I can give away a secret. Last year, Neal suggested that it would be fun if a binary picture was hidden in the answers. So this year I hid one. If you write all the answers in binary, with each answer below the previous and colour in the 1s black, you will see this:\nI also had a lot of fun this year making up the names, locations, weapons and motives for the final murder mystery puzzle. In case you missed them these were:\n # Murder suspect Motive 1 Dr. Uno (uno = Spanish 1) Obeying nameless entity 2 Mr. Zwei (zwei = German 2) To worry others 3 Ms. Trois (trois = French 3) To help really evil elephant 4 Mrs. Quattro (quattro = Italian 4) For old unknown reasons 5 Prof. Pum (pum = Welsh 5) For individual violent end 6 Miss. Zes (zes = Dutch 6) Stopping idiotic xenophobia 7 Lord Seacht (seacht = Irish 7) Suspect espied victim eating newlyweds 8 Lady Oito (oito = Portuguese 8) Epic insanity got him today 9 Rev. Novem (novem = Latin 9) Nobody in newsroom expected\n\n # Location Weapon 1 Throne room Wrench (1 vowel) 2 Network room Rope (2 vowels) 3 Beneath reeds Revolver (3 vowels) 4 Edge of our garden Lead pipe (4 vowels) 5 Fives court Neighbour's sword (5 vowels) 6 On the sixth floor Super banana bomb (6 vowels) 7 Sparse venue Antique candlestick (7 vowels) 8 Weightlifting room A foul tasting poison (8 vowels) 9 Mathematics mezzanine Run over with an old Ford Focus (9 vowels)\nFinally, well done to Scott, Matthew Schulz, Michael Gustin, Daniel Branscombe, Kei Nishimura-Gasparian, Henry Hung, Mark Fisher, Jon Palin, Thomas Tu, Félix Breton, Matt Hutton, Miguel, Fred Verheul, Martine Vijn Nome, Brennan Dolson, Louis de Mendonca, Roni, Dylan Hendrickson, Martin Harris, Virgile Andreani, Valentin Valciu, and Adia Batic for submitting the correct answer but being too unlucky to win prizes this year. Thank you all for taking part and I'll see you next December for the next competition.\n\n### Similar posts", null, "Christmas (2019) is coming!", null, "Christmas (2018) is over", null, "Christmas (2018) is coming!", null, "Christmas (2017) is over\n\nComments in green were written by me. Comments in blue were not written by me.\nI got my prize in the mail today. I really liked the stories from Gustave Verbeek; I thought that was pretty clever. I really appreciate you being willing to send the prizes internationally.\n\nThanks for setting this all up; I had a lot of fun solving the puzzles every day (and solving half them again when my cookie for the site somehow got deleted). I'll be sure to participate next time too!\nSC\nThanks, Matthew! The puzzles were really fun, and piecing the clues was very interesting too!\nJack" ]

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[ "# Publications 2020\n\n• ## Actin modulates shape and mechanics of tubular membranes – Archive ouverte HAL\n\n### A. Allard 1 M. Bouzid 2 T. Betz 3 C. SimonM. Abou-GhaliJ. Lemiere 4 F. Valentino 5 J. Manzi 4 F. Brochard-Wyart 6 K. Guevorkian 6 J. Plastino 6 M. Lenz 2 C. Campillo 7 C. Sykes 6\n\n#### A. Allard, M. Bouzid, T. Betz, C. Simon, M. Abou-Ghali, et al.. Actin modulates shape and mechanics of tubular membranes. Science Advances , American Association for the Advancement of Science (AAAS), 2020, 6 (17), pp.eaaz3050. ⟨10.1126/sciadv.aaz3050⟩. ⟨hal-02565199⟩\n\n• 1. LNE - Laboratoire National de Métrologie et d'Essais [Trappes]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 3. Atominstitut\n• 4. PCC - Physico-Chimie-Curie\n• 5. DTU Space - National Space Institute [Lyngby]\n• 6. PCC - Physico-Chimie-Curie\n• 7. inconnu\n• ## Archive ouverte HAL – Actin modulates shape and mechanics of tubular membranes\n\n### A. Allard 1 M. Bouzid 2 T. Betz 3 C. SimonM. Abou-GhaliJ. Lemiere 4 F. Valentino 5 J. Manzi 4 F. Brochard-Wyart 6 K. Guevorkian 6 J. Plastino 6 M. Lenz 2 C. Campillo 7 C. Sykes 6\n\n#### A. Allard, M. Bouzid, T. Betz, C. Simon, M. Abou-Ghali, et al.. Actin modulates shape and mechanics of tubular membranes. Science Advances , American Association for the Advancement of Science (AAAS), 2020, 6 (17), pp.eaaz3050. ⟨10.1126/sciadv.aaz3050⟩. ⟨hal-02565199⟩\n\n• 1. LNE - Laboratoire National de Métrologie et d'Essais [Trappes]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 3. Atominstitut\n• 4. PCC - Physico-Chimie-Curie\n• 5. DTU Space - National Space Institute [Lyngby]\n• 6. PCC - Physico-Chimie-Curie\n• 7. inconnu\n• ## Archive ouverte HAL – Asymptotic behavior of the length of the longest increasing subsequences of random walks\n\n### J. Ricardo G. Mendonça 1 Hendrik Schawe 2 Alexander K. Hartmann 3 Alexander Hartmann\n\n#### J. Ricardo G. Mendonça, Hendrik Schawe, Alexander K. Hartmann, Alexander Hartmann. Asymptotic behavior of the length of the longest increasing subsequences of random walks. Physical Review E , American Physical Society (APS), 2020, 101 (3), ⟨10.1103/PhysRevE.101.032102⟩. ⟨hal-02512208⟩\n\nWe numerically estimate the leading asymptotic behavior of the length $L_{n}$ of the longest increasing subsequence of random walks with step increments following Student's $t$-distribution with parameter in the range $1/2 \\leq \\nu \\leq 5$. We find that the expected value $\\mathbb{E}(L_{n}) \\sim n^{\\theta}\\ln{n}$ with $\\theta$ decreasing from $\\theta(\\nu=1/2) \\approx 0.70$ to $\\theta(\\nu \\geq 5/2) \\approx 0.50$. For random walks with distribution of step increments of finite variance ($\\nu > 2$), this confirms previous observation of $\\mathbb{E}(L_{n}) \\sim \\sqrt{n}\\ln{n}$ to leading order. We note that this asymptotic behavior (including the subleading term) resembles that of the largest part of random integer partitions under the uniform measure and that, curiously, both random variables seem to follow Gumbel statistics. We also provide more refined estimates for the asymptotic behavior of $\\mathbb{E}(L_{n})$ for random walks with step increments of finite variance.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. University of Oldenburg\n• 3. Institut für Physik\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Collective excitations of a one-dimensional quantum droplet\n\n### Marek TylutkiGrigori E. AstrakharchikBoris A. Malomed 1 Dmitry S. Petrov 2 Grigori Astrakharchik 3 Boris Malomed 4 Dmitry Petrov\n\n#### Marek Tylutki, Grigori E. Astrakharchik, Boris A. Malomed, Dmitry S. Petrov, Grigori Astrakharchik, et al.. Collective excitations of a one-dimensional quantum droplet. Physical Review A, American Physical Society 2020, 101 (5), ⟨10.1103/PhysRevA.101.051601⟩. ⟨hal-02881226⟩\n\nWe calculate the excitation spectrum of a one-dimensional self-bound quantum droplet in a two-component bosonic mixture described by the Gross-Pitaevskii equation (GPE) with cubic and quadratic nonlinearities. The cubic term originates from the mean-field energy of the mixture proportional to the effective coupling constant $\\delta g$, whereas the quadratic nonlinearity corresponds to the attractive beyond-mean-field contribution. The droplet properties are governed by a control parameter $\\gamma\\propto \\delta g N^{2/3}$, where $N$ is the particle number. For large $\\gamma>0$ the droplet features the flat-top shape with the discrete part of its spectrum consisting of plane-wave Bogoliubov phonons propagating through the flat-density bulk and reflected by edges of the droplet. With decreasing $\\gamma$ these modes cross into the continuum, sequentially crossing the particle-emission threshold at specific critical values. A notable exception is the breathing mode which we find to be always bound. The balance point $\\gamma = 0$ provides implementation of a system governed by the GPE with an unusual quadratic nonlinearity. This case is characterized by the ratio of the breathing-mode frequency to the particle-emission threshold equal to 0.8904. As $\\gamma$ tends to $-\\infty$ this ratio tends to 1 and the droplet transforms into the soliton solution of the integrable cubic GPE.\n\n• 1. Tel Aviv University [Tel Aviv]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 3. UPC - Universitat Politècnica de Catalunya [BarcelonaTech]\n• 4. Department of Interdisciplinary Studies\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Comment on “Effective Confining Potential of Quantum States in Disordered Media”\n\n### Alain Comtet 1 Christophe Texier 1\n\n#### Alain Comtet, Christophe Texier. Comment on “Effective Confining Potential of Quantum States in Disordered Media”. Physical Review Letters, American Physical Society, 2020, 124 (21), ⟨10.1103/PhysRevLett.124.219701⟩. ⟨hal-02881221⟩\n\nWe provide some analytical tests of the density of states estimation from the \"localization landscape\" approach of Ref. [Phys. Rev. Lett. 116, 056602 (2016)]. We consider two different solvable models for which we obtain the distribution of the landscape function and argue that the precise spectral singularities are not reproduced by the estimation of the landscape approach.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Critical energy landscape of linear soft spheres\n\n### Silvio Franz 1 Antonio Sclocchi 1 Pierfrancesco Urbani 2\n\n#### Silvio Franz, Antonio Sclocchi, Pierfrancesco Urbani. Critical energy landscape of linear soft spheres. SciPost Physics, SciPost Foundation, 2020. ⟨hal-02908534⟩\n\nWe show that soft spheres interacting with a linear ramp potential when overcompressed beyond the jamming point fall in an amorphous solid phase which is critical, mechanically marginally stable and share many features with the jamming point itself. In the whole phase, the relevant local minima of the potential energy landscape display an isostatic contact network of perfectly touching spheres whose statistics is controlled by an infinite lengthscale. Excitations around such energy minima are non-linear, system spanning, and characterized by a set of non-trivial critical exponents. We perform numerical simulations to measure their values and show that, while they coincide, within numerical precision, with the critical exponents appearing at jamming, the nature of the corresponding excitations is richer. Therefore, linear soft spheres appear as a novel class of finite dimensional systems that self-organize into new, critical, marginally stable, states.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. IPHT - Institut de Physique Théorique - UMR CNRS 3681\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Current fluctuations in noninteracting run-and-tumble particles in one dimension\n\n### Tirthankar Banerjee 1 Satya N. Majumdar 1 Alberto Rosso 1 Satya Majumdar 1 Gregory Schehr 1\n\n#### Tirthankar Banerjee, Satya N. Majumdar, Alberto Rosso, Satya Majumdar, Gregory Schehr. Current fluctuations in noninteracting run-and-tumble particles in one dimension. Physical Review E , American Physical Society (APS), 2020, 101 (5), ⟨10.1103/PhysRevE.101.052101⟩. ⟨hal-02565189⟩\n\nWe present a general framework to study the distribution of the flux through the origin up to time $t$, in a non-interacting one-dimensional system of particles with a step initial condition with a fixed density $\\rho$ of particles to the left of the origin. We focus principally on two cases: (i) when the particles undergo diffusive dynamics (passive case) and (ii) run-and-tumble dynamics for each particle (active case). In analogy with disordered systems, we consider the flux distribution both for the annealed and the quenched initial conditions, for the passive and active particles. In the annealed case, we show that, for arbitrary particle dynamics, the flux distribution is a Poissonian with a mean $\\mu(t)$ that we compute exactly in terms of the Green's function of the single particle dynamics. For the quenched case, we show that, for the run-and-tumble dynamics, the quenched flux distribution takes an anomalous large deviation form at large times $P_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho\\, v_0\\, \\gamma \\, t^2 \\psi_{\\rm RTP}\\left(\\frac{Q}{\\rho v_0\\,t} \\right) \\right]$, where $\\gamma$ is the rate of tumbling and $v_0$ is the ballistic speed between two successive tumblings. In this paper, we compute the rate function $\\psi_{\\rm RTP}(q)$ and show that it is nontrivial. Our method also gives access to the probability of the rare event that, at time $t$, there is no particle to the right of the origin. For diffusive and run-and-tumble dynamics, we find that this probability decays with time as a stretched exponential, $\\sim \\exp(-c\\, \\sqrt{t})$ where the constant $c$ can be computed exactly. We verify our results for these large deviations by using an importance sampling Monte-Carlo method.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Departing from thermality of analogue Hawking radiation in a Bose-Einstein condensate\n\n### M. Isoard 1 N. Pavloff 1\n\n#### M. Isoard, N. Pavloff. Departing from thermality of analogue Hawking radiation in a Bose-Einstein condensate. Phys.Rev.Lett., 2020, 124 (6), pp.060401. ⟨10.1103/PhysRevLett.124.060401⟩. ⟨hal-02317273⟩\n\nWe study the quantum fluctuations in a one-dimensional Bose-Einstein condensate realizing an analogous acoustic black hole. The taking into account of evanescent channels and of zero modes makes it possible to accurately reproduce recent experimental measurements of the density correlation function. We discuss the determination of Hawking temperature and show that in our model the analogous radiation presents some significant departure from thermality.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Dispersionless evolution of inviscid nonlinear pulses\n\n### M. Isoard 1 N. Pavloff 1 A. M. Kamchatnov 2\n\n#### M. Isoard, N. Pavloff, A. M. Kamchatnov. Dispersionless evolution of inviscid nonlinear pulses. EPL - Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2020. ⟨hal-02565206⟩\n\nWe consider the one-dimensional dynamics of nonlinear non-dispersive waves. The problem can be mapped onto a linear one by means of the hodograph transform. We propose an approximate scheme for solving the corresponding Euler-Poisson equation which is valid for any kind of nonlinearity. The approach is exact for monoatomic classical gas and agrees very well with exact results and numerical simulations for other systems. We also provide a simple and accurate determination of the wave breaking time for typical initial conditions.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. Institute of Spectroscopy\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Distribution of the time between maximum and minimum of random walks\n\n### Francesco Mori 1 Satya N. Majumdar 1 Satya Majumdar 1 Gregory Schehr 1\n\n#### Francesco Mori, Satya N. Majumdar, Satya Majumdar, Gregory Schehr. Distribution of the time between maximum and minimum of random walks. Physical Review E , American Physical Society (APS), 2020, 101 (5), ⟨10.1103/PhysRevE.101.052111⟩. ⟨hal-02881215⟩\n\nWe consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\\tau=t_{\\min}-t_{\\max}$ between the time $t_{\\min}$ of the global minimum and the time $t_{\\max}$ of the global maximum. We extend this result to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. In both cases, we compute analytically the first few moments of $\\tau$, as well as the covariance of $t_{\\max}$ and $t_{\\min}$, showing that these times are anti-correlated. We demonstrate that the distribution of $\\tau$ for Brownian motion is valid for discrete-time random walks with $n$ steps and with a finite jump variance, in the limit $n\\to \\infty$. In the case of L\\'evy flights, which have a divergent jump variance, we numerically verify that the distribution of $\\tau$ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event \"$\\tau = n$\" is exactly $1/(2n)$ for any finite $n$, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of $(1+1)$-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size $L$. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)].\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Dynamical Heart Beat Correlations during Running\n\n### Matti MolkkariGiorgio Angelotti 1 Thorsten Emig 1 Esa Rasanen 1\n\n#### Matti Molkkari, Giorgio Angelotti, Thorsten Emig, Esa Rasanen. Dynamical Heart Beat Correlations during Running. Sci.Rep., 2020, 10, pp.13627. ⟨10.1038/s41598-020-70358-7⟩. ⟨hal-02423731⟩\n\nFluctuations of the human heart beat constitute a complex system that has been studied mostly under resting conditions using conventional time series analysis methods. During physical exercise, the variability of the fluctuations is reduced, and the time series of beat-to-beat RR intervals (RRIs) become highly non-stationary. Here we develop a dynamical approach to analyze the time evolution of RRI correlations in running across various training and racing events under real-world conditions. In particular, we introduce dynamical detrended fluctuation analysis and dynamical partial autocorrelation functions, which are able to detect real-time changes in the scaling and correlations of the RRIs as functions of the scale and the lag. We relate these changes to the exercise intensity quantified by the heart rate (HR). Beyond subject-specific HR thresholds the RRIs show multiscale anticorrelations with both universal and individual scale-dependent structure that is potentially affected by the stride frequency. These preliminary results are encouraging for future applications of the dynamical statistical analysis in exercise physiology and cardiology, and the presented methodology is also applicable across various disciplines.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Extreme value statistics of correlated random variables: a pedagogical review\n\n### Satya N. Majumdar 1 Arnab PalGregory Schehr 1\n\n#### Satya N. Majumdar, Arnab Pal, Gregory Schehr. Extreme value statistics of correlated random variables: a pedagogical review. Physics Reports, Elsevier, 2020, ⟨10.10667⟩. ⟨hal-02512248⟩\n\nExtreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of uncorrelated' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this review, we will first recall the classical EVS for uncorrelated variables and discuss the three universality classes of extreme value limiting distribution, known as the Gumbel, Fr\\'echet and Weibull distribution. We then show that, for weakly correlated random variables with a finite correlation length/time, the limiting extreme value distribution can still be inferred from that of the uncorrelated variables using a renormalisation group-like argument. Finally, we consider the most interesting examples of strongly correlated variables for which there are very few exact results for the EVS. We discuss few examples of such strongly correlated systems (such as the Brownian motion and the eigenvalues of a random matrix) where some analytical progress can be made. We also discuss other observables related to extremes, such as the density of near-extreme events, time at which an extreme value occurs, order and record statistics, etc.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Few-body bound states of two-dimensional bosons\n\n### G. Guijarro 1 G. E. Astrakharchik 1 J. Boronat 1 B. BazakD. S. Petrov 2\n\n#### G. Guijarro, G. E. Astrakharchik, J. Boronat, B. Bazak, D. S. Petrov. Few-body bound states of two-dimensional bosons. Physical Review A, American Physical Society 2020, ⟨10.1103/PhysRevA.101.041602⟩. ⟨hal-02537195⟩\n\nWe study clusters of the type A$_N$B$_M$ with $N\\leq M\\leq 3$ in a two-dimensional mixture of A and B bosons, with attractive AB and equally repulsive AA and BB interactions. In order to check universal aspects of the problem, we choose two very different models: dipolar bosons in a bilayer geometry and particles interacting via separable Gaussian potentials. We find that all the considered clusters are bound and that their energies are universal functions of the scattering lengths $a_{AB}$ and $a_{AA}=a_{BB}$, for sufficiently large attraction-to-repulsion ratios $a_{AB}/a_{BB}$. When $a_{AB}/a_{BB}$ decreases below $\\approx 10$, the dimer-dimer interaction changes from attractive to repulsive and the population-balanced AABB and AAABBB clusters break into AB dimers. Calculating the AAABBB hexamer energy just below this threshold, we find an effective three-dimer repulsion which may have important implications for the many-body problem, particularly for observing liquid and supersolid states of dipolar dimers in the bilayer geometry. The population-imbalanced ABB trimer, ABBB tetramer, and AABBB pentamer remain bound beyond the dimer-dimer threshold. In the dipolar model, they break up at $a_{AB}\\approx 2 a_{BB}$ where the atom-dimer interaction switches to repulsion.\n\n• 1. UPC - Universitat Politècnica de Catalunya [Barcelona]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Finite-time adiabatic processes: Derivation and speed limit\n\n### Carlos Plata 1 David Guéry-Odelin 2 Emmanuel Trizac 3 Antonio Prados 4\n\n#### Carlos Plata, David Guéry-Odelin, Emmanuel Trizac, Antonio Prados. Finite-time adiabatic processes: Derivation and speed limit. Physical Review E , American Physical Society (APS), 2020, 101 (3), ⟨10.1103/PhysRevE.101.032129⟩. ⟨hal-02535447⟩\n\nObtaining adiabatic processes that connect equilibrium states in a given time represents a challenge for mesoscopic systems. In this paper, we explicitly show how to build these finite-time adiabatic processes for an overdamped Brownian particle in an arbitrary potential, a system that is relevant both at the conceptual and the practical level. This is achieved by jointly engineering the time evolutions of the binding potential and the fluid temperature. Moreover, we prove that the second principle imposes a speed limit for such adiabatic transformations: there appears a minimum time to connect the initial and final states. This minimum time can be explicitly calculated for a general compression/decompression situation.\n\n• 1. Padova University\n• 2. Atomes Froids (LCAR)\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 4. Universidad de Sevilla\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Locally quasi-stationary states in noninteracting spin chains\n\n### Maurizio Fagotti 1\n\n#### Maurizio Fagotti. Locally quasi-stationary states in noninteracting spin chains. SciPost Phys., 2020, 8, pp.048. ⟨10.21468/SciPostPhys.8.3.048⟩. ⟨hal-02423699⟩\n\nLocally quasi-stationary states (LQSS) were introduced as inhomogeneous generalisations of stationary states in integrable systems. Roughly speaking, LQSSs look like stationary states, but only locally. Despite their key role in hydrodynamic descriptions, an unambiguous definition of LQSSs was not given. By solving the dynamics in inhomogeneous noninteracting spin chains, we identify the set of LQSSs as a subspace that is invariant under time evolution, and we explicitly construct the latter in a generalised XY model. As a by-product, we exhibit an exact generalised hydrodynamic theory (including \"quantum corrections\").\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Mapping and Modeling the Nanomechanics of Bare and Protein-Coated Lipid Nanotubes\n\n### Guillaume Lamour 1 Antoine Allard 1, 2 Juan Pelta 1 Sid Labdi 1 Martin Lenz 3 Clément Campillo 1\n\n#### Guillaume Lamour, Antoine Allard, Juan Pelta, Sid Labdi, Martin Lenz, et al.. Mapping and Modeling the Nanomechanics of Bare and Protein-Coated Lipid Nanotubes. Physical Review X, American Physical Society, 2020, 10 (1), pp.011031. ⟨10.1103/PhysRevX.10.011031⟩. ⟨hal-02512272⟩\n\nMembrane nanotubes are continuously assembled and disassembled by the cell to generate and dispatch transport vesicles, for instance, in endocytosis. While these processes crucially involve the ill-understood local mechanics of the nanotube, existing micromanipulation assays only give access to its global mechanical properties. Here we develop a new platform to study this local mechanics using atomic force microscopy (AFM). On a single coverslip we quickly generate millions of substrate-bound nanotubes, out of which dozens can be imaged by AFM in a single experiment. A full theoretical description of the AFM tip-membrane interaction allows us to accurately relate AFM measurements of the nanotube heights, widths, and rigidities to the membrane bending rigidity and tension, thus demonstrating our assay as an accurate probe of nanotube mechanics. We reveal a universal relationship between nanotube height and rigidity, which is unaffected by the specific conditions of attachment to the substrate. Moreover, we show that the parabolic shape of force-displacement curves results from thermal fluctuations of the membrane that collides intermittently with the AFM tip. We also show that membrane nanotubes can exhibit high resilience against extreme lateral compression. Finally, we mimic in vivo actin polymerization on nanotubes and use AFM to assess the induced changes in nanotube physical properties. Our assay may help unravel the local mechanics of membrane-protein interactions, including membrane remodeling in nanotube scission and vesicle formation.\n\n• 1. LAMBE - UMR 8587 - Laboratoire Analyse, Modélisation et Matériaux pour la Biologie et l'Environnement\n• 2. PCC - Physico-Chimie-Curie\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Archive ouverte HAL – Multi-component colloidal gels: interplay between structure and mechanical properties\n\n### Claudia Ferreiro-CordovaMehdi Bouzid 1 Emanuela del GadoGiuseppe Foffi 2 Claudia Ferreiro-Córdova\n\n#### Claudia Ferreiro-Cordova, Mehdi Bouzid, Emanuela del Gado, Giuseppe Foffi, Claudia Ferreiro-Córdova. Multi-component colloidal gels: interplay between structure and mechanical properties. Soft Matter, Royal Society of Chemistry, 2020, 16 (18), pp.4414-4421. ⟨10.1039/C9SM02410G⟩. ⟨hal-02881157⟩\n\nWe present a detailed numerical study of multi-component colloidal gels interacting sterically and obtained by arrested phase separation. Under deformation, we found that the interplay between the different intertwined networks is key. Increasing the number of component leads to softer solids that can accomodate progressively larger strain before yielding. The simulations highlight how this is the direct consequence of the purely repulsive interactions between the different components, which end up enhancing the linear response of the material. Our work {provides new insight into mechanisms at play for controlling the material properties and open the road to new design principles for} soft composite solids\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. LPS - Laboratoire de Physique des Solides\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel\n\n### Naftali R. Smith 1 David S. Dean 2 Pierre Le Doussal 3 Satya N. Majumdar 1 Grégory Schehr 1\n\n#### Naftali R. Smith, David S. Dean, Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr. Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel. Phys.Rev.A, 2020, 101 (5), pp.053602. ⟨10.1103/PhysRevA.101.053602⟩. ⟨hal-02484003⟩\n\nWe study a system of N noninteracting spinless fermions in a confining double-well potential in one dimension. We show that when the Fermi energy is close to the value of the potential at its local maximum, physical properties, such as the average density and the fermion position correlation functions, display a universal behavior that depends only on the local properties of the potential near its maximum. This behavior describes the merging of two Fermi gases, which are disjoint at sufficiently low Fermi energies. We describe this behavior in terms of a correlation kernel that we compute analytically and we call it the inverted parabola kernel. As an application, we calculate the mean and variance of the number of particles in an interval of size 2L centered around the position of the local maximum, for sufficiently small L. We discuss the possibility of observing our results in experiments, as well as extensions to nonzero temperature and to higher space dimensions.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. LOMA - Laboratoire Ondes et Matière d'Aquitaine\n• 3. LPTENS - Laboratoire de Physique Théorique de l'ENS\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids\n\n### Alessandro Manacorda 1 Gregory Schehr 2 Francesco Zamponi 1\n\n#### Alessandro Manacorda, Gregory Schehr, Francesco Zamponi. Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids. Journal of Chemical Physics, American Institute of Physics, 2020, 152 (16), pp.164506. ⟨10.1063/5.0007036⟩. ⟨hal-02554137⟩\n\n• 1. Systèmes Désordonnés et Applications\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Archive ouverte HAL – Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States\n\n### John MartinStefan WeigertOlivier Giraud 1\n\n#### John Martin, Stefan Weigert, Olivier Giraud. Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States. Quantum, Verein, 2020. ⟨hal-02881098⟩\n\nCoherent and anticoherent states of spin systems up to spin j=2 are known to be optimal in order to detect rotations by a known angle but unknown rotation axis. These optimal quantum rotosensors are characterized by minimal fidelity, given by the overlap of a state before and after a rotation, averaged over all directions in space. We calculate a closed-form expression for the average fidelity in terms of anticoherent measures, valid for arbitrary values of the quantum number j. We identify optimal rotosensors (i) for arbitrary rotation angles in the case of spin quantum numbers up to j=7/2 and (ii) for small rotation angles in the case of spin quantum numbers up to j=5. The closed-form expression we derive allows us to explain the central role of anticoherence measures in the problem of optimal detection of rotation angles for arbitrary values of j.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Optimizing Brownian escape rates by potential shaping\n\n### Marie Chupeau 1 Jannes GladrowAlexei Chepelianskii 2 Ulrich F. KeyserEmmanuel Trizac 1 Ulrich Keyser\n\n#### Marie Chupeau, Jannes Gladrow, Alexei Chepelianskii, Ulrich F. Keyser, Emmanuel Trizac, et al.. Optimizing Brownian escape rates by potential shaping. Proceedings of the National Academy of Sciences of the United States of America , National Academy of Sciences, 2020, 117 (3), pp.1383-1388. ⟨10.1073/pnas.1910677116⟩. ⟨hal-02512216⟩\n\nBrownian escape is key to a wealth of physico-chemical processes, including polymer folding, and information storage. The frequency of thermally activated energy barrier crossings is assumed to generally decrease exponentially with increasing barrier height. Here, we show experimentally that higher, fine-tuned barrier profiles result in significantly enhanced escape rates in breach of the intuition relying on the above scaling law, and address in theory the corresponding conditions for maximum speed-up. Importantly, our barriers end on the same energy on which they start. For overdamped dynamics, the achievable boost of escape rates is, in principle, unbounded so that the barrier optimization has to be regularized. We derive optimal profiles under two different regularizations, and uncover the efficiency of N-shaped barriers. We then demonstrate the viability of such a potential in automated microfluidic Brownian dynamics experiments using holographic optical tweezers and achieve a doubling of escape rates compared to unhindered Brownian motion. Finally, we show that this escape rate boost extends into the low-friction inertial regime.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. LPCT - Laboratoire de Physico-Chimie Théorique\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Reversal of contractility as a signature of self-organization in cytoskeletal bundles\n\n### Martin Lenz 1\n\n#### Martin Lenz. Reversal of contractility as a signature of self-organization in cytoskeletal bundles. eLife, eLife Sciences Publication, 2020, 9, ⟨10.7554/eLife.51751⟩. ⟨hal-02518848⟩\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Archive ouverte HAL – Scalable quantum computing with qudits on a graph\n\n### E. O. Kiktenko 1 A. S. NikolaevaPeng XuG. V. Shlyapnikov 2 A. K. Fedorov 3\n\n#### E. O. Kiktenko, A. S. Nikolaeva, Peng Xu, G. V. Shlyapnikov, A. K. Fedorov. Scalable quantum computing with qudits on a graph. Physical Review A, American Physical Society 2020, 101 (2), ⟨10.1103/PhysRevA.101.022304⟩. ⟨hal-02512218⟩\n\nWe show a significant reduction of the number of quantum operations and the improvement of the circuit depth for the realization of the Toffoli gate by using qudits. This is done by establishing a general relation between the dimensionality of qudits and their topology of connections for a scalable multi-qudit processor, where higher qudit levels are used for substituting ancillas. The suggested model is of importance for the realization of quantum algorithms and as a method of quantum error correction codes for single-qubit operations.\n\n• 1. IPE - Schmidt United Institute of Physics of the Earth [Moscow]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 3. Russian Quantum Center\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – State transition graph of the Preisach model and the role of return-point memory\n\n### M. Mert Terzi 1 Muhittin Mungan\n\n#### M. Mert Terzi, Muhittin Mungan. State transition graph of the Preisach model and the role of return-point memory. Physical Review E, 2020, 102 (1), ⟨10.1103/PhysRevE.102.012122⟩. ⟨hal-02908545⟩\n\nThe Preisach model has been useful as a null-model for understanding memory formation in periodically driven disordered systems. In amorphous solids for example, the athermal response to shear is due to localized plastic events (soft spots). As shown recently by one of us, the plastic response to applied shear can be rigorously described in terms of a directed network whose transitions correspond to one or more soft spots changing states. The topology of this graph depends on the interactions between soft-spots and when such interactions are negligible, the resulting description becomes that of the Preisach model. A first step in linking transition graph topology with the underlying soft-spot interactions is therefore to determine the structure of such graphs in the absence of interactions. Here we perform a detailed analysis of the transition graph of the Preisach model. We highlight the important role played by return point memory in organizing the graph into a hierarchy of loops and sub-loops. Our analysis reveals that the topology of a large portion of this graph is actually not governed by the values of the switching fields that describe the individual hysteretic behavior of the individual elements, but by a coarser parameter, a permutation $\\rho$ which prescribes the sequence in which the individual hysteretic elements change their states as the main hysteresis loop is traversed. This in turn allows us to derive combinatorial properties, such as the number of major loops in the transition graph as well as the number of states $| \\mathcal{R} |$ constituting the main hysteresis loop and its nested subloops. We find that $| \\mathcal{R} |$ is equal to the number of increasing subsequences contained in the permutation $\\rho$.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Statistics of first-passage Brownian functionals\n\n### Satya N. Majumdar 1 Baruch Meerson\n\n#### Satya N. Majumdar, Baruch Meerson. Statistics of first-passage Brownian functionals. J.Stat.Mech., 2020, 2002 (2), pp.023202. ⟨10.1088/1742-5468/ab6844⟩. ⟨hal-02497830⟩\n\nWe study the distribution of first-passage functionals of the type where represents a Brownian motion (with or without drift) with diffusion constant D, starting at x 0  >  0, and t f  is the first-passage time to the origin. In the driftless case, we compute exactly, for all n  >  −2, the probability density . We show that has an essential singular tail as and a power-law tail as . The leading essential singular behavior for small A can be obtained using the optimal fluctuation method (OFM), which also predicts the optimal paths of the conditioned process in this limit. For the case with a drift toward the origin, where no exact solution is known for general n  >  −1, we show that the OFM successfully predicts the tails of the distribution. For it predicts the same essential singular tail as in the driftless case. For it predicts a stretched exponential tail for all n  >  0. In the limit of large Péclet number , where is the drift velocity toward the origin, the OFM predicts an exact large-deviation scaling behavior, valid for all A: , where is the mean value of in this limit. We compute the rate function analytically for all n  >  −1. We show that, while for n  >  0 the rate function is analytic for all z, it has a non-analytic behavior at z  =  1 for  −1  <  n  <  0 which can be interpreted as a dynamical phase transition. The order of this transition is 2 for  −1/2  <  n  <  0, while for  −1  <  n  <  −1/2 the order of transition is ; it changes continuously with n. We also provide an illuminating alternative derivation of the OFM result by using a WKB-type asymptotic perturbation theory for large . Finally, we employ the OFM to study the case of (drift away from the origin). We show that, when the process is conditioned on reaching the origin, the distribution of coincides with the distribution of for with the same .\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Stochastic growth in time-dependent environments\n\n### Guillaume Barraquand 1 Pierre Le Doussal 1 Alberto Rosso 2\n\n#### Guillaume Barraquand, Pierre Le Doussal, Alberto Rosso. Stochastic growth in time-dependent environments. Physical Review E , American Physical Society (APS), 2020, 101 (4), ⟨10.1103/PhysRevE.101.040101⟩. ⟨hal-02565202⟩\n\nWe study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance $c(t)$ depending on time. We find that for $c(t)\\propto t^{-\\alpha}$ there is a transition at $\\alpha=1/2$. When $\\alpha>1/2$, the solution saturates at large times towards a non-universal limiting distribution. When $\\alpha<1/2$ the fluctuation field is governed by scaling exponents depending on $\\alpha$ and the limiting statistics are similar to the case when $c(t)$ is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential (2) An exactly solvable discretization, the log-gamma polymer model (3) Numerical simulations.\n\n• 1. Champs Aléatoires et Systèmes hors d'Équilibre\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Swimmer Suspensions on Substrates: Anomalous Stability and Long-Range Order\n\n### Ananyo Maitra 1, 2 Pragya SrivastavaM. Cristina MarchettiSriram RamaswamyMartin Lenz 2, 3\n\n#### Ananyo Maitra, Pragya Srivastava, M. Cristina Marchetti, Sriram Ramaswamy, Martin Lenz. Swimmer Suspensions on Substrates: Anomalous Stability and Long-Range Order. Phys.Rev.Lett., 2020, 124 (2), pp.028002. ⟨10.1103/PhysRevLett.124.028002⟩. ⟨hal-02475283⟩\n\nWe present a comprehensive theory of the dynamics and fluctuations of a two-dimensional suspension of polar active particles in an incompressible fluid confined to a substrate. We show that, depending on the sign of a single parameter, a state with polar orientational order is anomalously stable (or anomalously unstable), with a nonzero relaxation (or growth) rate for angular fluctuations, not parallel to the ordering direction, at zero wave number. This screening of the broken-symmetry mode in the stable state does lead to conventional rather than giant number fluctuations as argued by Bricard et al., Nature 503, 95 (2013), but their bend instability in a splay-stable flock does not exist and the polar phase has long-range order in two dimensions. Our theory also describes confined three-dimensional thin-film suspensions of active polar particles as well as dense compressible active polar rods, and predicts a flocking transition without a banding instability.\n\n• 1. LJP - Laboratoire Jean Perrin\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 3. ESPCI ParisTech\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Symmetries in $B \\to D^* \\ell \\nu$ angular observables\n\n### Marcel AlgueróSébastien Descotes-Genon 1 Joaquim MatiasMartín Novoa-Brunet 2\n\n#### Marcel Algueró, Sébastien Descotes-Genon, Joaquim Matias, Martín Novoa-Brunet. Symmetries in $B \\to D^* \\ell \\nu$ angular observables. JHEP, 2020, 06, pp.156. ⟨10.1007/JHEP06(2020)156⟩. ⟨hal-02518081⟩\n\nWe apply the formalism of amplitude symmetries to the angular distribution of the decays B → D$^{∗}$ℓν for ℓ = e, μ, τ . We show that the angular observables used to describe the distribution of this class of decays are not independent in absence of New Physics contributing to tensor operators. We derive sets of relations among the angular coefficients of the decay distribution for the massless and massive lepton cases which can be used to probe in a very general way the consistency among the angular observables and the underlying New Physics at work. We use these relations to access the longitudinal polarisation fraction of the D$^{∗}$ using different angular coefficients from the ones used by Belle experiment. This in the near future can provide an alternative strategy to measure ${F}_L^{D\\ast }$ in B → D$^{∗}$τν and to understand the relatively high value measured by the Belle experiment. Using the same symmetries, we identify three observables which may exhibit a tension if the experimental value of ${F}_L^{D\\ast }$ remains high. We discuss how these relations can be exploited for binned measurements. We also propose a new observable that could test for specific scenarios of New Physics generated by light right-handed neutrinos. Finally we study the prospects of testing these relations based on the projected experimental sensitivity of new experiments.\n\n• 1. IJCLab - Laboratoire de Physique des 2 Infinis Irène Joliot-Curie\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – The convex hull of the run-and-tumble particle in a plane\n\n### Alexander K HartmannSatya N Majumdar 1 Hendrik Schawe 2 Gregory Schehr 1 Alexander Hartmann 2 Satya Majumdar 1\n\n#### Alexander K Hartmann, Satya N Majumdar, Hendrik Schawe, Gregory Schehr, Alexander Hartmann, et al.. The convex hull of the run-and-tumble particle in a plane. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2020, 2020 (5), pp.053401. ⟨10.1088/1742-5468/ab7c5f⟩. ⟨hal-02881103⟩\n\nWe study the statistical properties of the convex hull of a planar run-and-tumble particle (RTP), also known as the \"persistent random walk\", where the particle/walker runs ballistically between tumble events at which it changes its direction randomly. We consider two different statistical ensembles where we either fix (i) the total number of tumblings $n$ or (ii) the total duration $t$ of the time interval. In both cases, we derive exact expressions for the average perimeter of the convex hull and then compare to numerical estimates finding excellent agreement. Further, we numerically compute the full distribution of the perimeter using Markov chain Monte Carlo techniques, in both ensembles, probing the far tails of the distribution, up to a precision smaller than $10^{-100}$. This also allows us to characterize the rare events that contribute to the tails of these distributions.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. University of Oldenburg\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – The influence of the brittle-ductile transition zone on aftershock and foreshock occurrence\n\n### Giuseppe Petrillo 1 Eugenio Lippiello 1 François Landes 2 Alberto Rosso 3\n\n#### Giuseppe Petrillo, Eugenio Lippiello, François Landes, Alberto Rosso. The influence of the brittle-ductile transition zone on aftershock and foreshock occurrence. Nature Communications, Nature Publishing Group, 2020. ⟨hal-02908552⟩\n\nAftershock occurrence is characterized by scaling behaviors with quite universal exponents. At the same time, deviations from universality have been proposed as a tool to discriminate aftershocks from foreshocks. Here we show that the change in rheological behavior of the crust, from velocity weakening to velocity strengthening, represents a viable mechanism to explain statistical features of both aftershocks and foreshocks. More precisely, we present a model of the seismic fault described as a velocity weakening elastic layer coupled to a velocity strengthening visco-elastic layer. We show that the statistical properties of aftershocks in instrumental catalogs are recovered at a quantitative level, quite independently of the value of model parameters. We also find that large earthquakes are often anticipated by a preparatory phase characterized by the occurrence of foreshocks. Their magnitude distribution is significantly flatter than the aftershock one, in agreement with recent results for forecasting tools based on foreshocks.\n\n• 1. Department of Mathematics and Physics [Caserta]\n• 2. LRI - Laboratoire de Recherche en Informatique\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Three- and four-point connectivities of two-dimensional critical $Q-$ Potts random clusters on the torus\n\n### Nina Javerzat 1 Marco Picco 2 Raoul Santachiara 1\n\n#### Nina Javerzat, Marco Picco, Raoul Santachiara. Three- and four-point connectivities of two-dimensional critical $Q-$ Potts random clusters on the torus. J.Stat.Mech., 2020, 2005, pp.053106. ⟨10.1088/1742-5468/ab7c5e⟩. ⟨hal-02416915⟩\n\nIn a recent paper, we considered the effects of the torus lattice topology on the two-point connectivity of Q-Potts clusters. These effects are universal and probe non-trivial structure constants of the theory. We complete here this work by considering the torus corrections to the three- and four-point connectivities. These corrections, which depend on the scale invariant ratios of the triangle and quadrilateral formed by the three and four given points, test other non-trivial structure constants. We also present results of Monte Carlo simulations in good agreement with our predictions.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. LPTHE - Laboratoire de Physique Théorique et Hautes Energies\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Two anyons on the sphere: nonlinear states and spectrum\n\n### Alexios P. PolychronakosStéphane Ouvry 1\n\n#### Alexios P. Polychronakos, Stéphane Ouvry. Two anyons on the sphere: nonlinear states and spectrum. Nucl.Phys.B, 2020, 951, pp.114906. ⟨10.1016/j.nuclphysb.2019.114906⟩. ⟨hal-02340259⟩\n\nWe study the energy spectrum of two anyons on the sphere in a constant magnetic field. Making use of rotational invariance we reduce the energy eigenvalue equation to a system of linear differential equations for functions of a single variable, a reduction analogous to separating center of mass and relative coordinates on the plane. We solve these equations by a generalization of the Frobenius method and derive numerical results for the energies of non-analytically derivable states.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Universal gap statistics for random walks for a class of jump densities\n\n### Matteo Battilana 1 Satya N. Majumdar 1 Gregory Schehr 1\n\n#### Matteo Battilana, Satya N. Majumdar, Gregory Schehr. Universal gap statistics for random walks for a class of jump densities. Markov Processes And Related Fields, Polymat Publishing Company, 2020. ⟨hal-02518812⟩\n\nWe study the order statistics of a random walk (RW) of $n$ steps whose jumps are distributed according to symmetric Erlang densities $f_p(\\eta)\\sim |\\eta|^p \\,e^{-|\\eta|}$, parametrized by a non-negative integer $p$. Our main focus is on the statistics of the gaps $d_{k,n}$ between two successive maxima $d_{k,n}=M_{k,n}-M_{k+1,n}$ where $M_{k,n}$ is the $k$-th maximum of the RW between step 1 and step $n$. In the limit of large $n$, we show that the probability density function of the gaps $P_{k,n}(\\Delta) = \\Pr(d_{k,n} = \\Delta)$ reaches a stationary density $P_{k,n}(\\Delta) \\to p_k(\\Delta)$. For large $k$, we demonstrate that the typical fluctuations of the gap, for $d_{k,n}= O(1/\\sqrt{k})$ (and $n \\to \\infty$), are described by a non-trivial scaling function that is independent of $k$ and of the jump probability density function $f_p(\\eta)$, thus corroborating our conjecture about the universality of the regime of typical fluctuations (see G. Schehr, S. N. Majumdar, Phys. Rev. Lett. 108, 040601 (2012)). We also investigate the large fluctuations of the gap, for $d_{k,n} = O(1)$ (and $n \\to \\infty$), and show that these two regimes of typical and large fluctuations of the gaps match smoothly.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Universal Scaling of the Velocity Field in Crack Front Propagation\n\n### Clément Le Priol 1 Pierre Le Doussal 2 Laurent Ponson 3 Alberto Rosso 4 Julien Chopin 5\n\n#### Clément Le Priol, Pierre Le Doussal, Laurent Ponson, Alberto Rosso, Julien Chopin. Universal Scaling of the Velocity Field in Crack Front Propagation. Physical Review Letters, American Physical Society, 2020, 124 (6), ⟨10.1103/PhysRevLett.124.065501⟩. ⟨hal-02512228⟩\n\nThe propagation of a crack front in disordered materials is jerky and characterized by bursts of activity, called avalanches. These phenomena are the manifestation of an out-of-equilibrium phase transition originated by the disorder. As a result avalanches display universal scalings which are however difficult to characterize in experiments at finite drive. Here we show that the correlation functions of the velocity field along the front allow to extract the critical exponents of the transition and to identify the universality class of the system. We employ these correlations to characterize the universal behavior of the transition in simulations and in an experiment of crack propagation. This analysis is robust, efficient and can be extended to all systems displaying avalanche dynamics.\n\n• 1. LPENS (UMR_8023) - Laboratoire de physique de l'ENS - ENS Paris\n• 2. Champs Aléatoires et Systèmes hors d'Équilibre\n• 3. DALEMBERT - Institut Jean Le Rond d'Alembert\n• 4. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 5. IF-UFB - Instituto de Fisica, Universidade Federal da Bahia\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Universal Survival Probability for a d -Dimensional Run-and-Tumble Particle\n\n### Francesco Mori 1 Pierre Le Doussal 2 Satya N. Majumdar 1 Satya Majumdar 1 Gregory Schehr 1\n\n#### Francesco Mori, Pierre Le Doussal, Satya N. Majumdar, Satya Majumdar, Gregory Schehr. Universal Survival Probability for a d -Dimensional Run-and-Tumble Particle. Physical Review Letters, American Physical Society, 2020, 124 (9), ⟨10.1103/PhysRevLett.124.090603⟩. ⟨hal-02512214⟩\n\nWe consider an active run-and-tumble particle (RTP) in $d$ dimensions and compute exactly the probability $S(t)$ that the $x$-component of the position of the RTP does not change sign up to time $t$. When the tumblings occur at a constant rate, we show that $S(t)$ is independent of $d$ for any finite time $t$ (and not just for large $t$), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed $v$ of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. Champs Aléatoires et Systèmes hors d'Équilibre\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Archive ouverte HAL – Velocity and diffusion constant of an active particle in a one-dimensional force field\n\n### Pierre Le Doussal 1 Satya N. Majumdar 2 Satya Majumdar 2 Gregory Schehr 2\n\n#### Pierre Le Doussal, Satya N. Majumdar, Satya Majumdar, Gregory Schehr. Velocity and diffusion constant of an active particle in a one-dimensional force field. EPL - Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2020, 130 (4), pp.40002. ⟨10.1209/0295-5075/130/40002⟩. ⟨hal-02881224⟩\n\nWe consider a run an tumble particle with two velocity states $\\pm v_0$, in an inhomogeneous force field $f(x)$ in one dimension. We obtain exact formulae for its velocity $V_L$ and diffusion constant $D_L$ for arbitrary periodic $f(x)$ of period $L$. They involve the \"active potential\" which allows to define a global bias. Upon varying parameters, such as an external force $F$, the dynamics undergoes transitions from non-ergodic trapped states, to various moving states, some with non analyticities in the $V_L$ versus $F$ curve. A random landscape in the presence of a bias leads, for large $L$, to anomalous diffusion $x \\sim t^\\mu$, $\\mu<1$, or to a phase with a finite velocity that we calculate.\n\n• 1. Champs Aléatoires et Systèmes hors d'Équilibre\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Asymptotic behavior of the length of the longest increasing subsequences of random walks – Archive ouverte HAL\n\n### J. Ricardo G. Mendonça 1 Hendrik Schawe 2 Alexander K. Hartmann 3 Alexander Hartmann\n\n#### J. Ricardo G. Mendonça, Hendrik Schawe, Alexander K. Hartmann, Alexander Hartmann. Asymptotic behavior of the length of the longest increasing subsequences of random walks. Physical Review E , American Physical Society (APS), 2020, 101 (3), ⟨10.1103/PhysRevE.101.032102⟩. ⟨hal-02512208⟩\n\nWe numerically estimate the leading asymptotic behavior of the length $L_{n}$ of the longest increasing subsequence of random walks with step increments following Student's $t$-distribution with parameter in the range $1/2 \\leq \\nu \\leq 5$. We find that the expected value $\\mathbb{E}(L_{n}) \\sim n^{\\theta}\\ln{n}$ with $\\theta$ decreasing from $\\theta(\\nu=1/2) \\approx 0.70$ to $\\theta(\\nu \\geq 5/2) \\approx 0.50$. For random walks with distribution of step increments of finite variance ($\\nu > 2$), this confirms previous observation of $\\mathbb{E}(L_{n}) \\sim \\sqrt{n}\\ln{n}$ to leading order. We note that this asymptotic behavior (including the subleading term) resembles that of the largest part of random integer partitions under the uniform measure and that, curiously, both random variables seem to follow Gumbel statistics. We also provide more refined estimates for the asymptotic behavior of $\\mathbb{E}(L_{n})$ for random walks with step increments of finite variance.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. University of Oldenburg\n• 3. Institut für Physik\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Brownian flights over a circle – Archive ouverte HAL\n\n### Alexander VladimirovSenya ShlosmanSergei Nechaev 1\n\n#### Alexander Vladimirov, Senya Shlosman, Sergei Nechaev. Brownian flights over a circle. Physical Review E , American Physical Society (APS), 2020, 102 (1), ⟨10.1103/PhysRevE.102.012124⟩. ⟨hal-03009773⟩\n\nThe stationary radial distribution, $P(\\rho)$, of the random walk with the diffusion coefficient $D$, which winds with the tangential velocity $V$ around the impenetrable disc of radius $R$ for $R\\gg 1$ converges to the distribution involving the Airy function. Typical trajectories are localized in the circular strip $[R, R+ \\delta R^{1/3}]$, where $\\delta$ is the constant which depends on the parameters $D$ and $V$ and is independent on $R$.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Building an irreversible Carnot-like heat engine with an overdamped harmonic oscillator – Archive ouverte HAL\n\n### Carlos A. PlataDavid Guéry-Odelin 1 Emmanuel Trizac 2 Antonio Prados\n\n#### Carlos A. Plata, David Guéry-Odelin, Emmanuel Trizac, Antonio Prados. Building an irreversible Carnot-like heat engine with an overdamped harmonic oscillator. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2020, 2020 (9), pp.093207. ⟨10.1088/1742-5468/abb0e1⟩. ⟨hal-03017062⟩\n\nWe analyse non-equilibrium Carnot-like cycles built with a colloidal particle in a harmonic trap, which is immersed in a fluid that acts as a heat bath. Our analysis is carried out in the overdamped regime. The cycle comprises four branches: two isothermal processes and two \\textit{locally} adiabatic ones. In the latter, both the temperature of the bath and the stiffness of the harmonic trap vary in time, but in such a way that the average heat vanishes for all times. All branches are swept at a finite rate and, therefore, the corresponding processes are irreversible, not quasi-static. Specifically, we are interested in optimising the heat engine to deliver the maximum power and characterising the corresponding values of the physical parameters. The efficiency at maximum power is shown to be very close to the Curzon-Ahlborn bound over the whole range of the ratio of temperatures of the two thermal baths, pointing to the near optimality of the proposed protocol.\n\n• 1. Atomes Froids (LCAR)\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Chaos-assisted tunneling resonances in a synthetic Floquet superlattice – Archive ouverte HAL\n\n### Maxime Arnal 1 Gabriel Chatelain 1 Maxime Martinez 2 Nathan Dupont 1 Olivier Giraud 3 D. Ullmo 3 Bertrand Georgeot 2 Gabriel Lemarié 2 Juliette Billy 1 David Guéry-Odelin 1\n\n#### Maxime Arnal, Gabriel Chatelain, Maxime Martinez, Nathan Dupont, Olivier Giraud, et al.. Chaos-assisted tunneling resonances in a synthetic Floquet superlattice. Science Advances, 2020, 6 (38), pp.eabc4886. ⟨10.1126/sciadv.abc4886⟩. ⟨hal-02534927⟩\n\nThe field of quantum simulation, which aims at using a tunable quantum system to simulate another, has been developing fast in the past years as an alternative to the all-purpose quantum computer. In particular, the use of temporal driving has attracted a huge interest recently as it was shown that certain fast drivings can create new topological effects, while a strong driving leads to e.g. Anderson localization physics. In this work, we focus on the intermediate regime to observe a quantum chaos transport mechanism called chaos-assisted tunneling which provides new possibilities of control for quantum simulation. Indeed, this regime generates a rich classical phase space where stable trajectories form islands surrounded by a large sea of unstable chaotic orbits. This mimics an effective superlattice for the quantum states localized in the regular islands, with new controllable tunneling properties. Besides the standard textbook tunneling through a potential barrier, chaos-assisted tunneling corresponds to a much richer tunneling process where the coupling between quantum states located in neighboring regular islands is mediated by other states spread over the chaotic sea. This process induces sharp resonances where the tunneling rate varies by orders of magnitude over a short range of parameters. We experimentally demonstrate and characterize these resonances for the first time in a quantum system. This opens the way to new kinds of quantum simulations with long-range transport and new types of control of quantum systems through complexity.\n\n• 1. Atomes Froids (LCAR)\n• 2. Information et Chaos Quantiques (LPT)\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Chiral active hexatics: Giant number fluctuations, waves and destruction of order – Archive ouverte HAL\n\n### Ananyo Maitra 1 Martin Lenz 2, 3 Raphael Voituriez 1\n\n#### Ananyo Maitra, Martin Lenz, Raphael Voituriez. Chiral active hexatics: Giant number fluctuations, waves and destruction of order. Physical Review Letters, American Physical Society, 2020. ⟨hal-03085233⟩\n\nActive materials, composed of internally driven particles, have been shown to have properties that are qualitatively distinct matter at thermal equilibrium. However, most spectacular departures from equilibrium phase behaviour were thought to be confined to systems with polar or nematic asymmetry. In this paper we show that such departures are also displayed in more symmetric phases such as hexatics if in addition the constituent particles have chiral asymmetry. We show that chiral active hexatics whose rotation rate does not depend on density, have giant number fluctuations. If the rotation-rate depends on density, the giant number fluctuations are suppressed due to a novel orientation-density sound mode with a linear dispersion which propagates even in the overdamped limit. However, we demonstrate that beyond a finite but large lengthscale, a chirality and activity-induced relevant nonlinearity invalidates the predictions of the linear theory and destroys the hexatic order. In addition, we show that activity modifies the interactions between defects in the active chiral hexatic phase, making them non-mutual. Finally, to demonstrate the generality of a chiral active hexatic phase we show that it results from the melting of chiral active crystals in finite systems.\n\n• 1. LJP - Laboratoire Jean Perrin\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 3. PMMH (UMR_7636) - Physique et mécanique des milieux hétérogenes\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Collective excitations of a one-dimensional quantum droplet – Archive ouverte HAL\n\n### Marek TylutkiGrigori E. AstrakharchikBoris A. Malomed 1 Dmitry S. Petrov 2 Grigori Astrakharchik 3 Boris Malomed 4 Dmitry Petrov\n\n#### Marek Tylutki, Grigori E. Astrakharchik, Boris A. Malomed, Dmitry S. Petrov, Grigori Astrakharchik, et al.. Collective excitations of a one-dimensional quantum droplet. Physical Review A, American Physical Society 2020, 101 (5), ⟨10.1103/PhysRevA.101.051601⟩. ⟨hal-02881226⟩\n\nWe calculate the excitation spectrum of a one-dimensional self-bound quantum droplet in a two-component bosonic mixture described by the Gross-Pitaevskii equation (GPE) with cubic and quadratic nonlinearities. The cubic term originates from the mean-field energy of the mixture proportional to the effective coupling constant $\\delta g$, whereas the quadratic nonlinearity corresponds to the attractive beyond-mean-field contribution. The droplet properties are governed by a control parameter $\\gamma\\propto \\delta g N^{2/3}$, where $N$ is the particle number. For large $\\gamma>0$ the droplet features the flat-top shape with the discrete part of its spectrum consisting of plane-wave Bogoliubov phonons propagating through the flat-density bulk and reflected by edges of the droplet. With decreasing $\\gamma$ these modes cross into the continuum, sequentially crossing the particle-emission threshold at specific critical values. A notable exception is the breathing mode which we find to be always bound. The balance point $\\gamma = 0$ provides implementation of a system governed by the GPE with an unusual quadratic nonlinearity. This case is characterized by the ratio of the breathing-mode frequency to the particle-emission threshold equal to 0.8904. As $\\gamma$ tends to $-\\infty$ this ratio tends to 1 and the droplet transforms into the soliton solution of the integrable cubic GPE.\n\n• 1. Tel Aviv University [Tel Aviv]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 3. UPC - Universitat Politècnica de Catalunya [BarcelonaTech]\n• 4. Department of Interdisciplinary Studies\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Comment on “Effective Confining Potential of Quantum States in Disordered Media” – Archive ouverte HAL\n\n### Alain Comtet 1 Christophe Texier 1\n\n#### Alain Comtet, Christophe Texier. Comment on “Effective Confining Potential of Quantum States in Disordered Media”. Physical Review Letters, American Physical Society, 2020, 124 (21), ⟨10.1103/PhysRevLett.124.219701⟩. ⟨hal-02881221⟩\n\nWe provide some analytical tests of the density of states estimation from the \"localization landscape\" approach of Ref. [Phys. Rev. Lett. 116, 056602 (2016)]. We consider two different solvable models for which we obtain the distribution of the landscape function and argue that the precise spectral singularities are not reproduced by the estimation of the landscape approach.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Critical energy landscape of linear soft spheres – Archive ouverte HAL\n\n### Silvio Franz 1 Antonio Sclocchi 1 Pierfrancesco Urbani 2\n\n#### Silvio Franz, Antonio Sclocchi, Pierfrancesco Urbani. Critical energy landscape of linear soft spheres. SciPost Physics, SciPost Foundation, 2020. ⟨hal-02908534⟩\n\nWe show that soft spheres interacting with a linear ramp potential when overcompressed beyond the jamming point fall in an amorphous solid phase which is critical, mechanically marginally stable and share many features with the jamming point itself. In the whole phase, the relevant local minima of the potential energy landscape display an isostatic contact network of perfectly touching spheres whose statistics is controlled by an infinite lengthscale. Excitations around such energy minima are non-linear, system spanning, and characterized by a set of non-trivial critical exponents. We perform numerical simulations to measure their values and show that, while they coincide, within numerical precision, with the critical exponents appearing at jamming, the nature of the corresponding excitations is richer. Therefore, linear soft spheres appear as a novel class of finite dimensional systems that self-organize into new, critical, marginally stable, states.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. IPHT - Institut de Physique Théorique - UMR CNRS 3681\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Current fluctuations in noninteracting run-and-tumble particles in one dimension – Archive ouverte HAL\n\n### Tirthankar Banerjee 1 Satya N. Majumdar 1 Alberto Rosso 1 Satya Majumdar 1 Gregory Schehr 1\n\n#### Tirthankar Banerjee, Satya N. Majumdar, Alberto Rosso, Satya Majumdar, Gregory Schehr. Current fluctuations in noninteracting run-and-tumble particles in one dimension. Physical Review E , American Physical Society (APS), 2020, 101 (5), ⟨10.1103/PhysRevE.101.052101⟩. ⟨hal-02565189⟩\n\nWe present a general framework to study the distribution of the flux through the origin up to time $t$, in a non-interacting one-dimensional system of particles with a step initial condition with a fixed density $\\rho$ of particles to the left of the origin. We focus principally on two cases: (i) when the particles undergo diffusive dynamics (passive case) and (ii) run-and-tumble dynamics for each particle (active case). In analogy with disordered systems, we consider the flux distribution both for the annealed and the quenched initial conditions, for the passive and active particles. In the annealed case, we show that, for arbitrary particle dynamics, the flux distribution is a Poissonian with a mean $\\mu(t)$ that we compute exactly in terms of the Green's function of the single particle dynamics. For the quenched case, we show that, for the run-and-tumble dynamics, the quenched flux distribution takes an anomalous large deviation form at large times $P_{\\rm qu}(Q,t) \\sim \\exp\\left[-\\rho\\, v_0\\, \\gamma \\, t^2 \\psi_{\\rm RTP}\\left(\\frac{Q}{\\rho v_0\\,t} \\right) \\right]$, where $\\gamma$ is the rate of tumbling and $v_0$ is the ballistic speed between two successive tumblings. In this paper, we compute the rate function $\\psi_{\\rm RTP}(q)$ and show that it is nontrivial. Our method also gives access to the probability of the rare event that, at time $t$, there is no particle to the right of the origin. For diffusive and run-and-tumble dynamics, we find that this probability decays with time as a stretched exponential, $\\sim \\exp(-c\\, \\sqrt{t})$ where the constant $c$ can be computed exactly. We verify our results for these large deviations by using an importance sampling Monte-Carlo method.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Density scaling of generalized Lennard-Jones fluids in different dimensions – Archive ouverte HAL\n\n### Thibaud Maimbourg 1 Jeppe C. DyreLorenzo CostigliolaJeppe Dyre\n\n#### Thibaud Maimbourg, Jeppe C. Dyre, Lorenzo Costigliola, Jeppe Dyre. Density scaling of generalized Lennard-Jones fluids in different dimensions. SciPost Physics, SciPost Foundation, 2020, 9 (6), ⟨10.21468/SciPostPhys.9.6.090⟩. ⟨hal-03117941⟩\n\nLiquids displaying strong virial-potential energy correlations conform to an approximate density scaling of their structural and dynamical observables. This scaling property does not extend to the entire phase diagram, in general. The validity of the scaling can be quantified by a correlation coefficient. In this work a simple scheme to predict the correlation coefficient and the density-scaling exponent is presented. Although this scheme is exact only in the dilute gas regime or in high dimension d, a comparison with results from molecular dynamics simulations in d = 1 to 4 shows that it reproduces well the behavior of generalized Lennard-Jones systems in a large portion of the fluid phase.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Dispersionless evolution of inviscid nonlinear pulses – Archive ouverte HAL\n\n### M. Isoard 1 N. Pavloff 1 A. M. Kamchatnov 2\n\n#### M. Isoard, N. Pavloff, A. M. Kamchatnov. Dispersionless evolution of inviscid nonlinear pulses. EPL - Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2020. ⟨hal-02565206⟩\n\nWe consider the one-dimensional dynamics of nonlinear non-dispersive waves. The problem can be mapped onto a linear one by means of the hodograph transform. We propose an approximate scheme for solving the corresponding Euler-Poisson equation which is valid for any kind of nonlinearity. The approach is exact for monoatomic classical gas and agrees very well with exact results and numerical simulations for other systems. We also provide a simple and accurate determination of the wave breaking time for typical initial conditions.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. Institute of Spectroscopy\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Distribution of the time between maximum and minimum of random walks – Archive ouverte HAL\n\n### Francesco Mori 1 Satya N. Majumdar 1 Satya Majumdar 1 Gregory Schehr 1\n\n#### Francesco Mori, Satya N. Majumdar, Satya Majumdar, Gregory Schehr. Distribution of the time between maximum and minimum of random walks. Physical Review E , American Physical Society (APS), 2020, 101 (5), ⟨10.1103/PhysRevE.101.052111⟩. ⟨hal-02881215⟩\n\nWe consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\\tau=t_{\\min}-t_{\\max}$ between the time $t_{\\min}$ of the global minimum and the time $t_{\\max}$ of the global maximum. We extend this result to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. In both cases, we compute analytically the first few moments of $\\tau$, as well as the covariance of $t_{\\max}$ and $t_{\\min}$, showing that these times are anti-correlated. We demonstrate that the distribution of $\\tau$ for Brownian motion is valid for discrete-time random walks with $n$ steps and with a finite jump variance, in the limit $n\\to \\infty$. In the case of L\\'evy flights, which have a divergent jump variance, we numerically verify that the distribution of $\\tau$ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event \"$\\tau = n$\" is exactly $1/(2n)$ for any finite $n$, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of $(1+1)$-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size $L$. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)].\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Efficient generation of random derangements with the expected distribution of cycle lengths – Archive ouverte HAL\n\n### J. Ricardo G. Mendonça 1, 2\n\n#### J. Ricardo G. Mendonça. Efficient generation of random derangements with the expected distribution of cycle lengths. Computational and Applied Mathematics, Springer Verlag, 2020, 39 (3), ⟨10.1007/s40314-020-01295-4⟩. ⟨hal-03085042⟩\n\nWe show how to generate random derangements efficiently by two different techniques: random restricted transpositions and sequential importance sampling. The algorithm employing restricted transpositions can also be used to generate random fixed-point-free involutions only, a.k.a. random perfect matchings on the complete graph. Our data indicate that the algorithms generate random samples with the expected distribution of cycle lengths, which we derive, and for relatively small samples, which can actually be very large in absolute numbers, we argue that they generate samples indistinguishable from the uniform distribution. Both algorithms are simple to understand and implement and possess a performance comparable to or better than those of currently known methods. Simulations suggest that the mixing time of the algorithm based on random restricted transpositions (in the total variance distance with respect to the distribution of cycle lengths) is $O(n^{a}\\log{n}^{2})$ with $a \\simeq \\frac{1}{2}$ and $n$ the length of the derangement. We prove that the sequential importance sampling algorithm generates random derangements in $O(n)$ time with probability $O(1/n)$ of failing.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. USP - Universidade de São Paulo\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Engineered Swift Equilibration of brownian particles: consequences of hydrodynamic coupling – Archive ouverte HAL\n\n### Salambô Dago 1 Benjamin Besga 1 Raphaël Mothe 1 David Guéry-Odelin 2 Emmanuel Trizac 3 Artyom Petrosyan 1 Ludovic Bellon 1 Sergio Ciliberto 1\n\n#### Salambô Dago, Benjamin Besga, Raphaël Mothe, David Guéry-Odelin, Emmanuel Trizac, et al.. Engineered Swift Equilibration of brownian particles: consequences of hydrodynamic coupling. SciPost Physics, SciPost Foundation, 2020, 9 (5), ⟨10.21468/SciPostPhys.9.5.064⟩. ⟨ensl-02570537v2⟩\n\nWe present a detailed theoretical and experimental analysis of Engineered Swift Equilibration (ESE) protocols applied to two hydrodynamically coupled colloids in optical traps. The second particle slightly perturbs (10% at most) the response to an ESE compression applied to a single particle. This effect is quantitatively explained by a model of hydrodynamic coupling. We then design a coupled ESE protocol for the two particles, allowing the perfect control of one target particle while the second is enslaved to the first. The calibration errors and the limitations of the model are finally discussed in details.\n\n• 1. Phys-ENS - Laboratoire de Physique de l'ENS Lyon\n• 2. Atomes Froids (LCAR)\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Experimental Study of Collective Pedestrian Dynamics – Archive ouverte HAL\n\n### Cécile Appert-RollandJulien PettréAnne-Hélène OlivierWilliam WarrenAymeric Duigou-MajumdarEtienne PinsardAlexandre Nicolas 1\n\n#### Cécile Appert-Rolland, Julien Pettré, Anne-Hélène Olivier, William Warren, Aymeric Duigou-Majumdar, et al.. Experimental Study of Collective Pedestrian Dynamics. Collective Dynamics, 2020, 5, pp.A109. ⟨10.17815/CD.2020.109⟩. ⟨hal-02992406⟩\n\nWe report on two series of experiments, conducted in the frame of two different collaborations designed to study how pedestrians adapt their trajectories and velocities in groups or crowds. Strong emphasis is put on the motivations for the chosen protocols and the experimental implementation. The first series deals with pattern formation, interactions between pedestrians, and decision-making in pedestrian groups at low to medium densities. In particular, we show how pedestrians adapt their headways in single-file motion depending on the (prescribed) leader's velocity. The second series of experiments focuses on static crowds at higher densities, a situation that can be critical in real life and in which the pedestrians' choices of motion are strongly constrained sterically. More precisely, we study the crowd's response to its crossing by a pedestrian or a cylindrical obstacle of 74cm in diameter. In the latter case, for a moderately dense crowd, we observe displacements that quickly decay with the minimal distance to the obstacle, over a lengthscale of the order of the meter.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Extreme value statistics of correlated random variables: a pedagogical review – Archive ouverte HAL\n\n### Satya N. Majumdar 1 Arnab PalGregory Schehr 1\n\n#### Satya N. Majumdar, Arnab Pal, Gregory Schehr. Extreme value statistics of correlated random variables: a pedagogical review. Physics Reports, Elsevier, 2020, ⟨10.10667⟩. ⟨hal-02512248⟩\n\nExtreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of uncorrelated' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this review, we will first recall the classical EVS for uncorrelated variables and discuss the three universality classes of extreme value limiting distribution, known as the Gumbel, Fr\\'echet and Weibull distribution. We then show that, for weakly correlated random variables with a finite correlation length/time, the limiting extreme value distribution can still be inferred from that of the uncorrelated variables using a renormalisation group-like argument. Finally, we consider the most interesting examples of strongly correlated variables for which there are very few exact results for the EVS. We discuss few examples of such strongly correlated systems (such as the Brownian motion and the eigenvalues of a random matrix) where some analytical progress can be made. We also discuss other observables related to extremes, such as the density of near-extreme events, time at which an extreme value occurs, order and record statistics, etc.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Few-body bound states of two-dimensional bosons – Archive ouverte HAL\n\n### G. Guijarro 1 G. E. Astrakharchik 1 J. Boronat 1 B. BazakD. S. Petrov 2\n\n#### G. Guijarro, G. E. Astrakharchik, J. Boronat, B. Bazak, D. S. Petrov. Few-body bound states of two-dimensional bosons. Physical Review A, American Physical Society 2020, ⟨10.1103/PhysRevA.101.041602⟩. ⟨hal-02537195⟩\n\nWe study clusters of the type A$_N$B$_M$ with $N\\leq M\\leq 3$ in a two-dimensional mixture of A and B bosons, with attractive AB and equally repulsive AA and BB interactions. In order to check universal aspects of the problem, we choose two very different models: dipolar bosons in a bilayer geometry and particles interacting via separable Gaussian potentials. We find that all the considered clusters are bound and that their energies are universal functions of the scattering lengths $a_{AB}$ and $a_{AA}=a_{BB}$, for sufficiently large attraction-to-repulsion ratios $a_{AB}/a_{BB}$. When $a_{AB}/a_{BB}$ decreases below $\\approx 10$, the dimer-dimer interaction changes from attractive to repulsive and the population-balanced AABB and AAABBB clusters break into AB dimers. Calculating the AAABBB hexamer energy just below this threshold, we find an effective three-dimer repulsion which may have important implications for the many-body problem, particularly for observing liquid and supersolid states of dipolar dimers in the bilayer geometry. The population-imbalanced ABB trimer, ABBB tetramer, and AABBB pentamer remain bound beyond the dimer-dimer threshold. In the dipolar model, they break up at $a_{AB}\\approx 2 a_{BB}$ where the atom-dimer interaction switches to repulsion.\n\n• 1. UPC - Universitat Politècnica de Catalunya [Barcelona]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions – Archive ouverte HAL\n\n### Etienne GranetMaurizio Fagotti 1 Fabian H.L. Essler\n\n#### Etienne Granet, Maurizio Fagotti, Fabian H.L. Essler. Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions. SciPost Phys., 2020, 9 (3), pp.033. ⟨10.21468/SciPostPhys.9.3.033⟩. ⟨hal-02542815⟩\n\nWe consider the problems of calculating the dynamical order parameter two-point function at finite temperatures and the one-point function after a quantum quench in the transverse field Ising chain. Both of these can be expressed in terms of form factor sums in the basis of physical excitations of the model. We develop a general framework for carrying out these sums based on a decomposition of form factors into partial fractions, which leads to a factorization of the multiple sums and permits them to be evaluated asymptotically. This naturally leads to systematic low density expansions. At late times these expansions can be summed to all orders by means of a determinant representation. Our method has a natural generalization to semi-local operators in interacting integrable models.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Finite-time adiabatic processes: Derivation and speed limit – Archive ouverte HAL\n\n### Carlos Plata 1 David Guéry-Odelin 2 Emmanuel Trizac 3 Antonio Prados 4\n\n#### Carlos Plata, David Guéry-Odelin, Emmanuel Trizac, Antonio Prados. Finite-time adiabatic processes: Derivation and speed limit. Physical Review E , American Physical Society (APS), 2020, 101 (3), ⟨10.1103/PhysRevE.101.032129⟩. ⟨hal-02535447⟩\n\nObtaining adiabatic processes that connect equilibrium states in a given time represents a challenge for mesoscopic systems. In this paper, we explicitly show how to build these finite-time adiabatic processes for an overdamped Brownian particle in an arbitrary potential, a system that is relevant both at the conceptual and the practical level. This is achieved by jointly engineering the time evolutions of the binding potential and the fluid temperature. Moreover, we prove that the second principle imposes a speed limit for such adiabatic transformations: there appears a minimum time to connect the initial and final states. This minimum time can be explicitly calculated for a general compression/decompression situation.\n\n• 1. Padova University\n• 2. Atomes Froids (LCAR)\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 4. Universidad de Sevilla\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Fluctuations of the Product of Random Matrices and Generalized Lyapunov Exponent – Archive ouverte HAL\n\n### Christophe Texier 1\n\n#### Christophe Texier. Fluctuations of the Product of Random Matrices and Generalized Lyapunov Exponent. Journal of Statistical Physics, Springer Verlag, 2020, 181 (3), pp.990-1051. ⟨10.1007/s10955-020-02617-w⟩. ⟨hal-03017028⟩\n\nI present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products $\\Pi_n=M_nM_{n-1}\\cdots M_1$, where $M_i$'s are i.i.d.. Following Tutubalin [Theor. Probab. Appl. {\\bf 10}, 15 (1965)], the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group. The formalism is illustrated by considering products of random matrices from the group $\\mathrm{SL}(2,\\mathbb{R})$ where explicit calculations are possible. For concreteness, I study in detail transfer matrix products for the one-dimensional Schr\\\"odinger equation where the random potential is a L\\'evy noise (derivative of a L\\'evy process). In this case, I obtain a general formula for the variance of $\\ln||\\Pi_n||$ and for the variance of $\\ln|\\psi(x)|$, where $\\psi(x)$ is the wavefunction, in terms of a single integral involving the Fourier transform of the invariant density of the matrix product. Finally I discuss the continuum limit of random matrix products (matrices close to the identity ). In particular, I investigate a simple case where the spectral problem providing the generalized Lyapunov exponent can be solved exactly.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Formation of dispersive shock waves in a saturable nonlinear medium – Archive ouverte HAL\n\n### Sergey K. IvanovJules-Elémir SuchorskiAnatoly M. KamchatnovMathieu Isoard 1 Nicolas Pavloff 1\n\n#### Sergey K. Ivanov, Jules-Elémir Suchorski, Anatoly M. Kamchatnov, Mathieu Isoard, Nicolas Pavloff. Formation of dispersive shock waves in a saturable nonlinear medium. Physical Review E , American Physical Society (APS), 2020, 102 (3), ⟨10.1103/PhysRevE.102.032215⟩. ⟨hal-03017066⟩\n\nWe use the Gurevich-Pitaevskii approach based on the Whitham averaging method for studying the formation of dispersive shock waves in an intense light pulse propagating through a saturable nonlinear medium. Although the Whitham modulation equations cannot be diagonalized in this case, the main characteristics of the dispersive shock can be derived by means of an analysis of the properties of these equations at the boundaries of the shock. Our approach generalizes a previous analysis of step-like initial intensity distributions to a more realistic type of initial light pulse and makes it possible to determine, in a setting of experimental interest, the value of measurable quantities such as the wave-breaking time or the position and light intensity of the shock edges.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Freezing transition in the barrier crossing rate of a diffusing particle – Archive ouverte HAL\n\n### Sanjib Sabhapandit 1 Satya N. Majumdar 2\n\n#### Sanjib Sabhapandit, Satya N. Majumdar. Freezing transition in the barrier crossing rate of a diffusing particle. Physical Review Letters, American Physical Society, 2020. ⟨hal-03010264⟩\n\nWe study the decay rate $\\theta(a)$ that chracterizes the late time exponential decay of the first-passage probability density, $F_a(t|0) \\sim e^{-\\theta(a)\\, t}$, of a diffusing particle in a one dimensional confining potential $U(x)$, starting from the origin, to a position located at $a>0$. For general confining potential $U(x)$ we show that $\\theta(a)$, a measure of the barrier (located at $a$) crossing rate, has three distinct behaviors as a function of $a$, depending on the tail of $U(x)$ as $x\\to -\\infty$. In particular, for potentials behaving as $U(x)\\sim |x|$ when $x\\to -\\infty$, we show that a novel freezing transition occurs at a critical value $a=a_c$, i.e, $\\theta(a)$ increases monotonically as $a$ decreases till $a_c$, and for $a \\le a_c$ it freezes to $\\theta (a)=\\theta(a_c)$. Our results are established using a general mapping to a quantum problem and by exact solution in three representative cases, supported by numerical simulations. We show that the freezing transition occurs when in the associated quantum problem, the gap between the ground state (bound) and the continuum of scattering states vanishes.\n\n• 1. Raman Research Institute\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Human running performance from real-world big data – Archive ouverte HAL\n\n### Thorsten Emig 1 Jussi Peltonen\n\n#### Thorsten Emig, Jussi Peltonen. Human running performance from real-world big data. Nature Communications, Nature Publishing Group, 2020, 11 (1), ⟨10.1038/s41467-020-18737-6⟩. ⟨hal-03065483⟩\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Intermittent resetting potentials – Archive ouverte HAL\n\n### Gabriel Mercado-VásquezDenis BoyerSatya N. Majumdar 1 Grégory Schehr 1\n\n#### Gabriel Mercado-Vásquez, Denis Boyer, Satya N. Majumdar, Grégory Schehr. Intermittent resetting potentials. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2020, 2020 (11), pp.113203. ⟨10.1088/1742-5468/abc1d9⟩. ⟨hal-03010255⟩\n\nWe study the non-equilibrium steady states and first passage properties of a Brownian particle with position $X$ subject to an external confining potential of the form $V(X)=\\mu|X|$, and that is switched on and off stochastically. Applying the potential intermittently generates a physically realistic diffusion process with stochastic resetting toward the origin, a topic which has recently attracted a considerable interest in a variety of theoretical contexts but has remained challenging to implement in lab experiments. The present system exhibits rich features, not observed in previous resetting models. The mean time needed by a particle starting from the potential minimum to reach an absorbing target located at a certain distance can be minimized with respect to the switch-on and switch-off rates. The optimal rates undergo continuous or discontinuous transitions as the potential strength $\\mu$ is varied across non-trivial values. A discontinuous transition with metastable behavior is also observed for the optimal strength at fixed rates.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Ising model with stochastic resetting – Archive ouverte HAL\n\n### Matteo Magoni 1 Satya N. Majumdar 1 Grégory Schehr 1\n\n#### Matteo Magoni, Satya N. Majumdar, Grégory Schehr. Ising model with stochastic resetting. Physical Review Research, American Physical Society, 2020, 2 (3), ⟨10.1103/PhysRevResearch.2.033182⟩. ⟨hal-03010228⟩\n\nWe study the stationary properties of the Ising model that, while evolving towards its equilibrium state at temperature $T$ according to the Glauber dynamics, is stochastically reset to its fixed initial configuration with magnetisation $m_0$ at a constant rate $r$. Resetting breaks detailed balance and drives the system to a non-equilibrium stationary state where the magnetisation acquires a nontrivial distribution, leading to a rich phase diagram in the $(T,r)$ plane. We establish these results exactly in one-dimension and present scaling arguments supported by numerical simulations in two-dimensions. We show that resetting gives rise to a novel \"pseudo-ferro\" phase in the $(T,r)$ plane for $r > r^*(T)$ and $T>T_c$ where $r^*(T)$ is a crossover line separating the pseudo-ferro phase from a paramagnetic phase. This pseudo-ferro phase is characterised by a non-zero typical magnetisation and a vanishing gap near $m=0$ of the magnetisation distribution.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Large deviations of glassy effective potentials – Archive ouverte HAL\n\n### Silvio Franz 1 Jacopo Rocchi 1\n\n#### Silvio Franz, Jacopo Rocchi. Large deviations of glassy effective potentials. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2020, 53 (48), pp.485002. ⟨10.1088/1751-8121/ab9aeb⟩. ⟨hal-03017029⟩\n\nThe theory of glassy fluctuations can be formulated in terms of disordered effective potentials. While the properties of the average potentials are well understood, the study of the fluctuations has been so far quite limited. Close to the MCT transition, fluctuations induced by the dynamical heterogeneities in supercooled liquids can be described by a cubic field theory in presence of a random field term. In this paper we set up the general problem of the large deviations going beyond the assumption of the vicinity to $T_{MCT}$ and analyze it in the paradigmatic case of spherical ($p$-spin) glass models. This tool can be applied to study the probability of the observation of a dynamics with memory of the initial condition in regimes where, typically, the correlation $C(t,0)$ decays to zero at long times, at finite $T$ and at $T=0$.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Last-Passage Time for Linear Diffusions and Application to the Emptying Time of a Box – Archive ouverte HAL\n\n### Alain Comtet 1 Françoise Cornu 1 Grégory Schehr 1\n\n#### Alain Comtet, Françoise Cornu, Grégory Schehr. Last-Passage Time for Linear Diffusions and Application to the Emptying Time of a Box. Journal of Statistical Physics, Springer Verlag, 2020, ⟨10.1007/s10955-020-02637-6⟩. ⟨hal-02988500⟩\n\nWe study the statistics of last-passage time for linear diffusions. First we present an elementary derivation of the Laplace transform of the probability density of the last-passage time, thus recovering known results from the mathematical literature. We then illustrate them on several explicit examples. In a second step we study the spectral properties of the Schr\\\"{o}dinger operator associated to such diffusions in an even potential $U(x) = U(-x)$, unveiling the role played by the so-called Weyl coefficient. Indeed, in this case, our approach allows us to relate the last-passage times for dual diffusions (i.e., diffusions driven by opposite force fields) and to obtain new explicit formulae for the mean last-passage time. We further show that, for such even potentials, the small time $t$ expansion of the mean last-passage time on the interval $[0,t]$ involves the Korteveg-de Vries invariants, which are well known in the theory of Schr\\\"odinger operators. Finally, we apply these results to study the emptying time of a one-dimensional box, of size $L$, containing $N$ independent Brownian particles subjected to a constant drift. In the scaling limit where both $N \\to \\infty$ and $L \\to \\infty$, keeping the density $\\rho = N/L$ fixed, we show that the limiting density of the emptying time is given by a Gumbel distribution. Our analysis provides a new example of the applications of extreme value statistics to out-of-equilibrium systems.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers – Archive ouverte HAL\n\n### Stéphane Ouvry 1 Alexios Polychronakos\n\n#### Stéphane Ouvry, Alexios Polychronakos. Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers. Nucl.Phys.B, 2020, 960, pp.115174. ⟨10.1016/j.nuclphysb.2020.115174⟩. ⟨hal-02886896⟩\n\nExplicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walks, and in particular for the triangular lattice chiral walks recently introduced by the authors. A key element in the enumeration is the derivation of some identities involving some remarkable trigonometric sums –which are also important building blocks of non trivial quantum models such as the Hofstadter model– and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Apéry-like numbers, the cousins of the Apéry numbers which play a central role in irrationality considerations for ζ(2) and ζ(3) .\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Mapping and Modeling the Nanomechanics of Bare and Protein-Coated Lipid Nanotubes – Archive ouverte HAL\n\n### Guillaume Lamour 1 Antoine Allard 1, 2 Juan Pelta 1 Sid Labdi 1 Martin Lenz 3 Clément Campillo 1\n\n#### Guillaume Lamour, Antoine Allard, Juan Pelta, Sid Labdi, Martin Lenz, et al.. Mapping and Modeling the Nanomechanics of Bare and Protein-Coated Lipid Nanotubes. Physical Review X, American Physical Society, 2020, 10 (1), pp.011031. ⟨10.1103/PhysRevX.10.011031⟩. ⟨hal-02512272⟩\n\nMembrane nanotubes are continuously assembled and disassembled by the cell to generate and dispatch transport vesicles, for instance, in endocytosis. While these processes crucially involve the ill-understood local mechanics of the nanotube, existing micromanipulation assays only give access to its global mechanical properties. Here we develop a new platform to study this local mechanics using atomic force microscopy (AFM). On a single coverslip we quickly generate millions of substrate-bound nanotubes, out of which dozens can be imaged by AFM in a single experiment. A full theoretical description of the AFM tip-membrane interaction allows us to accurately relate AFM measurements of the nanotube heights, widths, and rigidities to the membrane bending rigidity and tension, thus demonstrating our assay as an accurate probe of nanotube mechanics. We reveal a universal relationship between nanotube height and rigidity, which is unaffected by the specific conditions of attachment to the substrate. Moreover, we show that the parabolic shape of force-displacement curves results from thermal fluctuations of the membrane that collides intermittently with the AFM tip. We also show that membrane nanotubes can exhibit high resilience against extreme lateral compression. Finally, we mimic in vivo actin polymerization on nanotubes and use AFM to assess the induced changes in nanotube physical properties. Our assay may help unravel the local mechanics of membrane-protein interactions, including membrane remodeling in nanotube scission and vesicle formation.\n\n• 1. LAMBE - UMR 8587 - Laboratoire Analyse, Modélisation et Matériaux pour la Biologie et l'Environnement\n• 2. PCC - Physico-Chimie-Curie\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Multi-component colloidal gels: interplay between structure and mechanical properties – Archive ouverte HAL\n\n### Claudia Ferreiro-CordovaMehdi Bouzid 1 Emanuela del GadoGiuseppe Foffi 2 Claudia Ferreiro-Córdova\n\n#### Claudia Ferreiro-Cordova, Mehdi Bouzid, Emanuela del Gado, Giuseppe Foffi, Claudia Ferreiro-Córdova. Multi-component colloidal gels: interplay between structure and mechanical properties. Soft Matter, Royal Society of Chemistry, 2020, 16 (18), pp.4414-4421. ⟨10.1039/C9SM02410G⟩. ⟨hal-02881157⟩\n\nWe present a detailed numerical study of multi-component colloidal gels interacting sterically and obtained by arrested phase separation. Under deformation, we found that the interplay between the different intertwined networks is key. Increasing the number of component leads to softer solids that can accomodate progressively larger strain before yielding. The simulations highlight how this is the direct consequence of the purely repulsive interactions between the different components, which end up enhancing the linear response of the material. Our work {provides new insight into mechanisms at play for controlling the material properties and open the road to new design principles for} soft composite solids\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. LPS - Laboratoire de Physique des Solides\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Non-Hermitian quantum impurity systems in and out of equilibrium: Noninteracting case – Archive ouverte HAL\n\n### Takato YoshimuraKemal Bidzhiev 1 Hubert Saleur\n\n#### Takato Yoshimura, Kemal Bidzhiev, Hubert Saleur. Non-Hermitian quantum impurity systems in and out of equilibrium: Noninteracting case. Physical Review B, American Physical Society, 2020, 102 (12), ⟨10.1103/PhysRevB.102.125124⟩. ⟨hal-03017010⟩\n\nWe provide systematic analysis on a non-Hermitian PT -symmetric quantum impurity system both in and out of equilibrium, based on exact computations. In order to understand the interplay between non-Hermiticity and Kondo physics, we focus on a prototypical noninteracting impurity system, the resonant level model, with complex coupling constants. Explicitly constructing biorthogonal basis, we study its thermodynamic properties as well as the Loschmidt echo starting from the initially disconnected two free fermion chains. Remarkably, we observe the universal crossover physics in the Loschmidt echo, both in the PT broken and unbroken regimes. We also find that the ground state quantities we compute in the PT broken regime can be obtained by analytic continuation. It turns out that Kondo screening ceases to exist in the PT broken regime, which was also previously predicted in the non-hermitian Kondo model. All the analytical results are corroborated against biorthogonal free fermion numerics.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids – Archive ouverte HAL\n\n### Alessandro Manacorda 1 Gregory Schehr 2 Francesco Zamponi 1\n\n#### Alessandro Manacorda, Gregory Schehr, Francesco Zamponi. Numerical solution of the dynamical mean field theory of infinite-dimensional equilibrium liquids. Journal of Chemical Physics, American Institute of Physics, 2020, 152 (16), pp.164506. ⟨10.1063/5.0007036⟩. ⟨hal-02554137⟩\n\n• 1. Systèmes Désordonnés et Applications\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States – Archive ouverte HAL\n\n### John MartinStefan WeigertOlivier Giraud 1\n\n#### John Martin, Stefan Weigert, Olivier Giraud. Optimal Detection of Rotations about Unknown Axes by Coherent and Anticoherent States. Quantum, Verein, 2020. ⟨hal-02881098⟩\n\nCoherent and anticoherent states of spin systems up to spin j=2 are known to be optimal in order to detect rotations by a known angle but unknown rotation axis. These optimal quantum rotosensors are characterized by minimal fidelity, given by the overlap of a state before and after a rotation, averaged over all directions in space. We calculate a closed-form expression for the average fidelity in terms of anticoherent measures, valid for arbitrary values of the quantum number j. We identify optimal rotosensors (i) for arbitrary rotation angles in the case of spin quantum numbers up to j=7/2 and (ii) for small rotation angles in the case of spin quantum numbers up to j=5. The closed-form expression we derive allows us to explain the central role of anticoherence measures in the problem of optimal detection of rotation angles for arbitrary values of j.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Optimizing Brownian escape rates by potential shaping – Archive ouverte HAL\n\n### Marie Chupeau 1 Jannes GladrowAlexei Chepelianskii 2 Ulrich F. KeyserEmmanuel Trizac 1 Ulrich Keyser\n\n#### Marie Chupeau, Jannes Gladrow, Alexei Chepelianskii, Ulrich F. Keyser, Emmanuel Trizac, et al.. Optimizing Brownian escape rates by potential shaping. Proceedings of the National Academy of Sciences of the United States of America , National Academy of Sciences, 2020, 117 (3), pp.1383-1388. ⟨10.1073/pnas.1910677116⟩. ⟨hal-02512216⟩\n\nBrownian escape is key to a wealth of physico-chemical processes, including polymer folding, and information storage. The frequency of thermally activated energy barrier crossings is assumed to generally decrease exponentially with increasing barrier height. Here, we show experimentally that higher, fine-tuned barrier profiles result in significantly enhanced escape rates in breach of the intuition relying on the above scaling law, and address in theory the corresponding conditions for maximum speed-up. Importantly, our barriers end on the same energy on which they start. For overdamped dynamics, the achievable boost of escape rates is, in principle, unbounded so that the barrier optimization has to be regularized. We derive optimal profiles under two different regularizations, and uncover the efficiency of N-shaped barriers. We then demonstrate the viability of such a potential in automated microfluidic Brownian dynamics experiments using holographic optical tweezers and achieve a doubling of escape rates compared to unhindered Brownian motion. Finally, we show that this escape rate boost extends into the low-friction inertial regime.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. LPCT - Laboratoire de Physico-Chimie Théorique\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed p -Spin Model – Archive ouverte HAL\n\n### Giampaolo Folena 1 Silvio Franz 1 Federico Ricci-Tersenghi\n\n#### Giampaolo Folena, Silvio Franz, Federico Ricci-Tersenghi. Rethinking Mean-Field Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed p -Spin Model. Physical Review X, American Physical Society, 2020, 10 (3), ⟨10.1103/PhysRevX.10.031045⟩. ⟨hal-03017024⟩\n\nThe spherical p-spin model is not only a fundamental model in statistical mechanics of disordered system, but has recently gained popularity since many hard problems in machine learning can be mapped on it. Thus the study of the out of equilibrium dynamics in this model is interesting both for the glass physics and for its implications on algorithms solving NP-hard problems. We revisit the long-time limit of the out of equilibrium dynamics of mean-field spherical mixed p-spin models. We consider quenches (gradient descent dynamics) starting from initial conditions thermalized at some temperature in the ergodic phase. We perform numerical integration of the dynamical mean-field equations of the model and we find an unexpected dynamical phase transition. Below an onset temperature, higher than the dynamical transition temperature, the asymptotic energy goes below the \"threshold energy\" of the dominant marginal minima of the energy function and memory of the initial condition is kept. This behavior, not present in the pure spherical p-spin model, resembles closely the one observed in simulations of glass-forming liquids. We then investigate the nature of the asymptotic dynamics, finding an aging solution that relaxes towards deep marginal minima, evolving on a restricted marginal manifold. Careful analysis, however, rules out simple aging solutions. We compute the constrained complexity in the aim of connecting the asymptotic solution to the energy landscape.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Reversal of contractility as a signature of self-organization in cytoskeletal bundles – Archive ouverte HAL\n\n### Martin Lenz 1\n\n#### Martin Lenz. Reversal of contractility as a signature of self-organization in cytoskeletal bundles. eLife, eLife Sciences Publication, 2020, 9, ⟨10.7554/eLife.51751⟩. ⟨hal-02518848⟩\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Rigorous bounds on dynamical response functions and time-translation symmetry breaking – Archive ouverte HAL\n\n### Marko Medenjak 1 Tomaz Prosen 2 Lenart Zadnik 3\n\n#### Marko Medenjak, Tomaz Prosen, Lenart Zadnik. Rigorous bounds on dynamical response functions and time-translation symmetry breaking. SciPost Physics, SciPost Foundation, 2020, 9 (1), ⟨10.21468/SciPostPhys.9.1.003⟩. ⟨hal-02935659⟩\n\nDynamical response functions are standard tools for probing local physics near the equilibrium. They provide information about relaxation properties after the equilibrium state is weakly perturbed. In this paper we focus on systems which break the assumption of thermalization by exhibiting persistent temporal oscillations. We provide rigorous bounds on the Fourier components of dynamical response functions in terms of extensive or local dynamical symmetries, i.e., extensive or local operators with periodic time dependence. Additionally, we discuss the effects of spatially inhomogeneous dynamical symmetries. The bounds are explicitly implemented on the example of an interacting Flo-quet system, specifically in the integrable Trotterization of the Heisenberg XXZ model.\n\n• 1. IPM - institut de Physique Théorique Philippe Meyer\n• 2. FMF - Faculty of Mathematics and Physics [Ljubljana]\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Scalable quantum computing with qudits on a graph – Archive ouverte HAL\n\n### E. O. Kiktenko 1 A. S. NikolaevaPeng XuG. V. Shlyapnikov 2 A. K. Fedorov 3\n\n#### E. O. Kiktenko, A. S. Nikolaeva, Peng Xu, G. V. Shlyapnikov, A. K. Fedorov. Scalable quantum computing with qudits on a graph. Physical Review A, American Physical Society 2020, 101 (2), ⟨10.1103/PhysRevA.101.022304⟩. ⟨hal-02512218⟩\n\nWe show a significant reduction of the number of quantum operations and the improvement of the circuit depth for the realization of the Toffoli gate by using qudits. This is done by establishing a general relation between the dimensionality of qudits and their topology of connections for a scalable multi-qudit processor, where higher qudit levels are used for substituting ancillas. The suggested model is of importance for the realization of quantum algorithms and as a method of quantum error correction codes for single-qubit operations.\n\n• 1. IPE - Schmidt United Institute of Physics of the Earth [Moscow]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 3. Russian Quantum Center\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Schrödinger approach to Mean Field Games with negative coordination – Archive ouverte HAL\n\n### Thibault Bonnemain 1 Thierry Gobron 2 Denis Ullmo 1\n\n#### Thibault Bonnemain, Thierry Gobron, Denis Ullmo. Schrödinger approach to Mean Field Games with negative coordination. SciPost Physics, SciPost Foundation, 2020, ⟨10.21468/SciPostPhys.9.4.059⟩. ⟨hal-02923105⟩\n\nMean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled agents in interaction. Here we consider such systems when the interactions between agents result in a negative coordination and analyze the behavior of the associated system of coupled PDEs using the now well established correspondence with the non linear Schr\\\"odinger equation. We focus on the long optimization time limit and on configurations such that the game we consider goes through different regimes in which the relative importance of disorder, interactions between agents and external potential varies, which makes possible to get insights on the role of the forward-backward structure of the Mean Field Game equations in relation with the way these various regimes are connected.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. LPP - Laboratoire Paul Painlevé - UMR 8524\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Self-isolation or borders closing: What prevents the spread of the epidemic better? – Archive ouverte HAL\n\n### O. ValbaV. AvetisovA. GorskyS. Nechaev 1\n\n#### O. Valba, V. Avetisov, A. Gorsky, S. Nechaev. Self-isolation or borders closing: What prevents the spread of the epidemic better?. Physical Review E , American Physical Society (APS), 2020, 102 (1), ⟨10.1103/PhysRevE.102.010401⟩. ⟨hal-03009765⟩\n\nPandemic distribution of COVID-19 in the world has motivated us to discuss combined effects of network clustering and adaptivity on epidemic spreading. We address the question concerning the choice of optimal mechanism for most effective prohibiting disease propagation in a connected network: adaptive clustering, which mimics self-isolation (SI) in local communities, or sharp instant clustering, which looks like frontiers closing (FC) between cities and countries. SI-networks are \"adaptively grown\" under condition of maximization of small cliques in the entire network, while FC-networks are \"instantly created\". Running the standard SIR model on clustered SI- and FC-networks, we demonstrate that the adaptive network clustering prohibits the epidemic spreading better than the instant clustering in the network with similar parameters. We found that SI model has scale-free property for degree distribution $P(k)\\sim k^{\\eta}$ with small critical exponent $-2<\\eta<-1$ and argue that scale-free behavior emerges due to the randomness in the initial degree distributions and is absent for random regular graphs.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Semiclassical evaluation of expectation values – Archive ouverte HAL\n\n### Kush Mohan MittalOlivier Giraud 1 Denis Ullmo 1\n\n#### Kush Mohan Mittal, Olivier Giraud, Denis Ullmo. Semiclassical evaluation of expectation values. Physical Review E , American Physical Society (APS), 2020. ⟨hal-03017036⟩\n\nSemiclassical Mechanics allows for a description of quantum systems which preserves their phase information, while using only the system's classical dynamics as an input. Over the time an identification has been developed between stationary phase approximation and semiclassical mechanics. Although it is true that in most of the cases in semiclassical mechanics the significant contributions come from the neighborhood of the stationary points, there are some important exceptions to it. In this paper we address one of these exceptions, occurring in the evaluation of the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions which are not in the neighborhood of stationary points and provide new semiclassical expressions for the evolution of the expectation values. For our analysis we employ and discuss two major semiclassical tools. The first one is the association of the quantum evolution of a wavefunction to the classical evolution of a Lagrangian manifold, as done by Maslov. The second one is the derivation of an expression for the semiclassical Wigner function whose properties under canonical transformation are made explicit. Using the canonical invariance of the formalism, we derive an expression for the expectation value of observables for the one-dimensional case and then generalize it to higher dimensions. We find that the expression can be written as the sum of a classical contribution which corresponds to what is referred to as the Truncated Wigner Approximation (TWA) in the cold-atoms physics context, or the Linearized Semiclassical Initial Value Representation(LSC-IVR) in chemical or molecular physics, and additional terms associated with interferences. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Spectral statistics of random Toeplitz matrices – Archive ouverte HAL\n\n### Eugene Bogomolny 1\n\n#### Eugene Bogomolny. Spectral statistics of random Toeplitz matrices. Physical Review E , American Physical Society (APS), 2020, 102 (4), ⟨10.1103/PhysRevE.102.040101⟩. ⟨hal-03017017⟩\n\nSpectral statistics of hermitian random Toeplitz matrices with independent identically distributed elements is investigated numerically. It is found that the eigenvalue statistics of complex Toeplitz matrices is surprisingly well approximated by the semi-Poisson distribution belonging to intermediate-type statistics observed in certain pseudo-integrable billiards. The origin of intermediate behaviour could be attributed to the fact that Fourier transformed random Toeplitz matrices have the same slow decay outside the main diagonal as critical random matrix ensembles. The statistical properties of the full spectrum of real random Toeplitz matrices with i.i.d. elements are close to the Poisson distribution but each of their constituted sub-spectra is again well described by the semi-Poisson distribution. The findings open new perspective in intermediate statistics.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## State transition graph of the Preisach model and the role of return-point memory – Archive ouverte HAL\n\n### M. Mert Terzi 1 Muhittin Mungan\n\n#### M. Mert Terzi, Muhittin Mungan. State transition graph of the Preisach model and the role of return-point memory. Physical Review E, 2020, 102 (1), ⟨10.1103/PhysRevE.102.012122⟩. ⟨hal-02908545⟩\n\nThe Preisach model has been useful as a null-model for understanding memory formation in periodically driven disordered systems. In amorphous solids for example, the athermal response to shear is due to localized plastic events (soft spots). As shown recently by one of us, the plastic response to applied shear can be rigorously described in terms of a directed network whose transitions correspond to one or more soft spots changing states. The topology of this graph depends on the interactions between soft-spots and when such interactions are negligible, the resulting description becomes that of the Preisach model. A first step in linking transition graph topology with the underlying soft-spot interactions is therefore to determine the structure of such graphs in the absence of interactions. Here we perform a detailed analysis of the transition graph of the Preisach model. We highlight the important role played by return point memory in organizing the graph into a hierarchy of loops and sub-loops. Our analysis reveals that the topology of a large portion of this graph is actually not governed by the values of the switching fields that describe the individual hysteretic behavior of the individual elements, but by a coarser parameter, a permutation $\\rho$ which prescribes the sequence in which the individual hysteretic elements change their states as the main hysteresis loop is traversed. This in turn allows us to derive combinatorial properties, such as the number of major loops in the transition graph as well as the number of states $| \\mathcal{R} |$ constituting the main hysteresis loop and its nested subloops. We find that $| \\mathcal{R} |$ is equal to the number of increasing subsequences contained in the permutation $\\rho$.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Statistics of the number of records for random walks and Lévy flights on a 1D lattice – Archive ouverte HAL\n\n### Philippe Mounaix 1 Satya Majumdar 2 Grégory Schehr 2\n\n#### Philippe Mounaix, Satya Majumdar, Grégory Schehr. Statistics of the number of records for random walks and Lévy flights on a 1D lattice. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2020, 53 (41), pp.415003. ⟨10.1088/1751-8121/abac97⟩. ⟨hal-02958283⟩\n\nWe study the statistics of the number of records R n for a symmetric, n-step, discrete jump process on a 1D lattice. At a given step, the walker can jump by arbitrary lattice units drawn from a given symmetric probability distribution. This process includes, as a special case, the standard nearest neighbor lattice random walk. We derive explicitly the generating function of the distribution P (R n) of the number of records, valid for arbitrary discrete jump distributions. As a byproduct, we provide a relatively simple proof of the generalized Sparre Andersen theorem for the survival probability of a random walk on a line, with discrete or continuous jump distributions. For the discrete jump process, we then derive the asymptotic large n behavior of P (R n) as well as of the average number of records E(R n). We show that unlike the case of random walks with symmetric and continuous jump distributions where the record statistics is strongly universal (i.e., independent of the jump distribution for all n), the record statistics for lattice walks depends on the jump distribution for any fixed n. However, in the large n limit, we show that the distribution of the scaled record number R n /E(R n) approaches a universal, half-Gaussian form for any discrete jump process. The dependence on the jump distribution enters only through the scale factor E(R n), which we also compute in the large n limit for arbitrary jump distributions. We present explicit results for a few examples and provide numerical checks of our analytical predictions.\n\n• 1. CPHT - Centre de Physique Théorique [Palaiseau]\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Stochastic growth in time-dependent environments – Archive ouverte HAL\n\n### Guillaume Barraquand 1 Pierre Le Doussal 1 Alberto Rosso 2\n\n#### Guillaume Barraquand, Pierre Le Doussal, Alberto Rosso. Stochastic growth in time-dependent environments. Physical Review E , American Physical Society (APS), 2020, 101 (4), ⟨10.1103/PhysRevE.101.040101⟩. ⟨hal-02565202⟩\n\nWe study the Kardar-Parisi-Zhang (KPZ) growth equation in one dimension with a noise variance $c(t)$ depending on time. We find that for $c(t)\\propto t^{-\\alpha}$ there is a transition at $\\alpha=1/2$. When $\\alpha>1/2$, the solution saturates at large times towards a non-universal limiting distribution. When $\\alpha<1/2$ the fluctuation field is governed by scaling exponents depending on $\\alpha$ and the limiting statistics are similar to the case when $c(t)$ is constant. We investigate this problem using different methods: (1) Elementary changes of variables mapping the time dependent case to variants of the KPZ equation with constant variance of the noise but in a deformed potential (2) An exactly solvable discretization, the log-gamma polymer model (3) Numerical simulations.\n\n• 1. Champs Aléatoires et Systèmes hors d'Équilibre\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Stochastic Resetting and Applications – Archive ouverte HAL\n\n### Martin Evans 1 Satya Majumdar 2 Grégory Schehr 2\n\n#### Martin Evans, Satya Majumdar, Grégory Schehr. Stochastic Resetting and Applications. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2020. ⟨hal-03010293⟩\n\nIn this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field. PACS numbers: 05.40.-a, 05.70.Fh, 02.50.Ey, 64.60.-i arXiv:1910.07993v2 [cond-mat.stat-mech]\n\n• 1. Université d'Edimbourg\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• ## Strong-coupling theory of counterions with hard cores between symmetrically charged walls – Archive ouverte HAL\n\n### Ladislav Šamaj 1 Martin Trulsson 2 Emmanuel Trizac 3\n\n#### Ladislav Šamaj, Martin Trulsson, Emmanuel Trizac. Strong-coupling theory of counterions with hard cores between symmetrically charged walls. Physical Review E , American Physical Society (APS), 2020, 102 (4), ⟨10.1103/PhysRevE.102.042604⟩. ⟨hal-03085023⟩\n\nBy a combination of Monte Carlo simulations and analytical calculations, we investigate the effective interactions between highly charged planar interfaces, neutralized by mobile counterions (salt-free system). While most previous analysis have focused on point-like counterions, we treat them as charged hard spheres. We thus work out the fate of like-charge attraction when steric effects are at work. The analytical approach partitions counterions in two sub-populations, one for each plate, and integrates out one sub-population to derive an effective Hamiltonian for the remaining one. The effective Hamiltonian features plaquette four-particle interactions, and it is worked out by computing a Gibbs-Bogoliubov inequality for the free energy. At the root of the treatment is the fact that under strong electrostatic coupling, the system of charges forms an ordered arrangement, that can be affected by steric interactions. Fluctuations around the reference positions are accounted for. To dominant order at high coupling, it is found that steric effects do not significantly affect the interplate effective pressure, apart at small distances where hard sphere overlap are unavoidable, and thus rule out configurations.\n\n• 1. SAS - Slovak Academy of Sciences\n• 2. Lund University [Lund]\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Superfluid transition in disordered dipolar Fermi gases – Archive ouverte HAL\n\n### S. I. MatveenkoV. I. YudsonB. L. AltshulerG. V. Shlyapnikov 1\n\n#### S. I. Matveenko, V. I. Yudson, B. L. Altshuler, G. V. Shlyapnikov. Superfluid transition in disordered dipolar Fermi gases. Physical Review A, American Physical Society 2020. ⟨hal-03017056⟩\n\nWe consider a weakly interacting two-component Fermi gas of dipolar particles (magnetic atoms or polar molecules) in the two-dimensional geometry. The dipole-dipole interaction (together with the short-range interaction at Feshbach resonances) for dipoles perpendicular to the plane of translational motion may provide a superfluid transition. The dipole-dipole scattering amplitude is momentum dependent, which violates the Anderson theorem claiming the independence of the transition temperature on the presence of weak disorder. We have shown that the disorder can strongly increase the critical temperature (up to 10 nK at realistic densities). This opens wide possibilities for the studies of the superfluid regime in weakly interacting Fermi gases, which was not observed so far.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Symmetries in $B \\to D^* \\ell \\nu$ angular observables – Archive ouverte HAL\n\n### Marcel AlgueróSébastien Descotes-Genon 1 Joaquim MatiasMartín Novoa-Brunet 2\n\n#### Marcel Algueró, Sébastien Descotes-Genon, Joaquim Matias, Martín Novoa-Brunet. Symmetries in $B \\to D^* \\ell \\nu$ angular observables. JHEP, 2020, 06, pp.156. ⟨10.1007/JHEP06(2020)156⟩. ⟨hal-02518081⟩\n\nWe apply the formalism of amplitude symmetries to the angular distribution of the decays B → D$^{∗}$ℓν for ℓ = e, μ, τ . We show that the angular observables used to describe the distribution of this class of decays are not independent in absence of New Physics contributing to tensor operators. We derive sets of relations among the angular coefficients of the decay distribution for the massless and massive lepton cases which can be used to probe in a very general way the consistency among the angular observables and the underlying New Physics at work. We use these relations to access the longitudinal polarisation fraction of the D$^{∗}$ using different angular coefficients from the ones used by Belle experiment. This in the near future can provide an alternative strategy to measure ${F}_L^{D\\ast }$ in B → D$^{∗}$τν and to understand the relatively high value measured by the Belle experiment. Using the same symmetries, we identify three observables which may exhibit a tension if the experimental value of ${F}_L^{D\\ast }$ remains high. We discuss how these relations can be exploited for binned measurements. We also propose a new observable that could test for specific scenarios of New Physics generated by light right-handed neutrinos. Finally we study the prospects of testing these relations based on the projected experimental sensitivity of new experiments.\n\n• 1. IJCLab - Laboratoire de Physique des 2 Infinis Irène Joliot-Curie\n• 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## The convex hull of the run-and-tumble particle in a plane – Archive ouverte HAL\n\n### Alexander K HartmannSatya N Majumdar 1 Hendrik Schawe 2 Gregory Schehr 1 Alexander Hartmann 2 Satya Majumdar 1\n\n#### Alexander K Hartmann, Satya N Majumdar, Hendrik Schawe, Gregory Schehr, Alexander Hartmann, et al.. The convex hull of the run-and-tumble particle in a plane. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2020, 2020 (5), pp.053401. ⟨10.1088/1742-5468/ab7c5f⟩. ⟨hal-02881103⟩\n\nWe study the statistical properties of the convex hull of a planar run-and-tumble particle (RTP), also known as the \"persistent random walk\", where the particle/walker runs ballistically between tumble events at which it changes its direction randomly. We consider two different statistical ensembles where we either fix (i) the total number of tumblings $n$ or (ii) the total duration $t$ of the time interval. In both cases, we derive exact expressions for the average perimeter of the convex hull and then compare to numerical estimates finding excellent agreement. Further, we numerically compute the full distribution of the perimeter using Markov chain Monte Carlo techniques, in both ensembles, probing the far tails of the distribution, up to a precision smaller than $10^{-100}$. This also allows us to characterize the rare events that contribute to the tails of these distributions.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n• 2. University of Oldenburg\n\nDownload PDF via arXiV.org\n\nDetails\n• ## The influence of the brittle-ductile transition zone on aftershock and foreshock occurrence – Archive ouverte HAL\n\n### Giuseppe Petrillo 1 Eugenio Lippiello 1 François Landes 2 Alberto Rosso 3\n\n#### Giuseppe Petrillo, Eugenio Lippiello, François Landes, Alberto Rosso. The influence of the brittle-ductile transition zone on aftershock and foreshock occurrence. Nature Communications, Nature Publishing Group, 2020. ⟨hal-02908552⟩\n\nAftershock occurrence is characterized by scaling behaviors with quite universal exponents. At the same time, deviations from universality have been proposed as a tool to discriminate aftershocks from foreshocks. Here we show that the change in rheological behavior of the crust, from velocity weakening to velocity strengthening, represents a viable mechanism to explain statistical features of both aftershocks and foreshocks. More precisely, we present a model of the seismic fault described as a velocity weakening elastic layer coupled to a velocity strengthening visco-elastic layer. We show that the statistical properties of aftershocks in instrumental catalogs are recovered at a quantitative level, quite independently of the value of model parameters. We also find that large earthquakes are often anticipated by a preparatory phase characterized by the occurrence of foreshocks. Their magnitude distribution is significantly flatter than the aftershock one, in agreement with recent results for forecasting tools based on foreshocks.\n\n• 1. Department of Mathematics and Physics [Caserta]\n• 2. LRI - Laboratoire de Recherche en Informatique\n• 3. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails\n• ## Topological effects and conformal invariance in long-range correlated random surfaces – Archive ouverte HAL\n\n### Nina Javerzat 1 Sebastian Grijalva 1 Alberto Rosso 1 Raoul Santachiara 1\n\nWe consider discrete random fractal surfaces with negative Hurst exponent $H<0$. A random colouring of the lattice is provided by activating the sites at which the surface height is greater than a given level $h$. The set of activated sites is usually denoted as the excursion set. The connected components of this set, the level clusters, define a one-parameter ($H$) family of percolation models with long-range correlation in the site occupation. The level clusters percolate at a finite value $h=h_c$ and for $H\\leq-\\frac{3}{4}$ the phase transition is expected to remain in the same universality class of the pure (i.e. uncorrelated) percolation. For $-\\frac{3}{4} • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques Download PDF via arXiV.org Details • ## Toward the full short-time statistics of an active Brownian particle on the plane – Archive ouverte HAL ### Satya N. Majumdar 1 Baruch Meerson #### Satya N. Majumdar, Baruch Meerson. Toward the full short-time statistics of an active Brownian particle on the plane. Physical Review E , American Physical Society (APS), 2020, 102 (2), ⟨10.1103/PhysRevE.102.022113⟩. ⟨hal-03017046⟩ We study the position distribution of a single active Brownian particle (ABP) on the plane. We show that this distribution has a compact support, the boundary of which is an expanding circle. We focus on a short-time regime and employ the optimal fluctuation method (OFM) to study large deviations of the particle position coordinates$x$and$y$. We determine the optimal paths of the ABP, conditioned on reaching specified values of$x$and$y$, and the large deviation functions of the marginal distributions of$x$, and of$y$. These marginal distributions match continuously with \"near tails\" of the$x$and$y$distributions of typical fluctuations, studied earlier. We also calculate the large deviation function of the joint$x$and$y$distribution$P(x,y,t)$in a vicinity of a special \"zero-noise\" point, and show that$\\ln P(x,y,t)$has a nontrivial self-similar structure as a function of$x$,$y$and$t$. The joint distribution vanishes extremely fast at the expanding circle, exhibiting an essential singularity there. This singularity is inherited by the marginal$x$- and$y$-distributions. We argue that this fingerprint of the short-time dynamics remains there at all times. • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques Download PDF via arXiV.org Details • ## Truncated moment sequences and a solution to the channel separability problem – Archive ouverte HAL ### Nadia Milazzo 1 Daniel BraunOlivier Giraud 1 #### Nadia Milazzo, Daniel Braun, Olivier Giraud. Truncated moment sequences and a solution to the channel separability problem. Physical Review A, American Physical Society 2020, 102 (5), ⟨10.1103/PhysRevA.102.052406⟩. ⟨hal-03017063⟩ We consider the problem of separability of quantum channels via the Choi matrix representation given by the Choi-Jamio{\\l}kowski isomorphism. We explore three classes of separability across different cuts between systems and ancillae and we provide a solution based on the mapping of the coordinates of the Choi state (in a fixed basis) to a truncated moment sequence (tms)$y$. This results in an algorithm which gives a separability certificate using semidefinite programming. The computational complexity and the performance of it depend on the number of variables$n$in the tms and on the size of the moment matrix$M_t(y)$of order$t$. We exploit the algorithm to numerically investigate separability of families of 2-qubit and single-qutrit channels; in the latter case we can provide an answer for examples explored earlier through the criterion based on the negativity$N$, a criterion which remains inconclusive for Choi matrices with$N=0$. • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques Download PDF via arXiV.org Details • ## Universal gap statistics for random walks for a class of jump densities – Archive ouverte HAL ### Matteo Battilana 1 Satya N. Majumdar 1 Gregory Schehr 1 #### Matteo Battilana, Satya N. Majumdar, Gregory Schehr. Universal gap statistics for random walks for a class of jump densities. Markov Processes And Related Fields, Polymat Publishing Company, 2020. ⟨hal-02518812⟩ We study the order statistics of a random walk (RW) of$n$steps whose jumps are distributed according to symmetric Erlang densities$f_p(\\eta)\\sim |\\eta|^p \\,e^{-|\\eta|}$, parametrized by a non-negative integer$p$. Our main focus is on the statistics of the gaps$d_{k,n}$between two successive maxima$d_{k,n}=M_{k,n}-M_{k+1,n}$where$M_{k,n}$is the$k$-th maximum of the RW between step 1 and step$n$. In the limit of large$n$, we show that the probability density function of the gaps$P_{k,n}(\\Delta) = \\Pr(d_{k,n} = \\Delta)$reaches a stationary density$P_{k,n}(\\Delta) \\to p_k(\\Delta)$. For large$k$, we demonstrate that the typical fluctuations of the gap, for$d_{k,n}= O(1/\\sqrt{k})$(and$n \\to \\infty$), are described by a non-trivial scaling function that is independent of$k$and of the jump probability density function$f_p(\\eta)$, thus corroborating our conjecture about the universality of the regime of typical fluctuations (see G. Schehr, S. N. Majumdar, Phys. Rev. Lett. 108, 040601 (2012)). We also investigate the large fluctuations of the gap, for$d_{k,n} = O(1)$(and$n \\to \\infty$), and show that these two regimes of typical and large fluctuations of the gaps match smoothly. • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques Download PDF via arXiV.org Details • ## Universal properties of a run-and-tumble particle in arbitrary dimension – Archive ouverte HAL ### Francesco Mori 1 Pierre Le Doussal 2 Satya N. Majumdar 1 Grégory Schehr 1 #### Francesco Mori, Pierre Le Doussal, Satya N. Majumdar, Grégory Schehr. Universal properties of a run-and-tumble particle in arbitrary dimension. Physical Review E , American Physical Society (APS), 2020, 102 (4), ⟨10.1103/PhysRevE.102.042133⟩. ⟨hal-03010271⟩ We consider an active run-and-tumble particle (RTP) in$d$dimensions, starting from the origin and evolving over a time interval$[0,t]$. We examine three different models for the dynamics of the RTP: the standard RTP model with instantaneous tumblings, a variant with instantaneous runs and a general model in which both the tumblings and the runs are non-instantaneous. For each of these models, we use the Sparre Andersen theorem for discrete-time random walks to compute exactly the probability that the$x$component does not change sign up to time$t$, showing that it does not depend on$d$. As a consequence of this result, we compute exactly other$x$-component properties, namely the distribution of the time of the maximum and the record statistics, showing that they are universal, i.e. they do not depend on$d$. Moreover, we show that these universal results hold also if the speed$v$of the particle after each tumbling is random, drawn from a generic probability distribution. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 124, 090603 (2020)]. • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques • 2. LPENS (UMR_8023) - Laboratoire de physique de l'ENS - ENS Paris Download PDF via arXiV.org Details • ## Universal Scaling of the Velocity Field in Crack Front Propagation – Archive ouverte HAL ### Clément Le Priol 1 Pierre Le Doussal 2 Laurent Ponson 3 Alberto Rosso 4 Julien Chopin 5 #### Clément Le Priol, Pierre Le Doussal, Laurent Ponson, Alberto Rosso, Julien Chopin. Universal Scaling of the Velocity Field in Crack Front Propagation. Physical Review Letters, American Physical Society, 2020, 124 (6), ⟨10.1103/PhysRevLett.124.065501⟩. ⟨hal-02512228⟩ The propagation of a crack front in disordered materials is jerky and characterized by bursts of activity, called avalanches. These phenomena are the manifestation of an out-of-equilibrium phase transition originated by the disorder. As a result avalanches display universal scalings which are however difficult to characterize in experiments at finite drive. Here we show that the correlation functions of the velocity field along the front allow to extract the critical exponents of the transition and to identify the universality class of the system. We employ these correlations to characterize the universal behavior of the transition in simulations and in an experiment of crack propagation. This analysis is robust, efficient and can be extended to all systems displaying avalanche dynamics. • 1. LPENS (UMR_8023) - Laboratoire de physique de l'ENS - ENS Paris • 2. Champs Aléatoires et Systèmes hors d'Équilibre • 3. DALEMBERT - Institut Jean Le Rond d'Alembert • 4. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques • 5. IF-UFB - Instituto de Fisica, Universidade Federal da Bahia Download PDF via arXiV.org Details • ## Universal survival probability for a correlated random walk and applications to records – Archive ouverte HAL ### Bertrand Lacroix-A-Chez-Toine 1 Francesco Mori 2 #### Bertrand Lacroix-A-Chez-Toine, Francesco Mori. Universal survival probability for a correlated random walk and applications to records. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2020. ⟨hal-03085067⟩ We consider a model of space-continuous one-dimensional random walk with simple correlation between the steps: the probability that two consecutive steps have same sign is$q$with$0\\leq q\\leq 1$. The parameter$q$allows thus to control the persistence of the random walk. We compute analytically the survival probability of a walk of$n$steps, showing that it is independent of the jump distribution for any finite$n$. This universality is a consequence of the Sparre-Andersen theorem for random walks with uncorrelated and symmetric steps. We then apply this result to derive the distribution of the step at which the random walk reaches its maximum and the record statistics of the walk, which show the same universality. In particular, we show that the distribution of the number of records for a walk of$n\\gg 1$steps is the same as for a random walk with$n_{\\rm eff}(q)=n/(2(1-q))$uncorrelated and symmetrically distributed steps. We also show that in the regime where$n\\to \\infty$and$q\\to 1$with$y=n(1-q)$, this model converges to the run-and-tumble particle, a persistent random walk often used to model the motion of bacteria. Our theoretical results are confirmed by numerical simulations. • 1. Weizmann Institute of Science [Rehovot, Israël] • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques Download PDF via arXiV.org Details • ## Universal Survival Probability for a d -Dimensional Run-and-Tumble Particle – Archive ouverte HAL ### Francesco Mori 1 Pierre Le Doussal 2 Satya N. Majumdar 1 Satya Majumdar 1 Gregory Schehr 1 #### Francesco Mori, Pierre Le Doussal, Satya N. Majumdar, Satya Majumdar, Gregory Schehr. Universal Survival Probability for a d -Dimensional Run-and-Tumble Particle. Physical Review Letters, American Physical Society, 2020, 124 (9), ⟨10.1103/PhysRevLett.124.090603⟩. ⟨hal-02512214⟩ We consider an active run-and-tumble particle (RTP) in$d$dimensions and compute exactly the probability$S(t)$that the$x$-component of the position of the RTP does not change sign up to time$t$. When the tumblings occur at a constant rate, we show that$S(t)$is independent of$d$for any finite time$t$(and not just for large$t$), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed$v$of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem. • 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques • 2. Champs Aléatoires et Systèmes hors d'Équilibre Download PDF via arXiV.org Details • ## Velocity and diffusion constant of an active particle in a one-dimensional force field – Archive ouverte HAL ### Pierre Le Doussal 1 Satya N. Majumdar 2 Satya Majumdar 2 Gregory Schehr 2 #### Pierre Le Doussal, Satya N. Majumdar, Satya Majumdar, Gregory Schehr. Velocity and diffusion constant of an active particle in a one-dimensional force field. EPL - Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Publishing, 2020, 130 (4), pp.40002. ⟨10.1209/0295-5075/130/40002⟩. ⟨hal-02881224⟩ We consider a run an tumble particle with two velocity states$\\pm v_0$, in an inhomogeneous force field$f(x)$in one dimension. We obtain exact formulae for its velocity$V_L$and diffusion constant$D_L$for arbitrary periodic$f(x)$of period$L$. They involve the \"active potential\" which allows to define a global bias. Upon varying parameters, such as an external force$F$, the dynamics undergoes transitions from non-ergodic trapped states, to various moving states, some with non analyticities in the$V_L$versus$F$curve. A random landscape in the presence of a bias leads, for large$L$, to anomalous diffusion$x \\sim t^\\mu$,$\\mu<1$, or to a phase with a finite velocity that we calculate. • 1. Champs Aléatoires et Systèmes hors d'Équilibre • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques Download PDF via arXiV.org Details • ## Wigner–Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity – Archive ouverte HAL ### Aurélien GrabschChristophe Texier 1 #### Aurélien Grabsch, Christophe Texier. Wigner–Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity. Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2020, 53 (42), pp.425003. ⟨10.1088/1751-8121/aba215⟩. ⟨hal-03017007⟩ We consider a multichannel wire with a disordered region of length$L$and a reflecting boundary. The reflection of a wave of frequency$\\omega$is described by the scattering matrix$\\mathcal{S}(\\omega)$, encoding the probability amplitudes to be scattered from one channel to another. The Wigner-Smith time delay matrix$\\mathcal{Q}=-\\mathrm{i}\\, \\mathcal{S}^\\dagger\\partial_\\omega\\mathcal{S}$is another important matrix encoding temporal aspects of the scattering process. In order to study its statistical properties, we split the scattering matrix in terms of two unitary matrices,$\\mathcal{S}=\\mathrm{e}^{2\\mathrm{i}kL}\\mathcal{U}_L\\mathcal{U}_R$(with$\\mathcal{U}_L=\\mathcal{U}_R^\\mathrm{T}$in the presence of TRS), and introduce a novel symmetrisation procedure for the Wigner-Smith matrix:$\\widetilde{\\mathcal{Q}} =\\mathcal{U}_R\\,\\mathcal{Q}\\,\\mathcal{U}_R^\\dagger = (2L/v)\\,\\mathbf{1}_N -\\mathrm{i}\\,\\mathcal{U}_L^\\dagger\\partial_\\omega\\big(\\mathcal{U}_L\\mathcal{U}_R\\big)\\,\\mathcal{U}_R^\\dagger$, where$k$is the wave vector and$v$the group velocity. We demonstrate that$\\widetilde{\\mathcal{Q}}$can be expressed under the form of an exponential functional of a matrix Brownian motion. For semi-infinite wires,$L\\to\\infty$, using a matricial extension of the Dufresne identity, we recover straightforwardly the joint distribution for$\\mathcal{Q}$'s eigenvalues of Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length$L$, the exponential functional representation is used to calculate the first moments$\\langle\\mathrm{tr}(\\mathcal{Q})\\rangle$,$\\langle\\mathrm{tr}(\\mathcal{Q}^2)\\rangle$and$\\langle\\big[\\mathrm{tr}(\\mathcal{Q})\\big]^2\\rangle$. Finally we derive a partial differential equation for the resolvent$g(z;L)=\\lim_{N\\to\\infty}(1/N)\\,\\mathrm{tr}\\big\\{\\big( z\\,\\mathbf{1}_N - N\\,\\mathcal{Q}\\big)^{-1}\\big\\}$in the large$N\\$ limit.\n\n• 1. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques\n\nDownload PDF via arXiV.org\n\nDetails" ]

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[ "# beyond calculus Calculus\n\n## [PDF] [PDF] 5/7/2004 CALCULUS BEYOND CALCULUS Math21a, O Knill\n\n7 mai 2004 · Topics beyond multi-variable calculus are usually labeled with special names like ”linear algebra”, ”ordinary differential equations”,\n\n## [PDF] [PDF] Practice Problems Developed by Jake Chipps - Beyond Calculus\n\nUse the Fundamental Theorem of Calculus to evaluate the following expressions: Carbon-?14, how many grams are left after 2000 years?\n\n## [PDF] [PDF] On Beyond Calculus - Houghton College\n\n9 jan 2016 · On Beyond Calculus Rebekah Yates Houghton College, Houghton, NY To Infinity and Beyond Getting an Angle on the Area of Spherical\n\n## [PDF] [PDF] Leibniz â•fl Beyond The Calculus - CORE\n\n5 jan 1991 · Grant, Hardy (1991) \"Leibniz — Beyond The Calculus,\" Humanistic Mathematics Network Journal: Iss 6, Article 5\n\n## [PDF] [PDF] Calculus AB Practice Problems - Beyond Calculus\n\nBeyond Calculus Practice Problems Developed by Jake Use the Fundamental Theorem of Calculus to evaluate the following expressions: 1\n\n## [PDF] [PDF] On Beyond Calculus poem\n\nOn Beyond Calculus Rebekah B Johnson Yates A retelling of the beginning and end of Dr Seuss's On Beyond Zebra Said calculus student Meredith McGrath,\n\n## [PPT] [PPT] Beyond Newton and Leibniz: The Making of Modern Calculus\n\nBeyond Newton and Leibniz: Early 17th Century Advances in Calculus 1684 - published his papers on Calculus- based on discoveries he made dating\n\n## [PPT] [DOC] Dual Enrollment Placement Test Requirements and Exemptions\n\nMath beyond Calculus- MTH 277, 285, 291, 292 Students taking AP English exams Mathematics, N/A, N/A, N/A, N/A, N/A, Score of 3 on AP Calculus BC exam\n\n## [PPT] [DOC] TENNESSEE MATHEMATICS TEACHERS' ASSOCIATION High\n\nThe count for Calculus and Advanced Topics is based on how many students are enrolled in calculus or courses beyond calculus (like differential equations)\n\n## [PPT] [DOC] CURRICULUM VITAE - Summer China Program, SCP International\n\nReceived teaching awards 1997, 2001, 2002 Co-Principal Investigator for NSF grant DUE-9851405, “Using Computers to Extend Reform Beyond Calculus ”\n\n## [PPT] [PPT] Math summer orientation - Illinois Math\n\nIf you have learned calculus but do not already have official University of Illinois credit for For students beyond calculus, 2 math/stat classes are recommended\n\n1\nPDF search" ]

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[ "Point Cloud Library (PCL)  1.10.1-dev\nsac_model_perpendicular_plane.h\n1 /*\n3  *\n4  * Point Cloud Library (PCL) - www.pointclouds.org\n5  * Copyright (c) 2010-2011, Willow Garage, Inc.\n6  * Copyright (c) 2012-, Open Perception, Inc.\n7  *\n9  *\n10  * Redistribution and use in source and binary forms, with or without\n11  * modification, are permitted provided that the following conditions\n12  * are met:\n13  *\n14  * * Redistributions of source code must retain the above copyright\n15  * notice, this list of conditions and the following disclaimer.\n16  * * Redistributions in binary form must reproduce the above\n17  * copyright notice, this list of conditions and the following\n18  * disclaimer in the documentation and/or other materials provided\n19  * with the distribution.\n20  * * Neither the name of the copyright holder(s) nor the names of its\n21  * contributors may be used to endorse or promote products derived\n22  * from this software without specific prior written permission.\n23  *\n24  * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS\n25  * \"AS IS\" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT\n26  * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS\n27  * FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE\n28  * COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,\n29  * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,\n30  * BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;\n31  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER\n32  * CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT\n33  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN\n34  * ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE\n35  * POSSIBILITY OF SUCH DAMAGE.\n36  *\n37  * $Id$\n38  *\n39  */\n40\n41 #pragma once\n42\n43 #include <pcl/sample_consensus/sac_model_plane.h>\n44 #include <pcl/common/common.h>\n45\n46 namespace pcl\n47 {\n48  /** \\brief SampleConsensusModelPerpendicularPlane defines a model for 3D plane segmentation using additional\n49  * angular constraints. The plane must be perpendicular to a user-specified axis (\\ref setAxis), up to a user-specified angle threshold (\\ref setEpsAngle).\n50  * In other words, the plane <b>normal</b> must be (nearly) <b>parallel</b> to the specified axis.\n51  * The model coefficients are defined as:\n52  * - \\b a : the X coordinate of the plane's normal (normalized)\n53  * - \\b b : the Y coordinate of the plane's normal (normalized)\n54  * - \\b c : the Z coordinate of the plane's normal (normalized)\n55  * - \\b d : the fourth <a href=\"http://mathworld.wolfram.com/HessianNormalForm.html\">Hessian component</a> of the plane's equation\n56  *\n57  *\n58  * Code example for a plane model, perpendicular (within a 15 degrees tolerance) with the Z axis:\n59  * \\code\n60  * SampleConsensusModelPerpendicularPlane<pcl::PointXYZ> model (cloud);\n61  * model.setAxis (Eigen::Vector3f (0.0, 0.0, 1.0));\n63  * \\endcode\n64  *\n65  * \\note Please remember that you need to specify an angle > 0 in order to activate the axis-angle constraint!\n66  *\n67  * \\author Radu B. Rusu\n68  * \\ingroup sample_consensus\n69  */\n70  template <typename PointT>\n72  {\n73  public:\n75\n79\n82\n83  /** \\brief Constructor for base SampleConsensusModelPerpendicularPlane.\n84  * \\param[in] cloud the input point cloud dataset\n85  * \\param[in] random if true set the random seed to the current time, else set to 12345 (default: false)\n86  */\n88  bool random = false)\n89  : SampleConsensusModelPlane<PointT> (cloud, random)\n90  , axis_ (Eigen::Vector3f::Zero ())\n91  , eps_angle_ (0.0)\n92  {\n93  model_name_ = \"SampleConsensusModelPerpendicularPlane\";\n94  sample_size_ = 3;\n95  model_size_ = 4;\n96  }\n97\n98  /** \\brief Constructor for base SampleConsensusModelPerpendicularPlane.\n99  * \\param[in] cloud the input point cloud dataset\n100  * \\param[in] indices a vector of point indices to be used from \\a cloud\n101  * \\param[in] random if true set the random seed to the current time, else set to 12345 (default: false)\n102  */\n104  const std::vector<int> &indices,\n105  bool random = false)\n106  : SampleConsensusModelPlane<PointT> (cloud, indices, random)\n107  , axis_ (Eigen::Vector3f::Zero ())\n108  , eps_angle_ (0.0)\n109  {\n110  model_name_ = \"SampleConsensusModelPerpendicularPlane\";\n111  sample_size_ = 3;\n112  model_size_ = 4;\n113  }\n114\n115  /** \\brief Empty destructor */\n117\n118  /** \\brief Set the axis along which we need to search for a plane perpendicular to.\n119  * \\param[in] ax the axis along which we need to search for a plane perpendicular to\n120  */\n121  inline void\n122  setAxis (const Eigen::Vector3f &ax) { axis_ = ax; }\n123\n124  /** \\brief Get the axis along which we need to search for a plane perpendicular to. */\n125  inline Eigen::Vector3f\n126  getAxis () const { return (axis_); }\n127\n128  /** \\brief Set the angle epsilon (delta) threshold.\n129  * \\param[in] ea the maximum allowed difference between the plane normal and the given axis.\n130  * \\note You need to specify an angle > 0 in order to activate the axis-angle constraint!\n131  */\n132  inline void\n133  setEpsAngle (const double ea) { eps_angle_ = ea; }\n134\n135  /** \\brief Get the angle epsilon (delta) threshold. */\n136  inline double\n137  getEpsAngle () const { return (eps_angle_); }\n138\n139  /** \\brief Select all the points which respect the given model coefficients as inliers.\n140  * \\param[in] model_coefficients the coefficients of a plane model that we need to compute distances to\n141  * \\param[in] threshold a maximum admissible distance threshold for determining the inliers from the outliers\n142  * \\param[out] inliers the resultant model inliers\n143  */\n144  void\n145  selectWithinDistance (const Eigen::VectorXf &model_coefficients,\n146  const double threshold,\n147  std::vector<int> &inliers) override;\n148\n149  /** \\brief Count all the points which respect the given model coefficients as inliers.\n150  *\n151  * \\param[in] model_coefficients the coefficients of a model that we need to compute distances to\n152  * \\param[in] threshold maximum admissible distance threshold for determining the inliers from the outliers\n153  * \\return the resultant number of inliers\n154  */\n155  std::size_t\n156  countWithinDistance (const Eigen::VectorXf &model_coefficients,\n157  const double threshold) const override;\n158\n159  /** \\brief Compute all distances from the cloud data to a given plane model.\n160  * \\param[in] model_coefficients the coefficients of a plane model that we need to compute distances to\n161  * \\param[out] distances the resultant estimated distances\n162  */\n163  void\n164  getDistancesToModel (const Eigen::VectorXf &model_coefficients,\n165  std::vector<double> &distances) const override;\n166\n167  /** \\brief Return a unique id for this model (SACMODEL_PERPENDICULAR_PLANE). */\n168  inline pcl::SacModel\n169  getModelType () const override { return (SACMODEL_PERPENDICULAR_PLANE); }\n170\n171  protected:\n174\n175  /** \\brief Check whether a model is valid given the user constraints.\n176  * \\param[in] model_coefficients the set of model coefficients\n177  */\n178  bool\n179  isModelValid (const Eigen::VectorXf &model_coefficients) const override;\n180\n181  /** \\brief The axis along which we need to search for a plane perpendicular to. */\n182  Eigen::Vector3f axis_;\n183\n184  /** \\brief The maximum allowed difference between the plane normal and the given axis. */\n185  double eps_angle_;\n186  };\n187 }\n188\n189 #ifdef PCL_NO_PRECOMPILE\n190 #include <pcl/sample_consensus/impl/sac_model_perpendicular_plane.hpp>\n191 #endif\nvoid setAxis(const Eigen::Vector3f &ax)\nSet the axis along which we need to search for a plane perpendicular to.\nbool isModelValid(const Eigen::VectorXf &model_coefficients) const override\nCheck whether a model is valid given the user constraints.\nunsigned int model_size_\nThe number of coefficients in the model.\nDefinition: sac_model.h:565\nvoid setEpsAngle(const double ea)\nSet the angle epsilon (delta) threshold.\ntypename PointCloud::Ptr PointCloudPtr\nDefinition: sac_model.h:74\nDefinition: bfgs.h:9\nstd::size_t countWithinDistance(const Eigen::VectorXf &model_coefficients, const double threshold) const override\nCount all the points which respect the given model coefficients as inliers.\nDefine standard C methods and C++ classes that are common to all methods.\nSampleConsensusModel represents the base model class.\nDefinition: sac_model.h:69\nstd::string model_name_\nThe model name.\nDefinition: sac_model.h:524\nEigen::Vector3f getAxis() const\nGet the axis along which we need to search for a plane perpendicular to.\ndouble getEpsAngle() const\nGet the angle epsilon (delta) threshold.\ntypename SampleConsensusModel< PointT >::PointCloudConstPtr PointCloudConstPtr\ntypename PointCloud::ConstPtr PointCloudConstPtr\nDefinition: sac_model.h:73\ndouble eps_angle_\nThe maximum allowed difference between the plane normal and the given axis.\nPointCloud represents the base class in PCL for storing collections of 3D points. ...\nSacModel\nDefinition: model_types.h:45\nvoid getDistancesToModel(const Eigen::VectorXf &model_coefficients, std::vector< double > &distances) const override\nCompute all distances from the cloud data to a given plane model.\nEigen::Vector3f axis_\nThe axis along which we need to search for a plane perpendicular to.\nshared_ptr< const SampleConsensusModel< PointT > > ConstPtr\nDefinition: sac_model.h:78\npcl::SacModel getModelType() const override\nReturn a unique id for this model (SACMODEL_PERPENDICULAR_PLANE).\nvoid selectWithinDistance(const Eigen::VectorXf &model_coefficients, const double threshold, std::vector< int > &inliers) override\nSelect all the points which respect the given model coefficients as inliers.\ntypename SampleConsensusModel< PointT >::PointCloudPtr PointCloudPtr\nSampleConsensusModelPerpendicularPlane(const PointCloudConstPtr &cloud, const std::vector< int > &indices, bool random=false)\nConstructor for base SampleConsensusModelPerpendicularPlane.\ntypename SampleConsensusModel< PointT >::PointCloud PointCloud\nA point structure representing Euclidean xyz coordinates, and the RGB color.\nshared_ptr< SampleConsensusModel< PointT > > Ptr\nDefinition: sac_model.h:77\nSampleConsensusModelPlane defines a model for 3D plane segmentation.\nboost::shared_ptr< T > shared_ptr\nAlias for boost::shared_ptr.\nDefinition: memory.h:81\nSampleConsensusModelPerpendicularPlane(const PointCloudConstPtr &cloud, bool random=false)\nConstructor for base SampleConsensusModelPerpendicularPlane.\nunsigned int sample_size_\nThe size of a sample from which the model is computed.\nDefinition: sac_model.h:562\nSampleConsensusModelPerpendicularPlane defines a model for 3D plane segmentation using additional ang..." ]

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[ "# BIRCH AND SWINNERTON-DYER CONJECTURE PDF", null, "Author: Kazratilar Zologami Country: Botswana Language: English (Spanish) Genre: Love Published (Last): 1 July 2004 Pages: 326 PDF File Size: 5.53 Mb ePub File Size: 13.71 Mb ISBN: 111-8-77346-340-1 Downloads: 71776 Price: Free* [*Free Regsitration Required] Uploader: Zululkis", null, "This was extended to the case where F is any finite abelian extension of K by. Combining this with the p-parity theorem of and and with the proof of the main conjecture of Iwasawa theory for GL 2 by, they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by, satisfy the Birch and Swinnerton-Dyer conjecture.\n\nNothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture. Assuming the Birch and Swinnerton-Dyer conjecture, is the area of a right triangle with rational side lengths a congruent number if and only if the number of triplets of integers satisfying is twice the number of triplets satisfying.\n\nThe interest in this statement is that the condition is easily verified. Admitting the BSD conjecture, these estimations correspond to information about the rank of families of elliptic curves in question. For example: suppose the generalized Riemann hypothesis and the BSD conjecture, the average rank of curves given by is smaller than.\n\nManjul Bhargava. Arul Shankar. Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0. Bryan John Birch. Peter Swinnerton-Dyer. Notes on Elliptic Curves II. Christophe Breuil. Brian Conrad. Fred Diamond. Richard Taylor mathematician. Book: J. John Coates mathematician. Kenneth Alan Ribet. Karl Rubin. Arithmetic Theory of Elliptic Curves.\n\nLecture Notes in Mathematics. Andrew Wiles. On the conjecture of Birch and Swinnerton-Dyer. Max Deuring.\n\nTim Dokchitser. Vladimir Dokchitser. On the Birch—Swinnerton-Dyer quotients modulo squares. Benedict H. Benedict Gross. Don B. Don Zagier. Heegner points and derivatives of L-series. Victor Kolyvagin. USSR Izv. Louis Mordell. On the rational solutions of the indeterminate equations of the third and fourth degrees. On the parity of ranks of Selmer groups IV. Christopher Skinner. The Iwasawa main conjectures for GL2. Jerrold B.\n\nSecond Series. Encyclopedia: Wiles. The Birch and Swinnerton-Dyer conjecture. Arthur Jaffe. The Millennium prize problems. American Mathematical Society. Book: Koblitz, Neal. Neal Koblitz. Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics. Roger Heath-Brown. Duke Mathematical Journal.\n\nIt uses material from the Wikipedia article \" Birch and Swinnerton-Dyer conjecture \". Except where otherwise indicated, Everything. Cookie policy.\n\nABHINANDAN PATRA PDF\n\n## Birch and Swinnerton-Dyer Conjecture", null, "Over the coming weeks, each of these problems will be illuminated by experts from the Australian Mathematical Sciences Institute AMSI member institutions. Elliptic curves have a long and distinguished history that can be traced back to antiquity. They are prevalent in many branches of modern mathematics, foremost of which is number theory. The reason for this historical confusion is that these curves have a strong connection to elliptic integrals , which arise when describing the motion of planetary bodies in space. The ancient Greek mathematician Diophantus is considered by many to be the father of algebra. His major mathematical work was written up in the tome Arithmetica which was essentially a school textbook for geniuses.\n\nEL PIANISTA DEL GHETTO DE VARSOVIA LIBRO PDF\n\n## What is the Birch and Swinnerton-Dyer conjecture?", null, "Its zeta function is where. Analogous to the Euler factors of the Riemann zeta function, we define the local -factor of When evaluating its value at , we retrieve the arithmetic information at , Notice that each point in reduces to a point in. So when tends to be small. Birch and Swinnerton-Dyer did numerical experiments and suggested the heuristic The is defined to be the product of all local -factors, Formally evaluating the value at gives So intuitively the rank of will correspond to the value of at 1: the larger is. However, the value of at does not make sense since the product of only converges when can be continued to an analytic function on the whole of , it may be reasonable to believe that the behavior of at contains the arithmetic information of the rank of." ]

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[ "", null, "# Flux store data into new measurement using HTTP API\n\nHello,\n\nHow can I store the output into a new measurement, let’s say “Total_Processor_Time”\nI am looking for capability like kapacitor’s InfluxDBOut node.\n\ncurl -XPOST localhost:8086/api/v2/query -sS\n-H ‘Accept:application/csv’\n-H ‘Content-type:application/vnd.flux’\n-d ‘from(bucket:“test”)\n|> range(start:-15m)\n|> filter(fn:® => r._measurement == “Processor” and\nr._field == \"Percent_Processor_Time\"and\nr.instance == “_Total”)\n|> group(columns: [“host”, “instance”])’\n\nNot sure how if this can be used and how to implement this:\nhttps://v2.docs.influxdata.com/v2.0/reference/flux/stdlib/built-in/outputs/to/#output-data-requirements\n\nThanks\n\n@Anaisdg is this possible?\n\nI also tried pandas with influxdb\n\nBut getting an error using ‘_write_client’ is not defined.\n\nNameError Traceback (most recent call last)\nin\n4 #system_stats.set_index(\"_time\")\n5\n----> 6 _write_client.write(bucket.name, record=df_after_timeindex, data_frame_measurement_name=‘Processor_1’,\n7 data_frame_tag_columns=[‘Processor_1’])\n\nNameError: name ‘_write_client’ is not defined\n\n1 Like\n\nWhoops you need to instantiate your write client like so:\n_write_client = client.write_api(write_options=WriteOptions(batch_size=1000,\nflush_interval=10_000,\njitter_interval=2_000,\nretry_interval=5_000))\n\n@Ashish_Sikarwar,\nYou can use the `to()` flux function to write the data to a new measurement like so:\n|> to(bucket: “test”, measurementColumn: “Total_Processor_Time” ) assuming that you already have a measurement with that name in that bucket. Otherwise you could just write the data to a new bucket.\n\n1 Like\n\nAwesome i will try that.\nThanks a lot @Anaisdg\n\n@Anaisdg one more thing.\nI need to query last 30 days data but only wants mean of data points with certain timestamp with five minutes of interval (not all the data of last 30 days)\nlike from\n11:00 am to 11:05 am, 11:05 am to 11:10 am\nThere is a function which help to get hourly data but i need hours+minute\n|> hourSelection(start: date.hour(t: now()), stop: date.hour(t: now()))\nWill “date” package be of any help and how can i use it.\n\nFor example take this:\n\n``````query= '''\nimport \"date\"\nfrom(bucket: \"yourdb\")\n|> range(start:-30d)\n|> filter(fn: (r) => r._measurement == \"Processor\")\n|> filter(fn: (r) => r._field == \"Percent_Processor_Time\")\n|> filter(fn: (r) => r.instance == \"_Total\")\n|> filter(fn: (r) => exists r._value)\n|> aggregateWindow(every: 10m, fn: mean)\n|> group(columns: [\"instance\", \"host\"])\n'''``````\n\n@Anaisdg We got this but we do not see timestamps _time being stored, not sure why?\n\n``````import \"date\"\n\nmin_duration = date.minute(t: now()) - 15\nmin_duration_end = date.minute(t: now())\nhour_duration_end = date.hour(t: now())\n\nhour_duration = if min_duration < 0 then date.hour(t: now())-1 else date.hour(t: now())\nhour_duration_start = if hour_duration == -1 then 23 else hour_duration\n\nmin_duration_start = if min_duration < 0 then 60 - min_duration else min_duration\n\nfrom(bucket:\"yourdb/autogen\")\n|> range(start:-30d)\n|> filter(fn: (r) =>\nr._measurement == \"Processor\" and\nr._field == \"Percent_Processor_Time\"and\nr.instance == \"_Total\")\n|> filter(fn: (r) =>\ndate.minute(t: r._time) > min_duration_start and\ndate.hour(t: r._time) == hour_duration_start)\n|> filter(fn: (r) =>\ndate.minute(t: r._time) <= min_duration_end and\ndate.hour(t: r._time) == hour_duration_end)\n|> filter(fn: (r) => exists r._value)\n//|> hourSelection(start: date.hour(t: now()), stop: date.hour(t: now()))\n//|> keep(columns: [\"_time\",\"_value\"])\n|> aggregateWindow(every: 5m, fn: mean)\n//|> mean()``````\n\n@Ashish_Sikarwar Is there not a `_time` column when you run the query as-is (with the `mean()` call commented out? `mean()` itself is an aggregate function, which does remove the `_time` column, but if you’re running the query as you pasted it in, it should still have a time column.\n\n1 Like\n\nThis topic was automatically closed 60 minutes after the last reply. New replies are no longer allowed." ]

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[ "# Graphics Class Methods\n\nWe can divide Graphics class methods into three categories: draw, fill, and miscellaneous. Draw methods are used to draw lines, curves, and outer boundaries of closed curves and images. Fill methods fill the interior area of graphics objects. There are also a few miscellaneous methods that fall in neither categoryfor example, MeasureString and Clear.\n\n3.2.1 Draw Methods\n\nThe draw methods of the Graphics class are used to draw lines, curves, and outer boundaries of closed curves and images. Table 3.2 lists the draw methods of the Graphics class.\n\n3.2.1.1 Drawing Lines\n\nThe DrawLine method draws a line beween two points specified by a pair of coordinates. DrawLines draws a series of lines using an array of points.\n\nTable 3.2. Graphics draw methods\n\nMethod\n\nDescription\n\nDrawArc\n\nDraws an arc (a portion of an ellipse specified by a pair of coordinates, a width, a height, and start and end angles).\n\nDrawBezier\n\nDraws a Bézier curve defined by four Point structures.\n\nDrawBeziers\n\nDraws a series of Bézier splines from an array of Point structures.\n\nDrawClosedCurve\n\nDraws a closed cardinal spline defined by an array of Point structures.\n\nDrawCurve\n\nDraws a cardinal spline through a specified array of Point structures.\n\nDrawEllipse\n\nDraws an ellipse defined by a bounding rectangle specified by a pair of coordinates, a height, and a width.\n\nDrawIcon\n\nDraws an image represented by the specified Icon object at the specified coordinates.\n\nDrawIconUnstretched\n\nDraws an image represented by the specified Icon object without scaling the image.\n\nDrawImage\n\nDraws the specified Image object at the specified location and with the original size.\n\nDrawImageUnscaled\n\nDraws the specified Image object with its original size at the location specified by a coordinate pair.\n\nDrawLine\n\nDraws a line connecting two points specified by coordinate pairs.\n\nDrawLines\n\nDraws a series of line segments that connect an array of Point structures.\n\nDrawPath\n\nDraws a GraphicsPath object.\n\nDrawPie\n\nDraws a pie shape specified by a coordinate pair, a width, a height, and two radial lines.\n\nDrawPolygon\n\nDraws a polygon defined by an array of Point structures.\n\nDrawRectangle\n\nDraws a rectangle specified by a coordinate pair, a width, and a height.\n\nDrawRectangles\n\nDraws a series of rectangles specified by an array of Rectangle structures.\n\nDrawString\n\nDraws the specified text string at the specified location using the specified Brush and Font objects.\n\nDrawLine has four overloaded methods. The first argument of all DrawLine methods is a Pen object, with texture, color, and width attributes. The rest of the arguments vary. You can use two points with integer or floating point values, or you can pass four integer or floating point values directly:\n\n1. public void DrawLine(Pen, Point, Point);\n2. public void DrawLine(Pen, PointF, PointF);\n3. public void DrawLine(Pen, int, int, int, int);\n4. public void DrawLine(Pen, float, float, float, float);\n\nTo draw a line, an application first creates a Pen object, which defines the color and width. The following line of code creates a red pen with a width of 1:\n\n```\n```\n```Pen redPen = new Pen(Color.Red, 1);\n```\n```\n```\n\nAfter that we define the endpoints of the line:\n\n```\n```\n```float x1 = 20.0F, y1 = 25.0F;\nfloat x2 = 200.0F, y2 = 100.0F;\n```\n```\n```\n\nFinally, we use the pen and points as input to DrawLine:\n\n```\n```\n```Graphics.DrawLine(redPen, x1, y1, x2, y2);\n```\n```\n```\n\nListing 3.1 shows how to use the different overloaded methods. We create four pens with different colors and widths. After that we call DrawLine with different valuesincluding integer, floating point, and Point structuresto draw four different lines. Three of them start at point (20, 20).\n\nListing 3.1 Drawing lines\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create four Pen objects with red,\n// blue, green, and black colors and\n// different widths\nPen redPen = new Pen(Color.Red, 1);\nPen bluePen = new Pen(Color.Blue, 2);\nPen greenPen = new Pen(Color.Green, 3);\nPen blackPen = new Pen(Color.Black, 4);\n// Draw line using float coordinates\nfloat x1 = 20.0F, y1 = 20.0F;\nfloat x2 = 200.0F, y2 = 20.0F;\ne.Graphics.DrawLine(redPen, x1, y1, x2, y2);\n// Draw line using Point structure\nPoint pt1 = new Point(20, 20);\nPoint pt2 = new Point(20, 200);\ne.Graphics.DrawLine(greenPen, pt1, pt2);\n// Draw line using PointF structure\nPointF ptf1 = new PointF(20.0F, 20.0F);\nPointF ptf2 = new PointF(200.0F, 200.0F);\ne.Graphics.DrawLine(bluePen, ptf1, ptf2);\n// Draw line using integer coordinates\nint X1 = 60, Y1 = 40, X2 = 250, Y2 = 100;\ne.Graphics.DrawLine(blackPen, X1, Y1, X2, Y2);\n// Dispose of objects\nredPen.Dispose();\nbluePen.Dispose();\ngreenPen.Dispose();\nblackPen.Dispose();\n}\n```\n\nThe output from Listing 3.1 is shown in Figure 3.1. We've drawn four lines starting at point (20, 20).\n\nFigure 3.1. Using DrawLine to draw lines", null, "3.2.1.2 Drawing Connected Lines\n\nSometimes we need to draw multiple connected straight line segments. One way to do this is to call the DrawLine method multiple times.\n\nThe Graphics class also provides the DrawLines method, which can be used to draw multiple connected lines. This method has two overloaded forms. One takes an array of Point structure objects, and the other takes an array of PointF structure objects:\n\n1. public void DrawLines(Pen, Point[]);\n2. public void DrawLines(Pen, PointF[]);\n\nTo draw lines using DrawLines, an application first creates a Pen object, then creates an array of points, and then calls DrawLines. The code in Listing 3.2 draws three line segments.\n\nListing 3.2 Using DrawLines to draw connected lines\n\n```PointF[] ptsArray =\n{\nnew PointF( 20.0F, 20.0F),\nnew PointF( 20.0F, 200.0F),\nnew PointF(200.0F, 200.0F),\nnew PointF(20.0F, 20.0F)\n\n};\ne.Graphics.DrawLines(redPen, ptsArray);\n```\n\nThe code in Listing 3.2 draws what is shown in Figure 3.2.\n\nFigure 3.2. Using DrawLines to draw connected lines", null, "3.2.1.3 Drawing Rectangles\n\nThe next basic drawing object is a rectangle. When you draw a rectangle through your applications, you need to specify only the starting point, height, and width of the rectangle. GDI+ takes care of the rest.\n\nThe Graphics class provides the DrawRectangle method, which draws a rectangle specified by a starting point, a width, and a height. The Graphics class also provides the DrawRectangles method, which draws a series of rectangles specified by an array of Rectangle structures.\n\nDrawRectangle has three overloaded methods. An application can use a Rectangle structure or coordinates of integer or float types to draw a rectangle:\n\n1. public void DrawRectangle(Pen, Rectangle);\n2. public void DrawRectangle(Pen, int, int, int, int);\n3. public void DrawRectangle(Pen, float, float, float, float);\n\nTo draw a rectangle, an application first creates a pen and a rectangle (location, width, and height), and then it calls DrawRectangle. Listing 3.3 draws rectangles using the different overloaded forms of DrawRectangle.\n\nListing 3.3 Using DrawRectangle to draw rectangles\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create pens and points\nPen redPen = new Pen(Color.Red, 1);\nPen bluePen = new Pen(Color.Blue, 2);\nPen greenPen = new Pen(Color.Green, 3);\nfloat x = 5.0F, y = 5.0F;\nfloat width = 100.0F;\nfloat height = 200.0F;\n// Create a rectangle\nRectangle rect = new Rectangle(20, 20, 80, 40);\n// Draw rectangles\ne.Graphics.DrawRectangle(bluePen,\nx, y, width, height);\ne.Graphics.DrawRectangle(redPen,\n60, 80, 140, 50);\ne.Graphics.DrawRectangle(greenPen, rect);\n// Dispose of objects\nredPen.Dispose();\nbluePen.Dispose();\ngreenPen.Dispose();\n}\n```\n\nFigure 3.3 shows the output from Listing 3.3.\n\nFigure 3.3. Drawing individual rectangles", null, "The DrawRectangles method draws a series of rectangles using a single-pen. It is useful when you need to draw multiple rectangles using the same pen (if you need to draw multiple rectangles using different pens, you must use multiple calls to DrawRectangle). A single call to DrawRectangles is faster than multiple DrawRectangle calls. DrawRectangles takes two parametersa pen and an array of Rectangle or RectangleF structuresas shown in Listing 3.4.\n\nListing 3.4 Using DrawRectangles to draw a series of rectangles\n\n```Pen greenPen = new Pen(Color.Green, 4);\nRectangleF[] rectArray\n{\nnew RectangleF( 5.0F, 5.0F, 100.0F, 200.0F),\nnew RectangleF(20.0F, 20.0F, 80.0F, 40.0F),\nnew RectangleF(60.0F, 80.0F, 140.0F, 50.0F)\n};\ne.Graphics.DrawRectangles(greenPen, rectArray);\ngreenPen.Dispose()\n```\n\nFigure 3.4 shows the output from Listing 3.4. As you can see, it's easy to draw multiple rectangles using the DrawRectangles method.\n\nFigure 3.4. Drawing a series of rectangles", null, "3.2.1.4 Drawing Ellipses and Circles\n\nAn ellipse is a circular boundary within a rectangle, where each opposite point has the same distance from a fixed point, called the center of the ellipse. An ellipse within a square is called a circle. Figure 3.5 shows an ellipse with its height, width, and center indicated.\n\nFigure 3.5. An ellipse", null, "To draw an ellipse, you need to specify the outer rectangle. GDI+ takes care of the rest. DrawEllipse draws an ellipse defined by a rectangle specified by a pair of coordinates, a height, and a width (an ellipse with equal height and width is a circle). DrawEllipse has four overloaded methods:\n\n1. public void DrawEllipse(Pen, Rectangle);\n2. public void DrawEllipse(Pen, RectangleF);\n3. public void DrawEllipse(Pen, int, int, int, int);\n4. public void DrawEllipse(Pen, float, float, float, float);\n\nTo draw an ellipse, an application creates a pen and four coordinates (or a rectangle), and then calls DrawEllipse. Listing 3.5 draws ellipses with different options.\n\nListing 3.5 Drawing ellipses\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create pens\nPen redPen = new Pen(Color.Red, 6 );\nPen bluePen = new Pen(Color.Blue, 4 );\nPen greenPen = new Pen(Color.Green, 2);\n// Create a rectangle\nRectangle rect =\nnew Rectangle(80, 80, 50, 50);\n// Draw ellipses\ne.Graphics.DrawEllipse(greenPen,\n100.0F, 100.0F, 10.0F, 10.0F );\ne.Graphics.DrawEllipse(redPen, rect );\ne.Graphics.DrawEllipse(bluePen, 60, 60, 90, 90);\ne.Graphics.DrawEllipse(greenPen,\n40.0F, 40.0F, 130.0F, 130.0F );\n// Dispose of objects\nredPen.Dispose();\ngreenPen.Dispose();\nbluePen.Dispose();\n}\n```\n\nFigure 3.6 shows the output from Listing 3.5.\n\nFigure 3.6. Drawing ellipses", null, "3.2.1.5 Drawing Text\n\nThis section briefly discusses the drawing of text. Chapter 5 covers this topic in more detail.\n\nThe DrawString method draws a text string on a graphics surface. It has many overloaded forms. DrawString takes arguments that identify the text, font, brush, starting location, and string format.\n\nThe simplest form of DrawString looks like this:\n\n```\n```\n```public void DrawString(string, Font, Brush, PointF);\n```\n```\n```\n\nwhere string is the text that you want to draw, Font and Brush are the font and brushes used to draw the text, and PointF is the starting point of the text.\n\nListing 3.6 uses the DrawString method to draw \"Hello GDI+ World!\" on a form.\n\nListing 3.6 Drawing text\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\ne.Graphics.DrawString(\"Hello GDI+ World!\",\nnew Font(\"Verdana\", 16),\nnew SolidBrush(Color.Red),\nnew Point(20, 20));\n}\n```\n\nNote\n\nYou might notice in Listing 3.6 that we create Font, SolidBrush, and Point objects directly as parameters of the DrawString method. This method of creating objects means that we can't dispose of these objects, so some cleanup is left for the garbage collector.\n\nFigure 3.7 shows the output from Listing 3.6.\n\nFigure 3.7. Drawing text", null, "The DrawString method has several overloaded forms, as shown here:\n\n• public void DrawString(string, Font, Brush, RectangleF);\n• public void DrawString(string, Font, Brush, PointF, StringFormat);\n• public void DrawString(string, Font, Brush, RectangleF, StringFormat);\n• public void DrawString(string, Font, Brush, float, float);\n• public void DrawString(string, Font, Brush, float, float, StringFormat);\n\nNow let's see another example of drawing textthis time using the StringFormat class, which defines the text format. Using StringFormat, you can set flags, alignment, trimming, and other options for the text. (Chapter 5 discusses this functionality in more detail.) Listing 3.7 shows different ways to draw text on a graphics surface. In this example the FormatFlags property is set to StringFormatFlags.DirectionVertical, which draws vertical text.\n\nListing 3.7 Using DrawString to draw text on a graphics surface\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create brushes\nSolidBrush blueBrush = new SolidBrush(Color.Blue);\nSolidBrush redBrush = new SolidBrush(Color.Red);\nSolidBrush greenBrush = new SolidBrush(Color.Green);\n// Create a rectangle\nRectangle rect = new Rectangle(20, 20, 200, 100);\n// The text to be drawn\nString drawString = \"Hello GDI+ World!\";\n// Create a Font object\nFont drawFont = new Font(\"Verdana\", 14);\nfloat x = 100.0F;\nfloat y = 100.0F;\n// String format\nStringFormat drawFormat = new StringFormat();\n// Set string format flag to direction vertical,\n// which draws text vertically\ndrawFormat.FormatFlags =\nStringFormatFlags.DirectionVertical;\n// Draw string\ne.Graphics.DrawString(\"Drawing text\",\nnew Font(\"Tahoma\", 14), greenBrush, rect);\ne.Graphics.DrawString(drawString,\nnew Font(\"Arial\", 12), redBrush, 120, 140);\ne.Graphics.DrawString(drawString, drawFont,\nblueBrush, x, y, drawFormat);\n// Dispose of objects\nblueBrush.Dispose();\nredBrush.Dispose();\ngreenBrush.Dispose();\ndrawFont.Dispose();\n}\n```\n\nFigure 3.8 shows the output from Listing 3.7.\n\nFigure 3.8. Drawing text with different directions", null, "3.2.1.6 Creating a Line Chart Application\n\nAs promised, the examples in this book not only show the use of GDI+, but also encourage you to use GDI+ practices in real-world applications, We will create one more real-world application, a line chart application. In this example we will use all the functionality we have discussed so far. Our line chart application will draw lines when a user clicks on a form.\n\nWe create a Windows application and add a check box and a button. Then we change the Text properties of the button and the check box to call them Clear All and Rectangle, respectively. Then we add code to draw two lines and some numbers (using the DrawString method). The initial screen of the line chart application looks like Figure 3.9.\n\nFigure 3.9. The line chart application", null, "When you click on the form, the application draws a line. The first line starts from the bottom left corner, where the values of our x- and y-axes are both 0. After a few clicks, the chart looks like Figure 3.10. Every time you click on the form, the application draws a line from the previous point to the current point and draws a small ellipse representing the current point.\n\nFigure 3.10. The line chart application with a chart", null, "The Clear All button removes the lines and initializes the first point to (0, 0). Now if you check the Rectangle box and click on the form, the chart looks like Figure 3.11. When you click the left mouse button for the first time, the application draws a line from point (0, 0) to the point where you clicked the button.\n\nFigure 3.11. The line chart with rectangles to mark points", null, "Now let's see the code. First we declare starting and ending points. These points will be used to draw a line when you click the left mouse button. The default values of both points are shown in the following code fragment, which represents position (0, 0) on the screen:\n\n```\n```\n```private Point startPoint = new Point(50, 217);\nprivate Point endPoint = new Point(50, 217);\n```\n```\n```\n\nThe next step is to draw vertical and horizontal axis lines with index numbers. We do this on the form's paint event handler with the help of the DrawString method. Listing 3.8 provides code for the form-paint event handler. As the listing shows, we simply draw a vertical line, a horizontal line, and the marks on these lines.\n\nListing 3.8 Drawing lines and marks\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\nGraphics g = e.Graphics;\nFont vertFont =\nnew Font(\"Verdana\", 10, FontStyle.Bold);\nFont horzFont =\nnew Font(\"Verdana\", 10, FontStyle.Bold);\nSolidBrush vertBrush = new SolidBrush(Color.Black);\nSolidBrush horzBrush = new SolidBrush(Color.Blue);\nPen blackPen = new Pen(Color.Black, 2);\nPen bluePen = new Pen(Color.Blue, 2);\n// Drawing a vertical and a horizontal line\ng.DrawLine(blackPen,50,220,50, 25);\ng.DrawLine(bluePen,50,220,250,220);\n// x-axis drawing\ng.DrawString(\"0\",horzFont,horzBrush,30, 220);\ng.DrawString(\"1\",horzFont,horzBrush,50,220);\ng.DrawString(\"2\",horzFont,horzBrush,70,220);\ng.DrawString(\"3\",horzFont,horzBrush,90,220);\ng.DrawString(\"4\",horzFont,horzBrush,110,220);\ng.DrawString(\"5\",horzFont,horzBrush,130,220);\ng.DrawString(\"6\",horzFont,horzBrush,150,220);\ng.DrawString(\"7\",horzFont,horzBrush,170,220);\ng.DrawString(\"8\",horzFont,horzBrush,190,220);\ng.DrawString(\"9\",horzFont,horzBrush,210,220);\ng.DrawString(\"10\",horzFont,horzBrush,230,220);\n// Drawing vertical strings\nStringFormat vertStrFormat = new StringFormat();\nvertStrFormat.FormatFlags =\nStringFormatFlags.DirectionVertical;\n\ng.DrawString(\"-\",horzFont,horzBrush,\n50, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n70, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n90, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n110, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n130, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n150, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n170, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n190, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n210, 212, vertStrFormat);\ng.DrawString(\"-\",horzFont,horzBrush,\n230, 212, vertStrFormat);\n// y-axis drawing\ng.DrawString(\"100-\",vertFont,vertBrush, 20,20);\ng.DrawString(\"90 -\",vertFont,vertBrush, 25,40);\ng.DrawString(\"80 -\",vertFont,vertBrush, 25,60);\ng.DrawString(\"70 -\",vertFont,vertBrush, 25,80);\ng.DrawString(\"60 -\",vertFont,vertBrush, 25,100);\ng.DrawString(\"50 -\",vertFont,vertBrush, 25,120);\ng.DrawString(\"40 -\",vertFont,vertBrush, 25,140);\ng.DrawString(\"30 -\",vertFont,vertBrush, 25,160);\ng.DrawString(\"20 -\",vertFont,vertBrush, 25,180);\ng.DrawString(\"10 -\",vertFont,vertBrush, 25,200);\n// Dispose of objects\nvertFont.Dispose();\nhorzFont.Dispose();\nvertBrush.Dispose();\nhorzBrush.Dispose();\nblackPen.Dispose();\nbluePen.Dispose();\n}\n```\n\nNote\n\nThe idea in Listing 3.8 is to show an extensive use of the DrawString method. Alternatively and preferably, you could replace DrawString with the DrawLine and/or DrawLines method.\n\nNow on the mouse-down event handler, we draw a line from the starting point (0, 0) to the first mouse click. We store the mouse click position as the starting point for the next line. When we click again, the new line will be drawn from the current starting position to the point where the mouse was clicked. Listing 3.9 shows the mouse-down click event handler. We create a new Graphics object using the CreateGraphics method. After that we create two Pen objects. We store the previous point as the starting point and the current point as the ending point. The X and Y properties of MouseEventArgs return the x- and y-values of the point where the mouse was clicked.\n\nNow we check to see if the Rectangle check box is checked. If so, we draw a rectangle to mark the connecting point of the two lines. If not, we draw an ellipse as the connecting point.\n\nListing 3.9 The mouse-down event handler\n\n```private void Form1_MouseDown(object sender,\nSystem.Windows.Forms.MouseEventArgs e)\n{\nif (e.Button == MouseButtons.Left)\n{\n// Create a Graphics object\nGraphics g1 = this.CreateGraphics();\n// Create two pens\nPen linePen = new Pen(Color.Green, 1);\nPen ellipsePen = new Pen(Color.Red, 1);\nstartPoint = endPoint;\nendPoint = new Point(e.X, e.Y);\n// Draw the line from the current point\n// to the new point\ng1.DrawLine(linePen, startPoint, endPoint);\n// If Rectangle check box is checked,\n// draw a rectangle to represent the point\nif(checkBox1.Checked)\n{\ng1.DrawRectangle(ellipsePen,\ne.X-2, e.Y-2, 4, 4);\n}\n// Draw a circle to represent the point\nelse\n{\ng1.DrawEllipse(ellipsePen,\ne.X-2, e.Y-2, 4, 4);\n}\n// Dispose of objects\nlinePen.Dispose();\nellipsePen.Dispose();\ng1.Dispose();\n}\n}\n```\n\nThe Clear All button removes all the lines by invalidating the form's client area and sets the starting and ending points back to their initial values. Code for the Clear All button click event handler is given in Listing 3.10.\n\nListing 3.10 The Clear All button click event handler\n\n```private void button1_Click(object sender,\nSystem.EventArgs e)\n{\nstartPoint.X = 50;\nstartPoint.Y = 217;\nendPoint.X = 50;\nendPoint.Y = 217;\nthis.Invalidate(this.ClientRectangle);\n}\n```\n\n3.2.1.7 Drawing Arcs\n\nAn arc is a portion of an ellipse. For example, Figure 3.12 shows an ellipse that has six arcs. An arc is defined by a bounding rectangle (just as an ellipse), a start angle, and a sweep angle. The start angle is an angle in degrees measured clockwise from the x-axis to the starting point of the arc. The sweep angle is an angle in degrees measured clockwise from the startAngle parameter to the ending point of the arc. So an arc is the portion of the perimeter of the ellipse between the start angle and the start angle plus the sweep angle.\n\nFigure 3.12. Arcs in an ellipse", null, "The DrawArc method draws an arc on a graphics surface. DrawArc takes a pen, a pair of coordinates, a width, and a height. There are many DrawArc overloaded methods. An application can use a Rectangle or RectangleF object and integer or float coordinates:\n\n• public void DrawArc(Pen, Rectangle, float, float);\n• public void DrawArc(Pen, RectangleF, float, float);\n• public void DrawArc(Pen, int, int, int, int, int, int);\n• public void DrawArc(Pen, float, float, float, float, float, float);\n\nThe Pen object determines the color, width, and style of the arc; Rectangle or RectangleF represents the bounding rectangle; and the last two parameters are the start angle and sweep angle.\n\nTo draw an arc, the application creates Pen and Rectangle objects and defines start and sweep angles. Then it calls the DrawArc method.\n\nLet's create an application that will draw an arc to match the values of the start and sweep angles. We create a Windows application, adding add two text boxes and a button control. The final form looks like Figure 3.13.\n\nFigure 3.13. A sample arc application", null, "We define two floating variables on the class level to store the start and sweep angles:\n\n```\n```\n```private float startAngle = 45.0f;\nprivate float sweepAngle = 90.0f;\n```\n```\n```\n\nNow let's draw an arc on the form's paint event handler. Listing 3.11 draws an arc. We first create a pen and a rectangle, and we use them in the DrawArc method with start and sweep angles.\n\nListing 3.11 The paint event handler\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\nPen redPen = new Pen(Color.Red, 3);\nRectangle rect =\nnew Rectangle(20, 20, 200, 200);\ne.Graphics.DrawArc(redPen,\nrect, startAngle, sweepAngle);\nredPen.Dispose();\n}\n```\n\nNow we add code for the Reset Angles button. Listing 3.12 simply sets the start and sweep angles by reading values from the text boxes and calls the Invalidate method, which forces GDI+ to call the form's paint event handler.\n\nListing 3.12 The Reset Angles button click event handler\n\n```private void ResetAnglesBtn_Click(object sender,\nSystem.EventArgs e)\n{\nstartAngle =\n(float)Convert.ToDouble(textBox1.Text);\nsweepAngle =\n(float)Convert.ToDouble(textBox2.Text);\nInvalidate();\n}\n```\n\nFigure 3.14 shows the default output from the application.\n\nFigure 3.14. The default arc, with start angle of 45 degrees and sweep angle of 90 degrees", null, "Now let's change the start and sweep angles to 90 and 180 degrees, respectively, and click the Reset Angles button. The new output looks like Figure 3.15.\n\nFigure 3.15. An arc with start angle of 90 degrees and sweep angle of 180 degrees", null, "Let's change angles one more time. This time our start angle will be 180 degrees, and the sweep angle will be 360 degrees. The new output looks like Figure 3.16.\n\nFigure 3.16. An arc with start angle of 180 degrees and sweep angle of 360 degree", null, "3.2.1.8 Drawing Splines and Curves\n\nA curve is a sequence of adjoining points with a tension. The tension of a curve provides its smoothness and removes corners. A cardinal spline is a sequence of multiple joined curves. Basically, in a curve there is no straight line between two points. To illustrate, Figure 3.17 shows two curves.\n\nFigure 3.17. Two curves", null, "There are two types of curves: open and closed. A closed curve is a curve whose starting point is the ending point. A curve that is not a closed curve is called an open curve. In Figure 3.18 the first curve is an open curve, and the second curve is a closed curve.\n\nFigure 3.18. Open and closed curves", null, "3.2.1.9 Drawing Open Curves\n\nProgrammatically, a curve is an array of connected points with a tension. A curve has a starting point and an ending point. Between these two points can be many intermediate points. The Graphics class provides two methods for drawing curves: DrawCurve and DrawClosedCurve. The DrawCurve method draws a curve specified by an array of Point structures. The DrawClosedCurve draws a closed curve specified by an array of Point structures. Both DrawCurve and DrawClosedCurve have overloaded methods.\n\nDrawCurve has the following overloaded forms:\n\n• public void DrawCurve(Pen, Point[]);\n• public void DrawCurve(Pen, PointF[]);\n• public void DrawCurve(Pen, Point[], float);\n• public void DrawCurve(Pen, PointF[], float);\n• public void DrawCurve(Pen, PointF[], int, int);\n• public void DrawCurve(Pen, Point[], int, int, float);\n• public void DrawCurve(Pen, PointF[], int, int, float);\n\nThe simplest form of DrawCurve is\n\n```\n```\n```public void DrawCurve(Pen pen, Point[] points);\n```\n```\n```\n\nwhere points is an array of points.\n\nTo test the DrawCurve methods, we create a Windows application and add Listing 3.13 to the form's paint event handler. It creates an array of points and draws a curve using the DrawCurve method.\n\nListing 3.13 Drawing a curve\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create a pen\nPen bluePen = new Pen(Color.Blue, 1);\n// Create an array of points\nPointF pt1 = new PointF( 40.0F, 50.0F);\nPointF pt2 = new PointF(50.0F, 75.0F);\nPointF pt3 = new PointF(100.0F, 115.0F);\nPointF pt4 = new PointF(200.0F, 180.0F);\nPointF pt5 = new PointF(200.0F, 90.0F);\nPointF[] ptsArray =\n{\npt1, pt2, pt3, pt4, pt5\n};\n// Draw curve\ne.Graphics.DrawCurve(bluePen, ptsArray);\n// Dispose of object\nbluePen.Dispose();\n}\n```\n\nFigure 3.19 shows the output from our Listing 3.13.\n\nFigure 3.19. Drawing a curve", null, "Note\n\nThe default tension is 0.5 for this overloaded version of DrawCurve.\n\nThe second form of DrawCurve is\n\n```\n```\n```public void DrawCurve(Pen pen,\nPoint[] points,\nfloat tension\n);\n```\n```\n```\n\nHere the tension parameter determines the shape of the curve. If the value of tension is 0.0F, the method draws a straight line between the points. The value of tension should vary between 0.0F and 1.0F.\n\nNow let's update the example in Listing 3.13. We add a text box, a label, and a button to the form. We change the properties of these controls, and the form looks like Figure 3.20.\n\nFigure 3.20. A curve-drawing application", null, "Now we will update our sample code to draw a curve using the tension value entered in the text box. We add a float type variable, tension, at the class level:\n\n```\n```\n```private float tension = 0.5F;\n```\n```\n```\n\nThen we update the form's paint event handler as shown in Listing 3.14. We provide tension as the third argument to the DrawCurve method.\n\nListing 3.14 Drawing a curve with tension\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create a pen\nPen bluePen = new Pen(Color.Blue, 1);\n// Create an array of points\nPointF pt1 = new PointF( 40.0F, 50.0F);\nPointF pt2 = new PointF(50.0F, 75.0F);\nPointF pt3 = new PointF(100.0F, 115.0F);\nPointF pt4 = new PointF(200.0F, 180.0F);\nPointF pt5 = new PointF(200.0F, 90.0F);\nPointF[] ptsArray =\n{\npt1, pt2, pt3, pt4, pt5\n};\n// Draw curve\ne.Graphics.DrawCurve(bluePen, ptsArray, tension);\n// Dispose of object\nbluePen.Dispose();\n}\n```\n\nNow we add code for the Apply button, which simply reads the text box's value and sets it as the tension, as in Listing 3.15.\n\nListing 3.15 The Apply button click event handler\n\n```private void ApplyBtn_Click(object sender,\nSystem.EventArgs e)\n{\ntension = (float)Convert.ToDouble(textBox1.Text);\nInvalidate();\n}\n```\n\nIf you enter \"0.0\" in the text box and hit Apply, the output looks like Figure 3.21, and if you enter the value \"1.0\" in the text box and hit Apply, the output looks like Figure 3.22.\n\nFigure 3.21. Drawing a curve with a tension of 0.0F", null, "Figure 3.22. Drawing a curve with a tension of 1.0F", null, "You can also add an offset and specify a number of segments for the curve:\n\n```\n```\n```public void DrawCurve(\nPen pen,\nPointF[] points,\nint offset,\nint numberOfSegments\n);\n```\n```\n```\n\nThe offset specifies the number of elements to skip in the array of points. The first element after the skipped elements in the array of points becomes the starting point of the curve.\n\nThe numberOfSegments property specifies the number of segments, after the starting point, to draw in the curve. It must be at least 1. The offset plus the number of segments must be less than the number of elements in the array of the points.\n\nThe following method skips the first element of the array of points and starts drawing a curve from the second point in the array, stopping after three segments:\n\n```\n```\n```int offset = 1;\nint segments = 3;\ne.Graphics.DrawCurve(bluePen, ptsArray,\noffset, segments);\n```\n```\n```\n\nThe final version of DrawCurve takes a pen, points array, offset, number of segments, and tension:\n\n```\n```\n```public void DrawCurve(\nPen pen,\nPoint[] points,\nint offset,\nint numberOfSegments,\nfloat tension\n);\n```\n```\n```\n\nHere's an example:\n\n```\n```\n```int offset = 1;\nint segments = 3;\ne.Graphics.DrawCurve(bluePen, ptsArray,\noffset, segments, tension);\n```\n```\n```\n\n3.2.1.10 Drawing Closed Curves\n\nAs stated earlier, a closed curve is a curve whose starting and ending points are the same. The Graphics class provides the DrawClosedCurve method to draw closed curves. It has the following overloaded forms:\n\n• public void DrawClosedCurve(Pen, Point[]);\n• public void DrawClosedCurve(Pen, PointF[]);\n• public void DrawClosedCurve(Pen, Point[], float, FillMode);\n• public void DrawClosedCurve(Pen, PointF[], float, FillMode);\n\nThe simplest form of DrawClosedCurve takes two parameters: a pen and an array of points. Listing 3.16 creates an array of points and a pen and calls the DrawClosedCurve method.\n\nListing 3.16 Drawing closed curves\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n\n// Create a pen\nPen bluePen = new Pen(Color.Blue, 1);\n// Create an array of points\nPointF pt1 = new PointF( 40.0F, 50.0F);\nPointF pt2 = new PointF(50.0F, 75.0F);\nPointF pt3 = new PointF(100.0F, 115.0F);\nPointF pt4 = new PointF(200.0F, 180.0F);\nPointF pt5 = new PointF(200.0F, 90.0F);\nPointF[] ptsArray =\n{\npt1, pt2, pt3, pt4, pt5\n};\n// Draw curve\ne.Graphics.DrawClosedCurve(bluePen, ptsArray);\n// Dispose of object\nbluePen.Dispose();\n}\n```\n\nFigure 3.23 shows the output from Listing 3.16. The result is a closed curve.\n\nFigure 3.23. Drawing a closed curve", null, "The second form of DrawClosedCurve takes as arguments the tension of the curve and FillMode. We have already discussed tension. FillMode specifies how the interior of a closed path is filled and clipped. The FillMode enumeration represents the fill mode of graphics objects. It has two modes: Alternate (the default mode) and Winding.\n\nAs the documentation says,\n\nTo determine the interiors of a closed curve in the Alternate mode, draw a line from any arbitrary start point in the path to some point obviously outside the path. If the line crosses an odd number of path segments, the starting point is inside the closed region and is therefore part of the fill or clipping area. An even number of crossings means that the point is not in an area to be filled or clipped. An open figure is filled or clipped by using a line to connect the last point to the first point of the figure.\n\nThe Winding mode considers the direction of the path segments at each intersection. It adds one for every clockwise intersection, and subtracts one for every counterclockwise intersection. If the result is nonzero, the point is considered inside the fill or clip area. A zero count means that the point lies outside the fill or clip area.\n\nWe will clarify these definitions with examples in the discussion of paths in Chapter 9.\n\nListing 3.17 uses DrawClosedCurve to draw a closed curve with a tension and fill mode.\n\nListing 3.17 Drawing a closed curve with a tension and fill mode\n\n```// Draw curve\nfloat tension = 0.5F;\ne.Graphics.DrawClosedCurve(bluePen, ptsArray,\ntension, FillMode.Alternate);\n```\n\n3.2.1.11 Drawing Bézier Curves\n\nThe Bézier curve, developed by Pierre Bézier in the 1960s for CAD/CAM operations, has become one of the most used curves in drawing. A Bézier curve is defined by four points: two endpoints and two control points. Figure 3.24 shows an example of a Bézier curve in which A and B are the starting and ending points and C and D are two control points.\n\nFigure 3.24. A Bézier curve", null, "The Graphics class provides the DrawBezier and DrawBeziers methods for drawing Bézier curves. DrawBezier draws a Bézier curve defined by four points: the starting point, two control points, and the ending point of the curve. The following example draws a Bézier curve with starting point (30, 20), ending point (140, 50), and control points (80, 60) and (120, 18).\n\n```\n```\n```e.Graphics.DrawBezier(bluePen, 30, 20,\n80, 60, 120, 180, 140, 50);\n```\n```\n```\n\nDrawBeziers draws a series of Bézier curves from an array of Point structures. To draw multiple beziers, you need 3x + 1 points, where x is the number of Bézier segments.\n\nListing 3.18 draws Bézier curves using both DrawBezier and DrawBeziers.\n\nListing 3.18 Drawing Bézier curves\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\nGraphics g = e.Graphics ;\n// Create pens\nPen bluePen = new Pen(Color.Blue, 1);\nPen redPen = new Pen(Color.Red, 1);\n// Create points for curve\nPointF p1 = new PointF(40.0F, 50.0F);\nPointF p2 = new PointF(60.0F, 70.0F);\nPointF p3 = new PointF(80.0F, 34.0F);\nPointF p4 = new PointF(120.0F, 180.0F);\nPointF p5 = new PointF(200.0F, 150.0F);\nPointF p6 = new PointF(350.0F, 250.0F);\nPointF p7 = new PointF(200.0F, 200.0F);\nPointF[] ptsArray =\n{\np1, p2, p3, p4, p5, p6, p7\n};\n// Draw a Bézier\ne.Graphics.DrawBezier(bluePen, 30, 20,\n80, 60, 120, 180, 140, 50);\n// Draw Béziers\ne.Graphics.DrawBeziers(redPen, ptsArray);\n// Dispose of objects\nredPen.Dispose();\nbluePen.Dispose();\n}\n```\n\nFigure 3.25 shows the output from Listing 3.18.\n\nFigure 3.25. Drawing Bézier curves", null, "3.2.1.12 Drawing a Polygon\n\nA polygon is a closed shape with three or more straight sides. Examples of polygons include triangles and rectangles.\n\nThe Graphics class provides a DrawPolygon method to draw polygons. DrawPolygon draws a polygon defined by an array of points. It takes two arguments: a pen and an array of Point or PointF strucures.\n\nTo draw a polygon, an application first creates a pen and an array of points and then calls the DrawPolygon method with these parameters. Listing 3.19 draws a polygon with five points.\n\nListing 3.19 Drawing a polygon\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\nGraphics g = e.Graphics ;\n// Create pens\nPen greenPen = new Pen(Color.Green, 2);\nPen redPen = new Pen(Color.Red, 2);\n// Create points for polygon\nPointF p1 = new PointF(40.0F, 50.0F);\nPointF p2 = new PointF(60.0F, 70.0F);\nPointF p3 = new PointF(80.0F, 34.0F);\nPointF p4 = new PointF(120.0F, 180.0F);\nPointF p5 = new PointF(200.0F, 150.0F);\nPointF[] ptsArray =\n{\np1, p2, p3, p4, p5\n};\n// Draw polygon\ne.Graphics.DrawPolygon(greenPen,ptsArray);\n// Dispose of objects\ngreenPen.Dispose();\nredPen.Dispose();\n}\n```\n\nFigure 3.26 shows the output from Listing 3.19.\n\nFigure 3.26. Drawing a polygon", null, "3.2.1.13 Drawing Icons\n\nThe DrawIcon and DrawIconUnstretched methods are used to draw icons. DrawIcon draws an image represented by a specified object at the specified coordinatesstretching the image to fit, if necessary. DrawIconUnstretched draws an image represented by an Icon object without scaling the image.\n\nDrawIcon and DrawIconUnstretched take two arguments: an Icon object and upper left corner coordinates of a rectangle. To draw an icon using these methods, an application first creates an icon and either a Rectangle object or coordinates to the upper left corner at which to draw the icon.\n\nAn Icon object represents a Windows icon. An application creates an Icon object using its constructor, which takes arguments of string, Icon, Stream, and Type. Table 3.3 describes the properties of the Icon class.\n\nTable 3.4 describes some of the methods of the Icon class.\n\nTable 3.3. Icon properties\n\nProperty\n\nDescription\n\nHandle\n\nRepresents the window handle of an icon.\n\nHeight\n\nRepresents the height of an icon.\n\nSize\n\nRepresents the size of an icon.\n\nWidth\n\nRepresents the width of an icon.\n\nTable 3.4. Icon methods\n\nMethod\n\nDescription\n\nClone\n\nClones an Icon object, creating a duplicate image.\n\nSave\n\nSaves an Icon object to the output stream.\n\nToBitmap\n\nConverts an Icon object to a Bitmap object.\n\nListing 3.20 draws icons. The application first creates two Icon objects, then creates a Rectangle object and calls DrawIcon and DrawIconUnstretched.\n\nListing 3.20 Drawing icons\n\n```Icon icon1 = new Icon(\"mouse.ico\");\nIcon icon2 = new Icon(\"logo.ico\");\nint x = 20;\nint y = 50;\ne.Graphics.DrawIcon(icon1, x, y);\nRectangle rect = new Rectangle(100, 200, 400, 400);\ne.Graphics.DrawIconUnstretched(icon2, rect);\n```\n\nFigure 3.27 shows the output from Listing 3.20.\n\nFigure 3.27. Drawing icons", null, "3.2.1.14 Drawing Graphics Paths\n\nA graphics path is a combination of multiple graphics shapes. For example, the graphics path in Figure 3.28 is a combination of lines, an ellipse, and a rectangle.\n\nFigure 3.28. A path", null, "GraphicsPath is defined in the System.Drawing.Drawing2D namespace. You must import this namespace using the following line:\n\n```\n```\n```using System.Drawing.Drawing2D;\n```\n```\n```\n\nThe Graphics class provides a DrawPath method, which draws a graphics path. It takes two arguments: Pen and GraphicsPath.\n\nTo draw a graphics path, first we create a GraphicsPath object, then we add graphics shapes to the path by calling its Add methods, and finally we call DrawPath. For example, the following code creates a graphics path, adds an ellipse to the path, and draws it.\n\n```\n```\n```GraphicsPath graphPath = new GraphicsPath();\ng.DrawPath(greenPen, graphPath);\n```\n```\n```\n\nLet's add more shapes to the graph. Listing 3.21 creates a graphics path; adds some lines, an ellipse, and a rectangle; and draws the path.\n\nListing 3.21 Drawing a graphics path\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create a pen\nPen greenPen = new Pen(Color.Green, 1);\n// Create a graphics path\nGraphicsPath path = new GraphicsPath();\n// Add a line to the path\n// Add an ellipse to the path\n// Create a rectangle and call\nRectangle rect =\nnew Rectangle(50, 150, 300, 50);\n// Draw path\ne.Graphics.DrawPath(greenPen, path);\n// Dispose of object\ngreenPen.Dispose();\n}\n```\n\nFigure 3.29 shows the output from Listing 3.21.\n\nFigure 3.29. Drawing a path", null, "3.2.1.15 Drawing Pie Shapes\n\nA pie is a slice of an ellipse. A pie shape also consists of two radial lines that intersect with the endpoints of the arc. Figure 3.30 shows an ellipse with four pie shapes.\n\nFigure 3.30. Four pie shapes of an ellipse", null, "The Graphics class provides the DrawPie method, which draws a pie shape defined by an arc of an ellipse. The DrawPie method takes a Pen object, a Rectangle or RectangleF object, and two radial angles.\n\nLet's create an application that draws pie shapes. We create a Windows application and add two text boxes and a button control to the form. The final form looks like Figure 3.31.\n\nFigure 3.31. A pie shapedrawing application", null, "The Draw Pie button will draw a pie shape based on the values entered in the Start Angle and Sweep Angle text boxes. Listing 3.22 shows the code for the Draw Pie button click event handler.\n\nListing 3.22 Drawing a pie shape\n\n```private void DrawPieBtn_Click(object sender,\nSystem.EventArgs e)\n{\n// Create a Graphics object\nGraphics g = this.CreateGraphics();\ng.Clear(this.BackColor);\n// Get the current value of start and sweep\n// angles\nfloat startAngle =\n(float)Convert.ToDouble(textBox1.Text);\nfloat sweepAngle =\n(float)Convert.ToDouble(textBox2.Text);\n// Create a pen\nPen bluePen = new Pen(Color.Blue, 1);\n// Draw pie\ng.DrawPie( bluePen, 20, 20, 100, 100,\nstartAngle, sweepAngle);\n// Dispose of objects\nbluePen.Dispose();\ng.Dispose();\n}\n```\n\nNow let's run the pie shapedrawing application and enter values for the start and sweep angles. Figure 3.32 shows a pie for start and sweep angles of 0.0 and 90 degrees, respectively.\n\nFigure 3.32. A pie shape with start angle of 0 degrees and sweep angle of 90 degrees", null, "Figure 3.33 shows a pie for start and sweep angles of 45.0 and 180.0 degrees, respectively.\n\nFigure 3.33. A pie shape with start angle of 45 degrees and sweep angle of 180 degrees", null, "Figure 3.34 shows a pie for start and sweep angles of 90.0 and 45.0 degrees, respectively.\n\nFigure 3.34. A pie shape with start angle of 90 degrees and sweep angle of 45 degrees", null, "Note\n\nWe will see a real-world pie chart application in Section 3.4.\n\n3.2.1.16 Drawing Images\n\nThe Graphics class also provides functionality for drawing images, using DrawImage and DrawImageUnscaled. DrawImage draws an Image object with a specified size, and DrawImageUnscaled draws an Image object without scaling it. The DrawImage method has many overloaded forms.\n\nNote\n\nHere we discuss simple images. Chapters 7 and 8 discuss the Image class, its members, and imaging-related functionality in detail.\n\nAn application creates an Image object by calling the Image class's static FromFile method, which takes a file name as an argument. After that you create the coordinates of a rectangle in which to draw the image and call DrawImage. Listing 3.23 draws an image on the surface with a size of ClientRectangle.\n\nListing 3.23 Drawing an image\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n\ntry\n{\n// Create an image from a file\nImage newImage =\nImage.FromFile(\"dnWatcher.gif\");\n\n// Draw image\ne.Graphics.DrawImage(newImage,\nthis.ClientRectangle);\nnewImage.Dispose();\n}\ncatch (Exception ex)\n{\nMessageBox.Show(ex.Message.ToString());\n}\n}\n```\n\nFigure 3.35 shows the output from Listing 3.23.\n\nFigure 3.35. Drawing an image", null, "3.2.2 Fill Methods\n\nSo far we have seen only the draw methods of the Graphics class. As we discussed earlier, pens are used to draw the outer boundary of graphics shapes, and brushes are used to fill the interior of graphics shapes. In this section we will cover the Fill methods of the Graphics class. You can fill only certain graphics shapes; hence there are only a few Fill methods available in the Graphics class. Table 3.5 lists them.\n\n3.2.2.1 The FillClosedCurve Method\n\nFillClosedCurve fills the interior of a closed curve. The first parameter of FillClosedCurve is a brush. It can be a solid brush, a hatch brush, or a gradient brush. Brushes are discussed in more detail in Chapter 4. The second parameter is an array of points. The third and fourth parameters are optional. The third parameter is a fill mode, which is represented by the FillMode enumeration. The fourth and last optional parameter is the tension of the curve, which we discussed in Section 3.2.1.10.\n\nThe FillMode enumeration specifies the way the interior of a closed path is filled. It has two modes: alternate or winding. The values for alternate and winding are Alternate and Winding, respectively. The default mode is Alternate. The fill mode matters only if the curve intersects itself (see Section 3.2.1.10).\n\nTo fill a closed curve using FillClosedCurve, an application first creates a Brush object and an array of points for the curve. The application can then set the fill mode and tension (which is optional) and call FillClosedCurve.\n\nListing 3.24 creates an array of PointF structures and a SolidBrush object, and calls FillClosedCurve.\n\nListing 3.24 Using FillClosedCurve to fill a closed curve\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create an array of points\nPointF pt1 = new PointF( 40.0F, 50.0F);\nPointF pt2 = new PointF(50.0F, 75.0F);\nPointF pt3 = new PointF(100.0F, 115.0F);\nPointF pt4 = new PointF(200.0F, 180.0F);\nPointF pt5 = new PointF(200.0F, 90.0F);\nPointF[] ptsArray =\n{\npt1, pt2, pt3, pt4, pt5\n};\n// Fill a closed curve\nfloat tension = 1.0F;\nFillMode flMode = FillMode.Alternate;\nSolidBrush blueBrush = new SolidBrush(Color.Blue);\ne.Graphics.FillClosedCurve(blueBrush, ptsArray,\nflMode, tension);\n// Dispose of object\nblueBrush.Dispose();\n}\n```\n\nTable 3.5. Graphics fill methods\n\nMethod\n\nDescription\n\nFillClosedCurve\n\nFills the interior of a closed cardinal spline curve defined by an array of Point structures.\n\nFillEllipse\n\nFills the interior of an ellipse defined by a bounding rectangle specified by a pair of coordinates, a width, and a height.\n\nFillPath\n\nFills the interior of a GraphicsPath object.\n\nFillPie\n\nFills the interior of a pie section defined by an ellipse specified by a pair of coordinates, a width, a height, and two radial lines.\n\nFillPolygon\n\nFills the interior of a polygon defined by an array of points specified by Point structures.\n\nFillRectangle\n\nFills the interior of a rectangle specified by a pair of coordinates, a width, and a height.\n\nFillRectangles\n\nFills the interiors of a series of rectangles specified by Rectangle structures.\n\nFillRegion\n\nFills the interior of a Region object.\n\nFigure 3.36 shows the output from Listing 3.24.\n\nFigure 3.36. Filling a closed curve", null, "3.2.2.2 The FillEllipse Method\n\nFillEllipse fills the interior of an ellipse. It takes a Brush object and rectangle coordinates.\n\nTo fill an ellipse using FillEllipse, an application creates a Brush and a rectangle and calls FillEllipse. Listing 3.25 creates three brushes and calls FillEllipse to fill an ellipse with a brush.\n\nListing 3.25 Filling ellipses\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\nGraphics g = e.Graphics ;\n// Create brushes\nSolidBrush redBrush = new SolidBrush(Color.Red);\nSolidBrush blueBrush = new SolidBrush(Color.Blue);\nSolidBrush greenBrush = new SolidBrush(Color.Green);\n// Create a rectangle\nRectangle rect =\nnew Rectangle(80, 80, 50, 50);\n// Fill ellipses\ng.FillEllipse(greenBrush,\n40.0F, 40.0F, 130.0F, 130.0F );\ng.FillEllipse(blueBrush, 60, 60, 90, 90);\ng.FillEllipse(redBrush, rect );\ng.FillEllipse(greenBrush,\n100.0F, 90.0F, 10.0F, 30.0F );\n// Dispose of objects\nblueBrush.Dispose();\nredBrush.Dispose();\ngreenBrush.Dispose();\n}\n```\n\nFigure 3.37 shows the output from Listing 3.25.\n\nFigure 3.37. Filling ellipses", null, "3.2.2.3 The FillPath Method\n\nFillPath fills the interior of a graphics path. To do this, an application creates Brush and GraphicsPath objects and then calls FillPath, which takes a brush and a graphics path as arguments. Listing 3.26 creates GraphicsPath and SolidBrush objects and calls FillPath.\n\nListing 3.26 Filling a graphics path\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create a solid brush\nSolidBrush greenBrush =\nnew SolidBrush(Color.Green);\n// Create a graphics path\nGraphicsPath path = new GraphicsPath();\n// Add a line to the path\n// Add an ellipse to the path\n\n// Create a rectangle and call\nRectangle rect =\nnew Rectangle(50, 150, 300, 50);\n// Fill path\ne.Graphics.FillPath(greenBrush, path);\n// Dispose of object\ngreenBrush.Dispose();\n}\n```\n\nFigure 3.38 shows the output from Listing 3.26. As the figure shows, the fill method fills all the covered areas of a graphics path.\n\nFigure 3.38. Filling a graphics path", null, "3.2.2.4 The FillPie Method\n\nFillPie fills a pie section with a specified brush. It takes four parameters: a brush, the rectangle of the ellipse, and the start and sweep angles. The following code calls FillPie.\n\n```\n```\n```g.FillPie(new SolidBrush(Color.Red),\n0.0F, 0.0F, 100, 60, 0.0F, 90.0F);\n```\n```\n```\n\nWe will discuss the FillPie method in the pie chart application in Section 3.4.\n\n3.2.2.5 The FillPolygon Method\n\nFillPolygon fills a polygon with the specified brush. It takes three parameters: a brush, an array of points, and a fill mode. The FillMode enumeration defines the fill mode of the interior of the path. It provides two fill modes: Alternate and Winding. The default mode is Alternate.\n\nIn our application we will use a hatch brush. So far we have seen only a solid brush. A solid brush is a brush with one color only. A hatch brush is a brush with a hatch style and two colors. These colors work together to support the hatch style. The HatchBrush class represents a hatch brush. We will discuss hatch brushes in more detail in Chapter 4.\n\nThe code in Listing 3.27 uses FillPolygon to fill a polygon using the Winding mode.\n\nListing 3.27 Filling a polygon\n\n```Graphics g = e.Graphics ;\n// Create a solid brush\nSolidBrush greenBrush =\nnew SolidBrush(Color.Green);\n// Create points for polygon\nPointF p1 = new PointF(40.0F, 50.0F);\nPointF p2 = new PointF(60.0F, 70.0F);\nPointF p3 = new PointF(80.0F, 34.0F);\nPointF p4 = new PointF(120.0F, 180.0F);\nPointF p5 = new PointF(200.0F, 150.0F);\nPointF[] ptsArray =\n{\np1, p2, p3, p4, p5\n};\n// Draw polygon\ne.Graphics.FillPolygon(greenBrush, ptsArray);\n// Dispose of object\ngreenBrush.Dispose();\n```\n\nFigure 3.39 shows the output from Listing 3.27. As you can see, the fill method fills all the areas of a polygon.\n\nFigure 3.39. Filling a polygon", null, "3.2.2.6 Filling Rectangles and Regions\n\nFillRectangle fills a rectangle with a brush. This method takes a brush and a rectangle as arguments. FillRectangles fills a specified series of rectangles with a brush, and it takes a brush and an array of rectangles. These methods also have overloaded forms with additional options. For instance, if you're using a HatchStyle brush, you can specify background and foreground colors. Chapter 4 discusses FillRectangle and its options in more detail.\n\nNote\n\nThe HatchBrush class is defined in the System.Drawing.Drawing2D namespace.\n\nThe source code in Listing 3.28 uses FillRectangle to fill two rectangles. One rectangle is filled with a hatch brush, the other with a solid brush.\n\nListing 3.28 Filling rectangles\n\n```private void Form1_Paint(object sender,\nSystem.Windows.Forms.PaintEventArgs e)\n{\n// Create brushes\nSolidBrush blueBrush = new SolidBrush(Color.Blue);\n// Create a rectangle\nRectangle rect = new Rectangle(10, 20, 100, 50);\n// Fill rectangle\ne.Graphics.FillRectangle(new HatchBrush\n(HatchStyle.BackwardDiagonal,\nColor.Yellow, Color.Black),\nrect);\ne.Graphics.FillRectangle(blueBrush,\nnew Rectangle(150, 20, 50, 100));\n\n// Dispose of object\nblueBrush.Dispose();\n}\n```\n\nFigure 3.40 shows the output from Listing 3.28.\n\nFigure 3.40. Filling rectangles", null, "FillRegion fills a specified region with a brush. This method takes a brush and a region as input parameters. Listing 3.29 creates a Region object from a rectangle and calls FillRegion to fill the region.\n\nListing 3.29 Filling regions\n\n```Rectangle rect = new Rectangle(20, 20, 150, 100);\nRegion rgn = new Region(rect);\ne.Graphics.FillRegion(new SolidBrush(Color.Green)\n, rgn);\n```\n\nNote\n\nChapter 6 discusses rectangles and regions in more detail.\n\n3.2.3 Miscellaneous Graphics Class Methods\n\nThe Graphics class provides more than just draw and fill methods. Miscellaneous methods are defined in Table 3.6. Some of these methods are discussed in more detail later.\n\n3.2.3.1 The Clear Method\n\nThe Clear method clears the entire drawing surface and fills it with the specified background color. It takes one argument, of type Color. To clear a form, an application passes the form's background color. The following code snippet uses the Clear method to clear a form.\n\n```\n```\n```form.Graphics g = this.CreateGraphics();\ng.Clear(this.BackColor);\ng.Dispose();\n```\n```\n```\n\nTable 3.6. Some miscellaneous Graphics methods\n\nMethod\n\nDescription\n\nAdds a comment to a Metafile object.\n\nClear\n\nClears the entire drawing surface and fills it with the specified background color.\n\nExcludeClip\n\nUpdates the clip region to exclude the area specified by a Rectangle structure.\n\nFlush\n\nForces execution of all pending graphics operations and returns immediately without waiting for the operations to finish.\n\nFromHdc\n\nCreates a new Graphics object from a device context handle.\n\nFromHwnd\n\nCreates a new Graphics object from a window handle.\n\nFromImage\n\nCreates a new Graphics object from an Image object.\n\nGetHalftonePalette\n\nReturns a handle to the current Windows halftone palette.\n\nGetHdc\n\nReturns the device context handle associated with a Graphics object.\n\nGetNearestColor\n\nReturns the nearest color to the specified Color structure.\n\nIntersectClip\n\nUpdates the clip region of a Graphics object to the intersection of the current clip region and a Rectangle structure.\n\nIsVisible\n\nReturns true if a point is within the visible clip region.\n\nMeasureCharacterRanges\n\nReturns an array of Region objects, each of which bounds a range of character positions within a string.\n\nMeasureString\n\nMeasures a string when drawn with the specified Font object.\n\nMultiplyTransform\n\nMultiplies the world transformation and the Matrix object.\n\nReleaseHdc\n\nReleases a device context handle obtained by a previous call to the GetHdc method.\n\nResetClip\n\nResets the clip region to an infinite region.\n\nResetTransform\n\nResets the world transformation matrix to the identity matrix.\n\nRestore\n\nRestores the state of a Graphics object to the state represented by a GraphicsState object. Takes GraphicsState as input, removes the information block from the stack, and restores the Graphics object to the state it was in when it was saved.\n\nRotateTransform\n\nApplies rotation to the transformation matrix.\n\nSave\n\nSaves the information block of a Graphics object. The information block stores the state of the Graphics object. The Save method returns a GraphicsState object that identifies the information block.\n\nScaleTransform\n\nApplies the specified scaling operation to the transformation matrix.\n\nSetClip\n\nSets the clipping region to the Clip property.\n\nTransformPoints\n\nTransforms an array of points from one coordinate space to another using the current world and page transformations.\n\nTranslateClip\n\nTranslates the clipping region by specified amounts in the horizontal and vertical directions.\n\nTranslateTransform\n\nPrepends the specified translation to the transformation matrix.\n\n3.2.3.2 The MeasureString Method\n\nMeasureString measures a string when it is drawn with a Font object and returns the size of the string as a SizeF object. You can use SizeF to find out the height and width of string.\n\nMeasureString can also be used to find the total number of characters and lines in a string. It has seven overloaded methods. It takes two required parameters: the string and font to measure. Optional parameters you can pass include the width of the string in pixels, maximum layout area of the text, string format, and combinations of these parameters.\n\nNote\n\nChapter 5 discusses string operations in detail.\n\nListing 3.30 uses the MeasureString method to measure a string's height and width and draws a rectangle and a circle around the string. This example also shows how to find the total number of lines and characters of a string.\n\nListing 3.30 Using the MeasureString method\n\n```Graphics g = Graphics.FromHwnd(this.Handle);\ng.Clear(this.BackColor);\n\nstring testString = \"This is a test string\";\nFont verdana14 = new Font(\"Verdana\", 14);\nFont tahoma18 = new Font(\"Tahoma\", 18);\nint nChars;\nint nLines;\n\n// Call MeasureString to measure a string\nSizeF sz = g.MeasureString(testString, verdana14);\nstring stringDetails = \"Height: \"+sz.Height.ToString()\n+ \", Width: \"+sz.Width.ToString();\nMessageBox.Show(\"First string details: \"+ stringDetails);\ng.DrawString(testString, verdana14, Brushes.Green,\nnew PointF(0, 100));\ng.DrawRectangle(new Pen(Color.Red, 2), 0.0F, 100.0F,\nsz.Width, sz.Height);\nsz = g.MeasureString(\"Ellipse\", tahoma18,\nnew SizeF(0.0F, 100.0F), new StringFormat(),\nout nChars, out nLines);\nstringDetails = \"Height: \"+sz.Height.ToString()\n+ \", Width: \"+sz.Width.ToString()\n+ \", Lines: \"+nLines.ToString()\n+ \", Chars: \"+nChars.ToString();\nMessageBox.Show(\"Second string details: \"+ stringDetails);\n\ng.DrawString(\"Ellipse\", tahoma18, Brushes.Blue,\nnew PointF(10, 10));\ng.DrawEllipse( new Pen(Color.Red, 3), 10, 10,\nsz.Width, sz.Height);g.Dispose()\n```\n\nFigure 3.41 shows the output from Listing 3.30.\n\nFigure 3.41. Using MeasureString when drawing text", null, "3.2.3.3 The FromImage, FromHdc, and FromHwnd Methods\n\nAs we discussed earlier, an application can use Graphics class members to get a Graphics object. The Graphics class provides three methods to create a Graphics object: FromHwnd, FromHdc, and FromImage.\n\nFromImage takes an Image object as input parameter and returns a Graphics object. We will discuss FromImage in more detail in Chapters 7 and 8. The following code snippet creates a Graphics object from an Image object. Once a Graphics object has been created, you can call its members.\n\n```\n```\n```Image img = Image.FromFile(\"Rose.jpg\");\nGraphics g = Graphics.FromImage(img);\n// Do something\ng.Dispose();\n```\n```\n```\n\nNote\n\nMake sure you call the Dispose method of the Graphics object when you're finished with it.\n\nFromHdc creates a Graphics object from a window handle to a device context. The following code snippet shows an example in which FromHdc takes one parameter, of type IntPtr.\n\n```\n```\n```IntPtr hdc = e.Graphics.GetHdc();\nGraphics g= Graphics.FromHdc(hdc);\n// Do something\ne.Graphics.ReleaseHdc(hdc);\ng.Dispose();\n```\n```\n```\n\nNote\n\nYou need to call the ReleaseHdc method to release resources allocated by a window handle to a device context, and also make sure you call the Dispose method of the Graphics object when you're finished with it.\n\nFromHwnd returns a Graphics object for a form. The following method takes a window handle.\n\n```\n```\n```Graphics g = Graphics.FromHwnd(this.Handle);\n```\n```\n```\n\nTo draw on a form, an application can pass this handle. Once an application has a Graphics object, it can call any Graphics class method to draw graphics objects.", null, "GDI+ Programming with C#\nISBN: 073561265X\nEAN: N/A\nYear: 2003\nPages: 145", null, "" ]

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[ "", null, "", null, "### Home > CCAA8 > Chapter 3 Unit 4 > Lesson CCA: 3.2.3 > Problem3-60\n\n3-60.\n\nSolve each equation below for $x$. Then check your solutions.\n\n1.  $\\frac{x}{8\\ }=\\frac{3}{4}$\n\nMultiply both sides by $8$.\n\n$8\\cdot\\frac{\\textit{x}}{8}=\\frac{3}{4}\\cdot8$\n\n$x=\\frac{24}{4}$\n\n$x=6$\n\n1.  $\\frac{2}{5}=\\frac{x}{40}$\n\nWhat can you multiply both sides of the equation by to eliminate the denominators?\n\n1.  $\\frac{1}{8}=\\ \\frac{x}{12}$\n\n$x=1.5$\n\n1.  $\\frac{x}{10}=\\frac{12}{15}$\n\nUse the method from part (a)" ]

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", null ]

[ "# The First Law of Thermodynamics Practice Questions\n\nThe following physics revision questions are provided in support of the physics tutorial on The First Law of Thermodynamics. In addition to this tutorial, we also provide revision notes, a video tutorial, revision questions on this page (which allow you to check your understanding of the topic) and calculators which provide full, step by step calculations for each of the formula in the The First Law of Thermodynamics tutorials. The The First Law of Thermodynamics calculators are particularly useful for ensuring your step-by-step calculations are correct as well as ensuring your final result is accurate.\n\nNot sure on some or part of the The First Law of Thermodynamics questions? Review the tutorials and learning material for The First Law of Thermodynamics\n\nThermodynamics Learning Material\nTutorial IDTitleTutorialVideo\nTutorial\nRevision\nNotes\nRevision\nQuestions\n13.5The First Law of Thermodynamics\n\n## Physics Revision Questions for The First Law of Thermodynamics\n\n1) 170 J of heat energy are supplied to a gas sample. As a result, the internal energy of the gas increased by 200 J. What is the WORK DONE ON THE GAS by the surroundings?\n\n1. 30 J\n2. -30 J\n3. 370 J\n4. -370 J\n\n2) The internal energy of a gas inside a cylinder during an adiabatic process increases by 20 J. What happens to the 4-kg piston of the cylinder during this process? Take g = 10 m/s2 if needed.\n\n1. Rises up by 50 cm\n2. Lowers down by 50 cm\n3. Rises up by 5 m\n4. Lowers down by 5 m\n\n3) The graph below is an example of", null, "2. Free expansion process\n3. Constant volume process\n4. Cyclic process\n\n## Whats next?\n\nEnjoy the \"The First Law of Thermodynamics\" practice questions? People who liked the \"The First Law of Thermodynamics\" practice questions found the following resources useful:\n\n1. Practice Questions Feedback. Helps other - Leave a rating for this practice questions (see below)\n2. Thermodynamics Physics tutorial: The First Law of Thermodynamics. Read the The First Law of Thermodynamics physics tutorial and build your physics knowledge of Thermodynamics\n3. Thermodynamics Revision Notes: The First Law of Thermodynamics. Print the notes so you can revise the key points covered in the physics tutorial for The First Law of Thermodynamics\n4. Check your calculations for Thermodynamics questions with our excellent Thermodynamics calculators which contain full equations and calculations clearly displayed line by line. See the Thermodynamics Calculators by iCalculator™ below.\n5. Continuing learning thermodynamics - read our next physics tutorial: The Kinetic Theory of Gases. Ideal Gases" ]

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[ "# C program to print all odd numbers from 1 to n\n\nWrite a C program to print all odd numbers from 1 to n using for loop. How to print odd numbers from 1 to n using loop in C programming. Logic to print odd numbers in a given range in C programming.\n\nExample\n\nInput\n\n`Input upper limit: 10`\n\nOutput\n\n```Odd numbers between 1 to 10:\n1, 3, 5, 7, 9```\n\n## Logic to print odd numbers from 1 to n using `if` statement\n\nLogic to print odd numbers is similar to logic to print even numbers.\n\nStep by step descriptive logic to print odd numbers from 1 to n.\n\n1. Input upper limit to print odd number from user. Store it in some variable say N.\n2. Run a loop from 1 to N, increment loop counter by 1 in each iteration. The loop structure should look like `for(i=1; i<=N; i++)`.\n3. Inside the loop body check odd condition i.e. if a number is exactly divisible by 2 then it is odd. Which is `if(i % 2 != 0)` then, print the value of i.\n\n## Program to print odd numbers using `if` statement\n\n``````/**\n* C program to print all Odd numbers from 1 to n\n*/\n\n#include <stdio.h>\n\nint main()\n{\nint i, n;\n\n/* Input upper limit from user */\nprintf(\"Print odd numbers till: \");\nscanf(\"%d\", &n);\n\nprintf(\"All odd numbers from 1 to %d are: \\n\", n);\n\n/* Start loop from 1 and increment it by 1 */\nfor(i=1; i<=n; i++)\n{\n/* If 'i' is odd then print it */\nif(i%2!=0)\n{\nprintf(\"%d\\n\", i);\n}\n}\n\nreturn 0;\n}``````\n\n## Logic to print odd numbers from 1 to n without `if` statement\n\nThe above approach is not optimal approach to print odd numbers. Observe the above program for a while. You will notice that, I am unnecessarily iterating for even numbers, which is not our goal.\n\nStep by step descriptive logic to print odd numbers without using `if` statement.\n\n1. Input upper limit to print odd number from user. Store it in some variable say N.\n2. Run a loop from 1 to N, increment it by 2 for each iteration. The loop structure should look like `for(i=1; i<=N; i+=2)`.\n3. Inside the loop body print the value of i.\n\n## Program to display odd numbers without using `if` statement\n\n``````/**\n* C program to display all odd numbers between 1 to n without using if statement\n*/\n\n#include <stdio.h>\n\nint main()\n{\nint i, n;\n\n/* Input upper limit from user */\nprintf(\"Print odd numbers till: \");\nscanf(\"%d\", &n);\n\nprintf(\"All odd numbers from 1 to %d are: \\n\", n);\n\n/*\n* Start a loop from 1, increment it by 2.\n* For each repetition prints the number.\n*/\nfor(i=1; i<=n; i+=2)\n{\nprintf(\"%d\\n\", i);\n}\n\nreturn 0;\n}``````\n\nNote: In the above program I have used shorthand assignment operator `i+=2` which is equivalent to `i = i + 2`.\n\nOutput\n\n```Print odd numbers till: 100\nAll odd numbers from 1 to 100 are:\n1\n3\n5\n7\n9\n11\n13\n15\n17\n19\n21\n23\n25\n27\n29\n31\n33\n35\n37\n39\n41\n43\n45\n47\n49\n51\n53\n55\n57\n59\n61\n63\n65\n67\n69\n71\n73\n75\n77\n79\n81\n83\n85\n87\n89\n91\n93\n95\n97\n99```\n\n## Program to print odd numbers in given range\n\n``````/**\n* C program to display all odd numbers in given range\n*/\n\n#include <stdio.h>\n\nint main()\n{\nint i, start, end;\n\n/* Input lower and upper limit from user */\nprintf(\"Enter lower limit: \");\nscanf(\"%d\", &start);\nprintf(\"Enter upper limit: \");\nscanf(\"%d\", &end);\n\nprintf(\"All odd numbers from %d to %d are: \\n\", start, end);\n\n/* If start is not odd then make it odd */\nif(start%2==0)\n{\nstart++;\n}\n\n/*\n* Initialize loop from start, increment it by 2.\n* For each repetition print the number.\n*/\nfor(i=start; i<=end; i+=2)\n{\nprintf(\"%d\\n\", i);\n}\n\nreturn 0;\n}``````\n\nOutput\n\n```Enter lower limit: 10\nEnter upper limit: 20\nAll odd numbers from 10 to 20 are:\n11\n13\n15\n17\n19```\n\nHappy coding 😉", null, "" ]

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[ "VDOC.PUB\n\n### Variational Methods In Geosciences [PDF]\n\nThis document was uploaded by our user. The uploader already confirmed that they had the permission to publish it. If you are author/publisher or own the copyright of this documents, please report to us by using this DMCA report form.\n\n#### E-Book Overview\n\nThe last few decades have seen a spectacular growth in the use of variational methods, one of the most classic and elegant methods in physical and mathematical sciences, as powerful tools of optimization and numerical analysis. The tremendous accumulation of information on the use of variational methods in the area of the geosciences, which includes meteorology, oceanography, hydrology, geophysics and seismology, indicated the need for the first symposium on Variational Methods in Geosciences to be organized and held in Norman on October 15-17, 1985. The value of this symposium was enhanced by the number of stimulating and informative papers presented\n\n#### E-Book Content\n\nFurther titles in this series 1. F.P. AGTERBERG Geomathematics 2. M. DAVID Geostatistical Ore Reserve Estimation 3. S. TWOMEY Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements 4. P.M. GY Sampling of Particulate Materials\n\nDevelopments\n\nin Geomathematics\n\n5\n\nVARIATIONAL METHODS IN GEOSCIENGES Proceedings of the International Symposium on Variational Methods in Geosciences held at the University of Oklahoma, Norman, Oklahoma, on October 15-17,1985 edited by\n\nYOSHI K. SASAKI (editor-in-chief ) TZVI GAL-CHEN, LUTHER WHITE, M.M. ZAMAN, CONRAD ZIEGLER ( editors ) L.P. CHANG, DAN J . RUSK ( associate editors )\n\nELSEVIER\n\nAmsterdam — Oxford — New York — Tokyo 1986\n\nE L S E V I E R S C I E N C E P U B L I S H E R S B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 A E Amsterdam, The Netherlands Distributors\n\nfor the United\n\nStates and\n\nE L S E V I E R S C I E N C E P U B L I S H I N G C O M P A N Y INC. 52, Vanderbilt Avenue New York, N Y 10017, U.S.A.\n\nInternational Symposium on Variational Methods in Geosciences (.1985 : University of Oklahoma) Variational methods in geosciences. (Developments in geomathematics ; 5) Bibliography: p. 1 . Earth sciences—Mathematics—Congresses. 2. Numerical analysis—Congresses. I. Sasaki, Yoshi K. II. Title. III. Series. QE33.2.M3I57 1985 550\\l 51 86-1981*9 ISBN 0-1M-42697-3 (U.S.) f\n\nISBN 0-444-42697-3 (Vol. 5) ISBN 0-444-41609-9 (Series) © Elsevier Science Publishers B.V., 1986 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or other­ wise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./Science & Technology Division, P.O. Box 330, 1000 A H Amsterdam, The Netherlands. Special regulations for readers in the U S A — This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U S A . All other copyright questions, including photocopying outside of the U S A , should be referred to the copyright owner, Elsevier Science Publishers B.V., unless otherwise specified. Printed in The Netherlands\n\nV\n\nThe following organizations are deeply appreciated for co-sponsoring this symposium:\n\nAmerican Meteorological Society Society for Industrial and Applied Mathematics American Geophysical Union Universite de Clermont Institut National de Recherche en Informatique et en Automatique Institut National d Astrophysique at Geophysique College of Geosciences, College of Engineering, Department of Mathematics, Energy Resources Institute and Energy Center of The University of Oklahoma 1\n\nCOMMITTEE MEMBERS\n\nInternational Organizing Committee Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Mr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr. Dr.\n\nH.T. Banks E. Barker L.P. Chang G. Chavent R. Daley E. Donaldson J.T. Edwards R. Ewing H. Fujita T. Gal-Chen M. Ghil D.E. Hinsman E. Issacson K. Johnson A. Kasahara M. Kawahara T. Kawaii K. Kunisch F. LeDimet J.M. Lewis J.L. Lions J. McGinley I.M. Navon V.H. Nguyen J.T. Oden N.A. Phillips J. Purser M. Rasmussen P.S. Ray J.N. Reddy D. Russell Y.K. Sasaki R.S. Seaman M.P. Singh A.N. Staniforth F. Stehli J. Stephens 0 . Talagrand Toksoz J. Tribbia L. White Q. Xu\n\n(Chinese Liaison)\n\n(Japanese Liaison) (Austrian Liaison) (French Liaison)\n\n(South-African Liaisoi (Belgian Liaison)\n\n(British Liaison)\n\n(Australian Liaison) (Indian Liaison) (Canadian Liaison)\n\n(Chinese Liaison)\n\nDr. O.C. Zienkiewicz\n\nLocal Organizing Committee Dr. Y.K. Sasaki Dr. J. McGinley Dr. L. White Dr. K. Johnson Dr. L.P. Chang Dr. R.J. Mulholland Lynda McGinley\n\nVII\n\nPREFACE Variational physical\n\nmethods,\n\none\n\nand mathematical\n\nof\n\nthe\n\nmost\n\nclassic\n\nand\n\nsciences, have been developed\n\nelegant within\n\nmethods\n\nin\n\nthe last\n\nfew\n\ndecades as powerful tools of optimization and numerical analysis. In the area of the geosciences, which includes meteorology, oceanography, hydrology,\n\ngeophysics\n\nextensively simulate provide\n\nused\n\nand\n\nto\n\ninterpret\n\ngeoscientific an\n\nseismology, and\n\nphenomena.\n\nopportunity\n\nfor\n\nthe\n\nvariational\n\nassimilate It was\n\ngeoscientific\n\ntimely\n\nresearchers\n\nin\n\nmethods\n\nto call\n\ndifferent\n\nfor\n\nhave\n\ndata\n\nbeen\n\nand\n\nto\n\na meeting\n\nto\n\ndisciplines\n\nof\n\nthe\n\ngeosciences to discuss problems of mutual interest. The\n\nfirst\n\norganized\n\nand\n\npresented\n\nwere\n\nsymposium held\n\nin\n\non\n\nVariational\n\nNorman\n\nstimulating\n\nand\n\non\n\nMethods\n\nOctober\n\n15-17,\n\ninformative, with\n\nin\n\nGeosciences\n\n1985.\n\nPapers\n\na desire\n\nwas\n\nthus\n\nwhich\n\nwere\n\nto have a second\n\nsymposium in about two years expressed by some researchers who participated in the first symposium. There were a number of outstanding papers presented in the symposium. therefore\n\npublish\n\nthem\n\nso\n\nthat\n\nothers\n\nin\n\nthe\n\ngeosciences\n\nand\n\nin\n\nWe\n\nother\n\ndisciplines may benefit. The symposium was well organized, due local and\n\ninternational\n\norganizing\n\nto the remarkable efforts by the\n\ncommittee members\n\ncited\n\nearlier.\n\nWe\n\nowe\n\nthe success of the symposium to the organizations whose support was essential to bring together such a broad spectrum of scientists and engineers. Finally, we this whose\n\nproject. abilities\n\nConference\n\nthank\n\nThanks this\n\nthe are\n\nwork\n\npersonnel\n\nat\n\nparticularly would\n\nSpecialist, who was\n\nhave\n\nCIMMS who have due\n\nmuch\n\nto\n\nto M s . Constance White, without\n\nsuffered,\n\nresponsible\n\ncontributed\n\nand\n\nto M s . Lynda\n\nfor the preliminary\n\nMcGinley,\n\narrangements.\n\nWe also thank Ms. Chris Heath and Mrs. Judy Johnston for their time and enery.\n\nApril 29, 1986\n\nYoshi K. Sasaki Symposium Chairman and Editor-in-Chief\n\n3\n\nTHE APPLICATION OF VARIATIONAL METHODS TO INITIALIZATION ON THE SPHERE\n\nR.W. DALEY Canadian Climate Centre, Atmospheric Environment Service, 4905 Dufferin Street, Downsview, Ontario, M3H 5T4, Canada\n\nINTRODUCTION\n\nThe atmospheric initialization problem arises because of the need to modify objective analyses of the atmospheric state so that when they are used as initi­ al conditions for integrating atmospheric forecast or climate models, no high frequencies will be excited.\n\nThe high frequency oscillations in these models\n\nare generally due to internal or external gravity waves which are permissable solutions of the equations, but have a faster time scale than the motions of interest.\n\nThe process of initialization requires initial fields to be adjusted such that they satisfy certain dynamic constraints (multi-variate diagnostic rela­ tionships) which are thought to only reflect the slow timescales of interest and not the unwanted fast timescales.\n\nderived from scaling arguments and have, until recently been only strictly valid for middle and high latitudes - the so-called quasi-geostrophic constraints.\n\nVariational initialization, developed originally by Sasaki in the late 1960's, attempts to satisfy exactly or approximately the imposed constraint while at the same time minimizing in some sense the adjustments made to the initial conditions.\n\nThe variational formulation of initialization constraints\n\nis more flexible and general, but if used with traditional constraints, will suffer from the same drawbacks as a non-variational formulation.\n\nIn recent years, a method of deriving global initialization constraints has been developed.\n\nIt is usually referred to as normal mode initialization and is\n\nnow widely used in operational global weather forecasting models.\n\nIt correctly\n\nhandles the planetary scale flow and is also superior in purely tropical flows.\n\n4 The present paper describes how normal mode initialization procedures can be placed in a variational context, thus deriving the benefits of a variational formulation and also exploiting the correct constraints for the global problem. Before discussing the variational formulation, it is necessary to briefly review normal mode initialization itself.\n\nNORMAL MODE INITIALIZATION THEORY\n\nNormal mode initialization on the sphere customarily begins with the primitive (hydrostatic) equations.\n\nIn pressure coordinates, these equations can be writ­\n\nten,\n\n3V — V + V V V + u> — + f k x V 3t 9p\n\n+ V \\$ = F ,\n\n(2.1)\n\nil + S I - O , 3p\n\n(2.2)\n\np\n\nV . y + —\n\n= 0 ,\n\n(2.3)\n\n3p\n\n(\n\n2- y.v)i! + * i - ( p 2 ! _ +\n\n3t\n\n3p\n\np 3p\n\nK\n\n•) --Jsa\n\n3p\n\nt\n\n( 2\n\n.\n\n4 )\n\np\n\nwhere F = frictional force, Q = heating, u is the vertical motion, k is the up­ ward pointing unit vector, K - R/C , \\$ is the geopotential and V is the horizonp tal velocity vector.\n\nEquations\n\n*/*\n\n(2.1-2.4) are respectively the equation of mo­\n\ntion, the hydrostatic equation, the continuity equation and the thermodynamic equation. Next linearize the equations about a basic state at rest and with a horizon­ tally averaged static stability,\n\n3V ^ + at\n\nv .\n\nf\n\nk\n\nv + v » - \"v\n\nx\n\n=\n\nV\n\n0 ,\n\n(2.5)\n\n(2.6)\n\n3p 3\n\n34\n\n3t\n\n3p\n\n+ a) r\n\n(2.7)\n\n5 where r\n\n= — - — (p — - K *) dS is the horizontally averaged static stability and p 3p 3p\n\nand R\n\nare the remaining terms in (2,1 and 2 . 4 ) .\n\nSet Ry, R of\n\n\\$\n\n- 0 and assume an exponential time behaviour.\n\nThe left hand sides\n\n(2.5-2.7) separate into two eigenvalue problems, a vertical structure equa­\n\ntion\n\nand\n\na\n\nhorizontal\n\nstructure\n\nequation.\n\nThese\n\nsolved to yield the normal modes of (2.5-2.7).\n\neigenvalue\n\nproblems\n\ncan be\n\nThey can be written symbolically\n\nfor the spherical case as,\n\nStructure\n\nUj\n\n(X, )\n\nv\n\n(X, \\$)\n\nj\n\n(X, 4>)\n\nFig. 1\n\nFrequency\n\nZ,(p)\n\n=\n\no*\n\nZ.(p)\n\n(2.8)\n\n6 where u, v are the eastward and northward velocity components, n and j are the horizontal and vertical mode indices, Z.(p) is the vertical structure, u), i\n\nn\n\nv ?, 1\n\ni\n\nj\n\nn are the horizontal structures and a_. is the frequency and X, 4> are the longi­\n\ntude and latitude. The vertical structures are shown in Fig. 1 for the four gravest vertical modes.\n\nThe frequencies a*j are shown in Fig. 2 for two vertical scales.\n\nThe top\n\npanel is for a\"large vertical scale, while the bottom panel is for a smaller vertical scale.\n\nThe modes separate into two groups on the basis of frequency.\n\nThose modes with low frequency are the Rossby modes, while the inertia-gravity modes have a much faster timescale. suppress the inertia-gravity modes.\n\nFig. 2\n\nNormal mode initialization attempts to\n\n7 The next step is to project the equations (2,5-2.7) on the normal modes. Arbitrary fields u,v and \\$ can be represented as a sum of the normal modes,\n\nu (X, , p)\n\nr\n\nZ I X. (t) n J J\n\nv (X, , p)\n\nu*j (X,\n\nVj\n\n\\$ (X, , p)\n\n( X , )\n\nZj(p)\n\n(2.9)\n\n•j (X, •)\n\nwhere X. (t) is an expansion coefficient.\n\nEquations (2.5-2.7), when projected\n\non the modes can be written,\n\n(2.10)\n\nwhere ft is the earth's rotation rate, X_. is the projection of the time tendency terms of (2.7-2.9), 2 fii a*! X ? is the projection of the remaining linear terms n on the left hand side and R_. is the projection of and R . 1\n\n\\$\n\nThe equations can be separated into fast and slow equations,\n\ny + 2 Q± o y = R y y\n\nSlow\n\nz + 2 Q± o\n\nFast\n\nz\n\nz = R\n\nz\n\nwhere (n,j) notation has been dropped.\n\n(2.11)\n\nThe slow modes (y) correspond to the\n\nRossby modes of Fig. 2, while the fast modes (z) correspond to the inertiagravity modes.\n\nNormal mode initialization proceeds as follows.\n\nLinear normal mode initiali­\n\nzation simply requires,\n\nz = 0\n\n(2.12)\n\nNon-linear normal mode initialization (at least to lowest order) requires\n\nz = 0\n\nor\n\nz = R 12 fli a z z\n\nIn both cases y remains fixed.\n\n(2.13)\n\n8 VARIATIONAL NORMAL MODE INITIALIZATION\n\nNormal mode Initialization can be illustrated by Fig. 3, which is known as a slow manifold diagram. modes respectively.\n\nZ and Y stand for the amplitudes of the fast and slow\n\nM is the locus of all points (atmospheric states) where z •\n\n0 and is known as the slow manifold.\n\nPoint A stands for some observed/analyzed\n\nstate of the atmosphere before initialization.\n\nIt is clear that any atmospheric\n\nstate which satisfies (2.12) will lie on the Y axis, while any state which satisfies (2.13) will lie on the slow manifold.\n\nZ\n\nFig. 3 The procedures described in the last section did not change the amplitudes of the slow modes.\n\nThus, equation (2.12) corresponds to the point ( L ) , while\n\n(2.13) corresponds to the point ( N ) .\n\nThis procedure suffers from the same limitations as all non-variational ini­ tialization.\n\nThere is no way to weight the observations according to their\n\npresumed accuracy. that u^, v^ and * Similarly U , V\n\nc\n\nA\n\nThe variational formulation proceeds as follows.\n\nare the observed/analyzed values of wind and geopotential.\n\nand *\n\nc\n\nAssume\n\nare the initialized fields.\n\nThus, the usual functional\n\nc\n\nis constructed,\n\n1\n\n\"ti\"\n\nwhere\n\n^ ^ \" +\n\n^\n\n+\n\n\"\n\nd S d P\n\n'\n\n(3#1>\n\n9\n\n(\n\nA\n\ns\n\n) cos d d\n\nI\n\nThe side conditions that are imposed are that U^, manifold (z = 0 ) .\n\nand W\n\n\\$\n\nand \\$^ are on the slow\n\nare specified weight functions which are inversely\n\nproportional to the presumed observation error.\n\nThe procedure is shown schematically in the slow manifold diagram on the top panel of Fig. 4 .\n\nZ, Y, M and A are as in Fig. 3 .\n\nThe lines \\$ and V are the\n\nlocus of all points (atmospheric states) which have the same geopotential (\\$) or wind field (V) as the observed/analyzed state ( A ) . The dashed elliptical lines are isopleths of constant I.\n\nThe solution to the variational problem is point\n\n(C), which is on the slow manifold and yet manages to minimize the value of I.\n\nv\n\nFig. 4\n\nx\n\n1= Constant\n\n10 There are several ways of approaching this problem.\n\nOne way is to create a\n\nnew functional by adding the constraints multiplied by appropriate Lagrange multipliers.\n\nThe use of a penalty method has also been proposed.\n\ntechnique is perhaps the simplest.\n\nThe following\n\nIt is illustrated in the lower panel of Fig.\n\n4.\n\nThe final state must satisfy,\n\nR Z\n\nC \" — Z ~ 2 fii a\n\ny\n\nC * A\n\ny\n\nZ\n\nC * A\n\n< '\n\nZ\n\n3\n\n2\n\nz\n\nwhere A indicates the point (A) on Fig. 4.\n\nNote that R\n\nz\n\nis a non-linear func­\n\ntion of all the y and z.\n\nStarting from point ( A ) , move to point ( 1 ) ,\n\nR (A) z\n\n(3.3)\n\n2 fli a\n\nR ( A ) is the value of the non-linear term calculated at point ( A ) . Z\n\nThe next step is to move horizontally to point ( 2 ) , which is a point where a horizontal line is also tangent to the ellipse.\n\nMathematically, this is ob­\n\ntained by minimizing,\n\nff\n\nh \"\n\nt< 2 \" A u\n\nU\n\n) 2 W\n\nV\n\n+\n\n( V\n\n2 \" A V\n\n) 2 W\n\nV\n\n+\n\n(\n\n*2 \"\n\nW\n\nd\n\nS\n\nd\n\nP\n\n»\n\n(\n\n3\n\n'\n\n4\n\n)\n\nP S subject to the side condition that z\n\n2\n\n= Z]_.\n\nIt is straightforward to show that the changes in the slow mode amplitudes during this step are given by,\n\n*j\n\n( u\n\n2 \" l y\n\n}\n\n\" j\n\n»\n\nu\n\n+ v\n\nf\n\n(\n\nwhere\n\nj\n\n/ fiiu\n\n1\n\nJ J P\n\ns\n\ny\n\nj\n\ny\n\n1\n\ny\n\nv )W„ + * j\n\ny\n\nv\n\ni\n\ny\n\n*\n\nj\n\ny\n\n3\n\n-\n\n5\n\n)\n\n11\n\nP s th and u , v , • are the u, v and \\$ components of the i slow mode y y y\n\nj\n\nand simi-\n\nlarly for u* etc. y The next step from point (2) to point (3) changes z, but not y and the process continues until (hopefully) it converges to the point (C^-).\n\nThe points (C) and (C^) are close, but not identical, so the final solu­ tion, while on the slow manifold does not exactly minimize (3.1).\n\nThe distance\n\nbetween (C) and ( C ) depends on the slope of the slow manifold with respect to 1\n\nthe slow mode (Y) axis.\n\nIn general, this slope is small, except for large Ross-\n\nby number, so points (C) and (C -) are not usually far apart. 1\n\nDISCUSSION\n\nIf W\n\nis large com-\n\npared with Wy in the tropics, the procedure will not converge.\n\nThere are some restrictions on the use of the method.\n\nThis non-con­\n\nvergence is related to the so-called ellipticity condition for the non-linear balance equation.\n\nIf W\n\nT\n\nV\n\nand W\n\n\\$\n\nare fully spatially variable, then the problem is usually too\n\nlarge to solve on the sphere.\n\nHowever, solutions can be obtained for the case\n\nwhen the weights are at most functions of latitude and pressure. There remain problems in the tropics. in the tropics are difficult to derive.\n\nAppropriate initialization constraints They appear to be highly implicit,\n\nrendering convergence difficult.\n\nREFERENCE\n\nDaley, R., 1981: Normal mode initialization. 450- 468.\n\nREV. GEOPHYS. SPACE PHYS., 19,\n\n13\n\nAPPLICATION OF OPTIMAL CONTROL TO METEOROLOGICAL PROBLEMS 0. TALAGRAND\n\nLaboratoire de Meteorologie Dynamique, ENS, 75231 Paris (France)\n\nINTRODUCTION A very general question with any kind of numerical modeling is the question of the s e n s i t i v i t y of the results to the input parameters of the model. A numerical model can be described as a process which, starting from a set of input parameters, produces a set of output parameters. In the case of a meteorological model, which integrates the equations governing the temporal evolution of the atmospheric flow, the input parameters are the i n i t i a l and lateral boundary conditions, and also the various physical and numerical parameters of the model. The output parameters are the meteorological\n\nfields\n\nproduced at successive times by the integration, and also the various diagnostic quantities which can be computed from these f i e l d s , such as for instance climatological means and variances, or transports of momentum, energy, e t c , effected by the model's c i r c u l a t i o n . There are many reasons why one can be interested in the s e n s i t i v i t y of a numerical model with respect to i t s input parameters, such as ( i ) The study of aspecific physical problem. For instance, what w i l l be the climatological consequences of an increase of the C0£ atmospheric content? ( i i ) The determination of the uncertainty on the model's output\n\nresulting\n\nfrom the uncertainty on the input. For instance, what is the uncertainty on a numerical forecast due to the uncertainty on the i n i t i a l conditions? ( i i i ) The determination of the values of the input parameters which maximize, or minimize, or \"optimize\" in some sense the value of some particular output parameter. For instance, the problem of assimilation of meteorological observations can be stated, as w i l l be shown l a t e r , as an optimization\n\nproblem\n\nof this type. In such problems, s e n s i t i v i t i e s are not required for themselves, but only as intermediaries which link the variations of the input parameters to the variations of the output parameter to be optimized, and are used in an appropriate algorithm which determines the optimizing values of the input parameters. One f i r s t approach for determining s e n s i t i v i t i e s is what can be called direct perturbation. A basic integration of the model having f i r s t been performed, the value of one of the input parameters is modified and a new\n\n14\n\nintegration is performed. The required s e n s i t i v i t i e s , or partial derivatives, are then computed by f i n i t e difference between the two integrations. At the cost of two integrations of the model, one can thus obtain the s e n s i t i v i t i e s of al1 output parameters of the model with respect to one input parameter. I t is in this way for instance that numerical studies of climatic s e n s i t i v i t y to the CO^ atmospheric content have been performed. In many situations however one w i l l not so much be interested in determining the s e n s i t i v i t i e s of a l l (or a large number of) the output parameters with respect to one input parameter, but rather in determining the s e n s i t i v i t i e s of one output parameter with respect to a l l (or a large number of) the input parameters. A typical example of such a situation would be the following\n\n: a\n\nnumerical forecast having f a i l e d in some particular respect, for instance in not predicting the deepening of a depression, one wants to determine what in the model's i n i t i a l conditions (and possibly also in i t s physical parameters) was at the origin of that particular f a i l u r e . What w i l l be necessary to know in this case are the s e n s i t i v i t i e s of the erroneously predicted parameter with respect to the model's i n i t i a l conditions. In such a s i t u a t i o n , i t would of course be possible to determine the required s e n s i t i v i t i e s through direct perturbation. But this would require as many explicit\n\nintegrations of the model\n\nas there are parameters with respect to which s e n s i t i v i t i e s are sought, and the numerical cost of these integrations would rapidly become prohibitive for large dimension models. This a r t i c l e is devoted to the theory and to the meteorological applications of the adjoint equations, through which the s e n s i t i v i t y problem which has just been discussed can be solved at a much lower cost than through direct\n\npertur­\n\nbation. Once the so-called adjoint of a numerical model has been developed, the s e n s i t i v i t i e s of one output parameter with respect to a J J input parameters can be obtained by performing one integration of the basic model over the time interval under consideration, followed by one backward integration of the adjoint model over the same time i n t e r v a l . The numerical cost of one adjoint integration w i l l usually be comparable to the cost of one integration of the basic model, and the determination of the required s e n s i t i v i t i e s w i l l basically cost, as direct perturbation studies, two model integrations over the time interval under consideration. Adjoint equations are tools of the theory of optimal control, which has been developed in the l a s t twenty years, and which generally deals with questions of how to \"control\" the input parameters of a numerical process in order to \"optimize\" i t s output parameters. The idea of applying adjoint equations to meteorological problems is by no means new, and was apparently f i r s t suggested by Marchuk (1974). Since then, several authors ( s e e , e . g . , Penenko and\n\n15\n\nsituations.\n\nI t is however extremely convenient, and i t saves cumbersome calculations with hosts of indices, to present the theory in a general abstract form without using explicit components. The two properties of Hilbert spaces which are important for the theory of adjoint equations are the following (i)\n\nLet\n\nbe a scalar continuous and differentiable function defined on a\n\nHilbert space y ,\n\nwith inner product noted (\n\nthere exists a uniquely defined vector\n\nwhich, by the very definition of a gradient, shows that the gradient v ^ with respect to u is equal to\n\nof\n\n17\n\n\\1 - K* \\1f\n\n{2A)\n\nThis expression shows that, i f a program is available which computes for given w,\n\nG^* w\n\ncan be e x p l i c i t l y determined by performing the following\n\noperations ( i ) starting from the value of u under consideration, compute\n\nv = G(u) and\n\nv\"Jf at point v. (ii)\n\ncompute\n\nG^* v\n\nI t is seen that vj^\n\ny\n\n^ .\n\ncan thus be determined, for given u, at the cost of one\n\nintegration of the direct model model\n\nu -> G(u)\n\nand one integration of the adjoint\n\nw -> G^* w . The adjoint model is I inear (contrary to the direct model,\n\nwhich w i l l not usually be l i n e a r ) , with coefficients which w i l l usually depend on the particular point u under consideration. Remark. One may wonder i f formula (2.4) can be really useful, in view of the fact that\n\nmay not be simpler to compute than\n\nsituation however, ^\n\n. In any particular\n\nw i l l always be a \"simple\" function of some set of\n\narguments, with respect to which analytical differentiation w i l l be easy.\n\nIt\n\nthen only suffices to choose v as being precisely such a simple set of arguments. Let us suppose that the operation\n\nu -> G(u)\n\ncan be described as the\n\ncomposition of a number of successive operations G =\n\nC ....\n\nC0\n\nn\n\n2\n\n]\n\nThis w i l l be the case, for instance, when G represents the temporal v\n\nintegration\n\nof a dynamical model, i , e . the composition of a number of elementary timestep integrations\n\n(each of which can also be described as the composition of a\n\nnumber of more elementary operations). The rule of differentiation of the composition of a number of operators leads for G^ to\n\n= c ; . . . . c< c; where, for each m, C is the linear operator obtained by differentiation of C . m\n\nm\n\nNow, a basic result on adjoint operators, easily obtainable from ( 2 . 2 ) , is that the adjoint of a product of operators is the product of their adjoints, taken in reverse order G\n\nu*\n\n=\n\nC\n\n1*\n\nC\n\n2*\n\nC\n\nn*\n\n( 2\n\n*\n\n5 )\n\nwhich shows that, in the adjoint computations, the adjoints of the operations which make up G w i l l have to be performed in reverse order. In p a r t i c u l a r ,\n\nif\n\n18\n\nG represents a temporal integration, the corresponding adjoint integration w i l l be performed backwards in time. In ( 2 . 5 ) , C^* i s , for any m, a linear operator whose coefficients depend on the results of the corresponding m-th step in the direct integration. This shows that, in order to perform the adjoint integration, i t is in principle necessary to store in memory the results of al1 the intermediary computations leading from u to G(u). This of course is costly and is the price to be paid for the gain in computing time afforded by the adjoint equations.\n\nAPPLICATION TO SENSITIVITY OF A DYNAMICAL MODEL WITH RESPECT TO INITIAL CONDITIONS We w i l l now describe how the general principle which has just been presented must be implemented in the case when the input vector u is the set of i n i t i a l conditions of a dynamical model and the operation\n\nu + v = G(u)\n\nis the\n\ntemporal integration of the model. The l a t t e r w i l l be defined by the equation dx\n\n=\n\nSt\n\nF(x)\n\n(3.1)\n\nwhich describes the temporal evolution of a state vector x ( t ) belonging to a Hilbert s p a c e d , with inner product noted function of £ t\n\nQ\n\n< , >. In ( 3 . 1 ) , F is a regular\n\ninto i t s e l f . Any i n i t i a l condition\n\nx(t ) = u Q\n\nat a given time\n\ndefines a unique solution x ( t ) to ( 3 . 1 ) , and the output vector v w i l l be the\n\nsolution x ( t ) , for a l l t ' s belonging to a given interval The scalar function ^f(v)\n\nft ,tj]. Q\n\nwhose gradient with respect to u is sought w i l l\n\nbe defined as\n\n2(v)\n\n=\n\nf\n\nH[x(t) , t ] dt\n\nwhere\n\nH[x,tJ is a scalar function defined on £x[t ,t^j\n\nt]\n\n(3.2)\n\nQ\n\n. Expression (3.2) is\n\nof course not the most general expression for a scalar function of v = fx(t), t\n\nQ\n\n^ t ^ t^\"J , but i t covers a very large range of functions which\n\nmay have to be considered in practical situations. For instance, if\n\nH[x,t]\n\nrepresents for any t the average of some meteorological f i e l d over some spatial domain, (3.2) w i l l represent the corresponding spatial and temporal average of the same f i e l d . For a given i n i t i a l condition u and for |*he corresponding solution x ( t ) of ( 3 . 1 ) , the f i r s t order variation equal to\n\n6 ^ resulting from a variation\n\n6u of u is\n\n19\n\n< VH(t) , 6 x ( t ) > dt\n\n(3.3)\n\no where vH(t) is the gradient of Hfx,t] with respect to x, taken at point [ x ( t ) , t ] , and\n\n6x(t)\n\ni s the f i r s t order variation of x ( t ) resulting at time t\n\nfrom the i n i t i a l perturbation 6u = 6 x ( t ) . The variation 6 x ( t ) i s obtained from Q\n\n6u by temporal integration of the tangent linear equation r e l a t i v e to the solution x ( t ) ^\n\n=\n\nwhere\n\nF'(t)\n\nF'(t)\n\n(3.4;\n\n6x\n\nis the linear operator obtained by differentiating\n\nF with\n\nrespect to x, and taken at point x ( t ) . Equation (3.4) i s l i n e a r , and i t s solution at a given time t can be written as 6x(t)\n\n=\n\nR(t,t )\n\nwhere\n\nR(t,t ) Q\n\n6u\n\ni s a perfectly defined linear operator of £\n\ncalled the resolvent of (3.4) between times t\n\nand t . The resolvent R ( t , t ' )\n\ndefined more generally for any two times t and t ' and possesses the following R(t,t) = I\n\ninto i t s e l f ,\n\ncomprised between t\n\n(3.5a)\n\nwhere I is the unit operator of £ =\n\n,\n\nproperties\n\nfor any t\n\n|^ R(t,f)\n\nis\n\nand t^\n\n, and\n\nF'(t) R(t,t'j\n\n(3.5b)\n\nfor any t and t ' . Equation (3.3) can now be rewritten\n\n< vH(t) , R ( t , t )\n\n/\n\n6u\n\n> dt\n\n< R * ( t , t ) vH(t) , 6u > dt\n\n< I\n\no\n\nJt\n\nR * ( t , t ) vH(t) dt , 6u >\n\nwhere we have introduced, for any t , the adjoint R * ( t , t ) of R ( t , t ) . Q\n\nseen from the l a s t line that the gradient V ^ J of ^ w i t h u\n\nI t is\n\nrespect to u is equal\n\n20\n\nto 1\n\nR * ( t , t ) vH(t) dt\n\n(3.6)\n\no We introduce at this point the adjoint equation of (3.4) ^\n\n=\n\n- F ' * ( t ) 6*x\n\n(3.7)\n\nwhich i s a linear d i f f e r e n t i a l\n\nequation whose variable 5*x also belongs to\n\n,\n\nand in which F ' * ( t ) is at any time t the adjoint of the operator F ' ( t ) . Let us denote S ( t ' , t ) the resolvent of (3.7) between times t and t ' .\n\nFor any two\n\nsolutions 6x(t) and 6 * x ( t ) of (3.4) and (3.7) respectively, the inner product < 6x(t) , 6 * x ( t ) > i s constant with time, as can be seen from the following equalities\n\n^\n\n, 6*x(t) >\n\n< 6x(t)\n\n= < - -J-(t)\n\n, 5*x(t) >\n\nd\n\n=\n\n+ < 6x(t) , ^ ( t )\n\n< F ' ( t ) 6 x ( t ) , 6*x(t) >\n\n-\n\n>\n\n< 6x(t) , F'*(t)\n\n6*x(t)>\n\n= 0 Let y and y\n\n1\n\nbe any two elements of £\n\n. The solution of the direct equation\n\n(3.4) defined by the i n i t i a l condition y at time t assumes at time t\n\n1\n\nthe value\n\nR ( t , t ) y , while the solution of the adjoint equation (3.7) defined by the 1\n\ni n i t i a l condition y\n\n1\n\nat time t\n\n1\n\nassumes at time t the value S ( t , t ' ) y\n\nl\n\n. The\n\ncorresponding equality between inner products therefore reads < R ( t ' , t ) y , y' >\n\n=\n\n< y , S(t,t')y' >\n\nwhich, being v a l i d for any y and y\n\n1\n\n(3.8)\n\n, shows that S ( t , t ' ) i s the adjoint\n\noperator of R ( t ' , t ) . Expression (3.6) accordingly becomes\n\nu\n\n1 -f\" \"o t\n\nS ( t , t ) vH(t) dt\n\n(3.9)\n\nWe now consider the inhomogeneous adjoint equation ^\n\n=\n\n- F ' * ( t ) 6*x\n\n- vH(t)\n\n(3.10)\n\n- vH(t) to the right-hand side of ( 3 . 7 ) . The\n\nsolution of (3.10) defined by the condition 6*x(t ) 1\n\n= 0\n\ni s equal to\n\n21\n\n6*x(t)\n\n= / J\n\nS(t,i) vH(t) dx\n\nt\n\nas is easily v e r i f i e d by using the resolvent properties ( 3 . 5 ) . Equation (3.9) now shows that\n\nis equal to 6 * x ( t ) . Q\n\nIn summary, the gradient V \"^pcan be obtained, for given u, by performing u\n\nthe following operations, which are in the present case the operations described in the previous section for the general case ( i ) Starting from u at time t , Q\n\nto ty\n\nintegrate the basic equation (3.1) from t\n\nStore the values thus computed for x ( t ) , t\n\nQ\n\nQ\n\n^ t ^ t^.\n\n( i i ) Starting from 6 * x ( t ^ ) = 0, integrate the inhomogeneous adjoint equation (3.10) backwards in time, the operator F ' * ( t ) and vH(t) being determined, at each time t , from the value of x ( t ) stored in the direct integration. The f i n a l\n\nresult 6 * x ( t ) obtained at time t Q\n\nQ\n\nis the gradient v ^ \" ^ .\n\nThe same general principle can also be applied i f one seeks the gradient of with respect to a set of parameters w ( e . g . , physical parameters) appearing in the evolution equation ( 3 . 1 ) . In order to make the dependence with respect to w e x p l i c i t ,\n\n&\n\n=\n\nF\n\n(3.1) must be rewritten as\n\n(*> > w\n\nof ^ w i t h\n\nrespect to w w i l l be properly defined only i f an\n\ninner product has been previously defined on the space of a l l possible w ' s . Application of the general principle described in the previous section then shows that, in order to e x p l i c i t l y\n\nfor given\n\ni n i t i a l condition x ( t ) and given parameter w , the steps ( i ) and ( i i ) Q\n\ndescribed s t i l l\n\nf\n\n1\n\nF\n\nw*\n\njust\n\nhave to be performed as above. Once the solution 6 * x ( t ) of\n\n(3.10) is known for\n\n\\ 7\n\nq\n\n( t )\n\n6\n\n*\n\nt\n\nQ\n\nx\n\n(\n\n^ t ^ t^\n\nt\n\n)\n\nd\n\nv w\n\n\"^\"'\n\ns\n\ngiven by the integral\n\nt\n\nwhere, for any t , F ' * ( t ) is the adjoint of the operator obtained by d i f f e r ­ entiating F(x,w) with respect to w at point ( x ( t ) , w ) . q\n\nI t therefore appears that the particular set of parameters with respect to which the gradient of ^\n\nis sought does not influence either step ( i ) or ( i i )\n\nabove, but only the computations which are performed on the adjoint\n\nsolution\n\n6*x(t) of (3.10), once i t has been computed in step ( i i ) . S i m i l a r l y , the p a r t i ­ cular function\n\nwhose gradient i s sought influences only the \"forcing\" term\n\n-vH(t) in the adjoint equation (3.10). The term\n\n- F ' * ( t ) 6*x\n\nin this equation,\n\n22\n\nwhich w i l l take the bulk of the adjoint computations (just as the\n\nintegration\n\nof (3.1) w i l l take the bulk of the direct computation of * J ) , depends neither on ^\n\nnor on the input parameters with respect to which the s e n s i t i v i t y\n\nof\n\nis to be determined. Once a program for computing this term and integrating (3.10) (with vH(t) as a parameter) has been developed, this program can be used for al1 s e n s i t i v i t y studies of output parameters of the direct model (3.1) with respect to i t s input parameters. This confers to the adjoint approach a genera­ l i t y and a f l e x i b i l i t y which are among i t s major advantages. Remark. The presentation of this section does not exactly follow the general principle described in the previous section, in that the adjoint equations (3.7) and (3.10) have been introduced without prior j u s t i f i c a t i o n ,\n\nand have\n\nonly been verified to lead to the required gradient. The interested reader w i l l find in Talagrand and Courtier (1985) an approach which, following the general principle of the previous section, rigorously leads to (3.7) and (3.10). APPLICATION TO THE VORTICITY EQUATION In order to make the previous developments more e x p l i c i t , we w i l l now derive the adjoint of the spherical v o r t i c i t y equation\n\nH\n\n=' J ( C + f ,\n\nA '\n\n1 C\n\n(4.1)\n\n)\n\nwhich expresses the lagrangian conservation of absolute v o r t i c i t y\n\nc + f\n\nin\n\na two-dimensional incompressible and inviscid flow along the surface of a rota­ ting sphere z. In ( 4 . 1 ) , c and f are the v o r t i c i t i e s of the r e l a t i v e flow and _i\n\nbasic rotation respectively, t is time, A\n\nis the inverse Laplacian operator\n\nalong z , and J is the Jacobian operator J ( a , b ) = va x vb = v x (a vb) where v now denotes f i r s t order differentiation along z . For a given solution c, of ( 4 . 1 ) , the tangent linear equation, analogous to ( 3 . 4 ) , reads =\n\nJ(\n\n,\n\nA '\n\n1\n\nt ) 7\n\n+ J( c + f ,\n\nA \"\n\n1\n\n5C )\n\n(4.2)\n\nwhere differentiation of the Jacobian has produced two terms. The determination of the adjoint of (4.2) requires the prior definition on an inner product on the space of a l l possible v o r t i c i t y f i e l d s , i . e . on the space of a l l\n\nregular\n\nfunctions on z with zero mean. The total kinetic energy K corresponding to a given v o r t i c i t y f i e l d r, is given by\n\n23\n\n2K\n\n=\n\n/\n\nV A ^ c . V A\n\n_\n\ndz\n\nc\n\n1\n\n(4.3)\n\nwhere the dot denotes scalar product of ordinary vectors in physical space. Expression (4.3) suggests the following definition for an inner product on the space of v o r t i c i t y < c , c' >\n\n=\n\nfields\n\n/\n\nV A \" V dz\n\nV A \" ' C .\n\n'z\n\nwhich can be rewritten after integration by parts\n\n< C\n\n, c' >\n\n=\n\n-\n\nJ\n\nc A \" V\n\n/\n\nA\"V\n\nC\n\ndz dz\n\n= -\n\n-j - /\n\nc / VtT\\ 1\n\ndz\n\n(4.4)\n\nI t immediately results from this expression that the Laplacian A and i t s _i inverse A are self-adjoint for this inner product, i . e . for any c and c < C , Ac' > < C ,\n\nA~V\n\n= >\n\n< AC =\n\n< A\n\n, c' _ 1\n\nC\n\n1\n\n> , C' >\n\nThe Jacobian J has the property t h a t , for any three scalar f i e l d s a,b,c jf J ( a , b ) c\n\ndz\n\n= j\n\na J(b,c)\n\ndz\n\n(4.5)\n\nas is seen from the following equalities ^J(a,b)\n\nc\n\ndz\n\n= J\n\n= j\n\nvx(avb) c\n\na J(b,c)\n\ndz\n\n= j\n\na vbxvc dz\n\ndz\n\nwhere integration by parts has been used for going from the second to the third term. Three scalar f i e l d s a , 6 c , 6 c ' being given, use of (4.4) and (4.5) yields < J(6c,a)\n\n,\n\n6c'\n\n>\n\n-j\n\nJ(6c,a)\n\n-i\n\n6c A\n\nA\" 6c'\n\ndZ\n\n1\n\n-1\n\n=\n\n-j\n\n6c J ( a , A \" 6 c ' ) 1\n\ndZ\n\n-1 Aj(a,A\n\n6c')\n\ndZ\n\n_ i\n\n=\n\n< 6c , A j ( a , A\n\n6c ) > 1\n\nwhich shows that, for given a , the adjoint of the operator the operator\n\n6c •> A J ( a , A 1\n\n_ 1\n\n6c')\n\n. Using the f a c t that A \"\n\n1\n\n6c •> J ( 6 c , a ) i s is self-adjoint\n\nthat the adjoint of the product of two operators is the product of their\n\n24\n\nadjoints taken in reverse order, we obtain the adjoints of both terms on the right-hand side of (4.2) and the following expression for the adjoint equation, analogous to (3.7) =\n\nA J ( A\" 6*r 1\n\n,\n\nA\n\n\"V\n\n)\n\n+\n\nJ ( £ + f\n\n, A\"Vc\n\n)\n\n(4.6)\n\nI t is seen that the same basic operators appear in this equation as in (4.1) and ( 4 . 2 ) . This means that a computer program for integrating\n\n(4.6) w i l l\n\nnormally be built from the same subroutines as a program for integrating ( 4 . 1 ) . This fact is very general and greatly f a c i l i t a t e s the development of adjoint models. APPLICATIONS TO METEOROLOGICAL PROBLEMS\n\nAt least three broad ranges of possible meteorological applications of adjoint equations can be identified ( i ) Diagnostic studies intended at tracing in the input parameters of a numerical model the origin of particular features observed in the results of that model. (ii)\n\nEstimation of the uncertainty on model results due to the uncertainty\n\non the input parameters. In this case, the estimated output uncertainty ( e . g . the standard deviation of the error on a particular output parameter) w i l l be obtained by multiplying the partial derivatives obtained through the adjoint equations by the known uncertainty on the input paramaters. ( i i i ) Optimization studies, intended at determining the values of the input parameters which \"optimize\" some output parameter. In such applications, gradients determined through the adjoint equations w i l l be used as interme­ diaries for computing successive approximations, through an appropriate optimization process, of the required values of the input parameters. In view of the broadness of possible applications, the works which have effectively been performed so far are s t i l l rather limited. Hall et al_. (1982) have determined the s e n s i t i v i t i e s of a few output parameters of a simple twolevel convective model with respect to i t s i n i t i a l conditions and physical parameters. A number of works have been performed on assimilation of meteoro­ logical observations, treated as an optimization problem. Assimilation is the process through which the i n i t i a l conditions of a numerical forecast are deter­ mined from observations distributed\n\nin both space and time. In meteorological\n\nservices, assimilation i s at present performed through sequential linear regression algorithms, which can be described as simplified forms of Kalman-\n\n25\n\nBucy f i l t e r i n g (see, e . g . , Bengtsson et aJL , 1981, for more d e t a i l s ) . Assimi­ lation can also be stated as a variationnal optimization problem in the following terms. The observations ^ ( t - ) having been performed at instants over a time interval\n\n[ t , t ^ , one f i r s t defines a scalar function 0\n\nwhich,\n\nfor any solution x ( t ) of the model, measures the discrepancy between that solution and the observations. This function\n\ncan be defined for instance as\n\na sum of squared deviations between the model and the observations 2=\n\nf <\n\nx(t.)\n\n- x\n\no b\n\n(\n\nT i\n\n)\n\n,\n\nx(x.) - X\n\nQ b\n\n(x ) i\n\n>\n\n(5.1)\n\nwhere the terms on the rigth-hand side must be limited to those components of x which have e f f e c t i v e l y been observed. Expression (5.1) is of type ( 3 . 2 ) , the integral being replaced by a f i n i t e sum. One then looks for the model solution which minimizes\n\n. This solution w i l l be defined by the corresponding i n i t i a l\n\ncondition at time t . In this context, adjoint equations w i l l be used for determining as described above the gradient of ^\n\nwith respect to the i n i t i a l\n\ncondition. Successive gradients, used in a descent algorithm implemented in the space of a l l possible i n i t i a l conditions, w i l l lead by successive approxima­ tions to the minimizing i n i t i a l x(t ) . This particular approach to the problem of data assimilation was apparently f i r s t suggested by Penenko and Obraztsov (1976). More recently, i t has been used by Lewis and Derber (1985) (see also the contributions by these two authors in this volume) and by Le Dimet and Talagrand (1986). S t i l l more recent results are reported in this volume by Courtier and Talagrand, and by Le Dimet and Nouailler. All these authors have so far used rather simple meteorological models, but the results they have obtained clearly show the numerical convergence of the minimization process and the meteorological quality of the f i e l d s i t produces. Courtier (pers. com.) has used the adjoint equations of a model for a diagnostic study of the f i r s t of the three types of applications considered above. Unacceptably large changes had been observed in the meteorological fields during the i n i t i a l i z a t i o n phase of the operational forecast at Direction de la Meteorologie, Paris (the i n i t i a l i z a t i o n is a process, which takes place between the assimilation and the forecast i t s e l f , and through which unrealistic high-frequency gravity waves are removed from the forecast's i n i t i a l condition). These large changes had in particular the effect of increasing the difference between the model f i e l d s produced by the assimilation and the observations. Using the adjoint of a shallow-water equation model (both this model and i t s adjoint are described in Courtier and Talagrand's contribution\n\nin this volume), Courtier determined the gradient of the global\n\nroot-mean-square change produced by the i n i t i a l i z a t i o n with respect to the geopotential f i e l d before the i n i t i a l i z a t i o n .\n\nHe obtained a c l e a r l y defined\n\n26\n\ndipolar structure which rotated in time with the sun. That structure was shown to be associated with a thermal wave which, being basically a gravity wave, was removed from the model by the i n i t i a l i z a t i o n process, although i t was r e a l l y present in the atmosphere and in the observations. Using the adjoint model was in this instance a very e f f i c i e n t way for identifying an imperfection in one of the stages of the forecasting procedure. Urban (1985) and Lacarra (pers. com.) have used adjoint equations for determining the perturbations on a given model state which w i l l amplify most rapidly in the ensuing evolution. In the approximation of the tangent linear equation ( 3 . 4 ) , this amounts to determining the perturbations 6u = 6 x ( t ) which, for given t , maximize the quantity A(6u t )\n\n=\n\n< xU)\n\n> fix(t) >\n\n6\n\n<\n\n6\n\nU\n\n'\n\n6\n\nU\n\n>\n\n=\n\n\"\n\n< R ( t , t ) 6u , R ( t , t ) 5u > Q\n\nQ\n\n< 6U ,\n\n6U >\n\n< 6u , R * ( t , t ) R ( t , t ) 6u > Q\n\n< 6U ,\n\nQ\n\n6U >\n\nThese perturbations are the eigenvectors corresponding to the dominant eigen­ values of the matrix\n\nC = R * ( t , t ) R ( t , t ) . These eigenvectors can be detero o\n\nmined, through an algorithm developed by Householder (see, e_^g., C i a r l e t , 1982) at the cost of the computation of CSu for a few appropriately chosen 6 u ' s . Noting that R * ( t , t ) is the resolvent of the homogeneous adjoint equation (3.7)\n\nbetween t and t\n\nQ\n\n(see equation ( 3 . 8 ) ) , C6u can be computed, for given 6 u ,\n\nby f i r s t integrating the direct model between t\n\nand t , and then the adjoint\n\nmodel between t and t . o Urban and Lacarra have implemented this approach on a two-level quasigeostrophic and a shallow-water model respectively, and for ranges t-t\n\nvarying\n\nbetween 12 and 48 hours. The dominant eigenvectors consist in both cases of quasi-geostrophic motions restricted to the smallest scales resolved by the model, and the corresponding eigenvalues of C vary t y p i c a l l y between 10 and 30 as the range t - t CONCLUSIONS\n\nincreases from 12 to 48 hours.\n\nThe examples which have been b r i e f l y discussed in this a r t i c l e are only a few of the many potential applications of the adjoint equations. The very general and systematic character of the adjoint approach makes i t an extremely powerful tool. However, although the adjoint equations are incomparably more e f f i c i e n t than direct perturbation computations for determining s e n s i t i v i t i e s , their numerical cost may in some cases remain high. This is especially true of variational data assimilation, which w i l l require at least a few descent\n\n27\n\nsteps for determining the i n i t i a l condition of one forecast. Each descent step w i l l i t s e l f require one direct integration and one adjoint\n\nintegration\n\nof the model over the time interval on which the observations are distributed. The length of that interval w i l l t y p i c a l l y be 24 hours, and i t i s clear t h a t , without appropriate simplifications, variational assimilation could not be implemented in practice with present models and computers. Now, such simpli­ fications are certainly possible. The diagnostic study performed by Courtier on the i n i t i a l i z a t i o n\n\nprocess shows for instance that instructive results can\n\nbe obtained with an adjoint which is much simpler and more economical than the direct model. In addition, one can confidently expect that future progress in computing power w i l l make feasible computations which could not be performed at present. For these various reasons, the study and development of adjoint methods certainly constitute a useful investment for the future progress of numerical modeling of the atmospheric flow. I t may be useful to add some final comments on the exact nature of the adjoint equations. I t is important to stress that an integration of the adjoint equation (3.10) is not a time-reversed integration of the basic equation ( 3 . 1 ) . The fields 6*x produced at time t by the integration of (3.10) are not physical fields at time t , but partial derivatives of the function ^\n\ndefined\n\nby (3.2) with respect to physical fields at time t . The difference between an adjoint integration and a time-reversed integration of the basic dynamical equation becomes particularly significant when the l a t t e r contains diffusive or dissipative terms, whose time-reversed integration usually is an ill-posed problem from a mathematical point of view. The linear diffusion equation, for instance, contains a Laplacian, which is s e l f - a d j o i n t . The presence of the minus sign in the adjoint equation (3.10) therefore insures that the integra­ tion of the adjoint diffusion equation is well-posed for integration into the past. More generally, whenever the integration of the basic dynamical equation is well-posed only for integration into the future, the integration of the corresponding adjoint equation w i l l be well-posed only for integration into the past. ACKNOWLEDMENTS The author thanks F.X. Le Dimet who f i r s t drew his attention to the interest of adjoint techniques for meteorological problems. Further contacts with F.X. Le Dimet, and also with J . Lewis, B. Urban, P. Courtier and J . F . Lacarra were extremely useful for the development of these techniques. P. Courtier and J . F . Lacarra in addition allowed the author to mention some of their yet unpublished r e s u l t s .\n\n28\n\nREFERENCES Bengtsson, L . , G h i l , M. and Kail en, E. ( E d i t o r s ) , 1981. Dynamic Meteorology. Data Assimilation Methods. Springer-Verlag, New-York, 330 pp. Cacuci, D.G., 1981. S e n s i t i v i t y theory for nonlinear systems. I.Nonlinear functional analysis approach. J . Math. Phys., 22: 2794-2802. C i a r l e t , P . G . , 1982. Introduction to matrix numerical analysis and optimiza­ tion (in French). Masson, P a r i s , 279 pp. H a l l , M.C.G., Cacuci, D.G. and Schlesinger, M . E . , 1982. S e n s i t i v i t y analysis of aradiative-convective model by the adjoint method. J . Atmos. S c i . , 39: 2038-2050. Le Dimet, F.X. and Talagrand, 0 . , 1986. Variational algorithms for analysis and assimilation of meteorological observations : theoretical aspects. Tellus, 38A: 97-110. Lewis, J . M . and Derber, J . C , 1985. The use of adjoint equations to solve a variationaladjustment problem with advective constraints. T e l l u s , 37A: 309322. Marchuk, G . I . , 1974. Numerical simulation of the problems of the dynamics of the atmosphere and of the ocean (in Russian). Gidrometeoizdat, Leningrad. Penenko, V. and Obraztsov, N.N., 1976. A variational i n i t i a l i z a t i o n method for the f i e l d s of the meteorological elements. MeteoroHogiya i Gidrologiya (English t r a n s l a t i o n ) , 11: 1-11. Talagrand, 0 . , 1985. The adjoint model technique and meteorological applica­ tions. I n : Proceedings of Workshop on High-Resolution Analysis, European Centre for Medium-Range Weather Forecast, Reading, United Kingdom, 325 pp. Talagrand, 0. and Courtier, P . , 1985. Formalization of the adjoint method. Meteorological applications (in French). Working note number 117, Etablissement d Etudes et de Recherches Meteorologiques, P a r i s , 10 pp. 1\n\nUrban, B . , 1985. Maximal error amplification in simple meteorological models (in French). Working note, Ecole Nationale de la Meteorologie, Toulouse, France.\n\n29\n\nA Review of V a r i a t i o n a l\n\nand O p t i m i z a t i o n\n\nMethods i n Meteorology\n\nby I.M.\n\nNavon\n\nSupercomputer Computations Research\n\nInstitute\n\nThe F l o r i d a S t a t e U n i v e r s i t y Tallahassee,\n\nFlorida\n\n32306-4052\n\nABSTRACT A condensed o v e r v i e w of v a r i a t i o n a l and o p t i m i z a t i o n methods i n Meteorology i s p r e s e n t e d . I t i s aimed at g i v i n g the reader a s h o r t c o n c i s e p e r s p e c t i v e of the developments i n the d i s c i p l i n e in the l a s t t h i r t y y e a r s and to present b r i e f l y recent developments in the a p p l i c a t i o n of o p t i m i z a t i o n and optimal c o n t r o l theory i n Meteorology. INTRODUCTION The f i r s t\n\napplications\n\npioneered by S a s a k i\n\nof\n\nvariational\n\n(1955, 1958)\n\nbased on the c a l c u l u s of v a r i a t i o n s . defines a functional\n\nmethods\n\ni n meteorology\n\nhave been\n\nwhen he developed an i n i t i a l i z a t i o n\n\nmethod\n\nI n t h i s general v a r i a t i o n a l formalism one\n\n- whose extremal\n\ns o l u t i o n minimizes the v a r i a n c e of\n\nd i f f e r e n c e between observed and analyzed v a r i a b l e v a l u e s ,\n\nthe\n\nin a l e a s t - s q u a r e s\n\nsense - s u b j e c t to a s e t of c o n s t r a i n t s which a r e s a t i s f i e d e x a c t l y or a p p r o x i ­ mately by the analyzed v a l u e s . A v a r i a t i o n a l f u n c t i o n a l i s f o r m u l a t e d , the m i n i m i z a t i o n of which g i v e s r i s e to a s e t of E u l e r - L a g r a n g e e q u a t i o n s , which a r e then s o l v e d n u m e r i c a l l y . brief\n\nA\n\nreview of a p p l i c a t i o n s of the v a r i a t i o n a l method i n meteorology w i l l be\n\npresented f o l lowed by a survey of the i n t r o d u c t i o n of non-1 i n e a r programming and optimization\n\nmethods i n meteorology and f i n a l l y\n\nthe i n t r o d u c t i o n of\n\noptimal\n\nc o n t r o l theory methods ( t h e a d j o i n t model t e c h n i q u e ) in meteorology wi 11 c l o s e the r e v i e w . VARIATIONAL METHODS I n the appl i c a t ion of v a r i a t i o n a l methods a d i f f e r e n t i a t i o n i s made between a strong c o n s t r a i n t ( i . e . t h a t an equal i t y c o n s t r a i n t should be i d e n t i c a l l y equal to zero)\n\nand a \" w e a k - c o n s t r a i n t \"\n\napproximately be equal to z e r o . a p e n a l t y method.\n\nwhere the e q u a l i t y c o n s t r a i n t should o n l y\n\nT h i s method i s e q u i v a l e n t to the f i r s t s t e p i n\n\nStephens (1965) a p p l i e d the v a r i a t i o n a l method to a n a l y s i s\n\nproblems, using f u n c t i o n a l s formulated w i t h weak c o n s t r a i n t s - and used w e i g h t s which i n a sense determine a low-pass f i l t e r . I n a s e r i e s of papers S a s a k i (1969, 1970a, 1970b) g e n e r a l i z e d h i s method to\n\n30 include\n\ntime-variations\n\nand dynamical\n\nequations\n\ni n order\n\nto f i l t e r\n\nhigh-\n\nfrequency n o i s e - and to o b t a i n d y n a m i c a l l y a c c e p t a b l e i n i t i a l v a l u e s in data void a r e a s . A m u l t i t u d e of papers a p p l y i n g these ideas appeared i n the 1970's using the variational\n\nmethod w i t h d i f f e r e n t\n\nconstraint\n\nequation Stephens (1970), B a r k e r e t a l .\n\nsuch as i n c l u d i n g the balance\n\n(1977).\n\nV a r i a t i o n a l s y n o p t i c - s c a l e a n a l y s i s was c a r r i e d out by Lewis (1972) and Lewis and Grayson (1972). analysis,\n\nS h e e t s (1973) a p p l i e d the v a r i a t i o n a l method to h u r r i c a n e\n\nw h i l e Lewis and Bloom (1978) and Bloom (1983) used a v a r i a t i o n a l\n\nadjustment using dynamic c o n s t r a i n t s i n the a n a l y s i s of mesoscale rawinsonde data.\n\n( S e e a l s o Thompson (1969), Ray e t a l . (1980) and Testud e t a l . (1983)\n\na p p l i e d the method f o r computing v e l o c i t y f i e l d s from Doppler Radar d a t a .\n\nJ. J .\n\nO ' B r i e n (1970) used a v a r i a t i o n a l f o r m u l a t i o n to o b t a i n r e a l i s t i c e s t i m a t e s of the v e r t i c a l\n\nvelocity.\n\nMiddle and l a r g e - s c a l e v a r i a t i o n a l adjustment of atmospheric f i e l d s was p e r ­ formed by Stephens and Johnson\n\n(1978).\n\nA variational\n\na n a l y s i s method was\n\nc a r r i e d o u t , appl ied towards the removal of S e a s a t S a t e l 1 i t e S c a t t e r o m e t e r winds by Hoffman (1982, 1984) using c o n j u g a t e - g r a d i e n t methods f o r the unconstrained mi nimi z a t i o n . The use of inequal i t y c o n s t r a i n t s i n v a r i a t i o n a l adjustment was introduced by Sasaki\n\nand McGinley (1982).\n\nS a s a k i and Goerss (1982) used the v a r i a t i o n a l\n\napproach f o r S a t e l l i t e data a s s i m i l a t i o n . A general ized v a r i a t i o n a l o b j e c t i v e a n a l y s i s based on a dual i t y between o p t i ­ mum i n t e r p o l a t i o n and v a r i a t i o n a l a n a l y s i s and using a*general ized c r o s s v a l i d a t i o n was developed by Wahba and Wandelberger\n\n(1980), and Wahba (1981,\n\n1982).\n\nSeaman e t a l . (1977) used a v a r i a t i o n a l blending technique over a l a r g e area based on a method of f i e l d s by i n f o r m a t i o n\n\nblending.\n\nUSE OF VARIATIONAL METHODS TO ENFORCE 'A POSTERIORI' CONSERVATION OF INTEGRAL INVARIANTS Sasaki\n\n(1975,\n\n1976)\n\nproposed\n\na variational\n\napproach\n\nfor\n\nenforcing\n\n'a\n\np o s t e r i o r i ' c o n s t r a i n t s of mass and t o t a l energy c o n s e r v a t i o n when s o l v i n g the shallow-water equations.\n\nB a y l i s s and I s a a c s o n (1975) and I s a a c s o n (1977) i n d e ­\n\npendently proposed to 1 i n e a r i z e the c o n s e r v a t i v e c o n s t r a i n t s about the p r e d i c t e d v a l u e s by a g r a d i e n t method a l s o w i t h the view of e n f o r c i n g i n t e g r a l conservation.\n\ninvariants\n\nThe two approaches have been t e s t e d and compared by Navon (1981).\n\nS a s a k i and Reddy (1980) used a s i m i l a r method f o r e n f o r c i n g p o t e n t i a l enstrophy conservation.\n\n31 VARIATIONAL NORMAL MODE I N I T I A L I Z A T I O N AND RELATED ISSUES Variational\n\nnormal mode i n i t i a l i z a t i o n was pioneered by Daley (1978), who\n\ncombined the Machenhauer (1977) non-1 i n e a r normal-mode i n i t i a l i z a t i o n (NMI) w i t h the v a r i a t i o n a l procedure of S a s a k i (1958) a l l o w i n g the adjustment of the wind f i e l d to the m a s s - f i e l d or v i c e - v e r s a based on presumed a c c u r a c y of o b s e r v a t i o n s (confidence weights). Use of t h i s concept was made by Daley and P u r i (1980) f o r data-assimilation. interpolation\n\nfour-dimensional\n\nP h i l l i p s (1981) proposed a slow-mode m u l t i v a r i a t e optimum\n\nand d i s c u s s e d the u s e f u l n e s s of a v a r i a t i o n a l\n\nanalysis.\n\nT r i b b i a (1982) general ized v a r i a t i o n a l NMI and used d i r e c t l y normal modes and longitude/latitude linear\n\nvariable\n\nweights.\n\nH i s approach r e q u i r e s s o l u t i o n of\n\na\n\nl e a s t - s q u a r e s problem.\n\nPuri\n\n(1982,\n\npressure (1982)\n\n1983)\n\napproach f o r\n\nvariational\n\nTemperton\n\nNMI to minimize the l o s s of\n\n(1982,\n\nthe v a r i a t i o n a l\n\n1984)\n\nNMI of\n\nthe\n\ngeneralized\n\nAugmented Lagrangian method\n\n(1986).\n\nLe-Dimet\n\napproach t o the problem of\n\net\n\nal.\n\nsurface\n\nthe\n\nECMWF g r i d p o i n t\n\nbounded d e r i v a t i v e method using a v a r i a t i o n a l\n\nas a d u a l i t y Semazzi\n\nused v a r i a t i o n a l\n\ninformation.\n\nTribbia\n\nmodel.\n\nA\n\napproach as w e l l\n\ni s now being t e s t e d by Navon and\n\n(1982)\n\nalso\n\nproposed\n\na\n\nvariational\n\ninitialization.\n\nI n t r o d u c t i o n of n o n - l i n e a r programming and o p t i m i z a t i o n methods in\n\nMeteorology\n\nThe Augmented Lagrangian combined m u l t i p l i e r p e n a l t y method was proposed by Navon\n\n(1982a,\n\n1982b)\n\nV i l l i e r s (1983, 1986).\n\nand implemented\n\nin d i f f e r e n t\n\nmodels by Navon and de\n\nLe-Dimet (1982a, 1982b), Le-Dimet and Talagrand (1985)\n\nand Le-Dimet and Segot (1985) a l s o employed the Augmented-Lagrangian method. A c o n s t r a i n t r e s t o r a t i o n method due to A. M i e l e e t a l . (1969) was implemented by Navon and de V i l l i e r s\n\n(1985).\n\nOptimal c o n t r o l methods ( t h e a d j o i n t\n\nmethod)\n\nThe a d j o i n t method i s an a p p l i c a t i o n of optimal c o n t r o l theory where a f u n c ­ t i o n a l i s minimized by f i n d i n g i t s g r a d i e n t w i t h r e s p e c t to one of the a n a l y s i s states\n\n(e.g.\n\nthe\n\ninitial\n\nstate)\n\nand then using unconstrained\n\nminimization\n\nmethods such as the c o n j u g a t e g r a d i e n t to i t e r a t e towards the optimal Finding\n\nthe\n\ni n v o l v e s use of\n\npioneered by Marchuk (1974, 1982) Cacuci\n\nequations.\n\nand d e s c r i b e d by Kontarev (1980).\n\nH a l l and\n\n(1984) used the method to study s e n s i t i v i t y of numerical models w i t h\n\nr e s p e c t to p h y s i c a l p a r a m e t e r s .\n\nRecent advances on t h i s t o p i c were done by\n\nTalagrand (1985), Le-Dimet and Talagrand (1985), C o u r t i e r Derber\n\nstate.\n\nThe method was\n\n(1985) and Derber\n\n(1985).\n\nLewis and Derber\n\n(1985), Lewis and\n\n(1985) used the\n\n32 method to s o l v e a v a r i a t i o n a l\n\nadjustment problem w i t h a d v e c t i v e c o n s t r a i n t s\n\nw h i l e Derber (1985) used the a d j o i n t method f o r a v a r i a t i o n a l 4-D data a s s i m i l a ­ t i o n using q u a s i - g e o s t r o p h i c models as c o n s t r a i n t s . to a s h a l l o w water e q u a t i o n s model.\n\nC o u r t i e r (1985) a p p l i e d i t\n\nLe-Dimet and Talagrand (1985) used the\n\nmethod f o r data a s s i m i l a t i o n w i t h a 1-D s h a l l o w water e q u a t i o n s model. REFERENCES B a r k e r , E . , G. H a l t i n e r and Y . S a s a k i , 1977: Three dimensional i n i t i a l i z a t i o n using v a r i a t i o n a l a n a l y s i s . P r o c . 3rd Conf. N. W. P . of the AMS, Omaha 169-181. B a y l i s s , A. and E . I s a a c s o n , 1975: How to make your a l g o r i t h m s c o n s e r v a t i v e . N a t . Amer. Math. S o c . Aug. A594-A595. Bloom, S . , 1983: The use of dynamical c o n s t r a i n t s i n the a n a l y s i s of mesoscale rawingsonde d a t a . T e l l u s , 35, 363-378. C o u r t i e r , P. 1985: Experiments i n data a s s i m i l a t i o n using the a d j o i n t model technique. Workshop on H i g h - R e s o l u t i o n A n a l y s i s ECMWF (UK) June 1985, 20pp. D a l e y , R . , 1978: V a r i a t i o n a l n o n - l i n e a r normal mode i n i t i a l i z a t i o n . Tellus, 30, 201-218. D a l e y , R and K P u r i , 1980: Four dimensional data a s s i m i l a t i o n and the slow manifold. Monthly Wea. R e v . , 108, 85-99. Derber J . C. 1985: The v a r i a t i o n a l 4-D a s s i m i l a t i o n of a n a l y s e s using f i l t e r e d models as c o n s t r a i n t s . P h . D. T h e s i s , U n i v . of W i s c o n s i n - M a d i s o n , 142pp. G l o w i n s k i , R . , 1984: Numerical methods f o r n o n - l i n e a r v a r i a t i o n a l problems. S p r i n g e r - S e r i e s i n Computational P h y s i c s , New Y o r k , 493pp. H a l l , M. C. G. and Cacuci D. G. 1983: P h y s i c a l i n t e r p r e t a t i o n of the a d j o i n t f u n c t i o n s f o r s e n s i t i v i t y a n a l y s i s of atmospheric models. J . Atmos. S c i . 40, 2537-2546. Hoffman, R. 1982: SASS Wind ambiguity removal by d i r e c t m i n i m i z a t i o n . Mon. Wea. R e v . , Vol 110, pp. 434-445. Hoffman, R. 1984: SASS Wind ambiguity removal by d i r e c t m i n i m i z a t i o n . P a r t I I : Use of smoothness and dynamical c o n s t r a i n t s . Mon. Wea. R e v . 112, 1829-1852. I s a a c s o n , E . , 1977: I n t e g r a t i o n schemes f o r Long-Term C a l c u l a t i o n s . In Advances i n Computer Methods f o r P D E ' S . A V i c h n e v e t s k y E d . AICA 251-255. Le-Dimet, F. X . , 1982: A g e n e r a l formal ism of V a r i a t i o n a l A n a l y s i s . CIMMS Report No.22. 34pp. Norman, OK 73019. Le-Dimet, F . X . , S a s a k i , Y . K. and L. W h i t e , 1982: Dynamic i n i t i a l i z a t i o n w i t h f i l t e r i n g of g r a v i t y . CIMMS Report No.40. Norman, OK 73019. Le-Dimet, F. X . and 0. T a l a g r a n d , 1985: V a r i a t i o n a l a l g o r i t h m s f o r a n a l y s i s and assimi l a t i o n of M e t e o r o l o g i c a l O b s e r v a t i o n s : T h e o r e t i c a l A s p e c t s . To appear T e l l u s 1986. Le-Dimet, F . X . and J . S e g o t , 1985: V a r i a t i o n a l a n a l y s i s of wind f i e l d and g e o p o t e n t i a l a t 500mb. P r o c . I n t Symp. on V a r i a t i o n a l Meth. i n G e o s c i e n c e s . Norman, OK, O c t . 15-17, 1985 pp. 13-15. L e w i s , J . M . , 1982: The o p e r a t i o n a l u p p e r - a i r a n a l y s i s using the v a r i a t i o n a l method. T e l l u s 24,514-530. L e w i s , J . M. and Bloom, S . C , 1978: I n c o r p o r a t i o n of t i m e - c o n t i n u i t y i n t o s u b s y n o p t i c a n a l y s i s by u s i n g dynamic c o n s t r a i n t s , T e l l u s 30, 496-515. L e w i s , J . M. and G r a y s o n , T. H . , 1972: The adjustment of s u r f a c e wind and p r e s s u r e by S a s a k i ' s V a r i a t i o n a l Matching Technique, J . A p p l . M e t e o r . , 1 1 , 586-597. L e w i s , J . M. and D e r b e r , J . C , 1985: The use of a d j o i n t e q u a t i o n s t o s o l v e a v a r i a t i o n a l adjustment problemwith a d v e c t i v e c o n s t r a i n t s . T e l l u s 37A, Vol . 4 ,\n\n33 pp. 309-322. 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M . , 1981: Implementation of a p o s t e r i o r i methods f o r e n f o r c i n g c o n ­ s e r v a t i o n of p o t e n t i a l enstrophy and mass i n d i s c r e t i z e d s h a l l o w - w a t e r e q u a t i o n models. Mon. Wea. R e v . , V o l . 109, 946-958. Navon, I . M . , 1982a: A p o s t e r i o r i numerical techniques f o r e n f o r c i n g s i m u l t a nious c o n s e r v a t i o n of i n t e g r a l i n v a r i a n t s upon f i n i t e - d i f f e r e n c e s h a l l o w water e q u a t i o n s models. Notes on Numerical F l u i d Dynamics, V o l . 5. H e n r i - V i v i a n d E d . V i e w e g . pp. 230-240. Navon, I . M . , 1982b: A Numerov-Galerkin technique a p p l i e d t o a f i n i t e - e l e m e n t s h a l l o w - w a t e r e q u a t i o n s model wi t h e x a c t c o n s e r v a t i o n o f i n t e g r a l i n v a r i a n t s i n in \"Finite-Element Flow.\" ( T . Kawai E d ) . U n i v . of Tokyo P r e s s pp. 75-86. Navon, I . 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T e s t u d , J . , and Chong, M. 1983: Three-dimensional wind f i e l d a n a l y s i s from d u a l - d o p p l e r radar d a t a . Part I : F i l t e r i n g i n t e r p o l a t i n g and d i f f e r e n ­ t i a t i n g the raw-data. J . C l i m a t e A p p l . Meteor. 22, 1204-1215. Thompson, P . D . , 1969: Reduction of a n a l y s i s e r r o r through c o n s t r a i n t s of dyna­ mical c o n s i s t e n c y . J . A p p l . Meteor. 8, 738-742. T r i b b i a , J . J . , 1982: On v a r i a t i o n a l normal mode i n i t i a l i z a t i o n . Mon. Wea. R e v . , 110, 6, 455-470. Wahba, G. and J . Wendelberger, 1980: Some new mathematical methods f o r v a r i a t i o n a l o b j e c t i v e a n a l y s i s using s p l i n e s and c r o s s - v a l i d a t i o n . Mon. Wea. R e v . , 108, 1122-1143. Wahba, G . , 1981: Some new t e c h n i q u e s f o r v a r i a t i o n a l o b j e c t i v e a n a l y s i s on t h e sphere using s p l i n e s . H o u g h - f u n c t i o n s , and sample s p e c t r a l d a t a . P r o c . 7th Conf. on P r o b a b i l i t y and S t a t i s t i c s i n t h e Atmos. S c i . Monterey C a l . Nov. 26, 1981, 213-216. Wahba, G . , 1982: V a r i a t i o n a l methods i n simultaneous optimum i n t e r p o l a t i o n and initialization i n \"The I n t e r a c t i o n between O b j e c t i v e A n a l y s i s and I n i t i a l i z a t i o n \" . P r o c . S t a n s t e a d Seminar. P u b l i c a t i o n i n M e t e o r o l o g y , No. 127, Dept. of Meteorology, M c G i l l U n i v . , M o n t r e a l , 178-185.\n\nUSE OF ADJOINT EQUATIONS FOR ASSIMILATION OF METEOROLOGICAL OBSERVATIONS BY BAROTROPIC MODELS Ph. Courtier, 0.\n\n1.\n\nDynamique, P a r i s , F r a n c e\n\nIntroduction The g e n e r a l\n\nformalism\n\npaper by 0 . T a l a g r a n d to\n\nM6t6orologie N a t i o n a l e , P a r i s , F r a n c e\n\nT a l a g r a n d , L a b o r a t o i r e de M e t £ o r o l o g i e\n\na s T86\n\nin\n\nthe\n\nof\n\ne q u a t i o n s i s presented i n\n\n(elsewhere i n these proceedings)\n\nfollowing.\n\nand i s\n\nthe\n\nrefered\n\nThe p r e s e n t one d e s c r i b e s two s e r i e s o f\n\nd a t a - a s s i m i l a t i o n experiments u s i n g two b a r o t r o p i c models based on the vorticity\n\nequation\n\nand on t h e\n\nshallow-water equations r e s p e c t i v e l y .\n\nF o l l o w i n g the g e n e r a l approach d e s c r i b e d i n T86, a r e used i n order to compute t h e g r a d i e n t conditions of\n\na scalar\n\nfunctional\n\nbetween t h e t r a j e c t o r y\n\nalgorithm\n\nminimize the\n\nis\n\nthen\n\ni n order to determine the i n i t i a l\n\nintroduced\n\nequation\n\nequation sphere r e a d s :\n\n- J(C+f,A\"io\n\n(2.1)\n\nwhere c and f a r e the r e l a t i v e\n\n(2.1),\n\nand b a s i c v o r t i c i t i e s ,\n\nJ and ^-l t h e\n\nand i n v e r s e l a p l a c i a n o p e r a t o r s . For a g i v e n s o l u t i o n\n\nthe tangent l i n e a r e q u a t i o n i s\n\n= J(6c,A\" C) 1\n\nC of\n\n:\n\n+ J(c+f . A - ^ e )\n\n(2.2)\n\nand using the k i n e t i c energy i n order to d e f i n e a s c a l a r p r o d u c t , have the a d j o i n t ^\n\nof equation (2.2)\n\n= AJU-VcA-ic) The\n\na\n\nc o n d i t i o n s which\n\nThe v o r t i c i t y e q u a t i o n at the s u r f a c e of a r o t a t i n g\n\njacobian\n\nin\n\nin\n\nfunctional.\n\n2.1 the v o r t i c i t y\n\n^\n\ninitial\n\ndiscrepancy\n\no f the model and the o b s e r v a t i o n s a v a i l a b l e\n\n2 . Experiments u s i n g the v o r t i c i t y\n\nff\n\nequations\n\nw i t h r e s p e c t to the\n\nwhich measures the\n\nthe p e r i o d o f time c o n s i d e r e d . The g r a d i e n t descent\n\nthe a d j o i n t\n\nexperiments\n\np s e u d o - s p e c t r a l model\n\n( s e e T86 f o r more d e t a i l s )\n\n:\n\n+ JU+f.A-^c) presented of\n\nthe\n\ns p h e r i c a l harmonics a t t r i a n g u l a r used h a s enough r e s o l u t i o n\n\nto\n\n(2.3)\n\nbelow\n\nvorticity\n\nare\n\nperformed\n\nequation (2.1)\n\nwith\n\nbuilt\n\nthe\n\naliasing\n\nerrors\n\nq u a d r a t i c terms o f ( 2 . 1 ) . Under t h i s c o n d i t i o n , we have o n l y to\n\na\n\non t h e\n\nt r u n c a t i o n 2 1 . The c o l l o c a t i o n prevent\n\nwe\n\nin\n\ngrid the\n\n36 replace\n\nin (2.3)\n\nused i n ( 2 . 1 )\n\nt h e o p e r a t o r s i n v o l v e d by the same d i s c r e t i z e d\n\nto obtained\n\nof the d i r e c t\n\nform\n\nmodel, ( s e e\n\nP . C o u r t i e r , 1985).\n\n2.2 The o b s e r v a t i o n s The s e t o f o b s e r v a t i o n s c o n s i s t s o f a l l the 500 mb o b s e r v a t i o n s of wind and g e o p o t e n t i a l a v a i l a b l e on t h e 24-hour p e r i o d s t a r t i n g 0:00Z,26\n\napril\n\n1984.\n\nThe g e o g r a p h i c a l d i s t r i b u t i o n\n\no b s e r v a t i o n of wind (Nh=1653 f o r g e o p o t e n t i a l ) 1.\n\no f the Nv=1913\n\ni s p r e s e n t e d on\n\nfigure\n\nThe main t o p i c i s t h a t the d i s t r i b u t i o n i s extremely i r r e g u l a r ,\n\nparticular\n\nthe A l e u t i a n\n\nlow c l e a r l y\n\nvisible\n\nat\n\nin the\n\nin\n\ngeopotential\n\na n a l y s i s p r o d u c e d by the o p e r a t i o n a l scheme at DMN, P a r i s f o r 0:00Z, 26 a p r i l 1986 was t o t a l l y v o i d of o b s e r v a t i o n s ( f i g u r e\n\n2.3 Numerical\n\n2).\n\nimplementation\n\nThe d i s t a n c e f u n c t i o n J used i n order to measure t h e d i s c r e p a n c y between a model s o l u t i o n and the o b s e r v a t i o n s i s d e f i n e d as : J = a.Jh + Jv\n\n(2.4)\n\nwhere J v and J h a r e t h e c o n t r i b u t i o n s o f t h e wind and g e o p o t e n t i a l o b s e r v a t i o n s r e s p e c t i v e l y . They a r e d e f i n e d as the sum o f the squares of the d i f f e r e n c e between t h e o b s e r v e d v a l u e s and t h e model v a l u e s which a r e o b t a i n e d\n\nfrom t h e s p e c t r a l components o f t h e\n\nvorticity\n\nthrough a b a l a n c e e q u a t i o n f o r g e o p o t e n t i a l . The\n\nintegration\n\nof\n\nthe\n\no b s e r v a t i o n time t h e e x p l i c i t respect\n\nequation\n\nrequires\n\nd e t e r m i n a t i o n o f the g r a d i e n t\n\nt o t h e s p e c t r a l components o f t h e v o r t i c i t y\n\nI n s t e a d of t r y i n g gradients,\n\nto\n\nfind\n\nat\n\neach\n\nof J with\n\nf i e l d ( s e e T86).\n\na n a l y t i c a l expressions for the\n\nrequired\n\nt h e y h a v e been computed by f o l l o w i n g t h e g e n e r a l approach\n\nd e s c r i b e d i n T86. T h i s r e q u i r e d t o t a k e t h e a d j o i n t s o f the p r o c e s s e s which s t a r t lead\n\nto\n\nfrom t h e s p e c t r a l components o f the v o r t i c i t y f i e l d and\n\nthe values of\n\nwind\n\nand g e o p o t e n t i a l\n\ncompared\n\nwith\n\nthe\n\nobservations.\n\n2.4 Numerical r e s u l t s The r e s u l t s p r e s e n t e d below a r e t h e most c h a r a c t e r i s t i c o f those obtained i n a s e r i e s o f e x p e r i m e n t s . F i g u r e 3 shows t h e h e i g h t\n\nfield\n\nproduced by t h e m i n i m i z a t i o n p r o c e s s using a l l a v a i l a b l e o b s e r v a t i o n s . The v a l u e o f t h e c o e f f i c i e n t a o f e q u a t i o n ( 2 . 4 ) was .03 m^s\"^ and the\n\nminimization\n\nhas been s t a r t e d\n\nComparison w i t h f i g u r e 2 are\n\nfrom an a t m o s p h e r e\n\nshows t h a t\n\nr e c o n s t r u c t e d by t h e m i n i m i z a t i o n\n\nprocess.\n\nremarkable t h a t t h e A l e u t i a n low i s p r e s e n t d e p r e s s i o n was absent from t h e i n i t i a l it\n\nin\n\nIt\n\nis\n\nfigure\n\nand\n\nminimization\n\nthrough\n\nheight\n\nnon-linear\n\nSince\n\nJ was 2 9 . 1\n\ndifference\n\nm and 8 . 0\n\nand w i n d v e c t o r ,\n\ndescent\n\nthat\n\nbalance\n\nequation\n\nvorticity that\n\nthe\n\na d e p r e s s i o n has\n\narea.\n\nroot-mean-square\n\nminimum o f\n\nflow\n\ns t a t e , of the descent p r o c e s s ,\n\np r o c e s s has been a b l e t o \"deduce\" t h a t\n\nto be present i n t h a t The\n\nthe\n\nthe\n\nparticularly\n\n3.\n\ni s n e c e s s a r i l y through the time c o n t i n u i t y imposed by the\n\nequation\n\nthe\n\na t r e s t {r, = 0 ) .\n\na l l major s t r u c t u r e s o f\n\nagainst\n\nprocess.\n\nTheses\n\ncorresponding\n\nms\"^ p e r\n\nto\n\nthe\n\nfinal\n\ni n d i v i d u a l o b s e r v a t i o n of\n\n185 m and 17.6 ms\"^ a t the s t a r t\n\nvalues\n\nalthough\n\nlarger\n\nthan\n\nof the\n\no p e r a t i o n a l ones a r e m e t e o r o g i c a l l y a c c e p t a b l e . One p r o c e s s through which i n f o r m a t i o n\n\nis\n\npropagated\n\na r e a d j u s t e d to o b s e r v a t i o n s i s a d v e c t i o n by the f l o w .\n\nand\n\nfields\n\nS i n c e one model\n\ne v o l u t i o n i s g l o b a l l y a d j u s t e d to the o b s e r v a t i o n s , t h e r e i s n o t downstream a d v e c t i o n\n\nin\n\nthe\n\nf u t u r e but a l s o upstream a d v e c t i o n i n t o\n\nthe p a s t . T h i s i s c l e a r l y v i s i b l e on f i g u r e 4 w h i c h shows t h e field\n\nproduced\n\nwith\n\nno\n\nhemisphere. The h e i g h t\n\nonly\n\nobservation\n\nin\n\nthe\n\nwestern\n\npart\n\nf i e l d i s reconstructed s a t i s f a c t o r i l y\n\nheight of\n\nthe\n\nupstream\n\nto the observed area up to l o n g i t u d e 15 W.\n\n3 . Experiments using the s h a l l o w - w a t e r e q u a t i o n s .\n\n3.1 The s h a l l o w - w a t e r Written geopotential\n\nin\n\nterms\n\nequations\n\nof\n\nvorticity\n\n«\n\nJ(c+f,A- C)\n\n\"\n\nV.(\n\n(C+OVA* !! )\n\n|3 at\n\n-\n\nJ(C+f,A n)\n\n+\n\nV.(\n\n(c+f)7A C )\n\nX\n\n- 1\n\nwith,K » we d e f i n e\n\njCVA'^.VA\" ; 1\n\na scalar\n\nn\n\n,\n\nand\n\n1\n\n- 1\n\n- A•J 6 r 1\n\n6\n\n=\n\nJ\n\ns\n\nThe Kronecker delta functions 6 elsewhere.\n\n1\n\n,\n\n0\n\nn\n\nl\n\n6 ^, equal 1 where r=i or s=j and are zero\n\nCarrying out the operations specified by (3) leaves 11 Euler-Lagrange equations some of which are complicated nonlinear partial differential equations. An iterative method is proposed for the solution so that at the first cycle level, the higher order terms are expressed with observed variables and are expressed by previously adjusted variables at subsequent cycles. At any particular solution cycle, these terms and the terms that are determined by observed variables are specified and can be treated as forcing functions. Following this approach, the Euler-Lagrange equations transform into a set of eleven simple algebraic or linear partial differential equations. Through reduction of variables, the number of equations is reduced to two diagnostic equations in geopotential and a velocity adjustment potential. These elliptic second order partial differential equations are easily solved by standard methods•\n\n51 SOME RESULTS OF THE ASSIMILATION The case study used for the test of the assimilation was a short wave over the Central Plains on 1200 GMT 10 April 1979. Shown in an objective analysis of the 500 mb heights (Fig. 1 ) , this disturbance was accompained by light precipitation (shaded patches) at 1235 GMT mostly from relatively shallow convective elements embedded within middle tropospheric clouds (6 k m ) . The model domain extends from a smoothed surface to 100 mb. Variables are located on a 100 km by 100 km horizontal staggered grid. We defined the precision modulus weights so that the heights, winds, and temperatures are approximately equally weighted. Other variables received smaller weights.\n\nFig. 1. The 500 mb height field at 1200 GMT, 10 April 1979 showing a weak short wave disturbance over the Central Plains.\n\nBecause this assimilation is not an initialization for a numerical prediction model, the often used procedure of determining the best initial analysis by finding the best forecast does not apply. We instead use three diagnostic criteria which, although they may be somewhat more subjective than measures of forecast skill, have found use in the verification of diagnostic analyses (Krishnamurti, 1968; Achtemeier, 1975; Otto-Bliesner et al, 1977). These criteria are measures of a) the extent to which the assimilated fields satisfy the dynamical constraints, b) the extent to which the assimilated fields depart from the observations, and c) the extent to which the assimilated fields are realistic as determined by pattern recognition. The last criterion requires that the signs, magnitudes, and patterns of the hypersensitive vertical velocity and local tendencies of the horizontal velocity components be physically consistent with respect to the larger scale weather systems. Adjusted variables at two successive cycles were averaged and reintroduced into the dynamic constraints. Residuals were computed as remainders of algebraic sums of individual terms of each constraint. The RMS error (Glahn and Lowry, 1972) for each level was then found. Residuals vanish (constraint satisfaction) when variables at two successive cycles are unchanged. A measure of the convergence of the variational method to constraint satisfaction is the difference between the initial RMS error of the residuals of the unadjusted variables substituted directly into the dynamic equations and the RMS values at each cycle. These differences are divided by the initial RMS errors, converted to percent and expressed in Table 1 as percent reduction of the initial RMS error.\n\n52 Table 1 Percent NOSAT RMS error reduction with respect to initial RMS residuals for the u- and v-horizontal momentum equations, the integrated continuity equation, and the hydrostatic equation after eight cycles through the solution sequence.\n\nEQUATION u-component v-component continuity hydrostatic\n\n2\n\n3\n\n4\n\n92 90 70 98\n\n94 93 68 98\n\n92 90 87 100\n\nLEVEL 5 92 90 90 100\n\n6\n\n7\n\n8\n\n9\n\n93 90 92 100\n\n93 90 93 100\n\n92 89 92 100\n\n90 86 91 100\n\nTable 1 shows how the reductions of the initial RMS error for the two horizontal momentum equations varies for the eight adjustable levels of the model. The solution stabilizes near 9 0 - 9 5 percent error reduction. The errors for the integrated continuity equation are reduced approximately by 70 percent at levels 2 and 3 and by approximately 90 percent at the upper levels. These improvements are, of course, dependent upon the magnitudes of the initial RMS errors. We first calculated the vertical velocity by the O'Brien ( 1 9 7 0 ) method and then determined the RMS errors for the integrated continuity equation. Had we assumed that the initial vertical velocity was zero, the initial RMS errors would have been much larger than the values used in Table 1 and the error reductions would have been 100 percent by the fourth cycle. The RMS errors for the hydrostatic equation are halved at each cycle and the percent error reduction increases monotonically to near 100 percent by the eighth cycle. The variational assimilation produced significant adjustments in height, temperature, and wind velocity in order that the values of these variables are solutions of the dynamic constraints. However, these modifications can cause large and physically unrealistic changes in other important variables such as vorticity, divergence, and vertical velocity and other quantities that involve derivatives of the basic variables. In addition, the local tendencies of the horizontal velocity components are sensitive to small errors in the basic variables when they are determined from the arithmetic sum of the other terms of the horizontal momentum equations. The patterns of these hypersensitive variables must be physically realistic when compared with other data sets such as cloud fields, precipitation, and independent measurements of the variable itself. Thus, the hypersensitive variables provide a critical test of the accuracy of the variational assimilation. The tendencies that are products of the assimilation are compared with the observed 3-hr tendencies of u and v calculated from the high frequency rawinsonde data collected over the central part of the U. S. as part of the NASA-AVE SESAME project and with the 3-hr tendencies calculated with values from the initial gridded fields substituted in place of the assimilated fields in the horizontal momentum equations. In making these comparisons, we assume that the observed 3-hr tendencies represent \"ground truth\" subject to the following qualifications. First, in keeping with the synoptic scale of the analysis, we have gridded only 3-hr tendencies taken from data collected at standard NWS observing sites. Second, the ground truth tendencies are calculated over the 3-hr interval from 1 2 0 0 - 1 5 0 0 GMT and are therefore centered at 1330 GMT. The tendencies found from the assimilation are centered at 1200 GMT. Therefore, some phase shift should be observed between the patterns. Third, to the extent that the tendencies calculated from the SESAME data suffer from mesoscale \"noise\", the patterns will not accurately represent the true pattern of synoptic scale tendencies.\n\n53 Strongest jet stream winds were located at level 8 (300 m b ) . Large magnitudes and gradients of the velocity can combine to create large tendencies if the terms of the horizontal momentum equations do not compensate. The 3-h tendency field obtained from the initial data (Fig. 2a) is a pattern of large magnitude centers of alternating sign spaced at approximately the average observation separation. These centers imply unrealistically large changes in v over three hours. With allowances for horizontal displacement of the pattern over 1.5 h, the only correspondence with the observed 3-h tendency field is the sign of the pattern along the eastern part of the domain. The SAT analysis (Fig. 2c) reproduces most of the features of the observed 3-h tendencies. The positive tendency center near the lower boundary of the grid (Texas-New Mexico border) in the SAT analysis appears over the Texas panhandle at 1330 GMT in the observed tendencies (Fig. 2 b ) . Furthermore, the relative minimum over Oklahoma is moved into southeastern Kansas. These displacements are in accord with the rapid northeastward movement of the weather systems through the area. Relative horizontal displacements were smaller within the weaker flow near the long wave ridge over the eastern part of the domain. Here the SAT analysis preserved the area of larger positive v-tendencies but located the maximum over Illinois rather than over Mississippi as found in the observed tendencies.\n\n54 DISCUSSION Now, does this variational assimilation method produce better hybrid data fields than other methods? Since intercomparison studies have not yet been performed, we cannot offer definitive answers to the question. However, we believe that the variational model should provide quality analyses if the following two criteria are satisfied. First, the variational assimilation method we have developed is a physical model. Four of the basic primitive equations that govern flow in free atmosphere subject to the assumptions that apply to hydrostatic and synoptic conditions have been used in the model derivation. Since the real atmosphere obeys these equations, it is expected that the three dimensional fields of meteorological variables should be reasonable approximations to the true atmosphere if they are solutions of the dynamic equations. Furthermore, advanced versions of this model that include the energy equation as a fifth constraint should provide analyses that are superior to the results presented here. Second, the dynamical equations permit many solutions. Therefore, the error characteristics of the observations and the horizontal distributions of the the precision moduli should be known with accuracy. The sensitivity of the variational model to the values given to these weights is currently not fully known and is the subject of investigation in the ongoing model development. Finally, we note from the results of the pattern recognition that the variational analysis produced physically reasonable fields of the hypersensitive tendency fields. These are the first relatively accurate diagnostic fields of local tendencies of the velocity components apart from initialization schemes for numerical prediction models. Our continued model developments should improve upon these results. ACKNOWLEDGEMENTS This research was supported by the National Administration (NASA) under contract NAS8-34902.\n\nAeronautics\n\nand\n\nSpace\n\nREFERENCES Achtemeier, G. L., 1975: On the Initialization problem: A variational adjustment method. Mon. Wea. Rev.. 103. 1090-1103. Glahn, H. R., and D, A. Lowry, 1972: The use of model output statistics (MOS) in objective weather forecasting. J. Appl. Meteor., 11.. 1203-1211. Krishnamurti, T. N., 1968: A diagnostic balance model for studies of weather systems of low and high latitudes. Rossby number less than one. Mon. Wea. Rev.. 96. 197-207. O'Brien, J.J., 1970: Alternative solutions to the classical vertical velocity problem. J. Appl. Meteor., 9, 197-203. Otto-Bliesner, B., D. P. Baumhefner, T. W. Schlatter, and R. Bleck, 1977: A comparison of several data analysis schemes over a data-rich region. Mon. Wea. Rev.. 105. 1083-1091. Sasaki, Y., 1958: An objective analysis based upon the variational method. J. Meteor. Soc. Japan. 36. 77-88. ^, 1970: Some basic formalisms in numerical variational analysis. Mon. Wea. Rev.. 98. 87 5-883. Whittaker, E., and G. Robinson, 1926: The Calculus of Observations (2nd Edition). London, Blackie and Son, LTD., pl76.\n\n55\n\nTHE VARIATIONAL INVERSE METHOD FOR THE GENERAL CIRCULATION IN THE OCEAN Christine PROVOST L . P . C M . , CNRS, Universite P. et Marie CURIE, Tour 24-25, 4 place Jussieu, 75230 Paris Cedex 05 (France). ABSTRACT Provost, C , 1986. International Symposium on Variational Methods in Geosciences. Norman, Oklahoma USA, Octobre 15-17, 1985. The variational inverse method is a three dimensional global optimisation for estimating the three dimensional f i e l d of geostrophic velocity from hydrographic data. Very simply, we ask for the smoothest velocity f i e l d ( i n the sense of an a r b i t r a r i l y defined norm) which is consistent with the data and with selected approximate dynamical constraints to within prescribed m i s f i t s , which, we w i l l argue, should never be zero. The misfits represent errors in the data and in the approximate dynamical constraints. By varying the misfits r e l a t i v e l y to one another, we explore the f u l l envelope of physically estima­ tes of the average geostrophic flow. Several applicatins are introduced of the method. >s\n\nn\n\nINTRODUCTION The variational inverse method addresses the c l a s s i c a l problem of estima­ ting the large-scale time-averaged circulation from hydrographic measurements of temperature, s a l i n i t y , pressure (hence density) and possibly various geochemical t r a c e r s . The density f i e l d determines only the v e r t i c a l shear of the horizontal geostrophic velocity through the well known thermal wind equation, and therefore the absolute velocity is undetermined by a constant of v e r t i c a l integration. This integration constant generally varies from one location to another. C l a s s i c a l l y , the indeterminacy i s removed by assuming that the geostropic velocity vanishes at some great and usually constant depth. However, direct measurements revealing large v e l o c i t i e s at depths have stimulated a search for new methods. The c l a s s i c a l assumption of a level-of-no-motion has been replaced by more sophisticated assumptions which incorporate more of the physics ; for example Stommel and Schott's beta spiral (1977), Wunsch's inverse method (1977). Our method also uses more of the physics. I t s advantages come from i t s more systematic and more general approach. Very simply, the variational inverse method seeks the s p a t i a l l y smoothest flow f i e l d ( i n the sense of an a r b i t r a r i l y defined norm) which is consistent with the data and dynamical constraints to within prescribed m i s f i t s , which, we w i l l argue, should never be zero. The misfits represent errors u\n\nl[\n\nin the data and dynamics\n\n(in the sense explained below) and can be estimated from a scaling a n a l y s i s . By varying the misfits r e l a t i v e l y to one another, we can explore the f u l l\n\n56\n\nenvelope of physically plausible estimates of the large-scale, time-averaged flow. Similar methods have been used by (for example) Wahba and Wendelberger (1980) for the interpolation of meteorological data ; by Bennett and Mcintosh (1982) to incorporate observations into t i d a l models with open boundaries and by Shure et a l . (1982) to estimate the magnetic f i e l d inside the Earth. An obvious advantage of the variational inverse method is t h a t , as a f u l l y three dimensional optimisation,\n\ni t can deal with data distributed at any locations\n\nand not necessarily aligned along sections. Indeed, the spatial distribution of h i s t o r i c a l hydrographic data is very inhomogeneous and somewhat chaotic. I . Methodology. Variational Inverse Method. We want to estimate the average velocity f i e l d . This flow f i e l d must sa­ t i s f y certain requirements ; i t should be smooth, i t should be consistent with the data, and i t should conform selected plausible dynamics. Variational c a l ­ culus offers a simple and f l e x i b l e method for obtaining such flow f i e l d s . In a general way, we seek a flow function ^(x_) which minimizes the functional\n\nfff*\\M\n\nJ[M =\n\n+ v\n\nD\n\nD [*] + ? y C j M d x\n\nJfrp] (1)\n\nc\n\nwhere R [ ] is a smoothness operator for ip, D [ W a data agreement operator, C j [ W dynamical constraints operators. The integration runs over the domain of the f i e l d x. in three dimensions. The y and y c . are Lagrange multipliers which D\n\ncorrespond to weights on the constraints. The function possesses the three required qualities : i t\n\nwhich minimizes J | > ]\n\nis smooth (according to the norm\n\nR ) , i t is consistent with the data and i t conforms to the selected dynamics Cj [ i p ] . We now give a simple example of choice of functionals R, D, C to\n\nillus­\n\ntrate the method. * A ///\n\nsimple (and arbitrary) measure of roughness is for example\n\n( V \\|0 dx 2\n\n2\n\nwhere the smoothing operator is R|>] = V ip 2\n\n(2)\n\nOf course, other measures of roughness are possible ( c f . section 4 ) . * If (10c)\n\n(continuity)\n\n(10d)\n\n10b\n\n_ = J L\n\n(9) (10a)\n\na p.f cos 6 3\"X\n\n=\n\nv\n\n(definition of the flow function)\n\ng\n\nd A\n\nlow latitudes these equations need to be modified, Provost 1986). Thus,\n\nu, v, w and (respectively the East-West, South-North, v e r t i c a l v e l o c i t i e s and density) are a l l expressible in terms of ^ . The hydrographic data are essen­ t i a l l y aliased measurements of ^ , thus : d • = l\n\ndz\n\n(^i,6i,zi) + aliasing e r r o r\n\nv\n\n(11) '\n\nare also expressible in terms of i>. Therefore, we can easily use the formalism described e a r l i e r . We apply the constraint (10d) in both the v e r t i c a l l y integrated form, and also in the non-integrated form in combination with the density equation. The v e r t i c a l l y integrated form is : Po a2 2\n\nf\n\n2ft 2\n\n/\n\n*\n\nw, -=\n\nwhere\n\nd\n\nL\n\nT\n\nZ\n\n=\n\n_ . t curl\n\nW\n\nT \"\n\nW\n\n(12)\n\nB\n\nand and w w^ == -- Ug i u . VH\n\nT\n\nB\n\nPo\n\nT\n\nThe last two equations are the traditional\n\napproximations for the boundary\n\nconditions on the v e r t i c a l v e l o c i t y , xis the wind stress and u and H ( X , 8 )\n\nB\n\n= u(X,6,-H)\n\nis the depth of the ocean. To apply the constraint (10d) in non-\n\nintegrated form, we invoke an approximation to the potential density equation, namely : a cos6\n\n3X\n\n36\n\na\n\n3z\n\nw\n\n^7\n\nA discussion of the errors in the approximations involved in (12) can be found in Provost and Salmon (1986). The constraint obtained by eliminating w between (10d)\n\nand (13) should be considered as rather weak (the neglected eddy fluxes\n\nin. (8) can be l a r g e ) . The v e r t i c a l velocity at any point is : w (x, e, )\n\nwj (A,e,- )\n\nz\n\nB\n\nH\n\n+\n\n^-| yH\n\ndz\n\n2\n\n< ) 1 4\n\nWe can increase the number of independant constraints by considering t r a ­ cers other than density. I f for example, a is the average value of a conserved tracer which has been measured, then we can use : \" a cosG\n\n3a 3X\n\n+\n\nV a\n\n3a 36\n\nW\n\n3a 3z\n\n=\n\n(\n\n1\n\n5\n\n)\n\nas an additional constraint. The dominant errors in (15) results from eddy fluxes. I f a is non conservative, then e x p l i c i t source terms must be added to (15),\n\nor the sizes of the source terms must be taken into account in deciding\n\nthe m i s f i t s .\n\n59\n\nFollowing the procedure described in section 1 , we seek the smoothest f i e l d of • I M l lobs\n\nwhere\n\n||.|| is some norm adapted to the definition of the problem. It may in­\n\nclude some information on the statistical structure of the fields through Gauss' precision moduli. The analysis (Problem A) may be defined as: \"Determine U* minimizing J and verifying (])\". Therefore the analysis is a problem of optimization with equal­ ity constraint. The next step is to give a numerical solution to it. (Sasaki, 1970) has given two formalisms. The first one said to be the weak formalism con­ sists to minimize the functional J ( U ) = J(U) + a | | A ( U ) | |\n\n(2)\n\n2\n\nW\n\nThe second term defining\n\nis used as a forcing term for verifying the\n\nconstraint. The optimal field is obtained by solving the equation giving the optimality condition\n\n72 VJ (U) = VJ(U) + 2 a B(U).A(U) = 0 w\n\n(3)\n\nTT\n\nwhere B(U) is the formal Jacobian of the operator A with respect to the state U. Many difficulties arise for solving ( 3 ) especially for limited area problems when the operator B gives some non standard boundary conditions. A common hypo­ thesis is to assume that the\n\nobservations are unbiased on the boundary if A is\n\nsome elliptical operator (a Laplacian for instance) this approximation is un­ fair because it leads to a solution depending only on the boundary terms and independent of the observations inside the domain. Another question is the estimation of the value of the parameter a and of the sensitivity of the opti­ mal solution to the chosen value, the penalty algorithm consists to solve a sequence of value\n\nproblems\n\nof a the\n\nwith a going\n\nto\n\nzero.\n\nUnfortunately\n\nfor\n\na\n\nsmall -\n\nproblem can become numerically ill conditioned.\n\nIf constraint ( 3 ) has to be solved with a good accuracy then the optimality condition is obtained by introducing the Lagrangian o£(U,A)\n\n(4)\n\n= J(U) + (A,A(U))\n\nand solving the Euler-Lagrange optimality conditions\n\n|f |f\n\n(U,A) = 0\n\n(5-a)\n\n(U,A) = 0\n\n(5-b)\n\nSystem ( 5 ) belongs to an infinite dimensional space i.e., it includes some boundary terms, a standard method consists to eliminate U between ( 5 - a ) and ( 5 - b ) to get an equation depending of A only, if A is nonlinear the solution may be very sensitive to the boundary term. Furthermore, an iterative solution of the Euler Lagrange equation needs a first guess of the Lagrange multipliers in the vicinity of the optimal solution and no physical estimation may be used to get it. Following\n\nSasaki (1970) this approach is labeled as the strong formalism.\n\nAUGMENTED LAGRANGIAN METHOD An alternative way using both duality and penalty methods (strong and weak formalism) is made o£ (U,A,a) A\n\nwith the augmented Lagrangian\n\n= J(U) + (A,A(U)) + |\n\n||A(U)||\n\n2\n\n(6)\n\nThe second term of the right hand side of ( 6 ) is a duality term the third a penalty one. The optimal solution (with some convexity assumptions) is a saddle point of the augmented Lagrangian i.e., it satisfies the following:\n\nc2? (u*,A,a) A\n\n=\n\n(u,v,z) . T\n\n(1)\n\nThe global baroclinic primitive equations are written symbolically in the vector form dW/dt\n\n= L(W).\n\nThe dependent variable vector W is now expressed as\n\n(2)\n\n79\n\nW(A,^p,0 = En E aE Er^8r ' ( a . » : * )\n\nH*(A,*|a,n)\n\nG„(p),\n\n(3)\n\nwhere Wf denotes the expansion coefficients, HJ represents the Hough harmonics, and the vertical normal mode functions. The summation of the three-dimensional nor­\n\nG\n\nn\n\nmal mode functions is made with respect to zonal wavenumber s, meridional modal in­ dex r, species of the normal modes a and vertical mode index n. As the species of nor­ mal modes for zonal w a v e n u m b e r s > 1, two kinds exist: high-frequency eastward and westward propagating inertial-gravity waves and low-frequency westward propagating rotational waves of the Rossby-Haurwitz type. For 5 = 0, low-frequency westward propagating waves of the Rossby-Haurwitz type are replaced by an orthogonal set of geostrophic modes (Kasahara, 1978). The spectral prediction equations for the expansion coefficients Wf are derived from the variational constraint: - L)\n\n6 J(dW/dt v\n\n2\n\ndV = 0,\n\n(4)\n\nwhere V denotes the three-dimensional global domain and the variation 8 is taken with respect to the expansion coefficients Wf.\n\nThis yields a set of ordinary differential equa­\n\ntions with respect to time for Wf: d\n\nW\n\n'^\n\nn\n\n:\n\nt\n\n)\n\n+ i„> (a,n)W > r\n\n= Nf{a,n : t)\n\nr\n\n(5)\n\nwhere\n\n2tt t) = ^ f\n\nrc/2 /\n\no\n\nN (A, n\n\n+ T\n\n(3)\n\nHere, Vtj) is the modifying divergent component, Y = - V x T is the modifying rotatio­ nal component. Boundary conditions are v. =0; this guarantees that the boundary separation\n\n(3) is unique. In fulfilling the first of the two conditions\n\n(2) we\n\nnow choose such that: V (j) 2\n\n+ 6 = 0\n\n(4)\n\nThis is the Poisson equation to be solved. In order to fulfill the second of the conditions (2) we consider:\n\n/ v d V = /(V4>) dV + J T d V 2\n\n2\n\n2\n\n(5)\n\nNote that the correlation term 2jV(})«rdV vanishes due to the boundary condi­ tions. The first term in (5) is fixed due to (4); thus (5) is minimum if and only if T=0.\n\nThis is equivalent to saying that the modifying field is irrota-\n\ntional.\n\nRESULTS Fig. 1 demonstrates the impact of the modification upon the wind field. We use a finite element grid (resolution 150 km in horizontal and 200 hPa in ver­ tical direction); the evaluation is actually not made with (u,v,u)) but with the mass flux vector across the elementary surfaces of the grid. Fig. 1 shows the divergence of this vector which is a particularly sensitive quantity. The left column shows the spurious 3D-noise divergence which is to be forced to zero by means of the modification algorithm. The center and right columns show the re-\n\n109\n\n3 D-AN A L Y S E D\n\n2 D-ANALYSED\n\n2 D-MOD I F I ED\n\nFig. 1 Horizontal patterns of 3 D - (left column) and 2D-mass flux divergence (center and right columns) for 5 pressure layers over ALPEX domain, in units lO**^s~V Left and center columns as analysed, right column after 3D-mass flux modifi­ cation with variational method. Convergent areas stippled, distance of isolines 4 units, date 5 March, 1 9 8 2 , 1 2 GMT.\n\n110 sponse of the 2 D - h o r i z o n t a l divergence with r e s p e c t t o the Comparison of the l e f t two columns i n Fig.\n\n1 reveals\n\nmodification.\n\ni s s m a l l e r than, and i n p a t t e r n f a i r l y independent upon, the cept f o r the l o w e s t l a y e r ) ; c a l v e l o c i t y component.\n\nt h i s demonstrates the s i g n i f i c a n t\n\nr o l e of the\n\nverti­\n\nComparison of the two r i g h t columns shows t h a t the modi­\n\nf i c a t i o n keeps the governing p a t t e r n of the s t r a t e s t h a t the m o d i f i c a t i o n logical\n\n3D-divergence 2D-divergence (ex­\n\nt h a t the\n\n2D-divergence\n\nunchanged;\n\nthis\n\ndemon­\n\nremoves the n o i s e b u t does n o t a l t e r the meteoro­\n\nsignal.\n\nFig.\n\n2 compares the modifying v e c t o r with the modified v e c t o r , i n a compo­\n\nn e n t - a v e r a g e d form, f o r 13 ALPEX-SOP d a t e s . The m o d i f i c a t i o n i s o f the o r d e r of 4%. This demonstrates the high q u a l i t y of contemporary s t a t i s t i c a l l y\n\nanalysed\n\n2000 1000\n\nModified 3 D - M a s s Flux Field\n\n500\n\nT\n\nf\n\n200\n\n«\n\n100\n\nat\n\nModification 20 h\n\no\n\n10 5\n\nApplied\n\n1\n\nAnalysed 3 D - R e s t Divergence 1/00 1/12 2/002/12 3/003/12 4/004/12 5/005/12 6/006/12 7/00 A L P E X - S O P (March,1982) - >\n\nFig. 2 3D-mass f l u x v e c t o r components, rms-averaged o v e r e n t i r e ALPEX-atmosphere (Fig. 1 ) , f o r 13 c o n s e c u t i v e d a t e s . Analysed f i e l d has small n o i s e d i v e r ­ gence (dashed c u r v e ; 1 0 k g / s of mass f l u x divergence corresponds t o 2 x l O ~ s \" of wind d i v e r g e n c e ) . This i s o b j e c t i v e l y removed by adding s m a l l modi­ f i c a t i o n (lower f u l l curve) t o y i e l d modified f i e l d (upper curve) which looks s i m i l a r t o a n a l y s e d f i e l d b u t i s e x a c t l y 3D-nondivergent. 7\n\n7\n\n1\n\nsynoptic f i e l d s\n\n(the a c t u a l r e s o l u t i o n of the a n a l y s e s used was 50 km i n h o r i ­\n\nz o n t a l and 50 hPa i n v e r t i c a l d i r e c t i o n - the d a t a were n u m e r i c a l l y i n t e g r a t e d o v e r the f i n i t e elements) . The impact of the mass f l u x m o d i f i c a t i o n\n\nupon the 3D-imbalance of the s e n s i ­\n\nb l e h e a t budget i s demonstrated i n the f o l l o w i n g\n\ntable:\n\nBefore mass-modif Leat Ion\n\nAf ter mass-modif Lcat Ion\n\n112\n\nrms-imb. of sensible heat budget 5 March, 1982,\n\n-24 GMT\n\n00-\n\nwithout mod.\n\n145 7 W / m\n\nwith mod.\n\n2\n\nZ\n\n49\n\nIt seems obvious from the table that the modification is material for sensible W/m heat budgets. The reduction by a factor of 20 in the sensible heat budget imba­ lance is typical for all cases investigated. As can be seen from Fig. 2 we have picked in Fig. 1 and in the table the case with the maximum modification of the record investigated. This dramatic improvement of heat budget accuracy through the moderate 4% modification of the mass flux is easy to understand in terms of the pertinent budget equation\n\n(Hantel and Emeis,\n\n1985).\n\nAnother example, from the FGGE data set as analysed by the European Centre in Reading, is shown in Fig. 3. zontal and 100\n\nThe resolution in this case was 267\n\nkm in hori­\n\nhPa in vertical direction. The rms-modifying field was 21x lO^kg/\n\ns equivalent to 2% of the observed/modified field.\n\nCONCLUSIONS This study should demonstrate the relevance of Sasaki's variational concept for synoptic budgets. It is understood that the modification can only be made if analysed estimates of both V (from an objective analysis) and a) (from the omega equation) are available; note that go carries valuable quasigeostrophic informa­ tion that is independent upon V. In this perspective the present approach can be considered the final solution to the classical vertical velocity problem (O'Bri­\n\nen, 1970). REFERENCES\n\nHantel, M. and S. Emeis, 1985: A diagnostic model for synoptic heat budgets. Arch. Met. Geoph. Biocl. , Ser. A Hantel, M. and S. Haase, 1983: Mass consistent heat budget of the zonal at­ mosphere. Bonner Meteorol. Abhandl., Heft 29, 84 pp. McGinley, J.A., 1984: Meteorological analysis using the calculus of varia­ tions (variational analysis). Riv. di Meteorologia Aeronautica, O'Brien, J.J., 1970: Alternative solutions to the classical vertical veloci­ ty problem. J. Appl. Meteorol., Sasaki, Y., 1958: An objective analysis based on the variational method. J. Met. Soc. Japan,\n\n33^, 407-420.\n\n36_, 77-88.\n\n44, 37-44.\n\n9_, 197-203.\n\nFig.\n\n600-800\n\n3 (opposite page) Horizontal patterns of 2D-mass flux divergence in layer hPa over the Equatorial Atlantic and Central Africa, in units Isolines every S units, solid=positive, dashed=negative, thick=zero. Date 2 January, 1979.\n\n10~7 -1.\n\n15\n\n113\n\nA FOUR-DIMENSIONAL ANALYSIS* ROSS N. HOFFMAN\n\nAtmospheric and Environmental Research, I n c . , Cambridge, MA 02139\n\nABSTRACT For a discretized deterministic model of the atmosphere, a single point in the model's phase space defines a complete t r a j e c t o r y . I t is possible to choose a point which minimizes the differences between the model trajectory starting at the chosen point and a l l data observed during an analysis period (-T < t < 0 ) . In this way data and model dynamics are combined to y i e l d a four-dimensional analysis exactly satisfying the model equations. This analy­ sis is the solution of the model's equations of motion defined by the optimal i n i t i a l conditions chosen at t = -T. Preliminary tests using a Gauss-Newton optimization method and simple spectral nonlinear models of the atmosphere demonstrate that the 4D analysis errors are much smaller than the measurement errors, the method is stable within an assimilation, and that observations of temperature alone are sufficient to maintain an accurate estimate of the veloc­ ity f i e l d . However, in these tests forecasts based on the 4D analyses are better than ordinary forecasts made from the observations at the end of the analysis interval (t = 0) only for the f i r s t 24 h. Beyond 24 h, both type of forecasts have the same s k i l l . INTRODUCTION A large part of our a priori knowledge of the atmosphere i s summarized by the equations governing the atmosphere's dynamics.\n\nThus an alternative to the\n\nusual analysis approach is to constrain the analysis to satisfy or nearly satisfy the governing equations using variational analysis methods (Sasaki, 1970).\n\nIn this study, Sasaki's variational analysis method i s examined for the\n\nspecial case when the model governing equations are used as strong exact) constraints.\n\nThe only restriction\n\n(i.e.,\n\non the type of observations which may\n\nbe used i s that i t must be possible to calculate a unique estimate of each ob­ servation from knowledge of the model evolution.\n\nIn this manner, the 4D analy­\n\nsis problem may encompass inverse problems associated with remotely sensed measurements.\n\nThis method combines data and dynamics, should eliminate the\n\nneed for i n i t i a l i z a t i o n and avoids rapid adjustments at the start of the f o r e ­ cast.\n\nThe analyses obtained are t r u l y 4D and use the model dynamics to achieve\n\nbalance.\n\nIn p r i n c i p l e , these analyses are obtained as the solution of a d i f f i ­\n\ncult nonlinear optimization problem with nonlinear constraints. much simpler unconstrained but equivalent problem may be stated:\n\nHowever, a Find the i n i -\n\n*Supported by Air Force Geophysics Laboratory, Air Force Systems Command, Contract F19628-83-C-0027.\n\n114 t i a l conditions at the start of the analysis period, such that the correspond­ ing model evolution best f i t s the data. RESULTS Results of our preliminary experiments in which a primitive equation model simulated nature and a quasigeostrophic model was used for forecasting are sum­ marized in F i g . 1 . These experiments are described in detail by Hoffman (1986).\n\n2000\n\n1\n\n1\n\n1\n\n1\n\n1\n\n1\n\n1\n\n1\n\n1\n\n1\n\n1\n\n:/\n\n/\n\n/\n\n1500 /: / • / / / / / /\n\n//\n\n/:\n\n/ / / /\n\nA\n\n1000\n\nV\n\n/ /\n\n/ / // //\n\n<\n\n/\n\n/\n\n/\n\n/\n\n/\n\n~\n\n/\n\n/ / / / /\n\n//\n\n/\n\n/\n\n/\n\n/\n\n/\n\n/\n\n/\n\n/. // // / /\n\nODF >•;'/ /\n\n500 /\n\n/\n\n/\n\n/\n\n/\n\n\\ 4/ DAF v\n\n0 -48\n\n\\\n\nL_\n\n\\\n\n7 x\n\n/\n\n/\n\n/ / /\n\n/\n\n1\n\n-24\n\n1\n\n/\n\n0\n\n/\n\n/ PIF/ /\n\n1\n\n|\n\n|\n\n1 48\n\n24\n\n1\n\n1 72\n\n1\n\n96\n\nTime (h)\n\nFig. 1 . Ensemble rms global error i/2 f the analysis/forecast experi­ ments as functions of time for the ODF (dotted l i n e ) , PIF (solid l i n e ) , and 4DAF (dashed l i n e ) . The measurement error (crosses) has an expected value of 580 x 10\"6. o r\n\n115\n\nFigure 1 shows the evolution of global analysis/forecast error for t = -42 h to t = 96 h.\n\nThe error is nondimensionalized and scaled:\n\nthe observational error\n\nlevel shown corresponds to rms temperature errors of ~ 1 K and rms wind errors of ~ 2 m s \" .\n\nFor comparison, forecasts made from the observations at t = 0\n\n1\n\nand from error-free values of the variables from the nature run at t = 0 are also displayed.\n\nThese three evolutions are termed the 4D analysis forecast\n\n(4DAF), the ordinary dynamical forecast (ODF), and the perfect i n i t i a l condi­ tions forecast ( P I F ) , respectively.\n\nBeyond about 24 h, the ensemble rms fore­\n\ncast error grows roughly linearly with time at the same rate for a l l three forecasts.\n\nAt t = 0, the analysis error for the 4D analysis is less than half\n\nthe expected rms measurement error.\n\nHowever, the forecast error for the PIF\n\nand 4DAF are already growing roughly linearly at t = 0 at the same rate ob­ served at later times.\n\nFor the ODF, on the other hand, there is no i n i t i a l er­\n\nror growth r a t e , and i t takes 24 h for this growth rate to build up. DISCUSSION The growth of error, as seen in F i g . 1 , is due to model error and to the growth of i n i t i a l observational errors.\n\nThe model error in these experiments\n\nis not due simply to an incorrectly specified parameter, rather the PE model has more degrees of freedom than the QG model.\n\nIn this section, we w i l l try to\n\nexplain some of the features seen in F i g . 1 , especially the slope of the error growth curve at t = 0.\n\nOn the basis of the experiments performed, i t\n\npossible to conclude whether i t\n\nis not\n\nis model or observational error which is most\n\nresponsible for these features since, as we shall see, both error sources should have the same qualitative e f f e c t s .\n\nExperiments with no model error\n\nwould have helped to separate these e f f e c t s .\n\nHowever, we do know from e a r l i e r\n\nexperiments with these models (Hoffman and Kalnay, 1983) that the error growth rate due to observational errors only, which was obtained by comparing pairs of QG forecasts, is generally about 3/4 of the error growth rate seen in F i g . 1 . Therefore, both types of error are probably contributing to the observed error growth. F i r s t we consider the case of a perfect model.\n\nEven without model error\n\nthe dynamical properties of the model insure a concave shape to the error growth curve.\n\nThis case is i l l u s t r a t e d schematically in F i g . 2, which is drawn\n\nin a frame of reference which moves with and is perpendicular to the nature trajectory.\n\nIn this frame of reference the origin is always nature and the na­\n\nture trajectory is the axis directed out of the page.\n\nThe \"A\" or \"attracting\"\n\naxis is tangent to the attractor of the system and the \"D\" or \"decaying\" axis is perpendicular to the attractor. error.\n\nDistance from the origin is the forecast\n\nThe axes are straight perpendiculars near the origin and become wavy\n\nand less than perpendicular away from the origin; this distinction symbolizes\n\n116\n\nF i g . 2. Schematic model phase space. The phase space shown may be imagined to be three-dimensional and the cross section shown moves with and is perpen­ dicular to the trajectory representing nature. This last trajectory appears as the large dot in the center of the attracting manifold ( A ) . As shown by the light arrows the motion in the cross section rapidly collapses onto the a t t r a c tor and then diverges away from the central t r a j e c t o r y . The trajectories for a 4DAF and an ODF are marked at regular intervals (x) by large dots, and by the integer t / i . the fact that dynamics linearized about nature are sufficient to explain the flow in phase space near the origin but further away nonlinearities are important. When viewed in this framework, the evolution along a typical traces a curve resembling a right hyperbola.\n\ntrajectory\n\nThis is due to exponential com­\n\npression towards the attractor and exponential divergence along the attractor away from the nature trajectory. important. 4DAF.\n\nError growth slows as nonlinearities become\n\nTwo trajectories are shown in F i g . 2, one for an ODF and one for a\n\nI f the minimization method succeeds in finding a trajectory which stays\n\nclose to nature during the analysis i n t e r v a l , i t succeeds by choosing i n i t i a l conditions close to the D-axis.\n\nBy t = 0, this trajectory is close to the\n\nA-axis and rapid error growth must follow.\n\nThat i s , the 4DAF is really a very\n\ngood but older (by an amount of time T) forecast, and i t s forecast error is a l ­ ready growing rapidly by t = 0.\n\nOn the other hand, the i n i t i a l error of the\n\nODF is random and t y p i c a l l y the i n i t i a l error growth is small because the t r a ­ jectory i n i t i a l l y approaches the attractor as fast as i t diverges from nature. Initially,\n\nerrors might even decay by this mechanism i f the approach to the\n\nattractor is rapid enough.\n\nNote that at t = 0 the 4D analysis error is mostly\n\nin the A-direction yet the difference between the analysis and the observations ( i . e . the start of the ODF) is mostly in the D-direction. vides a good f i t\n\nThus the VAM pro­\n\nto the data at t = 0 projected onto the growing modes.\n\nIn\n\nspite of t h i s , i t s actual error has a large projection on the growing modes.\n\n117\n\nOf course, the QG forecast model is not perfect.\n\nThe description of a PE\n\ntrajectory in terms of QG variables is a projection since the PE phase space has a higher dimension than the QG phase space.\n\nI t is for this reason that\n\nperfect i n i t i a l conditions are not optimal for making a forecast.\n\nIn F i g . 1 , a\n\nPIF curve for forecasts starting at t = -42 h would be nearly identical to the one drawn shifted to the l e f t by 42 h.\n\nCompared to this curve, the 4DAF are\n\nsuperior after the i n i t i a l 18 h, i . e . past t = -24 h.\n\nThat i s , there are QG\n\nt r a j e c t o r i e s that follow the PE evolution better than the PIF forecasts.\n\nThe\n\nprojection which most nearly obeys QG dynamics, we w i l l c a l l the shadow t r a j e c ­ tory.\n\nLeith (1980) has described how best to choose a QG i n i t i a l state to\n\nshadow the PE evolution; basically one desires a QG state which when nonlinearly balanced y i e l d s a PE state which agrees with the data. PIFs I have used a QG state which agrees with the data.\n\nIn contrast, in the One could use L e i t h ' s\n\nprocedure for each point on the PE trajectory to determine the shadow t r a j e c ­ tory.\n\nDifferences between the PE trajectory and the shadow trajectory for low\n\nvalues of the Rossby number should remain small and should not make a net con­ tribution to the growth of error.\n\n(For the low value of forcing parameter used\n\nin these experiments, the Rossby number is small (Hoffman, 1981).) The model error may, therefore, be understood in terms of the difference between the shadow trajectory and the QG forecast t r a j e c t o r y .\n\nThe model error\n\nin a similar experiment was found to be a complex mixture of deterministic and stochastic components (Hoffman, 1981, p. 526).\n\nThe deterministic component is\n\ndue to differences in the dynamics governing the two t r a j e c t o r i e s .\n\nIn part,\n\nthe stochastic component occurs because many PE model states project onto the same shadow s t a t e .\n\nThis model error combined with the dynamical properties of\n\nthe model - contraction of volumes in phase space and sensitive dependence on i n i t i a l conditions - insures that a QG trajectory can only closely approach the shadow trajectory for a short period of time.\n\nAs a r e s u l t , the 4DAF error\n\ncurve must be concave. REFERENCES Hoffman, R. N., 1981. Alterations of the climate of a primitive equation model produced by f i l t e r i n g approximations and subsequent tuning and stochastic forcing. J . Atmos. S c i . , 38: 514-530. Hoffman, R. N . , 1986. A four-dimensional analysis exactly satisfying equations of motion. Mon. Weather R e v . , 114:388-397. Hoffman, R. N. and E . Kalnay, 1983. Lagged average forecasting, an alternative to Monte Carlo forecasting. T e l l u s , 35A: 100-118. L e i t h , C. E . , 1980. Nonlinear normal mode i n i t i a l i z a t i o n and quasi-geostrophic theory. J . Atmos. S c i . , ^7_: 958-968. Sasaki, Y . , 1970. Some basic formalisms in numerical variational a n a l y s i s . Mon. Weather R e v . , 98: 875-883.\n\n119\n\nVARIATIONAL\n\nCHUNG-YI\n\nof P h y s i c s , A c a d e m i a\n\nof A t m o s p h e r i c\n\nTaiwan,\n\nAND DETERMINATION\n\nOF WEIGHTING\n\nFACTORS\n\nTSENG\n\nInstitute Dept.\n\nINITIALIZATION\n\nRepublic\n\nof\n\nSinica,\n\nNankang,\n\nSciences, National\n\nTaiwan\n\nTaipei, Univ.,\n\nTaiwan\n\nTaipei,\n\nChina\n\nABSTRACT T h e v a r i a t i o n a l o p t i m i z a t i o n t e c h n i q u e is u s e d t o d e v e l o p a n o p e r a t i o n a l s c h e m e for the i n i t i a l i z a t i o n of m e t e o r o l o g i c a l f i e l d s in n u m e r i c a l w e a t h e r p r e d i c t i o n . The linear balance equation is e m p l o y e d a s a s t r o n g c o n s t r a i n t t o m a i n t a i n t h e b a l a n c e b e t w e e n the m a s s a n d w i n d f i e l d s . The analysis equations are coupled elliptical partial differential equations which are solved by the r e l a x a t i o n m e t h o d . A method for determining a unique value o f t h e w e i g h t i n g f a c t o r s is p r o p o s e d . A case study has been made to i n v e s t i g a t e t h e a p p l i c a b i l i t y o f t h e a n a l y s i s s c h e m e t o s y n o p t i c d a t a in E a s t A s i a a r e a . V a r i a n c e s p e c t r u m a n a l y s i s of the height fields indicates that the short wave components have been c o m p l e t e ­ ly s u p p r e s s e d , w h i l e t h e l o n g w a v e c o m p o n e n t s r e m a i n i n t a c t .\n\nINTRODUCTION Variational 1969,\n\noptimization\n\n1 9 7 0 ) is a n e l e g a n t\n\nmeteorological (Lewis,1972),\n\nfields\n\noptimization\n\nable value balance mass\n\nHowever,\n\nof\n\nand wind\n\nfields\n\nthe o p t i m a l v a l u e\n\nDEVELOPMENT\n\n2\n\nfactors.\n\nsimultaneously the w e i g h t i n g\n\nformalism\n\n+ a ( v - v )\n\n2\n\n+\n\nJ\n\nof\n\nand data problem\n\nIn t h i s\n\npaper\n\nconstraint and a method\n\nfactors\n\nis\n\nanalyzing\n\nassimilation in a\n\nthe\n\nvari­ a\n\n5 ( 0 - 0 )\n\nwas\n\n2\n\n+ 2 ^ ' B (\n\nthe\n\ndetermining\n\nproposed.\n\nEQUATIONS used\n\nreason­\n\nlinear\n\n(1958,\n\nanalysis\n\nis h o w t o d e t e r m i n e\n\nas a strong\n\nOF THE ANALYSIS\n\nThe variational\n\n^ | { a ( u - u )\n\nof\n\nmethod\n\ndifficult\n\nscheme\n\nby Sasaki\n\nin o b j e c t i v e\n\n(Stephens,1970)\n\nanalysis\n\nis u s e d\n\nused\n\nthe most\n\nthe w e i g h t i n g\n\nequation\n\nas p r o p o s e d\n\nand sophisticated\n\nand has been\n\ninitialization\n\n(Ritchie,1975). ational\n\nanalysis\n\nu , v , 0 j } dxdy = 0\n\n120 with\n\nthe\n\nlinear\n\nbalance\n\nequation\n\nas\n\na strong\n\nconstraint\n\nB ( u,v,*) = - £ ( f v ) - ^ ( f u ) - V ^ = 0 2\n\n,t) A X + ( t )\n\n(3)\n\nT\n\nv\n\nwhere H ( Sx/ -j > T\n\nx/9Y-j»\n\n9\n\n9A\n\na n d\n\n9x/3P-j evaluated at longitude (j) and time t.\n\n3B\n\nThe symbol A i s\n\nused here to represent small departures from the f i r s t guess solution and the corresponding predicted values of the measurements. Equations (2) and (3) are solved recursively using Kalman f i l t e r i n g ( e . g . , Gelb, 1979).\n\nThe (simulated) measurement data is generally processed\n\nboth forwards and backwards in time.\n\nThe forward and backward solutions are\n\nthen combined optimally (Gelb, 1979) to produce a best estimate of the state vector at each timestep.\n\nUsing these procedures there are 4 steps involved\n\nin estimating _x. (a)\n\nAn estimate of x is f i r s t obtained based on the stationary wave\n\nmodel of the atmosphere.\n\nThis can be obtained using the above procedure by\n\nsetting those elements of the a priori covariance matrix used in the Kalman f i l t e r , corresponding to y^. and p^, as well as the state vector elements and p. themselves, equal to zero. (b)\n\nEstimates of y ^ and p. are obtained by linearly f i t t i n g the wave\n\namplitudes and phases over 2 day periods centered on each day. (c) H((j),t). (d)\n\nThe estimate, x_, is used to calculate the partial derivatives in All measurements are then reprocessed using the moving wave model. Iteration is employed, i f necessary, by returning to step ( b ) .\n\nSAMPLE RESULTS To test the analysis procedure, measurements have been simulated using equation (1) for a series of values of the state vector x_. shown in Figures 1 and 2 for B = 59, Q\n\n= 2, p^ = 0.1 radians/timestep and\n\nwith a l l other coefficients equal to zero. has been added to the simulated data.\n\nResults are\n\nOne percent measurement noise\n\nSampling was assumed to have occurred\n\n16 times/day at evenly spaced intervals in longitude and time (corresponding to timesteps of 90 minutes).\n\nAlthough this sampling rate i s s l i g h t l y more\n\nfrequent than i s typical of p o l a r - o r b i t t i n g s a t e l l i t e s\n\n(14 or 15 times/day),\n\nwe believe that the sampling r a t e , which was selected for mathematical con­ venience, should in no way l i m i t the interpretation of the r e s u l t s . As shown in Figure 1 the principal\n\nlimitation of the stationary waves\n\nprocedure, when applied to an atmospheric situation described by moving waves, i s that the amplitude of the waves tend to be underestimated, tional features of the solution are that there are i n i t i a l\n\n134\n\nperiods of approximately 24 hours duration during which the solution diverges away from the a priori information and becomes controlled by the measure­ ments.\n\nThe indicated variation in the inferred wave amplitude from one 12\n\nhour period to another is related both to measurement noise and to the phase of the wave at the beginning of each period.\n\nA value of p^ = 0.1 radians/90\n\nminutes corresponds to wave motion of approximately T T / 2 radians/day.\n\nThere\n\ni s thus a tendency for a small o s c i l l a t i o n to occur at multiples of 2 day periods. Figure 2 shows the phase r e t r i e v a l s for wave 3 in the forward and back­ ward directions in time individually for the stationary wave procedure.\n\nIt\n\nshould be expected that the phase estimated in the forward retrieval would be based on superimposing a stationary wave 3 on the moving wave centered at the middle of the averaging period, which is defined by the £ matrix. Since large values of q were used in this\" a n a l y s i s , the effective averaging period is 1 day (or somewhat less) and the phase should correspond to approx­ imately 1/2 day prior to the observation time.\n\nThis logic accounts for the\n\nphase difference of approximately TT/2 radians between the forward and back­ ward r e t r i e v a l s .\n\nMoreover, i t is this phase difference between the two r e ­\n\nt r i e v a l s which principally accounts for the underestimation of wave ampli­ tude in the combined estimate (Figure 1). The moving wave solution, on the other hand, exhibits no significant\n\ndif­\n\nference between the phases retrieved by processing the data forwards and backwards in time.\n\nThus the combined amplitude estimate is unbiased and\n\nvaries around the correct value of 2 units. The underestimation of wave amplitude using the stationary wave model worsens as p\n\n3\n\nincreases.\n\nThe energy of the moving wave is s p l i t between\n\nstationary wave 3 and the other Fourier components.\n\nThus, i f the atmosphere\n\nconsists of a spectrum of moving waves, the amplitudes of some wavenumbers are expected to be underestimated while others are l i k e l y to be overesti­ mated.\n\nOverall, however, we should expect that a large amplitude wave w i l l\n\nhave i t s amplitude consistently underestimated i f the stationary wave model is used.\n\nThe moving wave model, on the other hand, should produce an un­\n\nbiased estimate of wave amplitude. Figure 3 demonstrates that when the atmosphere i s characterized by a wave 3 whose amplitude increases from 1.2 to 2.8 units over the 10 day observa­ tion period (an increase of approximately 8%/day) and whose phase speed changes slowly with time ( p = 0.48 + 0.77t radians of longitude/day where 3\n\nt is measured in days from the beginning of the observation period), the moving wave solution adequately retrieves the amplitude of the moving wave. In contrast, the stationary wave model results in an underestimation of\n\n135\n\n2-5 r\n\n1.0 k .50 .00 r . . . . i . . . . i . . . . i . . . . i . . . . ' .00 5.0 10 15 20 25\n\nTime ( Holfdoy )\n\nTime Fig. 2. Wave 3 phase estimates (radians of longitude) obtained in the forward (squares) and backward (crosses) directions using the stationary wave model. The data is the same as that used to generate Fig. 1 .\n\nFig. 1 . Wave amplitude estimated using the stationary wave (triangles) and moving wave ( c i r c l e s ) models of the atmosphere. The data analyzed was simu­ lated from Bo = 59, B3 = 2 and p = 0.1 radians/90 minute timestep. 1% measure­ ment noise was added to the data. 3\n\n3.0 1\n\n.00\n\nr.... .00\n\n1\n\n• • • 1\n\n1 . . . . 1, . . . 1. . . . 1 5.0\n\n10\n\n15\n\n20\n\n... • i 25\n\nTime ( Holfdoy ) Fig. 3. Wave amplitude estimated usirvg the stationary wave (triangles) and moving wave ( c i r c l e s ) models of the atmosphere. The data was simulated from B = 59, B3 = 1.2 + 0.16t and p = 0.48 + 0.77t where t i s expressed in days. 1% measurement noise was added to the data. 0\n\n3\n\n136\n\nwave amplitude by an amount which becomes larger as the wave moves faster. CURRENT CONCLUSIONS A procedure for interpreting polar-orbitting s a t e l l i t e observations of the atmosphere in terms of large-scale moving and linearly growing plane­ tary waves has been coded and tested.\n\nThe tests have demonstrated the\n\nadvantages of this procedure over that currently in use and in which the atmosphere is assumed to consist of stationary planetary waves.\n\nThe prin­\n\ncipal advantage i s in the estimation of wave amplitude. The tests of the procedure have so far been restricted to highly i d e a l ­ ized situations in which the assumed model closely describes the state of the atmosphere.\n\nFigure 3 suggests that departures of the atmospheric state\n\nfrom this model may lead to some deficiencies in the solution.\n\nTests are\n\ntherefore underway to determine the dependence of the solution on the time over which the solution is being averaged and on other combinations of the forward and backward r e t r i e v a l s .\n\nOther tests are planned to investigate\n\nthe behavior of the solution as a function of the rate of change of phase speed and growth r a t e .\n\nREFERENCES Gelb, A . , 1979. Applied Optimal Estimation. Massachusetts Institute of Technology Press, Cambridge, MA. Kohri, W . J . , 1979. LRIR observations of the structure and propagation of the stationary planetary waves in the Northern Hemisphere during December, 1975. Thesis PB-82-156639, NTIS, Springfield, VA 22161. Rodgers, C D . , 1977. S t a t i s t i c a l principles of inversion theory. I n : Inver­ sion Methods in Atmospheric Remote Sensing, A. Deepak ( E d i t o r ) , Academic Press, New York.\n\n137\n\nIMPACT OF DOPPLER WIND ANALYSIS WEIGHTS DIAGNOSED PRECIPITATION IN A THUNDERSTORM\n\nON THREE DIMENSIONAL\n\nAIRFLOW AND\n\nC. L. ZIEGLER\n\n1\n\nNational Severe Storms Oklahoma 73069 U.S.A.\n\nLaboratory,\n\nNOAA,\n\n1313\n\nHal l e y\n\nCircle,\n\nNorman,\n\nABSTRACT The a i r flow in c o n v e c t i v e storms and t h e processes t h a t produce h y d r o meteors of v a r i o u s k i n d s , a r e being studied i n t e n s i v e l y by m e t e o r o l o g i s t s using Doppler radar o b s e r v a t i o n s . A v a r i a t i o n a l a n a l y s i s s i m u l t a n e o u s l y imposes two kinematic boundary c o n d i t i o n s and t h e mass c o n t i n u i t y equation on Doppler v e ­ l o c i t i e s t o d e r i v e t h e t h r e e - d i m e n s i o n a l thunderstorm a i r motions. Variable adjustment weights c o n t r o l t h e r e l a t i v e degree of h o r i z o n t a l d i v e r g e n c e a d j u s t ­ ment a t each l e v e l , which in t u r n changes the shape of t h e u p d r a f t p r o f i l e . Diagnoses of p r e c i p i t a t i o n and r e f l e c t i v i t y f i e l d s w i t h i n a t h u n d e r s t o r m , using t h e s e analyzed a i r motions in a t h r e e dimensional cloud model, demonstrate t h e s e n s i t i v i t y of r e t r i e v e d thunderstorm v a r i a b l e s t o t h e choice of v a r i a t i o n a l a n a l y s i s weights and t h e r e s u l t i n g v e r t i c a l v e l o c i t y f i e l d . INTRODUCTION Retrieval\n\nof the thunderstorm p r e c i p i t a t i o n\n\nt h r e e components\n\nfield\n\nr e q u i r e s knowledge of\n\nof a i r motion w i t h i n t h e storm volume\n\n(Ziegler,\n\n1985).\n\nthe\n\nThus,\n\nt h e a n a l y s i s of Doppler v e l o c i t y data from i n d i v i d u a l\n\nr a d a r s t o produce a i r v e ­\n\nlocity\n\nstep\n\ncomponents\n\nover\n\nthe\n\nthunderstorm p r e c i p i t a t i o n numerical\n\nmodel\n\nof\n\ncloud\n\nstorm\n\nvolume\n\ncontent\n\nwith\n\nprocesses\n\ntoward r e t r i e v a l\n\na three-dimensional based\n\non c o n t i n u i t y\n\ncloud\n\nretrieval\n\nmethod, which i s\n\nbased upon t h e cloud momentum e q u a t i o n s .\n\ndependent\n\nmodel\n\nof\n\nsnow mixing liquid defined\n\nand\n\nobservations\n\nvariations\n\ncloud\n\nice of\n\npotential\n\nratios,\n\nphase\n\nvelocity.\n\nand\n\nThe\n\nprecipitation\n\nsystem\n\nof\n\nas\n\nthe\n\ntotal\n\nconcentrations\n\nr a i n d r o p s , and graupel\n\nor h a i l .\n\nThe\n\nfrom t h e dynamic\n\nphysics,\n\ncontinuity\n\nof\n\nf o r heat\n\nThe t i m e -\n\nand t r a n s p o r t i v e p r o c e s s e s ,\n\nt e m p e r a t u r e , w a t e r vapor mixing\n\nas well\n\ndroplets,\n\ncloud\n\nand\n\nequations\n\nratio,\n\ncloud\n\nand mixing\n\nparam­ Doppler depicts ice\n\nratios\n\nand of\n\nRadar r e f l e c t i v i t y ,\n\nby the i n t e g r a l of t h e product of hydrometeor c o n c e n t r a t i o n d e n s i t y and\n\na power of p a r t i c l e diameter o v e r a l l output\n\nfundamentally d i f f e r e n t\n\ni n c o r p o r a t e s thermodynamic\n\nmodel.\n\nequations\n\nIt\n\nliquid\n\nthus\n\na vital\n\nand t h e w a t e r s u b s t a n c e .\n\neterized\n\nis\n\nis\n\nis\n\nfor rain,\n\ng r a u p e l , and snow.\n\npossible\n\nsizes,\n\nis\n\nThese model-derived\n\ncalculated\n\ncompared with the r e f l e c t i v i t i e s measured by r a d a r t o determine t h e of the modeled p r e c i p i t a t i o n\n\nfields.\n\nfrom model\n\nr e f l e c t i v i t i e s may be correctness\n\n138 VARIATIONAL WIND ADJUSTMENT The v a r i a t i o n a l wind a n a l y s i s derived least at\n\nfrom m u l t i p l e\n\nDoppler\n\nt h r e e independent\n\nany\n\npoint.\n\nimposes\n\ndata\n\nequations\n\nand\n\nconstraints\n\nthe\n\nequation\n\non an input wind of\n\nmass\n\nmust be used t o determine the wind\n\ninformation\n\nresults\n\nin\n\nan\n\nfield\n\ncontinuity.\n\nAt\n\nuniquely\n\noverdetermined\n\nproblem,\n\nwhich a l l o w s the most n e a r l y c o r r e c t answer t o be estimated through a method of least\n\nsquares\n\n(Ray and Sangren, 1 9 8 3 ) .\n\nthe horizontal vertical obtain cess\n\nvelocity\n\nestimate.\n\nimproved e s t i m a t e s\n\nis\n\nrepeated\n\nuntil\n\nmeasurement e r r o r ( After\n\nthis\n\nThe mass c o n t i n u i t y equation\n\nof the v e r t i c a l motion\n\nthe\n\ninitial\n\nmay d e p a r t\n\nnew v a l u e s .\n\nanalysis,\n\nhorizontal zero).\n\n+\n\np(\n\nfrom\n\ncomponents\n\ntwo r a d a r s .\n\nThis\n\nthan t h e\n\ndz\n\nto\n\npro­\n\npresumed\n\nare prescribed\n\nfrom\n\nis\n\nthe\n\nat\n\nall\n\nHowever, t h e v e r t i c a l\n\nreasonable\n\ndivergence\n\nvalues\n\nat\n\nlocations\n\nfar\n\nto\n\n\" -/ az\"\n\nair\n\ncomponents\n\na r e denoted\n\ndensity\n\nhorizontal\n\nZ\n\nto\n\nt\n\nbe\n\na\n\ndensity-\n\nconstant\n\nC\n\nequation\n\nwind components\n\n'\n\ndz\n\nand w*=pw. by u,\n\nv,\n\n(1)\n\nThe w e s t - e a s t ,\n\nand w,\n\ng r a t e d , both boundary c o n d i t i o n s variational\n\nfrom\n\nthe winds a r e a d j u s t e d\n\nt o r e q u i r e the i n t e g r a t e d\n\nsurface\n\ngrid veloc­\n\n( Z i e g l e r e t a l . , 1983)\n\nwhere p i s\n\nThe\n\nless\n\n1980).\n\nV e r t i c a l i n t e g r a t i o n of the a n e l a s t i c mass c o n t i n u i t y\n\ny i e l d s the e x p r e s s i o n\n\n'[ ^ ^\n\nare\n\nof\n\nand a\n\ni s then i n t e g r a t e d\n\nTo m i t i g a t e t h e s e e f f e c t s\n\nOne means of\n\n(usually\n\nwind\n\nby a t l e a s t\n\nsubstantially\n\nweighted\n\nC\n\nvelocities\n\n(Ray e t a l . ,\n\ni t e r a t i v e adjustments\n\nwhere t h e i n t e g r a t i o n b e g i n s . to\n\nprovide estimates\n\nfrom measured r a d i a l\n\n- 1 0 cm s * ) .\n\np o i n t s t h a t were sampled ities\n\nLinear equations\n\nc a r t e s i a n wind components\n\nare\n\nfunctional\n\nsouth-north,\n\nrespectively.\n\nand\n\nThe r e s u l t\n\nvertical\n\nis\n\nso t h a t when the components\n\nthat are\n\nthe\n\ninte­\n\nsatisfied.\n\nincorporating\n\nthe\n\nintegral\n\nconstraint\n\neq.\n\n(1)\n\nt a k e s the form\n\nE\n\n= J/{/[\n\n2 a\n\nwhere X i s denotes error\n\n(u-u ) 0\n\nthe\n\n2\n\npressed component\n\nby\n\nthe\n\n2\n\nv a r i a b l e ) Lagrange m u l t i p l i e r .\n\nquantity,\n\nand the weights\n\n,\n\nu\n\nAn analogous\n\na and 3 a r e determined\n\nuncertainty.\n\nu\n\nexpression\n\nrelates\n\n£\n\nThe r e s u l t of minimization\n\na d j u s t e d v a l u e s of t h e h o r i z o n t a l wind f i e l d\n\n(2)\n\nThe s u p e r s c r i p t o\n\nThese weights a r e r e l a t e d t o t h e Gauss p r e c i s i o n 2 2-1 2 formula a = ( 2 a ) , where o i s the v a r i a n c e\n\nuncertainty.\n\nthe v-component\n\n2\n\n(horizontally\n\nan observed\n\nanalysis.\n\n+ x [ / p ( ^ + ^ - ) d z - C]} dxdy\n\n+ 3 (v-v°) ]dz\n\nand the\n\nof eq.\n\nfrom an\n\nmoduli of\n\nthe\n\n(1) is\n\nu-\n\nv a r i a n c e of\n\n(2) i s t h a t the\n\nd e v i a t e as l i t t l e as p o s s i b l e\n\nt h e measured v a l u e s , w h i l e t h e i n t e g r a l c o n s t r a i n t eq.\n\nex-\n\nsatisfied.\n\nfrom\n\n139 The Euler-Lagrange e q u a t i o n s a s s o c i a t e d tional\n\nin eq.\n\nwith t h e minimization\n\nof t h e f u n c ­\n\n(2) t a k e t h e form\n\nand\n\nv = v °\n\nwhile\n\n+\n\nthe\n\npartial (4)\n\n- % f\n\n(4)\n\nintegral\n\nconstraint\n\ndifferential\n\nis\n\nrecovered\n\nby v a r i a t i o n of\n\nequation r e s u l t i n g from t h e s o l u t i o n\n\nX.\n\nThe\n\nof e q s .\n\nelliptic\n\n(1), ( 3 ) , and\n\nis\n\nwhich\n\nis\n\nsolved\n\nschemes a r e used followed tinuity\n\n0\n\nJ\n\nby\n\nsuccessive\n\nin e q s .\n\n(1),\n\noverrelaxation. (3),\n\n(4) and\n\n(5).\n\nConsistent\n\nby computation of u and v adjustments from e q s . equation\n\nfrom which w i s\n\nfinite\n\nS o l u t i o n of eq.\n\ndifference (5) f o r\n\n(3) and ( 4 ) .\n\nX is\n\nThe con­\n\ni s then r e i n t e g r a t e d t o o b t a i n a d j u s t e d v e r t i c a l mass f l u x w*, computed.\n\nTEST OF ANALYZED WIND FIELDS IN THE CLOUD MODEL The\n\ncloud\n\ntrieving\n\nthe\n\nmodel\n\ndescribed\n\ndistributions\n\nof\n\nin\n\nthe\n\nintroduction\n\nhas\n\nbeen\n\nemployed\n\nt e m p e r a t u r e and w a t e r substance\n\nin\n\na\n\nin\n\nnonsevere\n\nthunderstorm which occurred on 27 May 1979 in c e n t r a l Oklahoma ( Z i e g l e r , The Doppler\n\nobservations\n\nhave been obtained\n\nThe m u l t i p l e Doppler v e l o c i t i e s\n\noutlined\n\nin the p r e v i o u s s e c t i o n .\n\ntest,\n\nthe\n\nfirst\n\non assumed\n\nground\n\nvelocity\n\nThe second a n a l y s i s\n\nlevel\n\nto\n\nunity\n\nat\n\nerrors\n\nreflects\n\nthe\n\nbelief\n\nthat\n\nvariational errors\n\nof\n\nadjustment 0 . 5 m s~*\n\nThe r e s u l t i n g\n\nincreasing\n\nhorizontal\n\nand i n c r e a s e with h e i g h t . the\n\nwind\n\nerror\n\nweights\n\nfrom\n\nanalysis\n\ncontains\n\neach error\n\nfrom zero a t\n\ndivergence\n\npro­\n\nbias\n\nin upper l e v e l s of t h e storm t h a t a r e not p r o p e r l y accounted\n\nrently specified\n\nscanning\n\nt h e procedures\n\nthe standard d e v i a t i o n of t h e\n\nby a l i n e a r f u n c t i o n\n\nstorm t o p .\n\njustments a r e zero a t ground l e v e l cedure\n\nobservational\n\n(B) m u l t i p l i e s\n\nwind component\n\n1985).\n\nTwo wind a n a l y s e s a r e performed as a s e n s i ­\n\n( A n a l y s i s A) employing\n\nin each h o r i z o n t a l\n\nfrom t h r e e independently\n\na r e analyzed f o l l o w i n g\n\nre­\n\nf o r by c u r ­\n\nsources.\n\nThe observed wind f i e l d s south-north c r o s s - s e c t i o n s\n\nfrom a n a l y s i s A and B a r e i l l u s t r a t e d by v e r t i c a l\n\nthrough t h e u p d r a f t c o r e in F i g .\n\nla.\n\nThe maximum\n\n140\n\nSOUTH-NORTH DISTANCE (km)\n\nSOUTH-NORTH DISTANCE (km)\n\n(a)\n\n(b)\n\nFig. ] . V e r t i c a l s o u t h - n o r t h c r o s s - s e c t i o n of observed winds (a) and r e t r i e v e d and observed r e f l e c t i v i t i e s ( b ) , through t h e thunderstorm u p d r a f t in a n a l y s i s A and a n a l y s i s B. (a) V e l o c i t y s c a l e a t upper r i g h t . S t i p p l i n g i n d i c a t e s a w e s t ­ e r l y wind between 10 and 15 m s , w h i l e hatching i n d i c a t e s an e a s t e r l y wind exceeding 1 m s . (b) R e t r i e v e d r e f l e c t i v i t y contoured with s o l i d l i n e a t 5 dBZ i n t e r v a l , w h i l e observed r e f l e c t i v i t y (dBZ) i s r e p r e s e n t e d by a l t e r n a t e s t i p p l e d - u n s t i p p l e d r e g i o n s a t a 5 dBZ i n t e r v a l from o u t e r 20 dBZ boundary.\n\nupdraft are\n\nis\n\nthe\n\nzontal\n\n25 m s \"\n\nstrong, wind\n\nvergent\n\n1\n\na t 1 0 . 5 km above ground l e v e l\n\ndeep,\n\nmaxima\n\noutflow\n\nat\n\ncentrally\n\nexceeding\n\nlocated\n\n10 m s~*\n\nstorm t o p .\n\nweaker u p d r a f t in t h e v e r t i c a l\n\nstrength\n\nis\n\ndirectly\n\nrelated\n\nat\n\nmiddle\n\nThe modified column\n\nw h i l e t h e main u p d r a f t i s i n t e n s i f i e d .\n\n(AGL).\n\nu p d r a f t , the\n\nlocated\n\nKey v e l o c i t y\n\nflanking\n\nlevels,\n\nand\n\nthe\n\nweights\n\n11 km north of\n\namounts of cloud and p r e c i p i t a t i o n as well\n\nphysical\n\nas changes\n\nstrong\n\nt h e grid\n\ndi­ in\n\na\n\norigin,\n\ncirculation\n\nprocesses\n\nin modeled\n\nhori­\n\nresult\n\nThis d i f f e r e n c e in analyzed\n\nthrough t h e cloud\n\nfeatures\n\nwesterly\n\nto\n\nvarying\n\nreflec-\n\nti v i t y . A n a l y s i s A and B a r e s e p a r a t e l y i n s e r t e d i n t o t h e cloud model, which in each case\n\nis\n\ni n t e g r a t e d forward in t i m e . fields\n\nreflectivities Fig.\n\nlb,\n\nFig.\n\nla.\n\nis ity\n\nin\n\nof water substance\n\ndrafts, vertical\n\nobtained.\n\ni s held\n\nfixed,\n\nThe computed\n\nf o r each case a r e d i s p l a y e d along with measured r e f l e c t i v i t y the\n\nsame\n\nvertical\n\nThe primary d i f f e r e n c e s\n\nt h e reduced t o t a l is\n\nSince t h e input wind f i e l d\n\nare ultimately\n\ncross-section\n\nas\n\nthe\n\nwind\n\nfields\n\nshown\n\nin t h e l a t t e r c a s e .\n\nThis\n\nsensitiv­\n\nl a r g e l y due t o t h e decreased a r e a l e x t e n t and s t r e n g t h of low l e v e l\n\ncirculation\n\nto\n\nboth t h e supply hold\n\nin\n\nbetween t h e r e t r i e v a l s using a n a l y s i s A and B\n\np r e c i p i t a t i o n content\n\nwhich diminishes\n\nin\n\nof condensate\n\nprecipitation\n\naloft.\n\nand t h e a b i l i t y of\n\nThe reduced\n\nupthe\n\nprecipitation\n\n141 s t o r a g e c a p a c i t y of t h e wind f i e l d tical the\n\nvelocity origin\n\nas well\n\nas\n\nthe\n\nis\n\nentering tion\n\nrain fallspeed\n\neverywhere s i g n i f i c a n t l y this\n\n2 , which d e p i c t s\n\nver­\n\nof\n\nraindrop f a l l s p e e d .\n\nThe\n\nabove 1 km w h i l e t h e a n a l y s i s B up­\n\nl e s s than r a i n f a l l s p e e d .\n\nModeled r a i n drops\n\nregion of weak measured r e f l e c t i v i t y and low i n f e r r e d p r e c i p i t a ­\n\ncontent,\n\nare\n\nheld\n\nl a r g e r as they c o l l e c t erroneously\n\ni l l u s t r a t e d in Fig.\n\ncharacteristic profile\n\na n a l y s i s A u p d r a f t exceeds draft\n\nis\n\np r o f i l e s from both a n a l y s e s in a v e r t i c a l column 11 km north of\n\nhigh\n\naloft\n\nby\n\nthe\n\nstronger\n\nnumerous cloud d r o p l e t s .\n\ncalculated\n\nreflectivities\n\nanalysis A updrafts This s t o r a g e e f f e c t\n\nfrom a n a l y s i s A.\n\nweak a n a l y s i s B u p d r a f t s promote p r e c i p i t a t i o n\n\nfallout\n\nand\n\ncauses t h e\n\nConversely,\n\nand weaker\n\ngrow\n\nthe\n\nreflectivi­\n\nties.\n\nCONCLUSION It\n\nhas\n\nbeen demonstrated\n\nthunderstorm\n\nis\n\nsensitive\n\nDoppler wind a n a l y s i s . tant, of\n\ns i n c e most\n\nvertical\n\nof\n\nvelocity\n\ndistribution\n\nof\n\nstood,\n\nto\n\nthe\n\nlow\n\nat\n\nof each\n\nof\n\nprecipitation distribution the\n\nlevel\n\nvertical\n\nupdraft is\n\nvelocities\n\nwater vapor occurs t h e r e . height weights,\n\nis\n\nin\n\ncontrolled\n\nby the\n\nwhich d i r e c t l y\n\nvertical\n\nrelate to\n\nthe\n\nunder­\n\nand q u a n t i f i e d t o improve t h e accuracy of fields.\n\nFig. 2. Vertical profiles of u p d r a f t s (m s ) from a n a l y s i s A (solid curve) and analysis B (long d a s h - s h o r t dash c u r v e ) w i t h profile of c h a r a c t e r i s t i c rain­ drop t e r m i n a l fallspeed (dashed curve), in a vertical column 11 km north of the grid o r i g i n i n d i c a t e d in F i g . 1 .\n\n10 CD <\n\nUJ\n\n1\n\nthe\n\nThe degree\n\n5 10 VERTICAL VELOCITY (m s \" )\n\nin a\n\np a r t i c u l a r l y impor­\n\nSources of a n a l y s i s e r r o r a r e a p p a r e n t l y not well\n\nand need t o be b e t t e r defined\n\nv a r i a t i o n a l l y a d j u s t e d Doppler wind\n\n14\n\nstrength\n\nThe a r e a of\n\nvariational\n\nwind a n a l y s i s e r r o r .\n\nt h a t t h e modeled\n\n15\n\n142 REFERENCES Ray, P . S . , and K.L. Sangren, 1 9 8 3 : M u l t i p i e - D o p p l e r radar network d e s i g n . J . Clim. Appl. M e t e o r . , 3 2 , 1 4 4 4 - 1 4 5 4 . , C L . Z i e g l e r , W. Bumgarner, and R . J . S e r a f i n , 1 9 8 0 : S i n g l e - and m u l t i p l e Doppler radar o b s e r v a t i o n s of t o r n a d i c s t o r m s . Mon. Wea. R e v . , 1 0 8 , 1 6 0 7 1625. Z i e g l e r , C . L . , P . S . Ray, and N.C. Knight, 1 9 8 3 : Hail growth in an Oklahoma m u l t i c e l l storm. J . Atmos. S c i . , 4 0 , 1 7 6 8 - 1 7 9 1 . , 1 9 8 5 : R e t r i e v e l of thermal and microphysical v a r i a b l e s in observed convective storms. Part 1 : Model development and p r e l i m i n a r y t e s t i n g . J . Atmos. S c i . , 4 2 , 1 4 8 7 - 1 5 0 9 .\n\n145\n\nREMARKS ON SYSTEMS WITH UNCOMPLETE DATA\n\nJ.L.\n\nLIONS\n\nCollege de F r a n c e , 3 , rue d'Ulm, 75231 P a r i s Cedex 05 (France) and C.N.E.S.\n\n(Centre National\n\nd'Etudes S p a t i a l e s ) , 2 , Place M. Quentin,\n\n75039\n\nP a r i s Cedex 01 ( F r a n c e ) .\n\nABSTRACT\n\nD i s t r i b u t e d systems a r e s a i d \"with uncomplete data\" i f a l l the i n f o r m a t i o n a t our d i s p o s a l does not d e f i n e a unique s o l u t i o n but a s e t of c a n d i d a t e s o l u ­ t i o n s . Among t h e s e , one t r i e s t o choose a p a r t i c u l a r c a s e , minimizing a given criterion. We study t h i s s i t u a t i o n in two f a m i l i e s of examples, by using methods of optimal c o n t r o l t h e o r y .\n\nINTRODUCTION\n\nWe present in t h i s paper some remarks on the choice, oh a. pantlculaK solution among the s e t of a l l a r e not s u f f i c i e n t\n\n\"candidates\" as s o l u t i o n s of diA&Ubuted systems, when data\n\nt o uniquely d e f i n e a s o l u t i o n .\n\nMore p r e c i s e l y , we c o n s i d e r systems governed by paAtlal dlhk&ttntt&l equation*. The o p e r a t o r s a r e \"well behaved\" ; t o f i x i d e a s ,\n\nl e t us suppose t h a t we deal with\n\np a r a b o l i c o r h y p e r b o l i c e v o l u t i o n e q u a t i o n s . But tome initial conditions o r 6ome boundary conditions axe not knom. On the o t h e r hand, we have access t o otken inhumations : some v a l u e s of some components of the s t a t e of the system a r e g i ­ ven,\n\nsome a v e r a g e s a r e e s t i m a t e d , e t c . .\n\nuniquely d e f i n e a s o l u t i o n s e t of p o s s i b l e\n\nsolutions\n\n. But t h e s e data a r e not buhhicient t o\n\n: we have in f a c t an infinite dimensional s p a c e , o r a in some f u n c t i o n a l\n\nspace. This i s what we c a l l\n\na sys­\n\ntem with uncomplete data. We want t o choose among t h i s s e t a paAtlculasi solution which i s ,\n\nin\n\nsome\n\ns e n s e , \"optimal\". Problems of t h i s t y p e a r i s e in s e v e r a l s e t t i n g s . gical\n\nIn p a r t i c u l a r in m e t e o r o l o ­\n\nproblems. We r e f e r t o K.P. Bube, F.X. Le Dimet, R. Sadourny and t o the b i ­\n\nbliography t h e r e i n . One of the main d i f f i c u l t i e s\n\nconsists\n\nin the choice o£ the chltenta t h a t\n\nwant t o minimize, so as t o o b t a i n the \"optimal\" s o l u t i o n .\n\nIt is clear that,\n\nt h i s c o n t e x t , an e n d l e s s s e r i e s of d i f f e r e n t q u e s t i o n s a r i s e . We c o n f i n e\n\nves here t o tm example*.\n\nwe\n\nin\n\noursel­\n\n146 In the iin>t example fidR\n\nn\n\n(section\n\n, with a s t a t e y = { y y ) ; y l s\n\n2\n\n2 ) , we c o n s i d e r a p a r a b o l i c system in Q x ] 0 , T [ . 2\n\nU\n\nt i o n s on ( y ^ t . , ) , y ( t ) } , a t times 2\n\ni\n\nnot known a t time t=0 and we have informa­ t^...,\n\namong the s e t of candidate s o l u t i o n s ,\n\nt\n\nA\n\n° .\n\n=\n\ny\n\n^\n\n2\n\nIn {y,p,X^} , X..\n\noptimality system ,\n\nsatisfy\n\nthe condition ofi\"independence ofi meas­\n\nthen characterized , i = l,\n\nq,\n\nby the solution\n\no\\$ the\n\ngiven by\n\n'\n\n(2.21)\n\n+ A*)P =\n\n2\n\n+\n\ni\n\n=\n\nX. y\n\n1\n\n1\n\n5(t-t.)\n\n,\n\nar y\n\nl\n\ny\n\n=\n\n2\n\n=\n\nP\n\nl\n\n=\n\n0\n\no\n\nn\n\n(2.22)\n\n1\n\ny ( x o ) = 0 , P (x,T) = 0 , 1\n\n9\n\nP (x,T) = J L y ( , T ) >\n\nx\n\nP (x.o) = ^\n\ny ( x , o ) in\n\n2\n\n{(y^.),\n\n(y (t.),\n\n(2.23)\n\ni\n\n(2.24)\n\n2\n\nV h\n\n1\n\ni\n\nx\n\ny )}€K.\n\n2\n\n£ ^ (y .h -y(t )) < 0 i\n\n2\n\nfi\n\n2\n\ny j ),\n\n2\n\nL (fi) *L (fi) 2\n\ni\n\ne\n\n2\n\nsuch that\n\n(2.25) {(h\n\ni l ?\n\n2.4.\n\ny j ) , ( h . , y ) } . K.. 2\n\n2\n\nProof of the main result. We begin with a standard penalty approximation.\n\nfunctions az\n\n2\n\nz\n\nWe introduce the set of\n\nsuch that\n\n2\n\n\"af\n\ne\n\n(Q) .\n\nL\n\nAz L ( Q ) x ( Q ) 2\n\nz^ = z\n\n2\n\nL\n\n=\n\n{(z (t ), 1\n\ni\n\n0 nJ\n\non )\n\nz (x,o) = 0\n\nv ,\n\n(2.26)\n\n,\n\n2\n\ne\n\n1\n\n(z (t.) , 2\n\ny )) 2\n\ne\n\nin fi , K.\n\n(2.27)\n\n.\n\nWe v e r i f y , by an argument similar to the one used in Remark\n\n2.3,\n\nthat\n\n(2.27)\n\n152 makes sense f o r f u n c t i o n s\n\nz\n\nwhich s a t i s f y ( 2 . 2 6 ) .\n\nWe then c o n s i d e r\n\n• ie i o:\n\ndt\n\n- JJOo I\"lit\"\n\n3z + Az\n\ndt\n\nat\n\nJ\n\n(2.28)\n\nwhere in ( 2 . 2 8 ) , | | denotes the norm in L (ft) and\n\ndenotes the norm in\n\nH = L (ft) x l _ ( f t ) . 2\n\n2\n\nWe c o n s i d e r inf z\n\nJ (z)\n\n,\n\n(2.29)\n\nsatisfying\n\n(2.26)(2.27).\n\nProblem ( 2 . 2 9 ) admits a unique s o l u t i o n , denoted by\n\ny\n\n£\n\nWe d e f i n e h\n\n- - hit\n\n(2.30)\n\n+\n\nThe o p t i m a l i t y c o n d i t i o n i s given by\n\nT/3y\n\n8(z -y )\n\ne2\n\n2\n\nIT\" V z\n\ne 2\n\n) 'JO \"{J\n\nW—J\"*\n\n'\n\nsatisfying\n\ndt\n\n(p\n\ne>\n\n(\n\n^\n\n+\n\nA\n\n)( \"^))H Z\n\nD\n\nT\n\n*\n\n0\n\n(2.31)\n\n(2.26)(2.27).\n\nIf we t a k e in\n\n(2.31)\n\nz = y\n\n±\n\n*\n\nwhere\n\n(^(t^.yj)\n\n= o ,\n\n= p^ = 0\n\non £,\n\n(^(t^.yg) =\n\n0\n\n^(x,o) = 0 in ft and i = 1\n\n(2.32)\n\n. q >\n\nwe deduce t h a t\n\ne\n\n)\n\nA. e ]R is Moreover we have the\n\n+\n\nX^y\n\n1\n\n(2.33)\n\n6(t- ] t l\n\nwhere\n\nP\n\ne l\n\n(x,T) =\n\n?e2 < ' ) X\n\nT\n\nP 2 ( '°) x\n\n£\n\n0 , (2.34)\n\nll^2< > )\n\n=\n\n=\n\nconditions\n\nx\n\n^ ^ 2 (\n\nT\n\n' ° )\n\nx\n\n'\n\nIf we now t a k e the s c a l a r product of ( 2 . 3 3 ) with J/8y £ A)(z-y )) dt = J ( - # °\" \" 0\n\n£ 2\n\n+\n\n0\n\ne t\n\nH\n\nso t h a t ( 2 . 3 1 ) reduces t o\n\no\n\n3(z -y ) , - ^ g t ^ j d t . 2\n\n£ 2\n\nz-y\n\n£\n\n» we o b t a i n\n\n^ ( A ^ V ^ t ^\n\n153 X^CU ^ ( t ) - y ( t ) ) 1\n\ni\n\ne\n\ni\n\nH\n\n* 0.\n\n(2.35) 2\n\nWe now o b s e r v e t h a t given z\n\nsuch t h a t\n\nz(t.)\n\n= h.\n\n^ ^ V ^ i ^ H *\n\nV\n\nh\n\nas in\n\ni\n\n2\n\nh^e L (Q)x L ( f t ) , i = l ,\n\n, so t h a t ( 2 . 3 5 )\n\n...,q\n\nis equivalent\n\n, we can\n\nfind\n\nto\n\n0\n\n(2.36)\n\n(2.25).\n\nWe a r e now going to show t h a t l*i l\n\n- C\n\ne\n\nV i=l,\n\nq , where\n\n(2.37)\n\nC = constant.\n\nI f we i n t r o d u c e 0 > =\n\np\n\n- A. r\n\n-\n\n1\n\n(2.38)\n\nwe have - using the d e f i n i t i o n\n\n( 2 . 1 7 ) of\n\nr\n\n1\n\n0 ^\n\n(2.39)\n\n+ A )m = £\n\nand using ( 2 . 3 4 ) ^ 2 . 3 4 ) 2\n\n.mr\\)\n\n: (2.40)\n\n= 0.\n\nBut s i n c e\n\nz2\n\nremains in a bounded set o& L (Q), we have\n\n2 f 1 remains in a bounded s e t of L ( 0 , T ; D ( A ) ' ) 1\n\n9\n\nre2\n\nv\n\nI t then f o l l o w s\n\nfrom ( 2 . 3 9 ) ( 2 . 4 0 )\n\nJ\n\n(2.41)\n\nthat\n\np m remains £\n\nin a bounded s u b s e t of\n\n(2.42)\n\nL (Q,T ; H)\n\nremains in a bounded s u b s e t of L (0,T ; D ( A ) ' ) . Therefore m^o)\n\nBut using ( 2 . 3 4 )\n\nC)\n\n(2.43)\n\nremains in a bounded s u b s e t of\n\nSince\n\n3\n\n, i t follows\n\nH\"^(Q)\n\nfrom ( 2 . 3 8 )\n\nxH (ft). _1\n\nthat\n\nA e«i?(D(A) ; H), H = L (Q)x L ( f t ) , i t f o l l o w s 2\n\n2\n\nt h a t A* e^£(H ; D(A)\n\n1\n\n154\n\nso that using (2.43) we have that ^ie\n\nr\n\n2^°)\n\nremains in a bounded subset of H *(fi) •\n\nI t is now a simple matter to l e t\n\n(2.44)\n\nz -+ 0 and to complete the proof.\n\n2.5. Various remarks The method of proof is quite general and w i l l extend to a l l possible families of evolution equations - of course with appropriate function spaces setting. One can also obtain similar results for non linear systems - such as Navier Stokes systems - but this becomes much more technical and w i l l be presented in J . L . Lions (1985-1986). 3.\n\nPARABOLIC SYSTEMS WITH UNCOMPLETE BOUNDARY DATA\n\n3.1.Setting of the problem We consider the same system as in Section 2 , namely ay 1\n\n9 y\n\n2 (3.1)\n\nay?\n\ntyi\n\na2A^y2 + b.i — 3x. = 0\n\n3t\n\na\n\nbut\n\nhis\n\n9\n\n9\n\ntime initial\n\nconditions\n\nare known :\n\ny ( x , o ) = y ( x , o ) = 0 in fi , x\n\n(3.2)\n\n2\n\nand, on the contrary, boundary data arc uncomplete : y\n\n1\n\n= 0\n\non\n\n(3.3)\n\ne,\n\nbut no information is available on y I ^ • 2\n\nWe have the extra informations as in (2.6) : {y^V,\n\nuj),\n\n( y ( t ) , v )}eK. c R 2\n\ni u s\n\ni\n\n1\n\nwhere the\n\n(3.4)\n\n2\n\n2\n\nare such that (3.4) makes sense\n\n( )\n\nv\n\nAmong a l l the solutions of (3.1) . . . ( 3 . 4 ) , we want to find the solution which minimizes J i t )\n\ndxdt\n\n+\n\n3 J y ds, 2\n\n3>0\n\n.\n\nC) We shall not make precise here a l l the function spaces involved long and technical but without fundamental d i f f i c u l t y .\n\n(3.5)\n\nThis is\n\n155\n\nRemark 3 . 1 . If\n\n3=0\n\nin (3.5) (that would be the analogous of the problem in Section\n\n2) the existence of a solution is not clear (and i t i s dubious ! ) ; we have then no control .on the behaviour on the boundary of\n\ny^ ; the term 3j y\n\ncare of this d i f f i c u l t y .\n\n2\n\ntakes\n\n^\n\nWe want now to characterize the optimal solution of the above problem. We are going to obtain a result similar to Theorem 2 . 1 , under an appropriate hypo­ thesis on the \"independence of measurements\", as we now explain. 3.2. The condition of \"independence of measurements\" We introduce the functions\n\n(- jt rV)\n\n+\n\nV\n\nA\n\nr\n\nby\n\n1\n\nn V v .\n\n=\n\n(3.6)\n\n=0\n\nwith the boundary conditions [di^erent 3r\n\ni\n\nfrom those in Section 2)\n\n4\n\nr j = 0 on Z ,\n\n0 on\n\n(3.7)\n\nE .\n\nWe shall say that we have \"Independence the traces\n\nr l I of\n\nr\n\non\n\n1\n\nif\n\no£ measurements\"\n\n^ are l i n e a r l y independent in L ( I ) .\n\n(3.8)\n\n2\n\nRemark 3.2. We can v e r i f y , as in Remark 2.6, that given the y generally sati^^ied\n\n3.3.\n\ns , condition (3.8) i s\n\nt. ' s .\n\nwith respect to the\n\nOptimality system The optimality\n\nsystem is given by\n\nTHEOREM 3 . 1 . We assume that the { y , ^ . } satisfy 1\n\nurements\" as given\n\ny = {y^y^\n\nThe optimal solution the optimaliiy system\n\n(4+ A ) y = +\n\nA\n\nofa \"independence\n\n{y,p,A.}\n\nin\n\n,\n\n> { a\n\no^ meas­\n\ncharacterized\n\nby the solution\n\nA.e F , given by\n\n0,\n\n(3.9) 2\n\nwith the boundary M\n\nthe condition\n\nby ( 3 . 8 ) .\n\ny\n\n2\n\nf\n\n^\n\nx\n\ni ^ ^ \" V\n\nconditions\n\nThese traces make sense.\n\nActually\n\n2\n\nrl|\n\nof J . L . Lions and E. Magenes (1968).\n\neL (0,T ; H ( r ) ) z\n\n7\n\nwith the notations\n\n156\n\ny-, = 0 Pi P\n\n=\n\n'\n\n0\n\na\n\n= 0\n\n2\n\nz,\n\non\n\n3y\n\n2\n\non\n\non\n\n2\n\nZ,\n\n(3.10)\n\n2\n\n(3.11)\n\nZ,\n\nand tcc^i yi(x,o)\n\n= y (x,o)\n\n= 0\n\n?\n\np^xj)\n\n= 0 ,\n\nin fi, 3y\n\nP (x,T)\n\n=\n\n2\n\n-g^-(x,T)\n\nin fi, (3.12)\n\nU y ^ ) ,\n\ny j ) , ( y ( t . ) , y^)>€ K. , 2\n\n£ ^(y ,\n\nh -y(t ))\n\n1\n\n{(h\n\ni l 9\n\n3.4.\n\ni\n\nyj)\n\ni\n\n, (h\n\n< 0\n\nV h.\n\n,\n\ni 2\n\nL (fi) x L ( f i ) 2\n\n2\n\ne K..\n\nz\n\nsuch t h a t\n\n2\n\n?\n\nTSF\n\n(Q) ,\n\nE\n\n| | + Az c L ( Q )\n\n(3.14)\n\n2\n\nZj\n\n(3.13)\n\nSketch of the proof of the main r e s u l t . We i n t r o d u c e t h e s e t of f u n c t i o n s\n\n9z\n\nsuch t h a t\n\n0\n\n=\n\non z,\n\n1 {1)\n\ne\n\n2\n\nand such t h a t\n\n{ ( z ^ ) , \\),\n\n{z (t.),\n\nv\n\n2\n\nuJ)}eK\n\n1\n\n, i=l,\n\n(3.15)\n\nWe i n t r o d u c e then\n\nJT\n\n0 (z\n\n|\n\n9 z\n\n2 2\n\ne\n\ndt + \\$J z dZ + 2\n\n0\n\n1\n\nr\n\n3z\n\n+ Az\n\ndt\n\n(3.16)\n\nand we c o n s i d e r the problem infj (z), £\n\nz\n\nsatisfying\n\n(3.14)(3.15).\n\nProblem ( 3 . 1 7 ) admits a unique s o l u t i o n , denoted by (as in S e c t i o n 2)\n\np\n\n£\n\n(3.17) y . If we i n t r o d u c e\n\nby (3.18)\n\nwe f i n d t h a t\n\nl o h # >\n\nat\n\ndt\n\ngJ y\n\n+\n\nz\n\nT\n\n(\n\np\n\n0\n\nV\n\nz\n\nsatisfying\n\n(z -y\n\n£ 2\n\n£\n\n2\n\n'\n\n(\n\n^\n\n£ 2\n\n+\n\nA\n\n(3.14)(3.15).\n\nIt follows\n\nthat\n\ny\n\n3\n\n7 e 2\n\nwith the boundary c o n d i t i o n s\n\np\n\ne l\n\na\n\n2\n\nP\n\n0\n\n=\n\no\n\ne 2\n\n=\n\n= 0\n\n^\n\nn\n\n6\n\ny\n\non\n\ne2\n\n0\n\n= 0 ,\n\ne l\n\nZ\n\nZ\n\nand the \" i n i t i a l \" P (T)\n\nn\n\nconditions :\n\np (T) £ 2\n\n-5^(T)\n\n=\n\nin\n\nfi.\n\nTherefore\n\nwhere\n\n^\n\n)Vl\n\n+ A\n\n3t\n\n3t q\n\ne l\n\n= 0 aq\n\na\n\n2\n\n2 —\n\ne 2 =\n\n®e2\n\no\n\nn\n\n1\n\n3y q (T) £ l\n\n= 0 ,\n\nq (T) =\n\nBut l e t us r\n\n= f\n\nq (3)da. c\n\n£ 2\n\nintroduce\n\ne 2\n\n(T)\n\nin\n\nn.\n\n)dz-\n\n)\n\n(\n\nz\n\n~\n\ny\n\ne »\n\n158 Assuming to s i m p l i f y depend on\n\nt\n\n(but t h i s\n\nis\n\nby no means essential) t h a t\n\n, i t f o l l o w s from ( 3 . 2 4 ) ( 3 . 2 5 ) ( 3 . 2 6 )\n\nA\n\ndoes not\n\nthat\n\n0 3y,e2 at\n\n3r e2 'el\n\n=\n\n3\n\n0\n\nr (T)\n\ny 2^°\n\no\n\ne\n\nn\n\n(3.28)\n\n^'\n\n= 0.\n\ne\n\nI t f o l l o w s from standard r e s u l t s\n\nthat i\n\nr\n\nremains in a bounded s e t of L ( 0 , T ; H*(ft)x H ^ ) ) 2\n\n£\n\n(3.29)\n\n^ )\n\n1\n\ntherefore\n\nJ\n\nr\n\ni s bounded in L ( 0 , T ; 2\n\n(3.30)\n\nH*(r))\n\nand t h e r e f o r e\n\nq\n\ne 2\n\nl =^r z\n\ne 2\n\n|\n\nis\n\nbounded in\n\nBut ( 3 . 2 3 ) and ( 3 . 2 1 )\n\nH'^OJ ;\n\n(3.31)\n\nH2(r)) ^ ) .\n\nimply\n\n3\n\n(3.32) which, t o g e t h e r with ( 3 . 3 1 ) i m p l i e s t h a t i i _i ie 2 | ' bounded s e t of H (0,T r\n\nX\n\nr e m a i\n\nn s\n\ni\n\nn\n\na\n\nUsing the f a c t t h a t the follows\n\n|x | i E\n\nl\n\n2\n\nI\n\n(3.33)\n\nr | a r e l i n e a r l y independent 2\n\nthat\n\nin\n\nL (Z), 2\n\nit\n\nZ\n\n* c\n\nand the proof can be completed\n\n( )\n\n;H (r)) .\n\nby usual\n\nWe could o b t a i n more, but t h i s\n\n) H^(0,T ; X) = { 1, A'e Q\n\nL(a(T +A» ),T ) £ L ( T + A , T ) g\n\ng\n\ng\n\ng\n\nf g\n\n8\n\nFor illustration of the adaptive nature of the Bayesian method it is simplest to adopt the spike-loss which, we recall, selects the posterior mode. However, the qualitative behavior of the Bayesian analysis is typically rather insensitive to the particular choice of loss-model. It will be assumed that the observational errors are independent of the true state, T , so that, P (0 |T ) = P (0 -f(T )) s P (0 -T ) c o / o m\n\ng\n\nm\n\n8\n\nm\n\n(9)\n\nm\n\nAssuming differentiability, the optimal analysis must then obey:\n\nx\n\n( n \" ^n) \" n f\n\nx\n\nl\n\nx\n\nA\n\n(10b)\n\n( n - *n)\n\nn\n\nR\n\n(10a)\n\n= 0\n\nX\n\nn+i V i\n\ni\n\n°» \"=1,...,N\n\n=\n\n=\n\n=\n\n88\n\n0, n = l , . . . , N - l .\n\n(10c)\n\n(\n\n0\n\n° ' n l>---N. =\n\n1\n\n0\n\nd\n\n)\n\n(lOe)\n\n177 The Lagrange m u l t i p l i e r s play t h e same r o l e as t h e a d j o i n t v a r i a b l e s t h a t Le Dimet and Talagrand ( 1 9 8 5 ) and Lewis and Derber ( 1 9 8 5 ) use in\n\nconjunction\n\nwith t h e a d j o i n t a l g o r i t h m .\n\nEQUIVALENCE OF BEST-FIT TRAJECTORY TO KALMAN FILTERING To see t h a t Kalman f i l t e r i n g produces t h e same r e s u l t f o r\n\nas the\n\ndynamical t r a j e c t o r y , when both methods use t h e same d a t a , i t design a s e q u e n t i a l\n\nalgorithm f o r solving equations\n\nbest-fit\n\nis sufficient\n\n( 1 0 a ) through ( l O e ) .\n\ni s o l a t e the equations i n v o l v i n g x , x. , f , and A . and e l i m i n a t e x , A , Q\n\nto solve for x\n\nx\n\nas a function of A\n\nx\n\n£\n\nQ\n\nand of the data at t h e f i r s t\n\nX\n\nto\n\nFirst, f\n\nand\n\nl\n\ntwo time\n\nlevels: Pi\n\nx\n\n1\n\n= M;\n\nx\n\n1\n\n[A\n\nx\n\nl\n\nq\n\n+ f ]\n\n+ R\"\n\nx\n\nx\n\n1\n\n+ A| A ,\n\nx\n\n(11)\n\n2\n\nwhere [compare with e q u a t i o n s (4) and ( 6 ) ] : P\"\n\n= M\" + R^\n\n1\n\n1\n\n(12)\n\n1\n\nand M\n\ni\n\n\\ o I\n\n=\n\nR\n\nA\n\n+\n\nNote t h a t i t t\n\nQii s through A\n\nthrough t ^ .\n\n£\n\nfirst\n\n( 1 3 )\n\nIf A\n\nt h a t the solution\n\ntime step using a Kalman f i l t e r :\n\nfor x\n\nx\n\nQ\n\nx\n\nQ\n\ndepends on t h e data f o r time\n\nx\n\ncorresponds t o t h e s t a r t - u p i n i t i a l\n\nto t h e i r error-covariances; z =\n\nc o n d i t i o n s and R t.;\n\n£\n\n2\n\nwere z e r o , then t h e s o l u t i o n would be t h e same as f o r t h e\n\nA x 1\n\n+ f\n\nM. t o the e r r o r - c o v a r i a n c e of t h e f o r e c a s t ; x^ and R\n\nx\n\nassimilated\n\nat time t ; and P x\n\nto the f o r e c a s t f o r\n\nl\n\nQ\n\nt o the data\n\ni s the e r r o r - c o v a r i a n c e a f t e r the data have been\n\nx\n\nassimilated. Continue with t h e s e q u e n t i a l for x\n\nx\n\nobtained by s e t t i n g\n\nP^\n\nx\n\n= M\" [ A ^\n\n1\n\ny\n\n1\n\n+ f ] x\n\nA\n\n2\n\n+ R\"\n\nNow, t h e equation coupling x\n\n1\n\na l g o r i t h m by l e t t i n g y\n\nx\n\nx\n\nA\n\n2\n\nwhere y\n\n+ P\"\n\nx\n\n1\n\n(x\n\n-\n\nL\n\ny )\n\n£\n\n(and t h e r e b y t o data f o r t > t ) x\n\n(15) data and P\n\nx\n\nt h e i r error-covariance matrix. x\n\nP^\n\n1\n\nx\n\n2\n\n= l^\n\n1\n\n[A y 2\n\n2\n\nand f\n\n2\n\nand of t h e data f o r times up t o and i n c l u d i n g\n\nL\n\ncan be\n\n(lOd):\n\nRepeating t h e same p r o c e d u r e , now e l i m i n a t e x , A , 3\n\nl\n\n= 0,\n\nx\n\np l a y s t h e r o l e of i n i t i a l\n\na function of A\n\nsolution\n\nt ): (14)\n\nw r i t t e n in e x a c t l y t h e same form as\n\n-A*\n\nat\n\nl f\n\nto A\n\nx\n\nrepresent the\n\n= 0 (the Kalman-fiIter solution\n\n+ f ] 2\n\n+ R\"\n\n1\n\nx\n\n2\n\n+ Aj A , 3\n\nand e x p r e s s x\n\n2\n\nas\n\nt : 2\n\n(16)\n\n178 where P 2.\n\nand\n\n2\n\nNow, y\n\na r e defined\n\nby ( 1 2 ) and ( 1 3 ) w i t h i n d i c e s\n\ncan be defined as the s o l u t i o n\n\n2\n\nrecognized t o be i d e n t i c a l time s t e p s .\n\nthat the solution\n\nfor\n\nwhen A\n\n2\n\n3\n\nincremented from 1 t o = 0 and can be\n\nt o the r e s u l t given by Kalman f i l t e r i n g a f t e r two\n\nThen, by w r i t i n g A\n\nbe repeated a t h i r d t i m e .\n\nfor x\n\n3\n\nas a function of y\n\nWhen a l l will\n\n£\n\nand P , t h e procedure can £\n\nthe data have been a s s i m i l a t e d ,\n\nit\n\nis\n\nclear\n\nbe t h e same as the K a l m a n - f i I t e r s o l u t i o n .\n\nCONCLUSION For the Kalman f i l t e r t o g i v e t h e same r e s u l t as the b e s t - f i t both methods must use e x a c t l y the same i n f o r m a t i o n . conditions\n\ntrajectory,\n\nThe s t a r t - u p i n i t i a l\n\nand t h e i r e r r o r - c o v a r i a n c e s , which c o n s t i t u t e t h e p r i o r knowledge\n\nt h e Kalman f i l t e r , \"observations\". same v a l u e s x\n\nn\n\nmust be the same as the data x\n\nSimilarly all and f\n\np\n\nQ\n\nand R\n\nQ\n\nfor\n\nf o r the e a r l i e s t\n\ndata t h a t a r e t o be a s s i m i l a t e d\n\nmust have t h e\n\nas well as the same e r r o r - c o v a r i a n c e m a t r i c e s R\n\nand Q\n\nn\n\nn\n\nf o r the two methods. An important p a r t of the K a l m a n - f i l t e r i n g approach i s the i n c l u s i o n stochastic\n\nforces.\n\nThe c o u n t e r p a r t in the w e i g h t e d - l e a s t - s q u a r e s\n\nt h e allowance f o r the u n c e r t a i n t y of the f o r c i n g o b s e r v a t i o n s . data f ,\n\nwhich r e p r e s e n t the expected\n\nn\n\nv a l u e s of the f o r c i n g ,\n\nf\n\nn\n\n= 0 , allowance i s made f o r s t o c h a s t i c\n\nd e t e r m i n i s t i c a l l y unforced model. stochastic\n\nforcing\n\nis\n\nLikewise,\n\nforcing\n\nin t h e l i m i t of Q\n\nis\n\nThe f o r c i n g\n\na r e the\n\nt h a t a r e used by the f o r e c a s t model when advancing t h e s t a t e from t Thus, by s e t t i n g\n\nof\n\napproach\n\nn\n\nn\n\nquantities -\n\nto\n\n1\n\nt . n\n\nin a\n\n= 0,\n\nall\n\nremoved.\n\nAlthough the Kalman f i l t e r y i e l d s the same r e s u l t f o r the p r e s e n t s t a t e as the b e s t - f i t\n\ntrajectory,\n\ntwo approaches. filtering time s t e p ;\n\nt h e r e can be p r a c t i c a l d i f f e r e n c e t h a t d i s t i n g u i s h\n\nFor systems\n\nhaving only a few degrees of freedom, Kalman\n\nhas the advantage t h a t t h e r e is no need t o keep data f o r more than one all\n\npast o b s e r v a t i o n s a r e compressed\n\ni n t o t h e present f o r e c a s t and\n\nthe information about t h e i r accuracy has been compressed c o v a r i a n c e s of the f o r e c a s t . computing\n\ni n t o the e r r o r -\n\nThe p r i c e of t h i s convenience\n\nthe e r r o r - c o v a r i a n c e matrix at each time s t e p ;\n\nr e q u i r e s as much computational\n\ni s the n e c e s s i t y\n\nt o do t h i s p r o p e r l y\n\nFor l a r g e systems\n\nbecomes c o m p u t a t i o n a l l y i m p r a c t i c a l and f a s t e r methods a r e needed f o r the e v o l u t i o n of the s t a t e e r r o r - c o v a r i a n c e m a t r i x . approach has the advantage t h a t the s o l u t i o n compute i t s e r r o r - c o v a r i a n c e m a t r i x . as c o n j u g a t e - g r a d i e n t descent But i f\n\nThe b e s t - f i t\n\nmust be approximated i f\n\nthis\n\nmodelling\n\ntrajectory\n\ncan be obtained without having\n\nto\n\nFor example, an i t e r a t i v e a l g o r i t h m such\n\ncan be used t o s o l v e equations\n\nr e s u l t s from the end of one o b s e r v a t i o n a l\n\nas data at t h e beginning\n\nof\n\ne f f o r t as f o r t h e f o r e c a s t alone m u l t i p l i e d by\n\nt w i c e the number of degrees of freedom of the system.\n\n(lOe).\n\nthe\n\n( 1 0 a ) through\n\ni n t e r v a l a r e t o be used\n\nof the n e x t , then t h e weight matrix of t h e i t has not been computed.\n\nsolution\n\nThe l e a s t - s q u a r e s approach\n\n179 a l s o has the advantage of providing t h e best f i t throughout t h e o b s e r v a t i o n a l i n t e r v a l , which might be useful situations\n\nf o r hindcast\n\nstudies,\n\nand i t\n\nis better-suited\n\nin which the f o r c i n g must be recovered from o b s e r v a t i o n s of t h e\n\nas a function\n\nof\n\nto\n\nstate\n\ntime.\n\nREFERENCES Gandin, L . S . , 1 9 6 3 . O b j e c t i v e A n a l y s i s of Meteorological F i e l d s . T r a n s l a t e d by I s r a e l Program f o r Technical T r a n s l a t i o n . A v a i l a b l e from U.S. Department of Commerce Clearinghouse f o r Technical I n f o r m a t i o n . Kalman, R . E . , 1 9 6 0 . A new approach t o l i n e a r f i l t e r i n g and p r e d i c t i o n problems. T r a n s . ASME Journal of Basic Engineering, 8 2 : 3 5 - 4 5 . Le Dimet, F . - X . and Talagrand, 0 . , 1 9 8 5 . V a r i a t i o n a l a l g o r i t h m s f o r a n a l y s i s and a s s i m i l a t i o n of m e t e o r o l o g i c a l o b s e r v a t i o n s : T h e o r e t i c a l a s p e c t s . Submitted to Tellus. Lewis, J . M . , and Derber, J . C , 1 9 8 5 . The use of a d j o i n t equations t o s o l v e a v a r i a t i o n a l adjustment problem with a d v e c t i v e c o n s t r a i n t s . Submitted t o Tellus. Thacker, W . C , 1 9 8 5 . A c o s t - f u n c t i o n approach t o t h e a s s i m i l a t i o n of a s y n o p t i c data. Submitted t o T e l l u s .\n\n181\n\nASSIMILATION OF DYNAMICAL DATA IN A LIMITED AREA MODEL\n\nF.-X. LE DIMET\n\nand A. NOUAILLER\n\nApplied Mathematics Department, LAMP, University of Clermont II, B.P.45, 63170 Aubiere, (France)\n\nABSTRACT A method for recovering dynamical meteorological fields from sparse data is proposed. It is founded on variational techniques used in optimal control theory.. A description of the associate algorithm is given with an application with real data to the retrieval of a squall line on Western Africa.\n\nINTRODUCTION Retrieving meteorological fields from sparse observations in time and space needs to add some complementary information to the data. Most of the time this information is of statistical nature, unfortunately, useful statistics are not always available especially for small scales in space (regional forecasting) and time (rare or paroxismic events). Variational methods were firstly introduced in meteorology by Sasaki (1958, 1970), in this approach the information added to the data is the set of equations supposed to modelize the atmospheric flow. The fields are adjusted in such a way that (i)\n\nthey are as close as possible from the observation\n\n(ii) they verify the model used as a constraint. A major difficulty encountered in variational methods has been to analyse dynamical data i.e., to retrieve meteorological fields from observations dis­ tributed in time. For instance, (Lewis and Bloom 1978) give some techniques which are extensions of the steady state methods. Optimal control theory (Lions, 1971) provides a way to go through this obstacle due to the computational phase of the problem. A general formalism (Le Dimet, Talagrand (1986)) for the assimilation of dynamical data is given next and applied to real data on a limited area do­ main. GENERAL FORMALISM Let us consider a domain fi on which the meteorogical fields are represented by a variable X (X may include wind fields, geopotential, temperature, . . . ) .\n\nWe\n\nwill suppose that the evolution of the dynamical field is governed by the dif­ ferential system\n\nj£ =\n\nA(X)\n\nwhere A is some (nonlinear) partial differential operator with respect to the\n\n(1)\n\n182 space variables and that A is such that (1) provided with an initial condition Z has a unique solution\n\non the time interval\n\n[0,T].\n\nAn observation X of X is\n\ndone on [ o , t J X Q , for sake of simplicity we assume that it is continuous in space and time. The optimal analysis X * is defined as the closest solution of (1) from this observation. To X * is associated an initial condition Z * determined in such a way that J(Z*) = M i n J ^\n\n| |x (t)-X(t) | | d t .\n\n(2)\n\n2\n\nz\n\nTherefore, the problem of the optimal variational analysis is to determine Z * verifying (2). Let us explicit the algorithm on the spatially discretized dX — dt\n\nproblem\n\n= F(X )\n\n(3)\n\nn\n\nX\n\nbeing the discretized variable belonging to a finite dimensional space at\n\nr\n\neach time. In the next we will omit the subscript n. The optimality condition is written VJ(Z*)\n\n= 0\n\n(4)\n\nVJ being the gradient of the functional J with respect to the discretized initial condition\n\nZ .\n\nComputation of the gradient Deriving J with respect to the initial condition gives J(Z) = 2\n\n(X (t)-X(t).W)dt\n\n(5)\n\nz\n\nW is the derivative of the trajectory with respect to the initial condition. H being some admissible initial condition, a a scalar (3) is written with initial condition Z than with Z+aH d X\n\nZ\n\nd X\n\nd7~\n\n(a)\n\n=\n\nX(0)\n\nF\n\nV\n\n(\n\n= Z\n\n(b) \"\n\nG\n\n(\n\n=\n\nF ( X\n\nZ+aH\n\n}\n\n(6)\n\n= Z+aH\n\n(6a) from 6 b ) , and dividing by a , a goes to zero we get\n\nV -.ti\n\nU ^ A ; 7\n\ndt X(0)\n\nAfter substracting\n\ndw d t =-\n\nX+aH\n\nH\n\n(7)\n\nW(0) = H G is the Jacobian matrix of F. The adjoint system of (3) is introduced g\n\n^ G ( X ) .P = X - X\n\n(8)\n\nQ b s\n\nwith the condition P(T) = 0\n\n; G being the transpose of G. t\n\n(8) is multiplied by H after integrating by parts and using (5) and (7) we get VJ(Z) = 2P(0). Therefore, the gradient of J is twice the final value of the adjoint system\n\n183 integrated backwards from T to 0. This estimation of the gradient permits us to perform a classical method of optimization without constraint (gradient, con­ jugate gradient). APPLICATION TO A REAL CASE The domain of the experiment (Fig.]) is a 60 x 60 km square including 20 stations measuring wind and pressure each 30\" for someone (ALICE) and each 2 3 0 \" f\n\nfor the others (DELTA). The period of observation used, ranged from 3.00 to 8.00 on June 22nd corresponding to the passage of a squall line over the site. We have assumed that the equations governing the flow were\n\nin\n\na dX u\n\n+\n\n3t\n\nix\n\n3t at\n\n+\n\n^ c\n\n+\n\n+\n\n¥\n\n3u dy\n\nV77—\n\n+\n\n3v , dy + c\n\n3u dz\n\nfv +\n\nav dz\n\nfu +\n\nWr\n\n+\n\nk\n\np\n\n|f\n\n1\n\no\n\np\"\n\n+\n\nS|U|U\n\n(9-a)\n\n0\n\n=\n\n(9-b)\n\n1\n\no\n\n(9-c)\n\ndiv(U) = 0\n\nX dx where t is the time, w,y and z are space coordinates, u and v the horizontal components of the wind, w is the vertical one, p: pressure, p ^ : density ed to be constant), f: Coriolis parameter, C , C ^\n\nx\n\n(assum­\n\ncomponents of the squall line as estimated from the radar observations, C^ is the drag coefficient, | u | * (u^ + u^) The available data were u, v and p at the ground.The terms of vertical transport (wl^ and W T T ^ ) were estimated from the observation. For sake of simdz dz plicity\n\n| u | has been approximated by |\n\nu o\n\n^ | in the friction terms. The index s\n\n\"obs\" meaning observation and k estimated to be equal to 75. (S.I. units). The boundary values were prescribed on the inflow boundary and linearly ex­ trapolated from inside on the outflow boundary. Numerical Results and Conclusion The time integration used a leap frog scheme with a 30 second time-step. The optimization procedure was performed with a conjugate gradient algorithm. The cost of the method is proportional to the number of calls to the pro­ cedure evaluating J. After 100 estimations of J, its value has been decreased b\\ about 50%. Fig. 2 and Fig. 3 show the wind and pressure fields respectively between 3.50 and 4.20. The poor resolution in the wind field in the vicinity of the in­ flow boundary is due to a crude approximation of the drag forces. A sensible improvement could be done using the boundary term together with initial con­ dition as control variable. ACKNOWLEDGMENT This work was supported by contract INAG-ATP Recherches Atmospheriques. Computations were performed on the Cray IS of CCVR (Palaiseau, France) and figures realized\n\nusing the NCAR (National Center for Atmospheric Research)\n\n184 program which is supported by the National Science Foundation\n\n(U.S.A.)*\n\nThis method is a way for unifying analysis, data estimation and initiali­ zation furthermore it can be generalized including filtering of gravity waves (Le Dimet, Sasaki, White, 1983).\n\nREFERENCES Le Dimet, F.-X., Sasaki, Y.K. and White, L., 1983. Dynamic initialization with filtering of gravity waves. CIMMS, Report and Contribution N°40, University of Oklahoma U.S.A. Le Dimet, F.-X. and Talagrand, 0., 1986. Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects, accepted for publication by Tellus. Lewis, J.M. and Bloom, S.C., 1978. Incorporation of time continuity into subsynoptic analysis by using dynamical constraints. Tellus, 30: 496-5 16. Lions, J.L., 1971. Optimal control of systems governed by partial differential equations. Springer-Verlag Berlin, 396 pp. Sasaki, U., 1958. An objective analysis based on the variational method, J. Meteor. Soc. Japan, 36: 738-742. Sasaki, Y., 1970. Some basic formalisms in numerical variational analysis. Mon. Wea. Rev., 98: 875-883.\n\n185\n\nN\n\n10 k m\n\no °«\n\n#KORHOGO\n\nA\n\nC\n\nK\n\n?\n\n°7 A8\n\nw ——\n\nFig. 1\n\nFig.l: site of the experiment Fig.2(a-b) and 3(a-b): wind and pressure fields respect­ ively at 3.50 a.m. and 4.20 a.m. on 22nd June, 1981.\n\n7\n\nA\n\nO Alice Station\n\no\n\n15\n\no.°\"\n\nA Delta Station\n\ncontour from 96940 Pa. to 97170 Pa.\n\ncontour from 96980 Pa. to 97160 Pa.\n\n189\n\nVARIATIONAL PRINCIPLES AND ADAPTIVE METHODS FOR COMPLEX FLOW PROBLEMS\n\nJ.\n\nTINSLEY ODEN, T. STROUBOULIS, and PH. DEVLOO\n\nTexas\n\nInstitute\n\nTexas, 7 8 7 1 2 ,\n\nf o r Computational\n\nMechanics,\n\nThe U n i v e r s i t y of\n\nTexas,\n\nAustin,\n\nU.S.A.\n\nABSTRACT Oden, J . T . , S t r o u b o u l i s , T. and Devloo, P h . , 1 9 8 6 . V a r i a t i o n a l p r i n c i p l e s and adaptive methods for complex flow problems, Variational Methods in the Geosciences, Elsevier Science Publishers, N.Y.\n\nThis paper p r o v i d e s a b r i e f summary of s e v e r a l a d a p t i v e f i n i t e element methods t h a t a r e under development a t TICOM f o r the a n a l y s i s of complex problems in s o l i d and f l u i d mechanics.\n\nINTRODUCTION How good a r e the\n\nnumerical\n\nf i n i t e element g r i d ? and what polynomial\n\nsolutions?\n\nWhat t y p e s of elements\n\nThese a r e the t r a d i t i o n a l q u e s t i o n s\n\na posteriori\n\nplace\n\nshould one use?\n\nthe\n\nnodes\n\nin a\n\nHow many elements\n\nthe problem a t hand with\n\nresources a v a i l a b l e ?\n\ncomplex problems on modern computers.\n\ndeveloping\n\ndo we\n\ndegree a r e a p p r o p r i a t e f o r s o l v i n g\n\nthe l i m i t e d computational\n\nthese questions\n\nWhere\n\nt h a t a r i s e in p r a c t i c a l Recently,\n\nthe q u a l i t y of the s o l u t i o n\n\nerror\n\nestimates;\n\nthe\n\ncalculations\n\nof\n\nsome p r o g r e s s toward answering\n\ncorrect\n\ncan be e s t i m a t e d by\n\ns t r u c t u r i n g of\n\nthe mesh\n\ncan be determined by a d a p t i v e p r o c e d u r e s . In the these\n\npresent\n\nnote,\n\nwe o u t l i n e\n\na r e a s f o r problems in f l u i d\n\nefficient\n\na brief\n\nsummary of details,\n\nDemkowicz e t a l . ,\n\nA POSTERIORI\n\nresults\n\nvalue\n\nin i t s\n\nt h a t have been obtained\n\nThe i s s u e s of\n\nimplementation.\n\nobtained\n\nsee Oden e t .\n\nal.,\n\nby the\n\nauthors\n\nin\n\nimportance a r e the itself,\n\nThe p r e s e n t d i s c u s s i o n in\n\nr e c e n t months\n\n1985 and 1 9 8 6 ; Demkowicz\n\n(for\n\nand Oden, 1 9 8 6 ;\n\n1984 and 1 9 8 5 ; see a l s o , Babuska and R h e i n b o l d t ,\n\n1978a,\n\nb).\n\nERROR ESTIMATES\n\nWe begin by o u t l i n i n g estimates\n\nresults\n\nc o n s t r u c t i o n of r e l i a b l e e r r o r bounds, the a d a p t i v e a l g o r i t h m\n\nand the data s t r u c t u r e e s s e n t i a l is\n\nseveral dynamics.\n\nfor\n\nfinite\n\nproblems.\n\nWhile\n\nt h r e e general methods f o r o b t a i n i n g a-posteriori\n\nelement all\n\nof\n\napproximations the\n\nmethods\n\nof\n\ngeneral\n\ninvolve\n\nthe\n\nboundary and calculation\n\nerror initial\n\nof\n\nlocal\n\n190 element\n\nresiduals,\n\nthe f i r s t\n\nclass\n\ns i m i l a r t o those used in t i m e - s t e p\n\nfocuses\n\ncontrol\n\non a s t r a i g h t f o r w a r d\n\ncomputation\n\nin o r d i n a r y d i f f e r e n t i a l\n\nequations.\n\nEvolution of the Error Consider a l i n e a r p a r a b o l i c problem c h a r a c t e r i z e d by t h e v a r i a t i o n a l\n\nstate­\n\nment: Find |r-\n\n6 Hq(«)\n\nt •> u ( x , t )\n\nd> dx + a(u,)\n\n=\n\nf\n\nsuch\n\nthat\n\nt dx\n\nV 4 6 H*(n) where and\n\ncz K\n\n,\n\nn\n\nsmooth\n\nb\n\ndx = d X j d x £ . . . d x positive\n\nof t h e Sobolev s p a c e\n\nfunctions\n\n= span U j ,\n\nh\n\nFind\n\nJ\n\nt\n\nu\n\nN\n\nThus,\n\n(aVu«v + b-Vu)dx\n\nWe a p p r o x i m a t e\n\nx .\n\nbasis\n\nfl)\n\nover\n\nfunctions\n\nwith a\n\na\n\nsubspace\n\ndefined\n\nover\n\nelements:\n\nsuch\n\nh\n\nuJ d) dx + a ( u \\ d > ) N\n\nObviously,\n\nn\n\napproximation of ( 1 ) i s then: CH\n\nh\n\n9\n\n(J)^} d H g ( f i )\n\n2>\n\nThe s e m i d i s c r e t e\n\nof\n\na (u ) = /\n\nH q ( a ) spanned by polynomial\n\na r e g u l a r mesh of f i n i t e H\n\n,\n\n(1)\n\nthat\n\nf dx\n\n=\n\nV\n\nN\n\nthe e r r o r\n\ne*\n\n1\n\nis\n\nrelated\n\nto\n\nN = 1, 2,\n\nu\n\nand\n\nM\n\nu*\n\nby\n\n1\n\n(2)\n\nu = u^ + e* . 1\n\n(1) yields,\n\nel j dx + a(e ,) = - < r , d »\n\nVd>\n\nh\n\nh\n\nwhere\n\nr. n\n\n(3)\n\ni s the r e s i d u a l and\n\n^\n\nand\n\ne\n\nh by\n\nE\n\nto\n\nobtain\n\nthe\n\nsystem\n\nof\n\nordinary d i f f e r e n t i a l equations, dE(t) M\n\n+ KE(t) = R ( t )\n\nwhere\n\nE\n\n(5)\n\ni s the K - v e c t o r of nodal e r r o r v a l u e s\n\nE\n\nand\n\na\n\nM,\n\nK ,\n\nand\n\nR\n\nare\n\nm a t r i c e s with elements =\n\nI\n\nM\n\na6\n\n*a *6\n\nR\n\na = \" < V V\n\nfl\n\nWe s o l v e and o b t a i n typical\n\nd\n\nX\n\n;\n\nE*\n\na t each\n\nelement\n\nL - e r r o r over 2\n\nm\n\ne\n\n)\n\n=\n\na\n\nx\n\nin\n\nfinite\n\nan a p p r o p r i a t e temporal\n\nfi\n\nand a t each\n\nelement\n\nmesh.\n\ntime Then,\n\nt . for\n\nintegration Let\n\nfi e\n\nexample,\n\nscheme\n\ndenote a the\n\nlocal\n\ni s e s t i m a t e d by the e r r o r i n d i c a t o r ,\n\ng\n\nfn J\n\nin\n\nfi\n\nE\n\nj\n\na6\n\n=\n\nn u m e r i c a l l y using\n\n2\n\nI E VL (fi\n\na6\n\n(5) 1\n\n=\n\nK\n\nh 2\n\nI\n\ndx -\n\nM\n\ne\n\n*B\n\nf l\n\nd\n\nE\n\na\n\nm,\n\na fte\n\nE\n\n6\n\nX\n\ne A number of\n\nschemes can be developed which employ e v a l u a t i o n e q u a t i o n s\n\nthe type ( 5 ) but which d i f f e r\n\nin the way the f u n c t i o n s\n\n^\n\na\n\nof\n\na r e computed.\n\nResidual Methods We next c o n s i d e r the a b s t r a c t b o u n d a r y - v a l u e problem, Find\n\nu\n\nin\n\n=\n\nV\n\nsuch t h a t\n\nfor all\n\nv\n\nin\n\nV\n\n(6)\n\nwhere A\n\n= a (possibly\n\nn o n l i n e a r ) o p e r a t o r from a r e f l e x i v e Banach space\n\nV\n\ni n t o i t s dual\n\nv\n\n= an a r b i t r a r y t e s t f u n c t i o n in\n\nf\n\n= given data in\n\n= d u a l i t y p a i r i n g on\n\nof\n\nV\n\nV\n\nV\n\n• V\n\n*V\n\nThis problem i s e q u i v a l e n t t o the a b s t r a c t problem: A G a l e r k i n approximation of f i n i t e dimensional\n\nsubspace\n\n(6) of\n\nconsists V\n\nof\n\nsuch t h a t\n\nAu = f\n\nseeking\n\nin\n\nV\n\na function\n\n. u\n\nh\n\nin a\n\n192\n\n< A u\n\nv\n\n>\n\n=\n\n^\n\n<\n\nn\n\nh\n\n, v\n\n^\n\n>\n\nThe r e s i d u a l satisfy\n\no\n\nr^\n\n^\n\nr\n\nis\n\nthe o r i g i n a l\n\nv\n\nh\n\n1\n\nn\n\nt h e degree with which\n\nconditions\n\nthe approximation\n\nu^\n\nfails\n\nto\n\non the s o l u t i o n :\n\n• Since the r e s i d u a l\n\nbelongs\n\nt o the dual\n\nspace\n\nV\n\nand not n e c e s s a r i l y N\n\nmagnitude must be measured with r e s p e c t to the norm\n\nII\n\nI *! * 1\n\n1\n\nV , its\n\n*\n\non\n\nV\n\n:\n\n11 r\n\nh\"*\n\ns\n\n=\n\nu\n\n^vr-\n\np\n\nv\n\ng\n\nv\n\n(7) =\n\nsup !lv!' 1 , spanned by piecewise\n\nP\n\nh»* *\n\nc | l v\n\no\n\n\" h\" v\n\n+\n\nr\n\ns\n\nu\n\npolynomials of degree\n\nof t h e r e s i d u a l\n\nh\n\nelement\n\nspace\n\np .\n\ni s c o n s t r u c t e d according to\n\np\n\n(\n\n8\n\n)\n\nIvPLl\n\nC\n\nwhere element\n\nelements h =\n\nis\n\nof fi\n\nIf\n\nh\n\nVQ is is\n\nan element\n\nt h e mesh\n\nsize\n\nof\n\nV\n\n(i.e.,\n\nand\n\nv^\n\nis\n\nan a r b i t r a r y\n\nfor a partition\n\nT^\n\nof\n\nsup\n\nby\n\n),\n\ng\n\nmax e\n\na constant, VJjJ .\n\nIi\n\n,\n\nh\n\n= diameter (fi )\n\nh\n\nwe g e n e r a l l y have\n\nIv\n\n-\n\nV\n\nP|I\n\n=\n\n0(h)\n\nso t h a t i t makes sense a s y m p t o t i c a l l y sup h\n\nV\n\n.\n\n(as\n\nh\n\n0) t o approximate\n\nh\n\n193 I n t e r p o l a t i o n Error I t i s well\n\nEstimates\n\nknown ( s e e ,\n\ne.g.,\n\nOden and Carey, 2\n\nproblems the approximation e r r o r so c a l l e d\n\n!e l h\n\ninterpolation\n\n* C|u - v | |\n\nv\n\nh\n\n|\n\nu\n\n-\n\nh l l , n *\n\nu\n\nc\n\n1 v v\n\nIf\n\nu\n\nh\n\nu = 0\n\nu\n\n-\n\nh11\n\nv\n\nf\n\nb\n\nin \"\n\n| U v v\n\nh\n\ni s smooth enough,\n\ns\n\nC\n\ncan be bounded above by the\n\nV\n\n6 H\n\nh\n\n(9)\n\nh\n\nproblem\n\ndQ. , we have ^ l ^\n\n(\n\na local\n\n1 0\n\n)\n\ni n t e r p o l a t i o n e r r o r e s t i m a t e can be d e r i v e d\n\nQ^-elements)\n\nof the type ( f o r\n\nl\n\nin the case of the D i r i c h l e t\n\n2\n\nelliptic\n\nerror,\n\nVv\n\nq ,\n\nin\n\n= ||u - u j |\n\nv\n\nThus, f o r i n s t a n c e ,\n\n-au = f\n\n||ej|\n\n1981) that for linear\n\nh\n\nel l2,fi u\n\ne\n\nwhere\n\n-\\\n\nN^n\n\n0\n\n^'xx\n\n+\n\n^\n\ny\n\ny\n\n^\n\nThe b a s i c problem we f a c e when attempting t o make use of any of t h e s e mates\n\nis\n\nsolution\n\nt h a t we must using\n\ncalculate\n\nthe\n\nonly a v a i l a b l e i n f o r m a t i o n ,\n\na v a i l a b l e f i n i t e element s o l u t i o n for\n\nestimating\n\nhigher\n\nthe\n\nu\n\n.\n\nh\n\nsecond d e r i v a t i v e s\n\norder i.e.,\n\nthrough use of\n\nsomewhat i n t u i t i v e and not a l l\n\nu\n\n,\n\nu\n\nand M i l l e r ( 1 9 8 4 a , b ) .\n\nu\n\nesti­\n\nunknown\n\nthe c u r r e n t l y techniques\n\n, but many a r e ,yy Exceptions\n\nare\n\n\" e x t r a c t i o n formulas\" introduced by Babuska\n\nA discussion\n\ni s given in Demkowicz e t a l .\n\nor ,xy\n\na r e based on r i g o r o u s e s t i m a t e s .\n\nbased on s o - c a l l e d\n\nthe\n\nThere a r e numerous a priori j xx\n\nthe techniques\n\nd e r i v a t i v e s of\n\nof s e v e r a l methods f o r e s t i m a t i n g\n\n|u|\n\n(1985).\n\nADAPTIVE METHODS Once a l o c a l\n\nestimate\n\ni s a v a i l a b l e , the l o c a l\n\nq u a l i t y of the s o l u t i o n can be\n\nimproved by adapting the s t r u c t u r e of the method in one of the f o l l o w i n g ways: h-methods\n\n- - reducing\n\nthe\n\nmesh\n\nsize\n\nh\n\nby\n\nautomatically\n\nrefining\n\nthe\n\nmesh; r-methods - - d i s t o r t i n g the mesh by r e d i s t r i b u t i o n (moving) the p-methods - - i n c r e a s i n g\n\nthe l o c a l\n\nshape f u n c t i o n s on a f i x e d\n\npolynomial\n\nmesh.\n\nnodes;\n\ndegree of the f i n i t e\n\nelement\n\n194 While we have developed a l g o r i t h m s in a l l t h r e e c a t e g o r i e s , we s h a l l\n\noutline\n\nonly an h-method and an r-method h e r e .\n\nAn h-Method An e f f e c t i v e h-method i s c h a r a c t e r i z e d by the f o l l o w i n g a l g o r i t h m . 1)\n\nOn an i n i t i a l\n\neach element\n\nc o a r s e uniform mesh,\n\nin the mesh using\n\ntt\n\nQ\n\ncompute e r r o r\n\none of\n\nindicators\n\nthe techniques\n\ndescribed\n\nover\n\ne\n\nearlier.\n\nFor time-dependent problems, t h i s process i s done a t each time s t e p o r a f t e r a f i x e d number of time s t e p s . four element c l u s t e r s i s\n\nMAX\n\n9\n\n=\n\nm\n\na\n\nx\n\n0\n\ne\n\n5\n\n.k GR0UP\n\n*\n\nu\n\nwhere\n\nJc j U\n\n£\n\nParameters\n\na9\n\na\n\nMAX\n\nGR0UP\n\n9\n\nThe\n\n£\n\n8\n\n°MAX\n\nrefinement\n\nand\n\nr\n\ne\n\nf\n\ni\n\nn\n\ne\n\nu\n\nn\n\nr\n\ne\n\nf\n\ni\n\nn\n\noperation\n\nproblems)\n\ngroup.\n\nunrefinement\n\nThe\n\nin group\n\nk\n\nin the mesh.\n\n3\n\nare specified\n\nto d e f i n e when the mesh i s\n\nis\n\na\n\nof\n\nto be\n\nIf\n\n2-dimensional\n\n4)\n\nj\n\ni s the e r r o r i n d i c a t o r f o r element\n\n0.\n\n3)\n\ne\n\nrefined o r unrefined. e\n\nmesh f i n e enough to i d e n t i f y groups of\n\nCompute\n\n2)\n\n9\n\nAn i n i t i a l\n\nused.\n\ne\n\ninto\n\nbisection\n\nfour\n\nelements\n\noperation\n\na\n\nwhich\n\ncollapses\n\nsingle\n\ndefines a\n\nanother\n\ngroup\n\ninto\n\n4-element a\n\nsingle\n\nelement.\n\nA J a c o b i - c o n j u g a t e g r a d i e n t scheme can be used in applying t h e s e\n\nto e l l i p t i c\n\n(for\n\nproblems which p r o v i d e s f o r the\n\nrefinement process\n\nout with a r b i t r a r y node and element numbering. Demkowicz e t a l .\n\nto\n\nbe\n\nsteps\n\ncarried\n\nFurther d e t a i l s can be found in\n\n(1985).\n\nAn r-Method The idea here i s to e q u i d i s t r i b u t e the e r r o r on a mesh c o n s i s t i n g number. M of elements (see Diaz e t a l . , f(u)\n\n, where\n\nu\n\n1983).\n\nIndeed, i f\n\ni s the r e s t r i c t i o n of the e x a c t s o l u t i o n\n\ne s o l u t i o n t o the o p t i m i z a t i o n problem\n\ne\n\nto\n\nof a f i x e d\n\ni s of the form 0 , then the e\n\n195 J(h) - I f e. J ( h ) = PI Jo \\ © dx e fi e\n\nJ(h) , J(h) ,\n\nminimize subject\n\nto\n\ne\n\nthe\n\nconstraint,\n\n/ dx/h\n\n= M\n\n(for\n\nThus, we proceed as f o l l o w s : (1) Let 0 be the o r i g i n of a f i x e d global position\n\nv e c t o r from\n\n0\n\nt o the c e n t r o i d s\n\nf o u r q u a d r i l a t e r a l elements of area element\n\nIr B\n\ne\n\nis\n\n4\n\n.\n\ne\n\nh\n\n=\n\nj\n\n0\n\n/\n\nA\n\nj\n\nA. .\n\ndim\n\n= 2)\n\nis\n\n= CONST.\n\ne\n\nc o o r d i n a t e system and\n\nof elements\n\ni\n\ny.\n\nthe\n\nin a c l u s t e r\n\nk\n\nof\n\nThe e r r o r i n d i c a t o r per u n i t area\n\nin\n\nCompute,\n\nh\n\n4\n\nf'kfiW\n\n»\n\nIs\n\n(2) x\n\nk\n\nFor each c l u s t e r\n\n= C /B k\n\n(3)\n\nk , compute the a r e a - c e n t e r of e r r o r\n\nx\n\n, by\n\nx\n\nto\n\n.\n\nk\n\nMove the i n t e r i o r node of c l u s t e r\n\nt r i b u t e the e r r o r in c l u s t e r\n\nk\n\nt o c o i n c i d e with\n\nequidis-\n\nk .\n\n(4) Repeat t h i s process o v e r a l l 4-element c l u s t e r s in the mesh, and conL. t i nlue u e t h i s process u n t i l the l o c a t i o n s x converge to d e f i n e a f i n a l optimal mesh. A NUMERICAL EXAMPLE We\n\ncite\n\nabove.\n\none\n\nnumerical\n\nexample\n\nThe problem considered\n\nperformed\n\nhere i s\n\nusing\n\nof an i n v i s c i d compressible gas through a channel indicated.\n\nof compressible\n\n( r a t i o of s p e c i f i c\n\ny = 1.4 .\n\nThe f i n i t e bolic\n\nFull\n\nThe i n i t i a l\n\nmethod\n\nused t o\n\nis\n\ngas dynamics, with a gas\n\nused t o model\n\ndetails\n\nin\n\nintegrate\n\nthe governing\n\nthis\n\nthe e q u a t i o n s\n\na l g o r i t h m a r e given\n\nin time in\n\nsystem\n\nof\n\nhyper­\n\nA t w o - s t e p Laxto a\n\na forthcoming\n\nc o a r s e mesh i s shown in Figure 1 ( a ) and the computed\n\nshown in Figure 1 ( b ) .\n\nand w i d t h .\n\nprofiles.\n\ngov­\n\nconstant\n\nreport\n\nresults\n\nThere we see computed d e n s i t y p r o f i l e s f o r a u n i ­ through the l e f t\n\nA sequence of\n\nface.\n\nNote the computed\n\nf i n e r mesh s o l u t i o n s\n\ndetermined\n\nshock\n\nthrough an h-method a r e shown in Figures 2 and 3 t o g e t h e r with computed ty\n\nflow\n\n1986).\n\nform Mach 3 . 0 i n f l o w c o n d i t i o n location\n\ndescribed\n\nand o v e r a 2 0 - d e g r e e wedge as\n\nlaws with b i l i n e a r q u a d r i l a t e r a l e l e m e n t s .\n\nscheme i s\n\n(Oden e t a l . ,\n\nare\n\nh e a t s ) of\n\nelement\n\nconservation\n\nsolution.\n\nalgorithms\n\nsupersonic,\n\nThe gas i s assumed t o be a p e r f e c t gas so t h a t the problem i s\n\nerned by the Euler e q u a t i o n s\n\nWendroff\n\nthe\n\ntwo-dimensional\n\ndensi­\n\n196 The problem was a l s o ically\n\nenhanced\n\nvia\n\nan\n\nsolved\n\non a f i x e d\n\nr-method.\n\nThe\n\nmesh,\n\nand the s o l u t i o n\n\nresulting\n\ndistorted\n\nwas automat­\n\nmesh\n\nis\n\nshown\n\nin\n\nFigure 4 ( a ) with the corresponding d e n s i t y p r o f i l e s in Figure 4 ( b ) . These problems.\n\nresults Several\n\nshow\n\nthe\n\nutility\n\nof\n\nschemes\n\no t h e r , more complex examples a r e discussed\n\nfor in\n\ncomplex\n\nflow\n\n(Oden e t\n\nal.,\n\n1986).\n\nFig.\n\n1.\n\nA wedge-shaped channel f o r supersonic gas f l o w . (a) An i n i t i a l c o a r s e f i n i t e element mesh. (b) Density p r o f i l e s computed f o r the c o a r s e mesh with c o n c e n t r a t i o n of contour l i n e s around a shock l i n e .\n\n(a)\n\nFig.\n\n2.\n\nR e s u l t s of an h-method a d a p t i v e c a l c u l a t i o n . ( a ) A u n r e f i n e d / r e f i n e d mesh with a=0.2 and 3=0.5 (b) The d e n s i t y c o n t o u r s . c o n c e n t r a t i o n of contour l i n e s around a shock l i n e .\n\nFurther h - r e f i n e m e n t s . (a) The mesh with (b) The d e n s i t y c o n t o u r s . and e=0.2 .\n\na=0.15\n\n199\n\nFig.\n\n4.\n\nR e s u l t s of an r - t y p e a d a p t i v e scheme, which attempts t o c a p t u r e the shock,\n\n(a) A d i s t o r t e d mesh (b) Density c o n t o u r s .\n\n200 ACKNOWLEDGEMENT This work was supported in p a r t by the NASA Langley Research Center and in p a r t by the U.S. O f f i c e of Naval\n\nResearch.\n\nREFERENCES Babuska, I . and R h e i n b o l d t , W. C , 1 9 7 8 a . Error e s t i m a t e s f o r a d a p t i v e f i n i t e element computations. SIAM J n l . Numer. A n a l . , 1 5 : 4 , 7 3 6 - 7 5 4 . Babuska, I . and R h e i n b o l d t , W. C , 1 9 7 8 b . A - P o s t e r i o r i Error Estimates f o r the F i n i t e Element Method. I n t ' l . J n l . f o r Numer. Meth. in Eng., 1 2 : 1 5 9 7 - 1 6 1 5 . Demkowicz, L. and Oden, J . T . , 1 9 8 6 . On a mesh o p t i m i z a t i o n method based on a minimization of i n t e r p o l a t i o n e r r o r . I n t ' l . J n l . of Eng. S c i . , 2 4 : 5 5 - 6 8 . Demkowicz, Oden, J . T. and Devloo, P h . , 1 9 8 5 . On an H-type mesh refinement s t r a t e g y based on minimization of i n t e r p o l a t i o n e r r o r s . Comp. Meth. in Appl. Mech. and Eng., 5 3 : 6 7 - 8 9 . Demkowicz, L . , Oden, J . T. and S t r o u b o u l i s , T, 1 9 8 4 . a d a p t i v e methods f o r flow problems with moving b o u n d a r i e s . I . v a r i a t i o n a l p r i n c i p l e s and a - p o s t e r i o r i estimates. Comp. Meth. in Appl. Mech. and Eng., 4 6 : 217-251. Diaz, A. R., K i k u c h i , N. and T a y l o r , J . E . , 1 9 8 3 . A method of g r i d o p t i m i z a t i o n f o r f i n i t e element methods. Comp. Meth. in Appl. Mech. and Eng., 4 1 : 2 9 - 4 5 . Oden, J . T. and C a r e y , G. F . , 1 9 8 1 . F i n i t e Elements: Mathematical A s p e c t s . P r e n t i c e H a l l , Englewood C l i f f s , NJ. Oden, J . T . , Demkowicz, L . , S t r o u b o u l i s , T. and Devloo, P . , 1 9 8 5 . Adaptive methods f o r problems in s o l i d and f l u i d mechanics. In: I . Babuska and 0 . C. Zienkiewicz ( E d i t o r s ) , Adaptive Methods and Error Refinement in F i n i t e Element Computation. John Wiley and S o n s , L t d . , London. Oden, J . T . , S t r o u b o u l i s , T. and Devloo, P h . , 1 9 8 6 . Adaptive f i n i t e element methods f o r i n v i s c i d compressible f l o w , Part I . TICOM R e p o r t , 8 6 - 1 , The U n i v e r s i t y of Texas, Austin 1 9 8 6 .\n\n201\n\nPENALTY VARIATIONAL FORMULATION OF VISCOUS INCOMPRESSIBLE FLUID FLOWS\n\nJ.\n\nN. REDDY\n\nC l i f t o n C. Garvin P r o f e s s o r , Department of Engineering S c i e n c e and Mechanics, V i r g i n i a P o l y t e c h n i c I n s t i t u t e and S t a t e U n i v e r s i t y , B l a c k s b u r g , VA 24060 (USA)\n\nABSTRACT Reddy, J . N., 1 9 8 6 . P e n a l t y v a r i a t i o n a l f o r m u l a t i o n of v i s c o u s i n c o m p r e s s i b l e fluid flows. P r o c . I n t . Symp. on V a r i a t i o n a l Methods in G e o s c i e n c e s , U n i v e r s i t y of Oklahoma, Norman, OK 7 3 0 1 9 . A r e v i e w of the a p p l i c a t i o n s of the p e n a l t y f i n i t e element method t o v i s c o u s , incompressible f l u i d flows i s presented. The p e n a l t y v a r i a t i o n a l f o r m u l a t i o n of the e q u a t i o n s governing s t e a d y , laminar flow of i n c o m p r e s s i b l e v i s c o u s f l u i d s and a s s o c i a t e d f i n i t e - e l e m e n t model a r e d e s c r i b e d . Numerical r e s u l t s f o r a number of n o n t r i v i a l problems a r e p r e s e n t e d and d i s c u s s e d .\n\nINTRODUCTION A Historical\n\nReview\n\nBuoyancy d r i v e n f l o w s p l a y an important r o l e in many e n g i n e e r i n g practical\n\ninterest.\n\nThese i n c l u d e thermal i n s u l a t i o n of b u i l d i n g s\n\nproblems of (Ostrach,\n\n1 9 7 2 ; B a t c h e l o r , 1 9 5 4 ) ; heat t r a n s f e r through double glazed windows 1 9 6 5 ; G i l l , 1 9 6 6 ) ; c o o l i n g of e l e c t r o n i c equipment\n\n(Pedersen e t a l . ,\n\n(Elder, 1971);\n\nc i r c u l a t i o n of p l a n e t a r y atmosphere ( H a r t , 1 9 7 2 ) ; c r y s t a l growth from melt (Carruthers,\n\n1 9 7 5 ) ; c o o l i n g of n u c l e a r r e a c t o r c o r e s\n\ns t e r e l i z a t i o n of canned food fuel\n\n(Hiddink,\n\net a l . ,\n\n(Petuklov,\n\n1976);\n\n1 9 7 6 ) ; s t o r a g e of spent n u c l e a r\n\n( G a r t l i n g , 1 9 7 7 ) ; c o n v e c t i v e c o o l i n g of underground e l e c t r i c c a b l e\n\n(Chato and Abdulhadi, 1 9 7 8 ) ; and a n a l y s i s of s o l a r c o l l e c t o r systems Goldstein,\n\nsystems\n\n(Kuehn and\n\n1 9 7 6 ) a r e but a few examples.\n\nAlthough most of t h e s e f l o w s a r e f u l l y t h r e e - d i m e n s i o n a l\n\nand time\n\nthe l i m i t a t i o n s imposed by both experimental and t h e o r e t i c a l\n\ndependent,\n\ntechniques\n\nhave\n\nf o r c e d r e s e a r c h e r s t o a n a l y z e o n l y t h o s e f l u i d motions t h a t a r e b e l i e v e d render themselves t o approximation by two-dimensional\n\nmodels.\n\nto\n\nThe e q u a t i o n s\n\nd e s c r i b i n g t h e coupled c o n v e c t i v e heat t r a n s f e r and f l u i d flow a r e h i g h l y n o n l i n e a r , and the s t r o n g coupling obtain a n a l y t i c a l\n\nbetween the e q u a t i o n s make i t d i f f i c u l t\n\nA most p r a c t i c a l a l t e r n a t i v e t o t h i s the d i g i t a l\n\nto\n\nsolutions. limitation\n\ninvolves the e x p l o i t a t i o n\n\ncomputers and the use of numerical methods.\n\nnumerical s o l u t i o n of v i s c o u s\n\nflow e q u a t i o n s\n\nPerhaps t h e\n\nfirst\n\ni s due t o Thorn ( 1 9 3 3 ) , who used\n\nof\n\n202 t h e f i n i t e - d i f f e r e n c e method.\n\nMuch of the emphasis\n\ncomputers were not in e x i s t e n c e ,\n\nwas on e l l i p t i c\n\nin those d a y s , when\n\nequations.\n\nWith t h e advent of\n\ne l e c t r o n i c computers, big s t r i d e s were made in the numerical s o l u t i o n Navier-Stokes equations f o r viscous\n\nincompressible\n\nnumerical schemes used in computational d i f f e r e n c e methods.\n\nAn e x c e l l e n t\n\nfluids.\n\nA m a j o r i t y of\n\nf l u i d dynamics a r e based on f i n i t e -\n\ns u r v e y of t h e developments\n\nd i f f e r e n c e methods t o computational\n\nof\n\nin\n\nfinite-\n\nf l u i d dynamics can be found in Roache\n\n(1972). One of t h e major d i f f i c u l t i e s equations\n\nassociated\n\nin n o n - r e c t a n g u l a r g e o m e t r i e s\n\nwith t h e s o l u t i o n of\n\nNavier-Stokes\n\ni s the a p p l i c a t i o n of the boundary\n\nconditions.\n\nAlthough a t t e m p t s have been made t o r e c t i f y t h i s problem by\n\nconstructing\n\nbody f i t t e d c u r v i l i n e a r meshes in f i n i t e d i f f e r e n c e methods\n\net a l . ,\n\n1 9 8 1 ) , the f i n i t e element method has a d e f i n i t e\n\n(Ghosh\n\nadvantage in t h a t any\n\ncomplicated geometry can be s u i t a b l y r e p r e s e n t e d by using non-uniform and nonr e c t a n g u l a r meshes, imposed\n\nand a p p r o p r i a t e boundary c o n d i t i o n s\n\nof the model can be\n\nin a n a t u r a l way ( s e e Reddy 1 9 8 4 , 1 9 8 6 ) .\n\nThe remarkable success of the f i n i t e element method in s o l i d\n\nmechanics\n\ncoupled with i t s a b i l i t y t o model complex domains and handle boundary conditions\n\nhas i n s p i r e d r e s e a r c h e r s in computational\n\nthe f i n i t e element method.\n\nf l u i d mechanics t o employ\n\nMuch of the e a r l i e s t work in t h i s d i r e c t i o n was\n\np r i m a r i l y in t h e a r e a of porous media (Zienkiewicz potential\n\nf l o w s , which a r e considered\n\ndeveloped\n\nin l i n e a r e l a s t i c i t y .\n\nand Cheung, 1 9 6 5 ) and\n\nt o be simple e x t e n s i o n s\n\nof t h e p r o c e d u r e s\n\nEarly a p p l i c a t i o n s of the f i n i t e\n\nelement\n\nmethod in t h e numerical s o l u t i o n of t h e N a v i e r - S t o k e s e q u a t i o n s governing a viscous,\n\ni n c o m p r e s s i b l e f l u i d can be found in the works of Oden and h i s\n\ncolleagues\n\n(1969,\n\n1970, 1972), Argyris et a l .\n\n( 1 9 7 2 ) , Olson ( 1 9 7 2 ) , Baker ( 1 9 7 0 , et a l .\n\n(1969),\n\nTong ( 1 9 7 1 ) ,\n\nCheng\n\n1 9 7 3 ) , T a y l o r and Hood ( 1 9 7 3 ) , and Kawahara,\n\n( 1 9 7 4 ) , among o t h e r s .\n\nVarious Formulations of Fluid Flow The f i n i t e element models of the two-dimensional\n\nNavier-Stokes\n\nhave been based on f o u r b a s i c f o r m u l a t i o n s d e s c r i b e d Stream f u n c t i o n - v o r t i c i t y model.\n\nMost s t u d i e s\n\nequations\n\nin the paragraphs below.\n\nusing f i n i t e\n\nmethods f o l l o w the stream f u n c t i o n - v o r t i c i t y approach.\n\nThe\n\ndifference\n\nfinite-element\n\nmodel based on t h i s approach has been employed by Cheng ( 1 9 7 2 ) , Olson 1 9 7 4 ) , and by o t h e r s .\n\n(1972,\n\nIn t h i s model, boundary c o n d i t i o n s on the v o r t i c i t y a r e\n\ncomputed from the stream f u n c t i o n a t the boundary. l a r g e e r r o r s in the v o r t i c i t y ( s e e , Stream f u n c t i o n model.\n\nDavis,\n\nFinite-element\n\nto flows.\n\nmodels based on t h i s approach can be\n\nfound in the works of Olson and h i s c o l l e a g u e s h i g h e r - o r d e r n a t u r e of the e q u a t i o n ,\n\nHowever, t h i s\n\n1 9 6 8 ) f o r a d v e c t i o n dominated\n\n(1972,\n\nthe a s s o c i a t e d\n\n1974, 1976).\n\nDue t o the\n\nf i n i t e element model\n\nis\n\n203 a l g e b r a i c a l l y complex.\n\nDue t o the s i m i l a r i t y of t h e stream f u n c t i o n\n\nequation\n\nt o t h a t of t h e biharmonic e q u a t i o n governing t h e t r a n s v e r s e d e f l e c t i o n p l a t e , a p l a t e bending f i n i t e - e l e m e n t\n\nprogram can be modified\n\nof a\n\nto solve the\n\nflow\n\nproblem. V e l o c i t y - p r e s s u r e model.\n\nThis i s the most n a t u r a l\n\nwhich i s a l s o known as the mixed f o r m u l a t i o n .\n\nThe model\n\nN a v i e r - S t o k e s e q u a t i o n s and the c o n t i n u i t y e q u a t i o n s , the p r i m i t i v e v a r i a b l e s ( u , v , P ) 1 9 7 6 ; Reddy,\n\nand d i r e c t f o r m u l a t i o n , i s based on t h e\n\na l l expressed\n\nin terms of\n\n[see T a y l o r and Hood, 1 9 7 3 ; Olson and Tuann,\n\n1978].\n\nPenalty f u n c t i o n model.\n\nThe p e n a l t y f u n c t i o n model\n\nprimitive variable equations,\n\na c o n s t r a i n t on t h e v e l o c i t y f i e l d . finite-element\n\ni s a l s o based on the\n\nexcept t h a t the c o n t i n u i t y equation i s t r e a t e d as The c o n s t r a i n t i s introduced i n t o t h e\n\nmodel by means of the p e n a l t y f u n c t i o n method\n\n1 9 7 3 ; Hughes e t a l . , formulation w i l l\n\n1 9 7 6 , Reddy, 1 9 7 8 , 1 9 7 9 , 1 9 8 2 , 1 9 8 3 ) .\n\n(Zienkiewicz, D e t a i l s of\n\nthe\n\nbe d i s c u s s e d l a t e r in t h i s p a p e r .\n\nEach of the f o r m u l a t i o n s has c e r t a i n r e l a t i v e advantages and d i s a d v a n t a g e s . The v e l o c i t y - p r e s s u r e f o r m u l a t i o n i s the most d i r e c t and n a t u r a l one in t h a t all\n\nthe v a r i a b l e s a r e p h y s i c a l .\n\nnon-positive-definite. condition,\n\nHowever, t h e r e s u l t i n g f i n i t e element model\n\nThis i s a d i r e c t consequence of the\n\nwhich stands uncoupled from the momentum e q u a t i o n s .\n\nThe stream\n\nf u n c t i o n f o r m u l a t i o n i s a t t r a c t i v e in problems where d e s c r i p t i o n of t h e phenomena i s\n\nimportant.\n\nsystem of e q u a t i o n s .\n\nThe f o r m u l a t i o n a l s o r e s u l t s in\n\nflow\n\npositive-definite\n\nS i n c e the governing e q u a t i o n ( f o r the stream f u n c t i o n )\n\nof f o u r t h o r d e r , C* - c o n t i n u i t y of the approximating f u n c t i o n s (analogous t o the p l a t e bending e l e m e n t s ) . complex elements and hence,\n\nis\n\ncontinuity\n\nis\n\nis required\n\nThis r e s u l t s in a l g e b r a i c a l l y\n\nl a r g e computational\n\nefforts.\n\nThe stream\n\nfunction-\n\nv o r t i c i t y f o r m u l a t i o n s u f f e r s from the drawback of r e q u i r i n g boundary conditions\n\non the v o r t i c i t y , which i s unknown a p r i o r i\n\nthe stream f u n c t i o n ) .\n\nHowever,\n\n(and not independent\n\ni t i s convenient to describe the flow\n\nwith the aid of the stream f u n c t i o n and v o r t i c i t y .\n\nThe p e n a l t y f u n c t i o n\n\ni s a p r i m i t i v e v a r i a b l e model which r e s u l t s in a p o s i t i v e - d e f i n i t e e q u a t i o n s f o r Stokes f l o w .\n\nAnother advantage of t h e model\n\nof\n\nphenomena model\n\nsystem of\n\nis that the pressure\n\ndoes not appear as a primary unknown, and an approximation t o the p r e s s u r e can be obtained in\n\npostcomputation.\n\nP r e s e n t Study In the p r e s e n t paper the p e n a l t y f i n i t e element model f o r n a t u r a l\n\nconvection\n\ni s d e s c r i b e d and i t s a p p l i c a t i o n t o some n o n t r i v i a l problems i s p r e s e n t e d . p e n a l t y v a r i a t i o n a l f o r m u l a t i o n and a s s o c i a t e d\n\nfinite-element\n\ndescribed along with some of the computational\n\ndetails.\n\nmodel\n\nFinally,\n\nis\n\nnumerical\n\nr e s u l t s f o r a number of s t e a d y , v i s c o u s flow problems a r e p r e s e n t e d .\n\nWhile\n\nThe\n\n204 most of the t h e o r e t i c a l developments presented in t h e paper a r e of r e v i e w n a t u r e , the numerical r e s u l t s included should s e r v e as r e f e r e n c e s f o r f u t u r e investigators.\n\nEQUATIONS OF VISCOUS FLOW The equations d e s c r i b i n g the buoyancy d r i v e n flow of a v i s c o u s incompressible f l u i d , occupying domain a, can be w r i t t e n a s : u\n\ni\n\n.=0\n\ns\n\npu.u.\n\ns\n\n(1)\n\n+ p ,\n\nj\n\nr\n\nP\n\nf\n\nr\n\n0 g i\n\n[l - b(T - T )1 Q\n\np C ( u T , j ) - ( k T , j ) , j - u\\$ p\n\nj\n\nP\n\nq\n\ns\n\n[»(u\n\nUi\n\n+ u\n\nj\n\nf\n\n1\n\n)]\n\nf\n\nj\n\n= 0\n\n(2)\n\n= 0\n\n(3)\n\nwhere Cp\n\ni s the s p e c i f i c\n\nf.j\n\na r e body f o r c e\n\nheat a t c o n s t a n t p r e s s u r e\n\ng^\n\na r e the components of the g r a v i t a t i o n a l f o r c e\n\ncomponents\n\nk\n\ni s the thermal\n\np\n\ni s the p r e s s u r e\n\nq\n\ni s the heat source per u n i t mass\n\n\\$\n\nT T\n\nconductivity\n\ni s the temperature i s the r e f e r e n c e temperature f o r which buoyancy f o r c e s a r e z e r o\n\nQ\n\nu.j\n\na r e the v e l o c i t y\n\np\n\ni s the d e n s i t y\n\nu\n\ni s the\n\n8\n\ni s the volume expansion\n\n\\$\n\ni s the v i s c o u s d i s s i p a t i o n\n\nand,\n\nin 1\n\ncomponents\n\nviscosity\n\n= 3U./3X., etc.\n\n1\n\n9J\n\nEquations\n\ncoefficient function\n\nand summation on repeated s u b s c r i p t s\n\nis\n\nimplied.\n\nJ\n\n(l)-(3)\n\na r e t o be solved\n\nc o n d i t i o n s of a problem.\n\nin c o n j u n c t i o n with a p p r o p r i a t e boundary\n\nThese i n c l u d e a combination of p r e s c r i b e d\n\nt r a c t i o n s , t e m p e r a t u r e s and heat f l u x e s .\n\nvelocities,\n\nThe boundary r of the f l u i d\n\nregion\n\ncan be decomposed i n t o two p a i r s of d i s j o i n t p o r t i o n s : r = r\n\nu\n\nu\n\nr\n\n=\n\nt\n\nr\n\nu\n\nT\n\nr n r\n\nt\n\n= 4) (empty)\n\nr\n\nq\n\n= 4> (empty)\n\nu\n\nT\n\nO r\n\nHere r , r^, u\n\nvelocities,\n\nand\n\nq\n\n(4) r e p r e s e n t the p o r t i o n s of the boundary on which the\n\nstresses,\n\nThen the s p e c i f i e d\n\nr\n\ntemperature and heat f l u x , r e s p e c t i v e l y , a r e\n\nboundary c o n d i t i o n s\n\nof the type\n\nspecified.\n\n205 u\n\ni =* u\n\nt.\n\n,\n\no n\n\nr u\n\n= tt\n\na..n.\n\nT = T* on r q = (\n\nk T\n\n. on r _\n\n(5)\n\nt\n\nT\n\n»j) j n\n\n+ u\n\n[K 1 = 2y[S ) +\n\n[ K ] = AS ]\n\n,\n\n1 F\n\n,\n\n22\n\n2 2\n\npf\n\n2*i\n\ndA\n\n+\n\n6\n\nJ\" er 2 i t\n\n,p\n\nds\n\n+ [G]\n\n2 2\n\nI t has been f a i r l y e s t a b l i s h e d\n\nfrom convergence and s t a b i l i t y c o n s i d e r a t i o n s\n\n(Reddy, 1 9 8 6 ) t h a t reduced i n t e g r a t i o n technique i s t o be used t o e v a l u a t e t h e penalty terms.\n\nFor a b i l i n e a r element a 2 x 2 Gauss q u a d r a t u r e i s used\n\nevaluate a l l coefficient\n\nm a t r i c e s except the p e n a l t y t e r m s , and\n\nto\n\nl x l\n\nq u a d r a t u r e i s employed f o r t h e p e n a l t y t e r m s .\n\nS o l u t i o n Procedure The element e q u a t i o n s a r e assembled standard f a s h i o n\n\n(see Reddy, 1 9 8 4 ) .\n\ni n t o t h e global\n\nsystem m a t r i x in t h e\n\nBecause of t h e presence of t h e n o n l i n e a r\n\nc o n v e c t i v e t e r m s , t h e r e s u l t i n g system of a l g e b r a i c equations i s n o n l i n e a r , and an i t e r a t i v e s o l u t i o n scheme must be used t o s o l v e them. methods a r e :\n\nsuccessive\n\nsubstitution\n\nMost f r e q u e n t l y used\n\n( P i c a r d i t e r a t i o n ) and Newton-Raphson.\n\nIn the Picard i t e r a t i o n method, t h e n o n l i n e a r terms f o r the c u r r e n t a r e e v a l u a t e d using the s o l u t i o n from the p r e v i o u s i t e r a t i o n . a fairly\n\niteration\n\nThis scheme has\n\nl a r g e r a d i u s of convergence, but f o r many problems the r a t e of\n\nconvergence can be v e r y low. convergence.\n\nThe Newton-Raphson method has a s u p e r i o r r a t e of\n\nI t s convergence r a t e i s q u a d r a t i c as long as the i n i t i a l\n\nv e c t o r i s w i t h i n the r a d i u s of convergence.\n\nsolution\n\nU n f o r t u n a t e l y , the r a d i u s of\n\nconvergence of the Newton-Raphson method i s much s m a l l e r than t h a t of successive\n\nsubstitution.\n\nNUMERICAL RESULTS In-Line Bundle of C y l i n d e r s in Cross Flow Figure l a d e p i c t s the p h y s i c a l model of flow p a s t f i v e - r o w deep bundle of heated ( o r cooled)\n\ncylinders.\n\nby t h e t h i c k d o t t e d l i n e ACDB. i n f i n i t e bundle of c y l i n d e r s . 2a and 2 b .\n\nThe computational domain i s the r e g i o n\n\nenclosed\n\nFigure l b shows computational domain f o r an The boundary c o n d i t i o n s\n\nare indicated\n\nin\n\nFigs.\n\nIn the i n f i n i t e bundle c a s e , p e r i o d i c boundary c o n d i t i o n with\n\nregard t o v e l o c i t y i s a p p l i e d both a t the i n l e t and the o u t l e t of computational domain.\n\nthe\n\nIn o t h e r words, the v e l o c i t i e s obtained from p r e v i o u s\n\n208\n\nFlow .\n\n-A\n\n^ - V\n\no o o o o\n\nB\n\n6T\n\nc\n\np\n\nFig. l a . Geometry and computational domain for the problem of five-row deep i n - l i n e bundle of cylinders.\n\no o o o 0 W 0 O 0 VO O -F i g . lb. Geometry and computational domain for the problem of an i n f i n i t e cylinder bank.\n\n209\n\n210 i t e r a t i o n along the symmetry l i n e of the computational\n\ndomain ( F i g .\n\nlb) are\n\ntaken as t h e boundary c o n d i t i o n f o r both the i n l e t and t h e o u t l e t . t e m p e r a t u r e boundary c o n d i t i o n s\n\na t t h e i n l e t and the o u t l e t , the normalized\n\nt e m p e r a t u r e obtained a t s e c t i o n s\n\n1 - 1 and 2 - 2 in Fig.\n\nelement meshes f o r each case a r e shown in F i g s . meshes a r e designed\n\nFor\n\nl a a r e used.\n\n3a and 3 b .\n\nt o c a p t u r e the boundary l a y e r e f f e c t s\n\nThe\n\nThe f i n i t e finite-element\n\nnear the c y l i n d e r\n\nwalIs. V e l o c i t y v e c t o r s a t Re = 300 f o r f i v e rows of\n\ni n - l i n e c y l i n d e r bank and f o r\n\nan i n n e r row of an i n f i n i t e bundle a r e shown in F i g s . f o r a p i t c h t o diameter r a t i o of 1 . 8 . v e l o c i t y a t minimum flow c r o s s s e c t i o n . a d j a c e n t c y l i n d e r s (Fig.\n\n4a and 4 b , r e s p e c t i v e l y ,\n\nHere Re i s the Reynolds number based on The v e l o c i t y f i e l d\n\nin the gaps\n\nbetween\n\n4a) i n d i c a t e s t h a t t h e flow a f t e r the second c y l i n d e r\n\ni s almost f u l l y developed.\n\nThe d i f f e r e n c e between v e l o c i t y f i e l d\n\naround t h e\n\nt h i r d and the f o u r t h c y l i n d e r i s 1 . 8 p e r c e n t in e u c l e d i a n norm f o r t h e Re = 300 case.\n\nThis i s a l s o evidenced by almost i d e n t i c a l\n\ni n f i n i t e bundle in Fig.\n\n4b.\n\nc y l i n d e r and v e l o c i t y f i e l d\n\nv e l o c i t y f i e l d found f o r\n\nIn t h i s case the v e l o c i t y f i e l d around the f o u r t h around a c y l i n d e r f o r i n f i n i t e bundle has a\n\nd i f f e r e n c e of 1 . 2 p e r c e n t in e u c l e d i a n norm f o r Re = 3 0 0 .\n\nThe flow f i e l d\n\ns i m i l a r f o r v a r i o u s Re in the range Re = 100 - 600 s t u d i e d . c y l i n d e r s exemplify strong r e c i r c u l a t i n g r e g i o n s . a r e p r e s e n t behind the f i f t h row.\n\nThe gaps\n\nA p a i r of elongated\n\nThe s t r e a m l i n e s ,\n\nFor a d d i t i o n a l\n\nvortices\n\nisotherms and v o r t i c i t y\n\nl i n e s f o r the f i v e rows of c y l i n d e r bank a r e shown in F i g s . respectively.\n\nis\n\nbetween\n\n5a and 5 b ,\n\nr e s u l t s , see Dhaubhadel e t a l .\n\n(1986).\n\nConvection in an I n c l i n e d C a v i t y A s e r i e s of computations were performed with the c a v i t y t i l t e d a t 0 , 3 0 , 4 5 , 60 and 90 degrees\n\n(see\n\nP e l l e t i e r , et a l . ,\n\n1986).\n\nEach t i l t e d c a v i t y\n\nused a s o l u t i o n a t a s m a l l e r t i l t angle as an i n i t i a l 30 degrees\n\n(see Fig.\n\nguess.\n\nsimulation\n\nThe s o l u t i o n\n\nat\n\n6) c l e a r l y shows major changes from i t s 0 degree c o u n t e r\n\np a r t f o r Rayleigh number, Ra = 1 0 .\n\nThe c e n t r a l core i s becoming\n\nI t i s no longer s t a b l y s t r a t i f i e l d .\n\nThe thermal boundary l a y e r has\n\n6\n\nand r e s u l t s in a lower Nusselt number (see Fig.\n\nF u r t h e r i n c r e a s e of the t i l t angle a t 45 and 60 degrees\n\nf o r t h e u n i c e l l u a r - c h a r a c t e r i s t i c and an e s s e n t i a l l y (see Fig.\n\n7).\n\nthickened\n\n7).\n\nr e s u l t s in a c o n t i n u a t i o n of the flow p a t t e r n e s t a b l i s h e d\n\nNusselt number f u r t h e r d e c r e a s e s\n\nisothermal.\n\n(see F i g s .\n\n8 and 9 )\n\na t 30 degrees\n\nisothermal c o r e .\n\nThe v e l o c i t y f i e l d\n\nexcept The\n\nand\n\ns t r e a m l i n e s a r e approaching o v e r a l l symmetry. The v e l o c i t y f i e l d of the Benard s o l u t i o n (see Fig.\n\n(i.e.\n\na t 90 degrees)\n\n1 0 ) q u a l i t a t i v e l y resembles t h a t obtained a t 0 degrees\n\nCloser i n v e s t i g a t i o n , in the c o r n e r s .\n\na t low Ra inclination.\n\nhowever, r e v e a l s the p o s s i b l i t y of r e c i r c u l a t i o n e d d i e s\n\nAt Ra = 1 0 ^ no eddies a r e seen in the v e l o c i t y v e c t o r p l o t .\n\n211\n\n212\n\n213\n\nN\n\nA _Q E\n\noo\n\nwhere the\n\nTs\n\nfrom\n\nthe\n\nThe be\n\nis\n\nthe\n\ni n f i n i t e\n\nboundary\n\nboundary\n\nof\n\nand\n\ns t r u c t u r e s , r\n\nis\n\nthe\n\nis\n\ndistance\n\npole.\n\nfollowing\n\ns a t i s f i e d\n\nc o n t i n u i t y\n\non\n\nconditions\n\nshould waves|\n\nTc. fii fio\n\nn\n\n* n fii\n\nwhere\n\nsuperscripts\n\non\n\nside\n\nthe\n\nof\n\nfii\n\nTc\n\non\n\nfio\n\nfii\n\nand\n\nand\n\nfio\n\nfio\n\non\n\nmean\n\nthe\n\n(6)\n\nthe\n\nvalues\n\nboundary\n\nFigure\n\nTc,\n\n1.\n\nD e f i n i t i o n\n\nsketch\n\nrespectively.\n\nA COMBINATION V a r i a t i o n a l For be\n\nMETHOD\n\nOF\n\nBOUNDARY\n\nTYPE\n\nFEM\n\nAND\n\nBEM\n\nfunctional\n\nthe\n\nd i s c r e t i z a t i o n\n\nusefully\n\nintroduced.\n\nmethod\n\ni s\n\ndomain\n\nfio\n\napplied to\n\ndeal\n\nfunctional\n\nto\n\nbe\n\nof\n\nIn\n\nand\n\nthe\n\nwith\n\nthe\n\nthe\n\nbasic\n\ninner\n\nboundary\n\nthe\n\nelement\n\nr a d i a t i o n\n\nminimized\n\nfor\n\nequations,\n\ndomain\n\nthe\n\nfii,\n\nthe\n\nthe\n\nmethod\n\nis\n\nc o n d i t i o n . boundary\n\nv a r i a t i o n a l\n\nboundary\n\ntype\n\nintroduced Generally,\n\nvalue\n\nproblem\n\np r i n c i p l e f i n i t e in\n\nthe i s\n\ncan\n\nelement\n\nthe\n\nouter\n\nv a r i a t i o n a l expressed\n\nas\n\nfollows.\n\nn\n\n=\n\nI W\n\nc\n\nc\n\n8\n\n(\n\nV\n\nr\n\n°\n\n2\n\n~ %i ] u\n\n2\n\nd f i\n\n+ }/r\n\nc c § n n 0\n\n'n\n\nd r\n\n- / ccgnn dr r s\n\n> n\n\n(7)\n\n225 After the\n\nintegrating\n\nfollowing\n\nthe\n\nfirst\n\nt e r m by p a r t s ,\n\nthe\n\nfunctional\n\nis\n\ntransformed\n\ninto\n\nform.\n\nn =^/ .ccgnn, r - \\/ .cc n(v n d\n\nr\n\n2\n\nn\n\nf i\n\nk n)d£> 2\n\n+\n\ng\n\n+ |/ ccgnn, dr - / cc r,n, dr r o\n\nAssuming\n\nthat\n\nthe Helmholtz\n\nthe\n\nn\n\ninterpolation\n\nequation\n\nin\n\nr s\n\ng\n\nequation\n\neach element,\n\nfor\n\nthe\n\n(8)\n\nn\n\ns u r f a c e d i s p l a c e m e n t r| s a t i s f i e s\n\nfunctional\n\nc a n be s i m p l i f i e d\n\nas:\n\nn = ^ / . c c g n n , d r + ± J c c n n , d r -. / c c n n , d r r\n\nThis\n\nfunctional\n\nis\n\nn\n\nthe\n\nr o\n\nbasis for\n\ng\n\nderiving\n\nn\n\nr s\n\ng\n\n(9)\n\nn\n\nthe d i s c r e t i z e d\n\ncomputational\n\nequation.\n\nDiscretization For\n\nthe\n\nseries\n\nis\n\ninterpolation\n\nequation\n\nin\n\nthe\n\ninner\n\ndomain,\n\ne m p l o y e d b a s e d on t h r e e\n\nnode t r i a n g u l a r\n\nk k n = [ cos^^Ocos^y)\n\nk k cos(^x)sin(^-y)\n\nthe\n\ntrigonometric\n\nfunction\n\nelement a s : k k sin^pOcos^y)\n\n]\n\n\\\n\nou\n\nf\n\na\n\nwhere a\n\nare\n\ncentroid\n\nof\n\nconstants\n\nand\n\neach element.\n\nk is\n\nThis\n\nwavenumber w h i c h t a k e s\n\ninterpolation\n\nequation\n\nthe\n\nvalue\n\nsatisfies\n\nthe\n\n(10)\n\n2\n\nat\n\nthe\n\nHelmholtz\n\nequation. On\n\nthe other\n\nhand, the\n\nboundary\n\nUsing the Hankel function tal\n\nsolution\n\nfor\n\nof\n\nthe\n\noutgoing\n\ne l e m e n t method\n\nfirst\n\nkind\n\nscattered\n\nis\n\nzeroth\n\nwave,\n\nused i n\n\norder\n\nthe\n\nthe\n\nH?(kr)\n\nfollowing\n\nouter for\n\ndomain\n\nQo.\n\nthe\n\nfundamen­\n\nboundary\n\nintegral\n\ne q u a t i o n c a n be o b t a i n e d a s :\n\nnsc(p)(l where\n\np\n\nrotation the\n\nis\n\nthe\n\nof\n\nthe\n\nboundary\n\n- ^jr)\n\n=\n\ni/ (nsc(H?(kr)), r c\n\np o l e and r tangent\n\nat\n\nis\n\nH ? ( k r )r,sc\n\nthe d i s t a n c e from\n\npoint\n\np.\n\nthe\n\n(9),\n\nit\n\nfunctional\n\ni\n\nin\n\n-\n\nit\n\n, }dr\n\n(11)\n\nn\n\nand a d e n o t e s t h e\n\nLinear interpolation\n\nfunction\n\nis\n\nangle used\n\ntotal\n\nnumber o f\n\nnodal\n\nFrom e q u a t i o n\n\n(12),\n\na set\n\nthe\n\nmatrix\n\nform.\n\nfollowing\n\nof\n\n=\n\n1\n\nis\n\n'\n\nobtained\n\n2\n\n- -\n\nthat\n\nE\n\n( 1 2 )\n\npoints. complex\n\nlinear\n\nequations for\n\n{n.} c a n b e\n\nderived\n\n[K]{n) = ( F ) where motion. The\n\n[K]\n\nis\n\nstiffness\n\nThe f r o n t a l\n\ndetails\n\nof for\n\nelement.\n\nMinimizing\n\nwhere E i s\n\nn\n\nof\n\n(13) matrix\n\nand { F } i s\n\nsolution\n\ntechnique\n\ndiscretization\n\nthe external is\n\nsource to\n\nu s e d for s o l u t i o n\n\nprocess are given in\n\nexcite\n\nthe\n\nwave\n\nof e q u a t i o n ( 1 3 ) .\n\nKashiyama and Kawahara\n\n(1985).\n\n226\n\n0.77T\n\n0.5TT\n\nFigure\n\n3. F i n i t e e l e m e n t g r i d for elliptical island\n\nNUMERICAL In has\n\nt o show t h e v a l i d i t y\n\nbeen analyzed shows\n\ni n recent\n\nthat\n\nthe\n\ntest\n\nconsidered\n\n4\n\nshown The\n\nrespectively.\n\nnumber\n\nis\n\nobtained\n\ni n Figure total\n\n2.\n\nwave a n g l e\n\nby Yue e t a l . ( 1 9 7 6 ) ,\n\ncylindical\n\nthe\n\nand phase f u n c t i o n\n\nHowever,\n\no f many e n g i n e e r i n g\n\nstructures.\n\nFigure\n\nthe\n\nthese\n\nbase\n\npoints\n\nalong\n\na r e 288 and\n\nthe\n\nFigures coastline\n\nThe i n c i d e n t\n\nfinite could\n\nfrom\n\nelement\n\nwave those\n\nmethod.\n\nbe a t t r i b u t e d\n\nresults\n\nis\n\nelement\n\nassumption.\n\nare\n\nto\n\nacceptable\n\napplications.\n\nt o t h e wave d i f f r a c t i o n 6\n\nand\n\nwith\n\nfinite\n\nare different\n\nThese discrepancy\n\nmethod i s a p p l i e d\n\nisland\n\n1985),\n\non a c i r c u l a r\n\nthe mild-slope\n\nThe computed r e s u l t s\n\n10%,\n\nHomma\n\ncompared\n\nelements and nodal\n\nviolate\n\nassumption.\n\nrequirements\n\nthe present\n\nisland\n\nillustrates\n\nwho used a t h r e e d i m e n t i o n a l\n\ni s roughly\n\nof mild-slope\n\nthe accuracy Secondly, two\n\n1 . 5TT\n\nfunction.\n\nan e l l i p t i c 3\n\nwhen\n\ni s 0=TT, a n d 0 = 1 .5TT, r e s p e c t i v e l y .\n\nassumed t o be k a = l .\n\nthe v i o l a t i o n\n\naccurate\n\nFigure\n\nnumber o f f i n i t e\n\nthe classical\n\n(Kashiyama and Kawahara,\n\ninterpolation\n\nThe bottom slope\n\nThe maximum d i f f e r e n c e\n\nby\n\npaper\n\nmethod,\n\napproach i s\n\na n d 5 show t h e c o m p u t e d wave a m p l i t u d e\n\nwhen t h e i n c i d e n t\n\nfor\n\nlinear\n\n1\n\nt h e present method f u r t h e r , as\n\nidealization. 180,\n\nof present\n\nauthers\n\npresent\n\nc o n v e n t i o n a l method using To\n\nl . 3TT\n\n1.1TT\n\nEXAMPLES\n\norder\n\nwhich\n\n0.9TT\n\nF i g u r e . 5 . Computed wave a m p l i t u d e a n d phase f u n c t i o n (0=1.5TT)\n\nrepresents\n\nthe\n\nand s c a t t e r i n g\n\nfinite\n\nelement\n\n227\n\nAngle (degrees) F i g u r e 6.\n\nF i n i t e element g r i d\n\nF i g u r e 7.\n\nF i g u r e 8.\n\nComputed wave amplitude on c y l i n d e r\n\nComputed wave amplitude\n\ndistribution\n\nPresent method Umeda and Yano Experiment(Umeda and Yano)\n\nx/L\n\n-2.\n\n-1. F i g u r e 9.\n\nComputed e q u i - p h a s e\n\nline\n\n228 idealization.\n\nThe total number of finite elements and nodal points are 1608 and\n\n951, respectively. diameter\n\nof\n\ndistribution amplitude\n\naround\n\non\n\nrepresents\n\nthe\n\nthe\n\ncorresponds computed\n\nThe incident wave length is assumed to be L=D, where D is the\n\ncylindical structure. the structures.\n\nFigure 7 illustrates\n\ncylindical nodal points.\n\ncomputed\n\nresults\n\nIn this\n\nthe\n\nfigure,\n\nfor the upper cylinder and\n\nto the results in the case of single cylinder.\n\nequi-phase\n\nexperimental\n\nFigure 8 shows the computed wave\n\nline\n\nwhich\n\nis\n\ncompared\n\nresults by Umeta and Yano (1983).\n\nwith\n\ncan be seen that the computed result\n\ncomputed the\n\nthe\n\nwave\n\nsolid\n\nline\n\ndotted\n\nline\n\nFigure 9 shows\n\napproximation\n\nIn this figure,\n\nillustrates the shadow of diffracted waves in experiment.\n\namplitude\n\nthe\n\ntheory\n\nand\n\nthe black area\n\nFrom this figure,\n\nit\n\nis well in agreement with the approximation\n\ntheory and experimental results.\n\nCONCLUSION The combination method of boundary type finite elements and boundary is\n\npresented\n\nin\n\nthis\n\npaper.\n\nThe\n\nkey feature of this\n\nmethod\n\nis\n\nelements that\n\nthe\n\ninterpolation equation has been chosen so as to satisfy the Helmholtz equation in each element.\n\nThe variational functional to be minimized can be formulated only\n\nby the line integral of element. existing\n\nexperimental\n\nstudies,\n\nit\n\nand\n\nThe numerical results have been compared\n\nother numerical\n\nresults.\n\nFrom\n\nthese\n\nwith\n\ncomparative\n\nis concluded that the present method provides a useful tool for the\n\nanalysis of wave diffraction and refraction problems.\n\nREFERENCES Berkhoff, J.C.W., 1972. Computation of combined refraction and diffraction, Proc. 13th Conf. Coastal Eng., ASCE, 471-490. Bettess, P. and Zienkiewicz, 0 . C , 1977. Diffraction and refraction of surface waves using finite and infinite elements, Int. J. Numer. Methods Eng., 11; 1271-1290. Chen, H.S. and Mei, C C , 1974. Oscillations and wave forces in an offshore harbor, Ralph M. Persons Lab., Report No.190, MIT. Kawahara, M. and Kashiyama, K., 1985. Boundary type finite element method for surface wave motion based on trigonometric function interpolation, Int. J. Numer. Eng., 21: 1833-1852. Kashiyama, K. and Kawahara, M., 1985. Boundary type finite element method for surface wave problems, P r o c of JSCE, No.363/2 (Hydrauric and Sanitary E n g . ) : 205-214. Kawahara, M., Sakurai, H. and Kashiyama, K., (in press) Boundary type finite element method for wave propagation analysis, Int. J. Numer. Methods Fluids. Tsay, T.K. and Liu, P.L F,, 1983. A finite element model for wave refraction and diffraction, Applied Ocean Research, 5: 30-37. Umeta, S. and Yano, M,, 1983. A study of wave diffraction on multiple cylinders, Proc. JSCE, No.329, 93-103. (in Japanese) Yue, D.K.P., Chen, H.S. and Mei, C.C., 1976. A hybrid finite element method for calculating three dimensional water wave scattering, Ralph M. Persons Lab., report No.215, MIT. Zienkiewicz, O.C., Kelly, D.W. and Bettess, P., 1977. Marriage a la mode - the best of both worlds (finite elements and boundary integrals), In: R, Glowinski et al. (Editor), Energy Methods in Finite Element Analysis, 81-107.\n\n229\n\nTHE NUMERICAL ANALYSIS OF TWO-DIMENSIONAL STEADY FREE SURFACE FLOW PROBLEMS\n\nTsukasa NAKAYAMA and Mutsuto KAWAHARA Department of Civil Engineering, Chuo University 13-27, Kasuga 1-chome, Bunkyo-ku, Tokyo 112, Japan\n\nABSTRACT\n\nThe present paper deals with a numerical analysis of a two-dimensional steady free surface flow under gravity. In order to avoid complexity in computa­ tions due to the fact that the free surface profile is unknown a priori, the fluid region in the (x,y)-plane is transformed into a rectangular region in the complex potential plane, namely (faty)-plane. The problem is then formulated in terms of the vertical coordinate yCfaty. The governing equation is the Laplace equation and is solved by applying the boundary element method. The computa­ tional results have been compared with the available experimental data. Good agreements have been obtained,\n\nINTRODUCTION The a n a l y s i s of fluid f l o w with free s u r f a c e s is a d i f f i c u l t m a t h e m a t i c a l problem to be solved numerically as well as analytically, because the position of the free surface is unknown a priori and the s o l u t i o n domain c h a n g e s shape every computational\n\nstep.\n\nThe finite elment method and\n\nthe\n\nfinite\n\nference method have been applied with good results to some steady and\n\nits dif­\n\nunstesdy\n\nfree surface p r o b l e m s . H o w e v e r , these m e t h o d s require c o m p l e x a l g o r i t h m s adjust mesh or grid patterns to free surface profiles. effective scheme is developed\n\nby using a transformation\n\nto\n\nIn the present paper, an technique of\n\nvariables\n\nand the boundary element method.\n\nMATHEMATICAL FORMULATION OF A FREE SURFACE FLOW PROBLEM We consider a s o l i t a r y w a v e shown in Fig.l, which is t r a v e l l i n g in an open channel of uniform depth with a constant speed.\n\nA rectangular Cartesian coordi­\n\nnate system o-xy is so chosen that the x-axis coincides with the channel and that the y-axis coincides with the center line of the wave.\n\nbottom\n\nWe assume that\n\nthe solitary wave has a symmetric profile. The coordinate system moves together with the w a v e at the same speed, c, as that of the w a v e .\n\nSuch a choice\n\nof\n\nc o o r d i n a t e system reduces a t i m e - d e p e n d e n t p r o b l e m of w a v e p r o p a g a t i o n to a\n\n230\n\nFig. 1. A solitary wave in an open channel\n\nproblem.\n\nBecause of the symmetry of the solitary wave, the solu­\n\ntion domain is restricted\n\nto the half region of (x,y) plane.\n\nThe fluid\n\ndomain,\n\nV, is bounded by four b o u n d a r i e s ; the free surface boundary S^, the axis of symmetry\n\nthe channel bottom S3 and the far-downstream boundary S^.\n\nBy assuming the fluid to be inviscid and incompressible, and the flow to be i r r o t a t i o n a l , we can define both a v e l o c i t y ction vp. using\n\n4> and a stream\n\npotential\n\nfun­\n\nUsually, the problem under consideration is analysed in (x,y)-plane by\n\nvelocity\n\npotential\n\nor stream function as an unknown\n\nvariable. In such a\n\nphysical plane, a numerical approach as well as an analytical approach is rather difficult\n\nbecause the profile of the free surface is also an unknown\n\nvariable.\n\nThen, by making a change of variables, we transform the moving boundary to a fixed boundary\n\nproblem\n\nproblem.\n\nTRANSFORMATION OF THE SOLUTION DOMAIN The\n\nvalues of\n\nand\n\n^ are specified\n\non boundaries as shown in Fig.2.\n\nis the t o t a l f l o w rate per unit w i d t h of c h a n n e l .\n\nQ(=ch)\n\nBy regarding the v e l o c i t y\n\npotential and the stream function as independent variables and the coordinates,\n\nS : =\n\n= 0\n\ny = 0\n\non C\n\n2\n\non\n\n3\n\nC\n\nand\n\nC\n\n(3)\n\n4\n\n(4)\n\nThe condition (2) is derived from the dynamic boundary condition on the free surface.\n\nH Q ( = C\n\nthe assumption\n\n/2+gh) is the total head. that the velocity\n\nvanish on the boundaries S\n\n2\n\ncomponents\n\nThe condition (3) is derived under in the vertical\n\ndirection\n\nshould\n\nand S^.\n\nAlthough we can simplify the boundary geometry of the solution domain by the transformation of variables, we cannot eliminate the nonlinearity of the problem due to the nonlinear term in the equation (2).\n\nTherefore, the problem remains\n\nto be nonlinear in the ( a n d f i r s t t\n\nthe conservation\n\nof\n\nvariations.\n\n(1983) .\n\ntheorems\n\nA more\n\ngene­\n\no r d e r waves)\n\nf o r\n\ndiscontinuous\n\n1, the variational\n\nconditions\n\nof compatibility a r e\n\nsection.\n\nV A R I A T I O N A L CONDITIONS OF C O M P A T I B I L I T Y Consider valued,\n\na pair\n\n(Z , |V\" , t\n\nt\n\n( 6 )\n\nls=0 \"}\n\n4^ and +)\n\nrespectively, such that\n\nt\n\n|v^(s)=c|) (s) |v\"(s), and V=V^(s)uV^(s)u(Z (s)nV)\n\nt\n\nt\n\nt\n\nis as in the definition of a local parametrization\n\n(£ (s) divides V into two t\n\nsubdomains V. (s) and V. (s) and forms the common boundary between them) . ±\n\nin ( 6 ) , w e have used the Hadamard Lemma:\n\n(D4>) A\n\nIf w e define \\$ •\n\nA\n\n/ds| _ , which we call the d i s -\n\nn\n\nS—U\n\nAlso\n\n±\n\nn\n\nA\n\nS—U\n\nis also independent of the choice of parametrization by\n\nthen eq. (6) can be written in the form ±\n\n= ScjT -\n\n( C ^ n V \\$1\n\nwhere 6c|)Ed(J) (X,t (s) ,s)/ds|\n\n(7) Q\n\n.\n\nIn a manner similar to the derivation of the\n\nsecond order kinematical conditions of compatibility w e obtain (fi^)*\n\n=\n\nD {S4r - ( C ^ n V S I }\n\n+\n\nA\n\nn \\$(4), n ) b\n\na\n\nb\n\n±\n\nn (4), n n ) ^z b\n\na\n\nc\n\n±\n\nbc\n\n(8) (7) and (8)\n\nThe conditions\n\n(or the corresponding jumps) are called variational\n\nconditions of compatibility. Let us note that if we consider the following virtual deformation * * ^t+s'^t+s^ ^ S_J\n\nt\n\ni e n\n\n^\n\ne\n\n15\n\nkinematical conditions of compatibility. to the speed of propagation respectively.\n\n(-e,e)\n\n(8) are reduced to the corresponding\n\nJuroP °f\n\nIn this case 6 z and \\$4) are reduced\n\nand the displacement derivative 6/6t,\n\n238 F I R S T VARIATION OF ACTION FUNCTIONAL FOR DISCONTINUOUS MOTIONS C o n s i d e r a v i r t u a l deformation [t\n\n1\n\n(s)\n\n(s) ] and ( Z ( s ) ,i^ ( s ) ) t\n\n( - e , e ) « s ^ ( B , T ( s ) , Z ( s ) , \\ J j ( s ) ) where T ( s ) = t\n\nt\n\ni s g i v e n by a v i r t u a l deformation o f\n\nt\n\nt h e l a t t e r o f which has been d e f i n e d i n t h e INTRODUCTION ( i . e . , \\\\) ( X , t (s) , s ) i s motion o f B) . action integral\n\n(1): ( - e , e)e s ^ A ^ ^ ( s ) , i n t h e o b v i o u s way. i s d e f i n e d by 6 ^ ^ = 6 1 ^ ^ ( s ) / d s | _ Q .\n\ni n t e g r a l o v e r t h e time i n t e r v a l T ( s ) in ^ ^ ( s ) /\n\n1\n\n6 A\n\nBXT\n\n=\n\nV\n\nB\n\n\\ Z\n\nw\n\nt\n\nby\n\nt\n\nn\n\n9\n\nc n a r i\n\nof\n\nthe\n\ne\n\nto the i n t e g r a l over the o r i g i n a l change o f v a r i a b l e , then a f t e r\n\ne\n\nthe\n\nThe v a r i a t i o n\n\ne\n\ns\n\ninterval T=[t ,t2]\n\nt\n\nt\n\nT h i s deformation i n d u c e s t h e deformation o f\n\naction integral\n\ndifferentiation\n\n(Z ,\\p > ,\n\ni p ( s ) (X) =\n\ntime\n\nthe\n\no f i n t e g r a l we o b t a i n\n\ni\n\nt\n\nL\n\n(\n\nS\n\n\\ s = 0 V\n\n)\n\n\" '\n\n\\\n\nT\n\nM\n\nt\n\ne\n\nd\n\nZ\n\nN-l\n\nd\n\nt\n\nVBNZ\n\n+\n\nL t\n\nft\n\n6\n\nt\n\nd\n\nV\n\nN\n\nd\n\nt\n\n(9) where L ( s )\n\nt ( s ) ,ty (s)\n\n,ty (s)\n\n±\n\nA\n\n±\n\nexample I J J ( S ) E 8 \\ j j ( s ) / 8 t ( s ) .\n\nAlso in\n\ninduced ( E u c l i d e a n ) measure on\n\n(s)\n\n( s ) ip\n\nA\n\nf\n\n±\n\n^(s))\n\n(9), 6 t = d t ( s ) / d s | _ s\n\ni n which,\n\nand d Z\n\nQ\n\nN\n\nfor\n\nis\n\nthe\n\nZ..\n\nNow, l e t us d e f i n e new v a r i a t i o n s : t\n\na parallel family of curves\n\n1\n\nand recall that a moving wave front consists of is constant) then we have that the curvature\n\nfor the curves of this family are given by fi = fi(0)/(l-fi(0)a) for all sufficien­ tly small a .\n\n5(a)\n\nThe solution of (15) can be presented as\n\n= a(0) |l -\n\nfi(0)a|~\n\n1/2\n\nwhere a(0) and fi(0) are the amplitude and the curvature at time t=t^\n\n(16)\n\n(i.e.,\n\no=0) . Finally, let us note that the results of this section are complementary to that obtained by Cohen (1976), in the sense that they are valid in the differ­ ent range of approximation of the elastic plate.\n\nAPPENDIX We give here the geometrical and kinematical conditions of compatibility of this first and second order in the form they have been used in this work. These geometrical conditions are the following:\n\nU, ^ A\n\n= D II(j)I) + N ^ g N ! ! 5\n\nA\n\n241\n\n^ ' A B\n\n=\n\n1\n\nD\n\n( A\n\nD\n\nB )\n\n^\n\n+\n\n\" \" A B ^ ' C ^\n\nN\n\n+\n\n( A\n\nN\n\nF\n\nL\n\nB )\n\nD\n\nC\n\nA V * ' C D\n\nI* N\n\n^(AV*'/\n\n1 1 +\n\nC\n\nN\n\nD\n\n]\n\n1\n\n1\n\nI\n\nand these kinematical conditions are the following: I I I\n\n= ft^D\n\n^ ' A\n\n1\n\n-\n\n\" ( N J H ^ H\n\n= ° A 4\n\n\"\n\n(fiB = ^\n\n\"\n\n(\n\n^\n\nN\n\n^\n\n\" V ^ ' B ^\n\n'\n\nB\n\nC\n\n+\n\nN\n\nA B\n\n(\n\nN\n\nD\n\n(\n\nU\n\n(N)\n\n,\n\nD\n\nB\n\nI\n\n[\n\n*\n\n1\n\n1\n\n+\n\nN\n\nA It\n\n^\n\n'\n\nB\n\n^\n\n^\n\n- u^ff^ll}\n\n-\n\nU\n\n)\n\nD ( U A\n\n(\n\nN\n\n)\n\n) D I [ ^ A\n\n-u\n\n(N)\n\n^\n\n(T^yi\n\nThe higher order geometrical and kinematical conditions which have also been used in the work can be obtained from the above by the iteration process.\n\nREFERENCES Cohen, H., 1976. Waves propagation in elastic plates. J. of Elasticity 6:245248. Cohen, H., Wang, C.-C., 1982. On compatibility conditions for singular surfaces. Arch. Rational Mech. Anal. 80:205-261 Duvaut, G., Lions, J.-L., 1974. Problems unilateraux dans la theorie de la flexion forte des plaques; le case d'evolution. J. de Mecanique, 13:245-266. v. Karman, T., Biot, M.A., 1940. Mathematical Methods in Engineering, McGrawHill Book Company, Inc. New York and London. Oden, J.T., Reddy, J.N., 1983. Variational Methods in Theoretical Mechanics. (2nd edition), Springer-Verlag, Berlin and New York. Raoult, A., 1985. Construction d'un modele d'evolution de placques avec terme d'inerte de rotation. Ann. di Mat. Pura ed Appl. Ser. 4, 139:362-400. Thomas, T.Y., 1961. Plastic Flow and Fracture in Solids, Academic Press, New York and London. Toupin, R.A., 1962. Elastic materials with couple-stresses. Arch. Rational Mech. Anal. 11:385-414. Turski, J., 1984. Variational formulation of the singular surfaces propagation in nonsimple elastic materials. In: C. Rogers and T.B. Moodie (Editors), Wave Propagation: Modern Theory and Applications. North-Holland, Amsterdam. Turski, J., 1986. Calculus of variations for discontinous fields and its applications to selected topics in continuum mechanics. Ph.D. Thesis (submitted in an \"ad-hoc\" doctoral program) McGill University.\n\n243\n\nC O M P A R I S O N OF V A R I A T I O N A L GROUP-DIFFUSION PROBLEM:\n\nM E T H O D S FOR T H E ONE DIMENSIONAL\n\nSOLUTION CASE\n\nOF\n\nTHE\n\nDYNAMIC\n\nE. d e l V a l l e , J . C . D i a z , D. M e a d e 3 ^ E S F M - I P N , U n i d a d P r o f e s i o n a l Z a c a t e n c o , D e l e g a c i o n G u s t a v o A. Madero, 07300 Mexioc D.F., MEXICO P a r a l l e l P r o c e s s i n g I n s t i t u t e , O U - E E C S , 202 W. B o y d , S u i t e 2 1 9 , Norman, Oklahoma, U.S.A. 3 c e n t r o de E s t u d i o s N u c l e a r e s , U N A M , A. P o s t a l 7 0 - 5 3 t 0*1510 Mexico D.F., MEXICO 1\n\n2\n\n2\n\nABSTRACT Galerkin is a v a r i a t i o n a l method closely r e l a t e d to the c o l l o c a t i o n method. T h e i r c o n n e c t i o n h a s b e e n s t u d i e d by s e v e r a l authors. E f f i c i e n c y s t u d i e s of c o l l o c a t i o n and G a l e r k i n h a v e b e e n performed for a p p l i c a t i o n s to p r o b l e m s in v a r i o u s f i e l d s in engineering and applied sciences. The a d v a n t a g e s of the c o l l o c a t i o n and G a l e r k i n m e t h o d s are c o m b i n e d in t h e (hybrid) collocation-Galerkin method. H e r e i n , the c o l l o c a t i o n , G a l e r k i n and collocation-Galerkin m e t h o d s a r e a p p l i e d in t h e spatial c o o r d i n a t e s to s o l v e t h e d y n a m i c g r o u p - d i f f u s i o n , n e u t r o n - f l u x , and delayed-precursor concentration equations. Standard t e c h n i q u e s a r e e m p l o y e d to i n t e g r a t e on t i m e . Argonne's benchmark p r o b l e m s a r e u s e d in t h e s t u d y . THE\n\nPHYSICAL The\n\nMODEL\n\ncollocation,\n\napplied\n\nto\n\nprecursor delayed\n\nthe\n\ndynamic\n\n=\n\nv\n\ngroups\n\n- g * g - ^ D\n\nv\n\n[\n\n+\n\n^ C\n\nwith\n\ni\n\n3 i\n\n=\n\n(\n\nI I \\ i=1\n\nappropriate\n\nJ\n\nat\n\ng ~DgV(J) =\n\n~ e\n\n0\n\nP + d(S) I Y.(S) q j=1\n\n- Saturations equations JSC\n\n^.\n\n(3)\n\n• £ • iv K - 0 | (x,t, S(x,t)) = r(x,t) + f (x,t, S(x,t)) w\n\n(4)\n\nr(x.t)\n\n= -iKx) P ( x ) d(S(x,t)) grad\n\nf(x,t,k)\n\n= b (k) q (x,t) + I b (k) q (x) j=1\n\nd\n\n1 -\n\nRemarks :\n\n>\n\nC M\n\nis\n\nthe\n\nfield is J Y\n\n2 - More\n\nwater-flow\n\nfield,\n\nthe\n\na(S(x,t)\n\ncorresponding\n\noil-flow\n\n= + {b} T D i f f e r e n t i a t i n g Eq. {y(T)}\n\n=\n\nf o r 0 < T < At\n\n(13)\n\n( 1 1 ) w i t h r e s p e c t t o T,\n\ngives\n\n{b}\n\n,(14)\n\nThus, s u b s t i t u t i n g Eqs. ([K]T + [C])\n\n{b}\n\n(13) and (15) i n t o Eq.\n\n= {Q} -\n\n[K]\n\n( 1 0 ) , gives\n\n{ y ( 0 ) } f o r 0 < T < At\n\nNote t h a t t h e elements of m a t r i c e s\n\n(15)\n\n[K] and [C] i n Eq.\n\n(15) a r e f u n c t i o n s\n\nof k^^, u^, | e t c . , which i n t u r n a r e f u n c t i o n s of S^ and p^. ties\n\n(S\n\nand p^)\n\n2\n\nare\n\nadopted t o s o l v e Eq. {b} from\n\n(i.e.,\n\n{b)\n\nEq.\n\n(11).\n\n[C] _^ T\n\n{b}.\n\nt\n\nknown\n\n(15) .\n\na priori,\n\nan\n\nIn t h i s s t u d y , f i r s t T\n\nUsing\n\nThe p r o c e d u r e\n\nis\n\ncriterion\n\nnumerical computation.\n\npressure\n\nand Eq.\n\n(15)\n\nrepeated based\n\non\n\nDetails\n\nis\n\nuntil the\n\nand\n\nsaturation\n\nsolved\n\nto\n\ndesired\n\nnorm of\n\nbe\n\na t r i a l v a l u e i s assumed f o r { y ( T ) } a r e computed\n\nQ\n\nthese\n\nAs t h e s e q u a n t i ­\n\ni t e r a t i v e p r o c e d u r e must\n\n= { b } _ ) , and approximate v a l u e s of\n\nT\n\nare evaluated,\n\nconvergence\n\nnot\n\nvalues,\n\nK\n\no b t a i n modified\n\nconvergence\n\n{y}\n\n^ ^«p-At\n\nis\n\nis\n\ndefined\n\na r e omitted h e r e because\n\nand can be found i n r e f e r e n c e p u b l i s h e d by K u k r e t i e t a l . NUMERICAL RESULTS\n\nof\n\na n\n\n^\n\nvalues\n\nof\n\nachieved.\n\nA\n\nand\n\nspace\n\nused\n\nfor\n\nlimitation\n\n(1985).\n\nThe f i n i t e element p r o c e d u r e d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n has been implemented\n\nin a computer program.\n\nThis program has been used t o study\n\nc h a r a c t e r i s t i c s of\n\ntwo-phase flow through a r e g u l a r square domain, of\n\nsion 1,000 u n i t s .\n\nSome p e r t i n e n t i n p u t d a t a a r e g i v e n i n Table 1 .\n\nConsistent\n\nu n i t s were used f o r t h e i n p u t d a t a .\n\nThe porous media was c o n s i d e r e d\n\npressible\n\nflow\n\n(i.e.,\n\nj =\n\nconstant).\n\nNo\n\nwas\n\nconsidered\n\nacross\n\nboundaries of t h e domain and t h e p r e s s u r e and s a t u r a t i o n a t i n l e t were h e l d c o n s t a n t a t a l l t i m e s .\n\nthe\n\ndimen­\n\nthe\n\nincom­ outer\n\nand o u t l e t\n\nA s i m i l a r problem was s o l v e d by Gulbrandsen\n\n283 and W i l l i e\n\n(1985).\n\nTABLE 1 Input d a t a f o r\n\ntwo-phase flow i n a square domain\n\nK\n\nAbsolute p e r m e a b i l i t y .\n\n0.25,\n\n=\n\n- = - 100 Porosity\n\n,\n\nViscosity\n\nof\n\noil.\n\ny\n\ni\n\nV i s c o s i t y of w a t e r .\n\ny\n\n2\n\nI n i t i a l water s a t u r a t i o n\n\nS\n\n2\n\n0.25,\n\n=\n\n=\n\n1.5,\n\nk\n\nrl\n\n1.0,\n\nk\n\nr2 \" 2\n\n\"\n\nS\n\nl\n\n=\n\n\" 2»\n\nU\n\nS\n\n2\n\nS\n\n0.1,\n\nInitial reservior p r e s s u r e (water)\n\n14.7,\n\nPressure at\n\ninlet\n\n=\n\n44.7\n\nPressure at\n\noutlet\n\n=\n\n14.7,\n\nS a t u r a t i o n c o n t o u r s o b t a i n e d f o r t h e homogeneous domain a r e shown i n 1, f o r T = 30,000 and 60,000 u n i t s . units) 2(a)\n\n(b) , r e s p e c t i v e l y .\n\n(Fig.\n\nrespect\n\nto\n\n1) , the the\n\nIt\n\nis\n\nconnecting\n\npattern\n\ni n t h e domain\n\nlines\n\nwas\n\nthe\n\ndistorted\n\n(see F i g s .\n\nthe\n\nindicated\n\nt h a t a smooth,\n\nsuch\n\nKukreti\n\net\n\nflow c h a r a c t e r i s t i c s\n\npockets. al.\n\nsymmetrical\n\nAs\n\nthe\n\nthe\n\nNumerical\n\nflow p a t t e r n i s of\n\nreported\n\no b s e r v a t i o n s were a l s o concluded\n\nand to\n\noutlet,\n\npresence\n\nas\n\nwith\n\nexpected.\n\nof\n\nimpervious\n\nThe l o c a t i o n of impervious r e g i o n i s also.\n\nFurther d e t a i l s\n\n(1985).\n\ndue\n\nFigs.\n\nhomogeneous\n\n2 ) . The e x t e n t of d i s t o r t i o n was more f o r\n\nt h e case of two impervious p o c k e t s . to a f f e c t\n\na r e p r e s e n t e d in\n\na r e approximately symmetrical inlet\n\nFig.\n\n( f o r T = 60,000\n\no b s e r v e d t h a t f o r the case of\n\niso-saturation\n\ndiagonal\n\nsymmetrical\n\npocket(s)\n\nfrom\n\nresults\n\nf o r domains w i t h one and two impervious p o c k e t s and\n\ndomain\n\nThis\n\nCorresponding\n\nin\n\nthe\n\nresults\n\nresults\n\nreference\n\ni n an e x p e r i m e n t a l\n\nshown)\n\nseen also\n\nr e s t o r e d i n r e g i o n s away\n\nnumerical\n\nthis\n\n(not\n\nthe\n\nare\n\ngiven\n\nby\n\naforementioned\n\nstudy.\n\nCONCLUSION The proposed\n\nfinite\n\nelement\n\nthe m u l t i - p h a s e\n\nflow\n\noil\n\nAlso, the e f f e c t\n\nreservoirs.\n\np r o c e d u r e can be used t o model\n\ncharacteristics\n\nof\n\ncomplex,\n\nirregular\n\nand\n\neffectively heterogeneous\n\nof impervious p o c k e t s and dead-end p o r e s can\n\nbe a d e q u a t e l y modeled by t h e proposed p r o c e d u r e .\n\nACKNOWLEDGEMENT This study was sponsored by t h e Energy Resources I n s t i t u t e a t t h e U n i v e r ­ s i t y of Oklahoma, Norman. The support i s g r e a t l y\n\nacknowledged.\n\n284\n\n(a)\n\nAt\n\nT = 30,000\n\n(b)\n\nAt\n\nT * 60,000\n\nFig.\n\n1 . R e s u l t s of i s o - s a t u r a t i o n l i n e s f o r homogeneous domain w i t h o u t impervious p o c k e t s a t d i f f e r e n t time i n t e r v a l s . (At = 3 , 0 0 0 and (10x10) mesh)\n\n(a)\n\nOne impervious pocket\n\nFig.\n\n2 . R e s u l t s of i s o - s a t u r a t i o n l i n e s f o r domain w i t h impervious p o c k e t ( s ) . (At = 3 , 0 0 0 , T = 6 0 , 0 0 0 and (10x10) mesh)\n\n(b) Two impervious p o c k e t s\n\nREFERENCES Gulbrandsen, S . and W i l l i e , S . O . , 1 9 8 5 . A f i n i t e element f o r m u l a t i o n of t h e two-phase flow e q u a t i o n s f o r o i l r e s e r v o i r s . SPE 1 3 5 1 6 , Middle East O i l Tech. Conf. and E x h i b i t i o n , B a h r a i n , pp. 2 0 1 - 2 0 6 . K u k r e t i , A . R . , Zaman, M.M. and C i v a n , F . , 1 9 8 5 . Modeling of flow of immiscible f l u i d s i n heterogeneous i r r e g u l a r shaped r e s e r v o i r s for e f f i c i e n t o i l recovery. P r o j e c t Report No. CEES/PGE/ERI/84-85-1, Univ. of Oklahoma, Norman, 127p. Langsrud, 0 . , 1 9 7 6 . S i m u l a t i o n of two-phase flow by f i n i t e element methods. Paper No. SPE 5 7 2 5 , pp. 1 0 7 - 1 2 2 . Lewis, R.W. , V e r n e r , E.A. and Z i e n k i e w i c z , O . C , 1 9 7 4 . A f i n i t e element approach t o two-phase flow i n porous media. I n t . Symp. on F i n i t e Element Met. i n Flow Problems, Swansea. Todd, M.R., O'Dell, P.M. and H i r a s k i , G . J . , 1 9 7 2 . Methods f o r i n c r e a s e d accuracy i n numerical r e s e r v o i r s i m u l a t o r s . SPEJ, 1 2 : No.6: 5 1 5 - 5 3 0 . Z i e n k i e w i c z , O . C , 1 9 7 7 . The f i n i t e element method. 3rd E d i t i o n , McGraw-Hill Book C o . , New York.\n\n285\n\nDATA S T R U C T U R E S REFINEMENT\n\nAND\n\nALGORITHMS\n\nFOR\n\nSELF\n\nLOCAL\n\nGRID\n\nJ . C . D i a z and D . B . N o r t h P a r a l l e l P r o c e s s i n g I n s t i t u t e , U n i v e r s i t y of O k l a h o m a , E E C S 202 W. B o y d , Suite 2 1 9 , N o r m a n , O k l a h o m a 73019 (U.S.A)\n\nABSTRACT M a n y m o d e l s of i m p o r t a n t p h y s i c a l p e h n o m e n a a r e d e s c r i b e d u s i n g numerical schemes. O f t e n , t h e s e n u m e r i c a l m o d e l s are time dependent. Important active aspects of t h e p h e n o m e n a are localized in s m a l l a r e a s of t h e d o m a i n . These locations change often with time. U n i f o r m g r i d d i n g r e q u i r e s very small grid s i z e . V e r y l a r g e d o m a i n s w o u l d r e q u i r e l a r g e a m o u n t s of c o m p u t e r m e m o r y . S i n c e the i m p o r t a n t c h a n g i n g a r e a s a r e l o c a l i z e d , g r i d s i z e s h o u l d b e r e d u c e d o n l y in t h e a r e a s of h i g h a c t i v i t y . Local refinement permits implementation of the m o d e l w i t h s i g n i f i c a n t l y less storage allowing analysis of l a r g e r problems. Since the s i m u l a t i o n p r o c e d e s w i t h t i m e , the l o c a l r e f i n e m e n t m u s t a l s o be a b l e t o d y n a m i c a l l y a d a p t to r e f l e c t the m o v e m e n t of the a c t i v e areas. O u r aim is the d e v e l o p m e n t of h i g h q u a l i t y v a r i a t i o n a l s o f t w a r e c a p a b l e of d y n a m i c l o c a l grid r e f i n e m e n t for g e n e r a l d i s t r i b u t i o n . H e r e i n , w e d i s c u s s t h e d a t a s t r u c t u r e a n d a l g o r i t h m s n e e d e d to s u p p o r t t h e d y n a m i c p l a c i n g or r e m o v a l of l o c a l r e f i n e m e n t . The a b i l i t y of a p r o b l e m i n d e p e n d e n t g r i d a n a l y s i s to t r i g g e r the p l a c e m e n t or r e m o v a l o f l o c a l r e f i n e m e n t f o r a n a c c u r a t e l o c a l representation of t e m p o r a l changes in t h e s o l u t i o n w i l l be i l l u s t r a t e d in a m o v i n g f r o n t s i t u a t i o n . INTRODUCTION The\n\nneed\n\nfor\n\nhas\n\nbeen\n\nand\n\nRheinboldt,\n\nwidely\n\nefficient state\n\nproblems of\n\ninvestigated engineering. Rosenberg, to\n\ngeneral\n\n1982). grid\n\nby\n\nData\n\nthe\n\nrefinement been\n\nlaboratories\n\nintroduced\n\nthis\n\ntechnique differences\n\n1982);\n\nsome\n\nand,\n\na\n\nsimulators capable\n\n(Diaz\n\net\n\nsupporting\n\nefficient\n\nregeneration\n\nfixed\n\nfor\n\ndynamic\n\nal.,\n\n1984).\n\nthe\n\ngrid\n\nor at\n\ncommunity\n\nand\n\nhave\n\nand\n\nBesset,\n\nThis\n\nof\n\ntime\n\n1980).\n\nin\n\nalso\n\nreservoir\n\ndeveloped has\n\n1983,\n\nis\n\nlocal\n\nstep\n\nhave\n\nrefinement\n\nscheme\n\nremoval every\n\nfor\n\ncapability\n\ngrid\n\n(Babuska\n\nSherman,\n\nbeen\n\nmodels\n\nsupporting\n\ncorporations\n\ngrid\n\nlocal\n\nof\n\ntechniques\n\napplication\n\nlocal\n\n(Quandale\n\nplacement\n\noil\n\nformulae\n\nof\n\nof\n\n(Bank\n\nof\n\nnumerical\n\ncapable\n\nvariational\n\nsome\n\nin\n\nscientific\n\nstructures\n\nFinite\n\nscheme\n\ndefinition\n\nhave\n\nproposed\n\nwithout\n\nrefinement\n\nrecognized\n\nlocal\n\nResearchers\n\nlocal\n\nas\n\n(von been\n\n1985). has\n\nA\n\nbeen\n\ncapable\n\nof\n\nrefinement required\n\nby\n\n286 the\n\nother\n\nschemes.\n\nThe\n\nimplementation\n\none\n\nintroduced\n\nsimplification\n\nof\n\nuses\n\na\n\nthe\n\nsimilar\n\nsame\n\nconsists g r i d .\n\nSELF\n\ninformation on\n\nthe\n\nThis\n\nmanaging\n\nself\n\n1\n\ntree\n\nGrid\n\nin\n\nfour\n\nand\n\nand\n\nis\n\na\n\n1984).\n\nstores\n\nthe\n\nregularity\n\nmuch\n\nherein\n\net.al.,\n\nHowever, of\n\nt r a v e l\n\nare\n\nlocal\n\nmajor\n\ngrid\n\nIt\n\nbasically\n\nsimplification\n\nconditions\n\nthus\n\nthe\n\nfor\n\nthe\n\nalgorithms\n\nsimpler.\n\ngrid\n\nis\n\nfunction.\n\nrequiring\n\nrefinement\n\nrefinement\n\nc a p a b i l i t y\n\ncan\n\nbe\n\nsteps: local\n\nindicator\n\nlocations\n\nlocal\n\ntree\n\ndiscussed (Diaz,\n\nGRID REFINEMENT\n\nA n a l y s i s ~ The\n\nactivity of\n\nthe\n\nin\n\nstructure\n\ncell.\n\nstructure\n\nLOCAL\n\ndata\n\nassumption\n\ndecomposed\n\nper\n\ndirect\n\nl i m i t s\n\nthe\n\nA\n\nthe\n\nis\n\nfurther\n\nno\n\nanalyzed\n\nThis\n\nrefinement\n\nlonger\n\nusing\n\nanalysis\n\nneeded\n\nor\n\nand\n\nsome\n\nproduces\n\na\n\nlist\n\nlocations\n\nremoval\n\nwhere\n\nis\n\nrecommended . 2\n\n'\n\nGr£d_Management grid\n\nis\n\n-\n\nThe\n\nmodified\n\ndata\n\nstructure\n\naccording\n\nto\n\nthe\n\nsupporting\n\noutput\n\nof\n\nthe\n\nthe\n\nlocal\n\ngrid\n\nanalysis .\n\n^i:^El£ .^_^.£££E!^iZ\n\n~\n\nalgebraic\n\nspecified\n\nn\n\n3*\n\nthe ^•\n\nsystem\n\nT\n\nn\n\ne\n\nd\n\na\n\nt\n\nstructure\n\na\n\nby\n\nthe\n\nis\n\nused\n\nnumerical\n\nto\n\nassemble\n\nscheme\n\nused\n\nthe in\n\nmodel.\n\n^ y s t e m s ^ S o ^ u t i_on solution\n\n~\n\nmethod\n\nT\n\nn\n\nalgebraic\n\ne\n\nused\n\nexploits\n\nsystem\n\nis\n\nextensively\n\nsolved.\n\nThe\n\nthe\n\ntree\n\ndata\n\na\n\ndesired\n\nstructure . This\n\nsequence\n\naccuracy DATA A\n\nis\n\nof\n\nsteps\n\nachieved\n\nand\n\nis is\n\nrepeated\n\nuntil\n\ncoordinated\n\nwith\n\nthe\n\ntime\n\nlevel\n\nof\n\nstepping.\n\nSTRUCTURE complex\n\ndata\n\nr e f i n e m e n t .\n\nstructure\n\nThe\n\ncharacteristics. refinements placement easily\n\nof\n\nor\n\nIt\n\nthe\n\nfrom The\n\nis\n\nshould\n\nof\n\nthe\n\nrequired\n\nstructure\n\ngrid.\n\nremoval\n\nbuilt\n\nsupported.\n\ndata\n\naccurately\n\nIt\n\nshould\n\nlocal data\n\nstorage\n\nbe\n\ngrid.\n\nto\n\nsupport\n\nshould reflect\n\ndynamic\n\nThe\n\nstructure.\n\nrequired\n\nto\n\nused\n\nlocal\n\nhave the\n\ngrid\n\nseveral\n\ngrid\n\nand\n\nthe\n\nand\n\neasily\n\nsupport\n\nalgebraic\n\nsystem\n\nmust\n\nbe\n\nanalysis\n\nshould\n\nbe\n\nthe\n\nshould\n\nbe\n\nGrid\n\nrepresent\n\ngrid\n\nsmall . We\n\nhave\n\nmeeting\n\nthe\n\ncorresponds stored it\n\nis\n\nin the\n\nimplemented\n\na\n\nrequirements to the\n\na\n\npoint\n\nin\n\nstructure.\n\ncenter\n\nof\n\nan\n\ndata listed the\n\nstructure above. grid.\n\nA point undivided\n\nto\n\nsupport\n\nEach\n\nnode\n\nMinimal qualifies cell\n\nin\n\nin\n\nnumber to\n\nthe\n\nbe\n\nthe\n\nlocal\n\nthe of\n\ngrid\n\nstructure points\n\nare\n\nrepresented\n\ndomain\n\nor\n\nit\n\nis\n\nif a\n\n287 regular\n\npoint.\n\nundivided To\n\nfour\n\nthe\n\ncenter\n\nnew\n\nbecome parent\n\ntime\n\nthe\n\nregular\n\nis\n\nregular\n\nis\n\non\n\nof\n\nare\n\nthat\n\nalso\n\nit\n\nis\n\nboundary\n\nat\n\ngrid are\n\nis\n\na\n\ntree\n\nas\n\nAny\n\nchildren\n\nof\n\nfour\n\ndomain.\n\nintroduced\n\nrefined. as\n\ncorner\n\nthe\n\nrepresented\n\nwas\n\nthe\n\nof\n\nrefinement,\n\nnested cells\n\ncell\n\nif\n\nthe\n\nlocal\n\nthat\n\nundivided\n\nof\n\ndomain\n\nis\n\nstructure\n\nstructures would\n\nbe\n\nused\n\npoints macro\n\nTo m a k e\n\nthe\n\nrules\n\ngrid. big\n\nas\n\nreduce These\n\nare\n\nsome\n\nstructure the\n\nthe\n\nof\n\nchildren\n\nother\n\nto\n\nis\n\ncenters\n\npoints\n\nthe\n\nof that\n\nappropriate\n\nof\n\nFOR\n\nbe If\n\nto\n\nthis\n\na\n\nhappens\n\nby\n\nA forest\n\nmacro\n\ncell the\n\nbetter\n\nin\n\nof\n\na\n\ntree there\n\nthe\n\ntrees\n\n1985).\n\ncell\n\nthat\n\nis\n\nlarger\n\nare\n\nenforced.\n\nsize\n\nwithin\n\nmore\n\nthan\n\ncell\n\nis\n\nthe\n\ntwice\n\nrefined\n\napproximation\n\nfurther\n\nis\n\nSince\n\ncells,\n\net.al.,\n\nchanges to\n\nThere\n\ndomain.\n\n\"regularity rules\"\n\nprovides\n\nenforced\n\ncells.\n\ncell.\n\nwhole\n\n(Diaz,\n\nabrupt\n\nmacro\n\nmacro\n\nthe\n\nuniform,\n\nThis\n\nDYNAMIC three\n\nthe\n\nlocal\n\ngrid\n\nto\n\nproperties.\n\nrefinements\n\nand\n\nby\n\nnot\n\nto\n\nsearch\n\ncorresponding\n\nGRID REFINEMENT algorithms\n\nrefinement.\n\nand\n\nimportant The\n\nLOCAL\n\nmajor\n\nalgorithm;\n\nstructure.\n\neach\n\ngrafted\n\nallow\n\ninto\n\nunrefinements.\n\nare\n\nis\n\nnode\n\ni s .\n\nto\n\ncommon\n\nare\n\nmore\n\nnot\n\nrules\n\nderefine It\n\ngrid\n\nsize.\n\nThere\n\ndivided\n\nrepresent\n\nare\n\ncannot\n\nit\n\nALGORITHMS\n\nnature\n\nthat\n\nits\n\nallowing\n\nto\n\ncells\n\ndo\n\nA cell\n\ni n i t i a l l y\n\nassociated\n\nis\n\nthose\n\nThese\n\na\n\nit\n\nnode.\n\nThe\n\nas\n\nif\n\nnesting\n\nEach\n\nthe\n\nfor\n\nor\n\nsupport\n\nrequired.\n\ntree\n\nA point\n\ncells\n\na\n\nsearch\n\nbe\n\nable\n\nalgorithm to\n\nthe\n\nthat\n\nThey\n\nsupport\n\nare\n\na\n\nthe\n\nrefine\n\ndynamic\n\nalgorithm,\n\nalgorithm. to\n\nlocate\n\nlocates\n\npoint\n\nin\n\na\n\nthe\n\nnode data\n\nin\n\nthe\n\ntree\n\nstructure\n\nin\n\nthe\n\nphysical\n\nfor\n\nthe\n\nplacement\n\nthe\n\ncoordinated\n\ndomain. The\n\nrefine\n\nrefinement. representing be\n\nalgorithm The\n\nthe\n\nrefined.\n\npresent\n\nThe\n\nrefine\n\nthe\n\ntree\n\nstructure,\n\nthe\n\nfour\n\nchildren\n\nconsiders refinement The it\n\nis\n\nlocal\n\neach to\n\nno\n\nlonger\n\nchanging\n\nregeneration\n\nof\n\nthe\n\nneeded.\n\nfour\n\nIt\n\nthe\n\nedge\n\nis\n\na\n\nthe\n\ncan\n\nbe\n\nnested\n\npoints\n\nof\n\nfurther\n\nforest of\n\nof\n\ntrees\n\nthe\n\nnode\n\nto\n\nappropriate\n\nnode\n\nin\n\nrefined,\n\nintroduces\n\nrefinement,\n\ncreated\n\nby\n\nand\n\nthis\n\nthen local\n\nrules. removal\n\nof\n\nnested\n\nimportant\n\nto\n\ndo\n\ncan\n\ngrid.\n\nit\n\nof\n\ncoordinates\n\nfinds\n\nthat\n\nallows\n\nphenomena whole\n\nand\n\nalgorithm\n\nregularity\n\nalgorithm\n\nconsists\n\nrefinement\n\nrepresenting\n\nof\n\nthe\n\ninput\n\nverifies\n\nenforce\n\nunrefine\n\nallows\n\nrequired\n\nthis\n\nrefinement\n\nwhen\n\nd y n a m i c a l l y so\n\nbe\n\nfollowed\n\nwithout\n\nThe\n\nalgorithm\n\nrequires\n\nrequiring as\n\ninput\n\na\n\n288 tree\n\nforest\n\nrepresenting of\n\nlocates\n\ncorresponding\n\nthat\n\nthe\n\nthe\n\nthe\n\ncoordinates\n\nnode\n\nit\n\ncan\n\nbe\n\nremoves\n\nall\n\nchildren\n\nremoved\n\npresent\n\nthat\n\nis\n\nnode\n\nwithout\n\nand\n\nto\n\nnested be\n\nin\n\nthe\n\ntree\n\nviolating\n\nregular\n\nnodes\n\nrefinement\n\nremoved.\n\nthe\n\nThe\n\nand\n\nstructure, regularity\n\nintroduced\n\nby\n\nthe\n\nalgorithm verifies rules,\n\nthis\n\nand\n\npoint.\n\nGRID ANALYSIS The a\n\ntriggering\n\nproblem\n\nin\n\nthe\n\nanalysis\n\n1984;\n\nfor\n\nWeiser,\n\ntime\n\ntype\n\nFor\n\net\n\na l . ,\n\nthis\n\ngrid\n\nfunction being\n\nanalysis\n\nindicates\n\ncell.\n\nsumming\n\nassigned\n\nto\n\nall\n\ntrees\n\nup\n\nrecommends\n\nof\n\nindicator\n\nis is\n\nthe\n\nof\n\nnear\n\nFlaherty, 1980;\n\nand a\n\nBank\n\nBabuska, bisection-\n\nto\n\nreservoir\n\nactive\n\nwells\n\nherein\n\nmake\n\nan\n\nindicator\n\nactivity indicator\n\nnodes\n\nin\n\nforest\n\nby\n\nThe\n\nof\n\n(Diaz use\n\nfunction.\n\nthe\n\ntree\n\nan\n\nlocal\n\nof\n\nactivity from\n\nremoved\n\nif\n\nsibbling\n\ncell\n\nits\n\nvalue is\n\nbottoms\n\nstrategy\n\nvalues\n\ncells\n\nare\n\nwithin\n\nindicator.\n\nof\n\nthe\n\n25\\$\n\nits\n\nof a\n\nLocal\n\nvalue\n\nof\n\nof\n\nthen\n\nthe\n\nwhen\n\ntolerance\n\nevery\n\nall\n\nparent's\n\nundivided\n\nfor\n\nindicator the\n\nbisection-type be\n\nThis\n\nphenomena\n\ncalculated\n\nworking\n\ntheir\n\nspecified\n\nof is\n\nthe\n\nThus,\n\nundivided 2 5%\n\non\n\na\n\nan\n\nrefinement\n\nfour\n\nplaced above\n\nof\n\nroots.\n\ntolerance\n\nrefinement\n\nlevel\n\nin\n\nthe\n\napplied\n\nflow\n\ngrid\n\n1-dimensional\n\nproblems,\n\npresented\n\non\n\nactivity\n\nnested the\n\nbased\n\nchildren.\n\nnodes to\n\nresults\n\nancestor\n\nits\n\nthat\n\ni n d i c a t o r s specified\n\nthe\n\nThe\n\nover\n\nthe\n\nlocal\n\nand\n\n(Bieterman\n\ndependent\n\nused have\n\nand\n\nfor\n\nby\n\nauthors\n\nSherman,\n\nstrategies\n\nproposed\n\ncriterion\n\nChandra\n\nand\n\nis\n\nanalysis.\n\nis\n\nThis\n\nBank\n\ns u c c e s s f u l l y\n\nThe\n\nThe\n\nSeveral\n\n(Babuska,\n\nbeen\n\ngrid\n\nindicators\n\n1982;\n\nrepresent\n\ngrid\n\nmonitored.\n\nundivided\n\nthe\n\nbeen to\n\n1985).\n\nbisection-type\n\nThe\n\nby\n\nhas\n\nproblems\n\n1984,\n\nerror\n\ntime\n\nlocal\n\nalgorithm.\n\nrecently,\n\nhas\n\n2-dimensiona 1\n\ns t r a t e g y\n\nsimulation\n\nof\n\nof\n\nmodified.\n\nproblems\n\nMore\n\nproblems\n\nremoval\n\neasily\n\nRheinboldt,\n\n1985).\n\ndependent\n\n1 985 ) .\n\nstate\n\nand\n\nor\n\nanalysis\n\nbe\n\nstudies\n\nBabuska\n\ngrid\n\ncan\n\nt h e o r e t i c a l\n\nanalysis\n\nand\n\nplacement\n\nindependent\n\ngrid\n\nof\n\nof\n\nits\n\nparent\n\ni nd i c a t o r . APPLICATIONS It\n\nhas\n\nbeen\n\nalgorithm localized active\n\ncan\n\nfunction\n\nthat\n\nsuch\n\nreservoir a\n\nmoving was\n\nconcentration\n\nsimilar\n\nsuccessfully\n\nbehavior\n\noil\n\nconsider\n\nshown\n\nas\n\nwells\n\nfront\n\ng e n e r a t e d of\n\nan\n\ndata\n\nstructure\n\nrepresent that (Diaz\n\nrepresented et\n\nal.,\n\napplication. to\n\nby\n\ngrid\n\nThe\n\nflow\n\nregimes\n\nThus,\n\nHerein,\n\na c t i v i t y\n\nthe it\n\nanalysis\n\nconditions\n\n1984,1985).\n\nrepresent fluid.\n\nand\n\nchanging\n\nhigher\n\nfor near we\n\nindicator of\n\nthe\n\nnear\n\nthe\n\n289\n\n»»•\n\nTime = 600\n\nFigure\n\ninterphase display\n\nand\n\nthe\n\nindicator\n\n1:\n\nT\n\nDynamic\n\nGrid\n\nnegligible\n\nchanging\n\nfunction\n\nfor\n\naway\n\nlocal times\n\nfrom\n\ngrid 200,\n\nthe\n\nand 400,\n\nto\n\nm\n\ne\n\n= WO\n\na Moving\n\nfront.\n\nthe 600\n\ni\n\nIn\n\nlevel and\n\nFront.\n\nFigure\n\ncurves\n\n1,\n\nfor\n\nwe the\n\n800.\n\nCONCLUSIONS The\n\ndata\n\nimplemented Figure\n\n1\n\nalgorithms\n\nstructure for\n\na\n\nserial\n\ndemonstrate to\n\nand\n\nsupport\n\nthe\n\ncorresponding\n\narchitecture. a b i l i t y\n\ndynamic\n\nof\n\nplacement\n\nalgorithms The the and\n\nresults data or\n\nhave\n\nbeen\n\nimplied\n\nstructure removal\n\nof\n\nby and\n\nlocal\n\n290 grid\n\naccording\n\na b i l i t y\n\nto\n\nrepresenting\n\nto\n\ndirect the\n\nthe\n\ngrid\n\nthe\n\nchanging\n\nanalysis.\n\nautomatic of\n\nlocation\n\nThe\n\nchange of\n\ngrid of\n\na moving\n\nanalysis local\n\nhas\n\nthe\n\nrefinement\n\nfront.\n\nREFERENCES B a b u s k a , I . , J . C h a n d r a and J . E . F l a h e r t y ( E d i t o r s ) (1984). A d a p t i v e Computational Methods for P a r t i a l Differential E q u a t i o n s , SIAM P u b l i c a t i o n s , P h i l a d e l p h i a . B a b u s k a , I . a n d W. C . R h e i n b o l d t , (1982), A Survey of A - P o s t e r i o r i E r r o r E s t i m a t o r s and A d a p t i v e Approach i n t h e F i n i t e E l e m e n t M e t h o d , T e c h . R e p . BN 1 9 8 1 , U n i v e r s i t y o f M a r y l a n d , L a b . f o r Num. A n a l . B a n k , R. E. and A . H. S h e r m a n , ( 1 9 8 0 ) , A Refinement Algorithm and Dynamic Data S t r u c t u r e f o r F i n i t e E l e m e n t M e s h e s , T e c h n i c a l R e p o r t # 1 6 6 , U n i v e r s i t y of T e x a s , C e n t e r f o r Numerical Analysis . Bank, R.E. a n d A . W e i s e r , ( 1 9 8 5 ) , Some A - P o s t e r i o r i Error E s t i m a t o r s f o r t h e F i n i t e E l e m e n t M e t h o d , M a t h C o m p . To\n\nAppear.\n\nB i e t e r m a n , M. a n d I . B a b u s k a , ( 1 9 8 5 ) , An A d a p t i v e M e t h o d o f L i n e s w i t h E r r o r C o n t r o l f o r P a r a b o l i c E q u a t i o n s of t h e R e a c t i o n D i f f u s i o n T y p e , t o a p p e a r i n J o u r n a l of C o m p u t a t i o n a l P h y s i c s . 4\n\nDiaz, J . C , R.E. Ewing, R.W. J o n e s , A . E . M c D o n a l d , D.U. von R o s e n b e r g and L.M. U h l e r , ( 1 9 8 4 ) , S e l f - A d a p t i v e L o c a l G r i d Refinement A p p l i c a t i o n in Enhanced Oil R e c o v e r y , P r o c . 5th Int. S y m p . on F i n i t e E l e m e n t s a n d F l o w P r o b l e m s , A u s t i n , T e x a s , J a n u a r y 2 3 - 2 6 , pp. 4 7 9 - 4 8 4 . Diaz, J . C , R.E. Ewing, R.W. J o n e s , A . E . M c D o n a l d , D.U. von R o s e n b e r g and L.M. U h l e r , ( 1 9 8 5 ) , S e l f - A d a p t i v e Local Grid R e f i n e m e n t for Time D e p e n d e n t T w o - D i m e n s i o n a l S i m u l a t i o n , i n F i n i t e E l e m e n t s i n F l u i d s ( e d s . G a l l a h e r , C a r e y , Oden, Z i e n k i e w i c z ) J o h n W i l e y & S o n s , New Y o r k p p . 2 7 3 - 2 8 4 . D i a z , J . C. and R. E. E w i n g , ( 1 9 8 5 ) , Potential of HEP-like MIMD A r c h i t e c t u r e i n S e l f A d a p t i v e L o c a l G r i d R e f i n e m e n t for A c c u r a t e S i m u l a t i o n of P h y s i c a l P r o c e s s e s , i n P r o c e e d i n g s of t h e W o r k s h o p on P a r a l l e l P r o c e s s i n g U s i n g t h e Heterogeneous E l e m e n t P r o c e s s o r , M a r c h 2 0 - 2 1 , N o r m a n , OK. pp. 2 0 9 - 2 2 6 . D e n k o w i c z , L . , Ph. D e v l o o , and J . T . Oden, ( 1 9 8 5 ) , M e s h - R e f i n e m e n t S t r a t e g y B a s e d on M i n i m i z a t i o n E r r o r s . To a p p e a r . Q u a n d a l e , P. and for Improved on R e s e r v o i r\n\nOn of\n\na\n\nh-Type Interpolation\n\nP . B e s s e t , ( 1 9 8 3 ) , The Use of F l e x i b l e G r i d d i n g R e s e r v o i r M o d e l i n g , SPE # 1 2 2 3 9 , V I I SPE S y m p o s i u m S i m u l a t i o n , S a n F r a n c i s c o , November 1 6 - 1 8 .\n\nQ u a n d a l e , P. and P. B e s s e t , ( 1 9 8 5 ) , R e d u c t i o n of G r i d E f f e c t s Due to Local S u b - G r i d d i n g in S i m u l a t i o n s Using a Composite Grid, SPE# 1 3 5 2 7 , V I I I SPE S y m p o s i u m on R e s e r v o i r S i m u l a t i o n , D a l l a s , February 10-13. Rheinboldt, W.C a n d C K , M e s z t e n y i , ( 1 9 8 0 ) , On a D a t a Structure f o r A d a p t i v e F i n i t e E l e m e n t M e s h R e f i n e m e n t s , TOMS 6, p p . 1 6 6 187. von\n\nRosenberg, D.U., ( 1 9 8 2 ) , L o c a l Mesh R e f i n e m e n t f o r Finite D i f f e r e n c e M e t h o d s , SPE 1 0 9 7 4 p r e s e n t e d a t 1 9 8 2 SPE A n n u a l T e c h . C o n f . a n d E x h i b . , New O r l e a n s , LA, S e p t . 2 6 - 2 9 .\n\n291\n\nON THE MODELING OF SOIL LIQUEFACTION BY FINITE ELEMENT METHOD M.M. Zaman and J.G.\n\nLaguros\n\nSchool of Civil Engineering and Environmental Science, University of Oklahoma, Norman, Oklahoma, 73019, U.S.A.\n\nABSTRACT A numerical procedure based on the finite element technique is presented for evaluating the characteristics of soil liquefaction and the foundation response under seismic loading. Effects of soil-structure interaction on the on-set and propagation of liquefaction are studied. Application is demonstrated through solution of a numerical problem involving two partially embedded structures and underlying soil medium subjected to cyclic exci­ tation at the rigid bedrock. INTRODUCTION Excessive soil during\n\nsettlement\n\nof\n\nearthquakes\n\nfoundations\n\nhas been\n\ndue\n\nfound\n\nto\n\nliquefaction\n\nto be the major\n\ncause\n\nof of\n\nwidespread damage of numerous structures and foundations. In order to mitigate\n\nsuch hazards in the future, it is neces­\n\nsary to develop rational techniques to predict liquefaction char­ acteristics of natural of this paper\n\nsoil\n\nis to present\n\n(sand) deposits.\n\nfinite element method, to evaluate\n\naccurately\n\nof\n\nmodeling of\n\nsoil-structure soil\n\non\n\nApplication\n\nsolution\n\nembedded\n\non-set zones,\n\ninteraction\n\ndeposits.\n\nthrough\n\nthe\n\nliquefied\n\nobjective\n\nthe soil liquefaction\n\nfoundation response under seismic expansion\n\nThe main\n\na numerical procedure, based on the\n\nof\n\nof a numerical\n\nand\n\nliquefaction, identifying\n\nliquefaction the\n\nand\n\ntracing\n\nthe\n\nthe\n\nis given to the\n\neffects\n\nof\n\ncharacteristics\n\nof\n\nprocedure\n\nproblem\n\nstructures with soil medium\n\nEmphasis\n\nis\n\ninvolving\n\nsubjected\n\ndemonstrated two\n\npartially\n\nto cyclic\n\nexcita­\n\ntion at the rigid bedrock. REVIEW OF A\n\nLITERATURE\n\nseries\n\nof\n\nshaking\n\ntable\n\ntests\n\nperformed\n\nby\n\nYoshimi\n\nand\n\nTokimatsu\n\n(1979) have indicated\n\nthat the excessive pore\n\npressure\n\ndeveloped\n\nbelow\n\nsmaller\n\nfrom\n\nstructure\n\n(free-field), and the ratio of the excess pore pressure\n\na structure\n\nis\n\nthan\n\nthat\n\naway\n\nthe\n\n292 to\n\nthe\n\nthe\n\ninitial\n\nstructure\n\neffective becomes\n\ndation\n\nbecomes\n\nfield,\n\nwhile\n\nstructure\n\nRelatively\n\nzones\n\nfew\n\nstudies\n\nsettlement\n\nof\n\nbehavior\n\napproach.\n\nmixture. the\n\nground\n\nthe\n\nsubject\n\nFINITE\n\non\n\nfinite\n\nvicinity\n\nbeen\n\nwas\n\nformulation\n\nplane\n\nstrain\n\nstiffness, mass\n\nthe global\n\n{u} +\n\n{u} =\n\n[C]\n\n[K]\n\nand\n\nrespectively,\n\npore\n\nthe\n\nliterature\n\nbeen\n\nthe\n\nChang\n\nto\n\nanalyze in\n\neffective\n\nreported soil\n\nfootings A\n\nwhich stress\n\nrecently\n\nas\n\na\n\nby\n\ntwo-phase\n\nwere\n\nlocated\n\non\n\nreview\n\non\n\ndetailed\n\n[K]\n\n{R(t)}\n\n=\n\nindicates pressure\n\nare\n\nin\n\nstudy As\n\ndamping matrices equation\n\nof\n\nis\n\na\n\nbased\n\nstarting\n\nare\n\nevaluated\n\nmotion: (1)\n\ndamping\n\nvector,\n\nderivative\n\neffect\n\nthis\n\n(R(t)}\n\nmass,\n\nforce\n\nin\n\nidealization.\n\nand\n\nto o b t a i n\n\nsed\n\nfoundation/-\n\nthe\n\n{u} +\n\nclude\n\nfree-\n\n(1985).\n\nassembled\n\noverdot\n\nin\n\nearthquake\n\nunrealistic.\n\n[M]\n\nand\n\nas\n\nfoun­\n\nFORMULATION\n\nelement\n\n[M],\n\nthe\n\ntechnique\n\nconsidering\n\nand\n\nwhere\n\nthe\n\nthe\n\nobservations.\n\nusing\n\nstudies,\n\nBiswas\n\nsuch\n\nby\n\nhas\n\n(1984)\n\nby\n\nelement\n\n[C]\n\nof\n\nreported\n\nof\n\nelement\n\nstudy\n\nwhich\n\ntwo-dimensional\n\npoint,\n\nthe\n\nmodeled\n\nthese\n\nis g i v e n\n\nELEMENT\n\nThe\n\nof\n\nsurface\n\nthan\n\ninduced\n\nShiomi\n\nIn b o t h\n\nliquefaction\n\nhave\n\nwas\n\nsimilar\n\nand\n\ndecreases\n\nto\n\nfinite\n\nfoundation\n\nA\n\nZienkiewicz\n\nstructure\n\nunder\n\nmodeling\n\nthe\n\nthe\n\nsoil\n\nin\n\nthe\n\nzone directly\n\nsensitive.\n\nanalytical\n\nemployed\n\nbelow The\n\nsensitive\n\nbecome more\n\nconcerning (1984)\n\nless the\n\nstress\n\nheavier.\n\nwith\n\nthe\n\nand\n\nstiffness\n\nmatrices,\n\n{u} = d i s p l a c e m e n t respect\n\nanalysis\n\nto\n\nvector,\n\ntime.\n\nR(t)\n\ncan\n\nTo\n\nbe\n\nin­\n\nexpres­\n\nas\n\n(R(t) } = where (r (t)> B\n\n(R-^t) } +\n\n{R\n\n(t) } = /\n\ne\n\n=\n\n(2)\n\nB\n\n-\n\n[B]\n\nv\n\n(R (t) } [M]\n\n{ n } p»\n\n{u\n\n}\n\n(3a)\n\ndv\n\n(3b)\n\nNSL (R (t)} B\n\nIn E q . induced forces\n\n=\n\nE < e=l\n\nr\n\n( 2 ) , {R in the caused\n\n(\n\nt\n\n)\n\n}\n\nB\n\n(t)} =\n\nNSL\n\n{1,\n\n1,\n\nby\n\nexcess\n\nrepresents 0}\n\nsystem by\n\nstrain-displacement and\n\n(\n\n3\n\nC\n\n)\n\ne\n\nindicates\n\nvector\n\nbedrock\n\ndue\n\nresidual\n\ntransformation the\n\nnumber\n\nthat p\n\n1\n\nto\n\ninertia\n\nacceleration,\n\nof\n\npore\n\ne\n\nsubmerged\n\nis h y d r o s t a t i c\n\nin\n\na\n\nn\n\ng\n\npressure\n\nmatrix,\n\nforces\n\n{u } p . 1\n\nd\n\nat time {R (t)} f i\n\nAlso,\n\nrepresents soil\n\nelement;\n\nnature.\n\n[B]\n\nt = =\n\nelement {n}\n\n=\n\n293\n\n190\n\njr Loose Medium ~\"Dense\n\n1\n\nVery Dense 840\n\nFig. 1\n\nf\n\nPlane strain finite element idealization of soil-structure system.\n\nIt may be noted that in order to evaluate Eqs. it\n\nis\n\nnecessary\n\nto\n\ndetermine\n\nthe\n\nmagnitude\n\nof\n\n(3b) and (3c), pore\n\npressure\n\ndeveloped at all points in the submerged soil domain selected pore\n\npressure\n\ncomputation.\n\npore\n\npressure\n\nmodel\n\nployed\n\nfor\n\nthis\n\nIn this\n\nproposed\n\npurpose.\n\nby\n\nIt\n\nstudy,\n\na deterministic\n\nIshibashi\n\nis assumed\n\net\n\nal.\n\n(1977)\n\nthat\n\nthe\n\npore\n\ntype\n\nis\n\nem­\n\npressure\n\n(Ap')j, at any given instant of time, due to randomly\n\nrise\n\nfor\n\nvary­\n\ning shear stress history can be expressed as a (4) J\n\nN-l\n\nin which cycle,x\n\nn\n\n( g)\n\nrepresents\n\nNe\n\ntively.\n\n(for j = p) and negative\n\nC^,\n\nof\n\nshear\n\nstress\n\nand\n\nThe\n\n(Ap) ; n\n\nresidual\n\npore\n\npressure\n\nat\n\ntotal\n\nthat\n\nAp\n\nat\n\nis Ap =\n\npressure\n\nat\n\nany\n\n(Ap)\n\nNth\n\ngiven +\n\np\n\ncycle,\n\n(N-l)th cycle, that\n\ndure is employed elements.\n\n(for j = n ) , respec­\n\nand a are the associated material\n\nthe model.\n\n(Ap)p\n\nnumber\n\n= shear stress amplitude at Nth cycle, and the subscript\n\nj denotes positive for\n\nthe equivalent\n\nis P\n\nn\n\ncycle\n\n(Ap) Ap\n\n= V _± n\n\nat each time step for all submerged\n\nThe term N\n\nin Eq.\n\n(4)\n\nthe\n\nsum\n\nTo determine\n\nn #\n\nis\n\nis\n\nparameters\n\nis calculated\n\nto\n\nthe\n\nof the\n\npore\n\nThis proce­ soil\n\n(sand)\n\nfrom\n\n0\n\nwhere cycle\n\n=\n\napplied\n\nshear\n\n(1 < i < N) and i\n\nn\n\nstress\n\namplitude\n\ncorresponding\n\nto\n\nith\n\n= cyclic shear stress at the Nth cycle.\n\nFurther details of this model and its computer implementation\n\nare\n\n294\n\n(1985).\n\ngiven by Biswas In\n\nthe\n\npresent\n\nstudy,\n\ndissipation\n\nof\n\npore\n\nseismic shaking is neglected for simplicity. for short duration\n\nseismic\n\ninsignificant effect.\n\npressure\n\nduring\n\nIt is expected that\n\nshaking, this assumption will have an\n\nAlso, after the on-set of liquefaction\n\nin\n\nan element, the shearing modulus is reduced to zero and the bulk modulus close and\n\nis\n\nincreased\n\n0.5.\n\nto\n\ndamping\n\nupdated.\n\nBased\n\nmatrices\n\nby\n\nassigning\n\non\n\nthe modified\n\nare\n\nthe\n\nreevaluated\n\nAn implicit scheme\n\nPoisson's\n\nratio\n\nproperties, and\n\nthe\n\nthe\n\na\n\nvalue\n\nstiffness\n\nglobal\n\nmatrix\n\nis\n\n(Newmark - \\$ method) is used for the\n\nstep-by-step time integration of the global equation of motion. NUMERICAL\n\nEXAMPLE\n\nProblem Statement and Finite Element Mesh Used 1 depicts\n\nFigure\n\nthe plane strain\n\n(mesh) of two partially embedded\n\nfinite\n\nelement\n\nstructures.\n\nidealization\n\nThe underlying\n\nsoil\n\ndeposit consists of stratified sands with density increasing the\n\nground\n\nsurface\n\ntable is located\n\nto\n\nthe\n\nbottom\n\nat a depth of\n\nbedrock\n\nas\n\nshown.\n\nThe\n\n5 feet below the ground\n\nfrom water\n\nsurface.\n\nThe material properties used\n\nin the analysis are given by Biswas\n\n(1985).\n\nParameters\n\npore\n\nselected\n\nin\n\nfor\n\na manner\n\nthe\n\nthat\n\nthe\n\npressure\n\nsite\n\nprediction\n\npossesses\n\nmodel\n\nstrong\n\nare\n\npotential\n\nfor liquefaction under the applied uniform cyclic acceleration. Numerical Results The distribution of the maximum\n\nof shear\n\nshear\n\nstress history\n\nas at\n\nthe\n\nratio\n\na point\n\nto in\n\nbuildup\n\nof\n\nratio, defined\n\nthe initial mean effective stress at the same point, is shown Fig. 2.\n\namplitude\n\nstress\n\nThe shear stress ratio plays a key role in pore pressure and\n\nliquefaction\n\nsusceptability\n\nat\n\na point.\n\nThe\n\nhigher\n\nthe stress ratio, the faster is the rate of pore pressure buildup and the shorter\n\nis the time required\n\nfor liquefaction\n\nto\n\noccur.\n\nIt can be seen from Fig. 2 that the stress ratios are large near the foundation edges and decrease in regions away from the struc­ ture.\n\nThe\n\nstress\n\nratios\n\nbecause of the high ratio near\n\nthe\n\nof\n\nstress,\n\nthe\n\nfoundation\n\npore edge\n\neffective (SSTIN) and\n\nratio 1.6\n\nsmall\n\nunder\n\nstress.\n\nthe\n\nThe\n\ncan be attributed\n\nratios\n\npressure at\n\nvery\n\nedges\n\ninteraction\n\nlarge\n\nliquefaction.\n\ninitial\n\nfoundation\n\nsoil-structure Because\n\nare\n\neffects low\n\nreaches\n\nsec,\n\nin\n\nstructure\n\nlarge\n\nstress\n\nto the\n\nlarger\n\nthat\n\nregion.\n\ninitial\n\nmean\n\nunity\n\nfirst\n\nnear\n\nthe\n\non-set\n\nindicating\n\neffective the of\n\nAt about the same depth, a point in the free-field\n\n295 (away from the structure) undergoes time\n\nliquefaction\n\nat a much\n\nlater\n\n(6 sec. - 12 s e c ) .\n\nA pictorial representation of the temporal expansion of lique­ fied\n\nzone is shown\n\nin Fig.\n\n3, which clearly\n\nSSTIN on the soil liquefaction a\n\nlayered\n\nstudy, liquefy\n\nsite\n\nand\n\ntheoretically\n\ncoherent\n\ncharacteristics excitation\n\nall points\n\nat the same time.\n\nshows the effect\n\nlocated\n\nare\n\nat a site.\n\nconsidered\n\nstructure\n\nSince\n\nin\n\nat a given depth\n\nThe presence of\n\nof\n\nthis\n\nshould\n\nis seen\n\nto\n\ncause an appreciable change in this characteristic.\n\nLEGEND; LT1\n\nl i .2 6\n\nJ 3.2 hNV\\S 3 12.8 v\n\nmm\n\n6.u\n\n| 9.6\n\nlimnni 8 17.6\n\nFig. 2\n\nDistribution of shear stress ratio (max. shear stress/ initial eff. stress) (Base Excitation: Sinusoidal, Max Acceln=0.125g, Frequency=0.50 Hz)\n\nFig. 3\n\nPictorial representation of temporal expansion of liquefied zones. (Base Excitation: Sinusoidal, Max Acceln=0.125g, Frequency=0.50 Hz)\n\nCONCLUSIONS In this paper, a technique based on the finite element method is\n\npresented\n\nsaturated\n\nsand\n\nfor\n\nmodeling\n\ndeposits.\n\nliquefaction\n\nEmphasis\n\nwas\n\ncharacteristics\n\ngiven\n\nto\n\nof\n\nrealistically\n\n296 represent\n\nthe on-set\n\nliquefied\n\nzone(s), and to identify the effects of\n\ninteraction.\n\nFrom\n\nof liquefaction,\n\nthe\n\nnumerical\n\nto trace\n\nthe expansion\n\nof\n\nsoil-structure\n\nresults\n\npresented\n\nratios,\n\nthe\n\nherein,\n\nthe\n\nfollowing conclusions can be made: 1. Because the\n\nof\n\nlarge\n\nfoundation\n\nthese\n\nzones\n\nshear\n\nedge\n\nto\n\nstress\n\nliquefy\n\nzones\n\nsurrounding\n\nfirst indicating vulnerability\n\nliquefaction.\n\nThe\n\nliquefied\n\nzones\n\nof\n\npropagate\n\noutward as time increases. 2. The soil-structure\n\ninteraction phenomenon may have a signifi­\n\ncant effect on the liquefaction\n\ncharacteristics\n\nof a site and\n\nit should be considered in evaluating liquefaction potential. 3. The\n\nfinite\n\nelement\n\nmodeling soil\n\nmethod\n\ncan\n\nbe\n\nused\n\nvery\n\neffectively\n\nfor\n\nliquefaction.\n\nACKNOWLEDGEMENT Some\n\nof\n\nthe\n\nresults\n\nMr. G.C. Biswas.\n\nHis\n\nreported\n\nin this\n\ncontribution\n\npaper\n\nis greatly\n\nwere\n\nobtained\n\nappreciated.\n\nby The\n\nJunior Faculty Summer Fellowship awarded\n\nto the senior author by\n\nthe University of Oklahoma is gratefully\n\nacknowledged.\n\nREFERENCES Biswas, G . C , 1985. Modeling of soil liquefaction and foundation response under cyclic and earthquake loading. M . S . Thesis, University of Oklahoma, Norman. Chang, C.S., 1984. Analysis of earthquake induced footing settlement. Proc. 8th World Conf. on Earthq. Eng., San Francisco, III: 87-94. Ishibashi, I., Sherif, M.A. and Tsuchiya, C , 1977. Pore pressure rise mechanism and soil liquefaction. Soils and Foundations, 17: 2: 17-27. Whitman, R.V. and Lambe, P . C , 1982. Liquefaction: consequences for a structure. Proc. Soil Dyn. and Earthq. Eng. Conf. , Southampton, II: 941-949. Yoshimi, Y. and Tokimatsu, K., 1977. Settlement of buildings on saturated sand during earthquakes. Soils and Foundations, 17: 1: 23-38. Zienkiewicz, O.C. and Shiomi, T., 1984. Dynamic behavior of saturated porous media; the generalized biot formulation and its numerical solution. Int. J. for Num. and Anal. Met. in Geomech., 8: 71-96\n\n297\n\nRESPONSE OF CIRCULAR PLATES RESTING ON HOMOGENEOUS AND HALFSPACE\n\nISOTROPIC\n\nResearch\n\nAssistant,\n\nSchool\n\nof\n\nCivil\n\nEngineering\n\nEnvironmental Science, University of Oklahoma, Norman OK\n\nand\n\n73019\n\nABSTRACT This paper presents an analysis of circular plates resting on homogeneous, isotropic and elastic halfspace using the finite element method. Emphasis is given to modeling the nonlinear behavior of interface between plate and halfspace using a special interface/joint element. Parametric studies have been performed to assess the effects of several important factors. INTRODUCTION Circular plates are widely used as foundations of such\n\nstruc­\n\ntures as nuclear reactors, storage tanks and silos, among others. The analysis of flexural behavior of circular plates resting on a deformable soil medium per)\n\nconstitutes\n\n(also referred to as halfspace in this pa­\n\na problem\n\nof continued\n\ninterest\n\nand\n\nimportance\n\nto researchers as well as practicing engineers. Many\n\ninvestigators\n\nproblem.\n\nOne\n\nwho\n\nused\n\na\n\nmany\n\nother\n\nof\n\nthe\n\npower (1965),\n\nof and\n\nproblem. smooth\n\nMost\n\nto\n\npractical In\n\nthis\n\nbe\n\nof\n\nthese\n\nwas\n\non\n\nthe\n\n(1980)\n\nhalfspace.\n\nCheung\n\nand many\n\nflexural\n\nplate\n\nis neither\n\nahd\n\n(1979),\n\nthickness\n\nwhich\n\nis\n\nhalfspace.\n\nperfectly not\n\nsame\n\nperfectly smooth\n\nAlso most of the investigators assumed\n\nuniform\n\nand\n\nother\n\nto analyze the\n\nassumed\n\nbetween\n\n(1936),\n\nSubsequently,\n\nto analyze the\n\nenergy method\n\ncontact\n\ninteraction\n\nBorowicka\n\nsame problem.\n\nFaruque\n\nthe contact\n\nthis\n\nby\n\ntechnique.\n\ninvestigators\n\nbonded\n\nsituation, of\n\n(1979),\n\nresting\n\nnor perfectly bonded. plate\n\nexpansion\n\n(1983) used\n\nor perfectly\n\nIn an actual\n\nanalyzed\n\nworks\n\nfinite element method\n\nplate Zaman\n\npast\n\nanalyzed\n\nZaman\n\ninvestigators used Faruque\n\nthe\n\nseries\n\ninvestigators\n\nZienkiewicz behavior\n\nin\n\npioneering\n\nso\n\nin\n\nthe most\n\nsituations. paper,\n\nthe\n\neffects\n\nof\n\ninterface\n\nconditions\n\non\n\nbehavior of circular foundations of nonuniform thickness and resting on an elastic halfspace is\n\ninvestigated.\n\nthe\n\n298 PROPOSED ANALYSIS\n\nPROCEDURE\n\nFor finite element idealization, the plate-halfspace Fig. 1 is treated\n\nas an axisymmetric\n\nproblem.\n\nsystem in\n\nThin plate\n\ntheory\n\nis used to describe the flexural behavior of the plate. Displacements interface\n\nare assumed\n\nto be small.\n\nrelations.\n\nplastic\n\nof\n\nconstitutive\n\nPeak shear strength of the interface is assumed to be\n\nfunction of interface normal is also\n\nNonlinear behavior\n\nis idealized by elastic-perfectly\n\nassumed\n\nthat\n\nstress, cohesion and roughness.\n\ninterfaces\n\ntain any tensile stress.\n\nare nondilatant\n\nand\n\ncannot\n\nIt sus­\n\nThe thickness of interface element is\n\nconsidered to be small compared to its length. q/unit length\n\nAxis of symmetry \"73\n\nFigure 1\n\nAxisymmetric cirucular plate resting on isotropic homogeneous elastic halfspace.\n\nAnnular plate elements are used to model the flexural behavior of\n\nthe\n\nthe\n\nfoundation\n\ncentral\n\nplate.\n\nplate\n\nA\n\nregion.\n\nspecial Details\n\nformulation of\n\nthe\n\nis\n\nformulation\n\nare\n\nfor not\n\npresented here. A four-noded axisymmetric originally\n\nby Ghaboussi\n\net\n\ninterface element al.\n\n(1973)\n\n(Fig. 2) developed\n\nand modified\n\n(1981) is further modified\n\nsubsequently in the pre­\n\nsent study and employed to model the frictional behavior of platehalfspace\n\ninterface.\n\nUsing the notations of Fig. 2 and following the standard\n\nsteps\n\nof finite element approach, the element stiffness matrix, [ K ^ ] , for the interface element can be expressed in the [K ] A\n\n•T\n\n=\n\nwhere\n\n'v\n\n[ B\n\ni\n\n]\n\n(1)\n\n[B ] dv\n\n[D ] ±\n\n[B^] = strain\n\nform:\n\ni\n\n(relative) displacement transformation matrix\n\ngiven by - l\n\n\" 2\n\n2\n\n\" l\n\nB\n\n[B.]\n\n=\n\nB\n\n- 5 B\n\nB\n\n\" 3 B\n\nB\n\nB\n\n4\n\n0\n\nB\n\n6\n\n\" 4 B\n\nB\n\n3\n\n\" 3\n\n\" 4\n\n0\n\nB\n\nB\n\nB\n\n4\n\nl\n\n~ 2 B\n\nB\n\n6\n\nB\n\n0\n\nB\n\n5\n\nB B 0\n\n(2)\n\n299\n\nAxi9 of symmetry\n\nFigure 2\n\nThe four noded axisymmetric this study.\n\ninterface element used in\n\nand hj, cos\n\nh^ sin \\\\) B\n\n2 =\n\ncos i|> B\n\n3 =\n\nsin (3)\n\nt. 1 B\n\n2r Here h^ and h^ are\n\n6\n\n=\n\n2r\n\ninterpolation\n\nin terms of the nondimensional\n\nfunctions\n\nand may be\n\nexpressed\n\nlocal coordinate £ (varying from\n\n-1 to +1) as: 1\n\n2\n\n[D^] in Eq.\n\n(1) represents the constitutive relation matrix\n\nfor\n\nthe interface and is given by E\n\nwhere E\n\nss\n\ntively. as zero.\n\nss\n\n0\n\n0\n\nE\n\n0\n\n0\n\n0 nn\n\n(5)\n\n0 E\n\nand E\n\nare interface shear and normal modulii. respecnn As suggested by Ghaboussi et al. (1973) , E is assumed c\n\naa\n\n300 Simulation\n\nof deformation\n\nVarious tion\n\nand\n\nulated\n\nmodes\n\nof\n\nrebonding\n\nusing\n\nan\n\nis p r e s e n t e d\n\nelsewhere\n\nresults\n\nFigure\n\nshows\n\nthickness. of\n\nthe\n\nThe\n\nis t e r m e d\n\nas\n\n0.80.\n\nelastic\n\nsystem\n\nThe\n\nof\n\nand\n\nis d e f i n e d\n\nis\n\nundergoes\n\nDetails\n\ncircular\n\nof\n\nis\n\nsliding,\n\nof\n\nare\n\nthe\n\nthe\n\nplate\n\nas\n\nthe\n\nouter\n\nregion\n\nof\n\nsim­\n\nalgorithm\n\nto\n\nthat\n\nof\n\nof of\n\nanalysis, the value\n\nsupported acted\n\nupon\n\nby by\n\nnon-uniform\n\nratio\n\nregion\n\na a\n\nthickness the\n\nplate.\n\nouter\n\nregion\n\n6 is\n\nof\n\ndeep,\n\nfixed\n\nhomogeneous,\n\nuniform\n\nrigidity\n\nof\n\nthe p l a t e - e l a s t i c\n\nand m a y\n\nbe\n\nexpressed\n\nby\n\nsepara­\n\n1984).\n\ninner\n\nraft\n\nRelative\n\nstick,\n\nelement\n\nis d e f i n e d\n\nIn the p r e s e n t\n\nmedium\n\nq.\n\na\n\nthat\n\n\\$.\n\nsolid\n\nintensity\n\ninterface\n\n(Mahmood,\n\nto\n\nas\n\ntechnique.\n\nparameter\n\nthe\n\nequal\n\nan\n\nsuch\n\naxisymmetric\n\nregion\n\nof\n\nto\n\nan\n\nThe\n\ninner\n\nratio\n\nthat\n\ninterative\n\nNumerical 1\n\nmodes\n\ndeformation\n\npressure\n\nof\n\nhalfspace\n\nas\n\n3 (6)\n\nwhere\n\nE\n\nness,\n\nand\n\np\n\n, v\n\np\n\n, t\n\np\n\n, a\n\nare\n\np\n\nof\n\nYoung's\n\nplate,\n\nmodulus\n\nand\n\nPoisson's\n\nfluence\n\nof\n\ninterface\n\nmodulus, Poisson's\n\nrespectively.\n\nratio\n\nfor\n\nthe\n\ncondition,\n\nE\n\nK\n\n,\n\na\n\nand\n\ng\n\nelastic\n\nis\n\nFigure with of\n\nK\n\nis\n\ncompared smooth case\n\nshows\n\nthe variation\n\ninterface\n\nshown.\n\ndecreases\n\nand\n\nv\n\ng\n\n=\n\nwith to\n\nincreases\n\nis\n\ncase,\n\ndifference\n\nwith\n\ncentral\n\nResponse\n\nobserved\n\nincreasing\n\nbonded\n\nand\n\nof\n\ncondition.\n\nIt\n\nthe\n\nthe\n\ncontact\n\n0.0\n\nand\n\nthat\n\nvalue\n\nthe\n\nof\n\non\n\nfor\n\nshows\n\nsimilar\n\na\n\n4\n\ntribution face.\n\nthe\n\nis lower\n\n10,\n\n.01\n\nplot\n\nshows\n\nof\n\nIt\n\nare much\n\n=\n\ncondition\n\nFigure\n\nthe\n\nstress\n\nplate-soil\n\nplate ^\n\nin\n\nK\n\n.\n\nFor\n\nplate\n\nfor\n\nthe for\n\ndifferent\n\nv\n\n=\n\ng\n\na.\n\na\n\nFor is\n\nfor\n\nmore\n\ndifference\n\n.49.\n\nof\n\ncontact that,\n\nthose\n\nin\n\nsmooth\n\ninstance,\n\ngiven case\n\nand\n\nfor\n\nat\n\nfor\n\nthe plate\n\ncontact.\n\na,\n\na\n\nis\n\nof\n\nbonded =\n\n1.0, '\n\napproximately\n\n3.9% 3(b)\n\nis\n\n6.8%.\n\nAt\n\nhigher\n\nv , g\n\nFigure the\n\neffect\n\nof\n\nsmaller.\n\neffect\n\nnormal\n\nobserved\n\nvalues\n\ndisplacement\n\n'\n\ndifference\n\nis m u c h\n\nthe\n\nthan\n\nthis\n\ndisplacements\n\ninterface\n\nroughness\n\non\n\nstress\n\nthe\n\nsoil\n\nfor\n\nsmooth\n\nthe bonded\n\nat\n\ncases, case.\n\nplate\n\ncontact For\n\nK\n\n=\n\nthe\n\ncenter\n\nis a b o u t\n\n6.7%\n\nlower\n\ndis­\n\ninter­\n\nstresses .01, the\n\nr contact\n\nin­\n\ncentral\n\ndeflection\n\nin d e f l e c t i o n\n\ndecrease\n\n=\n\nwhereas\n\ninterface\n\nYoung's The\n\ns\n\nr v\n\nare\n\ng\n\nthick-\n\nexamined.\n\n3(a)\n\nand\n\nr\n\na\n\nv\n\nhalfspace.\n\nr' response\n\nratio,\n\n' for\n\nsmooth\n\n301 CONCLUSIONS The objective of this paper was to examine the flexural ior of circular plates resting on isotropic halfspace the plate-halfspace the\n\nnonlinear\n\ninteraction.\n\nbehavior\n\nof\n\nEmphasis was given to modeling\n\ninterface.\n\nBased\n\non\n\nthe\n\nparametric\n\nstudy, a certain trend of the plate response is noticed. eral,\n\nthe\n\ncentral\n\nplate-halfspace\n\ndeflection\n\ncontact.\n\nis\n\nContact\n\nsmooth plate-halfspace contact.\n\nbehav­\n\nconsidering\n\nmore\n\nin\n\nstresses\n\nare\n\ncase much\n\nof\n\nIn gen­ smooth\n\nlower\n\nplate responses is seen to be diminished with increasing a and\n\n(a) v\n\n=\n\ns Figure 3\n\n0.0\n\n(b) v\n\n= 0.49 s Effect of K , a and interface roughness on variation of central plate deflection.\n\nSmooth\n\n.2\n\nh I\n\n0.0\n\n.2\n\n1\n\n.4\n\ni\n\ni\n\ni\n\n.6\n\n.8\n\n1.0\n\n_r a\n\nFigure 4\n\nfor\n\nEffect of interface roughness on\n\nEffect of a and interface roughness on variation of contact stress at plate-soil interface for v = 0.0.\n\n302 REFERENCES Borowicka, H., 1936. Influence of rigidity of a circular foundation slab on the distribution of pressure over the contact surface. Proc. of 1st International Conf. in Soil Mech. and Found. Engng., 2: 144-149. Cheung, Y.K. and Zienkiewicz, O . C , 1965. Plates and tanks on elastic foundation - an application of finite element method. Int. J. for Solids and Struct., 1: 451-461. Faruque, M.O., 1980. The role of interface elements in finite element analysis of geotechnical engineering problems. M. Eng. Thesis, Carleton Univ., Canada. Faruque, M.O. and Zaman M.M., 1983. Approximate analysis of uniformly loaded circular plates on isotropic elastic half-space. Proc. of IX Congress of Natl. Academy of Engng. of Mexico, Leon, Mexico. Ghaboussi, J., Wilson, E. and Isenberg, J., 1973. Finite element analysis for rock joints and interfaces. J. of Soil Mech. and Found. Div., ASCE, 99: 833-848. Mahmood, I.U., 1984. Finite element analysis of cylindrical tank foundations resting on isotropic soil medium including soil-structure interaction. M.Sc. Thesis, Univ. of Oklahoma, Norman. Selvadurai, A.P.S., 1979. The interaction between a uniformly loaded circular plate and an isotropic elastic halfspace: variation method. J. of Struct. Mech., 7: 231-246.\n\nA\n\nSelvadurai, A.P.S. and Faruque, M.O., 1981. The influence of interface friction on the performance of cable jacking tests of rock masses. Proc. Impl. of Comp. Procedures and Stress-Strain Laws in Geotech. Engng., Chicago, I: 169-183. Zaman, M.M., 1979. Finite element analysis of interaction between an elastic circular plate and an isotropic elastic medium. M. Eng. Thesis, Carleton Univ., Canada.\n\n303\n\nEVOLUTION OF LOCAL AMPLITUDE IN TRAINS OF SHEAR FLOW INSTABILITY WAVES\n\nJ.M. RUSSELL School of Aerospace, Mechanical, and Nuclear Engineering, University of Oklahoma, 865 Asp Avenue, Rm 212, Norman, Oklahoma\n\n73019\n\nABSTRACT The equations of motion for small amplitude three-dimensional disturbances to an inviscid incompressible shear flow are manipulated to yield a single equation for the cross-stream displacement of a fluid particle. A variational principle satisfied by this displacement variable is derived and exploited to yield an equation similar to Whitham's law of conservation of wave action density (Whitham, 1974 Chap. 1 1 ) . The variational formalism recovers the familiar Rayleigh stability equation, thus establishing the compatibility between the present variational formalism and the more traditional theory of normal modes.\n\nFORMULATION ->-\n\nLet the ordered pair of functions (U,P) denote a reference\n\nflow\n\nsolution of\n\nthe following partial differential equation system, ->\n\n~\n\n+ U-VU = - — VP + g\n\n,\n\nV-U = 0\n\n,\n\n(la,b)\n\np\n\naV\n\nwhich we will abbreviate by the compact notation Eu(U,P) = {0}.\n\nThe above sys­\n\ntem is the set of equations of motion of an inviscid, incompressible, uniformdensity fluid with velocity U, pressure P, and mass density p, uniform gravitational acceleration g (Batchelor, 1967, p 380). pair of functions (U+u,P+p) denote a disturbed\n\nflow\n\nsubject to a Let the ordered\n\nsolution of Eu(U+u,P+p) =\n\n{0}, with lower case letters denoting the disturbances.\n\nForming the quantity\n\nEu(U+u,P+p) - Eu(U,P) = {0}, we obtain the equations of motion of the distur­ bances -> If\n\n+\n\n^' ^ V\n\n+\n\nu'VU + u-Vu = - ~ Vp\n\n,\n\nV-u = 0 -> ->\n\n.\n\n(2a,b)\n\n->\n\nWe restrict attention to the case in which u«u\n\n#### E-Book Information\n\n• Series: Developments in Geomathematics 5\n\n• Year: 1,986\n\n• Pages: 3-308\n\n• Pages In File: 293\n\n• Language: English\n\n• Identifier: 978-0-444-42697-0,0-444-42697-3,0-444-41609-9\n\n• Issn: 0167-5982\n\n• Org File Size: 28,263,518\n\n• Extension: pdf\n\n• Toc: Content:\nFurther titles in this series\nPage II\n\nFront Matter\nPage III\n\nPage IV\n\nPage V\n\nCommittee Members\nPage VI\n\nPreface\nPage VII\nYoshi K. Sasaki\n\nThe Application of Variational Methods to Initialization on the Sphere\nPages 3-11\nR.W. DALEY\n\nApplication of Optimal Control to Meteorological Problems\nPages 13-28\nO. TALAGRAND\n\nA Review of Variational and Optimization Methods in Meteorology\nPages 29-34\nI.M. Navon\n\nUse of Adjoint Equations for Assimilation if Meteorological Observations by Barotropic Models\nPages 35-42\nPh. Courtier, O. Talagrand\n\nThe Variational Four-Dimensional Assimilation of Analyses Using Quasigeostrophic Models as Constraints\nPages 43-48\nJOH. C. DERBER\n\nEvaluation of a Multivariate Variational Assimilation of Conventional and Satellite Data for the Diagnosis of Cyclone Systems\nPages 49-54\nGary L. Achtemeier, H.T. Ochs III, S.Q. Kidder, R.W. Scott\n\nThe Variational Inverse Method for the General Circulation in the Ocean\nPages 55-70\nChristine PROVOST\n\nVariational Analysis of Wind Field and Geopotential at 500 Mb\nPages 71-75\nF.-X. LE DIMET, J. SEGOT\n\nDesign of a Three-Dimensional Global Atmospheric Prediction Model by a Variational Method\nPages 77-80\nA. KASAHARA\n\nVariational Implicit Normal Mode Initialization for Nwp Models\nPages 81-85\nCLIVE TEMPERTON\n\nFormulation of Normal Modes and Nonlinear Initialization for Limited-Area Models\nPages 87-88\nR. JUVANON, DU VACHAT, B. URBAN\n\nSequential Estimation and Satellite Data Assimilation in Meteorology and Oceanography\nPages 91-100\nM. GHIL\n\nThe Use of Adjoint Equations to Solve Variational Adjustment Problems Subject to Vorticity Conservation Constraints\nPages 101-106\nJ.M. LEWIS\n\nVariational Modification of the 3D-Wind Field\nPages 107-112\nMichael Hantel\n\nA Four-Dimensional Analysis\nPages 113-117\nROSS N. HOFFMAN\n\nVariational Initialization and Determination of Weighting Factors\nPages 119-123\nCHUNG-YI TSENG\n\nPartial Spline Models for the Estimation of the Three Dimensional Atmospheric Temperature Distribution from Satellite Radiance Data and Tropopause Height Information\nPages 125-130\nGRACE WAHBA\n\nThe Retrieval of Moving Waves from Remotely-Sensed Atmospheric Data\nPages 131-136\nDerek M. Cunnold, Chowen Chou Wey\n\nImpact of Doppler Wind Analysis Weights on Three Dimensional Airflow and Diagnosed Precipitation in a Thunderstorm\nPages 137-142\nC.L. ZIEGLER\n\nRemarks on Systems with Uncomplete Data\nPages 145-159\nJ.L. LIONS\n\nTwo Dimensional Kalman Filtering and Assimilation of Wind Profiler Data\nPages 161-166\nD.F. PARRISH, S.E. COHN\n\nBayesian Optimal Analysis for Meteorological Data\nPages 167-172\nR.J. PURSER\n\nRelationships between Statistical and Deterministic Methods of Data Assimilation\nPages 173-179\nW.C. THACKER\n\nAssimilation of Dynamical Data in a Limited Area Model\nPages 181-185\nF.-X. LE DIMET, A. NOUAILLER\n\nVariational Principles and Adaptive Methods for Complex Flow Problems\nPages 189-200\nJ. TINSLEY ODEN, T. STROUBOULIS, PH. DEVLOO\n\nPenalty Variational Formulation of Viscous Incompressible Fluid Flows\nPages 201-221\nJ.N. REODY\n\nA New Combination Method of Boundary Type Finite Elements and Boundary Elements for Wave Diffraction and Refraction\nPages 223-228\nK. KASHIYAMA, M. KAWAHARA, H. SAKURAI\n\nThe Numerical Analysis of Two-Dimensional Steady Free Surface Flow Problems\nPages 229-234\nTsukasa NAKAYAMA, Mutsuto KAWAHARA\n\nVariational Principles in Continuum Mechanics and Their Application if the Study of Propagating Discontinuities\nPages 235-241\nJ. TURSKI\n\nComparison of Variational Methods for the Solution if the Dynamic Group-Diffusion Problem: One Dimensional Case\nPages 243-248\nE. del Valle, J.C. Diaz, D. Meade\n\nVariational Methods for Fluid Flow in Porous Media\nPages 251-263\nR.E. EWING\n\nA Finite Element Simulator for Incompressible Two-Phase Flow\nPages 265-277\nG. CHAVENT, G. COHEN, J. JAFFRE\n\nSimulation of Waterflooding in Heterogeneous, Compressible, and Irregularly Shaped Reservoirs\nPages 279-284\nA.R. Kukreti, M.M. Zaman, F. Civan, Y.R. Perera, G.C. Biswas\n\nData Structures and Algorithms for Self Adaptive Local Grid Refinement\nPages 285-290\nJ.C. Diaz, D.B. North\n\nOn the Modeling of Soil Liquefaction by Finite Element Method\nPages 291-296\nM.M. Zaman, J.G. Laguros\n\nResponse of Circular Plates Resting on Homogeneous and Isotropic Halfspace\nPages 297-302\nI.U. Mahmood\n\nEvolution of Local Amplitude in Trains if Shear Flow Instability Waves\nPages 303-308\nJ.M. RUSSELL\n\nIndex of Authors\nPage 309\n\n### Related Documents", null, "##### Variational Methods In Geosciences [PDF]\n\nYOSHI K. SASAKI (Eds.)", null, "##### Variational Methods In Imaging [PDF]\n\nOtmar Scherzer, Markus Grasmair, Harald Grossauer, Markus Haltmeier, Frank Lenzen (auth.)", null, "##### Variational Methods In Statistics [DJVU]\n\nJagdish S. Rustagi (Eds.)", null, "##### Variational Methods In Statistics [PDF]\n\nJagdish S. Rustagi (Eds.)", null, "##### Variational Methods In Economics [PDF]", null, "" ]

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[ "# C function argument and return values\n\n• Difficulty Level : Basic\n• Last Updated : 21 Dec, 2018\n\nPrerequisite : Functions in C/C++\n\nA function in C can be called either with arguments or without arguments. These function may or may not return values to the calling functions. All C functions can be called either with arguments or without arguments in a C program. Also, they may or may not return any values. Hence the function prototype of a function in C is as below:", null, "Want to learn from the best curated videos and practice problems, check out the C++ Foundation Course for Basic to Advanced C++ and C++ STL Course for foundation plus STL.  To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.\n\nThere are following categories:", null, "1. Function with no argument and no return value : When a function has no arguments, it does not receive any data from the calling function. Similarly when it does not return a value, the calling function does not receive any data from the called function.\nSyntax :\n```Function declaration : void function();\nFunction call : function();\nFunction definition :\nvoid function()\n{\nstatements;\n}\n```\n `// C code for  function with no``//  arguments and no return value`` ` `#include ``void` `value(``void``);``void` `main()``{``    ``value();``}``void` `value(``void``)``{``    ``int` `year = 1, period = 5, amount = 5000, inrate = 0.12;``    ``float` `sum;``    ``sum = amount;``    ``while` `(year <= period) {``        ``sum = sum * (1 + inrate);``        ``year = year + 1;``    ``}``    ``printf``(``\" The total amount is %f:\"``, sum);``}`\n\nOutput:\n\n```The total amount is 5000.000000\n```\n2. Function with arguments but no return value : When a function has arguments, it receive any data from the calling function but it returns no values.\n\nSyntax :\n\n```Function declaration : void function ( int );\nFunction call : function( x );\nFunction definition:\nvoid function( int x )\n{\nstatements;\n}\n```\n `// C code for function ``// with argument but no return value``#include `` ` `void` `function(``int``, ``int``[], ``char``[]);``int` `main()``{``    ``int` `a = 20;``    ``int` `ar = { 10, 20, 30, 40, 50 };``    ``char` `str = ``\"geeksforgeeks\"``;``    ``function(a, &ar, &str);``    ``return` `0;``}`` ` `void` `function(``int` `a, ``int``* ar, ``char``* str)``{``    ``int` `i;``    ``printf``(``\"value of a is %d\\n\\n\"``, a);``    ``for` `(i = 0; i < 5; i++) {``        ``printf``(``\"value of ar[%d] is %d\\n\"``, i, ar[i]);``    ``}``    ``printf``(``\"\\nvalue of str is %s\\n\"``, str);``}`\n\nOutput:\n\n```value of a is 20\nvalue of ar is 10\nvalue of ar is 20\nvalue of ar is 30\nvalue of ar is 40\nvalue of ar is 50\nThe given string is : geeksforgeeks\n```\n3. Function with no arguments but returns a value : There could be occasions where we may need to design functions that may not take any arguments but returns a value to the calling function. A example for this is getchar function it has no parameters but it returns an integer an integer type data that represents a character.\nSyntax :\n```Function declaration : int function();\nFunction call : function();\nFunction definition :\nint function()\n{\nstatements;\nreturn x;\n}\n```\n `// C code for function with no arguments ``// but have return value``#include ``#include `` ` `int` `sum();``int` `main()``{``    ``int` `num;``    ``num = sum();``    ``printf``(``\"\\nSum of two given values = %d\"``, num);``    ``return` `0;``}`` ` `int` `sum()``{``    ``int` `a = 50, b = 80, sum;``    ``sum = ``sqrt``(a) + ``sqrt``(b);``    ``return` `sum;``}`\n\nOutput:\n\n```Sum of two given values = 16\n```\n4. Function with arguments and return value\nSyntax :\n```Function declaration : int function ( int );\nFunction call : function( x );\nFunction definition:\nint function( int x )\n{\nstatements;\nreturn x;\n}\n```\n `// C code for function with arguments ``// and with return value`` ` `#include ``#include ``int` `function(``int``, ``int``[]);`` ` `int` `main()``{``    ``int` `i, a = 20;``    ``int` `arr = { 10, 20, 30, 40, 50 };``    ``a = function(a, &arr);``    ``printf``(``\"value of a is %d\\n\"``, a);``    ``for` `(i = 0; i < 5; i++) {``        ``printf``(``\"value of arr[%d] is %d\\n\"``, i, arr[i]);``    ``}``    ``return` `0;``}`` ` `int` `function(``int` `a, ``int``* arr)``{``    ``int` `i;``    ``a = a + 20;``    ``arr = arr + 50;``    ``arr = arr + 50;``    ``arr = arr + 50;``    ``arr = arr + 50;``    ``arr = arr + 50;``    ``return` `a;``}`\n\nOutput:\n\n```value of a is 40\nvalue of arr is 60\nvalue of arr is 70\nvalue of arr is 80\nvalue of arr is 90\nvalue of arr is 100\n```\n\nMy Personal Notes arrow_drop_up" ]

[ null, "https://media.geeksforgeeks.org/wp-content/cdn-uploads/Function-Prototype-in-c.png", null, "https://media.geeksforgeeks.org/wp-content/cdn-uploads/Types-Of-Functions-In-C.png", null ]

[ "# How do you graph the inequality x-2y> 4?\n\nOct 21, 2015\n\nDraw a dashed (to indicate it is not included in the solution) line for $x - 2 y = 4$;\nshade the side that does not include the origin (since $\\left(x , y\\right) = \\left(0 , 0\\right)$ is not a valid solution for $x - 2 y > 4$)\n\n#### Explanation:\n\nSolving $x - 2 y = 4$ for some arbitrary values of $x$\n(I chose $x = 0$ and $x = 4$)\n\n$x = 0 \\rightarrow y = - 2$\n$x = 4 \\rightarrow y = 0$\n\nDraw a line through the points $\\left(0 , - 2\\right)$ and $\\left(4 , 0\\right)$ (remember to make a \"dashed line\" because we do not want it included in the final solution.\n\nSince $\\left(x , y\\right) = \\left(0 , 0\\right)$ is not a valid solution for $x - 2 y > 4$ shade the side of the line that does not include $\\left(0 , 0\\right)$ to show the final solution area." ]

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[ "# 42.6 kg to lbs - 42.6 kilograms to pounds\n\nBefore we move on to the practice - that is 42.6 kg how much lbs conversion - we want to tell you a little bit of theoretical information about these two units - kilograms and pounds. So let’s start.\n\nHow to convert 42.6 kg to lbs? 42.6 kilograms it is equal 93.916923612 pounds, so 42.6 kg is equal 93.916923612 lbs.\n\n## 42.6 kgs in pounds\n\nWe are going to begin with the kilogram. The kilogram is a unit of mass. It is a base unit in a metric system, that is International System of Units (in short form SI).\n\nSometimes the kilogram can be written as kilogramme. The symbol of the kilogram is kg.\n\nThe kilogram was defined first time in 1795. The kilogram was defined as the mass of one liter of water. First definition was not complicated but totally impractical to use.\n\nLater, in 1889 the kilogram was defined using the International Prototype of the Kilogram (in short form IPK). The International Prototype of the Kilogram was prepared of 90% platinum and 10 % iridium. The International Prototype of the Kilogram was in use until 2019, when it was substituted by another definition.\n\nThe new definition of the kilogram is based on physical constants, especially Planck constant. Here is the official definition: “The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the metre and the second are defined in terms of c and ΔνCs.”\n\nOne kilogram is equal 0.001 tonne. It is also divided into 100 decagrams and 1000 grams.\n\n## 42.6 kilogram to pounds\n\nYou know a little about kilogram, so now let's go to the pound. The pound is also a unit of mass. We want to highlight that there are not only one kind of pound. What are we talking about? For instance, there are also pound-force. In this article we want to centre only on pound-mass.\n\nThe pound is used in the Imperial and United States customary systems of measurements. Naturally, this unit is in use also in other systems. The symbol of the pound is lb or “.\n\nThere is no descriptive definition of the international avoirdupois pound. It is defined as 0.45359237 kilograms. One avoirdupois pound could be divided to 16 avoirdupois ounces or 7000 grains.\n\nThe avoirdupois pound was implemented in the Weights and Measures Act 1963. The definition of this unit was placed in first section of this act: “The yard or the metre shall be the unit of measurement of length and the pound or the kilogram shall be the unit of measurement of mass by reference to which any measurement involving a measurement of length or mass shall be made in the United Kingdom; and- (a) the yard shall be 0.9144 metre exactly; (b) the pound shall be 0.45359237 kilogram exactly.”\n\n### How many lbs is 42.6 kg?\n\n42.6 kilogram is equal to 93.916923612 pounds. If You want convert kilograms to pounds, multiply the kilogram value by 2.2046226218.\n\n### 42.6 kg in lbs\n\nThe most theoretical section is already behind us. In next section we will tell you how much is 42.6 kg to lbs. Now you know that 42.6 kg = x lbs. So it is time to get the answer. Let’s see:\n\n42.6 kilogram = 93.916923612 pounds.\n\nThis is an exact result of how much 42.6 kg to pound. You can also round it off. After rounding off your outcome will be as following: 42.6 kg = 93.72 lbs.\n\nYou learned 42.6 kg is how many lbs, so let’s see how many kg 42.6 lbs: 42.6 pound = 0.45359237 kilograms.\n\nObviously, this time you may also round off this result. After rounding off your outcome is exactly: 42.6 lb = 0.45 kgs.\n\nWe are also going to show you 42.6 kg to how many pounds and 42.6 pound how many kg outcomes in charts. Look:\n\nWe will start with a table for how much is 42.6 kg equal to pound.\n\n### 42.6 Kilograms to Pounds conversion table\n\nKilograms (kg) Pounds (lb) Pounds (lbs) (rounded off to two decimal places)\n42.6 93.916923612 93.720\nNow see a table for how many kilograms 42.6 pounds.\n\nPounds Kilograms Kilograms (rounded off to two decimal places\n42.6 0.45359237 0.45\n\nNow you learned how many 42.6 kg to lbs and how many kilograms 42.6 pound, so it is time to go to the 42.6 kg to lbs formula.\n\n### 42.6 kg to pounds\n\nTo convert 42.6 kg to us lbs a formula is needed. We are going to show you a formula in two different versions. Let’s start with the first one:\n\nNumber of kilograms * 2.20462262 = the 93.916923612 outcome in pounds\n\nThe first formula give you the most correct outcome. In some cases even the smallest difference could be significant. So if you want to get a correct result - this version of a formula will be the best solution to convert how many pounds are equivalent to 42.6 kilogram.\n\nSo let’s move on to the second formula, which also enables calculations to learn how much 42.6 kilogram in pounds.\n\nThe second version of a formula is as following, have a look:\n\nNumber of kilograms * 2.2 = the outcome in pounds\n\nAs you see, this formula is simpler. It can be better solution if you want to make a conversion of 42.6 kilogram to pounds in fast way, for example, during shopping. Just remember that final outcome will be not so correct.\n\nNow we want to show you these two versions of a formula in practice. But before we are going to make a conversion of 42.6 kg to lbs we want to show you easier way to know 42.6 kg to how many lbs totally effortless.\n\n### 42.6 kg to lbs converter\n\nAnother way to learn what is 42.6 kilogram equal to in pounds is to use 42.6 kg lbs calculator. What is a kg to lb converter?\n\nCalculator is an application. Calculator is based on first formula which we showed you in the previous part of this article. Due to 42.6 kg pound calculator you can easily convert 42.6 kg to lbs. You only need to enter number of kilograms which you want to calculate and click ‘convert’ button. The result will be shown in a flash.\n\nSo let’s try to calculate 42.6 kg into lbs with use of 42.6 kg vs pound calculator. We entered 42.6 as an amount of kilograms. This is the outcome: 42.6 kilogram = 93.916923612 pounds.\n\nAs you see, our 42.6 kg vs lbs converter is user friendly.\n\nNow we are going to our chief topic - how to convert 42.6 kilograms to pounds on your own.\n\n#### 42.6 kg to lbs conversion\n\nWe are going to start 42.6 kilogram equals to how many pounds conversion with the first formula to get the most accurate result. A quick reminder of a formula:\n\nAmount of kilograms * 2.20462262 = 93.916923612 the result in pounds\n\nSo what have you do to learn how many pounds equal to 42.6 kilogram? Just multiply amount of kilograms, this time 42.6, by 2.20462262. It is exactly 93.916923612. So 42.6 kilogram is equal 93.916923612.\n\nYou can also round off this result, for example, to two decimal places. It is exactly 2.20. So 42.6 kilogram = 93.720 pounds.\n\nIt is high time for an example from everyday life. Let’s calculate 42.6 kg gold in pounds. So 42.6 kg equal to how many lbs? As in the previous example - multiply 42.6 by 2.20462262. It is equal 93.916923612. So equivalent of 42.6 kilograms to pounds, if it comes to gold, is 93.916923612.\n\nIn this example you can also round off the result. This is the outcome after rounding off, this time to one decimal place - 42.6 kilogram 93.72 pounds.\n\nNow let’s move on to examples calculated with a short version of a formula.\n\n#### How many 42.6 kg to lbs\n\nBefore we show you an example - a quick reminder of shorter formula:\n\nNumber of kilograms * 2.2 = 93.72 the result in pounds\n\nSo 42.6 kg equal to how much lbs? And again, you need to multiply amount of kilogram, in this case 42.6, by 2.2. Let’s see: 42.6 * 2.2 = 93.72. So 42.6 kilogram is 2.2 pounds.\n\nLet’s make another conversion using this version of a formula. Now convert something from everyday life, for instance, 42.6 kg to lbs weight of strawberries.\n\nSo convert - 42.6 kilogram of strawberries * 2.2 = 93.72 pounds of strawberries. So 42.6 kg to pound mass is equal 93.72.\n\nIf you know how much is 42.6 kilogram weight in pounds and can convert it with use of two different versions of a formula, let’s move on. Now we are going to show you all results in charts.\n\n#### Convert 42.6 kilogram to pounds\n\nWe know that results presented in tables are so much clearer for most of you. We understand it, so we gathered all these outcomes in charts for your convenience. Due to this you can quickly make a comparison 42.6 kg equivalent to lbs results.\n\nBegin with a 42.6 kg equals lbs table for the first formula:\n\nKilograms Pounds Pounds (after rounding off to two decimal places)\n42.6 93.916923612 93.720\n\nAnd now look 42.6 kg equal pound table for the second formula:\n\nKilograms Pounds\n42.6 93.72\n\nAs you can see, after rounding off, if it comes to how much 42.6 kilogram equals pounds, the results are not different. The bigger amount the more considerable difference. Keep it in mind when you need to do bigger amount than 42.6 kilograms pounds conversion.\n\n#### How many kilograms 42.6 pound\n\nNow you learned how to convert 42.6 kilograms how much pounds but we are going to show you something more. Do you want to know what it is? What about 42.6 kilogram to pounds and ounces calculation?\n\nWe will show you how you can convert it step by step. Let’s start. How much is 42.6 kg in lbs and oz?\n\nFirst things first - you need to multiply number of kilograms, in this case 42.6, by 2.20462262. So 42.6 * 2.20462262 = 93.916923612. One kilogram is equal 2.20462262 pounds.\n\nThe integer part is number of pounds. So in this case there are 2 pounds.\n\nTo calculate how much 42.6 kilogram is equal to pounds and ounces you need to multiply fraction part by 16. So multiply 20462262 by 16. It gives 327396192 ounces.\n\nSo final outcome is exactly 2 pounds and 327396192 ounces. You can also round off ounces, for instance, to two places. Then your outcome is equal 2 pounds and 33 ounces.\n\nAs you can see, calculation 42.6 kilogram in pounds and ounces quite simply.\n\nThe last calculation which we are going to show you is calculation of 42.6 foot pounds to kilograms meters. Both of them are units of work.\n\nTo calculate it you need another formula. Before we give you it, see:\n\n• 42.6 kilograms meters = 7.23301385 foot pounds,\n• 42.6 foot pounds = 0.13825495 kilograms meters.\n\nNow let’s see a formula:\n\nNumber.RandomElement()) of foot pounds * 0.13825495 = the outcome in kilograms meters\n\nSo to calculate 42.6 foot pounds to kilograms meters you need to multiply 42.6 by 0.13825495. It gives 0.13825495. So 42.6 foot pounds is exactly 0.13825495 kilogram meters.\n\nIt is also possible to round off this result, for example, to two decimal places. Then 42.6 foot pounds is exactly 0.14 kilogram meters.\n\nWe hope that this calculation was as easy as 42.6 kilogram into pounds calculations.\n\nWe showed you not only how to make a calculation 42.6 kilogram to metric pounds but also two other conversions - to know how many 42.6 kg in pounds and ounces and how many 42.6 foot pounds to kilograms meters.\n\nWe showed you also another solution to do 42.6 kilogram how many pounds calculations, this is using 42.6 kg en pound calculator. This will be the best choice for those of you who do not like converting on your own at all or this time do not want to make @baseAmountStr kg how lbs calculations on your own.\n\nWe hope that now all of you are able to do 42.6 kilogram equal to how many pounds conversion - on your own or using our 42.6 kgs to pounds calculator.\n\nIt is time to make your move! Convert 42.6 kilogram mass to pounds in the best way for you.\n\nDo you need to do other than 42.6 kilogram as pounds calculation? For example, for 10 kilograms? Check our other articles! We guarantee that conversions for other amounts of kilograms are so simply as for 42.6 kilogram equal many pounds.\n\n### How much is 42.6 kg in pounds\n\nAt the end, we are going to summarize the topic of this article, that is how much is 42.6 kg in pounds , we prepared for you an additional section. Here you can see the most important information about how much is 42.6 kg equal to lbs and how to convert 42.6 kg to lbs . Have a look.\n\nHow does the kilogram to pound conversion look? To make the kg to lb conversion it is needed to multiply 2 numbers. How does 42.6 kg to pound conversion formula look? . Have a look:\n\nThe number of kilograms * 2.20462262 = the result in pounds\n\nSee the result of the conversion of 42.6 kilogram to pounds. The accurate answer is 93.916923612 lbs.\n\nYou can also calculate how much 42.6 kilogram is equal to pounds with another, shortened version of the equation. Have a look.\n\nThe number of kilograms * 2.2 = the result in pounds\n\nSo this time, 42.6 kg equal to how much lbs ? The answer is 93.916923612 lb.\n\nHow to convert 42.6 kg to lbs in just a moment? You can also use the 42.6 kg to lbs converter , which will make all calculations for you and give you an exact answer .\n\n#### Kilograms [kg]\n\nThe kilogram, or kilogramme, is the base unit of weight in the Metric system. It is the approximate weight of a cube of water 10 centimeters on a side.\n\n#### Pounds [lbs]\n\nA pound is a unit of weight commonly used in the United States and the British commonwealths. A pound is defined as exactly 0.45359237 kilograms." ]

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[ "# Arrayformula using cells as parameters\n\nI would like to use the ARRAYFORMULA combined with two cells as range. For example:\n\nCell A1 value: A3\nCell A2 value: B7\n\nResulting formula: ARRAYFORMULA(A3:B7)\n\nIs it possible? How can I do?\n\nIt's possible with\n\n``````=arrayformula(indirect(A1 & \":\" & A2))\n``````\n\nIndeed, the string concatenation `A1 & \":\" & A2` produces the string \"A3:B7\", and indirect picks up the corresponding range.\n\nActually, `=indirect(A1 & \":\" & A2)` is already enough to return an array of values, but arrayformula may be needed if you are going to do some computation with those values.\n\nIf the addresses in A1 and A2 refer to another sheet, only the first of them should have the sheet name: i.e., put Sheet2!A3 in A1 and simply B7 in A2. This is because stating sheet name twice is redundant, and `indirect` won't like that.\n\n• Thanks for the answer. It works with data in the same sheet, but what about using data in another sheet (in the same spreadsheet). For example, data and the cells with the ranges in sheet1 and using the arrayformula in sheet2. I tried it but it returns nothing. Something like `=arrayformula(indirect(sheet1!A3 & \":\" & sheet1!B7))`\n– Rods\nDec 29 '17 at 12:32\n• @Rods Follow-up questions should be posted as new questions.\n– Rubén\nDec 29 '17 at 15:51\n• @Rods It works, see my edit\n– user135384\nDec 29 '17 at 16:38" ]

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[ "Microsoft Excel MCQ Questions and Answers - Set 03 - ObjectiveBooks\n\n# Practice Test: Question Set - 03\n\n1. Which of the following is a correct order of precedence in formula calculation?\n(A) Multiplication and division exponentiation positive and negative values\n(B) Multiplication and division, positive and negative values, addition and subtraction\n(C) Addition and subtraction, positive and negative values, exponentiation\n(D) All of above\n\n2. Which of the following options is not located in the Page Setup dialog box?\n(A) Page Break Preview\n(B) Page Orientation\n(C) Margins\n\n3. When you want to insert a blank imbedded excel object in a word document you can\n(A) Click the object command on the insert menu\n(B) Click the office links button on the standard toolbar\n(C) Click the create worksheet button on the formatting toolbar\n(D) Click the import excel command on the file menu\n\n4. You can use the drag and drop method to\n(A) Copy cell contents\n(B) Move cell contents\n(D) Both ‘a’ and ‘b’\n\n6. Which of the following is an absolute cell reference?\n(A) !A!1\n(B) \\$A\\$1\n(C) #a#1\n(D) A1\n\n7. A worksheet can have a maximum of _______ Number of rows\n(A) 256\n(B) 1024\n(C) 32000\n(D) 65535\n\n8. Getting data from a cell located in a different sheet is called _________\n(A) Accessing\n(B) Referencing\n(C) Updating\n(D) Functioning\n\n9. Excel uniquely identifies cells within a worksheet with a cell name\n(A) Cell names\n(B) Column numbers and row letters\n(C) Column letters and row numbers\n(D) Cell locator coordinates\n\n10. All worksheet formula\n(A) Manipulate values\n(B) Manipulate labels\n(C) Return a formula result" ]

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[ "The intersection() method returns a new set with elements that are common to all sets. The intersection of two or more sets is the set of elements which are common to all sets. For example:\n\nSymPyは記号計算を行うためのPythonパッケージです。最近、勾配法などの最適化手法を勉強中なのですが、関数の微分や、関数とその接線を図示するといったことが簡単にできるようなパッケージはないのかな、とさがして見つけたものです。※まだ使い始めたばかりなので、今後追記していくと ... Find the points at which two given functions intersect¶. Consider the example of finding the intersection of a polynomial and a line: In SymPy, any expression not in an Eq is automatically assumed to equal 0 by the solving functions. Since \\(a = b\\) if and only if \\(a - b = 0\\) , this means that instead of using x == y , you can just use x - y . .\n\nPlanet SymPy Guidelines. Planet SymPy is one of the public faces of the SymPy project and is read by many users and potential contributors. The content aggregated at Planet SymPy is the opinions of its authors, but the sum of that content gives an impression of the project.\n\nIn this article, we will see how to solve it with Excel. To find intersection of two straight lines: First we need the equations of the two lines. Then, since at the point of intersection, the two equations will have the same values of x and y, we set the two equations equal to each other. This gives an equation that we can solve for x sympy.geometry.util.intersection contains ( other ) [source] ¶ Subclasses should implement this method and should return True if other is on the boundaries of self; False if not on the boundaries of self; None if a determination cannot be made.\n\nHow to parametrize the curve of intersection of two surfaces in \\$\\Bbb R^3\\$? 2. Finding a line integral along the curve of intersection of two surfaces. 1.\n\nPlanet SymPy Guidelines. Planet SymPy is one of the public faces of the SymPy project and is read by many users and potential contributors. The content aggregated at Planet SymPy is the opinions of its authors, but the sum of that content gives an impression of the project. May 10, 2011 · How to find the intersection of two functions Previously we have seen how to find roots of a function with fsolve , in this example we use fsolve to find an intersection between two functions, sin(x) and cos(x):\n\nSolve each equation so that they are both equations with the y variable on one side of the equation by itself and the x variable on the other side of the equation with all the functions and numbers. For example, the two equations below are in the format that your equations need to be in before you begin. That can't be the fastest way to do it, I must be missing something. Does anyone know of any better ways I could implement parabola to parabola intersection point calculations? The function I wrote is below: def parabola_to_parabola_poi(a1, b1, c1, a2, b2, c2): \"\"\" Calculate the intersection point(s) of two parabolas.\n\nQuadrilateral formed by connecting the vertices of a convex quadrilateral to midpoints of non-adjacent sides ... The coordinates of the interior intersection points ... delayed assignment. How to assign an expression to a variable name. The expression is re-evaluated each time the variable is used. mathematica: GNU make also supports assignment and delayed assignment, but = is used for delayed assignment and := is used for immediate assignment. Suppose we want to find the area of the middle region here. We would need to know the points of intersection of the curves and use these as boundaries for our definite integral. We can find these with sympy, update the plot, and evaluate the integral.\n\nJan 23, 2014 · sympy example – finding tangent lines to a function ... Confirm your estimates of the coordinates of the second intersection point by solving the equations for the ... delayed assignment. How to assign an expression to a variable name. The expression is re-evaluated each time the variable is used. mathematica: GNU make also supports assignment and delayed assignment, but = is used for delayed assignment and := is used for immediate assignment. May 10, 2011 · How to find the intersection of two functions Previously we have seen how to find roots of a function with fsolve , in this example we use fsolve to find an intersection between two functions, sin(x) and cos(x): SymPy offers several ways to solve linear and nonlinear equations and systems of equations. Of course, these functions do not always succeed in finding closed-form exact solutions. In this case, we can fall back to numerical solvers and obtain approximate solutions.\n\nA Computer Science portal for geeks. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions. Suppose we want to find the area of the middle region here. We would need to know the points of intersection of the curves and use these as boundaries for our definite integral. We can find these with sympy, update the plot, and evaluate the integral. Quadrilateral formed by connecting the vertices of a convex quadrilateral to midpoints of non-adjacent sides ... The coordinates of the interior intersection points ... sympy.geometry.util.intersection contains ( other ) [source] ¶ Subclasses should implement this method and should return True if other is on the boundaries of self; False if not on the boundaries of self; None if a determination cannot be made.\n\nA computer algebra system written in pure Python. Contribute to sympy/sympy development by creating an account on GitHub. Sympy is able to solve a large part of polynomial equations, and is also capable of solving multiple equations with respect to multiple variables giving a tuple as second argument. To do this you use the solve() command: >>>\n\nSymPy: symbolic computing in Python Aaron Meurer 1 , Christopher P. Smith 2 , Mateusz Paprocki 3 , Ond°ej …ertík 4 , Sergey B. Kirpichev 5 , Matthew Rocklin 3 , AMiT Kumar 6 , Sergiu Ivanov 7 , Solveset and Solver Module. This wiki contains ideas to improve solveset and solver module.. Trigonometric Equation. I come across an idea (during discussion in google group) that can solve the trigonometric equation g = 0 where g is a trigonometric polynomial.\n\nSymPy has equation solvers that can handle ordinary differential equations, recurrence relationships, Diophantine equations, 10 and algebraic equations. There is also rudimentary support for simple partial differential equations. There are two functions for solving algebraic equations in SymPy: solve and solveset. SymPy offers several ways to solve linear and nonlinear equations and systems of equations. Of course, these functions do not always succeed in finding closed-form exact solutions. In this case, we can fall back to numerical solvers and obtain approximate solutions.\n\nSympy is able to solve a large part of polynomial equations, and is also capable of solving multiple equations with respect to multiple variables giving a tuple as second argument. To do this you use the solve() command: >>> Solve each equation so that they are both equations with the y variable on one side of the equation by itself and the x variable on the other side of the equation with all the functions and numbers. For example, the two equations below are in the format that your equations need to be in before you begin. Find the points at which two given functions intersect¶. Consider the example of finding the intersection of a polynomial and a line:\n\nSympy is able to solve a large part of polynomial equations, and is also capable of solving multiple equations with respect to multiple variables giving a tuple as second argument. To do this you use the solve() command: >>> Conversion from Python objects to SymPy objects Optional implicit multiplication and function application parsing Limited Mathematica and Maxima parsing: example on SymPy Live SymPy: symbolic computing in Python Aaron Meurer 1 , Christopher P. Smith 2 , Mateusz Paprocki 3 , Ond°ej …ertík 4 , Sergey B. Kirpichev 5 , Matthew Rocklin 3 , AMiT Kumar 6 , Sergiu Ivanov 7 , SymPy offers several ways to solve linear and nonlinear equations and systems of equations. Of course, these functions do not always succeed in finding closed-form exact solutions. In this case, we can fall back to numerical solvers and obtain approximate solutions.\n\nConversion from Python objects to SymPy objects Optional implicit multiplication and function application parsing Limited Mathematica and Maxima parsing: example on SymPy Live\n\nMay 10, 2011 · How to find the intersection of two functions Previously we have seen how to find roots of a function with fsolve , in this example we use fsolve to find an intersection between two functions, sin(x) and cos(x): Calling linsolve for numeric matrices that are not symbolic objects invokes the MATLAB ® linsolve function. This function accepts real arguments only. This function accepts real arguments only. If your system of equations uses complex numbers, use sym to convert at least one matrix to a symbolic matrix, and then call linsolve . 平面・空間ベクトル. 行列で表示する方法もあるみたいだが、高校数学の範囲では Point でなんとかなる？\n\nCar parts for sale by owner on craigslist in fresno california\n\nQuadrilateral formed by connecting the vertices of a convex quadrilateral to midpoints of non-adjacent sides ... The coordinates of the interior intersection points ...\n\nFinding Points of Intersection of Two Lines. Suppose that we have two lines. If these two lines intersect, then sometimes it might be important to find the coordinates of this intersection.\n\nsolve(x**4 - 4*x**3 + 2*x**2 - x, x) Solve the equations system: x + y = 4, xy = 3: solve([x + y - 4, x*y - 3], [x, y]) Calculate limit of the sequence n p n: limit(n**(1/n), n, oo) Calculate left-sided limit of the function jxj x in 0: limit(abs(x)/x, x, 0, dir=’-’) Calculate the sum ∑100 n=0 n 2: summation(n**2, (n, 0, 100)) Calculate ...\n\nNotes ===== The intersection of any geometrical entity with itself should return a list with one item: the entity in question. An intersection requires two or more entities. If only a single entity is given then the function will return an empty list. Quadrilateral formed by connecting the vertices of a convex quadrilateral to midpoints of non-adjacent sides ... The coordinates of the interior intersection points ...\n\nSolve each equation so that they are both equations with the y variable on one side of the equation by itself and the x variable on the other side of the equation with all the functions and numbers. For example, the two equations below are in the format that your equations need to be in before you begin.\n\nsympy.geometry.util.intersection contains ( other ) [source] ¶ Subclasses should implement this method and should return True if other is on the boundaries of self; False if not on the boundaries of self; None if a determination cannot be made.\n\nIntersection of two given sets is the largest set which contains all the elements that are common to both the sets. Intersection of two given sets A and B is a set which consists of all the elements which are common to both A and B.\n\nsolve(x**4 - 4*x**3 + 2*x**2 - x, x) Solve the equations system: x + y = 4, xy = 3: solve([x + y - 4, x*y - 3], [x, y]) Calculate limit of the sequence n p n: limit(n**(1/n), n, oo) Calculate left-sided limit of the function jxj x in 0: limit(abs(x)/x, x, 0, dir=’-’) Calculate the sum ∑100 n=0 n 2: summation(n**2, (n, 0, 100)) Calculate ... SymPy 0.7.2 documentation » Module code » sympy » Source code for sympy.geometry.line \"\"\"Line-like geometrical entities. In this article, we will see how to solve it with Excel. To find intersection of two straight lines: First we need the equations of the two lines. Then, since at the point of intersection, the two equations will have the same values of x and y, we set the two equations equal to each other. This gives an equation that we can solve for x SymPyは記号計算を行うためのPythonパッケージです。最近、勾配法などの最適化手法を勉強中なのですが、関数の微分や、関数とその接線を図示するといったことが簡単にできるようなパッケージはないのかな、とさがして見つけたものです。※まだ使い始めたばかりなので、今後追記していくと ... .\n\nSolve each equation so that they are both equations with the y variable on one side of the equation by itself and the x variable on the other side of the equation with all the functions and numbers. For example, the two equations below are in the format that your equations need to be in before you begin. Quadrilateral formed by connecting the vertices of a convex quadrilateral to midpoints of non-adjacent sides ... The coordinates of the interior intersection points ..." ]

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[ "## My Integer Factorization Study: Progress, 29 August 2022 (Pretty Dry Stuff)\n\nI’ve updated the testing “scratchpad” for the first example I’ve been working, and have added a second one. I’ll add a third one if I need more help in teasing a general algorithm out of all of this. That’s entirely possible.\n\nFirst, for m=1501, with more of the process written in:\n\n``````M = 1501 = 39^2 - 20, remainder between 4^2 and 5^2,\ngiving Ceiling Root C0=39 and putting M between\nA0 = 1496 = 39^2 - 5^2 and\nB0 = 1505 = 39^2 - 4^2.\nM = A0 + 5 = B0 - 4.\nB0 - A0 = 9.\nSetting A0 as origin, perfect square marks at 0 9 16 ...\nRow polynomial is x(10-x).\nNow calculate 2 or 3 \"higher remainders\" as follows:\n(C0+1)^2-M = 1600-1501 = 99, between 9^2 and 10^2.\nA1 = 1500 = 40^2 - 10^2.\nB1 = 1519 = 40^2 - 9^2.\nM = A1 + 1 = B1 - 18.\nB1 - A1 = 19.\nSetting A1 as origin, perfect square marks at 0 19 36 ...\nRow polynomial is x(20-x).\n(C0+2)^2-M = 1681-1501 = 181, between 13^2 and 14^2.\nIf we choose these two perfect squares, A2 = 1485 = 41^2 - 14^2.\nHowever, we can choose instead A2 = 1456 for uniformity with A0 and A1.\nA2 = 1456 = 41^2 - 15^2.\nB2 = 1485 = 41^2 - 14^2.\n(Currently doubting how much the B values matter aside from B-A span.)\nM = A2 + 45 = B2 + 16.\nB2 - A2 = 29.\nSetting A2 as origin, perfect square marks at 0 29 56 ...\nRow polynomial is x(30-x).\n(C0+3)^2-M = 1764-1501 = 263, between 16^2 and 17^2.\nA3 = 1364 = 42^2 - 20^2.\nB3 = 1403 = 42^2 - 19^2.\nM = A3 + 137 = B3 + 98.\nB3 - A3 = 39.\nSetting A3 as origin, perfect square marks at 0 39 76 ...\nVerified that the differences between A's and M constitute an integer\nsequence corresponding to the values of quadratic polynomial:\nF(X) = 24X^2-28X+5, where X is the integer added to C0.\nDiscriminant of F(x) is 304=2^4*19.\n\nSearching for the positive integer value of X so that F(X) is of the form\ny(10+10X-y), for some positive integer y.``````\n\nAnd then, for m=172451, with a tad more hand-waving:\n\n``````M = 172451 = 416^2 - 605, putting it between\nA0 = 172431 = 416^2 - 25^2 and\nB0 = 172480 = 416^2 - 24^2.\nC0 = 416.\nM = A0 + 20 = B0 - 29.\nPerfect square marks at 0 49 96 ...\nRow polynomial is x(50-x).\n(C0+1)^2-M = 1438, between 37^2 and 38^2.\nA1 = 172445 = 417^2 - 38^2.\nB1 = 172520 = 417^2 - 37^2.\nM = A1 + 6 = B1 - 69.\nPerfect square marks at 0 75 148 ...\nRow polynomial is x(76-x).\nNow we progress linearly rather than using C0+2.\nA2 = 172123 = 418^2 - 51^2.\nB2 = 172224 = 418^2 - 50^2.\nM = A2 + 328 = B2 + 227.\nPerfect square marks at 0 101 200 ...\nRow polynomial is x(102-x).\n\nF(X) = 168X^2-182X+6.\nDiscriminant is 29092 = 2^2*7*1039.\n\nSearching for the positive integer value of X so that F(X) is of the form\ny(50+26X-y), for some positive integer y.``````" ]

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[ "# surveyCV\n\nThe R package `surveyCV` carries out cross-validation for complex sample survey data.\nIt is a companion R package to our SDSS 2021 presentation, and to our Stat article “K-fold cross-validation for complex sample surveys” (published online Jan 12, 2022).\n\n`surveyCV` is designed to work with the `survey` package to specify the sampling design (strata, clusters, sampling weights, etc.), and to account for this design when forming CV folds and estimating the CV test error.\n\nThe package currently handles the entire CV process for linear and logistic regression models. For other models, users can generate design-based CV folds with the `folds.svy()` function, then use these folds in their own custom training/testing CV loop.\n\n## Installation\n\nInstall a stable version of the package from CRAN:\n\n``install.packages(\"surveyCV\")``\n\nOr, for the latest development version, install directly from GitHub:\n\n``````# install.packages(\"remotes\")\nremotes::install_github(\"ColbyStatSvyRsch/surveyCV\", build_vignettes = TRUE)``````\n\n## Usage: automated survey CV for linear or logistic regression\n\nThe function `cv.svy()` carries out K-fold CV on a dataset for a set of linear or logistic regression formulas, using specified strata, clusters, weights, and FPCs. For each model under consideration, it reports the design-based mean CV loss and a design-based estimate of its SE.\nUse `nest = TRUE` only if cluster IDs are nested within strata (i.e., if clusters in different strata might reuse the same names).\n\n``````library(surveyCV)\nlibrary(splines)\ndata(NSFG_data)\ncv.svy(NSFG_data, c(\"income ~ ns(age, df = 2)\",\n\"income ~ ns(age, df = 3)\",\n\"income ~ ns(age, df = 4)\"),\nnfolds = 4,\nstrataID = \"strata\", clusterID = \"SECU\",\nnest = TRUE, weightsID = \"wgt\")\n#> mean SE\n#> .Model_1 22616 756.02\n#> .Model_2 22536 748.01\n#> .Model_3 22559 766.89\n\n# The 2nd model (spline with 3 df) had lowest survey CV MSE,\n# although it's well within one SE of the other models.``````\n\nFor convenience, the function `cv.svydesign()` only needs a `svydesign` object and a formula, and will parse the relevant survey design information before passing it to `cv.svy()`.\nSimilarly, the function `cv.svyglm()` only needs a `svyglm` object, and will parse both the formula and the survey design.\n\n``````NSFG.svydes <- svydesign(id = ~SECU, strata = ~strata, nest = TRUE,\nweights = ~wgt, data = NSFG_data)\ncv.svydesign(formulae = c(\"income ~ ns(age, df = 2)\",\n\"income ~ ns(age, df = 3)\",\n\"income ~ ns(age, df = 4)\"),\ndesign_object = NSFG.svydes, nfolds = 4)\n#> mean SE\n#> .Model_1 22576 744.59\n#> .Model_2 22436 739.81\n#> .Model_3 22577 752.62\n\nNSFG.svyglm <- svyglm(income ~ ns(age, df = 3), design = NSFG.svydes)\ncv.svyglm(glm_object = NSFG.svyglm, nfolds = 4)\n#> mean SE\n#> .Model_1 22411 741.93``````\n\n## Usage: survey CV folds for other models\n\nThe function `folds.svy()` generates design-based fold IDs for K-fold CV, using any specified strata and clusters.\n(Briefly: For a stratified sample, each fold will contain data from each stratum. For a cluster sample, a given cluster’s rows will all be assigned to the same fold. See our Stat paper for details.)\n\nUsing these fold IDs, you can write your own CV loop for models that our packages does not currently handle.\n\nHere is an example of tuning the bin size for a design-based random forest, using the `rpms_forest()` function from the `rpms` package. Note that while `folds.svy()` accounts for the clustering, we also need to pass the cluster IDs and survey weights to `rpms_forest()` for design-consistent model-fitting, and we need to use the survey weights in the MSE calculations.\n\n``````library(rpms)\ndata(CE)\n\n# Generate fold IDs that account for clustering in the survey design\n# for a subset of the CE dataset\nnfolds <- 5\nCEsubset <- CE[which(CE\\$IRAX > 0), ]\nCEsubset\\$.foldID <- folds.svy(CEsubset, nfolds = nfolds, clusterID = \"CID\")\n\n# Use CV to tune the bin_size parameter of rpms_forest()\nbin_sizes <- c(10, 20, 50, 100, 250, 500)\nSSEs <- rep(0, length(bin_sizes))\nfor(ff in 1:nfolds) {\ntrain <- subset(CEsubset, .foldID != ff)\ntest <- subset(CEsubset, .foldID == ff)\nfor(bb in 1:length(bin_sizes)) {\nrf <- rpms_forest(IRAX ~ EDUCA + AGE + BLS_URBN,\ndata = train,\nweights = ~FINLWT21, clusters = ~CID,\nbin_size = bin_sizes[bb], f_size = 50)\nyhat <- predict(rf, newdata = test)\nres2 <- (yhat - test\\$IRAX)^2\n# Sum up weighted SSEs, not MSEs yet,\n# b/c cluster sizes may differ across folds and b/c of survey weights\nSSEs[bb] <- SSEs[bb] + sum(res2 * test\\$FINLWT21)\n}\n}\n# Divide entire weighted sum by the sum of weights\nMSEs <- SSEs / sum(CEsubset\\$FINLWT21)\n# Show results\ncbind(bin_sizes, MSEs)\n#> bin_sizes MSEs\n#> [1,] 10 204246617270\n#> [2,] 20 202870633392\n#> [3,] 50 201393921358\n#> [4,] 100 201085838446\n#> [5,] 250 201825549231\n#> [6,] 500 204155844501\n\n# Bin size 100 had the lowest survey-weighted CV MSE estimate,\n# though sizes 50 and 250 were quite similar.\n# Now we could fit a random forest with bin size 100 on full `CEsubset` dataset.``````\n\n## Stat paper\n\nOur GitHub repo includes R code to reproduce figures for our Stat article “K-fold cross-validation for complex sample surveys” (published online Jan 12, 2022).\n\nScripts for the PPI and NSFG examples are in the `data-raw` folder, in the `PPI_Zambia_plot.R` and `NSFG_plot.R` scripts. We cannot share the proprietary PPI dataset, but the preprocessed NSFG dataset is included in the package as `NSFG_data`, and instructions for preprocessing the NSFG data are in the same folder in the `NSFG_data.R` script.\n\nSimulation code is in the `plots-for-Stat-paper` vignette." ]

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